Dynamics of Flexible Multibody Systems: Rigid Finite Element Method (Foundations of Engineering Mechanics)

Foundations of Engineering Mechanics Series Editors: V.I. Babitsky, J. Wittenburg

Edmund Wittbrodt · Iwona Adamiec-Wó jcik Stanisław Wojciech

Dynamics of Flexible Multibody Systems Rigid Finite Element Method

Series Editors: V.I. Babitsky Department of Mechanical Engineering Loughborough University Loughborough LE11 3TU, Leicestershire United Kingdom

J. Wittenburg Institut f u¨ r Technische Mechanik Universit¨at Karlsruhe (TH) Kaiserstraße 12 76128 Karlsruhe Germany

Authors: Edmund Wittbrodt Gda n´ sk University of Technology Faculty of Mechanical Engineering ul. Narutowicza 11/12 80 952 Gdan´ sk Poland e-mail: [email protected] ail: [email protected]

Stanis ław Wojciech University of Bielsko-Biala Department of Mechanics and Computer Methods ul. Willowa 2 43 309 Bielsko-Biala Poland e-mail: [email protected] e-mail: [email protected]

Iwona Adamiec-Wó jcik University of Bielsko-Biala Department of Mechanics and Computer Methods ul. Willowa 2 43 309 Bielsko-Biala Poland e-mail: [email protected] ISSN print edition: 1612-1384 ISBN-10: 3-540-32351-1 ISBN-13: 978-3-540-32351-8

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Homogenous Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Transformation of Coordinates and Homogenous Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Velocity and Acceleration of a Rigid Body . . . . . . . . . . . . . . . . . . 2.3 Description of Geometry of Rigid Links . . . . . . . . . . . . . . . . . . . . 2.4 Kinetic Energy and Lagrange Operators . . . . . . . . . . . . . . . . . . . . 2.5 Potential Energy of Gravity Forces . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Generalised Forces and Equations of Motion . . . . . . . . . . . . . . . . 2.7 Generalisation of the Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3

5 15 17 19 25 26 27

The Rigid Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Division of the Flexible Link into Rigid Finite Elements and Spring–Damping Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Kinetic Energy of the Flexible Link . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Energy of Deformation and Dissipation of Energy of Link p . . . 3.4 Synthesis of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 37 45 53 58 61

4

Modification of the Rigid Finite Element Method . . . . . . . . . . 4.1 Generalised Coordinates and Transformation Matrices . . . . . . . 4.2 Kinetic Energy of the Flexible Link and Its Derivatives . . . . . . . 4.3 Potential Energy of the Flexible Link . . . . . . . . . . . . . . . . . . . . . . 4.4 Synthesis of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 86 90 92 94

5

Calculations for a Cantilever Beam and Methods of Integrating the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 103 5.1 Equations of the Free Vibrations of a Beam . . . . . . . . . . . . . . . . . 103 5.1.1 Classical Rigid Finite Element Method Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

69

VI

Contents

5.1.2 Classical Rigid Finite Element Method Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.1.3 Modified Rigid Finite Element Method Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.1.4 Modified Rigid Finite Element Method Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2 Integrating the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 119 5.2.1 Newmark Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.2 Euler and Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . 123 5.2.3 Step-Size Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2.4 Stiff Systems of Differential Equations . . . . . . . . . . . . . . . 131 5.3 Numerical Effectiveness of Models and Methods of Integrating the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6

Verification of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.1 Vibrations of Whippy Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.1.1 Frequencies of Free Vibrations for a Uniform Beam . . . . 143 6.1.2 Linear and Non-linear Vibrations of a Viscoelastic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.1.3 Kane’s Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1.4 Analysis of Large Deflections . . . . . . . . . . . . . . . . . . . . . . . 163 6.2 Experimental Verification of the Method . . . . . . . . . . . . . . . . . . . 166 6.2.1 Large Amplitude Vibrations of a Fixed Whippy Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.2.2 Sandia Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.1 Offshore Crane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.1.1 Discretisation of Flexible Links and the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 183 7.1.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.2 Telescopic Rapier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.2.1 Discretisation of the Internal Rapier and the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 194 7.2.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.3 A-Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.3.1 Classical Rigid Finite Element Method Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.3.2 Modified Rigid Finite Element Method Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.3.3 Description of Programmes and Results of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.4 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

1 Introduction

Modelling dynamics of multibody systems has been a subject of interest in research centres for many years. Not only machines and mechanisms themselves but also many of their constituent parts are multibody systems; therefore modelling these is essential in design. Computer programmes using these models must enable us to take into account complex phenomena connected with flexibility of links and friction and clearance in joints. The difficulty in describing such problems lies in the fact that global phenomena such as motion of a mechanism or machine, which last a few seconds, have to be considered simultaneously with local phenomena, such as contact of links in joints or vibrations of flexible links, the duration of which is tenths or hundredths of seconds. This causes problems both at the stage of model formulation and during integration of equations of motion of multibody systems. One of the most challenging problems in modelling multibody systems is flexibility of links. The occurrence of large base motion (for example rotation of a crane body, motion of a vehicle, translation of manipulator arms) can cause vibrations of flexible links. There is also an opposite effect: vibrations of flexible links can disturb desired base motion. This is especially important when positioning is a concern. In order to compensate flexibility of links, drive systems have to be equipped with additional control systems. For many years commercial packages such as MSC.Adams or Dads have been used for modelling multibody systems. They allow not only kinematic analysis of complex mechanisms to be carried out, but also calculations of the dynamics of complex multibody systems, and they enable flexibility of links to be taken into account by means of special interfaces. To this end, additional models of flexible links have to be formulated, usually employing the finite element method (for example in Ansys or MSC.Nastran); then modal analysis necessary for the reduction of generalised coordinates has to be carried out; and finally a simplified model of a flexible link obtained in such a way has to be transferred to the package for dynamic analysis of multibody systems. Commercial packages are used mainly in large research and computational centres because they require considerable training in the methodology and

2

1 Introduction

software. Moreover, it is necessary to know the appropriate, often very detailed data for calculations. Thus the use of those commercial packages is limited and many research centres are still looking for simple yet effective methods for analysing the dynamics of multibody systems. One such method is the rigid finite element method, which is the subject of this book. It can serve both as an initial analysis before more traditional and complex methods are used and as an independent method. Its advantages are numerous: first, it is simple (the basic idea of the method is a division of flexible links into rigid elements connected with spring-damping elements); secondly, it adopts a uniform approach to describe rigid and flexible links; thirdly, it is numerically effective; fourthly, it can be applied to analyse both small and large deformations; finally, it can be widely and successfully used in industrial practice. Therefore the authors believe that many engineers and researchers will benefit from acquaintance with the rigid finite element method. The method was formulated at the Technical University of Gdansk and its foundations were first described by Kruszewski et al. (1975). The idea of the method is to discretise flexible links into rigid elements containing inertial features of bodies; these rigid elements are connected by massless and non-dimensional spring-damping elements. In Kruszewski et al. (1975) the mathematical models and all information necessary for computer implementation of the method are presented. The models and their application in practice are limited to analysis of deflections and vibrations in a given position, which means that only systems with a stable configuration have been considered. Wittbrodt (1983) presented a generalisation of the method of rigid and flexible elements for planar systems with changing configuration, and its application in dynamic analysis of structures was described by Kruszewski et al. (1984). The method has been applied in Poland for the dynamic analysis of mechanisms, machine tools, cranes, ship drive systems and even to vibration analysis of hulls. Wojciech (1984) presented a modification of the method which enabled large deflections of flexible links of planar linkage mechanisms with changing configuration to be analysed. An approach in which the system analysed is divided into subsystems, and flexible links are discretised by means of the rigid finite element method, is applied in both monographs (Wittbrodt, 1983; Wojciech, 1984). Equations of the dynamics of subsystems taking into account reactions in subsystem connections have been formulated on the basis of Lagrange equations and then the subsystems have been connected by means of constraint equations. Such an approach is standard when absolute coordinates are used (Gronowicz, 2003). The classical rigid finite element method as well as its modified version were generalised for spatial systems with changing configuration in Wojciech (1990); Adamiec-W´ojcik (1992), Wittbrodt and Wojciech (1995) and Adamiec-W´ojcik (2003). In these the rigid finite element method was combined for the first time with the method of homogenous transformations. This method, widely used in robotics (Paul, 1981; Craig, 1988), enables us to

1 Introduction

3

represent the transformation of coordinates, for both translation and rotation of a rigid body, by means of one matrix operation. Adamiec-W´ ojcik (2003) also presents a general algorithm for formulation of equations of motion of multibody systems using joint coordinates and homogenous transformations. This book presents a new, different formulation of the rigid finite element method. It is assumed that, for both the classical and modified formulations, homogenous transformations will be consistently used together with joint coordinates for the kinematic description of multibody systems. Joint coordinates enable us to reduce considerably the number of generalised coordinates of the system as compared to methods using absolute coordinates. The models and methods presented allow large deformations of flexible links to be considered. Simplified versions of models (called linear), which can be used when deflections of links are small, are also discussed. The models formulated give a unified approach both in cases when open and closed kinematic chains are considered as well as when the system consists of either only rigid links or when rigid and flexible links alternate. We think that this is one of most important features of the method and the description of multibody dynamics. We assume that the reader knows theoretical mechanics at the level of a mechanical engineering graduate and is able to use some elements of analytical mechanics, especially techniques concerned with derivation of equations of motion using the Lagrange equations. As for mathematics, we expect the reader to be competent in dealing with matrix calculations and differential calculus. In Chap. 2 the basics of transformations of coordinates and homogenous transformations are presented. In addition the equations of motion of rigid multibody systems are formulated using joint coordinates and homogenous transformations. The equations of motion of a new link attached to an existing kinematic chain are formulated and it is shown how they modify the equations of preceding links. The equations formulated in this chapter are then used throughout the following chapters. The formulation of the classical rigid finite element method is presented in Chap. 3. The equations of motion of a flexible link divided into rigid elements with six degrees of freedom (three translations and three rotations) are derived. The energy of spring deformation and the dissipation of energy in spring-damping elements are calculated. A linear model with simpler formulae, which is useful for analysis of small vibrations, is also discussed. At the end of the chapter the methods and formulae for calculations of the parameters of both rigid (rfe) and spring-damping elements (sde) are given. Chapter 4 deals with the modification of the rigid finite element method used to discretise beam-like links with bending and torsional flexibility. Nonlinear and linear models for analysis of large and small vibrations, respectively, are discussed. In the modified formulation of the method each rfe has only three degrees of freedom in relative motion, which are rotation angles. Thus, in relation to the classical formulation, the number of degrees of freedom is considerably smaller.

4

1 Introduction

The problems of numerical calculations are discussed in Chap. 5. At the beginning we present the application of the methods described in Chaps. 3 and 4 (both linear and non-linear formulation) to analysis of the free and forced vibrations of a beam. The reader can follow detailed formulations of various models. Computer simulations show the influence of the model on the results of calculations. Problems concerning the integration of equations of motion of systems discretised using the rigid finite element method are also considered, with special attention to the integration of systems of stiff differential equations. Chapter 6 is concerned with verification of the method. The results of numerical simulations obtained by means of the rigid finite element method are compared with those obtained by other authors who used different methods, and with results of experimental measurements. The method has been verified both for small and large deformations. An example of vibration analysis of a viscoelastic beam shows how the rigid finite element method can be used to analyse large deflections of whippy beams when complex physical relationships describe material features of flexible links. This chapter demonstrates that the rigid finite element method in both formulations gives results compatible with those published by other authors and with those obtained from experimental measurements. Practical applications of the method in dynamic analysis of machines and mechanisms are given in Chap. 7. They concern dynamic analysis of a crane, the telescopic rapier in textile machine, and the A-frame of a ship. The chapter shows not only how to proceed with a particular machine but also the many applications of the method. We would like to thank Krzysztof Augustynek and Andrzej Urbas for their considerable editorial help and our colleagues, co-authors of publications, whose research results we used in this book.

2 Homogenous Transformations

Displacement of a body from one position to another requires two operations: translation and rotation. In classical mechanics general motion of a body can be treated as a combination of translation and rotation about a fixed point. In robotics joint coordinates and homogenous transformations are generally used for description of rigid body motion (Craig, 1988). Joint coordinates enable us to describe the motion of a system of rigid bodies, which form open or closed kinematic chains, by using the least number of generalised coordinates. This leads to a reduction in the dimension of equations of motion describing the dynamics of multibody systems as compared to absolute coordinates, which are used more often. However, the equations are more complex and their derivation requires a specific approach, which will be described in this chapter. Homogenous transformations allow us to present two operations (translation and rotation) in the form of one complex operation. The consequence is that the transformation of coordinates from one system to another can be expressed by means of only one multiplication of a transformation matrix by a position vector.

2.1 Transformation of Coordinates and Homogenous Transformations In order to describe the position and orientation of a body in space, coordinate systems (called “frames of reference” by some authors) are defined and the rules of coordinate transformations are set out. Mathematical relations are formulated so that coordinates of a point in any coordinate system can be defined if the coordinates of this point in a given coordinate system and the parameters defining the reciprocal relation of the two systems are known. Let us assume that two coordinate systems {A} and {B} (Fig. 2.1) are given. The method of describing the relation of those systems is as follows.

6

2 Homogenous Transformations {B } B

B

Xˆ 2

B

Xˆ 1

Xˆ 3

{A} A

Xˆ 3

Ar B org

A

A

Xˆ 2

Xˆ 1

Fig. 2.1. Coordinate systems {A}, {B}. A rBorg is the vector defining the coordinates ˆ 1 ,A X ˆ 2 ,A X ˆ 3 ,B X ˆ 1 ,B X ˆ 2 ,B X ˆ3 of the origin of system {B} with respect to {A}. A X are the unit vectors of coordinate systems {A} and {B}, respectively

The position of the origin of coordinate system {B} with respect to {A} is defined by the components of the vector:  A xB,1   A rBorg =  A xB,2  , (2.1) A xB,3 while the orientation of {B} in {A} is defined by the elements of the rotation matrix:  B TA    B B B ˆ X ˆ 1 BX ˆ TAX ˆ 1 BX ˆ TAX ˆ1 X 1 2 3 A r11 A r12 A r13    A B B B  B ˆ TA ˆ B X ˆ TAX ˆ 2 BX ˆ TAX ˆ2 =  B R =  X1 X2 2 3   A r21 A r22 A r23  , (2.2) B B B B ˆ TA ˆ ˆ TAX ˆ 3 BX ˆ TAX ˆ3 X1 X3 B X A r31 A r32 A r33 2 3 ˆ TB ˆ is a scalar product of A ˆ and B. ˆ where A A Matrix B R is a matrix of direction cosines of the axes of coordinate system {B} with respect to system {A}. Its columns and rows are orthonormal vectors, which results in the following relation: A BR

−1 T =B =B AR AR .

(2.3)

This means that the inverse matrix to the rotation matrix is its transpose. It is important to note that the position of the body to which coordinate system

2.1 Transformation of Coordinates and Homogenous Transformations

7

{B } P

Br

{A} Ar

Ar

Borg

Fig. 2.2. Transformation of coordinates

{B} is assigned is defined by vector A rBorg , and A B R describes its orientation. If we know the position and orientation of {B} with respect to {A}, then, according to Fig. 2.2, the following can be written: A

r=A BR

B

r+

A

rBorg ,

(2.4)

where B r is the vector of coordinates of point P with respect to {B} and A r is the vector of coordinates of point P with respect to {A}. Among the nine components of rotation matrix A B R only three are independent because six relations resulting from the orthonormality of the matrix can be defined, which are as follows. If matrix A B R is written in the following form:  A A ˆ 1 AX ˆ 2 AX ˆ3 , (2.5) X BR = B

where

 ˆ TAX ˆ1 X 1   Aˆ  B ˆ TA ˆ  B X1 =  X1 X2  , B ˆ TA ˆ X X3 

B

1

B

B

 ˆ TAX ˆ1 X 2   Aˆ  B ˆ TA ˆ  B X2 =  X2 X2  , B ˆ TA ˆ X X3 

B

2

 B ˆ TA ˆ  X3 X1  B ˆ TA ˆ  Aˆ B X3 =  X3 X2  , B ˆ TA ˆ X X3 3

the following takes place: A ˆ TA ˆ B X1 B X1

= 1,

A ˆ TA ˆ B X2 B X2

= 1,

A ˆ TA ˆ B X3 B X3

= 1,

(2.6a)

A ˆ TA ˆ B X1 B X2

= 0,

A ˆ TA ˆ B X1 B X3

= 0,

A ˆ TA ˆ B X2 B X3

= 0.

(2.6b)

This means that the reciprocal location of the axes of systems {B} and {A} can be uniquely described by means of three parameters. There are many

8

2 Homogenous Transformations

methods of choosing those parameters (Blajer, 1998; Jurewiˇc, 1984; Shabana, 1998). Here we discuss only one, in which those parameters are called Euler rotation angles ZYX (Blajer, 1998). Let us consider in detail the case when the origins and axes of coordinate systems {B} and {A} coincide (Fig. 2.3a), and the problem is to define the rotations of system {B} about its axes, which leads to the situation as in Fig. 2.3b. In order to convert the system from position (a) to (b), we will proceed as follows: ˆ3 = – First, coordinate system {B} is rotated by angle ϕ3 about axis A X Bˆ  X3 and coordinate system {B} is obtained. – Secondly, the coordinate system obtained is rotated by angle ϕ2 about

ˆ 2 , which results in coordinate system {B} . axis B X
ˆ 1 to – Finally, the system obtained is rotated by angle ϕ1 about axis B X achieve the position as in Fig. 2.3b. This procedure is shown in Fig. 2.4. In order to transform the body from position (b) to (a) (Fig. 2.3), one should proceed in the reverse order, which means the following should be performed: – Rotate the system by −ϕ1 about axis – Rotate the system by −ϕ2 about axis – Rotate the system by −ϕ3 about axis

ˆ1 X ˆ2 X B

ˆ X3 B

B


The rotation matrix parameterised by means of Euler angles ZYX can be written in the form:

{A}

(b)

{A}, {B}

(a)

A

A

{B}

Xˆ 3, B Xˆ 3

Xˆ 3 B

B

Xˆ 3

Xˆ 2

A A

Xˆ 2, Xˆ 2 A

A

Xˆ 1, B Xˆ 1

Xˆ 2

B

Xˆ 1

B

Xˆ 1

Fig. 2.3. Initial (a) and final (b) location of coordinate systems {A} and {B}

2.1 Transformation of Coordinates and Homogenous Transformations (a)

A

(b)

Xˆ 3 = B0 Xˆ 3

B'

j3 A

B0

Xˆ 3

Xˆ 3

j2

Xˆ 1

B0

Xˆ 2 B0

j3 j2 A A

9

Xˆ 1

Xˆ 2 = B9 Xˆ 2

Xˆ 2

j3 B0

B0

Xˆ 2

Xˆ 1

j2 B'

Xˆ 1

B' ˆ

(c)

X3

B

Xˆ 3

B

j1

Xˆ 2

j1 B' ˆ

X2

j1

B'

Xˆ 1 = B Xˆ 1

Fig. 2.4. Euler angles ZYX

A BR








B B =A B

R B
R B R = R3 (ϕ3 ) R2 (ϕ2 ) R1 (ϕ1 ) =     cϕ2 0 sϕ2 1 0 0 cϕ3 −sϕ3 0     1 0   0 cϕ1 −sϕ1  =  sϕ3 cϕ3 0   0 (2.7) 0 0 1 −sϕ2 0 cϕ2 0 sϕ1 cϕ1   cϕ3 cϕ2 cϕ3 sϕ2 sϕ1 − sϕ3 cϕ1 cϕ3 sϕ2 cϕ1 + sϕ3 sϕ1   =  sϕ3 cϕ2 sϕ3 sϕ2 sϕ1 + cϕ3 cϕ1 sϕ3 sϕ2 cϕ1 − cϕ3 sϕ1  ,

−sϕ2

cϕ2 sϕ1

cϕ2 cϕ1

where sϕi = sin ϕi , cϕi = cos ϕi . Angles ϕ3 , ϕ2 , ϕ1 are called yaw, pitch and roll angles, respectively. Matrix A B R defined by (2.7) can also be interpreted as the rotation operator. If the coordinates of point P with respect to coordinate system {B}

10

2 Homogenous Transformations {B}

B

Xˆ 3 Br p

P B

x p,3 B B

B

Bx

Xˆ 1

Xˆ 2

x p,1

p,2

Fig. 2.5. Point P and its coordinates

B9 B

Xˆ 3

Xˆ 3 B9

x p,2

Bx

P Bx

p,2

p,3 B

B9

x p,3

j1

Xˆ 2 B9

Xˆ 2

-j1 B

Xˆ 1 = B9 Xˆ 1

Fig. 2.6. Rotation by −ϕ1

define vector B rP (Fig. 2.5), then the coordinates of this point with respect to {A} can be calculated by carrying out successive rotations described in (2.7) as follows: 1. Rotation by −ϕ1 Coordinate systems {B} and {B  } obtained as a result of rotation by −ϕ1 ˆ 1 are shown in Fig. 2.6. about axis B X Coordinates of point P with respect to coordinate system {B  } are as follows: B
B B







xP,1 = B xP,1 ,

(2.8a)

xP,2 = B xP,2 cϕ1 − B xP,3 sϕ1 ,

(2.8b)

xP,3 = B xP,2 sϕ1 + B xP,3 cϕ1 ,

(2.8c)

2.1 Transformation of Coordinates and Homogenous Transformations B9 B0

11

Xˆ 3

Xˆ 3 B9 B0

x p,1

B9 ˆ

X 2 = B0 Xˆ 2

P

x p,1

B0 B9x

x p,3

-j2

p,3 B0

Xˆ 1

j2

B9 ˆ

X1

Fig. 2.7. Rotation by −ϕ2

which can be written as: B


rP = R1 (ϕ1 )B rP .

(2.9)

2. Rotation by −ϕ2 Figure 2.7 presents coordinate systems {B  } and {B  } obtained by
ˆ 2 = B

X ˆ 2. having rotated {B  } by angle −ϕ2 about axis B X Coordinates of point P with respect to coordinate system {B  } are defined by formulae: B

B

B









xP,1 = B xP,1 cϕ2 +B xP,3 sϕ2 , xP,2 =

B


xP,2 ,

(2.10a) (2.10b)

B


B


xP,3 = − xP,1 sϕ2 + xP,3 cϕ2 ,

(2.10c)

or in the vector form: B






rP = R2 (ϕ2 )B rP = R2 (ϕ2 )R1 (ϕ1 )B rP .

(2.11)

3. Rotation by angle −ϕ3 Coordinate systems {B  } and {A} are shown in Fig. 2.8. Coordinate sys

ˆ 3 = AX ˆ 3 by tem {A} is the result of rotation of {B  } about axis B X angle −ϕ3 . Coordinates of point P with respect to coordinate system {A} are now as follows:

A A A









B



B



xP,1 = B xP,1 cϕ3 −B xP,2 sϕ3 , xP,2 = xP,3 =

B



xP,1 sϕ3 + xP,3

xP,2 cϕ3 ,

(2.12a) (2.12b) (2.12c)

12

2 Homogenous Transformations A B0

Xˆ 2

Xˆ 2 A

B0

x p,1

B0x

x p,1 A

P

x p,2

p,2

B0

Xˆ 1

A

Xˆ 1

j3

-j3

B0

Xˆ 3 = A Xˆ 3

Fig. 2.8. Rotation by angle −ϕ3

and in the vector form: A





rP = R3 (ϕ3 )B rP = R3 (ϕ3 )R2 (ϕ2 )R1 (ϕ1 ) B rP .

(2.13)

It should be noted that in many applications angles ϕ1 and ϕ2 (rotations ˆ 2 axes of the body) are assumed to be ˆ 1 and lateral X about longitudinal X A small and thus the form of matrix B R from (2.7) can be considerably simplified. However, linearisation of trigonometrical functions of angles ϕ1 and ϕ2 results in lack of orthonormality by matrix A B R, and thus the property (2.3) does not hold. The transformation of coordinates from one coordinate system to another, defined by (2.4), has the mathematical disadvantage that it requires two operations: multiplication of a matrix by a vector and summation of vectors. In robotics and manipulator mechanics homogenous transformations are used (Paul, 1981; Craig, 1988), which enable the transformation of coordinates to be presented by means of only one operation: multiplication of a matrix by a vector. According to this approach the following position vectors are introduced: A

T  r3 A r4 = A x1 A x2 A x3 1 = , (2.14a) 1 B

T  r3 B , (2.14b) r4 = B x1 B x2 B x3 1 = 1 where A r3 , B r3 are vectors A r and B r from (2.4), and formula (2.4) is written in the form:

2.1 Transformation of Coordinates and Homogenous Transformations

 A

r4 =

A

r3



1

 

 =

A A B R rBorg

0



Ar 4



1

B

r3



1   

AT B

13

=A BT

B

r4 .

(2.15)

Br 4

In further considerations we will use vectors with four components A r4 and r4 , unless stated otherwise or evident from the context, and thus subscript 4 will be omitted and formula (2.15) will take the form:

B

A

r=A BT

where A BT

A =

BR

A

0

B

r,

(2.16)

rBorg . 1

Use of this formula is especially convenient when successive transformations have to be carried out. Let B C T be the transformation matrix from coordinate system {C} to {B}, and A B T be the transformation matrix from coordinate system {B} to {A}. Then the transformation matrix from coordinate system {C} to {A} has the following form: A CT

=A BT

B CT

(2.17)

and coordinates from system {C} are transformed to {A} according to the formula: C A A B C r=A (2.18) C T r = B T C T r. It is important to point out that by using formula (2.4) we obtain a much more complicated relation:  A   B C A B (2.19) r= A B R C R r + rCorg + rBorg . The elements of matrix T have a simple geometrical interpretation. Figure 2.9 shows the local coordinate system { } assigned to a body and the global coordinate system denoted { }. Transformation of coordinates is then carried out as follows: (2.20) r = Tr , where r are coordinates of a point with respect to the local coordinate system { } , r are coordinates of a point with respect to the global coordinate system { },   cϕ3 cϕ2 cϕ3 sϕ2 sϕ1 − sϕ3 cϕ1 cϕ3 sϕ2 cϕ1 + sϕ3 sϕ1 x0,1  sϕ cϕ sϕ sϕ sϕ + cϕ cϕ sϕ sϕ cϕ − cϕ sϕ x  3 2 1 3 1 3 2 1 3 1 0,2   3 2 T= .  −sϕ2 cϕ2 sϕ1 cϕ2 cϕ1 x0,3  0 0 0 1

14

2 Homogenous Transformations

{}

Xˆ 93

Xˆ 3

j3

{ }⬘

Xˆ 92

A r9

j2 r j1 Xˆ 91

x 0,3 Xˆ 2 x 0,1

x 0,2

Xˆ 1

Fig. 2.9. Transformations of coordinates from the local { }
to global { } coordinate system

Inverse transformation from system { } to { } can be defined according to the formula: (2.21) r = T−1 r and matrix T−1 takes the form: T−1 = where R is defined by (2.7),



RT −RT r0 , 0 1

(2.22)

 x0,1 r0 =  x0,2  . x0,3 

Matrix T from formulae (2.20), (2.21) can also be presented in the form: T = T1 T2 T3 T6 T5 T4 , where



1 0 T1 =  0 0  1 0 T4 =  0 0

   0 x0,1 1 0 0 0   0 0   , T =  0 1 0 x0,2  , T 1 0  2 0 0 1 0  3 0 0 0 1 0 1   0 0 0 cϕ2 0 sϕ2  0 1 0 cϕ1 −sϕ1 0  ,T =  sϕ1 cϕ1 0  5  −sϕ2 0 cϕ2 0 0 1 0 0 0

0 1 0 0

(2.23)

 1 0 0 0 0 1 0 0   =  0 0 1 x0,3  , 0 0 0 1   0 cϕ3 −sϕ3  sϕ3 cϕ3 0 ,T =  0 6  0 0 1 0 0 

0 0 1 0

 0 0 . 0 1

2.2 Velocity and Acceleration of a Rigid Body

15

It can be seen that each matrix T1 to T6 depends on only one parameter: T1 = T1 (x0,1 ), T2 = T2 (x0,2 ), T3 = T3 (x0,3 ), T4 = T4 (ϕ1 ), T5 = T5 (ϕ2 ), T6 = T6 (ϕ3 ). In certain problems of dynamics some parameters describing rigid body motion can be assumed to be constant or known functions of time and in such a case the motion is described by a smaller number of parameters. For example, in the case of planar motion the number of parameters decreases from six (x0,1 , x0,2 , x0,3 , ϕ1 , ϕ2 , ϕ3 ) to three (x0,1 , x0,2 , ϕ3 ). In the case of rotation about a point the motion can be described by angles (ϕ1 , ϕ2 , ϕ3 ).

2.2 Velocity and Acceleration of a Rigid Body In certain applications it is necessary to know the relations which allow velocities and accelerations (both linear and angular) of a rigid multibody system to be calculated (Gronowicz, 2003). In order to solve such a task, let us consider again a general problem of coordinate transformations which was discussed at the beginning of this chapter. The situation to be examined is shown in Fig. 2.10. The task is to describe the motion of vector B r with respect to coordinate system {A}. According to the general rules of kinematics (Craig, 1988), if the angular velocity of system {B} with respect to {A} equals A ΩB the following can be written: d A
A A ( R Q) = A (2.24) B R Q + ΩB × (B R Q), dt B

{B } AW B

Br

{A } Ar

Ar

Borg

Fig. 2.10. Position vectors A r, B r; vector of angular velocity A ΩB ; reference coordinate system {A}; local coordinate system assigned to the rigid body {B}

16

2 Homogenous Transformations

where A B R is the orientation matrix of {B} with respect to {A}, defined by (2.2), and Q is any vector in {B}. Having taken into account that: A

r = A rBorg +

A B B R r,

(2.25)

d A B ( R r). dt B

(2.26)

one obtains: A

r˙ = A r˙ Borg +

Taking into account (2.24), we can write: A

B B ˙ +A ΩB ×A r˙ = A r˙ Borg +A B R r B R r.

(2.27)

Acceleration can be calculated according to the formula: A

¨r =

A

¨rBorg +

d A B ( R r˙ ) + dt B

A

˙B× Ω

A B BR r

+

A

ΩB ×

d A B ( R r) dt B

B B ˙ B ×A R B r ˙ +A Ω r +A ΩB ×A = A ¨rB +A B R ¨ B R r B

(2.28)

B B ˙ +A ΩB ×A +A ΩB × (A BR r B R r),

˙ B is the vector of angular acceleration. where A Ω Relation (2.28) can be written as follows: A ˙ B × A RB r + A ΩB × (A ΩB × ¨r = A ¨rBorg + A Ω B

A B B R r)

B B ˙ +A r. ×A BR r B R ¨

+ 2A ΩB (2.29)

In the formula above we can name all the components of acceleration known in mechanics: A¨ rBorg

˙ B × Ar Ω ΩB × ( ΩB × A r) 2A ΩB × A r˙ A B¨ BR r A

A

A

acceleration of the origin of system {B} in the global system {A}, tangential component of the acceleration, normal component of the acceleration, Coriolis acceleration, local component of acceleration.

Relation (2.24) can also be used for calculation of angular acceleration. Let {C} rotate about {B} with angular velocity B ΩC , and {B} rotate about {A} with angular velocity A ΩB . In order to calculate A ΩC , vectors of angular velocities with respect to {A} are summed and one obtains: A

ΩC = A ΩB +

A B

B ( ΩC ) = A ΩB +A B R ΩC .

(2.30)

Differentiation of (2.30) with respect to time leads to: ˙ C = AΩ ˙ B + d (A RB ΩC ). (2.31) Ω dt B Having applied (2.24) to the second component of the right side of the above equation, we obtain the following: A ˙ ˙ B +A RB Ω ˙ C +A ΩB ×A RB ΩC . (2.32) ΩC = A Ω B B A

2.3 Description of Geometry of Rigid Links

17

2.3 Description of Geometry of Rigid Links Homogenous transformations based on the Denavit–Hartenberg notation are popular among researchers in the field of robotics, where they are used to describe the relative position of links connected by rotary and prismatic joints (with one degree of freedom). These matrices are functions of four parameters (Fig. 2.11): two constant parameters depend on the geometry of the rigid links (link length and link twist) and two other parameters are used for description of relative motion of links (joint angle or link offset). In the case of translational motion of link p in relation to link p − 1, dp is a generalised coordinate (θp = const.), while for a rotary joint the generalised coordinate describing the relative motion of link p with respect to p − 1 is θp (dp = const.). In order to describe the relative position of link p with respect to link p−1, ˆ 1, P X ˆ 2, P X ˆ 3 assigned to link p is introduced in the the coordinate system P X following way: ˆ 3 is along the axis of joint p. – Axis P X – The origin of the coordinate system {p} is placed at the point where the perpendicular to the axes of neighbouring joints intersects with the axis of joint p. p qp

p -1

p p

p -1 ˆ

X3

Xˆ 2

Xˆ 3

ap

Op p -1 ˆ

X2

p

Xˆ 1

dp a p -1

Op -1 p -1 ˆ

X1

a p -1

Fig. 2.11. Parameters describing the relative position of two successive links: ap−1 link length; αp−1 link twist; θp joint angle; dp link offset

18

2 Homogenous Transformations

– Axis – Axis

P P

ˆ 1 is along this perpendicular in the direction of the next joint. X ˆ 2 completes the right-hand coordinate system. X

Thus the transformation matrix from coordinate system {p} to system {p − 1} takes the following form (Craig, 1988):   cθp −sθp 0 ap−1  sθ cα   p p−1 cθp cαp−1 −sαp−1 −sαp−1 dp  p−1 (2.33) T =  , p  sθp sαp−1 cθp sαp−1 cαp−1 cαp−1 dp  0 0 0 1 where sθp = sin θp , cθp = cos θp , sαp−1 = sin αp−1 , cαp−1 = cos αp−1 ; and the generalised coordinate q˜p describing the relative motion of link p with respect to link p − 1 is defined as follows:  dp for a prismatic joint q˜p = . (2.34) θp for a rotary joint The assumption that joints are either prismatic or rotary (link p has one degree of freedom in relation to link p − 1) does not limit considerations. If a real kinematic pair has more than one degree of freedom, it is treated as a set ˜ p − 1 links have of n ˜ p kinematic pairs with one degree of freedom each, and n no length (˜ np is the number of degrees of freedom of the kinematic pair). The generalised coordinates of the pth kinematic pair are then the components of the vector: (p) (p) ˜ p = [˜ q1 · · · q˜n˜ p ]T , (2.35) q (p)

where q˜j (j = 1, . . . , n ˜ p ) are the generalised coordinates of the set of kinematic pairs substituting the real kinematic pair according to formula (2.34). Transformation from coordinate system {p}, assigned to link p, to system {p − 1} of the previous link is defined by the following formula: ˜ (p) = B

n ˜p  j−1

jT

(p)

,

(2.36)

j=1

where

j−1 (p) jT

 

(p)

q˜j



(p)

cθj

(p)

−sθj

0

(p)

aj−1



 (p) (p) (p) (p) (p) (p) (p)   sθ cα  j j−1 cθj cαj−1 −sαj−1 −sαj−1 dj  . =  (p) (p) (p) (p) (p) (p) (p)   sθj sαj−1 cθj sαj−1 cαj−1 cαj−1 dj  0 0 0 1

It is important to note that if n ˜ p > 1, matrix 01 T(p) defines transformation (p) from the first substitute link of kinematic pair p, and a0 is the length of link (p) p − 1, while parameters aj for j > 0 equal zero.

2.4 Kinetic Energy and Lagrange Operators

19

2.4 Kinetic Energy and Lagrange Operators In order to formulate the equations of motion by means of homogenous transformations, it is assumed that the motion of a link is described by independent parameters which are components of the vector: T  q = q1 . . . qk . . . qn ,

(2.37)

where qk (k = 1, . . . , n) are generalised coordinates. When functionals E and V defining kinetic and potential energies of the link are known, the equations of motion can be formulated using the Lagrange equations (Leyko, 1996): d ∂E ∂E ∂V − + = Qk dt ∂ q˙k ∂qk ∂qk

for k = 1, 2, . . . , n,

(2.38)

where q˙k are generalised velocities and Qk are non-potential generalised forces. These equations can also be written in the form: εk (E) + where εk (E) =

∂V = Qk , ∂qk

(2.39)

d ∂E ∂E − dt ∂ q˙k ∂qk

are Lagrange operators, or in the matrix form: εq (E) + where

∂V = Q, ∂q

(2.40)



 d ∂E ∂E εq (E) = − , dt ∂ q˙k ∂qk k=1,...,n   ∂V ∂V = , ∂q ∂qk k=1,...,n Q = (Qk )k=1,...,n .

For derivation of the equations of motion we assume that the body is placed on a moving base {A} (Fig. 2.12), the motion of which is known with respect to the inertial coordinate system { }. This is the case when the dynamics of working machines placed on a drilling platform or ship deck is analysed. It is assumed that the motion of the coordinate system {A} with respect to the inertial system is defined by the transformation matrix: B0 (t) = B0 (xA,1 (t), xA,2 (t), xA,3 (t), ϕA,3 (t), ϕA,2 (t), ϕA,1 (t)),

(2.41)

20

2 Homogenous Transformations {A}

AX ˆ

Xˆ 39

{ }9

3

Xˆ 29

j A,3 {}

r9

Xˆ 3 r

Xˆ 19

j A,1

j A,2

Aˆ

X1

AX ˆ

2

x A,3

x A,2

x A,1

Xˆ 2

Xˆ 1

Fig. 2.12. Coordinate system assumed: { } inertial coordinate system; {A} coordinate system of the moving base; { }
local coordinate system assigned to the body

which in the general case can be presented in the form: B0 (t) = T1 T2 T3 T6 T5 T4 ,

(2.42)

where Ti is defined as in (2.23), having assumed ϕi = ϕA,i , x0,i = xA,i . If r is the vector defining the position of particle dm in the local coordinate system { } assigned to the body, then the coordinates of particle dm in the global system { } can be defined according to the formula: r = B0 (t) B(q) r = B r ,

(2.43)

where r = [ x1 x2 x3 1 ]T , r = [ x1 x2 x3 1 ]T , B0 (t) is the transformation matrix from system {A} to coordinate system { } defined by (2.41), B = B(q1 , . . . , qn ) is the transformation matrix from the local coordinate system { } to system {A} dependent on generalised coordinates (q1 , . . . , qn ) which are components of vector (2.37), B = B0 (t)B(q). In the special case when there is no moving base {A}, one can assume: B0 (t) = I,

(2.44)

where I is the identity matrix, which necessarily yields the following relation: B = B(q).

(2.45)

In order to calculate kinetic energy E of the body, the concept of the trace of a matrix is used (Jureviˇc, 1984), and the following can be written:

2.4 Kinetic Energy and Lagrange Operators

  x˙   1 1  1  T x˙ 2   [ x˙ 1 x˙ 2 dm = tr  dE = tr r˙ r˙  x˙ 3  2 2    0  x˙ 1 x˙ 1 x˙ 1 x˙ 2     1  x˙ 2 x˙ 1 x˙ 2 x˙ 2 = tr  2   x˙ 3 x˙ 1 x˙ 3 x˙ 2    0 0 =

21

    x˙ 3 0 ]

x˙ 1 x˙ 3 x˙ 2 x˙ 3 x˙ 3 x˙ 3 0

  

dm

 0     0   dm 0     0

(2.46)

1 2 1 (x˙ 1 + x˙ 22 + x˙ 23 ) dm = ν 2 dm, 2 2

where tr(A) =

p "

aii

i=1

is the trace of matrix Ap×p = (aij )i,j=1, ..., p , ν 2 = x˙ 21 + x˙ 22 + x˙ 23 . Since components of vector r are constant (with respect to time), then: ˙  r˙ = Br and the kinetic energy of the body with mass m is: # # % $   1 1 ˙ T dm ˙ r r T B tr r˙ r˙ T dm = tr B E= 2 2 m m     #   1 ˙ HB ˙ T }. ˙  r r T dm B ˙ T = 1 tr{ B = tr B  2 2 

(2.47)

(2.48)

m

Matrix H from the above formula is known as a pseudo-inertia matrix and its elements are calculated as follows:   J(Xˆ
Xˆ
) JXˆ
Xˆ
JXˆ
Xˆ
JXˆ
2 3 1 2 1 3 1 #  J ˆ
ˆ
J ˆ
ˆ
J ˆ
ˆ
J ˆ
 T  X1 X2 (X1 X3 ) X2 X3 X2  H = r r dm =   = (hij )i,j=1,...,4 ,  JXˆ
Xˆ
JXˆ
Xˆ
J(Xˆ
Xˆ
) JXˆ
 1 3 2 3 1 2 3 m JXˆ
JXˆ
JXˆ
m 1 2 3 (2.49) where J(Xˆ
Xˆ
) , J(Xˆ
Xˆ
) , J(Xˆ
Xˆ
) are moments of inertia with respect to respec2 3 1 3 1 2 ˆ X ˆ X ˆ X ˆ, X ˆ  and X ˆ  in local coordinate system { } , J ˆ
ˆ
, tive planes X 2

3

1

3

1

2

X1 X2

JXˆ
Xˆ
, JXˆ
Xˆ
are products of inertia with respect to local coordinate sys1 3 2 3 tem { } , JXˆ
, JXˆ
, JXˆ
are static moments with respect to local coordinate 1 2 3 system { } .

22

2 Homogenous Transformations

These moments are calculated as: # # 2 2 J(Xˆ
Xˆ
) = x 1 dm, J(Xˆ
Xˆ
) = x 2 dm, 2

3

1

3

m

#

JXˆ
Xˆ
= 1

2

JXˆ
= 1

m

1

2

m

x1 x2

dm,

x1 dm,

x1 x3

JXˆ
Xˆ
= 1

3

# JXˆ
= 2

x 3 dm, 2

m

#

m

#

# J(Xˆ
Xˆ
) = #

dm,

m

JXˆ
Xˆ
= 2

#

x2 dm,

JXˆ
= 3

m

3

x2 x3 dm,

m

x3 dm.

(2.50)

m

˙ occurring in (2.47) and (2.48) can be calAccording to (2.43), matrix B culated as follows:   ˙ ˙ = dB = d B0 (t) B(q) = B ˙ 0 B + B0 B. B (2.51) dt dt Since ˙ = dB = " ∂B q˙ = " B q˙ , B j j j dt ∂qj j=1 j=1 n

n

(2.52)

where Bj =

∂B , ∂qj

then ˙ =B ˙ 0B + B

n "

Bj q˙j ,

(2.53)

j=1

where Bj = B0 Bj . In view of (2.43) and (2.53), the following relations occur: ∂B = B0 Bk = Bk , ∂qk

(2.54a)

˙ ∂B = Bk . ∂ q˙k From (2.54a) and (2.54b) follows:

(2.54b)

˙ ∂B ∂B = . ∂qk ∂ q˙k

(2.55)

In view of (2.51) the kinetic energy can be written in the form:    T    n n   $ % " " 1 1 T ˙ ˙ ˙ ˙     B0 B + Bj q˙j H B0 B + Bj q˙j . E = tr B H B = tr  2 2    j=1

j=1

(2.56)

2.4 Kinetic Energy and Lagrange Operators

23

In order to calculate the derivatives of the kinetic energy with respect to generalised coordinates and velocities, the following features of the trace of a matrix will be used: tr (A + B) = tr (A) + tr (B) , tr (λA) = λtr (A) ,   tr AT = tr (A) .

(2.57)

Differentiating the kinetic energy (2.56) with respect to generalised velocities, and using (2.55), the following is obtained: & & ' ' ˙ ˙T ∂B ∂B 1 1 ∂E T ˙ ˙ tr = HB + tr B H ∂ q˙k 2 ∂ q˙k 2 ∂ q˙k (2.58) % % $ $ 1 1 T T ˙ ˙ HB . + tr B tr Bk H B = k 2 2 In view of the symmetry of matrix H and features (2.57) of the trace, formula (2.58) takes the form: % $ ∂E ˙T . (2.59) = tr Bk H B ∂ q˙k Differentiating the above with respect to time, we obtain: % $ % $ d ∂E ˙ T + tr Bk H B ¨T . ˙ k HB (2.60) = tr B dt ∂ q˙k ˙ k can be calculated as follows: Matrix B  dB0 dBk d ∂B d  B0 Bk = = Bk + B0 dt ∂qk dt dt dt n n " ∂Bk " ˙ 0 Bk + ˙ 0 Bk + B0 q˙j = B Bk,j q˙j , = B ∂qj j=1 j=1

˙k= B

(2.61)

where Bk,j =

∂B ∂Bk = B0 = B0 Bk,j = B0 Bj,k = Bj,k . ∂qj ∂qk ∂qj

The partial derivatives of the kinetic energy with respect to generalised coordinates are given by: & & ' ' ˙ ˙T ∂B ∂ B T ∂E 1 1 ˙ ˙ = tr HB + tr BH , (2.62) ∂qk 2 ∂qk 2 ∂qk which, in view of the symmetry of matrix H, gives: & ' ˙ ∂E ∂B T ˙ = tr HB . ∂qk ∂qk

(2.63)

24

2 Homogenous Transformations

˙ The elements of ∂ B/∂q k according to (2.53) can be calculated as follows:   n n " " ˙ ∂ ˙ ∂B ∂ dB ˙ 0 Bk + = B0 B + = Bj q˙j  = B Bj,k q˙j . (2.64) ∂qk ∂qk dt ∂qk j=1 j=1 ˙ k , and thus: In view of (2.61), the expression on the right of (2.64) equals B $ % ∂E ˙ k HB ˙T . = tr B ∂qk

(2.65)

Therefore the expression ∂E/∂qk equals the first component of the right side of (2.60), and the Lagrange operator defined in (2.39) takes the form: εk (E) =

$ % d ∂E ∂E ¨ . − = tr Bk HB dt ∂ q˙k ∂qk

(2.66)

¨ is calculated by differentiating (2.51): Matrix B ˙ + B B. ¨ ˙ 0B ¨ =B ¨ 0 B + 2B B 0

(2.67)

˙ is defined by (2.52) and B ¨ obtained by differentiating (2.52) with where B respect to time is defined as follows: ( n )

" n n " " ∂Bj dBj ¨ q˙j + Bj q¨j = B= q˙i q˙j + Bj q¨j dt ∂qi j=1 j=1 i=1 =

n " n "

Bj,i q˙i q˙j +

j=1 i=1

n " j=1

Bj q¨j =

n " n " i=1 j=1

Bi,j q˙i˙ q˙j +

n "

Bi q¨i . (2.68)

i=1

Substituting (2.68) into (2.67) and then into (2.66) yields:  T     n n n   "" " ˙ ˙ ¨ εk (E) = tr Bk H B0 B + 2B0 B + Bi,j q˙i q˙j + Bi q¨i      i=1 j=1 i=1 =

n "

aki q¨i + ek

for k = 1, 2, . . . , n,

(2.69)

i=1

where $

ak,i = tr Bk HBT i

%

 T     n n   "" ˙ ˙ ¨   , ek = tr Bk H B0 B + 2B0 B + Bi,j q˙i q˙j ,     i=1 j=1

˙ is defined by (2.52). B

2.5 Potential Energy of Gravity Forces

25

The above equations can be presented in the matrix form: q + e, εq (E) = A¨

(2.70)

where A = A(t, q) = (ak,i )k,i=1,...,n ˙ = (ek )k=1,...,n . e = e(t, q, q)

and

It should be noted that matrix A is symmetrical. Forms (2.69) and (2.70) of the Lagrange operators will be used in the following chapters in formulating the equations of motion of multibody systems.

