Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems
Jan Awrejcewicz
Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems State-of-the-Art, Perspectives and Applications
Prof. Dr. Jan Awrejcewicz, Phd, DSc. Technical University Lodz Department of Automatics and Biomechanics 1/15 Stefanowski St. 90-924 Lodz Poland
[email protected] ISBN 978-1-4020-8777-6
e-ISBN 978-1-4020-8778-3
Library of Congress Control Number: 2008939501 c 2009 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
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This volume contains the invited papers presented at the 9th International Conference “Dynamical Systems – Theory and Applications” held in Ł´od´z, Poland, December 17–20, 2007 dealing with nonlinear dynamical systems. The conference gathered a numerous group of scientists and engineers, who deal with widely understood problems of dynamics met also in engineering and daily life. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automatics and Biomechanics of the Technical University of Ł´od´z. The patronage over the conference has been taken by the following scientific institutions: Mechanics and Machine Dynamics Committees of the Polish Academy of Sciences, Polish Society of Theoretical and Applied Mechanics, Polish Association for Computational Mechanics, and Technical Committee of Nonlinear Oscillations of IFToMM. The financial support has been given by the Department of Education at the Ł´od´z City Hall, Ministry of National Education and the Polish Association for Computational Mechanics. We welcomed nearly 100 persons from 13 countries all over the world. They decided to share the results of their research and many years of experience in a discipline of dynamical systems by submitting many interesting papers. The Scientific Committee includes the following members: Igor V. Andrianov – Aachen; Jan Awrejcewicz – Ł´od´z; Jose M. Balthazar – Rio Claro; Denis Blackmore – Newark; Iliya Blekhman – Sankt Petersburg; Roman Bogacz – Warsaw; Tadeusz Burczy´nski – Gliwice; Dick van Campen – Eindhoven; Czesław Cempel – Pozna´n; Lothar Gaul – Stuttgart; J´ozef Giergiel – Cracow; Katica Hedrih – Niˇs; Janusz Kowal – Cracow; Vadim A. Krysko – Saratov; Włodzimierz Kurnik – Warsaw; Claude-Henri Lamarque – Lyon; Nuno M. Maia – Lisbon; Leonid I. Manevitch – Moscow; Krzysztof Marchelek – Szczecin; Yuriy Mikhlin – Kharkov; Jan Osiecki – Warsaw; Wiesław Ostachowicz – Gda´nsk; Victor Ostapenko – Dnepropetrovsk; Zbigniew Peradzy´nski – Warsaw; Ladislav P˚ust – Prague; Giuseppe ´ Rega – Rome; Tsuneo Someya – Tokyo; G´abor St´ep´an – Budapest; Jerzy Swider – Gliwice; Ryszard Tadeusiewicz – Cracow; Hans True – Lyngby; Andrzej Tylikowski
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– Warsaw; Ferdinand Verhulst – Utrecht; J´ozef Wojnarowski – Gliwice; Klaus Zimmermann – Ilmenau, and the invited talks follow: 1. Igor V. Andrianov (Aachen, Germany), “Continuous models for discrete media valid for micro- and nano-scales” 2. Vincenzo P. Castelli (Bologna, Italy), “Kinematic and kinetostatic modelling of human articulations: knee and ankle joints” 3. David Y. Gao (Blacksburg, USA), “New way to understand and control chaos: canonical duality approach and triality theory” 4. Alexander P. Seyranian (Moscow, Russia), “Multiparameter stability theory with mechanical applications” The following topics have been covered by the oral presentations: – – – – – – – – –
Bifurcations and chaos in mechanical systems Control in dynamical systems Asymptotic methods in nonlinear dynamics Stability of dynamical systems Lumped and continuous systems vibrations Original numerical methods of vibration analysis Man-machine interactions Dynamics in life sciences, bioengineering, medicine Other problems
It has been observed that an extensive thematic scope comprising dynamical systems stimulates a wide exchange of opinions among researchers dealing with different branches of dynamics, and results in effective solutions of many problems of dynamical systems in mechanics and physics, both in terms of theory and applications. A brief description of the volume content follows. I.V. Andrianov et al. study rod and beam having two different elasticity moduli and having different cross-sections. First governing equations are derived, and then free frequencies of vibrations are estimated using analytical and numerical approaches. In the first case the solution form is formulated, the compatibility conditions are introduced, and then a standard perturbation procedure is applied. The obtained set of recurrent equations yields the being sought free frequencies of the studied rod and beam vibrations. The Pad´e approximations are applied for matching both estimated frequencies of rod and beam vibrations. The obtained frequencies are compared with those yielded via numerical results, and high efficiency of the Pad´e approximation is exhibited. The Lyapunov exponents assess the sensitivity to initial conditions of dynamical systems. This means that it gives the rate of exponential divergence for perturbed initial conditions when time tends to infinity. If all exponents are strictly negative, stability of the system is ensured. On the contrary, if the real part of one exponent is not strictly negative, chaos is possible. Moreover, although obtained through an asymptotic study of the dynamical system, this constant is often used in order to assess finite-time convergence. C.-H. Lamarque and F. Schmidt use a finite-time
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Lyapounov exponent. In the spirit of its definition, they study the maximum value of tangent increment according to normal or non-normal case for the associated linear operator. The definition of a quasi Lyapunov exponent to quantify (estimate) the stability of a dynamical system in the transient regime is proposed. A simple non-linear system is analysed and this definition is applied to it. It is then obvious that, this constant may be much greater to the Lyapunov exponent than it is normally known due to the non-orthogonality of the eigenvectors of the Jacobian matrix. Previous investigations on the nonlinear dynamics of shape memory oscillators, based on a thermomechanically consistent model with four state variables, highlighted the occurrence of chaotic responses in some ranges of the model parameters. The numerical characterization of chaos in smooth dynamical systems is often carried out via the computation of Lyapunov exponents. In the study of D. Bernardini and G. Rega, however, the phase transformations in the shape memory material induce discontinuities in the vector field and are described by an internal variable, the fraction of martensite that belongs to a complex state space. The computation of the classical Lyapunov exponents does not seem to be a convenient strategy. In the present work the attention is focused on the simpler direct numerical tool represented by the method of wandering trajectories. The method is used to distinguish between responses and quantify them in the present thermo-mechanical framework. Numerical simulations in the excitation frequency-amplitude plane, as well as domains of chaotic behavior built in various model parameter planes are presented in order to characterize the richness and robustness of the non-regular dynamical behavior of shape memory oscillators. L.I. Manevitch and V.V. Smirnov present a study of energy exchange in the system of two weak coupled oscillatory chains. The mechanism of energy transfer appears as the interchain hopping of localized nonlinear excitations – breathers. It is supposed that along side with coupling, the amplitudes of vibration are also small enough to restrict ourselves by third and fourth degrees in the power expansion of elastic potential. To reveal spatially localized nonlinear excitations with oscillatory degree of freedom one can derive nonlinear equations with respect to complex combinations of the displacements and velocities. These coupled nonlinear equations were analyzed using multiple scale expansion. As this takes place, three characteristic times may be discriminated. The fastest of them corresponding to higher eigenfrequency of linearized system, intermediate time – to interchain energy transfer, and the slow time – to modulate vibrations. As a result, we have obtained asymptotic solution for the breather transferring energy between the chains. Such a transfer turns out to be impossible when amplitude of vibration grows. In this case localization of the breather on the initially excited chain becomes possible. The condition of the transition from interchain energy exchange to confinement of breather on one chain have been formulated. Analytical results are confirmed by computer simulation data. The model of dynamics of a mechanical system mounted on a moving platform is presented by A. Urba and S. Wojciech. Description of such systems can be used to design control systems which allow us to compensate waving. In the paper dynamic analysis of a gantry crane used to transport BOP (Blow Out Preventer) is presented.
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The crane is placed on a drilling platform. Waves cause motion of the platform and the load. The load is considered as a rigid body with 6-DOF with respect to the platform. The load is connected with the frame by means of two flexible ropes. Homogenous coordinates and transformations are used to describe behavior of the system. Equations of motion are derived using the Lagrange equations of the second order. Numerical calculations present the influence of amplitude, frequency and direction of waves on the system displacements and contact forces between the load and the frame guides. K. Zimmermann et al. present some theoretical investigations of the motion of a straight chain of three (equal) point masses interconnected with kinematical constraints. The ground contact can be described by dry (discontinuous) or viscous (continuous) friction. The controls are assumed in the form of periodic functions with zero average, shifted on a phase concerning each other. Thus, there is a spreading wave along the chain of point masses. In the case of small friction a condition for the locomotion of the center of the mass with the help of an average method is derived. In the case of smooth control and control with impacts explicit expressions for the stationary velocity of the motion of the center of mass are obtained. It is shown that, using specified control motion, this is possible not only in the case of isotropic friction, but also in the direction of the non-isotropic friction. Without shift of the phases in the control law and with a linear friction model the locomotion is impossible. Comparisons of these analytical relations with numerical results are carried out. The received theoretical results can be used for the development of mobile robots applying the principles of the motion outlined above. In machine tools of planar parallel structure with two translatory degrees of freedom, a rotatory degree of freedom is kinematically locked. Yet due to geometric faults, for example, assembly errors or different geometries due to production tolerances, such machine tools exhibit an additional rotational behavior. Stresses within the structure occur leading to deflections of the tool center point, and thus, reducing the quality of the workpiece. For compensating these errors an adaptronic strut, which can be implemented within such a machine tool, has been developed by C. Rudolf and J. Wauer. The strut comprises a piezoceramic sensor-actuator unit for controlled correction of those static and quasi-static deflections. Piezoceramic elements were chosen due to their high positioning accuracy and the small installation space required. In addition, a control design for the compensation is presented. Examining the controlled adaptronic strut initially as stand-alone system under external loads which represent process loads or constraint forces, the strut is eventually implemented within the machine tool. The effect of the controlled strut on the operating behavior of the machine tool is investigated. An approximate method for calculation of rigid rotor motion, supported on two tilting pads aerodynamic bearings oscillating with large amplitudes, limited only by the bearing clearance, is presented by L. P˚ust and J. Kozanek. Dynamic characteristics of such type of bearings are very strongly influenced by inertia properties of tilting pads – the stiffness and dynamic matrices are non-symmetric and their elements are non-monotonous functions of angular frequency. The approximate models of stiffness and damping forces – valid for the entire area of journal motion in the
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aerodynamic bearing’s clearance – are based on the transformation of linear stiffness and damping characteristics into the entire area. The assumption that the rheological properties of three-tilting pads bearing are centrally symmetric is also used. Linear dynamic characteristics of aerodynamic bearings, given for the selected positions of journal in the form of linear stiffness and damping matrices were extended onto strongly nonlinear characteristics by using special correction function and by the plane rotation matrix on the entire area of bearing clearance, and for the common angular positions. Used correction function consists of two parts, the first one proportional to the properties of unloaded bearing, the second one to the properties of loaded bearing. Rigid rotor with generally distributed mass along the axis of rotation produces coupling between motions of both bearings. Differential equations of motion of the rotor are numerically solved and resulting time histories of journal motion and their plane trajectories are presented and analysed. Strongly nonlinear stiffness and damping characteristics of aerodynamic bearings cause various kinds of rotor oscillations. Application of this approximate mathematical model of rotor supported on aerodynamic bearings is presented on examples, where the influence of rotor eccentricity and its shift along the axis of rotation is shown. T. Burczy´nski et al. study selected identification problems of dynamical mechanical structures that some parameters are uncertain and modelled in the framework of fuzzy sets. The identification problem is formulated as the minimization of some objective functionals (fitness functions) which depend on measured and computed dynamical fields such as displacements, strains or natural frequencies. In order to obtain the unique solution of the identification problem one should find the global minimum of the objective functional. In the majority engineering dynamical cases it is not possible to determine exactly all parameters of the system. It is necessary to introduce some uncertain parameters which describe granular character of data. The paper deals with identification of the fuzzy parameters of material and shape of the structure. In order to solve the identification problem the fuzzy evolutionary algorithm is applied. Evaluation of the fitness function is performed by means of the fuzzy finite element method. Several numerical examples of identification of elastic and composite structures are presented. Shape memory alloys (SMA) are materials that even when they are submitted to the some type of deformation, possess the ability to recuperate their original form through adequate thermal procedures. The main phenomena associated with these alloys are the effect of pseudoelastic and shape memory. Such phenomena happen due to martensite phase transformation that occur in these alloys. The dynamical response of systems composed by shape memory alloy presents nonlinear characteristics and a very rich nonlinear dynamic behavior. V. Piccirillo et al. present analytical and numerical investigations of a shape memory oscillator, where the restitution force is described by a polynomial constitutive model. The governing equations of the motion through a perturbation multiple scale method, in the case of the primary resonance are derived and analysed. They are studied numerically by means of phase portrait, Lyapunov exponents, frequency power and Poincar´e maps. Frequency–response curves are constructed for shape memory oscillators by various excitation levels and detuning parameter. A rich class of solutions and bifurcations,
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including jump phenomena, saddle-node bifurcations, is found. Good agreement between numerical and analytic solutions is obtained in the considered resonance region. A. Okni´nski and B. Radziszewski study dynamic properties of a material point moving in a gravitational field and colliding with a moving motion-limiting stop (representing unilateral constraints). The motion of the limiter is assumed as periodic with piecewise constant velocity. The Poincar´e map, describing evolution from an impact to the next impact, is derived. Grazing motion as well as periodic solutions are computed analytically and their stability is determined in analytic form. An investigation of transient is important in engineering, in particular, in the problem of absorption. Over the past years different new devices have been used for the vibration absorption and for the reduction of the transient response of structures. It seems interesting to study nonlinear passive absorbers for this reduction. Yu.V. Mikhlin et al. consider the transient in a system containing a linear oscillator, linearly coupled to an essentially nonlinear attachment with a comparatively small mass. A damping is taken into account. It is assumed that some initial excitation implies vibrations of the linear oscillator. The multiple scales method is used to construct a process of transient in the system under consideration. Numerical simulation confirms an efficiency of the analytical construction. A transfer of energy from initially perturbed linear subsystem to the nonlinear absorber can be observed. A similar construction is made to describe the transient in a system which contains a linear oscillator and a vibro-impact absorber with a comparatively small mass. Both an exact integration with regards to impact conditions, and the multiple scales method are used for this construction. The transient in such system under the external periodical excitation was considered too. Numerical simulation confirms an efficiency of the analytical construction. Most formulations on normal impacts of multibody systems use the momentum balance equations. Newton’s hypothesis is usually employed, where the restitution coefficient is defined by the relative normal velocities of the impacting bodies before and after the collision. When friction is present in the impact, different impact modes are possible: sliding, sticking or reverse sliding. The variables in the momentum balance equations are the changes in the velocity and the two components of the impulse, one in the normal direction to the common tangent of the contact surfaces and other in the tangent direction. To solve the equations two additional conditions are needed, one comes from the Coulomb law, and the other from the definition of the restitution coefficient. The use of the Newton hypothesis leads to wrong results in the simulation of impacts with friction. The use of the Poisson hypothesis can be found in the literature. There are situations, where the direction of slip varies during collision, the only energetically consistent definition is the so-called energetic coefficient of restitution. J.M. Mayo addresses problems related to impacts with friction of planar flexible multibody systems. The floating frame of reference formulation is used to model the flexible bodies. The normal and tangential impulses in the contact point are calculated by a computational algorithm based on the graphics techniques developed by Routh. Numerical results using both Poisson and Stronge definitions of the restitution coefficient are shown.
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Inverse simulation techniques are computational methods, in which control inputs to a dynamic system that produces desired system outputs are determined. Such techniques can be powerful tools for the analysis of problems associated with manoeuvring flight. The problem with the inverse simulation problem at hand is, however, that we deal with an underactuated mechanical system – the six-degree-offreedom aircraft is traditionally controlled by at most four control inputs: the aileron, elevator and rudder deflections and, optionally, by thrust changes. The same number of motion restriction can then be imposed on the aircraft motion. In the study of W. Blajer et al. the following complex problem is studied: a specified trajectory in space (two constraints on aircraft position), a demand on fuselage attitude with respect to the trajectory, and optionally a specification on the flight velocity (motion on the trajectory). A tangent realization of trajectory constraints is observed, which yields two additional constraints on fuselage attitude with respect to the desired trajectory. The consequent governing equations of the prescribed trajectory flight arise then as a set of differential-algebraic equations, and an effective method for solving the equations is developed. The solution consists of time-variations of aircraft state variables in the prescribed motion and the demanded control that ensures the realization of the motion. This gives a unique opportunity to study the simulated control strategies and evaluate feasibility of the modeled aircraft maneuvers. Some results of numerical simulations are reported. C. Behn contributes to the adaptive control of nonlinearly perturbed multi-input u, multi-output y, minimum phase systems with strict relative degree two. The author deals with systems, which are not known exactly, only structural information about the system (like relative degree or the minimum phase condition) are available. The consideration of these uncertain systems leads to the use of adaptive control. The aim is to design universal adaptive controllers, which learn from the behaviour of the system, so automatically adjust their parameter and achieve a pre-specified control objective: stabilization of the system or tracking of a given reference signal with any pre-specified, feasible accuracy lambda >0 (so-called lambda-tracking). Almost all already existing controllers in the literature offer the same drawback: though the controllers consist of a feedback strategy and a simple parameter adaptation law, this adaptation law is of the type, that the gain parameter can only monotonically increase. With respect to limited resources in applications, it is necessary to design adaptation laws, which let the gain parameter increase while the control objective is not achieved, and let him decrease, while it is achieved. But, from the analytical point of view and with respect to the mathematical background (theorems and their proofs) the arbitrary adaptation laws (gain parameter models) cannot be chosen. Some simulations of gain parameter models, which exhibit the properties mentioned above, in application to bio-inspired sensors with unknown system parameters to adaptively compensate unknown ground excitements are presented and the results are discussed. In the paper by J. Awrejcewicz et al. the dynamic instability and nonlinear vibrations of isotropic plates with complex form subjected to in-plane periodic compressive load are analyzed. The proposed approach is based on application of R-function method with variational one. The von K´arm´an governing equations
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are reduced to an ordinary nonlinear differential equation regarding time by the Bubnov-Galerkin method. To apply the Bubnov-Galerkin method for plates of an arbitrary shape and with different boundary conditions, a complete system of basis functions is constructed using R-functions theory. To find instability regions linear equations with periodic coefficients, the so-called Mathieu-Hill equation, are used. The amplitude – frequency characteristics and instability regions for plates with cutouts are obtained. The effect of static and dynamic factors of load action, cutouts parameters and different boundary conditions are studied. L.A. Kov´acs et al. present an analytical investigation on the unconditional stability of robots subjected to digital force control. As a benchmark example, a one degree-of-freedom robot model is considered with proportional plus differential force feedback. The results of the passivity based analysis are compared to the exact/analytical stability limits of the investigated controller in the form of stability charts. In addition, the optimization aspects of slightly damped mechanical structures with digital force control are discussed. Simple closed form results include the largest stable proportional gain and the least steady state force error as well as the optimal control parameters that provide the shortest transients for the controlled force signal. The stability charts based on the analytical results are measured and verified by a series of experiments. G.V. Kostin and V.V. Saurin are aimed at modeling and optimization of controlled dynamical systems with distributed elastic and inertial parameters. The general integro-differential method for solving wide class of boundary value problems is developed and criteria of solution quality are proposed. The numerical algorithm for discrete approximation of controlled motions is carried out and applied to design the optimal control law steering an elastic system to the terminal position and minimizing the given objective function. The polynomial control of plane motions of a homogeneous cantilever beam is investigated. Such type of system disturbances can induce essential elastic deflections and lead to sufficient computational difficulties when the conventional approaches are used. The optimal control problem of beam transportation from the initial rest position to given terminal state, in which the full mechanical energy of the system reaches its minimal value, is considered. The obtained numerical results are analyzed and compared with the conventional Fourier’s solution. L.A. Klimina et al. analyse the problem of behavior of aerodynamic pendulum with vertical axis of rotation in connection with the study of dynamics of small-scale vertical axis wind power generator. It is assumed that the aerodynamic load is comprised of quasi-steady part (determined on the basis of wind tunnel experiments), and unsteady part described with the help of the so-called added masses. Mathematical model of free rotation of such pendulum is constructed and includes nonlinear ODEs and transcendental algebraic equations. Qualitative analysis of the phase portrait is performed: all equilibrium positions are found, their stability is studied, characteristics of stable rotational regime are determined (basing on this regime, the working regime of the wind power generator is formed), attraction domains of equilibrium positions and of this rotational regime are found. Numerical investigation confirms the obtained results. The developed mathematical model is used for
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investigation of working regimes of the system “wind turbine + generator”. Estimations of input power depending on external load in the circuit are obtained, optimal power and load are found. A pitch angle control is proposed in order to increase the output power. E.M. Jarze˛ bowska and P.C. Szklarz present a new strategy to design a kinematic feedback controller, which is based on the error function. The error function is predefined by a designer and the resulting feedback controller ensures the convergence of the error to zero or to some specified bound. Usually in a control problem either at a kinematic or dynamic level, a tracking or following error is defined as a difference between desired and actual values of coordinates, or as a distance form a curve, respectively. It concerns both state and output control. A specific controller has to be subsequently designed in such a way that it has to ensure some kind of convergence or boundedness of the error. A few works present tracking control strategy designs, for which the tracking error is defined in a different way, e.g. due to uncertainties in a system dynamics. However, these approaches do not change the controller design process. In authors’ approach this is the error function dynamics, which ensures the convergence to the predefined system motion and the controller is designed based on this function. The feedback control problem is as follows for a given system kinematics find control inputs that ensure a desired ordinary differential equation on the error function value to be satisfied. The error function value is the value on the actual location of the system. The ordinary differential equation ensures the convergence of the error value to the specified bound. The problem is then precisely defined in a set of ordinary differential equations. According to the formulation of the feedback control problem, a definition of the error function is essential. It may reflect a way in which a system converges to a desired curve in order to achieve a desired error function value behavior. Comparing the authors’ design with typical tracking control designs, the differences start from defining the error function for the desired motion and designing of the controller based on this function. This implies that the convergence of the controlled motion to the desired one is guaranteed in the controller design process. The proposed control strategy offers several advantages significant from the practical point of view. The theoretical development is illustrated with an example of a desired motion control of a two-wheeled mobile platform. In the approach this motion control may be either tracking or following, or both. During the last two decades or so there has been a gradually increasing interest in the detection of damage based upon techniques that take into consideration changes in the dynamic properties of the structures, leading to thousands of articles and entire conferences specifically dedicated to the subject. Those techniques occupy quite a wide spectrum in terms of diversity as recent surveys reveal. They do not only address the problem of detection, but often also include the localization and the quantification. In many cases the main difficulty relies on the fact that the damage indicators that are proposed are not sensitive enough to detect damage in a sufficiently early stage. N.M. Maia and R.P. Sampaio have pursued such a sensitive indicator and found out that a variation of a known measure, known as the Frequency Domain Assurance
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Criterion, could give a good contribution to the objective of detecting damage. The simple technique that the authors have been developing recently and that is based on a correlation factor between dynamic responses taken along a period of time and the main objective of which is to serve as an indicator for the existence of damage, although – as it is discussed – it can also provide information about the relative damage quantification. That indicator has been named as The Detection and Relative damage Quantification indicator (DRQ) and some variations of it have been developed to improve its performance. Some numerical simulations are presented as examples and experimental tests in the laboratory as well as in a real working structure are shown to illustrate the method. V.-F. Duma studies the complete angular and linear scanning functions of a 2-D scanner comprising two individually driven galvoscanners. The characteristic functions of a 1-D scanner were developed and the command function was obtained. It has been shown that with a proper programming of the laser utilized in the system, there exist a wide spectrum of industrial, art and/or scientific applications. A. Krysko et al. are aimed on comparing numerical results of two different and widely used in engineering approaches, namely FDM (Finite Difference Method) and FEM (Finite Element Method) on the example of regular and chaotic dynamics of the Euler-Bernoulli beams. Transition scenarios are illustrated and studied, among other. Systems with a pendulum are often applied for reduction of vibrations in many mechanical and civil structures. Special pendulums mounted in buildings and working against earthquake as dynamical dampers or mounted on bridges tower working against river vortexes are classical examples of such systems. Motions of the system can be both regular or, for some parameters, chaotic. Then, the pendulum may worsen the system response, and it can be required to include a control, to fulfill the protection condition. The paper by K. Ke˛ cik and J. Warmi´nski proposes to use a magneto-rheological damper (MR), installed between oscillator and the ground, to provide controllable damping of the system. The effectiveness of the proposed smart base isolation system is studied numerically and experimentally. The results show that MR fluids can be designed for performing a controllable damping force and can be a very effective tool for vibration control. Common mechanical systems in rotor dynamics are rotating shafts of different shapes joined with special, mostly axi-symmetric bodies, which can be bladed disks, geared wheels, fans, etc. Designed rotating systems and operating conditions are still becoming more and more complex and therefore it is necessary to create advanced mathematical and computer models of the studied dynamical systems. This problem is addressed by M. Hajˇzman et al. The contribution is intended to the modeling of the flexible rotors that can be decomposed into shaft and disk subsystems mutually joined together. Rotating shaft subsystems are considered as one dimensional continuum. The original shaft finite element in the rotating coordinate system is presented. Because disks can be of a complex shape they are modeled as three dimensional continuum also in the rotating coordinate system. The coupling matrix is used for the connection of the disk and shaft subsystems. The presented methodology is advantageous mainly due to the possibility of considering
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various effects of rotation that cannot be introduced in the commercial FEM codes. The described model of the flexible rotor is usable especially for the analysis of high-frequency vibrations, where the common assumption of rigid disks in rotor dynamics is not correct. The possibility of a model reduction by the modal synthesis method is a big benefit too. Stochastic dynamics of the systems composed of hereditary elements is analyzed by K.R. Hedrih. First, the definition of a hereditary system is introduced, and then examples of light standard thermo-rheological elements are provided. The stochastic dynamics of the thermo-rheological double pendulum system putting emphasis on its parametric resonance is studied. S.-M. Cretu focused on a study of achievements of the tensegrity theory regarding living forms and some tensegrities obtained from platonic polyhedra. She presents the calculus for the form-finding of a tensegrity system that approximates the human forearm. Problems related to stability, bifurcation and chaotic behavior of a buck converter controlled by lateral pulse-width modulation and zero average dynamics are studied by F. Angulo et al. The authors are mainly focused on the duty cycle computation using a linear approximation of the sliding surface. M. Popescu and A. Dumitrache study stability problems regarding periodic solutions of the quasi-linear control systems in a critical case. The problem is transformed to that of an orbital stability, and then the theorem associated with stability estimation is formulated and proved. Finally, it should be emphasized that the selected invited papers are mainly oriented toward dynamics and control of engineering systems. They have been also reviewed by two independent referees to satisfy the Springer publisher standards. My sincere gratitude is expressed to the numerous referees, the conference participants and the Department of Automatics and Biomechanics staff. Special thanks go to Ms. Nathalie Jacobs for her help and encouragement to the publication of this volume. Poland November 2008
Jan Awrejcewicz
Contents
Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Igor V. Andrianov, Jan Awrejcewicz, and Dieter Weichert 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Asymptotic Simplification of Boundary Conditions . . . . . . . . . . . . . 4 4 A Single Fibre Embedded in the Half-Space . . . . . . . . . . . . . . . . . . . 5 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents . . . . Franziska Schmidt and Claude-Henri Lamarque 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A General Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Duffing-Like Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Lorenz Attractor and Other Dynamical Systems . . . . . . . . . . . . 3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Characterization of the Chaotic Nonregular Dynamics of Pseudoelastic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Davide Bernardini and Giuseppe Rega 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Description of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Method of the Wandering Trajectories . . . . . . . . . . . . . . . . . . . . 4 Characterization of Typical Trajectories . . . . . . . . . . . . . . . . . . . . . . . 4.1 Symmetric Period 1 Solution . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Chaotic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Unsymmetric Period 1 Solution . . . . . . . . . . . . . . . . . . . . . . 5 Comparison with Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . .
