Dynamics of Solid Structures: Methods using Integrodifferential Relations 9783110516449, 9783110516234

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Georgy V. Kostin, Vasily V. Saurin Dynamics of Solid Structures

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Georgy V. Kostin, Vasily V. Saurin

Dynamics of Solid Structures |

Methods using Integrodifferential Relations

Mathematics Subject Classification 2010 35K51, 35L53, 49S05, 65K10, 70Q05, 74B05, 80A20 Authors Dr Georgy V. Kostin Russian Academy of Sciences Institute for Problems in Mechanics Prospect Vernadskogo 101-1 Moscow 119526 Russia [email protected] Dr Vasily V. Saurin Russian Academy of Sciences Institute for Problems in Mechanics Prospect Vernadskogo 101-1 Moscow 119526 Russia [email protected]

ISBN 978-3-11-051623-4 e-ISBN (PDF) 978-3-11-051644-9 e-ISBN (EPUB) 978-3-11-051625-8 Set-ISBN 978-3-11-051645-6 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck Cover image: Stockbyte/Stockbyte/thinkstock ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

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To our families

Preface The manuscript is a continuation of the monograph “Integrodifferential Approaches in Linear Elasticity” published by De Gryuter, the series “Studies in Mathematical Physics”, Volume 10, 2012. The new book is a result of the authors’ activity in solid dynamics and control theory in recent years at the Institute for Problems in Mechanics of the Russian Academy of Sciences. Dynamics of solids has been actively studied in the scientific community over the past decades, and its applications are very actual and attractive nowadays. However, many fundamental questions in this area still remain open for discussion. Just take the conventional Hamilton principle in dynamics, for instance, that is formulated only for boundary value problems with respect to time which are not so substantive in practice. Although various numerical algorithms, among which Galerkin ones play a key role, have been intensively developed, a significant limiting factor in their evolution is the lack of convenient variational principles for initial-boundary value problems. Even more sophisticated barriers appear in formulating and solving inverse problems in dynamics, for which the very existence of a solution is often not grounded. All of this was the main reason for the authors to work out new variational and projection approaches to dynamic phenomena such as vibrations, forced motions of elastic or viscoelastic bodies and structures, heat and mass transfer processes in solids, and so on. The key idea of the proposed approaches is that the state variables introduced can be always divided into two groups. The first group consists of the so-called measured quantities, e.g., displacements, strains, velocities, temperature. The second one includes unmeasured values: stresses, momenta, heat fluxes, etc. At the same time, governing equations can be split into three types: firstly, initial and boundary conditions, secondly, balance and continuity laws, and thirdly, constitutive relations. The first type of equations reflects the influence of the environment on the considered system. The second one describes the fundamental physical phenomena and hypotheses of continuum; all of these laws do not depend on media properties. In contrast, the constitutive relations connect measured and unmeasured unknowns and contain information about intrinsic properties of the object under study. In physics, generalized statements suppose that some of the governing equations are weakened; these are typically balance equations. The essence of the method of integrodifferential relations (MIDR), which was grounded in the authors’ book mentioned above, is that equations of the third type are represented in the integral form, whereas the other equations must be considered as strict constraints. The initialboundary value problem modified in accordance with this scheme can be reduced to the minimization of a non-negative functional over all admissible variables. Such a reformulation became a starting point to develop advanced numerical techniques to state analysis, solution quality estimation, and optimization in dynamics of solids. https://doi.org/10.1515/9783110516449-201

VIII | Preface Tentatively, the book can be divided into three principal parts. The first part deals with different variational formulations of initial-boundary value problems that provides the foundation for the other two parts. In Chapter 1, the state of the art in the dynamics of solids is presented. A major emphasis in the survey is placed on a variety of direct as well as inverse dynamic problems and on the method of integrodifferential relations in statics and dynamics. The basic ideas of this approach are discussed with the example of deformations of elastic bodies. In Chapter 2, generalized statements of conventional parabolic and hyperbolic problems are presented and their variational properties are analyzed. Methodological aspects of the MIDR are outlined by involving firstly the Cauchy problem for finitedimension mechanical systems with elastic elements. Then the initial-boundary value problems describing vibrations of elastic structures and processes of heat transfer or diffusion in solids are considered. The classical and novel, following the MIDR, variational principles in linear elasticity are introduced in the next chapter for dynamic and static cases. Special attention is paid to the relation between the novel formulations and the dual Hamilton principles. Chapter 4 is devoted to variational statements in structural mechanics. In its beginning, the motions of an elastic beam and viscoelastic rod are discussed in detail. In the end of this chapter, variational properties of minimized functionals describing the behavior of mechanical structures with lumped as well as distributed parameters are discussed. In the middle part of the book (Chapters 5–8), suitable variational and projection procedures based on the MIDR to solve initial-boundary and boundary value problems are described. Various ways to weaken the constitutive relations between balanced stresses and compatible strains as well as between momentum density and velocities of material points, which are given usually in the local form, are presented in this part. In Chapter 5, usefulness of the Ritz method, conventionally applied in statics, in solving the initial-boundary value problems is shown. Specific features of this approach and its efficiency in dynamics are illustrated by numerical solutions on the basis of polynomial approximation and the finite element method. Possible advantages and shortcomings of semi-discrete approximation to obtain numerical results for dynamic problems in mechanics of structures are discussed in Chapter 6. Productivity of such kind of approximations for variational and projection algorithms is analyzed with the example of elastic rod motions. After that, the comparison of the Ritz and Galerkin methods in beam dynamics is performed. Chapter 7 is devoted to numerical analysis of eigenvalue problems based on variational as well as projection techniques and the MIDR. Firstly, eigenvalue properties are studied for elementary parabolic and hyperbolic differential equations. Then the semi-discretization developed in the previous chapter for initial-boundary value problems is extended to the case of natural in-plane vibrations of rectangular plates.

Preface

| IX

In Chapter 8, harmonic motions of elastic rectilinear beams with convex cross sections are investigated in the frame of the 3D model in the linear theory of elasticity. It is shown that the natural vibrations of the beams with cross sections having more than one axis of symmetry are decomposed into at least four independent motions, namely, longitudinal, torsional, and two types of lateral forms. Vice versa, the lack of symmetry leads to complex interdependent free and forced motions, as demonstrated for beams with asymmetric triangle cross sections. The last part of the book deals with one type of inverse dynamic problems, namely, control problems. In accordance with the MIDR, a numerical procedure of double minimization is proposed in Chapter 9 to design optimal strategies of mechanical system motions. The key point of this approach is that a solution quality functional and an objective index are sequentially minimized over state and control parameters, respectively. Accordingly, finite element algorithms for the optimization of elastic rod or body motions controlled by boundary displacements and forces are described. Questions concerning the incorporation of semi-discrete approximations and polynomial control laws into numerical schemes for inverse dynamic problems are considered in Chapter 10. Based on the MIDR and the variational formulation, a modification of objective functions aimed to regularize numerically ill-posed inverse problems is discussed. Additionally, Pontryagin’s maximum principle combined with model reduction is applied to design a feedforward strategy for the optimal locomotion of a 3D beam. Chapter 11 is devoted to various applications of the above-mentioned methods in mechatronics. The following optimal control problems are studied: rotations of an electromechanical manipulator with a flexible link, modeling and control of a high bay rack feeder with viscoelastic elements. It is shown that the proposed methodology is applicable to control of real-world technical systems and gives one the possibility to develop new efficient algorithms. The Appendix contains the most important definitions and necessary information on the vector and tensor algebra as well as functional analysis. Potential readers of this book will be mathematicians, engineers, as well as graduate and postgraduate students who are interested in learning the mathematical basis, modeling, and numerical technique in solid mechanics.

Basic notation The following notation is used throughout the book unless otherwise specified.

Acronyms BVP DAE EVP FEM IBVP LSM MIDR ODE PDE

boundary value problem differential-algebraic equation eigenvalue problem finite element method initial-boundary value problem least squares method method of integrodifferential relations ordinary differential equation partial differential equation

Functions and constants ℕ = {0, 1, 2, …} ℤ+ = {1, 2, 3, …} a(x), b(x) ∈ ℝn A(t) ∈ ℝn×n B(t) ∈ ℝn×m 4 C(x) ∈ ℝd d ∈ {1, 2, 3} f (t, x) ∈ ℝd {i, j, k, l, m, n} ∈ ℤ i, j ∈ ℤn J ∈ℝ K ∈ ℝn×n L>0 M ∈ ℝn×n n(x) ∈ ℝd p(t, x) ∈ ℝd p(t) ∈ ℝn q(t, x) ∈ ℝd s(t) ∈ ℝn t∈ℝ T >0 u(t) ∈ ℝm

set of natural numbers set of positive integers basis vectors state matrix of a linear ODE system control matrix of a linear ODE system elastic modulus tensor space dimension (volume) force vector integer indices multi-indices cost function in optimal control problems stiffness matrix beam or rod length mass matrix unit outward normal vector momentum density vector generalized momenta of a mechanical system vector of surface stress/heat flux generalized force vector time terminal time instant control vector

https://doi.org/10.1515/9783110516449-202

XII | Basic notation v(t, x) ∈ ℝd V ⊂ ℝd w(t, x) ∈ ℝd w(t) ∈ ℝn W(t) ∈ ℝ x ∈ ℝd x(t) ∈ ℝn γ Γ Δ ε(t, x) ∈ ℝd×d κ(x) > 0 λ∈ℂ ν(x) ∈ ℝ ξ (t, x) ∈ ℝd×d ρ(x) > 0 σ(t, x) ∈ ℝd×d Υ∈ℝ Φ≥0 Ψ≥0 Ω ω≥0

(residual) velocity vector spatial domain displacement vector generalized coordinates of a mechanical system mechanical energy spatial coordinate vector state vector of an ODE system weighting coefficient boundary of a spatial domain relative integral error strain tensor stiffness coefficient eigenvalue Poisson’s ratio residual strain tensor material volume density stress tensor functional of action constitutive functional time integral of energy domain of a function frequency

Contents Preface | VII Basic notation | XI 1

Introduction | 1

2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.4.3

Generalized formulations of parabolic and hyperbolic problems | 9 Finite-dimensional mechanical systems | 9 Forced motions of a mechanical system with elastic elements | 10 Variational statement of mechanical problems | 11 Relation with conventional variational principles | 13 Longitudinal motions of elastic rods | 15 Equations of rod dynamics | 16 Variational statement of the IBVP | 17 Membrane vibrations and acoustic waves | 20 Classical statement | 21 Weakened formulation relying on the MIDR | 23 Conditions of stationarity | 24 Heat transfer in solids | 25 Equations in linear thermodynamics | 26 Generalized formulation based on the MIDR | 26 Variational properties of the minimization problem | 28

3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3

Variational principles in linear elasticity | 31 Dynamics of elastic bodies | 31 Problems of elastodynamics | 31 Hamilton principles | 33 Integrodifferential statement in elasticity | 35 A family of constitutive functionals | 36 Comparative analysis of variational problems | 38 Dynamic variational principle in displacements and stresses | 41 Spectral problems in elasticity | 43 Harmonic vibrations of elastic bodies | 43 Natural vibrations of solids | 45 Variational statements of harmonic problems | 45 Variational formulations in elastostatics | 47 Static problems in linear elasticity | 47 Relationship of static variational principles | 48 Bilateral energy estimates | 50

XIV | Contents 4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2 4.2.1 4.2.2 4.3 4.3.1 4.3.2

Variational statements in structural mechanics | 53 Lateral motions of elastic beams | 53 Dynamic equations for beam bending | 53 Complementary Hamilton principles | 55 Method of integrodifferential relations in beam theory | 57 Energy estimates of solution quality | 58 A family of variational problems | 59 Comparison of variational formulations | 62 Longitudinal motions of viscoelastic rods | 64 Models of deformation with viscosity and rheology | 64 Minimization with integral constitutive relations | 65 Structures with lumped and distributed parameters | 67 Motions of a rod weighted at the ends | 67 Variational statement for hybrid system dynamics | 69

5 5.1 5.1.1 5.1.2 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.4 5.4.1 5.4.2 5.4.3 5.4.4

Ritz method for initial-boundary value problems | 71 Finite-dimensional dynamic problems | 71 Chain of linear oscillators | 71 Polynomial approximation of time functions | 72 Bivariate polynomials in rod and beam modeling | 74 Longitudinal motions of an elastic homogeneous rod | 74 Conventional Galerkin method | 77 Ritz method and the MIDR | 78 Lateral elastic displacements | 83 Numerical simulation of beam bending | 85 FEM modeling of elastic rod dynamics | 88 Modified minimization problem | 88 Piecewise polynomial approximations | 89 Continuity of kinematic and dynamic fields | 93 Constitutive relations in the FEM | 97 Spline representation of elastic body motions | 101 Approximation to a problem of elastodynamics | 101 Forced motions of an elastic body | 104 Approximations of displacement and stress fields | 105 Numerical example | 107

6 6.1 6.1.1 6.1.2 6.1.3 6.2

Variational and projection techniques with semi-discretization | 113 Variational approach to elastic structure dynamics | 113 Variational statements of a dynamic problem | 113 Approximating system of ODEs | 114 Numerical analysis of elastic beam motions | 118 Projection approach to dynamic problems | 122

Contents | XV

6.2.1 6.2.2 6.3

Projection statement for elastic beam motions | 122 Modification of the Petrov–Galerkin method | 123 FEM realization of the Petrov–Galerkin method | 125

7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6 7.3 7.3.1 7.3.2 7.3.3 7.3.4

Integrodifferential approach to eigenvalue problems | 129 Modification of the Galerkin method for elastic structures | 129 Longitudinal motions of an elastic rod | 129 Comparison of Galerkin and variational approaches | 131 Projection approach based on the MIDR | 136 Semi-discretization in problems of natural beam vibrations | 139 Natural vibrations of elastic plates | 139 Variational approach to the eigenvalue problem | 142 Projection approach to the eigenvalue problem | 144 Variational versus projection approaches | 146 Longitudinal plate vibrations | 147 Lateral in-plane vibrations | 150 Special models for plate motions | 152 Statement of the eigenvalue problem | 152 Simplified model of longitudinal vibrations | 156 Refined model of 2D rod vibrations | 159 Lateral vibrations of a free 2D beam | 161

8 8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.1.5 8.1.6 8.1.7 8.2 8.2.1 8.2.2 8.2.3 8.2.4

Spatial vibrations of elastic beams with convex cross-sections | 165 Natural motions of a cuboid beam | 165 Projection approach to a 3D eigenvalue problem in elasticity | 165 System of DAEs approximating the beam vibrations | 167 Decomposition of vibration equations for a homogeneous beam | 168 Breathing of a body with the square cross section | 170 Torsion of the body | 173 Longitudinal vibrations | 174 Lateral vibrations | 177 Natural vibrations of beams with triangular cross sections | 180 Dynamics of an elastic triangular prism | 180 Semi-discretization of displacement and stress fields | 181 Integral projections for the triangular cross section | 184 Natural vibrations of a beam with the isosceles cross section | 189

9 9.1 9.1.1 9.1.2 9.1.3

Double minimization in optimal control problems | 197 Optimization of beam motions with polynomials | 197 Statement of an optimal control problem | 197 Discretization based on the MIDR | 198 Parametric optimization of the beam motions | 199

XVI | Contents 9.1.4 9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.2.6 9.2.7 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5

Numerical examples of controlled motions | 200 Polynomial control in dynamic problems of linear elasticity | 206 Statement of an inverse dynamic problem | 206 Time–space discretization based on the MIDR | 208 Optimization of motion parameters | 208 Structure of polynomial approximations | 209 Longitudinal motions of a controlled elastic rod | 210 Controlled motions of a cuboid body with the square bases | 212 Comparison of 2D and 4D models | 215 FEM in control problems of elastic rod displacements | 220 Statement of an optimal control problem | 220 Parameterization of state and control functions | 221 Control optimization for approximate systems | 223 Regularization of integral error | 224 Exact solution of the control problem | 226

10 Semi-discrete approximations in inverse dynamic problems | 229 10.1 Projection approach to optimization in elastodynamics | 229 10.1.1 Semi-discretization and FEM in controlled beam dynamics | 229 10.1.2 Optimization and regularization of approximate solutions | 231 10.1.3 Solution quality in parametric optimization | 232 10.2 Optimal control of elastic body motions | 235 10.2.1 Variational formulation of a direct dynamic problem | 235 10.2.2 Projection formulation of the problem on body motions | 237 10.2.3 Statement of an optimal control problem | 238 10.2.4 Algorithm of discretization | 239 10.2.5 Spectral boundary value problem | 242 10.2.6 System of ODEs with respect to time | 244 10.2.7 Finite-dimensional control problem | 245 10.3 Variational approach to optimization of parabolic systems | 248 10.3.1 Statement of a control problem | 248 10.3.2 Fourier method in heat transfer problems | 250 10.3.3 Optimal control problem of rod heating | 252 10.3.4 Variational formulation of the IBVP | 253 10.3.5 Spatial discretization with polynomials | 253 10.3.6 Discretization error in the variational approach | 255 10.3.7 Numerical results of heat control | 255 11 Modeling and control in mechatronics | 259 11.1 Optimal rotations of an electromechanical manipulator | 259 11.1.1 Motion of a flexible link by a drive | 259 11.1.2 Optimal angular rotation of the link | 261

Contents | XVII

11.1.3 11.2 11.2.1 11.2.2 11.2.3 11.2.4

Projection approach to the problem on link motions | 263 Control of a flexible structure with viscoelastic links | 268 Controlled mechanism with flexible links | 268 Optimal control problem of structure motion | 269 Finite element algorithm | 270 Numerical simulation of structure dynamics | 271

A

Vectors and tensors | 275

B

Sobolev spaces | 279

Bibliography | 281 Index | 287

1 Introduction One of the actual challenges in solving direct and inverse problems of dynamics, describing the behavior of physical systems, is the development of special approaches to effective modeling of processes and the optimal choice of control laws governing these processes. To do this, the theory and methods of mathematical physics are actively used, which are now intensively applied to different fields of science, for example, in chemistry, biology, economics, etc. These methods are also widely presented in the technology of modeling in engineering for various systems and devices. The goals of mathematical physics are closely related to the study of controlled processes in systems with distributed properties, which usually occupy a certain space region (continuous media, structures, etc.). Parameters that characterize the state of a dynamic system and its behavior usually depend on spatial coordinates and time. The models that describe the behavior of such systems can be divided into three hierarchical levels: description of the influence of the environment on distributed elements, characteristics of the interaction of elementary system volumes, and properties of a volume element. Relations at the first level are defined by external factors, including, in general, the boundary and initial conditions. The second level corresponds to the interaction of elementary volumes in accordance with the constitutive laws and takes into account, for example, the transfer of material particles in space, which makes it possible to obtain equations for the processes of an interelement interaction. Finally, the third level determines the properties of the medium in the elementary volume. As a rule, systems with distributed parameters are described by equations in partial derivatives, or equations of mathematical physics, and in some cases, integral or integrodifferential relations. These models can also include functionals with respect to unknown variables or functions. It is often required in variational approaches that such functionals reach their stationary value on an admissible set of functions. These functions are often the desired solution of the problem. In this case, the formulation of the problem is ordinarily based on the corresponding variational principle, which has a certain physical meaning. Sometimes, the solution gives the extremum of the functional. The classification of systems of partial differential equations and their relation to the calculus of variations can be found in the classic books of Courant and Hilbert [22], as well as Morse and Feshbach [70]. The variety of natural phenomena stipulates a wide range of approaches to solving problems in mathematical physics. A detailed description of methods in the field of computational mechanics is presented in [91]. Among these techniques, special attention should be paid to three areas that have subjected, especially in recent years, to significant development, namely, calculus of variations, projection approaches and the least squares method (LSM). These methods have their own advantages as well as some shortcomings that we would like to discuss in this book. https://doi.org/10.1515/9783110516449-001

2 | 1 Introduction Variational principles and their application to various areas of physics have a long history. Nevertheless, the significance of these principles became clearer only thanks to the achievements in the finite element method (FEM), which goes back to the works of Courant [21] and Turner [101]. Since then, numerous confirmations have been received that the theory of calculus of variations is a reliable tool in the mathematical justification of the FEM. Conversely, the rapid spread of this method has stimulated the development of variational approaches. The basic ideas of the FEM can be found, for example, in the books [7, 80, 93, 104]. Recently, numerical algorithms based on variational approaches have been intensively improved. Applications of these methods to various fields of mechanics, physics, and engineering are considered, for instance, in [32]. The variational technique described by the authors [56] made it possible to efficiently obtain a rather high number of new numerical results. Other approaches, such as the Petrov–Galerkin method [6, 11] or the LSM [60], have been also actively developed at the present time to include numerical modeling of dynamic processes. To improve the efficiency and reliability of calculations, various a priori and a posteriori estimates of the solutions quality are applied [90]. Among the variational statements in mathematical physics, one can outline the principles of minimum potential energy and complementary energy, the Hamilton principle (see [103]). An alternative approach to solving initial-boundary value problems of mathematical physics is based on the Laplace transforms [59]. It is possible to mention a systematic way to formulate new variational problems by using the Friedrichs transformation. An important feature of the variational principles is that the basic equations describing the behavior of a physical system follow directly from the necessary conditions of stationarity for the corresponding functional. In addition, variational statements have a number of advantages in comparison with the formulations given by differential equations in ordinary or partial derivatives. Firstly, the calculus of variations is suitable to convert the problem originally specified in derivatives to an equivalent problem, which is often solved in a more simple way than the original one. In the variational formulation under additional constraints, this transformation is as a rule performed by using the Lagrange multiplier method, which is a very effective and regular procedure. In the frame of this approach, families of variational principles that are equivalent to each other were obtained [37]. Secondly, if an exact solution of the problem cannot be found, the variational method often leads to different finite-dimensional formulations to get an approximate solution. Thirdly, the numerical implementation of the variational principles usually guarantees the stability of computations and the optimality of approximate solutions. In this case, the matrices defining a finite-dimensional system of equations are commonly symmetric and positive definite.

1 Introduction

| 3

One of the disadvantages of this approach is that variational principles have not been formulated for all problems of mathematical physics. It should be emphasized here that new methods are constantly proposed to extend the range of applicability for the calculus of variations. For example, new minimization problems with nonstandard boundary conditions in the linear theory of elasticity were justified in [89]. As a rule, certain mathematical difficulties also arise in constructing the quality estimates of obtained approximations [94]. When a variational problem is derived by using the Lagrange multipliers, it often loses the properties of positive definiteness and symmetry. This is the case for the Hu–Wasidzu principle in the theory of elasticity [103]. Projection approaches, such as the methods of Galerkin [11], Petrov–Galerkin [6], etc., are deprived of some of the drawbacks, which are inherent in variational approaches. Firstly, these methods are applicable to problems for which the variational principles have not been formulated yet. Secondly, the projection technique is a more flexible apparatus to compose the system of governing equations. Recent studies and a review of modifications concerning the Galerkin methods based on a variety of ways to select test and trial functions can be found in [38]. Certainly, projection approaches also have their drawbacks. In particular, the choice of test and trial functions is a procedure that is not always unambiguous and simple. It is difficult sometimes to ensure the stability of numerical algorithms and their convergence, especially in nonlinear problems. As well as for variational approaches, it is not so simple to construct reliable quality estimates of an approximate solution. The third approach, which is also attributed to methods of mathematical physics, is the LSM. A general review of the LSM, including FEM procedures, is presented in [13]. Indeed, it looks rather attractive to compose a nonnegative functional in the following way: all equations describing a physical phenomenon under study are squared, summed, and integrated in space and time. In addition, it is known in advance that the global minimum of this integral is equal to zero. Conventional FEM strategies are applied in LSM to find approximate solutions. At the same time, implicit bilateral estimates of the solution quality can be respectively constructed. The lower bound of the functional is known, and the value of the functional on the approximate solution can always be chosen as the upper bound. Nevertheless, it should be noted that the Euler–Lagrange equations (necessary conditions of stationarity) for the minimization problems formulated in the conventional way, in general, differ from the system of partial differential equations that generate this functional. In other words, the problem based on the LSM is a variational principle for another boundary value problem or an initial-boundary value problem. Thus, the questions of the existence and uniqueness of the solution for this system require a more detailed investigation. One of the common characteristics inherent in all of the above methods is some ambiguity in the formulation of finite-dimensional approximations to the solution. It

4 | 1 Introduction is not clear which of the relations need to be weakened, and which must be exactly satisfied. As an example, let us consider the equations of the linear theory of elasticity. In the initial formulation, there are 15 variables, namely, 12 components of stress and strain tensors, as well as three components of the displacement vector, which correspond to nine partial differential equations (equilibrium equations and kinematic relations) and six algebraic equations that define the constitutive (Hooke’s) law [103]. If all these equations, including the boundary conditions, are considered in the integral (weak) form, this leads to the Hu–Wasidzu principle, which is formulated with respect to 18 variables (three Lagrange multipliers are added). This statement does not contain any restrictions on unknown functions. The physical interpretation of the Lagrange multipliers follows from the stationarity conditions of the corresponding functional. In the variational problem, the number of independent variables is reduced by requiring a priori strict implementation of some constitutive equations. For example, one can derive the Hellinger–Reissner principle, in which there are 12 unknown functions. After the successive elimination of a part of these variables, the classical principle of minimum potential energy is obtained, in which only three variables remain, namely, the components of the displacement vector. The equivalence of these principles was theoretically grounded, for example, in [12]. But, it is clear from a practical point of view that it is much easier to solve the problem if there are only three variables instead of 18. Such uncertainty characterizes projection approaches, for instance, the Galerkin method. If the integral projections of the system equations are calculated, the appropriate choice of spaces of trial and test functions is of great importance. In addition to the ambiguity mentioned above, the LSM is very sensitive to the choice of weighting factors. The presence of such factors is due to the fact that the constitutive relations have different dimensions. Equilibrium equations, for instance, have the physical dimension of the force per unit volume, the relations of the Hooke’s law can be, in some cases, dimensionless like kinematic conditions; the boundary conditions are given in units of length or force per unit area. Note that defining the corresponding weighting coefficients for a system of equations is not a simple problem. An approach that retains a number of advantages and takes into account the shortcomings inherent in variational and projection techniques as well as the LSM is discussed in this book. We refer to it as the method of integrodifferential relations (MIDR) [43]. The essence of this approach is that some of the constitutive equations must be satisfied exactly, whereas the other relations are considered in integral form. Relations that need to be weakened are determined a priori, often from the physical point of view. For example, only the Fourier law is generalized in heat transfer problems, while the first law of thermodynamics, the initial and boundary conditions are strictly satisfied [87]. In computer simulation in elasticity, approximated stress and displacement

1 Introduction

| 5

fields obey the equilibrium equations, the kinematic relations, and boundary conditions, whereas Hooke’s law are weakened, i.e., are given integrally [43] or projected onto some finite-dimensional functional space [86]. An approximate solution of the resulting integrodifferential problem is possible to find by minimizing the corresponding quadratic functional under differential constraints in the form of equilibrium equations, kinematic relations, and boundary conditions. The value of the functional on an admissible solution can be used directly to estimate the quality of the approximations obtained, while the integrand serves as the local quadratic residual of the solution. This formulation is completely consistent with the ideas of the least-squares method, but it is simultaneous, as shown, as a variational principle. Thus, the calculus of variations and the LSM is combined in this case. Studies have shown [47] that there are other positive definite quadratic forms representing Hooke’s law, which are not necessarily complete squares, but they can also be the basis for variational principles. Thus, an energy error functional was constructed that allows us to divide the integrodifferential problem, originally formulated in terms of stresses and displacements, into two independent subproblems: one in displacements (the principle of minimum potential energy), the other in stresses (the principle of minimum complementary energy) [48]. For variational formulations in accordance with the MIDR, the bilateral energy estimates of the solution quality [47] were proposed. Finite element algorithms were worked out that allow not only to calculate the magnitude of the integral error, but also to elaborate strategies of the adaptive mesh refinement in order to improve the solution quality [51]. In accordance with the ideas of the MIDR, a projection approach was developed, as a modification of the Petrov–Galerkin method. If the semi-discrete polynomial approximations and projection techniques are used, three-dimensional static and dynamic problems of the theory of elasticity can be solved with rather high accuracy [83, 56]. The beam theory, based on intuitive hypotheses proposed by Bernoulli [25], occupies a special place among appoaches applied to approximate description of motions of elongated bodies. Despite the fact that this theory is suitable to a wide class of problems, it does not take into account the influence of such factors as shear displacements, deplanation, deformation of the cross section, and the coupling of longitudinal and lateral motions through Poisson’s ratio, etc. The refinement formulae were proposed that allow one to consider the tension– compression of the cross sections at longitudinal displacements (the Rayleigh correction: [77, 68]), and also the shears and rotations of the sections under elastic bending (Timoshenko’s beam model [99, 97]). In the classical model of beam torsion [98], a deplanation is taken into account, which can be found by solving the plane Poisson problem. In the model proposed by Reissner [81], a variational approach was used to derive equations describing the elastic bending of a thin plate under the predetermined distribution of displacement fields in the transverse direction. Variational

6 | 1 Introduction formulations are also applied to obtain joint higher-order beam equations that suppose specific distribution of displacements and stresses in an elastic body [64]. Beam models for structures composed of anisotropic or composite materials were proposed in [10, 79]. The local relations of Hooke’s law can be weakened not only integrally, but also by using an asymptotic approach [44]. This approach makes it possible to develop refining beam models that serve as a reliable tool for analysis of complex structures [50]. Separate approximations, including finite-dimensional representations with respect to a part of the coordinates and variable coefficients depended on one selected coordinate, were applied in the MIDR to reduce the original system of equations to the system of ordinary differential equations (ODE) [48, 52]. Note that the MIDR is primarily aimed at more efficient numerical procedures based on the ideas of the calculus of variations and discrete approximations of the unknown functions. The approaches discussed in this book were applied not only to static and spectral problems of the theory of elasticity, but also to the direct and inverse initial-boundary problems of the mechanics of solids [54], hydro- [57] and thermodynamics [86]. The design of control strategies for dynamic systems with distributed parameters has been actively studied in recent years. In the theoretical study of complex technical objects, such as manipulators, large space vehicles, etc., a number of control problems arise for dynamic systems consisting of coupled elastic and rigid bodies. The desire to increase the efficiency of the system leads to the neediness of the design weight reduction, which can lead to substantial structural flexibility and, as a result, unexpected vibrations that impair the quality of the dynamic process (accuracy, speed, energy, etc.) [73]. Processes such as oscillations, heat transfer, diffusion, and convection are a part of a large variety of applications in science and engineering. The theoretical foundation for optimal control problems with linear partial differential equations (PDEs) and convex functionals was established by Lions and others [66, 65, 100, 39]. Linear hyperbolic equations are treated, as in Lions’ book, in [2, 16]. An introduction to the control of vibrations can be found in [58]. Oscillating elastic networks are investigated in [35, 61, 63]. Different statements of control problems for dynamical systems and methods to obtain their solutions are discussed, for example, in [14]. There are basically two different approaches to the control design for distributed parameter processes. In the first approach, often called late lumping, the control laws are directly designed for the distributed parameter models and then converted to a finite approximation. It is worth noting that the control strategies for infinite-dimensional systems often rely on specific spectrum analysis of linear operators [9, 23]. The control method considered in [17, 18] enables us to analytically construct a constrained distributed control and ensures that the system is brought to a given state in a finite time. This method is based on a decomposition of the original system into several subsystems by

1 Introduction

| 7

the Fourier approach. Questions concerning the design of control laws for the elastic and viscoelastic systems with the help of boundary actions were considered in [40, 82]. A numerical approach for the solution of PDE-constrained optimal control problems is adapted to hyperbolic equations in [31]. The method of choice proposed there is either a full discretization method, in the case of small-sized problems, or the vertical method of lines, in the case of medium-sized problems. In applications, the second approach, so-called early lumping, is broadly used for numerical control design if the mathematical models are given in the form of PDEs. In this way, the initial-boundary value problem is first discretized and reduced to a system of ODEs by means of the Rayleigh–Ritz or the Galerkin methods. On the other hand, finite difference or finite element procedures as well as other model approximation techniques can be applied as shown in [8, 19]. The direct discretization is also well known in control problems (see [62], for instance). A family of Galerkin approximations based on solutions of the homogeneous beam equation was constructed and sufficient conditions for stabilizability of such finite-dimensional systems were derived in [105]. One of the drawbacks of the early lumping approach is that it is rather difficult to know the connection between the original distributed parameter model and its discretized version a priori. However, this relationship can be quantified by the explicit error estimates following directly from the MIDR formulation for inverse problems as shown in [4, 46, 75]. These estimates allow one to verify the quality of the finitedimensional modeling, to refine numerical solutions, and to make corresponding corrections of the control laws. The algorithms to the parametric optimization of the motion control for elastic structures with beam-type elements have been developed (see [49, 51, 54]). The main idea in this approach is two successive minimization procedures. After discretization of the system and control parameters, the minimum of a solution quality functional is found for arbitrary values of the control function. After this, the optimal control is designed by using the condition of stationarity of the objective functional.

2 Generalized formulations of parabolic and hyperbolic problems 2.1 Finite-dimensional mechanical systems Only for a restricted range of applications, control theory provides the exact solutions of inverse dynamic problems in analytic form. The implementation of effective numerical procedures based on the variational technique is limited by the lack of convenient generalized formulations of Cauchy problems in mechanics. The first attempts to use the calculus of variations in physics were made by Maupertuis at the beginning of the eighteenth century. In turn, the Hamilton principle of ‘least’ action has been widely spread throughout the scientific community, especially to derive the equations of motion [33]. A deep relationship between both variational formulations was displayed with exploiting the duality properties and the Friedrichs transformation [95]. These statements were cleared up to be true only for boundary problems in time. Therefore, any attempts are important to generalize the results to a more substantial case when the initial positions and speeds of system elements are given, and also to apply the variational approach to the dynamics of non-conservative systems [28]. In accordance with the Hamilton principle, conjugate (kinetic) foci appear, as usual, in variation of unknown functions of a dynamic system. This means that no extreme values of the functional action exist, but only its stationary points [34], what greatly complicates the verification of numerical methods and the estimation of approximation quality. Any variational formulation of Cauchy problems is of great interest since its numerical solution can be obtained by modifications of mathematically well-grounded techniques, such as, for example, the Ritz method, the Galerkin method, the FEM, etc. One drawback of said approaches is that it is difficult to determine a priori the relation between the original system with distributed parameters and its finite-dimensional approximate model. Such a relation can be established with explicit estimates of solution quality, following directly from the variational formulations of mechanical problems based on the MIDR [40]. The basic idea of this method is that the constitutive equations are satisfied in an integral form, and the other governing relations, such as Newton’s second law, are considered as differential or algebraic constraints. The variational approaches presented here allow us also to develop advanced algorithms of control design by taking into account local and integral quality criteria and to carry out online parametric optimization of complex dynamic processes. In this section, two variational statements of a Cauchy problem in terms of analytical mechanics will be given for controlled motions of a linear mechanical system with elastic elements (springs), which has a finite number of degrees of freedom. https://doi.org/10.1515/9783110516449-002

10 | 2 Generalized formulations of parabolic and hyperbolic problems The first variational formulation is reduced to constrained minimization of a nonnegative quadratic functional. This functional has the dimension of action and is an integral residual of the constitutive equations, which defines the relation between generalized momenta and velocities of system particles as well as between elastic forces in the springs and relative displacements of these particles. The second variational problem is associated with the Hamilton principle when certain conditions are given at both initial and terminal time instants of system motion. In what follows, extremal properties of these two generalized statements are studied for the considered type of mechanical systems. Bilateral accuracy estimates for approximate solutions of the original Cauchy problem are also proposed. 2.1.1 Forced motions of a mechanical system with elastic elements Let us consider a mechanical system with a finite number of degrees of freedom. The motion of the system is described by linear differential equations represented in the vector form: ̈ + K ⋅ w(t) = B ⋅ u(t) . M ⋅ w(t)

(2.1)

In equation (2.1), the unknown vector of generalized coordinates is defined as w(t) ∈ ℝn , and the vector of external loads is given by u(t) ∈ ℝm . The mass matrix M ∈ ℝn×n is positive-definite and symmetric, whereas the stiffness matrix K ∈ ℝn×n is symmetric and non-negative. The control matrix B ∈ ℝn×m as well as M and K are constant. The values of generalized coordinates and velocities are set at the initial time instant: w(0) = w 0

̇ and w(0) = ẇ 0 .

(2.2)

The dot between two vectors denotes the inner product whereas the dot between a matrix and a vector is explained in Appendix A. For example, if w is considered as a column vector then the product w ⋅ w ∶= w T w gives a scalar value whereas the operations K ⋅ w ∶= Kw and w ⋅ K ∶= KT w = Kw have the same resulting vector, by definition, due to the matrix symmetry. The aim of the study is to find the vector w ∗ (t) in the interval t ∈ [0, T], which is the solution of the Cauchy problem (2.1), (2.2). The vector functions of the generalized momenta and forces are represented respectively as p = M ⋅ ẇ

and

s = −K ⋅ w + B ⋅ u .

(2.3)

By taking into account equation (2.3), the equality (2.1) is expressed through the variables p and s in the equivalent form: ̇ = s(t) . p(t)

(2.4)

2.1 Finite-dimensional mechanical systems | 11

Introduce two vectors, further constitutive functions, defining the relationship between the velocities and the momenta as well as the coordinates and the forces so that q = M ⋅ ẇ − p and r = K−1 ⋅ s + w − K−1 ⋅ B ⋅ u .

(2.5)

It is worth noting that these functions are identically equal to zero on the exact solution. Given Newton’s second law (2.4), the vector of forces s can be excluded. Then the original problem (2.1), (2.2) is reduced to the ODE system of the first order ̇ − p(t) = 0 { q(t) = M ⋅ w(t) { −1 ̇ + w(t) − K−1 ⋅ B ⋅ u(t) = 0 r(t) = K ⋅ p(t) {

(2.6)

with the initial conditions w(0) = w 0

and p(0) = p0 = M ⋅ ẇ 0 .

(2.7)

After a linear transformation, the system (2.6) is brought to the canonical form [29] ̇ = M−1 ⋅ p(t) w(t)

and

̇ = −K ⋅ w(t) + Bu(t) . p(t)

(2.8)

2.1.2 Variational statement of mechanical problems To generalize the Cauchy problem (2.6), (2.7), let us compose a quadratic form consisting of the constitutive functions q and r as 1 φ+ = (q ⋅ M−1 ⋅ q + r ⋅ K ⋅ r) 2

with φ+ (t) ≥ 0 .

(2.9)

The physical dimension of this expression is energy. After integrating the function φ+ (t) over the interval of motion [0, T], the following functional is obtained: T

̇ ̇ Φ+ = ∫ φ+ (t, w(t), p(t), w(t), p(t)) dt ≥ 0 . 0

(2.10)

The integral (2.10) defines a quadratic form with respect to w, p, and their first derivatives. This form is the discrepancy for the ODE system (2.6). It should be emphasized that the value of Φ+ is equal to zero only for the actual motion. As the integral Φ+ reaches its absolute minimum at the exact solution of the system (2.6), the following variational problem can be formulated.

12 | 2 Generalized formulations of parabolic and hyperbolic problems Problem 2.1. Find the functions of coordinates q∗ (t) and momenta p∗ (t) such that Φ+ [w ∗ , p∗ ] = min Φ+ [w, p] = 0 w, p

(2.11)

subject to the constraints (2.7). In order to show that Problem 2.1 is a mathematically strict generalization of the original problem, let us derive the first variation of the introduced functional Φ+ T

δΦ+ = ∫ (q ⋅ M−1 ⋅ δq + r ⋅ K ⋅ δr) dt 0 T

= ∫ (q ⋅ δẇ − q ⋅ M−1 ⋅ δp + r ⋅ δṗ + r ⋅ K ⋅ δw) dt . 0

(2.12)

Suppose that the function of external loads u(t) is continuous in the interval t ∈ [0, T]. After integrating the expression (2.12) by parts and taking into account the initial constraints (2.7), the necessary condition for the stationarity of the functional Φ+ is obtained: T

T

0

0

δΦ+ = ∫ (K ⋅ r − q)̇ ⋅ δw dt − ∫ (M−1 ⋅ q + r)̇ ⋅ δp dt + [q ⋅ δw + r ⋅ δp]t=T = 0 .

(2.13)

Then the Euler–Lagrange equations with the terminal transversality conditions have the form: ̇ = K ⋅ r(t) q(t)

̇ = −M−1 ⋅ q(t) and r(t)

with q(T) = r(T) = 0 .

(2.14)

When expressing one constitutive function through the derivative of the other, for example, q = −Mr,̇ the ODE system of second order results as ̈ + K ⋅ r(t) = 0 with r(T) ̇ M ⋅ r(t) = r(T) = 0 .

(2.15)

It is possible to prove that the solution of equation (2.14) is r(t) ≡ 0

and

q(t) ≡ 0

(2.16)

in the given time interval. This means that the Euler–Lagrange equations of Problem 2.1 together with the initial conditions (2.7) and the equalities (2.4) are equivalent to the original ODE system (2.1), (2.2). If the function u(t) has discontinuities of the first kind at the time instants t = ti (ti > ti−1 , i = 1, … , k, t0 = 0), then the Weierstrass–Erdman conditions q(ti − 0) = q(ti + 0)

and r(ti − 0) = r(ti + 0) ,

(2.17)

2.1 Finite-dimensional mechanical systems | 13

have to be satisfied. But in our case, these conditions automatically hold in accordance with equation (2.16). Represent the admissible functions of generalized coordinates w and momenta p by means of the exact solution w ∗ , p∗ and the variations δw, δp in the form: w = w ∗ + δw

and p = p∗ + δp .

(2.18)

Given the exact solution, the minimum of the integral Φ+ and also its first variation are equal to zero. Due to the quadratic form of the functional, it follows that Φ+ [w, p] = Φ+ [w ∗ , p∗ ] + δΦ+ + δ2 Φ+ = δ2 Φ+ = Φ+ [δw, δp] ≥ 0 .

(2.19)

That is, the necessary condition of minimum for the second variation δ2 Φ+ is valid. Let us show that the functions q and r are the conjugate variables of the generalized coordinates w and momenta p, respectively. Indeed, it follows from the equalities (2.6) and (2.9) [30, 69, 72] that 𝜕φ+ 𝜕q = ⋅ M−1 ⋅ q = q 𝜕ẇ 𝜕ẇ

and

𝜕φ+ 𝜕r = ⋅K⋅r =r. 𝜕ṗ 𝜕ṗ

(2.20)

The Hamiltonian for the variational problem (2.7), (2.11) has the form ̇ w, r) − φ+ (q, r) , ̇ p) + r ⋅ p(t, H(t, p, w, q, r) = q ⋅ w(q,

(2.21)

where the derivatives are obtained from invertible linear relations introduced by equation (2.5) according to ẇ = M−1 ⋅ (q + p) and ṗ = K ⋅ (r − w) + B ⋅ u .

(2.22)

By taking into account the discrepancy function φ+ in the form (2.9), the explicit expression for the Hamiltonian (2.21) can be derived as 1 H = (q ⋅ M−1 ⋅ q + r ⋅ K ⋅ r) + q ⋅ M−1 ⋅ p − r ⋅ K ⋅ w + r ⋅ B ⋅ u(t) . 2

(2.23)

The direct differentiation of equation (2.23) confirms that Hamilton’s equations ẇ =

𝜕H , 𝜕q

ṗ =

𝜕H , 𝜕r

q̇ = −

𝜕H , 𝜕w

ṙ = −

𝜕H 𝜕p

(2.24)

are equivalent to the system (2.9), (2.11). 2.1.3 Relation with conventional variational principles Consider another quadratic integral: T

Φ− = ∫ φ− (t, w, p, w,̇ p)̇ dt 0

1 with φ− = (q ⋅ M−1 ⋅ q − r ⋅ K ⋅ r). 2

(2.25)

In contrast to Φ+ , the functional Φ− is not of a fixed sign. Therefore, the only possible generalized problem is to seek a stationary value of Φ− .