2.5 Potential Energy of Gravity Forces It is assumed that the coordinates of the centre of mass in the local coordinate system { } are defined by the vector:  T (2.71) rc = xc,1 xc,2 xc,3 1 . ˆ 3 of the global system { } is perpendicular to the earth’s surface, If axis X the potential energy of gravity forces is defined as: Vg = mgxc,3 ,

(2.72)

where m is the mass of the body, g is the acceleration of gravity, xc,3 is the coordinate of the centre of mass, which is a component of the vector defining the position of the centre of mass in the inertial coordinate system, T  rc = xc,1 xc,2 xc,3 1 . If transformation matrix B from the local coordinate system to the global system is known, the vector of coordinates of the centre of mass in the inertial coordinate system can be defined from the relation: (2.73) rc = B rc .   Having defined vector θ3 = 0 0 1 0 , coordinate xc,3 can be calculated as: xc,3 = θ3 B rc ,

(2.74)

and it follows then that the potential energy of gravity equals: Vg = mg θ3 B rc .

(2.75)

The derivative of (2.75) with respect to the generalised coordinate qk yields: ∂Vg = mg θ3 Bk rc ∂qk where Bk is defined in (2.53).

for k = 1, . . . , n,

(2.76)

26

2 Homogenous Transformations

This can be written in matrix form as follows: ∂Vg = g, ∂q

(2.77)

where g = g(q) = (gk )k=1,...,n , gk = ∂qkg = mg θ3 Bk rc . The elements of vector g depend on Bk and thus on the vector of generalised coordinates q. ∂V

2.6 Generalised Forces and Equations of Motion When external forces and moments act on the body under discussion, they have to be considered in equations of motion by means of generalised forces. Unlike in vectors of coordinates, the fourth component of vectors of forces and moments equals zero:   P1  P2   (2.78a) P=  P3  , 

0

 M1  M2   M=  M3  . 0

(2.78b)

In Fig. 2.13 force P acts on the body at point N in the local coordinate system { } . According to (2.43) any vector from the system { } can be defined in the system { } by using transformation matrix B, which depends on the { } Xˆ 3 { }9

N

Xˆ 93

P9

r9

Xˆ 92

r Xˆ 2 Xˆ 91 Xˆ 1

Fig. 2.13. Force P
acting at point N in the local coordinate system { }


2.7 Generalisation of the Procedure

27

generalised coordinates describing the motion of the body, and thus the following can be written: r = B r ,

(2.79a)



P = BP ,

(2.79b)

where B is a transformation matrix defined in (2.43). The generalised force associated with the k-th generalised coordinate is given by (Leyko, 1996): Qk (P) = PT

∂r ∂qk

for k = 1, . . . , n,

(2.80a)

where P is the force expressed in the global system { } and r is the vector of coordinates of the point at which force P acts in the system { }. Bearing in mind (2.79), (2.80a) can be then rewritten as: Qk (P ) = PT BT Bk r .

(2.80b)

If an external moment acting on the body is defined by components of the following vector in the system { } : T  M = M1 M2 M3 0

(2.81)

then, after necessary transformations (Grzego˙zek et al., 2003) the generalised force associated with the k-th generalised coordinate due to moment M can be written in the following form: Qk (M ) = M1

3 " l=1

bl,3 bk,l,2 + M2

3 "

bl,1 bk,l,3 + M3

l=1

3 "

bl,2 bk,l,1 ,

(2.82)

l=1

where (bi,l )i,l=1,...,3 are respective elements of B and (bk,i,l ) k=1,...,n are rei, l=1,...,3 spective elements of Bk . Finally, Lagrange equations lead to the following equations of motion: A¨ q = Q − e − g,

(2.83)

where A, e are defined in (2.70), g is the vector defined in (2.77) and Q = (Qk )k=1,...,n , Qk = Qk (P ) + Qk (M ) are calculated from formulae (2.80) and (2.82).

2.7 Generalisation of the Procedure Further considerations lead to the derivation of the equations of motion of an open kinematic chain of rigid bodies (Fig. 2.14).

28

2 Homogenous Transformations link p ∼(p )

link p -1

q

∼(p -1) (p -1) q ,q =

,q(p ) =

q(p -1) ∼(p )

q

q(p -2) ∼

q (p -1)

link 2 ∼(2)

q {A}

A

Xˆ 3

link 1

∼(1) q(1) =q

A A

q(1) ,q(2) = ∼ q (2)

Xˆ 2

Xˆ 1

Fig. 2.14. Open kinematic chain q *(p) vector of generalised coordinates describing the motion of link p with respect to link p − 1; q(p) vector of generalized coordinates describing the motion of link p with respect to {A}

It is assumed that the vector of generalised coordinates, in terms of which the motion of body p is defined, can be presented in the form:  (p−1) q , (2.84) q(p) = *(p) q where q(p−1) is the vector of generalised coordinates describing the motion of *(p) = body p − 1 preceding body p in the inertial coordinate system {0} and q (p) (p) T [* q1 . . . q* ] is the vector of generalised coordinates defining the relative * np motion of body p in relation to body p − 1. Thus (2.43) can be written as follows: * (p)p* r = B(p)p* r = B(p−1) B r,

(2.85)

where B(p) = B0 (t)B

(p)

B(p−1) = B0 (t)B

= B0 (t)B

(p−1)

(p−1)

* (p) , B

(q(p−1) ),

* (p) (* * (p) = B q(p) ), B p*

r is the coordinate vector in the local coordinate system of body p.

2.7 Generalisation of the Procedure

29

Bearing in mind considerations from Sect. 2.4, the kinetic energy of body p takes the form: ˙ (p) H(p) B ˙ (p)T }, * (p) = 1 tr{B E (2.86) 2 where H(p) is defined by (2.49). It follows from formula (2.84) that the vector of generalised coordinates of body p has: p " *p = n *j (2.87) np = np−1 + n j=1

components, and the Lagrange operators for body p in accordance with (2.70) can be written as follows: * (p) q * (p) ) = A ¨ (p) + * εq(p) (E e(p) , where

(2.88)

(p) * (p) = (* A ak,j )k,j=1,...,np , (p)

* ek )k=1,...,np , e(p) = (* (p)

(p)

(p)T

* ak,j = tr{Bk H(p) Bj },   T    np np   (p) (p) ˙ (p) + " " B(p) q˙(p) q˙(p)  ¨ 0 B(p) + 2B ˙ 0B e*k = tr Bk H(p) B , i,j i j     i=1 j=1 (p)

Bk =

∂B(p) (p)

∂qk

= B0 (t)

(p)

(p)

Bk,j =

∂Bk

(p)

∂qj

= B0 (t)

∂B

(p)

(p)

,

∂qk

∂ 2 B(p) (p)

(p)

.

∂qk ∂qj

Having partitioned matrices and vectors, the above formula can be written in one of the following forms:  (p)   (1)   (p)  ¨ * * (p) * (p) * A * e1 1,1 . . . A1,j . . . A1,p  q  .   .  .  . .   .. .. ..   ..    ..      (p)     (j) * (p) ) =  A q  + * * (p) . . . A * (p) * (p) . . . A ¨ , (2.89) εq(p) (E  v,p * v,1 v,j   ev      .  .. ..    ..  . . .   . . .   .   .  (p) (p) (p) (p) (p) * * * * ep ¨ Ap,1 . . . Ap,j . . . Ap,p * q where

  (p) * (p) = * A a , v, j = 1, . . . , p, v,j nv−1 +l,nj−1 +s l=1,...,* nv nj s=1,...,*   (p) * e(p) = e * , v = 1, . . . , p; v nv−1 +l l=1,··· ,* n
v

30

2 Homogenous Transformations

or

 * (p)

εq(p) (E

)=

(p)

(p)

Ap−1,p−1 Ap−1,p (p)

Ap,p−1

(p)

 

¨ (p−1) q



¨ (p) * q

Ap,p

 +

(p)

ep−1



e(p) p

,

(2.90)

where 

* (p) A 1,1  (p) .. Ap−1,p−1 =  .  (p) * Ap−1,1 + (p) * (p) · · · Ap,p−1 = A p,1

   * (p) * (p) ··· A A 1,p−1 1,p  (p)   .. .. ..  , Ap−1,p =  , . . .    (p) (p) * * · · · Ap−1,p−1 Ap−1,p , (p) * (p) * (p) A p,p−1 , Ap,p = Ap,p ,



(p)

ep−1

 (p) * e1  .  (p)  = e(p) p .  ..  , ep = * (p) * ep−1

When using these formulae, the following has to be assumed: *(1) q(1) = q

n1 = n *1 .

(2.91)

Following the procedure from Sect. 2.5, one can calculate the potential energy of the gravity forces of body p, and then its derivatives with respect to the generalised coordinates, thus obtaining:   ∂ V*g (p) = g * , l l=1,...,np ∂q(p) (p)

(p)

(2.92)

(p)

rc and p* rc is the vector of the centre of mass of where g*l = mp g θ3 Bl p* body p in system {p}. This relation can be rewritten as:  (p)  *1 g  .   ..  (p)   ∂ V*g  (p)  (p) * (2.93) = g =  , * g  v  ∂q(p)  ..   .  (p) *p g where

  (p) *v(p) = g*nv−1 +l g

l=1,...,* nv

or ∂ V*g *(p) = =g ∂q(p) (p)

, for v = 1, . . . , p, 

(p) gp−1 , g(p) p

(2.94)

2.7 Generalisation of the Procedure

where

31

 (p) *1 g  .  (p)  *p(p) . =  ..  , gp = g (p) *p−1 g 

(p)

gp−1

The generalised forces resulting from external forces and moments are calculated as in Sect 2.6, which yields:  (p)  * Q 1  .   ..     * (p) =  * (p) (2.95) Q Q ,  v   ..   .  * (p) Q p

where

  * (p) = Q * (p) ) + Q - (p) ) * (p) * (p) Q v nv−1 +l (P nv−1 +l (M

or



where

(p)

Qp−1

* (p) = Q

(p)

l=1,...,nv

, v = 1, . . . , p,

,

Qp

(2.96)

 * (p) Q 1  .  (p) * (p)  =  ..  , Qp = Qp . * (p) Q 

(p)

Qp−1

p−1

Finally, the equations of motion of body p can be written in analogy to (2.83) as: * (p) q ¨ (p) = * f (p) , (2.97) A (p) (p) (p) (p) (p) (p) (p) (p) * * * * * , A ,Q ,* * = Q −* e −g e ,g are defined by (2.88), where f (2.93) and (2.95), respectively. Equation (2.97) describe the motion of body p and demonstrate how this motion depends on the generalised coordinates of the preceding links (* q(1) · · · (p−1) (p) * . They can be written in an equivalent * ) and its own coordinates q q partitioned form:    (p)  (p) (p) ¨ p−1 q Ap−1,p−1 Ap−1,p f   p−1  ,  (2.98) = (p) (p) (p) (p) ¨ * q Ap,p−1 Ap,p fp (p)

(p)

(p)

(p)

(p)

(p)

(p) where f p−1 = Qp−1 − ep−1 − gp−1 , f p = Qp − e(p) p − gp . These equations take into account the kinetic and potential energies of

body p and force P (p) and moment M (p) acting on this link.

32

2 Homogenous Transformations

When the system of p bodies is considered, the kinetic energy and the potential energy of gravity forces of all links have to be calculated as follows: E (p) = Vg(p) =

p " i=1 p "

* (i) = E (p−1) + E * (p) , E

(2.99a)

V*g(i) = Vg(p−1) + V*g(p) .

(2.99b)

i=1

Assuming that the equations of motion of p−1 bodies, which take into account

(p−1) energies T (p−1) and Vg as well as forces P (1) · · · P (p−1) and moments

M (1) · · · M (p−1) , take the form: ¨ (p−1) = f (p−1) A(p−1) q

(2.100)

in view of (2.98) this leads to the equations of motion of all p links (together with link p): ¨ (p) = f (p) , (2.101) A(p) q where



(p)

A(p−1) + Ap−1,p−1 A

 A(p) = 

(p) Ap,p−1



(p)

f (p−1) + f p−1

f (p) = 

(p)

(p)

p−1,p  (p)



 ,

Ap,p

.

fp

These equations can also be written as:     (p)  (p) (p)  ¨ (1) (p) * q A1,1 · · · A1,j · · · A1,p f1    . .. ..  ..   ..    .. .   . .  .         (p)  (p)  (p) (p)   ¨ (j)  = ,   Ai,1 · · · Ai,j · · · Ai,p   q f *   i      .. ..   ..   ..   ..  . . .  .   .  (p) (p) (p) (p) fp ¨ (p) Ap,1 · · · Ap,j · · · Ap,p * q where p "

(p)

Ai,j =

* (l) , A i,j

l=max{i,j} (p)

fi

=

p "

(l) * fi ,

l=i

* (l) is defined in (2.98), A i,j (l)

*i is defined in (2.93), g * (l) is defined in (2.95). Q i

(2.102)

2.7 Generalisation of the Procedure

33

Equations (2.101) and (2.102) demonstrate the equations of motion of a kinematic chain of p bodies and how connecting a new body p to the kinematic chain of p − 1 bodies changes the equations of motion of the preceding links. It should be noted that the connection of a new body p to a kinematic chain of p−1 bodies causes not only an increase in the vector of generalised coordinates (p−1)

q (p−1) (p) to q = and an increase in the system of equations from q *(p) q of motion, but also essentially modifies the equations of motion of preceding bodies. This is a result of using of joint coordinates, which means that the motion of body p is defined with respect to the preceding body p − 1. This is also an important feature of models created by means of joint coordinates, which differentiates them from those obtained using absolute coordinates.

3 The Rigid Finite Element Method

The idea of the rigid finite element method is to discretise a system of many links, including flexible links, and replace it with a system consisting of rigid finite elements (rfes) which are connected with each other and with a base by means of spring–damping elements (sde). The position of the system is described by generalised coordinates which are displacements of the elements resulting from discretisation (Fig. 3.1). Division of the system considered into rfes and sdes may be natural or virtual Kruszewski et al. (1975), or both; in such a case part of the system is naturally divided, while another part is virtually divided into rfes and sdes. Natural division occurs when we can isolate rigid parts and flexible parts (for which mass and dimensions are negligible) of the system. Rigid parts are treated then as rigid bodies and are called rigid finite elements (rfe), while flexible parts are treated as spring and damping elements and are called spring–damping elements (sde). As an example of natural division one can consider a floating platform, used in shipbuilding, in which a generating set is elastically placed on a platform by means of a rubber cushion with spring and damping features. The platform, in turn, is mounted on a hull by means of a spring washer (Fig. 3.2). In this system both the generating set and the platform can be treated as rfes while all mounting elements are sdes. Another example is a machine tool in which all units can be treated as rigid bodies and slide connections as flexible elements (Fig. 3.3). The virtual division takes place when the system considered is a material continuum, in which mass, spring and damping features are distributed in a continuous manner (Fig. 3.4a). Discretisation of the material continuum is carried out in two stages. First, the continuum is divided into relatively simple elements with finite dimension, and their spring and damping features are concentrated at one point (the sde is obtained in this way); this is called primary division. In the secondary division, rfes are isolated between sdes from the primary division and in this way a system of rfes connected by sdes

36

3 The Rigid Finite Element Method Xˆ 93

{ }9 Xˆ 92 rfe

sde

Xˆ 91

Aˆ

X3

{A}

AX ˆ

2

Aˆ

X1

Fig. 3.1. Division of a system into rfes and sdes with a common moving base {A} and the local coordinate system {}’ of a rfe

Fig. 3.2. Generating set mounted on a floating platform

is obtained. Such a discretisation can be carried out for beams or plates and this is presented in Fig. 3.4. The principles for discretisation of flexible links into rfes and sdes, methods of defining their parameters, and the choice of the coordinate systems will be discussed in this chapter. We will also formulate appropriate transformation

3.1 Division of the Flexible Link

37

Fig. 3.3. Division of a machine tool into rfes and sdes

formulae and equations of motion of a flexible link discretised by means of the rfe method, as well as those of the whole kinematic chain to which the flexible link is appended.

3.1 Division of the Flexible Link into Rigid Finite Elements and Spring–Damping Elements Let us first consider the simple case when link p is a prismatic beam. Figure 3.5 presents the model considered. First, the beam of length L(p) is divided into m(p) sections of equal length ∆(p) . Flexible features of elements are concentrated in sdes, which are placed in the middle of the elements of length ∆(p) (Fig. 3.5a). In the secondary division a flexible link is replaced by m(p) + 1 rfes numbered from 0 to m(p) , and m(p) sdes numbered from 1 to m(p) (Fig. 3.5b). The methods of calculating parameters characterising rfes and sdes will be presented in Sect. 3.6. In the general case, however, the link p can be a body with a complex geometrical shape (curved beam with a changing cross-section, plate, shell, or any combination of those). Therefore, we discuss later the case in which flexible link p is divided into m(p) + 1 rigid elements connected with κ(p) sdes.

38

3 The Rigid Finite Element Method

(a)

element with finite dimensions

sde in which flexible and damping features are concentrated (b)

sde

X⬘3

sde sde

rfe

rfe

X⬘2 X⬘1 A

X3

A A

X2

X1

Fig. 3.4. Virtual division of a beam and a plate into rfes and sdes: (a) primary division into elements with finite dimension in order to determine parameters of sdes; (b) secondary division: isolation of rfes connected by sdes defined in primary division

L(p) (a)

D(p)

D(p)

(b) rfe(p,0) rfe(p,1)

sde(p,1) sde(p,2)

D(p)

D(p)

rfe(p,i)

sde(p,i) sde(p,i+1)

D(p)

rfe(p,m (p) )

sde(p,m(p) )

Fig. 3.5. Discretisation of a flexible link: (a) primary division; (b) secondary division

3.1 Division of the Flexible Link

E

(p)

{p,m }

(p)

E3(p,m )

E3(p,m

39

(p)

)

C (p)(p) E

E

m

E3(p,i)

{p,i} j3(p,i)

E

Ci(p)

(a)

E1(p,m

E2(p,i)

E

(p)

)

j2(p,i)

j1(p,i)

E1(p,i)

rfe (p,i) 0

{p,0}

X3(p,0)

(p,i)

S E 3

X2(p,0)

{p,k}Y Y3(p,k) Y

E Y (p,kr) 3

E {p,k }Y l

E Y (p,kl) 3

E {p,k 0 E

0 E

r}

Y2(p,k) E

(p,i)

S1

E

Y2(p,kr)

Y2(p,kl)

(p,i)

S2

X1(p,0) link p -1

(b)

E Y (p,kl) 1

sde (p,k) rfe (p,kr) rfe (p,k1)

E

Y1(p,kr) Y1(p,k)

Fig. 3.6. Link p in the undeformed state: (a) rfes (b) sde (p, k) connecting rfe (p, kl ) with rfe (p, kr )

Let us consider the system presented in Fig. 3.6, which is link p divided into rfes and sdes in the undeformed state. The elements of the calculational model (rfes and sdes) are in the reference position defined by a lack of load (external forces and moments and inertial forces) acting on link p. It is assumed that rfe (p, 0) is the base, in relation to which the position of the rest of elements is described. The position of rfe (p, i) of the undeformed link with respect to rfe (p, 0) is defined by the position of the coordinate system E {p, i} with respect to the coordinate system {p, 0}, and thus by the transformation matrix with constant components: 0 (p,i) 0 (p,i)

Θ 0 (p,i) Es , (3.1) = E ET 0 1

40

3 The Rigid Finite Element Method

where 0E Θ(p,i) is the matrix of cosines of the system E {p, i} with respect to the axes of system {p, 0} and 0E s(p,i) is the vector of coordinates of the origin of system E {p, i} in system {p, 0}. It is assumed that the axes of coordinate system E {p, i} are the principal central axes of inertia of rfe (p, i). If the axes of systems E {p, i} and {p, 0} are parallel, then matrix 0E Θ(p,i) is the identity matrix. The coordinate system {p, i} is assigned to each rfe in such a way that its axes coincide with the axes of system E {p, i} when link p is undeformed. System {p, i} moves together with rfe (p, i) when link p undergoes deformation. Its position in the reference system E {p, i} is defined by the generalised coordinates of rfe (p, i), which are the components of the vector: (p,i)

x (p,i) ˜ q = , (3.2) ϕ(p,i) (p,i) (p,i) (p,i)

where x(p,i) = [x1 x2 x3 ]T is the vector of coordinates of the origin of system {p, i}, rigidly assigned to rfe (p, i), with respect to system E {p, i}, (p,i) (p,i) (p,i) ϕ(p,i) = [ϕ1 ϕ2 ϕ3 ]T is the vector of the rotation angles of the axes of system {p, i} with respect to the axes of the reference system E {p, i}. (p,i) A geometrical illustration of coordinates q˜j (j = 1, . . . , 6) is presented in Fig. 3.7. Since the axes of the system {p, i} are chosen as the principal central axes of inertia of rfe (p, i), mass and inertial features of the rfe are defined by four parameters:

{p,i }

X3(p,i ) j3(p,i) X2(p,i ) (p,i)

j2 E

{p,i } E (p,i ) 3

Ci(p)

(p,i)

j1

X1(p,i )

E2(p,i )

Ci(p)

x3(p,i) x1(p,i) x2(p,i) E1(p,i ) (p,i)

(p,i)

(p,i)

Fig. 3.7. Coordinate system {p, i} rigidly assigned to rfe (p, i), x1 , x2 , x3 (p,i) (p,i) (p,i) are the coordinates of the centre of mass in E {p, i} and ϕ3 , ϕ2 , ϕ1 are the E Euler angles ZY X with respect to the axes of system {p, i}

3.1 Division of the Flexible Link

41

m(p,i) − mass, (p,i)

J2

ˆ (p,i) , − mass moment of inertia with respect to axis X 1 ˆ (p,i) , − mass moment of inertia with respect to axis X

(p,i) J3

ˆ (p,i) . − mass moment of inertia with respect to axis X 3

J1

(p,i)

2

(3.3)

The transformation matrix from the system of rfe (p, i) to the reference system E {p, i} can be presented in the form: (p,i) (p,i)

R x T(p,i) = , (3.4) 0 1 where x(p,i) is defined in (3.2),  (p,i) (p,i) (p,i) (p,i) (p,i)

(p,i) (p,i) c1   s(p,i) c(p,i) s(p,i) s(p,i) s(p,i) + c(p,i) c(p,i)  3 2 3 2 1 3 1  (p,i) (p,i) (p,i) −s2 c2 s1

c3

R(p,i) =

(p,i)

cj

c2

(p,i)

= cos ϕj

;

c3

(p,i)

sj

s2

s1

(p,i)

= sin ϕj

− s3



(p,i) (p,i) (p,i) (p,i) (p,i) s2 c1 + s3 s1  (p,i) (p,i) (p,i) (p,i) (p,i)  s3 s2 c1 − c3 s1 ,  (p,i) (p,i) c2 c1

c3

for j = 1, 2, 3.

The transformation matrix from the local system of rfe (p, i) to the system of rfe (p, 0), in view of (3.1) and (3.4), takes the following form: * (p,i) =0 T(p,i) T(p,i) . B E

(3.5)

As a result of the displacement of the rfe, which is caused by external and inertial forces, sdes undergo deformations. Sde (p, k) connecting rfe (p, kl ) with rfe (p, kr ) is presented in Fig. 3.6b. When link p is undeformed, the position of sde (p, k) is characterised by the coordinate system {p, k}Y . The axes of this system are called the principal axes of the sde, which means that forces acting on the sde along any of the axes of system {p, k}Y cause deformations of the element in the direction of this axis (direction of the acting force). Similarly, moments acting about the principal axes of the sde cause rotations only about those axes. On the other hand, the coordinate systems E {p, kl }Y and E {p, kr }Y are rigidly assigned to rfes (p, kl ) and (p, kr ) respectively, and they coincide with the reference system {p, k}Y when link p is undeformed. The position of sde (p, k) in the local coordinate systems of rfes (p, kl ) and (p, kr ) is defined by the transformation matrices with constant components:  E (p,kl ) E (p,kl ) YΘ Ys E (p,kl ) = , (3.6a) YT 0 1  E (p,kr ) E (p,kr ) Θ s Y Y E (p,kr ) = , (3.6b) YT 0 1

42

3 The Rigid Finite Element Method

(p,kl ) E (p,kr ) where E ,YΘ are cosine matrices of the principal axes of sde (p, k) YΘ (p,kl ) E (p,kr ) , Ys in the local coordinate systems of rfes: E {p, kl }, E {p, kr }, E Ys E are coordinates of sde (p, k) in coordinate systems {p, kl } and E {p, kr }. Due to the load imposed on the system, rfes (p, kl ) and (p, kr ) undergo displacements which are defined by the vectors: (p,k )

x l *(p,kl ) = q , (3.7a) ϕ(p,kl ) (p,k )

x r *(p,kr ) = . (3.7b) q ϕ(p,kr )

As a result, reciprocal translation and rotation of systems E {p, kl }Y and {p, kr }Y shown in Fig. 3.8 take place. The coordinates of the origins of systems E {p, kl }Y and E {p, kr }Y , after *(p,kl ) and rfe (p, kr ) by vector q *(p,kr ) , with displacement of rfe (p, kl ) by vector q respect to the reference systems of rfes, can be calculated from the formulae:   E (p,kl ) (p,kl ) y = R(p,kl ) ϕ(p,kl ) E + x(p,kl ) , (3.8a) Ys

E

E

  y(p,kr ) = R(p,kr ) ϕ(p,kr )

E (p,kr ) Ys

+ x(p,kr ) .

(3.8b)

In view of (3.6) and (2.22), and since it has been assumed that in the undeformed state the coordinate systems {p, k}Y , E {p, kl }Y and E {p, kr }Y coincide, the vectors from (3.8) can be presented in relation to system {p, k}Y as:

{p,k}Y Y3(p,k)

EY (p,kl ) E{p,k }Y 3 l (p,kr) jY,3

EY (p,kr ) 3

E

{p,kr}Y

(p,kr) jY,3

EY (p,kl ) 2 (p,kr) jY,1

EY (p,kr ) 2

(p,kr) jY,2

(p,kl ) jY,2

(p,kl) jY,1

y3(p,kl )

EY (p,kr ) 1

y3(p,kr ) Y2(p,k )

EY (p,kl ) 1

y1(p,kl ) y1(p,kr )

y2(p,kl ) Y1(p,k)

y2(p,kr ) Fig. 3.8. Deformation of sde k of link p

3.1 Division of the Flexible Link

y(p,kl ) = y(p,kr ) =

+ +

E (p,kl ) YΘ

,T +

E (p,kr ) YΘ

(p)

(p,kl ) R(p,kl ) E + xk l Ys

,T +

,

(p)

(p,kr ) R(p,kr ) E + xk r Ys

−

,

+

−

E (p,kl ) YΘ

+

,T

E (p,kr ) YΘ

43

E (p,kl ) , Ys

,T

(3.9a)

E (p,kr ) . Ys

(3.9b)

Therefore: y(p,kl ) = y(p,kr ) =

+ +

E (p,kl ) YΘ

,T +

E (p,kr ) YΘ

, (p,kl ) (p,kl ) s + x (R(p,kl ) − I) E , Y

,T +

, (p,kr ) (R(p,kr ) − I) E + x(p,kr ) . Ys

(3.10a) (3.10b)

Deformations of sde (p, k), caused by displacements (3.10), occur as a result of forces with the potential: (p,k)

VY

=

3 ,2 1 " (p,k) + (p,kl ) (p,k ) cY,j yj − yj r , 2 j=1

(3.11)

(p,k)

where cY,j , (j = 1, 2, 3) are coefficients of translational stiffness of sde (p, k). Equation (3.11) can now be presented in the matrix form: (p,k)

VY

=

1 + (p,k) ,T (p,k) ∆y CY ∆y(p,k) , 2

(3.12)

where ∆y(p,k) = y(p,kl ) − y(p,kr ) ,   (p,k) cY,1 0 0   (p,k) (p,k) = CY 0   0 cY,2 . (p,k) 0 0 cY,3 The function of dissipation of translational energy of sde (p, k) caused by displacements of rfes (p, kl ) and (p, kr ) can be written in a similar way: (p,k)

WY

=



where (p,k)

DY (p,k)

1 + (p,k) ,T (p,k) ∆y˙ DY ∆y˙ (p,k) , 2

 = 

(p,k)

(3.13)



dY,1

0

0

dY,2

0

0

0

dY,3

(p,k)

0 (p,k)

 , 

dY,j , (j = 1, 2, 3) are coefficients of translational damping of sde(p, k). Displacements of rfes (p, kl ) and (p, kr ) also cause rotations of the axes of the coordinate systems E {p, kl }Y and E {p, kr }Y with respect to the reference

44

3 The Rigid Finite Element Method

system {p, k}Y of rfe (p, k). The rotation angles of rfes (p, kl ) and (p, kr ), which are the result of displacements (3.7), can be written in system {p, k}Y as follows: ,T + (p,k ) (p,kl ) ϕY l = E ϕ(p,kl ) , (3.14a) YΘ (p,kr )

ϕY

+ =

E (p,kr ) YΘ

,T

ϕ(p,kr ) .

(3.14b)

Thus, the potential energy of rotational deformation of sde (p, k) can be presented in the form: ,2 1 " (p,k) + (p,kl ) (p,k ) = cϕ,j ϕY,j − ϕY,j r , 2 j=1 3

Vϕ(p,k)

(3.15)

(p,k)

where cϕ,j (j = 1, 2, 3) are coefficients of rotational stiffness of sde (p, k) or in the matrix form: ,T 1+ (p,k) ∆ϕ(p,k) Cϕ Vϕ(p,k) = ∆ϕ(p,k) , (3.16) 2 where (p,k )

(p,kr )

∆ϕ(p,k) = ϕY l − ϕY  (p,k) 0 cϕ,1  (p,k) (p,k) Cϕ =  0 cϕ,2 0

0

, 

0

 0 .

(p,k)

cϕ,3

Similarly, the function of rotational energy dissipation of sde (p, k) can be written in the following form: ,T 1+ (p,k) ˙ (p,k) Dϕ ˙ (p,k) , ∆ϕ ∆ϕ (3.17) Wϕ(p,k) = 2 where

  (p,k) Dϕ =

(p,k)

(p,k)

dϕ,1

0

0

dϕ,2

0

0

0

dϕ,3

(p,k)

0

  ,

(p,k)

dϕ,j are coefficients of rotational damping of sde (p, k). It should be noted that the numbers m(p) (m(p) + 1 is the number of rfes of link p) and κ(p) (number of sdes of link p) are arbitrary, which means that rigid elements can be connected with any number of sdes. The only simplifying assumption made is that sde (p, k) connects only two rigid elements. This assumption could be easily omitted, yet it seems natural in as far as the (p,k) (p,k) number of sdes is arbitrary and not all coefficients cx,j , cϕ,j of sde (p, k) have to be nonzero.

3.2 Kinetic Energy of the Flexible Link

45

3.2 Kinetic Energy of the Flexible Link The generalised coordinates describing the motion of rfe (p, 0) with respect to the link preceding the flexible link depend on the joint between link p − 1 and p, and according to (2.84) can be presented as components of the vector: + ,T (p) (p) *(p,0) = q*1 , . . . , q*n˜ p,0 , (3.18) q where n *p,0 is the number of degrees of freedom of rfe (p, 0) in relative motion with respect to link p − 1. The transformation matrix from system {p, 0} to the coordinate system of link p − 1 can be written in the form: * (p,0) = B



n ˜ p,0

j−1 (p) . jT

(3.19)

j=1

The vector of generalised coordinates of rfe (p, 0) and the transformation matrix from system {p, 0} to global systems can be written as:  (p−1) q , (3.20a) q(p,0) = *(p,0) q * (p,0) , B(p,0) = B(p−1) B

(3.20b)

where np,0 = np−1 + n *p,0 is the number of elements of vector q , q(p−1) (p−1) and B are respectively the vector of generalised coordinates and transformation matrix from the coordinate systems of link p − 1 preceding link p to the global system. Proceeding as in Sect. 2.7 according to (2.90), one can write: (p,0)

* (p,0) ) = A ¨ (p,0) + e(p,0) q εq(p,0) (E    (p,0)  (p,0) (p,0)  (p−1) Ap−1,p−1 Ap−1,(p,0) q e    p−1  =  (p,0) +  , (p,0) (p,0) (p,0) A(p,0),p−1 A(p,0),(p,0) * q e (p,0)

(3.21)

(p,0)

* (p,0) is the kinetic energy of rfe (p, 0), εq(p,0) is the Lagrange operator. where E The remaining rfes of link p, which are rfe (p, 1) to rfe (p, m(p) ), are treated as elements of the kinematic chain appended to rfe (p, 0). The vectors of the generalised coordinates and transformation matrices to the global coordinate system can be written as follows:  (p−1)  (p,0)

q q *(p,0)  , (3.22a) q(p,i) = = q (p,i) (p,i) * q * q (p,i) (p,0) * (p,i) * (p,0) B * (p,i) , B =B = B(p−1) B (3.22b) B

46

3 The Rigid Finite Element Method

* (p,i) is defined in (3.5) and i = 1, . . . , m(p) . *(p,i) is defined in (3.2), B where q The kinetic energy of rfe (p, i) can be calculated from the formula: % $ ˙ (p,i)T . ˙ (p,i) H(p,i) B * (p,i) = 1 tr B (3.23) E 2 Since the axes of the coordinate system assigned to rfe (p, i) are the principal central axes, matrix H(p,i) is diagonal and its components are as follows: , 1 + (p,i) (p,i) (p,i) (p,i) (p,i) , (3.24a) −J1 + J2 + J3 h1,1 = h1 = 2 , 1 + (p,i) (p,i) (p,i) (p,i) (p,i) h2,2 = h2 = , (3.24b) J1 − J2 + J3 2 , 1 + (p,i) (p,i) (p,i) (p,i) (p,i) h3,3 = h3 = + J2 − J3 , (3.24c) J1 2 (p,i)

h4,4 = m(p,i) , (p,i)

(p,i)

(3.24d)

(p,i)

where J1 , J2 and J3 are mass moment of inertia of rfe (p, i) with respect to the axes of coordinate system {p, i}. Bearing in mind that vector q(p,i) has: np,i = np−1 + n *p,0 + 6 = np,0 + 6

(3.25)

components, and proceeding analogically as in Sect. 2.7, one can write: ˜ (p,i) q ¨ (p,i) + * εq(p,i) = A e(p,i) ,

(3.26)

where (p,i) * (p,i) = (˜ al,s )l,s=1,...,np,i , A (p,i)

* e(p,i) = (˜ el )l=1,...,np,i , % $ T (p,i) (p,i) , * al,s = tr Bl H(p,i) B(p,i) s (p,i) e˜l

=

np,i np,i " "

% $ (p,i) (p,i)T (p,i) q˙s(p,i) q˙j , tr Bl H(p,i) Bs,j

i = 0, 1, . . . , m(p) ,

s=1 j=1

or

 εq(p,0) =  

(p,0)

  (p,0)  e ¨ (p−1) q   p−1  , + (p,0) (p,0) ¨˜ (p,0) q A(p,0),(p,0) e(p,0) (p,0)

Ap−1,p−1 Ap−1,(p,0) (p,0)

A(p,0),p−1

(3.27a)



 (p−1)   (p,i)  ep−1 ¨ q    (p,i)   ¨ (p,0)   (p,i) (p,i) (p,i)  =  A(p,0),p−1 A(p,0),(p,0) A(p,0),(p,i)   q ˜ e(p,0)  + ,    (p,i) (p,i) (p,i) (p,i) ¨˜ (p,i) q e(p,i) A(p,i),p−1 A(p,i),(p,0) A(p,i),(p,i) (p,i)

(p,i)

(p,i)

Ap−1,p−1 Ap−1,(p,0) Ap−1,(p,i)

εq(p,i)

(3.27b)

3.2 Kinetic Energy of the Flexible Link

47

where (p,i)

(p,i)

(p,i)

(p,i)T

Ap−1,p−1 = (˜ al,s )l,s=1,...,np−1 , (p,i)

Ap−1,(p,0) = A(p,0),p−1 = (˜ al, np−1 +s ) l=1,...,n

, p−1

s=1,...,˜ np,0 (p,i) Ap−1,(p,i)

=

(p,i)T A(p,i),p−1

(p,i)

= (˜ al, np,0 +s ) l=1,...,n

, p−1

s=1,...,6 (p,i) A(p,0),(p,0)

(p,i)

= (˜ anp−1 +l, np−1 +s )l,s=1,...,˜np,0 , (p,i)T

(p,i)

(p,i)

A(p,0),(p,i) = A(p,i),(p,0) = (˜ anp−1 +l, np,0 +s ) l=1,...,˜n

,

p,0

(p,i) A(p,i),(p,i)

 =

(p,i) a ˜np,0 +l, np,0 +s

  (p,i) (p,i) ep−1 = e˜l

l=1,...,np−1

  (p,i) (p,i) e(p,0) = e˜np−1 +l   (p,i) (p,i) e(p,i) = e˜np,0 +l

l,s=1,...,6

,

,

l=1,...,˜ np,0

l=1,...,6

s=1,...,6



,

. (p,i)

It is important to note that calculation of quantities a ˜l,s

˜ (p,i) is very and h l (p,i)

(p,i)

time-consuming since it requires calculations of matrices Bl , Bl,j . Products of matrices and their traces also have to be calculated, which considerably lengthens the time of computations. Some of those calculations can be eliminated by using features of transformation matrices B(p,i) , and examples are presented later. Having taken into account (3.22b) and (3.5), the following can be written: (p,i) (3.28) B(p,i) = B0 T(p,i) , where (p,i) B0

=

(p,i) B0

 q

(p,0)



 =

R(p,i) x(p,i) . 0 1

T(p,i) = T(p,i) (˜ q(p,i) ) =

(p,i)

Θ0 = 0

B(p,0) 0E T(p,i)

(p,i)

s0 1

,

Let us assume that the described choice of generalised coordinates of links (p,i) 1 to p − 1 and rfe (p, 0) ensure orthonormality of matrix Θ0 , which means that: ,T + (p,i) (p,i) Θ0,l Θ0,j = δl,j for l, j = 1, 2, 3, (3.29)

48

3 The Rigid Finite Element Method (p,i)

(p,i)

where Θ0,l is the l-th column of matrix Θ0 and δl,j is the Kronecker delta. The derivatives of matrix T(p,i) with respect to the generalised coordinates ˜ (p,i) , are as follows: of rfe (p, i), which are the components of vector q  (p,i)  0 bl   if l = 1, 2, 3,   0 0 ∂T(p,i) (p,i) = =  (p,i) Tl (3.30) (p,i)  R ∂ q˜l  0  l  if l = 4, 5, 6,  0 0 where (p,i)

b1

(p,i)

b2

(p,i)

b3

(p,i)

R4

(p,i)

R5

 T = 100 ,  T = 010 ,  T = 001 ,   (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) 0 c1 s2 c3 + s1 s3 −s1 s2 c3 + c1 s3     =  0 c1(p,i) s2(p,i) s3(p,i) − s1(p,i) c3(p,i) −s1(p,i) s2(p,i) s3(p,i) − c1(p,i) c3(p,i)  ,   (p,i) (p,i) (p,i) (p,i) 0 c1 c2 −s1 c2  (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i)  −s c3 s1 c2 c3 c1 c2 c3  2   (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i)  =  −s2 s3 s1 c2 s3 c1 c2 s3  ,   (p,i) (p,i) (p,i) (p,i) (p,i) −c2 −s1 s2 −c1 s2 

(p,i)

R6



(p,i) (p,i) s3

−c2

=  c(p,i) c(p,i) 2 3

(p,i) (p,i) (p,i) s2 s3

−s1

(p,i) (p,i) (p,i) s2 c 3

s1

0

Bearing in mind that matrix H

(p,i)

H

(p,i)

h1

0

  0 =   0

(p,i)

h2

(p,i)

(p,i)

0

0

0

0

, h2

 .

0

   (p,i)  0 = H ,  0 m(p,i) 

(3.31)

(p,i)

}, it can be proved that:  (p,i) + ,T Dl,j 0 (p,i) (p,i) (p,i) = Tl H , Tj 0 0

= diag{h1

(p,i) (p,i) s3

− s1



m(p,i)

0 (p,i)

(p,i) (p,i) (p,i) s2 c 3

c1

(p,i) (p,i) c3

+ s1

can be presented in the form:

h3

0

(p,i) (p,i) (p,i) s2 s3

−c1

0

(p,i)

0

0 where H

(p,i) (p,i) s3

− c1

0 (p,i)



(p,i) (p,i) c3

− c1

, h3

(3.32)

3.2 Kinetic Energy of the Flexible Link

49

where

(p,i)

= 0 for l = 1, 2, 3, j = 4, 5, 6 and l = 4, 5, 6, j = 1, 2, 3, + ,T (p,i) (p,i) (p,i) Rj = Rl H for l, j = 4, 5, 6.

(p,i)

Dl,j

(p,i)

,T

= m(p,i) bl

Dl,j

(p,i)

+

(p,i)

Dl,j

bj

for l, j = 1, 2, 3,

Taking into account (3.32) and (3.28), one can calculate:   + ,T . + ,T . (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) (p,i) B0 Tj tr B0 Tl H = tr Θ0 Dl,j Θ0 . (3.33) (p,i)

Since matrix Θ0 following occurs:

is orthonormal, and thus relations (3.29) are fulfilled, the

 $ + ,T . % (p,i) (p,i) (p,i) (p,i) tr Θ0 Dl,j Θ0 = tr Dl,j .

(3.34)

Thus, in view of (3.27), (3.28) and (3.34), we obtain:  + ,T . (p,i) (p,i) (p,i) (p,i) a ˜np,0 +l,np,0 +j = tr B0 Tl H(p,i) B0 Tj % $ (p,i) , = tr Dl,j

l, j = 1, . . . , 6.

(3.35) (p,i)

Expressions obtained after calculating the traces of matrices Dl,j

are pre(p,i)

sented in Table 3.1. It should be noted that only elements of matrix A(p,i),(p,i) with subscripts (4, 6), (6, 4), (5, 5), (5, 6), (6, 5) and (6, 6) are variables since (p,i) (p,i) they depend on ϕ1 and ϕ2 . The remaining elements are constant and most of them equal zero. Of all 36 elements only six change. (p,i) Proceeding in a similar way, it can be shown that many components e˜np,0 +l occurring in (3.26) equal zero. Table 3.2 contains nonzero values of the traces of matrix products which are connected with rfe (p, i). Thus, for l = 1, . . . , 6 and s, j = 1, . . . , 6, the following should be assumed:  expression from Table 3.2,     + ,T .  if the respective combination (p,i) (p,i) tr Bnp,0 +l H(p,i) Bnp,0 +s, np,0 +j =  of l, s and j is specified there,    0, otherwise (3.36) It is important to note that for 216 possible combinations of subscripts (l, s, j), according to (3.36), only in 16 cases is the trace nonzero. Taking this into account considerably reduces the number of operations necessary for the generation of the equations of motion for flexible links.

1

m 0 0 0

0

0

l\j

1 2 3 4

5

6

(p,i)

0

0

0 m(p,i) 0 0

2

0

0

0 0 m(p,i) 0

3

(p,i)



(p,i)

− h2

0

(p,i)

+ h3

0 0 0 (p,i) (p,i) h2 + h3

4

 (p,i)

s2

(p,i)

(p,i)

,

,2

(p,i)

s1 (p,i)

c1 − h2

+

+

s1

(p,i) (p,i) (p,i) c2 c1

h3

+

(p,i)

(p,i)

+ h2

+h3

(p,i)

h1

0 0 0 0

5

,2

Table 3.1. Elements a ˜np,0 +l, np,0 +j of matrix A(p,i),(p,i)

+

(p,i)

h3

,

(p,i)

− h2

,

(p,i)

+h3

(p,i)

+h2

s1

+

+

(p,i)2

c1

(p,i)2

s1

(p,i) (p,i) (p,i) c2 c1 2 (p,i) (p,i) h1 c2

+

(p,i)2

s2

(p,i)2

s2

0 0 0 (p,i) (p,i) (p,i) −(h2 + h3 ) s2

6

(p,i)2

+ s1

(p,i)2

+ c1

,

,

50 3 The Rigid Finite Element Method

3.2 Kinetic Energy of the Flexible Link

  l

s

j

4

5

5

5

6

4

5

6

6

4

5

5 5

4

4

6

+ +

6

4

6

6

4

5

− h3



(p,i)

(p,i)

+

(p,i) s1

(p,i)

h1

4

6

6 6

4

5

5

5

6

+ +

6

,

(p,i)

− h3

(p,i)

(p,i)

, 2

c1

+

(p,i)

c2

,2

(p,i)

c2

(p,i) (p,i) c1

(p,i) h3

+

+

(p,i)

,2

s1

,2

,

+

(p,i) c1

,

(p,i) h2

, 2

(p,i)

− h3

(p,i)

+

+ h3

(p,i) (p,i) c1

s1

+

+

s1

(p,i)

(p,i)

h2

(p,i) (p,i) c1

c1

− h2

(p,i) −h1

6

+

(p,i)

h3

(p,i)

+ h3

s1

(p,i)



,2

,2

− h2

,T .