13 13 14 15 17 22 23 25 25 26 27 28 28 29 29 31
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Overall Characterization of the Non-regular Solutions and Effect of the Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Localized Nonlinear Excitations and Interchain Energy Exchange in the Case of Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leonid I. Manevich and Valeri V. Smirnov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear Chains with Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nonlinear Chains with Weak Nonlinearity . . . . . . . . . . . . . . . . . . . . . 4 Chains with Nonlinearity, Compatible with Coupling . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Analysis of the Gantry Crane Used for Transporting BOP . . . . . Andrzej Urba´s and Stanisław Wojciech 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Model of the System . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Kinetic Energy of the Systems . . . . . . . . . . . . . . . . . . . . . . . 2.2 Potential Energy of Gravity Forces . . . . . . . . . . . . . . . . . . . 2.3 Energy of Deformation and Dissipation Energy of Sde . . 2.4 Energy of Deformation and Dissipation Energy of the Ropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Reaction Forces of the Support . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motion of a Chain of Three Point Masses on a Rough Plane Under Kinematical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klaus Zimmermann, Igor Zeidis and Mikhail Pivovarov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Smooth Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Dry (Discontinuous) Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Viscosity (Continuous) Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compensation of Geometric Errors in a PKM Machine Tool . . . . . . . . . . . Christian Rudolf and J¨org Wauer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Control Concept for Adaptronic Strut . . . . . . . . . . . . . . . . . . . . . . . . 3 Implementation into PKM Machine Tool . . . . . . . . . . . . . . . . . . . . . . 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37 37 38 41 43 46 47 49 49 49 52 52 52 54 54 56 59 59 61 61 62 64 65 68 69 70 71 71 73 74 77
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5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ladislav P˚ust and Jan Koz´anek 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Motivation of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mathematical Model of the Rotor Motion at Small Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Rotor Motion at Large Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81 81 82 83 87 89 93 94
Identification of Dynamical Systems in the Fuzzy Conditions . . . . . . . . . . . 95 Tadeusz S. Burczy´nski, Witold Beluch and Piotr Orantek 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2 Formulation of the Identification Problem of the Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3 The Two-Stage Fuzzy Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.1 The First Stage – Global Optimization . . . . . . . . . . . . . . . . 99 3.2 The Second Stage – Local Optimization . . . . . . . . . . . . . . . 100 4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1 The Identification of Geometrical Parameters of a Void . . 102 4.2 The Identification of Laminate’s Elastic Constants . . . . . . 102 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 On Nonlinear Response of a Non-ideal System with Shape Memory Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 V. Piccirillo, J. M. Balthazar, B. R. Pontes Jr. and J. L. P. Felix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2 (SMA) Constitutive Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3 Mathematical Model of the Non-ideal System . . . . . . . . . . . . . . . . . 109 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Dynamics of a Material Point Colliding with a Limiter Moving with Piecewise Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Andrzej Okni´nski and Bogusław Radziszewski 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2 Motion with Impacts: A Simple Motion of the Limiter . . . . . . . . . . 118 3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
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Periodic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Chattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Multiple Solutions and Discontinuous Dependence on Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Transient in 2-DOF Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Yuri Mikhlin, Gayane Rudnyeva, Tatiana Bunakova and Nikolai Perepelkin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2 Transient in a System Containing an Essentially Nonlinear Oscillator as Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3 Transient in the Vibro-Impact System . . . . . . . . . . . . . . . . . . . . . . . . 133 3.1 Free Oscillations in the Vibro-Impact System . . . . . . . . . . 134 3.2 Transient in a Case of Forced Oscillations . . . . . . . . . . . . . 136 4 Transient in 2-DOF Nonlinear System with Limited Power Supply 137 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 On the Use of the Energetic Coefficient of Restitution in Flexible Multibody Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Juana M. Mayo 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 2 Floating Frame of Reference Formulation . . . . . . . . . . . . . . . . . . . . . 143 3 Generalized Impulse-Momentum Balance Equations . . . . . . . . . . . . 144 4 Coefficients of Restitution and Routh’s Diagrams . . . . . . . . . . . . . . 145 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Modeling of Aircraft Prescribed Trajectory Flight as an Inverse Simulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Wojciech Blajer, Jerzy Graffstein and Mariusz Krawczyk 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2 Modeling Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3 Prescribed Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4 Governing Equations and the Solution Code . . . . . . . . . . . . . . . . . . . 158 5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Improved Adaptive Controllers for Sensory Systems – First Attempts . . . 163 Carsten Behn and Joachim Steigenberger 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 2 General System Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3 Control Objective & Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
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3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.2 Current Control Strategy & Theorem . . . . . . . . . . . . . . . . . 167 4 New Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5 Improved Gain Adaptation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1 Comparative Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 Further Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Research of Stability and Nonlinear Vibrations by R-Functions Method . . 179 Jan Awrejcewicz, Lidiya Kurpa and Olga Mazur 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3 Method of Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5 R-Functions Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Experiments on the Stability of Digital Force Control of Robots . . . . . . . . 191 L´aszl´o L. Kov´acs, P´eter Galambos, Andr´as Juh´asz, and G´abor St´ep´an 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2 Model of Digital Force Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5 Theoretical vs. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 197 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Motion Analysis and Optimization for Beam Structures . . . . . . . . . . . . . . . 201 Georgy Kostin and Vasily Saurin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 3 An Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Parametrical Analysis of the Behavior of an Aerodynamic Pendulum with Vertical Axis of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Liubov Klimina, Boris Lokshin and Vitaly Samsonov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 2 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3 Existence of Auto-Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
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Numerical Analysis of Rotational Modes and Domains of Attraction for β = 0 and Various Values of c . . . . . . . . . . . . . . . . 214 5 Average Trapped Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6 Pitch Angle Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Error Function Based Kinematic Control Design for Nonholonomic Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 El˙zbieta Jarze˛ bowska and Paweł Cesar Sanjuan Szklarz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 2 A Control Theoretic System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 223 3 An Abstract Feedback Control Strategy Architecture . . . . . . . . . . . . 223 4 Design of a Kinematic Control Strategy Based on the Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 5 Control of a Two-Wheeled Mobile Robot . . . . . . . . . . . . . . . . . . . . . 226 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 A Simple Correlation Factor as an Effective Tool for Detecting Damage . . 233 Rui Sampaio and Nuno Maia 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 3.1 Sensitivity to Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.2 Quantification of Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 4 Experimental Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Mathematical Functions of a 2-D Scanner with Oscillating Elements . . . . 243 Virgil-Florin Duma 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2 Equations of the 2-D Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 3 Angular and Linear Scanning Functions . . . . . . . . . . . . . . . . . . . . . . 245 3.1 Scanning Functions of the 2-D System . . . . . . . . . . . . . . . . 245 3.2 θ and x Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.3 ϕ and y Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Analysis of Regular and Chaotic Dynamics of the Euler-Bernoulli Beams Using Finite-Difference and Finite-Element Methods . . . . . . . . . . . 255 Anton Krysko, Jan Awrejcewicz, Maxim Zhigalov, and Olga Saltykowa 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
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On the Numerical Solution to Vibration and Stability Beam Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 3.1 FDM with Approximation O(c2 ) . . . . . . . . . . . . . . . . . . . . . 257 3.2 FEM with the Bubnov-Galerkin Approximation . . . . . . . . 258 4 Numerical Results Obtained via FDM and FEM . . . . . . . . . . . . . . . . 259 5 Transition Scenarios into Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Regular and Chaotic Motions of an Autoparametric Real Pendulum System with the Use of a MR Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Jerzy Warmi´nski and Krzysztof Ke˛ cik 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 2 Model of the Vibrating System and Equations of Motions . . . . . . . . 267 3 Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 4 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 270 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Modeling of Flexible Rotor Vibrations in the Rotating Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 ˇ sek and Vladim´ır Zeman Michal Hajˇzman, Jakub Saˇ 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 2 FEM Model of a Rotating Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3 FEM Model of a Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 4 Mathematical Model of the Whole Disk-Shaft System . . . . . . . . . . 282 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Stochastic Dynamics of Hybrid Systems with Thermorheological Hereditary Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Katica R. (Stevanovi´c) Hedrih 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 2 Light Standard Thermo-Rheological Hereditary Element . . . . . . . . 290 3 Thermo-Rheological Double Pendulum System – System of the Averaged Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4 Stochastic Dynamics of the Thermo-Rheological Double Pendulum system – Parametric Resonance . . . . . . . . . . . . . . . . . . . . 296 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Tensegrity as a Structural Framework in Life Sciences and Bioengineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Simona-Mariana Cretu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
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Contents
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Regular Polyhedra and Their Applications to Life Sciences and Bioengineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 3 Tensegrity and Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 4 Applications of Tensegrity Concepts to Living Forms . . . . . . . . . . . 304 5 Form-Finding Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 6 Transformations of the Tensegrity Systems . . . . . . . . . . . . . . . . . . . . 308 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Study of Nonlinear Dynamics in a Buck Converter Controlled by Lateral PWM and ZAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Fabiola Angulo, Jorge E. Burgos, and Gerard Olivar 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 2 Modeling the DC-DC Buck Converter with LPWM and ZAD . . . . 314 3 Computation of the Duty Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 4 Bifurcational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 4.1 Period-Doubling Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 318 4.2 Border Collision Bifurcation and Chaos . . . . . . . . . . . . . . . 323 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Stability of the Periodic Solutions for Quasi-linear Controlled Systems in Critical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Mihai Popescu and Alexandru Dumitrache 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 2 Quasi-linear Controlled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 3 Orbital Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating Igor V. Andrianov, Jan Awrejcewicz, and Dieter Weichert
1 Introduction We study the problem of load-transfer from fiber inclusion to matrix. Many papers are devoted to the infinite fibre in an elastic space. 3D analog of Melan problem is analysed by Muki and Sternberg [1]. They regard the original fibre as made of two superimposed elastic fibres, the first with the same characteristics as the matrix and treated in the framework of 3D elasticity, the latter with the elastic coefficient equal to difference between those of the actual fibre and of the matrix considered as a 1D continuum. The governing integral equation is obtained by imposing the same average axial strain in the two fictitious bars. Many researches used as asymptotic parameters ratios λ1 = R/L, λ2 = E/E1 or λ3 = EE1 ( RL )2 ln( 2L R ), where E, E1 are the Young modulus of matrix and fibre, respectively; and R, L are the radius and length of the circular fibre, respectively. Freund [2] studied a model describing sliding of circular cylindrical fibre along a hole in an elastic solid, and obtained asymptotic solutions for the cases when the fibre is very stiff or very weak in comparison with the matrix material (λ2 > 1, respectively). Eshelby [3] and Argatov and Nazarov [4] used parameters λ1 1, then the interfacial shear stress remains almost constant, for λ3 4(b + 3cx20) then: γ −β 0 γ + β −1 1 γ −β 1 −1 , , P= ,P = A= 0 γ +β 1 γ +β 2β −(γ − β ) 1
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and divergence at time t can be assessed through (δ X (t) = PetA P−1 ): e−γ t −γ sinh(β t) + β cosh(β t) sinh(β t) δ X (t) = δ X0 , (β 2 − γ 2)sinh(β t) γ sinh(β t) + β cosh(β t) β where: γ = a2 , β =
1 2
(10)
a2 − 4(b + 3cx20).
A similar formula for δ X (t) can be obtained in the other cases: a2 < 4(b + 3cx20) and a2 = 4(b + 3cx20). We decide to search the maximum of this divergence (10), according to X0 , δ X0 δ x0 . In order to and t. For this, we solve the system of equations (8), with δ X0 = δ x˙0 ρ cos(θ ) , which does not reduce the facilitate our calculations, we take: δ X0 = ρ sin( θ) f1 (x0 ,t, θ ) generality of our study. So if we write δ X (t) = , then the finite-time f2 (x0 ,t, θ ) pseudo-Lyapunov exponents can be assessed by searching the maximum of:
λt = g(x0 ,t, δ X0 ) =
1 ln[ f12 (x0 ,t, θ ) cos(θ )2 + f22 (x0 ,t, θ ) sin(θ )2 ]. 2t
(11)
This finite-time pseudo-Lyapunov exponent (11) may be superior to the Lyapunov exponent. Indeed, for example with a = 0.15, b = −1, c = 1, this last one is negative λ = −0.12 < 0 (see Fig. 1), whereas λt = 0.252718 for x0 = 2.189, t = 6.44065s, θ = −9.4247 (by solving Eq. (7)). 1
0.5
0
−0.5
−1
−1.5
0
20
40
60 t
80
100
120
Fig. 1 Lyapunov exponent time convergence of the dynamical system (9) calculated with Wolf’s algorithm [3] a = 0.15, b = −1, c = 1
On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents
17
3 2 1 1.2
1.4
1.6
x0
1.8
2
2.2
2.4
0 -1 -2 -3
Fig. 2 Maximum of the finite-time pseudo-Lyapunov exponents (Eq. (11)) of the dynamical system (9) with a = 0.15, b = −1, c = 1 along the initial condition x0 . Thus the finite-time pseudoLyapunov exponent may take negative or positive values (decreasing or growing of the divergence), whereas the Lyapunov exponent is negative
This can be generalized by varying the initial condition x0 . To do this, we solve once again Eq. (7) with fixed initial condition X0 . The results are plotted in Figs. 1–3. Thus, we see that the numerical value of the finite-time pseudo-Lyapunov exponents may be superior to the “classical”, infinite-time Lyapunov exponent. Thus, using this last exponent for assessing the evolution of the divergence at finite time is not adapted for quantifying worst divergence. In this case, the calculations are rather simple and it is possible to solve them analytically. This is generally not the case. That’s why we now study some more challenging dynamical systems.
2 The Lorenz Attractor and Other Dynamical Systems The behavior of the Lorenz attractor is given by the formula: ⎧ ⎪ ⎨x˙ = σ (y − x), y˙ = ρ x − y − xz, ⎪ ⎩ z˙ = xy = bz.
(12)
We apply the same procedure to this dynamical system: here the Jacobian matrix is of size (3 × 3). So the potential eigenvalues are solutions of a three-degrees polynomial function, whose coefficients are real functions of the parameters (σ , ρ , b) and the initial conditions (x0 , y0 , z0 ).
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80 60 40 20 0 -20
1.2
1.4
1.6
1.8
2
2.2
2.4
x0
-40
Fig. 3 Angle θ of the initial infinitesimal divergences δ X0 = ρ cos(θ ), δ Y0 = ρ sin(θ ) giving the maximum of λt along the initial condition x0 , obtained by solving system of equations (8).
Nevertheless the conclusions are the same as for the case of the Duffing-like oscillator: the finite-time pseudo-Lyapunov exponents may be superior to the classical one, see Figs. 4–6. Untill now we do not clearly analyze a underlying structure. There are several reasons for this behavior: first, this pseudo-Lyapunov exponent does depend on time and initial conditions whereas the classical Lyapunov exponent does not (ergodicity). Moreover, the Jacobian matrix is just an approximation at first order of the phenomenon. Then, the initial divergence taken for calculations is not infinitesimal. Finally, the eigenvectors of the Jacobian matrix may not be orthogonal, what is assumed computing the Lyapunov exponent via Wolf’s algorithm. It is impossible to distinguish the importance of each of these reasons in the final result. But one fact is clear: each of these causes acts. Indeed, if we take a linear oscillator, the finite-time pseudo-Lyapunov exponent appears to be the superior to the infinite-time one even if the initial condition does not intervene in the calculations (Jacobian matrix depends only on the parameters of the system). For example, for a linear oscillator x¨ + ax˙ + bx = 0 with a = 0.25 and b = 0.01, the Lyapunov exponent is equal to −0.05, whereas the finite-time pseudoLyapunov exponent varies and reaches a maximum λt = 0.1309 of t = 8.9971s and (x0 , x˙0 ) = (ρ cos(θ ), ρ sin(θ )) with θ = −4.9597◦. Moreover, this conclusion is also correct if we take a system of which the nonorthogonality of the system eigenvectors 3 not satisfied. For example, we may propose a logistic map whose Jacobian matrix is (1 × 1): xn+1 = kxn (1 − xn).
(13)
On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents
19
5
Lyapunov exponents
0
-5
-10
-15
-20 0
200
400
600
800
1000
Time t
Fig. 4 Spectrum of the Lyapunov exponent along time of the dynamical system (12) with σ = 10.0, ρ = 28.0, b = 83 , calculated with Wolf’s algorithm [3]. The behavior is thus chaotic, with λ ≈ 2.66 > 0
6.2 6 5.8 5.6 5.4 5.2 5 4.8 -12
-11
-10
-9
-8
-7
-6
z0
Fig. 5 Maximum of the finite-time pseudo-Lyapunov exponents of the dynamical system (12) with σ = 10.0, ρ = 28.0, b = 83 along the initial condition z0 = z(0)
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8
6
4
2
-12
-11
-10
-9
-8
-7
-6
z0
Fig. 6 Time t when the maximum of λt (see Fig. 5) is reached along the initial condition z0 = z(0)
1 0.8 0.6 0.4 0.2 0 20
40
60
80
100
-0.2 -0.4 -0.6
Fig. 7 Finite-time pseudo-Lyapunov exponent of system (13) with k = 3.56
For k = 3.56, this system is stable (λ = −0.0653929), but the finite-time pseudoLyapunov exponent shows a somewhat different behavior, see Fig. 7. Finally, we want to highlight the fact that these results can be generalized to non-smooth systems. Indeed, a generalized Lyapunov exponent can be defined
On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents
21
x k m
g cos (wt)
c
x xmax
Fig. 8 The linear impact oscillator
(see [7, 11–15]) and thus also a generalized finite-time pseudo-Lyapunov exponent ( [16, 17]). This Lyapunov exponent can be defined using matrices of passage and matrices of jump, see [13, 16]. This makes it possible to define and calculate the spectrum of finite-time pseudo-Lyapunov exponents. For example, considering an impact linear oscillator (see Fig. 8), the equations of motion can be written: x¨ + 2ε w1 x˙ + w21 x = f cos(wt), (14) x = xmax : x+ = x− , x˙+ = −ex˙− . The matrices of passage and jump can thus be written, see [16]. eα t −α sin(bt) + b cos(bt) sin(bt) J= , −(α 2 + b2) sin(bt) α sin(bt) + b cos(bt) b −e 0 δ x− δ t1 = − −1 , S = − e+1 (w2 x − f cos(wt )) −e , max 1 − 1 x2 x2 where α = −ε w1 , b = w1 1 − ε 2.
(15)
In these equations. J is the matrix of passage, S is the matrix of jump and δ t1 is the difference of time between the two impacts: that of the original path and that of the perturbated one. The divergence between the two paths and after n impacts is as follows:
δ X (t) = Jn Sn Jn−1 Sn−1 . . . J0 δ X0 .
(16)
Here some work has to be done in order to define how to calculate the finitetime pseudo-Lyapunov exponent, that is to say the maximum of the spectrum on the whole path. It is possible to bound this divergence on every part of the path, that is to say every matrix Ji or Si . But in this case, the bound obtained may be too coarse (see [17]).
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F. Schmidt, C.-H. Lamarque 15
1000
10
500
5
0
1
0
2
3
4
5
-500 1
-5
2
3
4
5
-1000 -1500
-10 -15
-2000 -2500
Fig. 9 Motion and finite-time pseudo-Lyapunov exponent along time for the system of Eq. (15), with ε = 0.01, a = 2ε w1 , w1 = 0.05, w = 2.6, f = 20, xmax = 16, x(0) = x(0) ˙ =0
But it is possible for a given initial condition to calculate its
path and behavior of δ x(t,x0 ) divergence along time, according to the formula 1t ln , see Fig. 9: δ x0 The spectrum of the finite-time pseudo-Lypunov exponent can thus be estimated and the conclusion is the same as in the smooth cases. Here, the consequences of this phenomenon are even worse than in the smooth case, because of the grazing bifurcation. Indeed, this case where the speed of impact on the wall is equal to zero, is challenging: the matrix of jump has got a term that tends to infinity and the behavior of the system is complicated (see [18, 19]).
3 Conclusion Lyapunov exponents are a convenient way to define the behavior of a dynamical system if time tends to infinity. The method is also often used to quantify the finitetime response of a dynamical system if uncertainties exist. For a transient behavior and the practical study of divergence or stability, we look for another estimation that could lead to bounds for tangent behavior. That is why finite-time pseudo-Lyapunov exponents have been computed in continuous cases and discrete ones. In all these studies, the conclusions have always been that they may be superior to the Lyapunov exponent. Particularly, they may be positive whereas the Lyapunov exponent is negative. This does not mean that the domain of parameters concerned by chaos is extended. Indeed, chaos is predicted in an infinite time. Here, we just deal with finite-time and we highlight the fact that this indicator called a finite-time pseudo-Lyapunov exponent makes it possible to bound the tangent finite-time divergence of a dynamical system with initial conditions known with finite accuracy, and numerical calculations made with strictly positive small step of time. This indicator is mainly a bound to quantify finite-time maximum of instability. A way to go further in this study would be to examine the influence of time and initial conditions on the Lyapunov exponent of higher derivatives [20].
On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents
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References 1. A. M. Lyapunov. The general problem of the stability of motion. International Journal of Control, 55(3):531–773, 1992. 2. V. I. Oseledec. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Transactions of the Moscow Mathematical Society, 19:197–231, 1968. 3. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3):285–317, 1985. 4. F. E. Udwadia and H. F. von Bremen. An efficient and stable approach for computation of Lyapunov characteristic exponents of continuous dynamical systems. Applied Mathematics and Computation, 121(2–3):219–259, 2001. 5. K. Ramasubramanian and M. S. Sriram. A comparative study of computation of Lyapunov spectra with different algorithms. Physica D: Nonlinear Phenomena, 139(1–2):72–86, 2000. 6. G. Tancredi, A. S`anchez, and F. Roig. A comparison between methods to compute Lyapunov exponents. Astronomical Journal, 121(2):1171–1179, 2001. 7. R. Ding and J. Li. Nonlinear finite-time Lyapunov exponent and predictability. Physics Letters A, 364(5):396–400, 2007. 8. E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani. Growth of non-infinitesimal perturbations in turbulence. Physical Review Letters, 77(7):1262–1265, 1996. 9. J. M. Nese. Quantifying local predictability in phase space. Physica D: Nonlinear Phenomena, 35(1–2):237–250, 1989. 10. C. Nicolis, S. Vannitsem, and J. -F. Royer. Short-range predictability of the atmosphere: mechanisms for superexponential error growth. Quarterly Journal – Royal Meteorological Society, 121(523):705–722, 1995. 11. N. Hinrichs, M. Oestreich, and K. Popp. Dynamics of oscillators with impact and friction. Chaos, Solitons & Fractals Nonlinearities in Mechanical Engineering, 8(4):535–558, 1997. 12. L. Jin, Q. -S. Lu, and E. H. Twizell. A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems. Journal of Sound and Vibration, 298(4–5):1019–1033, 2006. 13. P. C. Muller. Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos, Solitons & Fractals Some Nonlinear Oscillations Problems in Engineering Sciences, 5(9):1671–1681, 1995. 14. A. Stefa´nski and T. Kapitaniak. Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos, Solitons & Fractals, 15(2):233– 244, 2003. 15. Q. Wu and N. Sepehri. On Lyapunov’s stability analysis of non-smooth systems with applications to control engineering. International Journal of Non-linear Mechanics, 36(7):1153– 1161, 2001. 16. F. Schmidt and C. -H. Lamarque. Computation of the solutions of the Fokker-Planck equation for one and two dof systems. Communications in Nonlinear Science and Numerical Simulation, 74(2008):529–542. 17. F. Schmidt and C. -H. Lamarque. Un indicateur pour optimiser les calculs trajectographiques. Bulletin de Liaison des Ponts et Chauss´ees, BLPC No. 263–264, znillet - Qoˆut - septembre 2006. 18. A. P. Ivanov. The dynamics of systems near to grazing collision. Journal of Applied Mathematics and Mechanics, 58(3):437–444, 1994. 19. O. Janin and C. H. Lamarque. Stability of singular periodic motions in a vibro-impact oscillator. Nonlinear Dynamics, 28(3–4):231–241, 2002. 20. U. Dressler and J. D. Farmer. Generalized Lyapunov exponents corresponding to higher derivatives. Physica D: Nonlinear Phenomena, 59(4):365–377, 1992.
Numerical Characterization of the Chaotic Nonregular Dynamics of Pseudoelastic Oscillators Davide Bernardini and Giuseppe Rega
1 Introduction Previous studies on the nonlinear dynamics of pseudoelastic oscillators showed the occurrence of chaotic responses in some ranges of the system parameters [1,2]. The restoring force was modeled by a thermomechanically consistent model with four state variables [3]. In comparison with the simpler polynomial constitutive laws considered for example in [4], the present model is characterized by more governing parameters and it is therefore interesting to understand whether nonregular responses only occur in isolated zones or are actually robust outcomes. The relevant analyses need to be carried out through some synthetic measure of non-regularity that has to be reliable and computationally simple in order to allow for systematic investigations in meaningful parameter spaces. Whereas the numerical characterization of chaos in smooth dynamical systems is often carried out via the computation of Lyapunov exponents, in the present case the computation of such exponents, following, for example [5], does not seem to be a convenient strategy. The attention has thus been focused on the simpler direct numerical tool represented by the method of wandering trajectories [6]. The method is based on the numerical evaluation of the separation between pairs of neighboring trajectories normalized with respect to a suitable measure of the motion amplitude. Such normalized perturbations provide a tool to detect the occurrence of nonregular and chaotic responses. The method has been successfully applied in the literature to estimate regular and chaotic responses for non-smooth mechanical oscillators with up to two degrees of freedom [7].
D. Bernardini and G. Rega Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Universit`a di Roma, Via Antonio Gramsci 53, 00197 Roma, Italy, e-mail:
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
25
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D. Bernardini, G. Rega
The purpose of this paper is to calibrate and validate the method of wandering trajectories within a thermomechanical framework and to present some results on the overall characterization of the chaotic response of pseudoelastic oscillators.
