Ultrasonics: data, equations, and their practical uses

ULTRASONICS Data, Equations, and Their Practical Uses CRC_DK8307_FM.indd i 11/14/2008 11:58:03 AM CRC_DK8307_FM.ind...

Author: Dale Ensminger | Foster B. Stulen

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ULTRASONICS Data, Equations, and Their Practical Uses

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ULTRASONICS Data, Equations, and Their Practical Uses Edited by

Dale Ensminger Foster B. Stulen

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-0-8247-5830-1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Ultrasonics : Data, Equations, and Their Practical Uses / editors, Dale Ensminger and Foster B. Stulen. p. cm. “A CRC title.” Includes bibliographical references and index. ISBN 978-0-8247-5830-1 (alk. paper) 1. Ultrasonic waves--Industrial applications. 2. Ultrasonics. I. Ensminger, Dale. II. Stulen, Foster B. III. Title. TA367.U35 2009 620.2’8--dc22

2008034102

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents Preface .............................................................................................................................. vii Editors ............................................................................................................................... ix Contributors ...................................................................................................................... xi 1

Oscillatory Motion and Wave Equations ..............................................................1 Dale Ensminger

2

Ultrasonic Horns, Couplers, and Tools ................................................................27 Dale Ensminger

3

Advanced Designs of Ultrasonic Transducers and Devices Using Finite Element Analysis ...........................................................................129 Foster B. Stulen and Robert B. Francini

4

Piezoelectric Materials: Properties and Design Data ...................................... 185 Dale Ensminger

5

Magnetostriction: Materials and Transducers .................................................235 Dale Ensminger

6

Pneumatic Transducer Design Data ................................................................... 273 Dale Ensminger

7

Properties of Materials ........................................................................................285 Dale Ensminger

8

Ultrasonics-Assisted Physical and Chemical Processes .................................323 Dale Mangaraj, B. Vijayendran, and Dale Ensminger

9

Advances in Generation and Detection of Ultrasound in the Field of Nondestructive Testing/Evaluation ..........................................365 Allan F. Pardini, Gerald J. Posakony, and Theodore T. Taylor

10

Medical Ultrasound: Therapeutic and Diagnostic Imaging ...........................407 Foster B. Stulen

11

Mechanical Effects of Ultrasonic Energy ..........................................................447 Dale Ensminger

12

Criteria for Choosing Ultrasonics ...................................................................... 471 Dale Ensminger

Index ................................................................................................................................481

v

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Preface Ultrasonics is a form of acoustical energy, generally pitched above the audible range of frequencies. The interesting phenomena attributable to ultrasonics, particularly in its early history, elicited many suggestions for its use. Some of these suggestions seem to imply that ultrasonics has a mysterious and magical power. Many suggestions, however, have proved within the past century to be practical in various areas of science and industry. All these practical applications have benefited greatly from recent technological advances in electronics and computer science. However, the ultimate effectiveness of any process depends on the knowledge available to the developer and user about the principles on which the process is based. In ultrasonic applications, the basic principles to consider are those related to both ultrasonics and its application. Ultrasonics processes involve considerations of phenomena such as wave propagation, chemical reactions, thermal effects resulting from absorption of energy, effects attributable to cavitation, and other stress-, friction-, and momentum-related factors. Any successful application of ultrasonics must be related in a practical way to some acoustic property or properties of the media being irradiated ultrasonically. This book includes data and functions gathered from several areas of science and technology for the purpose of facilitating the use of ultrasonic energy. The objectives of this work are to evaluate the practicality of new ideas and to make important data available so that the design of ultrasonic systems meets the needs of present and new developments. These developments include applications in both standard and harsh environments. Chapters 1 and 2 discuss basic ultrasonic equations that relate to factors affecting wave motion and the application of these factors to the design of ultrasonic systems such as vibrating bars, horns, plates, and rings. Chapter 3 goes more deeply into designs that apply finite elements to ultrasonic drivers (transducers), large horns, and blades. Chapter 4 includes properties and design data applicable to piezoelectric materials and the uses of piezoelectric transducers. Chapter 5 provides information relative to the design of magnetostrictive transducers. Chapter 6 covers pneumatic and liquid transducers and their design and uses. Chapter 7 examines mechanical and physical properties of materials, including those properties necessary for the welding and forming of materials. Chapter 8 reviews the chemical properties and compatibilities of materials, and discusses the chemical effects of ultrasound. Several practical applications are presented. Chapter 9 is an update of nondestructive testing applications. This chapter provides examples of modern equipment that can be readily procured and configured for use in many demanding ultrasonic nondestructive testing applications. Electromagnetic acoustic transducers (EMATs) and lasers and their uses are described. Chapter 10 presents inclusive coverage of medical uses of ultrasound today. Medical ultrasound, therapies, and diagnostic imaging as well as low-intensity and high-intensity applications are discussed. Low-intensity areas include imaging devices and blood flow measurements such as Doppler methods. Surgery has benefited considerably from ultrasonic applications in such areas as removal of cataracts, lithotripsy, the treatment of prostate cancer, atrial fibrillation treatment, wrinkle reduction, fat removal, wound cleaning and healing, sonophoresis, ultrasonic welding in vivo, and the dissolving of clots.

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viii

Preface

Chapter 11 discusses the mechanical effects of ultrasonic energy. This chapter reviews some of the mechanical factors previously discussed, while emphasizing some new processes as well. Chapter 12 summarizes the history of ultrasonics. It discusses how prior problems that faced the advancement of the industry have been overcome and projects further advances. We appreciate the cooperation that we have received from all the experts who have contributed their chapters to this book. We also appreciate the help and cooperation of the publisher who has waited patiently for the manuscript to be completed. We hope that the reader will find this book to be a time-saving and dependable source of valuable information.

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Editors Dale Ensminger joined the Battelle staff in 1948 while a student at the Ohio State University. He began active research in ultrasonics and acoustics in February 1950 and since then has been responsible for or participated in more than 980 projects in acoustics, ultrasonics, and related areas. He received a bachelor’s degree in mechanical engineering in March 1950, and a bachelor’s degree in electrical engineering in June 1950. Between 1950 and 1953, he took additional courses in mathematics and physics at the Ohio State University on a part-time basis in support of the needs presented by ultrasonics research. Ensminger is the author of approximately 150 articles and holds several patents on applications of ultrasonics. He has written a book Ultrasonics—The Low- and High-Intensity Applications, published in February 1973. A second edition of this book, Ultrasonics: Fundamentals, Technology, Applications, was published in 1988. This book is presently being updated. Ensminger is the author of the chapter “Acoustic Dewatering” in the book Advances in Solid–Liquid Separation, edited by H. S. Muralidhara and published by Battelle Press in 1986. He was an editorial reviewer for Nondestructive Testing Handbook, 2nd Edition, Volume Seven, Ultrasonic Testing, published by the American Society for Nondestructive Testing in 1991. He is the author of A Handbook on Ultrasonic Methods of Nondestructive Testing written for the U.S. Army through the Watertown Arsenal in 1973. He is also the author of a major section on nondestructive testing by ultrasonics appearing in a handbook used by the U.S. Air Force and published in 1971. Ensminger is a member of the Acoustical Society of America, Ultrasonics Industry Association, the Society for Nondestructive Testing, and ASM International. Since his retirement, Ensminger has continued as a staff member (senior research scientist) of the Battelle Memorial Institute, where he has been employed since his college days (June, 1948 until May, 2008). Dr. Foster B. Stulen received his PhD from the Massachusetts Institute of Technology in 1980. His dissertation was on the use of frequency analysis of the myoelectric signal to quantify muscle fatigue. In 1979, he joined Battelle in Columbus, Ohio, one of the world’s largest contract R&D firms. There, he was mentored by Dale Ensminger and Dr. Naga Senapati in the field of ultrasonics. Dr. Stulen spent 18 years at Battelle developing and testing concepts ranging from large ultrasonic degassers for oil field applications to ultrasonic mosquito chasers. He even worked on the proverbial kitchen sink, acoustic emission monitoring from the spalling of porcelain. At Battelle, Dr. Stulen developed considerable expertise in modeling and analyzing power ultrasonic systems. One of his last projects there was to model an ultrasonic surgical system for Ethicon Endo-Surgery (EES), a Johnson & Johnson company. As a result, he accepted a position with that company in 1997. Since then, Dr. Stulen has made significant contributions to the Harmonic™ product line as it is known worldwide today. Dr. Stulen left EES for 1 year to join a medical device start-up company as the chief technology officer. The business strategy was to identify innovations at universities and academic health centers and develop them through NIH Small Business Innovation Research grants. During two grant cycles, the team of engineers working with him submitted nearly 40 grants at a success rate approaching 50%. ix

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x

Editors

Dr. Stulen is a guest lecturer at Miami University in Oxford, Ohio, where he created a senior technical elective course—Medical Device Design. He is also an associate editor for the Journal of Medical Devices, a publication of the ASME. He has numerous publications and has been awarded more than 21 patents with several additional patents pending.

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Contributors Dale Ensminger Battelle Memorial Institute Columbus, Ohio Robert B. Francini Kiefner and Associates, Inc. Columbus, Ohio Dale Mangaraj Innovative Polymer Solutions Battelle Memorial Institute Columbus, Ohio Allan F. Pardini Applied Physics and Materials Characterization Sciences Pacific Northwest National Laboratory Richland, Washington

Gerald J. Posakony Applied Physics and Materials Characterization Sciences Pacific Northwest National Laboratory Richland, Washington Foster B. Stulen Ethicon Endo-Surgery Cincinnati, Ohio Theodore T. Taylor Applied Physics and Materials Characterization Sciences Richland, Washington B. Vijayendran Battelle Memorial Institute Columbus, Ohio

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1 Oscillatory Motion and Wave Equations Dale Ensminger

CONTENTS 1.1 Introduction ............................................................................................................................1 1.2 Elementary Vibratory Systems .............................................................................................2 1.2.1 The Pendulum............................................................................................................. 3 1.2.1.1 Simple and Compound Pendulums ..........................................................4 1.2.1.2 Torsional Pendulums ..................................................................................7 1.2.2 The Simple Spring/Mass Oscillator ........................................................................8 1.2.2.1 Effect of the Mass of the Spring on the Spring/Mass Oscillator ..............................................................................9 1.2.2.2 Effect of Losses on the Spring/Mass Oscillator .....................................9 1.3 Impedance, Resonance, and Q ........................................................................................... 10 1.3.1 Electrical and Mechanical Q ................................................................................... 11 1.4 Acoustic Wave Equations .................................................................................................... 12 1.4.1 Plane Wave Equation ............................................................................................... 12 1.4.2 The General Wave Equation ................................................................................... 14 1.4.3 Transverse Wave Equation for Flexible String ..................................................... 14 1.4.4 Transverse Wave Equation for a Membrane ........................................................ 15 1.4.5 Transverse Wave Equation for Bars ....................................................................... 16 1.4.6 The Plate Wave Equation......................................................................................... 17 1.4.6.1 Lamb Waves ............................................................................................... 17 1.4.7 Love Waves (5) .......................................................................................................... 19 1.4.8 Laplace Operators .................................................................................................... 19 1.5 Horns...................................................................................................................................... 20 1.6 Effects Due to Material Geometry and Elastic Properties .............................................22 1.7 Summary of Physical Factors in Ultrasonic Technology ................................................ 24 Further Readings .......................................................................................................................... 25

1.1

Introduction

The two properties of a medium that enable it to conduct acoustic waves are (1) its mass and (2) its elastic properties. The simple presentation in this chapter of the manner in which each of these two properties influence acoustic wave propagation is intended to assist the reader in understanding how to apply the principles and data presented in later chapters. The temptation to offer a more extensive treatise has been suppressed in the interest of providing a useful reference book, rather than a textbook. 1

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Ultrasonics: Data, Equations, and Their Practical Uses

2

In this chapter, the functions of mass and elasticity are demonstrated in the presentation of fundamental wave equations germane to the field of ultrasonics. In general, specific solutions to these equations are presented in later chapters, where they are most appropriately correlated with principles and data. The concept of equivalent circuits is introduced for its value in the design of acoustic systems, including transducers and active elements of the systems.

1.2

Elementary Vibratory Systems

Ultrasonic energy is acoustic energy, and acoustic principles apply to any aspect of its study or use. There are many types of acoustic waves. They propagate through a medium as a succession of mechanical actions and reactions. The particle motion at a fixed position x in a wave is oscillatory. Although the general wave equation does not imply linear conditions and harmonic particle motion, common concepts and usage of ultrasonic energy involve harmonic motion. An acoustic wave is generated by any force that produces oscillatory vibrations within a continuous medium. The vibrations are passed from element to element, due to its inertial and elastic properties, at a rate corresponding to the velocity of sound in the medium. The mass of the element is determined by its volume and the density of the medium. The amplitude of the element motion is determined by the forces exerted upon it, its mass, and the elastic conditions surrounding it. These principles are fundamental to the development and understanding of wave motion equations. Force is defined as follows: an influence that, if applied to a free body, results chiefly in an acceleration of the body and sometimes in elastic deformation and other effects. The force, F, required to produce an acceleration, a, in a mass, m, is given by F = ma

(1.1)

A force, F, applied to any mass (or object) equals exactly the sum of all opposing forces, that is, the sum of all components of force acting in a direction opposite that of the applied force. (According to Newton’s 3rd law of motion for every action, there is an equal and opposite reaction.) The energy in an acoustic system also obeys the law of conservation of energy. The law of conservation of energy states that energy cannot be created or destroyed. A wave passing through a medium loses energy to the medium by various absorption mechanisms. The dissipated energy is not destroyed, but it is changed; that is, kinetic energy is changed to thermal energy. Equation 1.1 may be written in the form of a differential equation, that is F ma m

d 2x dt 2

The opposing forces in a linear system may be represented by a proportionality constant, k, multiplied by the displacement, x, that is F− = kx

(1.2)

as F− is equal in magnitude and opposite F+.


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Oscillatory Motion and Wave Equations

3

Therefore, we can write m

d 2x kx dt 2

or d 2x k x0 2 dt m

(1.3)

Equation 1.3 is the differential equation for harmonic vibration. The acceleration is proportional to the displacement and always directed toward the origin. Its general solution is well-known, that is x A cos t B sin t (1.4)

C e jt D ejt

where A, B, C, and D are constants to be determined from the initial conditions of motion, j2 is –1, ω is angular frequency (=2πf ) , ω2 = k/m, and f is frequency. The constants A, B, C, and D are related as follows: A=C+D B = j(C − D) The vibratory motion of simple mechanisms, such as pendulums and mass-loaded springs, operating within a linear range are described in a relationship similar to Equation 1.3. The frequency equation is ω2 equated to the coefficient of the term (k/m) in Equation 1.3, that is 2

1.2.1

k m

(1.5)

The Pendulum

The pendulum is a simple example of oscillatory motion. Basic configurations of pendulums are (1) the simple pendulum (Figure 1.1a), (2) the compound pendulum (Figure 1.1b), and (3) the torsional pendulum (Figure 1.1c). In accordance with the law of conservation of energy, the motion of the pendulum is sustained by the alternating exchange of energy from one form to another (potential and kinetic) until the energy of the system is fully dissipated by various loss mechanisms.

ᐉ

ᐉ

M (a)

θ

ᐉ

d

θ

θ (b)

(c)

FIGURE 1.1 Common pendulums: (a) simple pendulum, (b) compound pendulum, and (c) torsional pendulum.


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4 1.2.1.1

Simple and Compound Pendulums

The simple pendulum is illustrated (Figure 1.1a) by a mass, m, on the free end of a string oscillating about a fixed support (or axis). The string is assumed to be inextensible and of negligible weight. A body that vibrates like a pendulum but with its mass distributed in any geometrical manner from or about its axis and not concentrated is called a physical or compound pendulum (Figure 1.1b). The simplest example of a compound pendulum is a uniform rod suspended at one end and free to vibrate like a pendulum if displaced and released. Another familiar form of the compound pendulum is a mass located at any position between the axis of rotation and the free end of a uniform bar. The simple torsional pendulum (Figure 1.1c) is represented as a rigid disk attached to one end of a light shaft, the other end of which is fixed. The rigid disk of Figure 1.1c may be replaced by any other configuration. The motions of each of these systems is described by the general relationship d 2 T() 0 dt 2 Ι

(1.6)

Therefore, the frequency relationships for any type of pendulum are dependent upon the ratio T(θ)/I, where T is the torque applied about the fixed point (or axis) of the pendulum, I the moment of inertia of the mass of the system about the fixed axis, and θ the angle in a vertical plane between the equilibrium position and the line through the support axis and the center of gravity of the pendulum (Figure 1.1a). The pull of gravity on the mass, m, and the distance between the point of support and the center of gravity determines the torque for the simple and the compound pendulums. Torque in the torsional pendulum is related to the elastic properties of the shaft plus gravitational factors attributable to the geometry of the oscillating body. For the simple and compound pendulums, the torque is the force of gravity resolved normal to a straight line through the support axis and the center of gravity, mg sin θ, multiplied by the distance, , between the support axis and the center of gravity (Figure 1.2). This torque is always in a direction that tends to reduce θ, and is T() mg sin

(1.7)

I m 2

(1.8)

For the simple pendulum,

where m is the mass attached to the inextensible string and the length of the string plus the distance to the center of the mass. Substituting these quantities into Equation 1.6 gives d 2 g sin 0 dt 2

(1.6a)

for the simple and the compound pendulums. Equation 1.6a is nonlinear because of the term sin θ. Exact solutions are obtainable for only a few nonlinear differential equations. A common approach is to substitute θ for sin θ,


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5

ᐉ

θ mg sin θ

m

θ (T = mgᐉ sin θ)

mg FIGURE 1.2 Torque on a simple pendulum due to the pull of gravity.

ignoring the higher order terms in the sin θ series. This changes Equation 1.6a into the form of the linear Equation 1.3, for which the solution is applicable only to a very small θ. Substituting higher order terms from the sin θ series sin

3 5 7 9 3! 5! 7 ! 9!

into Equation 1.6a would increase its accuracy for larger θ, but the resulting equation is nonlinear again, calling for an approximate method of solution. When a pendulum is released from a large angle θ and allowed to vibrate freely, the amplitude of swing will decrease gradually toward zero as energy is lost to the atmosphere and sin θ approaches θ. Substituting θ for sin θ (for very small θ) into Equation 1.6a gives d 2 g 0 dt 2

(1.6b)

which is similar in form to Equation 1.3. Therefore

g

(1.9)

which is the frequency equation commonly used for the pendulum in elementary physics. For the purposes of this book, the conventional method leading to Equation 1.6b is adequate. Finding the frequency of vibration of a pendulum consisting of any combination of physical features involves summing all components of torque acting on the system and dividing this sum by the total of the moments of inertia of all elements in the system.


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6

O ᐉ

L

θ ms r

FIGURE 1.3 Uniform bar with end mass load.

For the uniform bar mounted at one end and freely vibrating as a compound pendulum (Figure 1.1b), these quantities are 1 I m 2 3

(1.10)

T mg sin 2

(1.11)

for long slender rod about one end and

where the center of gravity is located at a distance /2 from one end of the bar. For very small θ

3g 2

(1.12)

or f

1 2

3g 2

(1.12a)

Adding a mass to the end of a uniform bar (Figure 1.3) increases the torque and the moment of inertia by amounts attributable to the weight of the added mass and its distance from the axis O. If is the length of the bar (extending from the axis O to the surface of the mass) and r the distance from the same surface point to the center of mass, the values of total torque and moment of inertia for very small θ are T = −(msLgθ + mrg θ/2) I = (mr 2/3) + msL2 L=+r


(m sL m r (L r)/2)g 1 m sL2 m r (L r)2 3

(1.13)

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7

For the special case in which ms is negligible, r = 0 and 3

gh L2

(1.13a)

g L

(1.13b)

If mr is negligible,

or the angular frequency of a simple pendulum. 1.2.1.2

Torsional Pendulums

The simple torsional vibrator is represented in Figure 1.1c by a rigid round plate of uniform thickness attached to the end of a light, cylindrical, elastic rod, or bar of length L. The opposite end of the bar is mounted rigidly to the support. The axes of the plate and the bar coincide. Within the elastic range, rotating the plate through an angle θ applies a torque, kθ, to the bar or T = kθ

(1.14)

where k is a torsional spring constant related to the dimensions of the bar, its geometry, and the shear modulus, G, of the material of the bar. For a round bar of length L and diameter d, assuming that the applied torque produces strictly shear stress in the bar without axial deformation, the spring constant, k, is k

d 4 G 32L

(1.15)

For a body rotating about a fixed axis, the moment of inertia of the body with respect to the axis multiplied by the angular acceleration is equal to the moment of the external forces acting on the body with respect to the axis of rotation, or I

d 2 k dt 2

(1.16)

For a round plate of uniform thickness and diameter D, I

m pD 2 8

(1.17)

where mp is the mass of the plate. Therefore, rearranging Equation 1.16, substituting for I and k, and simplifying gives d 2  d 4 G    0 dt 2  4m pD 2L 

(1.16a)

for the round plate on the end of a cylindrical rod.


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8

Equation 1.16a is similar in form to Equation 1.3. Therefore, the angular frequency of a torsional pendulum consisting of a round plate of uniform thickness mounted coaxially on a cylindrical rod and oscillating within the linear elastic range of the rod is

d 4 G 4m pD 2L

(1.18)

The torsional pendulum may consist of any of numerous geometrical structures other than the coaxial uniform bar and disk. The resonance frequency for any combination of such elements is determined according to the general rule derived from Equations 1.6 and 1.13b: (1) total all torques acting upon the pendulum element and divide this sum (gravitational and elastic quantities) by the sum of all moments of inertia of the components of the oscillatory system and (2) equate ω2 to the coefficient corresponding to k/m of Equation 1.3 if linear assumptions are applicable. Nonlinear equations are usually solved by approximate methods. 1.2.2

The Simple Spring/Mass Oscillator

The torsional pendulum is a type of spring/mass oscillator, because the rod is like a spring. The principles leading to Equation 1.3 are illustrated by the simple spring/mass oscillator of Figure l.4. A force, F, moves the mass toward the right to a distance x from the equilibrium position where the spring exerts an opposing force equal to F on the mass. For a linear spring, this reactive force is Fs = −kx

(1.2b)

where k is the spring constant in newtons per meter. If losses in the spring and between the mass and the slide are negligible, the force exerted on the mass by the compressed spring will accelerate the mass toward the neutral position when the force, F, is removed. These are the exact conditions assumed in the derivation of Equation 1.3. Thus the frequency of the undamped, or lossless, spring/mass oscillator is f

1 k 2 m

(1.19)

FS = −kx 1

2 m

x Frictionless surface FIGURE 1.4 Simple spring/mass oscillator.


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Oscillatory Motion and Wave Equations 1.2.2.1

9

Effect of the Mass of the Spring on the Spring/Mass Oscillator

In practical cases, the mass of the spring affects its natural resonance frequency. If the mass of the spring is uniformly distributed along its length, the natural resonance frequency of the mass-loaded spring is f

1 k m 2 m s 3

(1.19a)

where ms is the mass of the spring. 1.2.2.2

Effect of Losses on the Spring/Mass Oscillator

A spring/mass oscillator loses energy by mechanisms such as friction, internal absorption, and air resistance. The equilibrium equation for such a system is m

d 2x dx Rm kx Ft dt 2 dt

(1.20)

where Rm(dx/dt) represents the sum of all losses in the system and Ft is a driving force. When Ft suddenly drops to zero, the amplitude of vibration decays at a rate determined primarily by the loss factor, Rm. The resonant frequencies of vibration and the rate of decay are determined by solutions to Equation 1.20 for which Ft = 0, or m

d 2x dx Rm kx 0 dt 2 dt

(1.20a)

for which the solution is  k k  x et K1 cos t 2 sin t M′ M′  

(1.21)

The damped angular frequency is d

k t 02 2 M′

(1.22)

where ω 0 is the undamped angular frequency, α = Rm/2m is the damping factor, K1 and K2 are the amplitudes of displacement, and M′

4m 2k 2 4km R m

Here, m also includes the corrections for the mass of the spring. A second method of writing the solution to Equation 1.20a is x = K e−t cos(dt + φ)


(1.21a)

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10

where K and φ are amplitude and phase constants, respectively. Equation 1.21a provides a means of determining α and, therefore, Rm, by measuring the rate of decay of the freely vibrating system.

1.3

Impedance, Resonance, and Q

The relationships between forces, motions, and losses within a mechanical vibratory system, and transmission of acoustic waves in general, are similar to those that exist between voltages, current, and impedances within electrical circuitry. Equivalent circuits and acoustic impedances are very useful and important concepts in analyzing and designing acoustic systems. In more general Equation 1.20, the vibrational amplitude of the system caused by the force, Ft, is determined by the amplitude and frequency of Ft and by the reactions of m, R m, and k. When Ft = 0, as in Equation 1.20a, the energy stored within the system is dissipated and the amplitude of vibration decays according to Equations 1.21 and 1.21a. Note the comparisons between the mathematical relationships describing the motions in the mechanical system and the electrical functions of Figure 1.5, a closed-loop series RLC circuit. The integro-differential equation that relates the loop current, i, with the source voltage, Vt, in Figure 1.5, is Vt V1 V2 V3 L

(1.23)

di 1 R ei ∫ idt dt C

where V1 is the voltage across the inductor, L, due to current, i; V2 the voltage across the resistor, Re, due to the current, i; and V3 the voltage across the capacitor, C, due to the current, i. Substituting v = dx/dt converts Equation 1.20 to Ft m

V1

i Vt

R

dv R m v k ∫ vdt dt

(1.24)

V2

L C

V3

FIGURE 1.5 Simple RLC electrical circuit.


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11

which is identical in form with Equation 1.23. Comparing Equations 1.23 and 1.24 shows equivalences between the two systems, which are: Mass, m, is equivalent to inductance, L. Force across a mass precedes the motion by 90°. Voltage across a coil precedes the current by 90°. Mechanical resistance, Rm, is equivalent to electrical resistance, Re. Force across Rm is in phase with the motion. Voltage across Re is in phase with current. Particle velocity, v, is equivalent to current, i. Elastic constant, k, is equivalent to the inverse of capacitance (1/C). Force across a lossless spring lags the motion of the spring by 90°. Voltage across a capacitor lags current by 90°. These comparisons show the logic behind equivalent circuits for use in designing and analyzing acoustic systems. Electrical impedance, Ze, is the vectorial sum of the electrical resistance and reactances, or Ze = Re + j(XL − Xc)

(1.25)

The mechanical impedance, Zm, is the vectorial sum of the mechanical resistance and reactances, or Zm = Rm + j(Xm − Xs)

(1.26)

where XL = ωLe is the inductive reactance of the electrical circuit, Xc = 1/ωCe the capacitive reactance of the electrical circuit, Xm = ωLm the reactance due to the density or mass of the material of the medium, and Xs = k/ω the reactance corresponding to the elastic properties of the material of the medium. Resonance occurs when |XL|=|Xc| and when |Xm|=|Xs|. 1.3.1

Electrical and Mechanical Q

The quality factor, Q, is a measure of the sharpness of resonance of an oscillatory system. In a series resonance circuit, it is defined as Q = Xm/Rm = rm/Rm = r/(2 − 1) = r/2

(1.27)

where ωr is the angular frequency at resonance, ω1 the angular frequency below resonance at which the amplitude of displacement in a driven system is 0.707 times its amplitude at resonance, ω2 the angular frequency above resonance at which the amplitude of displacement is 0.707 times its amplitude at resonance, and ω2 − ω1 the bandwidth of the system. In electrical series resonant circuitry, the voltage that appears across either the inductor or capacitor is Q times the voltage inserted in series with the circuit. In a mechanically vibrating system, it is convenient to relate the ratio of mechanical displacement amplitude at a position of maximum displacement at resonance (such as a free end of a longitudinally resonant uniform bar) to the amplitude at that same position under the same driving force, but at a frequency well removed from resonance. This ratio is the Q of the mechanical system. The stresses are affected similarly. Their distribution within the system depends upon the geometry and physical properties of the material of the system.


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12

Xm

E

Xs

Rm

FIGURE 1.6 Equivalent parallel circuit for a simple mechanically resonant system.

Most common ultrasonic systems are represented by equivalent parallel circuits or combinations of series and parallel circuits. Figure 1.6 is an equivalent parallel circuit for a simple, mechanically resonant system. For levels of Q in a parallel circuit higher than 10, Zr = QX

(1.28)

where Zr is resistive impedance at resonance. X is the reactance in ohms of either the total equivalent inductor or the total equivalent capacitor at resonance. Therefore, the relative impedance of a high-Q circuit is maximum at resonance. The shapes of the curves are identical to those of the series RLC circuit in which current peaks at resonance. When the Q of a parallel resonant circuit falls below 10, two different conditions may be called resonance. In maximum impedance parallel resonance, the impedance is maximum, but is not resistive. In resistive impedance parallel resonance, the parallel impedance is a pure resistance, but the impedance is not maximum. The characteristic acoustic impedance is a resistive component of the acoustic impedance of a material, the product of density and velocity of sound. Characteristic acoustic impedances are listed with other acoustic properties of materials in Chapter 6. In an active ultrasonic system, an electrical source supplies the energy to drive a mechanical system. In a passive device, electrical energy is produced in a system by reaction to mechanical forces. Electromechanical coefficients relate these parameters, so that both mechanical and electrical quantities are included within one equivalent circuit for analytical and design purposes. Use of equivalent circuits in the design of transducers is demonstrated in Chapters 4 and 5. Other practical uses of equivalent circuits are demonstrated in other chapters.

1.4 1.4.1

Acoustic Wave Equations Plane Wave Equation

The general method of deriving the plane wave equation is to consider the motion of plane waves in a medium in which the velocity of sound is c. Attenuation of the wave is assumed to be zero. The waves are free to move in both the positive and negative direction of x described by the equation f1(x ct ) f2 (x ct )


(1.29)

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13

where f1(x − ct) refers to waves moving in the positive direction of x and f2(x + ct) to waves moving in the negative direction of x. If f1(x − ct) and f2(x + ct) are continuous, Equation 1.29 may be differentiated twice with respect to x (keeping t constant), giving 2 f1′′(x ct ) f2′′(x ct ) x 2

(1.30)

Similarly, differentiating Equation 1.29 twice with respect to t (keeping x constant) gives 2 c 2f1′′(x ct ) c 2f2′′(x ct ) t 2

(1.31)

or, by comparing Equations 1.30 and 1.31, 2 2 2 c t2 x 2

(1.32)

As a premise for its derivation, Equation 1.29 is an obvious general solution to the plane wave equation in Equation 1.32. The characteristic of a true plane wave is that pressures and motions at every position in a plane normal to the direction of propagation are equal in amplitude and phase. These conditions are seldom, if ever, met in actual practice. The type of wave is not defined by the velocity, c, in Equation 1.32. The equation describing a longitudinal wave in a thin bar (d ℓ/2 from the small end of the horn. The stress distribution is s j

YV1 ( e

)

( 2/c 2 ) ( 2/c ′ 2 ) x

  2 (r22 r12 ) 2 2  x 2  cos  2 2 c c′  c′   r2 c′  2 (r22 r12 ) 2 2  2 r22 c2 c′  c′


 x   sin  c′  

(2.93)

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Ultrasonic Horns, Couplers, and Tools

55

where Y is Young’s modulus of elasticity for the material of the horn. Maximum stress occurs where tan

r22c 2 4 (r22 r12 ) c c′ 2 c 2 x c′ 2r22c c′ 2 c 2 2(c′ 2 2c 2 ) (r22 r12 )

(2.94)

At the position of maximum stress, x < ℓ/2 from the small end of the horn. The mechanical impedance, which is necessary in matching unlike segments for a resonant system, is   x2 ja 2Y  2 1 sS  b Zm 2 2  v r r 2 ( ) x  x 2 2 1 sin  cos c′ r2 c′  

(2.95)

 2 (r 2 r 2 ) 1  x  1 1  x  1 2 (r22 r12 ) 1  2 2 1 2 2  cos 2  sin   2 2 c c′  c′  c′ r2 c c′  c′   r2 c′  where s is the stress as a function of x, S the cross-sectional area as a function of x, and v the velocity as a function of x. The ratio of displacements at the ends of the horn is 1 v1 e( 2 v2

2.4

)

( 2/c 2 ) ( 2/c ′ 2 )

(2.96)

Horn Design and Performance Factors

Several factors enter into the choice of materials and the design of ultrasonic horns. The basic considerations fall under two major headings: 1. Application objectives and requirements 2. Power source flexibility and capability Application objectives and requirements determine the specifications for the ultrasonic system and the environment in which it will operate. Power source flexibility and capability determine what the designer has at his disposal (whether commercially available, designable, or readily on hand) for driving the horn to perform the requirements of the project. The primary factors affecting horn design and performance include: 1. 2. 3. 4.