14 | 2 Generalized formulations of parabolic and hyperbolic problems Problem 2.2. Find such functions w ∗ (t) and p∗ (t) that equate to zero the first variation of the functional T

T

0

0

δΦ− = ∫ (ṙ − M−1 ⋅ q) ⋅ δp dt − ∫ (K ⋅ r + q)̇ ⋅ δw dt + [q ⋅ δw − r ⋅ δp]t=T = 0

(2.26)

subject to the constraints (2.7). The corresponding Euler–Lagrange equations under the terminal conditions can be written as q̇ = −K ⋅ r

and ṙ = M−1 ⋅ q

with q(T) = r(T) = 0 ,

(2.27)

which are equivalent to the ODE system (2.15). This proves the variational nature of this problem statement, too. Note that the integral Φ− has a special structure Φ− = Θ1 − 2Θ0 + Θ2 + Ξ0

with Ξ0 =

1 T ∫ (B ⋅ u) ⋅ K−1 ⋅ (B ⋅ u) dt 2 0

(2.28)

allowing us to reveal its correlations with the mechanical functional of action. The following notation T

Θ0 [w, p] = ∫ L0 dt , 0

1 ̇ L0 = (p ⋅ ẇ + w ⋅ p), 2

T

Θ1 [w] = ∫ L1 dt , 0

T

Θ2 [p] = ∫ L2 dt , 0

1 L1 = (ẇ ⋅ M ⋅ ẇ − w ⋅ K ⋅ w + 2w ⋅ B ⋅ u), 2

(2.29)

1 L2 = (p ⋅ M−1 ⋅ p − ṗ ⋅ K−1 ⋅ ṗ + 2ṗ ⋅ K−1 ⋅ B ⋅ u) 2 is used in (2.28). The functions L1 and L2 in equation (2.29) are quadratic forms of either w and ẇ or p and p.̇ The function L0 is a bilinear form of the variables w and p, which does not depend explicitly on any mechanical properties of the system. The last term Ξ0 in the expression of Φ− from equation (2.28) does not affect the result, since it depends exceptionally on the given vector of external loads u. For the system (2.1), the functions L1 and L2 represent the Lagrangian expressed in terms of the coordinate vector w or the momentum vector p, respectively. After integration by parts, the functional Θ0 has the form: Θ0 = [w ⋅ p]t=T t=0 .

(2.30)

Given the initial system state (2.7), the difference Θ0 depends on the terminal values of both the coordinates w and the momenta p.

2.2 Longitudinal motions of elastic rods | 15

Suppose that either a pair of the boundary conditions from Table 2.1 (all the rows except for the last) are given at the ends of the time interval t ∈ [0, T], or the periodic conditions can be set in accordance with the last row. By taking into account equation (2.30), the functional Φ− is representable in these cases as the sum of two terms, one of which depends only on the coordinates w while the other depends only on the momenta p according to Φ− = Θ1 [w] + Ξ1 [w] + Θ2 [p] + Ξ2 [p] − Ξ0 .

(2.31)

Here, the expressions of Ξ1 and Ξ2 are introduced in Table 2.1 for the conditions of types A and B defined at the initial and terminal time instants. Table 2.1: Functional terms for different time conditions. Condition A

Condition B

Term Ξ1

Term Ξ2

w(0) = w 0

w(T ) = w 1

0

w 1 ⋅ p(T ) − w 0 ⋅ p(0)

p(0) = p0

p(T ) = p1

p1 ⋅ w(T ) − p0 ⋅ w(0)

0

w(0) = w 0

p(T ) = p1

p1 ⋅ w(T )

−w 0 ⋅ p(0)

p(0) = p0

w(T ) = w 1

−p0 ⋅ w(0)

w 1 ⋅ p(T )

w(0) = w(T )

p(0) = p(T )

0

0

It follows from the structure of the functional Φ− that the original variational problem for all five types of boundary conditions in Table 2.1 splits into two independent subproblems: the first over the displacements w and the second over p as δΦ− [w, p] = 0

⇔

δΘ1 [w] + δΞ1 [w] = 0

and

δΘ2 [p] + δΞ2 [p] = 0 .

(2.32)

The first subproblem is to find the stationary value of the action in accordance with the conventional Hamilton principle [30], whereas the second subproblem corresponds to the complementary (dual) principle [95] formulated with respect to the generalized momenta p. All the above-mentioned statements can be generalized to the case of linear mechanical systems with time-varying parameters M(t), K(t), B(t), and also to the case when rank K ≤ rank M = n under the constraints w ⋅ K(t) ⋅ w ≥ 0 and w ⋅ M(t) ⋅ w > 0 for any time instant t ∈ [0, T] and any displacements w ≠ 0.

2.2 Longitudinal motions of elastic rods The extension of variational approaches to systems with spatially distributed parameters is better to illustrate by model problems, perhaps, rather simple in structure but

16 | 2 Generalized formulations of parabolic and hyperbolic problems still reflecting basic properties of variational formulations common for more meaningful objects. As an example, longitudinal displacements of a thin rectilinear elastic rod are studied.

2.2.1 Equations of rod dynamics Introduce a coordinate axis Ox directed along the elastic rod. The origin of the axis is placed at the point O and shown in Figure 2.1. The position of any rod point in deformed state is defined by the function X(t, x) = x + w(t, x) of the time t and the coordinate x. Here, x is the position of this point in an undeformed state and u is the function of elastic deformations. If the strain of the rod is set with respect to the variable X, then it is said that the Euler description of rod motion is given, else all governing relations are expressed through the variables x and the Lagrange description is chosen.

Figure 2.1: Thin rectilinear elastic rod.

Let κ(x) ∶= A(x)E(x) denote the stiffness coefficient of the rod with Young’s modulus E(x) > 0 of material and the cross-sectional area A(x) > 0. The rod linear density ρ(x) ∶= A(x)ρv (x) is proportional to the material volume density ρv (x) > 0. Let also f (x) is the linear density of external forces; κ0 and κ1 are the coefficients of elastic supports at the rod ends. In the absence of deformations and velocities, the rod occupies the segment x ∈ [x0 , x 1 ]. The motion under study starts at the time instant t = t 0 and finishes at t = t 1 . The problem time-space domain is (t, x) ∈ Ω = (t 0 , t 1 ) × (x 0 , x 1 ). In accordance with the MIDR [56], the unknown functions, which should be found, are the longitudinal displacements w(t, x) of rod points, the linear density of momentum p(t, x), and the normal forces s(t, x) ∶= A(x)σ11 (t, x) acting in the cross section of the rod. Here, σ11 is the component of the stress tensor, which is normal to the rod’s cross section. In turn, the only component of the strain tensor considered in the model is ε11 . Divide all governing equations describing the stress-strain state of the rod into three groups: (a) kinematic relations ε11 (t, x) = wx (t, x) and v1 (t, x) = wt (t, x)

for (t, x) ∈ Ω ;

(2.33)

2.2 Longitudinal motions of elastic rods | 17

(b) constitutive relations (Hooke’s law and the velocity-momentum relation) σ11 (t, x) = E(x)ε11 (t, x)

and p(t, x) = ρv1 (t, x) for (t, x) ∈ Ω ;

(2.34)

(c) balance equations (Newton’s second law of motion) pt (t, x) = sx (t, x) + f (t, x) for (t, x) ∈ Ω .

(2.35)

By expressing the functions of strain, velocity, stress, and momentum in equation (2.35) through the displacements w in accordance with equations (2.33) and (2.34), the following second-order PDE is obtained: ρ(x)wtt (t, x) = κ(x)wxx (t, x) + f (t, x)

for (t, x) ∈ Ω .

(2.36)

The boundary conditions at the rod ends x = x0 and x = x1 can be sorted into three categories: (i) kinematic constraints (Dirichlet conditions in the homogeneous case) w(t, x 0 ) = u0 (t) ,

w(t, x 1 ) = u1 (t) ;

(2.37)

(ii) dynamic constraints (Neumann conditions in the homogeneous case) s(t, x0 ) = u0 (t) ,

s(t, x1 ) = u1 (t) ;

(2.38)

s(t, x1 ) + κ1 w(t, x 1 ) = u1 (t) .

(2.39)

(iii) mixed constraints s(t, x0 ) − κ0 w(t, x0 ) = u0 (t) ,

Here, u0 (t) and u1 (t) are either prescribed displacements or forces. Only one boundary condition of the three types presented in equations (2.37)–(2.39) must be given at each rod end. As a result, nine different types of boundary constraints are able to be chosen. To complete the formulation of an IBVP, the initial conditions on displacements and momentum density have to be added: w(t0 , x) = w0 (x)

and p(t0 , x) = p0 (x) ,

(2.40)

where the functions w0 (x) and p0 (x) are given initial displacements and momentum density, respectively.

2.2.2 Variational statement of the IBVP The solution w∗ (t, x), p∗ (t, x), s∗ (t, x) of the IBVP (2.36)–(2.40) may not exist in the classical sense depending on the smoothness properties of the functions ρ(x), κ(x),

18 | 2 Generalized formulations of parabolic and hyperbolic problems f (t, x), w0 (x), p0 (x), u0 (t), and u1 (t). To generalize the problem, an integral statement of the constitutive laws proposed in [48] is considered instead of their local formulation (2.34). To relate the momentum density with the velocities as well as the normal stresses with the strains according to equations (2.33) and (2.34), introduce two auxiliary constitutive functions: v(t, x) ∶= wt (t, x) − ρ−1 (x)p(t, x)

and q(t, x) ∶= wx (t, x) − κ −1 (x)s(t, x) .

(2.41)

For the exact solution, these functions must be equivalent to zero as: v(t, x) = 0

and q(t, x) = 0 .

(2.42)

The generalized statement for a problem of elastic rod motions can be formulated as follows. Problem 2.3. Find such functions w∗ (t, x), p∗ (t, x), s∗ (t, x) that obey an integral equation Φ[w, p, s] = ∫ φ(t, x) dΩ = 0

(2.43)

φ(t, x) ∶= aρ(x)v2 (t, x) + bκ(x)q2 (t, x)

(2.44)

Ω

with

as well as the balance relation (2.35), two of the boundary conditions (2.37)–(2.39), and the initial conditions (2.40). Here, Φ is a functional, and φ is a function of quadratic discrepancy for the constitutive equations (2.42). It is worth noting that the integrand φ defined in (2.44) is non-negative at any values a > 0 and b > 0. It directly follows from the last property that the functional Φ is non-negative, too. This allows us to reformulate the integrodifferential Problem 2.3 as a variational one. Problem 2.4. Find those functions w∗ (t, x), p∗ (t, x), s∗ (t, x) that minimize the functional Φ[w∗ , p∗ , s∗ ] = min Φ[w, p, s] = 0 w, p, s

(2.45)

subject to the constraints (2.35), (2.37)–(2.39), and (2.40). If a < 0 and b < 0, the problem (2.37) is also reducible to the minimization after plain multiplication of Φ by the factor −1. Problem 2.4 is equivalent to the following formulation.

2.2 Longitudinal motions of elastic rods | 19

Problem 2.5. Find the functions w∗ (t, x), p∗ (t, x), s∗ (t, x) that are a stationary point for Φ, that is, δΦ[w, p, s] = 0

(2.46)

subject to the same constraints (2.35), (2.37)–(2.39), and (2.40). This statement is valid even if ab < 0. Denote the actual and arbitrarily chosen admissible displacement, momentum, and force fields via w∗ , p∗ , s∗ and w, p, s, respectively. Define that w = w∗ + δw ,

s = s∗ + δs ,

p = p∗ + δp ,

(2.47)

where δp, δs, δw are variations of displacements, momentum density, and forces, respectively. Then Φ[p, s, w] = Φ[w∗ , p∗ , s∗ ] + δΦ[w∗ , p∗ , s∗ , δw, δp, δs] + δ2 Φ[δw, δp, δs] .

(2.48)

Here, Φ[w∗ , p∗ , s∗ ] = 0 in accordance with (2.42). It follows from the quadratic structure of the functional Φ that the second variation δ2 Φ[δw, δp, δs] = Φ[δw, δp, δs] is nonnegative if a(x) > 0 and b(x) > 0 for x ∈ (x 0 , x1 ). Derive the system of Euler–Lagrange equations together with the corresponding natural conditions for the problem (2.44). The first variation of the functional can be represented as the sum δΦ = δw Φ + δp Φ + δs Φ of the variations with respect to the unknowns w, p, s. The relation between the momentum function p and the force function s according to Newton’s second law (2.35) has to be taken into account in the corresponding relation between their variations: δpt = δsx .

(2.49)

After integrating the expression δΦ by parts and taking into account the constraints of Problem 2.5, the necessary condition of stationarity is obtained: δw Φ + δp Φ + δs Φ = 0 , δw Φ = − ∫ (aρ(x)vt + (bκ(x)q)x )δw dΩ Ω

x1

t1

x0

t0

1

+ a ∫ [ρ(x)vδw]t=t 1 dx + b ∫ [κ(x)qδw]x=x dt , x=x0

(2.50)

δp Φ = −a ∫ vδp dΩ , Ω

δs Φ = −b ∫ qδs dΩ . Ω

It can be seen from equation (2.50) that δΦ = 0 over all admissible variations δw, δp, δs if the equalities (2.42) hold.

20 | 2 Generalized formulations of parabolic and hyperbolic problems Introduce an auxiliary function t

g(t, x) ∶= − ∫ v(τ, x) dτ

(2.51)

t0

and get the expression for the first variations with respect to p and s after some equivalent transformations as follows: δp Φ + δs Φ = a ∫ gt δp dΩ − b ∫ qδs dΩ = ∫ (agx − bq)δs dΩ Ω

Ω

x1

t1

Ω

x=x

+ a ∫ [gδp]t=t 1 dx + a ∫ [gδs]x=x10 dt . x0

t0

(2.52)

By using equations (2.50)–(2.52), it is possible to derive the Euler–Lagrange system with the corresponding terminal conditions: ρ(x)gtt − (κ(x)gx )x = 0 , 1

q = gx ,

1

g(t , x) = gt (t , x) = 0 .

(2.53)

For the sake of simplicity, let the boundary conditions of the first category (2.37) and/or the second category (2.38) be given in Problem 2.5. It means that either the displacements w or the forces s are prescribed at the points x = x 0 and x = x0 . Then the natural conditions at the rod ends following from four types of boundary constraints as w(t, x0 ) = u0 (t) ∧ w(t, x1 ) = u1 (t)

⇒

g(t, x 0 ) = 0 ∧ g(t, x1 ) = 0 ,

w(t, x0 ) = u0 (t) ∧ s(t, x1 ) = u1 (t)

⇒

g(t, x 0 ) = 0 ∧ gx (t, x1 ) = 0 ,

s(t, x 0 ) = u0 (t) ∧ w(t, x1 ) = u1 (t)

⇒

gx (t, x0 ) = 0 ∧ g(t, x1 ) = 0 ,

s(t, x 0 ) = u0 (t) ∧ s(t, x1 ) = u1 (t)

⇒

gx (t, x0 ) = 0 ∧ gx (t, x1 ) = 0 .

(2.54)

The homogeneous system (2.53), (2.54) determines a terminal-boundary value problem with respect to the variable g(t, x). It can be shown that there is only a trivial solution of this problem, and hence q(t, x) ≡ 0 and v(t, x) ≡ 0. In other words, if the solution w∗ (t, x), p∗ (t, x), s∗ (t, x) of the IBVP (2.35), (2.37), (2.38), (2.40), and (2.42) exists in the classical sense, then the system of necessary stationarity conditions (2.53) together with the essential and natural boundary relations (2.54), Newton’s second law (2.40), as well as the initial constraints (2.40) is equivalent to the original problem of elastic rod motion. This means that Problems 2.3, 2.4, 2.5 are formulated strictly in terms of the calculus of variations.

2.3 Membrane vibrations and acoustic waves As a natural extension of the problem on elastic rod motions discussed in the previous section, hyperbolic IBVPs governed by a scalar PDE with the multidimensional

2.3 Membrane vibrations and acoustic waves | 21

Laplace operator is considered. Such systems of equations describe, for instance, forced vibrations of a tense membrane spanned on a rigid frame, propagation of acoustic wave in a tank filled with air, and so on. 2.3.1 Classical statement Restrict ourselves to the case when a linearly elastic medium occupies a volume x ∈ V ⊂ ℝd of the dimension d = 1, 2, 3. To be more specific, let small d-dimensional motions of a compressible ideal fluid inside a solid thin-walled tank covering the volume V as its boundary Γ = 𝜕V be described in a Cartesian coordinate system. The functions ̂ x) ∶= p0 + p(t, x) ∈ ℝ , p(t,

̂ x) ∶= ρ0 (x) + r(t, x) ∈ ℝ , r(t,

̂ x) = q0 (x) + q(t, x) ∈ ℝd q(t, denote in this section the absolute pressure, the volume density, and the mass flux, respectively. Here, p0 denotes the average of pressure over the volume V at the initial time instant t = t0 , ρ0 is the nominal density, which can vary over the spatial coordinates x, and q0 is the initial flow distribution. It is supposed that the values p(t, x), r(t, x), and q(t, x) change only slightly in the time interval t ∈ [t0 , t1 ]. Thus, the functions p, r, and q represent small deviations from the corresponding reference values p0 > 0 and ρ0 (x) > 0, and q0 (x) for x ∈ V . The turbulent and vortex fluctuations of the fluid parameters are not considered. In what follows, the system are derived in terms of the pressure p and the mass flow q as dynamic unknowns. In the adiabatic model of compressible ideal fluids, the pressure p is represented as a linear function of the volume density r according to p(t, x) = c02 r(t, x)

with c02 =

κ0 (x) ρ0 (x)

for (t, x) ∈ Ω ,

(2.55)

where c0 is the speed of sound and κ0 is the bulk modulus. The equation of momentum balance for the media is defined in the form qt (t, x) + ∇p(t, x) = 0

for (t, x) ∈ Ω ,

(2.56)

where Ω = (t 0 , t 1 ) × V is the space–time domain of the unknown functions and ∇ denotes the gradient operator. The volume V has to meet the requirements of smoothness, namely, to be Lipschitz continuous or even piecewise smooth. The reference mean pressure in the medium is supposed to be big enough to guarantee its continuity according to rt (t, x) + ∇ ⋅ q(t, x) = f (t, x)

for (t, x) ∈ Ω .

(2.57)

Here, the function f is responsible for distributed mass in- and outflow of different nature.

22 | 2 Generalized formulations of parabolic and hyperbolic problems Usually the velocity potential is introduced to derive the wave equation [70]. To compose a hyperbolic system, which conforms to the governing relations for elastic rod or spring vibrations considered in the previous section, a time potential w(t, x) is defined on the domain Ω in accordance with the following constitutive laws: r(t, x) = −ρ(x)wt (t, x) and q(t, x) = κ(x) ⋅ ∇w(t, x)

for (t, x) ∈ Ω .

(2.58)

By choosing the corresponding material constants as ρ(x) =

κ̄ 0 ρ0 (x) κ0 (x)

and κ(x) = [

κ̄ 0 0

0 ] κ̄ 0

with κ̄ 0 = ∫ κ0 (x) dV , V

the equality for acoustic motions ρ(x)wtt (t, x) − ∇ ⋅ (κ(x) ⋅ ∇w(t, x)) = 0

for (t, x) ∈ Ω

(2.59)

is obtained after substituting equation (2.58) into equation (2.57). The momentum balance (2.56) is automatically satisfied if the constitutive relations (2.58) are valid. Note that the positive defined symmetric tensor κ of second rank is involved here instead of a scalar to take into account, in the general case, a possible spatial anisotropy of the dynamic equation (2.59). In acoustics, either the pressure or the mass flux along the outer unit normal n(x) is often given on the boundary Γ. The generalized linear condition can be written down with respect to the time potential w as α(x)w(t, x) + β(x)n(x) ⋅ κ(x) ⋅ ∇w(t, x) = u(t, x)

for x ∈ Γ ,

(2.60)

where the coefficients α and β characterize the type of this condition. For example, if α = 1 and β = 0 then the pressure is defined through the function ut . In contrast, if α = 0 and β = 1 then the mass flux is prescribed by u. In the case when α = 1 but β ≠ 0, certain flexibility of the limiting tank shell can be taken into account. To obtain a unique solution of equations (2.59)–(2.61), the time potential and its time derivative have to be defined at the initial time instant according to w(t0 , x) = w0 (x) and wt (t0 , x) = v0 (x)

for x ∈ V .

(2.61)

Then the corresponding IBVP can be stated as follows. Problem 2.6. Find a scalar function w∗ (t, x) that obeys the governing equations (2.55)– (2.59) for given functions f , u, w0 , and v0 .

2.3 Membrane vibrations and acoustic waves | 23

2.3.2 Weakened formulation relying on the MIDR The constitutive laws (2.58) defining a linear relation between the relative density r and the time derivative of the potential w as well as between the flux vector q and the gradient of w can be rewritten through the constitutive functions v ∶= wt + ρ−1 r

and s ∶= ∇w − κ −1 ⋅ q

(2.62)

according to v(t, x) = 0

and s(t, x) = 0

for (t, x) ∈ Ω .

(2.63)

The balance relation (2.57) remains as a PDE constraint, whereas the boundary condition (2.60) is possible to be represented through the potential w and the flux q in the form: α(x)w(t, x) + β(x)n(x) ⋅ q = u(t, x)

for x ∈ Γ .

(2.64)

The only second expression in the initial conditions (2.61) should be changed as follows: w(t0 , x) = w0 (x)

and r(t0 , x) = r0 (x) = −ρv0 (x)

for x ∈ V .

(2.65)

To generalize Problem 2.6 in accordance with the MIDR, the local constitutive equations (2.63) are replaced by one integral equality Φ = ∫ φ(t, x) dΩ = 0 Ω

1 with φ ∶= (aρv2 + b(s ⋅ κ ⋅ s)) , 2

(2.66)

where the functions v and s are used in compliance with equation (2.62). After that, Problem 2.6 is reformulated as follows. Problem 2.7. Find scalar functions w∗ (t, x) and r ∗ (t, x) as well as a vector function q∗ (t, x) such that the integral equality (2.66), the PDE (2.57), the boundary condition (2.64), and the initial constraints (2.65) are satisfied. For any potential w and flux q at positive weighting coefficients a > 0 and b > 0 in equation (2.66), the quadratic integrand φ is a non-negative function of the time t and the coordinate x. Therefore, the functional Φ is also non-negative and reaches its absolute minimum at the solution. Thus, Problem 2.7 can be reduced to a variational one. Problem 2.8. Find such unknown time potential w∗ (t, x), volume density r ∗ (t, x), and mass flux q∗ (t, x) that minimize the functional according to Φ[w∗ , q∗ ] = min Φ[w, r, q] = 0 w,r,q

subject to the constraints (2.57), (2.64)–(2.66).

(2.67)

24 | 2 Generalized formulations of parabolic and hyperbolic problems If a ≠ 0 and b ≠ 0, Problems 2.6 and 2.7 should be replaced by a variational problem, in which only stationary values of the constitutive functional Φ are of interest. Problem 2.9. The unknowns w∗ (t, x), r ∗ (t, x), q∗ (t, x) are sought as a stationary point of the functional Φ, that is, δΦ[w∗ , r ∗ , q∗ ; δw, δr, δq] = ∫ (av(ρδwt + δr) + bs ⋅ κ ⋅ (∇δw − κ −1 ⋅ δq)) dΩ = 0 (2.68) Ω

under the constraints (2.57), (2.64)–(2.66).

2.3.3 Conditions of stationarity Let w, r, q be respectively any admissible potential, density, and mass flux, whereas w∗ , r ∗ , q∗ be the actual solution. The corresponding variations δw, δr, δq are introduced in agreement with w = w∗ + δw, r = r ∗ + δr, and q = q∗ + δq. By taking into account equations (2.66) and (2.68), it is possible to show that the relation Φ[w, r, q] = Φ[w∗ , r ∗ , q∗ ] + δΦ[w∗ , r ∗ , q∗ ; δw, δr, δq] + Φ[δw, δw, δq] = δ2 Φ is valid. Here, the first variation of the functional Φ consists of three parts: δw Φ, δr Φ, and δq Φ. The second variation coincides with the functional, where the unknown functions are replaced with their variations δw, δr, and δq. Thus, δ2 Φ is non-negative. After integration by parts with accounting the essential constraints (2.57), (2.64)–(2.66), the terms of the first variation δΦ have the form: δw Φ = − ∫ (aρvt + b∇ ⋅ (κ ⋅ s))δw dΩ Ω

t1

0

+ a ∫ [ρvδw]t=t t=t 1 dV + b ∫ ∫ n ⋅ (κ ⋅ s)δw dΓ dt , t0

V

δr Φ = a ∫ vδr dΩ , Ω

Γ

(2.69)

δq Φ = −b ∫ s ⋅ δq dΩ . Ω

It can be seen from the structure of the first variation δΦ that the constitutive relations (2.63) must be satisfied for the solution. The admissible volume density r(t, x) and mass flux q(t, x) as well as their variations are correlated to each other through the balance equation (2.57). It means that ∇ ⋅ δq = −δrt .

(2.70)

Introduce the auxiliary function h(t, x) according to t

h ∶= − ∫ v(τ, x) dτ . t0

(2.71)

2.4 Heat transfer in solids | 25

The variation δr Φ is integrated by parts successively with respect to t and x by using the expression of δrt given in equation (2.70). Then the resulting variation of Φ is representable as δw Φ = ∫ (aρhtt − b∇ ⋅ (κ ⋅ s))δw dΩ Ω

t1

0

+ a ∫ [ρht δw]t=t t=t 1 dV + b ∫ ∫ n ⋅ (κ ⋅ s)δw dΓ dt , V

t0

Γ

δr Φ + δq Φ = ∫ (a∇h − bs) ⋅ δq dΩ Ω

(2.72)

t1

1

− a ∫ [hδr]t=t t=t 0 dV + a ∫ ∫ (hn ⋅ δs) dΓ dt . V

t0

Γ

By taking into account the equality (2.68) together with the boundary and initial constraints αδw + βn ⋅ δq = 0

for x ∈ Γ

and δw = δr = 0

for t = t0 ,

the following equations are obtained: ρht + ∇ ⋅ (κ ⋅ ∇h) = 0 [αh + β∇n ⋅ h]x∈Γ = 0,

for (t, x) ∈ Ω , h(t 1 , x) = ht (t 1 , x) = 0 .

(2.73)

The system (2.73) with the homogeneous boundary and terminal conditions has the only trivial solution h ≡ 0. So, the condition of stationarity (2.68) together with equation (2.71) is equivalent, in generalized sense, to the constitutive relations (2.63). The conditions (2.71) and (2.73) with the essential constraints (2.57), (2.64)–(2.66) describe correctly, from the mathematical point of view, the original acoustic Problem 2.6. As for dynamic problems of membrane vibrations [22] at d = 2, it is necessary to consider the variable w(t, x1 , x2 ) as the function of lateral displacements. Then r(t, x1 , x2 ) corresponds to the momentum density, whereas the vector q(t, x1 , x2 ) takes into account the lateral component of the membrane tension, which appears due to inclination of the membrane under its deformation.

2.4 Heat transfer in solids The subject of this section is heating systems with distributed parameters. The dynamic model of the systems under consideration is described by a parabolic partial

26 | 2 Generalized formulations of parabolic and hyperbolic problems differential equation. By taking into account boundary and initial conditions, an IBVP can be formulated for heat transfer processes in solids. The variational approaches based on the MIDR, applied earlier to elastic systems, are extended here on this type of dynamic problems. 2.4.1 Equations in linear thermodynamics Consider a heat-conducting body occupied a volume x ∈ V ⊂ ℝd of the dimension d ∈ {1, 2, 3}. In accordance with thermodynamics laws [102], the linear PDE describing the temperature distribution w(t, x) at different body points x during some time interval t ∈ (t 0 , t 1 ) has the form: ρ(x)cp (x)wt (t, x) = ∇ ⋅ (λ(x)∇w(t, x)) + f (t, x)

for (t, x) ∈ Ω .

(2.74)

Here, ρ denotes the volume density of material, cp is the specific heat capacity, λ is the thermal conductivity, f is the volume power density supplied by external heat sources at a point x, Δ is the Laplace operator, Ω = V × (t 0 , t 1 ) is the time-space domain. Similar to the acoustic problems discussed in the previous section, the generalized linear condition on the boundary Γ = 𝜕V is given as follows: α(x)w(t, x) + β(x)λ(x)∇w(t, x) ⋅ n(x) = u(t, x) for x ∈ Γ ,

(2.75)

where n denotes the unit outer normal, ∇ is the gradient operator. The coefficients α and β characterize the type of boundary conditions, and if α = 1 and β = 0 then the temperature is defined through the function u. In contrast, if α = 0 and β = 1 then the heat flux is fixed at this boundary patch. In the case when β = 1 but α ≠ 0, heat exchange between the body and the environment can be taken into account. To obtain a unique solution of equation (2.74), the temperature distribution has to be prescribed at the initial time instant as w(0, x) = w0 (x) for x ∈ V .

(2.76)

The corresponding IBVP can be stated as follows. Problem 2.10. Find a function w∗ (t, x) that obeys the constraints (2.74)–(2.76) for given functions f , u, and w0 . 2.4.2 Generalized formulation based on the MIDR First of all, physical laws underlying thermodynamics should be recalled. The equality (2.74) follows from Fourier’s law combined with the first law of thermodynamics, i.e., conservation of energy.

2.4 Heat transfer in solids | 27

Fourier’s law defines a linear relation between the rate of flow of heat energy per unit area through a surface, so-called heat flux, q(t, x) and the gradient of the temperature according to s(t, x) = 0

for (t, x) ∈ Ω

with s(t, x) ∶= q + λ∇w ,

(2.77)

where the constitutive vector s(t, x) ∈ ℝd is introduced. The first law of thermodynamics, in turn, leads to the balance relation ρ(x)cp (x)wt (t, x) + ∇ ⋅ q(t, x) = f (t, x)

for (t, x) ∈ Ω .

(2.78)

The boundary condition (2.75) expressed through the temperature w and the flux q is rewritten as α(x)w(t, x) + β(x)n(x) ⋅ q(t, x) = u(t, x)

for x ∈ Γ .

(2.79)

The initial condition is given in equation (2.76). By applying the MIDR to solve Problem 2.10, let us replace the local constitutive equation (2.77) by one integral equality Φ = ∫ φ(t, x) dΩ = 0 Ω

with φ ∶= s ⋅ s ,

(2.80)

where the function s is utilized in agreement with equation (2.77). Thus, the original IBVP is reformulated as follows. Problem 2.11. Find a scalar function w∗ (t, x) and a vector function q∗ (t, x) such that the integral equality (2.80), the balance law (2.78), the boundary condition (2.79), and the initial constraint (2.76) hold. It is worth noting that the quadratic integrand φ defined in equation (2.80) is a non-negative function of the time t and the coordinate x for any temperature and flux fields w, q. Therefore, the functional Φ is also non-negative and attains its absolute minimum at the exact solution of Problem 2.10 if any [4]. That is why, similar to the problems of the elastic systems named in this chapter, Problem 2.11 can be reduced to a variational one. Problem 2.12. Find the unknowns w∗ (t, x) and q∗ (t, x) that minimize the functional Φ[w∗ , q∗ ] = min Φ[w, q] = 0 w, q

subject to the constraints (2.76), (2.78), (2.79), (2.80).

(2.81)

28 | 2 Generalized formulations of parabolic and hyperbolic problems 2.4.3 Variational properties of the minimization problem Similar to the previous variational problems, w, q denote any admissible temperature and flux vector, whereas w∗ and q∗ are the actual fields. Join them via the corresponding variations δw, δq like w = w∗ + δw and q = q∗ + δq. According to equation (2.81), the relation Φ[w, q] = Φ[w∗ , q∗ ] + δΦ + δ2 Φ = δw Φ + δq Φ + δ2 Φ holds. Here, the first variation of the functional Φ splits into two parts: δw Φ and δq Φ. The second variation is given by δ2 Φ = Φ[δw, δq]. As φ(δw, δq) ≥ 0, so this variation is non-negative. After integration by parts with accounting equations (2.76)–(2.80), the first variation of Φ takes the form: t1

δw Φ = −2λ ∫ (∇ ⋅ s)δw dΩ + 2λ ∫ ∫ (s ⋅ n)δw dΓ dt , t0

Ω

Γ

δq Φ = 2 ∫ s ⋅ δq dΩ . Ω

If the variation δw Φ + δq Φ is equal to zero over any admissible δw and δq, then the constitutive relation (2.77) is valid. The temperature w and the heat flux q as well as their variations are related by Fourier’s law (2.78) as ∇ ⋅ δq = −ρcp δwt . Let us introduce the auxiliary function r(t, x) so that s = ∇r .

(2.82)

Then the necessary condition of stationarity for Φ after integration by parts with respect to t and x is able to be represented by δΦ = 0

1

t 1 with δΦ = − ∫ (ρcp rt + λΔr)δw dΩ + ∫ ∫ (r(n ⋅ δq) − λ(n ⋅ ∇r)δw) dΓ dt 2 Ω t0 Γ

+ ∫ ρcp rδw|t=t 1 dΩ = 0 .

(2.83)

Ω

The equality (2.83) together with the boundary conditions αδw + βn ⋅ δq = 0

for x ∈ Γ

implies that the following relations must be satisfied: ρcp rt + λΔr = 0

for (t, x) ∈ Ω ,

[αr + βλn ⋅ ∇r]x∈Γ = 0,

r(t 1 , x) = 0 .

(2.84)

2.4 Heat transfer in solids | 29

It is possible to prove that the system (2.84) with the homogeneous boundary and terminal conditions has the only trivial solution r ≡ 0. Thus, the condition of stationarity (2.83) together with equation (2.82) is equivalent, in some sense, to the constitutive relations (2.77). The equations (2.84) with the essential constraints (2.76), (2.78), and (2.79) give a strict mathematical description of the original heat transfer Problem 2.10.

3 Variational principles in linear elasticity 3.1 Dynamics of elastic bodies The aim of this chapter is to describe modifications of the MIDR for static and dynamic linear elasticity problems and discuss possible applications of this approach for the analysis of 3D elastic body behavior. In this section, the statements of dynamic problems and its conventional variational formulations are given. A parametric family of quadratic functionals is considered and their conditions of stationarity, which are equivalent to the governing relations, and are derived. After that, new variations in displacements, momentum density, and stresses are defined for the initial-boundary value problem of linear elasticity. 3.1.1 Problems of elastodynamics Consider an elastic body occupying a bounded spatial domain V with an external piecewise smooth boundary Γ in the d-dimensional space (d ∈ {1, 2, 3}). By taking into account the assumptions of the linear theory of elasticity about the smallness of elastic deformations and relative velocities, the body motions can be described by a PDE system [98]. First, let us introduce kinematic variables, namely, the displacement vector w(t, x) and the strain tensor of the second order ε(t, x). Then dynamic variables are represented by p(t, x) for the momentum density vector and by σ(t, x) for the stress tensor of the second order as well. All of these functions completely characterize the behavior of the elastic body and depend on the time t and the spacial coordinate vector x = x1 e1 + ⋯ + xd ed . The Cartesian coordinate system Ox is determined by the origin O and the standard unit basis vectors ei , such that ei ⋅ ej = δij . In the linear theory of elasticity, differential constitutive relations between the momentum density p and the velocity distribution w t as well as between the stresses σ and strains ε can be written as v(t, x) = 0 and ξ (t, x) = 0

for (t, x) ∈ Ω ,

(3.1)

where Ω = (t 0 , t 1 ) × V is the time–space domain. To be short in equation (3.1), the constitutive vector function v relating the velocity with the momentum density and the constitutive tensor function ξ determining Hooke’s law are defined as v ∶= w t − ρ−1 p and ξ ∶= ε − C−1 ∶ σ .

(3.2)

Here, the strain tensor ε depends linearly on the displacement vector according to 1 ε ∶= (∇w + ∇w T ) . 2 https://doi.org/10.1515/9783110516449-003

(3.3)

32 | 3 Variational principles in linear elasticity The volume density ρ(x) and the elastic modulus tensor C(x) introduced in equation (3.2) are given by functions of the coordinates x. The components Cijkl ∶= [C]ijkl of the tensor C in the coordinate system Ox are characterized by the following symmetry property: Cijkl = Cijlk = Cklij .

The superscript T denotes the transposition operator. The dots and colons between vectors and tensors point out to their scalar and double scalar products, respectively. The gradient operator ∇ can be represented componentwise in the Cartesian coordinate system Ox as ∇ = (𝜕/𝜕x1 , … , 𝜕/𝜕xd ). By using the relations (3.1)–(3.3), the vector equation of elastic body motions ρ(x)w tt (t, x) = ∇ ⋅ C(x) ∶ ε(t, x) + f (t, x)

for (t, x) ∈ Ω

(3.4)

can be rewritten in terms of the stresses σ and the momentum density p as pt (t, x) − ∇ ⋅ σ(t, x) − f (t, x) = 0 for (t, x) ∈ Ω .

(3.5)

Here, f (t, x) is the vector of external volume forces. The divergence operator is denoted by ∇⋅ and results in the vector with the components: [∇ ⋅ σ]i =

𝜕σi,1 𝜕σi,2 𝜕σi,3 + + 𝜕x1 𝜕x2 𝜕x3

with i = 1, … , d .

The relation (3.5) reflects the balance of elastic, external, and inertia volume forces inside the body. The surface stress vector imposed on the boundary Γ is defined by q ∶= σ ⋅ n

(3.6)

with the normal n(x) to the boundary Γ. Let us constrain ourselves to the case of the linear boundary conditions expressed with respect to the displacements and stresses in the form: α(x) ⋅ w(t, x) + β(x) ⋅ q(t, x) = u(t, x)

for (t, x) ∈ (t 0 , t 1 ) × Γ .

(3.7)

The given function u(t, x) introduced in equation (3.7) is either the boundary vector of external loads or displacements in dependence on the problem. The second-order tensors α(x) and β(x) define the type of boundary conditions. Let us restrict ourselves to the case when the principal axes Oei for i = 1, … , d are common for the tensors α and β. At that, αi (x) and βi (x) are the corresponding eigenvalues of these tensors relating with the basic vector ei (x). If the conditions αi (x) = 1 and βi (x) = 0 are valid on some part Γ1i ⊂ Γ of the boundary, then the displacement wi = ui = u ⋅ ei is given. Here, ui is the component of the vector u, which is collinear with the basis vector ei . If αi = 0 and βi = 1 hold true on the boundary patch Γ2i ⊂ Γ, then the external

3.1 Dynamics of elastic bodies | 33

stress qi = ei ⋅ σ ⋅ n = ui is defined. The constraint (3.7) also covers the special case of elastic foundation if the pair αi and βi are not equal to zero for some index i (usually, αi βi > 0). As initial conditions, we give the distribution of the displacements w and momentum density p at the fixed time instant t0 , as sufficiently smooth functions of the coordinates x according to w(t 0 , x) = w 0 (x)

and p(t 0 , x) = p0 (x)

for x ∈ V .

(3.8)

Note that the boundary conditions (3.7) and the initial conditions (3.7) should be consistent. Summarizing, we can formulate the IBVP. Problem 3.1. Find such a displacement vector w(t, x), a momentum vector p(t, x), and a stress tensor σ(t, x) that satisfy the constitutive and kinematic relations (3.1)–(3.3), the balance equation (3.5), as well as the boundary and initial conditions (3.7)–(3.8). 3.1.2 Hamilton principles Consider two conventional dual approaches generalizing Problem 3.1 discussed in the previous subsection. In accordance with the first method, suppose that the distribution of displacements at both the beginning (t = t 0 ) and the end (t = t 1 ) of body motion is prescribed as w(t 0 , x) = w 0 (x)

and w(t 1 , x) = w 1 (x)

for x ∈ V .

(3.9)

It is assumed also that there exist functions of kinetic energy K(t) and potential energy U(t) depending on the displacement vector w(t, x). Additionally, the variation δw does not lead to the change of the boundary loads, i.e., Γ̄ 1i ∪ Γ̄ 2i = Γ for i = 1, … , d. Under this assumption, the Hamilton principle of ‘least’ action follows from the principle of virtual work [103]. Then the corresponding generalized statement of the dynamic BVP (3.1)–(3.3), (3.5)–(3.7), (3.9) can be given. Problem 3.2. Find such a displacement vector w ∗ (t, x) that satisfies the condition of stationarity δΥ[w ∗ ] = 0

t1

with Υ = ∫ (K(t) − U(t)) dt , t0

K=

1 ∫ ρw t ⋅ w t dV , 2 V

U=

1 ∫ ε ∶ C ∶ ε dV − ∫ f ⋅ w dV − ∑ ∫ ui wi dΓ 2 2 V V i=1 Γi

(3.10) d

34 | 3 Variational principles in linear elasticity subject to the initial and terminal constraints (3.9) as well as the boundary conditions in the displacements wj (t, x) = uj (t, x)

for x ∈ Γ1j

with j = 1, … , d ,

(3.11)

where wj = ej ⋅ w and uj = ej ⋅ u, Υ is the functional of action. In this problem, the displacement field w(t, x) must rigorously satisfy the kinematic relation (3.3). As follows from equation (3.11), the components wj of the displacement vector w must strictly satisfy the boundary condition (3.7) on Γ1j , where αj = 1 and βj = 0. The equation of motion (3.4) and the stress boundary conditions on Γ2j , where αj = 0 and βj = 1, appear in the condition of stationarity (3.10). In accordance with the approach dual to the previous one, suppose that the initial and terminal distribution of momentum is fixed by p(t 0 , x) = p0 (x)

and p(t 1 , x) = p1 (x)

for x ∈ V .

(3.12)

Assume also that there exist the complementary kinetic energy Kc (t) and potential energy Uc (t) expressed via the momentum density p and the stresses σ. If the variations δp and δσ obeying the balance equation (3.5) does not cause any changes in the boundary displacements given on Γ1j , then the dual principle can be formulated for the complimentary functional of action. Problem 3.3. Find such a momentum vector p∗ (t, x) and a stress tensor σ ∗ (t, x) that are a stationary point of the following functional: δΥc [p∗ , σ ∗ ] = 0

t1

with Υc = ∫ (Kc (t) − Uc (t)) dt ,

1 Kc = ∫ ρ−1 p ⋅ p dV , 2 V Uc =

t0

(3.13) d

1 ∫ σ ∶ C−1 ∶ σ dV − ∑ ∫ ui qi dΓ 1 2 V i=1 Γi

subject to equation (3.5), the initial and terminal constraints (3.12), as well as the boundary conditions in the stresses qj (t, x) = uj (t, x)

for x ∈ Γ2j

with j = 1, … , d ,

(3.14)

where qj = ej ⋅ q and uj = ej ⋅ u. In agreement with this principle, the fields of momentum density p(t, x) and stresses σ(t, x) must rigorously satisfy the equilibrium equation (3.5) and the boundary conditions (3.14) on Γ2j . It can be proved that the constitutive relations (3.1) and the

3.1 Dynamics of elastic bodies | 35

kinematic equation (3.3) together with the boundary conditions on Γ1j are equivalent to the conditions of stationarity (3.13). It is important to emphasize that the Hamilton principle 3.2 and its dual counterpart 3.3 is only formulated under boundary constraints in time, e.g., (3.9) or (3.12), and cannot be extended to the initial-boundary value Problem 3.1.

3.1.3 Integrodifferential statement in elasticity Similar to Chapter 2, let us generalize the IBVP 3.1 by introducing an integral equation instead of the differential constitutive relations (3.1). Consider the following integral: Φ+ [w, p, σ] = ∫ φ+ (t, x) dΩ

(3.15)

1 φ+ ∶= (ρv ⋅ v + ξ ∶ C ∶ ξ ). 2

(3.16)

Ω

proposed in [54] with

Here, v is the residual velocity vector, which joins the momentum density p and the velocities w t , ξ is the residual strain, which links together the stresses σ and the strains ε. Both functions v and p are defined in equation (3.2). Problem 3.1 of elastic body motions can be reformulated with the help of the nonnegative functional Φ+ as follows. Problem 3.4. Find such displacements w ∗ (t, x), momentum density p∗ (t, x), and stresses σ ∗ (t, x) that Φ+ [w ∗ , p∗ , σ ∗ ] = 0

(3.17)

subject to the balance relation (3.5) as well as the boundary and initial constraints (3.7), (3.8). The same as for the hyperbolic problems in Chapter 2, the integrand φ+ has the dimension of energy. Since the functional Φ+ is non-negative, Problem 3.4 can be reduced to the constrained minimization. Problem 3.5. Find optimal functions w ∗ (t, x), p∗ (t, x), and σ ∗ (t, x) such that the functional Φ+ attains its absolute minimum Φ+ [w ∗ , p∗ , σ ∗ ] = min Φ+ [w, p, σ] = 0 w,p,σ

(3.18)

subject to Newton’s second law (3.5) as well as the boundary and initial conditions (3.7), (3.8).