2 Description of the System The system under consideration is a simple oscillator where the restoring force is provided by a device with pseudoelastic behavior. The model used for the restoring force fits into the family of models introduced in [3] that are derived from the assignment of two constitutive functions: the free energy and the dissipation function. The specific version used in this work has been introduced in [1] and is characterized by a new form for the dissipation function with respect to those in [3]. The evolution of the system is described, besides by the displacement, velocity and temperature (x, v, ϑ ), also by the martensitic phase fraction ξ ∈ [0, 1]. The typical pseudoelastic loops occur as a consequence of the Forward (FwT) and Reverse (RvT) Transformation, respectively associated to increasing and decreasing of ξ . The nondimensional dynamics of the oscillator is modeled by the following piecewise smooth system of four ordinary differential equations in the variables x := [x, v, ξ , ϑ ] (for details see [1]) x˙ = v, v˙ = −x + (sλ )ξ − ζ v + γ cos ατ , ξ˙ = H[sv − Jh(ϑe − ϑ )],
(1)
Λ + Jλ ϑ ϑ˙ = L H[sv − Jh(ϑe − ϑ )] + h(ϑe − ϑ ), Jλ where H and Λ are constitutive functions that take different expressions depending on the kind of transformation [2] as well as on the value ξ0 of the martensite fraction at the end of the last transformation process. The vector field (1) can thus take three different forms depending on the expressions of H and Λ. However, once activated, each kind of behavior is smooth. The parameters λ , L, J, h represent respectively: the length of the pseudoelastic plateaus, the latent heat of transformation, the linear temperature dependence of the transformation forces and the coefficient of convective heat exchange with the environment. Moreover s = sgn(x) whereas ζ , γ , α denote respectively the viscous damping and the excitation amplitude and frequency. The initial conditions (i.c.) required to compute the evolution of the system is a 5-ple of numbers Σ := (x, v, ξ , ϑ , ξ0 ). The time evolution of ξ0 is almost everywhere constant as it jumps from a value to another whenever there is a switch between different kinds of behavior. For the following analyses it is important to note that not every 5-ple of numbers Σ represents a physically admissible state of the system. A procedure to determine admissible values of Σ is described in [2].
Numerical Characterization of the Chaotic Nonregular Dynamics
27
3 The Method of the Wandering Trajectories The Method of Wandering Trajectories (MWT) is a tool for the characterization of the asymptotic behavior of dynamical systems under periodic forcing excitation. The basic idea is very simple: a motion is classified as non-regular if the separation with a neighboring trajectory starting from an admissible state overcomes a given threshold. Of course, a key issue is the proper definition of the auxiliary trajectory and of the threshold. Let the fiduciary trajectory to be characterized be denoted as u(t) with initial condition u(0) = u0 . For any other trajectory u(t) ˜ such that u(0) ˜ = u˜ 0 initial and current separations are defined as follows h0 := | u0 − u0 | ,
h(t) := | u(t) − u(t)| .
(2)
The MWT proceeds as follows: • Integrate u(t) over T := [0, T] and compute, over the subinterval T1 := [t1 , T] where transients expired, for each component, the vibration amplitude of the fiduciary trajectory, defined as follows Ai :=
1 |ai − bi | , 2
where ai := max ui (t) and bi := min ui (t); t∈T1
t∈T1
(3)
this yields a vector A measuring the scale of the motion. • Another trajectory u(t) is defined by the initial condition u0 derived from u0 by perturbing each component proportionally to the corresponding vibration amplitude h0 := ε A. • Define the normalized separations relative to the vibration amplitudes
αi :=
|hi (t)| . Ai
(4)
Provided they correspond to an admissible state of the system, the initial values αi (0) are thus equal to the chosen parameter ε . If the motion is regular the normalized separations either take values of the same order of magnitude as ε or decay to zero. By contrast, non-regular motions may lead, after the transients, to normalized separations much higher than ε . More specifically, trajectories initiated from two nearby points on a chaotic attractor separate away from each other until the separation levels off at the size of the attractor. The main assumption of the MWT is that, with some preliminary knowledge of the system dynamics, it is possible to determine a threshold level α¯ for the normalized separation that characterizes the occurrence of non-regular motions. In particular, a trajectory u(t) is characterized as non-regular if, at some t ∈ T1 , the normalized separation with respect to a trajectory with initial separation h0 = ε A, exceeds the chosen threshold αi (t) > α . (5)
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D. Bernardini, G. Rega
This test certainly detects the sensitivity to initial conditions of the trajectory. However this is only a necessary but not sufficient condition for the motion to be chaotic. In facts, the sensitivity to initial conditions alone only indicates that the perturbation may have taken the trajectory outside the basin of attraction of the attractor. A chaotic motion, besides being sensitive to initial conditions, is also wandering in the sense that it attempts to fill a bounded region K of the phase space (for any q ∈ K there is a time t such that u(t) = q). Therefore the MWT tends to overestimate the number of non-regular trajectories, which turns out to be in favor of safety from an engineering viewpoint.
4 Characterization of Typical Trajectories The constitutive model for the restoring force covers a great variety of situations. The following set of parameters, as in [1], is considered as reference for the validation procedure and in the following is referred to as RMP Λ = 8.125, J = 3.1742, L = 0.124, q1 = 0.98, q2 = 1.2, q3 = 1.0246,
h = 0.08,
together with a = 0.03 and ζ = 0.03. These parameters correspond to a typical pseudoelastic cycle in a mildly convective environment. For the physical meaning of the parameters see [1].
4.1 Symmetric Period 1 Solution This is the most common type of solution. The system exhibits two pseudoelastic loops for positive and negative displacements. To show an example, the system, with RMP, is integrated for 100 periods with excitation amplitude γ = 1 and frequency α = 0.6 starting from the admissible initial condition u0 = [0, 0, 0, 1] and ξ0 = 0 corresponding to the system at rest in austenitic phase at the environment temperature. The time histories show that after about 20 periods transients expire and the system reaches a periodic attractor. Maxima, minima and vibration amplitudes for each component are computed for the fiduciary trajectory by taking the time interval T1 = [125.6, 628] x
v
Max 5.835 3.782 Min −5.835 −3.782 A 5.835 3.782
Ξ
ϑ
0.571 0 0.286
1.054 0.981 0.036
Numerical Characterization of the Chaotic Nonregular Dynamics 0.06
0.06
aX
0.04
0.04
0.02
0.02
0
29
ax
0 0
125.6
time 251.2 376.8
502.4
628
0
125.6
time 251.2 376.8
502.4
628
Fig. 1 Normalized separations in x and ξ or a periodic solution
In order to determine the perturbed trajectory it is important to ensure that the perturbed initial condition is again an admissible state. An admissible perturbed initial condition obtained by considering ε = 0.01 on all variables except for ξ , is (see [2]) u˜ 0
= [0.05835, 0.03782, 0.0, 1.00036]ξ0 = 0.0.
Integrating along the same time interval T it turns out that the perturbed trajectory u(t) ˜ is again a symmetric period 1 solution with features analogous to the fiduciary one. The absolute values of separation of the x and ξ components normalized relative to the respective vibration amplitudes are plotted in Fig. 1. The other variables exhibit analogous behavior. During the transient the normalized separations slightly overcome the assumed initial values, but then they rapidly decay to zero since the periodic attractor is asymptotically stable. The method of wandering trajectories classifies this solution as regular for any reasonable choice of α¯ .
4.2 Chaotic Solution As an example of chaotic trajectory the system is tested, as above, but at a lower frequency α = 0.245. Perturbing it again by 0.01Ai, the force-displacement cycles in the time interval [314, 628] for the fiduciary and the perturbed trajectories are drawn in Fig. 2 . It turns out that the fiduciary is chaotic whereas after the perturbation the response is periodic. The normalized separations are very high (Fig. 3), see in particular the ξ component, and the MWT classifies the solution as non-regular since, immediately after the transients, separations much higher than the initial values occur.
4.3 Unsymmetric Period 1 Solution It is rather frequent to find symmetry breaking bifurcations. In this case asymmetric periodic solutions occur and it is interesting to test the method on such solutions. To show an example, the system is integrated, again in the same conditions as in
30
D. Bernardini, G. Rega
Fig. 2 Pseudoelastic cycles for a chaotic trajectory and its perturbation
Fig. 3 Normalized separations in x and ξ or a chaotic solution
Fig. 4 Pseudoelastic cycles for an asymmetric periodic solution
previous examples but at a frequency α = 0.27. The fiduciary trajectory involves complete but highly non symmetric transformations (Fig. 4). Perturbing again, the ensuing solution turns out to be still periodic and non symmetric but mirrored. The normalized separations are large (Fig. 5) so the MWT classifies the solution as non-regular. However, by plotting two components of the separation against each other, a regular evolution is observed as opposed to what happens in the chaotic solution [2]. This is a situation in which the MWT may classify a periodic solution as nonregular. However, the same conclusion would be reached by looking at the maximum Lyapunov exponent that turns out to be positive, about 0.0046, as shown in [2].
Numerical Characterization of the Chaotic Nonregular Dynamics
31
Fig. 5 Normalized separations in the asymmetric periodic solution
Fig. 6 Bifurcation diagram and normalized separations
5 Comparison with Bifurcation Diagrams In order to have an overall picture of the system behavior, a constant excitation amplitude (γ = 1) bifurcation diagram with the frequency α as control parameter has been computed. The diagram is obtained by decreasing frequency with variable initial conditions taken from the adjacent computation point, and the region α ∈ (0.15, 0.3) is reported in Fig. 6. In the same frequency interval a systematic application of the MWT has been done for comparison. In particular, for each frequency, the response has been computed for T = 200 periods while checking the normalized separations with respect to trajectories perturbed by ε = 0.01 on T1 = 100 periods.
32
D. Bernardini, G. Rega
Fig. 7 Details of Fig. 6
On the same figure a curve depicting the results of the MWT is superposed. For each frequency the curve (to be read with respect to the right vertical axis) shows the maximum value over T1 of the normalized separation of the displacement. It turns out that, whenever the trajectories are periodic, the separation practically vanish. On the contrary, when the separation overcomes values of about 0.1, a slightly chaotic behavior is already observed. Values of the separation above 0.3 are definitely associated with the consolidated chaos. In Fig. 7 two more detailed pictures in the zones denoted above by R and L show that the normalized separation closely follows the occurrence of non-regular responses and that 0.3 may be taken as a reasonable estimate of the threshold α¯ to detect some consolidated chaotic responses.
6 Overall Characterization of the Non-regular Solutions and Effect of the Hysteresis The robustness of the chaotic response within the overall behavior of the system can now be investigated by computing behavior charts in which some control parameters are varied and the MWT is systematically applied to distinguish between regular and nonregular responses. A natural choice for the control parameters is the pair excitation frequency-amplitude at fixed initial conditions and material parameters. In particular, the analysis has been carried out for the above mentioned set of material parameters RMP as well for another set, called MP1, obtained from RMP by decreasing q2 from 1.2 to 1.02. The parameters MP1 correspond to a pseudoelastic loop with lower hysteresis with respect to RMP. The comparison between the two provide information about the effect of the hysteresis on the chaotic response. According to the previous analyses the threshold level for the normalized separations has been chosen as α¯ = 0.3. Integration of the trajectories has been carried out for 200 excitation periods, while restricting the interval T1 to the last 100 periods. Due to the complexity of the trajectories occurring in some parameter regions,
1
1
0.5
0.5 v_0
v_0
Numerical Characterization of the Chaotic Nonregular Dynamics
0
33
0
a = 0.21
a = 0.245
-0.5
-0.5
-1
-1 -1
-0.5
0 x_0
0.5
1
-1
-0.5
0 x_0
0.5
1
Fig. 8 Regions of nonregular response in initial conditions plane (white: regular, black dot: non-regular) 2
1.8
amplitude
1.6
1.4
1.2
1
0.8 0.1
0.2
0.3
0.4
frequency
Fig. 9 Behavior chart in excitation frequency-amplitude plane for RMP (white: regular, black dot: non-regular)
the application of the method requires a rather fine numerical integration. After calibration of various explicit and implicit integration algorithms, a reasonable compromise between accuracy and computational time has been reached by using a standard fourth-order Runge-Kutta algorithm with 2,000 steps per period. Preliminarily, an investigation has been carried out in the initial conditions domain. More specifically, the MWT has been first applied to build a section of
34
D. Bernardini, G. Rega
a kind of basin of attraction of chaotic responses in the plane of initial displacement x0 and velocity v0 . Initial conditions x0 ∈ [−1, 1] and v0 ∈ [−1, 1] have been considered together with ξ = ξ0 = 0 and ϑ = ϑ0 = 1. These values can be shown to be all admissible and correspond to the device in elastic, purely austenitic, phase. Two sample domains corresponding to the excitation amplitude γ = 1 and different frequencies α = 0.245 and α = 0.21 are shown in Fig. 8 (with RMP). At both frequencies, nonregular responses occur for various initial conditions. Analogous responses occur at the other frequencies where chaos is found. From consideration of such analyses, the pair (x0 , v0 ) = (−1.0, −1.0) has been selected as fixed initial condition, together with ξ0 = 0 and ϑ0 = 1, for the subsequent investigations. The frequency-amplitude behavior chart for the basic set of parameters RMP is shown in Fig. 9. For γ = 1, two clearly separated regions of non-regular motion are found, a compact one on the right, a more scattered one on the left. They are likely to correspond with the two kinds of chaotic motions highlighted in [1] by bifurcation diagrams. The presence of scattered points, especially at the higher excitation amplitudes, can be eliminated by a finer numerical integration. The same chart has then been computed with MP1 material parameters (Fig. 10). It turns out, as expected, that decreasing hysteresis leads to a significant increase of the size of the regions of irregular motion, with the intermediate region tending to cluster in nearly vertical stripes at lower frequencies. In-depth understanding of the kind of non-regular motion with respect to the neighbouring regular one would require complementing the chart with a number
2
1.8
amplitude
1.6
1.4
1.2
1
0.8 0.1
0.2
0.3
0.4
frequency
Fig. 10 Behavior chart in excitation frequency-amplitude plane for MP1 (white: regular, black dot: non-regular)
Numerical Characterization of the Chaotic Nonregular Dynamics
35
of bifurcation diagrams with frequency as control parameter (this is left for future investigations). Overall, the charts show that the chaotic motions are robust and persist in significant regions of the excitation frequency-amplitude plane.
7 Conclusions The Method of Wandering Trajectories has been shown to be effective in detecting the sensitivity to initial conditions of the orbits of a thermomechanically based pseudoelastic oscillator. The occurrence of chaotic responses has been characterized via excitation frequency-amplitude charts for two sets of material parameters. The results confirm that, although an increase of the hysteresis in the system tends to reduce chaotic motions, even in the reference case the occurrence of chaos is a robust outcome taking place in large regions of the frequency-amplitude plane.
References 1. Bernardini D, Rega G (2005) Thermomechanical modeling, nonlinear dynamics and chaos in shape memory oscillators, Mathematical and Computer Modelling of Dynamical Systems 11, 291–314. 2. Bernardini D, Rega G (2007) On the characterization of the chaotic response in the nonlinear dynamics of pseudoelastic oscillators, Proceedings of the 18th AIMETA Conference, Brescia (Italy), September 11–14. 3. Bernardini D, Pence TJ (2002) Models for one-variant shape memory materials based on dissipation functions, International Journal of Non-linear Mechanics 37, 1299–1317. 4. Savi MA, Pacheco PMCL (2002) Chaos and hyperchaos in shape memory systems, International Journal of Bifurcation and Chaos 12, 645–667. 5. M¨uller PC (1995) Calculation of Lyapunov exponents for dynamic systems with discontinuities, Chaos Solitons and Fractals 5, 1671–1681. 6. Awrejcewicz J, Dzyubak L, Grebogi C (2004) A direct numerical method for quantifying regular and chaotic orbits, Chaos Solitons Fractals 19, 503–507. 7. Awrejcewicz J, Dzyubak L, Grebogi C (2005) Estimation of chaotic and regular (stick-slip and slip-slip) oscillations exhibited by coupled oscillators with dry friction, Nonlinear Dynamics 42, 383–394.
Localized Nonlinear Excitations and Interchain Energy Exchange in the Case of Weak Coupling Leonid I. Manevich and Valeri V. Smirnov
1 Introduction The problem of energy exchange between weakly coupled nonlinear oscillators is actually far-reaching extension of classical beating problem in linear vibrations theory. Its modern stage goes back to the paper [1] in the field of nonlinear optics (the problem of nonlinear couplers). Then this problem was considered in [2] and [3]. Extension on the case of quasi-harmonic waves in two coupled sin-Gordon chain has been performed in series of papers [4, 5]. A new approach to the problem of non-linear energy exchange was proposed in [6]. In this paper the conception of limiting phase trajectory (LPT), corresponding to complete energy transfer between nonlinear oscillators has been introduced. It was shown that in the framework of this conception an adequate understanding and description of the problem can be obtained with the use of a pair of non-smooth basic functions of time. The proposed approach was also extended on the case of interchain energy exchange by quasiharmonic waves in weakly coupled oscillatory chains [6]. Recently we presented an efficient use of LPT conception in the case of “small” periodic Fermi-Pasta-Ulam (FPU) chains [7]. Contrary to previous papers, both symmetric and asymmetric interparticle potentials of interaction were considered. The interchain energy exchange by breathers was first studied both analytically and numerically in [8, 9]. A real possibility of this phenomenon has been shown. However, assumptions were made in that paper which require a justification. In the present paper we reconsidered the problem of interchain energy exchange using the LPT conception and considering subsequently the cases of weakly coupled chains with various degrees of nonlinearity (linear chains, weakly nonlinear chains, chains with nonlinearity compared with coupling). L.I. Manevich and V.V. Smirnov N.N. Semenov Institute of Chemical Physics, RAS 4 Kosygin street, 119991, Moscow, Russia, e-mail:
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
37
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L.I. Manevich, V.V. Smirnov
2 Linear Chains with Weak Coupling Let us consider a system of weakly coupled (FPU) chains with potential energy, which contains the terms of fourth order alongside with parabolic ones. The respective Hamilton function is: dqn, j 2 c2 β H = ∑ ∑ [( ) + (qn+1, j − qn, j )2 + (qn+1, j − qn, j )4 ] dt 2 4 n j=1, 2 γ (1) + ε (qn, 1 − qn, 2)2 , 2 where qn, j is the dimensionless displacement of particle “n”-th in “j”-th chain, c, β , and γ are dimensionless parameters of interaction and ε is a small coefficient of interchain coupling. It is easy to show that a modulation of particle displacements at the right edge of spectrum of the linearized system (un, j = (−1)n qn, j ) leads to the following continuum equations for envelope functions uj :
∂ 2u j ∂ 2u j + + u j + 16β u3j − εγ u3− j = 0, ∂ τ2 ∂ x2 τ = ω t, ω 2 = 4 + εγ . It is convenient to use the complex variables: 1 ∂uj 1 ∂uj √ √ Ψj = + iu j , Ψ j = − iu j . 2 ∂τ 2 ∂τ
(2)
(3)
The equations of motion (2) are converted to the form: i
1 ∂2 ∂ ¯ j ) − 4β (Ψ j − Ψ ¯ j )3 − ε γ (Ψ3− j − Ψ ¯ 3− j ) = 0. Ψj + Ψj + (Ψ j − Ψ ∂τ 2 ∂ x2 2
(4)
Let us assume the parameter of nonlinearity β is equal to zero. Now we obtain two linear chains with linear coupling. Using the multiscale expansion in ε: Ψ j = ε (ψ j + εψ j, 1 + ε 2 ψ j, 2 + . . .),
τ0 = τ , ξ = εx
τ1 = ετ ,
τ2 = ε 2 τ,
(5)
we obtain the following equations of different orders by small parameter ε :
ε 1: i∂τ0 ψ j + ψ j = 0, ψ j = χ j eiτ0 .
(6)
Localized Nonlinear Excitations and Interchain Energy Exchange
39
ε2:
γ i∂τ0 ψ j,1 + i∂τ1 ψ j + ψ j − (ψ3− j − ψ¯ 3− j ) = 0, 2 ψ j,1 = χ j,1 eiτ0 , γ i∂τ0 χ j,1 + i∂τ1 χ j − (χ3− j − χ¯ 3− j e−2iτ0 ) = 0. 2 The last equations lead to the following important relationships: γ i∂τ1 χ j − χ3− j = 0, 2 γ χ j,1 = χ¯ 3− j e−2iτ0 . 4
(7)
(8)
Now we can get the solution of Eqs. (8) in the form: 1 γ γ χ1 = √ [X1 cos( τ1 ) − iX2 sin( τ1 )] 2 2 2 1 γ γ χ2 = √ [X2 cos( τ1 ) − iX1 sin( τ1 )], 2 2 2
(9)
ε3: γ 1 i∂τ0 ψ j, 2 + i∂τ1 ψ j, 1 + i∂τ2 ψ j + ∂ξ2 (ψ j − ψ¯ j ) − (ψ3− j, 1 − ψ¯ 3− j, 1) = 0, 2 2 (10) 1 2 −2i ωτ 0) i∂τ0 χ j, 2 + i∂τ1 χ j, 1 + i∂τ2 χ j + ∂ξ (χ j − χ¯ j e 2 γ −2iωτ0 ¯ − (χ3− j, 1 − χ3− j, 1e ) = 0. 2 It is easy to see that after integrating with respect to fast time τ0 using of second relation (8), equations for different chains turn out to be uncoupled. The main point in the analysis of Eqs. (10) is that the unknown functions depend on “intermediate” time τ1 . The adequate procedure to remove this dependence is the averaging over time τ1 . After simple algebraic manipulations we obtain: γ2 1 i∂τ2 X1 + ∂ξ2 X1 − X1 = 0, 2 8 γ2 1 i∂τ2 X2 + ∂ξ2 X2 − X2 = 0, 2 8
(11)
where X1 and X2 are the functions of slow time τ2 . Eqs. (11) have solution in the form of plane wave X j = A j exp(i(kξ − ωτ2 )) (12) with dispersion relation
γ 1 ω = (k2 + ( )2 ). 2 2
40
L.I. Manevich, V.V. Smirnov
a
Energy exchange 1st chain
2nd chain
100
100
80
80
60
60
40
40
20
20
t
0
0 50
100 150 200 250 300 350 400
n
50
100 150 200 250 300 350 400
n
b
Fig. 1 (a) “Map” of total energy of linear chains – bright bands correspond to high energy value, dark bands – correspond to low energy, t – time, n – number of particle in the chain; (b) energy of 200-th particles in the different chains versus time. The plane wave was initiated in left chain at the time t = 0 only
It is very important that the structure of Eqs. (11) allows the wave localization on one chain only. This case corresponds to full energy exchange between the chains, if the solution (12) is considered as initial conditions for the first of Eqs. (8) in the “intermediate” time τ1 . It is obvious that the plane waves (12) migrate from one chain to other in accordance with Eqs. (9). Figure 1 shows an example of full exchange between the chains for the initial conditions A1 = 0.10, A2 = 0.0.
Localized Nonlinear Excitations and Interchain Energy Exchange
41
3 Nonlinear Chains with Weak Nonlinearity For parameter of nonlinearity β = 0 in Eqs. (4), we can study the influence of nonlinearity on the process of energy exchange. Using the series like (5) for a small parameter ε, we obtain the weak nonlinearity asymptotics, because the order of coupling terms is equal to ε2 , while the nonlinear terms give a contribution ∼ε3 . Thus, Eqs. (6–9) are the same as for linear and nonlinear systems, but Eqs. (11) resulting from averaging are changed:
γ2 1 3β i∂τ2 X1 + ∂ξ2 X1 − X1 + (3|X1 |2 X1 + 2|X2|2 X1 − X22X¯1 ) = 0, 2 8 8 (13) γ2 1 2 3β 2 2 2¯ (3|X2 | X2 + 2|X1| X2 − X1 X2 ) = 0. i∂τ2 X2 + ∂ξ X2 − X2 + 2 8 8 Equations (13) describe the pair of nonlinear oscillatory chains with nonlinear coupling contrary to the initial system with the linear coupling. It is very interesting that the structure of nonlinear terms is similar to the case of small FPU-system [7, 8]. These equations allow both anharmonic plane wave solution and a solution in the form of localized vibrations (breathers). The plane wave solution has the form: X j (ξ , τ2 ) = A j exp(−i(ωτ2 − kξ ))
(14)
with dispersion relation
γ 1 ω = (k2 + ( )2 ) − 6β (3A2j + A23− j ). (15) 2 2 Like the case of linear chains Eqs. (13) allow for a wave solution, localized on one chain only. This solution leads to the full energy exchange between chains. Figure 2 shows an example of small amplitude anharmonic plane wave in the weakly nonlinear system. Analysis of “phase plane” in the terms of “angle variables” [2, 5] does not show bifurcation both in-phase and anti-phase stationary point. The solution in the form of plane wave has the phase shift which is equal to π /2. Thus, this trajectory corresponds to the limiting phase trajectory (LPT) in the case of two nonlinear oscillators. Let us consider a localized solution of Eqs. (13): X j (ξ , τ2 ) = A j (ξ − vτ2 ) exp[−i(ωτ2 − qξ )], j = 1, 2.
(16)
Here, Aj are real functions. Substitution of this form into the equations of motion give the relation between wave number q and velocity of the wave v: v = −q. The equations for amplitudes Aj can be written as follows: A j + (ω −
q2 γ 2 − )A j + 6β (3A j 3 + A3− j 2 A j ) = 0, 2 8
(17)
42
L.I. Manevich, V.V. Smirnov
a
Energy exchange 1st chain
2nd chain
100
100
80
80
60
60
40
40
20
20
t
0
0 50
100 150 200 250 300 350 400 n
50
100 150 200 250 300 350 400 n
b
Fig. 2 Plane wave with full energy exchange in the weakly nonlinear system. (a) Energy “map”– bright and dark bands correspond to high and low energy values, respectively. The plane was initiated in right chain at the time t = 0 only. (b) The energy profiles of 200-th particles in both chains versus time
where primes denote differentiation with respect to argument. Let us suppose A1 = kA2 . The conditions of compatibility of Eqs. (17) lead to following values of k: k = +1; k = −1; k = 0. The last value has a principal importance, because of the existence of full energy exchange. Thus, the solution of Eqs. (13) describing the localized oscillations takes the following form:
Localized Nonlinear Excitations and Interchain Energy Exchange 500
400
400
300
300
200
200
100
100
t
500
43
0
0 50
100
150
200 n
250
300
350
400
50
100
150
200 n
250
300
350
400
Fig. 3 The energy “map” for moving small amplitude breather in the case of weak nonlinearity
(q2 + (γ /2)2 ) − 2ω exp[−i(ωτ2 − qξ )] 6(3 + κ 2)β ! (q2 + (γ /2)2) − 2ω (ξ − qτ2 − ξ0 )]X1 , × sch[ 2 X2 (ξ , τ2 ) = κ X1 (ξ , τ2 ), κ = 0, ±1. X1 (ξ , τ2 ) =
(18)
It is worth mentioning that the shape of small-amplitude solution is formed in the time scale that is slower, than the characteristic time of energy transfer between different chains. This statement is valid for both linear and nonlinear systems. An example of energy transfer in the case of moving breather is shown in Fig. 3.