(p,i) (p,i) c1

,



(p,i)

4

(p,i)

,T .

s1

(p,i)

(p,i)

h2

,

s1

− h3

− h2 5

+

(p,i)

h2

(p,i) h2

6

6

(p,i)

− h2



5

5

(p,i)

h3

(p,i)

6

+

(p,i)

− h2

4

(p,i)

Bnp,0 +l H(p,i) Bnp,0 +s, np,0 +j

tr

+

+

(p,i)

Bnp,0 +l H(p,i) Bnp,0 +s, np,0 +j

Table 3.2. Traces of matrix products tr

51

(p,i)

c2

+

(p,i)

(p,i)

, 2

s1

+

(p,i)

, 2

c1

(p,i) (p,i) c2

s2 (p,i)

c2

,2

c2

(p,i) (p,i) (p,i) c1 s2

s1

+

(p,i) s1

,2 +

(p,i) h3

+

(p,i) c1

, 2

(p,i) (p,i) c2

s2

5

The kinetic energy of the flexible link p divided into m(p) +1 rfes is defined as follows: (p) m " (p) E = E (p,i) . (3.37) i=0

If we present the vector of generalised coordinates of link p in the form:    q(p−1) (p,0) q  (p,0)  q(p) = ˜ = q (3.38) , (p) ˜ q ˜ (p) q where

+ ,T T (p) T ˜ (p) = q q , ˜ (p,1) . . . q ˜ (p,m )

52

3 The Rigid Finite Element Method

in view of (3.21) and (3.26) the following can be written: 

(p)

(p)

   (p) (p) εq(p) E (p) =   A(p,0),p−1 A(p,0),(p,0) (p)

(p)

Ap,p−1

Ap,(p,0)



  (p)  ¨ (p−1) q ep−1   ¨ (p,0)   (p)  (p) q  + e  ˜ A(p,0),p     (p,0)  , (p) ¨˜ (p) (p) q Ap,p ep (3.39) (p)

Ap−1,p−1 Ap−1,(p, 0) Ap−1,p

where (p) Ap−1,p−1

=

(p) m "

(p,i)

Ap−1, p−1 ,

i=0 (p) Ap−1,(p,0) (p) Ap−1,p (p) A(p,0),(p,0)

=

(p)T A(p,0), p−1

=

(p)T Ap, p−1

=

(p) m "

=

A(p) p,p

(p) ep−1

(p,i)

, + (p)T (p,1) (p,m(p) ) = Ap, (p,0) = A(p,0),(p,1) . . . A(p,0),(p,m(p) ) ,   (p,1) A(p,1), (p,1) 0 ... 0 ... 0   (p,2)   0 A(p,2), (p,2) . . . 0 ... 0     . . . .   .. .. .. ..   = , (p,i)   A . . . 0 0 0 . . . (p,i), (p,i)     . . . . .. .. .. ..     (p,m(p) ) 0 0 ... 0 . . . A(p,m(p) ),(p,m(p) ) =

e(p,0) =

(p) m "

(p) m "

(p,i)

ep−1 , (p,i)

e(p,0) ,

i=0



e(p) p

i=0

A(p,0), (p,0) ,

i=0 (p)

(p,i)

Ap−1, (p,0) ,

, (p,1) (p,m(p) ) = Ap−1,(p,1) . . . Ap−1,(p,m(p) ) , +

i=0

(p) A(p,0),p

(p) m "

(p,1)

e(p,1)



 (p,2)    e  (p,2)    .   ..   =  (p,i)  .  e(p,i)      ..   .   (p) (p,m ) e(p,m(p) )

3.3 Energy of Deformation and Dissipation of Energy of Link p

53

From a numerical point of view, it is important and advantageous that matrix (p,i) (p) Ap,p has the form presented above, where 6 × 6 sub-matrices A(p,i),(p,i) are placed on the diagonal. This is because the generalised coordinates describing the motion of each rfe with respect to rfe (p, 0) are independent. The potential energy of gravity forces of the flexible link p can be determined according to the formula: Vg(p)

=

(p) m "

m(p,i) g θ3 B(p,i) ˜rc(p,i) ,

(3.40)

i=0 (p,i)

are the coordinates of the centre of mass of rfe (p, i) in the local where ˜rc coordinate system  T ˜rc(p,i) = 0 0 0 1

for i = 1, . . . , m(p) .

From (3.40) one can obtain: 



(p)

gp−1

 (p)   =  g(p,0)  ,

(p) ∂Vg ∂q(p)

(3.41)

(p)

gp where  gp−1 =  (p)

(p) m "

 (p,i) (p,i)  ˜rc

m(p,i) g θ3 Bj

i=0

 g(p,0) =  (p)

(p) m "

,

j=1,...,np−1



m(p,i) g θ3 Bnp−1 +j ˜rc(p,i)  (p,i)

i=0

+

T

T

T

(p) (p) (p) gp(p) = gp,1 . . . gp,i . . . gp,m(p)   (p) (p,i) gp,i = m(p,i) g θ3 Bnp,0 +j ˜rc(p,i)

, j=1,...,˜ np,0

,T ,

j=1,...,6

for i = 1, . . . , m(p) .

3.3 Energy of Deformation and Dissipation of Energy of Link p The relations presented in Sect. 3.1 defining the energy of spring deformation of sde (p, k), which connects rfe (p, kl ) and rfe (p, kr ), can be written as follows: V (p,k) =

,T 1 + (p,k) ,T (p,k) 1+ (p,k) CY ∆y(p,k) + ∆ϕ(p,k) , ∆y ∆ϕ(p,k) Cϕ 2 2

(3.42)

54

3 The Rigid Finite Element Method (p,k)

where CY (p, k),

(p,k)

, Cϕ

are diagonal matrices of the stiffness coefficients of sde

∆y(p,k) = y(p,kl ) − y(p,kr ) =

+

E (p,kl ) YΘ

− (p,kl )

∆ϕ(p,k) = ϕY

(p,kr )

− ϕY

+ =

+

,T +

E (p,kr ) YΘ

E (p,kl ) YΘ

,T

R(p,kl ) − I

,T +



E (p,kl ) Ys

R(p,kr ) − I

ϕ(p,kl ) −

+



+ x(p,kl )

E (p,kr ) Ys

E (p,kr ) YΘ

,T

,

, + x(p,kr ) ,

ϕ(p,kr ) .

Before we calculate the derivatives of the potential energy of sde (p, k) with respect to the generalised coordinates of rfe (p, kl ) and rfe (p, kr ), first the relations concerning differentiation of quadratic forms will be discussed. Let us assume that: 1 (3.43) V = yT C y, 2 where T  y = y(q) = y1 (q) . . . yn (q) , C = diag {c1 , . . . , cn } , T

q = [q1 , . . . , qn ] . Then

where

∂y ∂q

 =

∂yi ∂qj

∂V = ∂q

 i,j=1,...,n



∂y ∂q

T C y,

(3.44)

is the gradient matrix.

One can consider two particular cases of (3.44): Case 1 y = A(q)s, where A is a square matrix n × n, s is a vector n × 1. In this case the following takes place: ( n ) " ∂ail ∂y = sl . ∂q ∂qj l=1

(3.45)

(3.46)

i,j=1,...,n

Case 2 y = A q, where A is a matrix with constant coefficients. In this case:  T ∂y ∂y =A≡ = AT . ∂q ∂q

(3.47)

(3.48)

3.3 Energy of Deformation and Dissipation of Energy of Link p

55

In view of (3.44) and (3.48), one can calculate from (3.42) the following: ∂V (p,k) (p,kl ) (p,k) =E CY ∆y(p,k) , YΘ ∂x(p,kl )

(3.49a)

∂V (p,k) (p,kr ) (p,k) = −E CY ∆y(p,k) . YΘ ∂x(p,kr )

(3.49b)

Having used (3.44), (3.46) and (3.42), we can also write: T ∂V (p,k) (p,kl ) (p,k) (p,kl ) (p,k) = G(p,kl ) E CY ∆y(p,k) + E Cϕ ∆ϕ(p,k) , YΘ YΘ ∂ϕ(p,kl )

(3.50a)

T ∂V (p,k) (p,kr ) (p,k) (p,kr ) (p,k) = −G(p,kr ) E CY ∆y(p,k) − E Cϕ ∆ϕ(p,k) , (3.50b) YΘ YΘ ∂ϕ(p,kr )

where (

(p,k ) 3 " ∂Ri,α l

E (p,kl ) s (p,kl ) Y α

G(p,kl ) =

α=1

( G(p,kr ) =

(p,k) ∂ϕ1

∂R(p,k) (p,k) ∂ϕ2

∂R(p,k) (p,k)

∂ϕ3 (p,i)

R4

(p,i)

, i,j=1,2,3

)

E (p,kr ) s (p,kr ) Y α

( = ( = ( =

∂ϕj

(p,k ) 3 " ∂Ri,α r

α=1

∂R(p,k)

)

∂ϕj

(p,k)

∂Ri,α

(p,k) ∂ϕ1 (p,k)

∂Ri,α

(p,k) ∂ϕ2 (p,k)

∂Ri,α

)

(p,k)

,

(p,k)

,

(p,k)

,

= R4 )

i,α=1,2,3

= R5 )

i,α=1,2,3

= R6

(p,k)

∂ϕ3

, i,j=1,2,3

i,α=1,2,3

(p,i)

, R5 and R6 are defined in (3.30). The potential energy of spring deformation of link p is the sum of the energies of the deformations of sdes (p, 1) to (p, κ(p) ): (p)

Vs(p)

=

κ "

V (p,k) .

(3.51)

k=1

Taking into account (3.49) and (3.50), one can obtain: κ(p)

" ∂Vs = v(k,l) (p,l) ˜ ∂q (p)

k=1

for l = 1, . . . , m(p) ,

(3.52)

56

3 The Rigid Finite Element Method

where 

(p,k)

vl

  0       0



1 2 3

   4   5  6    . ..    (p,k)   ∂Vs    ∂x(p,l)     =    (p,k)    ∂Vs   ∂ϕ(p,l)   ..  .     1  0 2    3    4      5 0 6

1 2 3 4 5 6

 – vector corresponding to q ˜ (p,l) ,                 – vector corresponding to q ˜ (p,kl ) (when l = kl ),   (p,k ) ˜ r (when l = kr ), and q             (p)  – vector corresponding to q ˜ (p,m ) .

If we denote: (p)

v

(p,l)

=

κ "

(p,k)

vl

,

(3.53)

k=1

the following can be written:  v(p,1)   ..   .    (p,i)  .  = v    ..   .   

(p)

∂Vs = v(p) ˜ (p) ∂q

v(p,m

(p)

(3.54)

)

If the flexible link p is correctly divided into rfes and sdes, each rfe becomes at least once element kl or kr when k = 1, . . . , κ(p) . This means that the components of vectors occurring in (3.54) are not zeros. It remains to discuss the case of rfe (p, 0) as in Fig. 3.9. Since the position of rfe (p, i) for i = 1, . . . , m(p) is defined with respect to rfe (p, 0), vectors y(p,kl )

3.3 Energy of Deformation and Dissipation of Energy of Link p

57

{p,0}Y ={p,kr}Y

rfe (p,kr)

rfe (p,0) sde (p,k)

Fig. 3.9. Position of rfe (p, kr ) and rfe (p, 0)

and y˙ (p,kl ) contain components which equal zero when sde (p, k) connects rfe (p, kr ) with rfe (p, 0). As a result the derivatives in (3.49b) and (3.50b) do not (p,k) are have to be calculated and thus the only nonzero elements of vector vl those in rows corresponding to l = kr . A similar procedure can be applied when the derivatives of the dissipation ˜˙ (p,l) when of energy of the flexible link are calculated with respect to vectors q (p) l = 1, . . . , m : κ + " (p)

Ws(p)

=

(p,k) WY

+

Wϕ(p,k)

,

(p)

=

k=1

κ "

W (p,k) ,

(3.55)

k=1

(p,k)

(p,k)

where WY is defined by (3.13) and Wϕ is defined by (3.17). Because of ˙ (p,i) ∂R ∂R(p,i) = , ∂ϕ(p,j) ˙ (p,j) ∂ϕ expressions for derivatives ∂W (p,kr ) ˙ (p,kr ) ∂ϕ

∂W (p,kl ) ˙ (p,kl ) ∂x

and

∂W (p,kr ) ˙ (p,kr ) ∂x

as well as (p,k)

will be analogous to (3.49) and (3.50), but CY (p,k) DY and

and

(3.56)

∂W (p,kl ) and ˙ (p,kl ) ∂ϕ (p,k) Cϕ should

(p,k) Dϕ ,

be replaced with respectively. Having omitted unnecessary transformations one can write:  w(p,1)   ..   .     (p,l)  w = ,    .   ..   

(p)

∂Ds = w(p) ˜˙ (p) ∂q

w(p,m

(p)

)

(3.57)

58

3 The Rigid Finite Element Method

where w(p,l) =

(p) κ3

k=1



(p,k)

wl

  0       0

, 

1 2 3

   4   5  6    ..  .    (p,k)  ∂Ws    ∂ x˙ (p,l)   = (p,k)      ∂Ws  (p,l)  ∂ϕ ˙   ..  .  1     2 0     3    4     5 0 6

∂W (p,k) = ∂ x˙ (p,l) ∂W (p,k) ˙ ∂ϕ

(p,k)

wl

(p,l)

E (p,l) YΘ

1 2 3 4 5 6

(p,k)

DY

  – vector corresponding to q ˜˙ (p,1) .             – vector corresponding to q ˜˙ (p,kl ) or q ˜˙ (p,kr ) ,                (p)  – vector corresponding to q ˜˙ (p,m ) , 

∆y˙ (p,k) ,

, + T (p,k) (p,l) (p,k) ˙ (p,k) l ∈ {kl , kr } . = G(p,l) E DY ∆y˙ (p,k) + Dϕ ∆ϕ YΘ

3.4 Synthesis of Equations Appending a flexible link p, discretised by means of the method described, to a chain of links 1 to p − 1, the equations of motion of which have the form: ¨ (p−1) = f (p−1) (t, q(p−1) , q˙ (p−1) ) A(p−1) q yields the equations of motion of all p links in the form:   ¨ (p) = f (p) t, q(p) , q˙ (p) , A(p) q

(3.58)

(3.59)

3.4 Synthesis of Equations

where



(p)

(p)

(p)

A(p−1) + Ap−1,p−1 Ap−1,(p,0) Ap−1,p

59



 (p)  (p) (p)   A(p) =  A(p,0),p−1 A(p,0),(p,0) A(p,0),p  ,   (p) (p) (p) Ap,p−1 Ap,(p,0) Ap,p   (p) (p) (p) f (p−1) −ep−1 −gp−1 +Qp−1   (p) (p) (p)  f (p) =  −e(p,0) −g(p,0) +Q(p,0)  ,  (p)

−ep (p)

(p)

(p)

(p)

(p)

−gp − v(p) − w(p) +Qp

(p)

(p)

(p)

(p)

(p)

Ap−1,p−1 , . . . , Ap,p and ep−1 , e(p,0) , ep are defined by (3.39), gp−1 , g(p,0) , gp

are defined by (3.41), v(p,0) , v(p) are defined by (3.54), w(p,0) , w(p) are defined (p) (p) (p) by (3.57), and Qp−1 , Q(p,0) , Qp are generalised forces resulting from loads acting on elements of link p. The generalised forces in the above equation can be determined using the formulae given in Sec. 2.6. (p,i) applied at the point with Let us assume (Fig. 3.10) that both force P (p,i) (p,i) coordinates defined by vector ˜rP
and a moment M (both defined with respect to coordinate system {p, i}) are acting on rfe (p, i): ,T + (3.60a) P(p,i) = P1(p,i) P2(p,i) P3(p,i) 0 , ,T + (3.60b) M(p,i) = M1(p,i) M2(p,i) M3(p,i) 0 . with respect to the The coordinates of the point of application of force P inertial coordinate system are calculated according to the formula: (p,i)

(p,i)

rP


(p,i)

= B(p,i) ˜rP


and the generalised forces resulting from force P form:

(3.61) (p,i)

can be presented in the

{p,i } P⬘(p,i ) rfe (p,i )

M3⬘

(p,i )

~ (p,i )

rP⬘

M2⬘(p,i )

M1⬘(p,i )

Fig. 3.10. Force P
(p,i) and moment M
(p,i) acting on rfe (p, i)

60

3 The Rigid Finite Element Method

 (p,0) QP


= 

(p,i)

QP


(p,0)

QP
,p−1 (p,0)

QP
,(p,0) (p,i)

QP
,p−1

 ,

(3.62a)



   (p,i)  =  QP
,(p,0)    (p,i) QP
,(p,i)

for i = 1, . . . , m(p) ,

(3.62b)

where QP
,p−1 = (P(p,i) B(p,i) Bj T

(p,i)

T

(p,i) (p,i) ˜rP
)j=1,...,np−1 ,

QP
,(p,0) = (P(p,i) B(p,i) Bnp−1 +j ˜rP
)j=1,...,˜np,0 , T

(p,i)

QP
,(p,i) = (P (p,i)

(p,i)T

T

(p,i)

T

(p,i)

(p,i)

(p,i)

B(p,i) Bnp,0 +j ˜rP
)j=1,...,6 .

The generalised forces resulting from moment M(p,i) are calculated directly from the formulae given in (2.82):   (p,0) Q
,p−1 M (p,0) , (3.63a) QM
=  (p,0) QM
,(p,0)  (p,i)  QM
,p−1    (p,i)  (p,i) QM
=  QM
,(p,0)  for i = 1, . . . , m(p) , (3.63b)   (p,i) QM
,(p,i) where ( (p,i) QM
,p−1


(p,i)

=

M1

3 "

(p,i)

(p,i)

bl,3 bk,l,2

l=1
(p,i) M2

+

3 "

(p,i) bl,1

(p,i) bk,l,3

+


(p,i) M3

l=1

(
(p,i)

QM,(p,0) =


(p,i)

M1

3 "

3 "

) (p,i) bl,2

(p,i) bk,l,1

l=1

(p,i)

k=1, ..., np−1 ,

(p,i)

bl,3 bnp−1 +k,l,2

l=1
(p,i) + M2

3 " l=1

(p,i) bl,1

(p,i) bnp−1 +k,l,3

+


(p,i) M3

3 " l=1

) (p,i) bl,2

(p,i) bnp−1 +k,l,1 k=1, ..., n ˜ p,0 ,

3.5 Linear Model

( (p,i) QM
,(p,i)

=


(p,i)

M1

3 "

(p,i)

(p,i)

bl,3 bnp,0 +k,l,2

l=1

+


(p,i) M2

61

3 "

(p,i) bl,1

(p,i) bnp,0 +k,l,3

+


(p,i) M3

3 "

l=1

) (p,i) bl,2

(p,i) bnp,0 +k,l,1

l=1

k=1, ..., 6.

Having summed the generalised forces from (3.62) and (3.63), the following is obtained: (p) Qp−1

=

(p) m "

(p,i)

(3.64a)

(p,i)

(3.64b)

QP
,p−1 ,

i=0 (p) Q(p,0)

=

(p) m "

QP
,(p,0) ,

i=0



Q(p) p

    =    

(p,1)

(p,1)

QP
,(p,1) + QM
,(p,1) .. . (p,l) (p,l) QP
,(p,l) + QM
,(p,l) .. . (p,m(p) )

(p,m(p) )

     .    

(3.64c)

QP
,(p,m(p) ) + QM
,(p,m(p) ) Equation (3.59) demonstrate the equations of motion of link p discretised by means of the rfe method and the influence of the flexible link p on the equations of motion of preceding links. The procedure presented in this section can be easily algorithmised and programmed. The routine, similar to this one, has been presented by Adamiec (2003). It can be especially easily implemented for computers by means of object-oriented programming techniques.

3.5 Linear Model In the model of flexible link p discretised by means of the rfe method, the transformation matrix of rfe (p, i) has the form (3.4). This means that the (p,i) (p,i) (p,i) defining rotations about the principal central axes angles ϕ1 , ϕ2 , ϕ3 of rfes may be large. However, when defining the expression for the spring deformation energy of sdes by (3.11) and (3.15), it has been assumed that those deformations are small and the relation between forces and moments on the one hand and displacements (translational and rotational) on the other is linear. In fundamental research for the rfe method (Kruszewski et al. 1984, 1999) the assumption that the components of the vector of generalised coordinates ˜ (p,i) are small always applies. q

62

3 The Rigid Finite Element Method

Later, we present the formulation of the equations of motion of the flexible link p, assuming that the rotation angles which are the components of vector ϕ(p,i) are small and using homogenous transformations. According to (2.23), the transformation matrix from coordinate system {p, i} to system E {p, i} can be presented as a product of six matrices: (p,i)

T(p,i) = T1

(p,i)

T2

(p,i)

T3

(p,i)

T6

(p,i)

T5

(p,i)

T4

,

(3.65)

where  (p,i) T1

 0 1 0 = 0 0 1 

0 0

0 0 0

1

1 0 0

0

 (p,i)

T3

(p,i)

1 0 0 x1

  ,  

(p,i)

c2   0 =  (p,i)  −s2 0 (p,i)

(p,i) T2



(p,i)

0 s2 1 0

(p,i)

0 c2 0 0 (p,i)

1 0 0

0

0 0 0

1



   0 1 0 x2(p,i)   , = 0  0 0 1 



  0  0 1 0  =  0 0 1 x(p,i)  ,   3 0 0 0 1 

(p,i) T5





1

0

0

0



   0 c1(p,i) −s1(p,i) 0  , =   (p,i) (p,i) c1 0  0 s1 0 0 0 1  (p,i) (p,i) c3 −s3 0  (p,i) (p,i) s (p,i) c3 0 3 T6 =   0 1  0 0 0 0

(p,i)

T4

 0  0 ,  0 1

0



 0 .  0 1

(p,i)

When angles ϕ1 , ϕ2 , ϕ3 are small, one can linearise the trigonometric functions of those angles by assuming the following: (p,i)

= cos ϕj

(p,i)

= sin ϕj

cj sj

(p,i)

≈ 1,

(p,i)

≈ ϕj

(p,i)

(3.66a) .

(3.66b) (p,i)

Substituting (3.66) into (3.65) and neglecting the second-order terms ϕl (p,i) ϕj for l, j = 1, 2, 3, transformation matrix T(p,i) can be written as follows:  T

(p,i)

1

  ϕ(p,i)  = 3  −ϕ(p,i)  2 0

(p,i)

−ϕ3 1

(p,i)

ϕ1

0

(p,i)

ϕ2

(p,i)

−ϕ1 1 0

(p,i)

x1



 6 "  (p,i) Dj q˜j , =I+ (p,i)  x3  j=1 1 (p,i) 

x2

(3.67)

3.5 Linear Model

where I is the identity matrix     0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1     D1 =  D2 =  , , 0 0 0 0 0 0 0 0 0 0 0 0



 0 0 0 0  0 0 −1 0    D4 =  , 0 1 0 0 0 0 0 0

0 0 0 0



0 0 0 0   D3 =  , 0 0 0 1

0 0 0 0

0 0 0 0

 0 0 1 0  0 0 0 0   D5 =  ,  −1 0 0 0  0 0 0 0



63



 0 −1 0 0 1 0 0 0   D6 =  . 0 0 0 0 0 0 0 0 

As a result of using transformation matrix T(p,i) in the form of (3.67), the following relations occur: (p,i)

Tj

(p,i)

Tl,j

=

∂T(p,i) (p,i)

∂ q˜j

= Dj for j = 1, . . . , 6,

∂ 2 T(p,i)

=

(p,i)

∂ q˜l

(p,i)

(3.68a)

= 0 for l, j = 1, . . . , 6.

(3.68b)

∂ q˜j

This leads to considerable simplifications in the expressions for the derivatives of the kinetic energy of rfe (p, i) and thus the whole link p. It can be easily checked that the mass matrix of element (p, i) is in this case a diagonal matrix with constant components: 

(p,i) A(p,i),(p,i)

     =    

m(p,i)

0

0

0

0

0

0

m(p,i)

0

0

0

0

0

0

m(p,i)

0

0

0 0 0

0

0

0

0 (p,i) h2

0

0

+

(p,i) h3

0

0

0 (p,i) h1

0

+ 0

          

0 (p,i) h3

0 (p,i) h1

(p,i)

+ h2

(3.69) for i = 1, . . . , m(p) . One should also note that: (p,i)

+ h3

(p,i)

+ h3

h2 h1

(p,i) h1

+

(p,i)

= J1

(p,i)

= J2

(p,i) h2

=

(p,i)

,

(3.70a)

(p,i)

,

(3.70b)

(p,i) J3 ,

(3.70c)

64

3 The Rigid Finite Element Method (p,i)

where Jj are moments of inertia of rfe (p, i) with respect to the central principal axes of inertia of this element defined by (3.3). (p) Form (3.69) of the mass matrix of rfe (p, i) means that matrix Ap,p occurring in (3.39) also becomes diagonal with constant components; this can be used in order to speed up the numerical computations. As a consequence of relation (3.68b), all the elements listed in Table 3.2 become zero. The components of the remaining matrices from (3.27b) and (3.39) are also considerably simplified since assuming transformation matrix T(p,i) in the form of (3.67) enables us to define:   6 " (p,i) (p,i) (p,i) (3.71) B(p,i) = B(p,0) D0 + Dj q˜j  , j=1 (p,i)

(p,i)

where D0 = 0E T(p,0) and Dj = 0E T(p,i) Dj are matrices with constant components. In the case considered the formulae for the derivatives of the potential energies of gravity forces as well as of deformation of spring damping elements will also become much simpler. For calculations of the derivatives of the potential energy of gravity forces in (3.41), the following has to be assumed because of (3.71): (p,i)

Bnp,0 +j = B(p,0) Dj for i = 1, . . . , m(p) ; j = 1, . . . , 6.

(3.72)

For calculations of the potential energy of deformation of the sde of a flexible (p,i) link when we assume that angles ϕj are small, an ingenious procedure presented in (Kruszewski et al., 1999) can be used. Having assumed that rotation matrices R(p,kl ) and R(p,kr ) of rfes (p, kl ) and (p, kr ), which are connected by sde (p, k), have the form:   (p,k ) (p,k ) 1 −ϕ3 l ϕ2 l  (p,k )  (p,k ) l (3.73a) R(p,kl ) =  1 −ϕ1 l   ϕ3 , (p,kl ) (p,kl ) −ϕ2 ϕ1 1   (p,k ) (p,k ) 1 −ϕ3 r ϕ2 r  (p,k )  (p,k ) r (3.73b) R(p,kr ) =  1 −ϕ1 r   ϕ3 . (p,kr ) (p,kr ) −ϕ2 ϕ1 1 Equations (3.10) and (3.14) can be written as: , (p,kl ) (p,kl ) x(p,kl ) +E , ϕ Y S + ,T + , (p,kr ) (p,kr ) (p,kr ) = E ϕ x(p,kr ) +E , YΘ Y S

y(p,kl ) = y(p,kr )

+

E (p,kl ) YΘ

,T +

(3.74a) (3.74b)

3.5 Linear Model

(p,kl )

ϕY

(p,kr )

ϕY where

+ = + =

 E (p,kl ) YS

,T

E (p,kr ) YΘ

ϕ(p,kl ) ,

(3.75a)

ϕ(p,kr ) ,

(3.75b)

,T

E (p,kl ) Y s3

0

 E (p,kl ) =  −Y s3

(p,kl )

−E Y s2

E (p,kl ) Y s1

0

E (p,kl ) Y s2

 E (p,kr ) YS

E (p,kl ) YΘ

(p,kl )

−E Y s1

 E (p,kr ) =  −Y s3

 , 

(p,kr )

−E Y s2

E (p,kr ) Y s1

0

E (p,kr ) Y s2



0

E (p,kr ) Y s3

0

65

(p,kr )

−E Y s1

  . 

0

Let us introduce vectors, with six components, of the coordinates of sdes with respect to the coordinate systems of rfes (p, kl ) and (p, kr ) as follows:  y(p,kl ) (p,kl ) = u (3.76a) (p,k ) , ϕY l  y(p,kr ) (p,kr ) u = (3.76b) (p,k ) , ϕY r then according to (3.74) and (3.75) one can write: ˜ (p,kl ) , u(p,kl ) = K(p,kl ) q (p,kr )

u

=K

(p,kr ) (p,kr )

˜ q

(3.77a)

,

(3.77b)

where  K(p,kl ) = 

Y EΘ

 (p,kl ) T



Y EΘ

E

0  K(p,kr ) = 

 E (p,kr ) T YΘ 0

  (p,kl ) T E (p,kl ) YS

Y

Θ

E

(p,kr ) YΘ

E Y

Θ

,

 (p,kl ) T

T E

(p,kr ) YS

 (p,kr ) T

 .

It should be noted that the components of matrices K(p,kl ) and K(p,kr ) are (p,kl ) E ,Y Θ(p,kr ) are conconstant because the elements of rotation matrices E YΘ (p,kl ) E (p,kr ) S and S , which are created stant and as well as those of matrices E Y Y (p,kl ) E (p,kr ) s and s . from vectors E Y Y

66

3 The Rigid Finite Element Method

The energy of spring deformation of sde k of link p, and the function of energy dissipation can be written now as follows: ,T + , 1 + (p,kl ) (3.78a) u − u(p,kr ) C(p,k) u(p,kl ) − u(p,kr ) , Vs(p,k) = 2 , + , (p,kr ) T (p,kr ) 1 + (p,kl ) (p,k) , (3.78b) u˙ WS = − u˙ D(p,k) u˙ (p,kl ) − u˙ 2 where  (p,k) CY 0 (p,k) = C (p,k) , 0 Cϕ  (p,k) DY 0 (p,k) D = (p,k) , 0 Dϕ (p,k)

(p,k)

(p,k)

(p,k)

are defined by (3.12) and (3.16), and DY , Dϕ are defined CY , Cϕ by (3.13) and (3.17). By applying (3.44) and (3.48), one can obtain from (3.78) the formulae for (p,k) ˜ (p,kl ) and q ˜ (p,kr ) : with respect to q the derivatives of Vs (p,k) + , ∂Vs (p,kl )T (p,k) (p,kl ) (p,kr ) u , (3.79a) = K C − u ˜ (p,kl ) ∂q (p,k) + , ∂Vs (p,kr )T (p,k) (p,kl ) (p,kr ) u . (3.79b) = −K C − u ˜ (p,kr ) ∂q In view of (3.77) we obtain: (p,k)

∂Vs (p,k) (p,kl ) (p,k) (p,kr ) ˜ ˜ = Ckl ,kl q + Ckl ,kr q , ˜ (p,kl ) ∂q

(3.80a)

(p,k)

∂Vs (p,k) (p,kl ) (p,k) (p,kr ) ˜ ˜ = Ckr ,kl q + Ckr ,kr q , ˜ (p,kr) ∂q

(3.80b)

where (p,k)

T

Ckl ,kl = K(p,kl ) C(p,k) K(p,kl ) , T

(p,k)

Ckl ,kr = −K(p,kl ) C(p,k) K(p,kr ) , (p,k)

(p,k)T

T

Ckr ,kl = Ckl ,kr = −K(p,kr ) C(p,k) K(p,kl ) , (p,k)

T

Ckr ,kr = K(p,kr ) C(p,k) K(p,kr ) . In a similar way, from (3.78b) one can obtain: (p,k)

∂Ws (p,k) ˙ (p,kl ) (p,k) ˙ (p,kr ) ˜ ˜ = Dkl ,kl q + Dkl ,kr q , ˜˙ (p,kl ) ∂q

(3.81a)

(p,k)

∂Ws (p,k) ˙ (p,kl ) (p,k) ˙ (p,kr ) ˜ ˜ = Dkr ,kl q + Dkr ,kr q , (p,k ) ˙ r ∂q ˜

(3.81b)

3.5 Linear Model

67

where T

(p,k)

Dkl ,kl = K(p,kl ) D(p,k) K(p,kl ) , T

(p,k)

Dkl ,kr = −K(p,kl ) D(p,k) K(p,kr ) , (p,k)T

(p,k)

T

Dkr ,kl = Dkl ,kr = −K(p,kr ) D(p,k) K(p,kl ) , T

(p,k)

Dkr ,kr = K(p,k,kr ) D(p,k) K(p,k,kr ) . (p,k )

If sde (p, k) connects rfe (p, 0) with rfe (p, kr ) then vector uY l occurring in (p,k ) (3.78a) has constant components and thus u˙ Y l = 0. Having differentiated ˜ (p,kr ) and q ˜˙ (p,kr ) , the following can be written: (3.78) with respect to vectors q (p,k)

∂Vs (p,k) (p,kr ) ˜ = Ckr ,kr q , ˜ (p,kr ) ∂q

(3.82a)

(p,k)

∂Ws (p,k) ˙ (p,kr ) ˜ = Dkr ,kr q , (p,k ) ˙ r ˜ ∂q (p,k)

(3.82b)

(p,k)

where Ckr ,kr , Dkr ,kr are defined by (3.80) and (3.81). If the stiffness and damping matrices of sde (p, k) are defined in the forms: 

C(p,k)

D(p,k)

0 . . .   0   =  ...   0  . . . 0  0 . . .   0   =  ...   0  . . .

···

0 .. .

···

0 .. .

(p,k)

(p,k)

···

(p,k) Ckr ,kl

···

(p,k) Ckr ,kr

···

.. . 0

···

.. . 0

· · · Ckl ,kl · · · Ckl ,kr .. .. . .

···

0 .. .

···

(p,k)

0 .. . (p,k)

· · · Dkl ,kl · · · Dkl ,kr .. .. . . (p,k)

(p,k)

· · · Dkr ,kl · · · Dkr ,kr .. .. . .

0 ···

0

···

0

 ··· 0 ..   .   ··· 0  ..  , .   ··· 0  ..   . ··· 0  ··· 0 ..   .   ··· 0  ..  , .   ··· 0  ..   . ··· 0

(3.83a)

(3.83b)

68

3 The Rigid Finite Element Method

one can write (p,k)

∂Vs ˜ (p) , = C(p,k) q ˜ (p) ∂q

(3.84a)

(p,k)

∂Ws ˜˙ (p) ∂q

˜˙ (p) = D(p,k) q

(3.84b)

˜ (p) is the vector of the generalised coordinates of rfes (p, 1) to (p, m(p) ). where q When sde (p, k) connects rfe (p, kr ) with rfe (p, 0), it has to be assumed that: (p,k)

(p,k)

(p,k)

(p,k)

Ckl ,kr = Ckl ,kl = Dkl ,kr = Dkl ,kl = 0. In view of (3.51) and (3.55) we can write the following: (p)

∂Vs ˜ (p) , = C(p) q ˜ (p) ∂q

(3.85a)

(p)

∂Ws ˜˙ (p) , = D(p) q ˜˙ (p) ∂q

(3.85b)

where (p)

(p)

C

=

κ "

C(p,k) ,

k=1 (p)

(p)

D

=

κ "

D(p,k) .

k=1

The stiffness and damping matrices of link p, discretised by means of the rfe method, are symmetrical matrices with constant components. The equations of motion of links from 1 to p, written previously in the form of (3.59), now can be written as follows: , + (p) (p) ¨˜ (p,0) + A(p) q ¨ (p) ¨ (p−1) + Ap−1,(p,0) q A(p−1) + Ap−1,p−1 q p−1,p ˜ (p)

(p)

(p)

= f (p−1) − ep−1 − gp−1 − Qp−1 ,

(3.86a)

(p) (p) ¨˜ (p,0) + A(p) q ¨˜ (p) ¨ (p−1) + A(p,0),(p,0) q A(p,0),p−1 q (p,0),p (p)

(p)

(p)

= −e0 − g0 + Q0 , (p)

(3.86b)

(p)

¨˜ (p,0) + A(p) ¨˜ (p) + C(p) q ¨ (p−1) + Ap,(p,0) q ˜ (p) + D(p) q ˜˙ (p) Ap,p−1 q p,p q = −gp(p) + Q(p) p .

(3.86c)

The specific case of equations (3.86) is when p = 1. Having omitted in (3.86c) the components connected with the motion of preceding links and rfe (p, 0), the equations of motion of the flexible link can be written in the form: A¨ q + Dq˙ + Cq = −g + Q,

(3.87)

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements

69

where ˜ (p) , q=q A = A(p) p,p , D = D(p) , C = C(p) , g = gp(p) , Q = Q(p) p . From (3.87) follow the equations which enable us to calculate the free frequencies and free forms of vibrations of the flexible link discretised by the rfe method. The frequencies of the free vibrations of the link are the solution of the eigenvalue problem of a matrix in the form:   (3.88a) det C − ω 2 A = 0, where ω is a frequency of free vibrations and the forms of free vibrations (eigenvectors) are calculated from the equation:   C − ω 2 A Φ = 0, (3.88b) where Φ are the vectors of the forms of free vibrations of a link. It should be noted that matrix A is diagonal with constant elements, which means that (3.88) can be transformed into the classical eigenvalue problem for matrix A−1 C: det (M − λI) = 0, (M − λI) Φ = 0,

(3.89a) (3.89b)

where M = A−1 C and λ = ω 2 .

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements Basic parameters of an rfe are the mass and inertial moments. These parameters occur in pseudo-inertia matrix H (2.49) and in the formulae defined by (3.3), and they are: – mass

# m(p,i) =

dm, m

(3.90)

70

3 The Rigid Finite Element Method

– second area moments # (p,i) J(Xˆ Xˆ ) = x21 dm, 2

(p,i) ˆ 1 X3 )

J(Xˆ

3

# x22 dm,

=

m

(p,i) ˆ 1 X2 )

J(Xˆ

# x23 dm, (3.91)

=

m

m

– moments of inertia (p,i)

= J(Xˆ

(p,i)

= J(Xˆ

J1 J3

(p,i) ˆ 1 X2 )

+ J(Xˆ

(p,i) ˆ 1 X3 )

+ J(Xˆ

– products of inertia # (p,i) JXˆ Xˆ = x1 x2 dm, 1

2

(p,i) ˆ , 1 X3 )

(p,i)

J2

(p,i) ˆ 2 X3 )

= J(Xˆ

(p,i) ˆ , 1 X2 )

+ J(Xˆ

(p,i) ˆ , 2 X3 )

(p,i) ˆ 1 X3

JXˆ

(3.92)

# =

m

x1 x3 dm,

(p,i) ˆ 2 X3

JXˆ

m

# =

x2 x3 dm, m

(3.93) – static moments # (p,i) JXˆ = x1 dm, 1

(p,i)

JXˆ

# =

x2 dm,

2

m

(p,i)

JXˆ

# =

x3 dm,

3

m

(3.94)

m

where x1 , x2 , x3 are coordinates of mass dm with respect to the local coordinate system of rfe (p, i). Usually in calculations the local coordinate system of an rfe is chosen in such a way that its axes coincide with principal and central axes of inertia of the rfe. In this case, since the origin of such a system is placed in the centre of mass of the rfe, the products of inertia and static moments are zero. For rfes which are the result of discretisation of a beam-like system (Fig. 3.11), when the beam is homogenous (constant density ρ) with constant ˆ 2, X ˆ 3 of the local coordinate ˆ 1, X cross-section (A=const) and when axes X system are principal central axes of inertia, mass and moments of inertia can be calculated according to the formulae given in Table 3.3. When the axes of the local coordinate system of the rfe are translated or rotated in relation to the principal central system of inertia, mass moments of inertia can be calculated according to the Steiner theorem and relations describing rotation of the coordinate system. Xˆ 2 rfe (p,i ) Xˆ 1 C

Xˆ 3

Fig. 3.11. Beam-like rfe (p, i)

d

h

A

a

Xˆ 2

Xˆ 2

C

Xˆ 2

shape of rfe

l

l

l Xˆ 3

Xˆ 3

Xˆ 3

V = lA

V =

π d2 l 4

V = ahl

Xˆ 1

Xˆ 1

Xˆ 1

volume V

m=ρ

π d2 l 4

m = ρahl

m = ρlA

mass m

JXˆ 3

JXˆ 2

md2 8 m = (3d2 + 4l2 ) 48 m = (3d2 + 4l2 ) 48 JXˆ 1 =

JXˆ 3

JXˆ 2

m 2 (a + h2 ) 12 m 2 = (l + a2 ) 12 m 2 = (l + h2 ) 12 JXˆ 1 =

JXˆ 1 = 0 ml2 JXˆ 2 = 12 ml2 JXˆ 3 = 12

moment of inertia

Table 3.3. Mass and moments of inertia for a beam-like rfe

J(Xˆ 3 Xˆ 1 )

J(Xˆ 2 Xˆ 1 )

ml2 12 md2 = 16 md2 = 16 J(Xˆ 2 Xˆ 3 ) =

J(Xˆ 3 Xˆ 1 )

J(Xˆ 2 Xˆ 1 )

ml2 12 ma2 = 12 mh2 = 12 J(Xˆ 2 Xˆ 3 ) =

J(Xˆ 1 Xˆ 3 ) = 0

J(Xˆ 1 Xˆ 2 )

ml2 12 =0 J(Xˆ 2 Xˆ 3 ) =

second area moments

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements 71

d

D

d

Xˆ 2

Xˆ 2

shape of rfe

l D

l

Xˆ 3

Xˆ 3

Xˆ 1

Xˆ 1

V =

V =

π h2 2 (D +Dd+d2 ) 12

π l (D2 − d2 ) 4

volume V

πh m = ρ (D2 + 12 Dd + d2 ) 2

π m = ρ l (D2 − 4 d2 )

mass m

J(Xˆ 3 Xˆ 1 )

J(Xˆ 2 Xˆ 1 )

C = D4 + 4D3 d + 10 D2 d2 + 4D d3 + d4

C 3 m l2 2 80 A 3 B = m 80 A 3 B = m 80 A J(Xˆ 2 Xˆ 3 ) =

J(Xˆ 3 Xˆ 1 )

J(Xˆ 2 Xˆ 1 )

ml2 12 m 2 = (D + d2 ) 16 m 2 = (D + d2 ) 16 J(Xˆ 2 Xˆ 3 ) =

second area moments

B = D4 + D3 d + D2 d2 + D d3 + d4

where: A = D 2 + D d + d2

JXˆ 3

JXˆ 2

3 B m 40 A  C 3m = B + l2 80 A A   C 3m = B + l2 80 A A JXˆ 1 =

JXˆ 3

JXˆ 2

m 2 (D + d2 ) 8 m = (3D2 + 3d2 + 4l2 ) 48 m = (3D2 + 3d2 + 4l2 ) 48

moment of inertia JXˆ 1 =

Table 3.3. Continued...

72 3 The Rigid Finite Element Method

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements

73

The basic parameters of the sde are values characterising spring and damping features. These are, for spring features of sde (p, k), coefficients of (p,k) (p,k) (p,k) (p,k) (3.12), and translational stiffness cY,1 , cY,2 , cY,3 , occurring in matrix CY (p,k)

(p,k)

(p,k)

(p,k)

(3.16). coefficients of rotational stiffness cϕ,1 , cϕ,2 , cϕ,3 , from matrix Cϕ The damping features of sde (p, k) are defined by coefficients of translational (p,k) (p,k) (p,k) (p,k) (3.13), and coefficients damping dY,1 , dY,2 , dY,3 , occurring in matrix DY (p,k)

(p,k)

(p,k)

(p,k)

(3.17). of rotational damping dϕ,1 , dϕ,2 , dϕ,3 , in matrix Dϕ The coefficients of stiffness and damping of the sde, in which all such features of a beam segment with length ∆l are concentrated, are determined on the basis of the assumption that a real segment of the beam will deform in the same way and with the same velocities of deformation as the equivalent sde under the same load. Considerations below are carried out using the Kelvin– Voigt rheological model (Fig. 3.12) (Niezgodzi´ nski and Niezgodzi´ nski, 1984). (p,k) (p,k) The coefficients of translational stiffness cY,1 and damping dY,1 , which are the coefficients of stretching (Fig. 3.13), are derived by assuming that stresses along segment ∆l are constant and equal: σ(t) = Eε(t) + η ε(t), ˙

(3.95)

where ε(t) is a unit elongation with respect to time, σ(t) is normal stress as a function of time, E is Young’s modulus and η is the normal damping material constant. Substituting the following relations into (3.95): σ(t) =

P1 (t) , A

ε(t) =

∆w1 (t) , ∆l

ε(t) ˙ =

∆w˙ 1 (t) , ∆l

(3.96)

where P1 (t) is the force of tension of the element, ∆w1 (t) is deformation (elongation) of the element and ∆w˙ 1 (t) is velocity of the deformation, we obtain

s (t ) e (t )

E

h

s (t )

Fig. 3.12. Kelvin–Voigt rheological model

74

3 The Rigid Finite Element Method Xˆ 3 (a)

Dw1(t )

Dl

P1(t )

Xˆ 1

P1(t )

Xˆ 3

Xˆ 1

Dw1(t )

(b) P1(t ) P1(t )

Fig. 3.13. Deformation during tension: (a) segment of a beam; (b) equivalent sde

the dependence of the force on deformation and the velocity of deformation for the element considered: P1 (t) =

EA ηA ∆w1 (t) + ∆w˙ 1 (t). ∆l ∆l

(3.97)

The analogical formula for the equivalent sde is: (p,k)

(p,k)

P1 (t) = cY,1 ∆w1 (t) + dY,1 ∆w˙ 1 (t).

(3.98)

Since the left-hand sides of (3.97) and (3.98) are identical, the coefficients at ∆w1 (t) and ∆w˙ 1 (t) have to be also equal, and thus: (p,k)

EA , ∆l ηA . = ∆l

cY,1 = (p,k)

dY,1

(p,k)

(3.99) (3.100) (p,k)

The coefficients of translational stiffness cY,2 and damping dY,2 , which are in this case coefficients of shear, are calculated on the basis that tangential stresses are constant not only along segment ∆l considered, but also for the whole area of cross-section; this can be assumed (though not corresponding in reality) since an influence of tangential stresses on bending vibrations is very small and thus even a large error in the value of the coefficient of shear has a negligible influence on the accuracy of calculations of bending vibrations. Moreover, this error can be decreased by introducing the shape coefficient of the cross-section, which takes into account a nonuniform tangential stress pattern (to be discussed later). In order to derive the coefficients of stiffness and damping for shear, we assume that under the shear force the deformations and velocity of deformation

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements (a)

(b)

Xˆ 2

Xˆ 2

Dw2 Xˆ 1 P2(t )

P2(t )

75

Xˆ 1 P2(t ) Dw2

P2(t )

Dl

Fig. 3.14. Shear deformations of: (a) beam-like element; (b) equivalent sde

of the beam segment and the equivalent sde are the same (Fig. 3.14). Thus, using the Kelvin–Voigt model, we obtain: τ (t) = Gγ(t) + η γ(t), ˙

(3.101)

where γ(t) is the angle of nondilational strain, τ (t) is tangential stress, G is shear modulus (Kirchhoff’s modulus) and η is a material constant of tangential damping. Having substituted the following relations into (3.101): ∆w2 (t) ∆w˙ 2 (t) P2 (t) , γ(t) = γ(t) ˙ = , (3.102) A ∆l ∆l where P2 (t) is the shear force in the element considered and ∆w2 (t) is the transverse strain of the element (Fig. 3.14), one can determine the relation of the tangential force with respect to the deformation and velocity of deformation: ηA GA ∆w2 (t) + ∆w˙ 2 (t). (3.103) P2 (t) = ∆l ∆l For the sde substituting the respective features of the element, force P2 (t) takes the form: (p,k) (p,k) (3.104) P2 (t) = cY,2 ∆w2 (t) + dY,2 ∆w˙ 2 (t). τ (t) =

Having compared the coefficients of ∆w2 (t) and ∆w˙ 2 (t) in (3.103) and (3.104), we obtain: GA (p,k) , (3.105) cY,2 = ∆l ηA (p,k) dY,2 = . (3.106) ∆l Accuracy of the above formulae can be increased by introducing the coefficient of cross-section shape χ, which takes into account the nonuniform tangential stress pattern as follows: (p,k)

GA , χ∆l ηA . = χ ∆l

cY,2 = (p,k)

dY,2

(3.107) (3.108)

76

3 The Rigid Finite Element Method

The shape coefficient can be calculated from the following relation (Podrzucki and Kalata, 1971): # 2 S A dA, (3.109) χ= 2 J2 y2 A

where S is the static moment of cross-section area, shown in Fig. 3.15, with ˆ 2 , A is the area of the cross-section, J2 is the second moment of respect to X ˆ 2 , y is the width of the cross-section (Fig. 3.15). For inertia with respect to X a rectangular cross-section this coefficient is χ = 1, 2. Having omitted possible differences resulting from different shape coeffiˆ 3 are the same cients, the coefficients of stiffness and damping in direction X ˆ as in direction X2 , and thus: (p,k)

(p,k)

cY,3 = cY,2 ,

(p,k)

(p,k)

dY,3 = dY,2 .