36 | 3 Variational principles in linear elasticity Denote any admissible displacement, momentum, stress fields, respectively, as w, p, σ. Then express them through the actual fields w ∗ , p∗ , σ ∗ and their variations δw, δp, δσ according to w = w ∗ + δw ,

p = p∗ + δp ,

σ = σ ∗ + δσ .

After taking into account (3.13), the integral Φ+ can be represented as Φ+ [w, p, σ] = Φ+ [w ∗ , p∗ , σ ∗ ] + δΦ+ [w, p, σ] + δ2 Φ+ [w, p, σ] = Φ+ [δw, δp, δσ] ≥ 0 . The necessary condition of stationarity states that δΦ+ = δw Φ+ + δp Φ+ + δσ Φ+ = 0 . The second variation δ2 Φ+ is, at least, non-negative in accordance with the necessary condition of minimality. 3.1.4 A family of constitutive functionals As it was shown in Chapter 2, a family of quadratic integrals is derived to generalize a hyperbolic problem of mathematical physics. In a similar manner, the following oneparametric family of constitutive functionals Φ can be proposed in linear elasticity by Φ = ∫ φ(t, x) dΩ

(3.19)

φ ∶= aρ(v ⋅ v) + b(ξ ∶ C ∶ ξ ) ,

(3.20)

Ω

with

which stationarity conditions combined with balance, boundary, and initial essential constraints (see equations (3.5), (3.7), (3.8)) are equivalent, in some sense, to the whole system of governing equations for solid body motions. In equation (3.20), a and b are real numbers like in the previous chapter. Without loss of generality, let a2 + b2 = 21 and a ≥ 0. At a = b = 1/2, the integral Φ coincides with the functional Φ+ introduced in equation (3.15). If a ≠ 0 and b > 0, then each functional Φ(b) corresponds its own problem of constrained minimization similar to Problem 3.5. For arbitrary non-zero values of a and b (ab ≠ 0), it is possible to give a common variational formulation of Problem 3.1. Problem 3.6. Find the functions w ∗ (t, x), p∗ (t, x), σ ∗ (t, x) that provide the stationarity of the functional Φ introduced in equations (3.19) and (3.20), i.e., δΦ[w, p, σ] = 0 subject to the constraints (3.5), (3.7), and (3.8).

(3.21)

3.1 Dynamics of elastic bodies | 37

After integrating by parts and taking into account equations (3.2)–(3.6), the terms of the first variation δΦ = δw Φ + δp Φ + δσ Φ are obtained in the form: δw Φ = − 2 ∫ (aρv t + b∇ ⋅ (C ∶ ξ )) ⋅ δw dΩ Ω

t1

1

+ 2a ∫ [ρv ⋅ δw]t=t t=t 0 dV + 2b ∫ ∫ n ⋅ (C ∶ ξ ) ⋅ δw dΓ , V

t0

δp Φ = − 2a ∫ Ωv ⋅ δp dΩ ,

(3.22)

Γ

δσ Φ = −2b ∫ ξ ∶ δσ dΩ . Ω

It can be seen from equation (3.22) that the first variation δΦ equals zero for the function w(t, x), p(t, x), σ(t, x), which satisfies equation (3.2). Let us derive in an explicit form the necessary conditions of stationarity for the functional Φ (Euler–Lagrange equations). For that, proper allowance must be made for the relation between the vector of momentum density p and the tensor of inner stresses σ in accordance with Newton’s second law (3.5), which is expressed in variations as δpt = ∇ ⋅ δσ .

(3.23)

Similar to Section 2.2, let us introduce the auxiliary displacement vector: t

r(t, x) ∶= − ∫ v(τ, x) dτ . t0

(3.24)

Then the variation terms from equation (3.22) constrained by (3.23) can be found according to δw Φ = 2 ∫ (aρr tt − b∇ ⋅ (C ∶ ξ )) ⋅ δw dΩ Ω

t1

1

− 2a ∫ [ρr t ⋅ δw]t=t t=t 0 dV + 2b ∫ ∫ n ⋅ (C ∶ ξ ) ⋅ δw dΓ dt = 0 , V

t0

Γ

δp Φ + δσ Φ = 2 ∫ (aεr − bξ ) ∶ δσ dΩ

(3.25)

Ω

1

t1

+ a ∫ [r ⋅ δp]t=t t=t 0 dV − a ∫ ∫ r ⋅ δσ ⋅ n dΓ dt = 0 , V

t0

Γ

where the tensor 1 εr = (∇r + ∇r T ) 2 has the structure of the Cauchy strain tensor.

(3.26)

38 | 3 Variational principles in linear elasticity By taking into account the boundary and initial constraints (3.7) and (3.8) written down in variations, compose the system of Euler–Lagrange equations and natural boundary and terminal conditions given by ρr tt − ∇ ⋅ (C ∶ εr ) = 0 α ⋅ r + β ⋅ qr = 0 r(t 1 , x) = 0

and

for x ∈ Γ

aεr + bξ = 0 , with qr ∶= (C ∶ εr ) ⋅ n ,

(3.27)

and r t (t 1 , x) = 0 .

The homogeneous PDE system (3.27) with respect to the vector r is a terminalboundary value problem of linear elasticity that possesses an unique zero solution. The necessary conditions v(t, x) ≡ 0 and ξ (t, x) ≡ 0 follow directly from this property. In other words, if there exists a classical solution w ∗ (t, x), p∗ (t, x), σ ∗ (t, x) of Problem 3.6 than the system (3.27) together with the constraints (3.5), (3.7), and (3.8) is equivalent to the original IBVP (Problem 3.1). 3.1.5 Comparative analysis of variational problems Consider one functional from the parametric family (3.19), namely, at the values a = −b = 21 in more detail. It can be represented as Φ− = Φ|b=− 1 = ∫ φ− (t, x) dΩ = Θ1 + Θ2 − 2Θ0 , 2

Ω

1 φ− ∶= (ρv ⋅ v − ξ ∶ C ∶ ξ ) , 2 1 Θ0 = ∫ (p ⋅ w t − σ ∶ ε) dΩ , 2 Ω 1 Θ1 = ∫ (ρw t ⋅ w t − ε ∶ C ∶ ε) dΩ , 2 Ω 1 Θ2 = ∫ (ρ−1 p ⋅ p − σ ∶ C−1 ∶ σ) dΩ . 2 Ω

(3.28)

Note that the integral Θ1 depends only on displacements, whereas the expression for the functional Θ2 does not include the vector function w. In turn, the bilinear form Θ0 does not contain explicitly elastic or inertial material constants. After integration by parts, this functional is reduced according to t1

1

2Θ0 = ∫ [p ⋅ w]t=t t=t 0 dV − ∫ f ⋅ w dΩ − ∫ ∫ q ⋅ w dΓ dt . V

Ω

t0

Γ

(3.29)

As well as in Problems 3.2 and 3.3, let the displacement component wi be given on the boundary part Γ1i by the function ui (t, x), whereas the stress component qi is defined on Γ2i by the same function ui as discussed in Subsection 3.1.1. It means that

3.1 Dynamics of elastic bodies | 39

either αi = 1 and βi = 0 or αi = 0 and βi = 1. Additionally, the surfaces Γ1i and Γ2i do not intersect, i.e., Γ̄ 1i ∪ Γ̄ 2i = Γ ,

Γ1i ∩ Γ2i = ∅ ,

i = 1, … , d .

After substituting the expression (3.29) into equation (3.28), the functional Φ− can be rewritten in the form: d

t1

Φ− = Θ1 + ∫ f ⋅ w dΩ + ∑ ∫ ∫ ui wi dΓ dt 0 i=1 t

Ω

d

t

Γ2i

1

1

+ Θ2 + ∑ ∫ ∫ ui qi dΓ dt − ∫ [p ⋅ w]t=t t=t 0 dV Γ1i

0 i=1 t

V

(3.30)

If the fields of displacements or momentum density are given at the initial and terminal time instants, then the integral Φ− is decomposed into two independent terms: Φ− [w, p, σ] = Υ1 [w] + Υ2 [p, σ] , d

t1

Υ1 [w] = Θ1 [w] + ∫ f ⋅ w dΩ + ∑ ∫ ∫ ui wi dΓ dt + Ξ1 [w] , Ω

d

t

Γ2i

0 i=1 t

1

(3.31)

1

Υ2 [p, σ] = Θ2 [p, σ] + ∑ ∫ ∫ ui qi dΓ dt − ∫ [p ⋅ w]t=t t=t 0 dV + Ξ2 [p] . 0 i=1 t

Γ1i

V

The functional Υ1 depends exceptionally on the displacements w, whereas Υ2 relates to both the momentum density p and the stresses σ. The integrals Ξ1 and Ξ2 over the spatial domain V contain the kinematic and dynamic vectors w and p, respectively. Table 3.1 presents five types of time boundary conditions at t = t 0 and t = t 1 (columns A, B). The corresponding expressions for integrals Ξ1 [w] and Ξ2 [p] are shown in the last two columns of the table. The four upper rows in the table define different initial-terminal problems with respect to time, and the lowest row corresponds Table 3.1: Functional terms for different time conditions of elastic body motions. Condition A 0

Condition B 1

Term Ξ1

Term Ξ2

w(t , x) = w 0 (x)

w(t , x) = w 1 (x)

0

Ξ12 − Ξ02

p(t 0 , x) = p0 (x)

p(t 1 , x) = p1 (x)

Ξ11 − Ξ01

0

w(t 0 , x) = w 0 (x)

p(t 1 , x) = p1 (x)

Ξ11

−Ξ02

p(t 0 , x) = p0 (x)

w(t 1 , x) = w 1 (x)

−Ξ01

Ξ12

w(t 0 , x) = w(t 1 , x)

p(t 0 , x) = p(t 1 , x)

0

0

40 | 3 Variational principles in linear elasticity to a time periodic problem. The following notation is used in Table 3.1: Ξi1 = ∫ pi (x) ⋅ w(t i , x) dV

and

Ξi2 = ∫ w i (x) ⋅ p(t i , x) dV

with i = 0, 1 .

V

V

(3.32)

The condition of stationarity for the functional Φ− under the balance relation (3.5), the spacial boundary equations (3.7), and time constraints placed in Table 3.1 can be written in accordance with equation (3.31) as δΦ− = δw Υ1 [w] + (δp Υ2 [p, σ] + δσ Υ2 [p, σ]) = 0 . Consequently, each BVP mentioned in Table 3.1 is decomposed into two independent subproblems. Problem 3.7. Find such admissible displacements w ∗ (t, x) that are a stationary point of the functional Υ1 , i.e., δw Υ1 [w] = 0 subject to the boundary condition (3.7) on the parts Γ1i with i = 1, … , d and the time conditions from Table 3.1, which constrain the displacement vector w. Problem 3.8. Find such admissible momentum density p∗ (t, x) and stresses σ ∗ (t, x) that provide the functional Υ2 with its stationary value, i.e., δp Υ2 [p, σ] + δσ Υ2 [p, σ] = 0 subject to the balance equation (3.5), the boundary condition (3.7) on Γ2i with i = 1, … , d, and the time conditions from Table 3.1 restricting the momentum vector p. It is important that Problem 3.7 coincides with Problem 3.2, i.e., Υ1 = Υ, if the displacement vector is fixed at both initial and terminal time instants (the first row in Table 3.1). If the momentum density is given at the beginning and at the end of the process (the second row in Table 3.1), then Problem 3.8 is equivalent to Problem 3.3, Υ2 = Υc . It means that these new statements closely relate with the Hamilton principles. Both principles of ‘least’ action follows from Problems 3.7 and 3.8 in the periodic case (the lowest row in Table 3.1). In contrast to the conventional variational principles (Problems 3.2 and 3.3), which are only formulated for time–space boundary value problems, the functional Φ− is applicable to the initial-boundary value problem (Problem 3.1) as well. It is important to emphasize that Problem 3.6 with the initial conditions (3.8) cannot be decomposed into two independents subproblems. This is stipulated by the fact that

3.1 Dynamics of elastic bodies | 41

the last term in the functional (3.30) depends explicitly on both the displacements w and the momentum density p taken at the terminal time. Such decomposition is also impossible in the case with mixed boundary conditions (3.7) (αi (x)βi (x) ≠ 0 given on some part of boundary for at least one index i) even if the time conditions are taken from Table 3.1. In all of these cases, the variational problem has to be solved simultaneously with respect to unknown displacement, momentum, and stress functions.

3.1.6 Dynamic variational principle in displacements and stresses As it was shown in the previous subsections, the variational formulations of the IBVP for forced motions of an elastic body can be given with respect to the displacement vector, the momentum density vector, and the stress tensor. It implies that a large number of variables, namely, six components of the displacement and momentum vectors as well as six components of the stress tensor should be found. Even after introducing the auxiliary tensor χ according to t

σ(t, x) = χt (t, x) and p(t, x) = ∇ ⋅ χ(t, x) + ∫ f (τ, x) dτ t0

(3.33)

to satisfy Newton’s second law (3.5), nine independent functions (components of this tensor and a vector w) remain in the system. To decrease the number of unknown functions in the variational formulation and to raise the effectiveness of numerical computation, a special element Φ at a = 0 and b = 1/√2 of the functional family (3.19) should be chosen as Φ0 [w, σ] =

1 Φ| = ∫ φ0 dΩ , √2 a=0 Ω

(3.34)

where 1 φ0 ∶= ξ ∶ C ∶ ξ 2

1 with ξ = (∇w + ∇w T ) − C−1 ∶ σ . 2

(3.35)

In order to correctly formulate a problem of constrained minimization relating to the functional Φ0 introduced in equation (3.34), the relation between the momentum density and velocities defined in equations (3.1), (3.2), i.e., v(t, x) = w t (t, x) − ρ−1 p(t, x) = 0 ,

(t, x) ∈ Ω

(3.36)

has to be treated as an additional constraint imposed on the displacement and momentum fields. After taking into account equation (3.36), the new variational statement sounds as follows.

42 | 3 Variational principles in linear elasticity Problem 3.9. Find those admissible displacement and stress fields w ∗ (t, x), σ ∗ (t, x) that minimize the functional Φ0 [w ∗ , σ ∗ ] = min Φ0 [w, σ] = 0

(3.37)

w,σ

subject to the balance, boundary, and initial constraints: ρ(x)w tt (t, x) = ∇ ⋅ σ(t, x) + f (t, x) for (t, x) ∈ Ω , α(x) ⋅ w(t, x) + β(x) ⋅ q(t, x) = u(t, x) w(t 0 , x) = w 0 (x)

and

with q = σ ⋅ n for x ∈ Γ ,

(3.38)

w t (t 0 , x) = ρ−1 (x)p0 (x) .

Let actual and admissible unknowns be denoted respectively as w ∗ and σ ∗ , w and σ. By using the variational notation, w = w ∗ + δw

and σ = σ ∗ + δσ ,

write down the expansion of the quadratic functional Φ0 [w, σ] = Φ0 [w ∗ , σ ∗ ] + δΦ0 [w, σ] + δ2 Φ0 [w, σ] = Φ0 [δw, δσ] . By construction, the second variation is non-negative (δ2 Φ0 ≥ 0), and the first variation is equal to zero if ξ (t, x) ≡ 0. In Problem 3.9, the displacement vector w and stress tensor σ are interdependent due to the first equation in (3.38) and, maybe, due to the boundary conditions of the third kind. Such dependency is valid also for their variations δw and δσ according to δw tt = ∇ ⋅ δσ . Derive the necessary condition of stationarity for the functional Φ0 under the given constraints. Let us introduce an auxiliary second-order tensor χ(t, x) (see equation (3.33)) that defines equilibrium fields of displacements and stresses t

τ2

t0

t0

w(t, x) = ∇ ⋅ χ(t, x) + ∫ ∫

f (τ1 , x) dτ1 dτ2 , ρ(x)

σ(t, x) = ρ(x)χtt (t, x) .

(3.39)

After identical transformations, the first variation of the functional Φ0 expressed through the tensors ξ and χ looks like 1

δχ Φ0 = ∫ (Δξ − ρξtt ) ∶ δχ dΩ + ∫ ρ[ξt ∶ δχ − ξ ∶ δχt ]t=t t=t 0 dV V

Ω

t1

+ ∫ ∫ ((n ⋅ C ∶ ξ ) ⋅ (∇ ⋅ δχ) − (∇ ⋅ C ∶ ξ ) ⋅ (δχ ⋅ n)) dΓ dt = 0 , t0

Γ

(3.40)

3.2 Spectral problems in elasticity |

43

where 1 Δξ = (∇(∇ ⋅ C ∶ ξ ) + ∇(∇ ⋅ C ∶ ξ )T ) . 2 By taking into account the boundary and initial constraints for the tensor χ, the Euler–Lagrange equations with the boundary and terminal natural conditions can be obtained from equation (3.40) according to ρξtt (t, x) = Δξ (t, x)

for (t, x) ∈ Ω ,

α ⋅ (∇ ⋅ C(x) ∶ ξ (t, x)) + ρβ ⋅ (n(x) ⋅ C(x) ∶ ξtt (t, x)) = 0

for x ∈ Γ ,

(3.41)

ξ (t 0 , x) = ξt (t 0 , x) = 0 . It can be proved that the only trivial solution of the terminal-boundary value Problem (3.41) with respect to the tensor ξ is valid. It means that the conditions of stationarity (3.41) together with the minimization problem constraints (3.38) are equivalent to the governing equations of linear elasticity (3.1)–(3.8).

3.2 Spectral problems in elasticity In this section, boundary value problems on harmonic vibrations in linear elasticity, which directly follow Problem 3.1 are considered. Two types of elastic vibrations are usually distinguished, namely, forced and natural harmonic motions. The former relate with external loads oscillating at a fixed frequency, whereas the latter appear in an elastic body because of its initial deformations and velocities. Both types of motions are able to be analyzed by using the variational approaches mentioned in the previous section.

3.2.1 Harmonic vibrations of elastic bodies In accordance with Problem 3.1, the displacement vector w(t, x), the momentum vector p(t, x), and the stress tensor σ(t, x) have to satisfy the constitutive and kinematic relations (3.1)–(3.3), the balance equation (3.5), as well as the boundary and initial conditions (3.7)–(3.8). The volume forces f (t, x) and the boundary disturbances u(t, x) in the case of harmonic motions can be expressed in the form: f (t, x) = f c (x) cos(ωt) + f s (x) sin(ωt)

for x ∈ V ,

u(t, x) = uc (x) cos(ωt) + us (x) sin(ωt) for x ∈ Γ , where ω denotes the frequency of the external loads and deformations.

(3.42)

44 | 3 Variational principles in linear elasticity It can be proved that the problem of harmonic vibrations is decomposed into two subproblems either for the external functions f c and uc or for f s and us . Both subproblems have similar solutions, which is why we confine ourselves to studying only the first subproblem in which external loads are described by cosine functions. To exclude the time variable, let us introduce the harmonic unknown functions: ̄ w(t, x) = w(x) cos(ωt) , (3.43)

̄ sin(ωt) , p(t, x) = p(x) ̄ cos(ωt) . σ(t, x) = σ(x)

The bar symbol and the subscribe c appearing in equation (3.42) are further omitted. This substitution is not always correct. In the case of resonance frequencies, it is necessary to find a solution of a special kind. Such a case should be considered separately and it is out of the scope of the book. The constitutive relations (3.1) are reduced to the form: v(x) = 0

and

ξ (x) = 0

for x ∈ V ,

(3.44)

where the amplitudes of the functions (3.2) are given by v = −ωw − ρ−1 p and ξ = ε − C−1 ∶ σ

1 with ε ∶= (∇w + ∇w T ) . 2

(3.45)

Newton’s second law (3.5) is replaced by ωp(x) − ∇ ⋅ σ(x) − f (x) = 0

for x ∈ V .

(3.46)

Although the formal expression for the boundary conditions (3.5) is the same, they are defined on the boundary of the volume V , i.e., α(x) ⋅ w(x) + β(x) ⋅ q(x) = u(x)

for x ∈ Γ ,

(3.47)

where q(x) is the boundary stress vector according to equation (3.6). Since the originally initial-boundary value problem degenerates to a boundary value problem, the initial conditions (3.8) are eliminated from consideration. The resulting BVP can be formulated as follows. Problem 3.10. Find such amplitudes of the displacements w(x), the momentum density p(x), and the stresses σ(x) that satisfy the constitutive relations (3.44) with the residual functions (3.45), the balance equation (3.46), and the boundary conditions (3.6).

3.2 Spectral problems in elasticity | 45

3.2.2 Natural vibrations of solids When the external loads and boundary displacements are absent, in other words, f (x) ≡ 0

and u(x) ≡ 0 ,

Problem 3.10 has a non-trivial solution only if the frequency ω equals to eigenvalues {ωi }i∈ℤ+ . Without loss generality, the non-negative eigenfrequencies ωi ≥ 0 are only taken into account. The function of momentum density can be excluded algebraically via equation (3.46). As a result, the constitutive functions (3.45) take the form: v = −ωw −

∇⋅σ ρω

and ξ = ε − C−1 ∶ σ .

(3.48)

The boundary conditions (3.46) are rewritten with the zero right-hand side as α(x) ⋅ w(x) + β(x) ⋅ q(x) = 0 for x ∈ Γ .

(3.49)

The corresponding eigenvalue problem can be given in displacements and stresses. Problem 3.11. Find the set (specter) of eigenfrequencies ωi with i ∈ ℤ+ and respective eigenforms w(x, ωi ), σ(x, ωi ) obeying the constitutive relations (3.44) defined by equation (3.48) and the boundary conditions (3.8). Notice that the eigenforms are found up to an arbitrary real constant, which can be chosen, for example, by normalizing the resulting functions. It is possible to reduce the PDEs (3.44) to one equation with respect either to the displacement vector 2ρω2 w + ∇ ⋅ C ∶ (∇w + ∇w T ) = 0

and σ = C ∶ ε

(3.50)

or to the stress tensor ∇(ρ−1 ∇ ⋅ σ) + (∇(ρ−1 ∇ ⋅ σ))T + 2ω2 C−1 ∶ σ = 0

and

w=−

∇⋅σ . ρω2

(3.51)

To reformulate Problem 3.10 in terms of either displacements or stresses, the corresponding replacement must be made in the boundary condition (3.49). 3.2.3 Variational statements of harmonic problems The variational problems on forced or natural vibrations of an elastic body can be formulated based on the conventional Hamilton principles – either Problems 3.2 or 3.3

46 | 3 Variational principles in linear elasticity and correspond to Problems 3.10 and 3.11. As in Section 3.1, either the displacements w(x) or the stresses q(x) are given on the boundary surfaces Γij ⊂ Γ in accordance with equations (3.11) and (3.14). The direct principle of the ‘least’ action for the forced harmonic vibration is stated as follows. Problem 3.12. Find such displacement amplitudes w ∗ (x) that satisfies the condition of stationarity δΥ[w ∗ ] = 0 U=

with Υ = K − U ,

K=

1 ∫ ρω2 w ⋅ w dV , 2 V

d

1 ∫ ε ∶ C ∶ ε dV − ∫ f ⋅ w dV − ∑ ∫ ui wi dΓ , 2 2 V V i=1 Γi

(3.52)

subject to the boundary conditions in displacements wj (x) = uj (x)

for x ∈ Γ1j

with j = 1, … , d .

(3.53)

The complementary principle is formulated accordingly. Problem 3.13. Find such a stress tensor σ ∗ (x) that is a stationary point of the functional δΥc [σ ∗ ] = 0

with Υc = Kc − Uc ,

Kc = d

(∇ ⋅ σ) ⋅ (∇ ⋅ σ) 1 dV , ∫ 2 V ρω2

1 Uc = ∫ σ ∶ C−1 ∶ σ dV − ∑ ∫ ui qi dΓ , 1 2 V i=1 Γi

(3.54)

subject to the boundary conditions in stresses qj (x) = uj (x)

for x ∈ Γ2j

with j = 1, … , d .

(3.55)

Similar to the general dynamic case considered in Section 3.1, a family of constitutive functionals can be proposed for harmonic motions of an elastic body as Φ = ∫ (aρ(v ⋅ v) + b(ξ ∶ C ∶ ξ )) dV V

(3.56)

where the constitutive functions v and ξ are defined according to equations (3.46). In equation (3.56), a and b are such real numbers that a2 + b2 = 21 and a ≥ 0. The integral Φ denotes by Φ+ if a = b = 1/2 and by Φ− if a = −b = 1/2. The variational formulation based on the MIDR is the following. Problem 3.14. Find the functions w ∗ (t, x), and σ ∗ (t, x) that guarantee the stationarity of the functional (3.56), i.e., δΦ[w, σ] = 0 subject to the boundary constraints (3.6).

(3.57)

3.3 Variational formulations in elastostatics | 47

In the case when the boundary conditions are given either in the displacements (3.53) or in stresses (3.55), the functional Φ− can be decomposed after integration by parts into two terms Υ[w] and Υc [σ]. As a consequence, Problem 3.14 at a = −b = 1/2 splits into two independent Subproblems 3.12 and 3.13. Problem 3.14 is reduced to a minimization problem in the case when a > 0 and b > 0, particularly, for Φ+ . This functional 1 ∫ (ρv ⋅ v + ξ ∶ C ∶ ξ ) dV 2 V ρω2 w ⋅ w p⋅p ε∶C∶ε σ ∶ C−1 ∶ σ =∫( + ωw ⋅ p + + −σ∶ε+ ) dV 2 2ρ 2 2 V

Φ+ =

(3.58)

is a linear combination of the mechanical energies expressed via displacements and stresses according to Φ+ [w, σ] = W1 [w] − 2W0 [w, σ] + W2 [σ] ,

(3.59)

where W0 = ∫ (σ ∶ ε − (∇ ⋅ σ) ⋅ w − w ⋅ f ) dV , V

W1 = ∫ ( V

W2 = ∫ ( V

ρω2 w ⋅ w ε ∶ C ∶ ε + ) dV , 2 2

(3.60)

(∇ ⋅ σ) ⋅ (∇ ⋅ σ) σ ∶ C−1 ∶ σ (∇ ⋅ σ) ⋅ f f ⋅f + + + ) dV . 2 2 2ρω 2 ρω 2ρω2

In Problem 3.11, one of the energies (3.60) can be scaled because the found solution has an arbitrary multiplier.

3.3 Variational formulations in elastostatics In accordance with the linear theory of elasticity, the static stress-strain state follows directly from Problem 3.10 if the frequency ω is equated to zero. Of course, the same result is obtained by using the governing equations of Problem 3.1 after removing all partial derivatives with respect to time. After that, the first constitutive relation from equation (3.1) gives us that the momentum density function p is equivalent to zero. The initial conditions should be also excluded from consideration. 3.3.1 Static problems in linear elasticity In the static case, the local constitutive relation between the stresses σ and strains ε can be rewritten as ξ (x) = 0

for x ∈ V ,

(3.61)

48 | 3 Variational principles in linear elasticity where the constitutive tensor function ξ has the form: ξ = ε − C−1 ∶ σ

1 with ε = (∇w + ∇w T ) . 2

(3.62)

The vector-valued equilibrium equation is expressed through the stresses σ according to ∇ ⋅ σ(x) + f (x) = 0

for x ∈ V .

(3.63)

The boundary conditions, which are linear with respect to the displacement vector w and the stress vector q, is given by α(x) ⋅ w(x) + β(x) ⋅ q(x) = u(x)

for x ∈ Γ .

(3.64)

Thus, the BVP can be formulated. Problem 3.15. Find such a displacement vector w(t, x), and a stress tensor σ(t, x) that satisfy the constitutive relation (3.61) with the residual tensor (3.62), the equilibrium equation (3.63), and the boundary condition (3.64).

3.3.2 Relationship of static variational principles The classical variational principles in the linear theory of elasticity are well studied [103], and their relationships were discussed in detail, e.g., in [56]. Two of them, namely, the principles of minimum total potential energy and of minimum complementary potential energy, can be immediately derived from Problems 3.12 and 3.13. Given the volume and surface forces potential, the direct variational statement sounds as follows. Problem 3.16. Among all admissible displacements w(x) of an elastic body, find the actual displacements w ∗ (x) giving the absolute minimum of the total potential energy U[w ∗ ] = min U[w] , w

U=

d

1 ∫ ε ∶ C ∶ ε dV − ∫ f ⋅ w dV − ∑ ∫ ui wi dΓ , 2 2 V V i=1 Γi

(3.65)

subject to the boundary conditions in displacements (3.53). The dual principle is formulated in a proper way under assumption that the boundary displacements do not change during any stress variation.

3.3 Variational formulations in elastostatics | 49

Problem 3.17. Among all admissible stresses σ(x) of an elastic body, find the actual stresses σ ∗ (x) giving the absolute minimum to the total complementary energy Uc [σ ∗ ] = min Uc [σ] , σ

Uc =

d

1 ∫ σ ∶ C−1 ∶ σ dV − ∑ ∫ ui qi dΓ , 1 2 V i=1 Γi

(3.66)

subject to the conditions in stresses (3.55), (3.63). To apply the MIDR, let a=0

and

b=

1 2

in the constitutive functional (3.56), i.e., Φ=

1 ∫ ξ ∶ C ∶ ξ dV . 2 V

(3.67)

This integral makes it possible to formulate the corresponding minimization problem. Problem 3.18. Find such displacements w ∗ (x) and stresses σ ∗ (x) that minimize the functional (3.67) according to Φ[w ∗ , σ ∗ ] = min Φ[w, σ] w,σ

(3.68)

subject to the equilibrium condition (3.63) and the boundary constraint (3.64). The functional Φ given in equation (3.67) can be represented as σ∶ε dV , 2

Φ = W1 − 2W0 + W2 ,

W0 = ∫

ε∶C∶ε W1 = ∫ dV , 2 V

σ ∶ C−1 ∶ σ W2 = ∫ dV , 2 V

V

(3.69)

where W1 and W2 are the strain and stress elastic energies. It is seen from equation (3.69) that the values of W1 and W2 depend only on the kinematic variable w and the dynamic variable σ, respectively. At the same time, the term W0 relates the strains ε(w) to the stresses σ and does not depend explicitly on the properties of elastic material. After integration by parts, the elastic energy W0 takes the form: 1 1 W0 = − ∫ ∇ ⋅ σ ⋅ w dV + ∫ q ⋅ w dΓ . 2 V 2 Γ

(3.70)

It is possible to show with the equilibrium equation (3.63) that the following equality ∫ ∇ ⋅ σ ⋅ w dV = − ∫ f ⋅ w dV V

V

is valid. Thus, the important property is correct.

(3.71)

50 | 3 Variational principles in linear elasticity Corollary 3.1. The elastic energy W0 stored in a body is equal to a half of the mechanical work done by external forces in agreement with W0 =

1 1 ∫ f ⋅ w dV + ∫ q ⋅ w dΓ . 2 V 2 Γ

(3.72)

It is worth noting that this property holds for any equilibrium stress fields and corresponds to Clapeyron’s theorem. By using Corollary 3.1, the following theorem formulated in [47, 83] can be proved (see [56]). Theorem 3.1. For any kinematically admissible displacement field w under the boundary constraints (3.53) and any equilibrium stress field σ under the boundary constraints (3.55), the sum of the total potential and complementary energies defined respectively in equations (3.65) and (3.66) is non-negative: Φ[w, σ] = U[w] + Uc [σ] ≥ 0 .

(3.73)

As follows from the structure (3.73) of the functional Φ, Problem 3.18 is equivalent to independent minimization of the potential energy U over the vector function w and the complementary energy Uc over the tensor function σ according to min Φ[w, σ] = min U[w] + min Uc [σ] = Uc [w ∗ ] + Uc [σ ∗ ] . w, σ

w

σ

(3.74)

In addition to such decomposition, the following equality for the minima of the functionals Φ, U, and Uc Φ[w ∗ , σ ∗ ] = U[w ∗ ] + Uc [σ ∗ ] = 0

(3.75)

holds at the solution. Corollary 3.2. It follows from equation (3.74) that the existence of the minima for the total potential energy U and the total complementary energy Uc are the only necessary conditions for the existence of a static solution in linear elasticity. It is necessary to prove for sufficiency that the sum of these energies is equal to zero as in equation (3.75). 3.3.3 Bilateral energy estimates The MIDR enables to construct bilateral estimates of the total elastic energy W0 [w ∗ , σ ∗ ] = W1 [w ∗ ] = W2 [σ ∗ ] stored in a solid body [83]. These estimates are defined for any admissible displacements w(x) and stresses σ(x), which can be obtained, for example, on the basis of the Ritz method [83] as approximation to the exact solution w ∗ (x) and σ ∗ (x).

3.3 Variational formulations in elastostatics | 51

In particular, let uj (x) ≡ 0

for x ∈ Γ1j

with j = 1, … , d

(3.76)

in equation (3.53), i.e., the corresponding zero displacement components are given on all the boundary surfaces Γ1j . It follows then from the minimization statement (3.74) that Uc [σ] = W2 [σ] ≥ W2 [σ ∗ ] = W0 [w ∗ , σ ∗ ] . At the same time, equation (3.75) results in the expression max −U[w] = Uc [σ ∗ ] = W2 [σ ∗ ] , w

which leads to the inequality −U[w] ≤ W2 [σ ∗ ] = W0 [w ∗ , σ ∗ ] . Hence, the following theorem has been proved. Theorem 3.2. For any kinematically admissible displacement field w satisfying the boundary conditions (3.53) with zero right-hand sides (3.76) and any equilibrium stress field σ obeying the boundary condition (3.55), the bilateral estimates of the elastic energy stored in the body − U[w] ≤ W0 [w ∗ , σ ∗ ] ≤ Uc [σ]

(3.77)

hold. Vice versa, if the volume force is absent, i.e., f (x) ≡ 0 and uj (x) ≡ 0

for x ∈ Γ2j

with j = 1, … , d

in equation (3.55), i.e., no stresses are applied to the boundary parts Γ2j then U[w] = W1 [w] ≥ W1 [w ∗ ] = W0 [w ∗ , σ ∗ ] . It follows from equations (3.74) and (3.75) that max −Uc [σ] = U[w ∗ ] = W1 [w ∗ ] σ

and, therefore, −Uc [σ] ≤ W1 [w ∗ ] = W0 [w ∗ , σ ∗ ] . Consequently, the following theorem is valid.

(3.78)

52 | 3 Variational principles in linear elasticity Theorem 3.3. For any equilibrium stress field σ satisfying the homogeneous boundary conditions (3.53) with the right-hand sides (3.78) and any kinematically admissible displacement field w obeying the boundary conditions (3.53), the estimates − Uc [σ] ≤ W0 [w ∗ , σ ∗ ] ≤ U[w] are valid.

(3.79)

4 Variational statements in structural mechanics 4.1 Lateral motions of elastic beams 4.1.1 Dynamic equations for beam bending In this section, plane lateral motions of a rectilinear elastic beam are studied in the frame of the Euler–Bernoulli model. The following PDE, which is defined over the time–space domain Ω = (t 0 , t 1 ) × (x 0 , x1 ), describes the dynamic bending of the beam according to ρ(x)wtt (t, x) + (κ(x)wxx (t, x))xx = f (x, t)

for (t, x) ∈ Ω ,

(4.1)

where w(t, x) denotes lateral displacements of the beam midline, ρ(x) > 0 is the linear mass density, κ(x) > 0 is the bending stiffness coefficient, f (t, x) is external loads distributed along the beam of length L = x 1 − x 0 (see Figure 4.1). At the initial time instant t = t 0 , some deformed shape of the central line and velocities of its points are given together with two pairs of boundary conditions at each beam end x = x0 and x = x 1 . These constraints will be discussed further.

Figure 4.1: Rectilinear elastic beam.

By defining new coordinates and variables as x = Lx∗ ,

t = τ0 t ∗ ,

ρ(x) = ρ0 ρ∗ (x) , ρ τ0 = L √ 0 , κ0 2

w(t, x) = Lw∗ (t, x) ,

κ(x) = κ0 κ ∗ (x) , 1

1 x ρ0 = ∫ ρ(x) dx , L x0

f (t, x) = L−3 κ0 f ∗ (t, x) , 1

1 x κ0 = ∫ κ(x) dx , L x0

(4.2)

let us proceed to a dimensionless form of beam dynamic equation (4.1). Here, τ0 is the characteristic time value, whereas ρ0 and κ0 are respectively the mean linear density and the bending stiffness of the beam. In the case when ρ = const and κ = const, ρ∗ = κ ∗ = 1 at such change of variables. By substituting the expressions (4.2) into equation (4.1) and scaling with the factor L3 κ0−1 , a dimensionless equation is obtained similar to equation (4.1). The star superscript is omitted in what follows. https://doi.org/10.1515/9783110516449-004

54 | 4 Variational statements in structural mechanics Two dimensionless dynamic variables p(t, x) and s(t, x) are introduced into consideration. They are the linear momentum density in the lateral direction and bending moment in the cross section, respectively, which along with the kinematic variable w(t, x) characterize the behavior of the system. In the linear beam theory, the constitutive relations can be written in the form: v(t, x) = 0

and q(t, x) = 0

for (t, x) ∈ Ω .

(4.3)

Here, the first equation joins the momentum density p(t, x) and the velocities wt (t, x), whereas the second one links the moments s(t, x) with the bending curvature wxx (t, x) through the auxiliary functions: v ∶= wt − ρ−1 p and

q ∶= wxx − κ−1 s .

(4.4)

By involving equations (4.3) and (4.4), the equation of motion (4.1) given in displacements can be expressed via the function s and p according to pt (t, x) + sxx (t, x) = f (t, x)

for (t, x) ∈ Ω .

(4.5)

Define the linear boundary conditions, which are expressed through the values of the displacements w, the bending angles wx , the moments s, and the shear forces in cross sections −sx , as follows: α00 w(t, x0 ) + β00 sx (t, x0 ) = u00 (t)

and

α10 wx (t, x 0 ) − β10 s(t, x 0 ) = u01 (t) ,

α01 w(t, x1 ) − β01 sx (t, x1 ) = u10 (t)

and α11 wx (t, x 1 ) + β11 s(t, x 1 ) = u11 (t) .

(4.6)

Here, uij (t) for i, j = 0, 1 denote the boundary functions of external disturbances. The constant coefficients αji ≥ 0 and βji ≥ 0 define the type of boundary conditions. For example, the functions ui0 (t) give the displacements w(t, x i ) at the ends of the beam if the corresponding coefficients are equal to αi0 = 1 and βi0 = 0. Vice versa, if αi0 = 0 and βi0 = 1 for i = 0 or i = 1, then the shear forces −sx (t, x i ) are applied to the selected end cross section. If the constants αi1 = 1 and βi1 = 0 are chosen, then the slope of the beam midline wx (t, xi ) is fixed at the boundary point x i . In contrast, the bending moment s(t, xi ) is given at this point if αi1 = 0 and βi1 = 1. Such boundary conditions can be used for various types of elastic supports in the case when αji βji ≠ 0. To uniquely describe the dynamics of the beam, the initial distributions of the lateral displacements w and the momentum density p must be chosen as functions of x according to w(0, x) = w0 (x),

p(0, x) = p0 (x) .

(4.7)

The compatibility of the initial fields (4.7) and the boundary conditions (4.6) have to be addressed. Thus, the following IBVP can be formulated. Problem 4.1. Find such displacements w(t, x), momentum density p(t, x), and moments s(t, x) that obey the constitutive relations (4.3) and (4.4), the balance equation (4.5), as well as the boundary and initial conditions (4.6)–(4.7).

4.1 Lateral motions of elastic beams | 55

4.1.2 Complementary Hamilton principles To solve Problem 4.1, it is necessary to take into account that the functions ρ(x) and κ(x) might have jumps at some beam points, the boundary functions uij (t) for i, j = 0, 1 can be instantaneously changed, and the external loads f (t, x) are possibly discontinuous and have singularities on special manifolds in Ω. As a result, there can exist singularities of the sought kinematic and dynamic fields. For such problems, variational techniques are widely applied to find generalized solutions [103]. Similar to the theory of linear elasticity discussed in Chapter 3, conventional variational principles are formulated only for problems in which periodic or boundary (not initial) conditions are imposed with respect to the time. Let us describe two complementary Hamilton principles, which are suitable for analysis of beam dynamics. It is supposed in the principle of ‘least’ action that the displacements are fixed at the beginning and at the end of the time interval t ∈ [t 0 , t 1 ] as w(t 0 , x) = w0 (x)

and w(t 1 , x) = w1 (x)

for x ∈ (x 0 , x1 ) .

(4.8)

Additionally, the kinetic energy K(t) and the potential energy U(t) expressed through the displacement function w(t, x) should be integrable over the time interval t ∈ (t 0 , t 1 ). Furthermore, the variation of the displacements δw does not cause any change of the bending moment s and the shear force −sx at the ends of the spacial interval x = x0 and x = x 1 , i.e., αji βji = 0 with αji + βji = 1 for i, j = 0, 1 (see equation (4.6)). Then the principle of virtual work together with all the above-mentioned preconditions leads to the Hamilton principle. Problem 4.2. Find such a displacement field w∗ (t, x) that guarantees the stationarity of the action functional Υ[w] as follows: δΥ[w∗ ] = 0 , K= U=

1

t1

where Υ = ∫ (K(t) − U(t)) dt

1 x ∫ ρwt2 dx 2 x0

t0

with

and

1

(4.9) 1

1

x 1 x 2 dx − ∫ fw dx − ∑ (β0i ui0 (t)w(t, x i ) − β1i ui1 (t)wx (t, x i )) ∫ κwxx 2 x0 x0 i=0

subject to the initial and terminal constraints (4.8) as well as the boundary conditions in displacements: α0i (w(t, xi ) − ui0 (t)) = 0

and

α1i (wx (t, xi ) − ui1 (t)) = 0

for i = 0, 1 .

(4.10)

In accordance with this Hamilton principle, the boundary conditions in displacements, when αji = 1 and βji = 0, are essential in Problem 4.2. It means that these constraints have to be exactly satisfied in accordance with equation (4.10). As seen in

56 | 4 Variational statements in structural mechanics equation (4.9), the corresponding boundary functions uij (t) are not included into the linear part of the functional Υ. In contrast, the boundary conditions are natural constraints on bending moments in Problem 4.2, when αji = 0 and βji = 1. They are taken into account via the linear terms of the potential energy U. As a consequence, the constitutive relations (4.3) together with the time conditions (4.8) and the boundary conditions (4.10) are satisfied a priori. In contrast, the equation of motion (4.1) and the boundary constraints imposed both on the bending moments s and on the shear forces −sx constitute the system of Euler–Lagrange equations together with the conditions of transversality followed from the variational principle. To formulate the problem which is dual to Problem 4.2, let us define the initial and terminal momentum densities by p(t 0 , x) = p0 (x)

and p(t 1 , x) = p1 (x)

for x ∈ (x 0 , x 1 ) .

(4.11)

It is supposed also that functions of the complementary kinetic and potential energies, Kc (t) and Uc (t), respectively expressed through the momentum density p and bending moment s exist. Each of the four conditions at the end points (4.6) is given either in displacements or in moments. Then the corresponding variational statement is the following. Problem 4.3. Find the momentum density p∗ (t, x) and the bending moment s∗ (t, x) that are a stationarity point of the complementary functional of action Υc , so that δΥc [p∗ , s∗ ] = 0 , x1

Kc = ∫

x0

x1

U =∫

x0

p2 dx 2ρ

t1

where Υc = ∫ (Kc (t) − Uc (t)) dt t0

and

with (4.12)

s2 dx + [α0i ui0 (t)sx (t, xi ) − α1i ui1 (t)s(t, x i )]i=1 i=0 2κ

is subject to the balance equation (4.5), the initial and terminal constraints (4.11), as well as the boundary conditions in moments β0i (sx (t, xi ) − (−1)i ui0 (t)) = 0 and β1i (s(t, xi ) + (−1)i ui1 (t)) = 0 for i = 0, 1 .

(4.13)

The boundary conditions (4.13) of Problem 4.3 are essential if αji = 0 and βji = 1 for some indices i and j. The initial and terminal constraints (4.11) fixing the momentum distribution p belong to the group of essential constraints, too. Furthermore, the momentum density p and the bending moments s are differentially related via Newton’s second law (4.5). It means, in particular, that the functions of distributed loads f (t, x)

4.1 Lateral motions of elastic beams | 57

as well as of boundary forces and moments uij (t) are absent in the complementary potential energy. The boundary constraints are natural conditions when they limit the displacements and angles in the original BVP (αji = 1 and βji = 0). The corresponding right-hand side functions uij (t) form the linear part of the complementary action functional Υc . In the complementary principle, these conditions together with compatibility equations κ(x)pxx (t, x) + ρ(x)st (t, x) = 0

for (t, x) ∈ Ω

(4.14)

constitute the necessary conditions of stationarity for Υc . It is worth mentioning that both principles discussed in this subsection (Problems 4.2 and 4.3) relate only to the PDE systems with time boundary conditions and cannot be applied directly to solve Problem 4.1 (IBVP).