4 Chains with Nonlinearity, Compatible with Coupling Let us return to Eqs. (4) for the case of amplitudes providing compatibility √ of nonlinear and coupling terms by parameter ε. If their values reach a magnitude ε , the expansion (5) turns out to be invalid. Therefore we will use the following multi-scale expansion: √ Ψ j = ε (ψ j + εψ j, 1 + ε 2 ψ j, 2 + . . .),
τ0 = τ , τ1 = ετ , √ ξ = ε x,
τ2 = ε 2 τ ,
(19)
that leads to following equations for different orders of small parameter ε:
ε 1/2 : i∂τ0 ψ j + ψ j = 0, ψ j = χ j eiτ0 ,
(20)
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L.I. Manevich, V.V. Smirnov
ε 3/2 : 1 γ i∂τ0 ψ j,1 + i∂τ1 ψ j + ψ j + ∂ξ2 (ψ j − ψ¯ j ) − (ψ3− j − ψ¯ 3− j ) 2 2 3 − 4β (ψ j − ψ¯ j ) = 0,
ψ j,1 = χ j,1 eiτ0 ,
(21)
γ 1 i∂τ0 χ j,1 + i∂τ1 χ j + ∂ξ2 (χ j − χ¯ j e−2iτ0 ) − (χ3− j − χ¯ 3− j e−2iτ0 ) 2 2 − 4β (χ j eiτ0 − χ¯ j eiτ0 )3 e−iτ0 = 0. Integrating last equations (21) with respect to “fast” time τ0 , we get two coupled equations: γ 1 i∂τ1 χ j + ∂ξ 2 χ j − χ3− j + 12β |χ j |2 χ j = 0. (22) 2 2 First of all, we can see, that there are two symmetric solutions of Eqs. (22). In the class of localized soliton-like solutions they have the form: in-phase solution
χ1 (ξ , τ1 ) =
1 4
2ω + q2 + γ 1 sch( 3β 4
2ω + q2 + γ (ξ + qτ1 ) 6β
× exp(i(ωτ1 − qξ ),
(23)
χ1 (ξ , τ1 ) = χ2 (ξ , τ1 ) and the anti-phase solution
χ1 (ξ , τ1 ) =
1 4
2ω + q2 − γ 1 sch( 3β 4
2ω + q2 − γ (ξ + qτ1 ) 6β
× exp(i(ωτ1 − qξ ), χ1 (ξ , τ1 ) = −χ2 (ξ , τ1 ).
(24)
The Hamiltonian corresponding to Eqs. (22) is
γ 1 h = − (χ1 χ¯ 2 + χ¯ 1 χ2 ) + (|∂ξ χ1 |2 + |∂ξ χ2 |2 ) 2 2 + 6β (|χ1|4 + |χ1|4 ).
(25)
Similarly to symmetric solutions (23–24 ) we can suppose that localized soliton-like solutions of Eqs. (22) can be represented in the form:
χ j = A(ξ )X j (τ1 ),
(26)
Localized Nonlinear Excitations and Interchain Energy Exchange
45
where a space-dependent amplitude A has the same profile for both chains. Thus, integrating Eq. (25) with respect to space variables ξ, we get the “energy” of the system as a function of time variable:
γ 1 H = − N(X1 X¯2 + X¯1 X2 ) + μ N(|X1 |2 + |X2|2 ) 2 2 + 6β ν N 2 (|X1 |4 + |X2|4 ),
(27)
where new parameters are defined by a soliton profile: N=
ν=
A2 d ξ ,
μ=
(∂ξ A)2 d ξ /
A2 d ξ , (28)
A4 d ξ /( A2 d ξ )2 .
In such a case we get an analog of two nonlinear oscillators, described by functions Xj , which were studied recently in detail [6] by one of authors. Both beating with full energy exchange and confinement of initial excitation in the one of chain can be observed when the value of “occupation number” N grows. It was shown, that the process of energy exchange is defined by trajectories, closed to LPT [6] and pertinent to attractive area of one of two stationary points of the system. There are two stationary points of Eqs. (24) at a small “occupation number” N and four ones exist if N is large enough. It is easy to see from analysis of “phase plane” in the terms of “angle variables”, that the following can be introduced [2]: X1 = cos θ eiδ1 , X2 = sin θ eiδ2 . The parameter controlling a structure of phase plane, is κ = 6βν N/γ . Four typical cases are shown in Fig. 4. The structure of phase trajectories and the conditions of LPT existence were described in detail in [6]. If the parameter κ is smaller than 0.5, only two stationary points exist: in-phase (δ = 0, θ = π /4) and anti-phase (δ = π , θ = π /4) ones (see Fig. 4). Closed trajectories near the LPT describe full energy exchange. At κ = 0.5 anti-phase mode becomes unstable one, that leads to separatrix creation (Fig. 4b). So, if we start from the state near the new asymmetric modes, we can not transfer energy effectively from one chain to another one. But a possibility of full exchange near the LPT is well preserved. The total prohibition of energy exchange appears when κ reaches 1. Then the separatrix coincides with LPT and trajectories closed around anti-phase mode are broken (Fig. 4c). So, full confinement of excitation occurs in the one of chain. The main conclusion is that the full energy exchange is possible up to values of parameter κ does not exceeding unity (Fig. 4d). After that only partial exchange can occur near the asymmetric modes. The computer simulation data, an example of which is shown in Fig. 5, demonstrate a confinement of initial excitations in one the chains at κ ∼1.2.
46
L.I. Manevich, V.V. Smirnov 2p
a
b
c
d
q p
0 2p q p
0
p D
0
p D
2p 0
2p
Fig. 4 Transformation of “phase plane” of Eqs. (24) in the terms of angle variables. θ characterizes the amplitude ratio and Δ = δ1 − δ2 – the phase shift. The occupation number N increases from (a) to (d) fragments: (a) κ < 0.5, (b) 0.5 < κ < 1, (c) κ = 1.0, (d) κ > 1.0 (see text) 1st chain
2nd chain 500
400
400
300
300
200
200
100
100
t
500
0
50 100 150 200 250 300 350 400
n
0
50 100 150 200 250 300 350 400
n
Fig. 5 Confinement of breather in first chain. Breather was initiated in the first chain at t = 0. After exchange with the second chain, breather returns to “parent” chain
5 Conclusions Analytical and numerical studies of wandering excitation both in linear and nonlinear chains coupled by weak linear interaction show an existence of two asymptotic limits of energy transfer between different chains. The first of them is characterized
Localized Nonlinear Excitations and Interchain Energy Exchange
47
by quick energy transfer in comparison to processes of excitation formation. In such a case the waves in the different chains exhibit the phase shift which is equal π /2. It means that the respective trajectory is closed to the LPT. In contrast, excitations with large amplitudes can show both full energy exchange near LPT and partial one near stationary points up to full confinement of excitation in one of the chains. Acknowledgement The work was supported by Program of OXN, Russian Academy of Sciences.
References 1. Jensen SM (1982) The nonlinear coherent coupler, IEEE J Quantum Elect 18, 1580–1583. 2. Kosevich AM, Kovalyov AS (1989) Introduction to Nonlinear Physical Mechanics, Naukova Dumka, Kiev (in Russian). 3. Uzunov IM, Muschall R, G¨olles M, Kivshar YS, Malomed BA, Lederer F (1995) Pulse switching in nonlinear fiber directional couplers, Phys Rev E 51, 2527–2537. 4. Khusnutdinova KR (1992) Non-linear waves in a double row particle system, Vestn MGU Math Mech 2, 71–76. 5. Khusnutdinova KR, Pelinovsky DE (2003) On the exchange of energy in coupled Klein-Gordon equations, Wave Motion 38, 1–10. 6. Manevich LI (2007) New approach to beating phenomenon in coupled nonlinear oscillatory chains, Arch Appl Mech 77, 301–312. 7. Manevich LI, Smirnov VV (2007) Discrete breathers and intrinsic localized modes in small FPU systems, Proc APM 293, St-Petersburg. 8. Manevich LI, Smirnov VV (2007) Intrinsic Localized Modes Mobility in Small Fermi-PastaUlam Systems, this issue. 9. Kosevich YuA, Manevich LI, Savin AV (2007) Wandering breathers and self-trapping in weakly coupled nonlinear chains: classical counterpart of macroscopic tunneling quantum dynamics, e-ArXiv: 0705.1957(1).
Dynamic Analysis of the Gantry Crane Used for Transporting BOP Andrzej Urba´s and Stanisław Wojciech
1 Introduction Dynamic analysis of offshore systems mounted on platforms or vessels is especially difficult since it is necessary to consider phenomena connected with a sea waves [1]. The waving causes additional impulse forces in the system which should be taken into account in design process. The suitable description of dynamic behavior of the system allows its using in design of control systems. The algorithm for generating the equations of motion of the gantry crane using homogenous transformations and joints coordinates is presented in the paper [2, 3]. The Lagrange equations of the second order are used to derive the equations of motion.
2 Mathematical Model of the System The gantry crane used to transport the load called BOP (Blowout Preventer) is analysed. The system is considered as a system of two rigid bodies with 12 DOF (Fig. 1). Frame {F} is treated as a rigid body (6 DOF) connected with platform {D} by means of spring-damping elements (sde). The load is also treated as a rigid body with 6 DOF with respect to platform {D}. The load is connected with the frame by means of two flexible ropes. The motion of the load is limited by guides which are modelled as spring-damping elements with backlash. It is assumed that the motion of the platform is known. Consequently the position of coordinate system {D} with respect to global coordinate system {G} is known.
A. Urba´s and S. Wojciech Department of Mechanics and Computer Science, University of Bielsko-Biała, Poland, e-mail:
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
49
50
A. Urba´s, S. Wojciech ˆ (F) {F } Z y (F) A2
frame
A1 B2 C (F)
(
x (F),y (F),z (F) C C C
)
j (F) ˆ (F) X ˆ (L) Y
y (L)
C (L) (x (L),y (L) ,z (L)) C
ˆ (F) Y q (F)
C (F ) ˆ (L) {L} Z
guides C
B1
C(L) {D} ˆ (D) Z
C
load (BOP)
q (L) j (L) ˆ (L) X
y (D)
ˆ (D) Y moving platform
q (D) {G} ˆ Z j (D) ˆ (D) X ˆ Y
ˆ X a heading
Fig. 1 Model of the gantry crane
The position of coordinate system {D} with respect to {G} is defined by: ⎫ x(D) = x(D) (t) ⎬ y(D) = y(D) (t) the coordinates of the origin of coordinate system in, ⎭ z(D) = z(D) (t)
(1)
⎫ ψ (D) = ψ (D) (t) ⎬ θ (D) = θ (D) (t) the Euler angles ZYX [1] which describe ⎭ ϕ (D) = ϕ (D) (t) any possible orientation of frame {D} with respect to {G}. The transformation matrix from coordinate system {D} to {G} has the following form: ⎡
⎤ cψ (D) cθ (D) cψ (D) sθ (D) sϕ (D) − sψ (D) cϕ (D) cψ (D) sθ (D) cϕ (D) + sψ (D) sϕ (D) x(D) ⎢ sψ (D) cθ (D) sψ (D) sθ (D) sϕ (D) + cψ (D) cϕ (D) sψ (D) sθ (D) cϕ (D) − cψ (D) sϕ (D) y(D) ⎥ ⎥, A(D) (t) = ⎢ ⎣ −sθ (D) cθ (D) sϕ (D) cθ (D) cϕ (D) z(D) ⎦ 0 0 0 1
(2)
Dynamic Analysis of the Gantry Crane Used for Transporting BOP
51
where cψ (D) = cos ψ (D) , cθ (D) = cos θ (D) , cϕ (D) = cos ϕ (D) , sψ (D) = sin ψ (D) , sθ (D) = sin θ (D) , sϕ (D) = sin ϕ (D) . In the specific case when there is no motion of platform {D}, transformation matrix A(D) is the matrix with constant coefficients. It is assumed that the motion frame {F} and load {L} with respect to platform {D} is described by independent parameters which are components of the vectors: T
q(F) = x(F) y(F) z(F) ψ (F) θ (F) ϕ (F) , T
q(L) = x(L) y(L) z(L) ψ (L) θ (L) ϕ (L) .
(3)
Frame {F} is connected with platform {D} by means of spring-damping elements. Since angles ϕ (F) , θ (F) , ψ (F) , ϕ (L) , θ (L) , ψ (L) are assumed to be small the transformation matrices from coordinate systems {F} and {L} to {D} can be presented in the form: ⎡ ⎤ 1 −ψ (F) θ (F) x(F) ⎢ (F) 1 −ϕ (F) y(F) ⎥ ⎥, ˜ (F) = ⎢ ψ B (4) ⎣ −θ (F) ϕ (F) 1 z(F) ⎦ 0 0 0 1 ⎡ ⎤ (L) (L) (L) 1 −ψ θ x ⎢ ψ (L) 1 −ϕ (L) y(L) ⎥ ⎥. B˜ (L) = ⎢ ⎣ −θ (L) ϕ (L) 1 z(L) ⎦ 0
0
0
1
The transformation of coordinates from local coordinate systems {F} and {L} to {D} can be written in the form: (D)
(F) = B˜ (F) r˜ P ,
(D) rP
(L) B˜ (L) r˜ P ,
rP
(F)
=
(5)
(L)
where r˜ P , r˜ P are the position vectors of point P in local coordinate systems {F} and {L}. The transformation matrices from local coordinates systems {F} and {L} to global coordinate system {G} are defined as follows: B(F) = A(D) B˜ (F) (q(F) ), B(L) = A(D) B˜ (L) (q(L) ).
(6)
The equations of motion are formulated using the Lagrange equations of the second order: d ∂E ∂E ∂V ∂D − + + = Qk , k = 1, . . . , 12, (7) dt ∂ q˙k ∂ qk ∂ qk ∂ q˙k
52
A. Urba´s, S. Wojciech
where: E is the kinetic energy of the system, V is the potential energy of gravity forces, D is a form describing the dissipation of energy of the system, Qk are non-potential generalised forces, qk , q˙k are generalised coordinates and velocities, respectively.
2.1 Kinetic Energy of the Systems The kinetic energy of the system can be written in the form: E = E (F) + E (L) , (8) , , + + T ˙ (L) H(L) B˙ (L)T are kinetic energy where E (F) = 12 tr B˙ (F) H(F) B˙ (F) , E (L) = 12 tr B of the frame and the load, respectively, H(F) , H(L) are pseudo inertia matrices [3].
2.2 Potential Energy of Gravity Forces The potential energy of gravity forces can be described as follows: (F)
Vg = Vg (F)
(F)
(L)
(L)
+ Vg ,
(9) (L)
where Vg = m(F) g θ3 B(F) rC(F) , Vg = m(L) g θ3 B(L) rC(L) are the potential energies of the frame and the load, respectively: m(F) , m(L) are masses of the frame and
the load, g is the acceleration of gravity, θ3 = 0 0 1 0 , (F)
(L)
r (F ) , r (L) are the vectors of coordinates of the centre of mass of the frame and C C the load in local coordinate systems.
2.3 Energy of Deformation and Dissipation Energy of Sde It is assumed that the load can be in contact with the guides only along lines A(k) , B(k) , where k = 1, 2, 3, 4 (Fig. 2). The guides are modelled as spring-damping ˆ (D) elements (sde) with backlash (E (k, p) ), which limit the motion of the load in X (k) ˆ (D) directions. The number of sdes at each edge is denoted as ne . and Y Sde E (k, p) is described by the following parameters: ˆ and Y ˆ directions, cxE (k, p) , cyE (k, p) – coefficients of stiffness in X
Dynamic Analysis of the Gantry Crane Used for Transporting BOP
53
A (k)
{D} Zˆ (D)
b y(k,p) E
ΔEy (k,p)
cEy (k,p) E (k,p)
ΔEx (k,p)
x c x(k,p) bE (k,p) E
B (k) Yˆ (D) Xˆ (D)
Fig. 2 Spring-damping elements with backlash
ˆ and Y ˆ directions, bxE (k, p) , byE (k, p) – coefficients of damping in X ˆ and Y ˆ directions. Δx (k, p) , Δy (k, p) – backlash in X E
E
Generalised forces arising from sdes can be written as follows: (k)
(F) Qe
=
4 ne
(F)T
(D)
∑ ∑ UE (k, p) FE (k, p) ,
(10)
k=1 p=1
⎡
where
⎢ (F) UE (k, p) = ⎢ ⎣0 1 00 ⎡ 10 ⎢ (L) UE (k, p) = ⎢ 0 ⎣ 1 00 (F)
(F)
(F)
(L)
(F)
(F)
(F) E (k, p)
0
1 0 0 −yE (k, p) zE (k, p)
(L)
0 x 1
0 (L)
(F)
−xE (k, p) (L)
0 −yE (k, p) zE (k, p) (L)
0 xE (k, p) 1
0
0 (L)
⎤ 0 (F) E (k, p) (F) yE (k, p)
−z
⎥ ⎥, ⎦ ⎤
0
(L) ⎥ −zE (k, p) ⎥ ⎦, (L)
−xE (k, p) yE (k, p)
(L)
xE (k, p) , yE (k, p) , zE (k, p) , xE (k, p) , yE (k, p) , zE (k, p) are coordinates of sde E (k, p) in the local coordinate systems {F} and {L}, .T (D) x) (D, y) F (k, p) = F (D, is a force in sde E (k, p) , (k, p) F (k, p) 0 E (D, x)
E (D, x)
E
(D, x)
FE (k, p) = FS,E (k, p) + FT,E (k, p) ,
(D, y)
(D, y)
(D, y)
FE (k, p) = FS,E (k, p) + FT,E (k, p) are stiffness and dampˆ and Y ˆ directions. ing forces seting in each sde E (k, p) in X
54
A. Urba´s, S. Wojciech
2.4 Energy of Deformation and Dissipation Energy of the Ropes The load is connected with the frame by means of two flexible ropes (p = 1, 2). Energy of spring deformation and dissipation of energy of the ropes can be expressed as: (p)
Vs (p)
(p)
where cr , br
1 (p) (p) (p) 2 ΔlA p B p , = c r δr 2
1 (p) (p) ˙ (p) 2 (p) ΔlA p B p , D s = b r δr 2
(11)
are coefficients of stiffness and damping of rope p, (p)
ΔlA p B p is the elongation of rope p, 0 when ΔlA p B p ≤ 0, (p) δr = 1 when ΔlA p B p > 0.
2.5 Reaction Forces of the Support Let P(k) (k = 1, 2, 3, 4) denote the connection points of the frame and the platform. Sdes are placed in these points (Fig. 3). It is assumed that stiffness and damping ˆ (D) , Y ˆ (D) and coefficients have large values and limit the motion of the frame in X (D) ˆ Z directions. Generalised forces arising from those elements are as follows: (F)
Qp =
4
(F)T
(F)
∑ UP(k) FP(k) ,
(12)
k=1
ˆ (D) Z
{D}
y (D) (k)
ˆ (D) Y
P (k) (x (F ) , y (F) , z (F ) )
P
P (k)
P (k)
F (F,(k)y) P
F (F,(k)z) P
x (D) P (k)
ˆ (D) X
Fig. 3 Flexible connecting the frame and the platform
F (F,(k)x) P
P (k)
Dynamic Analysis of the Gantry Crane Used for Transporting BOP SF
y
D
55
[T]
10 t [s] 0 0
5
10
15
20
25
-10 -20 survival operation
-30 -40
a
-50
SF
yD
[T]
60 50 40 30
survival operation
20 10 t [s] 0
b
0
5
10
15
20
25
-10
Fig. 4 The influence of working conditions on sum of forces ∑ FyD : (a) the edge 1 and 2, (b) the edge 3 and 4
(F)
where UP(k) is defined similarly to (10), .T (F) x) (F, y) (F, z) FP(k) = F (F, , F F (k) (k) (k) P P P (F) (F, x) (F, x) = F (k) + F (k) stiffness and P(k) S, P T, P (F, y) (F, z) FP(k) and FP(k) are defined in (10).
F
ˆ direction, damping forces in P(k) , in X
The equations of the system can be written in the following form: ˙ Aq¨ = f(t, q, q),
(13)
where A = A(t, q)is the mass matrix. The model of dynamics of the gantry crane considered is described by 12 ordinary differential equations of the second order. In order to integrate the equations of motion the Runge-Kutta method with constant step-size is used.
56
A. Urba´s, S. Wojciech Fl [T] 280 survival operation
279 278 277 276 275 274 273 272 271
t [s]
a
270
0
5
10
15
20
25
Fl [T] 286 284 282 280 278 276 274 272
survival operation
270 268 266 264
b
t [s] 0
5
10
15
20
25
Fig. 5 The influence of conditions of work on forces in the ropes: (a) the rope 1, (b) the rope 2
3 Numerical Calculations For the mathematical model of the gantry crane the computer program has been developed. The following input data are taken from the technical documentation [4]: the mass of the load and the frame: m(L) = 550 000 kg, m(F) = 110 000 kg, dimensions of the load 4.8 × 5.5 × 20.3 m. Stiffness coefficients of sdes in contact points between the guides and the load were obtained using the simple model of the guides prepared by means of the finite element method package. It has been assumed that damping coefficients are proportional to stiffness coefficients [5]. The large values of the stiffness and damping coefficients of sdes in points P(k) were assumed. These large values caused some numerical problems when integrating the equations of motion of the system. For that reasons we applied the Runge-Kutta method of fourth order with time step equal to 10−4 s. Additionally the damping coefficients were taken of the order which compensates potential problems due to large stiffness.
Dynamic Analysis of the Gantry Crane Used for Transporting BOP SFyD
57
10 0
0
5
25
20
15
10
-10 -20 -30
quasi-statics dynamics
-40 -50 -60 -70
a SFyD
70 60 50 quasi-statics dynamics
40 30 20 10 0
b
0
5
10
15
20
25
-10
Fig. 6 Comparison of results of quasi-static and dynamic analysis - forces ∑ FyD (a) the edge 1 and 2, (b) the edge 3 and 4
Figures 4–7 present calculation results in the case when the motion of the plat(D) form is described in the following way: x(D) = y(D) = ψ (D) = 0, z(D) = z0 + (D)
a3 sin(2π t/T ), θ (D) = a5 sin(2π t/T), ϕ (D) = a6 sin(2π t/T ), where z0 = 36 m, T = 10 s, and a3 , a5 , a6 are amplitudes depending on heading and weather conditions (Table 1). Figure 4 presents a comparison of forces acting between the guides and the load when different conditions of work are considered. The results have been obtained for heading angle equal to 45◦ . The influence of conditions of work on forces in ropes can be observed in Fig. 5. In order to express how dynamic state of the system considered influences the reaction forces, a quasi-static problem has been solved. For the quasi-static analysis it is assumed that in Eq. (13), q¨ = 0 and q˙ = 0. Also the backlash is omitted in that analysis. The resulting set of nonlinear algebraic equations was solved using iterative Newton’s method. Figure 6 presents a comparison of results for quasi-static and dynamic analysis. Forces acting between the guides and the load are presented. It is assumed the heading angle is equal to 90◦ and survival conditions of work.
58
A. Urba´s, S. Wojciech SF
yD
[T]
a = 90, a3 = 0, a5 = 0, a6 = 0.0138 (roll)
10 0
t [s] 0
5
10
15
20
25
-10 -20 -30
p) b(k, = by(k, p) = 0 x
-40
p),b(k, p) - 50% b(k, x y p),b(k, p) - 100% (m = 0.1) b(k, x y
-50
p),b(k, p) - 150% b(k, x y
-60
k = 1, 2
p = 1, 2, 3, 4
-70
a
-80
SF
yD
[T]
a = 90, a3 = 0, a5 = 0, a6 = 0.0138 (roll)
80 70 60
p) = b(k, p) = 0 b(k, x y p),b(k, p) - 50% b(k, x y
50
p),b(k, p) - 100% (m = 0.1) b(k, x y
40
p),b(k, p) - 150% b(k, y x
30 k = 3, 4
p = 1, 2, 3, 4
20 10 t [s] 0
b
0
5
10
15
20
25
-10
Fig. 7 The influence coefficients of damping of sdes on sum of forces ∑ FyD : (a) edges 1 and 2, (b) edges 3 and 4 Table 1 Conditions of work for BOP Heading
Operational
Survival
Heave z(D)
Pitch θ (D)
Roll ϕ (D)
Heave z(D)
Pitch θ (D)
Roll ϕ (D)
α
a3
a5
a6
a3
a5
a6
0◦ 45◦ 90◦
0.1343 0.1115 0.1140
0.0023 0.0008 0
0 0.0023 0.0045
0.4458 0.3521 0.3724
0.0061 0.0023 0
0 0.0077 0.0138
The influence of coefficients of damping of sdes on forces acting between the guides and the load is shown in Fig. 7. The motion of the platform is described only by function ϕ (D) in survival conditions of work.
Dynamic Analysis of the Gantry Crane Used for Transporting BOP
59
4 Conclusions The numerical simulations presented prove that dynamic analysis of the BOP is important in the design process. The conditions of work of such an offshore system cause that impulse forces arise which exceed forces obtained in the static analysis. Acknowledgement The investigation presented in the paper has been partially supported by grant 4 T07A049 2B founded by Polish Committee of Science.
References 1. Maczyski A (2005) Positioning and Stabilization of Loads of Luffing Jib Cranes, TechnoHumanistic Academy Press, Theses 14, Bielsko-Biaa (in Polish). 2. Craig JJ (1995) Introduction to Robotics. Mechanics and Control, WNT, Warsaw (in Polish). 3. Wittbrodt E, Adamiec-W´ojcik I, Wojciech S (2006) Dynamics of Flexible Multibody Systems. Rigid Finite Element Method, Springer, Berlin/Heidelberg/New York. 4. Technical documentation for BOP (2007) PROTEA, Gda´nsk-Olesno. 5. Uhl T (1997) Computer-Aided Identification of Models of Mechanical Structures, WNT Warsaw (in Polish).