Effects related to Poisson’s ratio Effects related to internal losses and thermal conductivity Effects related to loading variations Effects related to design anomalies


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56

The primary factor to consider in designing ultrasonic horns for power applications is velocity of sound in the material of the horn. Velocity of sound in solid media is variously defined elsewhere in this book as follows: 1. Bar velocity of sound. The rate at which an acoustic wave in which the motion is strictly rarefaction and compression in the axial direction travels along a thin homogeneous bar in which the lateral dimensions are very small compared to the wavelength of the disturbance. This velocity is given by co

Y

(1.36)

where Y is Young’s modulus of elasticity and ρ is the density of the material of the bar. This is the basic quantity, velocity of sound, c, appearing in the solutions to the horn equation of Section 2.3.1 through 2.3.7. Its derivation does not take into account the effects of Poisson’s ratio on the velocity of sound. 2. Bulk velocity of sound. The rate at which an acoustic wave, compressional in the direction of propagation, travels through an infinite, solid medium having lateral dimensions that are much larger than the length of the wave. This velocity is cB co

1 (1 )(1 2 )

(1.56)

where σ is Poisson’s ratio of the medium. The bulk velocity, cB, is always >co due to the effects of Poisson’s ratio. 3. Shear velocity of sound. The rate at which an acoustic wave consisting of shear, or transverse, motion only travels through a medium. This velocity is cs co

1 2(1 )

(1.57)

where µ is shear modulus of elasticity in the medium. 2.4.1

Effects of Poisson’s Ratio on Performance

The equations of Section 2.3 are useful for designing horns if the lateral dimensions of the horns and tools are small compared with the wavelength. The slenderness of the horns is considered to be sufficient reason to ignore the effects of Poisson’s ratio. A stress applied along the axis of a rod produces a strain parallel to the axis and a corresponding strain of opposite sign in a direction normal to the axis. Poisson’s ratio is defined as the ratio of the normal strain to the axial strain and it is a characteristic of all solid metals. The effect is to divide the total kinetic energy of the vibrating system between the desired longitudinal mode and a lateral component of motion. The effects may range from (a) that attributable to the inertia of the mass associated with the lateral dimensions of the horn to (b) actual spurious structural resonances that may coincide, or nearly coincide, with the resonance frequency of the intended mode of vibration and thus disrupt the performance of the horn. The latter condition is a factor in the design of wide and large area horns considered in Section 2.8. Lord Rayleigh [5] analyzed theoretically the effect of Poisson’s ratio on the period of vibration of a longitudinally resonant solid cylinder or wire. The analysis is based upon


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57

comparing the total kinetic energy of the system due to the longitudinal displacement plus that due to the lateral displacement caused by Poisson’s ratio with the kinetic energy of the longitudinal motion alone. The effect of the lateral inertia on the period of vibration, T, is equivalent to adding a mass to the end of a slender bar if Poisson’s ratio can be ignored: it increases the period of vibration. Rayleigh assumes a simplified lateral function of displacement but the correction is valid for cylindrical rods to ratios of diameter to wavelength of ∼0.4, according to Derks [6]. Rayleigh defined the kinetic energy associated with the longitudinal component of displacement as 2

kinetic energy U

S  d  dx 2 ∫o  dt 

(2.97)

where ρ is the density of the medium, S the cross-sectional area of the transmission line, the displacement at x, d/dt the particle velocity at x, and x the direction of propagation. For a round wire or bar, the corresponding kinetic energy associated with radial expansions and contractions due to Poisson’s ratio, σ, is 2

dη  S 2r 2 U ρ∫ ∫   dxr dr 0 0  dt  4

r



2

d 2  ∫0  dt dx  dx

(2.98)

where η is the lateral displacement of the particle at distance r from the axis. Particle motion in the round, uniform, homogeneous bar, resonating in a free–free longitudinal mode is described by o cos

x sin t c

(2.99)

d x o cos cos t dt c and d 2 2o x sin cos t dt dx c c

(2.100)

Using these quantities in Equations 2.99 and 2.100 gives 2So2 sin 2 t 4

(2.101)

4S 2r 2o2 cos 2 t 8c 2

(2.102)

U and U

The total absolute value of kinetic energy is U U


2So2 4

2 2r 2   1 2c 2   

(2.103)

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58

For the cylindrical half-wave resonator ω 2 2d 2  2 2r 2    U U U 1 U 1 2  2c  8c 2   

(2.104)

In terms of Y and c, the total kinetic energy is U U

S 2Yο2  2 2r 2  1  4c 2  2c 2 

(2.105)

For the uniform bar in longitudinal half-wave resonance c 2 ( f )2 Y/ρ

S 2Yo2 4 2 2 T 4U

(2.106)

Therefore, T2 ∝ U T2 c* 2 U c2 U U (T T)2

(2.107)

where T is the period of vibration when the effect of Poisson’s ratio, σ, is ignored and T + δT is the period of vibration corresponding to taking σ into account. The corrected value of velocity of sound in the uniform bar due to Poisson’s ratio is, therefore c* 2

S 2Yo2 4[U U]

(2.108)

The ratio of corrected value due to Poisson’s ratio to bar value of velocity of sound is c* 2 1 1 2 2r 2 2 2d 2 c2 1 1 2c 2 8c 2

(2.109)

As long as the wavelength of the propagating wave is large compared with the lateral dimensions of the rod, the effects of Poisson’s ratio on bar velocity are insignificant. By assuming that σ 2k2d2/8 is very small, Derks [6] linearized the Rayleigh correction to c* 1 1 2k 2d 2 c 16

(2.110)

where k = ω/c. 2.4.1.1

Corrections to Velocity of Sound in Tapered Horns

2.4.1.1.1 Exponential Horns The procedures of the previous section for correcting velocity of sound for effects due to Poisson’s ratio in uniform bars can be extended to other geometrical configurations. The procedure will be illustrated for exponential horns only. For slender horns, the corrections due to σ are very slight and are usually ignored, because other factors have greater


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59

influence on the performance of these horns. However, Poisson’s ratio cannot be ignored in the design of horns having at least one significantly large lateral dimension compared with a wavelength. Not only does it influence the velocity of sound in horns of large cross section, wide blade-types, and special designs such as flared or cup shaped horns, but it also is an important cause of spurious, deleterious modes of vibration in these designs. For the round, exponentially tapered horn x c ′ x  x/2  sin e o cos 2 c c′  ′ 

(2.26)

The kinetic energy associated with longitudinal motion alone is 2

U ex

S  d    dx 2 ∫0  dt 

(2.111)

d x c′ x  x/2  e o cos sin cos(t ) dt c′ 2 c′   S = Soe−γx U ex

S 2o2 2

r = roe−γx 2

x c ′ x  ∫ᒌ cos c′ 2 sin c′  ex dx 

c ′ x x 2 c ′ 2 x  2 x cos cos sin sin 2 dx ∫ᒌ  2 c′ c′ c′ 4 c′  S 2o2  2c ′ 2  o 1  4 4 2  

U ex

S o 2o2 2

(2.112)

(2.113)



(2.113a)

Comparing the exponential horn with the uniform bar of equal length (λ/2), diameter (do), and material T2 U c′ 2 2 2 Tex U ex c

(2.114)

where c′ is the velocity corresponding to an exponentially tapered horn, ignoring σ, which leads to U c ′ 2 2c ′ 2 2 1 U ex c 4 2

(2.115)

When the lengths are identical and the maximum diameter of the exponential horn equals the diameter of the uniform bar, the lateral inertia of the uniform bar is greater than that of the exponentially tapered horn. The apparent velocity of sound (ignoring σ) in the tapered horn is greater than that of the uniform bar according to Equation 2.25, that is c′ 2


c2 2c 2   1 4 2   

(2.25)

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60

The effect of Poisson’s ratio is determined from the relationship U ex c* 2 ex U ex U ex c′ 2

(2.116)

where Uex is given by Equation 2.113a, and c′, the apparent velocity of sound in the exponential horn ignoring Poisson’s ratio, is given by Equation 2.25. S 2r 2 U ex 4

2



d 2  ∫o  dt dx  dx

S 2r 2 2  2c′ 2  U ex o o o   c′  4  4

2

∫o ex sin 2

x dx c′

d 2 x  2c′ 2o  x/2 sin  e dt dx c c′ 2 ′   2 c*ex c′ 2

2c 2  2c 2   2c 2  2ro 2 2 [1 e ] 1 4 2   

or U ex

S oo2 2ro2 2 4c ′ 2

e   2c′ 2 4 2   1     8  

(2.117)

The total kinetic energy is U ex U ex

S o 2o2  2c′ 2  S oo2 2ro2 2 1  4 4 2  4c ′ 2 

 2c′ 2 4 2   1 e     8   

(2.118)

from which 2 c*ex U ex 2c′ 2 c′ 2 U ex U ex 2γc′ 2 2ro2 2 [1 e ]

(2.119)

where c*ex is the velocity of sound in an exponential horn corrected for Poisson’s ratio and c′ the velocity of sound in an exponential horn taking only the taper into consideration. In relation to the bar velocity of sound, c (which does not take σ into consideration), 2 c ex c′ 2

1 2c 2 ]   [1 e ] 8c 2

 2ro2 [4 2

1  

(2.120)

and 2 c ex c2 2c 2   2ro2  1 4 2   1 8c 2  


1   4 2 2c 2  [1 e ]   2   

(2.121)

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Ultrasonic Horns, Couplers, and Tools 2.4.2

61

Effects of Losses on Horn Performance

Losses that affect horn performance fall into two general categories: 1. Those that are associated with the load against or into which the horn works 2. Internal damping 2.4.2.1

Losses Associated with the Load into which the Horn Works

This condition is illustrated in a very simplified manner by the equivalent circuit of Figure 2.11. The transformer, Tr, represents the multiple degrees of coupling between the horn and the variety of loads to which the horn might be subjected. In general, the impedance reflected by the load into the horn is ignored in designing a horn for commercial use. The load impedance may vary between (a) that of another member resonant at the resonance frequency of the driving horn, (b) that of a nonresonant member that can be coupled to the horn with modification in such a manner that the two resonate as a single unit (half-wave), and (c) that of a highly viscous load (tightly coupled horn with tip immersed in an extremely viscous liquid). Conditions described by case (c) may cause significant effects on the performance of the horn, such as increasing the dynamic period of the horn and causing excessive heating in high-power applications of ultrasonics. High internal losses produce similar effects, as discussed later. In most commercial applications of power ultrasonics involving treating high-loss types of loads the horn is either energized under no load condition and the energy stored is dissipated in the load as the horn is suddenly pressed against or into it or the damping aspect of the load impedance is insignificant as far as the performance of the horn is concerned. In either case, the horn is designed as a “no-load” item. Impedance matching of horns and elements is discussed in Section 2.5. 2.4.2.2

Internal Damping Factors

The design of horns for power applications of ultrasonic energy is based upon tabulated properties (elastic constants, densities, velocity of sound, fatigue limits) of selected metals. Chemical compatibility with materials to be treated and elastic properties as functions of temperature are often critical considerations in choosing materials. Thermal conductivity and Q affect the performance of the horn. High thermal conductivity is especially important to the performance of horns of large cross sections. When velocity of sound, c, in a bar material is determined dynamically, the result includes the effect of the internal loss factor, R m. It is this value of c that is used to design

Lh

Tr RL Ch

CL

LL

FIGURE 2.11 Equivalent circuit of load with losses.


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62

Rh

Lh

Ch

FIGURE 2.12 Equivalent circuit of a horn with internal losses.

a horn. The primary interests in Rm are (a) quality control: identification of materials of quality inferior to that expected of an alloy and (b) monitoring performance: discerning between incipient damage (fatigue) in operating horns and downward shifts in frequency due only to thermoelastic effects. Figure 2.12 is an equivalent circuit of a horn with internal losses (damping). The horn equation may be modified to account for internal damping by adding the term Rm/ρc2 as follows: 1 2 R m 2 1 S ξ 2 0 c 2 t 2 c 2 xt S x x x 2

(2.122)

2 v 1 S v R m v 2 j v0 x 2 S x x c 2 x c 2

(2.123)

or

assuming harmonic motion and homogeneous material (i.e., the intrinsic loss mechanisms are distributed uniformly throughout the volume of the horn). Equations 2.122 and 2.123 may be used wherever the narrow horn equations are applicable, such as (a) uniform bars or (b) tapered horns. The first step in solving either Equation 2.122 or 2.123 is to identify the type of horn and thereby determine the quantity (1/S)∂S/∂x. 2.4.2.2.1 Uniform Bars with Losses In a uniform bar, the taper is zero; therefore, dS/dx = 0, and Equation 2.123 may be written d2v R m dv 2 v0 j 2 dx c 2 dx c 2

(2.123a)

for which the solution is   R2  2 v e j(R mx/2c ) V1 cos  1 2m 2  x 4 c  c  j


 Rm R2 R2   1 2m 2 sin  1 2m 2  x  2c 4 c 4 c   c

(2.124)

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63

Therefore, the effect of damping is to change the apparent velocity of sound to 1 c 2 2 2 R 2c 2 R m 1 2 m2 c

c′d c

(2.125)

The stress in a half-wave uniform bar in which Rm ≠ 0 is s j j

Y dv dx   Y j(R m x/2c2 )  R3 R2  R2 e V1 j 3m 3 cos 1 2m 2  2m 2 1 sin  c 8 c c 4 4 c    

(2.126)

R2 x 1 2m 2 c 4 c

(2.127)

where

When Rm = 0 s j

Y x x x V1 sin jcV1 sin jcm sin jy c c c c

S m c

Xsin

x sin(t) c

(2.6a)

The complex impedance, Zmd, due to considering Rm ≠ 0 is Zmd = sS/v or   S Y   1 Zmd  o 2   4 4 2 2 2  2 2  2c   16 c R m ( 4 c R m ) tan 

(2.128)

3 2 2 4 2 2 c j[R m 4 2c 2 (R m 4 2c 2 )] 4 2c 2 R m tan

{4R m 4 R m [R m 16 4c 4 ]tan 2 }

which reduces to Zm j

SoY  x  tan jspc tan   c  c

(2.7)

when Rm = 0. 2.4.2.2.2 Exponentially Tapered Horns with Losses For the exponentially tapered horns, for which S = Soe−γx, Equation 2.123 becomes d2v  R m  dv 2 j v0  dx 2  c 2  dx c 2


(2.123b)

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64 for which the solution is v e(j(R m/c

2 ))( x/2 )

1/ 2  (c 2 jR m )2   Vo cos 1  x c 4 2 2c 2  

2 1/ 2

c 2 jR m 2 1/ 2

  c 2 jR m   c 2  1    2c   

sin

 c 2 jR m    1    c 2c  

   x   

(2.129)

The apparent phase velocity, cd′, of an exponentially tapered horn with losses is therefore 1/2

2   (c 2 jR m )   c′d c 1    2ρc   

(2.130)

which converts to c′

c 2c 2 1 4 2

(2.25)

when Rm = 0. 2.4.2.2.3 Conically Tapered Horns with Losses For the conically tapered horn in which the area decreases with increasing x, Sx

D x2 2 [D1 (D1 D 2 )x]2 4 4

Taking the derivatives of S and inserting the function (1/S)∂S/∂x into Equation 2.123 gives 2v  2(D1 D 2 ) j R m  v 2  v0  2 c 2  x c 2 x  [D1 (D1 D 2 )x]

(2.123c)

The solution to Equation 2.123c is of the form (D1 D 2 )V1e j(R m/c )K1x [D1 (D1 D 2 )x] 2

v

 D1K 2 x c x  cos (1 K 3 )sin   c  c  ( D1 D 2 )

(2.131)

The constants K1, K2, and K3 are determined by applying the boundary conditions when x = 0 where v = V1


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65 v 0 x 2v 2 2 V1 2 x c

and the condition described by the horn equation, Equation 2.123c. The final solution is  (D D 2 )V1e( jR m /c2 )x   D1 j R m D 1  x c  x  v  1 cos 1 sin    2  c 2c (D1 D 2 )  c   [D1 (D1 D 2 )x]   (D1 D 2 )

(2.132)

Equation 2.132 reduces to the undamped solution, Equation 2.60, when Rm = 0. 2.4.2.2.4 The Meaning of Rm The quantity Rm represents a summation of mechanically related loss mechanisms within the horn material. The significant mechanisms include thermoelastic effects, interaction with thermal phonons, and dislocation damping. Plastic deformation and fatigue damage are irreversible effects associated with dislocation movement. Therefore, dislocations are probably the most significant of the internal loss mechanisms in horns used in power applications of ultrasonic energy. Fatigue strength (endurance limit) is of major importance in choosing materials for horns to be used in power applications of ultrasonic energy. In most cases, Rm is considered only intuitively in designing ultrasonic horns. For the materials most commonly used (titanium, aluminum alloys, steel alloys, monel), its effect is usually insignificant and has already entered into the measurement of bar velocity of sound. Heat generated by losses in the horn and the performance of the horn in manners related to Q (such as damping, resonance amplitudes, and bandwidth) are usually anticipated in the design by experience and available performance data. Occasionally, one may wish to measure Rm, and for this purpose, modern instrumentation provides relatively simple means. Such measurements are particularly useful for quality control of horn materials. Increased Rm and decreased velocity of sound (determined by accurate measurements of dimensions and half-wave resonance frequency) are indicators of a decrease in the quality of a horn material. The fact that ultrasonic horns typically vibrate in longitudinal half-wave resonance suggests logical methods of making the measurement. Because Rm represents the energy absorption losses within the material of a vibrating member, it must have a direct relationship to the rate at which the amplitude of displacement of the member in resonance decays. It also must affect the quality factor, Q, of the system. Thus, R m may be determined either (a) by measuring the damping factor of a freely suspended bar in longitudinal half-wave resonance or (b) by measuring the Q of this suspended bar. Q is also determined by the damping factor. The damping capacity is affected by the magnitude of strain, frequency of vibration, temperature, composition, grain size, heat treatment, aging, cold work, and state of magnetization (for ferroelectric materials) [7]. The bar is suspended in a nonconstrained manner, usually by means of extremely light wires. The bar is energized by means of a suitable noncontacting source (such as electromagnetic means). The amplitude of vibration is measured at one end by means of a noncontacting device (such as capacitance gage). Fiber optic techniques provide very accurate measurements of very low-amplitude displacements. Electronic instrumentation is provided for displaying and recording amplitude, time, and frequency (Figure 2.13).


10/29/2008 3:48:14 PM


66

Amplitude, frequency, time recorder

Light wire suspensions Bar

Energizer

L/4

L/4 L

Noncontacting displacement gage (capacitance, fiber optic)

FIGURE 2.13 Electronic instrumentation for determining Q, R m, and damping factor of a resonating bar.

The relationship between Rm, Q, and damping factor may be explained by using the electrical equivalent tuned circuit consisting of an inductance, resistance, and capacitance in series. In the electrical circuit, the total voltage across the inductors is given by the equation Erms = Irms Ze = Irms(Re + jXL ) = Irms(Re + jLe)

(2.133)

where Erms is the RMS voltage across the coil; Irms the RMS current through the coil; Re the electrical resistance across the coil, ohms; XL the inductive reactance of the coil at frequency, f, ohms; and Le the inductance of the coil. The Q of the coil in an electrical circuit is Q = XL/Re = tan θ

(2.134)

where θ is the phase angle between current and voltage. In the mechanical system, letting Force, F, correspond to voltage Ems Velocity, v, correspond to current, Irms Resistance, Rm, correspond to electrical resistance, Re Mechanical reactance, Xm, correspond to electrical reactance gives the correlation Q = Xm/Rm = tan θ = f/∆f

(Q ≥ 10)

(2.135)

The Q of the resonant bar is determined by measuring frequency at resonance, fo, the frequency above resonance, f2, and the frequency below resonance, f1, at which the power is one-half the power at the resonance peak. Because power is proportional to the square of the velocity, hence of the square of amplitude of displacement, the half power points occur at the frequencies at which the amplitude is 1/√ 2 (=0.707) times the amplitude at resonance. The Q is determined using Equation 2.135 when ∆f ℓ1 > λ1/4), and consists of a metal with density, ρ1, and bar velocity, c1. The length of the segment on the right is ℓ2 (ℓ2 ≤ λ 2/4), and the bar consists of material of density, ρ2, and bar velocity of sound, c2. In practice, ℓ2 is generally specified. The cross-sectional area of both elements is identical. The matching condition is that Zx1 = Zx2 or jSρ1c1 tan

1 jS 2c 2 tan 2 c1 c2 Material B

Material A

xb

xa λ/2 FIGURE 2.14 Resonant bars of uniform cross section but different materials.


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71

or tan

1 2c 2 tan 2 1c1 c1 c2

(2.145)

All terms of Equation 2.145 are specified with the exception of ℓ1. By definition, ℓ1 ≥ λ1/4, which places it in the second quadrant (ωℓ1/c1 ≥ 90°). 2.5.2

Horn and Lumped Mass

A lumped mass is an element having dimensions that are very small compared with a wavelength of sound. The stiffness of any material overlapping the lateral dimensions of the driving horn is assumed to be infinite and the motion of the entire mass is assumed to be everywhere in phase. The mass may be a change in dimensions of a horn near the load end or it may be an entirely different material attached for specific types of applications. According to these specifications, the length of the horn, ℓ, is less than λ1/2 and greater than λ1/4. The mechanical impedance of the mass is ZM = −jωM = −jωρ × Volume 2.5.2.1

(2.146)

Lumped Mass Attached to a Uniform Bar

In this case, the impedance of the uniform bar, Equation 2.7 is set equal to ZM of the mass (Figure 2.15), that is  x  Zm Zm jSc tan   c  or tan

Μ ρ (Volume of mass) 2 c1 S 1c1 S 1c1

(2.147)

S

M

L FIGURE 2.15 Mass-loaded resonant bar.


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72

When the mass is a change in dimensions but not of material at the end of a bar (either reduction or increase in cross-sectional area, S) tan

( Volume of mass) c1 Sc1

(2.147a)

Similar procedures are used in determining dimensions of stepped horns driving a mass load. 2.5.2.2

Lumped Mass Attached to Exponential Horn

According to Equation 2.33, the mechanical impedance, Zm, of the exponential horn is  2c ′   x  x jS o Y  e tan    c′   c′ 4  Zm c ′ x   1 tan 2 c′  

(2.33)

Letting Zm = ZM leads to tan

c′

2Me  ω 2c′  Μc′e SoY   c′ 4  2 2 2e ( Volume of mass)  2c′  Μc′e SoY   c′ 4  2

(2.148)

2 2(Volume of mass)  c′  2c′( Volume of mass) SoY  e  c′ 2 4 

(2.148a)

or tan

c′

2

The length, ℓ of the exponentially tapered portion is determined by iteration using Equations 2.148 (ℓ > λ/4). The lengths of mass-loaded horns of any taper for which the mechanical impedance relationship is known are determined by following the procedures of Sections 2.5.2.1 and 2.5.2.2: equate the mechanical impedance, Zm, of the horn with that of the mass at the junction between the two and solve for the length of horn that satisfies the equation with ℓ > λ/4. To be treated as a lumped mass, all dimensions of M must be very small compared with λ in the material of the mass at the resonance frequency, fo. Otherwise, it must be considered as a distributed mass. 2.5.3

Horns with Distributed Loads

Seldom will two adjacent sections of a resonant structure be properly described by any term other than as a distributed mass. Its impedance is always complex, as illustrated in the previous sections dealing with bars and horns of various geometries. Several variations in


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73

impedance functions may be present within a half-wave structure, such as a piezoelectric transducer for power applications. (Piezoelectric transducer design data are presented in Chapter 4.) 2.5.3.1

Matching a Uniform Bar to a Horn

The matching principle is demonstrated in Section 2.5.1 for two bars of equal cross section but different materials. The same procedure may be followed for matching any distributed mass to any other distributed mass vibrator (horn). For example, to match a uniform bar to an exponentially tapered horn, note that the impedance of the horn at x = ℓ1 is  2c ′    jS o Y1  tan  1  e1   c′   c′ 4  Zh c ′   1 tan 1  c′  2 

(2.149)

and the impedance of the bar is Zh = −jSbρ2c2 tan(ℓ2/c2)

(2.150)

Impedance matching occurs when Zh = Zb. Usually the application requirements determine the dimensions of the bar. A horn is chosen that seems best for use with the specified bar. If the application calls for a long bar (λ/2 or greater), the most desirable condition is to make ℓ2 = nλ/2 where n = 1, 2, 3, 4, …. In this way, a standard horn may be used without altering its length. The stresses at the junction are minimum so that the two elements may be joined safely by threaded elements, silver brazing, or other acceptable means. It is not always possible to make the attachment a resonant length. Very often the application requires a short bar attachment. The length of a standard horn has to be adjusted so that its mechanical impedance matches that of a bar that is short compared with a half-wavelength, or of a bar for which the length exceeds n half-wavelengths by a fraction of a half-wavelength. When the length of the bar ℓ2 < λ/4, ℓ1 > λ/4, the impedance relationship for determining ℓ1 is 2 c2 2 1  c′  e S b2c 2c′ tan 2 S o Y1   c′ 4  2 c2

(2.151)

 2c ′    S o Y1  tan  1  e1  c′   c′ 4  S b2c 2 tan 2 c′ 1  c2  1 tan  c′  2 

(2.152)

tan 1 c′

e1 S b2c 2 tan

or

The bar length, ℓ2, is specified and the horn length, ℓ1, is determined using Equation 2.152.


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74

Stress at the junction between the horn and the bar may be determined using the equation for stress distribution in a uniform bar, letting x = ℓ2, that is S b jYb

m sin 2 j2c 2m sin 2 c2 c2 c2

(2.153)

where m is the maximum displacement at the free end of the bar. Often short bars of tool steel are silver brazed to a horn to perform certain functions such as materials forming. Silver brazed joints have fillets. The stress concentrations due to the fillets can be estimated using Equations 2.17, 2.18, and 2.153. If this concentrated stress exceeds the endurance limit of the soldered joint, the tool is too long and the joint will fail in fatigue very rapidly. Impedance matching of other geometrical combinations is accomplished in a manner similar to the examples given in this section. 2.5.4

Multiple Mode Systems

A system consisting of one or more elements of distributed elastic mass is capable of vibrating in more than one mode (such as longitudinal, flexural, or torsional motion). The design problem is to control the modes of vibration throughout the system to deliver the proper type of activity to the application zone. Design objectives might include any of the following: 1. Having strictly one specific mode of vibration, with all others suppressed within a vibratory member—preferably by designing the component so that no outside structural constraints are needed for this mode suppression. In the ultrasonics industry, horns are commonly designed to operate in the longitudinal mode. If a structure vibrates in more than one mode at the same frequency, the total vibrational energy of the structure is shared by all modes, thus reducing the energy available to the desired mode. 2. Sharing the available energy between modes. This condition is usually undesirable. However, there are certain exceptions where applications are best satisfied by the motion that combined modal action can provide at the load end of an ultrasonic transmission line. In the case of slender rods (or wires) that are several longitudinal half-wavelengths long, it is virtually impossible to prevent flexural modes from occurring simultaneously with longitudinal modes. In certain types of applications, the elliptical motion afforded by the combination of modes is beneficial, for example, in disintegrating friable substances obstructing small passageways. 3. Transferring energy from one element vibrating in one pure mode to a second element in the vibratory system vibrating in a different pure mode. A horn vibrating in a longitudinal mode may be used to drive a uniform bar in flexure to bond thin strips or to spot-weld thin plastic sheets. Other combinations might include a longitudinally vibrating horn driving a thin diaphragm, plate, or cylinder in flexure (e.g., transducers mounted on the bottom of an ultrasonic cleaning tank). In the present sense, a multiple-mode system is defined as one in which a source operating at resonance in one mode excites resonance in a second member vibrating in a different mode from that of the first. Coupled elements in such a system must be matched in mechanical impedance, as in any other coupled system. In addition, some thought should be given to the coupling conditions themselves, that is, as much as is possible, design so that the stress patterns in each of the connected elements conform at the junction to enhance rather


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75

than inhibit the intended modal patterns. Poor coupling conditions, including method and position, will produce poor output activity. In some cases, tabulations of theoretical data for bars and plates considered in the following sections may extend beyond practical limits from an applications standpoint. These are included for completeness, particularly with respect to frequency limits for conditions intermediate to those presented. For example, mass loading conditions, such as (a) a mass on the end of cantilever, (b) a mass loading a plate with any boundary conditions, or (c) a water load on any plate, are not included. In each case, the load would reduce the fundamental frequency, and, depending upon the loading conditions, would generally alter the modal patterns. Other partial or flexible constraints could lead to an infi nite number of conditions, complicating frequency and modal problems. One might apply fi nite element methods more practically to analyze these complicated systems. 2.5.4.1

Longitudinal Mode Horn Driving a Flexural Bar

The transverse wave equation for bars is given in Equation 1.40 as 4 2 YI 4 2 2 c o t 2 x 4 S x 4

(1.40)

where is the amplitude of lateral displacement from the rest position x; co the bar velocity of sound in the medium of the bar; κ the radius of gyration of the cross-sectional area S, or κ 2 = −RM/YS with R as the radius of curvature of the neutral axis or plane at position x, M the bending moment, Y Young’s modulus of the bar material, S the cross-sectional area at x; and I the moment of inertia of the area at x (=Sκ 2). In using Equation 1.40 for flexural bars, it is assumed that the displacement and slope at any position along the bar are small enough that variations in angular momentum may be neglected. Its derivation is based upon a balance between the bending moments (M) and shear forces (Fx) across a lateral element dx. These moments and shear forces across an element, dx, are represented by Mx − M(x+dx) − (Fy)(x+dx)dx = 0

(2.154)

where (M)(xdx) (M)x

M dx x

and (Fy )(xdx) (Fy )x

Fy x

dx

However, these terms for M(x+dx) and (Fy)(x+dx) are defined here by the first two terms of Taylor’s series, ignoring second-order and higher terms involving (dx)2, and so on, for small dx. Therefore, Equation 1.40 is still an approximation, even for small values of displacement and slope, but it is very nearly correct if the amplitude of vibration is small compared with the length of the bar. Because the amplitudes of flexural vibrations of bars as used in power applications of ultrasonics generally are very small compared with the lengths of the bars, use of Equation 1.40 is justified for the derivations that follow.


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76 2.5.4.2

Moments of Inertia

The moment of inertia of an area with respect to a given axis is the limit of the sum of the products of the elementary areas into which the area may be conceived to be divided and the square of their distance, y, from the given axis; that is I ∫ y 2 dA A 2

(2.155)

Referring to the rectangle of Figure 2.16, the area is A = 2by and dA = 2b dy. Therefore, the moment of inertia of the rectangle about the axis 0 − 0 is I∫

h 2

0

2 by 2 dy

2 by 3 3

h/2

0

bh 3 12

(2.156)

Equations for determining moment of inertia, section modulus (I/c), and radius of gyration of cross sections that might be used in flexural elements in ultrasonic applications are given in Table 2.3. Here c is the distance from the neutral axis to the outer fibers of the member. For elements other than those listed in Table 2.3 but for which the cross-sectional area can be defined in terms of their dimensions, use Equation 2.155. The performance of all beams in flexural vibration, whether they are uniform in cross section or complex in structure, is dependent upon the moment of inertia of a solid. The moment of inertia of a solid body with respect to a given axis is the limit of the sum of the products of the masses, dm, of each of the elementary particles into which the body may be conceived to be divided and the square of their distance from the given axis. Thus I m ∫ y 2 dm ∫ y 2

dm g

(2.157)

where dm = dw/g represents the mass of an elementary particle, dw the weight of the elementary particle, g gravity, and y the distance of the particle from the given axis. The moment of inertia of a solid of elementary thickness about an axis is equal to the moment of inertia of the area of one face of the solid about the same axis multiplied by the mass per unit volume of the solid times the elementary thickness of the solid. Polar moment of inertia is another important term in the design of vibrating structures. The polar moment of inertia is equal to the sum of the moments of inertia about any two axes at right angles to each other in the plane of the area and intersecting at the pole (Figure 2.17), that is Ip = Ix + Iy where Ix is the moment of inertia of area A about axis XX and IY the moment of inertia of area A about axis YY. y O

O

h

I = bh3/12

b FIGURE 2.16 Moment of inertia for rectangle.


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77

TABLE 2.3 Moment of Inertia (I), Section Modulus (I/c), and Radius of Gyration (κ = Î/(I/A)) of Various Cross Sections Relative to Flexural Elements Used in Ultrasonic Applications Cross Section

h

Moment of Inertia, I

Section Modulus, I/c

Radius of Gyration, κ

bh 3 12

bh 2 6

bh 3 3

bh 3 3

bh 3 3

b2 4 bb1 b12 3 h 36( b b1 )

b2 4 bb1 b12 2 h 12(2 b b1 )

h 2( b2 4 bb1 b12 )

d 4 r 4 64 4

d 3 r 3 32 4

r d 2 4

4 (D d 4 ) 64

D4 d4 D 32

(R 4 r 4 ) 4

R 4 r4 R 4

c b

c

h

h 0.577 h 3

b b1 c h

c

6( b b1 )

1 2 b b1 h 3 b b1

b

d r R D

d r c1

c2

8   r4    8 9 

r

R 2 r2 2

I 0.1908r 3 c2

D2 d2 4 9 2 64 6

I 0.2587 r 3 c1 c1 = 0.4244r a

a 3 b 4

a 2 b 4

5 3 4 R 16

5 3 R 8

a 2

b c

5 R 24

R

(continued)


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78 TABLE 2.3

(Continued)

Cross Section

Moment of Inertia, I

R

c

b c

R

φ

Section Modulus, I/c

Radius of Gyration, κ

5 3 4 R 16

5 3 3 R 16

5 R 24

[A(12b2 a 2 )] 48

[A(12b2 a 2 )] 48 b

(12 b2 a 2 ) 48

a c = a/2 tan φ A = nb2 tan φ n = number of sides

Iy = x2A (Moment of inertia of area A about the axis Y−Y)

Y

x r

A

Ix = y2A (Moment of inertia of area A about the axis X−X)

y

Ip = (x2 + y2) A = r2A

X

X

Y FIGURE 2.17 Polar moment of inertia.

2.5.4.3

General Solution to the Transverse Wave Equation for Thin Bars 4 1 2 2 2 2 0 4 x c o t

(1.40a)

Assuming harmonic motion, Equation 1.40a also may be written 4ξ 2 0 x 4 2c o2

(1.40b)

A solution to Equation 1.40b is of the form 4 p4 Aepx p4 x 4

Aepx so that Equation 1.40b may be written

p4


2 0 2c o2

(1.40c)

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79

or p4

2 0 2c o2

(1.40d)

Letting q4 = ω2/κ 2co2 , Equation 1.40d simplifies to p4 − q4 = 0

(1.40e)

The four roots of Equation 1.40e are p1 = q

p2 = −q

p3 = jq p4 = −jq

Therefore, the general solution to Equation 1.40b is = A1eqx + A2e−qx + A3ejqx + A4e−jqx

(2.158)

or replacing the exponential functions with their trigonometric equivalents = B1 sinh qx + B2 cosh qx + B3 sin qx + B4 cos qx

(2.159)

The phase velocity of a wave in the flexural bar is v c o Because q2 v

2 2c o2

q also may be written q = ω/v Equation 2.159 is a useful general solution to Equation 1.40b that can be made specific to any uniform slender bar typical of those driven in flexure for ultrasonic power applications. The end conditions are first defined. The functions characterizing these conditions are then applied to Equation 2.159, first, at x = 0 to determine (initially) the constants Bn, and, second, at x = ℓ, to determine the allowable frequencies of resonance for the fundamental mode and the overtones. The nodal positions and positions of maximum displacement are located by a similar process, by noting the characteristics of each type of node and maximum displacement amplitude. In most cases, the solutions arrived at following this procedure are sufficiently accurate for designing ultrasonic power systems. In the case of bars with one free end, the equations lead to a first approximation for the location of the node nearest the free end that is ∼3.5% in error. The driven motion is assumed to be normal to the axis of the bars. In the present sense, a slender bar is one that (a) provides a cross-sectional dimension in the direction of motion at the coupling position that is small compared with a wavelength of sound in the material of the bar at the driven frequency and (b) exhibits negligible compressional deformation in response to the motion of the driving horn at the coupling position. The bars may be hollow or solid. (Excitation of plates and cylinders into their natural modes by horns vibrating in longitudinal modes are considered in Sections 2.6 and 2.7.) Bars vibrating in flexure typically used in ultrasonic power applications include (a) those driven initially in a free–free mode in an unloaded condition being suddenly forced against a load while the driving force remains active for a short period of time and


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80

(b) those driven initially in a clamped–free mode being suddenly impressed against the load while the driving force remains active for a brief period of time. The conditions at the ends, nodes, and antinodes along a bar that is vibrating in flexural resonance are explained by basic principles of strength of materials. These are summarized as follows: 1. Where the bar experiences a bending moment, M, at the point under consideration (end or boundary, node, antinode) M YI

d 2 dx 2

(2.160)

where Y is Young’s modulus of elasticity, I the moment of inertia of the cross section of the beam parallel to the direction of motion, YI the flexural rigidity of the bar, the amplitude of displacement at position x (measured from the neutral position), and x the distance from one end of the bar. 2. The bar experiences a shear, σx, that is a function of bending moment given by x

dM d 3 YI 3 dx dx

(2.161)

3. At all nodal positions =0 4. At free ends, nothing is located at x < 0 or at x > ℓ to cause shear or bending moment so that at x = 0, ℓ d 2 d 3 0 dx 2 dx 3

(2.162)

5. At nodal positions nearest a free end =0 but the bar does experience bending moment and shear across the nodal position so that d d 3 3 0 dx dx

(2.163)

6. At nodal positions intermediate to those nearest free ends and all other nodes between constrained ends (pinned, clamped), the bending moment at the nodal position is zero, so that both d2 = 0 and ____2 = 0 (2.164) dx 7. A sliding end is assumed to be one at which the end plane of the bar slides along a plane surface and these two planes maintain perfect parallelism during the entire vibration cycle. Under these conditions, the slope of the axis is zero (axis of the displaced bar is parallel to the axis at the equilibrium position) and the axial shear is also zero, so that d d 3 3 0 dx dx


(2.165)

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81

8. Equation 2.165 is also applicable at positions of maxima other than at free ends, as shear as a function of x is also zero at these points. These conditions are used to solve the equations describing the modes of bars in flexural vibration and for locating the positions of nodes and antinodes of the vibrating bar. The flexural bars of Section 2.5.4 (as well as the plates of Section 2.6) are assumed to be perfectly elastic, homogeneous, isotropic materials of uniform thickness, unless otherwise designated. The thicknesses are considered to be small compared with the wavelength and all other dimensions. 2.5.4.4

Specific Solutions to the Flexural Bar

2.5.4.4.1 Clamped–Clamped Bar Two types of nodes occur in the clamped–clamped bar. The first type is associated with the clamping conditions at each end (x = 0, ℓ), characterized by = 0 and d/dx = 0. These are the only nodes associated with the fundamental frequency. An additional node located between the ends and characterized by = 0 and d2/dx2 = 0 occurs for each higher overtone; that is, one node corresponding with the first overtone, two nodes with the second overtone, and so on. When x = 0 = B1(0) + B2(1) + B3(0) + B4(1) = 0 d/dx = B1q(1) + B2q(0) + B3q(1) − B4(0) = 0 Then B1 = −B3 B2 = −B4 Therefore the solution for the clamped–clamped bar may be written = B1(sin h qx − sin qx) + B2(cos h qx − cos qx)

(2.166)

d/dx = qB1(cos h qx − cos qx) + qB2(sin h qx + sin qx)

(2.167)

and

The resonance frequencies are determined by applying the boundary conditions at x = ℓ, where = 0, and d/dx = 0. Thus = B1(sin h qℓ − sin qℓ) + B2 (cos h qℓ − cos qℓ) = 0

(2.166a)

d/dx = qB1(cos h qℓ − cos qℓ) + qB2 (sin h qℓ + sin qℓ) = 0

(2.167a)

B1(sin h qℓ − sin qℓ) = −B2(cos h qℓ − cos qℓ)

(2.166b)

B1(cos h qℓ − cos qℓ) = −B2(sin h qℓ + sin qℓ)

(2.167b)

and

from which


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82

Dividing Equation 2.166b by Equation 2.167b sinh q sin q cosh q cos q cosh q cos q sinh q sin q

(2.168)

sin h2 qℓ − sin2 qℓ = cos h2 qℓ − 2 cos h qℓ cos qℓ + cos2 qℓ

(2.169)

Cross multiplying

which, after substituting for q (=ωℓ/v), reduces to cos h (ωℓ/v) cos(ωℓ/v) = 1

(2.170)

Equation 2.170 is valid only at discrete values of ωℓ/v. The function cos h θ increases from 1 to ∞ as θ increases from 0 to ∞. The function cos θ fluctuates between 1 and −1 as θ increases from 0 to ∞. Therefore, Equation 2.151 is not in a satisfactory form for determining the allowed values of ωℓ/v and, consequently, of the allowed resonant frequencies. This problem is easily solved by using the identities cosh 2

1 tanh 2 1 tanh 2

cos 2

1 tan 2 1 tan 2

(2.171)

by which Equation 2.170 is converted to tan

tanh 2v 2v

(2.172)

The fact that tanh θ is nearly unity for all values of θ > π and that periodically tan θ will equal the value of ±tan h θ simplifies considerably identification of the discrete values of ωℓ/2v and, therefore, of the allowed frequencies. A simple means of determining these allowed values of ωℓ/2v and, therefore, the allowed frequencies is to plot ±tan h(ωℓ/2v) and tan(ωℓ/2v) to the same scale as functions of (ωℓ/2v) and locating the intercepts between the two functions (Figure 2.18). For the clamped–clamped bar, the intersections occur at [3.0112, 5, 7, 9,… ] 2v 2 c 4

(2.173)

2.0 tan (ωᐉ/2v) tanh (ωᐉ/2v)

1.0

0

π/2

π

−1.0 3.0112π/4

3π/2

2π

5π/2

−tanh (ωᐉ/2v)

−2.0

FIGURE 2.18 Graphical determination of allowed frequencies for flexural resonance in clamped–clamped bars.