4.1.3 Method of integrodifferential relations in beam theory In accordance with the ideas of the MIDR [45], the following generalized statement of Problem 4.1 for elastic beam bending is given. Problem 4.4. The actual displacements w(t, x), momentum density p(t, x), and moments s(t, x) are found that satisfy the integral constitutive relation Φ+ [w, p, s] = ∫ φ+ dΩ = 0 Ω

1 with φ+ ∶= (ρv2 + κq2 ) , 2

(4.15)

the balance equation (4.5), as well as the boundary and initial conditions (4.6), (4.7). The integrand φ+ (t, x), which is introduced in equation (4.15) through the residual functions v = wt − ρ−1 p and q = wxx − κ−1 s defined in equation (4.4), is non-negative as a perfect square and measured in the unit of linear energy density. Thus, the functional Φ+ [w, p, s] is also greater or equal to zero for any functions w, p, s and attains its absolute minimum at the actual motion. This fact allows one to reduce Problem 4.4 to a constrained minimization problem. Problem 4.5. Find such displacements w∗ (t, x), momentum density p∗ (t, x), and moments s∗ (t, x) that minimize the constitutive functional Φ+ according to Φ+ [w∗ , p∗ , s∗ ] = min Φ+ [w, p, s] = 0 w,p,s

(4.16)

subject to the balance relation (4.5), the boundary and initial conditions (4.6), (4.7).

58 | 4 Variational statements in structural mechanics The integral Φ+ can be rewritten in the form: Φ+ = Ψ1 + Ψ2 − 2Ψ0

with

Ψ0 [w, p, s] = ∫ ψ0 dΩ , Ω

ψ0 =

pwt swxx + , 2 2

ψ1 =

Ψ1 [w] = ∫ ψ1 dΩ , ρwt2 2

+

Ω 2 κwxx

2

Ψ2 [p, s] = ∫ ψ2 dΩ ,

ψ2 =

,

2

2

Ω

(4.17)

p s + . 2ρ 2κ

The integrands ψi (t, x) for i = 0, 1, 2 are different representations of linear energy density stored in the beam. The energy density ψ1 (t, x) depends on the lateral displacements w(t, x) and its partial derivatives, whereas the quadratic form ψ2 (t, x) include only p(t, x) and s(t, x). The function ψ0 (t, x) is linear over all the variables w(t, x), p(t, x), s(t, x) and does not include explicitly inertial and elastic beam parameters. The balance equation (4.5) implies that the functions of momentum density p and moments s have to be continuously differentiable in time and twice continuously differentiable in space, respectively. At that, these requirements are extremely stringent for Problem 4.4 where the existence of weak derivatives are only demanded. To avoid this incompatibility in the variational Problem 4.5, Newton’s second law should be understood in the following integral form: ∫ (pht − shxx ) dΩ̂ = − ∫ fh dΩ̂ + ∫ (nx shx − (nt p − nx sx )h) dΓ̂ . Ω̂

Ω̂

Γ̂

(4.18)

This equality is to hold for any open simply connected subdomain Ω̂ ⊂ Ω with piecewise smooth boundary Γ̂ = 𝜕Ω̂ and any rather smooth function h(t, x). Here, nt and nx are projections of the outer unit normal to Γ̂ onto the axes t and x, respectively. It is possible to formulate the resulting requirements to Problem 4.4. First, the displacement and moment functions, w and s, are continuously differentiable over x along any line t = const inside Ω. Second, the displacement and momentum fields, w and p, are continuous along any line x = const inside Ω. Third, the function nt p − nx sx crosses continuously any inclined patch of smooth curves in Ω, where |nt ||nx | ≠ 0. 4.1.4 Energy estimates of solution quality By using the constitutive functional Φ+ and the integrals Ψi , i = 0, 1, 2, introduced in (4.17), a set of quality criteria can be proposed for arbitrary lateral displacements w, momentum density p, and bending moments s, which obey the constraints (4.6), (4.7), (4.18). The integral quality of approximate functions w, p, s is able to be estimated via such dimensionless ratios like Δ1 =

Φ+ 2Ψ0 =1− , Ψ1 + Ψ2 Ψ1 + Ψ2

Δ2 =

Φ+ , 2Ψ1

Δ3 =

Φ+ . 2Ψ2

(4.19)

4.1 Lateral motions of elastic beams | 59

It is worth noting that the quotients Ψi /(t 1 − t 0 ), i = 0, 1, 2, express the average value of beam energy during the considered time interval. Thus, the smaller the magnitudes Δj [w, p, s] are from equation (4.17), the better the quality is of the corresponding trial functions w, p, s. Moreover, the local ‘energy’ error of some admissible motion w(t, x), p(t, x), s(t, x) can be characterized explicitly with the integrand φ+ (w, p, s) defined in equation (4.15). One more index of approximation accuracy is the difference between the total mechanical energy stored by the beam and the work of external forces. To estimate this imbalance, let us introduce a new functional Ψ[w, p] as t1

x1

Ψ = ∫ W dt

with W = ∫ ψ dx ,

t0

x0

where ψ ∶=

2 p2 κwxx + , 2ρ 2

(4.20)

where W(t) is the corresponding energy and ψ(t, x) is its linear density. In other words, W is the sum of kinetic energy expressed trough the momentum density and elastic energy in the terms of displacement derivatives. Due to the continuity of admissible displacements w(t, x) and momentum density p(t, x) along the time direction (x = const), the energy function W(t) is also continuous. After differentiation with respect to the time t and double integration by parts over the coordinate x, the following expression for power is obtained with taking into account the balance relation (4.18): x1

Ẇ = ∫ ( x0

x1

ppt + κwxx wtxx ) dx = Ȧ + Wṗ ρ 1

x=x Ȧ = ∫ fwt dx + [swtx − sx wt ]x=x 0 x0

with

and

(4.21)

x1

Wṗ = ∫ (−ηpt + κξwtxx ) dx . x0

Here, A is the work of an external force; Wp denotes parasitic energy. The integral of this energy over time is decreasing if the approximation tends to the exact solution, i.e., if one of the ratios Δi defined in equation (4.19) is to converge to zero. 4.1.5 A family of variational problems In Chapter 3, equivalent variational statements of the dynamic problem in linear elasticity have been introduced via a one-parametric family of quadratic constitutive functionals. Similarly, a proper family of functionals can be defined for the problem on lateral motions for Euler–Bernoulli beams as Φ = ∫ φ dΩ Ω

with φ ∶= aρv2 + bκq2

for a2 + b2 =

1 and ab ≠ 0 . 2

(4.22)

60 | 4 Variational statements in structural mechanics It will be shown below that the necessary conditions of functional stationarity together with all essential constraints implemented in the corresponding generalized problem constitute all the governing equations of beam motions. Without loss of generality, the real coefficient a(b) in equation (4.22) may be positive. The functional Φ at a = b = 21 coincides with the integral Φ+ in equation (4.15). If b > 0, then each functional Φ(b) corresponds to a constrained minimization problem equivalent to Problem 4.5. Let actual and arbitrary admissible displacement, momentum, as well as moment fields be denoted respectively by w∗ , p∗ , s∗ and w, p, s. Define also that w = w∗ + δw, p = p∗ + δp, s = s∗ + δs. In this notation, the varied functional Φ and its first variation equal zero at the exact solution w∗ , p∗ , s∗ . Thus, it holds for any admissible fields w, p, s that a b ∗ Φ = ∫ ( (ρwt∗ + ρδwt − p∗ + δp)2 + (κwxx + κδwxx − s∗ + δs)2 ) dΩ , κ Ω ρ Φ = Φ[w∗ , p∗ , s∗ ] + δΦ[w∗ , p∗ , s∗ , δw, δp, δs] + δ2 Φ[δw, δp, δs] = Φ[δw, δp, δs] , ∗ δΦ = ∫ (2av(wt∗ , p∗ )(ρδwt − δp) + 2bq(wxx , s)(κδwxx − δs)) dΩ .

(4.23)

Ω

Here, δΦ and δ2 Φ denote the first and second variations of Φ, respectively. It is possible to write down a common variational formulation of the IBVP (Problem 4.1) for any values b ≠ 0. Problem 4.6. Find the actual displacements w∗ (t, x), momentum density p∗ (t, x), and moments s∗ (t, x) at which the constitutive functional Φ attains its stationary value δΦ = δw Φ + δp Φ + δs Φ = 0

(4.24)

subject to the balance equation, and the boundary and initial conditions (4.5)–(4.7). In equation (4.24), δw Φ, δp Φ, δs Φ are the first variations of the functional Φ with respect to w, p, s. After integration by parts with taking care of the problem constraints (4.6), (4.7), (4.18), the following expressions δw Φ = − ∫ (aρvt − b(κq)xx )δw dΩ Ω

x1

t1

1

x=x + a ∫ [ρvδw]t=t t=t 0 dx + b ∫ [κqδwx − (κq)x δw]x=x0 dt , 1

x0

t0

(4.25)

δp Φ + δs Φ = − ∫ (avδp + bqδs) dΩ Ω

are derived. It is possible to conclude from equation (4.25) that the first variations of the functional Φ become equal to zero over admissible functions δw, δp, δs if the constitutive relations (4.3) hold in a generalized sense.

4.1 Lateral motions of elastic beams | 61

In order to obtain the necessary conditions of stationarity, like Euler–Lagrange equations, for the functional Φ, it needs to take into account a differential relation between the momentum density p and the bending moment s imposed by Newton’s second law (4.18). The similar relation is valid for the variations δp and δs, namely, δpt = −δsxx .

(4.26)

Introduce an auxiliary function: t

r(t, x) = ∫ v(τ, x) dτ .

(4.27)

t0

It follows from equation (4.25) that δw Φ = − ∫ (aρrtt − b(κq)xx )δw dΩ Ω

x1

t1

1

x=x + a ∫ [ρrt δw]t=t t=t 0 dx + b ∫ [κqδwx − (κq)x δw]x=x0 dt , 1

x0

t0

δp Φ + δs Φ = − ∫ (arxx + bq)δs dΩ

(4.28)

Ω

x1

1

t1

1

x=x + a ∫ [rδp]t=t t=t 0 dx + a ∫ [rδsx − rx δs]x=x0 dt . x0

t0

Due to the constraints (4.6) and (4.7) that influence on the variations δw, δp, δs, the resulting Euler–Lagrange equations with natural boundary and terminal conditions are given by ρrtt + (κrxx )xx = 0

and arxx + bq = 0

for (t, x) ∈ Ω

with

α00 r(t, x0 ) + β00 (κrxx (t, x0 ))x = 0 , α10 rx (t, x0 ) − β10 κrxx (t, x0 ) = 0 , α01 r(t, x1 ) − β01 (κrxx (t, x1 ))x = 0 ,

(4.29)

α11 rx (t, x1 ) + β11 κrxx (t, x1 ) = 0 , r(t 1 , x) = 0 ,

rt (t 1 , x) = 0 .

The homogeneous PDE system (4.29) corresponds to a terminal-boundary value problem with respect to the variable r(t, x) defined over the time–space domain Ω = (t 0 , t 1 ) × (x 0 , x1 ). This system has a unique zero solution r(t, x) ≡ 0, which directly follows two identities v(t, x) ≡ 0 and q(t, x) ≡ 0. In other words, if the classical solution w∗ , p∗ , s∗ of Problem 4.1 exists then the system (4.29) together with the essential constraints of Problem 4.6 is equivalent to the governing equations (4.1), (4.3), (4.6), and (4.7) of elastic beam motions.

62 | 4 Variational statements in structural mechanics 4.1.6 Comparison of variational formulations Let us consider a representative of the functional family Φ, which has been introduced in the previous subsection in accordance with equation (4.22). This is Φ− ∶= Φ|b=−1/2 . The functional Φ− can be given by Φ− [w, p, s] = ∫ φ− dΩ = Θ1 + Θ2 − 2Θ0 Ω 2

with

ρv2 κq − , where 2 2 1 1 2 Θ0 = ∫ (pwt − swxx ) dΩ , Θ1 = ∫ (ρwt2 + κwxx ) dΩ , 2 Ω 2 Ω 1 Θ2 = ∫ (ρ−1 p2 − κ −1 s2 ) dΩ . 2 Ω φ− ∶=

(4.30)

As for the integrals Ψi , i = 0, 1, 2, defined in (4.17), it can be seen that the functional Θ1 depends on the displacement function w, whereas this function does not appear in Θ2 . The bilinear functional Θ0 , in turn, does not depend explicitly on the inertial and elastic properties of the beam material. By taking into account the balance equality (4.18), the integral Θ0 is reduced to the expression: t1

1

x1

0

t=t 2Θ0 = − ∫ fw dΩ + ∫ [wx s − wsx ]x=x x=x0 dt + ∫ [wp]t=t 0 dx . Ω

x0

t0

(4.31)

We can study the same case as in Problems 4.2 and 4.3, when the boundary conditions are given either in displacements or in moments. It implies that αji βji = 0 and αji + βji = 1 for i, j = 0, 1 and there are no mixed conditions of the third type. After the substitution of the expression (4.31) into equation (4.30), the functional Φ− can be converted by allowing the initial and boundary constraints (4.7), (4.10), (4.13) into the form: Φ− [w, p, s] = Υ[w] + Υc [p, s] + Ξ[w, p]

x1

1

with Ξ = − ∫ [wp]t=t t=t 0 dx . x0

(4.32)

If the distribution of either lateral displacements or momentum density are prescribed both at the beginning and at the ending of the process, then the integral Φ− splits into two independent parts: Φ− = Υ1 [w] + Υ2 [p, s]

with Υ1 = Υ[w] + Ξ1 [w] ,

Υ2 = Υc [p, s] + Ξ2 [p] .

(4.33)

Here, Υ1 depends only on the displacements w, Υ2 includes both the momentum variable p, and the moment one s. In turn, the functionals Ξ1 and Ξ2 are sums of integrals over the spatial interval x ∈ [x 0 , x1 ], which relate to the functions w and p, respectively. The expressions for Ξ1 and Ξ2 are presented in Table 4.1 for different types of dynamic BVPs. The first four rows correspond to the fixed initial and terminal fields of

4.1 Lateral motions of elastic beams | 63

displacements or momentums. The lowest row concerns time-periodic problems. The following notation is used in the table: x1

Ξ01 [w] = ∫ p0 (x)w(t 0 , x) dx x0 x1

Ξ02 [p] = ∫ x0

0

w0 (x)p(t , x) dx

x1

and Ξ11 [w] = ∫ p1 (x)w(t 1 , x) dx , x0

x1

Ξ12 [p] = ∫ x0

and

1

(4.34)

w1 (x)p(t , x) dx .

It is worth reminding that w0 (x) and w1 (x) denote known initial and terminal displacements, whereas p0 (x) and p1 (x) are for given initial and terminal momentum densities. Table 4.1: Functional terms for different time conditions of beam motions. Condition A 0

Condition B 1

Term Ξ1

Term Ξ2

w(t , x) = w0 (x)

w(t , x) = w1 (x)

0

Ξ12 − Ξ02

p(t 0 , x) = p0 (x)

p(t 1 , x) = p1 (x)

Ξ11 − Ξ01

0

w(t 0 , x) = w0 (x)

p(t 1 , x) = p1 (x)

Ξ11

−Ξ02

p(t 0 , x) = p0 (x)

w(t 1 , x) = w1 (x)

−Ξ01

Ξ12

w(t 0 , x) = w(t 1 , x)

p(t 0 , x) = p(t 1 , x)

0

0

For the five BVPs with the time constraints put in Table 4.1, the first variation of the functional Φ− must be equal to zero on the solution according to δΦ− [w, p, s] = δw Υ1 [w] + δp Υ2 [p, s] + δs Υ2 [p, s] = 0 . Thus, the original variational problem is decomposed into two independent subproblems. Problem 4.7. Find such a displacement field w∗ (t, x) that provides the stationarity of the functional, i.e., δΥ1 [w∗ ] = 0 subject to the constraints A and B from Table 4.1 as well as the boundary conditions in displacements (4.10). Problem 4.8. Find such momentum density p∗ (t, x) and bending moments s∗ (t, x), which constitute a stationary point of Υ2 so that δΥ2 [p∗ , s∗ ] = 0 subject to the balance equation (4.5), the constraints A and B from Table 4.1, and the boundary conditions in moments (4.13).

64 | 4 Variational statements in structural mechanics It seems important that Problem 4.7 coincides with Problem 4.2 (Hamilton principle) if the displacements are fixed at the end points of the time interval in accordance with the first row in Table 4.1. Vice versa, Problem 4.8 duplicates Problem 4.3 if the momentum distributions are given, as is shown in the second row of Table 4.1. Both Hamilton principles follow from Problems 4.7 and 4.8 if two periodic conditions are considered as in the lowest row of the table. It should be also mentioned that the functional Φ− as well as the other functionals Φ can be used to solve the IBVP (Problem 4.1) in contrast to the conventional functionals of action Υ and Υc , which are adapted exceptionally for dynamic BVPs. This fact noticeably restricts their application. After all, only time periodic problems represented in Table 4.1 is of great practical importance. If the initial conditions (4.7) are given, then the last term Ξ in the equation (4.32) depends on the product of the functions w and p taken at the terminal time instant. This term does not allow decomposition of the functional Φ− ; therefore, the problem has to be solved with respect to all the variables w, p, s. The same might be said about the problems with mixed boundary conditions as considered in [56], when the functional cannot be subdivided even under the time conditions from Table 4.1. This case will be discussed in detail in Section 4.3.

4.2 Longitudinal motions of viscoelastic rods The standard linear viscoelastic model is a way to describe the behavior of a viscoelastic material based on the idea of a linear combination of springs and dampers. As simplified cases, the Maxwell and Kelvin–Voigt models are used in applications. These models are found out to be inaccurate for many material characteristics. For example, the Maxwell model does not reflect creep or recovery, while the Kelvin–Voigt model does not take into account stress relaxation. In contrast, the standard model discussed in this section is a model that characterizes such processes.

4.2.1 Models of deformation with viscosity and rheology Let us consider the dynamics of a rectilinear thin rod described in Section 2.2. The coordinates of the rod ends are x = x0 and x = x 1 as shown in Figure 2.1. The time interval of rod motion is t ∈ (t 0 , t 1 ). There are three unknown scalar variables to be found: the displacements of the rod points w(t, x), the linear momentum density p(t, x), and the forces s(t, x) normal to the cross section. All the functions are defined in the domain Ω = (t 0 , t 1 ) × (x0 , x1 ). The balance equation remains the same as in equation (2.35), i.e., pt (t, x) − sx (t, x) − f (t, x) = 0

for (t, x) ∈ Ω .

(4.35)

4.2 Longitudinal motions of viscoelastic rods | 65

For simplicity, only kinematic and/or dynamic constraints are supposed at the rod ends in accordance with equations (2.37) and (2.38), so that w(t, x0 ) = u0 (t)

or s(t, x 0 ) = u0 (t) ,

w(t, x1 ) = u1 (t)

or s(t, x1 ) = u1 (t)

(4.36)

for t ∈ (t 0 , t 1 ). Rewrite for clarity the initial conditions (2.37) as w(t0 , x) = w0 (x)

and p(t0 , x) = p0 (x)

for x ∈ (x 0 , x 1 ) .

(4.37)

In agreement with the standard viscoelastic model, the constitutive relations between the momentum density and the velocities as well as between the force and deformations can be represented in the form: s(t, x) + μ(x)st (t, x) = κ(x)(wx (t, x) + η(x)wtx (t, x)) , p(t, x) = ρ(x)wt (t, x) for (t, x) ∈ Ω .

(4.38)

Here, μ denotes the relaxation time, κ is the stiffness of the rod, the coefficient η corresponds to the retardation time, and ρ is the linear material density. In such a way, the classical statement of the IBVP for the viscoelastic rod’s motions is the following. Problem 4.9. Find functions w∗ (t, x), p∗ (t, x), s∗ (t, x) that satisfy equations (4.35)– (4.38).

4.2.2 Minimization with integral constitutive relations There are several ways to proceed from Problem 4.9 to problems relying on the MIDR. First of all, the constitutive relations (4.38) can be rewritten in a generalized form by using the quadratic integral: Φ+ [w, p, s] =

1 ∫ (ρ(x)v2 (t, x) + κ(x)q2 (t, x)) dΩ ≥ 0 . 2 Ω

(4.39)

In the viscoelastic case, the residual functions v and q are defined as v ∶= wt − ρ−1 p ,

(4.40)

q ∶= wx + ηwtx − κ −1 (s + μst ) .

(4.41)

and

As a result, it is possible to formulate the following minimization problem.

66 | 4 Variational statements in structural mechanics Problem 4.10. Find w∗ (t, x), p∗ (t, x), s∗ (t, x) minimizing the functional Φ+ [w∗ , p∗ , s∗ ] = min Φ+ [w, p, s] = 0 w, p, s

(4.42)

subject to the constraints (4.35)–(4.37). All linear viscoelastic models are representable by a Volterra equation relating forces with strains according to t

s(t, x) = κr (x)wx (t, x) + ∫ R(t − τ, x)wtx (τ, x) dτ t0

= κr (x)wx (t, x) + Ir (w(t, x)) ,

(4.43)

or vice versa t

wx (t, x) = κc−1 (x)s(t, x) + ∫ C(t − τ, x)st (τ, x) dτ t0

= κc−1 (x)s(t, x) + Ic (s(t, x)) ,

(4.44)

where K is termed the creep function and F is named the relaxation function. For the standard model described by the differential relation from equation (4.38), the relaxation function is given by R(t, x) = κ ⋅

μ − η −t/μ ⋅e , μ

(4.45)

whereas the creep function is defined as C(t, x) =

μ − η −t/η ⋅e . κη

(4.46)

In this case, the elastic moduli for creep and relaxation are replaced by κr (x) = κc (x) = κ(x) .

(4.47)

It is natural to introduce two reciprocal constitutive functions for the residual forces as g ∶= s − κ −1 wx − Ir (w) ,

(4.48)

h ∶= wx − κ −1 s − Ic (s)

(4.49)

and for the residual strains as

Here, the relaxation integral Ir and the creep integral Ic are utilized according to equation (4.43).

4.3 Structures with lumped and distributed parameters | 67

The constitutive functions g(t, x) and h(t, x) must be equal to zero on the solution. This means that Φv [w, p, s] =

1 ∫ (ρ(x)v2 (t, x) − h(t, x)g(t, x)) dΩ = 0 , 2 Ω

(4.50)

where v is the residual velocity defined in (4.40). This functional, which includes the linear operators introduced in equations (4.43) and (4.44), is characteristic for the viscoelastic standard model. It is possible to prove that the integral Φv is non-negative if these two operators are mutually invertible. In this case, Problem 4.10 can be modified by implicating Φv instead of Φ+ . Problem 4.11. Find such w∗ (t, x), p∗ (t, x), s∗ (t, x) that minimize the functional Φv [w∗ , p∗ , s∗ ] = min Φv [w, p, s] = 0 w, p, s

(4.51)

subject to the equalities (4.35)–(4.37).

4.3 Structures with lumped and distributed parameters One important class of objects in solid mechanics is hybrid elastic structures including elements with both distributed and lumped parameters. For example, the dynamic behavior of a manipulator with flexible links that are connected by a spring or loaded with a point mass can be modeled by a combined PDE and ODE system. In such systems of equations, it is necessary to correctly take into account the interdependence of dissimilar elements. The focus in this section is on two types of lumped elements built into the beam mechanism, namely a linear spring and a point mass.

4.3.1 Motions of a rod weighted at the ends As in Section 2.2, let us consider a model of elastic rod motions but add to this system two point masses attached at the ends of the rod as shown in Figure 4.2. Only longitudinal displacements of the rod point are allowed in this model. The masses mi at points xi for i = 0, 1 can be connected with the fixed walls by an elastic spring with the stiffness coefficients κi and loaded respectively by the external point forces ui (t). Compared with the rod described in Chapter 2, additional forces distributed along the x axis are introduced and caused by a so-called Winkler foundation.

Figure 4.2: Rectilinear elastic rod weighted at the ends.

68 | 4 Variational statements in structural mechanics Define first the constitutive relations for the rod as v(t, x) = 0 ,

q(t, x) = 0 ,

r(t, x) = 0

for (t, x) ∈ Ω

(4.52)

with the domain Ω = (t 0 , t 1 )×(x 0 , x1 ). Three auxiliary residual functions are introduced in equation (4.52) according to v ∶= wt − ρ−1 p ,

q ∶= wx − κ−1 s ,

r ∶= w + μ−1 (g − f ) ,

(4.53)

where ρ(t, x) denotes the linear density, κ(t, x) is the coefficient of the rod stiffness, μ(t, x) is the coefficient of Winkler foundation, and f (t, x) defines external disturbances. An additional unknown g(t, x) is used in equation (4.53), which is the total distributed force applied to the rod. Functions w(t, x), p(t, x), and s(t, x) are respectively the displacements, the linear momentum density, and the elastic normal forces in the cross section. Instead of boundary conditions, the following lumped constitutive relations vi (t) = 0

and ri (t) = 0

with i = 0, 1

for t ∈ (t 0 , t 1 )

(4.54)

are given at both rod ends. The residual functions i −1 vi (t) ∶= wt (t, x i ) − m−1 i pi (t) and ri (t) ∶= w(t, x ) + κi (gi (t) − ui (t))

(4.55)

are introduced in equation (4.54) for convenience. Here, the momenta pi (t) and the forces gi (t) are new dynamic functions for the i-th point mass with i = 0, 1. Notice that the kinematic function w from equation (4.53) is used in the constitutive relations (4.55) due to the requirement of displacement continuity. If the spring stiffness κi tends to the infinity, the end mass is rather small, i.e., x1

mi ≪ ∫ ρ(x) dx , x0

and simultaneously ui (t) = κi−1 wi (t), then the prescribed displacement wi (t) is given at the point x = xi . In contrast, the force ui (t) is defined at this point if both κi → 0 and mi → 0. The mixed boundary condition of the third kind (2.39) appears only when the mass tends to zero. The momentum balance equation for the rod is given by pt (t, x) = sx (t, x) + g(t, x) for (t, x) ∈ Ω ,

(4.56)

whereas similar equations for point masses are written in the form: ṗ0 (t) = g0 (t) + s(t, x0 ) and ṗ1 (t) = g1 (t) − s(t, x1 ) for t ∈ (t 0 , t 1 ) .

(4.57)

The plus and minus signs before the function s in equation (4.54) arise due to the opposite directions of the normal forces at the rod ends.

4.3 Structures with lumped and distributed parameters | 69

To form a complete system of governing equations, the initial conditions w(t 0 , x) = w0 (x) p0 (t 0 ) = p00

and p(t 0 , x) = p0 (x)

for x ∈ (x 0 , x 1 ) ,

and p1 (t 0 ) = p01

(4.58)

are required. The classical statement of the IBVP for the rod with point masses and springs is the following. Problem 4.12. Find such functions w∗ (t, x), p∗ (t, x), s∗ (t, x), g ∗ (t, x), p∗0 (t), p∗1 (t), g0∗ (t), g1∗ (t) that satisfy equations (4.52)–(4.58). 4.3.2 Variational statement for hybrid system dynamics Generalize Problem 4.12 based on the MIDR by introducing the functional Φ+ = ∫ φ+ dΩ + Ω

1

1

t 1 ∑ ∫ (mi vi2 (t) + κi ri2 (t)) dt ≥ 0 , 2 i=0 t 0

1 φ+ = (ρ(x)v2 (t, x) + κ(x)q2 (t, x) + μ(x)r 2 (t, x)) dΩ ≥ 0 , 2

(4.59)

where the constitutive functions v, q, r are taken from equation (4.53) and the definition of the functions vi , ri is given in equation (4.55). As the functional Φ+ is non-negative for any unknown functions, it is possible to formulate a constrained minimization problem. Problem 4.13. Find such displacements w∗ (t, x), linear momentum density p∗ (t, x), normal forces s∗ (t, x), linear force density g ∗ (t, x), momenta p∗0 (t) and p∗1 (t), boundary forces g0∗ (t) and g1∗ (t) that lead the functional to its global minimum: Φ+ [w∗ , p∗ , s∗ , g ∗ , p∗0 , p∗1 , g0∗ , g1∗ ] =

min

{w, p, s, g, p0 , p1 , g0 , g1 }

Φ+ [w, p, s, g, p0 , p1 , g0 , g1 ] = 0

(4.60)

subject to the balance constraints (4.56) and (4.57), as well as the initial conditions (4.58). It is worth emphasizing that the boundary conditions are absent in this statement. These constraints are generalized and taken into account through the sum of the time integrals in the functional Φ+ . The variables g(t, x), g0 (t), g1 (t) can be algebraically eliminated from Problem 4.13 by using equations (4.56), (4.57). The corresponding expressions g(t, x) = pt (t, x) − sx (t, x) , g0 (t) = ṗ0 (t) − s(t, x0 ) and g1 (t) = ṗ1 (t) + s(t, x 1 ) .

(4.61)

70 | 4 Variational statements in structural mechanics must be substituted into the constitutive functions r, r0 , r1 . After that, these functions in the form r(t, x) = w + μ−1 (pt (t, x) − sx (t, x) − f (t, x)) , ri (t) = w(t, xi ) + κi−1 (pi̇ (t) + (−1)i s(t, x i ) − ui (t)) with i = 0, 1

(4.62)

are applied in equation (4.59). As a result, the functional Φ+ depends on a smaller number of variables. Thus, Problem 4.13 can be reformulated. Problem 4.14. Find w∗ (t, x), p∗ (t, x), s∗ (t, x), p∗0 (t), p∗1 (t) such that the functional attains its extremum value Φ+ [w∗ , p∗ , s∗ , p∗0 , p∗1 ] = under the initial conditions (4.58).

min

{w,p,s,p0 ,p1 }

Φ+ [w, p, s, p0 , p1 ] = 0

(4.63)

5 Ritz method for initial-boundary value problems 5.1 Finite-dimensional dynamic problems 5.1.1 Chain of linear oscillators In order to explain the prospective numerical application of the discussed variational formulations, consider a mechanical system consisting of n point particles with the masses mi , i = 1, … , n, connected one after another in a rectilinear chain by weightless springs (see Figure 5.1). The first particle is elastically attached to a movable base, which is displaced in agreement with a given law u(t). The only longitudinal motions of the chain along the x-axis are under study.

Figure 5.1: Chain of linear oscillators.

Let xi (t) denote the coordinate of the i-th particle in an inertial reference frame Ox. If u ≡ 0 in the equilibrium state, then this particle rests at the position xi (t) = xi0 = iL/N, where L is the length of the elastic chain. Choose the deviation of the particles from their equilibrium positions wi (t) = xi (t) − xi0 as generalized coordinates. Then the equations of system motions can be written as mi wï = ci (wi−1 − wi ) + ci+1 (wi+1 − wi )

with wn+1 = wn and w0 = u(t) ,

(5.1)

where i = 1, … , n and ci is the stiffness coefficient of the i-th spring. The generalized momenta and forces are introduced according to pi = mi wi̇

and si = ci (wi−1 − wi ) + ci+1 (wi+1 − wi ) .

(5.2)

The problem is to transfer the mechanical system in the given time T from the initial static state wi (0) = 0

and pi (0) = 0

with u(0) = 0

(5.3)

and pi (T) = 0

with u(T) = uT ,

(5.4)

to the terminal one wi (T) = uT

where uT denotes a prescribed chain shift. https://doi.org/10.1515/9783110516449-005

72 | 5 Ritz method for initial-boundary value problems The control input u is chosen from the space of polynomial functions as u(t) =

2n

uT (3T − 2t)t 2 + ∑ ui t i+1 (t − T)2 . T3 i=1

(5.5)

The coefficients ui are determined so that the terminal conditions (5.4) are satisfied. Of course, there exist many ways to transfer the studied system from the initial state to the final one. The presented example is selected only to illustrate the properties of numerical solutions of dynamic problems obtained on the basis of the proposed variational formulations. 5.1.2 Polynomial approximation of time functions To approximate the generalized coordinates w(t) ∈ ℝn and momenta p(t) ∈ ℝn in the numerical solution, the polynomial of a sufficiently high degree m

w̃ = ∑ w i t i , i=1

m

p̃ = ∑ pi t i , i=1

s̃ = ṗ̃

(5.6)

satisfying the initial conditions (5.3) are applied. By assembling all the coefficients w i and pi of the polynomials (5.6), a vector of design parameters m

m

i=1

i=1

y = ⨁ w i ⨁ pi ∈ ℝ2mn is obtained. Then the problem (2.7), (2.11) of constrained minimization for the func̃ ̃ tional Φ+ is reduced (after substitution of the polynomials w(t) and p(t) of equation (5.6) for the unknown functions w(t) and p(t)) to minimization of the quadratic function of the design parameter vector y: Φ+ (y, u) → min , y

(5.7)

where u = (u1 , … , u2n ) denotes the vector of control parameters introduced in equation (5.5). In turn, this optimization is equivalent to solving a linear algebraic system of the dimension 2mn with respect to the vector y. It is possible by using the polynomials (5.6) with rational coefficients to get an approximate solution analytically. Then its quality is characterized exceptionally by the discretization error. To numerically solve the control problem (5.2)–(5.4), the following algorithm is applied: – Firstly, the input function (5.5) and the approximations (5.6) are substituted in the functional (2.10).

5.1 Finite-dimensional dynamic problems | 73

– – –

After that, its minimum value is sought with respect to the design parameters y for arbitrary components of the control vector u. The resulting optimal vector y ∗ (u) determines the approximate phase trajectories ̃ y ∗ ). ̃ y ∗ ) and p̃ ∗ = p(t, w̃ ∗ = w(t, Only after that, the vector ũ is found satisfying the terminal conditions of equation (5.4), in other terms, ̃ w(T, y ∗ (u)) = 0 and

–

̃ y ∗ (u)) = 0 . p(T,

(5.8)

̃ ̃ = u(t, u). The resulting control law is u(t)

The relative error of the numerical solution can be defined through the ratio of the functional Φ+ and the time integral of the energy Ψ according to Δ=

Φ+ (y ∗ (u), u) Ψ(y ∗ (u))

T

with Ψ(y) = ∫ W̃ (t, y) dt , 0

(5.9)

where the approximate energy is introduced as 1 ̃ . W̃ = (p̃ ⋅ M−1 ⋅ p̃ + w̃ ⋅ K ⋅ w) 2

(5.10)

Consider the following system with the sample parameters: n = 3,

T = 4,

uT = 0.1 ,

mi = 1 ,

ci = 16 ,

i = 1, 2, 3 .

In this case, the exact solution w ∗ (t) and p∗ (t) of the Cauchy problem (5.2), (5.3) as well ̃ can be obtained for the fixed control u(t). ̃ and p(t) ̃ as its approximation w(t) Therefore, it is possible to get the relative error of the polynomials (5.6) according to Δ0 =

Σ[w,̃ p]̃ Ψ[w,̃ p]̃

with

1 T Σ = ∫ ((p̃ − p∗ ) ⋅ M−1 ⋅ (p̃ − p∗ ) + (w̃ − w ∗ ) ⋅ K ⋅ (w̃ − w ∗ )) dt . 2 0

(5.11)

Define the relative control error of ũ = u(t, u)̃ as Δu =

‖ũ − u∗ ‖22 ‖u∗ ‖22

T

with ‖u‖22 = ∫ u2 (t) dt . 0

(5.12)

Here, u∗ (t) is the control displacements exactly bringing the mechanical system (5.2) ̃ is the polynomial control obtained in accorto the terminal state (5.4), whereas u(t) dance with the numerical algorithm. The exact control u∗ (t) and the corresponding displacement of the particles xi∗ (t) for i = 1, 2, 3 are presented in Figure 5.2. For given parameters, the system is moved

74 | 5 Ritz method for initial-boundary value problems

Figure 5.2: Control u(t) and displacements xi (t).

from the initial state to the terminal one without large vibrations due to the smooth input function. In Figure 5.3, the errors Δ, Δ0 , and Δu defined in (5.9), (5.11), and (5.12), respectively, versus the approximation order m are shown in the logarithmic scale. As it can be seen from the chart, all the errors quickly tend to zero for the chosen approximations and control. To increase the solution quality, other finite-dimensional representations of the unknown functions are used, e.g., piecewise polynomial splines.

Figure 5.3: Relative errors Δ, Δ0 , and Δu .

5.2 Bivariate polynomials in rod and beam modeling 5.2.1 Longitudinal motions of an elastic homogeneous rod Let us return to the problem of longitudinal forced motions of thin rectilinear elastic rods considered in Section 2.2. To be more specific, one of the rod ends at x = x 1 is supposed to be free of external loads, while the other end with the coordinate x = x 0 is transferred along the axis x in agreement with a given time law Lu(t). The coordinates of the ends in some moving frame of reference can always be set as x 0 = 0 and x 1 = L

5.2 Bivariate polynomials in rod and beam modeling

| 75

for the undeformed rod. The initial time instant of the process is able to be shifted so that t 0 = 0, and terminal one is defined as t 1 = T. Suppose that the geometrical and mechanical parameters of the rod do not depend on the spatial coordinate x, that is, the linear density ρ(x) and the stiffness coefficient κ(x) are constant. In this case, the choice of the new dimensionless coordinates and the displacement function x∗ = L−1 x ,

t∗ = √

κ t, ρL2

w∗ (t, x) = L−1 w(t, x) − u(t)

allows us to reduce the original PDE to the form: wtt (t, x) − wxx (t, x) = a(t) ,

̈ where a(t) ∶= −u(t)

for (t, x) ∈ Ω .

(5.13)

Here, Ω = (0, T) × (0, 1) is the time–space domain. The dot over the symbol denotes the derivative with respect to the time t. No external distributed forces are assumed. The star superscripts are omitted for simplicity in what follows. The homogeneous boundary and initial conditions are given by w(t, 0) = 0 w(0, x) = w0 (x)

and wx (t, 1) = 0 ,

(5.14)

and wt (0, x) = p0 (x) .

(5.15)

For some prescribed law of motion u(t), the IBVP defined by the PDE (5.13) with the constraints (5.14) and (5.15) can be solved by the method of separation of variables [22]. For that, represent the displacements according to ∞

w(t, x) = ∑ yk (t)bk (x) .

(5.16)

k=1

Here, the basis functions bk (x) are eigenforms of the BVP: b″ (x) = λ2 b(x)

with b(0) = b′ (1) = 0 .

(5.17)

The prime symbol denotes the derivative with respect to the coordinate x. For this problem, the eigenvalues λk are found analytically as λk = −

π + kπ 2

for k ∈ ℕ .

Then the corresponding solutions of the BVP (5.17) have the form bk (x) = √2 sin λk x

(5.18)

and constitute an orthonormal set, i.e., 1

∫ bk (x)bl (x) dx = δk,l , 0

Here, δk,l is the Kronecker delta.

1

1/2

‖bk ‖2 = (2 ∫ sin2 (λk x) dx) 0

= 1.

(5.19)

76 | 5 Ritz method for initial-boundary value problems Let us substitute the series (5.16) in the equation of motion (5.13) and multiply this equality sequentially by each of the basis functions bk (x) defined in equation (5.18) for k ∈ ℕ. After integrating the resulting expressions over the interval x ∈ (0, 1) and taking into account equations (5.16) and (5.19), the countable ODE system yk̈ + λk2 yk = Ck a(t)

for k ∈ ℕ

1

with Ck = ∫ bk (x) dx = 0

√2 λk

(5.20)

is derived with respect to the Fourier coefficients yk (t). A particular solution of ODEs (5.20) for the homogeneous initial conditions can be presented as yk1 (t) =

Ck t ∫ sin(λk (t − τ))a(τ) dτ . λk 0

(5.21)

To compose a solution for arbitrary non-homogeneous initial conditions (5.15), the eigenfunction of the k-th mode needs to be added to the particular solution (5.21) according to yk = yk0 (t) + yk1 (t) with yk0 (t) = wk0 cos λk t +

p0k sin λk t , λk

(5.22)

where 1

wk0 = ∫ w0 (x)bk (x) dx 0

and

1

p0k = ∫ p0 (x)bk (x) dx . 0

An important characteristic of the dynamic process is the mean mechanical energy: Wf =

1 ∫ (w2 + wt2 ) dΩ . 2T Ω x

By taking into account the orthonormality of the basis functions in accordance with equation (5.19), the expression Wf =

T 1 ∑ ∫ (λk2 yk2 (t) + yk2̇ (t)) dt 2T k=0 0 ∞

(5.23)

is obtained after integration by parts. It means that the total energy of the rod is a direct sum of the energies of every eigenform. Consider the polynomial law of rod acceleration l

a(t) = ∑ am t m , m=0

(5.24)

5.2 Bivariate polynomials in rod and beam modeling

| 77

which will be used in Chapter 9 for parametric optimization of the elastic system locomotion. In this case, the integral (5.21) can be calculated analytically as l

yk1 (t) = ∑ ak,m [(−1)m+1 cos( m=0

τk = λk t ,

ak,m = j

{ ∑i=0 pm (t) = { j { ∑i=0

√2k!am , λkm+3

mπ + τk ) + pm (τk )] , 2 (5.25)

(−1)j−i t 2i (2i)!

for m = 2j and j ∈ ℕ ,

(−1)j−i t 2i+1 (2i+1)!

for m = 2j + 1 and j ∈ ℕ .

If the polynomial degree l in equation (5.24) is fixed, the functions of initial distribution w0 (x) and p0 (x) are rather smooth, and the motion time is limited, t ≤ T < ∞, then the terms of series (5.16) is decreasing with increasing the index k. This fact follows the finiteness of the eigenfunctions according to the estimates |bk (x)| ≤ √2

and |yk0 (t)| ≤ |u0k | +

|vk0 | λk

as well as the analyticity of the particular solution (5.25) over the closed-time interval t ∈ [0, T]. When the initial conditions are homogeneous (w0 (x) = p0 (x) ≡ 0), it is possible to estimate the convergence rate of the series (5.16) for a specific polynomial (5.24) and given process duration: |yk1 (t)bk (x)| ≤ Cλk−3 . Here, the coefficient C depends on the time T and the coefficients am of the acceleration (5.24). The convergence properties of the series (5.16) for the inhomogeneous initial displacements w0 (x) and momentum density p0 (x) are characterized by the smoothness degree of these functions [18]. 5.2.2 Conventional Galerkin method By extracting the first n equations in (5.20), the n-mode approximation n

w(n) (t, x) = ∑ yk (t)bk (x) k=1

(5.26)

to the exact solution of the problem (5.13)–(5.15) is obtained. Note that the integral (5.21) for an arbitrary function of acceleration a(t) does not always exist in a closed form. Moreover, the eigenvalues and eigenforms of rod vibrations can be usually derived by involving special numerical methods if the distribution of density ρ(x) or stiffness κ(x) is not uniform [3, 71].

78 | 5 Ritz method for initial-boundary value problems One of widespread approaches to solve dynamic problems is the Galerkin method. In the simplest modification of this approach, finite-dimensional approximations of unknown variables expressed via a specific set of basis functions are applied. For an elastic rod, its longitudinal displacements can be found, for example, in the form: n

w(n) = ∑ wk ψk (t, x) ,

(5.27)

k=1

where ψk are basis functions with certain properties of completeness [88], wk are unknown coefficients. At that, the approximate solution w(n) has to satisfy the homogeneous boundary conditions in displacements, i.e., the first equation in (5.14), and possibly, the first and second relations in the initial conditions (5.15). After that, the approximation (5.27) is substituted in equation (5.13), and the system of integral projections is assembled in accordance with T

(n) − a)ψk dΩ + ∫ [wx(n) ψk ]x=1 dt = 0 ∫ (wtt(n) − wxx 0

Ω

for k = 1, … , n .