Motion of a Chain of Three Point Masses on a Rough Plane Under Kinematical Constraints Klaus Zimmermann, Igor Zeidis, and Mikhail Pivovarov
1 Introduction A series of papers have analyzed the rectilinear motion on a rough plane of bodies (mass points) connected by viscoelastic elements (springs and dampers) in the case when the force of normal pressure is not changed. The system is moved by forces that changed harmonically and acting between the bodies. The asymmetry of the friction force, required for a motion in a given direction, is provided by the dependence of the friction coefficient on the sign of the velocity of the bodies which make up the system. This effect can be achieved if the contact surfaces of the bodies are equipped with a special form of scales (needle-shaped plate with a required orientation of scales (needles)). In [1–5], the dynamics of a system of two bodies joined by an elastic element with a linear characteristic were considered. The motion is excited by a harmonic force acting between the bodies. In [3], a magnetizable polymer was employed as an elastic element and the motion was excited by a magnetic field. In the case of small friction, the analytical expression for the average velocity of steady motion of the whole system was found and it is shown, that the motion with this velocity is stable. A similar investigation for a system of two bodies joined by a spring with a nonlinear (cubic) characteristic was shown in [5]. Algebraic equations were obtained for average velocities of the steady motion. It was shown that there exist up to three different motion modes, one or two of them are stable. The limiting case of asymmetric friction is the kinematic condition that admits the motion only in one direction. This condition was considered in [6] in connection with a computer model of an earthworm. In [7], a numerical solution of the motion equations of a chain of bodies joined by viscoelastic elements is presented for the case when each body can move only in one K. Zimmermann, I. Zeidis, and M. Pivovarov Technische Universitaet Ilmenau PF 100565, 98684 Ilmenau, Germany, e-mail: klaus.zimmermann @tu-ilmenau.de,
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
61
62
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direction. In [8] the rectilinear motion of a vibration-driven system on a horizontal rough plane consisting of a carrying body, which interacts with the plane directly, and of internal masses that perform harmonic oscillations relative to the carrying body, is considered. The vertical and horizontal oscillations of the internal masses have the same frequency, but they are shifted in phase. It is shown that by controlling the phase shift of the horizontal and vertical oscillations, it is possible to change the velocity of the steady motion of the carrying body, and it is not necessary to use scales in order to provide friction asymmetry. In [9] the motion of two mass points connected by a linear spring, when the coefficient of friction does not depend on the direction of motion, is discussed. Due to the change of the normal force in dependence on time asymmetry of friction is present. The change of normal force is realized by the rotation of two unbalanced rotors with various angular velocities. In [10], the rectilinear motion of a body with a movable internal mass moving along a straight line parallel to the line of the body motion on a rough plane was investigated. A periodic control mode was constructed for the relative motion of the internal mass for which the main body moves with a periodically changing velocity passing the same distance in a given direction. It is supposed that, at the beginning and the end of each period, the velocity of the main body is zero. The internal mass can move within fixed limits. The control modes relative to the velocity and acceleration of the internal mass were considered. The optimal parameters of both modes which lead to a maximum of the average velocity of motion of the main body for a period were found. In the present paper we consider the motion of a straight chain of three equal mass points interconnected with kinematical constraints. The ground contact can be described by dry (discontinuous) or viscous (continuous) friction. The controls are assumed in the form of periodic functions with zero average, shifted on a phase one concerning each other. Thus, there is a travelling wave along the chain of mass points. It is shown that, using special control algorithms motion is possible by isotropic coefficient of friction and by constant normal force. In the case of non-isotropic friction motion is possible in the direction of the greater friction.
2 Equations of Motion We consider the motion of a system of three mass points with the coordinates xi (i = 1, 2, 3) and with the masses m, connected by kinematical constraints along an axis OX(Fig. 1). The motion of the system is excited by the kinematical constraints setting the distances L1 (t) > 0 and L2 (t) > 0 between mass points L1 (t) = L0 + a1 (t) , where a1 (0) = a2 (0) = 0.
L2 (t) = L0 + a2 (t) ,
(1)
Motion of a Chain of Three Point Masses on a Rough Plane
63
Fig. 1 The schematic view of the system of three mass points
Let us consider L1 (t) and L2 (t) (hence as well a1 (t) and a2 (t)) as periodic functions with period T . The kind of functions a1 (t) and a2 (t) will be considered below. There is the force of friction F (Vi ) acting on each mass point from the surface, directed against motion and depending on the velocity Vi = x˙i (i = 1, 2, 3). The law of friction F (Vi ) will be discussed later. The velocity of the center of masses of system can be represented as V=
1 (x˙1 + x˙2 + x˙3) . 3
The equation of the motion of the center of mass is as follows 3mV˙ = F (x˙1 ) + F (x˙2 ) + F (x˙3 ) ,
(2)
x2 (t) − x1 (t) = L1 (t) , x3 (t) − x2 (t) = L2 (t) .
(3)
where
By substituting the expressions (1) and (3) in (2) the equation of the motion (2) takes the form 1 1 2 1 1 2 3mV˙ = F V − a˙1 − a˙2 + F V + a˙1 − a˙2 + F V + a˙1 + a˙2 . (4) 3 3 3 3 3 3 We assume that in the initial moment t = 0 the velocity of the center of mass V (0) = 0. Let us introduce dimensionless variables in according to the following formulas (the asterisk ∗ is a symbol of dimensional variables): T xi = x∗i / L (i = 1, 2, 3) , V = V ∗ , t = t ∗ / T, L ai = a∗i / L, Li = L∗i / L, (i = 1, 2) , F (V ) =
∗
∗
(5)
∗
F (V ) F (V · L / T ) = . Fs Fs
In the above L is the characteristic linear dimension (for example the greatest value a1 (t) or a2 (t) in period T ), Fs is the characteristic value of the friction force.
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Hereafter we use dimensionless variables. Introducing the dimensionless variables in Eq. (4) and denoting u1 (t) = a˙1 (t), u2 (t) = a˙2 (t), we rewrite Eq. (4) in dimensionless variables (the old symbols are hold) dV ε 2 1 1 1 1 2 = F V − u1 − u2 + F V + u1 − u2 + F V + u1 + u2 , dt 3 3 3 3 3 3 3 (6) 2 where ε = Fms TL . Let us notice, that since a1 (t) and a2 (t) are periodic functions with period T , therefore u1 (t) = a˙1 (t) and u2 (t) = a˙2 (t) are also periodic functions with period T and have zero average value. Further, we assume everywhere that ε 1. The smallness of the parameter ε shows that the value of the friction force Fs is small compared to the amplitude of the “driving” force mL / T 2 . Equation (6) has a so called “standard form” [11]. Averaging the right side of the Eq. (6) relative to the variable t in the period 1 yields
ε dV = G(V ), dt 3
(7)
where G (V ) =
1
2 1 1 1 1 2 F V − u1 − u2 + F V + u1 − u2 + F V + u1 + u2 d t. 3 3 3 3 3 3
0
Now it is necessary to define the functions u1 (t), u2 (t) and the law of friction.
3 Smooth Control Let us consider the functions a1 (t) and a2 (t) composed from the parabolas and shown in Fig. 2, and accordingly marked as a solid and as a dashed curve. These functions have continuous derivatives u1 (t) and u2 (t), and they are shown in Fig. 3. Applied controls have the form ⎧ 0, 0 ≤ t ≤ 1 / 3, ⎪ ⎪ ⎪ ⎨ 2(3t − 1), 1 / 3 < t ≤ 1 / 2, u1 (t) = ⎪ −2(3t − 2), 1 / 2 < t ≤ 5 / 6, ⎪ ⎪ ⎩ 6(t − 1), 5 / 6 < t ≤ 1. (8) ⎧ 6t, 0 ≤ t ≤ 1 6, / ⎪ ⎪ ⎪ ⎨−2(3t − 1), 1 6 < t ≤ 1 2, / / u2 (t) = ⎪ 2(3t − 2), 1 2 < t ≤ 2 / / 3, ⎪ ⎪ ⎩ 0, 2 / 3 < t ≤ 1.
Motion of a Chain of Three Point Masses on a Rough Plane
65
a1, a2 0.2
0.15
0.1
0.05
0
t 0
0.5
1
Fig. 2 The function a1 (t)(solid) and a2 (t)(dashed) u1, u2 1
0.5
0 -0.5 -1
t 0
0.5
1
Fig. 3 The function u1 (t) (solid), u2 (t) (dashed)
They are equal to zero on an interval of length 1 / 3 and are shifted on time for this magnitude one relatively to another.
4 Dry (Discontinuous) Friction We assume that the Coulomb dry friction acts on the mass point i (i = 1, 2, 3). The dimension force of dry friction F ∗ (V ) satisfies the Coulomb law ⎧ ⎪ if V < 0 ⎨F− = k− N, F ∗ (V ) = F0 , if V = 0, ⎪ ⎩ −F+ = −k+ N, if V > 0
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where N is the force of normal pressure (in this case N = m g, where g is the free fall acceleration), k− and k+ are the coefficients of dry friction at the motion in a negative and in a positive direction respectively, F− ≤ F+ (k− ≤ k+ ). Let us denote Fa the resultant of all forces applied to the mass point (except the force of dry friction). Now we have ⎧ ⎪ ⎨
F− , i f V = 0 F0 = −Fa , i f V = 0 ⎪ ⎩ −F+ , i f V = 0
and and and
Fa < −F− − F− ≤ Fa ≤ F+ . Fa > F+
The expression for dimensionless friction force takes the form (i = 1, 2, 3) ⎧ ⎪ ⎨
1, x˙i < 0 F (x˙i ) = μ0 , x˙i = 0. ⎪ ⎩ −μ , x˙i > 0
(9)
In the above the value Fs in formulas (5) is Fs = F− (F− is the magnitude of the friction force at the motion in a negative direction), μ = F+ / F− = k+ / k− ≥ 0, μ0 ∈ [−μ , 1] and the expression for μ0 (x˙i = 0) is ⎧ ⎪ ⎨
1, i f Fa < −1 μ0 = −Fa , i f − 1 ≤ Fa ≤ μ . ⎪ ⎩ −μ , i f Fa > μ
(10)
In what follows we prove the so far obtained result assuming that x˙i (t) is a piecewise continuous function of time. This assumption is quite sufficient for simulating a feasible motion. The first and third conditions (10) are satisfied in case x˙i (t) = 0 and x¨i (t) = 0. This conditions hold simultaneously only at isolated points, which does not effect the system motion. The second condition (10) is connected with sticking (“stick-slip” motions). This effect is characteristic for systems with dry friction. Let us notice that for the given control the velocity of each mass point could not be equal to zero on a finite time interval. Hence, the “stick-slip” effect is absent. After substituting the expression (8) and (9) in the Eq. (7) we obtain ⎧ ⎪ ⎪ ⎪ ⎨
3, V ≤ −2 / 3, dV ε 2 − μ − 3V (1 + μ )/2, −2 / 3 < V ≤ 0, = dt 3⎪ 2 − μ − 6V (1 + μ ), 0 < V ≤ 1 / 3, ⎪ ⎪ ⎩ −3μ , V > 1 / 3. We consider the solution of the Eq. (11) with the initial condition V (0) = 0.
(11)
Motion of a Chain of Three Point Masses on a Rough Plane
67
If μ = 2 (friction in the positive direction is twice more than friction in a negative direction) the system remains in rest. If μ < 2 the chain moves to the right with the velocity V=
. 2−μ · 1 − e−2ε (1+μ )t , 6 (1 + μ )
and tends to a stationary value Vs =
2−μ . 6 (1 + μ )
In case of isotropic friction one gets Vs = 1 / 12. Thus, under this control algorithms, motion is possible in the case of isotropic friction and in the case of non-isotropic friction in the direction of the greater friction. If μ > 2 the chain moves to the left with the velocity V=
. ε 2 (2 − μ ) · 1 − e− 2 (1+μ )t , 3 (1 + μ )
and tends to a stationary value Vs =
2 (2 − μ ) . 3 (1 + μ )
In Fig. 4 the results of the numerical integration of the exact and averaged equations in the case of symmetric friction ( μ = 1) and for parameter ε = 0.3 are shown.
V 0.1 0.08 0.06 0.04 0.02 0
0
5
t 10
Fig. 4 Solutions of the exact and the averaged equations for dry friction
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5 Viscosity (Continuous) Friction In this section we assume that the force of viscous friction is a power function of velocity. For the friction law in dimension we assume F(V ) = ν |V |α sgnV . The expression for dimensionless friction force follows F (V ) = |V |α sgnV,
(12)
where ν is the coefficient of viscous friction, and the value Fs in formulas of α transition to dimensionless variables (5) is Fs = ν VL α , α > 0. After substitution expressions (8) and (12) into Eq. (7), one obtains . dV ε 1+α 1+α 1+α = 4 (1 − 3V ) , − (3V + 2) − 3 |3V | dt 2 (1 + α ) 31+α − 2 / 3 < V < 1 / 3. If V ≤ −2 / 3, then dV / dt > 0 and if V ≥ 1 / 3, then dV / dt < 0. Hence the stationary solution is only possible on an interval −2 / 3 < V < 1 / 3. At linear viscous friction (α = 1) the chain on the average remains in rest. For the other α stationary velocity V = 0 can be found from the equation G1 (V ) = 4 (1 − 3V )1+α − (3V + 2)1+α − 3 |3V |1+α = 0.
(13)
Let us investigate roots of Eq. (13). Observe that G1 (−2 / 3) = 4 · 31+α − 3 · 21+α > 0, G1 (0) = 4 − 21+α , G1 (1 / 3) = −3
1+α
(14) − 3 < 0.
From expression (14) follows that G1 (0) < 0 for α > 1 and G1 (0) > 0 for 0 < α < 1. The expression for the first derivative of the function G1 (V ) for V = 0 has the form
dG1 = − (1 + α ) 4 (1 − 3V )α + (3V + 2)α + 3 |3V |α sgnV . dV On the interval 0 < V < 1 / 3 the derivative of the function G1 (V ) is negative and this means that this function monotonically decreases on this interval. Thus the function G1 (V ) has one positive roots for 0 < α < 1 on the interval 0 < V < 1 / 3. For α > 1 the value G1 (0) is negative and the function G1 (V ) has at the ends of the interval values of identical signs. Thus the function G1 (V ) has no positive roots for α > 1 on the interval 0 < V < 1 / 3. For α > 1 at the ends of the interval −2 / 3 < V < 0 the function G1 (V ) has values of different signs. This implies that the function G1 (V ) has at least one negative root for α > 1 on the interval −2 / 3 < V < 0. In Fig. 5 the results of the numerical integration of the exact and the averaged equations for the case of viscous friction with α = 2 and ε = 0.3 are shown.
Motion of a Chain of Three Point Masses on a Rough Plane
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V 0
-0.02
-0.04
-0.06
t 0
10
20
30
Fig. 5 Solutions of the exact and the averaged equations for viscous friction
Fig. 6 The prototype of the system of three mass points
6 Conclusions It is shown that using periodical control algorithms, motion is possible in the case of isotropic friction and in the case of non-isotropic friction in the direction of the greater friction. Without a shift of the phases in the control law and with a linear friction model, the locomotion is impossible. In the case of small friction we derived a condition for the locomotion of the center of the mass with the help of an average method. In the case of smooth control we received explicit expressions for the average velocity of the motion of the center of mass. Comparisons of these analytical expressions to numerical results are carried out. A prototype of this system was created (Fig. 6). Acknowledgement This paper is supported by Deutsche Forschungsgemeinschaft (DFG, ZI 540/6-1).
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References 1. Zimmermann K, Zeidis I, Steigenberger J (2002) Mathematical model of worm-like motion systems with finite and infinite degree of freedom, Theory and Practice of Robots and Manipulators, Proceedings of 14-th CISM-IFToMM Symposium. Springer, Wien, NY, 507–516. 2. Zimmermann K, Zeidis I, Steigenberger J, Pivovarov M (2004) An approach to wormlike motion, 21st International Congress of Theoretical and Applied Mechanics, Book of Abstracts. Warsaw, Poland, 371. 3. Zimmermann K, Zeidis I, Naletova VA, Turkov VA (2004) Modelling of worm-like motion systems with magneto-elastic elements, Phys. Status Solidi (c) 1, 3706–3709. 4. Zimmermann K, Zeidis I (2007) Worm-like locomotion as a problem of nonlinear dynamics, Journal of Theoretical and Applied Mechanics, 45, 179–187. 5. Zimmermann K, Zeidis I, Pivovarov M, Abaza K (2007) Forced nonlinear oscillator with nonsymmetric dry friction, Arch. Appl. Mech. 77, 353–362. 6. Miller G (1988) The motion dynamics of snakes and worms, Comput. Graph. 22, 169–173. 7. Steigenberger J (1999) On a Class of Biomorphic Motion Systems. Faculty of Mathematics and Natural Sciences, Technische Universitaet Ilmenau, Germany, preprint 12. 8. Bolotnik NN, Zeidis I, Zimmermann K, Yatsun SF (2006) Dynamics of controlled motion of vibration-driven systems, J. Comput. Sys. Sci. Int. 45, 831–840. 9. Bolotnik NN, Pivovarov M, Zeidis I, Zimmermann K, Yatsun SF (2007) Motion of vibrationdriven mechanical systems along a straight line, 14th International Workshop on Dynamics & Control, Moscow-Zvenigorod, Russia, 19. 10. Chernousko FL (2005) On a motion of a body containing a movable internal mass, Dokl. Akad. Nauk 405, 1–5. 11. Bogolyubov NN, Mitropolskii YuA (1961) Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York.
Compensation of Geometric Errors in a PKM Machine Tool Christian Rudolf and J¨org Wauer
1 Introduction In machine tools of parallel structure with two or three translatory degrees of freedom the rotatory degree of freedom is kinematically locked. Yet due to geometric faults, for example assembly errors or different geometries due to production tolerances, such machine tools exhibit an additional rotational behavior. Stresses within the structure occur leading to deflections of the tool center point, and thus, reducing the quality of the workpiece. For compensating these errors an adaptronic strut which can be implemented within such a machine tool has been developed. The strut comprises a piezoceramic sensor-actuator unit for controlled correction of those static and quasi-static deflections. Piezoceramic elements were chosen due to their high positioning accuracy and the small installation space required. The functional principle of a scale with a vibrating string is used for measuring the external load. A control concept for the adaptronic strut is introduced. Finally, after implementing the strut in a model of an exemplary machine tool the compensation of influences due to specified geometric errors is examined. Figure 1 shows a machine tool with parallel kinematics of three translatory degrees of freedom. Due to geometric errors, such as assembly errors or differing geometries due to production tolerances, stresses within the structure occur resulting in deflections of the tool center point (TCP) of the machine tool. Thus, the quality of the workpiece is reduced. An adaptronic strut as shown in Fig. 2 has been developed for compensating such errors. The strut, similar in shape to conventional struts in machine tools, is cut in two halves and a piezoelectric sensor-actuator unit is implemented in-between, giving the strut an additional degree of freedom. The geometric deflections in focus of this contribution are mostly static or quasistatic, and thus, only inducing static or quasi-static signals on the piezoelectric C. Rudolf and J. Wauer Universit¨at Karlsruhe (TH), Institut f¨ur Technische Mechanik Kaiserstraße 10/Wilhelm-NusseltWeg 4, 76131 Karlsruhe, Germany e-mail:
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
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struts pillars
headstock
Fig. 1 Parallel kinematics machine tool with three translational degrees of freedom [1]
upper half piezoelectric transducer
lower half
Fig. 2 Adaptronic strut [2]
sensor element. However, due to the internal leakage resistance of piezoceramic materials such signals are not measurable [3, 4]. By adapting the functional principle of a scale with a vibrating string a work-around for this problem was found. A string, which was mounted along the strut, as can be seen in Fig. 2, is excited by a solenoid. A dynamic signal is induced on the strut and onto the piezoelectric sensor. This signal can easily be acquired and, using frequency counters or phase-locked loops, its frequency course can be determined [5]. The equation f0 =
1 2lS
T Aρ
(1)
describes the relation between the eigenfrequency f0 of the string and pre-stress T on the string, with length lS, cross-sectional area A and density ρ of the string. Thus, by measuring the frequency the external load on the strut can be determined. Further information about this functional principle can be found in [2, 5, 6].
Compensation of Geometric Errors in a PKM Machine Tool x1
c2
c1 m1
d2
d1
73
x2
x3 c3
m2 a
m3
F
d3
Fig. 3 Three-body-oscillator as lumped mass approach for modeling the adaptronic strut
2 Control Concept for Adaptronic Strut The simplest mechanical model of the adaptronic shown in Fig. 2 is a three-body oscillator as illustrated in Fig. 3. The upper and lower halves of the strut are approached by lumped masses m1 and m3 , respectively, the piezoelectric element in-between is represented by m2 . The springs ci and dampers di represent corresponding material properties, the force F represents external influences such as constraint forces on the strut. With system matrix A, control vector b and disturbance vector bs , the equations of motion for the three-body oscillator in state space form read d z = Az + bu + bSuS dt
(2)
where state vector z represents positions and velocities of the lumped masses. The controller force of the actuator between bodies 2 and 3 is u = Fr and the external disturbance force is us = F. Using the principle of Least Quadratic Regulator (LQR) the parameters for a state controller for this single variable system can be determined. The controller force Fr becomes Fr = −rT z (3) with state vector z. The control vector r is chosen such that the quadratic cost functional ∞ 1 1 J= zT (t)Qz(t) + Fr2 dt (4) 2 κ 0
is minimized. The scalar κ > 0 is a value for the cost of the controller input whereas Q is a positive, semi-definite matrix weighting the system state. For more information on determining the controller parameters using LQR, see [7–11]. For realizing the state controller described above the system state must be known. However, since not all system states are measured an additional element must be introduced. The so-called Luenberger observer estimates the system states according to ∧ ∧ d ∧ z = A z +bu + l y −y . (5) dt
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+ -
C
G H
Fig. 4 Closed-loop system with position feedback with prefilter F = 1, integrator H(s) = 1/s, plant G with state controller and compensation element C [1]
Fig. 5 Simple model of adaptronic strut with flexible bodies [1]
Fig. 6 Model of adaptronic strut by use of CAD data [4]
∧
Here in, y is the measured system state, whereas y is the estimation of this system state by the observer. The observer matrix l is chosen such that the eigenvalues of the observed system are further on the left of the imaginary axis than the eigenvalues of the controlled system. This guarantees the observer to be faster than the controller. Since, according to F¨ollinger [7], the two sets of eigenvalues can be set independently, that can easily be achieved using the pole placement procedure according to Ackermann [8]. However, this control concept proves to be not sufficient. When the statecontrolled system experiences a constant external disturbance force F a steady-state error remains [1]. Therefore, the control concept is extended according to the block diagram depicted in Fig. 4. The controller additionally includes an integrating element H in the feedback path as well as a constant compensator element C. The value for C is determined using the root locus procedure such that the enhanced controlled system is endued with dominating zeroes far away from the imaginary axis and a large stability margin according to the Nyquist criterion [1]. As the simulation results presented in [1, 4] have shown the formerly remaining steady-state error vanishes when the extended control structure is used. This applies for all examined systems, the three-body oscillator shown in Fig. 3 as well as for the flexible multi-body systems shown in Figs. 5 and 6. In the following the influence of this strut when implemented into the machine tool with parallel kinematics shown in Fig. 1 is examined.
3 Implementation into PKM Machine Tool For the examination, the settings shown in Fig. 7 are used. The struts are numbered from 1 to 6 starting from the bottom left when seen from the top, the inertial reference frame is given by axes x and y.
Compensation of Geometric Errors in a PKM Machine Tool
75
y
3
4
2
5 x 1
6
Fig. 7 Inertial reference frame and numbering of struts within machine tool
The strut at position 3 is exchanged for a strut of variable length as described above. Starting from an otherwise ideal system one specified kind of geometric error is introduced into the machine tool at two different positions. The application point of the upper joint of one of the struts is shifted by 15e-6m along the axis of this strut, pointing away from the origin of the introduced inertial reference frame. First, this geometric error is placed on the application point on strut 4, then its influence when on strut 1 is examined. Within the simulations of this contribution, the headstock of the machine tool is at rest and no machining processes are taken into account. The results are shown in Figs. 8–10. The dashed lines represent the static deflections of the tool center point of the machine tool due to the examined error whereas the solid lines show the results when the piezoelectric element in the strut is actuated for compensating the influence of the error. When being on strut 4 the geometric error leads to translational deflections of the TCP of order 1e-8m in x- and of order 1e-5m in y-direction, as depicted in Fig. 8a and b, respectively. The influence of the deflection in x-direction is negligible whereas the deflection in y-direction might result in a reduction of quality of the workpiece during the machining process. Therefore, its compensation is required. For a chosen set value of length of strut 3 for the controller the deflection of the TCP in y-direction can be reduced close to zero, as depicted in Fig. 8b. This actuation has no influence on the x-deflection of the TCP. However, when regarding the rotational deflections of the TCP about these two axes, the influence of the geometrical error even increases when the compensating control unit is used. Looking at the magnitude of deflection, the rotation about the xaxis (1e-7 rad) is negligible. About the y-axis it is in the order of 1e-4 rad. Depending on the machining process this still might be negligible, e.g. for turning of short shafts where there is solely point contact between tool and workpiece. For milling processes, however, this deflection is not acceptable and a different approach should be tried. For completeness, Fig. 9 depicts the translational (a) and angular (b) deflections of the TCP along and about the z-axis. Due to the structure of the examined machine tool these deflections are easily taken into account. By adding an identical offset onto all three driving functions of the guiding skids equal to the occurring deflection
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y−deflection TCP [m]
x−deflection TCP [m]
10-8 2
0
−2
0
1 time [s]
0.5 0 −0.5 −1
2
x-deflection of TCP
a
1
b
10−7
0
1 time [s]
2
y-deflection of TCP
−0.5
10−4
0
yy [rad]
yx [rad]
2
−2
−1
−4 0
c
1 time [s]
angular deflection of TCP about global x-axis
−1.5
2
d
0
1 time [s]
2
angular deflection of TCP about global y-axis
Fig. 8 Influence of geometric error on strut 4 on position (a)/(b) and orientation (c)/(d) of the TCP with and without compensation by variable strut on 3
in z-direction, the translational deflection along z can be compensated. The rotational deflection about the z-axis is negligible since is identical to the rotation axis of the spindle. As shown in Fig. 10a–d, similar results occur when the same geometric error occurs in the application point of the upper joint on strut 1 with the machine tool being otherwise ideal. The compensating strut of variable length is again placed on position 3. Now, the geometric error leads to considerable deflections of the TCP in both, xand y-direction. Using the compensation unit of strut 3 the deflection in y-direction can be reduced to zero, shown in Fig. 10b, whereas the deflection in x-direction is mostly unaffected, as illustrated in Fig. 10a. The same applies for the corresponding rotational deflections about x- and y-axis, Fig. 10c and d, respectively. Again, deflections of the TCP along and about the z-axis do not have to be taken into account due to the structure of the machine tool.
Compensation of Geometric Errors in a PKM Machine Tool
2
10-5
8
0
6 4
-2 0
a
10
yz [rad]
z−deflection TCP [m]
10-6
77
1 time [s]
z-deflection of TCP
2
0
b
1 time [s]
2
angular deflection of TCP about global y-axis
Fig. 9 Influence of geometric error on strut 4 on vertical position (a) and orientation (b) of the TCP with and without compensation by variable strut on 3
4 Discussion In both cases examined it was not possible to reduce all deflections of the TCP which were induced by the implemented geometric error. Either one of the deflections was not affected by the actuation of the compensation unit or it even increased when the other one was reduced. Thus, regarding the symmetry of the machine tool under examination, implementing solely one of the presented controlled adaptronic struts into the machine tool is, in general, not sufficient for compensating occurring geometric errors. However, there are exemptions for special cases of geometric errors in combination with position an in dependence of the machining process when one compensation unit is sufficient. In turning processes, for example, there is only point contact between tool and workpiece. Thus, small angular deflections are not significant. Furthermore, assuming one of the struts, e.g. at position 5, to be of different length this can be taken into account by substituting strut 6 by a strut of variable length. Making this pair of struts equal in length and adjusting the driving function of the corresponding guiding skid by an additional offset, the ideal structure of the machine tool can be regained.