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83

and, therefore n2 2 fn 2 4 v 2n 2c

(2.173a)

where n = 1, 2, 3, 4, … is the number of the fundamental mode and the overtones. From Equations 2.173 and 2.173a fn

c [3.01122 , 52 , 7 2 , 92 , …] 8 2

(2.174)

The nodal positions for the overtones located between those at x = 0, ℓ are determined in a manner similar to that used in determining the allowed frequencies. As stated previously, those nodes lying between the ends are characterized by = 0 and d2/dx2 = 0. The second derivative of Equation 2.159 is d 2 q 2B1(sinh qx sin qx) q 2B 2 (cosh qx cos qx) dx 2

(2.175)

The conditions characterizing the nodes, that is d2 = ____2 = 0 dx B1(sin h qx − sin qx) = −B2(cos h qx − cos qx) B1(sin h qx + sin qx) = −B2(cos h qx + cos qx) lead to the equation which applies to overtones, which is tan qx = tan h qx or tan (ωx/v) = tan h (ωx/v)

(2.176)

Equation 2.176 is valid only at discrete values of (ωx/v) and only for values of (ωx/v) appearing in quadrants where tan(ωx/v) is positive. These values of ωx/v at the nodal positions corresponding to overtones of the clamped–clamped (fixed–fixed) bar are x [5, 9, 13, 17 ,…] v 4

(2.177)

The positions relative to length of the bar of these intermediate nodes corresponding to overtones are determined by comparing the allowed positions (ωx/v) with the corresponding allowed frequencies (ωℓ/2v). From Equation 2.173, those quantities corresponding to the overtones, [5, 7 , 9, 11, … ] v 2

(2.178)

For the first overtone x 5 v 4


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84

from the equation designating position (2.177) and 5 v 2 from the equation designating allowed frequencies (2.178). Then 5x 5 2 4 and x = 0.5 ℓ which is the only node occurring between the ends of the clamped–clamped bar at the first overtone (f2). The nodes corresponding to the higher-order modes are determined in a similar manner. The nodal positions, corresponding frequencies, positions of maximum displacement, and phase velocities are given in Table 2.5 for various conditions of flexural bars in resonance. The positions at which displacement is maximum for the clamped–clamped bar are characterized by = m, d/dx = 0, and d3/dx3 = 0. (Positions of maximum displacement and the locations of the nodes are important to the effective design of systems involving a horn coupled to a bar and driving the bar into flexural resonance by motion across its axis [usually normal to its axis].) Returning to Equation 2.166 and applying the conditions characterizing the maxima of displacement m = B1(sin h qx − sin qx) + B2(cos h qx − cos qx) d/dx = qB1(cosh qx − cos qx) + qB2 (sinh qx + sin qx) = 0 3 d /dx3 = q3B1(cos h qx + cos qx) + q3B2(sin h qx + sin qx) = 0

(2.166) (2.167) (2.179)

From Equations 2.167 and 2.179, tan h qx = −tan qx or tan h (ωx/v) = −tan(ωx/v)

(2.180)

Equation 2.180 is valid for values of (ωx/v) lying in quadrants where tan(ωx/v) is negative. The allowed values of ωx/v are x [3.0112, 7 , 11, 15, …] v 4

(2.181)

Following the procedure for locating the nodes, that is, by comparing Equation 2.181 with the equation for allowed frequencies (Equation 2.178), the position of maximum displacement for the fundamental frequency is determined to be x 3.0112x 3.0112 v 2 4

(2.182)

or x = 0.5ℓ This is the only position of maximum displacement at the fundamental frequency. Positions of maxima for the overtones are determined in the same manner. Table 2.4 gives


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85

TABLE 2.4 Frequency, Phase Velocity, Nodal Positions, and Positions of Maximum Displacement for Uniform Bars in Flexural Resonance Under Various Boundary Conditions Clamped–clamped bar Displacement equation: = B1[sin h ωx/v − sin ωx/v] + B2[cos h ωx/v − cos ωx/v] Frequency (fn), nodal positions (ωxn/v) other than clamped ends, and positions of maximum displacement (ωxm/v) xm = 0.3ℓ, 0.7ℓ x m [3.0112, 7.11, 15, …] v 4 x n [5, 9, 13, 17 , …] 4 v xm = 0.214ℓ, 0.5ℓ, 0.786ℓ c fn 2 [1.194 2 , 2.988 2 , 52 , 7 2 , …] 8 a. n = l; f = f1; v = v1; xn = 0, ℓ; xm = 0.5ℓ b. n = 2; f2 = 2.756f1; v2 = 1.66v1; xn = 0, 0.55ℓ, ℓ c. n = 3; f3 = 5.404f1; v3 = 2.32v1; xn = 0, 0.357ℓ, 0.643ℓ

Clamped–free bar Displacement equation: = B1[sin h ωx/v − sin ωx/v] + B2[cos h ωx/v − cos ωx/v] Frequency (fn), nodal positions (ωxn/v) other than clamped end and nearest free end, and positions of maximum displacement (ωxm/v) x m [3.0112, 7 , 11, 15, …] 4 v a. n = 1; f = f1; v = v1; xn = 0; xm = ℓ b. n = 2; f2 = 6.263f1; v2 = 2.50v1; xn = 0.837ℓ; xm = 0.504ℓ, ℓ c. n = 3; f3 = 17.536f1; v3 = 4.18v1; xn = 0, 0.5ℓ, 0.9ℓ; xm = 0.3ℓ, 0.7ℓ, ℓ Free–free bar Frequency (fn), nodal positions (ωxn/v) other than nearest free ends, and positions of maximum displacement (ωxm/v) fn

c [3.01122 , 52 , 7 2 , …] 8 2

x n [1, 5, 9, 13, 17 , …] v 4 x m [3.00112, 7 , 11, 15, …] v 4 a. n = 1; f = f1; v = v1; xn = 0.17ℓ, 0.83ℓ; xm = 0.5ℓ b. n = 2; f2 = 2.756f1; v2 = 1.66v1; xn = 0.1ℓ, 0.5ℓ, 0.9ℓ; xm = 0.3ℓ, 0.7ℓ c. n = 3; f3 = 5.40f1; v3 = 2.32v1; xn = 0.071ℓ, 0.357ℓ, 0.643ℓ; xm = 0.215ℓ, 0.5ℓ, 0.785ℓ (continued)


10/29/2008 3:48:18 PM


86 TABLE 2.4

(Continued)

Bar pinned, hinged, or simply supported at each end Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn

c 2 2 2 2 [1 , 2 , 3 , 4 , …] 2 2

x n [0, 1, 2, 3, 4, …] v x m [1, 3, 5, …] 2 v a. n = l; f = f1; v = v1; xn = 0, ℓ; xm = 0.5ℓ b. n = 2; f2 = 2f1; v2 = 1.414v1; xn = 0, 0.5ℓ, ℓ; xm = 0.25ℓ, 0.75ℓ c. n = 3; f3 = 3f1; v3 = 1.732v1; xn = 0, 0.33ℓ, 0.66ℓ, ℓ; xm = 0.167ℓ, 0.5ℓ, 0.833ℓ Clamped–hinged bar Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn

2 c 2 2 [5 , 9 , 13 2 , …] 32 2

x n [0, 5, 9, 13, …] 4 v x m [3.0112, 7 , 11, …] 4 v a. n = 1; f = f1; v = v1; xn = 0, ℓ; xm = 0, 0.667ℓ b. n = 2; f2 = 3.24f1; v2 = 1.8v1; xn = 0, 5ℓ/9, ℓ; xm = 0.346ℓ, 0.778ℓ c. n = 3; f3 = 6.76f1; v3 = 2.6v1; xn = 0, 5ℓ/13, 9ℓ/13, ℓ; xm = 0.2316ℓ, 0.5385ℓ, 0.846ℓ Pinned–free bar Frequency (fn), nodal positions (ωxn/v) other than at pinned end and nearest free end, and positions of maximum displacement (ωxm/v) c [0, 52 , 92 , 13 2 , …] fn 32 2 x n [1, 2, 3, …] v x m [1, 3, 5, …] 2 v a. n = 1; f = f1; v = v1; xn = 0.8ℓ; xm = 0.4ℓ, ℓ b. n = 2; f2 = 3.24f; v2 = 1.8v1; xn = 0, 0.444ℓ, 0.889ℓ; xm = 0.222ℓ, 0.667ℓ, ℓ c. n = 3; f3 = 6.76f1; v3 = 2.6v1; xn = 0, 4ℓ/13, 8ℓ/13, 12ℓ/13; xm = 2ℓ/13, 6ℓ/13, 10ℓ/13, ℓ


10/29/2008 3:48:18 PM

Ultrasonic Horns, Couplers, and Tools TABLE 2.4

87

(Continued)

Clamped–sliding bar Frequency (fn), nodal positions (ωxn/v) other than at clamped end, and positions of maximum displacement (ωxm/v) fn

c [3.01122 , 7 2 , 112 , …] 32 2

x n [0, 5, 9, 13, …] v 4 x m [3.0112, 7 , 11, 15, …] v 4 a. n = 1, f = f1; v = v1; xn = 0; xm = ℓ b. n = 2; f2 = 5.504f1; v2 = 5.404f1; xn = 0, 7143ℓ; xm = 0.4302ℓ, ℓ c. n = 3; f3 = 13.345f1; v3 = 3.653v1; xn = 0, 0.4545ℓ, 0.8182ℓ; xm = 0.2738ℓ, 0.6364ℓ, ℓ Free–sliding bar Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn

c 2 [7 , 112 , 152 , …] 32 2

x n [5, 9, 13, …] 4 v x m [3.0112, 7 , 11, 15, 19, …] v 4 f

1.14 2

Yb2 1.14cb 2 12 12

a. n = 1; f = f1; v = v1; xn = 0.143ℓ, 0.714ℓ; xm = 0, 0.430ℓ, ℓ b. n = 2; f2 = 2.469f1; v2 = 1.571v1; xn = ℓ/11, 5ℓ/11, 9ℓ/11; xm = 0, 0.274ℓ, 0.6364ℓ, ℓ c. n = 3; f3 = 4.592f1; v3 = 2.143v1; xn = ℓ/15, ℓ/3, 3ℓ/5, 13ℓ/15; xm = 0, 0.2008ℓ, 0.4667ℓ, 0.733ℓ, ℓ

Pinned–sliding bar Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn

c 2 2 2 2 [3 , 5 , 7 , 9 , …] 8 2

x n [0, 1, 2, 3, …] v x m [1, 3, 5, …] v 2 a. n = 1; f = f1; v = v1; xn = 0, 0.667ℓ; xm = 0.333ℓ, ℓ

(continued)


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88 TABLE 2.4

(Continued) b. n = 2; f2 = 2.778f1; v2 = 1.667v1; xn = 0, 0.4ℓ, 0.8ℓ; xm = 0.2ℓ, 0.6ℓ, ℓ c. n = 3; f3 = 5.444f1; v3 = 2.333v1; xn = 0, 2ℓ/7, 4ℓ/7, 6ℓ/7; xm = 0.1429ℓ, 0.4286ℓ, 0.7143ℓ, ℓ

Fundamental resonant frequencies of wedge-shaped and conical bars rigidly mounted at the large end may be determined by the following equations: a. Wedge-shaped bar vibrating in a direction normal to parallel sides, frequency, Hz b. Wedge-shaped bar vibrating in a direction parallel to parallel sides, frequency, Hz f c.

0.85 2

0.85 cb Yb2 2 12 12

Conical bar, frequency, Hz f

1.39 2

Ya 2 1.39 ca 4 2 2

ℓ = length of bar c = bar velocity of sound b = thickness of the bar at the base in the direction of vibration a = radius of the cone at the base Phase velocity in every case is determined by (vn)2 = ωncκ.

nodal positions, corresponding frequencies, positions of maximum displacement, and phase velocities for flexural bars subjected to various boundary conditions. The overtones of none of the conditions given in Table 2.4 are harmonics. Boundary conditions shown in Table 2.4 are those corresponding to flexural modes that might be of most interest in the design of systems for ultrasonic power applications. Only the lower three modes are included, but these are sufficient for identifying the boundary conditions and characteristics of nodal and antinodal positions applicable to all overtones. The accuracy of the data of Table 2.4 as applied to real conditions of ultrasonic applications depends upon how closely the real conditions match those of the table. It may be difficult to completely satisfy certain boundary conditions. For example, it is difficult to imagine an application wherein the sliding contact end conditions leading to the listed data are exactly applicable. Those data listed are for the purpose of completeness and should sufficiently accurate for initial design purposes. 2.5.4.4.2

Rayleigh Method of Determining Fundamental Frequencies of Systems with Distributed Mass The fundamental frequency and only the fundamental frequency of an elastic system with distributed masses can be determined with good accuracy by a method derived by Rayleigh [5]. The method is based upon the fact that the total energy in a lossless vibratory system remains constant throughout each cycle of vibration. The potential energy of an elastic beam in flexure depends upon its elastic properties and the deformation it experiences at any instant of time. The kinetic energy of the beam is a function of its mass and velocity distribution at any instant. Thus, obtaining an accurate expression for the fundamental frequency of the beam requires an accurate description of the deflection curve associated with the fundamental mode of vibration of the beam. An initial curve is assumed, considering the boundary conditions, to obtain a reasonable characterization of the actual


10/29/2008 3:48:19 PM


89

deflection curve. If the curve is accurate, the method gives an accurate frequency equation. If it is inaccurate, the correct curve usually can be obtained by iteration. The Rayleigh method may be explained in detail by considering the spring/mass oscillator, Equation 1.3, writing it in the form M

d 2 k 0 dt 2

(1.3a)

where M is the lumped mass attached to the end of a weightless spring (the spring can be of any form such as coil spring, cantilever, etc.), the displacement, and k the spring constant. Equation 1.3 is recognized as a force balance equation in which the force k is what the spring exerts on the mass to accelerate it at the rate of d2/dt2 and the force Md2/dt2 is the familiar F = Ma, which is the force required to accelerate the mass at rate a. The potential energy of a body is energy of position. It is the energy possessed by a body held in such a position that it can do work upon being released. The kinetic energy of a body is energy of motion. The potential energy of a mass on a spring is the amount of work without loss required to move the mass against the force of the spring to its current position, that is P.E. = (k/2) = k2/2 The kinetic energy of the mass is given by the familiar equation K.E. = Mv2/2 Because the total energy of a lossless vibratory system remains constant, Mv 2 k 2 C 2 2

(2.183)

where v = d/dt the instantaneous velocity of the mass at position . Differentiating both sides of Equation 2.183 with respect to t gives Mv

d dv k 0 dt dt

(2.184)

Substituting d/dt = v, d2/dt2 = dv/dt into Equation 2.184 leads to Equation 1.3a. Assuming harmonic motion, that is, = m sin ωt, ω2 = k/M

(1.5)

Assuming simple harmonic motion, the frequency of a lossless mass-loaded spring may also be determined by the fact that the energy of the system oscillates between maximum potential energy (when kinetic energy is zero) to maximum kinetic energy (when potential energy is zero): 2 Maximum potential energy (1/2)km

Maximum kinetic energy = (1/2) M where ωm = Vm, the maximum velocity of the mass.


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90

Equating these two maxima and simplifying also leads to Equation 1.5a. In this case, note that Equation 1.5a was derived by dividing the equation for maximum potential energy by the equation for maximum kinetic energy and, as these two are equal, the quotient is 1. The denominator contains the factor ω2; therefore, multiplying both sides by ω2 obtains 2

maximum potential energy 2 k maximum kinetic energy M

This equation is the basis of the Rayleigh method. To determine the total potential energy of an elastic system with distributed masses, such as a flexural bar, requires an accurate identification of the deflection curve associated with the vibration. In some cases, such as that of a simply supported elastic beam of uniform cross section, it is a simple matter to obtain the correct beam deflection curve. Other systems require iteration. Using a cantilever beam with a mass load at the free end for purposes of illustration, assume the following: 1. 2. 3. 4. 5. 6.

The beam is clamped at x = 0. The length, ℓ, lies along the x-axis. Bending is restricted to within the xy plane. The mass of the beam is m. The displacement along the beam is = (x). The mass at the end of the beam is M.

The boundary conditions are: 1. At x = 0, d/dx = 0. 2. At x = ℓ, the mass, M, is vibrating at the maximum displacement, m, at the lowest, or fundamental, mode. The potential energy of the bar is P.E. ∫ x

d 2 dx dx 2

(2.185)

where Mx is the bending moment at x in terms of x. The general differential equation of the elastic curve of a beam, or bar, is EI

d 2 Mx dx 2

(2.186)

Therefore, the potential energy of the beam is given by 2

P.E.

1  d 2  YI  2  dx ∫  dx  2

(2.187)

The kinetic energy of the beam is given by K.E.


1 2 2 dm 2∫

(2.188)

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91

where dm = ρS dx is the mass per unit length of the bar and S the cross-sectional area of the bar. The maximum potential energy of the bar at the lowest resonance frequency is 2

(P.E.)max

1  d 2  YI dx 2 ∫0  dx 2 

(2.189)

and the maximum kinetic energy of the bar is (K.E.)max

2 2

∫0 2 dm

(2.190)

The total energy of a lossless vibratory system remains constant, that is Total Energy = Potential Energy + Kinetic Energy = Constant For the mass-loaded beam, the potential energy of the beam is converted to kinetic energy, which is shared by both the beam and the mass load, M. Therefore, the frequency equation for the mass loaded bar is 2

 d 2  YI ∫0  dx 2  dx

2

∫0 2 dm Mm2 2

 d 2  YI∫  2  dx 0  dx 

S x ∫ 0

2

(2.191)

2 dx Mm

Equation 2.191 is applicable to beams of any taper. For the uniform, mass-loaded cantilever, Sx = S is constant, so that

2

 d 2  YI∫  2  dx 0  dx 

2 S∫ 2 dx Mm

(2.192)

0

The displacement curve, , usually assumed for the uniform cantilever is

m ( 3 x 2 x 3 ) 2 3

(2.193)

This curve meets the criteria of the boundary conditions—when x = 0, = 0, and d/dx = 0. Taking the derivatives of required by Equation 2.192 and solving for ω2 leads to 2

YI∫

0

S∫

0


2 9m ( 2 2x x 2 )dx 6

2 m 2 (9 2x 4 6x 5 x 6 )dx Mm 4 6

3YI  3  33m  140 M 

(2.194)

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92

For the unloaded flexural, uniform bar (M = 0), 2

3YI YI 12.7273 33S 4 S 4 140

(2.195)

or 3.56753

YI S 4

(2.195a)

In Table 2.4, the lowest mode resonance frequency listed for the fixed–free (cantilever) bar is given by f11

c (1.194)2 8

(2.196)

Equation 2.196 may be written  YI  YI 11 2f11 2  (1.194 2 ) 3.5176 4 S S 4 8  

(2.196a)

which is within 0.074% of what is usually claimed as the exact equation. As stated previously, Equation 2.195a would have been the exact equation if the assumed bending curve had been exact. A more accurate expression than that of Equations 2.195 and 2.195a for the cantilever with a mass, M, at the free end is obtained by rewriting Equation 2.194 in the form 2

3YI [Km M] 3

(2.194a)

and determining the value of the constant, K, by assuming M = 0 and utilizing the more accurate Equation 2.196a as follows: 02

3YI 3YI YI 2 11 (3.5176)2 3 4 Km KS S 4

(2.196b)

K = 0.2424534 Then the frequency of the mass-loaded, uniform cantilever beam is determined by the relationship 2 m

3YI [0.24245S M] 3

(2.196c)

For the special case in which M = ρSℓ = m, the mass of the beam,  YI  2 m 2.4146   M 3 


(2.196d)

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Ultrasonic Horns, Couplers, and Tools 2.5.4.5

93

Stresses in Bars in Flexure

The operating life of flexural bars used for high-intensity ultrasonic applications is a function of the maximum stresses to which the bar will be subjected during use. The maximum stresses in a uniform bar in flexure appear in the outer fibers of the bar. These stresses alternate between compression and tension, and because the frequency of vibration is ultrasonic, fatigue failure can occur rapidly if the displacement amplitude is excessive. Bending stresses in a beam are compressive on the side from the neutral axis nearest the center of curvature and tensile on the side of the neutral axis away from the center of curvature. In a uniform bar, the stress is s

My d 2 Yy 2 I dx

(2.197)

where s is the stress at y and y the distance from the neutral axis. The maximum stress occurs at the point where y = c, where c is the distance from the neutral axis to the outer fiber. Therefore, maximum stress, sm, is sm

Mc d 2 Yc 2 I dx

(2.198)

The steps to determine the maximum stress in flexural bars under any of the conditions discussed in the following section are: 1. 2. 3. 4. 5.

Determine the equation describing displacement, . Obtain the second derivative of displacement with respect to x. Determine the distance, c, between the neutral axis and the outer fibers. Obtain Young’s modulus of elasticity for the materials of the bar. Use these quantities in Equation 2.198 to obtain an equation of stress as a function of displacement, .

The resulting equation is useful for determining the safe operating conditions (displacement) for the bar based upon its endurance limit. For example, assume that a bar is operated in a pinned–pinned mode. The equation of motion at resonance is (Table 2.4) msin

x v

(2.199)

and d 2 2 x sin m dx 2 v2 v

(2.200)

Assume that the cross section of the bar is rectangular with thickness, ha, parallel to displacement. The distance, c, for this bar is c = ha/2. Therefore, the equation for the maximum (outer-fiber) stress at position x is sm

Yh 2 x m 2 sin 2 v v

(2.201)

for the uniform bar of rectangular cross section vibrating in a pinned–pinned mode.


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94 2.5.4.6

Coupling between Driver and Flexural Bar

Careful consideration should be given to conditions of coupling between a driving horn and a flexural bar. Conditions to consider include: (a) coupling position and (b) coupling method. Position and method of coupling should be designed so that they do not interfere with the natural, or designated, modal pattern. Interferences can be caused not only by restricting the freedom to conform to the modal pattern that the bar would exhibit if it were freely vibrating, but also by introducing distortions in stress patterns. For example, welding a horn to a flexural bar will alter the stiffness of the bar at the position affected by the weld and thus tend to alter the modal pattern. Placing this type of couple between a node and an antinode will restrict the slope of the displacement curve and thus alter the modal pattern. Bolt holes alter the stress distribution within the bar, and thus cause a change in the stiffness at the cross section in which the holes are located. The tendency is to lower the frequency by an amount that depends upon the location of the coupling—whether it is a position of high stress or low stress. The nodal positions and positions of maximum displacement given in Table 2.4 should be helpful in locating preferred positions for coupling between a horn and a flexural bar.

2.6

Plates

The plate wave equation may be written in the form ∇ 4

3(1 2 ) 2 0 Yh 2 t 2

(2.202)

where ∇4 = ∇2∇2 where ∇2 is the Laplace operator, Y the Young’s modulus of elasticity, the displacement amplitude, h the half thickness of the plate, σ Poisson’s ratio, ρ the density of the material of the plate, and t the time. Equation 2.202 may also be written in the form D∇ 4 2h

2 0 t 2

(2.202a)

where D is flexural rigidity or D

2Yh 3 3(1 2 )

(2.203)

Letting ha = 2h (the plate thickness) and ρa = 2hρ (the mass density per unit of area of the plate), Equation 2.202a may also be written as D∇ 4 ρa


2 0 t 2

(2.202b)

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95

where D

Yh a3 12(1 2 )

(2.203a)

Equation 2.202a resembles the other wave equations—for example, 4 2 YI 4 2 Y 4 2 2 c o t 2 x 4 S x 4 ρ x 4

(1.40)

which is the flexural bar equation. Rearranged 2Y

4 ξ 2 2 0 4 x t

(1.40a)

or D

4 2 0 x 4 t 2

(1.40b)

where D is the flexural rigidity of a uniform bar (=κ 2Y). The bar equation can be derived from the plate wave equation, Equation 2.202a. The stresses developed in plates in flexure are more complicated than those for the flexural bar. An incremental element in a plate under stress is subjected to constraints in all directions, which accounts for the term involving Poisson’s ratio in Equation 2.202. The following procedures are based upon the assumption that the motion (x, y) of a unit volume within a resonant plate is harmonic regardless of the mode of vibration, that is, the motion of the volume (or particle) at x, y is described by (x , y )sin t or (x , y )cos t

(2.204)

Either part of Equation 2.204 when inserted into Equation 2.202a leads to ∇ 4

2h 2 0 D

(2.205)

which may be written (∇4 − k4) = 0

(2.205a)

where k4

2h 2 3 2(1 2 ) D Yh 2

Equation 2.205a is a homogeneous linear equation. It can be factored and written in the form (∇2 + k2) (∇2 − k2) = 0

(2.205b)

The complete solution to Equation 2.205b is a combination of solutions to (∇2 + k2) = 0


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96 and

(∇2 − k2) = 0 following common procedures for solving homogeneous linear equations with constant coefficients. From Equations 1.45a and 1.45b, the Laplace operator may be written as follows: 1. In polar coordinates in a two-dimensional system, ∇2

1   1  2  2 1 1  2  2  2  r  2  2  2 r r r r   r r r r  

(1.45)

where r and φ are the polar coordinates of distance and direction. 2. In rectangular coordinates in a two-dimensional system, ∇2

2 2 2 2 x y

(1.45)

Morse and Ingard [10] provide solutions to Equation 2.205 in polar coordinates. Leissa [11] provides solutions in polar, elliptical, and rectangular coordinates. He also presents the Laplace operator in skew coordinates but finds no general solutions to Equation 2.205 in skew coordinates that allow a separation of variables. Timoshenko [12] and Ensminger [3] also offer solutions in polar and rectangular coordinates. The present purposes will be satisfied by including solutions in polar coordinates for circular plates and in rectangular coordinates for square or rectangular plates. Lamb wave solutions are presented in rectangular coordinates only. 2.6.1

General Solution to the Plate Equation in Polar Coordinates

Leissa [11] gives the general solution to the plate wave equation, Equation 2.205, in polar coordinates as follows: (r, )

∑ [Am Jm (kr) BmYm (kr) CmIm (kr) DmK m (kr)]cos(m )

m0

∑

(2.206) [A *

m Jm

(kr) B *

m Ym

(kr) C *

mI m

(kr) D *

m K m (kr)]sin(m )

0

where  2  k a   D 

1/ 4

 12a 2 (1 2 )    Yh a3  

1/ 4

 3 2(1 2 )    Yh 2  

1/ 4

where ρa is mass density per unit area of the material of the plate; ha the thickness of the plate; ρ the density of the material of the plate; H the half-thickness of the plate; Jm and Ym Bessel functions of the first and second kinds, respectively; Im and Km modified Bessel functions (hyperbolic Bessel functions) of the first and second kinds, respectively; Am. and Dm constants that determine the mode shape and are solved by applying the boundary conditions; and r and φ the polar coordinates of distance and direction.


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Ultrasonic Horns, Couplers, and Tools 2.6.2

97

General Solution to the Plate Equation in Rectangular Coordinates

The general solution to Equation 2.205 presented by Leissa [11] in rectangular coordinates is

(x, y) ∑ [A msin k 2 2 y Bmcos k 2 2 y m1

Cm sinh k 2 2 y D m cosh k 2 2 y]sin x

* sin k 2 2 y B * cos k 2 2 y ∑ [A m m m=1

* sinh k 2 2 y D * cosh k 2 2 y]cos x Cm m

(2.207)

where α = ω/v and v is the phase velocity in the plate. 2.6.3

Specific Solutions to the Plate Wave Equations

2.6.3.1

Circular Plate of Uniform Thickness and Radius a Clamped at the Outer Circumference [3,10]

From Equation 2.206, (a, φ) = [AJm(ka) + BYm(ka)]cos(mφ) = 0 = [AJm(ka) + BYm(ka)]sin(mφ) = 0

(2.206a)

from which  J (ka)  B A  m   Ym (ka) 

(2.208)

and Ym (ka)

d d J m (kr) J m (ka) Ym (kr) dr dr

(2.209)

These conditions can be satisfied only at discrete frequencies. These frequencies are fixed by the values βmn [11], from Table 2.5, where βmn = (a/π)kmn. k2

h

3(1 2 ) Y

Therefore fmn

h 2a 2

Y (mn )2 3ρ(1 2 )

mn n → n -

(2.210)

m 2

where h is the plate thickness, m the number of nodal diameters, and n the number of nodal circles, including the clamped edge.


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98 TABLE 2.5

Values of βmn for Clamped Circular Plate β01 = 1.0174 β11 = 1.4677 β21 = 1.8799 β31 = 2.2741 β41 = 2.6568 β51 = 3.0321 β61 = 3.4018 β71 = 3.7678 β81 = 4.1288 β91 = 4.4876 β07 = 7.0019 β17 = 7.4946 β27 = 7.8752 β37 = 8.4454

β02 = 2.0074 β12 = 2.4824 β22 = 2.9274 β32 = 3.3538 β42 = 3.7677 β52 = 4,1722 β62 = 4.5694 β72 = 4.9607 β82 = 5.3472 β92 = 5.7296 β08 = 8,0016 β18 = 8.4954 β28 = 8.9779 β38 = 9.4516

β03 = 3.0047 β13 = 3.4881 β23 = 3.9477 β33 = 4.3911 β43 = 4.8224 β53 = 5.2442 β63 = 4.6584 β73 = 6.0664

β04 = 4.0030 β14 = 4.4910 β24 = 4.9590 β34 = 5.4129 β44 = 5.8556 β54 = 6.2893

β05 = 5.0027 β15 = 5.4928 β25 = 5.9668 β35 = 6.4273

β09 = 9.0015 β19 = 9.4958 β29 = 9.9803 β39 = 10.4559

From the values of βmn, Y  h f01 0.9387  2   a  (1 2 )

(2.210a)

and 2

  fmn  mn  f01 0.9661(mn )2 f01  01 

(2.210b)

Values of βmn and radii of nodal circles in the clamped-edge circular plate are independent of Poisson’s ratio. The relative radii of nodal circles, rn/a, are determined by the equation  r   r  Jm  k n  Im  k n   a  a J m (ka) I m (ka)

(2.211)

Leissa [11] tabulates relative values of radii of nodal circles (r n/a) for the clamped plate. 2.6.3.2

Free Circular Plate of Uniform Thickness and Radius a

The vibrational modes and corresponding frequencies of resonance for a free circular plate with m nodal diameters and n nodal circles can be determined by applying the following values of βmn, from Table 2.6, to Equation 2.210. Values for relative radii of nodal circles (rn/a) for a freely vibrating circular plate (Poisson’s ratio = 0.33) are given by Leissa [11]. 2.6.3.3

Circular Plate with Fixed Center

The frequencies of vibration for circular plates with centers fixed corresponding to modes having nodal diameters are the same as those for vibrations of corresponding modes in a free plate. The values of βmn for a circular plate with its center fixed and with n nodal circles and the corresponding relative radii (rn/a) (with σ = 1/3) are listed in Table 2.7.


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99

TABLE 2.6 Values of βmn for Free Circular Plate of Uniform Thickness and Radius a (Poisson’s ratio = 0.33) β01 = 0.9594 β02 = 1.9763 β03 = 2.9826 β04 = 3.9884 β05 = 4.9915 β06 = 5.9940 β07 = 6.9963 β08 = 7.9959 β09 = 8.9953

β11 = 1.4419 β12 = 2.4627 β13 = 3.4724 β14 = 4.4813 β15 = 5.4847 β16 = 6.4868 β17 = 8.1266 β18 = 8.4894 β19 = 9.4886 β20 = 0.7283

β21 = 1.8899 β22 = 2.9156 β23 = 3.9501 β24 = 4.9589 β25 = 5.9618 β26 = 6.9680 β27 = 9.6915 β28 = 8.9728 β29 = 9.9728 β30 = 1.1132

β31 = 2.3154 β32 = 3.3581 β33 = 4.4118 β34 = 5.4272 β35 = 6.4327 β36 = 7.4392 β37 = 8.4415 β38 = 9.4442 β39 = 10.441 β40 = 1.4794

β41 = 2.7215 β42 = 3.8038 β43 = 4.8515 β44 = 5.8728 β45 = 6.8854 β46 = 7.8938 β47 = 8.9002 β48 = 9.9060 β49 = 10.911 β50 = 1.8313

β51 = 3.1155 β52 = 4.2108 β53 = 5.2747 β54 = 6.3054 β55 = 7.3246 β56 = 8.3395 β57 = 9.3585 β58 = 10.3683 β59 = l 1.3748

TABLE 2.7 Values of βmn and Relative Radii of Nodal Circles (rn/a) for a Circular Plate with Fixed Center

2.6.3.4

n

βmn

rn/a

0 1 2 3 4 5 6

0.6166 1.4556 2.4902 3.4956 4.5005 5.4967 6.4969

— 0.8682 0.9050, 0.5075 0.9288, 0.6447, 0.3615 0.9287, 0.7214, 0.5008, 0.2808 0.9549, 0.7719, 0.5907, 0.4100, 0.2299 0.9608, 0.8079, 0.6531, 0.4997, 0.3469, 0.1945

Circular Plates Simply Supported All Around

The boundary conditions for a simply supported circular plate are, at the circumference: 1. The displacement parallel to the axis is (a) = 0. 2. Bending and twisting moments are zero. 3. ∂2/∂θ2 = 0. These conditions, when applied to Equation 2.209, lead to AnJn(mn) + CnJn(mn) = 0         A n  J′′n (mn )  mn  J′n (mn ) C n I′′n (mn )  mn  I′n (mn ) 0        

(2.212)

where 2 2mn a 2

3(1 2 ) Yh 2

(2.213)

The frequency equation is J n1(mn ) I n1(mn ) 2mn J n (mn ) I n (mn ) 1


(2.214)

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100

The roots of Equation 2.214 for σ = 0.3 are β01 = 0.7101 β02 = 1.7365 β03 = 2.7419 β04 = 3.7439

β11 = 1.1885 β12 = 2.2170 β13 = 3.2274 β14 = 4.2329

β21 = 1.6121 β22 = 2.6658 β23 = 3.6892 β24 = 4.7024

These values of βmn may be applied to Equation 2.210 to obtain the allowed frequencies for the various modes of vibration. The relative radii of nodal circles (rn/a) of a simply supported circular plate for σ = 0.3 are given by Leissa [11]. 2.6.4

Annular Plates of Uniform Thickness

An annular plate is a circular plate containing a concentric hole. There are nine possible basic (or simple) boundary conditions for the annular plate: Center clamped

Center simply supported

Center free

Outer circumference clamped Outer circumference simply supported Outer circumference free Outer circumference clamped Outer circumference simply supported Outer circumference free Outer circumference clamped Outer circumference simply supported Outer circumference free

One may easily visualize innumerable combinations of these basic conditions, such as center clamped over only a small portion of its circumference and the outer circumference clamped over the same segment of the plate or over a different segment of the plate, and so on. Only a few conditions have been selected for discussion for their potential benefit to ultrasonic power applications. For example, an annular plate simply supported or clamped at the internal circumference and free at the external circumference might represent a wheel driven axially at an ultrasonic frequency. The area surrounding the inner circumference of such a wheel is subjected to concentrated stresses by displacements along the axis of the center hole, causing serious fatigue problems under the influence of high-power ultrasonic energy. Fatigue damage may be minimized by using materials of high fatigue resistance and by careful geometrical design. Consistent with the terminology used in the previous discussions, the radius of the outer edge of the plate is designated a. The inner radius is designated b. Values of the frequency constant, βmn, for various ratios, b/a, for various boundary conditions are presented in Tables 2.8 through 2.16. Frequencies of the various modes may be determined by Equation 2.210. 2.6.4.1

Annular Plates Clamped on Outside and Inside

The boundaries are nodal circles characterized by (r) = d/dr = 0 at r = a and r = b. These conditions applied to Equation 2.206 lead to 2

2

2 2 2 2    r    r    r   r  r  (r) A 1    1    ln   B 1    1     b    b    a   a  a         

2

(2.215)

The allowed frequencies are given in Equation 2.210.