(5.28)

If all of the functions ψk obey the boundary condition in displacements (5.14), then the integral equalities (5.28) are reduced to the form: ∫ (wx(n) ψk,x + (wtt(n) − a)ψk ) dΩ = 0 Ω

for k = 1, … , n

(5.29)

via integration by parts. The coefficients wk for k = 1, … , n are found from the system (5.29) or (5.28). A sophisticated question attributed to the Galerkin method is the proper choice of a basis {ψk } and a number n guaranteeing the required quality of a numerical solution. 5.2.3 Ritz method and the MIDR Based on the MIDR discussed in the previous chapters, it is possible to obtain explicit estimates of solution quality. These estimates allow us to develop effective numerical algorithms for static as well as dynamic problems in solid mechanics [56]. Describe one of the possible algorithms relating to the variational statement of the IBVP in linear elasticity formulated previously as Problem 2.4. In this case, the longitudinal motions of an elastic rod, which are described by the PDE system (5.13)–(5.15) with the boundary displacement u(t) given in equation (5.24), is under study. As in Section 2.2, the functions of normal forces s(t, x) and linear momentum density p(t, x) must be added to the displacement function w(t, x). The minimum of the mean-square residual is sought according to Φ+ = ∫ φ+ (w, p, s) dΩ → min , Ω

1 φ+ = (v2 + q2 ) 2

w, p, s

with v = wt − p and q = wx − s ,

(5.30)

5.2 Bivariate polynomials in rod and beam modeling

| 79

subject to the constraints pt − sx = a(t) w(t, 0) = 0

for (t, x) ∈ Ω , (5.31)

and s(t, 1) = 0 ,

w(0, x) = w0 (x)

and p(0, x) = p0 (x) .

The polynomial approximation to desired variables is utilized. For a numerical solution to this problem, the unknown variables are given by ̃ x) = w0 (x) + tx ∑ wij t i x j , w(t, i+j≤nw

̃ x) = p0 (x) + t ∑ pij t i x j , p(t,

(5.32)

i+j≤np

1

̃ x) = a(t)(1 − x) − ∫ p̃ t (t, y) dy s(t, x

with the indices {i, j} ⊂ ℕ. The finite-dimensional functions introduced in equation (5.32) obey all the constraints (5.31). After substituting the approximations (5.32) in the functional Φ+ [w, p, s] and introducing the vector of desired parameters y ∈ ℝn , which are composed of the unknown coefficients wi,j for 0 ≤ i + j ≤ nw and pi,j for 0 ≤ i + j ≤ np , the original problem is reduced to unconstrained minimization: Find such a vector y ∗ that ̃ + (y ∗ ) = min Φ ̃ + (y) , Φ y

̃ + (y) = Φ+ [w,̃ p,̃ s]̃ is a quadratic function. The dimension of the vector y is where Φ defined by 1 1 n = (nw + 1)(nw + 2) + (np + 1)(np + 2) . 2 2 Let the polynomial degrees np and nw be equal to each other. Then the minimization is equivalent to the solution of the algebraic system: ̃ + (y) 𝜕Φ = 0 for y ∈ ℝn 𝜕y

with n = (nw + 1)(nw + 2) .

Example 5.1. Assume that the rod is motionless and strainless at the beginning t = 0 of the process, i.e., w0 (x) = 0 ,

p0 (x) = 0 ,

u(0) = 0 ,

̇ = 0, u(0)

(5.33)

where u(t) is the shift of the rod end at x = 0. Suppose also that the position and velocity of this end are fixed at the time instant T. In other words, u(T) = uT

̇ and u(T) = 0.

(5.34)

80 | 5 Ritz method for initial-boundary value problems Let the acceleration a = ü be a linear function of the time t. Then it follows from equation (5.34) that a(t) = a0 + a1 t

with a0 =

6uT 12u and a1 = − 3T . T2 T

(5.35)

̇ (dashed curve), and the acThe displacement u(t) (solid curve), the velocity u(t) celeration a(t) (dash-dot curve) of the transferred end are presented in Figure 5.4 for T = 4 and uT = 1. In view of the system linearity, the solution is proportional to the coefficient uT . The other solutions are obtained simply by scaling. Therefore, it is enough to consider only the case when uT = 1. The trajectories of rod points are defined only by the parameter T. Let us analyze the quality of resulting polynomial approximations (5.32) versus the polynomial degree nw .

Figure 5.4: Displacement u, velocity u,̇ and acceleration a of the rod end at x = 0.

As it has been proposed in Subsection 4.1.4, the absolute solution error in the terms of ̃ + (y ∗ ). Then the relative error can be introduced as energy is defined by the functional Φ Δ=

̃+ Φ ̃ Ψ

T

̃ dt ̃=∫ W with Ψ 0

and

1 ̃ = 1 ∫ (w̃ 2x + p̃ 2 ) dx , W 2 0

(5.36)

̃ y) is the approximate mechanical energy of the elastic rod, expressed where W(t, ̃ x, y) given in ̃ x, y) and the momentum density p(t, through the displacements w(t, equation (5.32). Let the duration of motion be fixed (T = 4). The relative displacements w(t, x) − u(t) of rod points as a function of t and x for the polynomial degree nw = 15 are shown in Figure 5.5. It is seen that the displacement of the free end at x = 1 changes its sign once during the process. An important characteristic of the process is the approximate mean energy of ̃ as a function of the parameter nw . The solid curve in Figure 5.6 the system W = T −1 Ψ ∗ demonstrates strictly monotonic decreasing of the value W − W with the growth of ∗ the polynomial dimension. Here, the mean energy W ≈ 0.0406 is found with the help

5.2 Bivariate polynomials in rod and beam modeling

| 81

Figure 5.5: Displacements w(t, x) − u(t) of rod points for T = 4.

Figure 5.6: Mean energy W vs. polynomial degree nw .

of the Fourier method for a rather large number of modes n > 30. This value is close to the exact one. For comparison, the dashed curve reflects the change of the approx∗ imate mean energy W − W f (n) obtained for the n-mode solution in agreement with equation (5.26). In this example, the Fourier method gives us more accurate results in terms of energy. The relative error Δ for polynomial approximation as a monotonic function of the degree nw is depicted in Figure 5.7. As it can be seen from Figures 5.6 and 5.7, the greater is the dimension of the approximations, the closer is the numerical solution to the exact one. Introduce the new relative error function Δ0 (nw ) = 1 −

W(nw ) Wf

and compare it with Δ from equation (5.36) for the same number nw . This function at nw = 15 is equal to Δ0 ≈ 0.13%. The square of this error is comparable with the value of the error Δ (√Δ(nw ) ≈ 0.17%). This correlation can be explained by the fact that the functional Φ+ relates to the energy variation (its deviation from the exact quantity), but the value of Δ0 , in some sense, is proportional to the square of this variation.

82 | 5 Ritz method for initial-boundary value problems

Figure 5.7: Relative error Δ vs. polynomial degree nw .

To demonstrate the property, represent an approximate solution w(t, x), p(t, x), and s(t, x) of the problems (5.30), (5.31) as w = w∗ + δu ,

p = u∗t + δp ,

s = u∗x + δs .

The functionals Φ+ and Ψ are the quadratic form with respect to the variables w, p, and s and can be expanded in the series Φ+ [w, p, s] = Φ∗+ + δΦ+ + δ2 Φ+ , Ψ[w, p, s] = Ψ∗ + δΨ + δ2 Ψ , where Φ∗+ = 0 and Ψ∗ are the exact values of the functionals. The other terms of the expansions and their estimates are given by δ2 Φ+ = ∫ ((δwx − δs)2 + (δwt − δp)2 ) dΩ , Ω

δΦ+ = ∫ ((wx∗ − s∗ )(δwx − δs) + (wt∗ − p∗ )(δw − δp)) dΩ

⇒

Ω

|δΦ+ | ≤ C1 √Φ∗+ δ2 Φ+ = 0 and 1 ∫ (δwx2 + δp2 ) dΩ , 2 Ω 1 δΨ = ∫ (u∗x δux + wt∗ δp) dΩ 2 Ω δ2 Ψ =

Δ0 = |

⇒

|δΨ| ≤ C2 √δ2 ΨΨ∗ ,

2 2 Ψ∗ − Ψ √ δ Ψ + O( δ Ψ ) . | ≤ C 2 Ψ∗ Ψ∗ Ψ∗

The dependence of the relative error Δ on the time interval length T is presented in Figure 5.8. The displacement of the rod end wT = 1 as well as the approximation orders nw = np = 12 introduced in equation (5.32) are defined. It possible to see in the

5.2 Bivariate polynomials in rod and beam modeling

| 83

Figure 5.8: Relative error Δ vs. time T .

chart that the solution accuracy is decreasing with the elongation of the time interval. This deals with the fact that the elastic vibrations with relatively high frequencies may be excited in the rod. Such kind of vibrations is hardly interpolated with polynomials. When the duration of motion is comparable or shorter than the period of the first eigenmode (T ≤ 2πλ1−1 = 4), the relative error is sufficiently small despite the high energy stored in the rod. Note that it looks rather promising to use other basis functions, for instance splines, to raise the accuracy of the numerical solution over a long time interval (T ≫ 4 in Example 5.1). Such approximations will be considered in the next sections. 5.2.4 Lateral elastic displacements In Section 4.1, the dynamic problem on lateral displacements of a thin rectilinear beam has been formulated in the frame of the Euler–Bernoulli model. Consider here this problem in more detail. Let one end of the elastic beam at x = x 1 = 1 be free of loads. The other at x = x0 = 0 is rigidly attached to a base that moves across in accordance with prescribed law u(t). The stiffness κ and the density ρ are uniformly distributed along the beam. In this case, the equation of motion (4.1) in the coordinate system transferring together with the base is reduced after the transformation (4.2) to wtt (t, x) + wxxxx (t, x) = a(t)

with a(t) = −utt (t)

for (t, x) ∈ Ω ,

(5.37)

where a(x) denote the distributed inertial force. The external load is absent in the example. The boundary conditions are specified as w(t, 0) = wx (t, 0) = wxx (t, 1) = wxxx (t, 1) = 0 .

(5.38)

Some initial distribution of displacements and velocities is also given by w(0, x) = w0 (x)

and wt (0, x) = p0 (x) .

(5.39)

84 | 5 Ritz method for initial-boundary value problems As well as for the longitudinal motions of the rod discussed in Subsection 5.2.1, the solution of the PDE (5.37) with the boundary and initial constraints (5.38), (5.39) is found as a series (5.16). For the beam bending, bk (x) for k ∈ ℕ denote the eigenfunctions of the following BVP: d4 b(x) = λ4 b(x) dx 4

with b(0) = b′ (0) = b″ (0) = b‴ (0) .

The solution of this problem is bk (x) = cosh λk x − cos λk x −

cosh λk + cos λk (sinh λk x − sin λk x) , sinh λk + sin λk

(5.40)

where the eigenvalues λk , or wave numbers, are derived from the transcendental equation 1 + cosh λ ⋅ cos λ = 0 .

(5.41)

It is possible to prove that these functions are orthonormal. In other words, they obey the relations represented in equation (5.19). By projecting the dynamics equation (5.37) on the basis functions bk (x) and by taking into account equation (5.41), a countable ODE system similar to equation (5.20) follows from (5.40) as yk̈ (t) + ω2k yk (t) = Ck a(t) ωk = λk2

for k ∈ ℕ

with

1

and Ck = ∫ bk (x) dx ,

(5.42)

0

where ωk are the eigenfrequencies of the beam vibrations. The particular solution of the system (5.42) with the homogeneous initial conditions has the form: yk1 (t) =

Ck t ∫ a(τ) sin ωk (t − τ) dτ . ωk 0

(5.43)

The solution of equation (5.37) for the zero right-hand side with the non-homogeneous initial data (5.39) can be written according to yk0 (t) = wk0 cos ωk t + 1

p0k sin ωk t ωk

wk0 = ∫ w0 (x)bk (x) dx 0

with 1

and p0k = ∫ p0 (x)bk (x) dx . 0

The mean mechanical energy of the beam is Wf =

1 ∫ (w2 + wt2 ) dΩ . 2T Ω xx

5.2 Bivariate polynomials in rod and beam modeling

| 85

It is possible to rewrite it by taking into account the orthonormality of the basis in the form: Wf =

T 1 ∑ ∫ (λk4 yk2 (t) + yk2̇ (t)) dt . 2T k=0 0 ∞

(5.44)

When the beam base is transferred in accordance with the polynomial law l

a(t) = ∑ am t m , m=0

the integral (5.43) is represented explicitly by l

mπ + τk ) + pm (τk )] 2 m=0 m!Ck am τk = ωk t and ak,m = , ωm+2 k yk1 (t) = ∑ ak,m [(−1)m+1 cos(

with

where the function pm is defined in equation (5.25). 5.2.5 Numerical simulation of beam bending Let us extend the Ritz method based on variational formulations of dynamic problems with polynomial approximations to the case of lateral beam motions. In addition to the unknown displacements w(t, x), the bending moment s(t, x) and the linear momentum density p(t, x) are introduced. In accordance with Problem 4.5, the minimum of the constitutive functional Φ+ is sought as follows: Φ+ = ∫ φ(w, p, s) dΩ → min , w,p,s

Ω

1 φ = (v2 + q2 ) 2

with v = wt − p and q = wxx − s ,

(5.45)

subject to the constraints

pt + sxx = a(t)

for (t, x) ∈ Ω ,

w(t, 0) = wx (t, 0) = s(t, 1) = sx (t, 1) = 0 , w(0, x) = w0 (x)

(5.46)

and p(0, x) = p0 (x) .

For numerical simulation, the unknowns are taken from the set of polynomials with respect to the time and space coordinates: ̃ x) = w0 (x) + tx2 ∑ wi,j t i x j , w(t, i+j≤nw

̃ x) = p0 (x) + t ∑ pi,j t i xj , p(t, i+j≤nw

1 1 1 ̃ x) = a(t)(1 − x)2 − ∫ ∫ p̃ t (t, x1 ) dx1 dx2 . s(t, 2 x x2

(5.47)

86 | 5 Ritz method for initial-boundary value problems Note that such approximations satisfy all the constraints (5.46). Let us substitute w,̃ p,̃ s̃ for w, p, s in the functional Φ+ and introduce into consideration the vector of design parameters y ∈ ℝn , n = (nw + 1)(nw + 1), assembled from all the coefficients wi,j and pi,j for the indices i ∈ ℕ and j ∈ ℕ with i + j ≤ nw . After that, the unconstrained ̃ + (y) = Φ+ [w,̃ p,̃ s]: ̃ minimization problem is stated for the quadratic function Φ ∗ Find a vector y such that ̃ + (y ∗ ) = min Φ ̃ + (y) . Φ y

Example 5.2. It is supposed that the beam is undeformed and stays at rest at the beginning of the motion, that is, w0 (x) = 0

and

p0 (x) = 0 .

The beam acceleration a(t) is a linear function given by equation (5.35) with the parameter uT = 1. The relative error of the polynomial solution is defined as Δ=

̃ + (y ∗ ) Φ , ̃ ∗) Ψ(y

T

̃ y) dt , ̃ = ∫ W(t, Ψ(y) 0

1 ̃ = 1 ∫ (w̃ 2xx + p̃ 2 ) dx , W 2 0

(5.48)

̃ is the approximate mechanical energy. where W(t) In Figure 5.9, the displacements of the beam points w are shown as a function of the coordinates t and x at T = 4 and nw = 15. The distribution of the local error φ introduced in equation (5.45) is depicted at these parameters in Figure 5.10. The maximal values of the error appear at the beginning of the process.

Figure 5.9: Displacements of beam points w at T = 4.

Analyze the mean energy W = T −1 Ψ, where the functional Ψ is given in equation (5.48). ∗ The solid curve in Figure 5.11 displays the decreasing of the energy error W − W with increasing the order of the approximation nw . For comparison, the dashed curve ∗ demonstrates the change of the energy error W − W f (nw ) obtained for the nw -mode ∗ approximation in accordance with equation (5.44). The value W ≈ 5.25 × 10−3 is found by the Fourier method for nw > 20 and close to the exact energy.

5.2 Bivariate polynomials in rod and beam modeling

| 87

Figure 5.10: Local error distribution φ at Mw = 15.

Figure 5.11: Mean energy error W − W (nw ). ∗

Figure 5.12: Relative solution error Δ(nw ).

The relative error of the polynomial approximation Δ is shown in Figure 5.12 as a monotonic function with respect to the degree Mw . As it can be seen from Figures 5.11 and 5.12, the polynomials tend to the exact solution when the approximation order is increasing. It is worth noting that the time T = 4 chosen in this example is greater than the period of the first eigenmode T1 = 2πλ1−2 ≈ 1.787. It allows one to explain why a rather high degree nw is required to obtain a reliable solution.

88 | 5 Ritz method for initial-boundary value problems

5.3 FEM modeling of elastic rod dynamics 5.3.1 Modified minimization problem Let us return to the problem of longitudinal motions for the homogeneous elastic rod studied in Example 5.1. In agreement with the variational statement (5.30) and (5.31), the problem is to find the minimum of the constitutive functional Φ+ [w, p, s] = ∫ φ+ dΩ → min , w, p, s

Ω (5.49) 1 φ+ = ((wt − p)2 + (wx − s)2 ) 2 subject to the conventional constraints given in an inertial coordinate system as

pt (t, x) = sx (t, x) w(t, 0) = u(t) ,

for (t, x) ∈ Ω = (0, T) × (0, 1) , s(t, 1) = w(0, x) = p(0, x) = 0 .

(5.50)

To satisfy Newton’s second law, the PDE relation in equation (5.50), the momentum (p) and force (s) functions are expressed through an auxiliary dynamic variable r like s(t, x) = rt (t, x)

and p(t, x) = rx (t, x) for (t, x) ∈ Ω .

(5.51)

It is possible to prove that the function r(t, x) must be continuous on the space– time domain Ω in accordance with Newton’s laws, which join force and momentum fields. The minimized functional 1 (5.52) Φ+ [w, r] = ∫ ((wx − rt )2 + (wt − rx )2 ) dΩ 2 Ω

contains the first derivatives of the functions w(t, x) and r(t, x). This fact supposes the existence of generalized partial derivatives, i.e., w, r ∈ H 1 (Ω, ℝ). As a consequence, their continuity also follows [78]. The boundary and initial conditions (5.50) can be rewritten through w and r as w(t, 0) = u(t) ,

r(t, 1) = c ,

w(0, x) = 0 ,

r(0, x) = c .

(5.53)

For simplicity, the constant c will equate to zero in what follows. In this section, a modification of the FEM developed on the basis of the Ritz method is applied to numerical solution of the minimization problem. The unknown displacements w, momentum density p, and forces s are found in spaces of piecewise polynomial functions defined on the triangulated domain Ω [42]. It is important to mention that the FEM algorithm discussed in this section can be naturally extended to other types of boundary conditions as well as to the case of nonhomogeneous initial functions and distributed loads. The considered simplifications give us a possibility to focus on essential features of the algorithm without details complicating the issue.

5.3 FEM modeling of elastic rod dynamics | 89

5.3.2 Piecewise polynomial approximations Some nodal points tn and xm are singled out respectively in the coordinate axes t and x so that xm > xm−1 x0 = 0 ,

for m = 1, … , M xM = 1 ,

t0 = 0 ,

and tn > tn−1

for n = 1, … , N ,

tN = T .

With the help of the straight lines x = xm and t = tn (see Figure 5.13), the domain Ω is subdivided into MN rectangles Ωm,n = (tn−1 , tn ) × (xm−1 , xm ) , where m = 1, … , M and n = 1, … , N.

Figure 5.13: Uniform triangle mesh on the domain Ω.

The vertices of these rectangles with the coordinates (tn , xm ), which are nodes of the designed mesh, are denoted as Am,n . The open interval joining two points Ak,l and Am,n are identified as Ak,l Am,n . To be short, the following notation for the rectangle edges are introduced: Tm,n = Am,n−1 Am,n

and Xm,n = Am−1,n Am,n .

The diagonals of the rectangle Ωm,n (see Figure 5.14) divide it into four triangles: Δ1,m,n = Bm,n Am−1,n−1 Am−1,n ,

Δ2,m,n = Bm,n Am−1,n Am,n ,

Δ3,m,n = Bm,n Am,n Am,n−1 ,

Δ4,m,n = Bm,n Am,n−1 Am−1,n−1 .

(5.54)

with the central point Bm,n at the intersection of the diagonals Am−1,n−1 Am,n and Am,n−1 Am−1,n .

90 | 5 Ritz method for initial-boundary value problems

Figure 5.14: Substructure of the mesh on the rectangle Ωm,n .

Let the additionally generated edges of these four triangles (5.54) call Q1,m,n = Bm,n Am−1,n−1 ,

Q2,m,n = Bm,n Am−1,n ,

Q3,m,n = Bm,n Am,n ,

Q4,m,n = Bm,n Am,n−1 .

(5.55)

Complete bivariate polynomials of the degree K in the Bésier–Bernstein form [27] are utilized to approximate the unknown variables on each of the 4MN triangles Δl,n,m defined in equation (5.54). In accordance with this representation, the variables w and r on any triangle Δ ⊂ Ω with the vertices Ai = (ti , xi ) ∈ Ω, i = 1, 2, 3 (the local indexation is used with the counterclockwise traversal), can be approximated by ̃ x) = w(t, ̃ x) = r(t,

∑ wi,j bi,j,k (t, x) ,

i+j+k=K

∑ ri,j bi,j,k (t, x) ,

(5.56)

i+j+k=K

bi,j,k (t, x) =

(i + j + k)! i j b1 (t, x)b2 (t, x)bk3 (t, x) , i!j!k!

{i, j, k} ⊂ ℕ ,

where the basis linear functions, the so-called barycentric coordinates, are introduced like (x − x3 )(t − t3 ) − (t2 − t3 )(x − x3 ) b1 (t, x) = 2 , d (x − x1 )(t − t1 ) − (t3 − t1 )(x − x1 ) b2 (t, x) = 3 , d (x − x2 )(t − t2 ) − (t1 − t2 )(x − x2 ) (5.57) b3 (t, x) = 1 , d d = det T ,

t1 [ T = [ x1 [ 1

t2 x2 1

t3 ] x3 ] . 1 ]

5.3 FEM modeling of elastic rod dynamics | 91

Here, T denotes the extended vertex matrix, in which determinant d is equal to the doubled area of the triangle Δ. The linear functions bi in equation (5.57) have the following properties: bi (tj , xj ) = δij

for i, j = 1, 2, 3

with b1 + b2 + b3 = 1 .

It is often convenient to index the Bésier–Bernstein polynomials by the elements of the set: 𝒟K (Δ) = {Pi,j,k =

i(t1 , x1 ) + j(t2 , x2 ) + k(t3 , x3 ) ∶ i + j + k = K, i, j, k ∈ ℕ} K

of so-called domain points [24]. Such sets of points for K = 4 are shown in Figure 5.14. There is a useful expression for the integral of Bésier–Bernstein polynomials over the triangle Δ given by ∫ bijk (t, x) dΔ = Δ

(i + j + k)! d. (i + j + k + 2)!

The product of two functions bi,j,k and bl,m,n can always be reduced to a new Bésier–Bernstein polynomial according to bi,j,k (t, x)bl,m,n (t, x) =

j+m k+n (i+l i )( j )( k )

(i+j+k+l+m+n i+j+k+l )

bi+l,j+m,k+n (t, x) ,

where the binomial coefficients n! n ( )= i i!(n − i)! is used. The partial derivatives of such polynomials are also expressed explicitly as 𝜕 b = (i + j + k)(bi−1,j,k b1,t + bi,j−1,k b2,t + bi,j,k−1 b3,t ) , 𝜕t i,j,k 𝜕 b = (i + j + k)(bi−1,j,k b1,x + bi,j−1,k b2,x + bi,j,k−1 b3,x ) . 𝜕x i,j,k

(5.58)

The first partial derivatives of the barycentric linear function bi for i = 1, 2, 3 are obtained via the vertex coordinates (ti , xi ) of the triangle Δ: x2 − x3 , d t −t b1,x = 3 2 , d

b1,t =

x3 − x1 , d t −t b2,x = 1 3 , d b2,t =

x1 − x2 , d t −t b3,x = 2 1 . d b3,t =

(5.59)

It is necessary to zero the summands in the right-hand sides of equation (5.58) if one of the multi-index components becomes negative, i.e., at i = 0, j = 0, k = 0.

92 | 5 Ritz method for initial-boundary value problems For the polynomial degree K chosen uniformly over the whole triangle mesh, the unknown functions w(t, x) and r(t, x) on the triangle Δl,m,n are defined in accordance (l,m,n) (l,m,n) with equation (5.56) by the local parameters wj,k and rj,k , whose number for each triangular domain is equal to Nt =

(K + 1)(K + 2) . 2

(5.60)

Let us introduce a vector of local parameters ẑ ∈ ℝNl , in which components ẑ i are (l,m,n) (l,m,n) and rj,k of all the mesh triangles Δl,m,n . The the Bésier–Bernstein coefficients wj,k ̂ dimension of the vector z is Nl = 8MNNt ,

(5.61)

whereas its components ẑ i is ranged so that (l,m,n) ẑi1 = wj,k

i1 = j0 + k0

and and

(l,m,n) ẑi2 = rj,k

with

i2 = Nt + j0 + k0 ,

where

j0 = 8((N − n)M + m − 1)Nt + 2(l − 1)Nt , 2K − j + 3 + k + 1, 2 j = 0, … , K , k = 0, … , K − j , k0 = j

l = 1, 2, 3, 4 ,

m = 1, … , M ,

(5.62)

n = 1, … , N .

Here, j0 denotes the sequence number of the last coefficient for the previous triangle, (l,m,n) k0 defines a one-dimensional indexation the Bésier–Bernstein coefficients wj,k in the triangular element Δl,m,n . These elements are ranged so that the index l runs from 1 to 4 first, then the middle one m is changed from 1 to M. After all, the last index n varies in the reverse sequence from N to 1. This reverse order needs for a recursive computation on each time subinterval t ∈ (tn−1 , tn ). There is a vector a(t, x) ∈ ℝNl of piecewise discontinuous basis functions relating with the vector of local parameters z.̂ The components ai (t, x) of the vector a are given by { bj,k,K−j−k (t, x) , ai1 (t, x) = ai2 (t, x) = { { 0, (l,m,n)

(t, x) ∈ Δl,m,n , (t, x) ∉ Δl,m,n .

Here, b(l,m,n) j,k,K−j−k are the bivariate Bésier–Bernstein polynomials of the degree K, which are defined over the triangle Δl,m,n in accordance with equation (5.56). The indices i1 and i2 have been already introduced in equation (5.62). With the help of this vector notation, approximations of kinematic and dynamic fields are representable in the form: ̂ x) ⋅ ẑ ŵ = w(t,

̂ x) ⋅ ẑ and r ̂ = r(t,

ŵ = Ew a(t, x)

and

r̂ = Er a(t, x) ,

with

(5.63)

5.3 FEM modeling of elastic rod dynamics | 93

where the diagonal matrices E0w [ 0 Ew = [ [ [⋮ [ 0

0 E0w ⋮ 0

⋯ ⋯ ⋱ ⋯

0 0 ⋮ E0w

E0r [ 0 Er = [ [ [⋮ [ 0

0 E0r ⋮ 0

⋯ ⋯ ⋱ ⋯

0 0 ] ] ] ∈ ℝNl ×Nl ⋮] E0r ]

] ] ∈ ℝNl ×Nl ] ]

with E0w = [

INΔ 0

with E0r = [

0 0

0 ], 0

]

and 0 ] INΔ

are used to extract required basis functions. The unit matrices of the dimension n × n are denoted by In . 5.3.3 Continuity of kinematic and dynamic fields An important step of the finite element algorithm is the implementation of the contĩ x). For that, the gluing ̃ x) and r(t, nuity condition for the approximate functions w(t, of polynomials (5.56) over the edges, which are common for conjugate triangles, has to be guaranteed. In addition, the approximations have to be fit for the boundary and initial conditions (5.53). The chosen Bésier–Bernstein basis of piecewise polynomial splines allows one to explicitly satisfy all interelement constraints. Note an essential property of the polynomials bi,j,k (t, x), which are defined on any triangle Δ in agreement with equation (5.56). The values of polynomial functions ŵ and r ̂ at the vertices A1 , A2 , and A3 are influenced by the only basis functions bK,0,0 (t, x), b0,K,0 (t, x), and b0,0,K (t, x), respectively. All the other basis functions are equal to zero at those points. At that, w(t1 , x1 ) = wK,0 ,

w(t2 , x2 ) = w0,K ,

w(t3 , x3 ) = w0,0 ,

r(t1 , x1 ) = rK,0 ,

r(t2 , x2 ) = r0,K ,

r(t3 , x3 ) = r0,0 .

That is a reason why the above mentioned functions and coefficients might be attributed to appropriate triangle vertices. Almost any mesh node (vertex) in the domain Ω belongs simultaneously to more than one triangle. It follows from the continuity condition that the coefficients of the (l,m,n) (l,m,n) corresponding variable, either wj,k or rj,k , which are related to the same vertex but originate from different triangles, must be equal to a unique nodal value. The only polynomials bj1 ,j2 ,j3 (t, x) with at least one index ji = 0 for i = 1, 2, 3 do not vanish in the edge Qi opposite to the vertex Ai of the triangle Δ. Here, j1 + j2 + j3 = K.

94 | 5 Ritz method for initial-boundary value problems Besides two basis functions related to the vertices that are incident to the edge Qi , there are (K − 1) more polynomials, which are not equal to zero along Qi . Thus, let the function b0,K−k,k with the coefficients w0,K−k , r0,K−k for k = 1, … , K − 1 be referred to the side Q1 ; bk,0,K−k with wk,0 and rk,0 to Q2 ; bK−k,k,0 with wK−k,k and rK−k,k to Q3 . It should be reminded that the local indexation is performed counter-clockwise around the triangle contour. Any edge of some specific triangle Δ1 either belongs to the outer boundary 𝜕Ω or connects this triangle with an adjoining one Δ2 . For a piecewise polynomial function to be continuous in the tetragon (t, x) ∈ Δ1 ∪ Δ2 , it is enough to equate the coefficients of these triangles which are attributed to their common edge and vertices. The opposite directions of traversal along the common edge for each triangle have to be taken into account. For example, if the first side Q(1) 1 (2) (1) (2) (1) of the element Δ1 is the second side Q(2) 2 of Δ2 , then wk,0 = w0,k and rk,0 = r0,k for k = 1, … , K − 1 and so on. The basis polynomials bj,k,K−j−k (t, x) with both jk ≠ 0 and j + k ≠ K are equal to zero on the triangle boundary and can be defined as inner functions. The number of such ̂ x) is ̂ x) or r(t, inner polynomials for w(t, Mt =

(K − 1)(K − 2) . 2

(5.64)

̂ x) The corresponding coefficients wj,k , rj,k influence merely on the inner values of w(t, ̂ x) in the element Δ. They do not take part in the interelement joining of the and r(t, spline. Thus, these coefficients correspond exclusively to the considered triangle. It should be taken into account that the parameters related to a certain node or an edge of the mesh are equated to one another, and in such a way, vary concurrently ̃ x) over more than one triangle. Thus, ̃ x) and r(t, with the values of the splines w(t, it is necessary to renumber all remaining independent parameters and to assemble them into one vector. This vector, in turn, has to be linked with local indices z.̂ Such a connection can be established in various manners, but special attention has to be paid to the numerical stability of the FEM algorithm based on chosen ordering. Let us describe, in an outline, one of the possible sequences for the global parameters of the triangle mesh under consideration. The superscripts a and b means that such parameters are attributed to some vertices Ai,j and Bi,j , respectively (see Figure 5.13). These points belong to the set uniting all domain points 𝒟K (Ω) = ⋃ 𝒟K (Δi ) , i∈ℐΩ

where i = (i1 , i2 , i3 ) ∈ ℐΩ and ℐΩ = {i ∈ ℤ3 ∶ 1 ≤ i1 ≤ 4 , 1 ≤ i2 ≤ M , 1 ≤ i3 ≤ N} . There are two types of coefficients associated with the vertices:

5.3 FEM modeling of elastic rod dynamics | 95

–

wia and ria with i = (i1 , i2 ) ∈ ℐa , where ℐa = {i ∈ ℤ2 ∶ 0 ≤ i1 ≤ N , 0 ≤ i2 ≤ M} ;

–

wib and rib with i = (i1 , i2 ) ∈ ℐb , where ℐb = {i ∈ ℤ2 ∶ 1 ≤ i1 ≤ N , 1 ≤ i2 ≤ M} .

The superscripts t, x, and q refer to the corresponding edges Ti,j , Xi,j , and Qi,j,k . The global parameters belonging to each edge are enumerated, for example, in accordance with the growth of either the temporal coordinate t or the spacial coordinate x of the corresponding domain point from the set 𝒟K (Ω). Thus, there are several edge parameters with different multi-index notation: – wit and rit for Ti2 ,i1 with i = (i1 , i2 , i3 ) ∈ ℐt , where ℐt = {i ∈ ℤ3 ∶ 1 ≤ i1 ≤ N , 0 ≤ i2 ≤ M , 1 ≤ i3 < K}; –

wix and rix for Xi2 ,i1 with i = (i1 , i2 , i3 ) ∈ ℐx , where ℐx = {i ∈ ℤ3 ∶ 0 ≤ i1 ≤ N , 1 ≤ i2 ≤ M , 1 ≤ i3 < K};

–

wiq and riq for Qi3 ,i2 ,i1 with i = (i1 , i2 , i3 , i4 ) ∈ ℐq , where ℐq = {i ∈ ℤ4 ∶ 1 ≤ i1 ≤ N , 1 ≤ i2 ≤ M , 1 ≤ i3 ≤ 4 , 1 ≤ i4 < K}.

Finally, the index Δ is used for the inner parameters of mesh triangles. The following indexation can be proposed for these inner degrees of freedom: (l,m,n) (l,m,n) – wiΔ = wj,k and riΔ = rj,k with i = (i1 , i2 , i3 , i4 ) ∈ ℐΔ . Here, i1 = n, i2 = m, i3 = l, i4 = k + (j − 1)(j − 2)/2, 0 < k < j < K, and ℐΔ = {i ∈ ℤ4 ∶ 1 ≤ i1 ≤ N , 1 ≤ i2 ≤ M , 1 ≤ i3 ≤ 4 , 1 ≤ i4 ≤ Mt } with the number Mt of inner degrees of freedom defined in equation (5.64). Let all of the above mentioned parameters settle themselves in the following order: (1) The free coefficients related with the domain points for the time layer tn−1 < t ≤ tn appear as a one-dimensional array in a loop of n from N to 1 with the step −1. (a) The parameters for the domain points at t = tn is posed in an inner loop of m from 1 to M with the step +1. a (i) First, the dynamic parameter r(n,m−1) is put. x x stay for 1 ≤ k < K. (ii) Then the edge parameters w(n,m,k) and r(n,m,k) a (iii) Last, the kinematic parameter w(n,m) is placed. For the current number n, all these unknowns form a vector y 0,n ∈ ℝMs with Ms = M(K + 1).

96 | 5 Ritz method for initial-boundary value problems (b) In a new inner loop of m from 1 to M with the step +1, the unknown coefficients for the domain points at tn−1 < t < tn is chosen. t run for 1 ≤ k < K. (i) First, the dynamic parameters r(n,m−1,k) b b follow. (ii) Then the vertex parameters w(n,m) and r(n,m) (iii) After that, the inner loop of the third level with l from 1 to 4 is the organizer for the remaining degrees of freedom in the rectangle Ωm,n with q q (A) w(n,m,l,k) and r(n,m,l,k) for 1 ≤ k < K, Δ Δ (B) w(n,m,l,k) and r(n,m,l,k) for 1 ≤ k ≤ Mt if any. a (iv) Last, the parameters w(n,m,k) are placed for 1 ≤ k < K. The parameters of this loop are added to the vector y 0,n to create a new vector of the current time layer y n ∈ ℝNs with the dimension Ns = 4M ⋅ K 2 . In turn, the vectors y n obtained in the course of the outer loop with respect to n are combined into one vector y = ⨁Nn=1 y n ∈ ℝNy of design parameters with Ny = N ⋅ Ns . (2) The boundary coefficients at x = 0 are given in a loop for n from N to 1 with the step −1 as follows: a (a) w(n,0) , t (b) w(n,0,k) for 1 ≤ k < K. This group generates a vector u ∈ ℝNu of control parameters with the dimension Nu = N ⋅ K. (3) Thereafter, the boundary parameters at x = 1 are placed in a loop for n from N to 1 with the step −1: a (a) r(n,M) , t (b) r(n,M,k) for 1 ≤ k < K. a a (4) The vertex parameters w(0,0) and r(0,0) are next. (5) The other initial conditions go after all in a loop of m from 1 to M: x x are put for 1 ≤ k < K. and r(0,m,k) (a) First, the parameters w(0,m,k) a a (b) Then the parameters w(0,m) and r(0,m) are posed. The boundary and initial conditions ordered in such a way give a vector of right-hand side parameters c ∈ ℝNc with Nc = 2(MK + NK + 1) . Merge the column vectors y and c into one column vector of global parameters z = y ⊕ c ∈ ℝNg

with Ng = Ny + Nc .

(5.65)

5.3 FEM modeling of elastic rod dynamics | 97

By taking into account the chosen order of the components in the vector z, the local and global parameters can be linearly related according to ẑ = Qz ,

Q ∈ ℝNl ×Ng ,

(5.66)

where the components of the matrix Q are either zeros or ones so that el = Qeg ,

el = (1, … , 1) ∈ ℝNl ,

eg = (1, … , 1) ∈ ℝNg .

Finally, the continuous kinematic and dynamic fields defined over the considered mesh are expressed with equations (5.63) and (5.66) in the matrix form: w̃ = w(t, x) ⋅ z

and r ̃ = r(t, x) ⋅ z

w = QT Ew a(t, x) ,

with

r = QT Er a(t, x) .

(5.67)

5.3.4 Constitutive relations in the FEM ̃ x, z) and dynamic It turns out after substitution of the kinematic approximation w(t, ̃ x, z) from equation (5.67) into the formula (5.52) of the functional approximation r(t, Φ+ that ̃ + (z) = Φ+ [w,̃ r]̃ = 1 z ⋅ F ⋅ z , Φ 2 F = Fww + Fwr + Frw + Frr , Fww = ∫ (w x w Tx + w t w Tt ) dΩ , Ω

Fwr = − ∫

Ω

(w t r Tx

+ w x r Tt

+ r t w Tx

Frr = ∫ (r x r Tx + r t r Tt ) dΩ ,

(5.68)

Ω

+ r x w Tt ) dΩ ,

Frw = FTwr .

Take into account the structure (5.65) of the global parameter vector z and express ̃ + introduced in equation (5.68) through the vectors of both design and the function Φ system parameters y, c as T F ̃ + (z) = Φ ̃ + (y, c) = 1 [ y ] [ y Φ 2 c FT [ c

Fy = FTy ∈ ℝNy ×Ny ,

Fc ∈ ℝNy ×Nc ,

Fc

][ y ], c F0 ] F0 = FT0 ∈ ℝNc ×Nc .

̃ + (y, c) is reached if its first derivative with respect to y The minimum of the function Φ is equal to zero: ̃+ 𝜕Φ = Fy y + Fc c = 0 . 𝜕y

(5.69)

98 | 5 Ritz method for initial-boundary value problems In accordance with equations (5.51) and (5.67), the solution of the linear algebraic system (5.69) is the vector ỹ = −F−1 y Fc c . This solution gives us the displacement, momentum, and force fields as ̃ x) = w(t, x) ⋅ z̃ , w(t,

̃ x) = r x (t, x) ⋅ z̃ , p(t,

̃ x) = r t (t, x) ⋅ z̃ , s(t,

z̃ = ỹ ⊕ c .

The relative error Δ of the obtained approximation is expressed in the matrix form: ̃ 1−1 (z)̃ , ̃ + (z)̃ Ψ Δ=Φ

̃ + (z)̃ = 1 c ⋅ (F0 − FTc F−1 Φ y Fc ) ⋅ c , 2 ̃ 1 (z)̃ = Ψ1 [w]̃ = 1 ∫ (w̃ 2x + w̃ 2t ) dΩ = 1 z̃ ⋅ Fww ⋅ z̃ . Ψ 2 Ω 2 Here, Ψ1 is the integral of rod energy over the time, which is a functional of displacements, and Fww is the matrix introduced in equation (5.68). The distribution of the ̃ x, c) = φ(w,̃ p,̃ s)̃ is found in accordance with equation (5.49). solution error φ(t, Example 5.3. Similar to Example 5.2, let us consider a forced longitudinal motion of the homogeneous elastic rod. The boundary and initial conditions are given according to equation (5.50), where the displacement of the rod end at x = 0 is defined by the polynomial: u(t) =

3t 2 2t 3 − . T2 T3

(5.70)

̃ x) are taken on the triangles as polynomials of If the spline displacements w(t, degree K ≥ 3, then the boundary condition at x = 0 can be implemented exactly. All components ci of the vector c for i > NK is equal to zero at that. These parameters characterize forces applied at the end x = 1 as well as displacements and momentum density at the initial time instance t = 0. The other components ci for i ≤ KN, which are responsible for the displacements at x = 0 and combined into the vector u(t), are found on the basis of the canonic representation (5.70) of the polynomial function u ̃ 0). and the Bernstein form of the one-dimensional spline w(t, (1,1,n) The only polynomial b0,K−k,k (t, 0) for k = 0, … , K and n = 1, … , N do not vanish ̃ x) are on the domain boundary 𝜕Ω at x = 0. These function in the approximation w(t, multiplied by the coefficient cj for j = (n − 1)K + k. Let the function K

̃ 0) = ∑ cj(k,n) b(1,1,n) w(t, 0,K−k,k (t, 0) = bn (t) ⋅ c n , k=0

bn (t), cn ∈ ℝK+1 ,

5.3 FEM modeling of elastic rod dynamics | 99

be expanded over the interval t ∈ [tn−1 , tn ] in the Taylor series: K

̃ 0) = ∑ [ w(t, k=0

K

tk dk bn ⋅ c ] = u(t) = ∑ dk t k . n dt k t=0 k! k=0

The formula to define the parameters ck in agreement with equation (5.70) is given by cn = B−1 n d,

[Bn ]j,k =

1 dj (1,1,n) [ b (t, 0)] . j! dt j 0,K−k,k t=0

Here, d = (d0 , d1 , … , dK ) is the vector of the polynomial coefficient. It is worth emphasizing that non-zero components of the vector d depend nonlinearly on the sole system parameters T. The node lines of the finite element mesh are placed equidistant from the neighbors with the steps Δt = N −1 T and Δx = M −1 with respect to t and x. The vertices Am,n are placed at the points (tn , xm ) = (nΔt, mΔx) for m = 0, … , M and n = 0, … , N, whereas the vertices Bmn are at ((n − 21 )Δt, (m − 21 )Δx). The terminal time is fixed at T = 4. The polynomial degree K and the mesh parameters M = N (see Figure 5.13) are varied. The relative error Δ versus the number of design parameters Ny is shown in Figure 5.15 for the test motion. The solid marked lines demonstrate the convergence of numerical solutions with the mesh parameter M at the fixed polynomial degrees. This is the so-called h-convergence. The decreasing of the error Δ(Ny ) with the growth of the degree K = 3, 4, … , 7, p-convergence, is portrayed by the dashed line.

Figure 5.15: Relative error of approximation Δ vs. polynomial degree K = 3 − 7 (dashed) and node number M = N = 1 − 7 (solid).

Note that a much more accurate solution is succeeded at the same approximation order Ny by increasing the polynomial degree K. Unfortunately, the numerical solution at such refinement gets quickly computer intensive and the conditionality number of the matrix Fy inverted is rapidly increasing. This fact limits the effectiveness of higher degree polynomials in the algorithm proposed.

100 | 5 Ritz method for initial-boundary value problems It is urgent for a given class of motions to find reasonable numbers K, M, and N so that consuming SPU time will be cut down while approximation accuracy will be raised. A promising direction on increasing the effectiveness of the FEM algorithm is application of adaptive mesh with a varying polynomial degree for different elements and nonuniform node scattering [56]. The node coordinates tn and xm , for example, are rather easy to change in the frame of the considered scheme. ̃ x) − u(t) are presented in FigThe relative elastic displacements of rod points w(t, ure 5.16 as a function of the time t and coordinate x. This distribution is approximately the same as obtained in Section 5.2 (see Figure 5.5).

Figure 5.16: Distribution of the displacements w(t, x).

In Figure 5.17, the distribution of the local error in the domain Ω is shown for the fixed ̃ x) reaches highmesh parameters K = 3 and M = N = 5. As seen, the error function φ(t, est values at the beginning of the motion. This fact relates to transients in the elastic rod that is due to the finite velocity of mechanical impact propagation bounded by

Figure 5.17: Distribution of the local error ̃ x) at K = 3 and M = N = 5. φ(t,

5.4 Spline representation of elastic body motions | 101

Figure 5.18: Momentum density p(t, x).