(3.110)

(p,k)

(p,k)

The coefficients of rotational stiffness cϕ,1 and damping dϕ,1 , which are torsional coefficients, are derived from the assumption that torsional stress along segment ∆l is constant. The torsional moment transferred by the beam element is (Fig. 3.16): Xˆ 3

y Xˆ 2

Fig. 3.15. Cross-section of the beam segment (a)

(b)

dA

M1(t )

M1(t ) g (t )

M1(t ) Dw4(t )

Xˆ 1 Dw4(t )

M1(t )

Xˆ 1

Dl

Fig. 3.16. Torsional deformations for: (a) segment of the beam; (b) equivalent sde

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements

77

# M1 (t) =

τ (t)ρ dA

(3.111)

τ (t) = Gγ(t) + η γ(t). ˙

(3.112)

A

and, as before, we assume that:

Substituting (3.112) into (3.111) and taking into account that: ρ∆w4 (t) , ∆l

γ(t) =

(3.113)

where ∆w4 (t) is the angle of torsion of the element ∆l, we can write: # # Gρ2 ∆w4 (t) ηρ2 ∆w˙ 4 (t) dA + dA. (3.114) M1 (t) = ∆l ∆l A

A

Having integrated the above, we obtain: M1 (t) =

ηJ0 GJ0 ∆w4 (t) + ∆w˙ 4 (t). ∆l ∆l

(3.115)

The same torsional moment M1 (t) transferred by the equivalent sde is defined by the formula: (p,k)

(p,k)

M1 (t) = cϕ,1 ∆w4 (t) + dϕ,1 ∆w˙ 4 (t).

(3.116)

Having compared (3.115) and (3.116) we obtain: (p,k)

GJ0 , ∆l ηJ0 . = ∆l

cϕ,1 = (p,k)

dϕ,1

(3.117) (3.118)

The formulae derived for the coefficients of torsional stiffness and damping are true only for circular symmetrical cross-sections. In different cases torsional stiffness GJ0 and the respective expression ηJ0 have to be replaced with appropriate formulae calculated according to principles of the theory of elasticity. For beams with rectangular cross-section (b × h) (3.117) and (3.118) take the following form (Niezgodzi´ nski and Niezgodzi´ nski, 1984): (p,k)

cϕ,1 = (p,k)

dϕ,1 =

µ2 Gb3 h , ∆l

(3.119)

µ2 ηb3 h , ∆l

(3.120)

where µ2 is a coefficient dependent on the ratio of the sides of the rectangular cross-section (Fig. 3.17). In the case of thin-walled beams with open profiles one can assume for the analysis of torsional vibrations that unbounded torsion takes place

78

3 The Rigid Finite Element Method m2

0.30

0.25

0.20

0.15

0.10

4

2

8

6

bh

Fig. 3.17. Shape coefficient µ2 for beams with rectangular cross-section

(Niezgodzi´ nski and Niezgodzi´ nski, 1984), and the stiffness and damping coefficients can be calculated according to the formulae: GJs , ∆l ηJs , = ∆l

(p,k)

(3.121)

cϕ,1 = (p,k)

dϕ,1 while

Js = α

s " ai h3 i

i=1

3

(3.122)

,

(3.123)

where s is the number of elements into which the cross-section is divided, α is the experimental coefficient dependent on the shape of the cross-section (Table 3.4), and ai , hi is the width and thickness of element i of the crosssection division. For example, for a cross-section in the form of a welded T-bar diaph  with a1 h31 +a2 h32 . ragms presented in Fig. 3.18, the second area moment Js = 1.40 3 (p,k)

(p,k)

The coefficients of rotational stiffness cϕ,2 and damping dϕ,2 (coefficients of bending) are derived in an analogical way to the coefficients of shear. The bending moment transferred by the beam element is: # (3.124) M2 (t) = σ(t)y dA. A

Having assumed, as before, that: σ(t) = Eε(t) + η ε(t) ˙

(3.125)

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements

79

Table 3.4. Shape coefficients α for torsion of thin-walled open profiles type of profile

name

rolled profiles

angle bar

1.00

I-bar

1.20

T-bar

1.15

channel bar

1.12

Z-bar

1.15

complex profiles

scheme

welded I-bar diaphragms

with

welded T-bar with diaphragms

α

1.50

1.40

and substituting (3.125) to (3.124) as well as taking into account the following (Fig. 3.19): y∆w5 (t) , (3.126) ∆l where ∆w5 (t) is the angle of deflection of the segment ∆l, we obtain: ε(t) =

80

3 The Rigid Finite Element Method

a2

h1

a1

h2

Fig. 3.18. Welded T-bar with diaphragms as an example of an open profile

ˆ X 3

(a)

(b)

DW 5(t )

ˆ2 X

DW 5(t )

dA y Xˆ 1

M2(t )

M2(t )

M2(t )

M2(t )

Dl Fig. 3.19. Bending deformations for: (a) beam-like element; (b) equivalent sde

# M2 (t) =

Ey 2 ∆w5 (t) dA + ∆l

A

#

ηy 2 ∆w˙ 5 (t) dA. ∆l

(3.127)

A

Having integrated the above over the area of the cross-section, we obtain: M2 (t) =

EJ2 ηJ2 ∆w5 (t) + ∆w˙ 5 (t). ∆l ∆l

(3.128)

The bending moment transferred by the equivalent sde is described by the relation: (p,k)

(p,k)

M2 (t) = cϕ,2 ∆w5 (t) + dϕ,2 ∆w˙ 5 (t).

(3.129)

Since the left-hand sides of (3.128) and (3.129) are identical, the coefficients by ∆w5 (t) and ∆w˙ 5 (t) have to be equal and thus:

3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements (p,k)

EJ2 , ∆l ηJ2 . = ∆l

(3.130)

cϕ,2 = (p,k)

dϕ,2

81

(3.131)

4 1X 4 3 are calcuThe stiffness and damping coefficients for bending in plane X lated in the same way. They are: (p,k)

EJ3 , ∆l ηJ3 . = ∆l

(3.132)

cϕ,3 = (p,k)

dϕ,3

(3.133)

The second area moments J1 , J2 , J3 are calculated according to the wellknown formulae used in the strength of materials (Niezgodzi´ nski and Niezgodzi´ nski, 1984). Some basic constants for the most frequently used construction materials are given in Table 3.5. The following are listed: density ρ, Young’s modulus E, shear modulus G, Poisson ratio ν and material coefficient Q−1 . The coefficients of normal and tangential damping are calculated when the coefficient of material loss is known since there is a relation between this coefficient and the material constant for harmonic vibrations η in the form: Q−1 =

ωη , E

(3.134)

where ω is the frequency of vibrations and moreover for isotropic materials the following relation between the material constants takes place (Podrzucki

Table 3.5. Constants for frequent construction materials (Niezgodzi´ nski and Niezgodzi´ nski, 1984) Type of material

Density Young’s ρ × 103 modulus (kg m−3 ) E × 105 (MPa)

Shear mod- Poisson ulus G×104 ratio ν (MPa)

carbon steel 0.11-0.56C grey cast iron 3.4C 2Si spheroidal iron 3.4C 2Si aluminium light alloys Ak20 (80Al 20Si)

7.86

2.1

8.1

0.3

6.7–7.4

0.95–1.1

3.8–4.2

0.23–0.27 1.6–160

7.0–7.4

1.0

4.1–4.2

0.23–0.27 0.16–0.64

2.7 2.7

0.7 0.67–0.74

2.7 2.7

0.32–0.36 unknown 0.32–0.35 unknown

Coefficient of material loss at room temperature Q−1 × 10−3 0.03–0.1

82

3 The Rigid Finite Element Method

and Kalata, 1971): G η = . η E

(3.135)

The formulae given in this section are limited to beams with constant or tapered cross-sections. Formulae necessary to calculate parameters of rfes and sdes resulting from division of beams with complex cross-sections (also curvilinear) or plates are described by Kruszewski et al. (1975).

4 Modification of the Rigid Finite Element Method

The rigid finite element method in its classical formulation enables any displacements of rigid elements of the flexible link to be considered and thus all possible deformations (bending, torsion, shear and extension) to be analysed. It is characteristic that displacements of each rfe are considered with respect to the coordinate system assigned to rfe 0. In this chapter a modification of the rigid finite element is used to discretise a flexible beam-like link with dominant bending and torsional flexibility. Unlike in the classical formulation, displacements of each rfe are considered in relation to the preceding element. The modification allows large bending and torsional deformations of flexible links to be analysed and is easily interpreted.

4.1 Generalised Coordinates and Transformation Matrices Figure 4.1 presents a beam-like link p, with bending and torsional flexibility, divided into m(p) + 1 rfes and m(p) sdes. The generalised coordinates of rfe (p, i) for i = 1, . . . , m(p) are chosen in a very different way in comparison to the classical formulation of the rigid finite element method. (Only in the case of rfe (p, 0) is the procedure identical to that in Chap. 3; the number of degrees of freedom of rfe (p, 0) depends on the joint between links (p − 1) and p.) Let us consider rfe (p, i − 1) and rfe (p, i) connected by a spherical joint at which sde (p, i) is placed as in Fig. 4.2. The coordinate system {p, i}Y is assigned to each sde (p, i) in such a way that its axes coincide with the principal axes of this spring-damping element. The position and orientation of the coordinate system {p, i}Y assigned to the sde are constant with respect to the coordinate system assigned to rfe (p, i − 1) and are defined by the following transformation matrix: (p,i) (p,i)

(p,i) ΘY sY , (4.1) TY = 0 1

84

4 Modification of the Rigid Finite Element Method rfe (p, m (p)) rfe (p, m (p) -1) sde (p, m (p)) sde (p, m (p) -1) link p

rfe (p,0)

rfe (p,1)

sde (p,2) sde (p,1) link p -1

Fig. 4.1. Flexible link p divided into rfes and sdes (p,i)

where ΘY is the direction cosine matrix of the axes of the system {p, i}Y with respect to the axes of the coordinate system {p, i − 1} assigned to rfe (p,i) (p, i − 1) and sY is the coordinate vector of the origin of the coordinate Y system {p, i} with respect to the coordinate system {p, i − 1}. The coordinate system {p, i} is assigned to rfe (p, i) in such a way that its axes coincide with the principal axes of sde (p, i) (which means that {p, i} = {p, i}Y ), when link p is in an undeformed state. As a result of external loads and inertial forces rfe (p, i) can rotate about a fixed point with respect to rfe (p, i − 1). It can be assumed that the motion of rfe (p, i) with respect to rfe (p, i − 1) is defined by Euler angles ZYX, defined in Fig. 4.2 as (p,i) (p,i) (p,i) ϕ3 , ϕ2 , ϕ1 , which are the components of the vector:  (p,i)  ϕ  1(p,i)  (p,i) ϕ =  ϕ2  . (4.2) (p,i) ϕ3 Transformation of coordinates from the system of rfe (p, i) to the system assigned to sde (p, i) can now be carried out according to the formula: (p,i)

rY

* (p,i) ˜r(p,i) , =B

(4.3)

4.1 Generalised Coordinates and Transformation Matrices {p, i }

Xˆ 2(p,i )

j3(p,i ) {p, i }y

Xˆ 3(p,i )

85

j2(p,i )

) Yˆ (p,i 3

Yˆ 2(p,i )

Xˆ 1(p,i ) j1(p,i ) Yˆ 1(p,i )

rfe (p, i )

sde (p, i ) rfe (p, i -1)

Fig. 4.2. Spring-damping element (p, i) connecting rfe (p, i − 1) with rfe (p, i)

(p,i)

where rY is the vector of coordinates of a point with respect to the coordinate system {p, i}Y , ˜r(p,i) is the vector of coordinates of a point with respect to the coordinate system {p, i}, * (p,i) = T(p,i) T(p,i) T(p,i) , B 6 5 4  (p,i)

T4

(p,i)



(p,i)



ϕ1

= T4

1

0

0

(p,i)

(p,i)



(p,i)



ϕ2

= T5

(p,i)

(p,i)

= T6



(p,i)

ϕ3





0

0

1

(p,i)

0 s2

(p,i)

0

0

0

1

(p,i)

−s3

c1



   0 1 0 0 , =  (p,i)  (p,i)  −s2 0 c2 0 

T6

0

   0 c1(p,i) −s1(p,i) 0  , =   (p,i) (p,i)  0 s1 c1 0 

T5

0

c3

 (p,i) s 3 =   0 0

0

(p,i)

0 0



0

 0 0 .  1 0

0

0 1

(p,i)

c3

In view of (4.1) the coordinates of a point of rfe (p, i) can be written with respect to the coordinate system {p, i − 1}, assigned to the preceding rfe, according to the formula: (p,i) i−1 r

(p,i)

= TY

(p,i) (p,i) * (p,i) * ˜r , r(p,i) = B B

(4.4)

86

4 Modification of the Rigid Finite Element Method

where (p,i)

B

=B

(p,i)

(p,i) * (p,i)

(ϕ(p,i) ) = TY

B

.

Having used (4.4) recursively, one can calculate coordinates 0 r(p,i) of a point of rfe (p, i) with respect to the coordinate system of rfe (p, 0), and then with respect to the global coordinate system {} according to the formulae: (p,i) = 0r

i 

(p,j)

Bj−1 ˜r(p,i) =0 B(p,i) ˜r(p,i) ,

(4.5a)

j=1

r(p,i) = B(p,0) 0 B(p,i) ˜r(p,i) = B(p,i) ˜r(p,i) ,

(4.5b)

where 0B

(p,i)

=

i 

B

(p,j)

,

j=1

* (p,0) 0 B(p,i) . B(p,i) = B(p,0) 0 B(p,i) = B(p−1) B It is characteristic that coordinates of the point of rfe (p, i) depend not only on the generalised coordinates of link p − 1 and rfe (p, 0) and (p, i) as in the classical formulation of the method, but also on the generalised coordinates of all rfes between rfe (p, 0) and rfe (p, i). This means that the vectors of generalised coordinates defining the motion of rfe (p, i) with respect to rfe (p, 0) and in the global coordinate system take the following forms, respectively:  (p,1)  ϕ  ..  (p,i) ˜ =  . , q (4.6a) ϕ(p,i)   q(p−1) q(p,0)  (p,0)  =q . ˜ = ˜ (p,i) q (p,i) ˜ q 

q(p,i)

(4.6b)

4.2 Kinetic Energy of the Flexible Link and Its Derivatives Because the local coordinate systems {p, i} are assigned to rfes in the way described, the axes of those systems do not have to be central principal axes as is the case in the classical formulation of the method. The equations of the flexible link p, discretised in the way presented, can be derived by proceeding

4.2 Kinetic Energy of the Flexible Link and Its Derivatives

87

as described in Chap. 2, when a rigid link is attached to an existing kinematic chain. The kinetic energy of rfe (p, i) of the flexible link can be expressed as: E (p,i) =

1 $ ˙ (p,i) (p,i) ˙ (p,i)T % , tr B B H 2

(4.7)

where B(p,i) is defined by (4.5) and H(p,i) is the pseudo-inertial matrix of rfe (p, i). Bearing in mind (4.6), the following can be written: " (p,i) (p,i) ∂E (p,i) d ∂E (p,i) (p,i) − = a ˜k,l q¨l + e˜k , (p,i) (p,i) dt ∂ q˙ ∂q l=1 k k np,i

i = 1, . . . , m(p) ; k = 1, . . . , np,i ,

(4.8)

where np,i = np,i−1 + 3 = np,0 + 3i = np−1 + n ˜ p,0 + 3i, % $ (p,i) (p,i) (p,i)T , H(p,i) Bl a ˜k,l = tr Bk (p,i) e˜k

=

np,i np,i " "

% $ (p,i) (p,i)T (p,i) (p,i) q˙ j q˙ l . tr Bk H(p,i) Bj,l

j=1 l=1

Relation (4.8) can now be written in the vector form: ˜ (p,i) q ˜(p,i) , εq(p,i) (E (p,i) ) = A ¨(p,i) + e

(4.9)

where ˜ (p,i) A 

˜ (p,i) ˜ (p,i) ˜ (p,i) ˜ (p,i) ˜ (p,i) A p−1,p−1 Ap−1,(p,0) Ap−1,(p,1) . . . Ap−1,(p,j) . . . Ap−1,(p,i)

  (p,i) A ˜  (p,0),p−1   (p,i) ˜ A  (p,1),p−1   ..  = .   (p,i) A  ˜ (p,k),p−1    ..  .   ˜ (p,i) A (p,i),p−1



  (p,i) (p,i) (p,i) (p,i) ˜ ˜ ˜ ˜ A(p,0),(p,0) A(p,0),(p,1) . . . A(p,0),(p,j) . . . A(p,0),(p,i)     ˜ (p,i) ˜ (p,i) ˜ (p,i) ˜ (p,i)  . . . A . . . A A A (p,1),(p,0) (p,1),(p,1) (p,1),(p,j) (p,1),(p,i)    .. .. .. ..  . . . . ,   (p,i) (p,i) (p,i) (p,i) ˜ ˜ ˜ ˜ A(p,k),(p,0) A(p,k),(p,1) . . . A(p,k),(p,j) . . . A(p,k),(p,i)      .. .. .. ..  . . . .   (p,i) (p,i) (p,i) (p,i) ˜ ˜ ˜ ˜ A(p,i),(p,0) A(p,i),(p,1) . . . A(p,i),(p,j) . . . A(p,i),(p,i)

88

4 Modification of the Rigid Finite Element Method



(p,i)

˜p−1 e



  (p,i)  e  ˜(p,0)     (p,i)  ˜(p,1)  e    .   .  (p,i) ˜ e =  . ,    (p,i)  e ˜(p,k)     .   .   .    (p,i) ˜(p,i) e   (p,i) ˜ (p,i) A = a ˜ p−1,p−1 l,s

,

l,s=1,...,np−1

T    (p,i) ˜ (p,i) ˜ (p,i) = = a ˜ A A l,np−1 +s l=1,...,np−1 , p−1,(p,0) (p,0),p−1 s=1,...,˜ np,0



˜ (p,i) ˜ (p,i) A p−1,(p,j) = A(p,j),p−1



T



(p,i)

= a ˜l,np,0 +3(j−1)+s

  (p,i) ˜ (p,i) = a ˜ A np−1 +l,np−1 +s (p,0),(p,0)

l,s=1,...,˜ np,0

l=1,...,np−1 s=1,2,3

,

j = 1, . . . , i,

,

,T + (p,i) ˜ (p,i) ˜ (p,i) = = (˜ anp−1 +l,np,0 +3(j−1)+s )l=1,...,˜np,0 , A A (p,0),(p,j) (p,j),(p,0) s=1,2,3

j = 1, . . . , i, + ˜ (p,i) = A

˜ (p,i) A (p,k),(p,j)

(p,i)

˜p−1 e

(p,i)

,T

(p,i)

= (˜ anp,0 +3(k−1)+l,np,0 +3(j−1)+s )l,s=1,2,3 ,

(p,j),(p,k)

j = 1, . . . , i;  (p,i) = e˜l ,

k = 1, . . . , i,





l=1,...,np−1

(p,i)

˜(p,0) = e˜np,0 +l e



l=1,...,˜ np,0

  (p,i) (p,i) ˜(p,k) = e˜np,0 +3(k−1)+l e

,

l=1,2,3

,

k = 1, . . . , i.

The kinetic energy of the flexible link p is expressed as: E (p) =

(p) m "

E (p,i) .

(4.10)

i=0

In view of (4.9) and (4.10), one can obtain the following relation: ˜ (p) q ˜(p) , εq(p) (E (p) ) = A ¨(p) + e

(4.11)

4.2 Kinetic Energy of the Flexible Link and Its Derivatives

where



q(p)





(p−1)

q  (p,0)  ˜   q    ϕ(p,1)      ..   , . =    ϕ(p,i)      ..     .   (p,m(p) ) ϕ

˜(p) e

(p)

ep−1

89



 (p)    e  (p,0)   (p)    e  (p,1)      .. , = .    (p)    e  (p,i)    ..     .   (p) e(p,m(p) )

˜ (p) A



        =        

˜ (p) A p−1, p−1

˜ (p) A p−1, (p,0)

˜ (p) A p−1, (p,1)

...

˜ (p) A p−1, (p,j)

...

˜ (p) A

˜ (p) A (p,0), p−1

˜ (p) A (p,0), (p,0)

˜ (p) A (p,0), (p,1)

...

˜ (p) A (p,0), (p,j)

...

˜ (p) A

˜ (p) A (p,1), p−1

˜ (p) A (p,1), (p,0)

˜ (p) A (p,1), (p,1)

...

˜ (p) A (p,1), (p,j)

...

˜ (p) A

. . .

. . .

. . .

˜ (p) A (p,k), p−1

˜ (p) A (p,k), (p,0)

˜ (p) A (p,k), (p,1)

. . .

. . .

. . .

˜ (p) A

˜ (p) A

(p,m(p) ), p−1

˜ (p) A p−1, p−1

=

(p,m(p) ), (p,0)

(p) m "

(p,0), (p,m(p) ) (p,1), (p,m(p) )

. . .

. . . ...

˜ (p) A (p,k), (p,j)

...

˜ (p) A

(p,k), (p,m(p) )

. . .

. . .

˜ (p) A

(p,m(p) ), (p,1)



p−1, (p,m(p) )

˜ (p) ... A

(p,m(p) ), (p,j)

˜ (p) ... A

(p,m(p) ), (p,m(p) )

˜ (p,i) A p−1, p−1 ,

i=0 (p) T m  " (p,i) ˜ (p) ˜ ˜ (p) = = A A A p−1, (p,0) (p,0), p−1 p−1, (p,0) ,

i=0 (p) T m  " (p,i) (p) (p) ˜ ˜ ˜ = Ap−1, (p,j) = A(p,j), p−1 A p−1, (p,j) ,

j = 1, . . . , m(p) ,

i=j

˜ (p) A (p,0), (p,0) =

(p) m "

˜ (p,i) A (p,0),(p,0) ,

i=0 (p) T m  " (p,i) (p) (p) ˜ ˜ ˜ = A A(p,0), (p,j) = A(p,j), (p,0) (p,0), (p,j) ,

j = 1, . . . , m(p) ,

i=j

 T ˜ (p) ˜ (p) = A = A (p,k), (p,j) (p,j), (p,k)

(p) m "

i=max{k, j} (p)

j = k, . . . , m

,

˜ (p,i) A (p,k), (p,j) ,

k = 1, . . . , m(p) ,

        ,        

90

4 Modification of the Rigid Finite Element Method (p) m "

(p)

˜p−1 = e

(p,i)

˜p−1 , e

i=0 (p) ˜(p,0) e

=

(p) m "

(p,i)

˜(p,0) , e

i=0 (p)

˜(p,k) = e

(p) m "

(p,i)

˜(p,k) , e

k = 1, . . . , m(p) .

i=k

4.3 Potential Energy of the Flexible Link The potential energy of gravity forces acting on rfe (p, i) can be presented as follows: (4.12) Vg(p,i) = m(p,i) gθ3 B(p,i) ˜rc(p,i) , where m(p,i) is the mass of the element, g is the acceleration of gravity, θ3 is (p,i) defined by (2.73), B(p,i) is defined by (4.5), ˜rc are the coordinates of the centre of mass of the rfe with respect to system {p,i}. From (4.12) one can obtain: (p,i)

∂Vg

(p,i) ∂qk

(p,i) (p,i) ˜rc

= m(p,i) gθ3 Bk

and in the vector form:

(p,i)

= gk

for k = 1, . . . , np,i ,

(p,i)

∂Vg = g(p,i) , ∂q(p,i) where



˜ (p,i) g

(p,i)

˜p−1 g

(p,i)

˜(p,0) g (p,i)

˜(p,j) g

(4.13)

(p,i)

˜p−1 g

(4.14)



 (p,i)   g  ˜(p,0)   (p,i)   g  ˜(p,1)     ..   = .  ,  (p,i)   g  ˜(p,j)   .   .   .    (p,i) ˜(p,i) g   (p,i) = gk , k=1,...,np−1   (p,i) = gnp−1 +k , k=1,...,˜  np,0 (p,i) = gnp,0 +(j−1)3+k

k=1,2,3

j = 1, . . . , m(p) .

4.3 Potential Energy of the Flexible Link

91

Taking into account that the potential energy of gravity forces of link p is defined by the formula: (p) m " (p) Vg(p,i) , (4.15) Vg = i=0

the following can be written: (p)

∂Vg ˜ (p) , =g ∂q(p) where



˜ (p) g

(p) ˜p−1 g

(p)

˜p−1 g

(4.16)



 (p)    g  ˜(p,0)   (p)    g  ˜(p,1)    ..   , . =    (p)   g ˜(p,j)      ..     .   (p) ˜(p,m(p) ) g =

(p) m "

(p,i)

˜p−1 , g

i=0 (p)

(p) m "

(p)

(p) m "

˜(p,0) = g

(p,i)

˜(p,0) , g

i=0

˜(p,i) = g

(p,j)

˜(p,i) , g

i = 1, . . . , m(p) .

j=i

The method of assigning the coordinate systems {p,i} results in a very simple form of expressions for the potential energy of spring deformation of the sde and the function of energy dissipation: Vs(p)

=

(p) m "

3 "

(p,i)

cϕ,j

+

(p,i)

,2

ϕj

,

(4.17a)

i=1 j=1

Ws(p)

=

(p) m "

3 "

(p,i)

dϕ,j

+

(p,i)

ϕ˙ j

,2 ,

(4.17b)

i=1 j=1 (p,i)

where cϕ,j (j = 1, 2, 3) are the coefficients of rotational stiffness of sde (p, i) (p,i)

and dϕ,j (j = 1, 2, 3) are the coefficients of rotational damping of sde (p,i).

92

4 Modification of the Rigid Finite Element Method

One can note that, just as in the classical formulation of the method, ˜ (p,0) . Having the expressions defined by (4.17) do not depend on q(p−1) and q differentiated (4.17), the following is obtained: (p)

∂Vg = C(p) ϕ(p) , ∂ϕ(p)

(4.18a)

(p)

∂Ws

˙ (p) ∂ϕ

˙ (p) , = D(p) ϕ

(4.18b)

where  ϕ(p,1) ..     .   (p,i)   = ϕ ,   ..   . 

ϕ(p)

(p)

C(p) D(p)

ϕ(p,m ) % $ (p) = diag C(p,1) . . . C(p,m ) , % $ (p) = diag D(p,1) . . . D(p,m ) ,

C(p,i)

 (p,i) 0 0 cϕ,1   (p,i) =  0 cϕ,2 0 , (p,i) 0 0 cϕ,3

D(p,i)

 (p,i) 0 0 dϕ,1   (p,i) =  0 dϕ,2 0 . (p,i) 0 0 dϕ,3





Matrices C(p) and D(p) are then diagonal matrices with constant elements.

4.4 Synthesis of Equations If a pair of forces with moment M and force P are acting on rfe (p,i) (p, i) of the flexible link, and the vector ˜rP
defines the coordinates of the point in the local coordinate system {p,i} where the force is applied, then the generalised forces resulting from the force and the moment can be calculated according to the formulae: (p,i)

(p,i)

= P

(p,i)T

B(p,i) Bk

Qnp,i−1 +k = P

(p,i)T

B(p,i) Bnp,i−1 +k ˜rP
+ Mk

(p,i)

Qk (p,i)

T

(p,i) (p,i) ˜rP


T

(p,i)

for k = 1, . . . , np,i−1 , (p,i)

(p,i)

(4.19a)

for k = 1, 2, 3. (4.19b)

4.4 Synthesis of Equations

93

This enables us to write the vector of generalised forces resulting from (p,i) (p,i) force P and moment M in the following form:  (p,i)  Qp−1   (p,i)  Q  (p,0)     Q(p,i)   (p,1)     .  ..  , (4.20) Q(p,i) =     (p,i)   Q  (p,j)     ..   .    (p,i)

Q(p,i) where

  (p,i) (p,i) Qp−1 = Qk , k=1,...,np−1   (p,i) (p,i) Q(p,0) = Qnp−1 +k , k=1,...,˜ np,0   (p,i) (p,i) Q(p,j) = Qnp,0 +(j−1) 3+k

k=1,2,3

,

j = 1, . . . , i.

If we assume that a force and a moment act on each rfe of link p, the resulting vector of generalised forces can be presented as follows:   (p) Qp−1    Q(p)   (p,0)     Q(p)   (p,1)      .. , (4.21) Q(p) =  .      Q(p)  (p,j)     ..     .   (p)

Q(p,m(p) ) where (p) Qp−1

=

(p) m "

(p,i)

Qp−1 ,

i=0 (p)

Q(p,0) =

(p) m "

(p,i)

Q(p,0) ,

i=0 (p) Q(p,i)

=

(p) m "

j=i

(p,j)

Q(p,i) ,

i = 1, . . . , m(p) .

94

4 Modification of the Rigid Finite Element Method

Having taken into account (4.11), (4.16) and (4.18) and assuming that the equations of motion of links 1 to p − 1 are defined by (3.58), the equations of motion of links 1 to p can be written in the form: ¨ (p) = f (p) , A(p) q

(4.22)

 q(p−1) =q ˜ (p,0)  ϕ(p) 

where q(p)

is a vector with np = np,0 + 3m(p) elements, 

(p)

(p)

 (p)  A(p) =  A(p,0),p−1  (p) Aϕ,p−1    f (p) =  

(p)

A(p−1) + Ap−1,p−1 Ap−1,(p,0) Ap−1,ϕ



 (p) (p) A(p,0),(p,0) A(p,0),ϕ  ,  (p) (p) Aϕ,(p,0) Aϕ,ϕ

(p)

(p)

(p)

(p)

(p)

(p)



f (p−1) −˜ ep−1 −gp−1 +Qp−1

  , 

−˜ e(p,0) −g(p,0) +Q(p,0) (p)

(p)

−˜ eϕ

−gϕ

(p)

˙ (p) +Qϕ − C(p) ϕ(p) − D(p) ϕ

T + ,  (p) (p) ˜ (p) ˜ (p) . . . A Ap−1,ϕ = Aϕ,p−1 , = A p−1,(p,1) p−1,(p,m(p) ) 

(p) A(p,0),ϕ

=

(p) Aϕ,(p,0)

T



˜ (p) ˜ (p) . . . A A = (p,0),(p,1) (p,0),(p,m(p) ) ,

 ˜ (p) ˜ (p) A (p,1),(p,1) · · · A(p,1),(p,m(p) )   .. .. , = . .   ˜ (p) (p) ˜ (p) (p) · · · A A (p) (p,m ),(p,1) (p,m ),(p,m ) 

A(p) ϕ,ϕ

 (p) ˜(p,1) e   .. , = .   (p) ˜(p,m(p) ) e 

e(p) ϕ

 (p) g(p,1)   .. , = .   (p) g(p,m(p) ) 

(p) gϕ

 (p) Q(p,1)   .. . = .   (p) Q(p,m(p) ) 

Q(p) ϕ

4.5 Linear Model (p,i)

If we assume that rotation angles ϕj (i = 1, . . . , m(p) ; j = 1, 2, 3) are small, then after linearisation of the trigonometrical functions of those an(p,i) from the coordinate system of rfe (p, i) to gles transformation matrix B

4.5 Linear Model

the coordinate system of rfe (p, i − 1) can be presented as:  (p,i) (p,i)  (p,i) 0 R ΘY sY (p,i) , B = 0 1 0 1 (p,i)

where θY

95

(4.23)

(p,i)

and sY

are defined by (4.1),   (p,i) (p,i) ϕ2 1 −ϕ3  (p,i)  R(p,i) =  ϕ3(p,i) 1 −ϕ1  . (p,i) (p,i) −ϕ2 ϕ1 1 (p,i)

can be presented in the form: It follows that transformation matrix B   3 " (p,i) (p,i) (p,i) (4.24) B = L0 I + Lj ϕj  , j=1 (p,i)

where L0 Lj =

(p,i)

= TY

Rj 0 , 0 0

, 



 00 0 R1 =  0 0 −1  , 01 0

 0 01 R2 =  0 0 0  , −1 0 0



 0 −1 0 R3 =  1 0 0  . 0 0 0

This can also be written as: B

(p,i)

(p,i)

= L0

+

3 "

(p,i)

Lj

(p,i)

ϕj

,

(4.25)

j=1 (p,i)

where L0

and

(p,i) Lj

=

(p,i) L0 Lj



(p,i)

ΘY = 0

(p,i)

sY 1





(p,i) Rj 0 ΘY Rj 0 = 0 0 0 0

are matrices with constant elements. Transformation matrices 0 B(p,i) from coordinate system {p,i} to the coordinate system of rfe 0 can now be presented as follows:   i i 3   " (p,k) (p,k) (p,k) (p,i) L0(p,k) + (4.26) = B = Lj ϕj . 0B k=1

j=1

k=1

To make further considerations easier, the following recurrent formula replacing (4.26) is defined: 0B

(p,i)

(p,i)

= L0

+

i " 3 " k=1 j=1

(p,i)

(p,k)

Lk,j ϕj

for i = 1, 2, . . . , m(p) .

(4.27)

96

4 Modification of the Rigid Finite Element Method

If we assume: (p,1)

(p,1)

L0

= L0

(p,1) L1,j

,

(4.28a)

(p,1) Lj

=

for j = 1, 2, 3,

(4.28b)

the proof of identity (4.27) for i = 1 is obvious. Let us assume that (4.27) is true for i − 1; thus we obtain:   3 " (p,i) (p,i) (p,i) (p,i) = 0 B(p,i−1) L0 + Lj ϕj  0B j=1

 = L0

(p,i−1)

+

i−1 " 3 "

 (p,i−1)

Lk,j

(p,k) 

L0(p,i) +

ϕj

k=1 j=1 (p,i−1)

= L0

(p,i)

L0

+

+

 (p,i)

Lj

(p,i) 

ϕj

j=1

i−1 " 3 "

(p,i)

(p,i−1)

Lk,j

L0

(p,k)

ϕj

+

k=1 j=1 i−1 " 3 " 3 "

3 "

(p,i−1)

Lk,j

3 "

(p,i−1)

L0

(p,i)

Lj

(p,i)

ϕj

j=1

(p,i)

Ls

(p,k)

ϕj

ϕs(p,i) .

k=1 j=1 s=1

(4.29) Ignoring the fourth component on the right of (4.29), as an expression of the second order, we will obtain (4.27) when: (p,i)

L0

(p,i) Lk,j (p,i) Lk,j

(p,i−1)

= L0

(p,i)

L0

,

(4.30a)

=

(p,i−1) (p,i) Lk,j L0

when k < i for j = 1, 2, 3,

(4.30b)

=

(p,i−1) (p,i) L0 Lj

when k = i for j = 1, 2, 3.

(4.30c)

(p,i)

(p,i)

The elements of matrices L0 and Lk,j occurring in (4.27) are constant and those matrices take the following form: (p,i)

L0

=

i 

(p,i)



L

=

(p,i)

(p,i)

k=1

 (p,i) Lk,j

=

Θk,j sk,j 0 0

(p,i)

Θ0 0

(p,i)

s0 1

,

.

(4.31a)

(4.31b)

(p,i)

The property of matrix Lk,j that all the elements of the fourth row are zero will be used in further considerations. Differentiating (4.27) we obtain: ˙ (p,i) =

0B

i " 3 " k=1 j=1

(p,i)

(p,k)

Lk,j ϕ˙ j

,

(4.32a)

4.5 Linear Model

¨ (p,i) =

0B

i " 3 "

(p,i)

(p,k)

Lk,j ϕ¨j

.

97

(4.32b)

k=1 j=1

The transformation matrix from the local coordinate system of rfe (p,i) to the inertial coordinate system can be presented in the form: B(p,i) = B(p,0) 0 B(p,i) .

(4.33)

Matrix B(p,0) is, as previously, the transformation matrix from the local coordinate system of rfe (p, 0) to the global coordinate system, and its elements depend on (p−1)

q (p,0) = . q ˜ (p,0) q From (4.33) it follows that: ˙ (p,0) 0 B(p,i) + B(p,0) 0 B ˙ (p,i) , ˙ (p,i) = B B

(4.34a)

¨ (p,0) 0 B(p,i) + 2B ˙ (p,0) 0 B ˙ (p,i) + B(p,0) 0 B ¨ (p,i) . ¨ (p,i) = B B

(4.34b)

The kinetic energy of rfe (p,i) is defined by the formula: 1 $ ˙ (p,i) (p,i) ˙ (p,i)T % , H E (p,i) = tr B B 2

(4.35)

and the Lagrange operators, in accordance with (4.26), have the form:   % $ d ∂E (p,i) ∂E (p,i) (p,i) (p,i) ¨ (p,i) , εl E (p,i) = − = tr B H B l (p,i) dt ∂ q˙(p,i) ∂ql l

(4.36)

where l = 1, . . . , np,i ,   q (p,0) for l = 1, . . . , np,0 , l (p,i) ql =  ϕ(p,i) for l = n + (i − 1)3 + j; i = 1, . . . , m(p) , j = 1, 2, 3. p,0 j Having taken into account (4.34), we obtain:   $ + (p,i) ˙ (p,0) 0 B ˙ (p,i) ¨ (p,0) 0 B(p,i) + 2B εl E (p,i) = tr Bl H(p,i) B ¨ (p,i) +B(p,0) 0 B

,T . .

(4.37)

Since: "

k=1

""

np,0 np,0

np,0

¨ (p,0) = B

(p,0) (p,0) q¨k

Bk

+

k=1 s=1

(p,0) (p,0) (p,0) q˙s ,

Bk,s q˙k

(4.38)

98

4 Modification of the Rigid Finite Element Method

then according to (4.32a) we can write the following: & np,0   " (p,i) (p,0) (p,i) εl E (p,i) = tr Bl H(p,i) Bk q¨k k=1

T    (p,i)  (p,0) (p,i) (p,i) + B Lk,j ϕ¨j + ul ,   k=1 j=1 i " 3 "

(4.39)

where ""

np,0 np,0 (p,i) ul

=

˙ (p,0) = B

(p,0) (p,0) Bk,s 0 B(p,i) q˙k q˙s(p,0)

˙ (p,0) + 2B

k=1 s=1 np,0 " (p,0) (p,0) Bk q˙k , k=1

3 i " "

(p,i)

(p,i)

Lk,j ϕ˙ j

,

k=1 j=1

and thus: p,0   n" % $ (p,i) (p,i)T (p,0) q¨k εl E (p,i) = tr Bl H(p,i) Bk

k=1

+

i " 3 "

 + ,T . (p,i) (p,i) (p,i) (p,0) (p,i) B tr Bl H Lk,j ϕ¨j

(4.40)

k=1 j=1

% $ (p,i) (p,i)T . + tr Bl H(p,i) ul Relation (4.40) can be written in a different form: n ˜ p,0 p−1 i " 3   n" " " (p,i) (p−1) (p,i) (p,0) ˜b(p,i) ϕ¨(p,i) + e˜(p,i) , εl E (p,i) = a ˜l,k q¨k + a ˜l, np−1 +k q¨˜k + l, k, j j l k=1

k=1 j=1

k=1

(4.41) where

% $ (p,i) (p,i) (p,i)T , a ˜l,k = tr Bl H(p,i) Bk   T . ˜b(p,i) = tr B(p,i) H(p,i) B(p,0) L(p,i) , l, k, j l k,j $ % (p,i) (p,i) (p,i)T = tr Bl H(p,i) ul e˜l . (p,i)

It can be easily proved that elements ˜bl, k, j occurring in this formula are constant for l = np,0 + 1, . . . , np,i when rotation matrix Θ(p,0) in the transformation matrix: (p,0) (p,0)

Θ s (p,0) (4.42) = B 0 1

4.5 Linear Model

99

is orthonormal. Let us consider l corresponding to element (p, α)(α = 1, . . . , m(p) ) and direction β( β = 1, 2, 3): l = np,0 + 3 (α − 1) + β,

(4.43)

then, because of the relation tr(A B) = tr(B A): $ % T ˜b(p,i) = tr B(p,0) L(p,i) H(p,i) L(p,i) B(p,0)T l, k, j α, β k, j $ % T T (p,i) (p,i) (p,i) = tr Lα, β H Lk, j B(p,0) B(p,0) .

(4.44)

When we present the pseudo-inertia matrix of element (p, i) in the form: (p,i) (p,i)

a J (p,i) H = , (4.45) (p,i)T a m(p,i) then, having taken into account (4.31b) and (4.42), we obtain:



 T (p,i)T

(p,i)

Lα, β H(p,i) Lk, j

 =

(p,i)

=

(p,i)

(p,i)

θα, β J(p,i) + sα, β a(p,i)

T



(p,i)

J(p,i) a(p,i) T a(p,i) m(p,i)

(p,i)

Θα, β sα, β 0 0

(p,i)T

θk,j



θk, j

0

sk, j

0

(p,i)T

(p,i)

(p,i)

+ θα, β a(p,i) + m(p,i) sα, β



0

 B

(p,0)T

B

(p,0)

=

T

Θ(p,0) 0 T s(p,0) 1



 =

Θ(p,0) s(p,0) 0 1 T

T

T

0

, 0 (4.46a)



Θ(p,0) s(p,0)

I

(p,i)T

sk,j



(4.46b)

.

s(p,0) Θ(p,0) s(p,0) s(p,0) + 1 Therefore, substituting (4.46) in (4.44), we can write:  $ T (p,i) (p,i) (p,i) (p,i)T bl, k, j = tr Θα, β J(p,i) + sα, β a(p,i) Θk,j  %  (p,i) (p,i) (p,i)T . + Θα, β a(p,i) + m(p,i) sα, β sk,j

(4.47) (p,i)

Since the elements of matrices and vectors in (4.47) are constant, bl, k, j for l defined by (4.43) is also constant. The kinetic energy of the whole flexible link can be expressed by the formula: (p) m " (p) E (p,i) . (4.48) E = i=0

100

4 Modification of the Rigid Finite Element Method

From (4.41) it follows that:  εl E

(p)

 =

(p) p−1 m " n"

(p,i) (p−1) al,k q¨k

+

i=0 k=1

+

(p) m "

(p) ˜ p,0 m " n"

(p,i)

(p,0)

al, np−1 +k q¨˜k

i=0 k=1

3 i " "

˜b(p,i) ϕ¨(p,i) l, k, j j

+

i=1 k=1 j=1

(p) m "

(4.49) (p,i) e˜l .

i=0

Having used the following identity: (p) m "

3 i " "

(p,i)

(p,i)

bl, k, j ϕ¨j

=

i=1 k=1 j=1

(p) m "

3 "

 

i=1 j=1



(p) m "

bl, k, j  ϕ¨j (p,i)

(p,i)

,

(4.50)

k=i

we can write:  εl E

(p)



"

"

np−1

=

n ˜ p,0 (p) (p−1) al, k q¨k

+

k=1

(p) (p,0) al, np−1 +k q¨˜k

+

(p) m "

3 "

(p)

(p,i)

bl, i, j ϕ¨j

(p)

+ el ,

i=1 j=1

k=1

(4.51) where (p) al, k

=

(p) m "

(p,i)

for k = 1, . . . , np,0 ,

(p,i)

for i = 1, . . . , m(p) , j = 1, 2, 3,

al,k

i=0 (p) bl, i, j

=

(p) m "

bl,k,j

k=i (p)

el

=

(p) m "

(p,i)

el

.

i=0

If vector q(p) is defined as in (4.22), then: 

(p)

(p)

(p)

Ap−1, p−1 Ap−1, (p,0) Ap−1, ϕ



¨ (p−1) q





(p)

ep−1



     (p, 0)   (p)  (p) (p) (p)   ¨˜  + e  εq(p) E (p) =    (p,0)  ,  A(p,0), p−1 A(p,0), (p,0) A(p,0), ϕ   q (p) (p) (p) (p) ¨ (p) ϕ Aϕ, p−1 Aϕ, (p,0) Aϕ, ϕ eϕ (4.52) where   (p) (p) Ap−1, p−1 = al,k , (p) Ap−1, (p,0)

+ =

l,k=1, ..., np−1

(p) A(p,0), p−1

,T

  (p) = al,np−1 +k

l=1, ..., np−1 k=1, ..., n ˜ p,0

,

4.5 Linear Model

,T   + (p) (p) (p) Ap−1, ϕ = Aϕ, p−1 = al,np,0 + 3 (i−1)+j 

(p)

i=1, ..., m



(p)

,

l=1, ..., np−1 (p)

, A(p,0), (p,0) = anp−1 +l, np−1 +k l,k=1, ..., n ˜ p,0   (p) (p) A(p,0), ϕ = anp−1 +l, np,0 +3 (i−1)+j l=1, ..., n˜

101

, j=1, 2, 3

,

p,0



(p)

A(p) ϕ, ϕ = bnp,0 +l, i, j (p)



(p)

i=1, ..., m

 l=1, ..., 3 m(p) i=1, ..., m



(p)

(p)

, j=1, 2, 3

,

, j=1, 2, 3

ep−1 = el l=1, ..., np−1 ,   (p) (p) e(p,0) = enp−1 +l , l=1, ..., n ˜ p,0   e(p) ϕ = enp,0 +3 (i−1)+j i=1, ..., m(p) . j=1, 2, 3

The potential energy and the function of dissipation of the deformation energy of the sde and the generalised forces can be presented in the form given for the nonlinear model. The equations of motion also take the analogical form to that given in (4.22). (p) The basic difference is that the elements of matrix Aϕ, ϕ are constant, which enables us to formulate the equations of the frequencies and forms of the free vibrations of the flexible beam-like link in the following form:   (4.53a) det C(p) − ω 2 A(p) ϕ, ϕ = 0, +

, (p) C(p) − ωi2 A(p) = 0. ϕ, ϕ φi

(4.53b)

The most apparent difference between the classical and modified formulations of the rigid finite element method is that: – In the classical formulation the mass matrix of the flexible link is diagonal, while the stiffness and damping matrices are full. – In the modified formulation the mass matrix is full and the stiffness and damping matrices are diagonal.