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101

TABLE 2.8 Frequency Parameters, βmn, for Annular Plates Clamped at the Inner and the Outer Circumferences Nodal diameters m 0 1 2 3 0 1 2 3

βmn for Values of b/a

Nodal circles n

0.1

0.3

0.5

0.7

0.9

2 2 2 2 3 3 3 3

1.6632 1.6963 1.9283 2.2776 2.7622 2.8220 3.0281 3.3687

2.1400 2.1729 2.2732 2.4657 3.5588 3.5572 3.6847 3.8330

3.0063 3.0231 3.0746 3.1671 4.9925 5.0127 5.0630 5.1227

5.0127 5.0228 5.0430 5.0930 8.3370 8.3370 8.3553 8.3855

15.0551 15.0584

24.9970 24.9970

The values of βmn for various ratios, b/a, that apply to the annular plate clamped at both the inner and the outer circumferences are given in Table 2.8. The values of n given in Table 2.8 include the nodal circles corresponding to the clamped conditions at the inner and the outer circumferences. 2.6.4.2

Annular Plate Clamped on Outside and Simply Supported on the Inside

The nodal circle corresponding to the clamped outer circumference is characterized by (r)

d 0 dr

The nodal circle corresponding to the simple support at the inner circumference is characterized by (r) = 0 The corresponding displacement equation is 2 2 2   r 2   r   r   r   r (r) A 1    1    ln   B 1    1     b    a   a   a   b       

2

(2.215a)

The frequency parameters, βmn, for various ratios of b/a that apply to the annular plate clamped at the outer circumference and simply supported at the inner circumference are given in Table 2.9. The values of n listed in Table 2.9 include both the clamped outer circumference and the simply supported inner circumference. 2.6.4.3

Annular Plates Clamped on Inside and Simply Supported on the Outside

The nodal ring corresponding to the clamped inner circumference is characterized by d = ___ = 0 dr The outer nodal ring corresponding to the simply supported circumference is characterized by =0


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102 TABLE 2.9

Frequency Parameters, βmn, for Annular Plates Clamped on the Outside and Simply Supported on the Inside βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

2 2 2 2 3 3 3 3

1.5132 1.5947 1.8939 2.2732 2.5781 2.6727 2.9639 3.3536

1.8478 1.9045 2.0824 2.3542 3.2461 3.2926 3.4283 3.6293

2.5445 2.5742 2.6632 2.8130 4.5240 4.5352 4.6127 4.6998

4.2108 4.2108 4.2468 4.3295 7.5191 7.5326 7.5527 7.5995

12.5319 12.5359 12.5440 12.5642 22.5169 22.5169 22.5237 22.5349

TABLE 2.10 Frequency Parameters, βmn, for Annular Plate Clamped on the Inside and Simply Supported on the Outside βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

2 2 2 2 3 3 3 3

1.3430 1.3875 1.6474 2.0132 2.4677 2.5225 2.7511 3.1074

1.7405 1.7837 1.9152 2.1448 3.1831 3.2148 3.3233 3.4869

2.4615 2.4861 2.5584 2.6821 4.4790 4.5016 4.5575 4.6237

4.1258 4.1503 4.1746 4.2348 7.4786 7.4854 7.5124 7.5527

12.4711 12.4751 12.4833 12.4954 22.4831 22.4831 22.4899 22.5012

The displacement equation is 2

2 2    r   r  r  (r) A 1    ln   B 1     b   a  b     

2

2   r  1    a  

(2.215b)

Frequency parameters, βmn, for various ratios, b/a, for annular plates clamped on inside and simply supported on outside are given in Table 2.10. The listed values of n include the nodal circle corresponding to the simple outer support and that corresponding to the clamped inner support. 2.6.4.4

Annular Plate Simply Supported on Inside and Outside Circumferences

The displacement of the annular plate, simply supported at inside and outside circumferences, is described by the equation (Leissa [11]) 2 2 2    r   r   r  r  (r) A 1    ln   B 1    1     a   b   a  b       

(2.215c)

The nodal circles corresponding to the simple supports at the inside and the outside circumferences are both characterized by =0


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103

Frequency constants, βmn, corresponding to various ratios, r/a, for the simply supported annular plate are given in Table 2.11. The values of n include the nodal circles corresponding to the inner and outer circumferences. 2.6.4.5

Annular Plate Clamped on Inside Circumference and Free on Outside Circumference

The one nodal circle associated with the clamped inner circumference is characterized by (r)

d 0 dr

The displacement, (r), is described by 2 2    r   r  r  (r) A 1    ln   B 1     b   b  b     

2

(2.215d)

The frequency constants, βmn, corresponding to various ratios, b/a, for the annular plate clamped at the inner circumference and free at the outer circumference are given in Table 2.12. The values of n listed in Table 2.12 include the nodal circle associated with the clamped inner circumference. TABLE 2.11 Frequency Parameters, βmn, for a Simply Supported Annular Plate βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

2 2 2 2 3 3 3 3

1.2121 1.3008 1.6199 2.0132 2.2887 2.3926 2.6953 3.0976

1.4621 1.5365 1.7493 2.0629 2.8789 2.9278 3.0746 3.3080

2.0132 2.0580 2.1845 2.3820 4.0137 4.0389 4.1135 4.2348

3.3385 3.3687 3.4283 3.5158 6.6693 6.6845 6.7072 6.7748

10.0053 10.0053 10.0306 10.0558 20.0004 20.0004 20.0105 20.0257

TABLE 2.12 Frequency Parameters, βmn, for Annular Plate Clamped at the Inner Circumference and Free at the Outer Circumference βmn for Values of b/a


m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

1 1 1 1 2 2 2 2

0.6547 0.5640 0.7546 1.1209 1.6011 1.6632 1.9362 2.3217

0.8215 0.8009 0.8981 1.1595 2.0776 2.1258 2.2710 2.5084

1.1477 1.1608 1.2204 1.3691 2.9364 2.9639 3.0481 3.1831

1.9362 1.9492 1.9955 2.0776 4.9210 4.9415 4.9925 5.0630

2.2843 5.9123 5.9295 5.9720 9.9137 14.8927 14.9097 14.9301

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104 2.6.4.6

Annular Plate Simply-Supported at the Inner Circumference and Free at the Outer Circumference

The nodal circle corresponding to the simple support at the inner circumference is characterized by =0 The displacement is described by 2 2 2   r  r   r   r  (r) A ln   B 1    C   1     b  b   a   b     

(2.215e)

Frequency constants, βmn, corresponding to various ratios, b/a, for the annular plate simply supported at the inner circumference and free at the outer circumference are given in Table 2.13. The values of n listed in Table 2.13 include the nodal circle associated with the simple support at the inner circumference. 2.6.4.7

Free–Free Annular Plate

Table 2.14 lists frequency constants, βmn, corresponding to various ratios, b/a, for the free–free annular plate. TABLE 2.13 Frequency Parameters, βmn, for Annular Plate Simply Supported on the Inner Circumference and Free on the Outer Circumference βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

1 1 1 1 2 2 2 2

0.5912 0.4827 0.7411 1.1209 1.4517 1.5626 1.9045 2.3173

0.5887 0.5800 0.7849 1.1299 1.7893 1.8696 2.0873 2.3969

0.6453 0.7017 0.8992 1.1910 2.4861 2.5325 2.6575 2.8524

0.7913 0.9192 1.1652 1.4412 4.1503 4.1746 4.2348 4.3295

1.3201 1.6199 2.0776 2.4942 12.4711 12.4751 12.4954 12.5238

TABLE 2.14 Frequency Constants, βmn, Corresponding to Various Ratios, b/a, for a Free–Free Annular Plate βmn for Values of b/a


m

n

0.1

0.3

0.5

0.7

0.9

2 3 0 1 2 3 0 1

0 0 1 1 1 1 2 2

0.7328 1.1209 0.9426 1.4412 1.8805 2.3173 1.9674 2.4450

0.7053 1.1145 0.9204 1.3617 1.8286 2.2732 2.2598 2.4408

0.6585 1.0747 0.9718 1.3201 1.7751 2.1915 3.0581 3.1237

0.6014 0.9995 1.1565 1.4930 1.9570 2.3756 5.0430 5.0630

0.5458 0.9082 1.8805 2.3756 3.0828 3.6984 15.0584 15.0652

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105

TABLE 2.15 Frequency Parameters, βmn, Corresponding to Various Ratios, b/a, for Annular Plates Clamped at the Outer Circumference and Free at the Center βmn for Values of b/a

2.6.4.8

m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 0

1 1 1 1 2 2 2 3

1.0166 1.4621 1.8696 2.2732 2.0005 2.4656 2.9069 3.0265

1.0747 1.4056 1.8146 2.2304 2.2887 2.4615 2.8292 3.6571

1.3392 1.4930 1.8006 2.1542 3.0828 3.1398 3.3080 5.0630

2.0897 2.1424 2.2843 2.4922 5.0630 5.0730 5.1227 8.3734

6.0395 6.0523 6.0813 6.1228 14.9943 14.9978 15.0146 25.0294

Annular Plate Clamped at the Outer Circumference and Free at the Inner Circumference

The displacement for this condition is described by 2 2    r   r  r  (r) A 1    ln   B 1     a   a  a     

2

(2.215f)

d 0 dr

The nodal circle associated with the clamped outer circumference is characterized by the frequency constants, βmn, corresponding to various ratios, b/a, for an annular plate clamped at the outer circumference and free at the inner circumference are given in Table 2.15. The values of n given in Table 2.15 include the nodal circle associated with the clamped outer circumference. 2.6.4.9

Annular Plate Simply Supported at the Outer Circumference and Free at the Center

The displacement for this condition is described by 2 2 2   r  r   r   r  (r) ln   B 1    C   1     a  a   a   a     

(2.215g)

The nodal circle corresponding to the simply supported outer circumference is characterized by (a) = 0 Frequency constants, βmn, corresponding to various ratios, b/a, are given in Table 2.16. The values of n listed in Table 2.16 include the nodal circle corresponding to the simple outer support.


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106 TABLE 2.16

Frequency Constants, βmn, Corresponding to Various Ratios, b/a, for Annular Plates Simply Supported at the Outer Circumference and Free at the Center βmn for Values of b/a

2.6.5

m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 0

1 1 1 1 2 2 2 3

0.7017 1.1867 1.6042 2.0132 1.7259 2.2053 2.6479 2.7530

0.6871 1.1388 1.5626 1.9827 1.9362 2.1542 2.5683 3.2926

0.7167 1.0841 1.5032 1.9019 2.5820 2.6613 2.8666 4.5352

0.8379 1.1608 1.5691 1.9414 4.2108 4.2468 4.3295 7.5191

1.3392 1.7347 2.2776 2.7474 12.5319 12.5440 12.5642 22.5169

Rectangular Plates

Equation 2.216 is the general solution to the plate wave equation in rectangular coordinates, that is

(x, y ) ∑ [A m sin k 2 2 y Bm cos k 2 2 y m1

C*m sinh k 2 2 y D*m cosh k 2 2 y]cos x

∑ [A*m sin k 2 2 y B*m cos k 2 2 y m0

Cm sinh k 2 2 y D m cosh k 2 2 y] sin x

(2.216)

There are 21 combinations of simple boundary conditions for rectangular plates, that is, any designated side may be clamped, simply supported, or free for its full length. Obviously, the ultrasonics engineer will encounter numerous mounting conditions that do not conform to these simple boundary conditions. Leissa [11] discusses the case of “a plate supported by (or embedded in) a massless elastic medium (or foundation).” He then modifies Equation 1.43 to include a constant K representing the stiffness of the foundation measured in units of force per unit length of deflection per unit of area of contact, as follows: D∇ 4 K a

2 0 2

(2.217)

The parameter k then is defined by k4

a 2 K D

where ρa = 2ρh.


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107

The results pertaining to the classical plate equation also apply to the case of the mounting in an elastic medium when Equation 2.217 is used instead of Equation 2.205b, that is k4

3 2(1 2 ) 2h 2 Yh 2 D

The case of an elastic mount having significant mass requires incorporating the conditions of the mounting coupled to the plate into the differential equation. The problem is related to impedance matching of elements in a combined resonant system. Solutions to equations for plates subjected to the various edge conditions range from fairly easy to very complicated. Assumptions in common with those made for flexural bars and circular plates used in obtaining the solutions that follow are that the materials are perfectly elastic, homogeneous, isotropic, and uniform in thickness. The thickness is considered to be small compared with all other dimensions and with the wavelength. In all of the following special solutions, the length dimension a of the plate is parallel to x and the width dimension b is parallel to y in an xy plane. For square plates, a equals b. The flexural rigidity is the same as it was for the circular plates, D

2Yh 3 3(1 2 )

(2.203)

The important stress–strain-related factors as functions of displacements in a resonant plate are as follows, in rectangular coordinates: 1. Bending and twisting moments:  2 2  M x D  2 2  y   x  2 2 M y D  2 2  x   y M xy D(1 )

(2.218)

2 xy

2. Transverse shearing forces: Qx D

(∇ 2) x

Qy D (∇ 2) y

(2.219)

3. The Kelvin–Kirchoff edge reactions: Μxy y Μxy Vy Qy x Vx Qx


(2.220)

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108

The solutions to the equation for rectangular plates depend upon the effects of the boundary conditions (clamped, simply supported, or free) on these quantities. The simple boundary conditions for rectangular plates are as follows: 1. For any side that is clamped: Assuming (a) that the edge is perfectly clamped so that the edge cannot move, (b) that the tangent plane to the deflected middle surface along this edge coincides with the initial position of the middle plane of the plate, and (c) that the x-axis coincides with the clamped edge and this clamping occurs at y = 0, ()y=0 = 0    y 

0 y0

2. For any side that is simply supported, for example, at y=0 Here, the edge is perfectly constrained so that ()y=0 = 0 The edge can rotate freely with respect to the x-axis, that is, there are no bending moments My along the edge or  2 2  (M y )y0 D  2 2  0 x   y 3. For any side that is free, at the free edge, the plate is assumed to experience no bending and twisting moments and no vertical shearing forces, therefore (My)y=0 = (Mxy)y=0 = (Qx)x=0 = 0 where Qx is vertical shearing force. Only two of these conditions are necessary for the complete determination of deflections satisfying the plate wave equation. The two boundary conditions are: a. From the requirement that bending moments along the free edge are zero,  2 2  (M y )x0 D  2 2  0 x  x0  y

(2.221)

b. The distribution of twisting elements Mxy is statically equivalent to a distribution of shearing forces of the intensity  M xy  Qx   y  x0


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109

Also the edge reactions Vx = 0. From Equations 2.220, M xy   Vx  Q x 0 y  xa  These requirements lead to the second boundary condition:  3 3   3 ( 2 )  0 xy 2  x0  x

(2.222)

4. For any side that is elastically supported and elastically built-in: Timoshenko [12] provides the boundary conditions for the “elastically supported and elastically built-in edge” which includes the edge x = a of a rectangular plate rigidly joined to a supporting beam. One of the boundary conditions of the plate along the edge x = a relates to the deflection curve of the beam described by the equation  4  2   2 B 4  D  2 ( 2 ) 2  x  x y  xa  y  xa

(2.223)

The second boundary condition relates to the twisting of the beam leading to C

 2 2   2  D 2 2    y  xy  xa y  xa  x

(2.224)

where B is the flexural rigidity of the beam and C the torsional rigidity of the beam. The following solutions, and approximate solutions, to the plate wave equation are intended to represent those boundary conditions of most value to the designer of ultrasonic power equipment. Some boundary conditions, including reactions from mounting structures, present very difficult mathematical conditions for analysis. Finite element analysis and other methods are available to the engineer who has the need to perform such studies. For example, Donnell’s model has been used successfully in analyzing modal conditions and frequencies in plates and cylinders. This method is discussed with examples of applications in Chapter 3. Approximate methods of matching systems often prove to be satisfactory, but experimentally determined parameters may be necessary to arrive at an acceptable design, particularly in the absence of a satisfactory finite element approach. 2.6.5.1

Rectangular Plate Simply Supported on All Sides

The rectangular plate having all sides simply supported (SSSS) leads to the simplest solution to the rectangular plate equation. According to Warburton [13], this is the only boundary condition for which the frequency factor can be expressed exactly by a simple formula. Timoshenko [12] gives the displacement function for this case in the form

(x, y ) ∑

∑ mn sin

m1 n1


ny mx sin a b

(2.225)

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110 The boundary conditions are

2 2 0 x 2 y 2

so that Μx 0 (for x 0, a) Μy 0 (for y 0, b) which, when applied to Equation 2.225 leads to mn A mn sin

ny mx sin a b

(2.226)

where Amn is an amplitude coefficient determined from the initial conditions of the problem and m and n are integers. For any mode of vibration, the nodal pattern is defined by m and n, the number of nodal lines in the x and y directions respectively, that is, the nodal lines m are normal to the x direction (side a) and nodal lines n are normal to the y direction (side b). The numbers m and n include also constrained boundaries. The frequency equation is [3] mn 2h

Y  m2 n2  2 2  3(1 )  a 2 b 

(2.227)

Because a equals b, the frequency equation for the square plate simply supported on all sides is [3] mn 2h

Y  m2 n2  2  3(1 )  a 2 

(2.227a)

which, for the lowest mode, is 11

2 2 h a2

Y 3(1 2 )

(2.227b)

The nodes for rectangular plates, in general, are straight lines parallel to the edges. In the case of square plates simply supported on all sides, the nodal lines follow the general pattern of rectangular plates. The nodal lines are always parallel to the edges. For square plates with other boundary conditions, the nodal lines may or may not be parallel to the sides depending upon the modes and method of excitation. Variations in nodal patterns are attributable to the fact that two mode shapes having the same frequency can exist simultaneously in the square plate. The relative amplitudes of these two modes depend upon the initial conditions. 2.6.5.2

Rectangular Plate Simply Supported on Two Opposite Sides and Clamped on the Remaining Two Sides

Figure 2.19 illustrates the SS-C-SS-C condition (simply supported on two opposite sides and clamped on the remaining two sides). Here the clamped edges are parallel to the x-axis (side a) and the simply supported edges are parallel to the y-axis (side b).


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111 F

F b

a

−F FIGURE 2.19 Plate clamped on two opposite sides and simply supported on the remaining two edges.

As stated previously, the boundary conditions at the simply supported edges are ()x=0,b = 0

(Displacement at x = 0, a)

Bending moments, Mx, along y are  2 2  (M x )x0 ,a D  2 2  0 x   y where D

2Yh 3 3(1 2 )

and the boundary conditions along the clamped edges are   ()y0 , b   0  y  y0,b

1 k 2 2

2 k 2 2 Applying these conditions to the general solution to the plate wave equation in rectangular coordinates (Equation 2.216) leads to the four homogeneous equations Bm + Dm = 0 Am λ1 + Cm λ 2 = 0


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112

Am sin λ1b + Bm cos λ1b + Cm sin h λ 2b + Dm cos h λ 2b = 0 Am λ1 cos λ1b − Bm λ1 sin λ1b + Cm λ 2 cos h λ 2b + Dm λ 2 sin h λ 2b = 0

(2.228)

where, as Leissa [11] points out, “for a nontrivial solution the determinant of the coefficients of Equations 2.228 must vanish”, that is 0

1 0

0

1 0

1

2 0 sin 1b cos 1b sinh λ 2 b cosh 2 b

1 cos 1b 1 sin 1b 2 cos h 2 b 2 sinh 2 b

(2.229)

leading to the characteristic equation 2 λ1λ 2(cos λ1b cos h λ 2b − 1) + (λ1λ 2) (sin2 λ1b − sin h2 λ 2b) = 0

(2.229a)

Various investigators have solved this problem for rectangular and square plates. Table 2.17 provides a summary of frequency constant data given in terms of ωa2√(ρh/D) for square plates for ratios of a/b.

TABLE 2.17 Frequency Parameters, λmn = ωa2√(ρh/D) for SS-C-SS-C Rectangular Plates ωa2√(ρh/D) for Values of n a/b

m

1

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0

1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6

12.139 13.718 15.692 18.258 20.824 24.080 28.946 56.25 95.28 42.58 54.743 78.975 115.6 91.7 102.21 123.3 156.0 160.7 170.3 189.23 219.20 249.43 258.5 275.85 303.6 358.0 366.8


2

3

4

23.65

38.7

58.63

69.32 146.03 253.6 51.65 94.584 170.1 276.8 100.23 140.19 212.85 318.0 168.98 206.6 275.18 378.4 257.5 293.8 360.0 458.8 366.0 400.9

129.1 280.13 492.0 66.3 154.77 305.325 516.4 114.35 199.8 348.3 558.0 182.63 265.2 410.85 619.2 271.0 351.1 493.425 698.4 379.25 457.4

208.4 458.33 808.4 86.15 234.58 483.98 834.4 133.78 279.63 527.63 877.2 201.73 344.6 590.63 938 289.75 429.8 672.525 1018.8 398 535.1

5 83.475

307.3 680.4 1202.8 110.95 333.93 706.74 1229.6 158.43 379.27 751.28 1272.8 226.05 443.8 814.5 1335.6 314.25 529.0 896.625 1416.4 421.5 633.7

6 113.225

425.9 947.03 1676.0 140.88 452.88 973.8 1704.4 188.05 498.5 1019.0 1748.4 252.25 563.5 1082.5 1811.6 343.75 647.9 1165.5 1893.2 450.5 752.2

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Ultrasonic Horns, Couplers, and Tools 2.6.5.3

113

Other Rectangular Plate Conditions

Previously, it was mentioned that there are 21 combinations of simple boundary conditions for rectangular plates, that is, any designated side may be clamped, simply supported, or free for its full length. Additional conditions may be encountered by the ultrasonics engineer that do not conform to these simple boundary conditions for the full length of any or all sides (e.g., clamping along only a portion of one side, and so on). Also, the ultrasonics engineer is always confronted with the types of loads to which the plates are subjected (air, vacuum, liquid), method of mounting driving elements (welding, bolting, etc.) and other geometrical (thickness variations) and structural differences (anisotropy, etc.) within the plates themselves. The engineer should be aware of the different methods available for solving with sufficient accuracy the many conditions that can be encountered in ultrasonics applications. Vibrating plates will react with the supporting structures. Two of the 21 combinations of simple boundary conditions for rectangular plates have been presented, relying heavily upon the fi ne work of Leissa [11]. The material presented in this chapter has been reworked for conciseness and convenience in use. Those previously presented are (a) rectangular plate having all sides simply supported (SSSS) and (b) rectangular plate having two opposite edges simply-supported and the remaining two edges are clamped (SS-C-SS-C), which was also fairly easy to solve. The remaining combinations of simple boundary conditions are: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Two opposite sides simply supported with the remaining two sides free (SS-F-SS-F) Simply supported-clamped-simply supported-simply supported (SS-C-SS-SS) Simply supported-clamped-simply supported-free (SS-C-SS-F) Simply supported on three sides and free on the remaining side (SS-SS-SS-F) Clamped on all four sides (C-C-C-C) Clamped on three sides with the fourth simply supported (C-C-C-SS) Clamped on three sides and free on the fourth (C-C-C-F) Clamped along two adjacent sides and simply supported on the remaining two sides (C-C-SS-SS) Clamped on two adjacent sides, simply supported on the third, and free on the fourth side (C-C-SS-F) Clamped on two adjacent sides and free on the two remaining sides (C-C-F-F) Clamped along two opposite sides, simply supported along one side, and free on the remaining side (C-SS-C-F) Clamped along one side, simply supported along two adjacent sides, and free on the fourth side (C-SS-SS-F) Clamped along one side, simply supported along one side adjacent to the clamped side, and free on the remaining two adjacent sides (C-SS-F-F) Clamped along two opposite sides and free on the remaining two sides (C-F-C-F) Clamped along one side, free along the two sides adjacent to the clamped edge, and simply supported along the remaining edge (C-F-F-SS) Clamped along one edge and free along the remaining three edges (C-F-F-F) (Cantilever) Simply supported along two adjacent edges and free along the remaining two edges (SS-SS-F-F)


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114

18. Simply supported along one edge and free along the remaining three edges (SS-F-F-F) 19. Free along all edges (F-F-F-F) The remaining 19 combinations of boundary conditions will not be treated here, for several reasons. Should the reader have need for solutions to any of these conditions, Leissa [11] offers sets and tables of solutions that are as complete as one might find in one volume. His book, Vibration of Plates, has been revised, reprinted, and published by the Acoustical Society of America through the American Institute of Physics (1993, ISBN 1-56396-294-2). Using the same basic equations, boundary conditions, and methods presented by Leissa and other investigators, modern computer techniques and programs available commercially make possible solutions to meet specific needs of the designer of ultrasonic systems. Finite element methods and Donnell’s model (Chapter 3) are examples of techniques available to the average engineer. A more complete discussion of these problems goes beyond the objectives of this chapter. Including all of the various conditions, as one can imagine, would require a large volume.

2.7

Rings and Hollow Cylinders

The vibration principles of rings, hollow cylinders, and bells are closely related. Elementary design principles are presented in the following sections. See Chapter 3 for a better technique for designing and analyzing the more complicated modes of vibration possible in large tubular sections. A ring may be considered as an element of a hollow cylinder, although the cross section of the material from which the ring is formed may be of any geometrical shape with dimensions small compared with the radius, r, of a centerline circle identified with the crosssections about the ring. This centerline lies in a center plane normal to the axis of the ring. The material of the ring is elastic and the cross section is constant about the circumference. The possible motions in a vibrating ring are: 1. 2. 3. 4.

Radial motion (Figure 2.20) Circumferential displacements similar to those of bars Flexural motion in a radial direction Flexural motion in a direction parallel with the axis of the ring and including twist 5. Torsional or twisting action about the centerline of the cross-section of the ring

2.7.1

Pure Radial Vibration

If the ring material can be made to expand and contract in phase about its circumference, the result is a pure radial mode of vibration. The ring vibrates uniformly about the centerline of the cross-sections having a mean or equilibrium radius of r. The cross-sectional area of the ring is S and Young’s modulus of elasticity is Y, its density is ρ, and its Poisson’s ratio is µ: F


SYc SYc 2r

(2.230)

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115

r +

FIGURE 2.20 Vibrating ring in pure radial motion.

As the ring vibrates in the radial direction at an amplitude of ±r it produces a strain within the material of the ring (in the circumferential direction) equal to c/2πr. The maximum force within the material of the ring occurs when the radial displacement is at its maximum or minimum value, and is where ℓ is the circumference of the centerline circle and S the cross-sectional area of the ring material. In terms of r, c = 2πr. The potential energy in the ring at maximum displacement is P.E.

F c SYc2 SY(2r )2 2 2 2(2r)

(2.231)

With no losses, this potential energy is converted into kinetic energy. At the instant the ring passes through the equilibrium position, the total energy within the ring is kinetic energy given by K.E.

S(2r)  dr    2  dt 

2

(2.232)

As kinetic energy plus potential energy is a constant value during a vibration cycle (no losses), K.E. + P.E. = constant so that d d (K.E.) (P.E.) 0 dt dt

(2.233)

Thus   dr   d 2r 2SY dr S(2 r) r 0    dt   dt 2 r dt 


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116 or

d 2r Yr 2 0 dt 2 r

(2.234)

r = A cos t + B sin t

(2.235)

The solution to Equation 2.234 is

where

Y r 2

(2.236)

or, the frequency of a ring vibrating in a pure radial mode is f

1 Y 2 r 2

(2236a)

When a ring vibrates in a pure radial mode, the entire mass of the ring moves in phase as the material undergoes alternately tension and compression. When the ring vibrates in a pure circumferential mode in a manner equivalent to longitudinal modes in bars with the radius, r, of the centerline circle remaining constant, no radiation occurs from any surface. The stresses also are pure compression and tension, assuming that the dimensions of the cross section of the ring are small compared with the radius, r, of the centerline circle and that the cross-sectional area, S, is continuous as a function of θ. The frequency equation of vibrations is given by n2 (1 n 2 )

Y r 2

(2.236b)

or fn

1 Y (1 n 2 ) 2 r 2

(2.236c)

where n is the number of wavelengths in the circumference for the fundamental frequency and all overtones (n = 1, 2, 3, … ) [5,12]. For n = 0, the equation is the same as that for pure radial modes, as in Equation 2.236. 2.7.2

Flexural Modes of Rings

If the motion is flexural and restricted to the plane of the ring containing the crosssectional axis (Figure 2.21), the frequency equation is n2

1 n 2 (n 2 1)2 Yrr2 4 (1 n 2 )r 4

(2.237)

where rr is the radius of the cross section (assuming a circular cross section) or n2


n 2 (n 2 1)2 Y 2 (1 n 2 )r 4

(2.238)

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117

FIGURE 2.21 Flexural mode of ring vibrating in plane containing centerline of cross section.

where κ is the radius of gyration, assuming the cross section to be uniform about the ring. Therefore n

n 2 (n 2 1)Y 2 (1 n 2 )r 4

(2.238a)

and fn

1 n 2 (n 2 1)Y 2 2 (1 n 2 )r 4

(2.238b)

For a ring vibrating in flexure but in a direction normal to the plane of the ring (plane containing the axis of the cross section of the ring), the frequency equation is n

n 2 (n 2 1)Yrr2 4(1 σ n 2r 4 )

(2.239)

n 2 (n 2 1)Y 2 1 n 2r 4

(2.239a)

for a ring of circular cross section or n

geometrical forms in which the radius of gyration, κ, is parallel to the direction of motion. Then fn

1 n 2 (n 2 1)Y 2 2 1 n 2r 4

(2.239b)

where σ is Poisson’s ratio. 2.7.3

Hollow Cylinders

The number of potential modes of vibration of a cylinder increases as the axial length increases from the extremely short lengths usually associated with rings to lengths corresponding to several average diameters. The vibrations associated with theses modes,


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118

whether they appear in rings or in cylinders, fall into two classes: (a) longitudinal or extensional and (b) flexural. In the ring, the longitudinal modes involve circumferential displacements. The flexural modes involve displacements either in the plane of the ring, perpendicular to the plane of the ring, or torsionally twisting with motion parallel to the axis of the ring. In the cylinder, the longitudinal modes may also be circumferential, but they can include particle displacements parallel to the axis of the cylinder or at any angle to the axis of the cylinder. Likewise, flexural modes may correspond to waves traveling in axial, circumferential, or skewed directions. Discussion of cylinders in this section is limited to basic principles. Chapter 3 discusses and illustrates extension of these principles to predict performance of tubes and cylinders using Donnell’s model. Some special cases of cylindrical vibrations are 1. As rings—considered in previous sections. 2. In longitudinal motion equivalent to uniform bar. In this case, the radius of the outer surface is small compared with the axial length and the wavelength. Unless the wall is extremely thin, use conditions for uniform bars. 3. In flexural motion equivalent to uniform bar. The conditions are the same as in case 2. 4. Longitudinal motion regardless of wave direction in which the radial dimensions are significantly large compared with a wavelength. 5. Flexural motion comparable to case 4.

2.8

Wide and Large-Area Horns

The design of an ultrasonic system for high-power applications is usually complex. A mode of vibration is specified as being the most appropriate for a given application. The designer’s problem is to deliver the specified mode without generating deleterious spurious modes. Spurious modes of vibration rob energy from and can completely overwhelm the intended mode of operation. Energy is most efficiently transferred from the source transducer into a load when the impedance of each element is properly matched to that of each adjacent element within the active system. For example, an exponentially tapered horn coupled to a short cylindrical tool can resonate as a half-wave system in a longitudinal mode when the mechanical impedance of the horn equals that of the tool at the junction between them. The combination can be designed to vibrate at a specified frequency by using equations describing impedances of the various geometrical configurations. The impedance of the horn is always assumed to be distributed. The impedance of a long tool also is assumed to be distributed, but when the dimensions are very small compared to a wavelength, the tool may be considered as a lumped mass. Impedance matching is important to optimum performance of any high-power ultrasonic system. The various elements in the system may vary in the planned modes (e.g., longitudinally vibrating horn driving a bar, a ring, a plate, or a cylinder in flexure). Also, reflected impedances due to loading conditions are factors in the performance of the system. These should be analyzed to determine their significance to the overall design of the system. These mode and impedance considerations are especially important to the design of wide and large-area horns. For example, if the lateral dimensions of a wide or large-area


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119

horn are equivalent to or near a multiple of half-wavelengths at the design frequency, they present the danger of producing corresponding lateral modes of vibration that can damage products being processed. Usually these horns are designed with slots parallel to the longitudinal motion. The elements between slots are also sources of additional spurious modes of vibration. These slots maximize heat dissipation to prevent creation of thermal hot spots in operation and help control modal characteristics. The following sections include sufficient information for designing effective wide, large-area and other special types of horns without resorting to finite-element methods. Finite-element procedures and Donnell’s model for designing special horns and more complex systems such as cylinders in wall flexural modes are presented in Chapter 3. 2.8.1

Wide Blade-Type Horns

Wide blade-type horns have found their most common application in seam welding of plastic sheet or similar products. Their designs may represent arrays of various geometrical designs, such as uniform bars, stepped horns, exponentially tapered horns, wedge-shaped horns, and catenoidal horns. Figure 2.22 is a typical stepped, blade-type horn. The design procedure includes (using this design as an example): 1. 2. 3. 4.

Selecting horn material Specifying the dimensions of the two end faces Specifying the dimensions and positions of the longitudinal slots Determining impedance relationships for matching solid end elements to the elements created by the slots 5. Calculating longitudinal dimensions based upon the properties of the material of the horn and the impedances

R1 d

λ/4

ᐉ

B

R2 FIGURE 2.22 Wide blade-shaped horn.