5 Conclusions An adaptronic strut for compensating static and quasi-static errors in machine tools with parallel kinematics has been presented. The functional principle of a scale with vibrating string was used for measuring the occurring loads with piezoelectric elements. A control concept for the developed strut was presented. The strut
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C. Rudolf, J. Wauer 10−6
10
y−deflection TCP [m]
x−deflection TCP [m]
-8.18
-8.19
-8.2
-8.21
a
0
0.5 time [s]
-4.96
5
0
-5
1
b
x-deflection of TCP
10-6
0
0.5 time [s]
1
y-deflection of TCP
10-5
10-5 2
yy [rad]
yx [rad]
-4.98 -5 -5.02 -5.04
c
0
-2
0
0.5 time [s]
1
angular deflection of TCP about global x-axis
0
0.5 time [s]
1
angular deflection of TCP about global y-axis
Fig. 10 Influence of deflection of application point of upper joint on strut 1 on the position and orientation of TCP with and without compensation
was implemented into a machine tool with parallel kinematics. Starting from an initially ideal machine tool structure, one of the original struts was substituted by the controlled adaptronic strut. An offset of an application point of a joint connecting the strut with the corresponding guiding skid was implemented into the model of the machine tool, representing a geometric error and resulting in a translational and a rotational deflection of the tool center point. Using the controlled strut, a reduction of this erroneous structure was aspired. However, while compensating deflections along or about a specified axis, deflections along and about perpendicular axes either increased or were not affected by the compensation. Thus, further studies have to be conducted regarding the substitution of one or two more controlled struts of variable length. Additionally, motions of the headstock and, thus, changing positions of the TCP in workspace as well as typical machining processes like turning or milling shall be included in future examinations.
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79
Acknowledgement Financial support of this research in the frame of the Priority Program No. 1156 “Adaptronik in Werkzeugmaschinen” by the German Research Foundation (DFG) is gratefully acknowledged.
References 1. Rudolf C, Wauer J, Munzinger C, Weis M (2007) Piezoelectric control of a machine tool with parallel kinematics. In: Proceedings of SPIE – Industrial and Commercial Applications of Smart Structures Technologies, 65270G. 2. Rudolf C, Wauer J, Fleischer J, Munzinger C (2005) An approach for compensation of geometric faults in machine tools. In: Proceedings of IDETC/CIE Conference, ASME (DETC 2005–84241). 3. Bill B (2002) Messen mit Kristallen: Grundlagen und Anwendungen der piezoelektrischen Messtechnik, Verlag Moderne Industrie, Landsberg/Lech. 4. Rudolf C, Wauer J, Martin T (2007) Piezoelectric control of a machine tool with parallel kinematics. In: Proceedings of III Eccomas Thematic Conference on Smart Structures and Materials. 5. Rudolf C, Wauer J, Fleischer J, Munzinger C (2006) Measuring static and slowly changing loads using piezoelectric sensors. In: Borgmann H (ed) Actuator 2006 Conference Proceedings, 540–543, HVG, Bremen. 6. Fleischer J, Kn¨odel A, Munzinger C, Weis M (2006) Designing adaptronical components for compensation of static and quasi-static loads. In: Proceedings of IDETC/CIE Conference, ASME (DETC 2006–99461). 7. F¨ollinger O (1994) Regelungstechnik, 8th ed., H¨uthig, Heidelberg. 8. Lunze J (2004) Regelungstechnik 2, 3rd ed., Springer, Heidelberg. 9. Preumont A (2002) Vibration Control of Active Structures – An Introduction, 2nd ed., Kluwer, Dordrecht. 10. D’Azzo JJ, Houpis CH, Sheldon SN (2003) Linear Control System Analysis and Design with Matlab, 5th ed., Taylor & Francis, New York. 11. Lunze J (2003) Regelungstechnik 1, 4th ed., Springer, Heidelberg.
Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics Ladislav Pust ˚ and Jan Koz´anek
1 Introduction Rotor motion in rotating machines supported on oil or gas bearings has been studied in many papers and many theoretical solutions have been published in a number of books, proceedings and journal articles (e.g. [1–5]). During the last decades, great attention has been given to nonlinearities in the entire rotor systems, particularly to the most common sources of nonlinearities in oil-, and also in gas-lubricated bearings. The problem of tilting pad bearings was also solved, but mainly in orientation to oil bearings. One of the first attempts of analytical approximations of stiffness and damping nonlinear characteristics was done in [4], examples of other methods for non-linear dynamic analysis are presented in [5–10]. New trends in developing of fluid dynamics bearings are in the last years focused on elaboration of more exact investigation of dynamic properties of gas and tilting pad bearings, both pressurized and self-acting (aerodynamic), e.g. [6, 11]. In spite of this, there are still many unsolved problems particularly in the field of new bearings, e.g. tilting pad aerodynamic bearings, for which new methods of solution must be elaborated. These bearings work on the aerodynamic principle, where the air at high revolutions is automatically sucked into the bearing clearance and there it forms the supporting layer. They are simpler than classical oil- or airbearings, which need additional equipment of oil or pressed air supply. The properties of aerodynamic bearings with three tilting pads were numerically solved and described by linear matrix expressions in Techlab Ltd., Prague, but only for several loads and at discrete revolutions. Due to the linear form these characteristic matrices can be used for solution of motion only in a small surrounding of stationary equilibrium state. L. P˚ust and J. Koz´anek Institute of Thermomechanics, Academy of Sciences of Czech Republic, Prague, 18200 Dolejskova 5, e-mail:
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
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In presented contribution, an approximate method for calculation of rigid rotor motion supported on two tilting pad aerodynamic bearings oscillating either with small or with large amplitudes limited only by bearing clearance rh is presented. Dynamic characteristics of such type of bearings are very strongly influenced by inertia properties of tilting pads – the stiffness and dynamic matrices are nonsymmetric and their elements are non-monotonous functions of angular frequency ω . The approximate models of stiffness and damping forces – valid for the entire area of journal motion in the aerodynamic bearing’s clearance – are based on the transformation of linear stiffness and damping characteristics given in two positions of rotor journal into the entire area. The assumption that rheological properties of three-tilting pads bearing are centrally symmetric is also used. Space motion of unbalanced rotor and mutual interaction of two identical aerodynamic bearings is then derived for small oscillations near the equilibrium position. The second part of the article is given to the elaboration of this linear dynamic model to the system modeling the nonlinearity of aerodynamic tilting pad bearings. Several examples show the existence of different types of vibrations forced by rotating unbalance and loaded also by various vertical forces.
2 Motivation of Research The application of air as lubricants in aerodynamic bearings is a modern trend in rotating machinery particularly in chemical and food industry, as it excludes the undesirable damage of products by oil. The main advantage of these bearings is very simple aerodynamic principle, which does not need any auxiliary apparatus (pressed air, oil pumps, etc.) and which is suitable for very high revolutions e.g. for high-speed compressors, working for long time periods with constant revolutions. Therefore several types of aerodynamic bearings were designed and realized in the Institute of Thermomechanics of AS CR in cooperation with CKD – New Energo Ltd. and applied in new rotor structures. An example of this application is the laboratory prototype of rotor supported on two aerodynamic tilting pad bearings shown in Fig. 1.
Fig. 1 Laboratory prototype
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This experimental set is based on high-frequency synchronous motor running up to 50,000 rpm. Its rigid rotor is supported in two identical radial aerodynamic bearings (diameter 50 mm, clearance rh = 0.05 mm) with three tilting pads in each bearing. The tilting pads have antifriction layers for safety running up and down. The greater machinery – turbo compressor equipped with aerodynamic bearings (120 mm, 100 kW, 18,000 rpm) is developed in CKD – New Energo Ltd. Theoretical background of this experimental research is provided in the project GACR No.101/06/1787 where the analytical and numerical investigation of simplified mathematical models of rotor excited by centrifugal force from the unbalanced rotor is solved.
3 Mathematical Model of the Rotor Motion at Small Oscillations Schema of experimental rotor IT ASCR is shown in Fig. 2. The rigid rotor is symmetric to the axis of revolution, but slightly asymmetric to the mid-span plane. Its mass is m = 7.6 kg and central inertia moment to the axes x and y is I = 0.10024 kgm2 . Distance between the centres of bearings is l = 0.32 m. Inertia properties defined by mass m and moment of inertia I can be replaced by effects of three masses [12], two of them m1 , m2 in the centres of bearings, the third mass m3 in the mid-span of length l. The centrifugal force meω 2 and the centre of gravity mg are shifted in distances a, b to the right from this mid-span point. The properties of aerodynamics tilting pad bearings (diameter 50 mm) were ascertained by numerical solution in the form of evolutive stiffness K(ω ) and damping B(ω ) matrices calculated in TECHLAB Ltd. For discrete values of revolutions in steps of 2,500 rpm (approx. 250 1/s) up to 50,000 rpm and for different values of loading from F = 0 up to F = 70 N. These values vary very strongly at different revolutions. There are no monotone functions, because the inertia of tilting pads causes resonance phenomena, superimposed on monotone increase or decrease at variation of revolutions. In Fig. 3, there are examples of stiffness and damping properties for unloaded journal (F = 0), when its position is in the centre of bearing. The main stiffness kxx is 1 a kyy m1
m3
m,I
y
m2 z
b x bxx
kxx
Fig. 2 Scheme of rotor IT AS CR
mg mew2 bxx
bxx kxx
x kxx
byy
STIFFNESS K [N/mm]
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L. P˚ust, J. Koz´anek STIFFNESS, LOAD=0
15000 10000 kxx=kyy
5000
kxy
0 −5000
kyx 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
4000
4500
5000
DAMPING B [Ns/m]
ANGULAR VELOCITY ω [1/s] DAMPING, LOAD=0
6000 bxx=byy
4000
bxy 2000 byx
0 −2000
0
500
1000
1500
2000
2500
3000
3500
ANGULAR VELOCITY ω [1/s]
Fig. 3 Stiffness and damping coefficients of unloaded journal (F = 0)
the same as kyy , the cross stiffness elements are of opposite sign: kyx = −kxy ; matrix K(ω ) is anti-symmetric. The same anti-symmetric properties has also damping matrix B(ω ), which elements are plotted in the bottom part of Fig. 3. The anti-symmetric properties are lost, when the position of the journal is shifted into another equilibrium position due to vertical load, e.g. F = 40 N. The properties of both stiffness and damping matrices change. It is seen from Fig. 4, where the curves kxx , kyy differ each from the other as well as damping curves bxx (ω ) and byy (ω ). The cross stiffness kxy (ω ), kyx (ω ) and cross damping bxy (ω ), byx (ω ) are not yet of opposite sign but they are of general form. Due to the increase of loading from 0 to 40 N, the peaks of curves caused by tilting pads inertia and aerodynamic forces shift from ω ∼ = 1,700 1/s to higher velocities ω ∼ = 2,600 1/s. Knowledge of stiffness matrices K(ω ) and B(ω ) suffices for ascertaining of rotor motion at small oscillations around the equilibrium position. This motion is described by linear differential equation in the form of two matrix equations [7, 8]: ¨ 1 + M3 X ¨ 2 + B1 X ˙ 1 + K1 X1 = meω 2 (1/2 − a/l) O + F, M1 X ¨ 2 + M3 X ¨ 1 + B2 X ˙ 2 + K2 X2 = meω 2 (1/2 + a/l) O + F, M2 X where:
(1)
DAMPING B [Ns/m]
STIFFNESS K [N/mm]
Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 10−4
85
STIFFNESS, LOAD=40N
2 kyy
1 0
kyx
−1
8000 6000 4000 2000 0 −2000 −4000
kxx
kxy
0
500
1000
1500 2000 2500 3000 3500 ANGULAR VELOCITY ω [1/s]
4000
4500
5000
4000
4500
5000
DAMPING, LOAD=40N bxx byy byx 0
500
1000
bxy
1500 2000 2500 3000 3500 ANGULAR VELOCITY ω [1/s]
Fig. 4 Stiffness and damping coefficients at vertical load (F = 40 N)
mi + m3 /4 m3 /4 , i = 1, 2, m3 /4 mi + m3 /4 m3 /4 0 M3 = 0 m3 /4
Mi =
(2)
Xi = [xi , yi ]T i = 1, 2 O = [cos ω t, sin ω t]T , F = [40, 0]T , kxxi (ω ) kxyi (ω ) bxxi (ω ) bxyi (ω ) , Bi = . Ki = kyxi (ω ) kyyi (ω ) byxi (ω ) byyi (ω )
(3) (4)
Indexes 1, 2 express the values belonging to the different journal equilibrium positions in bearings 1, 2. Multiplying Eqs. (1) by suitable combinations of mass ¨ 1 in the first matrices (2) we get the equations with separate second derivatives X ¨ 2 . SimEq. (1). By means of similar way we get the separate second derivatives X pler expressions are then reached by left multiplying these equations with inverse matrix (M1 M2 − M3 2 )−1 . Derived equations can be used for calculation of small oscillations near equilibrium positions, i.e. for small unbalance e, where the linear stiffness and damping matrices are acceptable. Examples for b = 0 and a = 0; 0.2, are shown in Figs. 5 and 6. Example of oscillations of symmetrical rotor (b = 0, m1 = m2 ) at excitation caused by the unbalance e = 10μ m in the mid-span (a = 0) is shown in Figs. 5a and b. The exact symmetry of rotor and of excitation produces the same motion trajectories in both bearings; as it is seen from time history of motion shown in Fig. 5a. The polar trajectories x1 , y1 and x2 , y2 overlap each other as seen in Fig. 5a. The trajectories in state planes are ellipses, which are influenced by linear properties of bearing characteristics.
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TIME HISTORY OF MOTION x, y, v, vy, F
10-4
MOTION x, y, v, vy, F
1 v1==dx1/dt=v2=dv2/dt,
vy1=dy1/dt=vy2=dy2/dt
0.5
0 x1=x2,
y1=y2
-0.5 -1 F(t) -1.5
0
0.002
0.004
0.006
0.008
0.01 0.012 TIME t [s]
0.014
0.016
0.018
0.02
Fig. 5a Time history of motion
5
10 - 5
JOURNAL MOTION x - y e=35e-6 [m]
4
DISPLACEMENT x [mm]
3 rh
2 1
y
0 -1 x1,y1=x2,y2
-2 -3 -4 -5 -6
x
-4
-2 0 2 DISPLACEMENT y [mm]
4
6 10 -5
Fig. 5b Motion trajectory
Excitation in another point different from the geometric centre changes the form of oscillations. Influence of small shift of unbalance position from the centre (a = 0.04) is shown in Fig. 6a where the time history of stationary oscillations in both bearings x1 (t), y1 (t) (second record) and x2 (t), y2 (t) (forth record) are plotted. Differences are of course also in velocity records first record for bearing 1, third one for bearing 2. Asymmetry of excitation causes different amplitudes and different types of oscillations in both bearings. These properties are markedly seen in x − y plane trajectories (Fig. 6b), where x1 − y1 trajectory is approximately straight line, x2 − y2 trajectory is ellipse.
Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 1.5
TIME HISTORY OF MOTION x, y, v, vy, F
10-4
1 MOTION x, y, v, vy, F
87
v1=dx1/dt, vy1=dy1/dt)
0.5
x1y1
0 v2=dx2/dt, vy2=dy2/dt) -0.5 x2y2
-1 -1.5
F(t) 0
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 TIME t [s]
0.02
Fig. 6a Time histories of x, y, v, vy and F 10 -5 JOURNAL MOTION x1,y1 and x2,y2, e=40e-6 [m] 5 4
DISPLACEMENT [mm]
3 2 x2,y2
1
y
0 x1,y1
-1 -2 -3 -4 -5
x
-6
-4
-2 0 2 DISPLACEMENT y [mm]
4
6 10 -5
Fig. 6b Planes: (x1 , y1 ), (x2 , y2 ) and (x, y)
4 Rotor Motion at Large Oscillations The method used in previous chapter can be applied due to the linearity of differential equations (1) only for small displacements from the equilibrium position [8,10]. For larger excitations and oscillations, the linear stiffness and damping matrices are not sufficient and therefore a nonlinear form of motion equations has to be derived. No suitable exact mathematical model of tilting pad aerodynamic bearing properties exists as well as appropriate algorithms of calculation. Therefore only approximate methods can be applied for ascertaining large oscillations. The first attempt of extension of force and damping properties on the entire bearing field was done in [7, 8], where the hyperbolic correction function was proposed and applied for solution. This correction function gives zero value in the centre of bearing in order to ensure continuity in this point. The more detailed calculation of stiffness and damping at various loading and various equilibrium positions shows that in the central bearing position at zero loading the stiffness matrix has
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nonzero elements, but it is anti-symmetric (see Fig. 3) and that the application of only one hyperbolic correction function is insufficient. Therefore the more exact correction has to be realized by expression consisting of two parts [9, 10], one of them fcor1 (r)K(ω )o describing properties in the centre of bearing (r = 0) i.e. at zero loading (F = 0), the second part fcor2 (r)K(ω )ro depends on stiffness matrix K(ω ) in the equilibrium point r0 at given loading (e.g. F = 40 N). The first correction function is monotone decreasing from value 1 in the bearing centre fcor1 (r) = (1 − r/rh)2 , (5a) where rh is bearing clearance. The second correction function is selected in the form of hyperbolic function fcor2 = s/(p + rh − r), (5b) where parameters s, p, h are suitably ascertained. This second correction function enables to express well the increase of restoring force at large displacements from the centre of bearing near the contact of journal with bush. More detailed derivation of these correction functions is in [9], where a model of rubbing is also mentioned. The resulting stiffness K(ri , ω ) and damping B(ri , ω ) matrices are calculated either according to the formulae K(ri , ω ) = fcor1 (ri )K(ω )o + fcor2 (ri )K(ω )ro , i = 1, 2, B(ri , ω ) = fcor1 (ri )B(ω )o + fcor2 (ri )B(ω )ro ,
(5c) (5d)
or separately with different correction functions for single elements of stiffness and damping matrices. This latter individual method enables to fit mathematical model better to the experimentally gained data. In this contribution, the transformations (5c) and (5d) of the whole matrices K(ri , ω ), B(ri , ω ) are used. But this r-transformation is not enough. To describe bearing’s properties in the entire plane area of clearance, it is needed to extend the stiffness and damping description from the clearance vertical radius 0 ≤ r < rh onto general position given by the angle ϕ ∈ (0, 2π). This can be realized multiplying stiffness and damping matrices by plane rotation matrix cos(ϕ ) sin(ϕ ) x y /r, r = x2 + y2 . C= , or C = (6) −y x − sin(ϕ ) cos(ϕ ) The motion of rotor journal in the aerodynamic bearing is given in a concise form by ¨ + CT (x, y)K (ri , ω ) C(x, y)X + MX (7) ˙ = O (ω t) meω 2 + F(t), + CT (x, y)B (ri , ω ) C(x, y)X where the matrices and vectors M,
X,
C(x, y),
fcor (r),
K (ri , ω ) ,
B (ri , ω ) ,
must be defined for each bearing (i = 1, 2) individually.
O (ω t) ,
F(t)
Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics
89
5 Examples The Eq. (7) enables to investigate the properties of mathematical model of aerodynamic bearing at various conditions. The next all records are for eccentricity e = 80e-6 m and for the mass distribution (in kg) given by the matrix M = [2.875 0.925; 0.925 2.875] corresponding to the experimental rotor IT AS CR (Figs. 1, 2). If the excitation of symmetrical rotor acts in its mid-span (a/l = 0), both journals vibrate in the same way. Plane trajectories x1 , y1 and x2 , y2 are identical and draw the same curve, see Fig. 7, again for ω = 3, 000 1/s. The vibration is identical in both bearings and periodic with nearly harmonic components x1 (t) = x2 (t) and y1 (t) = y2 (t). Central nonlinearity causes distortions from elliptic trajectory. Area limited by a circle with radius rh depicts the possible position of journal centre. Small shift of exciting unbalance from the mid-span of rotor to the right bearing (a/l = 0.1) causes different vibration of journal 1 and 2. Corresponding plane trajectories x1 , y1 and x2 , y2 differ each from other (see Fig. 8). Increase of eccentricity shift to a/l = 0.2 results in greater differences between trajectories of both journals. Figure 9 shows the transition of x1 , y1 trajectory into slender ellipse; trajectory x2 , y2 of right journal changes its form only in a low degree. The same feature has x2 , y2 trajectory (Fig. 10) also at rising shift of eccentricity position to a/l = 0.3. However the journal trajectory x1 , y1 in the left bearing changes its form markedly into a nearly straight line with opposite inclination in comparison with case a/l = 0.2 (Fig. 9). Small decrease of both trajectories against the centre of bearing (x = 0, y. = 0) is caused by a small static force Fstat = 40 N.
5
JOURNAL MOTION x - y
10 -5
4 rh
DISPLACEMENT x1,x2 [mm]
3 2 1
y
0 -1 x1,y1 = x2,y2
-2 -3 -4 -5
x
-6
-4
Fig. 7 Plane trajectories at a/l = 0
-2 1 2 DISPLACEMENT y1, y2 [mm]
4
6 10 -5
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L. P˚ust, J. Koz´anek 5
JOURNAL MOTION x - y
10-5
4
a/l = 0.1
rh
DISPLACEMENT x1,x2 [mm]
3 2 1 y
0 x1,y1
-1
x2,y2
-2 -3 -4 -5
-6
-4
x -2 0 2 DISPLACEMENT y1,y2 [mm]
4
6 10-5
Fig. 8 Plane trajectories at a/l = 0.1 5
JOURNAL MOTION x - y
10-5
4
a/l = 0.2
rh
DISPLACEMENT x1,x2 [mm]
3 2
x2,y2
1 0
y
-1
x1,y1
-2 -3 -4 -5
x -6
-4
-2 0 2 DISPLACEMENT y1, y2 [mm]
4
6 10
-5
Fig. 9 Plane trajectories at a/l = 0.2
Let us now see the influence of static load of the individual journals. This load is the same on both journals. The five-times higher static force Fstat = 200 N in comparison with the previous case, results in a great lowering of journal trajectory x1 , y1 and in very imperceptible increasing of its width (Fig. 11a). The second trajectory x2 , y2 comes down only very little and also its deformation does not change. All following records are done with shift of eccentricity position to a/l = 0.3.
Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 5
91
JOURNAL MOTION x - y
10-5
4
a/l = 0.3
3
Fstat = 38 N
DISPLACEMENT x1,x2 [mm]
rh 2 x2,y2
1
y
0 x1,y1
-1 -2 -3 -4 -5
x -6
-4
-2 0 2 DISPLACEMENT y1, y2 [mm]
4
6 -5 10
Fig. 10 Plane trajectories at a/l = 0.3
DISPLACEMENT x1,x2 [mm]
5
JOURNAL MOTION x - y
10-5
4
Fstat = 200 N
3
a/l = 0.3
rh
2
x2,y2
1 y
0 -1
x1,y1
-2 -3 -4 -5
x -6
-4
-2
0
2
DISPLACEMENT y1, y2 [mm]
4
6 -5 10
Fig. 11a Plane trajectories at Fstat = 200 N
The further increase of static load to Fstat = 400 N changes the form of both trajectories much more, as it is shown in Fig. 11b. The width and also the shift down of trajectory x1 , y1 increases, its length decreases. Trajectory x2 , y2 of the right journal changes its form in a small, but recognisable measure. If the vertical loads on journals increase to Fstat = 700 N, the both plane trajectories of motion change their forms, as shown in Fig. 11c. The upper part of trajectory x2 , y2 shifts down, journal trajectory x1 , y1 shifts only a little, but it changes its form into the drop-shaped one.
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DISPLACEMENT x1,x2 [mm]
5
JOURNAL MOTION x - y
10-5
4
Fstat = 400 N
3
a/l = 0.3
rh
2
x2,y2
1 y
0 -1 -2
x1,y1
-3 -4 -5
x -6
-4
-2
0
2
4
6
DISPLACEMENT y1, y2 [mm]
10
-5
Fig. 11b Plane trajectories, Fstat = 400 N
DISPLACEMENT x1,x2 [mm]
5
JOURNAL MOTION x - y
10-5
4
Fstat = 700 N
3
a/l = 0.3
rh
2 x2,y2
1
y
0 -1 -2
x1,y1
-3 -4 -5
x -6
-4
-2
0
2
DISPLACEMENT y1, y2 [mm]
4
6 -5
10
Fig. 11c Plane trajectories, Fstat = 700 N
The further increase of static load to Fstat = 1,000 N changes the shape of both trajectories fundamentally as it is shown in Fig. 11d. This can be explained by the strongly nonlinear stiffness characteristics near the bearing surface.
Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 5
93
JOURNAL MOTION x - y
10-5
4
Fstat = 1000 N
3
a/l = 0.3
DISPLACEMENT x1,x2 [mm]
rh
2 1 y
0 -1
x2,y2
-2 x1,y1
-3 -4 -5
x -6
-4
-2
0
2
DISPLACEMENT y1, y2 [mm]
4
6 -5 10
Fig. 11d Plane trajectories, Fstat = 1, 000 N
6 Conclusions The mathematical model of journal motion in aerodynamic bearings including properties of three tilting pads was derived for large amplitudes in the entire domain of clearance of bearing. Dynamic characteristics of such type of bearings are very strongly influenced by inertia properties of tilting pads – stiffness and dynamic matrices are non-symmetric and their elements are non-monotonous functions of angular frequency ω . Dynamic characteristics, given for selected positions of journal in the form of linear stiffness and damping matrices were extended by using special correction function fcor (r) on the entire area of bearing clearance 0 < r < rh , and by plane rotation matrix C for all angular positions −π < ϕ < π. The correction function consists of a part proportional to the stiffness and damping of unloaded bearing and of the second part proportional to the properties of loaded bearing. The approximate mathematical model of stiffness and damping properties of aerodynamic bearing is applied on examples, where influence of various positions of exciting eccentricity and of static loading is shown. Time history of journal motion and its x, y trajectories show the great influence of nonlinearity at large displacements. Due to the sufficient aerodynamic damping, the system was stable in all studied cases. Acknowledgement This work was supported by the Grant Agency of CR No.101/06/1787 “Dynamic properties of gas bearings and their interaction with rotor”.