5 Calculations for a Cantilever Beam and Methods of Integrating the Equations of Motion

The formulations of the rigid finite element method presented in the previous chapters have assumed that rfes have the maximal number of degrees of freedom. In the classical formulation of the method each rfe has six degrees of freedom, while the modification of the method assumes that each rfe has three degrees of freedom in relative motion. Such a general approach can result in a large number of degrees of freedom when multibody systems are considered, and certain numerical problems can occur. In this chapter we discuss those problems by taking the simple example of a cantilever beam, in which the free vibrations caused by initial bending will be analysed. In further considerations we will use the following denotations for four formulations of the method presented in Chaps. 3 and 4: – Prefixes C and M mean classical (Chap. 3) and modified (Chap. 4) formulation of the rigid finite element method, respectively. – Suffixes N and L mean nonlinear and linear formulations of the method. According to this the results will be denoted as follows: – CRFEN when the classical formulation of the nonlinear model is considered. – CRFEL for the classical formulation and the linear model (assumption (p,i) are small). that angles ϕj – MRFEN when the modification of the method is used for the nonlinear model. – MRFEL for the modified formulation and linear model.

5.1 Equations of the Free Vibrations of a Beam The aim of this section is to demonstrate a simple example of practical use of the equations and models formulated in the previous chapters. For clarity of discussion, the free vibrations of the cantilever beam presented in Fig. 5.1a

104

5 Calculations for a Cantilever Beam

(a)

3 2

b

a

L (b)

1

3 2

L 2 (c)

rfe0

1

L 2

sde1

sde 2

rfe1

rfe 2

L 2

L 4

L 4

Fig. 5.1. Division of a cantilever beam into rfes and sdes when m = 2: (a) general view, (b) primary division, and (c) secondary division

are the subject of numerical analysis. The beam is divided into three rfes and two sdes as shown in Fig. 5.1b,c. Later, we discuss the detailed procedure which leads to formulation of the equations of motion of the system, first for CRFEN and CRFEL, and then for MRFEN and MRFEL (omitting index p designating the number of the link and assuming m = 2). At the end of this section, having discussed the methods of integrating the equations of motion, we will present a comparison of their numerical effectiveness. 5.1.1 Classical Rigid Finite Element Method Nonlinear Model In Fig. 5.2 the following coordinate systems are presented: {0} assigned to rfe 0, E {1} and E {2} assigned to rfe 1 and 2, respectively, in the undeformed state, {1}Y and {2}Y assigned to sde 1 and 2, respectively. Transformation matrices 0E T(1) and 0E T(2) defining the position of rfes 1 and 2 in relation to rfe 0 of the undeformed beam have the forms defined by (3.1): 0 (i) E (i)

0 (i) 0s EΘ i = 1, 2, (5.1) T = E 0 1 where 0 (1) EΘ

= I,

0 (1) Es

=

0 (2) EΘ

= I,

0 (2) Es

=

1

T

2L

00

8L

00

7

T

, .

5.1 Equations of the Free Vibrations of a Beam Yˆ 3(1)

Xˆ 3(0) Xˆ 2

Yˆ 2(1)

Xˆ 1(0)

Yˆ 3(2)

Eˆ 3(1)

(0)

Eˆ 2(1)

Yˆ 1(1)

Eˆ 1(1)

105

Eˆ 3(2)

Yˆ 2(2)

Eˆ 2(2)

Eˆ 1(2)

Yˆ 1(2)

L /4

L /8

L /4

L /2

L /4

Fig. 5.2. Coordinate systems assigned to rfes and sdes in the undeformed state (1) Xˆ 3

j

j 2(1)

(1) 3

(1) Xˆ 2

(1) Xˆ 1

j1(1)

x3(1) (1)

ˆ (1) E 2

x2(1)

j 3(2)

j 2(2)

(2) Xˆ 2

j

ˆ (2) E 3

ˆ (1) E 3 x1

(2) Xˆ 3

ˆ (1) E 1

x1(2)

(2) Eˆ 2

(2) 1

(2) Xˆ 1

x (2) 3 (2)

x2

(2) Eˆ 1

Fig. 5.3. Local coordinate systems {1} and {2} together with generalised coordinates of rfes

Coordinate systems {1} and {2} assigned to rfes and the generalised coordinates of rfes 1 and 2 are presented in Fig. 5.3. The vectors of the generalised coordinates of rfes 1 and 2 defining their orientation and position with respect to systems E {1} and E {2} take the following form: (1)

x (1) q = , (5.2a) ϕ(1) (2)

x q(2) = , (5.2b) ϕ(2) where ,T + (1) (1) , x(1) = x(1) 1 x2 x3 + ,T (2) (2) , x(2) = x(2) 1 x2 x3

,T + (1) (1) ϕ(1) = ϕ(1) , 1 ϕ2 ϕ3 + ,T (2) (2) ϕ(2) = ϕ(2) . 1 ϕ2 ϕ3

106

5 Calculations for a Cantilever Beam

The transformation matrices from the local coordinate systems {1} and {2} to systems E {1} and E {2} take analogical forms to those from (3.4): (i) (i)

R x , (5.3) T(i) = 0 1 where



(i) (i)

(i) (i) (i)

(i) (i)

(i) (i) (i)

(i) (i)

c3 c2 c3 s2 s1 − s3 c1 c3 s2 c1 + s3 s1



 (i) (i) (i) (i) (i)  (i) (i) (i) (i) (i) (i) (i)  R(i) =   s3 c2 s3 s2 s1 + c3 c1 s3 s2 c1 − c3 s1  , (i) (i) (i) (i) (i) −s2 c2 s1 c2 c1 (i)

(i)

(i)

(i)

cj = cos ϕj , sj = sin ϕj ; i = 1, 2; j = 1, 2, 3. Coordinate systems {1}Y and {2}Y , presented in Fig. 5.3, are assigned to sdes 1 and 2, respectively. Sde 1 connects rfe 0 with rfe 1, while sde 2 connects rfe 1 and 2. Therefore, using the formulae from Sect. 3.1, the following has to be assumed: kl = 10 = 0, kr = 11 = 1 when k = 1,

(5.4a)

kl = 21 = 1, kr = 22 = 2 when k = 2,

(5.4b)

where k is the number of the sde. The position of sde 1 and 2 in the local coordinate systems of rfes E {1} and E {2} is defined by the following matrices: E (1 ) E (1 )

Θ 0 Ys 0 E (10 ) , (5.5a) = Y YT 0 1 E (1 ) E (1 )

Θ 1 Ys 1 E (11 ) , (5.5b) = Y YT 0 1  E (21 ) E (21 ) E (21 ) Ys = YΘ , (5.5c) YT 0 1 E (2 ) E (2 )

2 2 E (22 ) YΘ Ys , (5.5d) T = Y 0 1 where

E (10 ) YΘ

(11 ) =E = I, Y Θ E (21 ) YΘ

(22 ) =E = I, Y Θ



E (10 ) Ys E (21 ) Ys

T 1 = L00 , 4

T 1 = L00 , 4

E (11 ) Ys E (22 ) Ys



T 1 = − L00 , 4

T 1 = − L00 . 8

(kl ) (kr ) Matrices E and E are identity matrices and this results in simYΘ YΘ plification of (3.10). Moreover, if we assume that rfe 0 is fixed and thus

5.1 Equations of the Free Vibrations of a Beam

107

R(0) = I and x(0) = 0, the displacements of the ends of rfes connected by sdes 1 and 2 can be presented with respect to coordinate systems {1}Y and {2}Y as follows:   (10 ) y(10 ) = R(0) − I E + x(0) = 0, (5.6a) Ys   (11 ) y(11 ) = R(1) − I E + x(1) , (5.6b) Ys   (21 ) + x(1) , y(21 ) = R(1) − I E Ys   (22 ) y(22 ) = R(2) − I E + x(2) . Ys

(5.7a) (5.7b)

Therefore the deformations of sdes 1 and 2 caused by the earlier displacements can be calculated, according to (3.12), as: , +   (11 ) ∆y(1) = y(10 ) − y(11 ) = 0 − R(1) − I E (5.8a) − x(1) , Ys ∆y(2) = y(21 ) − y(22 ) +    (21 ) = R(1) − I E + x(1) − R(2) − I Ys

E (22 ) Ys

,

− x(2) .

Taking into account (5.3) and (5.5), we obtain:    (1) (1) − L4 c3 c2 − 1     (11 ) R(1) − I E =  − L s(1) c(1) , Ys 2 4 3 L (1) 4 s2    (1) (1) L c c − 1 3 2   4   (21 ) =  L s(1) c(1) R(2) − I E , Ys 2 4 3 (1) − L4 s2    (2) (2) L c − 1 − c 3 2   8   (22 ) =  − L s(2) c(2) R(2) − I E , Ys 8 3 L (2) s 8 2

and then:  ∆y(1) =

∆y(2) =



(1) (1) L 4 c3 c2  L (1) (1)  s3 c2 4 (1) − L4 s2   (1) (1) L 4 c3 c2  L (1) (1)  s3 c2 4 (1) − L4 s2

 −1

 −1

(1)

−x1

(5.9a)

(5.9b)

(5.9c)

2



(1)  −x2  , (1) −x3 (1)

+x1

(1)

+x2 (1) +x3

(5.8b)

(5.10a)

  (2) (2) + L8 c3 c2 − 1 (2) (2)

+ L8 s3 c2 (2) − L8 s2

(2)

−x1

(2)

−x2 (2) −x3

   .(5.10b)

108

5 Calculations for a Cantilever Beam

The energy of spring deformation of sdes 1 and 2, according to (3.11), (3.12), (3.15) and (3.16), can be presented in the form: (1)

V = VY

(2)

+ Vϕ(1) + VY

+ Vϕ(2) ,

(5.11)

where (1)

VY

(2)

VY

,T + , 1 + (1) ,T (1) + (1) , (1) 1+ , Vϕ = ∆ϕ(1) , ∆y ∆ϕ(1) C(1) CY ∆y ϕ 2 2 , + , ,T + , + T 1 1+ (2) = ∆ϕ(2) , ∆y(2) CY ∆y(2) , Vϕ(2) = ∆ϕ(2) C(2) ϕ 2 2 =

∆y(1) , ∆y(2) are defined by (5.10),     (1) (1) (2) −ϕ1 ϕ1 − ϕ 1    (2)  ∆ϕ(1) =  −ϕ(1) , ∆ϕ(2) =  ϕ(1) , 2  2 − ϕ2  (1) (1) (2) −ϕ3 ϕ3 − ϕ 3 $ $ % % (1) (1) (1) (1) (1) (1) (1) CY = diag cY,1 , cY,2 , cY,3 , C(1) ϕ = diag cϕ,1 , cϕ,2 , cϕ,3 , % % $ $ (2) (2) (2) (2) (2) (2) (2) = diag c , c , c CY = diag cY,1 , cY,2 , cY,3 , C(2) ϕ ϕ,1 ϕ,2 ϕ,3 , (1)

(2)

(1)

(2)

cY,1 = cY,1 = cϕ,1 = cϕ,1 =

E A (1) GA (2) (1) (2) , c = cY,2 = cY,3 = cY,3 = , L/2 Y,2 L/2 G J0 (1) E J3 (1) E J2 (2) (2) , cϕ,2 = cϕ,2 = , cϕ,3 = cϕ,3 = , L/2 L/2 L/2

E is Young’s modulus, G is Kirchhoff’s modulus, A = ab, J2 = A

b2 a2 , J3 = A , J0 = J2 + J3 , 12 12

a, b are dimensions of beam cross-section (Fig. 5.1a). According to (3.49) and because of (5.11), one can obtain: ∂V (1) (2) = −CY ∆y(1) + CY ∆y(2) , ∂x(1) ∂V (2) = −CY ∆y(2) . ∂x(2)

(5.12a) (5.12b)

In order to use (3.50), first the elements of matrices G(11 ) , G(21 ) and G(22 ) have to be calculated. We obtain the following:

5.1 Equations of the Free Vibrations of a Beam

( T

G(11 ) = 

G(21 )

G(22 )

T

T

)T

(1) 3 " ∂Ri,l

E (11 ) s (1) Y l l=1 ∂ϕj

0

109

i,j=1,2,3

0

 0 L (1)  4 c2  , 0

 (1) (1) (1) (1) =  L4 s2 c3 L4 s2 s3 L (1) (1) L (1) (1) 4 c2 s3 − 4 c2 c3 )T ( 3 (1) " ∂Ri,l E (21 ) = s (1) Y l l=1 ∂ϕj i,j=1,2,3   0 0 0  (1) (1) (1) (1) (1)  =  − L4 s2 c3 − L4 s2 s3 − L4 c2  , L (1) (1) L (1) − 4 c2 s3 0 4 c2 ( 3 ) T (2) " ∂Ri,l E (22 ) = s (2) Y l l=1 ∂ϕj i,j=1,2,3   0 0 0  (2) (2) (2) (2) (2)  =  L8 s2 c3 L8 s2 s3 L8 c2  . (2) L (2) (2) − L8 c2 0 8 c2 s3

(5.13a)

(5.13b)

(5.13c)

Thus, having differentiated (5.10) with respect to ϕ(1) and ϕ(2) , we can write: T T ∂V (2) (2) (1) = G(11 ) CY ∆y(1) + G(21 ) CY ∆y(2) + C(1) ϕ ∆ϕ ∂ϕ(1) (2) +C(2) , (5.14a) ϕ ∆ϕ T ∂V (2) (2) = −G(22 ) CY ∆y(2) − C(2) . (5.14b) ϕ ∆ϕ ∂ϕ(2)

Taking into account (5.11), we can calculate:  ∂V ∂V ∂x(1) = , (1) ∂V ∂q ∂ϕ(1)  ∂V ∂V ∂x(2) = , ∂V ∂q(2) (2)

(5.15a)

(5.15b)

∂ϕ

where ∂V /∂x(1) and ∂V /∂x(2) are defined by (5.12), ∂V /∂ϕ(1) and ∂V /∂ϕ(2) are defined by (5.13). Bearing in mind considerations from Sect. 3.2, the kinetic energy of rfe 1 and 2 can be presented in the form: 1 $ ˙ (1) (1) ˙ (1)T % 1 $ ˙ (2) (2) ˙ (2)T % H B + tr B H B , (5.16) E = tr B 2 2

110

5 Calculations for a Cantilever Beam

where B(i) = 0E T(i) , T(i) , i = 1, 2; % $ (i) (i) (i) H(i) = diag h1 , h2 , h3 , m(i) , i = 1, 2; L2 L2 L L (1) (2) m(1) = ρ A , m(2) = ρ A , h1 = m(1) , h1 = m(2) , 2 4 48 192 a2 a2 b2 b2 (1) (2) (1) (2) h2 = m(1) , h2 = m(2) , h3 = m(1) , h3 = m(2) . 12 12 12 12 The Lagrange operators are as follows: εq(i) (E) = A(i) q ¨(i) + h(i) ,

(5.17)

(p,i)

where A(i) is defined as A(p,i),(p,i) in Table 3.1 after omission of index p, ,T + e(i) = e1(i) e2(i) e3(i) e4(i) e5(i) e6(i) , (i)

el

=

6 6 " "

% $ (i) (i)T (i) tr Bl H(i) Bs,j q˙s(i) q˙j

for l = 1, . . . , 6.

s=1 j=1

From Table 3.2 it follows that: (i)

(i)

(i)

e1 = e2 = e3 = 0, 6 6 " % $ " (i) (i) (i)T (i) el = tr Bl H(i) Bs,j q˙s(i) q˙j

(5.18a) for l = 4, 5, 6.

(5.18b)

s=4 j=4

Moreover, in the sums from (5.18b), only 16 components are nonzero. It should be noted that vector e(i) can be presented in the form: e(i) = L(i) q˙ (i) , where

  (i) L(i) = Ll,j (i)

Ll,j =

6 "

l,j=1,...,6

(5.19)

,

% $ (i) (i)T tr Bl H(i) Bs,j q˙s(i) , for l, j = 1, . . . , 6.

s=1

Taking into account (5.15), (5.17) and (5.19), having omitted forces of gravity, and assuming lack of external loads, the equations of motion for CRFEN can be written in the form: ∂V = 0, (5.20) ¨ + LC ˙+ AC N q Nq ∂q where AC N



∂V



(1)

 ∂q(1)  ∂V q A(1) 0 L(1) 0 C   ,q = = = , LN = , . ∂V  q(2) 0 A(2) 0 L(2) ∂q







(2)

∂q

5.1 Equations of the Free Vibrations of a Beam

111

5.1.2 Classical Rigid Finite Element Method Linear Model (i)

In this case we assume that angles ϕj (i = 1, 2; j = 1, 2, 3) are small and, according to Sect.3.5, the following form of the Lagrange operator is obtained: ¨(i) , εq(i) (E) = A(i) q

where A(i)

(5.21) % $ (i) (i) (i) (i) (i) (i) = diag m(i) , m(i) , m(i) , h2 + h3 , h1 + h3 , h1 + h2 ,

(i)

m(i) , hj are defined by (5.16). The potential energy of spring deformation of the sdes and the derivatives of the potential energy with respect to the generalised coordinates can be calculated, differently from CRFEN, by using (3.74)–(3.85). In the linear case analysed, vectors y(10 ) , y(11 ) , y(21 ) and y(22 ) from (5.6) and (5.7) can be presented as follows: y(10 ) = 0, (11 ) y(11 ) = x(1) +E ϕ(1) , Y s

(5.22a) (5.22b)

(21 ) y(21 ) = x(1) + E ϕ(1) , Ys (22 ) (2) E (22 ) y = x +Y s ϕ(2) ,

(5.23a) (5.23b)



where E kj Ys

 kj E kj −E 0 Y s3 Y s2  kj E kj  , = −E 0  Y s3 Y s1 E kj E kj s − s 0 Y 2 Y 1 kj ∈ {(11 ), (21 ), (22 )} .

kj Bearing in mind the forms of vectors E given in (5.5), we obtain: Ys   0 0 0 E (11 ) =  0 0 −L/4  , (5.24a) Ys 0 L/4 0   0 0 0 E (21 ) =  0 0 L/4  , (5.24b) Ys 0 −L/4 0   0 0 0 E (22 ) =  0 0 −L/8  . (5.24c) Ys 0 L/8 0

When 6 × 6 matrices are defined, as in (3.77), in the form: E k

I Ys j kj K = , 0 I

(5.25)

112

5 Calculations for a Cantilever Beam

we can write: u(11 ) = K(11 ) q(1) ,

(5.26a)

u(21 ) = K(21 ) q(1) , u(22 ) = K(22 ) q(2) ,

(5.26b) (5.26c)



where (11 )

u

(2 )

(2 )

y(11 ) y 1 y 2 (21 ) (22 ) = = = , u , u . (1) (1) ϕ ϕ ϕ(2)

The potential energy of the spring deformation of sdes 1 and 2 can now be written as follows: ,T + , 1 + (11 ) ,T (1) (11 ) 1 + (21 ) V = C u + − u(22 ) C(2) u(21 ) − u(22 ) , (5.27) u u 2 2 where  (k) CY 0 (k) C = (k) , k = 1, 2. 0 Cϕ In view of (5.26), differentiating (5.27) with respect to q(1) and q(2) , according to (3.79), yields: + , ∂V (11 )T (1) (11 ) (21 )T (2) (21 ) (22 ) u , (5.28a) = K C u + K C − u ∂q(1) + , T ∂V (5.28b) = −K(22 ) C(2) u(21 ) − u(22 ) . (2) ∂q Taking (5.26) into account again, we obtain:

(21 )T ∂V (11 )T (1) (11 ) (2) (21 ) = K C K + K C K q(1) ∂q(1) T

−K(21 ) C(2) K(22 ) q(2) ,

(5.29a)

∂V = −K(22 ) C(2) K(21 ) q(1) + K(22 ) C(2) K(22 ) q(2) . ∂q(2) T

T

(5.29b)

In view of (5.21) and (5.29) the equations of motion of the beam, according to the CRFEL model, can be written in the form: AC ¨ + CC Lq L q = 0, where

 AC L

=

A(1)

0

0 A(2)



, CC L

T

=

(5.30)

C CC L,11 CL,12 C CC L,21 CL,22 T

,

(11 ) CC C(1) K(11 ) + K(21 ) C(2) K(21 ) , L,11 = K T

C (21 ) (2) (22 ) C K , CC L,12 = CL,21 = −K T

(22 ) C(2) K(22 ) . CC L,22 = K

5.1 Equations of the Free Vibrations of a Beam

113

5.1.3 Modified Rigid Finite Element Method Nonlinear Model For the modified formulation of the rigid finite element method we will use the notation and procedure described in Chap. 4. Having discretised the beam, we obtain the system shown in Fig. 5.4 when m = 2. Rigid elements 1 and 0 as well as 2 and 1 are connected by means of rotational sdes 1 and 2, respectively, the orientation and position of which with respect to preceding rfes are defined by matrices with constant elements (4.1): (i) (i)

(i) ΘY sY i = 1, 2, (5.31) TY = 0 1 where (1)

(2)

ΘY = ΘY = I, T  (1) sY = L4 0 0 , T  (2) sY = L2 0 0 . The vectors of the generalised coordinates of rfes 1 and 2, defining their motion in relation to the previous rfe, are as follows:  (1)   (2)  ϕ1 ϕ    1(2)  (2) ϕ(1) =  ϕ(1) = , ϕ (5.32)  ϕ2  , 2  (1) (2) ϕ3 ϕ3 (i)

where ϕj are defined in Fig. 5.4. According to (4.3) and (4.4), the transformation matrices from coordinate systems {1} and {2} to coordinate systems {1}Y and {2}Y , respectively, take the form: (i)

* (i) = R 0 , B (5.33) 0 1 ˆ (2) ˆ (2) Y3 X 3

ˆ (1) ˆ (1) Y X 3 3 ˆ (0) X 3

(1)

ˆ (0) X 2

j3

(1)

j2

(2)

(2)

j3

ˆ (1) X 2 (1)

j1

ˆ (0) X 1

ˆ (1) Y 2 ˆ (1) X 1

j2

ˆ (2) X 2

ˆ (2) Y 2 (2)

j1

ˆ (2) X 1 ˆ (2) Y 1

ˆ (1) Y 1

Fig. 5.4. Rigid finite elements and their generalised coordinates for the modified formulation

114

5 Calculations for a Cantilever Beam

where R(i) is defined by (5.3), and the transformation to the coordinate system of rfe 0 is carried out by using the following matrices:

(1) (1) * (1) R(1) sY (1) B = TY B = , (5.34a) 0 1



(1) (2) (2) R(2) sY R(1) sY B(2) = B(1) TY B(2) = 0 1 0 1 (5.34b)

(2) (1) R(1) R(2) R(1) sY + sY = . 0 1 It is important to note that these matrices depend on: q(1) = ϕ(1) − elements of matrix B(1) , (1)

ϕ (2) q = − elements of matrix B(2) . ϕ(2)

(5.35a) (5.35b)

The kinetic energy of the system considered is defined by the expression: 1 $ ˙ (1) (1) ˙ (1)T % 1 $ ˙ (2) (2) ˙ (2)T % + tr B H B , (5.36) H B E = tr B 2 2 where 

H(i)

(i)  0 0 m(i) xc,1  (i) h2 0 0   i = 1, 2, (i)  0 h3 0 0 0 m(i) L L m(1) = ρA , m(2) = ρA , 2 4 2 L L2 (1) (2) h1 = m(1) , h1 = m(2) , 12 48 2 a a2 (1) (2) h2 = m(2) , h2 = m(1) , 12 12 2 b a2 (1) (2) h3 = m(2) , h3 = m(1) , 12 12 L L (1) (2) xc,1 = , xc,1 = , 4 8 A = a b. (i)

h1  0  =  0 (i) m(i) xc,1

Having proceeded as in (4.7)–(4.11), the Lagrange operator can be written in the form: εq (E) = AM ¨ + eM Nq N,

(5.37)

5.1 Equations of the Free Vibrations of a Beam

where

115



AM N (1)

a1,1,i,j (2)

a1,1,i,j (2)

a1,2,i,j (2)

a2,2,i,j

 (1) (2) (2) (1) (2) A11 + A11 A12 e1 + e1 M = , q = q(2) , (2) (2) , eN = (2) A21 A22 e2 % $ (1) (1)T , = tr Bi H(1) Bj $ % (2) (2)T = tr Bi H(2) Bj , % $ (2) (2) (2)T = a2,1,j,i = tr Bi H(2) B3+j , $ % (2) (2)T = tr B3+i H(2) B3+j , for i, j = 1, 2, 3;

(1)

3 3 " "

(2)

6 " 6 "

(2)

6 " 6 "

e1,i =

$ % (1) (1)T tr Bi H(1) Bj,l q˙j q˙l ,

j=1 l=1

e1,i =

$ % (2) (2)T tr Bi H(2) Bj,l q˙j q˙l ,

j=1 l=1

e2,i =

$ % (2) (2)T tr B3+i H(2) Bj,l q˙j q˙l ,

for i = 1, 2, 3.

j=1 l=1

Just as for the CRFEN model, vector hM N can be presented in the form: M hM ˙ N = L q,

(5.38)

where LM = LM (q, q) ˙ is the matrix, the elements of which can be calculated in the way described by Adamiec (1992). The potential energy of the spring deformation of sdes 1 and 2 can be written as follows: V =

1 (1)T (1) (1) 1 (2)T (2) (2) ϕ Cϕ ϕ + ϕ Cϕ ϕ , 2 2

and thus:

∂V = CM q, ∂q 

where M

C

=

(5.39)

(5.40)

(1) Cϕ 0 (2) . 0 Cϕ

The equations of motion of the system analysed, after omitting forces of gravity and external loads, take the form: M M AM N q + eN + C q = 0.

(5.41)

116

5 Calculations for a Cantilever Beam

5.1.4 Modified Rigid Finite Element Method Linear Model (i)

Since angles ϕi are assumed to be small, the rotation matrices R(i) are given in (4.23):   (i) (i) 1 −ϕ3 ϕ2  (i)  R(i) =  ϕ3(i) (5.42) 1 −ϕ1  for i = 1, 2. (i) (i) −ϕ2 ϕ1 1 * (i) , defined in (5.33), as: This enables us to write matrices B * (i) = B

3 "

(i)

Lj ϕj ,

(5.43)

j=1

    

00 0 0 01 0 −1 Rj 0 , R1 =  0 0 −1  , R2 =  0 0 0  , R3 =  1 0 where Lj = 0 0 01 0 −1 0 0 0 0

 0 0 . 0

(i)

Matrices B , defined as in (4.4) and describing the transformation from rfe i to the preceding rfe, can also be presented in the form: (i)

B

(i) (i) * (i) = TY B = L0 +

3 "

(i)

(i)

Lj ϕj ,

(5.44)

j=1 (i)

(i)

(i)

(i)

where L0 = TY , Lj = TY Lj . (1)

(2)

Having omitted products ϕj ϕj of the second order, the transformation matrices from the coordinate systems of rfes 1 and 2 to the coordinate system of rfe 0, according to (4.27), are matrices with constant elements and take the forms: 3 " (1) (1) (1) (1) L1,j ϕj , (5.45a) B(1) = 0 B(1) = B = L0 + j=1

 B(2) = 0 B(2) = B

(1)

B

(2)

=  L0 + (1)

3 "



(2) (1) (1) L1,j ϕj  L0 +

j=1 (2)

≈ L0 +

2 " 3 "

3 "

 (2) (2) Lj ϕj 

j=3 (2)

(k)

Lk,j ϕj ,

k=1 j=1

(5.45b) where (1)

L(1) s = Ls (2)

(1)

for s = 0, 1, 2, 3, (2)

L0 = L0 L0 ,

5.1 Equations of the Free Vibrations of a Beam

L1,j

 (1) (2)  =L L

(2) L2,j

=

(2)

1,j 0 (1) (2) L0 Lj 

117

for j = 1, 2, 3.

In the case considered, when rfe 0 is fixed, the components of the equations of motion resulting from the kinetic energy can be derived in a simpler way than that presented in Sect. 4.5 (4.33)–(4.52). The kinetic energy of rfes 1 and 2 can be expressed as: 1 $ ˙ (i) (i) ˙ (i)T % i = 1, 2, (5.46) H B E (i) = tr B 2 where H(i) is defined by (5.36). From (5.45) it follows that: ˙ (1) = B

3 "

(1)

(1)

L1,j ϕ˙ j ,

(5.47a)

j=1

˙ (2) = B

2 " 3 "

(2)

(k)

Lk,j ϕ˙ j ,

(5.47b)

k=1 j=1

and thus: E (1) =

3 3 1 $ ˙ (1) (1) ˙ (1)T % 1 " (1) " (1) $ (1) (1) (1)T % = tr B H B ϕ˙ i ϕ˙ j tr L1,l H L1,j 2 2 j=1 l=1

= E (2) = =

=

1 (1)T (1) (1) ˙ , ˙ A ϕ ϕ 2 1 $ ˙ (2) (2) ˙ (2)T % tr B H B 2 3 2 3 2 1 " " (s) " " (k) $ (2) (2) (2)T % ϕ˙ l ϕ˙ j tr Ls,l H Lk,j 2 s=1 l=1 k=1 j=1  3 3 1 " (1) " (1) $ (2) (2) (2)T % ϕ˙ l ϕ˙ j tr L1,l H L1,j 2 j=1

(5.48a)

l=1

+

3 "

(1)

ϕ˙ l

+

3 "

$ % (1) (2) (2)T ϕ˙ j tr L2,l H(2) L1,j

l=1

j=1

3 "

3 " (2)

 % $ T (2) (2) (2) ϕ˙ j tr L2,l H(2) L2,j 

3 "

(2)

ϕ˙ l

ϕ˙ l

j=1

l=1

=

% $ (2) (2) (2)T ϕ˙ j tr L1,l H(2) L2,j

j=1

l=1

+

3 "

T 1 (1) (2) (1) (2) (2) ˙ ˙ (1) A1,2 ϕ ˙ [ϕ A1,1 ϕ˙ + ϕ 2 T

T

(2)

T

(2)

˙ (2) A2,1 ϕ ˙ (2) A2,2 ϕ ˙ (1) + ϕ ˙ (2) ], +ϕ

(5.48b)

118

5 Calculations for a Cantilever Beam

where

% $ (1) (1) (1)T al,j = tr L1,l H(1) L1,j , $ % (2) (2) (2)T a1,1,l,j = tr L1,l H(2) L1,j , % $ (2) (2) (2)T a1,2,l,j = tr L1,l H(2) L2,j , $ % (2) (2) (2)T a2,1,l,j = tr L2,l H(2) L1,j , % $ (2) (2) (2)T a2,2,l,j = tr L2,l H(2) L2,j , for l, j = 1, 2, 3. (2)

(2)

In view of (5.48), since matrices A(1) , A1,1 and A2,2 are symmetrical, we obtain: ∂E (1) ˙ (1) ∂ϕ ∂E (2) ˙ (1) ∂ϕ ∂E (2) ˙ (2) ∂ϕ

˙ (1) , = A(1) ϕ

(5.49a)

1 (2) (2)T ˙ (2) , A1,2 + A2,1 ϕ 2 , 1 + (2) (2) (2) (2) ˙ , ˙ (1) + A2,2 ϕ = A1,2 + A2,1 ϕ 2 (2)

˙ (1) + = A1,1 ϕ

(5.49b) (5.49c)

and then: εϕ(1) (E (1) ) =

d ∂E (1) ∂E (1) ¨ (1) , − = A(1) ϕ dt ∂ ϕ ∂ϕ(1) ˙ (1) (2)

(2)

¨ (1) + A1,2 ϕ ¨ (2) , εϕ(1) (E (2) ) = A1,1 ϕ εϕ(2) (E

(2)

)=

(2) (1) ¨ A2,1 ϕ

+

(2) (2) ¨ , A2,2 ϕ

(5.50a) (5.50b) (5.50c)

where (2)

(2)

(2)

(2)T

(2)

(2)

A1,1 = A1,1 , A1,2 = A2,1 =

1 2

  (2) (2)T A1,2 + A2,1 ,

A2,2 = A2,2 . Because the kinetic energy is the sum: E = E (1) + E (2)

(5.51)

εq (E) = AM ¨, Lq

(5.52)

in view of (5.50) one obtains:

5.2 Integrating the Equations of Motion



where AM L =

119

(1)

(2) (2) A(1) + A1,1 A1,2 ϕ . (2) (2) , q = ϕ(2) A2,1 A2,2

The potential energy of deformation of the sde is expressed in the same way as in the nonlinear model MRFEN, by (5.39), and its derivatives by (5.40). Therefore the equations of motion for the MRFEL model take the form: ¨ + CM q = 0. AM Lq

(5.53)

5.2 Integrating the Equations of Motion In order to demonstrate numerical problems which can occur when the rigid finite element method is used, let us consider the free vibrations of the beam from Fig. 5.1 and assume that m = 1. In this case the system shown in Fig. 5.5b or c is obtained, and thus rfe 1 with length L/2 is connected with a fixed rfe 0 by a spring-damping element. Parameters of the beam are given in Table 5.1. If we assume that angles ϕ1 , ϕ2 , ϕ3 are small, then according to Sects. 3.5 and 4.5 we can formulate the problem of the free vibrations for rfe 1 connected by the sde with the fixed rfe 0. Both formulations of the problem, classical and modified, are presented later. (1) Classical Formulation of the Method – CRFEL Proceeding as in Sect. 3.5, the equation of the free vibrations can be written in the form: ¨C + CC qC = 0, AC q (a)

(5.54)

3 2

b 1

a L j2 (b) x1

j3

x3 L/4

j1

x2 j2

(c)

j3

j1

Fig. 5.5. Division of a cantilever beam into rfe 0 and rfe 1: (a) primary division, (b) generalised coordinates of rfe 1 for the classical formulation, and (c) generalised coordinates of rfe 1 for the modified formulation

120

5 Calculations for a Cantilever Beam Table 5.1. Parameters of the beam parameter length, L width, a height, b Young’s modulus, E Kirchhoff’s modulus, G density

unit m m m MPa MPa kgm−3

value 1 0.020 0.002 2.1 × 105 0.8 × 105 7,850

 (C) (C) where AC = diag{m, m, m, h2 +h3 , h1 +h3 , h1 +h2 }, qC = x1 T ϕ1 ϕ2 ϕ3 ,  cY,1 0 0 0 0 0  0 cY,2 0 0 0 −cY,2 L4   0 0 cY,3 0 cY,3 L4 0 C C =  0 0 0 0 0 cϕ,1  2  0 0 cY,3 L4 0 cϕ,2 + cY,3 L16 0 2 L 0 0 cϕ,3 + cY,2 L16 0 −cY,2 4 0

x2

x3

    .   

Since the beginning of the local coordinate system of rfe 1 is placed in the centre of mass, the elements of the mass matrix can be calculated from the formulae: L m=ρab , 2

h1 = m

L2 , 48

h2 = m

a2 , 12

h3 = m

b2 12

(5.55)

and coefficients cY,1 , cY,2 , cY,3 and cϕ,1 , cϕ,2 , cϕ,3 according to the relations given in Sect. 3.6: cY,1 =

EA GA GJ0 EJ2 EJ3 , cY,2 = cY,3 = , cϕ,1 = , cϕ,2 = , cϕ,3 = , L L L L L (5.56)

where A = a b, J2 = A(b2 /12), J3 = A(a2 /12), J0 = J2 + J3 . (2) Modified Formulation of the Method – MRFEL The equation of the free vibrations of the system, according to (4.5), takes the following form: ¨M + CM qM = 0, AM q (M)

(M)

(5.57) T  = ϕ1 ϕ 2 ϕ 3 ,

where AM = diag{h2 + h3 , h1 + h3 , h1 + h2 }, qM CM = diag{cϕ,1 , cϕ,2 , cϕ,3 }. The elements of matrices AM and CM are calculated as in (5.55) and (M) (5.56) except for element h1 , which, since the beginning of the local coordinate system of rfe 1 coincides with the sde, is: (M)

h1

=

mL2 . 12

(5.58)

5.2 Integrating the Equations of Motion

121

Table 5.2. Frequencies of free vibrations i 1 2 3 4 5 6

ωi (Hz) classical formulation modified formulation 14.6 14.6 146.2 146.3 4,514.7 4,514.7 7,314.6 9,027.5 9,029.3

type of vibrations bending in plane (1,3) bending in plane (2,3) torsional longitudinal

For both formulations of the method (classical and modified) the frequencies of the free vibrations are calculated from the equation: −1

det([Aα ]

Cα − ω 2 I) = 0,

(5.59)

where α ∈ {C, M }. The numerical values of the frequencies calculated using both formulations of the method are given in Table 5.2. The analytically evaluated frequencies of the free vibrations of the cantilever beam in Fig. 5.1a differ from those in the table earlier by as much as 40%, which is the result of assuming that m = 1. The influence of number m on the accuracy of results obtained will be discussed in Chap. 6. However, analysis of the results from Table 5.2 shows that preserving all degrees of freedom of rfes, both in the classical and modified formulations of the method, leads to frequencies which may be one or two orders of magnitude larger than the lowest frequency. In analysis of forced vibrations, frequencies differing by orders of magnitude in the system analysed result in so-called stiff differential equations (Press et al., 2002; Chapra and Canale, 2002). In order to solve such equations, special methods have to be used, which enable us to eliminate the influence of high frequencies. Let us call this the numerical approach. The problem of eliminating the influence of high frequencies can also be addressed from the point of view of mechanics. The analysis of the numerical parameters of the beam shows that bending flexibility in plane (1, 3) is dominant. Thus we could have reduced the system considered so that only bending vibrations in plane (1, 3) were analysed. For the classical formulation of the method the following could have been assumed: T  (5.60) qC = x3 ϕ2 , while for the modified formulation: qM = [ϕ2 ] .

(5.61)

In the classical formulation of the method two values of frequencies which correspond to ω1 and ω5 from Table 5.2 are obtained, while for the modification of the method only one ω1 is obtained. In each case the number of

122

5 Calculations for a Cantilever Beam

degrees of freedom decreases to a third of the maximum number, which for large values of m can result in a considerably shorter time of calculation. For the classical method the mechanical approach does not eliminate the problem of high frequencies. Thus the numerical approach has to be used. Later we discuss methods of integrating stiff differential equations and the reduction of the generalised coordinates. The equations of motion of multibody systems containing flexible links, when using linear models, can be written in the form: A¨ q + Dq˙ + Cq = g(t, q, q), ˙

(5.62)

where q is the vector of generalised coordinates, A, D, C are mass, damping and stiffness matrices, respectively, g = g(t, q, q) ˙ is the vector of forces and moments. When large deformations are considered, the equations of motion can also be written in the form (5.62), but matrices A, D, C may depend on generalised coordinates and generalised velocities (Wojciech, 1984; AdamiecW´ ojcik, 1992). In this case the equations of motion are more often written as follows: A¨ q = F(t, q, q), ˙ (5.63) where F = g − Dq˙ − Cq. Equations (5.62) or (5.63) are usually supplemented with initial conditions in the form: q|t=t0 = q0 , q| ˙ t=t0 = q˙ 0 ,

(5.64a) (5.64b)

where t0 is the time calculations begin, q0 , q˙ 0 are initial values of the vectors of generalised coordinates and velocities. Equations (5.63) of the second order can be transformed to the classical problem of solving sets of differential equations of the first order in the form: y˙ = f (t, y), where

(5.65)



q y= , q˙

q˙ . f = A−1 F

Initial conditions (5.64) can then be written as:

q0 y|t=t0 = y0 = . q˙ 0

(5.66)

5.2 Integrating the Equations of Motion

123

Below the Newmark method (Bathe and Wilson, 1976) is described, as representative of a family of methods used when the equations of motion in the form (5.62) are integrated. Subsequently we discuss the following: Euler and Runge–Kutta methods, problems with the choice of the integration step, and integration of stiff systems (Press et al. 2002). 5.2.1 Newmark Method When the equations of motion are given in the form (5.62), and the following vectors are known: qt = q(t), ˙ q˙ t = q(t), ¨(t), q ¨t = q

(5.67)

where h is the integration step, then (5.62) at time t + h is solved by assuming that:   1 − α h2 q ¨ t + α h2 q ¨t+h , (5.68a) qt+h = qt + h q˙ t + 2 q˙ t+h = q˙ t + (1 − δ) h q ¨t + δ h q ¨t+h , (5.68b) where α, δ are coefficients introduced to increase accuracy and stability of the method. Substituting (5.68) into (5.62), we can calculate q ¨t as the solution of the set of algebraic equations:   ¨t+h = gt+h , (5.69) A + δ h D + α h2 C q where gt+h = g − D [q˙ t + (1 − δ) h q ¨t ] − C

 

1 − α h2 q qt + h q˙ t + ¨t . 2

Then qt+h and q˙ t+h can be calculated from (5.68). This method is unconditionally stable when α = 1/4 and δ = 1/2. The procedure presented requires a certain modification when the elements of matrices A, D, C and vector g depend on q and q. ˙ In such a case the iterative procedure presented by Wojciech (1984) can be used. 5.2.2 Euler and Runge–Kutta Methods Numerical methods for integrating differential equations are usually discussed when one differential equation and the corresponding initial problem in the form: dy = f (t, y) , dt = y0

y˙ = y|t=t0

(5.70a) (5.70b)

124

5 Calculations for a Cantilever Beam

are considered because of easy geometrical interpretation of the method of solution. Generalisation to the system in the form (5.63) is normally not difficult. The initial problem considered is to calculate: y1 = y (t0 + h) ,

(5.71)

where h is the step of integration, if y0 from (5.70b) is known. In the Euler method, the simplest but least accurate, the following is assumed:   (5.72) y1 = y0 + hf (t0 , y0 ) + O h2 .  2 The errors of the method are of O h , and only a derivative at the beginning of the interval t0 , t0 + h is used. The graphical interpretation of the method is presented in Fig. 5.6a. Another method, whose accuracy is higher by one order, is the midpoint method in which one calculates: k1 = hf (t0 , y0 ) ,   1 1 k2 = hf t0 + h, y0 + k1 , 2 2  3 y1 = y0 + k2 + O h . (a)

(5.73a) (5.73b) (5.73c)

y y(t)

y1

y ⬘= tga = f (t 0, y 0) y0 h

hf (t 0, y 0) t

t0 (b)

t1 = t0 + h hf (t 0, + 0.5h, y 0 + 0.5k1)

y

0.5k1 = h f (t 0, y 0) 2

y1

y0 h 2 t0

h 2

t t1 = t0 + h

Fig. 5.6. Calculation of y1 = y(t0 + h) by means of: (a) Euler method and (b) midpoint method

5.2 Integrating the Equations of Motion

125

Graphical interpretation of the method is shown in Fig. 5.6b. In this method, a tentative step in the middle of interval t0 , t0 + h is made, and then values t and y at the middle point are used in order to carry out calculation for the step over the whole interval (Legras, 1973). The Euler method is a method of the first order while the midpoint method is of the second order. This means that the results are accurate when function y (t) is linear or parabolic respectively. The midpoint method is also called the second-order Runge–Kutta method (Ralston, 1971). It can be easily seen that calculation of the solution y1 = y (t0 + h) requires the value of function f to be evaluated: – Only once in the Euler method. – Twice in the midpoint method. The observation that adding combinations of different coefficients in calculating the right-hand side y (t0 + h) eliminates the error terms order by order has become the basis for the family of methods of different orders which are called Runge–Kutta methods. The most popular is the Runge–Kutta method of the fourth order, in which the following is calculated: k1 = hf (t0 , y0 ) ,   1 1 k2 = hf t0 + h, y0 + k1 , 2 2   1 1 k3 = hf t0 + h, y0 + k2 , 2 2 k4 = hf (t0 + h, y0 + k3 ) , and then y1 = y0 +

  k2 k3 k4 k1 + + + + O h5 . 6 3 3 6

(5.74a)

(5.74b)

In this method, the error of which is of h5 order, the value of function f has to be evaluated four times when calculating the solution at t0 + h. This can be a serious drawback of the method, when processor time connected with calculating f is significant. There are many methods belonging to extrapolation interpolation methods (Legras, 1973; Ralston, 1971) in which calculations of the value of function f are reduced to a minimum. However, those methods require that we know a solution not only at one point (like the methods described here) but at t0 and at a certain number of earlier points, and although these methods are more and more popular we will not discuss them in this book. 5.2.3 Step-Size Control When differential equations describing dynamics of multibody systems are integrated, it is often the case that the course of a function contains both smooth and abruptly changing regions. Such a situation is presented in Fig. 5.7.