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120

The dimensions of the horn of Figure 2.22 are identified as follows: • • • • • • • • • • • •

B is the total width of the horn ℓ is the total length of the horn d is the width of each slot b is the width of the portion of the faces of the horn corresponding to each element n is the number of slots S1 is the cross-sectional area at the large end of the horn (=R1B) R1b is the cross-sectional area of the larger end of the horn between slots R2b is the cross-sectional area of the smaller end of the horn between slots ρ is the density of the horn material (constant throughout the horn) c is the bar velocity of sound in the horn a1 is the distance between the end of the slot and the larger face of the horn a2 is the distance between the end of the slot and the smaller face of the horn

The nodal position at resonance is assumed to be located at the large dimension of the step of the horn. The length of each slot is less than a half-wavelength of a stepped horn by an equivalent length x of a (a1 + a2), or ℓ = λ/2 − x + (a1 + a2) x = x1 + x2

(2.240)

where x1 is a length equivalent to a1 and x2 a length equivalent to a2. From the bar impedance Equation 2.7 tan

x 1 h a tan 1 c b c

tan

x 2 h a tan 2 c b c

(2.241)

Then x

c  1  h  a1    a 2    1  h tan  tan    tan  tan    c  c     b b

(2.242)

When a1 = a2 = a x2

c h  a   tan1  tan     c  b

From Figure 2.22, h = B/(n + 1) = b + d b = [B − (n + l)d]/(n + 1)


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121

The total length, ℓ, is, therefore

c c h h  a    a    (a1 a 2 ) tan1  tan  1   tan1  tan  2      c    2f c   b b

(2.243)

when a1 ≠ a2 and

c 2c  a   h 2a tan1  tan  2f c    b

(2.243a)

when a1 = a2. Because h/b = B/[B − (n + 1)d],

 c c  B a  (a1 a 2 ) tan1  tan 1  2f c    B (n 1)d  B  a    tan1  tan  2     c    B (n 1)d 

(2.243b)

when a1 ≠ a2 or

 2c B c a  2a tan1  tan  c  2f  B (n 1)d

(2.243c)

when a = a1 = a2. Thus, ignoring the effect of the fillet at the junction between the large thickness and the smaller thickness sections of the horn, a stepped blade-type horn is the same length as one of uniform thickness for the full length when all other dimensions are the same. The distance between slots is assumed to be sufficiently small that bar velocity is applicable. The effect of Poisson’s ratio on lateral expansion and contraction cannot be ignored, however, in choosing the total width of the horn and determining the lengths of the slots. Width dimensions equivalent to multiples of half-wavelengths at frequencies close to the design frequencies are likely to lead to vibration in a lateral mode rather than the design mode. The corresponding vibrations may damage products to be processed. 2.8.2

Large-Area Block-Type Horns

Figure 2.23 is a typical rectangular block-type horn. The dimensions of the horn of Figure 2.23 are End dimensions, R and B Distances between the ends of the slots and the end surfaces, b1 (top) and b2 (bottom) Width of each slot, d (assuming the same dimensions in all directions) Length of the horn, L The dimensions of the large-area horn are such that the effect of Poisson’s ratio on the velocity of sound in the material of the horn cannot be neglected. From Equation 2.110,


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122

b1 L

d b2

R

B

FIGURE 2.23 Large-area block-type horn.

the corrected velocity of sound in a large area block relative to bar velocity in the same material is approximately 2

c′ 1   2 2 1   [B R ] c 6 

(2.244)

Derks [6] applies this formula to determine a corrected value of velocity for a large area rectangular horn containing slots as follows: 2 2 2  R n 2d   c′ 1  f   B n1d   1    n 1   co 6  c   n1 1   2 

(2.245)

where c′ is the corrected value of the velocity of sound, n1 and n2 are the number of slots through sides B and R, respectively, and σ is Poisson’s ratio. The length of the horn is given by L

  c′ 1  n1d   n 2d  b1 b2 tan1  1 1 tan(k ′b1 )   2f k′  B n1d   R n1d        n1d   n 2d  tan1  1 1 tan(k ′b2 )    B n1d   R n1d    

(2.246)

where k′ = ω/c′. 2.8.3

Other Designs—Large, Cup-Shaped Horns

For applications to processes such as continuous seam welding of plastic sheet or strips, ultrasonic tools can be made to rotate while applying energy to the work through a


10/29/2008 3:48:23 PM


123

D2

D3

Coupling bolt hole x ᐉ FIGURE 2.24 Horn with cylindrical shell and cavity in which the cross-sectional area function is equivalent to that of a conical horn.

wheel-like surface. The first approach considered might be to drive a disc-shaped plate in flexure using a longitudinal mode horn coupled to its axis. However, a plate driven in such a manner at an amplitude sufficient to perform its intended function is likely to experience such severe stresses near its axis that it will fail in fatigue within a very short time. There are means of applying ultrasonic energy to work on a continuous basis using a rotating horn. Figure 2.24 illustrates one such tool. This horn was chosen to illustrate a method. Other designs may be used as well. The horn of Figure 2.24 is designed so that the area as a function of position x corresponds to that of a solid cone. The dimensions of the horn are described by d1 is the smallest diameter of the cavity located in the plane of the heavy end of the horn d2 is the diameter of the cavity at the large, open end d3 is the outside diameter of the horn (d3 ≤ 0.4λ) ℓ is the length of the horn x = 0 is located at the large diameter end of the cavity For illustrative purposes, assume that the vertex of the cavity coincides with the plane of the large area end, that is, d1 = 0. Because the horn is assumed to be equivalent to a conical horn, solutions to the horn equation should be similar. However, the outer diameter of the horn is sufficiently large that effects of Poisson’s ratio on the velocity of sound in the material of the horn should be considered. A close approximation for the velocity, c′, can be obtained using Equation 1.61, that is, 2 1  d′  c′ 1  d′  1  1   4  co 4  2c o 

2

(1.61)

where co is the bar velocity of sound in the material of the horn, c′ the corrected velocity of sound, and d′ the major diameter (=d3).


10/29/2008 3:48:23 PM


124

Using the previous assumptions, the area relationship as a function of x (measured from the open end of the horn) is Sx

(

)

2

2  2 x d 3 d 22 d 3 d 32 d 22  2 ( x) d 23 d 22 d 3x      4 4

(2.247)

Using the solution to the horn equation for a cone, one can write the velocity relationship as  (d 3 d 32 d 22 )V2    d 3 c′ v  cos (x ) sin (x )   c′ c′  ( x) d 23 d 22 d 3x   (d 3 d 32 d 22 ) 

(2.248)

where c′ is the corrected value of velocity of sound for the horn, V2 the amplitude of particle velocity at the heavy end of the horn, and ℓ the length of the horn. The half-wave resonant length is determined by tan c′ ω 2 2d

(

c′ d 3 d 32 d 22 3

(

)

2

d 32 d 22 c′ 2 d 3 d 32 d 22

)

(2.249)

2

The velocity node occurs where tan

d3 ( x) c′ c′ d 3 d 23 d 22

(2.250)

The amplification factor is 2 2  V1 1 d3 c ′  d 3 d 3 d 2  sin cos 2 2 2 2 V1 2 c′ c′ d3 d2 d 3 d 2

(2.251)

where 1 is the displacement amplitude at x = 0 and 2 the displacement amplitude at x = ℓ. The stress, s, at x is given by s j

(d 3

)

d 32 d 22 V2Y

( x) d 23 d 22 d 3x   

2

{(

)

d 3 d 32 d 22 (x )cos

(x ) c′

  c′ 2  2d 3 d 32 d 22 2 d 3 d 32 d 22 d 3x  sin (x )  c′ c′  d 3 d 32 d 22  

(

(2.252)

)

The position of maximum stress is given by tan

(x ) c′

4 2d 3

2 2 d 3 4c ′  ( x) d 23 d 22 d 3x  d 3 d 23 d 22  (x )     c′ d 3 d 23 d 22    2 4c ′ 2 d 23 d 22 6d 23x 6d 3 d 23 d 22 x 2 (d 23 d 22 ) 2d 22x (d 23 d 22 ) x 2 2 d 3 d 23 d 22   

(2.253)


10/29/2008 3:48:23 PM


125

The horn of Figure 2.24 can be adapted to a rolling contact type of operation, which can be done in variations of either of two ways: 1. Machining a bevel at the working (or open) end of the horn to contact the workpiece surface 2. Forming a raised section (wheel-rim) of dimensions suitable for coupling to the work surface The work coupling surface of the first method is an element of a cone of either method. In either case, the overall length of horn and tool section must be adjusted so that the combination resonates at a frequency that matches the frequency of the driving transducer. The system is broken down into the horn section and the tool section. The impedance of the horn at the junction between the two sections must match that of the tool at that position. The impedance at x of the horn of Figure 2.25 is Z = F/v = sS/v or

Z j

( x) d 23 d 22 d 3x  Y   x d 3 d 32 d 22  (x )     d c ′ 3 4 2  tan (x ) c′  d 3 d 32 d 22 

{

  c′ 2  2d 3 d 23 d 22 2 d 3 d 23 d 22 d 3x  tan (x )  c′ c′  d 3 d 23 d 22  

(

)

(2.254) where F is the force across the cross-sectional area at x. In designing a horn, as previously discussed, to drive a wheel-like tool section, the critical design dimensions are first specified—that is, d1, d2, d3, frequency, material, and

Fillet

Dw

D3

D2

Coupling bolt hole ᐉT L FIGURE 2.25 Wheel-loaded rolling-type ultrasonic horn.


10/29/2008 3:48:24 PM


126

dimensions of the wheel section, dw, and L, where dw is the outer diameter of the wheel portion, d1 is assumed to equal zero, and L is the axial length of the cylinder at the outer diameter of the wheel. Because one objective in selecting the horn design is to amplify the longitudinal displacement over that at the driven end, the enlargement will appear at the open end of the horn. This is the end corresponding to x = 0 in the design equations previously presented. The next step is to determine the length, ℓ, of the horn as a half-wave resonator without the wheel section, using Equation 2.254. The final total length, ℓT, of the horn plus wheel section will be ℓT = ℓ − X + L

(2.255)

where ℓ is the length calculated using Equation 2.249 and X is a length equivalent to that consumed by the wheel section. The value of X is determined by equating the mechanical impedance of the wheel to that of the horn at the junction between them and substituting X for x. Because the wheel is much shorter than a wavelength, its impedance may be determined fairly accurately by assuming that the wheel section is a lumped mass, that is, 2 Zm jM jL (d w d 22 )

4

(2.256)

Because X is measured from the origin and is 115

Y11E = 11.9 Y33E = 11.3

>300

>140

Y11E = 11.6 Y33E = 11.0

Y11E = 8.2 Y33E = 6.5

>115

500

Y11E = 11.7 Y33E = 11.1

3.8

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point (°C) kg/m3) (1010 N/m2) Q Thin Disk (Hz m)

2.0

2.0

2.0

0.4

0.8

0.3

0.8

1.0

Dissipation Factor (% 102 tan δ)

Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111

Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111

American Piezo Ceramics Inc. (APC), Duck Run Road, PO Box 189, Mackeyville, PA 17750

Source

198 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:24 AM


1050

5804 Navy III

1220

640

EC-55

EC-57

1300

1725

2125

1100

1050

2750

EC-64

EC-65

EC-66

EC-67

EC-69

EC-70

Lead zirconate titanate EC-63 1250

1170

EC-31

Barium titanate EC-21 1070

1100

5800

1340

1400

−105 240 [d15 = 382]

−107 245 [d15 = 390]

0.38

−32

87

−95 220 [d15 = 330]

0.37 0.74 −230 490 [d15 = 670] [k15 = 0.67; kp = 0.63]

0.31 0.62 [k15 = 0.55; kp = 0.5]

0.33 0.66 −107 241 [d15 = 362] [k15 = 0.59; kp = 0.56]

0.36 0.72 −198 415 [d15 = 626] [k15 = 0.68; kp = 0.62]

0.36 0.72 −173 380 [d15 = 584] [k15 = 0.69; kp = 0.62]

0.35 0.71 −127 295 [d15 = 506] [k15 = 0.72; kp = 0.60]

0.34 0.68 −120 270 [d15 = 475] [k15 = 0.69; kp = 0.58]

0.15

0.19 0.46 −58 150 [d15 = 245] [k15 = 0.48; kp = 0.31]

0.19 0.48 −59 152 [d15 = 248] [k15 = 0.49; kp = 0.32]

0.17 0.38 −49 117 [d15 = 191] [k15 = 0.37; kp = 0.26]

0.32 0.66 [k15 = 0.59; kp = 0.54]

−0.32 0.67 [k15 = 0.60; kp = 0.55]

16.2

900

−10.9 24.8 [g15 = 28.7]

−9.8 20.9 [g15 = 35.0]

75

960

80

−10.6 23.0 [g15 = 36.6]

−10.2 23.7 [g15 = 28.9]

100

400

500

600

550

400

1400

1050

1100

−11.5 25.0 [g15 = 38.2]

−10.7 25.0 [g15 = 39.8]

−10.3 24.1 [g15 = 37.0]

−5.5

−5.6 14.3 [g15 = 20.1]

−5.8 14.8 [g15 = 20.4]

−5.2 12.4 [g15 = 15.7]

−11.3 25.8 [g15 = 32.2]

−11.0 25.2 [g15 = 31.5]

1727

2181

2141

1752

1778

2026

2069

2845

2868

2845

2818

2110 [Np = 2310; N31 = 1570]

2110 [Np = 2260; N31 = 1570]

7.45

7.5

7.5

7.45

7.5

7.5

7.5

5.3

5.55

5.55

5.7

7.55

7.55

6.3

9.9

9.3

6.2

6.6

7.8

8.9

12.5

11.6

10.7

220

300

300

270

350

320

320

140

115

115

130

>300

Y11E = 8.6 Y33E = 7.1

11.4

>300

Y11E = 8.6 Y33E = 7.1

2.0

0.3

0.3

2.0

2.0

0.4

0.4

0.6

0.5

0.7

0.5

0.4

0.4

(continued)

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111

Piezoelectric Materials: Properties and Design Data 199

10/27/2008 10:44:24 AM


236

Lead titanate EC-97

ε11/εo

1150

HS-21

k33

d31

d33

Piezoelectric Strain Constant (10−12 m/V)

0.35 0.72 −312 730 [d15 = 825] [k15 = 0.67; kp = 0.61]

0.01 0.53 −3 68 [d15 = 67] [k15 = 0.35; kp = 0.01]

k31

Coupling Factor

2100

250

1700

2600

1800

G-1500

G-1512

HST-41

Lead metaniobate G-2000 250

2200

1000

G-1408

0.04 0.38 −10 80 [d15 = 115] [k15 = 0.33; kp = 0.066]

0.35 0.66 −157 325 [d15 = 625] [k15 = 0.69; kp = 0.59]

0.37 0.72 −232 500 [d15 = 680] [k15 = 0.78; kp = 0.63]

0.35 0.66 −120 280 [d15 = 360] [k15 = 0.59; kp = 0.57] 1250 0.26 0.60 −80 200 [d15 = 315] [k15 = 0.60; kp = 0.50] 1700 0.34 0.69 −166 370 [d15 = 540] [k15 = 0.65; kp = 0.58]

1350

1300

HDT-31

0.29 0.60 −84 190 [d15 = 300] [k15 = 0.64; kp = 0.50]

0.16 0.45 −30 86 [d15 = 125] [k15 = 0.42; kp = 0.26] 1050 0.18 0.51 −50 148 [d15 = 225] [k15 = 0.48; kp = 0.30]

570

Lead zirconate titanate G-53 720 960

600

Barium titanate HD-11

Lead magnesium niobate EC-98 5500

ε33/εo

Free Dielectric Constant

(Continued)

Materials

TABLE 4.1

g33

−4.5 [g15 = 50]

−11 [g15 = 37]

−9.3 [g15 = 35]

−11 [g15 = 29] −9.0 [g15 = 29] −11 [g15 = 36]

−13 [g15 = 36]

−5.5 [g15 = 25] −5.2 [g15 = 25]

36

22

20

25

22

23

30

16.0

16.0

−6.4 15.6 [g15 = 17.0]

−1.7 32 [g15 = 33.5]

g31

Piezoelectric Stress Constant (10−3V m/N)

—

>1200

15–20

70

70

—

—

—

—

—

>500

80

—

— Y33E = 13.0 —

1803

2210

140

300

>800

70

950

>5.8

Y11E = 4.0 Y33E = 4.7

>400

>270

Y11E = 7.0 Y33E = 5.9 >7.6

1.5

>360

0.6

2.2

1.8

0.3

>300

>240

0.6

2.2

1.5

0.6

2.0

0.9

Dissipation Factor (% 102 tan δ)

>330

Y33E = 6.3 Y33E = 5.4

Y11E = 8.1 Y33E = 6.7 Y11E = 9.0 Y33E = 8.2 Y11E = 6.3 Y33E = 4.9

>330

>125

Y11E = 11.1 Y33E = 11.5 Y11E = 8.1 Y33E = 6.5

>135

170

240

Y11E = 13.7

6.1

12.8

>7.4

>7.6

>7.5

>7.6

>7.6

−5.6

>5.6

7.85

6.7

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point (°C) kg/m3) (1010 N/m2) Q Thin Disk (Hz m)

Gulton Industries Inc., Metuchen, NJ 08840

Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840

Gulton Industries Inc., Metuchen, NJ 08840

Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Source


10/27/2008 10:44:24 AM


1050

1000

1160

1800

3000

2600

Sonox P5

Sonox P51

Sonox P52

—

—

—

Type materials not identified Sonox P2 480 800

Sonox P8

Lead zirconate titanate Sonox P4 1300 1480

Barium titanate Sonox P-1 960

−180

−210

−230

0.71 −0.33 [kp = −0.60]

0.71 −0.33 [kp = −0.60]

−45

530

510

400

125

−95 215 [d15 = 290]

−130 280 [d15 = 450]

−40 125 [d15 = 205]

−0.33 0.70 [k15 = 0.67; kp = −0.59]

−0.26 0.60 [k15 = 0.65; kp = −0.46]

−0.31 0.60 [k15 = 0.50; kp = −0.50]

−0.31 0.68 [k15 = 0.64; kp = −0.55]

−0.15 0.43 [k15 = 0.42; kp = −0.25]

−10.0

−8.0

−11.0

−11.0

−11.0 [g15 = 28]

−11.0 [g15 = 34]

−5.0 [g15 = 22]

23

19

25

30

25

25

14

70

100

90

400

1000

400 (radial)

350 (radial)

1320

1400

1500

1650

1600

1570

2300

7.5

7.4

7.7

7.8

7.7

7.8

5.3

360

340

200

220

Y11E = 6.3 Y33E = 5.3

Y11E = 6.7 Y33E = 5.3

Y11E = 6.7 Y33E = 5.3

300

Y11E = 9.1

Y11E = 10.0 Y33E = 7.7

325

120

Y11E = 7.7 Y33E = 5.9

Y11E = 11.1 Y33E = 11.1

1.5

2.0

2.0

0.5

0.2

0.5

0.7

(continued)

Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148

Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148

Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148

Piezoelectric Materials: Properties and Design Data 201

10/27/2008 10:44:25 AM


570

1070

140

300

175

Sonox P6

Sonox P62

K15

K81

K83

—

—

—

—

ε11/εo

425

1300

K180

K270

—

—

Lead-zirconate titanate K85 800 —

ε33/εo

Free Dielectric Constant k31

0.38

0.15

—

0.33 [k15 = 0.71; kp = 0.58]

0.30 [k15 = 0.68; kp = 0.52] −11 [g15 = 38]

270 −120 [d15 = 490]

0.70

—

—

−15.0 [g15 = 50]

180

65

85

—

—

−8.0

g31

26

41

27

42

−7.0

14.5

—

20

g33

Piezoelectric Stress Constant (10−3V m/N)

180 60 [d15 = 350]

—

—

≥18

205

100

−40

—

d33

d31

Piezoelectric Strain Constant (10−12 m/V)

0.67

0.43 — [kt =0.35; kp = 0.35]

[kt = 0.30; kp = 3t or (b) the width is >3t but 1. Boucher finds an improvement in efficiency by making the ratio D2/D1 ≥ 1.3. Ratios from 1.0 to 1.55 have proven to give good performance. 6.3.3.1

The Stem-Jet Whistle

The stem-jet whistle consists of a nozzle and a cavity connected by a stem holding the nozzle and the cavity in perfect axial alignment. The statement holding the nozzle and the cavity in perfect axial alignment is not an exaggeration. A slight misalignment will reduce the ability of the whistle to function at maximum efficiency. Any device added to correct the condition will at best interfere with the output of the whistle. The parts must be accurately machined and polished. The equations governing the design of the regular Hartmann whistle may also be applied to the stem-jet whistle. However, Boucher finds that over the interval a0 to am, dλ/da = 1.88. The stem-jet whistle operates at much lower pressures than the Hartmann whistle (output of 160 dB at a distance of 25 cm with an input pressure of 30 psi). Sonic radiation is proportional to (p − 0.3) for the stem-jet whistle. A whistle designed according to Figure 6.4 operates at approximately 10 kHz when the critical dimensions are Nozzle ID = 4.7625 mm (3/16 in.) Cup ID = 7.3152 mm (0.288 in.) Cavity depth = 7.3152 mm (0.288 in.) Cup OD = 9.525 mm (3/8 in.) Smaller diameter of stem = 2.3812 mm (3/32 in.)


10/24/2008 7:42:11 AM


278

Stem Cup

Air hose connector

Body

Reflector/Nozzle

FIGURE 6.4 Adjustable stem-jet whistle.

Length of smaller diameter of stem = 30.1626 mm (1.1875 in. = 1–3/16 in.) Larger diameter of stem = 3.175 mm (1/8 in.) Total stem length = 61.9126 mm (2.4375 in. = 2–7/16 in.) 6.3.3.2

Construction of the Stem-Jet Whistle

The stem-jet whistle requires very careful and accurate machining, especially with respect to the alignment of the cavity, stem, and nozzle. The reflector and nozzle are in one solid member. The outer, large diameter is first machined to size. Then the reflecting surface is machined to the desired shape. The first step is to machine the back side of the reflector before cutting the threaded section. In Figure 6.4, the threaded section is 1.27 cm (1/2 in.) in diameter and 1.5875 cm (5/8 in.) long. It is best to thread this section before boring out the inner air passageway. The reflector may include any of several geometries. Those commonly used are the spherical section, the hyperbolic section, the elliptical section, and the catenoidal section. The general positions of the reflecting surfaces relative to the sources are a minimum distance of approximately one wavelength, according to Equation 6.2, from the center of the spherical section or less and from the foci of the elliptical and hyperbolic sections. The catenoidal reflectors have no foci, and the position of the source relative to the reflecting surfaces is determined more by judgment relative to the wavelength and the reflecting angle and the beams projected as sketched geometrically. These are all general guidelines, and the designer has some freedom in designing each type. The body consists of a cylindrical shell threaded inside to match the threads of the reflector and nozzle section. It contains a block to which the stem is screwed using fine threads (such as 5–40 threads) on the stem. The passage holes for the airflow are also drilled through this block, close to the inner wall of the body. The cup of Figure 6.4 is machined with the cavity depth equal to the cavity diameter. The small end of the stem is lightly silver-soldered from the outside end of the cup. The parts are then screwed together and the alignment is very carefully checked. A suitable air hose connector is silver-soldered to the open end of the cylindrical section. The connection is carefully machined to the diameter of the body. After tuning the system, the setscrew is screwed in snugly to prevent any movement in the parts. The whistle is now ready to test and use. The individual parts are shown in Figure 6.5.


10/24/2008 7:42:11 AM

Pneumatic Transducer Design Data

279

Stem

Cup Reflector Setscrew

Body

Air hose connector FIGURE 6.5 Individual parts of the basic stem-jet whistle.

Nozzle

Stem Cavity

Nozzle cavity FIGURE 6.6 Added jet-edge aspect to the nozzle of the Hartmann whistle.

The previous design procedure is given only as an example. Changes in dimensions are possible, but similar care in construction must be taken to be certain that the components are well aligned. 6.3.3.3

Modification of the Hartmann Whistle

In this case, Hartmann whistle refers to modifications of the stem-jet whistle. One such modification refers to an addition to the nozzle section, in which a recess is machined to increase the jet-edge aspect of the sound emitted (see Figure 6.6). It is hoped that this addition will increase the efficiency of the whistle.


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280

If the nozzle diameter and the cavity diameter are equal, the diameter of the recess might be on the order of 1.3 D. For larger cavity diameters, a good choice might be to increase the diameter of the recess by at least 1.3 times the diameter of the cavity. 6.3.3.4

The Pin-Jet Whistle

The pin-jet whistle is another modification of the Hartmann whistle claimed to give higher efficiencies. In this case, a pin extending through the resonator and into the open end of the nozzle replaces the stem. The end of the pin is beveled with an included angle of 20–30°. Again, alignment of the axes of the pin, the nozzle, and the resonator is very important for efficient operation of the whistle. However, efficiencies claimed for properly designed and assembled pin-jet whistles are higher than efficiencies obtained using the stem-jet whistles. Equations used in designing the stem-jet whistle are also applicable to the pin-jet whistle. 6.3.3.5

Some Special Applications for the Whistle

6.3.3.5.1 Coating Very Fine Particles Tubing, such as copper, may be placed through the reflector of a whistle for supplying materials for coating small particles with polymeric materials. In Figure 6.7, using the same design as that of Figure 6.4, feed tubes are located on two circular centerlines through the reflector. These centerlines are of two different diameters. The inner-diameter centerline locates the feed tubes for a liquid polymer to be atomized for coating the particles and the outer ones are for the particles to be coated. Both sets of feed tubes are aimed axially at the outer circumference of the resonant cavity. The sonic energy atomizes the fluid into a very fine mist, which surrounds and coats the suspended solid particles and enables the polymer to quickly set before the coated particles settle out of suspension. This sonic method has been used to produce individually coated particles of very fine size at a reasonable rate. An example of past experience in coating individually spherical particles, 5 μm in diameter, with a 1 μm coating of polymer (Figure 6.8). This method appears to be a unique way of accomplishing this requirement. 6.3.3.5.2 Breaking Down Foams The stem-jet whistle is effective in breaking down certain foams. Foams consist of bubbles. To break them down, the sonic intensity must produce a tension in the bubble surfaces exceeding the tension required to break each bubble. This calculation takes into account the surface tension of the liquid from which the bubble is formed and its viscosity. Pin A logical explanation of the effectiveness of ultrasonics for this purpose appears to be as follows: when an ultrasonic wave impinges on a foam, each bubble in its path experiences motion. The motion includes two types: tensile and shear. Tensile shear comes from the longitudinal motion within the wave. The bubble undergoes compression and rarefaction under the influence of this motion, which produces tension and compression within the bubble wall. These stresses Cavity Nozzle are parallel to the wall of the bubble. It also produces tension across the bubble wall, that is, normal to the FIGURE 6.7 bubble wall in the path of the impinging wave. Pin-jet whistle.


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281 Particle feed tube

Stem

Body Air hose connection

Resonant cavity Coating feed tube

FIGURE 6.8 Sonic whistle for individually coating very small solid particles with a thin layer of polymer.

Compression/rarefaction

Shear

Tension/rarefaction FIGURE 6.9 Stresses developed in bubbles in a foam by ultrasonic waves.

The shear motion is due to the movement of a bubble under the influence of the ultrasound relative to its neighbor, which otherwise may be less or not at all influenced by the ultrasound. The result is a shear force acting on the bubbles. The viscosity plays a prominent part here. Therefore, three stresses are tending to disrupt the continuity of the bubble wall, leading to the bubble’s collapse: two tensile and one shear. Collapse occurs when the total of these forces exceeds the strength of the wall. The action of these stresses is illustrated in Figure 6.9. The assumption is made that the air blast producing the sonic energy is deflected from the foam by a protective film and that only the sonic energy is acting on it. A typical output intensity of an efficient 10 kHz whistle is 150 dB. This is approximately 6600 dynes/cm2 in air. If the surface tension of the liquid exceeds the output stress of the whistle, it is unlikely that the stress of the sound wave will break up the foam. 6.3.3.5.3 Other Means of Increasing Power from Hartmann-Type Whistles The Hartmann-type whistles, including the stem-jets and the pin-jets are ideal for multipleunit applications, that is, two or more whistles mounted in parallel to increase the total output power.


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282 6.3.4

Vortex Whistles

Vortices are problems that engineers must consider in many areas of industry such as aircraft, turbines, and wind tunnels. The theory will be simplified by considering vortices of two types: forced and free. Forced vortices are the type observed in a cylindrical tube filled with a fluid when it is spun. Free vortices are the type observed in the surface of a flowing river or in a bathtub when it is being drained. A vortex whistle uses the free vortex. A vortex whistle is designed to produce a vortex in a gas or liquid and to take advantage of the sound it produces. Figure 6.10 shows schematically the construction of such a device. A free vortex is formed when a fluid escapes from a larger volume through a small opening. It is maintained by the conservation of angular momentum. Angular momentum is the product of the momentum of a particle by its distance from the axis of rotation. Momentum is defined as the mass times velocity, that is, Mv. The angular momentum is therefore Mvr, where M is the mass of a particle, v its velocity, and r the distance from the axis of rotation. To preserve its angular momentum, as a particle draws nearer to the axis, it increases speed. Velocity and radius are the only variables. Variations in design from Figure 6.10 to increase the output intensities of the vortex whistles are possible. One method might be to use a conical or exponential shape from the inlet volume to the exit (Figure 6.11). The main advantage of a vortex whistle appears to be the possibility of using this design with either gases or liquids. 6.3.5

Vibrating Blade

A system has been commercially available based upon creating cavitation in a liquid using a flexural resonant blade. A narrow jet of fluid impinges on the sharp edge of the blade, causing it to vibrate in flexural resonance. The basics of this design are shown schematically in Figure 6.12. The fluid is ejected from a slit in a nozzle. The slit is narrow and long enough that the fluid impacts the full height of the blade. The slit should be perfectly parallel with the blade. Spacing between the blade and the nozzle may range from as small as the width of the slit to about four times the thickness of the blade. A blade, either one-quarter wavelength or

Air in Sound out

Hollow cylinder FIGURE 6.10 Vortex whistle. Air in Sound out

Hollow cylinder

Funnel

FIGURE 6.11 Conical vortex whistle.


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283 Slotted nozzle

Resonant blade

Fluid in

a. Top view, quarter-wave plate

Resonant blade Fluid in λ /4

λ /4

b. Top view, full-wave unit Resonant blade

Fluid in

λ/4 c. Side view, quarter-wave plate FIGURE 6.12 Liquid blade whistle.

one full wavelength long, is rigidly mounted parallel to the direction of the fluid’s flow. The quarter-wavelength blade is mounted at the first nodal position. The mounting must impede the flow of the fluid in a minimal amount. The full-wavelength blade is mounted at the two nodal positions by heavy pins. In either case, the mounting must be strong enough to resist the flow pressures and the forces of vibration. Typically, the source pressures are 12.5–20.7 kg/cm2 (150–250 psi) at flow rates of 45–60 m/s (150–200 ft./s). The design of the blade is critical not only from the standpoint of resonance characteristics (low damping), but also for fatigue life. A well-designed blade type of whistle should be useful for emulsification and homogenization on an industrial scale.

References 1. T. Hartmann, Journal of Scientific Instruments, 16, 1939, 146. 2. R. M. G. Boucher, U.S. Patent 2,800,100, July 23, 1957. 3. E. Brun and R. M. G. Boucher, Research on the acoustic air-jet generator: A new development, Journal of the Acoustical Society of America, 29(5), 573, May 1957. 4. J. Blitz, Fundamentals of Ultrasonics, Butterworths, London, 1963, p. 72. 5. L. Bergmann, Ultrasonics, John Wiley & Sons, Inc., Newyork, 1938, p. 2.


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10/24/2008 7:42:13 AM

7 Properties of Materials Dale Ensminger

CONTENTS 7.1 Introduction ........................................................................................................................ 286 7.2 Significance of Physical Properties of Tables 7.3 through 7.5 ...................................... 286 7.3 Ultimate Stress and Fatigue Limit................................................................................... 299 7.4 Materials Properties for Processing by Ultrasonics......................................................300 7.4.1 Friction and Wear ...................................................................................................300 7.4.2 Welding and Staking ............................................................................................. 302 7.4.2.1 Properties Necessary for Ultrasonic Welding and Forming of Materials ...................................................................... 302 7.4.2.2 Joint Designs for Efficient Welding .......................................................304 7.4.3 Tools for Ultrasonic Welding ................................................................................308 7.4.4 Staking .....................................................................................................................309 7.4.4.1 Hydraulic Speed Control ........................................................................ 310 7.4.5 Five Basic Staking Designs .................................................................................... 310 7.4.5.1 The Standard Rosette Profile Stake ....................................................... 311 7.4.5.2 The Dome Stake ....................................................................................... 311 7.4.5.3 The Hollow Stake ..................................................................................... 311 7.4.5.4 The Knurled Stake ................................................................................... 312 7.4.5.5 The Flush Stake ........................................................................................ 312 7.5 Stud Welding ...................................................................................................................... 313 7.6 Insertion .............................................................................................................................. 313 7.7 Swaging and Forming ....................................................................................................... 314 7.8 Spot Welding ...................................................................................................................... 315 7.9 Degating .............................................................................................................................. 316 7.10 Scan Welding ...................................................................................................................... 316 7.11 Bonding and Slitting ......................................................................................................... 317 7.11.1 Ultrasonic Bonding ................................................................................................ 317 7.11.2 Ultrasonic Slitting .................................................................................................. 317 7.12 Other Tool Designs and Application Methods .............................................................. 317 7.12.1 Fixed-Strap Bonding .............................................................................................. 318 7.12.2 Continuous Rolling Contact ................................................................................. 318 7.13 Metals Properties for Choice of Horns [1–3,6] ............................................................... 318 7.13.1 Horn Materials........................................................................................................ 318 7.13.2 High-Strength Low-Alloy Steels .......................................................................... 319 7.13.3 Ductile (Nodular) Iron ........................................................................................... 319 7.13.4 Hard-Facing Wear Surfaces .................................................................................. 319 7.13.5 Stainless Steels (ASM Hdbk)................................................................................. 320 285


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286

7.14 Important Refractory Metals............................................................................................ 320 7.15 Comments and Conclusions ............................................................................................ 321 References .................................................................................................................................... 321

7.1

Introduction

Data are presented in this chapter for selecting materials to be used in the wide variety of applications of ultrasonic energy. These properties include moduli, Poisson’s ratio, densities, fatigue limits and ultimate stresses of the materials, sonic velocities, characteristic acoustic impedances, melting points, boiling points, electrical resistivities, thermal conductivities, compatibilities between materials, wear resistances, effects of oxide protective coatings, toughness or brittleness, resistances to acids and bases, corrosion resistance, protective atmospheres, chemical resistance, effects of stress risers, weldability, solderability, and effects of positions of welds and solders in an ultrasonically activated structure. Table 7.1 lists the values and symbols of prefixes used in weights and measures recommended by the National Bureau of Standards [1]. These symbols will be used in all later references to these quantities. Table 7.2 gives the atomic numbers of the elements listed in alphabetical order [2,6]. This table will be useful in locating various elements in Table 7.3 as needed. Table 7.3 gives the atomic numbers, atomic weights, densities, melting points, and boiling points of the elements listed according to their atomic numbers [2,6]. Tables 7.4 and 7.5 list the physical properties of engineering materials useful in the design of ultrasonic apparatus [2–4].

7.2

Significance of Physical Properties of Tables 7.3 through 7.5

The equations of Chapter 2 demonstrate the importance of the modulus of elasticity, Poisson’s ratio, density, velocity of sound, and the characteristic acoustic impedance of materials. These are the significant quantities needed in the design of ultrasonic horns and systems used generally in the high-intensity applications of ultrasonic energy. TABLE 7.1 Prefixes of Weights and Measures


Prefix

Symbol

Multiple

Tera Giga Mega Kilo Hecto Deka Deci Centi Milli Micro Nano Pico

T G M k h dk d c m µ n p

1012 109 106 103 102 10 10−1 10−2 10−3 10−6 10−9 10−12

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Properties of Materials

287

TABLE 7.2 Alphabetical Order and Atomic Numbers of Elements Element Actinium Aluminum Americium Antimony (stibium) Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium (caesium) Chlorine Chromium Cobalt Copper (cuprum) Curium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold (aurum) Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron (ferrum) Krypton Lanthanum Lawrencium Lead (plumbum) Lithium Lutelium Magnesium Manganese Mendelevium Mercury

Symbol Ac Al Am Sb Ar As At Ba Bk Be Bi B Br Cd Ca Cf C Ce Cs Cl Cr Co Cu Cm Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf He Ho H In I Ir Fe K La Lr Pb Li Lu Mg Mn Md Hg

Atomic Number 89 13 95 51 18 33 85 56 97 4 83 5 35 48 20 98 6 58 55 17 24 27 29 96 66 99 68 63 100 9 87 64 31 32 79 72 2 67 1 49 53 77 26 36 57 103 82 3 71 12 25 101 80

Element Molybdenum Neodymium Neon Neptunium Nickel Niobium (columbium) Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium (kalium) Praesodymium Promethium Protoactinium Radium Radon Rhenium Rhodium Rubidium (wolfram) Ruthenium Samarium Scandium Selenium Silicon Silver (argentum) Sodium (natrium) Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin (stannum) Titanium Tungsten Unipentium Unnihexium Unnilhexium Uniquadrium Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium

Symbol

Atomic Number

Mo Nd Ne Np Ni Nb N No On O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Sm Sc Se Si Ag Na Sr S Ta Tc Te Tb Tl Th Tm Sn Ti W Unp Uns Unh Unq U V Xe Yb Y Zn Zr

42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 62 21 34 14 47 11 38 16 73 43 52 65 81 90 69 50 22 74 105 106 107 104 92 23 54 70 39 30 40

Source: Dean, J.E., Lange’s Handbook of Chemistry, McGraw-Hill, Inc., New York, 1994; Weast, R.C., CRC Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985.