Figure 5.19: Distribution of the force s(t, x).

the sonic speed. In the subdomain Ω0 = {t, x ∶ t < x} ⊂ Ω, the displacements w, the momentum density p, and the force s must identically equal to zero for the exact solution. The obtained approximations up to certain error are close to this solution. See ̃ x) and p(t, ̃ x) to compare. It can be observed in Figures 5.18 and 5.19 for the fields s(t, the same figures that these approximation strictly obey the initial and boundary coñ 1) = 0. ̃ x) = 0 and s(t, ditions p(0,

5.4 Spline representation of elastic body motions 5.4.1 Approximation to a problem of elastodynamics Let us describe one of possible algorithms of constrained minimization in a dynamic problem of linear elasticity (Problem 3.8). The efficiency of the MIDR will be illustrated on the example of spatial motions for a homogeneous isotropic body (d = 3). We con-

102 | 5 Ritz method for initial-boundary value problems fine ourselves to the case when any volume forces f (t, x) and surface loads are absent and the boundary conditions are given either in displacements or in stresses. Then the minimization problem with respect to unknown functions of displacements w(t, x) and stresses σ(t, x) is formulated for the functional Φ0 introduced in equation (3.34) according to Φ0 [w ∗ , σ ∗ ] = min Φ0 [w, σ] = 0 , w,σ

Φ0 = ∫ φ0 dΩ , Ω

where

1 φ0 = ξ ∶ C ∶ ξ , 2

ξ = ε − C−1 ∶ σ ,

(5.71)

subject to the conventional constraints ρ(x)w tt (t, x) = ∇ ⋅ σ(t, x) w(t, x) = u(t, x) q(t, x) = 0

for (t, x) ∈ Ω ,

for x ∈ Γ1 ,

(5.72)

with q = σ ⋅ n for x ∈ Γ2 ,

w(0, x) = w 0 (x) and

w t (0, x) = ρ−1 (x)p0 (x) . 1

2

Here, Ω = (0, T) × V is the time–space domain, Γ = Γ ∪ Γ is the boundary of the volume V occupied by the body, n(x) is the unit outer normal to the boundary Γ, ρ = const is the volume density, T is the length of the time interval in which the process is studied. For an isotropic body, the elastic modulus tensor C = const is defined by two moduli, for example, Young’s modulus E and Poisson’s ratio ν. The displacement boundary function u(t, x) are prescribed for x ∈ Γ1 . In general, the proposed algorithm can be outlined as follows. Firstly, introduce a ̃ x), σ(t, ̃ x) to the solution w ∗ (t, x), positive integer Ng and choose approximations w(t, ∗ σ (t, x) of the problem (5.71) as the sum Ng

̃ x), σ(t, ̃ x)) = ∑ zk (bk (t, x), βk (t, x)) , (w(t,

(5.73)

k=1

where the outer direct sums (bk (t, x), βk (t, x)) for k = ℤ+ form a basis system of linearly independent vector–tensor functions, and zk are some parameters. These basis functions are chosen in such a way that the ordered vector–tensor ̃ x), σ(t, ̃ x)) is able, for some set of the coefficients zk , to obey Newton’s second pair (w(t, laws (first constraint in equation (5.72)) as well as the continuity, boundary, and initial conditions. It directly follows from equation (5.72) that the boundary vector u(t, x) for x ∈ Γ1 , the initial displacements w 0 (x), and the momentum density p0 (x) have to be rewritten as Ng

u(t, x) = ∑ uk bk (t, x) , k=1

Ng

w 0 (x) = ∑ wk0 bk (0, x) , k=1

where uk , wk0 , and p0k are given coefficients.

Ng

p0 (x) = ∑ p0k bk,t (0, x) , k=1

5.4 Spline representation of elastic body motions | 103

Secondly, the subspace of vectors z = (z1 , … , zNg ) satisfying all constraints are defined by a new parameter vector y = (y1 , … , yNy ) of dimension Ny . The obtained ad̃ x, y) and σ(t, ̃ x, y) are substituted into the constitutive missible approximations w(t, functional Φ0 . Finally, the problem (5.71) of constrained minimization of Φ0 [w, σ] is transformed to the problem of unconstrained minimization of the quadratic function: ̃ 0 (y) ∶= Φ0 [w,̃ σ]̃ . Φ ̃ ∗0 = Φ ̃ 0 (y ∗ ) ≥ 0, To find the optimal vector y ∗ ∈ ℝNy and the corresponding minimum Φ the linear algebraic system 𝜕Φ0 (y) =0 𝜕y has to be solved. ̃ 0 reaches its zero value if and only if the It is worth noting that the function Φ extremal point ̃ x, y ∗ ), σ(t, ̃ x, y ∗ )) (w̃ ∗ (t, x), σ̃ ∗ (t, x)) ∶= (w(t, coincides with the exact solution (w ∗ (t, x), σ ∗ (t, x)). The following integral and local errors Δ=

̃ ∗0 Φ 0 characterizes the required quality of the solution estimated by the ratio Δ, which is used as relative integral error. At the same time, the function φ̃ ∗0 of the time t and the coordinates x denotes distribution of local error ex̃ ∗ > 0 is the time integral of pressed in the units of energy density. In equation (5.75), Ψ ∗ ̃ (t) stored in the elastic body, ψ̃ ∗ (t, x) denotes the approximate mechanical energy W volume density of this energy. It is possible to show via differentiation and integration by parts that ∗

̃ ̇ = ∫ q̃ ∗ ⋅ w̃ ∗t dΓ + W0̇ (t) , W Γ1

W0̇ = ∫ ξ ̃ ∶ C ∶ ε̃∗t dV , ∗

V

q̃ ∗ = C ∶ σ̃ ∗ .

(5.76)

104 | 5 Ritz method for initial-boundary value problems ̃ ∗ contains the term W0 . The energy W0 is This means that the energy function W the difference between the work done by external forces on the approximate motion ̃ ∗ (t) − W ̃ ∗ (0) stored by the body during the same motion. It (w̃ ∗ , σ̃ ∗ ) and the energy W follows from the expression for the first derivative of W0 defined in equation (5.76) that ∗ this energy imbalance relates to the integral error Δ through the residual tensor ξ ̃ . 5.4.2 Forced motions of an elastic body Let us study forced motions of an elastic body which has the shape of an elongated rectangular parallelepiped (see Figure 5.20). The geometry of this bar is given by V = {x = (x1 , x2 , x3 ) ∶ 0 < x1 < L, | x2 | < a, | x3 | < a} , where L is the length and 2a is the size of the square cross section. The duration of the considered time interval is T. There are no elastic deformations in the body and all its points are at rest at the initial time instant t = 0. The side x1 = 0 is supposed to move as a rigid surface along the axis x3 .

Figure 5.20: Body shape and coordinate systems.

For the problem under study, the boundary conditions are defined componentwise as σ1,2 (t, x) = σ2,2 (t, x) = σ2,3 (t, x) = 0

for x2 = ±a ,

σ1,3 (t, x) = σ2,3 (t, x) = σ3,3 (t, x) = 0

for x3 = ±a ,

σ1,1 (t, x) = σ1,2 (t, x) = σ1,3 (t, x) = 0

for x1 = L ,

w1 (t, x) = w2 (t, x) = 0

and w3 (t, x) = u(t)

(5.77)

for x1 = 0 .

Let the component w3 of equation (5.77) coinciding with the time function u on the side x1 = 0 (cross section), belong to the space of polynomials of the degree Nu according to Nu

w3 (t, 0, x2 , x3 ) = u(t) = ∑ uk t k , k=1

(5.78)

where uk are given constants. The initial constraints in equation (5.71) have zero right-hand sides: w 0 (x) = p0 (x) = 0 .

5.4 Spline representation of elastic body motions | 105

The point O in Figure 5.20 indicates the initial position of the body point with the coordinate x = 0, whereas Ot , for instance, denotes the place of the same point at the time t during bar motion. 5.4.3 Approximations of displacement and stress fields To describe the finite dimensional approximation of the unknown variables, set first the order Mp of polynomials with respect to the cross-sectional coordinates x2 and x3 . Then the corresponding functions w̃ k (t, x), σ̃ k,l (t, x) with k, l = 1, 2, 3, which are components of the displacement vector w̃ and the stress tensor σ,̃ are introduced as 2j+1 (i,j) w̃ 1 (t, x) = ∑ x22i x3 w1 (t, x1 ) , i+j≤Mp

2j+1 (i,j) w̃ 2 (t, x) = ∑ x22i+1 x3 w2 (t, x1 ) , i+j 0,

n = 1, … , N .

Additionally, these eigenvalues coincide with ones obtained by using the same approximations of unknown variables for Problems 4.2 and 4.3 according to the Hamilton principle. The solution of the minimization Problem 6.1 is characterized by the following structure of eigenvalues: + λ±(2n−1) = −νn+ ± iω+n ,

νn+ > 0 ,

ω+n > 0 ,

+ λ±2n = νn+ ± iω+n ,

n = 1, … , N .

Several eigenfrequencies calculated for the cantilever beam ω∗k > 0 for k = 1, … , 5 are placed in Table 6.1 as the solution of the characteristic equation: 1 + cos √ω cosh √ω = 0 The details have been discussed in Section 5.1 (see also [46]). Some errors of the eigenfrequencies Δω−k = ω−k − ω∗k and Δω+k = ω+k − ω∗k are indicated in the table. This is a result of discretization performed at the fixed order of Table 6.1: Eigenvalues of cantilever beam vibrations. k ω∗k

1

2

3

4

5

3.516

22.034

61.697

120.903

199.871

Δω−k

0 ,

δm > 0 ,

λ±2m = δm ± iωm , m = 1, 2, … .

It is important to note that the equivalence of linear algebraic systems (7.15), (7.17), (7.19), and (7.21) can be directly proved by the corresponding matrix transformations. However, when the variational formulation (7.8) is used to find motion parameters, it is necessary to take into account numerical properties of such systems. Let us choose again a finite number of basis functions (7.10) of the Bernstein polynomials (7.7) and represent them componentwise in the form: [a1 ]j = bj,N−j (x) and [a2 ]j = bN−j,j (x) for j = 1, … , N .

(7.22)

For the eigenvalue problem (7.19), the frequency relative errors Δωm = |

|λ2m | − ω0m | ω0m

(7.23)

are shown in Figure 7.2 at ω0m , m = 1, … , 9. At the beginning, the error value Δωm rapidly decreases up to a certain value if the approximation order N increases. After that it keeps approximately at the same level. Such behavior can be explained by the poor computational condition of the problem (7.19) compared to the condition in the problem (7.5), solved by the Galerkin method for the same polynomials degree N.

7.1 Modification of the Galerkin method for elastic structures | 135

Figure 7.2: Relative error of eigenfrequencies according to the variational approach.

However, the approximate solution found by the variational approach within the computer-based errors is the best in terms of minimization of energy norms, which is generated by a non-negative error functional Φ introduced in (7.8). To estimate the solution quality, it is possible to use the following relative error: ̃ u,̃ r]̃ Ψ̃ −1 [r]̃ , Δ = Φ[

1 ̃ x, w,̃ r)̃ dΩ , Φ̃ = ∫ φ(t, 2 Ω

1 ̃ r)̃ dΩ . Ψ̃ = ∫ ψ(x, 2 Ω

(7.24)

Here, φ̃ is the function of distributed error from (7.13), ψ̃ is the linear energy density, which is determined through the dynamic function r ̃ given in equation (7.10) according to 1 1 ψ̃ = (r x̃ (t, x) − p0 (x))2 + r 2t̃ (t, x) . 2 2 The corresponding norm of error Φλ = z ∗ [

A B

−B ]z C

(7.25)

for the eigenvalue problem (7.19) under the isoperimetric condition (unit energy norm of the eigenvectors) Ψλ = z ∗ [

A 0

0 ]z = 1 C

(7.26)

can be used to estimate the quality of the approximations for the vibration forms. This norm is given by the vector z from equation (7.16). Here, z ∗ is the complex conjugate vector of z. The convergence of numerical approximations to the solution of the problem (7.19) is reflected in Figure 7.3 for the nine lowest modes of system natural vibrations. Similarly with the behavior of error of the eigenvalues λm , the energy error Φλ is a rapidly decreasing function with respect to the parameter of approximation N. At the same time, this value increases significantly with a growth of the mode number m.

136 | 7 Integrodifferential approach to eigenvalue problems

Figure 7.3: Integral error Φλ (N) according to the variational approach.

As seen in Figure 7.4 for approximations of sufficiently large orders N, the real part δ1 of the first eigenvalue λ1 becomes much smaller than its imaginary part ω1 . The ratio δm ω−1 m can be used as an implicit estimate of the distance between the number ωm and the corresponding frequency of the rod natural vibrations ω0m defined in (7.6). It is also noticeable that the round-off errors significantly influence the accuracy of calculations at high degrees of polynomials (N > 5).

Figure 7.4: Real and imaginary parts of the eigenvalue λ1 (N).

7.1.3 Projection approach based on the MIDR An alternative approach to solve dynamic problems for systems with distributed parameters is based on the modification of the Petrov–Galerkin method. It lies in the fact that zero integral projection of constitutive functions v(t, x, u,̃ r)̃ and q(t, x, u,̃ r)̃ from equation (7.13) are calculated with respect to a specially selected finite-dimensional set of test functions g1 (t, x) ∈ 𝒢1 and g2 (t, x) ∈ 𝒢2 . Let us obtain a consistent system of ODE by using the integrals: ̃ 1 (t, x) dΩ = 0 , ∫ v(t, x, w,̃ r)g Ω

̃ 2 (t, x) dΩ = 0 . ∫ q(t, x, w,̃ r)g Ω

(7.27)

7.1 Modification of the Galerkin method for elastic structures | 137

The equations defining approximately longitudinal motions of the rod points can be written in a vector form. To do so, expand the test functions g1 (x) and g2 (x) over the bases which are defined in the spaces 𝒢1 and 𝒢2 , respectively. For the finite approximation with respect to the coordinate x, suppose that the dimension of each of the test functions space 𝒢1 and 𝒢2 coincides with the dimension N of the test functions ũ and r.̃ Then it is possible to present g1 (x) and g2 (x) by N

gj = ∑ b̂ j,k (x)gj,k (t) ∈ 𝒢j k=1

for j = 1, 2 ,

where b̂ j,k are selected basis functions and gj,k are arbitrary time-dependent coefficients. Collect the basic test functions b̂ j,k into two vectors bj (x) ∈ ℝN with j = 1, 2. Then the condition for the vanishing of the integrals in equation (7.27) is equivalent to a system of two vector equations of the first order: 1

̇ − p0 (x))b1 (x) dx = 0 , ∫ (w̃ t (t, x) − r x̃ (t, x) + u(t) 0

1

0

∫ (w̃ x (t, x) − r t̃ (t, x) + (w (x)) )b2 (x) dx = 0 .

(7.28)

′

0

It is necessary to take into account 2N initial conditions to formulate an initial ̃ ỹ 2 (t)) defined in (7.10) must ̃ ỹ 1 (t)) and r(x, value problem. The trial functions w(x, strictly satisfy the homogeneous conditions at t = 0, which are equivalent to the vector equation (7.12). The eigenvalue problem corresponding to the system (7.12) and (7.28) can be given by (λF − G)y = 0 with y ∈ ℂ2N and λ ∈ ℂ ,

(7.29)

where the following matrices are introduced: F=[

F1 0 1

0 ] ∈ ℝ2N×2N F2

F1 = ∫ b1 (x)aT1 (x) dx 0

as well as G=[

0 G2 1

G1 ] ∈ ℝ2N×2N 0

G1 = ∫ b1 (x)(a′2 )T (x) dx 0

with 1

and F2 = − ∫ b2 (x)aT2 (x) dx , 0

with 1

and G2 = − ∫ b2 (x)(a′1 )T (x) dx . 0

Let us demonstrate the convergence of the proposed projection approach. The vectors of basis functions b1 (x) and b2 (x) are chosen in accordance with test functions a1 (x) and a2 (x) of (7.22) that are used in the variational approach. By using the notation introduced in (7.7) for Bernstein polynomials, the following vectors can be written

138 | 7 Integrodifferential approach to eigenvalue problems componentwise: [b1 ]j = [b2 ]j = bj−1,N−j (x)

for j = 1, … , N .

By taking into account the structure of the matrix F and G, it is possible to show that all eigenvalues of the problem (7.29) have pure imaginary values: λ±m = ±iωm

for m = 1, … , N

with ωm > 0 .

This is a main distinction between the proposed approach and the Ritz method applied to the minimization problem (7.8). The relative frequency detuning Δωm , introduced in (7.23), is shown in Figure 7.5 for the first nine vibration modes as a function of the problem parameter N. It is possible to see that numerical rounding errors do not influence on the behavior of Δωm , as it was in the variational approach (see Figure 7.2). The frequency convergence for selected approximations to the exact values for the original problem (7.1) similar to that which has been obtained in accordance with the Galerkin method (cf. Figure 7.1).

Figure 7.5: Relative error of eigenfrequencies in accordance with the projection approach.

This projection approach has significant advantages in comparison with the classical Galerkin one. For example, the approximations of unknown functions can be directly used in the integral and local quality criteria (7.24)–(7.26). As shown in Figure 7.6, it is possible to determine the quality of eigenforms by applying the functional Φλ introduced in equation (7.25). In spite of the fact that the resulting system of approximate equations (7.12), (7.28) obtained in frame of the projection algorithm does not satisfy an extremality property in the sense of minimization of the functional Φ, this approach has a number of important specialties: 1. The differential order of the system (7.29) is two times less than the order of the variational system (7.12), (7.14). 2. The main difference of the projection approach from the variational one is that an initial value problem is solved, but not the boundary one (7.12), (7.14). 3. The eigenvalues run only pure imaginary values.

7.2 Semi-discretization in problems of natural beam vibrations | 139

Figure 7.6: Integrated error Φλ (N) in accordance with the projection approach.

These benefits can be applied to design sufficiently effective simulation and optimization procedures in solving direct and inverse dynamic problems for systems with distributed parameters.

7.2 Semi-discretization in problems of natural beam vibrations 7.2.1 Natural vibrations of elastic plates Consider an isotropic plate occupied in the plane Ox1 x2 a rectangular area: V = {x ∶ 0 < x1 < L , −h < x2 < h}

with x = (x1 , x2 ) .

As it is seen in Figure 7.7, the boundary of the area V = Ω consists of the four parts: Γ = 𝜕V = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 . It is supposed that the parts of the boundary Γ2 and Γ4 at x2 = ±h are free of loads. At that, the boundaries Γ1 and Γ3 for x2 = 0, L are subjected either zero stresses or zero displacements. It is considered further that the length L of the plate (beam) is much greater than its height 2h.

Figure 7.7: Rectangular plate.

140 | 7 Integrodifferential approach to eigenvalue problems Let us introduce the vectors of displacements w(t, x) and momentum density p(t, x) as well as the tensors of stresses σ(t, x) and strains ε(t, x) as w = (w1 , w2 ) σ=[

σ11 σ12

σ12 ] σ22

and p = (p1 , p2 ) , and ε = [

ε11 ε12

ε12 ]. ε22

(7.30)

The strain tensor ε depends on beam displacements w according to ε11 =

𝜕w1 , 𝜕x1

ε22 =

𝜕w2 , 𝜕x2

2ε12 =

𝜕w2 𝜕w1 + . 𝜕x1 𝜕x2

(7.31)

Hooke’s law and the relation between velocity and momentum for plane stress state can be written as ξ (t, x) = [

ξ11 ξ12

ξ12 ] = 0, ξ22

v(t, x) = (v1 , v2 ) .

(7.32)

For an isotropic body, the components of the tensor ξ and the vector v are defined respectively as 1 (σ − νσ22 ) , E 11 1+ν ξ12 ∶= ε12 − σ E 12 ξ11 ∶= ε11 −

ξ22 ∶= ε22 −

1 (σ − νσ11 ) , E 22

(7.33)

and v1 ∶=

𝜕w1 p1 − , 𝜕t ρ

v2 ∶=

𝜕w2 p2 − . 𝜕t ρ

Material properties are given by Young’s modulus E(x) > 0, Poisson’s ratio ν(x) > 0, and volume density ρ(x) > 0. If external volume forces are absent, then the equation of dynamic equilibrium has the form: f (t, x) = (f1 , f2 ) = 0 ,

(7.34)

where the vector of elastic and inertial volume forces f with the components f1 =

𝜕σ11 𝜕σ12 𝜕p1 + − , 𝜕x1 𝜕x2 𝜕t

f2 =

𝜕σ12 𝜕σ22 𝜕p2 + − 𝜕x1 𝜕x2 𝜕t

is introduced. Consider the case when only the homogeneous boundary conditions are given on the long sides of the beam σ12 (x1 , ±h) = 0

and σ22 (x1 , ±h) = 0 ,

(7.35)

7.2 Semi-discretization in problems of natural beam vibrations | 141

as well as on its ends σ11 (0, x2 ) = 0 ∧ σ12 (0, x2 ) = 0

∨ w1 (0, x2 ) = 0

∧ w2 (0, x2 ) = 0 ,

σ11 (L, x2 ) = 0

∨ w1 (L, x2 ) = 0

∧ w2 (L, x2 ) = 0 .

∧ σ12 (L, x2 ) = 0

(7.36)

In order to formulate an eigenvalue problem, represent the unknown variables as ̃ w(t, x) = w(x) sin ωt ,

̃ cos ωt , p(t, x) = p(x)

̃ sin ωt , σ(t, x) = σ(x)

(7.37)

where ω > 0 is the eigenfrequency. The sign tilde is omitted further. After substituting the expressions (7.37) and eliminating time, the components of the constitutive vector v can be reduced to the form: vj (x) = ωwj −

pj ρ

for j = 1, 2 .

(7.38)

In turn, the vector of volume forces is written as f1 (x) =

𝜕σ11 𝜕σ12 + + ωp1 𝜕x1 𝜕x2

and f2 (x) =

𝜕σ12 𝜕σ22 + + ωp2 . 𝜕x1 𝜕x2

(7.39)

By taking into account the dynamic equilibrium equation (7.34) and equation (7.39), express the momentum density through the stresses. Then the relations v1 (x) = ωw1 +

1 𝜕σ11 𝜕σ12 ( + ), ρω 𝜕x1 𝜕x2

v2 (x) = ωw2 +

1 𝜕σ12 𝜕σ22 ( + ) ρω 𝜕x1 𝜕x2

(7.40)

follow from equation (7.38). Let us formulate a problem: Find eigenfrequencies ω, displacements w(x), and stresses σ(x), which satisfy the constitutive equation (7.32) taking into account (7.33) and (7.40), as well as the boundary conditions (7.35) and (7.36). Similarly as to the problem on the elastic body motions, considered in Chapter 3, we introduce two energy functionals: Φω ± [w, σ, ω] =

1 ∫ (ρv ⋅ v ± ξ ∶ C ∶ ξ ) dV , 2 V

(7.41)

where C is the elastic moduli tensor. The energy of the mechanical system can be expressed either through displacements Ψω w [u, ω] =

1 ∫ (ρω2 w ⋅ w + ε ∶ C ∶ ε) dV , 2 V

(7.42)

142 | 7 Integrodifferential approach to eigenvalue problems or through stresses Ψω σ [σ, ω] =

1 (∇ ⋅ σ) ⋅ (∇ ⋅ σ) + ω2 σ ∶ C−1 ∶ σ) dV . ∫( 2ω2 V ρω2

(7.43)

The double dot products of the tensors in equations (7.41)–(7.43) for the case of isotropic material have the form: E 2 2 (ξ 2 + 2νξ11 ξ22 + ξ22 + 2(1 − ν)ξ12 ), 1 − ν2 11 E 2 2 ε∶C∶ε= (ε2 + 2νε11 ε22 + ε22 + 2(1 − ν)ε12 ), 1 − ν2 11 1 2 2 2 σ ∶ C−1 ∶ σ = (σ11 − 2νσ11 σ22 + σ22 + 2(1 + ν)σ12 ). E

ξ ∶C∶ξ =

Since the solution of the problem is defined up to an arbitrary factor, it is natural to give only one of the isoperimetric conditions: Ψω w =1

or Ψω σ = 1.

7.2.2 Variational approach to the eigenvalue problem The relationship between the variational principles based on the MIDR and the classical Hamilton principles has been shown in Section 3.2 as an example of the problem on elastic body motions. Given the eigenfrequency ω, the displacements w ∗ (t, x) and stresses σ ∗ (t, x), which are a stationary point for the functional δΦω − [w, σ, ω] = 0 , the problem is reduced to the variational problem in displacements as δΥω [w, ω] = 0

with Υω =

1 ∫ (ρω2 w ⋅ w − ε ∶ C ∶ ε) dV , 2 V

(7.44)

and the complementary one formulated in stresses according to δΥω c [σ, ω] = 0

with Υω c =

1 (∇ ⋅ σ) ⋅ (∇ ⋅ σ) − σ ∶ C−1 ∶ σ) dV . ∫( 2 V ρω2

(7.45)

Note that the displacement and stress fields in these statements strictly satisfy the boundary conditions (7.35) and (7.36).

7.2 Semi-discretization in problems of natural beam vibrations | 143

Introduce a finite-dimensional representation of the unknown functions w and σ with respect to one of the coordinates x2 in the following form: N

N−1

w1 = ∑ x̂ k w1(k) (x1 ) ,

w2 = ∑ x̂ k w2(k) (x1 ) ,

(k) σ11 = ∑ x̂ k σ11 (x1 ) ,

(k) σ12 = ∑ x̂ k (1 − x̂ 2 )σ12 (x1 ) ,

k=0 N

k=0 N

(k) σ22 = ∑ x̂ k (1 − x̂ 2 )σ22 (x1 ) , k=0

k=0 N−1 k=0

x̂ =

(7.46)

x2 , h

where wi(k) and σij(k) for i, j = 1, 2 are unknown functions of the coordinate x1 . The stresses selected in such a way automatically satisfy the boundary conditions (7.35). Let us define vector functions of design parameters as z 1 = (w1(0) , w2(0) , … , w1(N−1) , w2(N−1) , w1(N) ), (0) (0) (N−1) (N−1) (N) z 2 = (σ11 , σ12 , … , σ11 , σ12 , σ11 ),

(7.47)

(0) (N) z 3 = (σ22 , … , σ22 ),

i.e., z 1 (x1 ), z 2 (x1 ) ∈ ℝ2N+1 , z 3 (x1 ) ∈ ℝN+1 . Consider the following integrals: Θ(z ′1 , z 1 , ω, x1 ) =

1 h ∫ (ρω2 w ⋅ w − ε ∶ C ∶ ε) dx2 , 2 −h

Θc (z ′2 , z 2 , z 3 , ω, x1 ) =

1 h (∇ ⋅ σ) ⋅ (∇ ⋅ σ) − σ ∶ C−1 ∶ σ) dx2 . ∫ ( 2 −h ρω2

(7.48)

Here, the derivatives with respect to x1 are denoted, as usually, by prime symbols. By taking into account equations (7.46)–(7.48), the first variations of the functionals Υω and Υω c defined in equations (7.44) and (7.45) can be represented as δΥω = [ δΥω c

x1 =L L 𝜕Θ d 𝜕Θ 𝜕Θ ⋅ δz ] − ( − ) ⋅ δz 1 dx1 , ∫ 1 ′ 𝜕z ′1 dx 𝜕z 1 0 x1 =0 1 𝜕z 1

x1 =L L L 𝜕Θ d 𝜕Θc 𝜕Θc 𝜕Θ c = [ ′c ⋅ δz 2 ] −∫ ( − ) ⋅ δz 2 dx1 + ∫ ⋅ δz 3 dx1 . ′ 𝜕z 2 𝜕z 2 0 dx1 𝜕z 2 0 𝜕z 3 x1 =0

(7.49)

For selected approximations (7.46), the necessary conditions of stationarity for the functional Υω [z 1 , ω] are the Euler–Lagrange system of differential equations d 𝜕Θ 𝜕Θ − = 0, dx1 𝜕z ′1 𝜕z 1 written in the vector form.

(7.50)

144 | 7 Integrodifferential approach to eigenvalue problems It needs to add to the system (7.50) transversality conditions, which appear on the free ends. Taking into account the boundary conditions (7.36), we have x1 = 0 ∶ x1 = L ∶

𝜕Θ =0 𝜕z ′1 𝜕Θ =0 𝜕z ′1

∨ ∨

z1 = 0 , z1 = 0 .

(7.51)

The conditions of stationarity for the functional Υω c [z 2 , z 3 , ω] in accordance with the complementary Hamilton principle constitute the linear system of DAEs: d 𝜕Θc 𝜕Θc − = 0, dx1 𝜕z ′2 𝜕z 2

𝜕Θc = 0. 𝜕z 3

(7.52)

By accounting equations (7.36) and (7.49), the 2N + 1 boundary conditions in the form x1 = 0 ∶

z2 = 0

∨

x1 = L ∶

z2 = 0

∨

𝜕Θc = 0, 𝜕z ′2 𝜕Θc =0 𝜕z ′2

(7.53)

must be added to equation (7.52) to get the approximate eigenfrequencies ω and the eigenvectors z 2 (x1 ), z 3 (x1 ). After solving the boundary value problem (7.50), (7.51) and taking into account the isoperimetric condition Ψω w = 1 as well as relations (7.46) and (7.47), the eigenfunctions for displacements are obtained. The corresponding stresses are found from the problem (7.52), (7.53) at Ψω σ = 1. 7.2.3 Projection approach to the eigenvalue problem The algorithm for aprroximate solution of dynamic problems has been described in Chapter 6 based on the projection approach and semi-discretization. Let us apply this technique to build approximate systems of equations describing the natural beam vibrations in the frame of a plane stress model for an elastic body. Consider the problem in displacements w(x) and stresses σ(x) to find eigenvalues ω, studied in Subsection 7.2.1. The constitutive relations (7.32) subject to equations (7.31), (7.33), (7.40) and the boundary conditions (7.35), (7.36) must be satisfied in conventional formulation. Approximations, which automatically obey the boundary constraints (7.35) on the lateral beam sides, are found as the trial functions (7.46). In a projection approach (modification of the Petrov–Galerkin method [6, 11]), it is required that the displacements and stresses provide zero values to the integral projections of the constitutive vector and tensor on the selected vector (𝒰) and tensor (𝒯)

7.2 Semi-discretization in problems of natural beam vibrations | 145

fields of the test functions u(x) and τ(x): L

Σv = ∫ Θv dx1 = 0 , 0

L

Σξ = ∫ Θξ dx1 = 0 , 0

h

Θv = ∫ ρωv(w, σ) ⋅ u dx2 ,

∀u ∈ 𝒰 ,

−h h

Θξ = ∫ ξ (w, σ) ∶ τ dx2 ,

(7.54) ∀τ ∈ 𝒯 .

−h

Here, the vector u(x) has the dimension of length, τ(x) is a stress tensor, Θv and Θξ are discrepancy integrals of constitutive equations with the dimension of linear power density. In the same way as for the trial fields (7.46), the test functions can be polynomials with respect to the variables x2 . Note that no restrictions are imposed on these functions. To obtain a consistent system of equations, present the virtual displacements and stresses as follows: N

N−1

u1 = ∑ x̂ k u(k) 1 (x1 ) ,

u2 = ∑ x̂ k u(k) 2 (x1 ) ,

(k) τ11 = ∑ x̂ k τ11 (x1 ) ,

(k) τ12 = ∑ x̂ k τ12 (x1 ) ,

(k) τ22 = ∑ x̂ k τ22 (x1 ) ,

x̂ =

k=0 N

k=0 N

k=0

k=0 N−1 k=0

(7.55)

x2 , h

(k) where u(k) i and τij for i, j = 1, 2 are variables having necessary smoothness over x1 . Similar to the design parameters (7.47), three vectors (0) (N−1) (N−1) (N) y 1 (x) = (u(0) , u2 , u1 ) ∈ ℝ2N+1 , 1 , u2 , … , u1 (0) (0) (N−1) (N−1) (N) y 2 (x) = (τ11 , τ12 , … , τ11 , τ12 , τ11 ) ∈ ℝ2N+1 ,

(7.56)

(0) (N) y 3 (x) = (τ22 , … , τ22 ) ∈ ℝN+1 .

are composed defining the test functions (7.55). By recalling the notation introduced in equations (7.47) and (7.56), substitute the expressions for the displacements (7.46) and for the stresses (7.55) into the integrals Θξ and Θv from equation (7.54). After integration over x2 and differentiation over y j for j = 1, 2, 3, the following system of DAEs are derived: 𝜕Θη (z ′2 , z 1 , z 2 , z 3 , y 1 , ω, x1 ) 𝜕y 1 ′ 𝜕Θξ (z 1 , z 1 , z 2 , z 3 , y 2 , y 3 , ω, x1 ) 𝜕y 2 ′ 𝜕Θξ (z 1 , z 1 , z 2 , z 3 , y 2 , y 3 , ω, x1 ) 𝜕y 3

= 0, = 0, = 0.

(7.57)

146 | 7 Integrodifferential approach to eigenvalue problems Here, the first two equations are 4N + 2 first-order ODEs written in vector form and the last equality is equivalent to N + 1 algebraic equations, which are linear with respect to the design parameters composing the vectors z j for j = 1, 2, 3. The system of equations (7.57) based on the approximations (7.46) together with the constraints x1 = 0 ∶

z2 = 0

∨

z1 = 0 ,

x1 = l ∶

z2 = 0

∨

z1 = 0

(7.58)

resulting from the boundary conditions (7.36) can be used to find the natural frequencies ω and the eigenfunctions z j (x1 ) for j = 1, 2, 3. After that, the amplitudes of displacements w(x) and stresses σ(x) are reconstructed in accordance with equation (7.46). 7.2.4 Variational versus projection approaches It is possible to prove for a chosen class of the trial and test functions w(x), σ(x), u(x), τ(x) that the approximate system of equations (7.50) and (7.57) obtained respectively with the help of the variational and the projection approaches are equivalent. For this purpose, let us represent the conditions for the first variations of the functionals Υω [w, ω] and Υω c [σ, ω], given in equations (7.44) and (7.45), according to δΥω = ∫ (ρω2 w ⋅ δw − ε(w) ∶ C ∶ ε(δw)) dV = 0 , V

δΥω c

= ∫ (ρ−1 ω−2 (∇ ⋅ σ) ⋅ (∇ ⋅ δσ) − σ ∶ C −1 ∶ δσ) dV = 0 . V

The components δw1 (x1 ) and δw2 (x1 ) of the variation vector δw can be found from the relations for displacements defined in equation (7.46). It is necessary to replace the (k) (k) unknown functions w1,2 (x1 ) for δw1,2 (x1 ), which satisfy the boundary conditions (7.51). It is follows from equations (7.46) and (7.55) that the variations δû = δw + ρ−1 ω−2 ∇ ⋅ δσ

and

δτ̂ = δσ − C ∶ ε(δw)

belong respectively to the same spaces as the virtual displacements u(x) and stresses τ. After substituting the vector û and the tensor τ̂ in the expression (7.54) and taking into account the structure of the constitutive functions (7.33) and (7.40), write the zero condition for the functional Σ[w, σ, u, τ] = Σv [w, σ, u] + Σξ [w, σ, τ] in the form Σv [w, σ, δw, −C ∶ ε(δw)] = δΥω [w, δw] = 0 , Σξ [w, σ, ρ−1 ω−2 ∇ ⋅ δσ, δσ] = δΥω c [σ, δσ] = 0 .

(7.59)

7.2 Semi-discretization in problems of natural beam vibrations | 147

In accordance with equation (7.59), the system of equations (7.57) can be reduced either to the system (7.50) with the same boundary conditions (7.51) in displacements or to the system (7.52) with the constraints (7.53). The projection approach is applied further to find approximate frequencies and mode shapes of the natural beam vibrations. It will be shown in Section 7.3 that the finite decomposition of the unknown functions (7.46) is not unique. In special case, such as a free beam, which boundary conditions are given only in stresses, the projection approach makes it possible to construct consistent models, where the number of displacement functions wj(k) is greater than the number of stress functions σ1j(k) for j = 1, 2 as in [48, 52].

7.2.5 Longitudinal plate vibrations Consider an elastic plate (see Figure 7.7), which is made of a homogeneous isotropic material, i.e., E = const, ρ = const, ν = const. The system (7.57), (7.58) can be rewritten in the dimensionless form with E = ρ = h = 1 if the new scales of time t = h√ρE −1 t ̃ and

length x = hx̃ are introduced. Further, the sign tilde is omitted. Take into account the symmetry properties of the domain V and the boundary conditions (7.35) and (7.36). As it was shown in [44], the original system (7.57) can be split into two independent subsystems, which describe either longitudinal or lateral in-plane vibrations of the plate. To do so, the unknown functions with the inappropriate powers of the variable x̂ = x2 (see Table 7.1) must be removed from the expressions for displacements and stresses in equation (7.46). The parity superscripts characterize the symmetry (antisymmetry) property of the displacement and stress functions with respect to the axis x1 . Due to orthogonality of the odd and even power functions, the corresponding projections in the integrals (7.54) are automatically zeroed. Table 7.1: Functions of longitudinal and lateral vibrations. Vibrations

w1

w2

σ11

σ12

σ22

Longitudinal

w1(2j)

w2(2j+1)

(2j) σ11

(2j) σ22

(2j+1) σ12

Lateral

w1(2j+1)

w2(2j)

(2j+1) σ11

(2j+1) σ22

(2j) σ12

Tension and compression of the central line (x2 = 0), which are given by the function w1(0) (x1 ), are the determinative displacements for longitudinal vibrations. In contrast, lateral in-plane motions are characterized by the transversal deflections w2(0) (x1 ) with respect to this line.

148 | 7 Integrodifferential approach to eigenvalue problems First, let us study longitudinal vibrations. The displacements and stresses can be written in agreement with equation (7.46) and Table 7.1 by M

2j (2j)

σ11 = ∑ x2 σ11 (x1 ) , j=0 M

2j

M

j=0

σ22 = ∑ x2 (1 − x22 )σ22 (x1 ) , j=0 M−1 j=0

(2j)

2j+1

σ12 = ∑ x2

2j

w1 = ∑ x2 w1 (x1 ) ,

(1 − x22 )σ12

(2j+1)

(2j)

M−1

2j+1

w2 = ∑ x2 j=0

w2

(2j+1)

(x1 ) ,

(x1 ) ,

where the parameter M defines the dimension of the approximations. As a result, for the tension–compression process, characterized by changing the length of the central line, the differential order of the system is equal to 4M + 2. For the lowest possible degree of approximation M = 0, the system (7.57) is reduced to a single differential equation of the second order: d2 w1(0) + (1 − ν2 )ω2 w1(0) = 0 , dx12 1 dw1(0) 3ν (0) (0) (0) σ11 = , σ22 = σ11 , 2 1 − ν dx1 2 which coincides with the classical equation of longitudinal vibrations for a thin elastic rod only if ν = 0. The solution of the eigenvalue problem is obtained in analytic form for any type of the boundary conditions (7.36). For M = 1, the system (7.57) can be reduced to a bicubic equation with respect to the function w1(0) (x1 ), which specifies the displacements of the midline along the axis Ox1 . The roots of the characteristic equation λj (ω) for j = 1, … , 6 are found analytically. The imaginary (|Im λj |, solid curves) and real (|Re λj |, dashed curves) branches of the wavefrequency characteristics λj (ω) for the selected approximations are shown in Figure 7.8 at ν = 0.3. There are three critical frequencies values ω∗1 = 1.787 ,

ω∗2 =

10 √ 273 = 1.816 , 91

ω∗3 =

5√ 39 = 2.402 , 13

for the model defining four frequency zones, which are divided in Figure 7.8 by dashed lines. In the first zone (ω < ω∗1 ), two roots have pure imaginary values, the other four are complex. In the second (ω∗1 < ω < ω∗2 ) and fourth (ω∗3 < ω) zones, all roots are purely imaginary. There are four imaginary and two real roots in the third zone (ω∗2 < ω < ω∗3 ). It is important to note that the second zone contains a monotonically decreasing branch |Im λj (ω)|. The dashed straight line shows the dependence of the wave numbers on the frequency of the longitudinal vibrations (|λ| = ω) for the conventional model of a thin

7.2 Semi-discretization in problems of natural beam vibrations | 149

Figure 7.8: Wave-frequency characteristics of longitudinal plate vibrations.

rod. This line is almost tangent to Im λ1 (ω), where λ1 is the root of the characteristic equation that satisfies the conditions Re λ1 (ω) = 0 and Im λ1 (ω) > 0 for all ω. Since the classical model of a thin elastic rod does not take into account cross-sectional motions of rod points, it gives a more rough estimate for the natural frequencies of the system. Let us consider a special case of natural vibrations for an elastic beam clamped at both ends, namely, x1 = 0, L ∶

(2) (1) u(0) 1 (x1 ) = u1 (x1 ) = u2 (x1 ) = 0 .

The eigenfrequencies ωn in the first zone for L = 20 and ν = 0.3 are presented in Table 7.2. The first three rows correspond to the antisymmetry (a) of the function u(0) 1 with respect to the midpoint x1 = 10 of the central plate line, whereas the last three rows relate to the symmetry case (s). The relative divergence Δωn =

ωcn − ωn ωcn

(7.60)

between the classical eigenfrequencies ωcn = 0.05πn and those obtained in accordance with the projection approach at M = 1 are given in the table for each mode. When the second zone is close this value increases and reaches nearly 20% for the 14th mode. Table 7.2: Longitudinal eigenfrequencies for the beam with the clamped ends. Mode (a)

1

3

5

7

9

11

13

ωn Δωn (%)

0.158 −0.25

0.471 0.04

0.780 0.73

1.076 2.14

1.344 4.96

1.554 10.1

1.705 16.5

Mode (s)

2

4

6

8

10

12

14

ωn Δωn (%)

0.315 −0.15

0.626 0.32

0.930 1.32

1.215 3.31

1.457 7.22

1.635 13.3

1.766 19.7

150 | 7 Integrodifferential approach to eigenvalue problems The wave numbers Im λ1 (ωn ) are shown in Figure 7.8 for the first zone by circles, for the second zone by triangles, and for the third by squares. Note that a few eigenfrequencies are located in the rather narrow second frequency zone. At that, the interchangeable order of even and odd modes is violated. The frequency behavior becomes even more irregular in the third zone. The frequencies of different parity may take very similar values. For clarity, the frequency in these three zones versus the mode number is presented in Figure 7.9 by circles.

Figure 7.9: Eigenfrequencies of the clamped beam.

7.2.6 Lateral in-plane vibrations Now, consider the problem on lateral vibrations. The parity of functions degrees in the expansions (7.46) with respect to the variable x2 is determined in Table 7.1. In this case, the corresponding approximation of displacements and stresses is given by M

2j+1 (2j+1) σ11 (x1 ) ,

σ11 = ∑ x2 j=0 M

2j

M

j=0

σ12 = ∑ x2 (1 − x22 )σ12 (x1 ) , j=0 M

2j+1

σ22 = ∑ x2 j=0

(2j)

(1 − x22 )σ22

2j+1

w1 = ∑ x2

(2j+1)

M

w1

(2j+1)

2j

(x1 ) ,

w2 = ∑ x2 w2 (x1 ) , (2j)

j=0

(7.61)

(x1 ) .

It follows from the expressions (7.61) that the differential order of the system of equations approximately describing the beam bending is equal to 4M + 4. For the lowest possible degree of approximation M = 0, the system (7.57) can be reduced to one biquadratic differential equation with respect to the displacement function w2(0) (x1 ) according to d4 w2(0) d2 w2(0) + a(ων) − b(ων)w2(0) = 0 dx12 dx14

a = (1 + ν)(3 − ν)ω2

with

and b = (1 − ν2 )(3 − 2(1 + ν)ω2 )ω2 .

(7.62)

7.2 Semi-discretization in problems of natural beam vibrations | 151

It should be noted that the equation (7.62) at sufficiently low frequencies ω ≪ 1 tends to the classical beam equation describing the bending (Euler–Bernoulli equation) only in the case ν = 0. The same as in the Timoshenko beam [97], the considering model takes into account not only bending of the center line, but also the shear displacements x2 w1(1) (x1 ). Figure 7.10 gives the wave-frequency characteristics of the transverse in-plane motions at Poisson’s ratio ν = 0.3. There is a critical frequency ω∗0 = 1.074 (dashed line) separating the two zones: ω ∈ (0, ω∗0 ) and ω ∈ (ω∗0 , +∞). The solid curves correspond to the imaginary part |Im λj (ω)| of the roots of the characteristic equation, whereas the dashed curve depicts the real parts |Re λj (ω)|.