126

5 Calculations for a Cantilever Beam y

tA

tB

tC

tD

t

Fig. 5.7. Exemplified course of y(t). t ∈ tA , tB  ∪ tC , tD  – smooth course and t ∈ tB , tC  – abruptly changing course

It is easy to find a physical interpretation for such courses, for example when a force or moment is acting only at a certain interval. Even rough analysis of the course presented in Fig. 5.7 shows that because of the smooth course of function y for t ∈ tA , tB  ∪ tC , tD  the step of integration of (5.70), which would ensure proper accuracy, can be considerably larger than that for t ∈ tB , tC . If we assume that the initial problem (5.70) is integrated with a constant step of integration, it has to be so small that the solution is accurate for all t ∈ tA , tD , although the integration of (5.70) for t ∈ tA , tB  ∪ tC , tD  could be carried out with a much larger step than that for t ∈ tB , tC . A possible solution is to use a changeable step of integration, larger when the function is smooth and smaller when it is “violent”, and the best situation is when the algorithm itself is able to decide the length of the integration step. Some methods of estimation of local errors and the choice of the integration step are described later, based on (Press et al. 2002). When estimating the integration step two approaches can be used: stepdoubling and Fehlberg methods. We discuss both approaches of examining truncation errors, using the fourth-order Runge–Kutta method as an example. We will also present the modified midpoint method, which allows the integration step to be estimated at the time of calculation. Step-Doubling Method In this method each step is calculated twice, first with a step hand then as two independent steps with half of the length of step h (Fig. 5.8). It can be seen from Fig. 5.8 that in order to calculate the solution with one large and two small steps, three separate steps in the fourth-order Runge– Kutta method have to be carried out, and each of them requires function f (t) to be evaluated four times. Since both large and small steps use the same initial value f (t0 , y0 ), function f (t) is evaluated 11 times. This number should be

5.2 Integrating the Equations of Motion

127

Large step

Two small steps

Fig. 5.8. Points at which values of the function in the fourth-order Runge–Kutta method are evaluated

referenced to eight (two small steps) in order to find the coefficient of increase in calculation time necessary for the comparison of results obtained with one large and two small steps, and thus the decision about decreasing, increasing or keeping the integration step. The coefficient connected with algorithms controlling the length of integration step in the case considered is 11/8 = 1.37. Let us denote: (5.75a) y (t0 + 2h) as an exact solution of (5.70) at t0 + 2h, y1 as an approximate solution at t0 + 2h obtained with one integration step of 2h, (5.75b) y2 as an approximate solution at t0 + 2h obtained with two integration steps of h. (5.75c) Since we consider the fourth-order Runge–Kutta method, the following relations hold:   5 (5.76a) y (t0 + 2h) = y1 + (2h) φ + O h6 ,  6 5 (5.76b) y (t0 + 2h) = y2 + 2 (h) φ + O h , where φ is a constant value for the integration step carried out, of y (5) (t0 )/5! order, which results from the Taylor series. In the expression (5.76a) (2h)5 occurs because y (t0 + 2h) is calculated with the integration step of length 2h, while in (5.76b) 2(h)5 occurs because two integration steps are carried out, each of length h and each with h5 φ error. The difference between approximations y1 and y2 : ∆ = y2 − y1

(5.77)

is a good coefficient for the truncation errors. During calculations this difference has to be kept at a certain level in order to retain the required accuracy. Having omitted expressions of order h6 and higher in (5.76), and subtracting by sides, we obtain: 0 = y2 − y1 − 30h5 φ,

(5.78a)

  y2 − y1 ∆ = . 2 h5 φ = 15 15

(5.78b)

and thus:

128

5 Calculations for a Cantilever Beam

Substituting the earlier value into (5.77), approximation y2 of the exact solution can be improved and the following is obtained: y (t0 + 2h) = y2 +

  ∆ + O h6 15

(5.79)

and thus one obtains the accuracy of the fifth order, which is one order higher than in the fourth-order Runge–Kutta method. Runge–Kutta–Fehlberg Method An important and interesting feature of the Runge–Kutta methods is that for formulae of order M (larger than 4) the values of function f have to be evaluated M + 1 or M + 2 times in order to calculate the solution at t0 + h. As was shown before, using the fourth-order Runge–Kutta requires the value of function f to be evaluated four times (thus order M of the method equals the number of calculations of function f , which is also M ). This is another reason for the popularity of this method. Fehlberg invented a Runge–Kutta method of the fifth order in which function f is evaluated at six points, but at the same time, by a different combination of those values, the Runge–Kutta method of the fourth order is obtained. The difference between approximations y (t0 + h) obtained using the methods of the fourth and fifth order can then be used to control the size of the integration steps. The general form of the fifth-order Runge–Kutta method is: k1 = hf (t0 , y0 ) , k2 = hf (t0 + α2 h, y0 + b21 k1 ) , k3 = hf (t0 + α3 h, y0 + b31 k1 + b32 k2 ) , k4 = hf (t0 + α4 h, y0 + b41 k1 + b42 k2 + b43 k3 ) ,

(5.80a)

k5 = hf (t0 + α5 h, y0 + b51 k1 + b52 k2 + b53 k3 + b54 k4 ) , k6 = hf (t0 + α6 h, y0 + b61 k1 + b62 k2 + b63 k3 + b64 k4 + b65 k5 ) ,   y (t0 + h) = y0 + c1 k1 + c2 k2 + c3 k3 + c4 k4 + c5 k5 + c6 k6 + O h6 . (5.80b) If the method of the fourth order, using the same values of f as the fifthorder Runge–Kutta, has the form:   y ∗ (t0 + h) = y0 + c∗1 k1 + c∗2 k2 + c∗3 k3 + c∗4 k4 + c∗5 k5 + c∗6 k6 + O h6 (5.80c) then the error can be calculated as follows: ∆ = y (t0 + h) − y ∗ (t0 + h) =

6 "

(ci − c∗i )ki .

(5.81)

i=1

The values of coefficients ai , bij , ci , c∗i have been calculated by Fehlberg. However, Cash and Karp have given different values of these coefficients, for which

5.2 Integrating the Equations of Motion

129

Table 5.3. Cash and Karp values of coefficients i

ai

bij

1 2 3 4

1 5 3 10 3 5

5

1

6

7 8

j=

1 5 3 40 3 10 11 − 54 1, 631 55, 296 1

9 40 9 − 10 5 2 175 512 2

6 5 70 − 27 575 13, 824 3

35 27 44, 275 110, 592 4

ci 37 378

c∗i 2, 825 27, 648

0

0

250 621 125 594

18, 575 48, 384 13, 525 55, 296 277 14, 336 1 4

0 253 4, 096 5

512 1, 771

the formulae are more efficient and are used in computer implementations of the method. The values of the coefficients are listed in Table 5.3 (Press et al. 2002). Thus, no matter which method for estimation of the error we use (doubling the step or Fehlberg) error ∆ of calculating the solution at t0 + h is known approximately. Now the problem is how to keep this error in given limits, and what the relation is between error ∆ and integration step h. Let us assume the following denotations: ∆0 , h0 are the actual error and the integration step, respectively, ∆1 is the assumed error of calculation, h1 is the new integration step. Therefore, since ∆ is of order h5 , the following should be assumed according to (Press et al. 2002): 5 5 5 ∆1 5α 5 , 5 h1 = h0 · 5 (5.82) ∆0 5  0.2 ∆0 ≤ ∆1 where α = if . 0.25 ∆0 > ∆1 The decision about the size of the new integration step h1 obviously depends on the assumed accuracy ∆1 . Chapra and Canale (2002) as well as Press et al. (2002) describe how the earlier algorithm of controlling the stepsize can be used for the set of equations (5.65), and how to proceed when y(t) is a function with very small values (those of truncation error order). Modified Midpoint Method This method enables us to calculate y (t0 + H) as a sequence of successive steps with length: h=

H . n

(5.83)

130

5 Calculations for a Cantilever Beam

In order to determine y (t0 + H), the following has to be calculated (Press et al. 2002): z0 = y(t0 ), z1 = z0 + hf (t0 , z0 ) , zm+1 = zm−1 + 2hf (t0 + mh, zm ) for m = 1, 2, . . . , n − 1,

(5.84a)

and then: y (t0 + H) ≈ yn =

1 [zn−1 + zn + hf (t0 + H, zn )] . 2

(5.84b)

Graphical illustration of this procedure is shown in Fig. 5.9. This method is based on the midpoint method Eq. (5.73) and is a method of the second order. The essential difference is that the derivative of function f at each point is evaluated only once instead of twice as it is in the classical midpoint method. Press et al. (2002) demonstrated that because of the features of the series approximating the truncation errors in this method, the approximation: 4yn − yn/2 , (5.85) y (t0 + H) = 3 where n is an even number, yn is the value of y (t0 + H) when step h from (5.83) is assumed, and yn/2 is the value of y (t0 + H) for step 2h, has an accuracy of the fourth order, the same as for the fourth-order Runge–Kutta method. An essential advantage of the method described is that in order to calculate y (t0 + H) according to (5.85), the value of function f has to be evaluated only n + 1/2n times, while those values have to evaluated 4n times when the fourth-order Runge–Kutta method is used. This method, after some small modifications, has become the basis for the Bulirsch–Stoer–Deuflhard method (Press et al. 2002), in which in order to cross the interval H, the even

y 2hf (t 0 + m,h,zm)

ti = t 0 + ih

z0 h t0

z3 h

t1

zn-1

z2

z1

zm-1

zm

h

h t2

hf (t 0 + H,zn)

t3

tm-1

h tm

zn

zm+1

h tm+1

Fig. 5.9. Modified midpoint method

t n - 1 tn = t 0 + H

5.2 Integrating the Equations of Motion

131

values of n (number of substeps) are used, defined by: nk = 2k {2, 4, 6, 8, 10, 12, 14, . . .} .

(5.86)

Values y (t0 + H) are calculated in separate attempts for the successive values from the above set, as long as the error between yn and yn/2 is appropriately small. Usually the maximal value of k is no bigger than 8. The details for choice of step H are quite complex (Press et al. 2002) and will not be discussed here. 5.2.4 Stiff Systems of Differential Equations A system of differential equations is stiff when it contains rapidly changing components together with slowly changing ones. In many technical problems (including dynamics) rapidly changing components occur initially, which subsequently decay and are dominated by slowly changing components. Although those rapidly changing components may influence the solution only in a short interval of integration, they can be decisive for the step-size control. Not only systems of differential equations but also a single equation can be stiff. Chapra and Canale (2002) consider the following equation: dy = −1, 000y + 3, 000 − 2, 000 e−t . dt

(5.87)

If we assume the initial condition as y (0) = 0, analytical solution of (5.87) has the form: (5.88) y = 3 − 0.998 e−1000t − 2.002 e−t . As shown in Fig. 5.10, for t ≤ 0.005, the solution is dominated by the first exponential term e−1000t and then this component fades and term e−t dominates the solution. Press et al. (2002) present a simple example of a stiff system of differential equations: dU = 998U − 1998V, dt dV = −999U − 1999V. dt

(5.89a) (5.89b)

For the following initial conditions: U (0) = 1, V (0) = 0,

(5.90a) (5.90b)

the following is the solution of system (5.89): U = 2 e−t − e−1000t , V = −e

−t

+e

−1000t

.

(5.91a) (5.91b)

132

5 Calculations for a Cantilever Beam y

3

2 1

1

0

1

0.01 2

0.02 3

0

4 x

Fig. 5.10. Solution of a single stiff equation (5.87)

As in the case of (5.88), rapidly and slowly changing components are easy to notice. When integrating (5.89), term e−1000t requires the integration step size h 0 is constant. In Sect. 5.2.2 the Euler method is discussed. This method is usually defined in two forms: (1) As an explicit Euler scheme, in which: yn+1 = yn + h yn = yn + h f (tn , yn ) ,

(5.93a)

where yn is treated as y0 from (5.72) and yn+1 is treated as y1 from (5.72). (2) As an implicit Euler scheme, in which:  = yn + h f (tn+1 , yn+1 ) . yn+1 = yn + h yn+1

(5.93b)

The basic difference between these schemes is that the explicit scheme (5.93a) allows us to calculate yn+1 on the basis of the value at tn , while the im = plicit scheme (5.93b) requires a solution of a nonlinear equation since yn+1 f (tn+1 , yn+1 ) depends on the actually calculated value yn+1 .

5.2 Integrating the Equations of Motion

133

Both schemes are also different in other ways. When we use the explicit scheme for (5.92), we obtain the following: yn+1 = yn + h yn = (1 − c h) yn .

(5.94)

In order to ensure stability of the solution the following condition must hold: |1 − ch| < 1, (5.95) which means that the integration step must be: h
2/c, limn→∞ yn = ∞. If we use the implicit scheme for (5.92), then:  = yn − hcyn+1 , yn+1 = yn + h yn+1

(5.97)

and thus:

yn . (5.98) 1 + ch The method is absolutely stable since the denominator on the right is larger than 1. Even when h → ∞, yn+1 tends to zero, which is the exact solution of (5.92). The earlier considerations can easily be generalised for systems of linear differential equations with constant coefficients in the form: yn+1 =

dy = −C y, dt

(5.99)

where C is a positive definite matrix, y = [ y1 · · · ym ]T . Having used the explicit scheme for (5.99), we obtain: yn+1 = (I − C h) yn .

(5.100)

yn is bounded when n → ∞ only if the largest eigenvalue of matrix I − C h is smaller than 1, which means that the stability condition for the explicit scheme is: 2 , (5.101) h< λmax where λmax is the largest eigenvalue of matrix C. Use of the implicit scheme for (5.99) leads to the relation: yn+1 = (I + C h)

−1

yn .

(5.102)

134

5 Calculations for a Cantilever Beam −1

−1

If eigenvalues of matrix C are λ, then eigenvalues of (I + C h) are (1 + λh) . However, 1 < 1 for any h, (5.103) 1 + λh since eigenvalues λ of a positive definite matrix are positive. Thus the implicit scheme is stable for all step-size h. The cost of this stability is the necessity of inverting matrix (I + C h). Systems of differential equations in form (5.99) are often met in mechanics of multibody systems. Now we will consider a system of nonlinear differential equations in the form: dy = f (y) . (5.104) dt The implicit scheme used for this equation gives: yn+1 = yn + hf (yn+1 ) .

(5.105)

In the general case this is a system of nonlinear equations which can be difficult to solve. Let us then linearise term f (yn+1 ) as: f (yn+1 ) = f (yn ) +

∂f (yn+1 − yn ) , ∂yn

(5.106)

where ∂f /∂y is the gradient matrix with the following elements:   ∂f ∂fi = , for i, j = 1, 2, . . . , m. ∂y i,j ∂yj Substituting (5.106) into (5.105), we obtain:  5 −1 ∂f 55 yn+1 = yn + h I − h f (yn ) . ∂y 5yn

(5.107)

Two problems arise when using the earlier formulae: (1) In order to solve (5.107), one needs to use an iterative procedure. (2) The gradient matrix has to be evaluated. Press et al. (2002) state that if h is not too large, only one iteration in the Newton method is enough to find an approximate solution. In order to calculate the gradient matrix, usually the finite difference method is used, assuming:   fi (yn,1 , . . . , yn,j + δ, . . . , yn,m ) − fi (yn,1 , . . . , yn,j , . . . , yn,m ) ∂f . = ∂y i,j δ (5.108) Calculation of the gradient matrix is time-consuming. When we use backward differences, as in (5.108), evaluation of elements of matrix ∂f /∂y requires

5.3 Numerical Effectiveness of Models and Methods

135

the function of calculating the components of vector f to be evaluated 2m times. There is no guarantee that the method is stable, yet usually it is, because the local behaviour of the gradient matrix ∂f /∂y is similar to that in the case of the constant matrix C described earlier. The following can be considered as higher-order methods for stiff systems: – Generalisation of the Runge–Kutta methods, the most useful of which are the Rosenbrock methods. – Generalisation of the Bulirsch–Stoer method (modification of the midpoint method). – Predictor–corrector Gear methods. Usually they are used with automatic step-size control. Latter, we will present applications of the methods discussed, using as an example vibration equations of a cantilever beam obtained by means of both formulations of the rigid finite element method described in Chaps. 3 and 4.

5.3 Numerical Effectiveness of Models and Methods of Integrating the Equations of Motion In order to compare the models and methods of integrating the equations presented, a computer programme in Pascal (Delphi 7.0) has been elaborated. Analysis of numerical effectiveness of models and methods is limited to the free vibrations of the beam presented in Sect. 5.1. It has been assumed that the vibrations are caused by the initial deflection which is the result of a T  acting as in Fig. 5.11. concentrated force P = P1 P2 P3 0 For the numerical simulations the following is assumed: P1 = 0 N, P2 = 1 N, P3 = 1 N, the remaining values of parameters of the beam being given in Table 5.1. The equations of static equilibrium obtained by omitting, in the equations of motion, the terms resulting from the kinetic energy of rfes take the form: ∂V + Q = 0, (5.109) − ∂q where V is the energy of spring deformation of sdes, Q is the vector of (m) (m) the generalised forces with elements: Qi = PT Bi rP ; i = 1, . . . , n,  6 m if the CRFE method is used n= 3 m if the MRFE method is used

P3

L

Fig. 5.11. Load on the beam for initial deflections

P2 u3 P1

136

5 Calculations for a Cantilever Beam (m)

Bi

=

∂B(m) ∂qi

i = 1, . . . , n,

(m)

rP is the vector of coordinates of the point at which force P acts with respect to the local coordinate system of rfe m(it is assumed that force P acts at the end of the beam in its axis as in Fig. 5.11). In the general case (5.109) is a system of n algebraic nonlinear equations with n unknowns, which are the components of vector q. The equations have been solved using the iterative Newton method, by increasing force P from zero to the final value with step ∆P. Having determined vector q0 , the initial conditions for the equations of motion have been assumed in the following form: q|t=0 = q0 , ˙ t=0 = 0. q|

(5.110a) (5.110b)

The equations of motion for the respective models are: (5.20) (5.30) (5.41) (5.53)

– – – –

CRFEN, CRFEL, MRFEN, MRFEL.

The Runge–Kutta method of the fourth order with constant step h = 10−5 s has been assumed as the basic method for integrating the equations of motion. The calculations are carried out for the interval t ∈ 0, 2 s. Table 5.4 contains initial deflections and times of calculation (PC with Pentium 2.8 GHz processor) for the models considered and for different m (number of sdes of the beam). The formulae in Sect. 5.1 are derived for m = 2, yet their generalisation for any m > 2 is not difficult. The Runge–Kutta method with the basic integration step h = 10−5 s has been used. Analysis of the results presented shows that the CRFE method is much more effective both in its linear and nonlinear formulations. This is a result of the particular form of the mass matrix, which is diagonal with constant elements in the linear case. As for the nonlinear model, the considerably shorter time of calculations is the result of using formulae given in Tables 3.1 and 3.2. Table 5.4. Influence of m on static deflections and time of calculations m 1 2 3 4 5

CRFEN t(s) u3 (L) (m) 0.08743 7 0.10988 19 0.11402 39 0.11547 68 0.11614 115

CRFEL u3 (L) (m) 0.08928 0.11160 0.11574 0.11718 0.11785

t(s) 3 7 11 16 22

MRFEN u3 (L) (m) t(s) 0.08743 36 0.10988 98 0.11403 195 0.11548 343 0.11616 556

MRFEL u3 (L) (m) t(s) 0.08928 45 0.11160 55 0.11574 60 0.11718 74 0.11785 95

5.3 Numerical Effectiveness of Models and Methods

137

The time of calculation was a sixth of that required when the traces of matrices were calculated by a computer (3.27). It can be easily seen that assuming m = 3 allows us to determine the value of static deflection of the end of the beam with an error smaller than 2% in relation to values obtained when m = 5. This is the case for both the CRFE and MRFE methods. The differences between the deflections obtained by means of linear and nonlinear models are also small, although deflections of 0.11 m should be considered as relatively large when the beam is 1 m long. Figure 5.12 shows the courses of deflection u3 (L, t) of the end of the beam in the direction of axis 3 for methods considered when m = 3. The results presented in Fig. 5.12 prove the conclusion formulated earlier, namely that results obtained using both classical and modified methods according to linear and nonlinear models are compatible. For the comparison of the methods of integrating the equations of motion the following denotations are used: NM for the Newmark method, RK for the Runge–Kutta method of the fourth order, EL for the Euler method (implicit scheme), RF for the Runge–Kutta–Fehlberg method, RS for the Rosenbrock method, BS for the Bulirsch–Stoer method. The first three methods were used with a constant step-size h, while for the next three the step-size was automatically adjusted.

CRFEN CRFEL MRFEN MRFEL

0.1

u3 [m]

0.05

0

-0.05

-0.1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t [s]

Fig. 5.12. Courses of deflection of the end of the beam

138

5 Calculations for a Cantilever Beam

In order to compare accuracy of the results obtained by means of different methods, we use a percentage error defined as follows: ,2 6T + (M, h, ε) (RK) u3 − u3 dt ε(M, h,ε) =

0

6T + (RK) , 2 u3 dt

100%,

(5.111)

0 (RK)

where u3 is the course of the end of the beam in the direction of axis 3 (M, h, ε) is the course of obtained by using method RK with step h = 10−5 s, u3 the end of the beam in the direction of axis 3 obtained by using method M with step h or truncation error ε, T is the end of time of calculation (T = 2 s). Because the initial deflections and courses of the end of the beam are almost the same for all the models (CRFEN, CRFEL, MRFEN, MRFEL), value (RK) for each of them is calculated separately. u3 The accuracy of the Newmark method has been examined for linear models (CRFEL, MRFEL) and the results are presented in Table 5.5. It can be seen that satisfactory accuracy (error < 1%) can even be obtained for h = 10−2 s. Both methods (CRFEL and MRFEL) give practically the same errors with respect to step-size h. Time of calculation for the modified method is half of that for the classical method. The results of numerical tests for effectiveness of the Runge–Kutta of the fourth order and Euler method are presented in Tables 5.6 and 5.7. In the Table 5.6 some of the boxes are empty, which means that for the assumed step-size h the method was unstable. Only step-size h = 0.5 × 10−4 s enabled us to obtain reasonable results. Table 5.5. Influence of the step-size h on the error of the Newmark method (NM) h(s)

CRFEL ε(%) t(s)

−2

10 0.5 × 10−2 10−3 0.5 × 10−3 10−4 0.5 × 10−4

0.5527 0.1484 0.0083 0.0013 3.32 × 10−5 8.05 × 10−6

5 5 5 5 6 6

MRFEL ε(%) 0.5529 0.1483 0.0082 0.0012 3.31 × 10−5 8.05 × 10−6

t(s) 3 3 3 3 3 3

Table 5.6. Influence of step-size h on errors in the RK method h(s)

CRFEN ε(%)

t(s)

CRFEL ε(%)

t(s)

−3

10 0.5 × 10−3 10−4 0.5 × 10−4

4.80 × 10−13

13

5.30 × 10−13

6

MRFEN ε(%) 8.42 × 10−7 1.72 × 10−8 2.02 × 10−12 5.00 × 10−14

t(s) 5 9 22 44

MRFEL ε(%) 1.44 × 10−7 5.67 × 10−10 5.69 × 10−14 2.51 × 10−30

t(s) 3 6 8 17

5.3 Numerical Effectiveness of Models and Methods

139

The results presented in Table 5.7 prove that the Euler method is unconditionally stable. An error smaller than 1% is obtained with step-size h = 0.5 × 10−3 s for the CRFE method and with h = 10−3 s for MRFE. Like for the Newmark method, errors are similar both when the methods (classical and modified) and models (linear and nonlinear) are compared. The methods with the automatic step-size adjustment were tested with the following assumptions: the maximal step-size is hmax = 10−2 s, the minimal step-size is hmin = 10−5 s, the truncation error estimate is εobc = 10−3 s. The results of tests are presented in Table 5.8. Having compared the numerical effectiveness of methods CRFEN and MRFEN with the adaptive step-size, it can be seen that shorter times of calculation are obtained when the modification of the method is used. The MRFE method, in which the mass matrix is full, requires half the number of generalised coordinates. In this case, moreover, higher frequencies of the free vibrations, which correspond to longitudinal deflections and shear, do not occur in the system. This allows us to use a larger step-size. In summary, it can be stated that when the constant step-size is used the models formulated according to the classical formulation of the RFE method are more effective numerically than the models formulated by means of the modified rigid finite element method. On the other hand, when adaptive stepsize methods are used for integrating the equations of motion there is an opposite relation, which means that the models obtained by means of the MRFE method are more effective. As for the numerical effectiveness of the methods for integrating the equations of motion in the case of the constant Table 5.7. Influence of step-size h on errors for the EL method h (s)

CRFEN ε(%) t(s)

10−2 0.5 × 10−2 10−3 0.5 × 10−3 10−4 0.5 × 10−4

93.35 51.07 1.0648 0.1808 0.0107 0.0040

5 6 9 12 40 74

CRFEL ε(%) t(s) 21.29 7.66 0.4604 0.1385 0.0106 0.0040

5 5 6 8 19 33

MRFEN ε(%) t(s) 21.25 7.63 0.4581 0.1376 0.0105 0.0040

3 4 12 21 93 182

MRFEL ε(%) t(s) 21.28 7.65 0.4608 0.1387 0.0106 0.0040

3 3 5 7 26 49

Table 5.8. Comparison of the effectiveness of the models and methods Method RF RS BS

CRFEN ε(%) t(s) 2.22 × 10−4 0.0024 1.7831

38 16 234

CRFEL ε(%) t(s) 2.33 × 10−4 0.0012 0.0015

9 8 15

MRFEN ε(%) t(s) 8.33 × 10−4 0.0011 0.0013

24 15 18

MRFEL ε(%) t(s) 8.56 × 10−4 0.0011 0.0013

6 6 8

140

5 Calculations for a Cantilever Beam

step-size methods, the Newmark method is unrivalled for the linear systems. The Euler method is also stable. However, in the case of the adaptive step-size methods, the results obtained were practically the same for almost all models. Other experience of the authors shows that the Runge–Kutta–Fehlberg method, effective for all models CRFEN, CRFEL, MRFEN and MRFEL, is a universal method and deserves recommendation.

6 Verification of the Method

The rigid finite element method has often been applied in the problems of dynamics of many machines, manipulators and even power transmission systems. Yet, like any other method, it has to be verified. Such a verification can be carried out in different ways, but the primary procedure is the comparison of the results obtained using the method, algorithms and programmes with the results of experimental measurements. Another approach is to compare our own results of calculations with those obtained by different authors who use different methods. Finally our own results can be compared with those from commercial packages. In the range of problems which the authors deal with, those may be the packages of the finite element method (for example Ansys, MSC.Nastran) or for dynamic analysis of multibody systems (for example Adams, Dads). All three approaches will be presented in this book in order to verify the rigid finite element method. Section 6.1 presents an application of the rigid finite element method to analysis of bending and torsional vibrations of beams. Results concerning frequencies and forms of the free vibrations are compared with analytical solutions and results obtained by different authors. Vibrations of a flexible link of Kane’s manipulator are also analysed and compared with results obtained by other researchers. Results of calculations and experimental measurements for a whippy beam and a Sandia manipulator are presented in Sect. 6.2. Comparison of our own results with the results obtained using MSC. Nastran, Ansys and Adams packages, concerning deflections of the supporting structure of an A-frame, will be presented in Chap. 7. In order to avoid any conflict of denotations, both in this chapter and Chap. 7, the number of rigid finite elements of flexible links will be denoted by n (not m like earlier).

6.1 Vibrations of Whippy Beams Let us first consider linear free vibrations of the prismatic beam discussed in Chap. 5 and presented in Fig. 6.1. The division of the beam into n + 1 rfes

142

6 Verification of the Method L

(a)

a a

E,G,r

D

D

D

D

D

(b)

(c)

sde 1

rfe 0

sde i + 1

sde i

sde 2

rfe 1

sde n

rfe i {i }

rfe n Xˆ 3{i } Xˆ 2{i }

j3(i )

(d) E{1}

j2(i )

Eˆ 3(1)

Xˆ 1{i } Eˆ 2(1)

j1(i )

x3(i ) x2(i )

x1(i ) Eˆ 1(1)

(e)

{i } Xˆ 3{i } j3(i )

{i }Y Yˆ 3{i } j2(i )

Xˆ 2{i }

Yˆ 2{i }

Xˆ 1{i } j1(i ) Yˆ 1{i }

Fig. 6.1. Discretisation of a fixed beam: (a) beam considered, (b) primary division, (c) secondary division, (d) generalised coordinates of rfe i-CRFE method, (e) generalised coordinates of rfe i-MRFE method

and n sdes, and the generalised coordinates of the rfe for both classical and modified formulations of the method, are also presented in this figure. As in Chap. 5, index p is omitted in Fig. 6.1 and in further discussion because the system considered is the system of one link. Stiffness coefficients of sdes are calculated according to formulae (5.11). Geometrical and mass parameters for both methods are also calculated as in Sect. 5.1.

6.1 Vibrations of Whippy Beams

143

6.1.1 Frequencies of Free Vibrations for a Uniform Beam In both formulations of the method (CRFE and MRFE) fixing of the beam is taken into account by assuming that rfe 0 is motionless. When we assume (i) that angles ϕj (j = 1, 2, 3; i = 1, . . . , n) are small, the free vibrations of the beam can be considered as described in Sects. 5.1.2 and 5.1.4 for the CRFEL and MRFEL methods, respectively. The equations for frequencies and forms of the free vibrations for the beam considered can be derived from (5.30) or (5.53), as presented in Sect. 5.1: det(Cα − ω 2 Aα ) = 0,

(6.1a)

(C −

(6.1b)

α

ωi2 Aα )Φi

= 0,

where α ∈ {C, M }, ωi is the ith frequency of the free vibrations of the beam and Φi is the ith form of the free vibrations of the beam. Courses of vibrations obtained by means of classical and modified formulations of the method are compared in Sect. 5.3. Here, the results obtained by using these methods are compared with the analytical results. Parameters of the beam assumed for calculations are listed in Table 6.1. The calculations have been carried out in order to define the influence of number n (number of elements into which the beam is divided) on the accuracy of frequencies determined. Frequencies of the free vibrations (when shear effect is neglected) can be calculated according to formulae given by Osi´ nski (1978), and the first three frequencies of bending vibrations are: ω1 = 52.50 s−1 , ω2 = 328.99 s−1 , ω3 = 921.01 s−1 .

In Table 6.2 we present the frequencies of the free vibrations of the beam with respect to number n for the CRFEL and MRFEL models, but also the percentage error in relation to the analytical values given earlier. The changes in the percentage error with respect to number n for the first two frequencies are shown in Fig. 6.2. Table 6.1. Parameters of the beam parameter unit value length of the beam, L m 1 cross-sectional area, A = a × a m2 10−2 × 10−2 Young’s modulus, E MPa 2.1 × 105 Kirchhoff modulus, G MPa 0.8 × 105 density, ρ kg m−3 7,800

144

6 Verification of the Method Table 6.2. Frequencies of bending vibrations

no. number frequencies of the of rfes free vibrations for the CRFEL model 1 n=1 73.37 1 n=2 56.87 2 464.74 1 n=3 54.04 2 361.95 3 1,163.13 1 n=4 53.43 2 347.11 3 992.55 1 n=5 53.15 2 340.93 3 970.29 1 n=6 53.00 2 337.36 3 956.40 1 n=7 52.91 2 336.36 3 947.76 1 n=8 52.85 2 334.08 3 942.09 1 n=9 52.81 2 333.20 3 938.10 1 n = 10 52.78 2 332.57 3 936.31

relative error with respect to analytical solution (%) −39.75 −6.42 −41.26 −2.93 −10.02 −26.29 −1.77 −6.51 −7.77 −1.24 −3.63 −6.35 −0.95 −2.54 −3.84 −0.78 −1.94 −2.90 −0.67 −1.55 −2.29 −0.59 −1.28 −1.86 −0.53 −1.09 −1.55

frequencies of the free vibrations for the MRFEL model 74.14 56.69 463.29 53.87 360.87 1,158.99 53.26 346.08 989.69 52.98 339.71 967.63 52.83 336.36 953.81 52.74 334.37 946.21 52.68 333.08 939.56 52.65 332.21 936.67 52.62 331.58 932.87

relative error with respect to analytical solution (%) −41.22 −6.08 −40.82 −2.61 −9.69 −26.84 −1.45 −6.19 −7.46 −0.92 −3.26 −6.06 −0.64 −2.24 −3.56 −0.47 −1.63 −2.63 −0.36 −1.24 −2.01 −0.26 −0.98 −1.59 −0.23 −0.79 −1.29

Having analysed the error, we can conclude that both methods allow us to obtain results close to the analytical solution. The larger the number n is, the smaller the error is. For the first frequency of bending vibrations the error does not exceed 1% when n = 6. For the second frequency the limit of 1% is crossed when the beam is divided into n = 10. The results obtained using the modification of the method (MRFEL) are closer to the analytical solution than those obtained using the classical method (CRFEL). This can be explained by the fact that both in the modification of the method and nski (1978) similar assumptions have when calculating ω1 –ω3 according to Osi´ been made, i.e. shear effect is neglected. Table 6.3 presents the frequencies of torsional free vibrations of the beam with respect to the number n of rigid elements into which the beam is

6.1 Vibrations of Whippy Beams

145

45 40 First frequency

35

Second frequency

% error

30 25 20 15 10 5 0 1

2

3

4

5 6 Number of rfes

7

8

9

10

Fig. 6.2. Relative error with respect to the analytical solution for the first two frequencies of the free vibrations dependent on number n

discretised, and the frequencies are also compared with the analytical solution given by Osi´ nski (1978). The first three frequencies of torsional vibrations according to Osi´ nski (1978) are: ω1 = 5,107.68

rad s−1 ,

ω2 = 15,323.05 ω3 = 25,538.42

rad s−1 , rad s−1 .

The percentage error in calculating the first two frequencies of the torsional vibrations with respect to the number of rfes is shown in Fig. 6.3. The comparison for the frequencies of the torsional vibrations is equally good. A relative error lower than 1% is achieved when the beam is divided into four rfes for the first frequency and into ten for the second frequency of the torsional vibrations. 6.1.2 Linear and Non-linear Vibrations of a Viscoelastic Beam The rigid finite element method enables large deformations of flexible links to be taken into account. In this section, based on Wojciech et al. (1990) and Wojciech and Adamiec-W´ ojcik (1993), we present equations of dynamic equilibrium of a planar viscoelastic beam and formulate a problem of analysis of vibrations with large amplitude using MRFE when the standard linear model of damping is assumed. Let us assume the following: (a) The beam has a lengthwise symmetry. (b) Cross-section of the beam is constant. (c) Displacements can cause large rotations but small stresses.

n=5

n=4

1.14 9.97 26.21 0.64 6.68 16.31 0.41 3.66 9.97

5, 049.54 13, 796.60 18, 846.13 5, 074.93 14, 452.17 21, 629.21 5, 086.71 14, 762.20 22, 992.66

n=3

1 2 3 1 2 3 1 2 3

2.55 21.58

4, 977.42 12, 016.54

n=2

1 2

9.97

n=1

1

relative error with respect to the analytical solution (%)

frequencies according to CRFEL and MRFEL (rad · s−1 ) 4, 598.53

number of rfes

no.

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

no.

n = 10

n=9

n=8

n=7

n=6

number of rfes

Table 6.3. Frequencies of torsional vibrations frequencies according to CRFEL and MRFEL (rad · s−1 ) 5, 093.11 14, 932.25 23, 753.77 5, 096.98 15, 036.34 24, 219.77 5, 099.48 15, 102.48 24, 526.10 5, 101.20 15, 148.62 24, 736.74 5, 102.44 15, 181.67 24, 887.08

0.29 2.55 6.90 0.21 1.88 6.16 0.16 1.44 3.97 0.13 1.14 3.14 0.10 0.92 2.55

relative error with respect to the analytical solution (%)

146 6 Verification of the Method

6.1 Vibrations of Whippy Beams

147

25 First frequency

% error

20

Second frequency

15

10

5

0 1

3

5

7

9

Number of r fes

Fig. 6.3. Relative error in relation to the analytical solution for the first two torsional vibrations

(d) Hypothesis of plane sections is fulfilled. (e) The beam is homogenous and its features are described by the standard linear model. (f) Inertia of the rotary motion is negligible. Figure 6.4 shows bending forces and moments acting on element dX of the beam at moment t. The equations of motion of the beam can be presented by using components N and Q as follows: ∂(N cos θ) ∂(Q sin θ) ∂2u − , = ∂t2 ∂X ∂X ∂2v ∂(N sin θ) ∂(Q cos θ) ρA 2 = + − q, ∂t ∂X ∂X ∂M + Q. 0= ∂X

ρA

(6.2a) (6.2b) (6.2c)

However, when components H and V are used, we obtain: ∂2u ∂H , = 2 ∂t ∂X

(6.3a)

∂2v ∂V − q, = 2 ∂t ∂X

(6.3b)

∂M + V cos θ − H sin θ, ∂X

(6.3c)

ρA ρA 0=

148

6 Verification of the Method y V+ V'X dX Q + Q'X dX x

M + M'X dX

N + N'X dX q + q 'X dX

dX

H

M

H + H'X dX

q

qdX

N Q y=v

u + u'X dX

V

v + v'X dX

u

X,x X

dX

Fig. 6.4. Forces and moments acting on element dX of the beam. N normal component; Q shearing force; M bending moment; H horizontal component of forces Q and N ; V vertical component of forces Q and N

where t is time, ρ is density, A is the area of the cross-section of the beam, u = x − X is the longitudinal displacement (shortening), v = y is the vertical displacement (deflection), θ is an angle and q is the load intensity. Taking into account the linear standard model which describes features of the material and the hypothesis of plane sections, the physical relation can be written in the form:  2  ∂ θ M m ∂θ ∂M + − EJ + = 0, (6.4) ∂t τ ∂X∂t τ ∂X where m is a non-dimensional coefficient ∈ 0, 1, τ is the relaxation time, E is Young’s modulus and J is the second moment of inertia of the cross-section. The above relation can also be written as follows:   #t t s m − 1 m − 1 e− τ e− τ ds . θ
X (6.5) M = EJ θ
X + τ τ 0

Below, there are geometrical relations which follow from Fig. 6.4: ∂u = cos θ − 1, ∂X ∂v = sin θ. ∂X

(6.6a) (6.6b)

6.1 Vibrations of Whippy Beams

149

Having differentiated the above formulae twice with respect to time, we obtain: u ¨
X = −θ˙2 cos θ − θ¨ sin θ, v¨
X = −θ˙2 sin θ + θ¨ cos θ.

(6.7a) (6.7b)

Wojciech et al. (1990) present the solution of the equations describing vibrations of the beam analysed (system of six equations: (6.2) or (6.3), (6.4) or (6.5) and (6.6) or (6.7)) by means of the following methods: – Finite difference method (FDM), – Bubnov–Galerkin method, (BGM), – Finite element method (FEM), and the modification of the rigid finite element (MRFEN) method described in Chap. 4. However, because of the standard linear model assumed, the expressions for the potential energy of spring deformation and the function of dissipation of energy have to be reformulated. Having assumed the linear standard damping model, values of moments (i) Mj transferred by sde i of the beam from Fig. 6.1 can be defined from the formula: (i) Mj

=

(i) µj − (i) (i) cϕ,j ϕj + (i) τj

1

−

e

t (i) τ j

#t

s

(i) (i) τ (i) cϕ,j ϕj e j

ds for i = 1, . . . , n; j = 1, 2, 3,

0

(6.8) (i)

(i)

(i)

where cϕ,j , µj , τj

are coefficients. (i)

If one assumes µj = 1, the case when damping is neglected is obtained. Wojciech et al. (1990) compare results obtained by using all four methods for a simply supported beam shown in Fig. 6.5.

y

X = -0.5

X = 0.5 x

X,x

Fig. 6.5. Simply supported beam. Parameters of the beam are given in Table 6.1

150

6 Verification of the Method

The free vibrations of the beam have been considered with the following assumptions: A. Boundary conditions: ν(−0.5, t) = ν(0.5, t) = 0, M (−0.5, t) = M (0.5, t) = 0, N (−0.5, t) = (Q tg θ)X=−0.5 , N (0.5, t) = (Q tg θ)X=0.5 .

(6.9)

B. Initial conditions: ν(0, 0) = 0.2 m, ν(X, ˙ 0) = 0.

(6.10a) (6.10b)

Figure 6.6 presents courses of the deflections of the middle point of the beam (i) ν(0, t) for non-damped vibrations (µj = 1), while in Fig. 6.7 the damped (i)

free vibrations when µj = 0.5 are shown. For both cases it is assumed that (i)

τj = 1. Courses presented in Figs. 6.6 and 6.7 show that the results obtained by means of the rigid finite element method are close to those obtained by means of the finite element method. Figure 6.8 demonstrates how the accuracy of the results is influenced by the number of rigid elements n in which the beam has been divided. Wojciech and Adamiec-W´ojcik (1993) continued the comparison of the rigid finite element method with the finite element method by examining the influence of number n on the accuracy of results of calculations. A comparison of the first three forms of free vibrations and corresponding courses of the bending moment M , for a cantilever beam with changing cross-section (Fig. 6.9a), when h(x) changes linearly, is shown in Fig. 6.9b, c. Geometrical parameters of the beam are given by Wojciech and AdamiecW´ ojcik (1993).

v 0.2 FEM MRFEN 0

BGM 0.2

0.4

0.6

-0.2

Fig. 6.6. Free vibrations, non-damped – comparison of methods

t FDM

6.1 Vibrations of Whippy Beams

(a)

151

v 0.2 FEM RFE BGM 0

0.2

0.4

0.6

t FDM

-0.2

(b)

v 0.2

0.1

0

FEM MRFEN 0.5

1.0

1.5

t BGM

-0.1

FDM

-0.2

Fig. 6.7. Free vibrations, damped – comparison of methods: (a) ∈ 0, 0.7, (b) t ∈ 0, 2.0

v 0.2 n=6 n = 8,10 0

0.2

0.4

0.6

-0.2

Fig. 6.8. The influence of number n on the results

152

6 Verification of the Method

(a)

Xˆ 3 b

h (x) Xˆ 1 x L (b) FEM MRFEL

1.0

5.0

0

0.25

0.5

0.75

m

-0.5 (c)

2.0 1.0

0

0.5

0.75

0.25

m

-1.0

-2.0

FEM MRFEL

-3.0

Fig. 6.9. Free vibrations of the beam with changing cross-section: (a) beam h(0) = 0.05 m; h(l) = 0.005 m; b = 0.02 m; remaining parameters as in Table 6.1, (b) first three forms of the free vibrations, (c) courses of the bending moment M corresponding to vibrations from (b)

6.1 Vibrations of Whippy Beams

153

0.5

Amplitude of vibrations [m]

F = 54 N F = 184 N

0.4

F = 432 N 0.3

0.2

0.1

2

3

4

5

6 n

7

8

9

10

Fig. 6.10. Influence of number n on accuracy of results

The influence of number n on the accuracy of calculations is presented in Fig. 6.10. The calculations concern the free vibrations of the cantilever beam with constant cross-section with parameters given in Table 6.1. The vibrations are the result of a static deflection caused by a concentrated force P applied at rfe n: T  (6.11) P = 0 23 F F 0 , where F = 54, 184 or 432 N. It can be seen that acceptable accuracy is obtained even for n = 6. Figure 6.11 demonstrates displacements of the end ˆ 3 and the shortening of the beam for ˆ 2, X of the beam in directions of axes X (i) (i) the free damped vibrations (τj = 1; µj = 0.5) when F = 184 N. Analysis of these results shows that the rigid finite element method gives results compatible with those obtained by using the finite element method, and acceptable accuracy is obtained when the beams are divided into only a few elements (n ≥ 5) in the case when vibrations with low amplitudes are dominant. 6.1.3 Kane’s Manipulator The manipulator subject to comparative analysis is described by Kane et al. (1987), and its dynamic analysis requires consideration of large displacements

154

6 Verification of the Method

[m] x 3(L) 0.2 L - x 1(L)

0

0.1

0.2

-0.2

0.3

0.4

0.5 t [s]

x 2(L)

Fig. 6.11. Spatial vibrations of the beam. x2 (L) displacement in the direction of ˆ 3 ; L − x1 (L) shortening of the ˆ 2 ; x3 (L) displacement in the direction of axis X axis X beam

y2 y3

y1

Fig. 6.12. Kane’s manipulator

and non-linear effects (Fig. 6.12). This manipulator was also analysed in Du et al. (1992); the authors presented their results and compared them with those obtained by Kane. In both cases the finite element method together with the modal method was used to discretise a flexible link. In this section the results mentioned will be compared with those obtained by using the methods described in Chaps. 3 and 4. The manipulator considered is shown in Fig. 6.12; it is a spatial manipulator with three rotary kinematic pairs. It is assumed that the two first links are rigid while the third one with a variable cross-section is flexible. The motion of the manipulator is realised by

6.1 Vibrations of Whippy Beams

155

change in angles ψ1 , ψ2 , ψ3 . Kane et al. (1987), Du et al. (1992), Wittbrodt and Wojciech (1995) and Adamiec-W´ojcik (2003) considered the motion of the last flexible link as the motion of a body with known base motion. Two different kinematic inputs were considered. First input, the deployment process, causes change of link position from a stowed to a fully operational configuration in such a way that over 15 s angles ψ1 , ψ2 , ψ3 change from 180 to 90, from 180 to 45 and from 180 to 0 degrees, respectively. The functions of angle changes are given as follows: Deployment Process  π − ψ1 (t) =

ψ2 (t) =

ψ3 (t) =

 1π 2  π −  1π 4  π − 0

π 2T



t−

T 2π

sin

2πt T



0T 0T 0T

[rad] .

Initial and final (after T =15 s) positions are shown in Fig. 6.13a. The second input called the spin-up manoeuvre causes non-linear effects due to centrifugal forces and large deflections. This kind of input is found in, for example, rotor blades or satellites with long antenna movements. During the spin-up manoeuvre the first link is at rest, the third stays at rest in relation to the second, and the angular velocity ψ˙ 2 of the second link increases from 0 to 6 rad s−1 over 15 s; the functions are given in “spin-up Maneouvre” Spin-Up Maneouvre

ψ1 (t) =

ψ2 (t) =

1 π [rad] 2  + 2    , T 2  cos 2πt − 1 0T

ψ3 t = 0 [rad]. The initial position and the position while realising the kinematic input are shown in Fig. 6.13b. The algorithms presented in Chaps. 3 and 4 enable us to derive the equations of motion for a kinematic chain with unknown base motion. In order to use these algorithms in analysis of the manipulator presented, angles ψ1 , ψ2 , ψ3 are treated as generalised coordinates and additional constraint equations in

156

6 Verification of the Method (a)

(b)

Fig. 6.13. Kinematic inputs for the manipulator: (a) initial and final positions for the deployment process, (b) initial position and position during the motion for the spin-up manoeuvre

forms (6.12) and (6.13) are imposed on the system. Such an approach enables us also to determine the torques acting in joints. The manipulator considered forms an open kinematic chain; its geometry is described using methods presented by Craig (1988). To this end local coordinate systems are assigned to links in the way presented in Fig. 6.14. Denavit–Hartenberg parameters necessary to define transformation matrices are presented in Table 6.4.

6.1 Vibrations of Whippy Beams

157

{r } B1

L2 2 y2 (2) Xˆ 3

L1

1 y1

(0) Xˆ 1

(3) Xˆ 2 (3) Xˆ 3

y3

(2) Xˆ 1

3

(2) Xˆ 2

(1) Xˆ 1

B2

(3) Xˆ 1

(0) Xˆ 2

(1) Xˆ 3 (0) ˆ X1

{0}

Fig. 6.14. Local coordinate systems for Kane’s manipulator Table 6.4. Denavit–Hartenberg parameters for Kane’s manipulator ai−1 αi−1 di−1 θi

i=1 0 0 0 ψ1

i=2 L1

i=3 L2 3 π 2 0 ψ3

π 2

0 ψ2

Transformation matrices then take the form:    c1 −s1 0 0 c2  s1 c1 0 0  (2)  0 (1) ˜   ˜ = B  0 0 1 0  , B =  s2 0 0 0 01   c3 −s3 0 L2  0 0 1 0  (3,0) ˜  = B  −s3 −c3 0 0  , 0 0 0 1

−s2 0 c2 0

0 −1 0 0

 L1 0  , 0  1

(6.14)

where Si = sin ψi , ci = cos ψi , i = 1, 2, 3. The flexible link is discretised by means of the rigid finite element method, and the primary division is shown in Fig. 6.15a. The flexible link is a beam consisting of two sections with different cross-sections, and the bending axis of the second section is offset by e in relation to the central axis (Fig. 6.15c).