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Hydrogen (H) Helium (He) Lithium (Li) Beryllium (Be) Boron (B) Carbon (C) Nitrogen (N) Oxygen (O) Fluorine (F) Neon (Ne) Sodium (Na) Magnesium (Mg) Aluminum (Al) Silicon (Si) Phosphorus (P) Sulfur (S) Chlorine (Cl) Argon (Ar) Potassium (K) Calcium (Ca) Scandium (Sc) Titanium (Ti) Vanadium (V) Chromium (Cr) Manganese (Mn) Iron (Fe) Cobalt (Co) Nickel (Ni) Copper (Cu) Zinc (Zn) Gallium (Ga) Germanium (Ge) Arsenic (As)

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Atomic Number 1.00794 4.0026 6.941 9.012 10.81 12.011 14.007 16 18.998 20.179 22.9898 24.305 26.9815 28.0855 30.9738 32.06 35.45 39.95 39.098 40.08 44.96 47.88 50.942 51.996 54.938 55.847 58.94 58.69 63.546 65.39 69.72 72.59 74.92

Atomic Weight lb/in.

g/cm

3

0.8376 × 10−4 1.76 × 10−4 0.5259 1.68213 2.2794 2.23 1.163 × 10−3 1.33 × 10−3 1.67 × 10−3 9.0 × 10−4 0.9688 1.74 2.699 2.233 1.821 2.071 3.214 × 10−3 1.782 × 10−3 0.8581 1.5501 2.985 4.540 6.007 7.197 7.4182 7.8611 8.8575 8.9129 9.8683 7.1414 5.9789 5.3145 5.7297

Density

3.03 × 10−6 6.36 × 10−6 0.019 0.0668 0.0887 0.0802 0.042 × 10−3 0.0516 × 10−3 0.06 × 10−3 3.25 × 10−5 0.035 0.063 0.098 0.084 0.0658 0.0748 0.116 × 10−5 0.644 × 10−4 0.031 0.056 0.1078 0.164 0.217 0.26 0.268 0.284 0.32 0.322 0.324 0.258 0.216 0.192 0.207

3

Physical Properties of Elements Listed According to Atomic Number

TABLE 7.3

°F −434.6 −458 356.97 2,332.4 3,812 6,605.6 −345.8 −361.12 −363.32 −415.6 208.06 1,202 1,220.4 2,570 111.38 235.04 −149.76 −308.56 145.85 1,542.2 2,802.2 3,020 3,434 3,374.6 2,271.2 2,795 2,723 2,647.4 1,982.1 787.2 85.6 1,719.3 1,497

−259.14 −272.2 180.54 1,278 2,100 3,826 −209.9 −218.4 −219.6 −248.2 97.8 650 660.37 1,410 44.1 112.8 −100.98 −189.2 63.25 839 ± 2 1,539 1,660 ± 10 1,929 ± 10 1,857 ± 20 1,244 ± 3 1,535 1,495 1,453 1,083.4 ± 2 419.58 29.78 937.4 603 (@28 atm)

°C

Melting Point

2,500 5,000 4,622 8,730 −320.4 −297.3 −306.7 −410.9 1,621.2 2,030 4,442 4,271 536 832.41 −30.28 −302.26 1,420 2,703.2 5,129.6 6,395 5,430 4,841.6 3,863.6 4,982 6,420 4,949.6 4,652.6 1,664.6 4,357.4 5,126 1,135.4

°F

1,371 2,771 2,550 4,832 −195.8 −183 −188.14 −246.05 882.9 1,110 2,518 2,355 280 444.7 −34.6 −185.7 771 1,484 2,832 3,535 2,999 2,672 2,129 2,750 3,549 2,732 2,567 907 2,403 2,830 613

°C

Boiling Point


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Xenon (Xe) Cesium (Cs) Barium (Ba) Lanthanum (La) Cerium (Ce) Praseodymium (Pr) (α-form) Neodymium (Nd) Promethium (Pm) Samarium (Sm) Europium (Eu) Gadolinium (Gd) Terbium (Tb) Dysprosium (Dy)

Selenium (Se) Bromine (Br) Krypton (Kr) Rubidium (Rb) Strontium (Sr) Yttrium (Y) Zirconium (Zr) Niobium (Nb) (Columbium) Molybdenum (Mo) Technitium-95 (Tc) Ruthenium (Ru) Rhodium (Rh) Palladium (Pd) Silver (Ag) Cadmium (Cd) Indium (In) Tin (Sn) Antimony (Sb) Tellurium (Te) Iodine (I) 131.29 132.91 137.33 138.906 140.12 140.91 144.24 (145) 150.3 151.96 157.25 158.93 162.5

60 61 62 63 64 65 66

95.94 (98) 101.07 102.906 106.42 107.868 112.48 114.82 118.71 121.75 127.60 126.91

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

78.96 79.904 83.80 85.468 87.62 88.91 91.224 92.906

34 35 36 37 38 39 40 41

0.2533 0.26084 0.270 0.1895 0.287 0.2793 0.3085

2.08 × 10−4 0.06787 0.13 0.223 0.244 0.2339

0.369 0.3974 0.4498 0.4495 0.434 0.379 0.313 0.264 0.264 0.245 0.225 0.178 (@25°C)

0.174 0.113 1.355 × 10−4 0.554 × 10−1 0.0954 0.1616 0.2355 0.310

7.01 7.22 7.52 5.244 7.9441 8.23 8.54 (@ 25°C)

5.61 × 10−3 1.8785 3.598 6.1726 6.770 6.475

10.2139 11.0 12.45 12.4421 12.0131 10.4967 8.6638 7.3075 7.3075 6.65 6.228 4.94

4.8163 3.12 3.749 × 10−3 1.532 2.64 4.472 6.52 8.5807

1,850 1,976 1,961.6 1,511.6 2,391.8 2,480 2,568.2

−183.2 83.12 1,337 1,688 1,468.4 1,707.8

4,742.6 3,941.6 4,190 3,569 2,829.2 1,763.5 609.62 313.9 449.54 1,167.33 841.1 236.3

422.6 19.04 −249.88 102 1,416.2 2,773.4 3,365.6 4,474.4

1,010 1,080 1,072 822 1,311 ± 1 1,360 ± 4 1,409

−111.9 28.4 725 920 798 ± 2 931

2,617 2,204 2,310 1,965 1,554 961.93 320.9 156.61 231.97 630.74 449.5 113.5

217 −7.2 −156.6 38.89 769 1,523 ± 8 1,852 2,468 ± 10

5,660.6 4,460 3,232.4 2,906.6 5,851.4 5,505.8 4,235

−160.8 1,236.74 2,984 8,000 5,894.6 5,813.6

8,333.6 8,810.6 7,052 8,100 7,200 4,013 1,409 3,776 4,118 2,620 2,530 363.83

1,264.8 137.8 −242.1 1,266.8 1,523.2 6,038.6 29,030 5,970

(continued)

3,127 2,460(?) 1,778 1,597 3,233 3,041 2,335

−107.1 669.3 1,640 4,427 3,257 3,212

4,612 4,877 3,900 4,482 3,982 2,212 765 2,080 2,270 1,438 1,388 184.4

685 58.78 −152.3 686 1,384 3,337 4,999 3,299

Properties of Materials 289

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67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

Osmium (Os) Iridium (Ir) Platinum (Pt) Gold (Au) Mercury (Hg) Thallium (Ti) Lead (Pb) Bismuth (Bi) Polonium (Po)

Astatine (At) Radon (Rn) Francium (Fr) Radium (Ra) Actinium (Ac) Thorium (Th)

Protactinium (Pa) Uranium (U) Neptunium (Np) Plutonium (Pu) Americium (Am)

Atomic Number

(Continued)

Holmium (Ho) Erbium (Er) Thulium (Tm) Ytterbium (Yb) Lutetium (Lu) Hafnium (Hf) Tantalum (Ta) Tungsten (W) Rhenium (Re)

Element

TABLE 7.3

231.036 238.029 237.048 244 (243)

(210) (222) 223 226.025 227.028 232.038

190.2 192.23 195.08 196.967 200.59 204.38 207.2 208.98 (209)

164.93 167.26 168.934 173.04 174.967 178.49 180.948 183.85 186.21

Atomic Weight lb/in.

0.687 0.740 0.700 0.4285

0.422

0.18

0.3956

0.813 0.813 0.775 0.698 0.4896 0.428 0.4097 0.354 — 0.339

2.45 0.330 0.336 0.161 0.356 0.473 0.600 0.697 0.765

3

15.37 19.0161 20.5 19.4 12

22.5038 22.5038 21.4519 19.32 13.5521 11.847 11.3405 9.7987 9.196α 9.398β 7.0 9.73 g/L 5.20 g/L 4.98 10.07 11.6509

3

g/cm 6.18 9.15 9.31 4.472 9.85 13.0926 16.6079 19.2929 21.1751

Density °F

2,912 2,070 1,184 1,229 1,821.2

575.6 −95.8 80.6 1,292 1,922 3,182 (Approximately)

4,900 4,370 3,221.6 1,948 −37.966 578.3 621.5 530.34 489.2

2,678 2,776.6 2,813 1,515.2 3,012.8 4,040.6 5,424.8 6,170 5,756

1,575 1,134 637 ± 1 640 1,176 ± 4

302 −71 (27) 700 1,050 1,758

3,033 2,449 1,769 1,064.4 −38.87 303.5 327.5 271.3 254

1,470 1,522 1,545 ± 15 824 ± 5 1,656 ± 5 2,227 ± 20 3,020 3,410 3,189

°C

Melting Point °F

6,904.4 7,055.6 5,849.6 4,724.6

638.6 −79.24 1,250.6 2,084 5,792 8,654

4,928 5,550 3,140.6 2,179.4 5,999 9,700 9,797 10,220 10,160.6 (Est) 9,080.6 9,600 7,970 5,380 673.8 2,654.6 3,164 2,590 1,763.6

3,818 3,902 3,232 2,011

337 −61.8 677 1,140 3,200 4,790

5,027 5,316 4,410 2,971 356.6 1,457 1,740 1,421 962

2,720 2,510 1,727 1,193 3,315 5,371 5,425 5,660 5,627

°C

Boiling Point


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98 99 100 101 102 103 104 105 106 107

96 97 (251) (252) (257) (258) (259) (260) (261) (262) (263) (262)

(247) (247) 0.18

0.31

7 14.78α 13.25β 4.9824 8.84

2,444 983 840 820 (1,527) (827) (827) (1,627)

1,340 ± 40

Source: Dean, J.E., Lange’s Handbook of Chemistry, McGraw-Hill, Inc., New York, 1994; Ensminger, D., Ultrasonics, Marcel Dekker, Inc., New York, 1988; Weast, R.C., CRC Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985.

Note: Equivalent values: F = (9C/5) + 32. C = 5(F − 32)/9. 1.0 in. = 2.54 cm. l.0 lb = 0.45359237 kg. 1 kg = 2.204622622 lb. 1 lb/in.2 = 70.306956 g/cm2 = 0.070306956 kg/cm2. 1.0 kpsi = 70.30695796 kg/cm2. 1 lb/in.3 = 27.67990471 g/cm3. 1 g/cm2 = 10 kg/m2. 1 L = 1000 cm3 = 0.001 m3 = 0.3048 gal. 1 gal = 3.785411784 L. 1 qt. = 0.946352946 L. Atomic number = the number of protons in an atomic nucleus. Atomic weight = the relative mass of an atom based on a scale in which a specific carbon atom (carbon-12) is assigned a value of 12. Molecular weight = the total of the atomic weights of the elements in a molecule. Fatigue limit (endurance limit) = maximum value of repeated stress which will not produce failure regardless of the number of applied cycles.

Californium (Cf) Einsteinium (Es) Fermium (Fm) Mendelevium (Md) Nobelium (No) Lawrencium (Lr) (Uniquadrium) (Unq) (Unnipentium) (Unp) (Unnihexium) (Uns) (Unnilhexium) (Unh)

Curium (Cm) Berkelium (Bk)


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Aluminum Aluminum 2SO 17ST 2024-T4 ALCLAD 2024-T4 2025-T6 6061-T6 7075-T6 XAPOO1 Antimony Arsenic Barium Beryllium Bismuth Boron Brass 70–30 Naval Bronze (phosphor 5%) Bronze (7.5% In) Bronze (7.7% Al) Cadmium Calcium Carbon Cerium Chromium Cobalt

Materials 0.703 0.724

0.731 0.745 0.724 0.724 0.7944 0.7733 0.127 2.95 0.3234

0.914 0.914

0.5624 0.211 0.049 2.5308

10.3

10.4

10.6 10.3 10.3 11.3 11 1.8 40–44 4.6

13 13

8 3 0.7

36 30

kg/cm (×106)

2

10

psi (×10 )

6

Young’s Modulus

5 5

4 4 4 4 4

4 4

6

0.352 0.352

1.85

0.024–0.030

0.35 0.331 0.33 0.35

0.33 0.33 0.33

0.33

0.33 0.33

0.33

Poisson’s Ratio

0.281 0.281 0.281 0.281 0.281

0.281 0.281

kg/cm2 (×106)

Shear Modulus psi (×10 )

Physical Properties of Engineering Materials

TABLE 7.4

1.55 2.22 6.92 7.197 8.86

6.70 5.73 3.626 33–51 9.80 2.297 8.50 8.60 8.10 8.86

2.77 2.77 2.70 2.79

2.71 2.80

2.7

Density g/cm3

844

12 8.60

1,476

490–1,406

7–20 7–20 7–20 21

2,320–3,586

2,883 1,266 949 1,687 37

1,406

20 41 18 13.5 24

562.5–1,266

kg/cm

2

8–18

kpsi

Fatigue Limit

60 35

64 58 45 82 2,601

22

kpsi

4,218 2,461

4,500 4,078 3,164 5,765

1,5467

kg/cm2

Ultimate Strength


10/27/2008 2:28:02 PM

Columbium (niobium) Constantan (60% Cu, 40% Ni) Copper Annealed Rolled Duralumin 17S Gadolinium Gallium Germanium German silver Gold, pure Hafnium Hastelloy X Hastelloy C Hydrogen Inconel X Wrought Indium Iridium Iron Electrolytic Armco Iron Cast Wrought Lanthanum Lead Pure Annealed Rolled Antimony 6% Lithium Magnesium Am35 Drawn Annealed


1.1248

0.0703 0.8014

1.406

0.1104 5.2725 2.0036

0.3515 0.1828

0.1195 0.35

17

1 11.4

10.8 20

1.57 75 28.5

5.0

2.0

1.7

6.4

1.0545

15

0.35

0.35

0.40–0.45

1.74

11.30 11.40 11.40 10.90 0.526 1.738 1.74

7.20 7.80 6.17

0.28 0.28

0.42

8.90 8.93 8.93 2.79 7.94 5.98 5.31 8.40 19.32 13.09 8.23 8.94 3.026 × 10−6 8.30 8.25 7.31 22.40 7.80–7.90 7.90 7.85

0.35

8.80

37

7–17 37

6–18

12–17 12–17 12–17

2,614

492–1,195 2,601

422–1,266

844–1,195 844–1,195 844–1,195

18

(continued)

1,266


10/27/2008 2:28:02 PM


— 21.3 14.0

Phosphorus Platinum Plutonium (alpha phase) Polonium

5.624

— 80

1.4763

1.1951

2.109

30

17

3.515

40–50

Palladium

1.6169

kg/cm (×106)

2

23

psi (×10 )

6

Young’s Modulus

(Continued)

Manganese Manganin (84% Cu) Manganin (12% Mn, 4% Ni) Mercury Molydenum Monel Metal (67% Ni, 29.2% Cu, 1.7% Fe, 1.0% Mn) Monel Wrought Neodymium Neon Neptunium Nickel, pure Nickel Silver (64% Cu, 17% Zn, 18% Ni) Nitrogen Osmium Oxygen

Materials

TABLE 7.4

psi (×10 )

6

kg/cm2 (×106)

Shear Modulus

0.39 0.15–0.21

0.31

0.32 0.315

Poisson’s Ratio

9.196α 9.398β

1.82 21.45 19.0–19.7

1.163 × 10−3 0.813 1.3286 × 10−3 12.013

8.83 7.01 0.8999 g/L 20.2 8.8–8.9

13.582 10.3 8.90

7.41 8.40

Density g/cm3

80 20–50

kpsi

2

5,625 1,406–3,515

kg/cm

Fatigue Limit

20–24 60

120–200

kpsi

1,406–1,687 4,218

8,437–14,060

kg/cm2

Ultimate Strength


10/27/2008 2:28:02 PM

Potassium Radium Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver, pure Silver–nickel 18% Sodium Steel 1% Carbon 1% C hardened HY80 Mild Stainless 302 Stainless 304 Stainless 304L Stainless 347 Stainless 410 Stainless 416 Steel 4130 Steel 4340 Strontium Sulfur α γ Tantalum Tellurium Terbium Thallium


0.5905 1.1248 0.7733

0.0914

8.4 16 11

1.3

— 1.8981 0.4219 0.0844

—

27 6

1.2

28

5.2725 3.7962

0.0352

0.5 — 75 54

0.35

0.298

0.29 0.29

0.28–0.29

0.37

0.9688 7.70 7.84 7.84 7.76 7.85–7.90 7.9 7.9 8.03 7.90 7.67 7.70 8.0 8.0 2.64 2.07 1.92 16.60 6.24 8.3 11.85

0.858 5.5 21.175 12.442 1.532 12.45 7.52 2.985 hex 4.82 2.325 10.40–10.50 8.75

50–145

3,515–10,195

87

25–75

18

(continued )

6,117

1,758–5,273

1,266


10/27/2008 2:28:02 PM


2.5308 1.1810

6 16.8

2.0879 1.2935

0.8436 0.7733

29.7 24 18.4

12

11

psi (×10 )

6

kg/cm2 (×106)

Shear Modulus

0.21 0.21

0.28

0.322 0.322 0.323 0.3

0.27

Poisson’s Ratio

19.30 19.30 19.1 18.97 6.01 5.761 g/L 6.90 4.472 7.14 7.10 6.52

19.10–19.25

7.30 4.50–4.85 4.43 4.54 4.85

11.68

Density g/cm3 kpsi

2

kg/cm

Fatigue Limit

16–18

56

18–600

200

32

kpsi

1,125–1,266

3,937

1,265–42,184

14,061

2,250

kg/cm2

Ultimate Strength

Source: Bauccio, M., ASM Metals Reference Book, ASM International, Materials Park, OH, 1993; Dean, J.E., Lange’s Handbook of Chemistry, McGraw-Hill, Inc., New York, 1994; Ensminger, D., Ultrasonics, Marcel Dekker, Inc., New York, 1988; Weast, R.C., CRC Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985.

3.515

50

14.8

0.8014

kg/cm (×106)

2

7–10

psi (×10 )

6

Young’s Modulus

(Continued)

Thorium (induction melt) Tin Titanium 6Al4V Ti150A B120VCA (aged) Tungsten Tungsten Annealed Drawn Uranium D-38 Vanadium Xenon Ytterbium Yttrium Zinc Rolled Zirconium

Materials

TABLE 7.4


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297

TABLE 7.5 Physical Properties of Engineering Materials Velocity of Sound Bulk cB Materials Aluminum Aluminum 2SO 17ST Al 2024-T3 6061-T6 7075-T6 Antimony Arsenic Barium Beryllium Bismuth Boron Brass 70–30 Brass (naval) Bronze (Phosphor 5%) (Sn 7.5%) (Al 7.7%) Cadmium Calcium Carbon Cerium Chromium Cobalt Columbium (niobium) Constantan (Cu 60%, Ni 40%) Copper Annealed Rolled Curium Duralumin 17S Gallium Germanium German silver Gold Hafnium Hastelloy X Hastelloy C Inconel X Inconel wrought Indium Iridium Iron Iron alnico Iron, cast Iron electrolytic Iron wrought

Characteristic Impedance ρco (106 g/(cm2−s))

Bar co

(105 cm/s)

Shear cs

5.06

2.655

1740

5.10 5.08

6.35 6.25

3.10 3.10

1.73 1.75

5.05 5.07 3.40

6.27 6.35

3.08 3.10

1.70 1.78

12.75–12.87 1.79

12.80–12.89 2.18

8.71–8.88 1.10

2.33–2.41 2.14

3.40–3.48 3.49

4.37–4.70 4.43

2.10 2.12

3.70–4.04 3.61

3.43

3.53

2.23

3.12

2.40

2.78

1.50

2.40

4.30

5.24

2.64

4.60

3.60–3.71 3.81 3.75

4.60–4.80 4.76 5.01

2.26–2.33 2.33 2.27

4.10–4.25 4.25 4.47

5.15

6.32

3.13

1.76

Electrical Resistivity µohm cm

Thermal Conductivity Btu/h/ft2/in./°F

1320 1080 39.0 35.0 50.0 5.9 106.8 1.8 × 1012

131

1100 58 1015 697

6.83 3.43 1,375.0 78.0 13.0 6.24 13.1

412 470 639 871 165 464 479

153 1.673

2730

2,444.0 56.8 60 × 104 3.58 2.03

4.76 3.24

2.16 1.20

4.00 6.26

— — — 5.08

5.79 5.84 5.94 7.82

2.74 2.90 3.12 3.02

4.77 5.22 4.93 6.45

4.79 5.17–5.18 5.20 3.0–4.7 5.12

— 5.96 5.96 3.5–5.6 5.95

— 2.05 3.24 2.2–3.2 3.24

— 4.68 4.69 2.5–4.0 4.70

2.19 32.4

765 232

2060

91 8.37 9.71

175 406 498

420 (continued)


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298 TABLE 7.5

(Continued) Velocity of Sound Bulk cB

Materials Lanthanum Lead Pure Annealed Rolled Antimony 6% Lithium Magnesium Magnesium Am 35 Magnesium drawn, annealed Manganin (Cu 64%, Zn 17%, Ni 18%) Mercury Molybdenum Monel metal Monel wrought Nickel Nickel–silver (Cu 64%, Zn 17%, Ni 18%) Nitrogen Osmium Oxygen Palladium Phosphorus Platinum Plutonium Potassium Radium Rhenium Rhodium Selenium Silicon Silver Sodium Steel Steel, 1%C Steel, 1%C hardened Steel HY80 Steel, mild Steel Stainless 302 Stainless 304 Stainless 304L Stainless 347 Stainless 410 Stainless 416


Bar co

(105 cm/s)

Shear cs




59 1.20–1.25 1.19 1.21 1.37

2.16–2.40 2.16 1.96 2.16

0.70–0.79 0.70 0.69 0.81

2.46–2.72 2.46 2.23 2.36

20.65

4.90 5.00

5.74 5.79

3.08 3.10

0.99 1.01

4.46

4.94

5.77

3.05

1.00

3.83

4.66

2.35

3.90

5.40–5.45 4.40 4.52 4.79–4.90 3.83

6.25–6.29 5.35 6.02 5.60–6.04 4.62

3.35 2.72 2.72 2.96–3.00 2.32

6.35 4.76 5.31 4.90–5.38 4.03

241

494

138

94.1 5.17

58 950 144

6.84

430 217

0.147 9.5

2.80

3.26–3.96

1.67–1.73

6.98–8.46

10.8 1017 9.83 150 6.15 21.0 4.5

2.64–2.68

3.60–3.70

1.59–1.70

3.80–3.90

5.05–5.17

5.85–6.10

3.23

4.56

5.18 5.07

5.94 5.854

3.22 3.15

4.66 4.59

— 5.05–5.20

5.88 5.96–6.10

— 3.24

4.56 4.68–4.82

4.90 4.92 4.93 5.00 5.03 5.20

5.66 — 5.64 5.79 5.76 6.02

3.12 — 3.07 3.10 2.99 3.23

4.55 — 4.45 4.57 4.42 4.64

1.59 4.2

0.191 468 494 60 697

1056 3 581 2400 929

315

10/27/2008 2:28:03 PM

Properties of Materials TABLE 7.5

299

(Continued) Velocity of Sound

Bar co

(105 cm/s)

Shear cs


5.04 —

5.83 5.85

— 3.24

— 4.82

3.35

4.10

2.90

5.48

— 2.73–2.74 5.08 5.08 5.08 4.31–4.60 4.62 4.32 —

2.94 3.32–3.38 5.99–6.07 6.23 6.10 5.17–5.46 5.22 5.41 3.37

1.56 1.61–1.67 3.12–3.125 — 3.12 2.62–2.87 2.89 2.64 1.98–2.02

3.32 2.42–2.47 2.70–2.73 — 2.77 9.98–10.42 10.07 10.44 6.30

3.81 3.85 —

4.17 4.21 4.65

2.41 2.44 2.22–2.30

2.96 2.99 1.32–3.01(8.0)

Bulk cB Materials Steel 4130 Steel 4340 Sulfur Tantalum Tellurium Thallium Thorium Tin Titanium Titanium 6Al4V Titanium 150A Tungsten Tungsten Tungsten drawn Uranium Vanadium Zinc Zinc rolled Zirconium

7.3



2 × 1023 12.4 2 × 105 18.0 18.6 11.5 47.8

1.83 377 41 348 204 464 198

5.5

1100

29.0

168

26.0 5.92

240 780 784 2.98–41.0(8.0)

Ultimate Stress and Fatigue Limit

Because ultrasonic waves are stress waves, the ultimate stress and fatigue limit are important in the design of the vibratory elements of an ultrasonic system. Ultimate stress is that stress at which rupture occurs. Therefore, every precaution is taken to avoid these intense stress levels in the design of an ultrasonically vibrating system. The fatigue limit is especially important. It is the limit of stress below which a vibrating element may operate continuously without failure. The figures given for these values in Table 7.4 apply to standard low-frequency measurements. Ultrasonically, the cycles accumulate at a much higher rate and therefore the figures given in Table 7.4 must be used as relative values for endurance. The value used depends upon the application. Elements used to construct transducers, a transducer attached to a driven part, horns and horn designs, flanges on horns, or flexural bars are all subjected to stresses that may lead to failure or disconnection. Bolts used in the assembly of transducers are also subjected to stress. The stresses at which these elements will be subjected at a given location and amplitude of vibration may be determined fairly accurately using the equations of Chapter 2. Stress risers play an important part in determining the fatigue life of a part vibrating at ultrasonic frequency. Pits, scratches, embedded defects, and small radius fillets at the junction between two sectors of different diameters of a double cylinder are stress risers. A machine scratch within the fillet of a double-cylinder or similar structure may lead to failure even when the fillet radius itself would otherwise be acceptable for the applied stress.


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300

12

11

Stress PSI × 104

10

Curve of steady-state stress Curve adjusted to compensate for peak

14 Peak 14

15 Peak 15

9

(11) (10)

8 7

(7) (8) (9)

(13)

6 5 4 3 2 1 0 104

105

106

107

108

109

1010

Cycles

FIGURE 7.1 S–N curve for SNC-631 steel alloy at 38.6 kHz.

Ultrasonic waves are stress waves. A half-wave resonant bar vibrating in the longitudinal mode represents a standing stress wave. Stress is distributed along the bar in accordance with the amount the bar stretches at each point of the length to maintain the amplitude of displacement at the opposite ends. As shown in Chapter 2, the stress wave is dependent upon the geometry of the bar. Weld and silver solder joints should be located at positions where the stress level will never exceed the strength of the joint. Careful attention should be given to the location of welds and silver solder joints with respect to stresses in all designs of resonant systems. One method of determining the fatigue limit of a material ultrasonically is to prepare a series of identical specimens, such as well-machined and polished double-cylinder horns. The fillet itself is a stress riser, and by using the equations for calculating the stress rise (Equations 2.16 through 2.18), the actual stress at that point for a given amplitude of vibration may be determined accurately. The transducer and horn are mounted in a rigid structure, so that a capacitance type of displacement gage can be used to monitor the displacement at the free end of the horn. The horns are then driven at different discrete levels of intensities to failure in a noncorrosive atmosphere. From the displacement amplitude at the end and the distance to the position of the break, one can now calculate the stress level to which the horn was subjected at that point. The time-on and time-off and the temperature (held at a constant value during the test) are monitored. Figure 7.1 was obtained at a frequency of 38.6 kHz in this manner for special steel alloy, Japanese SNC-631.

7.4 7.4.1

Materials Properties for Processing by Ultrasonics Friction and Wear

Friction and wear are consequences of two mating surfaces rubbing together. Friction is the resistance to relative motion between surfaces of objects in contact with each other under pressure. The coefficient of friction is the ratio of this resistive force to the


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301

force normal to the surfaces of contact. Various lubricants may be used to minimize or control friction between moving surfaces. Kragelskii states that the interaction between surfaces is of a dual molecular-mechanical nature. The molecular interaction is due to the attraction between the two solids, their adhesion; the mechanical interaction is due to the mutual inter-penetration of localized regions on the compressed surfaces [5]. In ultrasonic applications, the compatibilities of materials, the effects of relative velocity, temperature, and pressure, surface conditions or contours, relative hardness, and of material toughness are important factors. Relative velocity and pressure affect the temperature between two mating surfaces. Acoustic vibration is an oscillating movement. The velocity of an ultrasonic tool is a function of the frequency and the amplitude of vibration. As the pressure increases, the amount of mechanical energy transformed into heat increases under constant velocity. Under constant pressure, the heating rate increases as the velocity increases. The increased temperature changes the mechanical properties of the rubbing materials and the films forming at the surfaces. These are conditions of friction between mating surfaces whether the motion is due to acoustic vibration or not. Two aspects of motion must also be considered: (1) a system in which one surface is sliding against an ultrasonically driven surface like a belt moving against an ultrasonic tool in a fixed position and (2) a system in which one surface is stationary and the other is driven ultrasonically. What happens in each case depends upon the pressure, the amplitude of vibration, the dwell time, and the physical and mechanical properties of each surface. All surfaces have asperities, large or small. The size of asperities in the surfaces relative to the amplitude of the movement are important when ultrasonic energy is applied. Wear is the erosion of surfaces by frictional forces. These frictional effects are not restricted to the mating surfaces. They are a means of coupling shear energy into the materials as well. By special design, the absorption of energy can be localized within certain regions of the mating parts to produce the types of welds and forms desired in thermoplastic materials. These principles lead to a wide range of uses for ultrasonic energy. A polymeric material must be thermoplastic to be molded, extruded, or bonded (welded) ultrasonically. The following terms are defined with regard to their importance in the effectiveness of ultrasound in welding, molding, extruding, or forming of plastics or metals: Friction. As defined previously, the resistance to relative motion between surfaces of objects in contact with each other under pressure. Weld. The localized coalescence of material wherein the coalescence is produced by heating to suitable temperatures, with or without the application of pressure, and with or without the use of filler material. Welding. Joining two materials by applying heat to melt and fuse them, with or without filler material. Weldability. The capacity of a material to be welded under the fabrication conditions imposed with a specific, suitably designed structure and to perform satisfactorily in the intended service. Bond. The joining of the weld material and the base material, or the joining of the base material parts when weld material is not used. Glue. An adhesive material made of crude, impure, amber-colored form of commercial gelatin of unknown detailed composition produced by the hydrolysis of animal collagen; gelatinizes in aqueous solutions, and dries to form a strong adhesive layer.


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302

Adhesive. A substance used to bond two or more solids so that they act or can be used as a single piece; examples are resins, formaldehydes, glue, paste, cement, putty, and polyvinyl resin emulsions. Adhesive bond. The forces such as dipole bonds that attract adhesives and base materials to each other. Adhesive bonding. The fastening together of two or more solids by the use of glue, cement, or other adhesive. 7.4.2 7.4.2.1

Welding and Staking Properties Necessary for Ultrasonic Welding and Forming of Materials *

Plastic materials are of two types: thermoplastic and thermosetting. Thermoplastic materials soften or become liquid as they are heated. This makes them capable of being bonded together by fusion. Thermosetting plastics are hard, brittle materials that char or degrade when they are subjected to intense heat. They cannot be welded ultrasonically [6]. 7.4.2.1.1 Compatibility of Materials It is necessary that any two plastic materials to be welded together be chemically compatible. If they are not compatible, there can be no chemical bonds, even when they melt at the same temperature. Similar thermoplastic materials, such as two acrylonitrile butadiene styrene (ABS) parts, may be welded together. Two parts of incompatible materials, such as polyethylene and polypropylene, cannot be welded together, although both have a similar appearance and many common physical properties. Two materials of similar molecular structure may be bonded if their melt temperatures are within 40°F (6°C) of each other. Generally speaking, only similar amorphous polymers have an excellent probability of being welded to each other. The chemical properties of any semicrystalline material make each one compatible with itself. Other factors that affect the weldability of plastic parts include hygroscopicity, mold release agents, lubricants, plasticizers, fillers, flame retardants, regrind, pigments, and resin grades. 7.4.2.1.2 Hygroscopicity Hygroscopicity is the tendency of a material to absorb moisture. Resins such as polyamide (nylon), polycarbonate, polycarbonate/polyester alloy (xenoy), and polysulfone are hygroscopic (i.e., they absorb and retain moisture from the air). When moist parts are welded, the water inside the materials boils off when the temperature reaches the boiling point. This process creates a foamy condition at the joint interface, making it difficult to achieve a hermetic seal and giving the assembled parts a poor cosmetic appearance. The bond strength is also weakened. 7.4.2.1.3 Mold Release Agents Mold release agents are usually sprayed directly on the mold cavity surface and are used to make parts eject from the mold cavity more readily by reducing friction between the part and the cavity walls. Unfortunately, mold release agent on the molded parts reduces surface friction in the joint interface between the parts when they are being welded.

* Much of the following information has been obtained by permission from the Guide to Ultrasonic Plastics Assembly by the Dukane Ultrasonics Corporation.


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303

Because the ultrasonic assembly process depends on surface friction, the use of mold release agents can be detrimental to weldability. Furthermore, the chemical contamination of the resin by the release agent can inhibit the formation of the desired bond. Some agents can be removed from parts with a preassembly cleaning operation using suitable solvents. If a release agent must be used, paintable/printable grades that permit painting and silk screening are preferred, because they interfere the least with ultrasonic assembly and often require no preassembly cleaning. Use of zinc stearate, aluminum stearate, fluorocarbons, and silicones should be avoided if possible. 7.4.2.1.4 Lubricants Lubricants such as waxes, zinc stearate, stearic acid, aluminum stearate, and fatty esters are added to resins to improve flow characteristics and enhance resin processability. Internal lubricants reduce the coefficient of friction. They cannot be removed. Therefore, they have a negative effect on ultrasonic welding. 7.4.2.1.5 Plasticizers Plasticizers are used to increase the flexibility and softness of a material and have a tendency to migrate or return to the joint of a welded part after a period of time, resulting in a weakened bond or joint. FDA-approved plasticizers are preferred to the metallic plasticizers. Experimentation is advised before setting up for production by means of ultrasonic assembly. 7.4.2.1.6 Fillers Fillers, such as glass fiber, talc, carbon fiber, and calcium carbonate, are added to resins to alter their physical properties. For instance, a glass filler might be added to a resin to improve its dimensional stability or material strength. Common mineral fillers, such as glass or talc, can actually enhance the weldability of thermoplastics, particularly semicrystalline materials, because they improve the resin’s ability to transmit vibrational energy. A glass content of 10–20% can substantially improve the transmission properties of a resin. The increase of the ratio of fillers to the increase in weldability exists below a certain prescribed level. Levels exceeding 10–20% may cause the accumulation of filler at the joint to be so severe that there may not be enough resin in the joint interface to form an acceptable weld. The accumulation at the joint interface is known as agglomeration or filler enrichment. If the amount of the filler in the joint exceeds 40%, there is more unweldable material there than weldable material. The weldability is more difficult to achieve consistently and overall assembly strength suffers. Filler contents >20% can cause excessive horn and fixture wear that may require special tooling. Because of the presence of particles at the resin surface, heat-treated steel or carbide-faced titanium horns may need to be used. Higher-powered ultrasonic equipment may also be required to create sufficient heat at the joint. 7.4.2.1.7 Flame Retardants Flame retardants are used to alter the combustible properties of plastics. Retardants such as antimony, boron, halogens, nitrogen, and phosphorous are added to resins to keep temperatures below a combustion level or to prevent a chemical reaction between the resin and oxygen or other combustion-aiding gases. Flame retardants can directly affect thermoplastic weldability by reducing the strength of the finished joint. High-power equipment, operating at higher than normal amplitudes, is often required so that the parts can be overwelded to achieve adequate strength.