Figure 7.10: Wave-frequency characteristics of lateral vibrations.

Similar to the model of Euler–Bernoulli, there are two pure imaginary roots and two complex in the first zone. However, the absolute values of the imaginary and real roots do not coincide. In the second zone, all the eigenvalues have pure imaginary values. If ω = ω∗0 , then two roots vanish. For comparison, the dot-dash curve 4 shows the wave-frequency characteristics of the Bernoulli beam: |λc | = √3ω2 . When ω → 0, the curves |λj (ω)| clump together and come close (not adhere) to the classical curve. Consider lateral vibrations of elastic plate rigidly clamped at its both ends x1 = 0, L ∶

w1(1) (x1 ) = u(0) 2 (x1 ) = 0 .

As in Subsection 7.2.5, choose the length of the plate L = 20. There exist 12 natural frequencies ωn in the first zone (ωn < ω∗0 ) for the selected parameters and boundary conditions. These frequencies are shown in Table 7.3 with either the character s for even functions of midline bending or a for odd ones. The corresponding wave numbers |Im λj (ωn )| are presented by circles in Figure 7.10. The eigenvalues belonging to the second zone (ωn > ω∗0 with n > 12) are marked by squares in the figure. For comparison with the longitudinal vibrations, the dependence of the values of the natural frequency on the number of lateral mode n is put in Figure 7.9 by lines with diamonds.

152 | 7 Integrodifferential approach to eigenvalue problems Table 7.3: Eigenfrequencies of lateral in-plane vibrations for the clamped plate. Mode (a)

1

3

5

7

9

11

ωn Δωn (%)

0.0319 1.27

0.1489 14.7

0.3125 27.5

0.4976 37.9

0.6931 46.1

0.8935 52.6

Mode (s)

2

4

6

8

10

12

ωn Δωn (%)

0.0821 7.81

0.2269 21.3

0.4032 33.0

0.5945 42.2

0.7929 49.5

0.9942 55.3

Table 7.3 also contains the data of natural vibrations for the beam clamped at both ends. The results obtained in accordance with the MIDR are compared with the results for the corresponding classical problem: d4 w2(0) + 3ω2 w2(0) = 0 , dx4 dw2(0) = 0. x1 = 0, L ∶ w2(0) = dx The eigenvalues for this problem are found from the solution of the transcendental equation cos Lλc cosh Lλc = 1

4

with λc = √3ω2 .

It can be seen from the table that the relative discrepancy (7.60) between the natural frequencies obtained by using the projection approach and the Euler–Bernoulli beam model is more than 1%, even for the first mode. The relative mismatch for the 11th eigenfrequency is greater than 50%.

7.3 Special models for plate motions 7.3.1 Statement of the eigenvalue problem As in Section 7.2, natural oscillations of isotropic elastic plates (beams) occupying the rectangular area V = {x ∶ −a1 < x1 < a1 , −a2 < x2 < a2 } in the plane Ox1 x2 (see Figure 7.11) is analyzed in the frame of the two-dimensional linear theory of elasticity. The amplitudes of displacements w(x), stresses σ(x), and strains ε(x) are introduced in equations (7.3) and (7.10). Hooke’s law and the relationship between velocities

7.3 Special models for plate motions | 153

Figure 7.11: Free elastic plate.

and momentum density for a plane stress state can be written in the form: 1 (σ − νσ22 ) = 0 , E 11 1 ξ22 = ε22 − (σ22 − νσ11 ) = 0 , E 1+ν σ =0 ξ12 = ε12 − E 12

(7.63)

1 𝜕σ11 𝜕σ12 ( + ) = 0, ρω 𝜕x1 𝜕x2 1 𝜕σ12 𝜕σ22 v2 (x) = ωw2 + ( + ) = 0. ρω 𝜕x1 𝜕x2

(7.64)

ξ11 = ε11 −

and

v1 (x) = ωw1 +

Suppose that the boundary Γ of the area V = Ω is free of stresses. It means that some homogeneous stress components are only given on the sides of the beam so that σ12 (x1 , ±a2 ) = 0

and σ22 (x1 , ±a2 ) = 0

(7.65)

σ11 (±a1 , x2 ) = 0

and σ12 (±a1 , x2 ) = 0 .

(7.66)

as well as The problem is to find the eigenfrequencies ω together with the corresponding eigenforms of displacements w(x) and stresses σ(x) that satisfy the constitutive equations (7.63), (7.64) as well as the boundary conditions (7.65), (7.66). Since the displacements are not subject to any constraints, they can be algebraically resolved with respect to the stresses according to w=−

1 ∇⋅σ ρω2

(7.67)

by using the expression (7.64). After substitution of the displacements (7.67) in the constitutive equation (7.63), the following tensor-valued equation that defines the stress field in the area V is obtained: 1 ξ = − 2 (ζ + ζ T ) − C−1 ∶ σ = 0 with ζ = ∇(ρ−1 ∇ ⋅ σ) . 2ω

154 | 7 Integrodifferential approach to eigenvalue problems It is supposed in the projection approach that approximations of stresses σ ∈ 𝒯1 satisfy exactly the boundary conditions (7.65), (7.66) and provide zero values for integral projections of the constitutive tensor ξ into the chosen tensor field of test functions τ(x) ∈ 𝒯2 in agreement with Σξ = ∫

a1

−a1

Θξ dx1 = 0

with Θξ = ∫

a2

−a2

ξ (σ) ∶ τ(x) dx2

for ∀τ ∈ 𝒯2 .

(7.68)

As in the previous section, the finite-dimensional representation for the components of the stress tensor σ with respect to the coordinate x2 : N

(k) σ11 = ∑ x̂ k σ11 (x1 ) , k=0 N

σ22 = ∑

k=0

N−1

(k) σ12 = ∑ x̂ k (1 − x̂ 2 )σ12 (x1 ) , k=0

(k) x̂ k (1 − x̂ 2 )σ22 (x1 )

x for x̂ = 2 a2

(7.69)

is applied. Here, σij(k) for i, j = 1, 2 are unknown functions of the coordinate x1 . The stresses are chosen in such a way that they automatically obey the boundary conditions (7.65). In compliance with the Galerkin method [20], the space of test functions 𝒯2 should coincide with the space of trial functions 𝒯1 . Involve this approach and define the components of the tensor τ as N

(k) τ11 = ∑ x̂ k τ11 (x1 ) , k=0 N

̂k

̂2

τ22 = ∑ x (1 − x k=0

N−1

(k) τ12 = ∑ x̂ k (1 − x̂ 2 )τ12 (x1 ) , k=0

(7.70)

(k) )τ22 (x1 ) .

Collect then the trial and test functions into three vectors: (0) (0) (N−1) (N−1) (N) z 1 = (σ11 , σ12 , … , σ11 , σ12 , σ11 ) , (0) (N) z 2 = (σ22 , … , σ22 ), (0) (0) (0) (N−1) (N−1) (N−1) (N) (N) y = (τ11 , τ22 , τ12 , … , τ11 , τ22 , τ12 , τ11 , τ22 ) ,

where z 1 (x1 ) ∈ ℝ2N+1 , z 2 (x1 ) ∈ ℝN+1 , and y(x1 ) ∈ ℝ3N+2 . Substitute the stresses (7.69) and (7.70) expressed through these vectors into the integral Θξ given in equation (7.68). After integration over the variable x2 and differentiation with respect to the vector y, the system of differential equations ′ ′ 𝜕Θξ (z ″ 1 , z 1 , z 1 , z 2 , z 2 , y, ω, x1 )

𝜕y appears.

=0

(7.71)

7.3 Special models for plate motions | 155

Given the structure of the constitutive equations (7.63) and (7.64), it can be shown that the differential order of the system (7.71) coincides with the number of boundary conditions z 1 (±a1 ) = 0

(7.72)

following directly from the boundary relations (7.66). The eigenfrequencies ω and eigenvectors z j (x1 ) with j = 1, 2 are found for the system of ODEs (7.71) with the boundary conditions (7.72). At that, the amplitudes of displacements w(x) and stresses σ(x) are restored in accordance with (7.67) and (7.69). To estimate the quality of the approximations obtained by the algorithm proposed, the functional 1 Φω 0 [σ, ω] = ∫ (ξ ∶ C ∶ ξ ) dV 2 V is used as it is shown in Section 7.2. The system energy Ψω σ [σ, ω] is expressed in terms of stresses according to (7.41). In order to determine their eigenvectors z j (x1 ), the unique solution of system (7.71), (7.72) can be derived according to the isoperimetric condition: Ψω σ = 1. As a result, the relative error of the eigenforms coincides with the value of the energy functional as follows: Δ = Φω 0 [σ, ω] .

(7.73)

By using the symmetry of the area V , and the boundary conditions (7.65), (7.66) with respect to the coordinate axes Ox1 and Ox2 , the eigenvalue problem (7.71), (7.72) can be decomposed into four independent subproblems. Mentioned in the previous section, the longitudinal and lateral vibrations (see Table 7.1) are also divided into two subgroups according to the following parity properties of the unknown functions: (k) (k) σ11 (x1 ) = ±σ11 (−x1 ) , (k) (k) σ12 (x1 ) = ∓σ12 (−x1 ) ,

(7.74)

(k) (k) σ22 (x1 ) = ±σ22 (−x1 ) .

It is interesting to note the relationship between the projection approach proposed in this section and the well-known variational principle of Hellinger–Reissner [103]. Let us formulate this principle for eigenvalues as follows. The amplitudes of its displacements w(x) and stresses σ(x) should strictly satisfy the boundary conditions (7.65), (7.66) and provide a stationary values for the functional: ΦHR [w, σ, ω] = with ε = 21 (∇w + ∇w T ).

1 ∫ (ρω2 w ⋅ w + 2σ ∶ ε − σ ∶ C−1 ∶ σ) dV , 2 V

156 | 7 Integrodifferential approach to eigenvalue problems Write down the conditions of stationarity for this functional: δw ΦHR = ∫ (ρω2 w + ∇ ⋅ σ) ⋅ δw dV = 0 , V

δσ ΦHR = ∫ (ε − C−1 ∶ σ) ∶ δσ dV = 0 .

(7.75)

V

To find an approximate solution for this problem, the displacement w ∈ 𝒱 and stresses σ ∈ 𝒯0 are chosen so that ρ−1 ω−2 ∇ ⋅ σ ∈ 𝒱. Here, 𝒱 and 𝒯0 are some linear spaces of admissible displacements and stresses. In this case, the first condition in equation. (7.75) can be uniquely resolved with respect to the vector w. As a consequence, we have the expression for the displacements (7.67). Suppose that 𝒯0 = 𝒯1 = 𝒯2 , where 𝒯1 and 𝒯2 are the spaces of the approximations (7.69) and the test functions (7.70), respectively. Then replace the variation δσ for the tensor τ in equation (7.75). The result will be the integral equation (7.68). 7.3.2 Simplified model of longitudinal vibrations Let natural motions of the homogeneous elastic plate (beam) be under study. As in the previous section, fix the dimensionless parameters E = ρ = a2 = 1. Derive the simplest model of longitudinal vibrations of the beam with the lowest approximation degree N = 1 in agreement with equation (7.69). (0) In this case, the stress tensor depends only on the two unknown functions σ11 (x1 ) (0) and σ22 (x1 ). The system (7.71) can be reduced to one differential equation of the second order (0) d2 σ11 (0) + λ2 (ω)σ11 =0 dx2

(0) (0) with σ22 = α(ω)σ11 ,

(7.76)

where the wave number λ and the multiplier α are the functions of the frequency ω and Poisson’s ratio ν according to 5 νω2 2 . λ2 = ω2 (1 − να(ω)) with α = 3 2 2ω2 − 5

(7.77)

The boundary conditions (0) σ11 (±a1 ) = 0

(7.78)

follow from equation (7.72). In contrast to the classical equation for longitudinal vibrations of a thin rod, the ODE (7.76) in terms of stresses contains a parameter λ, which depends nonlinearly on the frequency ω of natural oscillations as well as on the problem parameter ν. Similar beam models can also be obtained in other ways, for example, by means of an asymptotic approach [48].

7.3 Special models for plate motions | 157

The values λ2 (ω) are positive at ω ∈ (0, ω∗1 ) ∪ (ω∗2 , ∞) and negative at ω ∈ (ω∗1 , ω∗2 ). The critical frequencies, which divide the frequency domain onto three zones, are defined as ω∗1 = √

15 , 6 − 5ν

5 ω∗2 = √ . 2

Note that the values ω∗1 do not exceed ω∗2 (ω∗1 = ω∗2 at ν = 0), because values of Poisson’s ratio for conventional materials belong to the interval 0 ≤ ν ≤ 0.5. It can be shown that non-trivial solutions for boundary value problem (7.76)–(7.78) do not exist for λ2 (ω) ≤ 0. If λ2 (ω) > 0, the solution has the form (0) σ11 = c cos(λx)

(0) ∨ σ11 = c sin(λx)

depending on the symmetry properties (7.74). The free motions of the beam as a rigid body are not taken into account here. To determine the natural frequencies ω, the characteristic equation can be written as follows: 2a1 λ(ω) = πn,

n ∈ ℤ+ .

(7.79)

Two positive roots ω− and ω+ (the first and third zones) of equation (7.79) are explicitly found as a function of λ ≥ 0: ω± (n) = √

15 + 6λn2 ± √(15 − 6λn2 )2 + 300ν2 λn2 2(6 − 5ν2 )

,

λn =

πn . 2a1

The wave-frequency characteristics λ(ω± ) for the first and third zones at ν = 0.3 are shown in Figure 7.12 by solid curves. The frequency zones are separated by the dashed lines. If Poisson’s ratio ν tends to zero, then the value of the wave number tends to classical dependency λ → ω± , and the interval of the second zone reduced: ω∗1 → ω∗2 . The functions ω± (λ) are monotonically increasing, ω− (0) = 0, ω+ (0) = ω∗2 , and the following asymptotic relations are valid: ω− |n→∞ → ω1 ,

ω+ |n→∞ → √

6 λ . 6 − 5ν2 n

At a1 = 10 and ν = 0.3, the eight lowest eigenfrequencies ω− (n) for the first zone (0) are listed in Table 7.4. The symmetry or antisymmetry of the function σ11 (x1 ) relatively to the middle point of the beam midline x1 = 0 is marked by (s) or (a). The same values ω− (λn ) together with the frequencies of the third zone ω+ (λn ) are presented in Figure 7.12 by circles and squares, respectively. An important characteristic of natural modes is the ratio between the normal stresses α defined in equation (7.77). The dependence of the coefficient α on the mode

158 | 7 Integrodifferential approach to eigenvalue problems

Figure 7.12: Wave-frequency characteristics of longitudinal vibrations at M = 0. Table 7.4: Lowest frequency branch of longitudinal vibrations. Mode (s) ω−n Δωn (%)

1 0.1570 3 ⋅ 10−5

3 0.4695 3 ⋅ 10−3

5 0.7762 0.034

7

Mode (a)

2

4

6

8

1.0667 0.22

ω−n

0.3137 5 ⋅ 10−4

0.6240 0.011

0.9246 0.090

1.1976 0.51

Δωn (%)

number n for the first and third frequency zones is reflected in Figure 7.13 by solid and dashed curves, respectively. The normal stresses in the first zone oscillate in antiphase (α < 0), the absolute value |α| for the lower frequencies increases monotonically (although with very small rate) with the increasing number n. In the third zone α(n) > 0, this dependence is strictly a decreasing function. The magnitude of the (0) (0) stresses σ22 (x1 ) for small n is much higher than the longitudinal stresses σ11 (x1 ).

Figure 7.13: Ratio of the normal stresses α(n).

As suggested in the previous subsection, the quality of the displacement fields obtained can be clearly estimated by the value of the relative energy error Δ introduced in equation (7.73). For given system parameters, this error for the first frequency zone is shown in Figure 7.14 by triangles connected by solid lines for clarity. The approximation error rapidly grows with the increasing mode number n.

7.3 Special models for plate motions | 159

Figure 7.14: Relative energy error Δ(n).

7.3.3 Refined model of 2D rod vibrations To improve the accuracy of the approximate solutions, it is necessary to increase the dimension of the approximation. If N = 2, the components of the stress tensor σ(x) for the case of beam longitudinal vibrations, which have the form: (0) (2) σ11 = σ11 (x1 ) + x22 σ11 (x1 ) ,

(1) σ12 = (1 − x22 )x2 σ12 (x1 ) ,

(0) (2) σ22 = (1 − x22 )(σ22 (x1 ) + x22 σ22 (x1 ))

are defined by five unknown functions. The corresponding homogeneous boundary conditions are (0) (2) (1) σ11 (±a1 ) = σ11 (±a1 ) = σ12 (±a1 ) = 0 .

As for the projection model considered in Section 7.2, the system (7.71) at N = 2 (0) can be reduced to a bicubic differential equation with respect to the function σ11 (x1 ), which defines the normal stresses along the axis Ox1 on the center line x2 = 0. The roots of the characteristic equation λj (ω), j = 1, … , 6, for the chosen approximation are shown in Figure 7.15 at ν = 0.3. The values |Im λj | are pictured by solid curves, the values |Re λj | are depicted by dashed curves. The bifurcation of the solution occurs at critical values of frequency: ω∗1 = 1.566 ,

ω∗2 = 1.647 ,

ω∗3 = √

105 = 2.010 . 26

This set defines four vibration zones. Their boundaries are presented in Figure 7.15 by dashed lines. Two purely imaginary and four complex roots are situated in the first zone (ω < ω∗1 ). All the roots are purely imaginary in the second (ω∗1 < ω < ω∗2 ) and fourth (ω∗3 < ω) zones. There are four imaginary and two real roots in the third zone (ω∗2 < ω < ω∗3 ). It is noticeable in comparison with the beam model in displacements and stresses described in Section 7.2 that the boundaries of the frequency zones are shifted to the left and the interval of the second zone increases (see Figure 7.8).

160 | 7 Integrodifferential approach to eigenvalue problems

Figure 7.15: Wave-frequency characteristics of longitudinal vibrations at M = 1.

The value of the eigenfrequencies of longitudinal vibrations located in the first and second zones versus the mode number are shown at a1 = 10 by the line with circles in Figure 7.16. These values are reflected in the Figure 7.15 by circles and squares. In the (0) first zone, the modes with either even or odd leading function σ11 (x1 ) alternate with each other. In the second zone, the sequence of eigenfrequencies belonging to both symmetry classes is more diverse. The natural frequencies with different properties of eigenforms come near to each other. It is worth noting that the decreasing of the minimal absolute value of the wave numbers with the increasing of the mode number is inherent in these types of vibrations.

Figure 7.16: Natural frequencies of the free elastic plate.

The accuracy of the frequency can be determined indirectly with the changing of its approximate values as compare with the simplified beam model for N = 1. Table 7.4 shows the relative variation of frequencies for two successive models. As seen, all eight frequencies are changed a little. As it is demonstrated in Figure 7.14 (solid line marked with diamonds), the accuracy of vibration forms increases by several orders of magnitude for these modes as compared with the more coarse model. The relative error Δ(n) rapidly increases simultaneously with the mode number.

7.3 Special models for plate motions | 161

7.3.4 Lateral vibrations of a free 2D beam To study lateral vibrations of a free elastic beam, let the approximation in equation (7.36) be of the minimal admissible order N = 1. By taking into account the symmetry properties of the solution with respect to the axis Ox2 , which are given in Table 7.1, the corresponding stress approximation can be written as (1) σ11 = x2 σ11 (x1 ) ,

(0) σ12 = (1 − x22 )σ12 (x1 ) ,

(1) σ22 = x2 (1 − x22 )σ22 (x1 ) .

For this degree of approximation, the system (7.70) is reduced to a biquadratic (0) differential equation with respect to one function of shear stresses σ12 (x1 ): a(ω, ν)

(0) (0) d2 σ12 d4 σ12 (0) − c(ω, ν)σ12 = 0, + b(ω, ν) 4 dx12 dx1

a=1−

4ω2 , 35

c = 3ω2 +

b=

17 + 10ν 2 2(30 + 20ν − 7ν2 ) 4 ω − ω , 5 175

(7.80)

94 + 84ν − 7ν2 4 4(1 + ν)(10 − 7ν2 ) 6 ω − ω . 35 175

It can be proved that equation (7.80) asymptotically tends to the classical Euler– Bernoulli beam equation at ν = 0 for low eigenfrequencies (ω ≪ 1). Figure 7.17 shows the wave-frequency characteristics of the lateral motions of an elastic beam for Poisson’s ratio ν = 0.3. The critical frequency ω∗0 = 0.9806 (dotted line) separates two zones: ω ∈ (0, ω∗0 ) and ω ∈ (ω∗0 , +∞). The solid curves correspond to the imaginary roots of the characteristic equation |Im λj (ω)|, and the dashed curves correspond to the real roots |Re λj (ω)|. Although two of these roots are imaginary in the first zone and the other two are complex, as for the model of Euler–Bernoulli, the absolute values of the imaginary and real roots do not coincide. The solutions are found depending on their parity (s

Figure 7.17: Wave-frequency characteristics of lateral vibrations.

162 | 7 Integrodifferential approach to eigenvalue problems for even and a for odd degrees, respectively) in the form either σ12 (x1 ) = c1 cos |λ1 |x1 + c2 cosh |λ2 |x1 or σ12 (x1 ) = c1 sin |λ1 |x1 + c2 sinh |λ2 |x1 . In the second zone, all eigenvalues have purely imaginary values. The function that defines the shear stresses has the form either σ12 (x1 ) = c1 cos |λ1 |x1 + c2 cos |λ2 |x1 or σ12 (x1 ) = c1 sin |λ1 |x1 + c2 sin |λ2 |x1 . Two of these roots vanish if ω = ω∗0 . The wave-frequency characteristic for the 4 Euler–Bernoulli beam with the wave numbers |λc | = √3ω2 is displayed by the dashdotted curve in Figure 7.17. The curves |λj (ω)| coalesce with this curve at small values of frequencies (ω → 0). Let a half-length of the beam equal to a1 = 10. For this geometrical parameter, the lowest eigenfrequencies ωn for n ≤ 12 are located in the first zone (ωn < ω∗0 ). These (0) frequencies for symmetric (s) or antisymmetric (a) shear stresses σ12 (x1 ) are given in Table 7.5. Table 7.5: Natural frequencies of lateral beam vibrations. Mode (s)

1

3

5

7

9

11

ωn Δ (%)

0.0312 0.23

0.1485 1.35

0.3125 2.58

0.4949 3.73

0.6816 4.84

0.8634 6.08

Mode (a)

2

4

6

8

10

12

ωn Δ (%)

0.0813 0.74

0.2269 1.98

0.4025 3.16

0.5883 4.28

0.7739 5.43

0.9458 7.02

The table also contains the relative error Δ in computations of their eigenforms. For comparison, the accuracy of solutions for the first eight modes versus the number n of the eigenfrequency is reflected in Figure 7.15 by the dashed line marked with circles. As it can be seen in this figure for the given approximation (N = 1), the errors in determining the eigenforms are bigger for lateral motions than the corresponding errors for longitudinal vibrations. The calculations performed show that the main contribution to this error makes the off-diagonal component of the constitutive tensor ξ12 (x)

7.3 Special models for plate motions | 163

(see equation (7.63)). It is necessary to increase the dimension of the approximations to improve the accuracy. The wave numbers |Im λj (ωn )| are marked in Figure 7.17 by circles for the first zone, and by squares for the second one (ωn > ω∗0 , n > 12). The dependence of the natural frequencies on the mode number n is shown in Figure 7.16 in comparison with longitudinal eigenfrequencies. Note that the natural frequencies of the longitudinal and lateral vibrations are almost identical to each other if n = 20 at the given parameters of the problem.

8 Spatial vibrations of elastic beams with convex cross-sections 8.1 Natural motions of a cuboid beam To derive reliable beam models in the frame of the linear theory of elasticity, the spatial distribution of the displacement and stress fields should be taken into account. From the practical point of view, a wide class of elastic beam-type bodies consists of prismatic bars, shafts, pipes, etc. The eigenvalue problems for such bodies may have special symmetry properties that give us the possibility to decrease the dimension of the original system of PDEs and to apply, after a certain modification, the variation and projection approaches discussed in the previous chapter of this book. The procedure described in Section 7.2 is used to obtain finite-dimensional systems of ODEs, describing the natural vibrations of rectilinear beams with the rectangular cross-section. This approach utilizes special integral projections of constitutive equations relating the stress and strain tensors as well as the vectors of velocities and momentum density. 8.1.1 Projection approach to a 3D eigenvalue problem in elasticity Consider an elastic body having the form of a rectangular cuboid with the length 2a1 and the cross-sectional size 2a2 × 2a3 . At that, it is assumed that a1 ≫ a2 + a3 (see Figure 8.1). Introduce a Cartesian coordinate system Ox1 x2 x3 , x = (x1 , x2 , x3 ), with the origin located in the middle of the body and the axes Oxk parallel to the edges with the lengths 2ak for k = 1, 2, 3. The volume occupied by the body is defined as V = {x ∶ xk ∈ (−ak , ak ), k = 1, 2, 3} . Let us analyze motions of the elastic body (beam) in the case of natural vibrations described by governing equations of linear elasticity [36]. Represent this system in the form pt (t, x) = ∇ ⋅ σ(t, x)

1 and ε(t, x) = (∇w(t, x) + ∇w T (t, x)) 2

(8.1)

as well as p(t, x) = ρ(x)w t (t, x) and

σ(t, x) = C(x) ∶ ε(t, x) .

(8.2)

The first expression in equation (8.1) characterizes the changing of the momentum density vector p in time via the divergence of the stress tensor σ (balance equation). The second relation defines a Cauchy strain tensor ε through the displacement vector w. The constitutive equations connecting the stresses σ and the strains ε (Hooke’s https://doi.org/10.1515/9783110516449-008

166 | 8 Spatial vibrations of elastic beams with convex cross-sections

Figure 8.1: Prismatic region occupied by the elastic beam.

law) as well as the momentum density p and the velocity of body points is represented by equation (8.2). In this equation, C is the elastic modulus tensor and ρ is the volume density of material. The case is considered when the long sides of the beam are free of stresses according to xk = ±ak ,

k = 2, 3 ∶

σ ⋅ n(k) ± = 0,

(8.3)

and the end cross-sections are either free of loads or clamped as x1 = ±a1 ∶

σ ⋅ n(1) ± =0

∨ w = 0.

(8.4)

Here, n(k) ± = (±δ1k , ±δ2k , ±δ3k ) for k = 1, 2, 3 are the outward unit normals to the boundary surface of the body Γ with the Kronecker symbol δjk . The unknown eigenfunctions are found in the form: ̃ w(t, x) = w(x) sin ωt ,

̃ cos ωt , p(t, x) = p(x)

̃ sin ωt σ(t, x) = σ(x)

(8.5)

with omitting further the sign tilde. Here, ω is the unknown eigenfrequency of the elastic beam. In accordance with the MIDR, the constitutive equations (8.2) are represented in the integral form. The following projection problem can be formulated. Problem 8.1. Find such fields w ∗ (x), p∗ (x), σ ∗ (x) that satisfy the equilibrium equation p(x) + ∇ ⋅ σ(x) = 0 for x ∈ V ,

(8.6)

the boundary conditions (8.3) and (8.4), as well as the integral relation ∫ (ρωv ⋅ u + ξ ∶ τ) dΩ = 0 V

with

for ∀u ∈ L2 (V; ℝ3 ), τ ∈ L2 (V; ℝ3×3 )

1 v ∶= ωw − ρ−1 p and ξ ∶= (∇w + ∇w T ) − C−1 ∶ σ . 2

(8.7)

(8.8)

The vector v links momentum density and velocity of the body points, the tensor ξ defines Hooke’s law, the vector u, and the tensor τ are the test functions integrable in squares over the volume V .

8.1 Natural motions of a cuboid beam

| 167

8.1.2 System of DAEs approximating the beam vibrations The finite-dimensional approximations for the components of the displacement vector w, momentum density vector p, and the stress tensor σ, which are polynomial with respect to the normalized coordinates x̂ k = a−1 k xk for k = 2, 3 according to w1 (x) = ∑ w1(k,l) (x1 )x̂ k2 x̂ l3 ,

wn (x) =

k l p1 (x) = ∑ p(k,l) 1 (x1 )x̂ 2 x̂ 3 ,

pn (x) =

k+l≤N

k+l≤N

(k,l) σ1,1 (x) = ∑ σ1,1 (x1 )x̂ k2 x̂ l3 ,

∑

k+l≤N−1

σ2,3 (x) = g2 g3

∑

∑

k l p(k,l) n (x1 )x̂ 2 x̂ 3 ,

k+l≤N+1

k+l≤N

(k,l) σ1,n (x1 )x̂ k2 x̂ l3 ,

k+l≤N−2

gn ∶= 1 − x̂ 2n

wn(k,l) (x1 )x̂ k2 x̂ l3 ,

(k,l) σn,n (x) = gn ∑ σn,n (x1 )x̂ k2 x̂ l3 ,

k+l≤N

σ1,n (x) = gn

∑

k+l≤N−1

(8.9)

(k,l) σ2,3 (x1 )x̂ k2 x̂ l3 ,

for n = 2, 3 ,

are used to obtain a system of approximate equations with boundary conditions at (k,l) (k,l) x1 = ±a1 . Here, wm , p(k,l) m , σm,n for m, n = 1, 2, 3 are the unknown functions of the coordinate x1 . Note that the approximations (8.9) satisfy automatically the homogeneous boundary conditions (8.3). The polynomial representations (8.9) allow us to solve the original vector-valued equation of the momentum balance (8.6) with respect to all the functions pj(k,l) . To do this, it is necessary to equate to zero all the coefficients at the corresponding monomials of the coordinates x2 and x3 in equation (8.6). In general, the constitutive equations given either in the differential form (8.2) or in the integral one (8.7) cannot be exactly resolved by using the approximation obtained in equation (8.9). Nevertheless, the integral formulation allows us to compose a finite-dimensional consistent system of DAEs with respect to the unknown displace(k,l) (k,l) ments wm and stresses σm,n . Let the equations of this system be zero projections of the vector v and tensor ξ on the space of polynomials of the coordinates x2 and x3 according to ̂ 2 , x3 )) dS = 0 ∫ (v(x) ⋅ u(x S

and

̂ 2 , x3 )) dS = 0 ∫ (ξ (x) ∶ τ(x S

for ∀û 1 , τ̂ m,m ∈ ℙ2N , ∀û n , τ̂ 1,n ∈ ℙ2N−1 , ∀τ̂ 2,3 ∈ ℙ2N−2

(8.10)

with n = 2, 3 and m = 1, 2, 3 . Here, S = (−a2 , a2 ) × (−a3 , a3 ) is the cross-sectional area, û j (x2 , x3 ) and τ̂ j,k (x2 , x3 ) are respectively the components of the vector û and the tensor τ,̂ ℙ2N is the complete space of bivariate polynomials of the degree N over x2 and x3 .

168 | 8 Spatial vibrations of elastic beams with convex cross-sections This choice of polynomial spaces for the test functions corresponds to the number of linearly independent equations Nt = 9N 2 /2 + 15N/2 + 4 following from the relations (8.10). The dimension of the system is equal to the total number of the unknown (k,l) (k,l) displacements wm and stresses σm,n . (k,l) (k,l) (k,l) The stress components σ2,2 , σ3,3 , σ2,3 can be algebraically expressed through the remaining unknown functions of displacements and stresses. Of course, the differential order of the system Nd = (N + 1)(3N + 2) is not changed after this transformation and equals to the doubled number of the unknown displacement functions. The resulting Nd differential equations are combined with the same number of bound(k,l) ary conditions following from (8.4) and imposed on the functions σ1,j and (or) wj(k,l) for j = 1, 2, 3. This one-dimensional boundary eigenvalue problem allows us to determine numerically the natural frequencies and spatial forms of elastic beam vibrations. These displacements w(x), the momentum density p(x), and stresses σ(x) will exactly satisfy the balance equation (8.6) and the boundary conditions (8.3), (8.4). Since the integral relation (8.7) is satisfied approximately, the ratio of the functionals Δ = ΦΨ−1 ≥ 0 for 1 Φ = ∫ (ρv ⋅ v + ξ ∶ C−1 ∶ ξ ) dV with 2 V 1 Ψ = ∫ (ρω2 w ⋅ w + σ ∶ C−1 ∶ σ) dV = 1 2 V can be regarded as a criterion of solution quality at the chosen degree of approximation N (Δ ≪ 1, see [49]). Here, Ψ[w, σ] is the total mechanical energy of the beam.

8.1.3 Decomposition of vibration equations for a homogeneous beam In view of the symmetry properties which are inherent in the boundary value problem (8.4), (8.6), (8.10) for a homogeneous isotropic beam of the cuboid shape and are not violated by the approximations (8.9), four independent subsystems can be identified. These subsystems approximately describe tension–compression, bending around the axes Ox2 and Ox3 , as well as torsion. As it was noted in [55], this decomposition is typical for any degree of approximation of displacements w(x) and stresses σ(x). It appears due to the mirror symmetry of the elastic body V with respect to the coordinate planes Ox1 x2 and Ox1 x3 . For different types of beam motions, the maximum degrees N2,3 (k) for variables x2,3 , which are present in the approximations of displacements and stresses (8.9), are given in Table 8.1. The index j takes the values 1, 2, and 3. The even number k = 2n ≥ 0 determines the differential order of the corresponding boundary value problem. The parity (oddness) of the numbers N2 and N3 characterizes the properties of symmetry (antisymmetry) for the functions of displacements and stresses with respect to the

8.1 Natural motions of a cuboid beam

| 169

Table 8.1: Symmetry properties of the displacement and stress fields. Functions

Tension N2 N3

Bending, Ox2 N2 N3

Bending, Ox3 N2 N3

Torsion N2 N3

w1 , σj,j w2 , σ1,2 w3 , σ1,3 σ2,3

k k−1 k−2 k−3

k k−1 k k−1

k+1 k k−1 k−2

k+1 k k+1 k

k k−2 k−1 k−3

k+1 k−1 k k−2

k k k−1 k−1

k+1 k+1 k k

coordinate planes x2 = 0 and x3 = 0. If at least one of the numbers N2 or N3 is negative, the function with the corresponding index does not exist. The orders of the ODE systems are equal to (n + 1)(3n + 2) for tension–compression, (n + 1)(3n + 4) for bending, (n + 1)(3n + 6) for torsion. The minimal dimensions of the approximations represented by equation (8.9) are respectively 2, 4, and 6. It should be noted that the finite-dimensional decomposition of the unknown functions is not unique. For some special cases, such as the beam motions discussed in Section 7.1, when displacements are free of boundary conditions, meaningful modk,l els can be composed with the number of displacement functions wm greater than the number of stress functions σ1,m for m = 1, 2, 3. Further, a special case of a homogeneous isotropic beam with a square cross section (a2 = a3 ), is under study. By introducing the characteristic length x̃ = a2 and time t ̃ = a2 √ρ/E, where E is Young’s modulus, all the equations of linear elasticity can be transformed to the dimensionless form. The resulting system has two parameters: the relative length of the beams a = a1 /a2 and Poisson’s ratio ν. The unknown quantity is ̃ To simplify the notation, the sign tilde is omitted the dimensionless frequency ω̃ = tω. hereafter. For an isotropic solid, the components of the constitutive tensor ξ take the form: ξj,j =

𝜕wj 𝜕xj

3

− (1 + ν)σj,j − ν ∑ σk,k k=1

𝜕wj 1 𝜕w ξj,k = ( k + ) − G−1 σj,k 2 𝜕xj 𝜕xk

for j, k = 1, 2, 3 , for j, k = 1, 2, 3 and j ≠ k .

Here, G = 1/(1 + ν)/2 denotes the dimensionless shear modulus. There are two additional planes of symmetry χ± = (x3 ±x2 )/√2 = 0 for a beam with a square cross section. It is possible in this case to decompose the problem on longitudinal vibrations into two subproblems. The unknown displacement functions, momentum density, and stresses are selected either symmetric or antisymmetric with respect to interchanging the coordinates x2 and x3 . Owing to this decomposition, the first type of motions is actually the longitudinal vibrations of the beam. At that, the displacement of the central line of the beam

170 | 8 Spatial vibrations of elastic beams with convex cross-sections w1(0,0) (x1 ) is not identically equal to zero. Antisymmetric vibrations is called ‘breathing’ modes, similar to motions found for cylindrical shells. Although the section shape deforms during such motions, the points of the central axis Ox1 is fixed to zero (w1(0,0) (x1 ) ≡ 0). A similar decomposition is also performed under the torsional vibrations by taking into account the symmetry (antisymmetry) with respect to the permutation of x2 and x3 . This operation extracts the true torsional vibrations, for which the local angle of cross-sectional torsion 1 ϕ(x1 ) = (w3(1,0) − w2(0,1) ) 2

(8.11)

is not identically zero, and ‘shear’ vibrations, for which ϕ ≡ 0. It can be shown that the shear motions define breathing modes in the coordinate system χ = (x1 , χ+ , χ− ), rotated through an angle π/4 around the axis Ox1 .

8.1.4 Breathing of a body with the square cross section Consider eigenforms of the beam with the longitudinal displacements w1 to be symmetric relatively to the coordinate planes Ox1 x2 , Ox1 x3 and antisymmetric with respect to the planes Ox1 χ± . The parity (even or odd) for the powers of the variables x2,3 in the approximations (8.9) are set in agreement with Table 8.1 (row tension). Regarding the interchange of x2 and x3 , we get w1 (x1 , x2 , x3 ) = −w1 (x1 , x3 , x2 ) ,

w2 (x1 , x2 , x3 ) = −w3 (x1 , x3 , x2 ) ,

σ1,1 (x1 , x2 , x3 ) = −σ1,1 (x1 , x3 , x2 ) ,

σ2,2 (x1 , x2 , x3 ) = −σ3,3 (x1 , x3 , x2 ) ,

σ1,2 (x1 , x2 , x3 ) = −σ1,3 (x1 , x3 , x2 ) ,

σ2,3 (x1 , x2 , x3 ) = −σ2,3 (x1 , x3 , x2 ) .

(8.12)

This means that the beam points belonging to the center line Ox1 remain fixed during the vibrations and the motions in the coordinate planes Ox1 x2 and Ox1 x3 take place in antiphase. Let the degree of approximation be N = 2 in equation (8.9). Then the displacements and stresses are completely determined in accordance with equation (8.12) by the following components: w1 = w1(2,0) (x22 − x32 ) ,

w2 = w2(1,0) x2 ,

(2,0) 2 σ1,1 = σ1,1 (x2 − x32 ) ,

(1,0) σ1,2 = σ1,2 g2 x2 ,

(0,0) (2,0) 2 (0,2) 2 σ2,2 = g2 (σ2,2 + σ2,2 x2 + σ2,2 x3 )

with g2 = 1 − x22 .

σ2,3 = 0 ,

(8.13)

8.1 Natural motions of a cuboid beam

| 171

By using the approximation (8.13) and following the algorithm described in Subsection 8.1.2, the obtained system of ODEs can be reduced to one differential equation: d2 w2(1,0) d4 w2(1,0) + a (ω, ν) + b1 (ω, ν)w2(1,0) = 0 , 1 dx12 dx14 a1 = b1 =

3 − ν − 4ν2 2 ω − 6, 1−ν

(8.14)

2 − 6ν2 − 4ν3 4 1 − ν − 2ν2 2 1 − 2ν ω − 21 ω + 45 , 1−ν 1−ν 1−ν

biquadratic with respect to the unknown function w2(1,0) . The roots λj (ω, ν) of the corresponding characteristic equation for j = 1, … , 4 are found analytically. The complex-valued functions λj (ω, ν) at a fixed value of Poisson’s ratio ν define the wave-frequency characteristics of the natural vibrations. These functions do not correlate with the beam length and the boundary conditions. The dependence of the absolute values of the imaginary parts κj = |Im λj | (solid curves) and the real parts δj = |Re λj | (dashed curves) of the eigenvalues λj on the frequencies ω are depicted in Figure 8.2 at ν = 0.3. There are three critical values of frequency in this case, namely, ω(1) 1 ≃ 1.493 ,

ω(1) 2 ≃ 1.519 ,

ω(1) 3 ≃ 2.402

shown in Figure 8.2 by dash-dot lines. These values define the four frequency zones that differ one from another with the type of roots κj .

Figure 8.2: Wave-frequency characteristics of ‘breathing’ vibrations.

For the first zone ω ∈ (0, ω(1) 1 ), the eigenvalues λj are two pairs of complex conjugate numbers: λ1,2 = δ1 ± iκ1

and λ3,4 = −δ1 ± iκ1 .

172 | 8 Spatial vibrations of elastic beams with convex cross-sections (1) (1) For the second and fourth zones at ω ∈ (ω(1) 1 , ω2 ) ∪ (ω3 , +∞), all the roots are purely imaginary

λ1,3 = ±iκ1

and λ2,4 = ±iκ2 .

Note that there is one imaginary root in the second (relatively narrow) zone, in which the absolute value is decreasing if the frequency ω is increasing. Finally, there are two real roots λ1,3 = ±δ1 and two imaginary roots λ2,4 = ±iκ2 in (1) the third zone at ω ∈ (ω(1) 2 , ω3 ). The analytical expressions are obtained for the second and third critical frequencies: ω(1) 2 = √3/(1 + ν) ,

ω(1) 3 = √15/(2 + 2ν) .

The frequency ω(1) 1 (ν) is also found in radicals, but it is not given because of its excessive size. We now turn to ‘shear’ beam motions when the longitudinal displacements w1 are antisymmetric relatively to the planes Ox1 x2,3 and are symmetric with respect to the planes Ox1 χ± . The necessary degree of expansion over the variables x2 and x3 defined by equation (8.9) are presented in Table 8.1 (torsion). The corresponding displacement and stress fields are obtained from the expressions given in (8.12) by replacement of the minus sign in the right-hand sides for the plus sign. The beam points in the center line Ox1 are fixed, and the points that are lying in planes Ox1 χ± before deformation remain in the same planes. Let N = 2 in equation (8.9), then w1 = w1(1,1) x2 x3 ,

w2 = w2(0,1) x3 ,

(1,1) σ1,1 = σ1,1 x2 x3 ,

(0,1) σ1,2 = σ1,2 g2 x3 ,

(0,0) σ2,3 = σ2,3 g2 g3 ,

(1,1) σ2,2 = σ2,2 g2 x2 x3

with g2 = 1 − x22 and g3 = 1 − x32 . The approximating equations following from equation (8.10) in this case are reduced to a biquadratic differential equation with respect to the unknown function w2(0,1) (x1 ). It differs from the equation (8.14) for the coefficient b1 (ν, ω) to b2 (ν, ω) =

2(1 − 2ν)(3 − (1 + ν)ω2 )2 . 1−ν

The coefficient a1 (ν, ω) remains the same at that. The amplitude characteristics for shear vibrations are shown in Figure 8.3 at ν = 0.3. More specifically, the absolute values of the imaginary parts of the eigenvalues κj = |Im λj | (solid curves) as well as of their real parts δj = |Re λj | (dashed curves) versus the frequency ω are presented.

8.1 Natural motions of a cuboid beam

| 173

Figure 8.3: Wave-frequency characteristics of ‘shear’ vibrations. (2) There are two critical frequencies ω(2) 1 ≃ 1.424, ω2 ≃ 1.493, which are marked in Figure 8.3 by dash-dot lines, and three zones with different types of roots λj at the given

value of Poisson’s ratio. For the first zone ω ∈ (0, ω(2) 1 ), the eigenvalues λj consist of

(2) two pairs of complex-conjugate roots. In the second and third zones at ω ∈ (ω(2) 1 , ω2 ) ∪ (2) (2) (ω2 , +∞), there are two pairs of imaginary roots. Note that the equality ω2 = ω(1) 2 is correct for any admissible values ν.