158

6 Verification of the Method

(a) (1) (2)

(b) sde i r fe 0

r fe 1

(k ) xe,3

...

r fe k= n 3 sde k =

...

sde n r fe i ci

...

r fe n 1

n 3

(i ) xc,3

3 (c) e 1 2

Fig. 6.15. Discretisation of the flexible link: (a) primary division, (b) secondary division, (c) bending axis

Secondary division isolating rigid and spring-damping elements is presented in Fig. 6.15b. The transformation matrix from system E {i} assigned to rfe i before deformation to the system assigned to rfe 0 takes the form:   100 0 0 1 0 0  0 (i)  (6.15) = ET  0 0 1 ai  , 000 1 where ai = ∆ 2 + (i − 1) ∆, ∆ is the length of the rfe. Data for the last link given in Kane et al. (1987) are the following: Young’s modulus E = 6.8950×1010 N m−2 , shear modulus G =2.6519×1010 N m−2 , mass density ρ = 2,766.67 kg m−3 . The remaining parameters for each segment (i) (i) of the beam are presented in Table 6.5. Coefficients a2 , a3 , γ (i) (i = 1, 2)

6.1 Vibrations of Whippy Beams

159

Table 6.5. Parameters of link 3 parameter length (m) second moment of inertia (m4 ) second moment of inertia (m4 ) polar moment of inertia (m4 ) cross-sectional area (m2 ) shear area ratio shear area ratio torsional ratio offset of bending axis (m)

first segment B1 = 2.667 (1) J2 = 1.50 × 10−7 (1) J3 = 1.50 × 10−7 (1) J0 = 6.00 × 10−7 A(1) = 3.84 × 10−4 (1) a2 = 2.09 (1) a3 = 2.09 γ (1) = 2.727 e(1) = 0

second segment B2 = 5.333 (2) J2 = 4.8746 × 10−9 (2) J3 = 8.2181 × 10−9 (2) J0 = 32.8724 × 10−9 A(2) = 7.30 × 10−5 (2) a2 = 3.174 (2) a3 = 1.520 γ (2) = 1, 351.1 e(2) = 0.01875

correcting shear and torsional stiffness depend on the cross-section of the beam. As mentioned, angles ψ1 , ψ2 , ψ3 are treated as generalised coordinates; thus the two first links have been attributed with mass properties. It is assumed that both links are 8 m long as in (Kane et al. 1987). The remaining values are: mass density ρ = 2,766.67 kg m−3 and a box cross-section with the internal and external dimensions hz = 0.05 m and hw = 0.0475 m, respectively. Parameters of rfes and sdes of link 3 were calculated as in Wittbrodt and Wojciech (1995) and PZlosa (1995). Numerical calculations are carried out using the algorithms presented in Chaps. 3 and 4, and the flexible link is discretised by means of the rigid finite element method both in its classical formulation and its modification. The results of numerical simulations are compared with those presented by Du et al. (1992) obtained by using the modal method. As in the papers mentioned, u2 andu3 denote deflections of the end of the beam in directions 2 and 3, respectively, and θ=u 4 is the torsion angle about axis 1. Figure 6.16 shows the comparison of results for the deployment process. Good compatibility of results is achieved and the maximal difference in amplitudes does not exceed 10% for the rigid finite element method in relation to results obtained by Du et al. (1992). Comparison of results for the spin-up manoeuvre is shown in Fig. 6.17. In this case the compatibility of results is not as good as previous, which is caused by the rapid input responsible for non-linear effects; however, the error in this case too is acceptable. All numerical calculations were carried out assuming the number of rigid elements in the primary division to be 6, which gives seven rfes and six sdes in the secondary division. Here reference to the paper by Wittbrodt and Wojciech (1995) should be made. There the rigid finite element method in its classical formulation is also used to discretise the flexible link and analyse Kane’s manipulator. The equations of motion, however, are formulated with the assumption of known base motion and small relative displacements of rfes; thus the linear form of transformation matrices (3.67) was used. With this assumption special simplified

160

6 Verification of the Method 0.1

0.08 0.06

u 3 [m]

u 2 [m]

0.05 0

-0.05 -0.1

0.04 0.02 0

0

5

10

-0.02

15

0

5

t [s]

10

15

t [s]

10 DH CRFEN

q [deg]

5

MRFEN 0 -5 -10 0

5

10

15

t [s]

Fig. 6.16. Deflections u2 , u3 and torsion angle θ of the end of the last link for the deployment process, DH Du et al. (1992); CRFEN classical rigid finite element method (Chap. 3); MRFEN modification of the rigid finite element method (Chap. 4)

formulae of the kinetic energy were obtained. The algorithm presented in Chap. 3 enables us to formulate the equations of motion using both linear and non-linear transformation matrices (with or without the assumption of lineari(p,i) sation of the trigonometrical functions of angles ϕj ). The results obtained using linear and non-linear models are almost identical for the deployment process when joint coordinates change smoothly. In the case of the spin-up manoeuvre the differences obtained for deflection u2 and torsion angle θ are shown in Fig. 6.18. The algorithm presented here does not require any assumption about the kinematic input; and the formulation of equations of motion is general and can be used for any open kinematic chains with rigid and flexible links (the flexible link does not have to be the last in the chain). The use of this algorithm also enables torques acting in joints of the manipulator to be calculated. Interaction of the vibrations of a flexible link and the base motion, and thus the acceptability of disregarding the influence of the vibrations of the third link on the base motion, were analysed by using the following procedure (Adamiec-W´ojcik, 2002):

6.1 Vibrations of Whippy Beams 0.1

161

0.02

0

0 u 3 [m]

u 2 [m]

-0.1 -0.2

-0.02 -0.04

-0.3 -0.4 0

5

10

15

-0.06 0

5

10

t [s]

15

t [s] 40 DH CRFEN MRFEN

q [deg]

30 20 10 0 -10

0

5

10

15

t [s]

Fig. 6.17. Deflections u2 , u3 and torsion angle θ of the end of the last link for the spin-up manoeuvre

0.01

40 a b

-0.01 -0.02 -0.03 -0.04

a b

30

q [deg]

u 2 [m]

0

20 10 0

0

5

10

t [s]

15

-10

0

5

t [s]

10

15

Fig. 6.18. Deflection u2 and torsion angle θ for: (a) nonlinear (3,4), (b) linear (3.67) transformation matrices

1. Torques necessary for realisation of the motion described by (6.12) were calculated, assuming that all links are rigid. The equations of motion of the rigid system are obtained from the general algorithm, assuming (n(3) = 0). 2. These torques were applied to the joints, assuming that the third link is flexible. Joint angles ψ1 , ψ2 , ψ3 and their velocities, and also deflections and torsion of the end of the third link, were calculated. In order to show the influence of large deflections of the flexible link on the base motion, the input defined by (6.12) was analysed, assuming T = 6 s. It

162

6 Verification of the Method 1

0.5 0

0

u 3 [m]

u 2 [m]

0.5

-0.5

-0.5 -1

-1 -1.5

0

2

4

6

-1.5

0

2

t [s]

t [s]

4

6

2

q [m]

1 a 0

b

-1 -2 0

2

4

6

t [s]

Fig. 6.19. Deflections u2 , u3 and torsion angle θ of the end of the third link: (a) kinematic input, (b) force input

follows that the deployment process becomes a movement with considerable velocities and accelerations. Figure 6.19 presents the comparison of deflections and torsion of the end of the flexible link for the kinematic input and force input when the torques calculated according to (1) above are applied to the joints. The torque which should be applied in the joint between the second and the third links in order to ensure realisation of the motion described by (6.12) is presented in Fig. 6.20 for two cases: when the third link is flexible, and when it is rigid. It can be seen that the difference is quite large, which means that the torque calculated for a rigid system can be applied to a flexible system only when deflections are small. However, the procedure presented enables us to estimate whether the influence of the flexible vibrations on the base motion is significant. Flexibility also affects velocities and accelerations of joint variables. In the case considered vibrations of the third link do not significantly influence ψ˙ 1 and ψ˙ 2 . The influence of vibrations of the flexible link on velocity ψ˙ 3 is shown in Fig. 6.21. The course of joint velocity is presented for both kinematic and force inputs.

6.1 Vibrations of Whippy Beams

163

120 100

a b

M3 [N m]

80 60 40 20 0 -20 0

1

2

3

4

5

6

t [s]

Fig. 6.20. Torque in the third joint required for realisation of motion: (a) for flexible link, (b) for rigd link 0 -0.2

y 3 [deg]

-0.4 -0.6 -0.8 a b

-1 -1.2 -1.4

0

1

2

3

4

5

6

t [s]

Fig. 6.21. Velocity in the third joint: (a) kinematic input, (b) force input

6.1.4 Analysis of Large Deflections The MRFEN method enables us to analyse large displacements of flexible links taking into account the effect of joint distance shortening. In order to illustrate this effect, let us consider a whippy simply supported beam 1 m long with a circular cross-section of diameter d = 0.005 m. The beam was initially bent as a result of gravity force acting (Fig. 6.22a). Then a horizontal force F acting at one end of the beam was applied and the value of the force was changed from 0 to 180 N with step of 2 N. Deflections of the beam when the force equals 120 and 180 N are shown in Fig. 6.22b, c,

164

6 Verification of the Method (a) l (b)

F

(c)

F

Fig. 6.22. Whippy beam loaded with horizontal force: (a) initial position, (b) from of deflection for F = 120 N, (c) from of deflection of F = 180 N

respectively. The results were obtained when the beam was divided into 16 elements. The MRFEN method has also been employed for analysis of a four-bar mechanism with flexible links (Fig. 6.23), and the results of calculation have been compared with those presented by Munteanu et al. (2004) for the data given in Table 6.6. In order to discretise the beams, Munteanu et al. (2004) use two or three nodal elements with four generalised coordinates (Euler–Rodrigues parameters) in each node. They assume that the mass of the element is concentrated in its nodes. The equations of motion, derived without consideration of shear effects and longitudinal flexibility so the analysis is limited to bending, are completed with constraint equations. The motion of the mechanism is effected by a torque acting on link 1 and defined as:  M0 tt0 t ≤ t0 . M (t) = 0 t > t0 Courses of the angular velocity of links 1, 2 and 3 in the case when all links are rigid (M0 = 1 N m and t0 = 0.1 s) are presented in Fig. 6.24.

6.1 Vibrations of Whippy Beams

165

mc 2

I2,A2,I2

I3,A3,I3 mc1 I1,A1,l1

M

L

Fig. 6.23. Four-bar merchanism considered and the data

Table 6.6. Data for the four-bar mechanism length (m) cross-sectional area (m2 ) second area moment (m4 ) lumped mass (kg) density (kg m−3 ) Young’s modulus (N m−2 ) Poisson ratio distance between supports (m)

link 1 l1 = 0.108

link 2 l2 = 0.2794

link 3 l3 = 0.2705

A1 = 1.077 × 10−4

A2 = 0.406 × 10−4

A3 = 0.406 × 10−4

I1 = 1.616 × 10−10 mc1 = 0.0428

I2 = 8.674 × 10−12 mc2 = 0.0428 ρ = 2, 660

I3 = 8.674 × 10−12 mc3 = 0

E = 7.1 × 1010 ν = 0.3 L = 0.254

Figure 6.25 shows some highly deformed positions of the mechanism with flexible links when M0 = 3 N m and t0 = 0.1 s. The results are obtained for n(1) = 0, n(2) = n(3) = 14 rfes. Comparison of deformed positions of the four-bar mechanism obtained using the rigid finite element method with those presented by Munteanu et al. (2004) is shown in Fig. 6.26. It can be seen that the results are acceptably compatible.

166

6 Verification of the Method

Angular velocity [1/s]

40

link 1 link 2 link 3

20 0 -20 -40 -60

0

0.1

0.2

0.3

0.4

0.5

t [s]

Fig. 6.24. Angular velocities of links

t = 0.12 s

t = 0.14 s

t = 0.16 s

t = 0.18 s

t = 0.19 s

t = 0.20 s

t = 0.22 s

t=0 s

M

t = 0.24 s

Fig. 6.25. Deformations of flexible links of the mechanism

6.2 Experimental Verification of the Method In Sect. 6.1 we discussed the results of indirect verification, in which the results of our own calculations are compared with the results presented in literature. However, the basic approach for verification of the method is to compare the results of calculations with those of experimental measurements.

6.2 Experimental Verification of the Method

167

MRFEN t = 0.12 s

t = 0.16 s

t = 0.14 s

Munteanu et al.

Fig. 6.26. Comparison of deformations of the mechanism obtained using the MRFEN method with those of Munteanu et al. (2004)

rfe n

Xˆ (i2 )

Yˆ 2(i ) j (i ) 2

j 3(i )

Xˆ 1(i ) Yˆ 1(i )

j 2(i )

rfe 0

Yˆ 3(i )

j 3(i )

Xˆ 3(i )

Fig. 6.27. A cantilever beam with circular cross-section

In this section we present results of calculations and experimental measurements carried out for two cases. First, we consider large amplitude vibrations of a homogenous beam which was bent and then released. Second, vibrations of the flexible link of a Sandia type manipulator are discussed. 6.2.1 Large Amplitude Vibrations of a Fixed Whippy Beam Let us consider a fixed beam with a constant circular cross-section presented in Fig. 6.27. In order to discretise the beam, the modified formulation of the rigid finite element method presented in Chap. 4 is used; however, torsion is neglected and thus the vector of generalised coordinates of rfes 1 to n takes the form:  (1)  ϕ  ..   .    (i)  (6.16) ϕ=  ϕ ,  .   ..  ϕ(n)

168

6 Verification of the Method

Xˆ 1(i )

p (i )

Yˆ 2(i )

Xˆ (i2 )

j (i ) 2

j 3(i ) j

rfe i -1

(i )

j2(i )

Yˆ 1(i ) Yˆ 3(i )

j 3(i )

Xˆ 3(i ) (i)

(i)

Fig. 6.28. Angles ϕ2 , ϕ3 and ϕ(i)



where ϕ

(i)

=

(i) ϕ2 (i) . ϕ3

Let us consider two neighbouring elements i−1 and i presented in Fig. 6.28. It can be seen that the moment transferred by sde i should depend on angles (i) (i) ϕ2 and ϕ3 . The energy of spring deformation of the beam can be presented in the form: n 1 " (i)  (i) 2 Vs = c ϕ , (6.17) 2 i=1 where c(i) is the coefficient of elasticity in plane π(i) and ϕ(i) is the angle ˆ (i) and X ˆ (i) . between axes Y 1 1 In the case considered the following relation between angle ϕ(i) and (i) (i) ϕ2 , ϕ3 takes place (Wojciech and Adamiec-W´ojcik, 1994): (i)

(i)

cos ϕ(i) = cos ϕ2 cos ϕ3 .

(6.18)

From (6.17) we now obtain: ∂Vs (i)

= c(i) ϕ(i)

∂ϕ2

∂Vs (i)

∂ϕ3

∂ϕ(i) (i)

(i)

sin ϕ2 cos ϕ2 , sin ϕ(i)

= c(i) ϕ(i)

cos ϕ2 sin ϕ3 . sin ϕ(i)

∂ϕ2 = c(i) ϕ(i)

∂ϕ(i) (i)

∂ϕ3

(i)

= c(i) ϕ(i)

(i)

(6.19a)

(i)

(6.19b)

6.2 Experimental Verification of the Method

169

The analysis of the above expressions shows that in the case of small angles (i) (i) ϕ2 and ϕ3 we obtain: ∂Vs (i)

∂ϕ2 ∂Vs

(i) ∂ϕ3

(i)

(6.20a)

(i)

(6.20b)

= c(i) ϕ2 , = c(i) ϕ3 ,

which are the relations presented in Chap. 4. However, (Adamiec-W´ ojcik, 1992) demonstrated that in the case of large deformations formulae (6.19) have to be used. Experimental measurements of the deflections of the beam were carried out in two stages. First, stiffness in the case of large displacements was determined. To this end several measurements of static deflections of the beam loaded as in Fig. 6.29 were taken. The results of these measurements are listed in Table 6.7. Because the calculational model deals with large displacements of the beam, it is especially important that coefficients c(i) occurring in (6.19) are properly chosen. In order to use the results presented in Table 6.7, a planar model of the beam was formulated using the MRFEN method (Fig. 6.29). Since the cross-section of the beam is constant, it can be assumed that stiffness characteristics of all sdes are the same and linear, and thus: (a)

(b)

Xˆ 2

Xˆ 2

l

r (i )

l (i )

m K K

x c2,j x

Xˆ 1

l (n)

(i )

e 2,j

F

Xˆ 1

K

F

Fig. 6.29. Static load of the beam: (a) realisation of the load, (b) discretisation

170

6 Verification of the Method Table 6.7. Measured deflections of points K of the beam

no. (j) 1 2 3 4 5 6 7 8 9 10

1, 000F/ g (N) 10 20 30 40 50 60 70 80 90 100

xe2,j (mm)

no.

1, 000F/g (N)

xe2,j (mm)

no.

29 58 86 117 145 174 202 230 258 285

11 12 13 14 15 16 17 18 19 20

110 120 130 140 150 160 170 180 190 200

310 334 358 383 409 432 452 471 490 512

21 22 23 24 25 26 27 28 29 30

1, 000F/ g (N) 210 220 230 240 250 260 270 280 290 300

c(i) (ϕ(i) ) = c = const.

xe2,j (mm) 535 554 573 588 604 619 634 650 665 677

(6.21)

Coefficient c was determined by means of the least square method, having assumed that the following expression should be minimal: Ω(c) =

m "  e 2 x2,j − xc2,j ,

(6.22)

j=1

where xe2,j are measured values of deflections, xc2,j are calculated values of deflections and m is the number of experimental measurements. According to Fig. 6.29b, it is assumed that the beam under force F bends ˆ 2 . The centres of mass of rfes can be calculated from: ˆ 1X in plane X (k)

xc,1 =

k−1 "

l(j) cos

=l

+

k "

(k)

k−1 "

=

j=1

j "

j "

ϕ

(p)

(6.23a)

,

ϕ(p) + r(k) sin

p=1

lkj sin

ϕ(p) + l(0)

p=1

l(j) sin

j=1 k "

k " p=1

lkj cos

j=1

xc,2 =

ϕ(p) + r(k) cos

p=1

j=1 (0)

j "

j "

k " p=1

ϕ(p) (6.23b)

ϕ(p) ,

p=1



where lkj =

l(j) r(k)

if j < k . if j = k

Similarly, the coordinates of point K, at which force F is applied, can be written as:

6.2 Experimental Verification of the Method

xK,2 =

n "

lKj sin

j "

ϕ(p) ,

171

(6.24)

p=1

j=1

where lKj = l(j) . The equation of static equilibrium of the beam takes the form (AdamiecW´ ojcik, 1992): cϕ

(i)

=−

(k) n " ∂xc,1 k=1

∂ϕ

(i)

mk g +

∂xK,2 F, i = 1, . . . , n. ∂ϕ(i)

(6.25)

Since: j n " " ∂yK = l cos ϕ(p) , Kj ∂ϕ(i) p=1 j=i (k)

∂xc,1

∂ϕ(i)

=−

k "

lkj sin

j=i

j "

(6.26a)

ϕ(p) ,

(6.26b)

p=1

then in order to define angles ϕ(i) for a given value of force F, the following set of non-linear algebraic equations has to be solved: (i)

cϕi =

n " k=1

mk g

k " j=i

lkj sin

j " p=1

ϕ(p) + F

n " j=i

lKj cos

j "

ϕ(p) i = 1, . . . , n.

p=1

(6.27) Determination of angles ϕ(i) enables us to calculate deflection xK,2 from (6.24) for a given force F. Taking values of the force from Table 6.7, we can calculate corresponding values xc2,j occurring in expression (6.22). The value of the stiffness coefficient c ensures that the deflections calculated will be as close as possible to those from Table 6.7. In the second stage of the experiment the trajectory of a chosen point of the beam during spatial vibrations was measured. To this end a test stand presented in Fig. 6.30 was built. One end of the bar was fixed while the second one was loaded with replaceable weights. A point light source was set at the moving end of the bar. Initial static deflection of the beam is the sum of deflections caused by the weight of ˆ 1X ˆ 3 ) and external load Q (acting in plane X ˆ 3 ). When ˆ 2X the bar (in plane X load Q is removed, the beam performs the free spatial vibrations caused by the initial deflection. The investigations were carried out in a dark room and the motion of the light point (the end of the moving bar) was recorded on one frame. A moving disk diaphragm enabled us to cover the lens briefly in order to determine time within the period of vibration. Figure 6.31 presents ˆ 2 for different ˆ 1X the registered trajectory of the end of the bar in plane X periods of vibration. In order to compare the results of experimental measurements and calculations, the algorithms defined in Chap. 4 for the MRFEN method were used.

172

6 Verification of the Method rs K La K⬘

L B

Xˆ 3

Z

Dx3 Z

Dx2

a Xˆ 1

K⬙

Xˆ 2

K Q

Fig. 6.30. Test stand

Fig. 6.31. Trajectory of the end of the bar

The modelled beam had a length L = 1.42 m and circular cross-section with radius r = 0.00205 m. A point light source was modelled as lumped mass ms placed at distance rs measured from the free end of the beam. The beam was fixed in such a way that the axis of the undeformed beam formed angle α with the horizontal plane. Table 6.8 presents values of coefficient c calculated using the experimental measurements in the way described by formulae (6.22–6.27). It is important to note that the differences between the values of coefficient c measured during the experiment and those obtained using the formulae given in Sect. 3.6 decrease when the number of rfes into which the beam is divided

6.2 Experimental Verification of the Method

173

Table 6.8. Comparision of values of the stiffiness coefficient c number of rfes 3 4 5 6

according to the experiment 6.642 7.468 9.758 11.533

according to formula c = GJ/∆ 6.927 7.903 9.879 11.855

Table 6.9. Influence of number n on the deflections of the end of the beam number of rfes 3 4 5 6 7 8

case I 0.1791 0.1712 0.1727 0.1710 0.1722 0.1716

case II 0.1341 0.1269 0.1286 0.1283 0.1283 0.1278

increases. This can be easily explained by the fact that angles ϕ(i) necessary for deflection xK,2 decrease with the increase of number of rfes since they are relative angles which describe the rotation of the rfe with respect to the preceding one. For the analysis two cases of the free vibrations were considered, and they differed in parameters ms , rs , α: Case I : ms = 0.0732 kg; rs = −0.0212 m; α = 24.63◦ , Case II : ms = 0.0418 kg; rs = −0.0158 m; α = 18.89◦ . Load Q was realised by mass ma = 0.054 kg applied at distance La = 0.164 m from the end of the beam. The beam was released by cutting the string. The accuracy of results obviously depends on number n of rfes. Table 6.9 demonstrates the influence of number n on the initial bending of the beam under consideration. Figure 6.32 shows the influence of the number n of rfes on the results of calculations. Small differences between the courses for n = 3 and n = 4 can be seen but the differences in results obtained for n = 4, 5, 6 are negligible. Therefore it was assumed that the acceptable accuracy is obtained for n ≥ 4. Figure 6.33 ˆ 2 obtained from the experiˆ 1X presents the trajectory of point A in plane X mental measurements and numerical calculations for n = 5. The analysis of results shows good compatibility of experimental measurements and numerical calculations. 6.2.2 Sandia Manipulator The Sandia manipulator presented in Fig. 6.34 has often been used for experimental measurements, mainly for control purposes (Eisler et al., 1993; Olejak, 2000; Harlecki, 2002; Martins et al., 2002).

174 (a)

6 Verification of the Method 10-1

xA [m]

0.8 0.4 0 1.0

2.0

t [s]

-0.4 n=3 n=4 n=5

-0.8

(b)

10-1

yA [m]

0.8 0.4 0 1.0

2.0

t [s]

-0.4 -0.8

n=3 n=4 n=5

Fig. 6.32. Deflections of the end of the beam: (a) xA,1 , (b) xA,2

The links are driven by electric motors E1 (motor RTMct-85-2,3 for link 1) and E2 (motor RTMct-85-1 for link 2) controlled by PID controllers SYD 106/TH. The controllers operate with AD control cards (Kethley DAS 1800) in PC Pentium. Any kinematic input defined by angular velocities ψ˙ 1 and ψ˙ 2 can be realised. These velocities are converted into voltage [V] and used by AD cards to control the motion of manipulator links. Displacements of chosen points were measured using a photographic method (Adamiec et al., 2003). Positions of light points (P1 and P2 in Fig. 6.35), placed at the ends of the links, are registered on a film. By using a UP device (Fig. 6.35) working as a movable diaphragm, time-calibrated trajectories of the light points were obtained, which enabled time characteristics of changes in coordinates x1 and x2 of points P1 and P2 to be plotted. Figure 6.36 presents the courses of functions ψ1 (t) and ψ2 (t) assumed, and their time derivatives.

6.2 Experimental Verification of the Method 10-1

175

xA,2 [m]

Calculations Experiment

1.0

0.6

0.2 10-1 0.0 -1.0

-0.5

0.5

xA,1 [m]

-0.2

-0.6

-1.0

Fig. 6.33. Comparison of experimental measurements and calculations

When the links of the manipulator are rigid (with the same lengths as for flexible links), trajectory T of points P1 and P2 and courses of function X1 = xP1 ,1 (t) and X2 = xP1 ,2 (t) are calculated for the input functions assumed and are shown in Fig. 6.37. Geometrical and mass parameters for the manipulator with flexible links are given in Table 6.10, using notations shown in Fig. 6.38. Assuming the link parameters as in the table and input functions ψ1 and ψ2 defined in Fig. 6.36, the trajectory of point P2 and courses of functions X1 = xP2 ,1 (t) and X2 = xP2 ,2 (t) are shown in Fig. 6.39. In this figure curve R denotes the course for the manipulator with rigid links and curve E represents the courses for the manipulator with flexible links obtained from the experimental measurements. Applying the models discussed in Chap. 3, which use the classical rigid finite element method to discretise flexible links, numerical calculations were carried out for the manipulator considered. Input functions as in Fig. 6.36 and parameters of links defined in Table 6.10 were assumed. Comparison of

176

6 Verification of the Method

Fig. 6.34. Sandia manipulator – general view

numerical results with those from experimental measurements is presented in Fig. 6.40. Courses obtained by numerical calculations are plotted as curve C. Calculations were carried out for m1 = m2 = n = 4, so the flexible links were divided into only five rigid elements connected by four spring-damping elements. It can be seen that courses T, X1 and X2 obtained from numerical calculations and experimental measurements are very similar. In order to evaluate the compatibility of these courses, the following percentage errors are calculated:

6.2 Experimental Verification of the Method

177

w = const.

UP

DC x

y

P1

AD

Y1

L1

PC

Y2 P2

E2

E1 L2

Fig. 6.35. Sandia manipulator – measurement system. UP moving diaphram; DC camera; AD Kethley’s cards; PC PC computer; L1, L2 links 1 and 2; E1, E2 motors rotating links 1 and 2; P1 , P2 point meausred

E1 E2 e1 e2

5 5 E 5 I1 − I1C 5 5 100% = 2.8%, 5 =5 I1E 5 5 5 E 5 I2 − I2C 5 5 100(%) = 4.4%, 5 =5 I2E 5 ∆1,max = 100% = 4.1%, x1,max ∆2,max = 100% = 6.9%, x2,max

where IjE =

1 T

#T xE P2 ,j dt, 0

IjC =

1 T

#T xC P2 ,j dt, 0

5 5 C 5 ∆j,max = max 5xE P2 ,j − xP2 ,j , 0≤t≤T 5 5 5 xj,max = max 5xE P2 ,j j = 1, 2, 0≤t≤T

(6.28a) (6.28b) (6.28c) (6.28d)

178

6 Verification of the Method

(a)

0.6

3

0.5

2.5

0.4

y2 [ ]

y1 [ ]

3.5

2 1.5

0.3

1

0.2

0.5

0.1

0 0

1

2

3

0

4

0

t [s]

(b)

1

1.5

2

2.5

t [s] 0.8

0 0

0.5

1

1.5

2

2.5

3

3.5

4

0.7

-0.5

0.6 0.5

y 2 [1/s]

-1

0.4 0.3

.

-1.5

.

y 1 [1/s]

0.5

-2

0.2

-2.5

0.1 0

-3

0

0.5

1

t [s]

2

2.5

2

2.5

4 3 2

0.5

1

1.5

2

2.5

3

3.5

4

..

50 40 30 20 10 0 -10 0 -20 -30 -40 -50

y 2 [1/s2]

..

y 1 [1/s2]

(c)

1.5

t [s]

1 0

-1

0

0.5

1

1.5

-2 -3 -4 t [s]

t [s]

Fig. 6.36. Angular displacements, velocities and accelerations of the links of the mainpulator: (a) ψ1 , ψ2 , (b) ψ˙ 1 , ψ˙ 2 and (c) ψ¨1 , ψ¨2

xE P2 ,j (j = 1, 2) are courses of functions xP2 ,j (t) obtained by measurements, xC P2 ,j (j = 1, 2) are courses of functionsxP2 ,j (t) obtained by calculations, and T is the calculation time. It can be seen that the errors do not exceed 7%. The reasons for the difference in results of calculation and measurements are the following: 1. A photographic method was used. On the one hand this method eliminates problems connected with the influence of high frequency vibrations in the drive system on vibrations of flexible links (this is especially inconvenient when using acceleration sensors for measurements). On the other hand, however, this method is not exact and requires a lot of work in result processing, which can cause additional errors. 2. Errors made in measurements and calculations of mass and geometrical parameters of flexible links. 3. Errors in realisation of the assumed input functions.

6.2 Experimental Verification of the Method (a)

179

800 600 400

X 2 [mm]

200

-1200

0 -1000

-800

-600

-400

-200

0

200

400

600

-200

P1 P2

-400 -600 -800 -1000

X1 [mm] 600

(b)

400 200

X1 [mm]

0 -200

0

0.5

1

1.5

2

2.5

3

3.5

4 P1 P2

-400 -600 -800 -1000 -1200

t [s]

(c)

800 600

X2 [mm]

400 200 0 -200

0

1

2

3

4

P1 P2

-400 -600 -800 -1000

t [s]

Fig. 6.37. Motion of points P1 and P2 of the manipulator with rigid links: (a) T trajectory, (b) X1 functon xPj ,1 (t) and (c) X2 function xpj ,2 (t)

In general, it can be stated that the results of measurements and calculations are acceptably compatible. The courses of relative displacements of points P1 and P2 (caused by deflections of links 1 and 2) are shown in Fig. 6.41. The results are shown for different numbers n = m1 = m2 of spring-damping elements connecting rigid elements. It can be seen that the difference between the results obtained for n = 3, n = 4 and n = 5 is very small. The more elements into which the flexible link is divided, the longer the calculation time and the better accuracy of results. Since there is not a significant dif-

180

6 Verification of the Method a

AX ˆ

3

AX ˆ

BX ˆ

∆

B

3

2

b

rAC

Xˆ 2

mB

rBC

mA

BX ˆ

Aˆ

X1

lA

LA

1

l

L

Fig. 6.38. Notations for link parameters k(k = 1, 2)

(a)

800 600

X2 [mm]

400 200

E R

0 -200 -400 -600 -800 -1200 -1000 -800 -600 -400 -200

0

X1 [mm] -200

(b)

(c) 1000 500

-600

X2 [mm]

X1 [mm]

-400

-800

-500

-1000 -1200

0

0

1

2 t [s]

3

-1000

0

1

2

3

t [s]

Fig. 6.39. Comparison of calculation results for rigid links and measurements for flexible links for point P2 : (a) T trajectory, (b) X1 function xP2 ,1 (t) and (c) X2 function xP2 ,2 (t)

6.2 Experimental Verification of the Method

181

Table 6.10. Parameters of the manipulator links (k – number of the link) quantity

notation

length of part A

(k) lA (mm) A (k) xC,1 (mm) A (k) xC,2 (mm) A (k) xC,3 (mm) (k) mA (kg) (k) Lf (mm)

distance of the centre of mass from the origin of the local coˆ 2X ˆ3 ˆ 1X ordinate system X mass length of the flexible part intersection of the flexible part

(k)

lB (mm)

distance of the centre of mass from the origin of the local coˆB ˆB ˆ ordinate system B X 1 X2 X3

B

mass

k=2

120

200

0

68

0

0

0

40.3

28

3.08

318

524

rectangular a = 2.65 mm b = 62.0 mm

length of part B

total length of the link

k=1

(k)

xC,1 (mm)

(k) xC,2 (mm) B (k) xC,3 (mm) (k) mB (kg) (k) B

L

(mm)

119

circular φ = 3.9 mm 61

60.0

30

0

0

−44.7

0

6.6

0.235

557

785

ference in the results presented, it is assumed that n = 4 for calculations carried out in this section in order to combine the accuracy of the results and numerical effectiveness. It is important to note that deflections of flexible links are large. Results presented in Fig. 6.41 show that they are about 7 and 12% of the length of links. This means that the rigid finite element method can be successfully applied in analysis of large displacements of flexible links of machines and mechanisms. The results presented prove that both the rigid finite element method and its modification can be used in dynamic analysis of systems with changing configuration with flexible links, and they enable large deflections and influence of centrifugal forces on link vibrations to be taken into account. Both methods (CRFE and MRFE) have a simple physical interpretation. It should be noted that the MRFEN method allows us to consider large deflections of flexible links and effects of centrifugal forces without formulating special non-linear

182

6 Verification of the Method (a)

800 600

X2 [mm]

400 200

C E R

0 -200 -400 -600 -800 -1200 -1000 -800 -600 -400 -200

0

X1 [mm] -200

(b)

(c) 1000 500

-600

X2 [mm]

X1 [mm]

-400

-800

-500

-1000 -1200

0

0

0.5

1

1.5 2 t [s]

2.5

3

-1000

3.5

0

0.5

1

1.5 2 t [s]

2.5

3

3.5

Fig. 6.40. Comparison of calculation results and measurements for point P2 : (a) T trajectory, (b) X1 function xP2 ,1 (t) and (c) X2 function xP2 ,2 (t) (a) 30

(b) 80 60

20

X2 [mm]

X2 [mm]

40 10 0

20 0

-20

-10

-40 -20 -30 0

-60 0.5

1

1.5

2

2.5

3

3.5

-80 0

0.5

1

t [s]

1.5

2

2.5

3

3.5

t [s]

n= 3

n= 4

n= 5

Fig. 6.41. Link deflections x2 = u2 (Lj ): (a) link 1 (point P1 ) and (b) link 2 (point P2 )

components of the potential energy. An important feature of the algorithms presented is also the possibility of considering the influence of vibrations of flexible links on the base motion, which, as the results show, can be significant in planning trajectories of manipulators.

7 Applications

In this chapter we will present some applications of the rigid finite element method in analysis of real mechanisms and machines. Because of the illustrative purpose of the problems presented we will omit detailed derivations of the equations of motion for each particular case, referring those who are interested to the quoted sources. The first two examples present planar models of an offshore crane and a telescopic rapier. A spatial model is exemplified by the model of an A-frame formulated using both the classical and modified rigid finite element method.

7.1 Offshore Crane Offshore cranes are mounted on floating platforms or vessels and are primarily used for material handling from and to the deck of a supply vessel. Dynamic analysis of such cranes is especially difficult since it is necessary to consider phenomena connected with sea waves. Osi´ nski et al. (2004) formulate a planar model of a crane, installed on a vessel, which lifts a load off or lowers it to the deck of a supply vessel (Fig. 7.1). The model formulated is used to analyse forces acting in joints, ropes and hydraulic cylinders and for analysis of the motion of the system during the two most critical phases of the reloading operation: lifting the load from or lowering it onto the deck of a supply vessel. The model takes into account flexibility of the hoist rope, jib and hydraulic cylinders and motion of the vessels caused by sea motion, which is described using functions of time. 7.1.1 Discretisation of Flexible Links and the Equations of Motion Among the flexible subsystems the following are distinguished: beams with changeable cross-sections with longitudinal and bending flexibility ({1} and {2}) and elements with longitudinal flexibility only ({3} and {4}).

184

7 Applications (a)

Lifting Capacity

SWL [tonnes]

AKTRO

30

Main hoist

25 20 15

Aux. hoist

Gantry hoist

10 5

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 17,7 4,4 5,6 9,8 Working radius [m]

Luffing cylinders

Superstructure

RO

AKT

Aux. hoist Main hoist Jib

Gantry hoist Pedestal Rmin = 5000

Rmax = 30000

(b) {3} {5}

{2} {4}

{1}

{6}

Fig. 7.1. The system considered: (a) TTS-Aktro off-shore crane; (b) model of the crane: {1} the ship with the crane pedestal and slewing platform, {2} jib, {3} hydraulic cylinder, {4} shock absorber, {5} hoisting winch with wire rope, {6} load

Figure 7.2 shows a beam with longitudinal and bending flexibility and its model obtained by means of the rigid finite element method as well as generalised coordinates of the i-th rfe. The motion of such a subsystem is described by 3(m(k) + 1) generalised coordinates, which are elements of the following vector:

7.1 Offshore Crane

185

rfe (k, m(k )) Xˆ 1

Xˆ 1 j

(k,i ) 3

(k )

mi(k ), Ii

j 3(k,i)

sde (k, m(k ))

(k,i )

x2

b

D

Xˆ 1

a

Ci

rfe (k,i )

(k )

(k,i )

x2

sde (k,i ) rfe (k,1)

Xˆ 2

j0(k) Xˆ 2

Xˆ 3

j0(k)

rfe (k,0) (k )

A(k) (x1(k), x 2(k) )

(k,i )

x1

x1( k ,i )

A

sde (k,1)

A(k )

Xˆ 1

Fig. 7.2. Discretisation of a flexible beam and generalised coordinates of the rfe (k, i)



q(k)

where

q(k,0) q(k,1) .. .



           =  (k,i)  ,   q   ..   .   (k) (k,m ) q

(7.1)

,T + (k,0) (k,0) q(k,0) = x(k,0) x ϕ 1 2 3

is the vector of coordinates describing the position of rfe (k, 0), + ,T q(k,i) = x1(k,i) x2(k,i) ϕ3(k,i) is the vector describing the position of rfe (k, i) with respect to rfe (k, 0). Hydraulic cylinder {3} connecting the platform with the jib, and shock absorber {4} smoothing the courses of forces induced when lifting the load, have only longitudinal flexibility and are treated as a system of two rigid elements connected by a spring-damping element (Fig. 7.3).

186

7 Applications rfe (k,1)

(k)

C1 m 1(k) , I 1(k) sde (k)

r1(k) C

(k) 0

(k)

l1

m 0(k) , I 0(k)

rfe (k,0)

j (k)

(k)

D

(k) 0

r (k)

A ( x 1(k), x 2(k))

(k)

l0

Fig. 7.3. Discretisation of elements with longitudinal flexibility (s)

R1

{3 } R2(3) A(3) R1(3)

A(3)

R1(3) R2(2)

j {5 } j D,

R2(s)

(3)

W0

R1(2)

(2)

A

p2 = n2 {2 }

R1(2)

W2

R1(s)

S

A(2) R2(3)

W1

R2(s)

p1

T

P3

B

(2)

R2

Xˆ 2

jB

j0(1)

B

{1 }

Xˆ 2

j (4)

{4 }

p0

A(4)(x1(4),x2(4)) L(xL,1,xL,2) B

{6 }

Xˆ 1

(1)

A (x1(1) ,x 2(1) )

Xˆ1

Fig. 7.4. Subsystems

The vector of the generalised coordinates describing the motion of such a subsystem has the form: (k)

q(k) = [x1 (k)

(k,0)

(k)

(k,0)

(k)

x2

(k)

ϕ3

∆(k) ]T ,

(7.2) (k)

where x1 = x1 , x2 = x2 are coordinates of point A(k) , ϕ3 = ϕ(k) is ˆ 1 of the inertial the angle of inclination of the cylinder in relation to axis X (k) coordinate system, ∆ is deformation. In order to derive the equations of motion of the whole system (Fig. 7.4), the following assumptions concerning subsystems are made:

7.1 Offshore Crane

187

Subsystem {1} (pedestal) is a beam with longitudinal and bending flexibility. It is assumed that motion of rfe (1,0), which consists of the vessel and the initial part of the pedestal, is known. This means that components of the vector: (1,0)

q(1,0) = [x1

(1,0)

x2

(1,0) T

ϕ3

(1)

(1)

(1)

] = [x1 (t) x2 (t) ϕ0 (t)]T

(7.3)

are known functions of time. The last element of the flexible subsystem, rfe (1, m(1) ), is not only the final part of the pedestal but also part of the slewing platform of the jib. Subsystem {2}, representing the jib, is treated as a beam with longitudinal and bending flexibility. It is assumed that the drum of the hoisting winch is (0) (0) located on element p0 at the point with coordinates (a1 , a2 ) with respect to the local coordinate system of rfe (2, p0 ). Likewise it is assumed that the rollers of the rope system are located on elements p1 and p2 = m(2) at the (1) (1) (2) (2) points with coordinates (a1 , a2 ) and (a1 , a2 ), respectively, in relation to the local systems. The influence of the rotating movement of the rollers on the whole system is neglected. Mass parameters of rfes (2, p0 ), (2, p1 ) and (2, p2 ) include masses and moments of inertia of the hoisting winch drum and sheaves. Hydraulic cylinder {3} is connected with rfe (2, p3 ) of the jib and coordi(3) (3) nates of the local connection are (a1 , a2 ). Both hydraulic cylinder {3} and shock absorber {4} are modelled as elements with longitudinal flexibility. The motion of subsystem {5}, which is the drum of the hoisting winch, is described by the angle of rotation of the drum ϕ(D) . Load {6} is treated as a lumped mass and its motion is described by the coordinates of the vector: q(6) = [xL,1

xL,2 ]T .

(7.4)

If the load lies on the deck of a floating unit, its coordinates are known functions of time (coordinates of point B). Having calculated the kinetic energy, the potential energy of gravity forces and spring deformation of flexible beams and the rope as well as the function of energy dissipation, the equations of motion of the whole system can be written in the form (Osi´ nski et al., 2004): A¨ q + BR = F,

(7.5a)

T

(7.5b)

¨ = G, B q where 

 ¯ (1) q (2)  q  ¯ (3)  q   q=  q(4)  ,    q(5)  q(6)

188

7 Applications

¯ (1) = [q(1,1) · · · q(1,n1 ) ]T , q ¯ (2) = [q(2,1) · · · q(2,n2 ) ]T , q q(l,i) is defined by (7.1), q(3) , q(4) are defined by (7.2), q(5) = ϕD , q(6) is defined by (7.4), (2) (2) (3) (3) (s) (s) R = [R1 R2 R1 R2 R1 R2 RB,1 RB,2 ]T is the vector of unknown joint reactions, A is a mass matrix with changing elements, B is matrix with changing elements, F is the vector including expressions resulting from the potential energy, the function of energy dissipation and generalised forces resulting from the torque MD acting on the drum of the hoisting winch. The equations of motion (7.5a) form a system of: n = 3m(1) + 3(m(2) + 1) + 4 + 4 + 1 + 2 = 3(m(1) + m(2) ) + 14

(7.6)

nonlinear differential equations of the second order with n+8 unknowns. These equations are completed with constraint equations (7.5b) in the acceleration form (Osi´ nski et al., 2004). 7.1.2 Numerical Calculations In order to carry out numerical simulations, the jib was discretised in two different ways. Figure 7.5a presents primary and secondary division according to the description in Sect. 3.1. The second approach (Fig. 7.5b) takes into account the changing stiffness characteristic of the jib, for which the second moment of inertia changes as shown in Fig. 7.6. The accuracy of the discretisation has been verified by calculating the frequency of the free vibrations and static deflections caused by the concentrated force P. The model of the jib used for this analysis is shown in Fig. 7.7. Table 7.1 presents the values of the first three frequencies of free vibrations and deflections of the jib calculated for a different number of rigid elements. The results obtained when the jib was divided into elements of various length, i.e. taking into account the changing stiffness characteristic, are presented in the last row of the table. For further considerations and calculations this model was adopted and applied. In the next figures results are presented which were obtained for the following data: – Mass of the load mL = 30, 000 kg – Load radius R = 17.7 m – Nominal value of angular velocity of the hoist drum wnom = 0.45 m s−1 or 0.80 m s−1 – Radius of the hoist drum r0 = 0.616 m

7.1 Offshore Crane

189

(a)

(b)

Fig. 7.5. Primary and secondary divisions of the jib: (a) into elements of the same length, (b) with consideration of the stiffness characteristic

9.E+10 (mm)4 8.E+10 7.E+10 6.E+10 5.E+10 4.E+10 3.E+10 2.E+10 1.E+10 0.E+00 0

5,000

10,000

15,000 x1

Fig. 7.6. Second area moment of the jib

20,000

25,000

190

7 Applications

P

Fig. 7.7. Support and load of the jib in analysis of linear vibrations and in static analysis

Table 7.1. Influence of the number of rfes on frequencies of free vibrations and static deflections m(2)

6 8 10 12 14 16 18 20 10

f1 (Hz)

3.731 3.763 3.731 3.763 3.700 3.763 3.763 3.731 3.731

f2 (Hz)

18.594 19.068 19.037 19.163 18.879 19.100 19.195 19.131 18.942

f3 (Hz)

25.551 25.393 28.239 27.227 27.385 27.986 27.700 28.175 27.195

xP,2 (mm) g = 9.81(m s−2 ) P =0 −18.81 −18.76 −18.94 −18.63 −19.28 −18.71 −18.49 −18.89 −18.95

g=0 P = 2 × 105 (N) −50.42 −49.46 −50.49 −49.58 −51.28 −49.94 −49.29 −50.21 −50.44

The assumed mechanical characteristic of the hoist drive is shown in Fig. 7.8. Figures 7.9 and 7.10 present vertical displacements of load xL,2 and the deck of the supply vessel xD,2 as well as the velocities of the load and the drum calculated with the assumption that the vessel on which the crane is mounted does not move, while the movements of the supply vessel are limited to heave (vertical motion) described as follows: xB,2 = A sin(αt + ϕ0 ),

(7.7)

where A = 1.5 m is heave amplitude, α = 2π/12 s−1 is angular frequency of the heave, ϕ0 is the phase angle which defines the motion of the heave assumed as 90◦ (Fig. 7.9) and 270◦ (Fig. 7.10). The following dynamic coefficient is defined in order to describe the dynamics of the rope: FL , (7.8) ηL = FLnom where FL is a force in the rope at point T (Fig. 7.4), FLnom = mL g is a nominal (static) force in the rope. Changes in the dynamic coefficient of the rope for different phase angles are presented in Fig. 7.11.