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304

7.4.2.1.8 Regrind Regrind is the term for plastic material that has been recycled or reprocessed and added to the original resin. Ultrasonic assembly is one of the few processing methods that permits regrinding of parts, as no foreign substance is introduced into the resin. Providing that the percentage of regrind is not excessive, and the plastic has not been degraded or contaminated, few problems should arise. However, for best results, it is advisable to keep the regrind percentage as low as possible. 7.4.2.1.9 Colorants Colorants—liquid or dry—or pigments which have very little effect on weldability, unless the percentage of colorant to resin is exceedingly high. White and black parts often require more pigments than other colors and may cause some problems. Different colors of the same part may result in different setup parameters. Experimentation is recommended prior to full production. 7.4.2.1.10 Resin Grade Resin grade can have a significant effect on an application’s weldability. Resin grade is important, because different grades of the same material can have very different melt temperatures, resulting in poor welds or apparent incompatibility. Whenever possible, materials of the same grade should be used in the ultrasonic assembly process. Table 8.7 lists several thermoplastic materials that can be bonded ultrasonically, giving ratings for the effectiveness of the various types of welding operations. 7.4.2.2

Joint Designs for Efficient Welding

The Dukane Corporation has published an excellent treatise (referenced earlier) on ultrasonic bonding, which contains joint designs for ultrasonic welding. The joint design of the mating pieces is critical in achieving optimum assembly results. The joint design of a particular part depends upon factors such as type of plastic, part geometry, and the requirements of the weld. There are many different joint designs, each with its own advantages. Some of these designs are discussed later in this section. There are three basic requirements in joint design as follows: 1. A uniform contact area 2. A small initial contact area 3. A means of alignment A uniform contact area means that the mating surfaces should be in intimate contact around the entire joint. The joint should also be in one plane, if possible. A small initial contact area should be established between the mating halves. Doing so means less energy, and therefore less time, is required to start and complete the meltdown between the mating parts. A means of alignment is recommended, so that the mating halves do not misalign during the welding operation. Alignment pins and sockets, channels, and tongues are often molded into parts to serve as ways to align them. It is best not to use the horn or the fixture to provide part alignment. The need to follow the basic requirements for any joint design can be demonstrated using a flat butt joint. Only the high points will weld on a flat butt joint, resulting in erratic, inconsistent welds. Extending the weld time to increase the melt simply enlarges the original weld points and causes excessive flash outside of the joint. When one of the surfaces is brought to a point, it may produce a weld with a good appearance but having


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305

little strength, or, if good strength is obtained, it produces excessive flash and thus ruins the appearance of the weld. 7.4.2.2.1 The Energy Director An energy director is a triangular-shaped bead molded into the part interface. It typically runs around the entire joint perimeter. The energy director was developed to provide a specific volume of material to be melted so that good bond strength could be achieved without excessive flash. It is the joint design that is generally recommended for amorphous polymers. When ultrasonic energy is transmitted through the part under pressure and over time, the energy concentrates at the apex of the energy director (i.e., where the apex of the triangular-shaped bead contacts the other mating surface), resulting in a rapid buildup of heat that causes the bead to melt. The molten material flows across the joint interface, forming a molecular bond with the mating surface. The energy director meets two of the three basic requirements of a joint design: it provides (1) a uniform contact area and (2) a small initial contact area. The energy director itself does not provide a means of alignment or provide a means to control material flash. These requirements must be incorporated into the part design. The basic energy director design for an amorphous resin is a right triangle with the 90° angle at the apex and the base angles each at 45° (Figures 7.2a and 7.2b). This makes the height one-half the width of the base. The size of this energy director can range from 0.005 in. (0.127 mm) to 0.030 in. (0.762 mm) high and from 0.10 in. (0.254 mm) to 0.060 in. (1.53 mm) wide. For polycarbonate, acrylics, and semicrystalline resins, the energy director is an equilateral triangle, with all three angles being 60°. This design makes the height 0.866 times the base width. The base width can range from 0.010 in. (0.254 mm) to 0.050 in. (1.27 mm). The most common and basic joint design is the butt joint with an energy director. The width of the base of the energy director is between 20% and 25% of the thickness of the wall (i.e., B = W/4 to W/5). When the wall is thick enough to produce an energy director

α

β

α = 90°

β = 60°

A B A = height of the energy director B = width of the energy director

B

B = W/4 to W/5

At the base = W/4 to W/5 W

(a)

(b)

FIGURE 7.2 Butt joint configurations: (a) energy director for amorphous resins and (b) energy director for semicrystalline resins.


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larger than the maximum size, two smaller parallel energy directors should be used. The height at the apex of the energy director is either half the base or 0.866 times the base, depending on the material. This design produces a weld across the entire wall section with a small amount of flash normally visible at the finished joint. As stated in Section 7.4.2.2, the parts should be designed to include a means of alignment. If this is not possible, the fixture can be designed to provide the locating features necessary to keep the parts aligned with respect to each other. Typically, hermetic seals are easier to achieve with amorphous rather than semicrystalline materials. If a hermetic seal is required, it is important that the mating surfaces be as close to perfectly flat and parallel to each other as possible. The butt joint with an energy director is well suited for amorphous resins, because they are capable of molten flow and gradual solidification. However, it is not the best design for semicrystalline resins. With semicrystalline resins, the material displaced from the energy director usually solidifies before it can flow across the joint to form a seal, which causes a reduction in overall strength and makes hermetic seals difficult to achieve. However, sometimes there are certain limitations imposed by the design or size of the part that make it necessary to use an energy director on semicrystalline parts. In situations where an energy director must be used with a semicrystalline resin, the energy director should be larger and have a steeper angle to give it a sharper point (apex). This design enables it to partially embed in the mating surface during the early stages of the weld, thereby reducing the amount of premature solidification and degradation caused by exposure to the air. The larger, sharper design improves the strength and increases the chances of obtaining a hermetic seal. Experimentation has shown that the larger, sharper energy director design is also superior when working with polycarbonates and acrylics, even though both materials are classified as amorphous materials. 7.4.2.2.2 The Step Joint The step joint is a variation of the energy director joint design. Like the energy director, it meets two of the basic requirements of joint design: it provides (1) a uniform contact area and (2) a small initial contact area. A step joint also provides the third requirement: a means of alignment (Figure 7.3). The strength of a step joint is less than that of the butt joint, because only part of the wall is involved in the welding. The recommended minimum wall thickness is 0.080 in. (2.03 mm) to 0.090 in. (2.29 mm). A step joint may be used when cosmetic appearance of the assembly is important. Use of a step joint can eliminate flash on the exterior and produce a strong joint, as material from the energy director will typically flow into the clearance gap between the tongue and the step. The energy director is dimensionally identical to the one used on the butt joint. The height and width of the tongue are each one-third of the wall thickness (T = W/3). The width of the groove is 0.002 in. (0.05 mm) to 0.004 in. (0.10 mm) greater than that of the tongue to ensure that no interference occurs (G = T + 0.002 to 0.004 in.). The depth of the groove should be 0.005 in. (0.13 mm) to 0.010 in. (0.25 mm) greater than the height of the tongue, leaving a slight gap between the finished parts (D = T + 0.005 to 0.010 in.). This design is done for cosmetic purposes, so that it will not be obvious if the surfaces are not perfectly flat or the parts are not perfectly parallel. 7.4.2.2.3 The Tongue-and-Groove Joint The tongue-and-groove joint is another variation of the energy director (Figure 7.4). Like the step joint, it provides the three requirements of a joint design (1) a uniform contact area, (2) a small initial contact area, and (3) a means of alignment. It also prevents internal and external flash, as there are flash traps on both sides of the interface.


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307

W

Slip fit

W/8

T

W/3 B

H

A C W/3 G

D

W FIGURE 7.3 Step joint.

FIGURE 7.4 Tongue-and-groove joint.

The tongue-and-groove joint is used primarily for applications where self-location and flash prevention are important. It is an excellent joint design with applications calling for low-pressure hermetic seals. The main disadvantage of the tongue-and-groove joint is that less weld strength is possible, because less area is affected by the joint. The minimum wall thickness recommended for use with the tongue-and-groove joint is 0.120 in. (3.05 mm) to 0.125 in. (3.12 mm). Again, the energy director is dimensionally identical to the one used in butt joint. The height and width of the tongue are both one-third of the thickness of the wall. Clearance should be maintained on each side of the tongue to avoid interference and provide space for the molten material. Therefore, the groove should be 0.004 in. (0.10 mm) to 0.008 in. (0.20 mm) wider than the tongue. The depth of the groove should be 0.005 in. (0.13 mm) to 0.010 in. (0.25 mm) less than the height of the tongue. As with the step joint, a slight gap designed into the finished part assembly proves advantageous for cosmetic reasons. 7.4.2.2.4 The Shear Joint The shear joint is used when a strong hermetic seal is needed, especially with semicrystalline resins. A certain amount of interference is designed into the part for a shear joint (Figure 7.5). Welding is accomplished by first melting the contacting surfaces. As the melting parts telescope together, they continue to melt with a controlled interference along the vertical walls. A flash trap, which is an area used to contain the material displaced from the weld, may be used. The smearing action of the two melt surfaces at the weld interface eliminates leaks and voids, as well as exposure to air, premature solidification, and possible oxidative degradation. The smearing action produces a strong structural weld. Rigid sidewall support is very important with shear joint welding to prevent part deflection during welding. The walls of the fixtured part need to be supported up to the joint interface by the fixture, which should closely conform to the shape of the part.


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308

Depth of weld

Minimum lead-in 0.020 in. (0.5 mm) Interference

Fixture

Before weld

After weld

FIGURE 7.5 Shear joint.

Horn

Parts

1/4 in. (6 mm) or less

Near field

Greater than 1/4 in. (6 mm)

Far field

FIGURE 7.6 Near-field and far-field welding.

In addition, to make it easier to remove the part from the fixture, the fixture itself should be split so that it can be opened and closed. A shear joint meets the three requirements of joint design. The lead-in provides a means of alignment and self-location of the parts to be welded. Properly designed and molded parts ensure a uniform contact area. The small initial contact area between the parts occurs at the base of the lead-in. 7.4.3

Tools for Ultrasonic Welding

Ultrasonic welding is done by bringing an ultrasonic horn into contact with one of the materials to be welded. The face of the horn must be parallel with the interfaces to be welded. Figure 7.6 illustrates near field and far field welding. Near field welding refers to


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309

having a spacing of 1/4 in. (6 mm) or less between the horn and the joint. Far field welding refers to having distances of >1/4 in. between the horn and the weld. It is always better to weld near field. Far field welding requires higher than normal amplitudes, longer weld times, and higher air pressures to achieve a weld comparable to a near field weld. Generally speaking, it is best to use far field welding only for amorphous resins, which transmit energy better than semicrystalline resins. Having the joint interface and the horn area of contact each on single, parallel planes is important to good welding. The energy travels uniformly between the source and the joint. When the energy has to travel different distances through the specimen, the weld joint becomes very inconsistent, making either weak structural bonds or overwelded bonds. Other structural conditions that affect the welding process include sharp corners, holes or voids, and appendages. Sharp corners localize stress. Plastic parts, when subjected to ultrasonic energy, may fracture or melt in these high-stress areas. It is good to use a generous radius in all corners and edges. Holes or voids interfere with the transmission of ultrasonic energy. Little or no welding will be achieved beneath these areas. Eliminate all sharp angles, bends, and holes, where possible. Appendages, tabs, or other protrusions molded onto plastic parts also focus stress when subjected to vibratory energy and have a tendency to fall off (or degate). These problems may be minimized by adding a generous radius to the areas where the appendages join the main part, applying a light force to the appendage(s) to dampen the flexure, making the appendages thicker, or using a higher frequency (40 kHz rather than 20 kHz), if possible. Diaphragming may be a problem in welding thin sections of flat, circular parts, as they may flex under the influence of ultrasonic energy. As the part flexes up and down, it absorbs ultrasound to the extent that heat is formed and causes melting or produces a hole in the part. Diaphragming will often occur in the center of a part or at the gate area. Making those sections thicker may prevent diaphragming. 7.4.4

Staking

Ultrasonic staking allows a thermoplastic material to be attached to a nonthermoplastic material or to a metal using thermoplastic stakes, which are formed into a locked position ultrasonically. It is the process of melting and reforming a stud to mechanically lock a material in place. It provides an alternative to welding when (1) the two parts to be joined are made of dissimilar materials that cannot be welded (e.g., metal and plastic) or (2) simple mechanical retention of one part relative to another is adequate (i.e., molecular bonding is not necessary). The advantages of staking include short cycle time, tight assemblies with no tendencies to spring back, the ability to perform multiple stakes with one horn, good process control and repeatability, simplicity of design, and elimination of consumables, such as screws or adhesives. Staking is most commonly used to attach metal to plastic. A hole in a metal part is designed to receive a stud or boss, which is molded into the plastic part. The stud should be designed with a generous radius at its base to prevent fracturing. A vibrating horn with a contoured tip contacts the stud and creates localized, frictional heat. As the stud melts, light pressure from the horn reforms the head of the stud to the configuration of the horn tip (Figure 7.7). When the horn stops vibrating, the plastic material solidifies and the metal and plastic parts are fastened together.


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310

Staking cavity

Dissimilar material Plastic Before

After

FIGURE 7.7 Staking.

General guidelines for staking applications include 1. Use of a high-amplitude horn with a small contact area to localize heat and increase the rapidity of the melt. 2. Light initial contact force with controlled horn descent velocity to concentrate the ultrasonic energy at the limited horn and stud contact area. 3. Pretriggering the ultrasonic energy to create an out-of-phase relationship, preventing horn and stud coupling. 4. A slow actuator down speed to prevent stud fracture while allowing plastic material to flow into the horn cavity. 5. A heavier hold force during the hold time to give the stud optimum strength to retain the attached material. The integrity of an ultrasonically staked assembly depends upon the geometric relationship between the stud and the horn cavity, and the ultrasonic parameters used when forming the stud. Proper stake design produces optimum stud strength and appearance with minimum flash. The design depends on the application and physical size of the stud or studs being staked. The principle of staking, however, is always the same—the initial contact area between horn and stud must be kept to a minimum, concentrating the energy to produce a rapid, yet controlled melt. 7.4.4.1

Hydraulic Speed Control

The Dukane Corporation offers a hydraulic speed control for their press/thruster systems. It is especially useful in staking applications, as it regulates the velocity at which the horn descends once it contacts the stud. The horn’s downstroke slows to match the stud’s melt rate, so that intimate contact between the horn and the stud is maintained, giving the stud a finished appearance when the weld cycle is completed. Downstroke speed can be rapid prior to stud contact. Then horn descent can be precisely controlled during the actual staking operation. 7.4.5

Five Basic Staking Designs

The five basic staking designs include (1) the standard rosette profile stake, (2) the dome stake, (3) the hollow stake, (4) the knurled stake, and (5) the flush stake.


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Properties of Materials 7.4.5.1

311

The Standard Rosette Profile Stake

The standard rosette profile stake satisfies most requirements. The finished head is twice the diameter of the original stud. It is designed to stake studs with flat heads and is recommended for studs 1/16 in. (1.6 mm) outside diameter (OD) or larger (Figure 7.8). This stake is ideal for staking nonabrasive rigid and nonrigid thermoplastics. 7.4.5.2

The Dome Stake

The dome stake is typically used for studs of 430°C (800°F). This is better than any of the other refractories, which suffer accelerated oxidation at lower temperature levels.

7.15

Comments and Conclusions

The information presented in this chapter is intended for easy selection of materials to be used in the wide variety of high-intensity applications of ultrasonic energy. Some of the materials listed may have never been used before and may never be considered again. But the reasons for presenting them evolve from some past experiences. For example, in one case there was a need to ultrasonically promote a chemical reaction in an acidic solution (of pH 3) at a high temperature and a high pressure. The requirements were that the material to be treated was placed in a high-pressure cell. The ultrasonic transmission line had to be designed to extend through the bottom of the cell. The line had to withstand the acidic environment. So many of the materials that withstand acids, even at room temperatures, dissolve rapidly after the cavitation of the liquid breaks through the protective coating. The transmission line also had to be long enough that the ultrasonic source was protected from overheating. The solution was to use tantalum for the transmission line. Tantalum was unique for this purpose. It was not a material that was commonly used for transmission lines for horns. The acoustic properties for tantalum were unknown to the designer at the time of this program. He calculated a bar velocity of sound for determining the length using Young’s modulus and density for the material. This was the only information available to him at that time. Fortunately, the calculated value proved to be correct. The material required for the delay line is expensive—$2000 for the single 20 kHz unit. It held up very well for the complete project, making the treatment and cost worthwhile. This example is presented for the purpose of illustrating the reason for giving the choice of materials that are available for use in applying high-intensity ultrasonic energy to specific needs.

References 1. 2. 3. 4.

M. Bauccio, ASM Metals Reference Book, 3rd edition, ASM International, Materials Park, OH, 1993. J.E. Dean, Lange’s Handbook of Chemistry, 15th edition, McGraw-Hill, Inc., New York, 1994. D. Ensminger, Ultrasonics, Marcel Dekker, Inc., New York, pp. 206–213, 1988. Dukane Ultrasonics Corporation, Guide to Ultrasonic Plastics Assembly, Dukane Corporation, St. Charles, IL, 1995.


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5. I.V. Kragelskii, Friction and Wear, Butterworths, Washington, pp. 15–17, 1965. 6. R.C. Weast, Ediotor-in Chief, CRC Handbook of Chemistry and Physics, 60th edition, CRC Press, Inc., Boca Raton, FL, 1985. 7. G.S. Brady and H.R. Clauser, Materials Handbook, 13th edition, McGraw-Hill, Inc., New York, 1991. 8. S.P. Parker, Editor-in-Chief, McGraw-Hill Dictionary of Scientific and Technical Terms, 5th edition, McGraw-Hill, Inc., New York, 1994.


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8 Ultrasonics-Assisted Physical and Chemical Processes Dale Mangaraj, B. Vijayendran, and Dale Ensminger

CONTENTS 8.1 Introduction ........................................................................................................................ 324 8.2 Physical Processes.............................................................................................................. 324 8.3 Chemical Processes ........................................................................................................... 325 8.4 Ultrasonic Effects ............................................................................................................... 325 8.5 Degassing ............................................................................................................................ 325 8.6 Cavitation ............................................................................................................................ 326 8.6.1 Cavitation Phenomena .......................................................................................... 327 8.6.2 Ultrasonic Cleaners ............................................................................................... 328 8.6.2.1 Principles of Ultrasonic Cleaning ......................................................... 328 8.6.3 Ultrasonic Cleaning Fluids................................................................................... 329 8.7 Enzymes and Ultrasonic Cleansers ................................................................................ 332 8.7.1 Enzymes and Cleansers ........................................................................................ 333 8.8 De-inking of Office Waste Paper ..................................................................................... 333 8.9 Homogenization and Emulsification .............................................................................. 335 8.10 Dispersion and Homogenization .................................................................................... 337 8.11 Coagulation, Precipitation, and Filtration ...................................................................... 337 8.12 Atomization ........................................................................................................................ 338 8.13 Preparation of Nanomaterials ..........................................................................................340 8.14 Crystallization .................................................................................................................... 341 8.14.1 Crystallization in Metals ...................................................................................... 341 8.15 Diffusion and Filtration through Membranes ............................................................... 341 8.16 Chemical Effects.................................................................................................................342 8.16.1 Sonochemical Reactions in Aqueous Solutions.................................................344 8.16.1.1 Gases .........................................................................................................344 8.16.1.2 Inorganic Compounds............................................................................344 8.16.1.3 Organic Compounds ..............................................................................345 8.16.1.4 Nonaqueous Sonochemical Reactions ................................................. 347 8.17 Polymerization ................................................................................................................... 350 8.18 Polymer Degradation ........................................................................................................ 353 8.19 Polymer Processing ...........................................................................................................354 8.19.1 Vulcanization..........................................................................................................354 8.20 Devulcanization ................................................................................................................. 356

323


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324

8.21 Large-Scale Sonochemical Processing ............................................................................ 357 8.22 Summary............................................................................................................................. 360 References .................................................................................................................................... 361

8.1

Introduction

As the previous chapters have revealed, ultrasound has many attributes to offer industry, medicine, and research. This chapter discusses some important applications; in many of these, cavitation plays an important role. Materials undergo physical and chemical processes under external stress, both physical and chemical. Stress is often used to change the nature of materials, to give them shape and size, and to make them useful in different applications. It is therefore essential to have a basic understanding of physical and chemical processes.

8.2

Physical Processes

Most materials exist in three different states: solid, liquid, and gas. In the gaseous state, the molecules of a material move with high speed and occupy any amount of space available. The density of gas is very low at standard atmospheric temperature and pressure but may equal that of light liquids under very high stresses. In the liquid state, molecules retain some of their kinetic motion so as to flow readily. Because of cohesive forces, a liquid does not expand indefinitely like a gas. Relative positions of molecules in both the gaseous and the liquid state are random. In the solid state, molecules or ions have very little freedom to move around their center of mass; they occupy a fixed volume. Materials having fixed configuration are crystalline and materials with random configuration are called amorphous. There are many materials that are semicrystalline: they have both crystalline and amorphous phases, intermingled with each other. Polymers, which are composed of long chain molecules, exhibit a rubbery or leather state, where segments of molecular chains exhibit such hybrid phases. On the other hand, some polymers are crystalline, but behave like liquids. Materials change their physical state under thermal stress. The transition temperature from solid to liquid is known as the melting point, Tm, and the transition temperature from liquid to gas is the boiling point, T b. The temperature at which the glassy materials become leathery is known as the glass transition temperature, Tg. Another important physical process is the mixing of two or more materials. All gases provide a molecularly uniform mixture. Liquids mix only when their intermolecular forces are alike. The degree of miscibility depends on temperature and pressure. Emulsification is a process in which two immiscible liquids form a stable, compatible mixture with the help of a third compound known as an emulsifier. Emulsifiers are chemical compounds that have both nonpolar and polar, or ionic, parts. They stay at the interface and facilitate the coexistence of immiscible liquids that differ in their polarities. Most solids do not mix with each other to provide a homogeneous mass. However, they can be mixed in the liquid, solution, or glassy state under shear stress and can be cooled down to give a homogenous mass. Like mixing, separation is also an important physical process. Typical examples of separation processes include precipitation, filtration, degassing, cleaning, and so on.


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Ultrasonics-Assisted Physical and Chemical Processes

8.3

325

Chemical Processes

Most materials undergo chemical reactions when they are subjected to a stress or they come in contact with appropriate chemicals under suitable physical conditions. Such reactions are activated processes and the rate is accelerated by increase in temperature. The change in the rate of a chemical reaction depends on the activation energy, that is, the amount of energy the reactants need to undergo a reaction on a molecular scale. As an approximation, for most reactions, the rate doubles for every 10°C increase in temperature. Details of different chemical reactions can be obtained from standard textbooks on inorganic and organic chemistry. Materials in general undergo degradation when exposed to heat, light, and chemicals. In other words, they become less functional and finally break down. Metals undergo corrosion, a combined result of oxidation and hydration; ceramics become brittle and crack; and molecular weights of polymers decrease substantially, leading to weakening, crack development, and failure.

8.4

Ultrasonic Effects

As discussed earlier, high-frequency ultrasonic waves are propagated as oscillatory motion through materials: solid, liquid, or gas. The oscillatory effect is attenuated by scattering, absorption, and other mechanisms. The high-intensity oscillations in liquids cause strong bubble formation and collapse (cavitation), producing a large increase in instantaneous local temperature and pressure. Increase in temperature leads to phase changes, acceleration of chemical reactions, and material decomposition. Material decomposition may result in generation of free radicals capable of initiating chemical reactions, including polymerization. High-frequency ultrasonics has therefore found many applications in a variety of chemical and physical processes involving both organic and inorganic materials. Details of these processes have been discussed by Suslik, Ensminger, and Duraswamy [1–3]. The following sections provide brief descriptions of these processes and their uses in both product and process development.

8.5

Degassing

When high-intensity ultrasonic energy is applied to liquids containing dissolved gases, the gases are released into pockets at intensity levels below that at which cavitation of the solution occurs. The bubbles that are formed are not caused by cavitation. They are transient, combining by coalescence, and rise to the surface at a rate dependent upon the sizes of the bubbles and the viscosity of the liquid. When higher intensities are applied with the intention of producing cavitation, degassing occurs first. The rate of removal depends upon the intensity of the ultrasound, the viscosity of the fluid, and the bubble size. If the intensity is too high, the larger bubbles are shattered and move out at a lower rate. This phenomenon is sometimes used to remove gases from liquids to be used later for other purposes.


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326

Spinosa and Ensminger [4] have studied the use of ultrasonic energy in processing glass. The energy was applied through molybdenum transmission lines protected from oxidation by a nitrogen atmosphere. The viscous modeling indicates that bubble coalescence does occur and that it can contribute to a 15–20% improvement in glass throughput rate, thereby accomplishing an energy reduction in glass throughput rate, thereby accomplishing an energy reduction per pound of melter output. Degassing of carbonated drinks, beer, photographic solutions, and others is carried out by using either ultrasonic cleaning tanks or ultrasonically fitted continuous flow equipment. Degassing of molten metals using ultrasonics has been used to decrease porosity on solidification [5,6]. Ultrasonics has been used to defoam materials during processing. One small whistle aimed into the foam over a tank of oil used in the manufacture of soap reduced the amount of foam in the tank after filling to a satisfactory amount for truck transportation. The whistle operated at 15 kHz.

8.6

Cavitation

Ultrasonic cleaning is especially associated with cavitation. Cavitation is also the key to homogenization and emulsification. It is associated with the dispersion of materials in solvents and fluids and sometimes in the coating of particles and materials. It may or may not be involved in the atomization of liquids and droplet formation depending particularly upon the necessary controls imposed on how the drops are formed. Control of foams may or may not be associated with cavitation depending upon the technique to be used; that is, whether the control is based upon breaking down foam forming above the surface of the liquid or whether it inhibits formation of the foam from inside the bulk. Cavitation refers to the formation or rapid enlargement and collapse of cavities, or bubbles, in a liquid medium being subjected suddenly to low pressures. Cavitation may occur when liquid is forced through certain constrictions or behind a high-speed propeller. In the present context, cavitation is produced by the presence of high-intensity ultrasonic waves in a liquid. When a liquid is subjected to a high-intensity ultrasonic wave, during the rarefaction portion of the cycle when the pressure in the wave is below ambient, gas pockets expand with the impressed field until the pockets collapse violently due to the high stresses developed in the walls. The source of these gas pockets is generally molecules of gas that are very finely dispersed throughout the liquid volume. These may be located at vacant sites of the quasicrystalline structure of the liquid or they may be contained in invisible bubbles of microscopic dimensions [7]. Cavitation is of two types: gaseous cavitation and vaporous cavitation. Gaseous cavitation involves gases dissolved or entrapped in the liquid or existing on surfaces in contact with the liquid. Vaporous cavitation involves gases from the vaporization of the liquid itself. Most liquids contain nuclei about which cavitation bubbles originate. These nuclei may consist of dispersed dust particles, prominences on immersed surfaces, and minute gas bubbles. In fact, unless especially treated, liquids contain dissolved or entrained gas. Various factors influence the onset and intensities of the cavitation bubbles. These factors include the sizes of the nuclei, ambient pressure, amount of dissolved gases, vapor pressure, viscosity, surface tension, and the frequency and duration of the ultrasonic energy.


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327

To be able to produce the effects associated with the expansion and the violent collapse of cavitation bubbles, the bubble must be capable of expanding with the rarefaction part of the cycle of the impressed field and of collapsing before the total pressure reaches its minimum value. That is, the bubble must reach the size where it will collapse catastrophically in less than one-quarter the cycle of the impressed wave. Therefore, generation of intense cavitation depends upon the relationship between the dimensions of the nuclei and the wavelength and the intensity of the sound field. Bubbles larger than a critical radius, Rc, will not expand to an unstable size for catastrophic collapse before the pressure in the wave starts to increase. Frederick [8; 2, p. 67] gives the following relationships for Rc: 3P0 326 1/2 P0 ⬇ f

Rc

1

Rc

1 3  3.9  ⬇  R 0  f 

P0

2 R0

(8.1)

P0

2 R0

(8.2)

2/3

where Rc is the critical bubble radius (cm), R0 the initial bubble radius, γ the ratio of specific heats of the gas in the bubble, σ the surface tension of the liquid (dyn/cm), P0 the hydrostatic pressure (atm.), ρ the density of liquid (g/cm3), and ω is 2πf where f is frequency. The violence of the collapse of the bubble in a cavitation field depends upon the ratio Rm/R0. The greater the ratio, the more violent the collapse. R m is the maximum bubble radius before collapse. Increasing the ambient pressure does not increase the intensity of the collapse. Equations 8.1 and 8.2 show that the diameter of a resonating bubble must be smaller than the wavelength of the sound in the liquid for the bubble to grow to collapsible size at the particular frequency. In fact, the wavelength of the ultrasonic field usually is approximately two orders of magnitude larger than the critical diameter of the resonating bubble. Only the bubbles that are smaller than critical size are capable of rupture and subsequent collapse before the end of the pressure cycle. This is why production of observable cavitation is increasingly difficult as frequency increases. For example, at 1.0 MHz, the wavelength in water at 20°C is on the order of 0.15 cm. The maximum critical diameter of a resonant bubble under these conditions is ≤15 μm. From Equation 8.1, letting P0 = 16 atm in water and f = 20,000 Hz, for P0 >> 2σ/R0, Rc is ∼0.0336 mm, and λ/Rc is ∼22,050. When P0 is very small compared with 2σ/R0, Rc = 0.652 mm and λ/Rc is 1137. Under similar conditions, the ratios of λ/Rc at 1.0 MHz is ∼113.7 for the high-pressure case and 598.5 for the low-pressure case. 8.6.1

Cavitation Phenomena

Many remarkable phenomena are associated with cavitation produced by ultrasound. Some of the chemical and physical effects are attributable to the tremendous pressures on the gases within the bubbles and the very high temperatures associated with these pressures. Pressures as high as 5000 atm have been reported, causing instantaneous temperatures of at least 7200°C (13,000°F). Some chemical and physical effects associated with high-intensity cavitation include production of OH and other ions, erosion of metal surfaces, disruption of aggregates, and other effects not producible by any other known means.


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328

When fresh eggs that have been exposed very briefly to ultrasonically produced cavitation within the shell are broken open immediately following treatment, the contents (including the lining) are so thoroughly homogenized that they pour out like water. When the contents are held within the unbroken shell for a short period of time after treatment, they develop a strong odor of rotten eggs indicating that the contents have not only been homogenized but that they have experienced a molecular change during the treatment, indicated by the strong rotten egg odor, which is indicative of a sulfurous acid content (H2SO3). An egg treated at emulsifying intensities for a few seconds within the shell will cook and be edible when it is opened. 8.6.2

Ultrasonic Cleaners

Perhaps one of the oldest and most useful applications of ultrasonic cavitation is cleaning. Ultrasonic cleaning takes many forms. The common cleaner consists of a tank filled with cleaning fluid into which the contaminated surfaces to be cleaned are fully immersed so that the ultrasonic energy can have adequate access to them. The transducers are attached to the bottom of the tank or to the sides. Small parts may be suspended in a basket. Hard surfaces must not be allowed to ride on the bottom. Parts may be rotated slowly or allowed to tumble to assure complete access by the ultrasonic energy to contaminated surfaces. The tank may be specially designed for specific items such as an elongated trough for window blinds or golf clubs. Materials can be cleaned in a hostile environment by means of clamp-on transducers. Special horns may be designed to reach into interior volumes. A cleaning job is limited only by the imagination of the operator. The compatibility of the cleaning fluid with the equipment used for cleaning and the parts to be cleaned is extremely important. Materials to be cleaned must be compatible in dimensions with the cleaner.

8.6.2.1

Principles of Ultrasonic Cleaning

Ultrasonic cleaning is used in a variety of industries, including medical, dental, electronic, optical, and industrial operations. Cleaning is carried out primarily by cavitation in the cleaning fluid. The cavitation activity not only produces kinetic motion but also brings fresh solvent close to the contaminants where they are either dissolved or dispersed as very fine particles. Water and many other solvents are used as cleaning media. Cleaning agents are selected based on their ability to combine cavitational activity with chemical action. The effectiveness of cleaning depends on the type of stress generated between the contaminant and the cleaning fluid, severity of agitation, increase of attraction between the contaminant and cleaning fluid, gas content of the liquid, the adhesive forces between the contaminant and the liquid, and the potential for promoting desirable chemical reaction at the interface. When a surface containing a contaminant is exposed to cavitation, intensity of stress generated depends on the vapor pressure of the cleaning fluid, the gas content of the liquid, and the adhesive forces between the liquid and the surface. The intensity of the ultrasonic energy must exceed the intensity needed to promote cavitation in the cleaning solvent. In most cases, this is 0.5–0.6 W/cm2. The frequencies used in commercial equipment are 20–60 kHz, with 40 kHz being the most common. The power levels are commonly 200 W per each gallon of tank capacity, regardless of the type of irradiating surface used. Conversion efficiency of electronic generator and transducer determines power available to the cleaning solution.