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References 1. Czolczynski K (1994) Stability of high stiffness journal bearings, Wear 172, 175–183. 2. Lalanne M, Ferraris G (1990) Rotordynamics Prediction in Engineering, Wiley, New York. 3. Strzelecki S (2005) Dynamic characteristics of tilting 5-pad journal bearing. In: Sawicki JT, Muszynska A (eds.) Proceedings of the 3rd ISCORMA 2005, Cleveland, OH, September, 242–249. 4. Tondl A (1965) Some Problems of Rotordynamics, Chapman & Hall, London. 5. Vania A, Tanzi E (2004) Analysis of non-linear effects in oil-film journal bearings, Eighth International Conference on Vibrations in Rotating Machinery, I Mech E, C6231041/2004, 101–110. 6. Martyna M, Kozanecki Z (2007) Non-linear model of a tilting pad gas journal bearing for the power MEMS microturbine. In: Bently DE, Sawicki JT (eds.) Proceedings of the 4th ISCORMA 2007, Calgary, Canada, CD-ROM 309, 1–8. 7. P˚ust L, Koz´anek J (2005) Vibrations of a rigid rotor supported on aerodynamic bearings as an evolutive system. In: Sawicki JT, Muszynska A (eds.) Proceedings of the 3rd ISCORMA 2005, Cleveland, OH, September, 299–308. 8. P˚ust L, Koz´anek J (2005) Nonlinear and evolutive vibrations of rotor supported on aerodynamic bearings. In: Awrejczewicz J, Sendkowski D, Mrozowski J (eds.) Proceedings of 8th Conference on Dynamical Systems Theory and Applications, Lodz, Poland, December, 785–792. ˇ 9. P˚ust L, Simek J, Koz´anek J (2007) Motion of rotor supported on aerodynamic bearings. In: Zolotarev I (ed.) Proceedings of Conference Engineering Mechanics, IT ASCR, Svratka, May, CD-ROM, 116–112. 10. P˚ust L, Koz´anek J (2007) Evolutive and nonlinear vibrations of rotor on aerodynamic bearings, Nonlinear Dynamics 50, 829–840. 11. Hatch ChT et al. (2007) Stiffness and damping results from perturbation testing of externally pressurized radial gas bearings. In: Bently DE, Sawicki JT (eds.) Proceedings of the 4th ISCORMA 2007, Calgary, Canada, CD-ROM 301, 1–11. 12. Brepta R, P˚ust L, Turek F (1994) Mechanical Vibrations, Sobot´ales, Praha.
Identification of Dynamical Systems in the Fuzzy Conditions ´ Tadeusz S. Burczynski, Witold Beluch, and Piotr Orantek
1 Introduction The paper deals with the identification of the fuzzy parameters of material and shape of structures. In many identification and optimization problems for the structures being under dynamical loads one should find some unknown parameters, e.g. materials properties, boundary conditions or geometrical parameters. An identification problem can be formulated as the minimization of some objective functions depending on measured and computed state fields, as displacements, strains, eigenfrequencies or temperature. In order to obtain the unique solution of the identification problem the global minimum of the objective function should be found. In many engineering dynamical cases it is not possible to determine the parameters of the system precisely, so it is necessary to introduce some uncertain parameters which describe the granular character of data. There exist different models of information granularity: interval numbers, fuzzy numbers, rough sets, random variables, etc. In the present paper the granularity of information is represented in the form of the fuzzy numbers. In order to solve an identification problem, some optimization methods have to be used. In the proposed approach the fuzzy version of the evolutionary algorithm (FEA) is used as the first step of the identification procedure. The fuzzy steepest descent method with multilevel artificial neural network (ANN) is used in the second step. The special type of fuzzy ANN for the approximation of the fuzzy fitness function value and the special type of fuzzy fitness function gradient are used. The usage of the ANN enables the reduction of the computation time. The fuzzy finite element method (FFEM) is employed to solve the boundary-value problem.
T.S. Burczy´nski, W. Beluch, and P. Orantek Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, Konarskiego 18a, 44–100 Gliwice, Poland, e-mail:
[email protected],
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
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2 Formulation of the Identification Problem of the Dynamical Systems Consider the first case of an elastic isotropic structure containing a void which size and position are unknown and are described by a vector x. The vector of displacements u(z, t) is described by equation:
μ ∇2 u + (λ + μ )grad div u + Z = ρ u¨ (z,t),
z ∈ Ω,t ∈ T ∈ [0,t f ],
(1)
where: μ , λ – Lame constants, Z – body forces. Equation (1) is supplemented by boundary conditions: u(z,t) = u(z,t), ¯ ¯ p(z,t) = p(z,t),
z ∈ Γ ≡ ∂ Ω, z ∈ Γ ≡ ∂ Ω,
(2)
and initial conditions: u(z,t)|t=0 = uo (z),
o u(z,t)| ˙ t=0 = v (z),
z ∈ Ω.
(3)
It is assumed that boundary conditions and material parameters have the fuzzy character. The identification problem is treated as the minimization of an objective functional (fitness function) J(x), depending on measured uˆ and computed displacements u at sensor points zi , with respect to x: n
min J(x), x
where J(x) = ∑
i=1 T
(u(z,t) − uˆ (z,t))2 δ z − zi dΓdt.
(4)
Γ
Consider the second case of a multi-layered laminate. Multi-layered laminates are the fibre-reinforced composites built of a definite number of the stacked, permanently joined plies [1]. Plies in laminates are usually built of the same material having different fibers’ angles. Laminates have high strength/weight ratio and it is easy to tailor the material properties by manipulating such parameters as: components material, stacking sequence, fibres orientation or layer thicknesses. They can be typically treated as two-dimensional structures with four independent elastic constant: two Young moduli, one shear modulus and one Poisson ratio. The constitutive equation for a laminate’s single layer can be written in the following form [2]: ⎤ ⎡ E1 ν 21 E1 ⎧ ⎫ ⎫ 0 ⎧ 1− ν ν 1− ν ν 12 21 12 21 ⎨ σ 11 ⎬ ⎢ ⎨ ε 11 ⎬ ⎥ ν 12 E 2 E2 σ 22 = ⎢ (5) 0 ⎥ ⎦ ⎩ ε 22 ⎭ , ⎩ ⎭ ⎣ 1−ν 12 ν 21 1−ν 12 ν 21 σ 12 ε 12 0 0 G 12
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where: σij – stress vector, εij – strain vector, E1 , E2 – axial and transverse Young’s moduli, G12 – axial-transverse shear modulus, ν12 , ν21 – axial-transverse and transverse-axial Poisson ratios, respectively (ν21 = ν12 E2 /E1 ). The laminates are often manufactured individually, so the nondestructive methods of determination of the elastic constants must be employed. Indirect identification methods, based on numerical and mixed numerical-experimental techniques, have been developed intensely [3]. The eigenvalue problem for a laminate plate of length a, width b and thickness h in directions x, y and z, respectively, can be presented as [4]:
ρ hω 2 w = D11 w , xxxx + 4D16w , xxxy + 2(D12 + 2D66)w , xxyy + + 4D26w , xyyy + D22 w , yyyy
(6)
where: w – deflection in the z direction, ω – eigenvalue vector, ρ – mass density. The bending stiffness Di j can be obtained from the formula: Di j =
h 2
− h2
(k) (z(k) )2 Q¯ i j dz,
(7)
where: z(k) – the distance from the middle plane to the top of layer k, (k) Q¯ i j – plane stress reduced stiffness component of the layer k. One assumes that material constants in laminates x = (E1 , E2 , G12 , ν) are unknown and have the fuzzy character. In order to find them the functional J(x) which depends on N measured ωˆ j and N calculated ω j natural frequencies has to be minimized [5]: J(x) =
min J(x), x
N
∑ [ω j − ωˆ j ]2 .
(8)
j=1
In the fuzzy representation the objective function is modified to the fuzzy form and the edges of the intervals
J(x) ∈ J(x), J(x) (9) are calculated as follows:
/ / / / J(x j ) = ∑ min /qi − qˆi / , /q¯i − q¯ˆi/ , i
/ / / / ¯ j ) = ∑ max /qi − qˆi/ , /q¯i − q¯ˆi / . J(x
(10)
i
To solve both identification problems, the optimization methods have to be used. In present paper the so called two-stage fuzzy strategy, which merges the evolutionary
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algorithm, gradient methods and artificial neural networks is used as the optimization tool.
3 The Two-Stage Fuzzy Strategy The two-stage fuzzy strategy (TSFS) couples the advantages of gradient optimization methods, evolutionary algorithms (EAs) and artificial neural networks (ANNs) [6]. Gradient methods are fast and give precise results but their application is significantly limited. They can be used for continuous problems and they usually lead to the local optima if some multimodal problems are considered. In such cases the evolutionary algorithms, which are the global optimization methods can be used. As the evolutionary algorithms are time-consuming, it is convenient to couple that both methods (if possible). The block diagram of the TSFS is presented in Fig. 1. In the proposed attitude the fuzzy EA works in the first stage. As a result a set (cloud) of fuzzy points, being the starting point for the next stage, is generated. It is assumed that the considered set of points is located close to the global optimum. The moment of the transition between stages is crucial and depends on the following parameters of the first stage [7]: – the size of the chromosomes’ population – the variability of the best chromosome’s fitness function value – the diversity of the population In second stage the gradient method is used. Sensitivity of the objective function is approximated by means of the ANN.
Fig. 1 The two-stage fuzzy strategy
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3.1 The First Stage – Global Optimization The evolutionary algorithms are global optimization methods which origin from the observation of the natural evolution process [8]. A set (population) of possible solutions (individuals or chromosomes) of the optimization process is processed. Each chromosome consists of genes. Each gene represents one design variable. In order to simulate the evolution process chromosomes are modified. Firstly they are subjected to the selection procedure. Chromosomes’ continuity to exist and reproduction probability depend on their fitness value. Secondly, the evolutionary operators, typically crossover and mutation, are applied to generate new chromosomes which are new possible solutions of the optimization task. The procedure is repeated until the termination condition is fulfilled. The fuzzy evolutionary algorithm (FEA) works on chromosomes consisting of fuzzy genes [9]. Each gene represents one fuzzy number and each chromosome is a potential fuzzy solution of the problem. The j-th fuzzy chromosome ch j in the population consists of N genes and has the following form: ch j (x) = [x1j , x2j , . . . , xij , . . . , xNj ].
(11)
The standard representation of the fuzzy number can be inconvenient from the fuzzy number arithmetic point of view. It is suitable to represent the fuzzy number x as a set of the interval values [x, x] ¯ lying on the adequate levels called α -cuts, as presented in Fig. 2 [10]. In the present paper each gene xij in the chromosome ch j is a vector of five real values representing trapezoidal fuzzy number (Fig. 3): j
j
j
j
j
j
xi = [aL (xi ), aU (xi ), cv(xi ), bL (xi ), bU (xi )],
(12)
where: cv(xij ) – the central value of a fuzzy number;
m
m [0, 1]
1
0.75 0.5 0.25 x
0 x1 x0.75 x0.5 x0.25 x0
Fig. 2 The fuzzy number and corresponding α -cuts
x1 x0.75 x0.5 x0.25 x0
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Fig. 3 The fuzzy gene
ak (xij ), bk (xij ) – distances between the central value and the left and right boundaries of the interval on lower (L) and upper (U) α-cuts, respectively. New fuzzy mutation and crossover operators are introduced to operate on the fuzzy chromosomes [11]. The first type of the fuzzy mutation (FM1) modifies only the central value cv(xij ) of the randomly chosen gene xij , while the second fuzzy mutation operator (FM2) concentrates/deconcentrates the fuzzy gene xij by changing the distances ak (xij ), bk (xij ) for a chosen α -cut. The fuzzy arithmetic crossover (FC) produces two descendants ch1 (x)∗ and ch2 (x)∗ from two parent chromosomes ch1 (x) and ch2 (x) changing selected α -cuts. The fuzzy selection is based on the tournament selection method [12]. The fuzzy fitness function values must be compared in order to select the best individual in the tournament. The better chromosome wins with a probability depending on the introduced parameter β . The fuzzy finite element method (FFEM) is employed in order to calculate fuzzy fitness function values.
3.2 The Second Stage – Local Optimization The block diagram of the second stage of TSFS in presented in Fig. 4. The local optimization method combines the steepest descent method and the artificial neural networks [13]. A set of training vectors is created on the basis of the cloud of points generated in the previous stage. A special multilevel ANN is used as the approximation tool of the fuzzy fitness function. Each level of the ANN multilevel corresponds with a selected parameter of the fuzzy number (Fig. 5). The central level of the multilevel ANN corresponds with j the central value cv(xi ) of the fuzzy number (black colour), while the other levels correspond with other parameters of the fuzzy number: the grey levels correspond with the parameters ai , the white levels correspond with the parameters bi . The coordinates of the points (central value and fuzzy parameters) play the role of the input values while the fuzzy fitness values of these points play a role of the network output.
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Fig. 4 The scheme of the local optimization method
Fig. 5 The scheme of the multilevel ANN
After training the multilevel ANN the optimization process is performed by means of the steepest descent method. As the last step the termination condition is checked. If it is not satisfied, the considered point is added to the training vector set and the next iteration is carried out, otherwise, the point is treated as a result of the optimization process.
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Fig. 6 The structure with identified circular defect
4 Numerical Examples 4.1 The Identification of Geometrical Parameters of a Void The aim of the identification problem is to find the parameters defining the circular defect: x, y and r in a square plate (0.2 × 0.2 m) made of an isotropic material (Fig. 6). The plate is loaded by the fuzzy continuous dynamical loading ¯ p(z,t) = po H(t), where po = 10 kN and H(t) is the Heaviside’s function. 200 time-steps with Δ = 1μs are considered. It is assumed that the Young’s modulus of the plate material E and loading po are also the fuzzy values. The displacements are measured in 21 sensor points on the boundary. The loading po and Young modulus E are described by two α –cuts with the same intervals: po = [99.8; 100.2] kN, E = [2e11 − 2%; 2e11 + 2%] MPa. – – – –
The parameters of the FEA (first stage) are as follows: the population size pop size= 20 the number of generations gen num= 40 arithmetic crossover probability pc = 0.2 Gaussian mutation probability pm = 0.4
In the second stage the local optimization method found the optimum after 156 iterations. The actual and found values of the identified parameters are collected in Tab. 1.
4.2 The Identification of Laminate’s Elastic Constants The aim is to find the elastic constants in a laminate plate consisting of 16 plies of the same thickness h = 0.002 m (Fig. 7).
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Table 1 The identification results for the circular defect Parameter (m)
a1
a2
cv
b2
b1
x y r x y r
0.001 0.001 0.001 0.001 0.001 0.001
0.0005 0.0005 0.0005 0.0005 0.0005 0.0005
0.03 0.03 0.02 0.03 0.03 0.02
0.0005 0.0005 0.0005 0.0005 0.0005 0.0005
0.001 0.001 0.001 0.001 0.001 0.001
Actual
Found
Fig. 7 The laminate plate – dimensions and bearing
The laminate plate with stacking sequence: (0/45/90/-45/0/90/0/90)s is made of the epoxy-glass material. The first N = 10 eigenfrequencies of the plate are taken as the measurement data. Each of identified parameters is described by two α -cuts of different widths. – – – –
The parameters of the FEA are: The population size pop size= 100 The number of generations gen num= 400 Arithmetic crossover probability pc = 0.2 Gaussian mutation probability pm = 0.4
The variations of the identified parameters (for both edges of each interval) in the function of the generation number during the first stage of the strategy are presented in Fig. 8. The number of the local optimization method’s iterations was equal to 1000. The actual and found values of the identified parameters are collected in Tab. 2.
5 Conclusions An intelligent strategy coupling the fuzzy evolutionary algorithm, the multilevel artificial neural networks, and local optimization methods has been presented. This approach can be applied in the optimization and identification of mechanical
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Fig. 8 The identification results during the first stage: (a) E1 , (b)E2 , (c)ν12 , (d)G12 Table 2 The identification results for the laminate plate Parameter (Pa) Actual
Found
E1 E2 G12 ν E1 E2 G12 ν
a1
a2
cv
b2
b1
0.04e10 0.04e9 0.02e9 0.003 0.04e10 0.04e9 0.02e9 0.003
0.02e10 0.02e9 0.04e9 0.001 0.02e10 0.02e9 0.04e9 0.001
3.86e10 8.27e9 7.17e9 0.26 3.86e10 8.27e9 7.17e9 0.26
0.02e10 0.02e9 0.02e9 0.001 0.02e10 0.02e9 0.02e9 0.001
0.04e10 0.04e9 0.04e9 0.003 0.04e10 0.04e9 0.04e9 0.003
structures under dynamic loads. The application of evolutionary algorithms in the first stage of the strategy reduces the possibility of finding the local minimum of the objective function. The gradient method applied in the second stage uses information about the objective functional sensitivity obtained by means of neuro-computing. The identified values and the fitness functions are in the form of fuzzy numbers represented by α -cuts. The fuzzy finite element method has been employed to calculate the fuzzy fitness function values in the first stage of the strategy. The application of the artificial neural networks for approximation of the boundary-value problem in the second stage enables the reduction of the computational time.
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The strategy gives positive results for the identification of various parameters in isotropic as well as in orthotropic (laminates) dynamical structures. As the uncertain conditions have the granular form [14], the models based on the rough sets and random variables can be used instead of the fuzzy and interval approach presented in this paper. Acknowledgement The research is partially financed from the Polish science budget resources as the research project and the Foundation for Polish Science (2005–2008).
References 1. Peters ST (1998) Handbook of Composites. 2nd Edition, Chapman & Hall, London/ Weinheim/New York/Tokyo/Melbourne/Madras. 2. German J (2001) The Basics of the Fibre-Reinforced Composites’ Mechanics, Publications of the Cracow University of Technology, Cracow (in Polish). 3. Bledzki AK, Kessler A, Rickards R, Chate A (1999) Determination of elastic constants of glass/epoxy unidirectional laminates by the vibration testing of plates, Composite Science and Technology 59, 2015–2024. 4. Adali S, Verijenko VE (2001) Optimum stacking sequence design of symmetric hybrid laminates undergoing free vibrations, Composite Structures 54, 131–138. 5. Beluch W, Burczy´nski T (2004) Distributed evolutionary algorithms in identification of material constants in composites. In: 7th National Conference on Evolutionary Computation and Global Optimization, Kazimierz Dolny, 1–8. 6. Burczy´nski T, Orantek P (2005) The fuzzy evolutionary algorithms in optimization problems. In: 8th National Conference on Evolutionary Computation and Global Optimization, Korbiel´ow, 23–30. 7. Orantek P (2005) An intelligent computing technique in identification problems, Computer Assisted Mechanics and Engineering Sciences 13, 351–364. 8. Arabas J (2001) Lectures on Evolutionary Algorithms, WNT, Warsaw (in Polish). 9. Pedrycz W (1998) Fuzzy evolutionary computing, Soft Computing 2, 61–72, Springer, Berlin/Heidelberg. 10. Kacprzyk J (1986) Fuzzy Sets in System Analysis, PWN, Warsaw. 11. Burczy´nski T, Orantek P (2005) The two-stage fuzzy strategy in identification of the uncertain boundary conditions. In: T. Burczyski, W. Cholewa, W. Moczulski (eds.) Methods of Artificial Intelligence, AI-METH 2005, full papers on CD-ROM. 12. Michalewicz Z (1992) Genetic Algorithms + Data Structures = Evolutionary Programs, Springer, New York. 13. Duch W, Korbicz J, Rutkowski L, Tadeusiewicz R (2000) Neural Networks 6, Exit, Warsaw (in Polish). 14. Bargiela A, Pedrycz W (2002) Granular Computing: An Introduction, Kluwer, Boston, MA/Dordrecht/London.
On Nonlinear Response of a Non-ideal System with Shape Memory Alloy Vinicius Piccirillo, Jos´e Manoel Balthazar, Bento R. Pontes Jr., and Jorge L.P. Felix
1 Introduction The study of problems that involve the coupling of several systems, was explored widely in the last year, in function of the change of constructive characteristics of the machines and structures. In this way, some phenomena are observed in composed dynamic systems supporting structure and rotating machines, where are verified that the unbalancing of the rotating parts is the great causer of vibrations. In the study of these systems, for a more realistic formulation is to consider an energy source with limited power supply (non-ideal), that is, to consider the influence of the oscillatory system on the driving force and vice versa. Recently a number of works have been done, in order to investigate the resonant conditions of non-ideal vibrating systems [1], and a number of several of non-ideal systems has been studied, for some examples see [2–5]. On the other hand, the phenomena related to the shape memory alloys (SMA) are associate to the transformations of the phase, which can be caused by the variation of the temperature, as well as, for the variation in the tension level. Basically, the (SMA) presents two stable phases: austenite and martensite. According to the mechanical behavior, the shape memory alloys, can be divided in two categories: shape memory (SME) and pseudoelastic effect. Shape memory effect is related to the ability of the material to recover a great quantity of the residual strain, caused for the action of a loading and unloading, through of the increase of the temperature of the material, in this situation the martensitic phase is stable. Already the pseudoelasticity behavior refers to the ability of the material to obtain V. Piccirillo and B.R. Pontes UNESP – S˜ao Paulo State University, Department of Engineering Mechanics, CP 473, 17033-360, Bauru, SP. e-mail:
[email protected],
[email protected] J.M. Balthazar and J.L.P. Felix UNESP – S˜ao Paulo State University, Department of Statistics, Applied Mathematical and Computation, CP 178, 13500-230, Rio Claro, SP. e-mail:
[email protected], jorgelpfelix@ yahoo.com.br J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
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Electric Motor
y Rotating Imbalance
f
m r
SMA Beam
k
x
L
M,J e
a
b
Fig. 1 (a) Diagram of physical model and (b) mathematical model with limited power supply
a very large strain upon loading and fully recover through a hysteresis loop upon loading and it always occurs in high temperatures, in this situation the austenitic phase is stable. Shape memory and pseudoelasticity effects making the (SMA) research very attractive, as material with great potential for diverse applications in engineering science, such as, vibration control, active buckling control, or more generally active structural modifications schemes can be used in these mentioned applications [6, 7]. Diverse approaches and techniques have been developed to describe the thermodynamic behavior of alloys [8]. Here we will analyze a non-ideal (SMA) problem by using a numerical simulations. The main purpose of this paper is to study the possibilities of the existence of regular and irregular motions in a (SMA) non-ideal vibrating problem schematically shown in Fig. 1.
2 (SMA) Constitutive Modeling To describe behavior of the oscillator with a shape memory, the constitutive model proposed by Falk [9] has been adopted in the mathematical modeling of the problem. This model is based on Devonshire theory and defines a free energy of Helmholtz (ψ), in a polynomial form and it is capable to describe the shape memory and pseudoelasticity effects. The polynomial model, as more it is known to deals with one – dimensional cases and it does not consider an explicit potential of dissipation, and no internal variable is considered. On this form, the free energy depends only on the state variable observations (temperature and strain), that is, ψ = ψ (ε, T). The free energy is defined in such a way that, for high temperatures (T > TA ), the energy possesses only one point of minimum corresponding to the null strain representing the stability of the austenite phase (A); for intermediate temperatures
On Nonlinear Response of a Non-ideal System with Shape Memory Alloy
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(TM < T < TA ) it presents three points of a minimum corresponding to the austenitic (A) and two other martensitic phases (M+ and M− ), which are induced by some positive and negative stress fields, respectively upto the low temperature (T < TM ) there are two points of a minimum representation of the two variants of martensite (M+ e M− ), corresponding to the zero value strain. Therefore, the restrictions below are given by the following polynomial equation 1 1 1 ρψ(ε, T) = a(T − TM)ε2 − bε4 + eε6 , (1) 2 4 6 where: a and b are constants of the material, TM corresponds to the temperature at which the martensitic phase is stable and ρ is the mass density. If TA is defined as the temperature above which the austenite is stable and the free energy has only one minimum at zero strain, it is possible to write the following condition TA = TM +
b2 , 4ae
(2)
where constant e may be expressed in terms of other constants of the material. By definition [10], the stress – strains relation is given by
σ = a(T − TM)ε − bε3 + eε5 .
(3)
3 Mathematical Model of the Non-ideal System The adopted model of the vibrating system and the source of disturbance with a limited power supply are illustrated in Fig. 1. The vibrating system, consists of a mass M, a (SMA) element and a linear damping with viscous damping coefficient c. On the object with mass M, a non-ideal DC motor is placed, with a driving rotor of a moment of inertia J and r is the eccentrically of the unbalanced mass. The governing equations of motion of the system have the form ¯ 3 + e¯ x5 = 0, (M + m) x + cx − mr ϕ cos ϕ − ϕ 2 sin ϕ + a¯ (T − TM)x − bx (4) J + mr2 ϕ − mrx cos ϕ = S ϕ − H ϕ , where:
aAr bAr eAr , b¯ = 3 , e¯ = 5 . (5) L L L Let us assume that φ is the angular displacement of the rotor, S (φ ) is the controlled torque of the unbalanced rotor, H (φ ) is the resistant torque of the unbalanced rotor and Ar is the area of the element with a shape memory. Equation (4) may be simplified when the model of the DC motor is taken by removing the effect of the inductance S φ − H φ = g − hφ , (6) a¯ =
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where g represents the control parameter (voltage or constant torque) and h is a constant motor type [11]. It is convenient to work with dimensionless position and time according to u=
x and τ = ω0 t L
in such a way, that Eq. (4) is rewritten in the following form ¨ cos φ − φ ˙ 2 sin φ = 0, u¨ + 2μ u˙ + (θ − 1) u − α u3 + γ u5 − λ φ ¨ − η u¨ cos φ = ξ1 − ξ2 φ, ˙ φ
(7)
(8)
where dots represent the succesive time differentiation. Our dimensionless variables are given below bAr aAr TM T c , α= , θ= , μ= , (M + m)L TM 2(M + m)ω0 (M + m)Lω20 eAr mrL mr γ= , η= , , λ= (M + m)L (J + mr2 ) (M + m)Lω20 g h ξ1 = , ξ2 = . (J + mr2 ) ω0 (J + mr2 ) ω20 ω20 =
(9)
We have assumed by the above that the characteristic curve of the DC motor (energy source) is a straight line. As the shape memory alloys (SMA) present some different temperature dependent properties, in this article we analyse the pseudoelastic behavior considering a higher temperature, where the austenitic phase is stable in the alloy (θ = 2). Therefore the Eq. (7) is cast into the following form ¨ cos φ − φ ˙ 2 sin φ = 0, u¨ + 2μ u˙ + u − α u3 + γ u5 − λ φ (10) ˙ ¨ − η u¨ cos φ = ξ1 − ξ2 φ, φ and finally u˙ 1 = u2 , u˙ 2 =
1 (λ((ξ1 − ξ2 u4 ) cos u3 − u24 sin u3 ) 1 − λη cos2 u3 + αu31 − γu51 − (θ − 1)u1 − 2μu2 ),
u˙ 3 = u4 , u˙ 4 =
ξ1 η cos u3 + 1 − λη cos2 u3 1 − λη cos2 u3 ξ2 u 4 αu31 − γu51 − (θ − 1) u1 − 2μu2 − − λu24 sin u3 . η cosu3
(11)
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Table 1 Material constants for a Cu-Zn-Al-Ni alloy [12] a(Mpa/K)
b(Mpa)
e(Mpa)
TM (K)
TA (K)
523.29
1.868 × 107
2.186 × 109
288
364.3
4 Numerical Results The objective of this section is to analyze the vibrating problem shown in Fig. 1 taking into account the linear torque defined by the Eq. (9). The numerical simulations were carried out by using the Matlab-Simulink. For numerical simulations we use the variable-step Adams-Bashforth-Moulton method. In all simulations, to analyze the behavior of the non-ideal dynamical system, the spring is assumed to be made of a Cu-Zn-Al-Ni alloy and the properties presents in Table 1. Furthermore, we consider μ = 0.01, η = 0.6, λ = 0.4 and ξ2 = 1.5. Note that the ˙ of the passage through the resonance is obtained by changes in angular velocity φ DC motor. Results presented below were obtained on the dependence of the non-ideal system after taking into account a suitable values of the parameter ξ1 . This fact is related to the greater or lower interaction depending of the value of voltage or constant torque, that is, the dimensionless values ξ1 = 0.3, 1, 1.5; or 2 that we set to investigate the passage through a resonance. Let ξ2 = 1.5 be fixed. The dynamic behavior becomes much richer because some periodic, quasi-periodic or chaotic motions are possible to observe in this non-ideal system. In order to illustrate the response of the non-ideal system, we analyzed the response at higher temperatures (θ = 2), when the alloy is fully austenitic. We also plotted the Poincar´e section which represents the surface of section (u1 (τn ) , u2 (τn )). The points (u1 (τn ) , u2 (τn )) are captured for τn = nT, where n = 1, 2, 3, ..., with period T = Ω2πM [13]. The average angular velocity ΩM is obtained numerically ΩM =
φ (τ1 ) − φ (0) u3 (τ1 ) − u3 (0) = , τ1 τ1
(12)
where τ1 is a long time period for numerical calculation. The highest interaction between the vibrating system and the energy source will occur at a resonance. We define the resonance region as follows dϕ − ω = O (ε) , dt
(13)
where dϕ dt is the angular velocity, ε is a small parameter of the problem of order −3 10 , and ω is the natural frequency of the system. Generally, for a wide range of physical parameters when the system was started from rest, the angular
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velocity of the rotor would increase until it reached the neighborhood of the natural frequency ω. Then depending upon physical parameters values dϕ dt would increase beyond the resonance region (pass through) or it remains close to ω (capture). A periodic solution in the case of the angular velocity taken below the resonance at ξ1 = 0.3 is illustrated in Fig 2. Figure 3a illustrates a case when angular velocity is below resonance (ξ1 = 1). The quasi-periodic motion is have reported. The nature of this motion is confirmed in Fig. 3c by the power spectrum.