7.1 Offshore Crane

191

MD MD nom 1.4 1.2 1 0.8 0.6

0.54

0.4 0.2 0 0

0.5

1

1.8

1.5

2

wD w D nom Fig. 7.8. Characteristic of the hoist drive MDnom = mL g rD (a) 5

load

wave -0.8 -1

4 coordinate x2 [m]

-1.2

3

-1.4

2

-1.6 0

0.3 0.6 0.9 1.2 1.5 1.8

1 0 -1 -2 0

1.875

(b)

3.75

5.625

7.5 time [s]

9.375

11.25

drum

13.125

15

load

speed [m s-1]

1.6 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 -1.6 -2 -2.4 0

1.875

3.75

5.625

7.5 time [s]

9.375

11.25

13.125

15

Fig. 7.9. Results of calculations: (a) vertical displacements xL,2 and xD,2 , (b) velocity of the load (x˙ L,2 ) and the drum (vD = ϕ˙ D rD ) for phase angle ϕ0 = 90◦ and nominal speed of the load 0.46 m s−1

192

7 Applications (a)

wave

load

5

coordinate x2 [m]

4 3 2 1 0 -1 -2 0

(b)

1.875

3.75

5.625

7.5 time [s]

9.375

drum

11.25

13.125

15

load

1.6 1.2

speed [m s-1]

0.8 0.4 0 -0.4 -0.8 -1.2 0

1.875

3.75

5.625

7.5 time [s]

9.375

11.25

13.125

15

Fig. 7.10. Results of calculations: (a) vertical displacements xL,2 and xB,2 , (b) velocity of the load (x˙ L,2 ) and the drum (vD = ϕ˙ D rD ) for phase angle ϕ0 = 270◦ and nominal speed of the load 0.8 m s−1

It can be seen that initial phase ϕ0 considerably influences the dynamic coefficient, especially during the phase of lifting the load from the deck of the supply vessel. Many more results of numerical simulations as well as a detailed description of the mathematical model are presented by Osi´ nski et al. (1999); Osi´ nski et al., (2004).

7.2 Telescopic Rapier In this section we will present an application of the rigid finite element method to discretisation of a telescopic rapier, the scheme of which is presented in Fig. 7.12.

7.2 Telescopic Rapier

193

initial phase j0 0

90

180

270

1.75 Hoist rope coefficient

1.5 1.25 1 0.75 0.5 0.25 0 0

0.3

0.6

0.9

1.2

1.5 time [s]

1.8

2.1

2.4

2.7

3

Fig. 7.11. Hoist rope coefficient ηL for various phases at nominal speed of the load 0.80 m s−1

8

9

Xˆ 2 10 Xˆ 3

Xˆ 1 7 6

3 4 5

2

1

Fig. 7.12. Kinematic scheme of a telescopic rapier with its drive. 1–6 parts of drive assembly: engine, gear, crank shaft, con rod, beater, connector, 7 slide, 8 external rapier, 9 internal rapier, 10 drive belt

194

7 Applications

Rapiers of this type, because of their telescopic construction, allow the width of the weaving to be increased without decreasing the speed of weaving. Since the external rapier is fairly rigid, the internal rapier can move in the shed without additional guidance. During motion the external rapier moves at a given speed and functions as a slide; therefore the internal rapier moves at twice the speed, if we assume ideal stiffness of the drive belt. Flexibility of each element, especially of the internal rapier and the belt drive, considerably influences the trajectory of the rapier head. Excessive flexibility of the internal rapier can cause serious damage to the mechanism. In order to analyse the motion of the rapier, the physical model is formulated with the following assumptions: – Slide movement (7) is known and described by function w(t), and thus the drive system can be omitted in further considerations (elements 1–6). ˆ 2 and ˆ 1X – The external rapier is treated as a rigid body in both planes X ˆ ˆ X1 X3 . – Bending flexibility of the internal rapier is taken into consideration in ˆ 1X ˆ 2. plane X – Longitudinal flexibility of the belt drive is taken into account by means of stiffness coefficient cA,1 and the displacement of the belt axis in relation to the axis of the internal rapier (Fig. 7.13d). – Stiffness coefficient cb,2 (Fig. 7.13a) takes into account lateral flexibility ˆ 2 axis) of the driving belt. (in the X – The weft-carrying head is treated as lumped mass M . – The internal and external rapiers can contact only at points A, B, C, D, E ˆ 1X ˆ 2 plane and at A , A , E  , E  points in the X ˆ 3 plane. ˆ 1X in the X     Because points A , A and E , E deviate little from the central axis of the internal rapier, they are treated as points A and E. – Clearance is allowed to occur during the motion of the internal rapier in ˆ 3 plane. ˆ1X the X 7.2.1 Discretisation of the Internal Rapier and the Equations of Motion The modification of the rigid finite element method, described in Chap. 4, is used for discretisation of the flexible internal rapier. The division of the rapier into n + 1 rigid elements and n massless and nondimensional spring-damping elements, the generalised coordinates and flexible connections are presented in Fig. 7.14. The generalised coordinates of the rapier can be presented in the form: (A)

q = [x2

(s)

(s)

ϕ(0) · · · ϕ(n) x1 x3 ϕ]T .

(7.9)

The equations of motion are derived in the way presented in Chap. 4. However, the potential energy of spring deformation at contact points also has to be taken into account. The energy can be written as follows:

7.2 Telescopic Rapier (a)

ˆ X 2 w (t )

195

lE

A cb,2 cB,2

0

E' cE,2

cA,2 C

B

cC,2

cD,2

D

ˆ X 1

lB lC lD xS,1

(b)

E" w (t )

M S

A" 0

cE,3

cA,3 A

A'''

cA,3

cE,3

xS,3

j Xˆ 1

E ''' lA

Xˆ 3 Xˆ 2

(c)

(d)

Xˆ 3

Xˆ 1

A 2w D 2

D 2

cA,1

a

Fig. 7.13. Model of connection between internal and external rapiers: (a) in plane ˆ 1X ˆ 1X ˆ 2 , (b) in plane X ˆ 3 (S is the centre of mass of the head), (c) cross-section, X (d) connection of the internal rapier with the drive belt

– At points A and E: 1 c¯P,2 [xP,2 ]2 , 2

2 1 ∆ Z = δP,3 cP,3 xP,3 − sgn xP,3 2 2

VP,2 = VP,3 where



cA,2 for xA,2 > 0 , cb,2 for xA,2 < 0 c¯E,2 =  cE,2 , 1 for |xP,3 | > ∆ 2 , δP,3 = 0 otherwise c¯A,2 =

(7.10) for P ∈ {A, E} ,

(7.11)

196

7 Applications l

(a) sde 1

rfe 0

sde i +1

sde i

sde 2

sde n

rfe i

rfe 1

sde i

l (i )

j (i )

sde i -1 c (i -1)

rfe i -1

rfe i +1

c (i )

rfe i

r (i )

rfe n

ji (i )

(i )

S (i )(x 1 ,x 2 ) (b)

(c) c (ep)

d (ep)

j (ep) P

A r (S)

S

Fig. 7.14. Discretisation of the rapier and models of connections: (a) internal rapier ˆ 1X ˆ 2 , (b) contact point P in plane X ˆ 2, ˆ 1X and its division into rfes and sdes in plane X ˆ 1X ˆ3 (c) contact point E in plane X

xP,2 = xA,2 + ϕ(i) =

i 3

eP −1 3

li sin ϕi + deP sin ϕeP ,

i=0

ϕ(j) ,

j=0

xP,3 = xA,3 + r(s) sin ϕ, eP is the number of the rfe of the internal rapier contacting point P (Fig. 7.14b). – At points B, C and D: VP,2 =  where δP =

1 δP cP,2 [xP,2 ]2 2

1 for xP,2 < 0 . 0 otherwise

P ∈ {B, C, D},

(7.12)

7.2 Telescopic Rapier

197

Moreover, the energy of longitudinal deformation of the drive belt has to be taken into account (Fig. 7.13d) and it can be written in the form: Vb,1 =

1 cA,1 (xS,1 − lA cos ϕ + a sin ϕ − 2w)2 . 2

(7.13)

After necessary transformations and determination of the generalised forces including friction forces at contact points: TP,d = −sgn νP,d µP,d |NP,d | for (P = A, E and d = 2, 3) and (P = B, C, D and d = 2),

(7.14)

where νP,d is the component of the internal rapier velocity in direction d at contact point P , µP,d is a coefficient of kinetic friction, NP,D is a normal component of the reaction force at contact points, the equations of motion can be presented in the general form (Wojciech, 1993): A¨ q + Bq˙ + Cq = F.

(7.15)

The equations form a system of n + 4 ordinary differential equations of the second order. 7.2.2 Numerical Calculations The results of numerical simulations are obtained after assuming that the following function describes a kinematic input causing the motion of the rapier: w(t) =

w0 (1 − cos ωt). 2

(7.16)

Initial conditions are determined by solving the equivalent static problem. An influence of the flexibility of the rapier on its movement is presented in Fig. 7.15a by comparing the displacements of the head of the rigid rapier (n = 0) with those for a flexible one (n = 19). Figure 7.15b shows the influence of the number of rigid elements on the accuracy of the results. Wojciech (1993) describes a model of a telescopic rapier in which the finite difference method is used in order to model the flexible internal rapier. Figure 7.16 shows the comparison of the results obtained using the MRFEN method described in this book and those obtained by means of the finite difference method. Very good compatibility of results can be seen. The results of numerical simulations presented here and by Wojciech (1993) show that an acceptable accuracy of the results is achieved when n ≥ 13. The reasons for such a large number of elements are that the clearance between the internal and external rapiers is taken into account and that values of stiffness coefficients of spring-damping elements at contact points are large. This leads to vibrations with high frequencies occurring in the system.

198

7 Applications xM,2[10 -4m]

(a)

2p 0 wt 1

rigid

2

3 flexible 4

5 xM,2 [10 -4 m] p

(b)

2p

0 wt 1

2

3

4

n = 15 n = 13

5

n = 11

Fig. 7.15. Deflections of the rapier: (a) displacements xM,2 of the head of the flexible and rigid rapiers, (b) influence of number n of rfes on the accuracy of results

7.3 A-Frame An A-frame is a kind of a crane with a portal construction mounted by an articulated joint to the deck of a ship somewhere close to the stern. Servo-

7.3 A-Frame

199

xM,2 [10-4m] p

2p

wt

0

1

2

3

4 b 5

a

Fig. 7.16. Comparison of deflections of the rapier: (a) the finite difference method, (b) the MRFEN method

motors mounted to the legs of the frame enable it to rotate, which makes it possible to transfer a load from the deck beyond the dimensions of the deck. The system is usually equipped with one or more hoisting winches mounted to the deck, which by a system of rope sheaves and guides allows the load to be lifted and lowered. Figure 7.17 demonstrates how an A-frame works. This type of cranes is used for transferring very heavy elements, for example for setting underwater pipelines, for research operations or for lifting and lowering bathyscaphes (Fig. 7.18). The conditions of work of the A-frame are very specific. Unlike for other cranes (port cranes or wheeled cranes), the movement of the load is caused not only by displacements generated by the machine operator (inclination of the jib, hauling of the load), but also by the movement of both the ship and a supplementary vessel (which supplies or receives the load) as a result of wave motion. Figure 7.19 presents a scheme of the A-frame with the most important elements and characteristic points. The basic elements of the A-frame are briefly described below. F – supporting structure The supporting structure of the A-frame is a box structure and consists of steel plates of different thicknesses welded together. Three beams forming a rectangle or trapezoid are the body of the structure (Fig. 7.20). Sometimes beams (1), (2) and (3) are strengthened with truss elements. They also have formers and several additional masses; for example on and

200

7 Applications

(a)

(b)

(c)

(d)

Fig. 7.17. Positions of an A-frame during work

Fig. 7.18. A-frame used for lowering and lifting a bathyscaphe

inside the structure there are ladders enabling inspections to be carried out. Lengths of the beams are up to 20 m, and dimensions of the cross-sections are 2 × 2 m. The frame is connected with the deck of the ship by articulated joints at points AR and AL , which are treated as spherical joints not transferring

7.3 A-Frame

S

F

ML

^ FX

3

^ DX MR 3

R

F^

X2

L

SL NL

AL ^ DX 2

H

SR

H

AR F^

X1

201

NR

D^ X1

Fig. 7.19. A-frame, denotations: F supporting structure, S sheave, R rope, H drum of the hoisting winch, L load, SR , SL starboard and port servomotors, MR , ML points of servomotor connections to the A-frame, NR , NL points of servomotor connections to the deck, AR , AL points of mounting the A-frame to the deck, F X1 F X2 F X3 coordinate system assigned to the supporting structure, D X1 D X2 D X3 coordinate system assigned to the deck of the ship

top beam (2)

B2

B3

X3 B1

X3

X3

FX 3

port leg (3) gR starboard leg (1)

gL xLR /2

FX

1

xLR /2 FX 2

Fig. 7.20. Supporting structure of the A-frame

202

7 Applications

moments of forces. Its movement is controlled by means of servomotors SR and SL connected with the deck and the frame by articulated joints. SR , SL – servomotors Servomotors ensure regulation of the inclination angle of the A-frame supporting structure and thus they determine crane reach. Angle of inclination ψ depends on the length of servomotors SR and SL . It is assumed that the servomotors are undeformable and the courses of length changes, LR (t) and LL (t), are known: (7.17) LR = LL = L(t).

L – load For further considerations it is assumed that the load can be treated as a lumped mass mL . It is possible to take into account geometrical dimensions of the load; however, this may complicate the problem especially when phases of detaching the load from the deck and putting the load on the deck are considered. Treating the load as a rigid or flexible body flexibly connected with the deck requires analysis of complex contact problems between the load and the deck (Ko´scielny, 1993), and we omit this problem here. R, S, H – rope, sheave, winch The rope is reeled on drum H, goes through sheave S, and at its end load L is attached. Its length varies from ten to several hundreds of meters, when the A-frame is used for underwater work close to the sea bottom. The rope is flexible and has damping features. The stiffness coefficient of the rope, which depends on material features and the length, can be calculated according to (Osi´ nski and Wojciech, 1998). Because of the relatively small dimensions of the sheave in comparison with the dimensions of the frame, it is assumed that the rope goes through its centre.

MH jH

IH

dH

Fig. 7.21. Drum of the winch

7.3 A-Frame D FX ˆ

SV

Xˆ 3

SV

SVX ˆ

DV

1

Xˆ 3

3

{F } DV

{SV }

Xˆ 3

Xˆ 2 Xˆ 1

FX ˆ

{DV }

DVX ˆ

2

2

{D }

DX ˆ

2

Xˆ 3

j3 F

Xˆ 1 = D Xˆ 1

Xˆ 2

{b }

SX ˆ

j1 S

Xˆ 1

203

Xˆ 1

3

j2 {S}

SX ˆ

2

Fig. 7.22. Coordinate systems

It is also assumed that the motion of the winch drum H can be effected by moment MH (Fig. 7.21): MH = MH (t, ϕ˙ H )

(7.18a)

or defined as a kinematic input: ϕH = ϕH (t).

(7.18b)

The assumed coordinate systems are presented in Fig. 7.22. The following notations are used: {b} is the inertial coordinate system which is motionless, and is the reference system in which the motion of all elements is analysed; {S} is the coordinate system of the ship; its position is determined by wave motion and drives of the ship, which are both known functions of time. Transformation from ship system {S} to inertial system {b} is described by matrix bS T defined as a composition of transformations, each of which corresponds to one parameter connected with ship motion: b b

b SR Ss , T = (7.19) S 0 1  c3 c2 c3 s2 s1 − s3 c1 c3 s2 c1 + s3 s1 b  s3 c2 s3 s2 s1 + c3 c1 s3 s2 c1 − c3 s1 , SR = −s2 c2 s1 c2 c1   x1 (t) b  x2 (t)  , s = S x3 (t) ci = cos ϕi (t); si = sin ϕi (t) for i = 1, 2, 3, x1 , x2 , x3 , ϕ1 , ϕ2 , ϕ3 are defined in Fig. 7.22.

where



204

7 Applications

{D} is the deck coordinate system isolated from the ship system (by translation along a constant vector), in order to ease definitions of structural parameters, especially positions of structural elements which are on the deck. Reactions between the frame and the deck are expressed with respect to this coordinate system (with axes parallel to that of system {S}). Transformation from {D} to {S} is described by matrix SD T with constant elements. {F } is the coordinate system of the frame chosen in such a way that axes 1 and 3 are always in the plane of the frame, and its position depends on inclination of the frame caused by change in length of servomotors. Geometrical dimensions of the frame and values calculated such as forces, displacements and stresses are defined with respect to system {F }. Transformation from system {F } to {D} depends on inclination angle ψ and is defined by the following matrix:   1 0 0 0  0 cos ψ(t) − sin ψ(t) 0  D   (7.20) F T =  0 sin ψ(t) cos ψ(t) 0  . 0 0 0 1 {Bk }, k = 1, 2, 3 are the coordinate systems of beams, the position of which with respect to system {F } of the frame does not change. The frame is divided into three parts (beams) presented in Fig. 7.20 (legs of the frame and the top beam). Respective coordinate systems and the methods of transformations are presented in Figs. 7.23 and 7.24. The transformation from the coordinate system of the k-th beam (k = 1, 2, 3) to the coordinate system of the frame can be defined by the matrix: B2

{B2} B2

Xˆ 2 B2

Xˆ 1

Xˆ 3

zLR

B3 F

B1

B1

Xˆ 2 {B1}

Xˆ 1

Xˆ 1 B1

Xˆ 3

B3

Xˆ 3 B3

gR

Xˆ 1

gL Xˆ 2

Xˆ 3

{B3}

Fˆ

X2

xLR

Fig. 7.23. Division of the frame into beams and assigned coordinate systems {B1 }, {B2 }, {B3 }

7.3 A-Frame F

F

205

Xˆ 3

Xˆ 1 Bk ˆ X3

ak

Bk

xk,3 ˆ X 1

xk,1

ak

(xk,1,0,xk,3)

F

Xˆ 2

Bk ˆ X2

Fig. 7.24. Transformation of the system of the k-th beam to the system of the frame



cos αk − sin αk 0  0 0 1 F  Bk T =  − sin α − cos α 0 k k 0 0 0 where

 xk,1 xk,2  , xk,3  1

(7.21)

α1 = γR + π, x1,1 = xLR x1,2 = 0, x1,3 = 0, 2 , x2,1 = − xBLR , x2,2 = 0, x2,3 = zAB , α2 = 0, 2 , x α3 = −γL , x3,1 = − xLR 3,2 = 0, x3,3 = 0. 2

All quantities occurring in (7.21) are constant. {SV } and {DV } are the coordinate systems assigned to the supplementary vessel. Vessel system {SV } and deck system {DV } are introduced in order to define the motion of this vessel and the position of the load on its deck. It is assumed that the appropriate transformation from system {SV } to the inertial system is defined by matrix bSV T(t), while the transformation from system {DV } is defined by constant matrix SV DV T. Beams forming the frame are discretised using both classical and modified formulations of the method, both for the linear model since deflections of the supporting structure are small.

7.3.1 Classical Rigid Finite Element Method Linear Model In this model beam k, which forms the frame, is divided into m(k) + 1 rigid finite elements and m(k) spring-damping elements (Fig. 7.25). Local coordinate system {k, i} is assigned to each element and the corresponding

206

7 Applications rfe(2,m (2))

rfe(2,i )

rfe(2,0)

rfe(3,0)

rfe(1,m (1))

sde(1,n1)

rfe(3,i )

rfe(1,i )

sde(1,1) rfe(3,m (3))

rfe(1,0)

Fig. 7.25. Division of the beam into rfes and sdes in the CRFEL method

transformation matrix from this system to the coordinate system of the beam is defined. The coordinates from the local coordinate system of rfe i are transformed to system {Bk } of beam k (Fig. 7.26), and then they can be transformed to the remaining coordinate systems by the transformations presented above. For the transformation from {k, i} to {Bk } it is assumed that the deformations of sdes are small, which enables us to linearise the trigonometrical functions from the transformation matrices. Therefore the matrices take the form:   (k,i) (k,i) (k,i) (k,i) 1 −ϕ3 ϕ2 x1 + d1  (k,i) (k,i) (k,i) (k,i)   ϕ3 1 −ϕ1 x2 + d2  Bk Bk =(k,i) T(q(k,i) ), (7.22) T =  (k,i) (k,i) (k,i)   −ϕ2(k,i) ϕ1(k,i) 1 x3 + d3  0 0 0 1 (k)

where di,j are constants defined in Fig. 7.26. The motion of rfe i of the k-th beam is described by the components of the following vector of the generalised coordinates: (k,i)

q(k,i) = [x1

(k,i)

x2

(k,i)

x3

(k,i)

ϕ1

(k,i)

ϕ2

(k,i) T

ϕ3

] ,

(7.23)

7.3 A-Frame

j 2(k,i )

{k,i }

j 1(k,i )

rfe(k,i )

x 2(k,i ) j 3(k,i )

Bk

{Bk }

ˆ2 X

d 2(k,i ) Bk

207

x 3(k,i )

x 1(k,i )

Xˆ 1 d 3(k,i )

Bk

Xˆ 3

d 1(k,i ) Fig. 7.26. Coordinates of rfe i of beam k

while the vector of the generalised coordinates of the frame has the form:  (1)  q (7.24) qF =  q(2)  , q(3)  q(k,1)  .  =  ..  q(k,nk ) 

where q(k)

for k = 1, 2, 3.

This form of the vector of generalised coordinates means that the frame is divided into three independent subsystems. The mutual interaction of the beams (as well as the interaction of the servomotors and the deck) have to be expressed by appropriate reaction forces and moments, and connections between subsystems are described by constraint equations. The kinetic and potential energies of the frame and other subsystems are calculated according to the procedure presented in Chap. 2 and 3. As assumed earlier, the load is modelled as a lumped mass, the generalised coordinates of which are the elements of the vector:   xL,1 (7.25) qL =  xL,2  xL,3 and the motion of the winch drum is defined by angle ϕH : qH = [ϕH ].

(7.26)

208

7 Applications B1

MBR,2

Xˆ 1

MBR,1

RBR,2

B2

F

RBL,3

Xˆ 1

RBR,1 RBR,3

MBR,3

- MBR,1

Xˆ 3

- M BL,1

MBL,1 RBL,3

RBR,2

-MBL, 2

RBL,1

RBR,1

M BL, 2

RBL,2

- MBR,2

- MBL,3

- MBR,3 B2

RAR,3

Xˆ 1

RBL,3

ˆ2 X

B2

MR B1

B3

RBL,1 RBL,2 z

MBL,3

ML

Xˆ 3

SL

SR

Xˆ 2

RAL,1 F

Xˆ 1

RAR,1

RAR,2

RAL,3 RAL,2

B3 B1

F

ˆ3 X

Xˆ 2

Xˆ 2 B3

Xˆ 3

Fig. 7.27. Reaction forces and moments – denotations

The generalised coordinates of the system of load and winch are the components of the following vector:

qL . (7.27) qR = qH The vector of the generalised forces F includes the generalised forces resulting from the drive moment of the winch, from forces and moments of mutual reactions between the beams 1, 2 and 3, as well as from the deck and the servomotors. Division of the structure into beams requires formulation of constraint equations in order to analyse the uniform structure. Reactions of servomotors and moments of reactions between the beams as well as reactions in the support of the frame also have to be taken into account (Fig. 7.27). The vector of the generalised forces F includes reactions of constraints presented in Fig. 7.27, especially: RAL = [RAL,1

RAL,2

RAL,3 ]T ,

RBL = [RBL,1

RBL,2

RBL,3 ]T

MBL = [MBL,1

MBL,2

MBL,3 ]T ,

MBR = [MBR,1

MBR,2

MBR,3 ]T ,

RAR = [RAR,1

RAR,2

RAR,3 ]T ,

and RBR = [RBR,1

RBR,2

RBR,3 ]T ,

and forces in servomotors SL and SR . Denotations listed in Table 7.2 are assumed for further considerations.

7.3 A-Frame

209

Table 7.2. Denotations of reactions in joints point

reaction forces and moments

AR AL BR

RAR RAL RBR , MBR

BL

RBL , MBL

MR ML

SR SL

number of beam rfe 1 3 1 2 3 2 1 3

(1)

pAR (3) pAL (1) pBR (2) pBR (3) pBL (2) pBL (1) pSR (3) pSL

denotation matrix D vector h DAR DAL (1) (1) DBR,R , DBR,M (2) (2) DBR,R , DBR,M (3) (3) DBL,R , DBL,M (2) (2) DBL,R , DBL,M DSR DSL

hAR hAL 0 0 0 0 hSR hSL

Detailed derivations of formulae for the elements of matrices occurring in Table 7.2 are presented by FaZlat (2004). Having omitted this stage, the dynamic model of the A-frame can be written in the form of a system of differential equations: A¨ q − DR = F − Cq, ¨ = W, DT q

(7.28a) (7.28b)

where D = D(q), ˙ F = F(t, q, q), ˙ W = W(q, q), C is the stiffness matrix,   SR  SL     RAR    R   AL  R= .  RBR     MBR     RBL  MBL Two sets of equations can be isolated in the above system: equations of motion (7.28a), which are formulated on the basis of the Lagrange equations when the kinetic and potential energies of rfes and sdes as well as the load and the winch are calculated, and (7.28b), which are the acceleration form of the constraint equations. In those equations there are: nq =

3 " k=1

6(1 + m(k) ) + 4 (components of vector q)

(7.29a)

210

7 Applications

and nR = 2 + 2 × 3 + 2 × 6 = 20 (components of vector R) and thus n = nq + nR =

3 "

m(k) + 42 unknowns.

(7.29b)

(7.29c)

k=1

The number of equations is the sum of nq equations of motion (7.28a), and nR constraint equations (7.28b), so the number of equations equals the number of unknowns. Expressions connected with reactions and constraint equations of the load are omitted in the above equations and they are introduced into the system only when the load lies on the deck of the supplementary vessel or the ship. In this case the number of equations of motion and thus unknowns increases by three; the dimension of matrix D also increases. Since the system of equations formulated has a large number of unknowns, which results mainly from consideration of all six degrees of freedom of rfes, the following procedure is used in order to shorten the time of calculations. The system of equations is written in the form: A¨ q − DR = −G, ¨ = W, DT q

(7.30a) (7.30b)

where G = −F + Cq. ¨ is calculated from (7.30a) and one obtains: First, q ¨ = A−1 (−G + DR). q

(7.31)

This vector is then substituted into (7.30b) and thus the system of linear algebraic equations for calculating reaction vector R is obtained: DT A−1 DR = W + DT A−1 G.

(7.32)

Having defined reaction vector R, one can calculate the vector of gener¨ from (7.32). This procedure is especially effective when alised acceleration q matrix A has constant elements. In such a case the inverse matrix A−1 can be calculated only once and can be used at each calculation step. In the CRFEL method presented, the mass matrix is indeed a diagonal matrix with constant elements. This method of solution also has the advantage that at each calculation step we have to solve a system of equations only with the number of unknowns equal to the dimension of vector R. Because calculation of the inverse matrix A−1 is resolved into calculation of the reciprocals of the elements from the diagonal of matrix A from (7.30), the dimension of which, as mentioned, may be large for large m(1) , m(2) and m(3) , the time of calculations is considerably shorter.

7.3 A-Frame rfe(i )

rfe(m (1)+1)

211

rfe(m (1) + m (2))

(1)

rfe(m ) sde(i )

rfe(eL)

rfe(eR)

SR

SL

sde(1)

R3

rfe(n = m (1) + m (2) + m (3))

rfe(0)

R1

AR

AL R2

Fig. 7.28. Division of A-frame into rfes

Y{i } (i ) Yˆ 3

j (i )

{i }

(i ) Yˆ 1

3

j (i )

j (i1 )

2

ˆ (i ) Y 2

Fig. 7.29. Generalised coordinates of rfe

7.3.2 Modified Rigid Finite Element Method Linear Model In order to avoid division of the A-frame structure into three subsystems (beams), as in the model formulated using the classical rigid finite element method, we present here formulation of the modification of the rigid finite element method. This enables us to treat the whole A-frame as a sequence of rigid elements (Fig. 7.28), each of which has three degrees of freedom, angles (i) (i) (i) ϕ1 , ϕ2 , ϕ3 (Fig. 7.29), with respect to the preceding rfe.

212

7 Applications

The following denotations are assumed: – {i − 1} is the coordinate system assigned to rfe i − 1 – {i} is the coordinate system assigned to rfe i – Y{i} is the coordinate system assigned to rfe i before deformation The vector of the generalised coordinates of rfe i takes the form: (i)

q(i) = [ϕ1

(i)

ϕ2

(i)

ϕ3 ]T .

(7.33)

Having taken into account all subsystems and constraint equations, the equations of motion can be written in form (7.30) and:  (1)    q SR  .  .   SL  , qF =   . , R = RAL q(n)  R1 =  R2  , R3 

where RAL

R1 , R2 , R3 are defined in Fig. 7.28. At this point, it is important to point out the differences between the equations obtained by using classical and modified formulations of the rigid finite element method. These differences are listed in Table 7.3. 7.3.3 Description of Programmes and Results of Calculations Computer programmes have been worked out on the basis of the mathematical models of the A-frame. In order to check the correctness of the models and programmes, they have been verified by comparison of the numerical results obtained with those obtained by using commercial packages for dynamic analysis of multibody systems (Adams) and for the finite element method (MSC.Nastran, Ansys). Table 7.3. Comparison of certain features of the classical and modified RFE method feature number of generalised coordinates

classical RFE method 6

3 3

(1 + m(k) ) + 4

k=1

number of elements of reaction vector consideration of longitudinal displacements consideration of shear mass matrix

modified RFE method



3 1+

3 3



m(k)

k=1

20

5

yes yes constant, diagonal

no no constant, full

+4

7.3 A-Frame

213

First, static analysis was carried out and the loads on the structure were calculated when the mother-ship was immobilised. A separate computer programme for the static analysis on loads of the A-frame was worked out according to the CRFEL model. This programme enables us to carry out the calculations for different geometrical parameters and kinds of loads. It is used in the initial design of such mechanisms in firms which produce offshore cranes. The results obtained were compared with those obtained from the finite element method package MSC.Nastran. Good compatibility of results is obtained, which is shown in Figs. 7.30 and 7.31 presenting forces in servomotors and stresses in the middle of the top beam. Detailed data used for calculations and comparison of the two methods are given by FaZlat (2001) and Adamiec et al. (2003). In order to check the correctness of calculations concerned with the dynamics of the A-frame, additional calculations using software packages Ansys and Adams were carried out. First, the supporting structure was modelled in 700.0

MSC Nastran Our programme

630.0 560.0

force [kN]

490.0 420.0 350.0 280.0 210.0 140.0 70.0 90

100

110

120

130

140

angle of frame indication Y[⬚]

Fig. 7.30. Forces in servomotors

4.5

MSC Nastran Our programme

4.0

stress [MPa]

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -6,000

-4,000

-2,000

0

2,000

4,000

6,000

coordinate {x1} B2 [mm]

Fig. 7.31. Stress in the top beam of the frame

214

7 Applications

Attachment node 1

Attachment node 5

Attachment node 4 Attachment Node 3 Z

Attachment node 2

Fig. 7.32. Model obtained in Ansys

Ansys as a spatial frame. 1536 shell63 elements were used for discretisation. A regular grid (Fig. 7.32) was obtained by using automatic meshing. 97 massless beam4 elements were used to connect the frame with the deck and servomotors. Modal analysis was applied to the model obtained and was transferred to Adams. Calculations took into account 30 modes (30 first frequencies and forms of the free vibrations of the frame). After transferring the model to Adams (Fig. 7.33), it was complemented with the necessary elements (such as servomotors and drive systems). It has to be said that the model created in Adams is very sensitive to the way of defining constraints, especially for statically indeterminate systems, which is the case for the problem considered. When taking into account the two servomotors and symmetrical loading of the structure in the Ansys-Adams model, calculations did not result in symmetrical courses of reactions. Therefore a virtual equivalent servomotor (Fig. 7.33) was introduced which realised inclination of the A-frame, and only then were the models formulated with the RFE method able to be verified. In addition modelling of the rope system is difficult in Adams (Maczy´ nski and Szczotka, 2002) and this is why the operations of lifting and lowering the load were omitted in the model for the package. This means that the analysis was limited to the load caused by the deadweight of the A-frame and by the change in length of the servomotors. Comparison of results obtained using the packages described and those obtained by using the programmes based on the classical rigid finite element method is presented in Figs. 7.34 and 7.35.

7.3 A-Frame

215

Fig. 7.33. Model of the A-frame in Adams

520,000.0

ANSYS-ADAMS 480,000.0

CRFE

440,000.0

[N]

400,000.0 360,000.0 320,000.0 280,000.0 240,000.0 200,000.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

t [s]

Fig. 7.34. Vertical component of the reaction in the support of the A-frame

216

7 Applications 210,000.0

ANSYS-ADAMS CRFE

180,000.0

[N]

150,000.0 120,000.0 90,000.0 60,000.0 30,000.0

0 1 2

3 4 5 6 7

8 9 10 11 12 13 14 15 16 17 18 19 20

t [s]

Fig. 7.35. Force in the servomotor

Good compatibility of the courses demonstrates that the basic models and our own programmes for discretisation of the A-frame by means of the rigid finite element method are correct.

7.4 Further Applications The examples of applications of the rigid finite element method in modelling the dynamics of machines and mechanisms presented in this chapter have only illustrative character. The method has been used on many occasions by the authors and their colleagues. Here is a list of papers in English using the RFEM, classified according to machine type and classical and/or modified formulations: – Planar link mechanisms (MRFEM): Harlecki and Wojciech (1992), Adamiec and Wojciech (1993). – Spatial manipulators (MRFEM): Adamiec-W´ ojcik (2003); Wojciech (1996), Posa and Wojciech (2000). – Textile machines (MRFEM): Wojciech and Stadnicki (1990), Stadnicki and Wojciech (1995), and Posa and Wojciech (1995). – Band saws (MRFEM): Wojnarowski and Wojciech (1993), Wojnarowski and Adamiec-W´ojcik (2005). – Port cranes (CRFEM): Wittbrodt and Le Quy Thuy (1991), Wittbrodt and Osi´ nski (1993), Osi´ nski et al. (1999). – Vehicles (both): Dzida et al. (1993), Wittbrodt and Lipi´ nski (1997), Szczotka and Wojciech (2003), Adamiec-W´ ojcik (2003) and Harlecki et al. (2004). These examples are merely a small collection of possible applications of the rigid finite element method. Authors have used the method in Polish papers for the solution of many other interesting problems in dynamics of multibody systems, mainly connected with control of robot manipulators, vehicles and cranes.

7.4 Further Applications

217

We have presented formulations of the method which enable the dynamics of multibody systems to be described and analysed by means of homogenous transformations and joint coordinates. These are general algorithms, uniform in description, for analysis of complex systems with alternate rigid and flexible links. Important problems of reciprocal influence of base motion and vibrations are included. The method makes it possible to analyse large deflections of beam-like links, taking into account both linear and nonlinear physical relations describing material properties. We hope that the readers will appreciate the two fundamental advantages of the rigid finite element method, first the simplicity of physical interpretation of a system discretised into rigid bodies connected by spring-damping elements, and secondly the ease both of the division into elements and the combination of the natural division into rfes and sdes with the virtual division, which is necessary for the discretisation of the flexible links of machines and mechanisms. We look forward to receiving feedback about the usefulness and applications of the methods presented.

References

Adamiec-W´ojcik I. 1992. Dynamic analysis of manipulators with flexible links. Ph.D. Thesis. Strathclyde University, Glasgow. Adamiec-W´ojcik I. 1993. Dynamics of a drilling vehicle. Mach. Vib. Vol. 2. 223–228. Adamiec-W´ojcik I. 2002. Interactions between vibrations of flexible links and base motion of manipulators. J. Theor. Appl. Mech. Vol. 40. No. 2. 1–15. Adamiec-W´ojcik I. 2003. Modelling Dynamics of Multibody Systems Using Homogenous Transformations. Bielsko-BiaZla University Press, Bielsko-BiaZla Monographs No. 3. Adamiec-W´ojcik I., FaZlat P., Gancarczyk T. 2003. Computer analysis of static loads of an A-frame. Bielsko-BiaZla University Press, Bielsko-BiaZla No. 6. 7–25. Adamiec-W´ojcik I., Wojciech S. 1993. Application of the rigid finite element method in dynamic analysis of a plane manipulator. Mech. Mach. Theory. Vol. 28. No. 3. 327–334. Bathe K.J., Wilson E.L. 1976. Numerical Methods in Finite Element Analysis. Prentince Hall, Englewood Cliffs. Blajer W. 1998. Metody Dynamiki Uk+lad´ ow Wielocz+lonowych (Methods of Dynamics of Multibody Systems). Radom Technical University Press, Radom Monographs. Chapra S.C., Canale R.P. 2002. Numerical Methods for Engineers: With Software and Programming Applications. McGraw-Hill, New York. Craig J.J. 1988. Introduction to Robotics. Addison-Wesley, Massachusetts. Du H., Hitchings D., Davies G.A.O. 1992. A finite element structural model of a beam with an arbitrary moving base, Part I: Formulations, Part II: Numerical examples and solutions. Finite Elem. Anal. Des. Vol. 12. 117–150. Dzida J., Romaniszyn K., Wojciech S. 1993. Vibration of driving system 4x4 type of cross country vehicle. Arch. Mech. Eng. Vo. 40. 137–154. Eisler G.R., Robinett R.D., Segalman D.J., Feddema J.D. 1993. Approximate optimal trajectories for flexible-link manipulator slewing using recursive quadratic programming. J. Dyn. Sys. Meas. Control. Vol. 115. 405–410.

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Index

A-frame, 183 absolute coordinates, 5 acceleration of gravity, 25 acceleration of the origin, 16 adaptive stepsize, 139 algebraic equations, 123 angular accelerations, 15 angular frequency, 190 approximations, 127 articulated joints, 200 backward differences, 134 beams, 36 bending moment, 78 bending vibrations, 74 Boundary conditions, 150 Bubnov-Galerkin method, 149 Bulirsch-Stoer method, 135 centre of mass, 25 centrifugal forces, 155 classical formulation, 83 clearance, 197 closed kinematic chains, 5 coefficients of rotational damping, 44 coefficients of rotational stiffness, 44 coefficients of shear, 74 coefficients of translational damping, 43 coefficients of translational stiffness, 43 concentrated force, 135 constraint equations, 155 constraints, 214 contact points, 197 coordinate system, 5

Coriolis acceleration, 16 crane, 183 cranes, 216 damping features, 35 damping matrices, 67 degree of freedom, 17 Denavit-Hartenberg notation, 17 density, 81 deployment process, 155 derivatives, 30 differential equations, 131 discretisation, 35 displacements, 43 dissipation energy, 43 Division of the system, 35 Drum, 202 dynamic coefficient, 190 dynamics of multibody systems, 5 eigenvalue problem, 69 eigenvectors, 69 equations of motion, 19 Euler and Runge-Kutta methods, 123 Euler angles ZYX, 8 Euler methods, 123 Euler-Rodrigues parameters, 164 explicit Euler scheme, 132 external forces, 26 external moment, 27 external rapier, 194 Fehlberg methods, 126 finite difference method, 149

224

Index

finite element method, 141 flexible links, 35 floating platforms, 183 free damped vibrations, 153 free forms of vibrations, 69 free frequencies, 69 free vibrations, 69 frequency, 69 functionals, 19 generalised coordinates, 5 generalised forces, 19 generalised velocities, 19 global system, 16 gradient matrix, 134 gravity forces, 25 harmonic vibrations, 81 heave, 190 high frequencies, 121 hoisting winch, 184 hydraulic cylinder, 184 identity matrix, 20 implicit Euler scheme, 132 inertial coordinate system, 19 initial conditions, 122 initial problem, 123 integration step, 123 internal rapier, 194 inverse matrix, 6 Inverse transformation, 14 isotropic materials, 81 jib, 184 joint angle, 17 joint coordinates, 5 joint distance shortening, 163 Kelvin-Voigt rheological model, 73 Kinetic energy, 19 Kirchhoff’s modulus, 75 Kronecker delta, 48 Lagrange equations, 19 Lagrange operators, 19 large bending, 83 large deflections, 155 large deformations, 122 least square method, 170

linear accelerations, 15 Linear model, 61 linear standard damping model, 149 linearisation of trigonometrical functions, 12 link length, 17 link offset, 17 link twist, 17 load, 184 lumped mass, 207 machine tool, 35 machines, 183 manipulator, 153 mass matrix, 101 material continuum, 35 matrix of cosines, 40 matrix of direction cosines, 6 mechanisms, 183 midpoint method, 124, 125 modal method, 154 modified formulation, 103 Modified midpoint method, 129 moments, 22 moments of inertia, 21 moving base, 19 multiplication of a matrix by a vector, 12 Natural division, 35 Newmark method, 123 Newton method, 136 non-linear effects, 153 nonlinear model, 103 normal component, 16 normal stress, 73 numerical analysis, 104 numerical effectiveness, 104 object-oriented programming, 61 offshore crane, 183 open kinematic chain, 27 open kinematic chains, 5 orientation, 5 orientation matrix, 16 origin of coordinate system, 6 orthonormality, 7, 12 partial derivatives, 23

Index particle dm, 20 partitioned matrices, 29 partitioned vectors, 29 pitch angle, 9 planar models, 183 planar motion, 15 plates, 36 platform, 35 Poisson ratio, 81 position, 5 position vectors, 12 potential energies, 19 predictor-corrector Gear methods, 135 primary division, 35 principal axes of the spring-damping element, 41 principal central axes of inertia, 40 prismatic beam, 37 prismatic joints, 17 products of inertia, 21 pseudo-inertia matrix, 21 quadratic forms, 54 reaction force, 197 reciprocal influence of base motion, 217 relative position, 17 rigid body motion, 5 rigid finite element method, 35 rigid finite elements, 35 robot manipulators, 216 roll angle, 9 Rosenbrock methods, 135 rotary joint the generalised coordinate, 17 rotary joints, 17 rotation, 5 rotation matrix, 6 rotation operator, 9 Sandia manipulator, 173 scalar product, 6 sea waves, 183 secondary division, 35 servomotors, 202 shape coefficient of the cross-section, 74 shear modulus, 75 shock absorber, 184 shortening, 148 slide movement, 194

225

spatial model, 183 spin-up manoeuvre, 155 spring deformation, 53 spring-damping elements, 35 square matrix, 54 stability, 123 standard linear model of damping, 145 static deflections, 188 static moments, 21 Steiner theorem, 70 Step-Doubling Method, 126 stiff differential equations, 121 stiffness characteristics, 169 stiffness matrices, 67 supporting structure, 199 tangential component, 16 tangential damping, 81 tangential stresses, 74 Taylor series, 127 telescopic rapier, 183 tension, 73 time of calculations, 210 torsional free vibrations, 144 torsional moment, 76 torsional stress, 76 trace of a matrix, 20 transformation matrix, 5 translation, 5 translational energy, 43 translational motion, 17 transpose, 6 truncation errors, 127 unconditionally stable, 139 undeformed state, 39 uniform beam, 143 unstable, 138 vector of coordinates, 7 vehicles, 216 velocities, 15 virtual division, 35 wave motion, 203 whippy beams, 141 yaw angle, 9 Young’s modulus, 73

Foundations of Engineering Mechanics Series Editors:

Vladimir I. Babitsky, Loughborough University, UK Jens Wittenburg, Karlsruhe University, Germany

Further volumes of this series can be found on our homepage: springer.com Wittbrodt, E., Adamiec-Wojcik, I., Wojciech, S. Dynamics of Flexible Multibody Systems, 2006 ISBN 3-540-32351-1 Aleynikov, S.M. Spatial Contact Problems in Geotechnics, 2006 ISBN 3-540-25138-3 Skubov, D.Y., Khodzhaev, K.S. Non-Linear Electromechanics, 2006 ISBN 3-540-25139-1 Feodosiev, V.I., Advanced Stress and Stability Analysis · Worked Examples, 2005 ISBN 3-540-23935-9 Lurie, A.I. Theory of Elasticity, 2005 ISBN 3-540-24556-1 Sosnovskiy, L.A., TRIBO-FATIGUE · Wear-Fatigue Damage and its Prediction, 2005 ISBN 3-540-23153-6 Andrianov, I.V., Awrejcewicz, J., Manevitch, L.I. (Eds.) Asymptotical Mechanics of Thin-Walled Structures, 2004 ISBN 3-540-40876-2 Ginevsky, A.S., Vlasov, Y.V., Karavosov, R.K. Acoustic Control of Turbulent Jets, 2004 ISBN 3-540-20143-2, Kolpakov, A.G. Stressed Composite Structures, 2004 ISBN 3-540-40790-1 Shorr, B.F. The Wave Finite Element Method, 2004 ISBN 3-540-41638-2 Svetlitsky, V.A. Engineering Vibration Analysis - Worked Problems 1, 2004 ISBN 3-540-20658-2 Babitsky, V.I., Shipilov, A.

Resonant Robotic Systems, 2003 ISBN 3-540-00334-7 Le xuan Anh, Dynamics of Mechanical Systems with Coulomb Friction, 2003 ISBN 3-540-00654-0 Nagaev, R.F. Dynamics of Synchronising Systems, 2003 ISBN 3-540-44195-6 Neimark, J.I. Mathematical Models in Natural Science and Engineering, 2003 ISBN 3-540-43680-4 Perelmuter, A.V., Slivker, V.I. Numerical Structural Analysis, 2003 ISBN 3-540-00628-1 Lurie, A.I., Analytical Mechanics, 2002 ISBN 3-540-42982-4 Manevitch, L.I., Andrianov, I.V., Oshmyan, V.G. Mechanics of Periodically Heterogeneous Structures, 2002 ISBN 3-540-41630-7 Babitsky, V.I., Krupenin, V.L. Vibration of Strongly Nonlinear Discontinuous Systems, 2001 ISBN 3-540-41447-9 Landa, P.S. Regular and Chaotic Oscillations, 2001 ISBN 3-540-41001-5 Alfutov, N.A. Stability of Elastic Structures, 2000 ISBN 3-540-65700-2 Astashev, V.K., Babitsky, V.I., Kolovsky, M.Z., Birkett, N. Dynamics and Control of Machines, 2000 ISBN 3-540-63722-2 Kolovsky, M.Z., Evgrafov, A.N., Semenov Y.A, Slousch, A.V. Advanced Theory of Mechanisms and Machines, 2000 ISBN 3-540-67168-4