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Ultrasonic cleaning is attributable to many factors Ref. [9], especially those which include cavitation. The main factors include (1) development of stresses between the cleaning fluid and the contaminated surface, (2) agitation and dispersion of contaminant throughout the cleaning fluid, (3) increase of the attractive forces between the contaminant and the cleaning fluid, (4) promotion of chemical reactions at the contaminated surfaces in some cases, and (5) effective penetration of pores and crevices. Development of stresses between the cleaning fluid and the contaminated surface is exemplified in nearly all ultrasonic cleaning processes. As cavitation bubbles form at the surface to be cleaned, every component of the forming bubble is stressed at a high repetition rate. The components of the bubble include the internal liquid surface of the entire bubble plus any portion of the solid surface in direct contact with it. The intensity of the stresses under these conditions is a function of vapor pressure of the liquid, the gas content of the liquid, and the adhesive force between the liquid and the surface. These stresses are sufficiently high under cavitation conditions to erode the solid with time, to break suspended solids, to disperse materials throughout the liquid, and, in connection with the instantaneous temperature produced, to accelerate various chemical reactions. These chemical reactions help accelerate the cleaning operation. Agitation occurs not only in the presence of cavitation but also from intensities that promote flow without cavitation. When a compression wave is generated from a solid surface in a fluid, the compression at the interface directs the fluid away from the surface. As the source surface moves to cause rarefaction, other fluid moves to fill in for the outward flowing fluid, thus providing a continuous movement of fluid away from the source (ultrasonic wind). Agitation provides a scrubbing action that promotes the removal of contaminants. Such contaminants may be loose, solid particles or materials that will dissolve or emulsify in the cleaning fluid. A cleaner should never be overloaded. A very important aspect of ultrasonic cleaning is its ability to draw debris out of pores and crevices. 8.6.3

Ultrasonic Cleaning Fluids

Choice of cleaning solvent depends on the chemical compatibility of fluids with materials to be cleaned and the container, as well as on the effectiveness of the cleaning media at removing the contaminant from the surface. The stresses caused by cavitational bubbles are controlled by the vapor pressure of the liquid and volatile impurities present. As vapor pressure increases, surface tension decreases and the maximum stress associated with cavitation decreases. Hence, the cleaning media need to be carefully selected to provide desirable cavitational stress. The intensity of the cavitational stress depends on the ratio of the critical bubble size to the minimum bubble size. High-intensity cavitation stresses can erode plated and coated surfaces. It is important that erosion is minimized or eliminated by the choice of high-vapor-pressure solvents or by blending good solvents with high-vapor-pressure liquids. Addition of a small amount of methyl or ethyl alcohol to water has been used to avoid the destructive effects of ultrasonic cavitation [2, pp. 426–429; 10]. Solubility of a contaminant material in a solvent largely depends on the intensity of their intermolecular interaction. For most nonpolar/organic solvents and organic contaminants such as oil, grease, and paint, the solubility parameter—which is the square root of cohesive energy density (energy of vaporization per unit volume of the solvent)—is a good criterion for evaluating solvent capacity. A solvent will dissolve contaminant, if the difference between the solubility parameter of the solvent and that of the contaminant is


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small [2, pp. 434–435; 11], very often of the order 1–2 units, (Cal/cc)1/2. In case of polymeric contaminants, this difference has to be even smaller. In general, polar contaminants are dissolved by polar solvents and vice versa. Alcohols, ketones, esters, amines, nitriles, etc. are polar solvents. Hydrocarbons and silicon fluids are examples of nonpolar solvents. Ionic materials such as salts are dissolved in solvents with high dielectric constants. Water, which hydrates most ions and has a high dielectric constant, is a good solvent for many ionic solids. Dilute acids are often used to dissolve metal oxides formed due to oxidative degradation and corrosion of metallic substrates. Addition of wetting agents, surfactants, and detergents help in dislodging the contaminants from the substrate and are used both in aqueous and nonaqueous cleaning processes [12]. Selection of solvents should also take into account harmful side effects and environmental compatibility. It is possible that some chemical reaction might take place under acoustic cavitation in certain solvents, resulting in hydrogen embrittlement or dissolution of the container material. Such solvents should be avoided, or if it is necessary to use them, careful planning of materials, seals, and the like should be made before using them. Solvents with high vapor pressure may fume due to temperature rise during ultrasonic application. Thus one has to be very careful in selecting a solvent or solvent system for a particular job. There are many fluids used for ultrasonic cleaning. The manufacturers of the ultrasonic cleaners may offer or recommend certain proprietary trade-name formulations of their own to use with specific classes of soils. We say “may offer,” because they are just as often reluctant to suggest cleaning fluids. They “do not promote or endorse particular products or products from any particular vendor. The selection of a chemistry must be based on an analysis of the substrate and the soil and, finally, by actual laboratory or use environment testing of the product on real case parts. Today’s chemistries are very specialized and offer a wide variety of options which cannot be addressed in generalized recommendations.” [13] Commonly used cleaning agents used in ultrasonic cleaning are listed in Table 8.1 [2, pp. 426–430]. Solubility parameters of organic solvents (δ) are given in (Cal/cc)1/2. Industrial ultrasonic cleaners are standalone units, using stainless steel vessels of 5–150 L capacity for tanks. They are used routinely in the semiconductor industry. Although some cleaners operate at a lower frequency (25 kHz), most cleaners operate at a higher frequency. Temperature-controlled devices are used to maintain constant temperature during the cleaning process. Computer-controlled robotic devices are used for automatic material handling. Additional cleaning functions such as presoaking or vapor rinsing are added to the cleaner to make the cleaning process cost-effective. Small cleaners used in laboratory, jewelry shops, and small part cleaning facilities and ultrasonic cleaning devices (such as sonic toothbrushes) come in one package, containing both the container and the electronics. They operate at 50–60 kHz with a power input of approximately 25–150 W. The effectiveness of a cleaning process is evaluated by measuring the power density inside the cleaner. However, the normal propagation of high-intensity waves is disrupted by the presence of cavitation bubbles, making the measurement by means of pressure-sensitive probes impossible. Calorimetric measurement of heat generated during cleaning can be a good measure of cleaning efficiency, provided that all the energy is absorbed in the cleaning process. The more popular method to evaluate ultrasonic cleaners is to measure sonochemical activity, erosion effects, and probe reactions. Measurement of the iodine released from a solution of potassium iodide in carbon tetrachloride is one of the commonly used methods used. Ultrasonically initiated cavitation releases chlorine from the carbon tetrachloride and iodine from the potassium iodide. The chlorine takes the place of the released iodine in the KI formula and the removed iodine remains free in the solution.


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TABLE 8.1 Commonly Used Solvents for Ultrasonic Cleaning Solventsa

Base Materials

HCl with inhibitor

Ferrous alloys

H2SO4 with inhibitor

Ferrous alloys

HNO3 with inhibitor

Stainless steel, some aluminum Iron, steel, brass, zinc, aluminum

Phosphoric acid with inhibitor Water Detergent and water Soap and water Trichloroethylene Tetrachloroethylene Acetone Benzene Xylene Chlorothene

Toluene Freonsb Tricholorotrifluoroethene (Freon 113)

Any material not damaged by water All metals and many other materials All metals and many other materials All metals polar organic, δ = 9.5 All metals polar organic, δ = 9.5 All metals All metals aromatic, δ = 9.2 All metals δ = 8.9 Most materials (bearing materials meters, circuit boards, electronic components, etc.) All metals aromatic, δ = 9.1 Most materials (circuit boards, etc.)

Tetrachlorodifluorethane (Freon 114)

Most materials (circuit boards, etc.)

1,1,1-Trichloroethane

Electrical and electronic assemblies

Contaminants

Remarks

Oxides, heat-treatment scale Oxides, heat-treatment scale Oxides, heat-treatment scale Light scale and oxides, some drawing compounds oil, grease Loose soils, watersoluble soils Oils, greases, loose soils, etc. Oils, greases, loose soils, etc. Oil, grease, buffing compounds, etc. Oil, grease, buffing compounds, paints Oils, some plastic base cements (polar) Oil, grease, adhesive paint Oil, grease (slightly polar) Grease, oils, dust, lint, flux, oxides, pigments, inks

Plastic containers preferred

Oil, grease, adhesive paint Fluxes, loose soils, oils, other organic compounds

Stainless steel containers

Fluxes, loose soils, oils, other organic compounds Flux, dust, etc.

Plastic containers preferred 316 stainless steel and plastic are acceptable materials 316 stainless steel is an acceptable container Stainless steel containers Stainless steel containers Stainless steel containers Stainless steel containers Stainless steel Stainless steel containers Stainless steel containers Stainless steel containers Stainless steel containers

Stainless steel, aluminum, or plastic containers can be used; easily distilled and recycled Stainless steel, aluminum, or plastic can be used; can be used to 150°F but expensive Stainless steel containers

The following are proprietary trade-name formulations (by sonicpro.com) 983 Sonic Booster AC-40

Water conditioner, other cleaning agents Electrical equipment, electronic soft metals

Grease

Medium-duty cleaning performance Ideal for cleaning delicate items and components and painted surfaces. Mildly alkaline; nonbutyl polymeric; environmentally safe, nontoxic, biodegradable, and super-concentrated for economical use. (continued)


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(Continued)

Attack blind cleaning powder BP-70

10-60 LN-42 SC-80

PVC, fabrics, glass, metal, and painted surfaces Alkaline nonbutyl degreaser, safe on most surfaces except highly polished aluminum Molds Porcelain, glass, plastic, and metals Water-damaged metal tools and parts, aluminum and other metals; iron, steel, aluminum, brass, copper, plastic, and painted surfaces

Formulated for use in blind cleaning equipment Grease, oil, carbon deposits, ink pigment (printing press rollers), on industrial soils Carbon residue Smoke and fire residue

Injection mold cleaner Alkaline cleaner/degreaser

Rust, heat scale, tarnish, and oxides

Nonhazardous, biodegradable, and has no harmful vapors. Made from nontoxic organic acids, detergents, and penetrants. Apply OB-300 after cleaning with SC-800 to prevent flash rusting. High pressure; clamp-on transducers

Liquid CO2

a

Wetting agents are recommended for most solvents used in ultrasonic cleaners. DuPont trademark. Source: Ensminger, D., Ultrasonics, Marcel Dekker, New York, 1988: pp. 418, 419.

b

The free iodine produces a blue color in a starch solution. The intensity of the blue color is measured by a colorimetric method. It has been found that the rate of chlorine release is linear with time and appears to be proportional to the intensity of cavitation, which is proportional to power density in the cleaning bath. Carbon tetrabromide may be used in place of carbon tetrachloride, particularly if the measurement has to be done above room temperature [2, pp. 431–438]. Another method used to measure cleaning effectiveness is to measure erosion. This method uses the hanging of aluminum foil in the cleaning bath and measuring the weight loss as a function of time or measuring the time required to make holes in the foil. It is necessary to make certain that the foils have the same metallurgical history from piece to piece and from tank to tank for this method to be effective. Pressure-sensitive probes are used to measure the noise intensity above the threshold to evaluate the effectiveness of a cleaning tank. Other methods of measuring cleaning effectiveness have been discussed by Ensminger [14].

8.7

Enzymes and Ultrasonic Cleansers

Enzymes are biological catalysts. Many of the chemical changes in living organisms are catalyzed by enzymes. They involve the formation of a complex between the enzyme and the substrate, which changes the conformation of the substrate molecule and expedites the reaction by putting a strain on the bond to be broken. This step, called activation, greatly reduces the amount of energy required to cause the reaction. The reaction then


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proceeds and the enzyme is regenerated, becoming available to complex with another molecule of the substrate. The reuse of the enzyme makes it effective in minute quantities. Enzymes are widely used in the food industry. “Certain enzyme-catalyzed reactions in food do not occur in the intact tissue because of the physical separation of the enzyme and the substrate. Mechanical damage to the tissue can bring the enzyme and the substrate into contact with each other and permit such a reaction to occur” [15]. The most important factor affecting enzyme activity is temperature. Each enzyme has an optimum temperature which, with a few exceptions, is within the range 35–40°C (95–104°F). At low temperatures, enzyme activity is low and increases with increasing temperatures, approximately doubling with every 10°C increase in temperature up to the optimum temperature. If the temperature increases above the optimum temperature, denaturation occurs at an increasing rate until finally inactivation is complete. Another factor affecting enzyme activity is pH, being in the vicinity of pH7 for all but a few enzymes. For some, the pH range is very narrow; for others, it is relatively broad. If the pH range for activity is narrow, the activity must be buffered to continue. Enzyme and substrate concentrations are also factors in reaction rates. 8.7.1

Enzymes and Cleansers

Because of their activity with respect to biological materials, enzymes are finding use in ultrasonic cleansers used with surgical equipment. Some individuals question this practice because of the thermal effect on the activity of the enzyme. If the enzyme is to be effective, the manufacturer of the cleanser must remind the user of the safe temperature range and its environment to protect the enzyme from inactivation. Ultrasound applied at intensities below cavitation level accelerates the activity by bringing the enzyme into close contact with the biological debris and dispersing it throughout the volume of the cleanser. After removing the enzymatic fluids, a second ultrasonic wash using a standard rinse at a higher temperature can be performed to assure that sterilization is complete. The enzyme to be used must be carefully chosen with respect to the material with which it is to react.

8.8

De-inking of Office Waste Paper

A new area of interest of ultrasonic application is de-inking, for recovering cellulose fibers from waste papers from copiers, printers, and fax machines. Toner is typically removed from office waste paper in a secondary fiber mill by pulping the waste paper and applying mechanical/thermal energy to dislodge the fused toner from the paper surface and separating it by floatation, washing, and screening [16]. However, this process does not work well, because the toner is fused to the fiber by thermoplastic binders such as styrene-butyl acrylate copolymers, polyester resins, and so on. Further, this process gives toner particles ranging from 40µm to 100 μm diameter, making it difficult to remove them by a floatation process. Table 8.2 lists several prominent manufacturers of ultrasonic cleaners and cleaning equipment and the corresponding cleaner types. Ultrasonic generators such as liquid whistles and piezoelectric transducers have been used in the de-inking process and have been very successful in reducing the size of toner particles in the slurry, without the use of chemicals [17–19]. Norman et al. found that the


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TABLE 8.2 Ultrasonic Cleaner Manufacturers Manufacturer

Typical Cleaner Types

Blackstone-Ney Ultrasonics Jamestown, NY Phone: 800-766-6606 FAX: 716-665-2480

A variety of industrial and professional cleaning equipment, including console cleaners, benchtop cleaning tanks, multistage systems, and more to serve a wide range of industries and markets.

Branson Ultrasonics Corporation Danbury, CT

Full line of precision batch cleaning systems for aqueous solvent cleaning, and provides ultrasonic degreasers and ultrasonics parts cleaners; frequencies cover from 20 kHz to 100 kHz includes an applications laboratory to develop processes to meet the customer’s needs.

Coletene/Whaledent (USA) http://www.coltenewhaledent.com

Biosonic Ultrasonic cleaning systems (dental industries).

Crest Ultrasonics Corporation Trenton, NJ Phone: 800-992-7378

Serves a broad range of industries and markets with ultrasonic cleaning equipment and machinery, including aqueous cleaning systems, ultrasonic cleaning tanks, and ultrasonic parts washers.

Dentronix Cuyahoga Falls, OH Phone: 800-523-5944 and Ivyland, PA Phone: 615-800-4220 FAX: 215-364-8607

Specializes in infection controls: manufactures ultrasonic cleaning machines and heat sterilizers.

Elma Ultrasonic Technology http://www.elma-ultrasonic.com

Ultrasonic cleaning equipment for special cleaning tasks.

Greco Brothers, Inc. (USA) http://www.sonicor.com

Manual and automated ultrasonic cleaning systems with aqueousbased solutions or solvents.

Jensen Aqueous Cleaning [JENFAB] Berlin, CT Phone: 860-828-6516

Automated and semiautomated cleaning equipment and machines, ultrasonic parts washers and ultrasonic degreasers, and aqueous cleaning systems supplied to a range of industries and markets.

Lewis Ultrasonics Kiel, WI Phone: 800-545-0661

Complete line of cleaning systems.

S. Morantz, Inc. Philadelphia, PA Phone: 800-695-4522

Full line of ultrasonic cleaning machines featuring True Digital technology. Applications include fire restoration, window-blind cleaning, parts cleaning, and golf-club cleaning. Tanks also built to customer specifications.

Omegasonics Simi Valley, CA Phone: 800-669-8227

Ultrasonic parts washers, ultrasonic blind cleaning equipment, ultrasonic degreasers, and aqueous cleaning systems. Serve industrial, automotive, and commercial markets with ultrasonic equipment.

Pressure Products Company, Inc. Charleston, WV Phone: 800-624-9043

Ultrasonic cleaners for stencils up to 29 in. × 29 in. Economical and environmentally friendly.

RAMCO Equipment Corporation Hillside, NJ Phone: 908-687-6700

Committed to health, safety, and the environment. Ultrasonic cleaning equipment and machinery, aqueous cleaning systems, and ultrasonic cleaning tanks to meet a range of needs.

Smart Sonic Corp. (USA)

Stencil cleaner. Removes solder paster from fine-pitch stencils (RMA no-clean, or water-soluble).

Sonicor Instrument Corporation (USA) http://www.sonicor.com

Ultrasonic cleaning.

Weber Ultrasonics Clarkston, MI (USA) Phone: 248-620-5142

Serving the finishing industries: Industrial parts cleaners and welders offers SONIC—digital generator technology for cleaning, welding, or special applications.


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ultrasonic method significantly decreased the particle sizes in xerographic ink paper slurry and that different ultrasonic frequencies affect particles of different size ranges [20]. Scott and Gerber have used piezoelectric transducers in batch processes and liquid whistles in continuous processes for de-inking xerographic waste paper and have successfully reduced particle size for easy removal by the floatation process. They have demonstrated that the performance is strongly affected by treatment pH, number of treatment cycles, and pulp consistency [21]. Ramasubramanian and Madansetty have developed a novel technique, Acoustic Coaxing Induced Microcavitation (ACIM), for removing xerographic ink particles from waste paper [22]. They postulated that the cavitation threshold of hydrated paper is much higher compared to that of hydrophobic xerographic ink. Hence the ink regions can be coaxed acoustically to induce cavitation sites, preferably at the ink–paper interface. Microcavitational implosions generated at high-frequency ultrasonic waves can chisel away the ink paper joints and set the ink particles free without affecting the fiber mat underneath. The authors used ceramic transducers in pulsed mode (short repeated tone modes) at the resonance frequency. In this mode, the transducer operates at low acoustic power, yet develops high peak pressures, producing microcavitation. Focusing the acoustic field led to gain in acoustic intensity and progressive waves and using short tone bursts, precluded standing wave formation. Further, the use of highfrequency fields helps in generating nucleation sites on all particle surfaces. This is called acoustic coaxing effect. The use of low-duty factor acoustics in sonification allows time between pulses for the focal region to relax, equilibrate, restore, and regenerate, thereby reenacting cavitation at every pulse. As de-inking occurs, a white spot is generated in the printed region and the surrounding area remains unaffected. The experimental setup used in this study consists of a transducer, 40 mm in diameter with a focal length of 61.5 mm. The depth of focus is about 1.5 mm and the sample is kept at the focal plane close to the transducer. The entire assembly is immersed in a bath of distilled water. Tone bursts generated by a function generator are fed to the amplifier that drives the focused cavitation generator. The de-inked spot is ∼4 mm in diameter. The residence time for de-inking threshold increases with increasing pressure amplitude and duty factor. Longer pulses at a constant acoustic dose do a good cleaning job. Excessively long pulses, however, often lodge the separated toner particles back onto the paper. The optimal pulse parameters are: pulse width 10 μs, 1 kHz pulse repetition rate (PRR), and acoustic pressure amplitude of 45 atm peak negative, for spot de-inking.

8.9

Homogenization and Emulsification

To emulsify means to make or convert two or more liquid substances into an emulsion. An emulsion is a system of globules or particles of one liquid finely and uniformly dispersed in another. To homogenize means to make or render a mixture homogeneous. It refers to breaking up fat globules and dispersing them uniformly throughout a liquid. In other words, homogenizing a mixture is essentially the same as emulsifying it. Ultrasonic energy provides an excellent means of producing emulsions for laboratory studies. The equipment is rugged and easy to use. The general laboratory device consists of a transducer driving a resonant horn usually made of a titanium alloy such as Ti6Al4V. It can range in size from a small, lightweight unit consuming a very small amount of


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power to large units requiring kilowatts of energy. In most cases, one unit can be used for various studies. A variety of horn tip designs exist, from which one may be selected to fit a particular need. A unit may be made small and light enough to allow free manipulation by hand. This is a quality that is being widely used in operations, such as emulsifying the lens of the eye before replacing it with a new artificial lens. Ultrasonics makes it possible to remove the entire old lens before introducing the new one, without damaging the eye. In the early 1960s, ultrasonic emission from liquid whistles was successfully used for emulsification of a variety of products in water with or without the use of surfactants. The whistle works by impinging a high-pressure, flat jet of liquid on the edge of a thin blade, which is supported at the nodal points and located in a thin cavity [2, pp. 167–170; 23]. The blade vibrates in resonance due to the unstable condition created by the jet and the intense vibration causes localized cavitation. The energy released by the collapse of bubbles accelerates and propels the particles into the media, thereby facilitating the dispersion and emulsification. A typical liquid whistle system handles from 300 gph to 350 gph. In a typical commercial application for preparing resin-bonded solid lubricants consisting of molybdenum disulfide, graphite, and epoxy resin, the production rate by a liquid whistle operated by a 3 hp motor was two-and-half times greater than that of a 12 hp colloid mill [24]. Liquid whistles have been used for the manufacture of waxes and polishes, DDT, margarine emulsions, lotions, and antibiotic dispersions, as well as emulsions of mineral and essential oils. Another method of emulsification is the use of a horn coupled to a transducer so that it can oscillate in a longitudinal mode to produce cavitation when immersed in a liquid containing all of the ingredients. Intensity of cavitation is controlled by the power delivered to the horn that is carefully selected for the specific process. Details of the apparatus, experimental conditions, and the results are given by Weinstein et al. [25] and Hislop [26] and Last [27]. Parabolically focusing bowl types of piezoelectric transducers have been used to concentrate longitudinal ultrasonic waves internally. The liquid mixture is passed through the focus of the parabola to provide good emulsification. Although good emulsions can be prepared by ultrasonics, their stability is enhanced by using surface active agents. Sulfite liquor is used as an emulsifier for DDT emulsification [28] and stearic acid in dispersing mercury in paraffin oil. Mercury stearate formed in the emulsification process acts as a surfactant and stabilizes the dispersion. Similarly, a good dispersion of mercury is accomplished only after aerating the water, possibly by air oxidation of mercury, forming an Hg ion on the surface. The addition of potassium chloride removes the Hg++ by complex formation and destroys the emulsion. Ultrasonic emulsification is being carried out at the present time using powerful piezoelectric and magnetostrictive devices. Ultrasonics has been used to make slurries of coal powder in oil in order to burn it in oil burning plants. Ultrasonic agitation of a slurry containing 40% coal powder and a small amount of water provides a stable dispersion that does not settle during storage or transportation. Heinz [29] has examined the mechanisms of emulsification, both theoretically and experimentally, and has concluded that the efficiency of emulsification is largely controlled by the interfacial tension and viscosities of the liquids. Deformation of larger droplets are caused by the velocity gradients formed by shearing stresses caused by acoustic streaming and droplets burst when the shearing stresses are greater than the restoring forces associated with interfacial tension. The process follows the Taylor relationship η 0G = 8Γ(P + 1)/b(19P + 16)


(8.3)

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where η 0 is the viscosity of the continuous medium, G the velocity gradient, P the ratio of the viscosity of the disperse medium to that of the continuous medium, Γ the interfacial tension, and b the radius of the undistorted drop. Efficiencies of emulsification processes have been studied by measuring particle size distribution. It has been shown that the efficiency is also proportional to the intensity of the sound used above the threshold intensity required for the onset of emulsification [30].

8.10

Dispersion and Homogenization

Intense ultrasound has been successfully used in breaking agglomerates and dispersing a variety of solids in liquid media [31–35]. The method requires the use of an ultrasonic horn with its irradiating tip extending into the volume of the mixture. Industrial applications include dye pigment preparations, dispersions of china clay, mica, titanium dioxide, and others used in the rubber and paper industry, and dispersion of magnetic oxides in the manufacture of audio, video, computer tapes and disks. Use of ultrasonic dispersion provides coatings with fewer imperfections and less information dropouts. This will spur greater use of the technique as the emphasis on higher information density grows. Hard-faced ultrasonic horns can be used to minimize wear and tear in dispersing abrasive particles. Ultrasonics has been successfully used in dispersing solid catalysts in liquid epoxy resins without premature cure and in the preparation of mineral slurries [36]. Other potential applications include particle size reduction, removal of paints and surface coats, particle wetting, and slime removal. Precipitation of ultrafine coal and metal dust and separation of sulfur and ash from coal are possible applications. Defibrillation of natural fibers such as flax, hemp, kenaf, and others is an important step in their application as polymer biocomposites. Senapati and Krishnaswamy have successfully defibrillated flax fibers using an ultrasonic water bath operated at 20 kHz and maintaining a flax to water ratio of 1–400 [37]. The radiating surface sometimes has to be designed for the application. For example, a horn with a flanged radiating surface had to be designed to delaminate jig mica. The mica is a heavy mineral that settles rapidly to the tank bottom. The flange throws the mica particles into suspension where the ultrasound can encompass them and peel the layers apart [38].

8.11

Coagulation, Precipitation, and Filtration

Ultrasonics has also been successfully used for coagulation and precipitation. Both the mechanisms and the effectiveness of these functions depend upon the media, the intensity of the sound wave, the frequency, the means of irradiating the materials, the reactions that occur, the materials to be precipitated, and the density (referring to the number of particles per unit of volume) of particles to be coagulated or precipitated. The materials to be coagulated may range from soft and fibrous such as paper pulps to crystalline solids such as metal powders. In some cases, the objective is to remove already existing particles from a volume of material such as ultrafine coal dust from the air or water. In other cases,


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the objective maybe to remove materials as they are produced by means of ultrasonic irradiation, such as extremely small crystals of TiO2 from reactants as they are formed. Continuous vibration brings fibrous particles such as paper pulp to impinge against one another and become entangled to form larger particles, thereby causing precipitation. Precipitation of ultrafine coal and metal dust and separation of sulfur from coal ash has been carried out using ultrasonics. The ultrasonic process has been used to coagulate solids from sewage discharge and to increase the amount of water removed in drying by simultaneously applying ultrasound and electrophoresis to plates during compacting the debris [39]. Ultrasonics has also been used to improve filtration rate as well as to enhance filter life. The filter element is vibrated sometimes by direct contact with the horn. At other times, the vibrations are applied to the fluid to irradiate in the direction of the flow by placing the horn in the vicinity of the filter [40,41]. Filters made from sintered brass, stainless steel, sandstone, and other materials have been used, with pore sizes from few to over 100 μm. Continuous vibration slows the rate of deposition on the filter surface, resulting in less need for cleaning.

8.12

Atomization

Ultrasonic atomization is accomplished by at least three different means: (1) by placing the liquid in a focusing transducer, a bowl-shaped or elongated half cylinder with an operating frequency of 0.4–2.0 MHz with the focal region at or near the surface of the liquid; (2) by passing the liquid over the surface of a horn vibrating ultrasonically in a direction normal to the surface [42]; and (3) by injecting the liquid into the active zone of a high-intensity stem-jet whistle (see Figure 6.8). The production of aerosols is attributed to at least two mechanisms: (1) production of capillary waves on the surface of the liquid and (2) cavitation. Theories based upon these mechanisms are applicable only within certain limits. The more general case is more complicated, involving various dynamic factors and material properties. To a degree, the particle sizes are functions of excitation frequency. If the conditions are correct for optimum atomization, the frequency dependence applies over a wide range of frequencies. In general, cavitation is not a major cause of atomization in industrial systems. It may be a major factor in high-frequency focused nebulizers that are used for atomizing medicines in inhalation therapy. However, in horn-type systems involving atomization from a surface, cavitation can have a deleterious effect on the rate at which mists can be formed. An equation proposed by Lang for the number–mean diameter particle diameter, d, of the droplets in the aerosol is [43] d = 0.34 c

(8.4)

where λc is the wavelength of capillary waves (cm) and, according to Rayleigh [44] is given by  8T 

c  2   f 


1/ 3

(8.5)

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where T is surface tension, ρ the density of the liquid, and f the ultrasonic frequency (Hz). This equation applies to very low atomization rates with materials such as very low viscosity oils, very low flow rates, and fairly uniform film thickness. Also the film thickness is large compared with the amplitude of vibration of the atomizing surface. Ensminger discusses additional equations for various conditions of atomization [2, pp. 467–474]. In any feed system, as feed rate increases, atomization will pass through three stages: 1. Low flow rate, at which atomization is attributable entirely to ultrasonic forces. Within this stage, the particles will range in size about a mean diameter that might be approximated by the method of Peskin and Raco, average size increasing with flow rate. 2. Intermediate flow rate, at which atomization is attributable to both ultrasonic forces and fluid dynamic forces. The particle sizes caused by the fluid dynamics are much larger than those produced by ultrasonic forces. 3. High flow rate, at which atomization is primarily a fluid dynamic phenomenon. In general, the efficiency of ultrasonic atomization—that is, ultrasonic energy required to atomize a unit volume of liquid—increases as the frequency decreases. The advantages of ultrasonic atomization include narrow droplet size distribution and the ability to atomize fairly high-viscosity liquids. Further the droplets can be projected in a unique pattern. Ultrasonic atomization has found a variety of applications, including medical inhalants, atomized fuel for efficient combustion, atomization of industrial paints for electrostatic spraying, dispersing cleaners in large tanks, drying fabrics, and producing ultrafine metals and ceramics. Metal powders have been produced by atomizing molten metal [45] and cooling rapidly. Uniform crystals of very small dimensions have been produced by atomizing supersaturated solutions followed by rapid cooling. Medical nebulizers work at frequencies of 1–3 MHz and produce droplets of 1–5 μm. One advantage of the medical devices working within this frequency range is that they produce a mist, without larger particles, that can enter the lungs directly. Absence of any other gas during inhalation makes the technique suitable for anesthetic devices. Ultrasonic humidifiers, which produce atomized water vapor, are commercially available for use in home and offices. Fuel atomization helps efficient combustion. Charpenet [46] found that when an acoustic field, operating at 5–22 kHz, is applied to a steel burner, using both pulverized coal and fuel oil, the flame temperature immediately rose by ∼80°C and the exhaust carbon dioxide increased by 0.8%. Atomization of diesel fuel also improves efficiency, eliminates combustion delay, and enables the engines to run more economically at high loads with low smoke and quieter operation. Use of ultrasonic atomization in liquid fuel burners, such as in home heating, allows the use of large-diameter nozzles, which are self-cleaning under ultrasound and are more fuel-efficient than conventional burners. However, ultrasonic atomizers have not found commercial application, because of the added cost of ultrasonic equipment and recent improvements in conventional burners. It has found some application in special-purpose electric power generation where the rate of dispersion is very low [47]. A small ultrasonic atomizer was designed and built at Battelle Columbus Laboratories for use in portable thermoelectric generators. This system uses a single transistor oscillator with feedback control of the frequency.


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The unit requires 2.5 W from a 12 V battery or a thermoelectric generator without the load but jumps to 5 W when atomizing fuel. The driving frequency is controlled by feedback from the transducer, a pair of piezoelectric crystals. Fuel is fed by gravity and the thermal output is ∼20,000 Btu/h. The types of fuels were gasoline, kerosene, fuel oil, and alcohol. Ultrasonic atomization also helps to improve the sensitivity of flame photometry [48]. Placing high-power ultrasonic horns in direct contact with fabric helps to evaporate a much higher percentage of moisture than mechanical squeezing and leaves a small amount to be dried by heat. The efficiency is higher than for nonabsorbing materials.

8.13

Preparation of Nanomaterials

Nanoparticles of metals and ceramics have found important applications in polymer composites, catalysis, and in the production of amorphous metals. Nanoparticles are a few nanometers in diameter and have very large specific surface. Naturally occurring nanomaterials such as nanoclay, when incorporated into resins such as nylon, polypropylene, and the like produce composites with high strength and impact resistance, at four to five percent loading. In addition, the composites are flame-retardant and have low permeability [49]. Hence a substantial amount of work is being carried out on processes for producing nanofillers. Efficiency of solid catalysts is greatly enhanced as the particle size is reduced to nanoscale. Recently, nanosize metal particles have found important application in magnetic recording media as well in the manufacture of permanent magnets. However, production of nanoparticles on a large scale is a difficult process. Ultrasonics has found important application in the synthesis of nanostructure materials [50]. Ultrasonic cavitation plays an important role in this synthetic procedure. As mentioned earlier, extreme high temperatures and pressures are reached during the implosion of the cavitational bubbles. However, at the end of each cycle, the high temperature is quenched in ∼100 ns with cooling rates of the order of 1010 degrees per second. When a metallic or ceramic solution is subjected to such rapid cooling, amorphous nanoparticles are formed since the small particles do not find enough time to agglomerate or crystallize. Suslik initially used an ultrasonic procedure to synthesize amorphous nanoparticles of iron, by subjecting an alkane solution of iron pentacarbonyl to ultrasonic irradiation [1, pp. 410,411; 51]. His apparatus consisted of a solid titanium rod attached to a piezoelectric ceramic material and powered by a 20 kHz, 500 V power supply. Supported catalysts were prepared by adding an oxide support material, such as finely divided silica, to the carbonyl solution and colloids were prepared by adding a polymeric ligand such as polyvinyl pyrrolidone. However, nanoparticles of iron produced in this process agglomerate readily. Boeing [52] has patented a process for forming amorphous metals, using ultrasonic irradiation of metal carbonyl solution in n-heptane or n-decane and by extracting the nanoparticles from the hydrocarbon solvent by adding polar solvents of high vapor pressure such as ethoxyethyl alcohol, followed by an in situ coating process based on the addition of a polymer such as polyvinyl pyrollidone, acrylics or urethane, or related polymer precursors. The thinly coated nanoparticles do not agglomerate and maintain their miniature size. As mentioned earlier, the nanosize iron particles are being used in magnetic recording media. Quest Integrated has also used this method for synthesizing magnetic nanoparticles [53].


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8.14

341

Crystallization

Ultrasonic energy produces smaller crystals in supersaturated liquids. Crystals of controlled sizes are produced in the pharmaceutical industry by applying high-intensity ultrasonic energy at the bottom of vertical columns in conjunction with fluidized beds. The small-size crystals tend to move to the top, while the larger crystals settle toward the bottom. These oversized crystals are ultrasonically shattered and rise in the columns. Uniform size crystals are collected at a certain height in the column [54]. Lower ultrasonic frequencies, 20–30 kHz, are used for crystallization from melts. Cavitation produces fi ner grain structure [1, p. 129], resulting in improvement of ductility and impact toughness, and specific elongation, in the case of metals. Two categories of crystallization are recognized: (1) crystallization from a solution and (2) crystallization from a melt [2, pp. 410,411; 55]. Crystallization from solution involves the formation and growth of solid crystals of a solute in a supersaturated liquid solution. Crystallization from a melt occurs as the temperature of a liquid is lowered below its freezing point and solid crystals of the substance are allowed to form. 8.14.1

Crystallization in Metals

Applying ultrasonic energy to solidifying metals produces beneficial effects such as homogeneous, distribution of nonmetallic inclusions, reduced gas content, uniform alloying, alloying of normally immiscible substances (such as tin, zinc, and aluminum) and uniform grain refinement. Apparently cavitation and sonic stirring can produce a number of crystal centers far exceeding those forming under solidification without ultrasonic agitation. Sufficient energy cannot be introduced to break metal crystals down into smaller sizes after they have begun to solidify. The major problem is to determine a means of continuously coupling energy into the melt safely and on a suitable production scale. Several methods have been proposed but none has resulted in an evolution from laboratory scale. Coupling elements between the source and the melt must be either highly resistant to melting and eroding into the melt or some means must be found for continuously energizing and feeding a material intended to become at least a part of the melt. The entire volume of melt needs to be exposed until near solidification. Thus, in the metallurgical area, very desirable effects have been seen to be produced ultrasonically, but there are almost formidable problems in applying ultrasonics to large production quantities. Benefits have been realized where treatment of small volumes is involved, such as in arc-welding aluminum [56,57].

8.15

Diffusion and Filtration through Membranes

Ultrasound accelerates diffusion of ions through membranes and cell walls. This effect has been attributed to the disruption of the boundary layer at the interface. Ultrasonic irradiation increases the rate of diffusion of water and sugar through sugar beet membrane. The increased transport continues even after the irradiation is stopped. Substantial increase in the flow rate of crude oil through porous sand stone and that of water through stainless steel filters, under 20 kHz ultrasound has been observed by Fairbanks and Chen [58]. The rate of diffusion of salt through a cellophane membrane has been increased up to 100% by applying ultrasound in the direction of diffusion and is greater when the irradiation is in the direction of gravitational force [59,60]. The effect was not due to temperature


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rise nor due to membrane breakdown. A 69% increase in the diffusion of potassium oxalate through parchment membrane was observed when ultrasonic irradiation was applied in the direction of diffusion [61]. The increase was smaller in both cases when the irradiation was in the opposite direction.

8.16

Chemical Effects

Because ultrasonic frequency varies roughly from 20 kHz to 10 MHz and sound velocity in water at 20°C is ∼1500 m/s, the wavelengths are of the order of 7.5–0.015 cm, and as such are incapable of coupling with small molecules. Hence their effect occurs largely through physical mechanisms, depending on the nature of the system. Suslick has discussed the mechanism and influence of various factors on sonochemical reactions [1, p. 142]. The most important effect of ultrasonic irradiation is cavitation, which takes place in three consecutive steps: nucleation, bubble growth, and implosion. It is generally accepted that as the gas entrapped in small angle crevices of particulate contaminants is subjected to negative acoustic pressure, the bubble volume grows ending in bubble collapse or fragmentation, releasing small free bubbles into solution. Bubble collapse generates microcavities, which serve as nucleation sites for the next cycle of bubble growth and collapse. In homogeneous liquids, there can be either stable cavitation, leading to bubble oscillation, or transient cavitation, in which small bubbles undergo large excursions in volume and then terminate in violent collapse. In practice both stable and transient cavitation takes place simultaneously. Two types of effects have been proposed for implosive collapse of the bubbles; namely, hot spot pyrolysis and electrical discharge. The former hypothesis is largely accepted and has been modeled. The simplest model of bubble collapse, which assumes zero heat transport (adiabatic) and absence of condensable vapor pressure, estimates transient temperature buildup up to 10,000 K and pressure buildup up to 10,000 atm. More sophisticated hydrodynamic models predict temperature and pressure build up at approximately 5000 K and 1000 atm, with residence times of

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