On Nonlinear Response of a Non-ideal System with Shape Memory Alloy 0.9
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Figure 4 shows an interesting dynamical behavior for ξ1 = 1.5. Figure 4a illustrates a case when angular velocity is captured into the resonance region, causing a chaotic motion (see Fig. 4b). Chaotic dynamics in these kinds of systems can be quantitatively characterized by means of the power spectra (Fig. 4c). A strange attractor on the Poincar´e section (see Fig. 4d) obtained for non-ideal system has the complicated fractal structure with features of chaotic motion. Figure 5a illustrates a case of passage through resonance region for ξ1 = 2. In this case we obtained chaotic motion. The broadband character observed in the power spectrum is a characteristic feature of a chaotic solution. We have used the Poincar´e section to characterize the dynamic of the system. In Fig. 5d a strange attractor behavior of the system for ξ1 = 2 is reported.
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5 Conclusions In this paper, we analyzed the influence of the (SMA) spring on the non-ideal system in the passage through resonance. We present the study on the pseudoelastic behavior, considering a higher temperature, where austenitic phase is stable in the alloy. The torque generated by DC motor is of limited power supply and, according to classical Kononenko theory, is assumed as a straight line. The numerical simulations were performed by using the linear torque as well as by consideration of the primary resonance. During passage through the resonance of the DC
On Nonlinear Response of a Non-ideal System with Shape Memory Alloy 2
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motor-structure system what is modeled as a (SMA) oscillator, with a non-ideal excitation some “severe” vibrations appear. The interaction between the motor and the oscillating system is confirmed by the different phase portraits. The numerical results presented in this paper show that it is possible to get any regular and chaotic motions for the variation of the control parameter. Rising ξ1 some chaotic motions appear as well.
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References 1. Balthazar JM, Mook DT, Weber HI, Brasil RMFLRF, Fenili A, Belato D, Felix JLP (2003) An overview on non-ideal vibrations. Meccanica 38, 613–621. 2. Belato D, Weber HI, Balthazar JM, Mook DT (2001) Chaotic vibrations of a non-ideal electromechanical system. International Journal of Solids and Structures 38, 1699–1706. 3. Dantas MJH, Balthazar JM (2007) On the existence and stability of periodic orbits in nonideal problems: general results. ZAMM Journal of Applied Mathematics and Mechanics 58, 940–956. 4. Felix JLP, Balthazar JM, Brasil RMFLRF (2005) On tuned liquid column dampers mounted on a structural frame under a non-ideal excitation. Journal of Sound and Vibration 282, 1285– 1292. 5. Pontes BR, Oliveira VA, Balthazar JM (2000) On friction-driven vibrations in a mass blockbelt-motor system with a limited power supply. Journal of Sound and Vibration 234, 713–723. 6. Pietrzakowski M (2000) Natural frequency modification of thermally activated composite plates. Mecanica 1, 313–320. 7. Rogers CA, Liang C, Fuller CR (1991) Modeling of shape memory alloy hybrid composites for structural acoustic control. Journal of Acoustic Society of America 1, 210–220. 8. Paiva A, Savi MA (2006) An overview of constitutive models for shape memory alloys. Mathematical Problems in Engineering 2006, 1–30. 9. Falk F (1980) Model free–energy, mechanical and thermodynamics of shape memory alloys. ACTA Mettalurgica 28, 1773–1780. 10. Savi MA, Braga AMB (1993) Chaotic vibration of an oscillator with shape memory. Journal of the Brazilian Society of Mechanical Science 15, 1–20. 11. Warminski J, Balthazar JM, Brasil RMFLRF (2001) Vibrations of a non-ideal parametrically and self-excited model. Journal Sound and Vibration 245, 363–374. 12. Savi MA, Pacheco PMCL, Braga AMB (2002) Chaos in a shape memory two-bars truss. International Journal of Non-Linear Mechanics 37, 1387–1395. 13. Zucovic M, Cveticanin L (2007) Chaotic response in a stable Duffing system of non-ideal type. Journal of Vibration and Control 13, 751–767.
Dynamics of a Material Point Colliding with a Limiter Moving with Piecewise Constant Velocity ´ Andrzej Okninski and Bogusław Radziszewski
1 Introduction Vibro-impacting systems are very interesting examples of non-linear dynamical systems with important technological applications [1]. Dynamics of such systems can be extremely complicated due to velocity discontinuity arising upon impacts. A very characteristic feature of impacting systems is presence of non-standard bifurcations such as border-collisions and grazing impacts appearing in the case of motion with low velocity after impact, which often leads to complex chaotic motion [1]. The main difficulty with investigating impacting systems is in gluing pre-impact and post-impact solutions. The Poincar´e map, describing evolution from an impact to the next impact, is thus a natural tool to study such systems. In the present paper we investigate motion of a material point in a gravitational field colliding with a moving motion-limiting stop. Typical example of such dynamical system, related to the Fermi model, is a small ball bouncing vertically on a vibrating table. Since evolution between impacts is expressed by a very simple formula the motion in this system is easier to analyze than dynamics of impact oscillators. It is possible to simplify the problem further assuming a special motion of the limiter. The paper is organized as follows. In the second section of this article a onedimensional motion of a material point in a gravitational field, colliding with a limiter, representing unilateral constraints, is considered. In Section 3 results of numerical simulations are described while in Section 4 analytical results are presented. We discuss our results in the last section.
A. Okni´nski and B. Radziszewski ´ ˛ tokrzyska, Wydział Zarza˛ dzania i Modelowania Komputerowego, 25-314 Politechnika Swie Kielce, Al. Tysia˛ clecia P.P 7, Poland, e-mail:
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
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2 Motion with Impacts: A Simple Motion of the Limiter We consider motion of a material point moving vertically in a gravitational field and colliding with a moving motion-limiting stop, representing unilateral constraints. We shall assume that limiter’s mass is so large that its motion is not affected at impacts. Motion of the material point between impacts is described by equation: mx¨ = −mg,
(1)
where x˙ ≡ dx/dt and motion of the limiter is adopted as: y = y (t),
(2)
with a known function y. We shall also assume that y (y) ˙ is continuous (piecewise continuous) function of time. Impacts are modeled as follows:
+
x˙ τi
x (τi ) = y (τi ), − y( ˙ τi ) = −R[x˙ τi− − y( ˙ τi )],
(3a) (3b)
where impact times are neglected with respect to time intervals of motion between impacts and impacting bodies are considered perfectly rigid. In Eqs. (3) τi stands + for time of i-th impact while x˙− i , x˙i are left-sided and right-sided limits of x˙i (t) for t → τi , respectively, and R is the coefficient of restitution, 0 ≤ R < 1 [2]. − Let us consider Eq. (1) for t ∈ τi+ , τi+1 . General solution of this equation reads: 1 (1) (2) x(t) = − gt 2 + ci t + ci . 2
(4)
Applying to Eq. (4) impact conditions (3) the Poincar´e map is obtained [3]:
γ Y (Ti+1 ) = γ Y (Ti ) − (Ti+1 − Ti )2 + (Ti+1 − Ti )Vi ,
(5a)
Vi+1 = −RVi + 2R (Ti+1 − Ti ) + γ (1 + R) Y˙ (Ti+1 ) ,
(5b)
where Ti , Y (Ti ) , Vi , and γ are nondimensional time, position, velocity and acceleration, respectively: Ti = ωτi , Y (Ti ) = y (τi ) /a, Vi = (2ω / g) x˙ τi+ , γ = 2ω 2 a / g (6) and ω and a determine time and length scales. In our previous paper we have assumed limiter’s motion in form y (t) = a sin (ω t) [3]. This choice leads to serious difficulties in solving Eq. (5a) thus making analytical investigations of dynamics hardly possible. Accordingly, we have decided to choose the limiter’s motion as simple as possible. Let us thus assume that the limiter moves up periodically in time intervals [kT, (k + 1) T), k = 0, 1, . . . , where T = ω −1 , with a constant velocity v from level y (0) = 0 to the level y (T− ) = a and
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Y(T) 1.0 0.8 0.6 0.4 0.2
0
1
2
3
4 T
Fig. 1 Motion of the limiter
then immediately falls back, y (T+ ) = 0. In units (6) motion of the limiter is given by: Y (Ti ) = Ti ( mod 1) (7) (see Fig. 1), and Eqs. (5) become:
γ (Ti+1 ( mod 1)) = γ (Ti ( mod 1)) − (Ti+1 − Ti )2 + (Ti+1 − Ti )Vi ,
(8a)
Vi+1 = −RVi + 2R (Ti+1 − Ti ) + γ (1 + R) .
(8b)
Since period of motion of the limiter is equal one the map (8) is invariant under the translation Ti → Ti + 1 and thus the phase space (T, V ) is topologically equivalent to the cylinder. The map (8) must meet several physical conditions to correspond exactly to the original physical problem. First of all, since Eq. (8a) may have multiple solutions, we have to choose the solution Ti+1 with the smallest nonnegative difference Ti+1 − Ti . Moreover, since Vi is the velocity of the material point just after the impact, it must not be smaller than the velocity of the limiter, i.e. the condition Vi ≥ γ must be fulfilled.
3 Computational Results We have computed bifurcation diagrams and basins of attraction for the dynamical system (8). It follows from numerical simulations that, after a stage of transient dynamics, only two modes of dynamics can establish for a given values of control parameters γ , R. Firstly, a periodic motion with m impacts per k periods, referred to as (k/m) attractor, can settle.
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Fig. 2 Bifurcation diagrams in (V, γ )
Fig. 3 Upper boundary for time of the first impact
Secondly, infinite number of impacts in one period of the limiter’s motion ending with state of zero relative velocity is also possible. Dynamics of this kind is referred to as chattering while the state of zero relative velocity is called grazing or sticking. In Figs. 2, 3 bifurcation diagrams in (V, γ ) and (T, γ ) planes are shown, respectively for R = 0.85. Vertical lines in both figures correspond to critical values of γ for 1, 2, 3, 4, ∞ numbers of impacts (from right to the left), horizontal lines in Fig. 2 show the corresponding velocities of the first impact while horizontal lines in Fig. 3 show upper boundary for time of the first impact. 3 Basins of attraction are shown in Fig. 4 for R = 0.85, γ = 37 . The whole rectangular region shown in Fig. 4, V ≥ γ , T ∈ [0, 1], is the basin of periodic attractor (1/1). Wedge like part of the phase space (darker shaded region) depicts initial conditions for which chattering is present in the first period of limiter’s motion.
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Fig. 4 Basins of attraction
4 Analytical Results Computational results described in the last section can be fully explained. In what follows we shall at first present analytical computations for periodic motion with m impacts per k periods, denoted as (k/m), then for chattering and, finally, we shall address the problem of finding the solution Ti+1 of Eq. (8a) with the smallest nonnegative difference Ti+1 − Ti in case when multiple solutions are possible.
4.1 Periodic Motion We shall first study periodic solutions of Eqs. (8), for which the following conditions (k/1) (k/1) must be fulfilled: Vn+1 = Vn = V∗ , Tn+1 = Tn + k = T∗ + k, k = 1, 2, . . . , i.e. one impact per k periods. Substituting these conditions into Eqs. (8) we obtain the manifold of fixed points in the (T,V ) space: (k/1)
V∗
(k/1)
= k, γcr
=k
1−R , k = 1, 2, . . . , 1+R
(k/1)
T∗
∈ (0, 1) and arbitrary. (k/1)
(9)
It is important that the solutions appear exactly for γ = γcr (analogous solutions are known for limiter’s motion given by y (t) = a sin (ω t) but are stable in some invariant of parameter γ [3]). (k/1) Stability of solutions (9) can be studied in the standard way. Let Vn = V∗ + (k/1) un , Tn = T∗ + dn where un and dn are small perturbations. Then the linearized equations are of form: 1 (1 + R)/2 dn+1 dn = . (10) 0 R2 un+1 un
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Let us note that one of eigenvalues of the stability matrix in (10) is one (in T eigendirection) while another is R2 < 1. It follows that fixed points (9) are stable for R < 1 and form a manifold (the whole unit interval) on the T axis (in the y (t) = a sin (ω t) case fixed points have sharp values [3]). The results obtained above stress differences between the limiter’s motion (7) and y (t) = a sin (ω t) [3]. However, in the case of the simpler motion of the limiter (7) it is possible to carry out analytical computations much further. Indeed, let us now consider the case of m impacts per one period. This means that the following conditions must be fulfilled: V∗m+1 = V∗1 , T∗m+1 = T∗1 + 1. Substituting these conditions into Eqs. (8) we arrive at the following set of equations: ⎧ ⎪ γ = − (T∗2 − T∗1 ) + V∗1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪γ = − T∗3 − T∗2 + V∗2 ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎨γ (T − T ) = − (T + 1 − T )2 + (T + 1 − T )V ∗1 ∗m ∗1 ∗m ∗1 ∗m ∗m (11) ⎪ V∗2 = −RV∗1 + 2R (T∗2 − T∗1 ) + γ (1 + R) ⎪ ⎪ ⎪ ⎪ ⎪ V∗3 = −RV∗2 + 2R T∗3 − T∗2 + γ (1 + R) ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎩ V∗1 = −RV∗m + 2R (T∗1 + 1 − T∗m ) + γ (1 + R) which can be solved to yield manifold of fixed points:
(1 − R) 1 + R + 2R2 + (1 − R) Rm (1 − R)2 (1 + Rm) (1/m) (1/m) γcr = = , V∗1 , (1 + R)2 (1 − Rm) (1 + R)2 (1 − Rm ) (1/m)
T∗1
∈ (0, τm ) .
(12a)
We still have to demand that all impacts take place in the same period of the limiter’s motion. This condition, T∗m < 1, leads to inequality T∗1 < τm where
τm =
(1 − R) (1 + Rm) . (1 + R) (1 − Rm)
(12b)
We check that for k = m = 1 Eqs. (9), (12) agree. There are also other possibilities of periodic motions, i.e. cycles with m impacts per k periods. For example, to obtain the (3/2) case periodicity conditions V∗i+2 = V∗i , T∗i+2 = T∗i + 3 are imposed in Eqs. (8) to get the following set of equations: ⎧ 2 γ T∗i+1 ( mod 1) = γ (T∗i ( mod 1)) − T∗i+1 − T∗i + T∗i+1 − T∗i V∗i ⎪ ⎪ ⎪ ⎨V ∗i+1 = −RV∗i + 2R T∗i+1 − T∗i + γ (1 + R) 2 , ⎪ T T V γ ( mod 1) = γ ( mod 1) − T − T + T − T ⎪ ∗ ∗ ∗ ∗ ∗ ∗ ∗ i+2 i+1 i+2 i+1 i+2 i+1 i+1 ⎪ ⎩ V∗i+2 = −RV∗i+1 + 2R T∗i+2 − T∗i+1 + γ (1 + R) (13a) (13b) V∗i+2 = V∗i , T∗i+2 = T∗i + 3, T∗i+1 ( mod 1) = T∗i+1 − 1,
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where we assumed that first impact occurred in the first period, the second impact occurred in the second period and there were no impacts in the third period. After substituting Eqs. (13b) into (13a) we obtain: 9 1 + R2 (3/2) 2 2 2 3 1 + R − 2 (2 + R ) (1 + 2R ) . γcr = (13c) (1 + R)2 (1 − R2) Analysis of the derived formulae and numerical experiments suggest that the following simple expression (k/m) ∼ k 1 − R (14) γcr = m 1+R yields for R ≈ 1 very good approximation for critical value of γ at which the attractor (k/m) arises.
4.2 Chattering Let us consider the case of periodic motion with m impacts per period, c.f. Eqs. (11), (12). In the m → ∞ limit we get: (1 − R) 1 + R + 2R2 (1 − R)2 1−R (1/∞) (1/∞) (1/∞) . γcr = , V = , T ∈ 0, ∗ ∗ 1 1 1+R (1 + R)2 (1 + R)2 (15) (1/∞) (1/∞) (1/∞) For γ = γcr ,V = Vcr and T1 = T∗1 the material point impacts infinite number of times with the limiter in finite time during one period of motion of the limiter (1/∞) and grazes at T∞ = 1 with zero relative velocity, V∞ = γcr , see Fig. 5. Then the limiter moves down (infinitely fast) and the process repeats periodically in time because the conditions T∞+1 = T1 ( mod 1) , V∞+1 = V1 are fulfilled. (1/∞) (1/∞) (1/∞) For γ = γcr ,V = Vcr and T1 < T∗1 the material point impacts infinite number of times with the limiter in finite time during one period of motion of the limiter, grazes and sticks. Then at time T = 1 the limiter escapes infinitely fast
Fig. 5 Material point impacting infinite number of times with the limiter
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and the material point has right initial conditions, T∞ = 1, V∞ = γcr , to perform periodic grazing shown in Fig. 5. Let us now consider chattering without more restrictive assumption that the process be periodic. Assumption that the material point impacts infinite number of times in a period makes possible to write Eqs. (8) in form:
γ = − (Ti+1 − Ti ) + Vi,
(16a)
Vi+1 = −RVi + 2R (Ti+1 − Ti ) + γ (1 + R) .
(16b)
It follows that: wi+1 = Rwi ,
wi = Vi − γ ,
(17)
where wi is the velocity of the material point relative to the limiter. Now for R < 1 we have wi → 0. However, we have to demand additionally that all impacts, which started at time instant T1 , took place within the same period. It follows from Eq. (16a) that: ∞ V1 − γ T∞ − T1 = (V1 − γ ) ∑ Rk = , (18) 1−R k=0 and since the time of grazing must be smaller than length of period of limiter’s motion, T∞ < 1, the condition of chattering ending with graze is obtained: V1 − γ < 1 − T1. 1−R
(19)
Since we have not demanded periodicity the fate of the material point in the next period of the limiter’s motion requires further analysis.
4.3 Multiple Solutions and Discontinuous Dependence on Velocity Equation (8a) may have multiple solutions. More exactly, these equations have one or two solutions for Ti+1 depending on the values of the control parameter γ > 0 and initial conditions Ti , Vi . Let Ti , Ti+1 ∈ [0, 1]. In this case Eq. (8a) takes the form:
γ (Ti + δi ) = γ Ti − δi2 + δi Vi ,
(20)
where δi = Ti+1 − Ti , and has solutions: (1)
δi
= 0,
(21a)
(2)
= Vi − γ .
(21b)
δi
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(1)
We note that δi is the trivial solution corresponding to the impact at T = Ti . We (2) thus accept δi as the correct solution provided that the consistency conditions are fulfilled: (2) δi = Vi − γ ≥ 0, Ti+1 = Ti + Vi − γ ≤ 1. (22) When Vi grows the critical condition is reached: Ti + Vi − γ = 1,
(23)
and then, at Vi still growing, the solution (21b) does not fulfill the second of the conditions (22). In this case we assume Ti ∈ [0, 1], Ti+1 ∈ [1, 2] and Eq. (8a) is written as: γ (Ti + δi − 1) = γ Ti − δ 2 + δi Vi . (24) Solutions of Eq. (24) read (3, 4)
δi (4)
where only δi
=
1 Vi − γ ± (Vi − γ )2 + 4γ , 2
(25)
(3)
is meaningful since δi < 0, provided that consistency conditions: 1 Vi − γ ± (Vi − γ )2 + 4γ ∈ [1, 2] (26) Ti+1 = Ti + 2 (2)
(4)
. out that the solutions δi , δi -hold. It turns (1) (2) Vcr ,Vcr where: (1)
Vcr = 1 − Ti + γ −
γ , 1 − Ti
coexist in the velocity interval: Vi ∈
(2)
Vcr = 1 − Ti + γ ,
(27)
(2)
i.e. Vcr fulfills (23). It follows that when the velocity grows from Vi < V (2) to (2) (4) Vi > V (2) the solution for δi changes discontinuously from δi to δi . In general, multiple solutions can exist for Ti+1 ∈ [k, k + 1] , k = 1, 2, . . .. In Fig. 6 the function F(δ , T,V ) was plotted: F(δ , T,V ) = γ [(T + δ ( mod 1)) − (T ( mod 1))]/δ + δ − V,
(28)
where T = 0.63 and V = 0.4 (thin solid line), V = 0.49 (thick solid line, critical value of velocity), V = 0.6 (dashed line). Discontinuity in the figure is at δ = 1 − T (vertical dotted line). Indeed, for increasing value of the velocity number of solutions of Eq. (28) changes from two to one and the smallest solution of this equation changes discontinuously.
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Fig. 6 The shape of function F (δ , T,V )
5 Conclusions In the present paper we have investigated dynamics of a material point moving in gravitational field and colliding with a moving limiter. The motion of the limiter has been assumed as simple as possible, i.e. periodic with piecewise constant velocity. On the basis of analytical as well as computational results described above we shall sketch the overall picture of dynamical behaviour of the system (8). For (1/∞) γ < γcr there is only one asymptotically stable attractor, the grazing manifold Y (T ) = T ( mod 1). For initial conditions T1 , V1 obeying inequality (19) the material point performs infinite number of impacts and sticks after time T∞ and this process repeats in subsequent periods. Otherwise, the material point will approach the attractor asymptotically. (1/∞) In the case when γ = γcr there is only one stable attractor – periodic chattering without sticking, see Fig. 5. (1/∞) When γ > γcr there are two possibilities. Firstly, the parameter γ is exactly equal critical value for which a periodic attractor (k/m) with m impacts per k peri(k/m) ods (with specified sequence of the impacts) is asymptotically stable, i.e. γ = γcr . Then the material point approaches the attractor asymptotically. For initial conditions fulfilling inequality (19) chattering ending with sticking occurs in the first period as a transient process. This has been shown in Fig. 4. Let us stress that also in this case there is only one asymptotically stable attractor (k/m). All these (k/m) (k/m) attractors have sharply defined velocities V∗i while impact times T∗i belong to some manifolds (i = 1, 2, . . . , m), see for example Eqs. (12). We have compared in Section 1 properties of the (1/1) attractor from this work and [3].
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On the other hand, number of periodic attractors (k/m) is infinite but countable. (k/m) Therefore, there is another possibility that γ = γcr for any integer m and k (for a specified sequence of impacts). In this case motion of the material point should be quasi-periodic.
References 1. di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P (2008) Piecewise-Smooth Dynamical Systems. Theory and Applications. Springer, London. 2. Stronge WJ (2000) Impact Mechanics. Cambridge University Press, Cambridge. 3. Okniski A, Radziszewski B (2007) Grazing dynamics and dependence on initial conditions in certain systems with impacts, arXiv:0706.0257.
Transient in 2-DOF Nonlinear Systems Yuri Mikhlin, Gayane Rudnyeva, Tatiana Bunakova, and Nikolai Perepelkin
1 Introduction An investigation of transient is important in engineering, in particular, in problem of absorption. Over the past years different new devices have been used for the vibration absorption and for the reduction of the transient response of structures [1–4]. It seems interesting to study nonlinear passive absorbers for this reduction. In presented paper the transient in a system containing a linear oscillator, linearly coupled to an essentially nonlinear attachment with a comparatively small mass, is considered. A damping is taken into account. It is assumed that some initial excitation implies vibrations of the linear oscillator. The multiple scales method [5] is used to construct a process of transient in some nonlinear systems. A transfer of energy from the initially perturbed linear subsystem to the nonlinear absorber can be observed. A similar construction is made to describe the transient in a system which contains a linear oscillator and a vibro-impact absorber with a comparatively small mass. Both an exact integration with regards to impact conditions, and the multiple scales method are used for this construction. The transient in such system under the external periodical excitation was considered too. Besides, a transient in a system which describes an interaction of some rotating subsystem and the elastic one is constructed by using the multiple scales method too. Such system is known as nonideal system. One has a transient to one of the stationary regimes. Numerical simulation confirms an efficiency of the analytical construction in all considered systems.
Yu. Mikhlin, G. Rudnyeva, T. Bunakova, and N. Perepelkin National Technical University, Kharkov Polytechnic Institute, Kharkov, Ukraine, e-mail:
[email protected],
[email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009
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Fig. 1 The system with two connected oscillators
2 Transient in a System Containing an Essentially Nonlinear Oscillator as Absorber Let us consider a system with two connected oscillators, namely one linear and one nonlinear with a comparatively small mass, which is considered as absorber of the linear oscillator vibrations (Fig. 1). Here M is a mass of the main linear subsystem, m is a mass of the nonlinear absorber, ω2 ,γ and c characterize elastic springs, δ characterizes a linear dissipation force. To emphasis a smallness of some inertial and elastic characteristics of the absorber, as well a smallness of the dissipation force, the next transformations will be used, namely: m → εm, c → εc, γ → εγ, δ → ε2 δ, where ε is the small parameter (ε