8.1.5 Torsion of the body Consider the problem on beam torsion. For such motions, the component of displacement vector w1 is antisymmetric relatively to the coordinate planes Ox1 x2 and Ox1 x3 as well as the planes Ox1 χ± formed by the diagonals of the cross section. The degree of approximating polynomials are given in Table 8.1 (torsion). The properties of parity (8.12) are valid for torsional displacements and stresses. Furthermore, the points of the beam midline Ox1 are fixed. After taking into account the prescribed polynomials parity and equation (8.12), the system with the degree N = 2 of approximations (8.9) is reduced to one differential equation of the second order with respect to the unknown function w2(0,1) . The roots of the characteristic equation are purely imaginary and the frequency is linearly dependent on the eigenvalue: ω = √Gλ. The classic solution for torsional vibration with the √ same parameters corresponds to the relation ω(3) c ≃ 0.920 Gλ (see [97]). The solution structure becomes more complicated if the degree of approximation in (8.9) increases. For instance, the displacement and stress fields at N = 4 are given in the form: w1 = w1(3,1) x2 x3 (x22 − x32 ) ,

w2 = (w2(0,1) + w2(21) x22 + w2(0,3) x32 )x3 ,

174 | 8 Spatial vibrations of elastic beams with convex cross-sections (3,1) σ1,1 = σ1,1 x2 x3 (x22 − x32 ) ,

(0,1) (2,1) 2 σ1,2 = (σ1,2 + σ1,2 (x2 + x32 ))g2 x3 ,

(0,1) (2,1) 2 σ2,2 = (σ2,2 + σ2,2 (x2 + x32 ))g2 x2 x3 ,

(0,0) σ2,3 = σ2,3 g2 g3 (x22 − x32 )

with g2 = 1 − x22 and g3 = 1 − x32 . The system of approximating DAEs is reducible, after a series of transformations, into a single differential equation of the eighth order, for example, with respect to the unknown function w2(0,1) . For this model, the dependence of the wave numbers κj = |Im λj | (solid curves) and δj = |Re λj | (dashed lines) versus the frequency ω at ν = 0.3 is depicted in Figure 8.4. For the roots λj of the characteristic equation, there are seven zones separated by some critical frequencies. The frequencies dividing these zones for Poisson’s ratio ν = 0.3 are marked with dash-dot lines and given by ω(3) 1 ≈ 2.955 ,

ω(3) 2 ≈ 3.019 ,

ω(3) 3 ≈ 3.066 ,

ω(3) 4 ≈ 4.000 ,

ω(3) 5 ≈ 4.127 ,

ω(3) 6 ≈ 4.160 .

Classical linear wave-frequency dependence is shown by the sloping dash-dot line for comparison, which almost touches one of the curves κj (ω) at ω → 0.

Figure 8.4: Wave-frequency characteristics of torsional vibrations.

8.1.6 Longitudinal vibrations The expansion degrees for the longitudinal vibrations are presented in Table 8.1 (tension). The functions of displacements and stresses are symmetric subject to the permutation of the coordinates x2 , x3 and satisfy the relations which are obtained from equations (8.12) by replacing the minus sign in the right-hand sides for the plus sign. For the degree of approximation N = 2, the approximate displacements and stresses are given in the form:

8.1 Natural motions of a cuboid beam

w1 = w1(0,0) + w1(2,0) (x22 + x32 ) ,

| 175

w2 = w2(1,0) x2 ,

(0,0) (2,0) 2 σ1,1 = σ1,1 + σ1,1 (x2 + x32 ) ,

(1,0) σ1,2 = σ1,2 g2 x3 ,

(0,0) (2,0) 2 (0,2) 2 σ2,2 = (σ2,2 + σ2,2 x2 + σ2,2 x3 )g2 ,

σ2,3 = 0

with g2 = 1 − x22 . As a result, the DAE system is reduced to a bicubic differential equation with respect to the function w1(0,0) (x1 ), which specifies the displacements of the center line points along the axis Ox1 . The roots of the characteristic equation λj (ω, ν) for j = 1, … , 6 are found analytically. Figure 8.5 shows the wave numbers κj = |Im λj | (solid curves) and δj = |Re λj | (dashed curves) as functions of ω at ν = 0.3.

Figure 8.5: Wave-frequency characteristics of longitudinal vibrations.

There are two critical frequency values ω(4) 1 = 2.291,

ω(4) 2 = 2.402 ,

defining three zones, which are separated in Figure 8.5 by dash-dot vertical lines. Note (1) that the equality ω(4) 2 = ω3 = √15/(2 + 2ν) is valid for any value ν. There exists a monotonically decreasing branch ω(λ) in the second zone. The sloping dash-dot line shows the dependency of the wave number on the frequency of the longitudinal vibrations (λc = iω) for the classical model of a thin rod. This line is tangent to the curve λ1 (ω) (λ1 = iκ1 ). Since the model of a thin elastic rod does not take into account any lateral motion of the points in the cross section, the linear characteristic gives an upper bound for the eigenfrequencies of longitudinal vibrations. The numerical values of 20 lowest eigenfrequencies are presented in Table 8.2 for longitudinal vibrations of an elastic beam with a = 10. It is supposed that the beam is

176 | 8 Spatial vibrations of elastic beams with convex cross-sections Table 8.2: Eigenfrequencies of longitudinal vibrations. n ω

1 0.16

2 0.32

3 0.47

4 0.63

5 0.78

6 0.92

7 1.07

8 1.20

9 1.33

10 1.45

n ω

11 1.56

12 1.67

13 1.76

14 1.86

15 1.94

16 2.03

17 2.11

18 2.19

19 2.26

20 2.30

rigidly clamped at both ends according to x1 = ±10 ∶

w1(0,0) (x1 ) = w1(2,0) (x1 ) = w2(1,0) (x1 ) = 0 .

The 19 lowest frequencies correspond to the first vibration zone whereas the 20th frequency corresponds to the second zone. The values of ω(n) are marked by light dots on one of the branches of λ(ω) in Figure 8.5. For conventional model of a thin rod clamped at both ends with the same geometrical and mechanical parameters, longitudinal eigenfrequencies have the analytical representation ωc (n) = nπ/2/a. The relative frequency error for the first mode as compared with the classical value is Δω = 1.6 ⋅ 10−3 %. This value increases with the mode number, for example, it equals to 2% for the 10th mode and reaches already 6% for the 19th mode. For a more comprehensive understanding of the solutions structure of the eigenvalue problem, let us investigate the dependence of vibration modes on the material parameters. There is only one parameter for a homogeneous isotropic beam, namely, Poisson’s ratio ν, that determines the structure of wave numbers λj versus the frequency ω. The location of the critical frequencies of the longitudinal vibrations, where drastic changes of eigenvalues occur, is depicted in Figure 8.6 for different values of ν ∈ [0, 0.5]. The eigenforms of elastic vibrations are significantly varied in different frequency zones. As it can be seen in Figure 8.6, the number of zones does also change if the value of Poisson’s ratio increases. In zone I separated by a dashed line, the four roots

Figure 8.6: Critical frequencies vs. Poisson’s ratio.

8.1 Natural motions of a cuboid beam

| 177

of the characteristic equation are complex whereas the other two are imaginary. In zones II and IV, all of the roots are purely imaginary. Two eigenvalues are real in zone III, and there are four real roots in zone V. There exist several critical values ν. If ν = ν1 = 3/10, then zone III shrinks to a point. If ν = ν2 = (23 − √89)/40 ≃ 0.339, then the area II vanishes but a fifth zone arises. If ν increases very little to ν3 = 4/11 ≈ 0.367, then the second part of the zone V appears near the zero value of ω (zone I is separated from zero). The first zone vanishes if ν4 ≃ 0.373. 8.1.7 Lateral vibrations Bending motions around the axes Ox2 and Ox3 have equivalent wave-frequency characteristics for the elastic beam with the square cross section. For definiteness, consider the bending around the axis Ox3 . Then the points of the beam center line will move in the plane Ox1 x2 . As for the other natural motions, let us write the approximation for displacements and stresses on the basis of the decomposition performed in Subsection 8.1.7. The parity of expansion powers with respect to the variables x2 and x3 in equation (8.9) is determined relying on Table 8.1 (Bending, Ox3 ). At the degree of approximation N = 2, displacements and stresses have the form: w1 = w1(1,0) x2 ,

w2 = w2(0,0) ,

(1,0) σ1,1 = σ1,1 x2 ,

(0,0) σ1,2 = σ1,2 g2 ,

σ1,3 = σ2,3 = σ3,3 = 0 ,

w3 = 0 ,

(1,0) σ2,2 = σ2,2 x2 g2

with g2 = 1−x22 . It is possible to show that only plane motions of the beam are described by this approximation. This model allows for not only the bending of the center line but also the shear deformations ε1,2 (x1 , 0, 0) ≠ 0. The system resulting from equations (8.6) and (8.10) is reduced to one biquadratic equation with respect to the lateral displacements of the center line v = w2(0,0) (x1 ): d4 v d2 v + (1 + ν)ω2 ((3 − ν) 2 − b3 v) = 0 , 4 dx1 dx1

b3 = (1 − ν)(3 − 2(1 + ν)ω2 ) .

(8.15)

(1,0) (0,0) (1,0) The unknowns σ1,1 , σ1,2 , σ2,2 , and w1(1,0) are uniquely expressed through this function. For example, the other displacement function is given by

u = w1(1,0) =

1 d3 v dv (2 3 − c3 ) b3 dx1 dx1

with c3 = 3(1 − ν) + 4(1 + ν)ω2 .

It can be shown that the characteristic equation related with equation (8.15) always has two pure imaginary roots (complex conjugate) for all ω. There is a critical

178 | 8 Spatial vibrations of elastic beams with convex cross-sections frequency ω(5) 1 =√

5 4(1 + ν)

(5) separating two zones ω ∈ (0, ω(5) 1 ) and ω ∈ (ω1 , +∞). In the first zone, the two remaining roots are real whereas the second zone contains two imaginary roots. Consequently, the general solution of equation (8.15) is

v = c1 sin(κ1 x1 ) + c2 cos(κ1 x1 ) + c3 sinh(δ2 x1 ) + c4 cosh(δ2 x1 )

(8.16)

for the first frequency zone and v = c1 sin(κ1 x1 ) + c2 cos(κ1 x1 ) + c3 sin(κ2 x1 ) + c4 cos(κ2 x1 )

(8.17)

for the second zone, where κ1 (ω), κ1 (ω), δ2 (ω) are unknown real numbers and cj for j = 1, 2, 3, 4 are arbitrary constants. The eigenvalues ω are found as non-trivial solutions, either in the form (8.16) or (8.17), after resolving the boundary conditions (8.4) with respect to the unknowns cj . The relation between the roots of the characteristic equation and the frequency is shown in Figure 8.7 at ν = 0.3. The solid curves reflect the frequency versus the imaginary roots, whereas the dashed curve is for the real ones. The dash-dot line corresponds to the critical value ω(5) 1 ≃ 0.981. The dash-dot curve relates to the classical solution of the eigenvalue problem for bending vibrations of the beam with the same geometrical and mechanical parameters. A boundary value problem for the chosen approximation (N = 2) is completed by conditions: u(x1 ) = v(x1 ) = 0 ,

x1 = ±10 .

(8.18)

Figure 8.7: Wave-frequency characteristics of lateral vibrations.

8.1 Natural motions of a cuboid beam

| 179

The numerical values of the 20 lowest eigenfrequencies ω(n) for lateral vibrations of the beam clamped at both ends (a = 10) are shown in Table 8.3. The first 12 frequencies correspond to the solution of equation (8.16) (the first zone), and the rest group is obtained via the relation (8.17) (the second zone). All these values are marked with light dots in Figure 8.7. It can be proved that the number of eigenfrequencies corresponding to the solution (8.16) is limited and increases with the parameter a. Table 8.3: Eigenfrequencies of lateral vibrations. n ω

1 0.032

2 0.082

3 0.15

4 0.23

5 0.31

6 0.40

7 0.50

8 0.59

9 0.69

10 0.79

n ω

11 0.89

12 0.99

13 1.09

14 1.09

15 1.14

16 1.19

17 1.22

18 1.29

19 1.32

20 1.39

The difference between these frequencies and calculated in frame of the Euler– Bernoulli model is significant and reaches 3.7% even for the first mode. For the second mode, this difference increases up to 11%, and up to almost 22% for the third. Note that the difference between the lower natural frequencies obtained on the basis of integrodifferential and classical approaches decreases if the parameter a increases. Figure 8.8 presents the eigenforms of beam motions in the plane x3 = 0 for different values of ω(n) at ν = 0.3. The maximal displacements of the beam outer points are shown by the solid curves, the amplitudes of the displacements of points belong to the beam central axis (x2 = x3 = 0) are depicted by dashed-dot curves, and the dashed lines indicate the position of points at x3 = 0 for some beam cross-sections (x1 = 0, ±2, ±4, ±6, ±8). The first vibration mode n = 1 is shown at the top of Figure 8.8. It is interesting to note that, unlike the Euler–Bernoulli model, the slope angle of the midline at the clamped ends of the beam is not rigorously equal to π/2. This can be explained by shear deformations occurring during the beam vibrations.

Figure 8.8: Eigenforms of lateral vibrations.

180 | 8 Spatial vibrations of elastic beams with convex cross-sections The shear deformation takes a growing influence on the shape of the natural vibrations while the number n increases. It is seen that Bernoulli hypothesis about the orthogonality of the beam cross section with respect to its midline is not relevant to the analysis of beam motions for the last frequency (n = 12). This fact becomes even more evident for the first mode in the second zone (the bottom part of Figure 8.8, n = 13). It is noticeable that all cross sections of the beam turn to one side. Therefore, such motions can be called as shear vibrations.

8.2 Natural vibrations of beams with triangular cross sections 8.2.1 Dynamics of an elastic triangular prism Consider a long rectilinear prismatic body (beam) with a triangular cross section as shown in Figure 8.9. The origin of the Cartesian coordinate system is placed at the barycenter of one prism base. The axis Ox1 is directed to the other base along the beam length. Hence, the axes Ox2 and Ox3 are parallel to the body cross sections. It is assumed that the beam is made of homogeneous isotropic material with the volume density ρ, Young’s modulus E, and Poisson’s ratio ν.

Figure 8.9: Prismatic beam with a triangular cross section.

The harmonic vibrations of the beam are described by the following PDE system: ε − C−1 ∶ σ = 0 ,

∇ ⋅ σ + ρω2 w = 0 .

(8.19)

Here, C is the elastic modulus tensor, σ is the stress tensor with the components σi,j for i, j = 1, 2, 3. The components of the displacement vector w are denoted as wi for i = 1, 2, 3, ω is the frequency of natural vibrations. The Cauchy strain tensor ε has the components: 1 𝜕w 𝜕wj εi,j = ( i + ). (8.20) 2 𝜕xj 𝜕xi The boundary constraints can be divided into two groups. The conditions on the lateral sides of the beam are attributed to the first part. In the frame of the projection approach discussed in the previous chapters, these relations have to be satisfied

8.2 Natural vibrations of beams with triangular cross sections | 181

before constituting an approximating ODE system. In contrast, the second group consisting of the conditions on the prism bases is implemented together with the system of ODEs. Let us represent first the boundary conditions which are referred to as the first group. To be more particular, only a free beam is studied here. This means that no displacements are determined on the prism faces. The boundary conditions in the stresses defined on the lateral sides of the beam can be divided, in turn, into two subgroups due to the fact that these sides are parallel to the axis Ox1 . The equations which relate to the shear stresses σ1,2 and σ1,3 have the form: (i) σ1,2 n(i) 2 + σ1,3 n3 = 0 .

(8.21)

The other relations combine the components σ2,2 , σ3,3 , and σ2,3 : (i) σ2,2 n(i) 2 + σ2,3 n3 = 0

(i) and σ2,3 n(i) 2 + σ3,3 n3 = 0

(8.22)

for the prism faces Γi with i = 1, 2, 3. Here, n(i) j are the components of the normal vectors n(i) to Γi . The following boundary conditions are given on the bases of the prism at x1 = 0 and x1 = l (the second group): σ1,1 (0, x2 , x3 ) = σ1,2 (0, x2 , x3 ) = σ1,3 (0, x2 , x3 ) = 0 , σ1,1 (L, x2 , x3 ) = σ1,2 (L, x2 , x3 ) = σ1,3 (L, x2 , x3 ) = 0 ,

(8.23)

where L is the length of the beam. 8.2.2 Semi-discretization of displacement and stress fields In accordance with the semi-discretization method, the unknown trial functions w(x) and σ(x) are found as complete polynomials with respect to the beam lateral coordinates x2 and x3 . Approximations of the displacement and stress fields are taken in the form w1 = w3 =

∑

w1 (x1 )bi,j,k ,

w2 =

∑

w3 (x1 )bi,j,k ,

σ1,1 =

∑

(i,j) σ1,2 (x1 )bi,j,k

,

σ1,3 =

∑

σ2,2 (x1 )bi,j,k ,

σ3,3 =

∑

σ2,3 (x1 )bi,j,k

i+j+k=N1 i+j+k=N3

σ1,2 = σ2,2 = σ2,3 =

(i,j)

i+j+k=N5 i+j+k=N7 i+j+k=N9

(i,j)

(i,j)

(i,j)

∑

i+j+k=N2

w2 (x1 )bi,j,k , (i,j)

∑

σ1,1 (x1 )bi,j,k ,

∑

σ1,3 (x1 )bi,j,k ,

∑

σ3,3 (x1 )bi,j,k ,

i+j+k=N4

i+j+k=N6 i+j+k=N8

(i,j)

(i,j)

(i,j)

(8.24)

182 | 8 Spatial vibrations of elastic beams with convex cross-sections with the bivariate Bésier–Bernstein polynomials bi,j,k =

(i + j + k)! i j b1 (x2 , x3 )b2 (x2 , x3 )bk3 (x2 , x3 ) . i!j!k!

The choice of the numbers Ni , i = 1, … , 9, in equation (8.24) is discussed in Subsection 8.2.3. The following system of linear basis functions depending on the coordinates x2 and x3 of the triangular cross section is introduced in equation (8.24) as x − x2,3 x3,2 − x3,3 (x2 − x2,2 ) − 2,2 (x3 − x3,2 ) , d d x − x2,1 x − x3,1 (x2 − x2,3 ) − 2,3 (x3 − x3,3 ) , b2 = 3,3 d d x − x3,2 x − x2,2 b3 = 3,1 (x2 − x2,1 ) − 2,1 (x3 − x3,1 ) . d d

b1 =

(8.25)

Here, x2,i and x3,i for i = 1, 2, 3 are the coordinates of the triangle vertices and d = x2,1 x3,2 + x2,2 x3,3 + x2,3 x3,1 − x2,2 x3,1 − x2,3 x3,2 − x2,1 x3,3

(8.26)

is the doubled area of the cross section. In literature [104], the functions bi , i = 1, 2, 3, given in equation (8.25) are referred to as barycentric coordinates, but it is worth noting that they are linear functions (b-functions in what follows). The function b1 (x) is equal to zero at the triangle edge defined by the coordinates (x2,2 , x3,2 ), (x2,3 , x3,3 ) and reaches its maximum value in the triangle at the vertex with the coordinates (x2,1 , x3,1 ). The same properties are inherent to the other functions b2 and b3 with the only difference in the permutation of indices: 1 → 2, 2 → 3, 3 → 1. To integrate expressions depending on the b-functions over the triangular element, it is useful to know the following analytical formula: ∫ bi,j,k dS = S

(i + j + k)! d, (i + j + k + 2)!

(8.27)

where the area d/2 of the cross section S is defined by equation (8.26). The basic idea in solving the problem (8.20)–(8.23) is to apply the approximations (8.24) and the projection approach discussed in the previous section. These approximations must satisfy exactly the boundary conditions (8.21)–(8.23). After that, an ODE system with respect to the unknown coefficients in equation (8.24) is composed through the components 𝜕σ1,1 𝜕σ1,2 𝜕σ1,3 + + + ρω2 w1 , 𝜕x1 𝜕x2 𝜕x3 𝜕σ 𝜕σ 𝜕σ r2 = 1,2 + 2,2 + 2,3 + ρω2 w2 , 𝜕x1 𝜕x2 𝜕x3 𝜕σ1,3 𝜕σ2,3 𝜕σ3,3 r3 = + + + ρω2 w3 𝜕x1 𝜕x2 𝜕x3 r1 =

(8.28)

8.2 Natural vibrations of beams with triangular cross sections | 183

of the equilibrium vector r and the components 𝜕w1 σ1,1 ν − + (σ2,2 + σ3,3 ) , 𝜕x E E σ ν 𝜕w2 − 2,2 + (σ1,1 + σ3,3 ) , ξ2,2 = 𝜕x2 E E σ 1 𝜕w 𝜕w2 ξ1,2 = ( 1 + ) − 1,2 , 2 𝜕x2 𝜕x1 2G ξ1,1 =

σ 𝜕w3 1 𝜕w ξ2,3 = ( 2 + ) − 2,3 , 2 𝜕x3 𝜕x3 2G σ 𝜕w3 ν ξ3,3 = − 3,3 + (σ1,1 + σ2,2 ) , 𝜕x3 E E σ 𝜕w3 1 𝜕w ξ1,3 = ( 1 + ) − 1,3 2 𝜕x3 𝜕x1 2G

(8.29)

of Hooke’s tensor ξ , where the shear modulus G = E/(2 + 2ν) is introduced. By using the notation from equations (8.28) and (8.29), the constitutive relations (8.20) can be rewritten in the compact form: ξ =0

and r = 0 .

(8.30)

Here, the tensor ξ reflects Hooke’s law, i.e., the linear dependence of the Cauchy strain tensor on the stress one. In turn, the equilibrium vector r characterizes the relation between the displacements and the momentum density. The second equation in (8.30) describes the momentum balance. All the components of ξ and r become zero (either strongly or weakly) on the exact solution. If the ansatz functions (8.24) are used, then the non-zero values of these components define the quality of the corresponding approximate solution. In the proposed approach, the polynomial approximations (8.24) of degrees Nm for m = 1, … , 9 have to possess the following properties. Firstly, it is necessary that such approximations are able to satisfy exactly the boundary conditions (8.21) and (8.22). Secondly, it is important to select correctly a specific space of test functions (8.30) corresponding to each component of the constitutive vector (r1 , r2 , r3 , ξ1,1 , ξ2,2 , ξ3,3 , ξ1,2 , ξ1,3 , ξ2,3 ), which is composed of the functions (8.28) and (8.29). This choice must guarantee that the system of DAEs, which results from the corresponding projections of the components on these polynomial spaces, is consistent. Notice that the choice of polynomial approximations (8.24) designates the way to determine a subspace of test functions, for which the integral projections should be calculated. The space of complete polynomials of degree Km > 0 (m = 1, … , 9) ℙ2Km = {

∑

i+j+k=Km

ci,j,k,m bi,j,k (x2 , x3 ) ∶ ci,j,k,m ∈ ℝ}

(8.31)

with the monomials bi,j,k defined in equation (8.24) is suitable for this purpose. It is also desirable in a numerical algorithm that the structure of these DAE systems does not change with the approximation order. The rather complicated form of the constitutive and equilibrium relations (8.20) as well as the boundary conditions

184 | 8 Spatial vibrations of elastic beams with convex cross-sections (8.21), (8.22) implies that, in general case, the parameters Ni and Ki may differ from one another. After implementation of the boundary conditions (8.21) and (8.22), the equilibrium equations of (8.20) contain Ñ d independent stresses (i,j,k)

(σ1,1

(x1 ), σ1,2 (x1 ), σ1,3 (x1 )) (i,j,k)

(i,j,k)

and, in accordance with equation (8.28), their derivatives with respect to the spatial coordinate x1 . Similarly, Ñ w derivatives of (i,j,k)

(w1

(x1 ), w2

(i,j,k)

(x1 ), w3

(i,j,k)

(x1 ))

are included in the components of Hooke’s tensor (8.29). It is possible to show that the total differential order of the DAEs is equal to Ñ d + Ñ w and the following inequality Ñ d ≠ Ñ w is valid. It is worth noting that only the above-mentioned stress functions can be used to satisfy the boundary constraints (8.23) (2Ñ d conditions altogether). Therefore, the DAE system would be consistent only if Ñ d = Ñ w . To improve the system, a certain number of the stress or displacement functions should be eliminate. The maximal differential order Nd of a desirable system is chosen according to Nd = 2 min{Ñ d , Ñ w } . This condition brings some complexity in the composing of such a system. Nevertheless, these difficulties can be eliminated, as it is shown below, by choosing appropriate displacements and stresses as well as corresponding projections of the constitutive relations.

8.2.3 Integral projections for the triangular cross section The choice of trial and test spaces is no unique procedure as shown, for example, in [85]. Let us restrict ourselves to the case when the test spaces ℙ2Km in equation (8.31) are identical to each other, that is, Km = N0 for all m = 1, … , 9. Here, N0 is a positive integer. This number is simultaneously the degree of approximations of σ1,1 in equation (8.24) (N4 = N0 ). Due to the fact that the components of displacements are not subject to any boundary conditions on the prism sides, it is suitable to define the related integers as N1 = N2 = N3 = N0 . Note that the projection of the components ξ1,1 , ξ1,2 , ξ1,3 of equation (8.29) (Hooke’s tensor) on the space ℙ2N0 gives the following ODEs of the first order:

8.2 Natural vibrations of beams with triangular cross sections | 185

∫ ξ1,1 bi,j,k dS = 0 { { { S { { { { { ∫ ξ1,2 bi,j,k dS = 0 { { S { { { { { { ∫ ξ b dS = 0 { S 1,3 i,j,k

for i + j + k = N0

with respect to all the displacement functions w1 mension of the system (8.32) is

(i,j,k)

(x1 ), w2

(i,j,k)

(8.32)

(x1 ), w3

(i,j,k)

(x1 ). The di-

Nd 3 = (N0 + 1)(N0 + 2) . 2 2 Thus, the final number of independent stress functions selected from the whole set (i,j,k)

{σ1,1

(x) , σ1,2 (x) , σ1,3 (x)} (ijk)

(ijk)

in equation (8.24) has to be equal to Nd /2 as well. The other projections of Hooke’s law ∫ ξ2,2 bi,j,k dS = 0 { { { S { { { { { ∫ ξ3,3 bi,j,k dS = 0 { { S { { { { { { ∫ ξ b dS = 0 { S 2,3 i,j,k

for i + j + k = N0

(8.33)

define Nd /2 algebraic equations with respect to stress and displacement functions. In the second step, the boundary conditions (8.21) on the lateral faces of the beam are satisfied. To make this and build the necessary number of differential equations, let us introduce the following integers: N5 = N6 = N7 = N8 = N9 = N0 + 2 . This means that it is necessary to fulfill N0 + 3 boundary conditions on each beam side (linear relations at each monomial), or 3(N0 + 3) as a whole, with respect to the (i,j,k) (i,j,k) stress functions σ1,2 (x1 ), σ1,3 (x). Consequently, the approximations of σ1,2 and σ1,3 contain only Ñ τ = (N0 + 3)(N0 + 4) − 3(N0 + 3)

(8.34)

independent coefficients σ1,2 (x1 ), σ1,3 (x1 ) after implementation of these equations. By introducing a new notation τ̃ m (x) with m = 1, … , Ñ τ for the remaining coeffi(ijk) (i,j,k) cients σ1,2 (x1 ) and σ1,3 (x1 ), the approximation of the shear stresses satisfying the boundary conditions (8.21) can be presented as (i,j,k)

[

(i,j,k)

(m) Ñ τ ϑ̃ 1,2 (x2 , x3 ) σ1,2 ]. ] = ∑ τ̃ m (x1 ) [ (m) ̃ (x , x ) σ1,3 ϑ m=0 [ 1,3 2 3 ]

(8.35)

186 | 8 Spatial vibrations of elastic beams with convex cross-sections (m) (m) Here, ϑ̃ 1,2 (x2 , x3 ) and ϑ̃ 1,3 (x2 , x3 ) are basis functions obtained in agreement with the boundary conditions (8.21). After resolving the relations in equation (8.22), the approximations of σ2,2 , σ3,3 , and σ2,3 contain

N 3 Nσ = (N0 + 3)(N0 + 4) − 3(2N0 + 5) = d 2 2 independent coefficients σ2,2 (x1 ), σ3,3 (x1 ), σ2,3 (x1 ). Renumbering these coefficients can provide the relevant stress approximation in the form: (i,j,k)

(i,j,k)

(i,j,k)

(m) ϑ2,2 (x2 , x3 ) σ2,2 Nσ ] [ (m) [ ] [ σ3,3 ] = ∑ σm (x1 ) [ ]. [ ϑ3,3 (x2 , x3 ) ] m=0 (m) σ [ 2,3 ] [ ϑ (x , x ) ] 2,3

2

(8.36)

3

(m) (m) (m) Here, ϑ2,2 (x2 , x3 ), ϑ3,3 (x2 , x3 ), and ϑ2,3 (x2 , x3 ) are new basis functions consistent with the boundary conditions (8.22). So, the number Nσ of the functions σm (x1 ) is enough to exactly solve the system (8.33) with respect to these coefficients. Approximation of the equilibrium equations implies the vanishing of the following complete projections of the vector components r1 , r2 , r3 in equation (8.28) according to

∫ r1 bi,j,k dS = 0 { { { S { { { { { ∫ r2 bi,j,k dS = 0 for i + j + k = N0 . { { S { { { { { { ∫ r b dS = 0 { S 3 i,j,k

(8.37)

It can be seen that two last relations of equation (8.37) define Nτ = Nd /3 differential equations, which only include the derivatives of the functions σ1,2 and σ1,3 . According to equation (8.34), the number of the available variables Ñ τ is bigger than Nτ and this difference is equal to Ñ τ − Nτ = N0 + 2 . To reduce the number of the variables τ̃ m (x1 ), m = 1, … Ñ τ , the corresponding approximation of equation (8.35) is transformed in the following way. First, the complete projections of the functions σ1,2 and σ1,3 on the space ℙ2N0 are calculated as ∫ σ1,2 bi,j,k dS = 0 S

and

∫ σ1,3 bi,j,k dS = 0 S

for i + k + l = N0 .

(8.38)

After that, the system (8.38) is resolved with respect to Nτ coefficients τ̃ m (x1 ) selected arbitrarily.

8.2 Natural vibrations of beams with triangular cross sections | 187

In assembling a system of consistent ODEs, it is necessary to solve underdetermined systems of algebraic equations over τ̃ m (x). If the calculations are performed analytically, then the choice of variables, for which the equations (8.38) are resolved, is not so essential. But in numerical computations, a special approach should be applied, e.g., based on the Gauss elimination method to diminish computational errors. At the beginning of this successive process, the equation is chosen that contains the coefficient of maximum absolute value. After that, the variable at the maximum coefficient is resolved from this equation. The procedure accompanied with an appropriate transformation is repeated Nτ − 1 times. After solving equation (8.38) and substituting the result into equation (8.35), the following expression is obtained: [

O σ1,2

O σ1,3

Ñ τ

] = ∑ τm (x1 ) [ m=Nτ +1

(m) ϑ1,2 (x2 , x3 ) (m) ϑ1,3 (x2 , x3 )

].

(8.39)

Here, τm (x1 ) for m = Nτ + 1, … , Ñ τ are new coefficients, which are linear combinations (m) (m) of τ̃ n (x1 ) for n = 1, … , Ñ τ , whereas ϑ1,2 (x2 , x3 ) and ϑ1,3 (x2 , x3 ) are new basis functions 2 orthogonal to the polynomial space ℙN0 . Let us find a representation of σ1,2 and σ1,3 equivalent to equation (8.38) through a (m) (m) new basis including the obtained functions ϑ1,2 (y, z) and ϑ1,3 (x2 , x3 ). For this purpose, compose the following system of equations: (m) (m) (x2 , x3 )) dS = 0 (x2 , x3 ) + σ1,3 ϑ1,3 ∫ (σ1,2 ϑ1,2 S

(8.40)

for m = Nτ + 1, … , Ñ τ and resolve it with respect to some coefficients τ̃ i (x1 ) by the Gauss elimination method. After substituting the solution of equation (8.40) into equation (8.35) and collecting similar terms, the following expression is obtained: [

P σ1,2

P σ1,3

Nτ

] = ∑ τm (x1 ) [ m=1

(m) ϑ1,2 (x2 , x3 ) (m) ϑ1,3 (x2 , x3 )

].

Here, similar to equation (8.39), τm (x1 ) for m = 1, … , Nτ are new coefficients, whereas (m) (m) O ϑ1,2 (x2 , x3 ) and ϑ1,3 (x2 , x3 ) are new basis components, which are orthogonal to σ1,2 O and σ1,3 . We are able to verify that these approximations [

P O σ1,2 σ1,2 σ1,2 ]=[ P ]+[ O ] σ1,3 σ1,3 σ1,3

satisfy the boundary conditions (8.21). Thus, the final consistent DAE system includes equations (8.32), (8.33), and (8.37) (i,j,k) (i,j,k) (i,j,k) with variables σx(i,j,k) , w1 , w2 , w3 , τm , σm . The systems of differential equations (8.32) and (8.37) can be resolved with respect to the first derivatives of the corre(i,j,k) (i,j,k) (i,j,k) (i,j,k) sponding variables σ1,1 , w1 , w2 , w3 , τm . It is necessary to do so by taking into account the solution of the algebraic system (8.33) relative to the variables σm .

188 | 8 Spatial vibrations of elastic beams with convex cross-sections At that, all the differential variables σ1,1

(i,j,k)

(x1 ) , w1

τm (x1 )

(i,j,k)

(x1 ) , w2

(i,j,k)

(x1 ) , w3

(i,j,k)

for i + j + k = N0

(x1 )

and

for m = 1, … , Nτ

can be collected into a vector y(x1 ) ∈ ℝNd of design parameters with the dimension Nd = 3(N0 + 1)(N0 + 2) . After assembling the differential equations, the resulting ODE system can be rewritten in the vector form dy + Ky = 0 , (8.41) dx1 where K ∈ ℝNd ×Nd is a square matrix. In this case, the characteristic equation takes as follows: det(K(ω) + λI) = 0

(8.42)

with the unit matrix I. Equation (8.42) does not contain the zero root λ(ω) = 0 at ω ≠ 0. In other words, the general solution of the eigenvalue problem is a linear combination of only exponentials. (i,j,k) (i,j,k) (i,j,k) Let us assemble a vector y 1 of the design parameters σ1,1 (x1 ), w2 (x1 ), w3 (x1 ) (i,j,k) and another vector y 2 of w1 (x1 ), τm (x1 ). The vector y can be rearranged in such a way that y = y 1 ⊕ y 2 . The components of the derivative y ′2 (x1 ) are included in the first relations of equation (8.32) and in the two last relations of equation (8.37) that include the components of the vector y 1 in accordance with equation (8.28), (8.29), (8.36). Vice versa, the components of y ′1 (x1 ) are included in the first relations of equation (8.37) and the two last relations of equation (8.32) that depend on the components of the vector y 2 (see equations (8.28) and (8.29)). This means that the vector equation (8.41) has the following form: 0 d y1 [ ]+[ K2,1 dx1 y 2

K1,2 y ][ 1 ] = 0 , 0 y2

K1,2 , K2,1 ∈ ℝ

Nd 2

×

Nd 2

.

(8.43)

By using the specific structure of the matrix K, it is always possible to reduce the ODE system of first order (8.41) to an equivalent system of Nd /2 differential equations including the purely second derivatives of stress and displacement functions. After identical transformations, the ODEs (8.43) can be presented as d2 y 1 − K1,2 K2,1 y 1 = 0 . dx12

(8.44)

Hence, a new eigenvalue μ = λ2 is introduced to effectively solve the boundary value problem (8.44), (8.23). The characteristic equation for equation (8.44) det(K1,2 (ω)K2,1 (ω) + μI) = 0

8.2 Natural vibrations of beams with triangular cross sections | 189

with respect to μ has the polynomial order twice less than equation (8.42). By exploiting the symmetry properties of the boundary value problem relatively to the cross section x = L/2 and the form of the general solution of equation (8.44) (the eigenvalues λ are always paired), the original problem can be decomposed onto two subproblems (symmetric and antisymmetric) with the dimensions twice less than the total one. The natural frequencies ωi for i = 1, … , Nd as well as the corresponding eigenforms of stresses and displacements are found under condition that the determinant of a boundary algebraic system is equal to zero. This system is obtained after substituting the general solution of the system (8.41) into the boundary conditions (8.23).

8.2.4 Natural vibrations of a beam with the isosceles cross section Consider the problem on free natural vibrations of the rectilinear beam shown in Figure 8.9 with the cross section in the form of an isosceles triangle. Let the base b be parallel to the y-axis and the height h be oriented along the axis Ox3 . The homogeneous isotropic elastic material is described by three-dimensional constants: Young’s modulus, Poisson’s ratio, and the volume density. By using the Buckingham π theorem [15], it is possible to redefine three-dimensional parameters for linear elasticity problems. Thus, Young’s modulus E = 1, the volume density ρ = 1, and the base of the isosceles triangle b = 1 are taken. The other system parameters are not chosen arbitrarily. Poisson’s ration ν = 0.3 given in the study is typical for many structural materials. The dimensionless height of the beam cross section h = 1 and the beam length L are chosen to underline that the considered elastic body has the shape of a thin beam. Due to the symmetry of the cross section relatively to the axis Ox3 , the governing ODE system can be decomposed into two independent subsystems. At that, one of the subsystems describes coupled bending–torsional (bt) and torsional–bending (tb) motions of the beam. This system involves, for example, only even functions σ1,1 of the variable x3 . The other subsystem describes the bending–longitudinal (bl) and longitudinal–bending (lb) beam vibrations. Only odd functions σ1,1 of equation (8.24) fit to this subsystem. The coupling of bending with either tension or torsion is caused by an asymmetry of the beam cross section with respect to the axis Ox2 . In this case, natural vibrations cannot be separated into four independent types of longitudinal, bending, and torsional motions as is supposed for beams with symmetric cross sections [55]. Nevertheless, only one type of displacement and stress fields makes the largest contribution in the corresponding amplitudes of vibrations. This is a reason to introduce the classification of eigenfrequencies and attendant eigenforms with two letters abbreviating corresponding fields. The first letter denotes the dominant type of motion. To obtain a reliable numerical solution, a sufficiently high degree N0 of polynomial projections should be exploited. In accordance with the MIDR, the constitutive functionals proposed in [56] can be applied to estimate the quality of the solution.

190 | 8 Spatial vibrations of elastic beams with convex cross-sections The convergence of six real eigenvalues μn ∈ ℝ, n = 1, … , 6, are alternatively analyzed to define a necessary approximation dimension. Only these eigenvalues of the ODE system (8.44) tend to zero if the frequency ω vanishes and mainly determine the beam eigenforms for lower frequencies. If ω ≪ 1, then the others μn , n = 7, … , Nd /2, govern only transient processes near the boundary at x1 = 0 and x1 = L (Saint-Venant’s effects). For example, the first six eigenvalues μn versus the polynomial degree N0 are shown in Figure 8.10 for ω = 1. The pair of eigenvalues μ1 < 0 and μ2 > 0 presented by the lines with squares determines the bending-longitudinal forms described above. The negative values μ3 < 0 marked by circles are for the longitudinal-bending motions. The pair of lines passing through the extremal values μ4 < 0 and μ5 > 0 marked by triangles corresponds to the bending-torsional vibrations. The coupled torsionalbending motions are related with the last eigenvalue μ6 < 0 marked by diamonds. All the values μn converge rather fast. At N0 = 4, the maximum relative error |

μn (N0 ) − μn (N0 − 1) | = 0.02 μn (N0 − 1)

is found for the torsional-bending form. This approximation order will be used in what follows to solve the eigenvalue problem of beam vibrations.

Figure 8.10: Six basic eigenvalues μn vs. approximation order N0 .

An important attribute of elastic structure dynamics is the wave-frequency characteristics of a system. The dependence of the eigenvalues μn , n = 1, … , 6, on the frequency ω is represented in Figures 8.11 and 8.12. As it has been mentioned above, the six basic values μn (ω) depicted by curves 1–3 start at the coordinate origin. When the frequency ω tends to zero, the asymptotic behavior of these curves agrees with the wave–frequency characteristics for longitudinal, torsional, and lateral vibrations derived in the frame of the Euler–Bernoulli beam model. As the quantity μ corresponds to the square of the wave number λ, the function μ3 (ω) related with the beam compression and tension congruences to the classical one μ3 = −ω2 + O(ω3 )

8.2 Natural vibrations of beams with triangular cross sections | 191

(curves 2 in Figure 8.11). The similar characteristic for torsion μ6 = −4.416ω2 + O(ω3 ) (curves 2 in Figure 8.12) is also in a good correlation with the conventional model [56].

Figure 8.11: Real parts of eigenvalues μ vs. frequency ω for longitudinal–bending vibrations.

Figure 8.12: Real parts of eigenvalues μ vs. frequency ω for torsional–bending vibrations.

For the Euler–Bernoulli beam bending around every principal axis of inertia, two real and two imaginary eigenvalues λn , n = 1, 2, 3, 4, are equal to each other in absolute magnitude and depend on the frequency ω as a square root. This characteristic holds asymptotically for the 3D beam model under study (curves 1 and 3 in Figures 8.11 and 8.12) so that ω ω μ1 = − + O(ω2 ) , μ2 = + O(ω2 ) , √Jy √Jy ω ω μ4 = − + O(ω2 ) , μ5 = + O(ω2 ) , (8.45) √Jz √Jz Jy = ∫ x32 dS = S

1 , 36

Jz = ∫ x22 dS = S

1 . 48

Here, Jy and Jz are the moments of inertia, respectively, about the axes x2 and x3 . In contrast to the conventional model, these bending–longitudinal and bending– torsional functions are convex. The negative eigenvalues μ1 (ω) and μ4 (ω) are strictly decreasing, whereas the positive ones μ2 (ω) and μ5 (ω) attain their maximal values at ω ≈ 1.104 and ω ≈ 1.232, respectively.

192 | 8 Spatial vibrations of elastic beams with convex cross-sections The function μ2 (ω) changes its sign at the critical frequency ω(1) l ≈ 1.233. It was found out [85] that the beam eigenforms change dramatically when passing through such critical points. At the next singular frequency ω(2) l ≈ 2.763, one complex conjugate pair of longitudinal–bending eigenvalues (their real part is shown in Figure 8.11 by curve 4) turns into two positive real values, one of which (curve 6) is strictly increasing, whereas the other (curve 5) decreases, changes its sign at the frequency ω(3) l ≈ 2.794, and quickly gets flatter after passing closely over the eigenvalue μ2 (ω) (curve 3). For the bending–torsional vibrations, the inter-reaction of the corresponding eigenvalues is more sophisticated. A positive eigenvalue (curve 4 in Figure 8.12) meets the value μ5 (ω) (curve 3) at the singular point ω(1) t ≈ 2.255. The newly formed func(2) (2) tions (curve 5) keep complex conjugate values for ω ∈ (ω(1) t , ωt ) with ωt ≈ 2.285 (2) where they turn into real and then diverge from each other for ω ≥ ωt . One of these (3) curves (6) crosses the ω-axis at ω(3) t ≈ 2.306; the other (7) do so at ωt ≈ 2.555. One more real eigenvalue (curve 8) appears in the chosen wave-frequency domain. The next step of the proposed algorithm is in resolving the homogeneous boundary constraints (8.23) on the basis of the general solution obtained for the system (8.41). In accordance with the dimension of the stress functions σ1,1 (x), σ1,2 (x), σ1,3 (x) of the cross-sectional coordinates x2 and x3 , this solution depends on Nd unknown coefficients, which are determined with Nd /2 conditions at the beam end x1 = 0 and the same number at the other end x1 = L. The linear algebraic system resulting from these conditions has a non-trivial solution if it is degenerate or, in other words, the determinant of the corresponding system matrix equals to zero. Another important characteristic can be defined by finding all the beam lengths l(ω) at which such a degeneration takes place for some fixed frequency. As seen in Figure 8.13 for the bending–longitudinal as well as longitudinal–bending vibrations, the function l(ω) generates two sets of curves (solid and dashed, respectively) on the plane (ω, l) ∈ ℝ2 . The curves of one set do not intersect with the other; the points of intersection correspond to the case of multiple determinant roots and require special attention. Only one such point (ω, l) = (0.233, 13.38) is presented in Figure 8.13. All the curves asymptotically converge to the axes ω = 0 and l = 0. Much of the same picture appears for the bending–torsional and torsional–bending subsystem (solid and dashed curves

Figure 8.13: Beam length l vs. frequency ω for longitudinal–bending vibrations.

8.2 Natural vibrations of beams with triangular cross sections | 193

Figure 8.14: Beam length l vs. frequency ω for torsional– bending vibrations.

in Figure 8.14, respectively). In the depicted frequency-space domain, as much as two multiple roots have a place at (ω, l) = (0.194, 7.56) and (ω, l) = (0.244, 12.26). By fixing the beam length (as an example, l = L = 10) in the Figures 8.13 and 8.14, the whole frequency spectrum of the elastic beam in the chosen range of ω < 0.4 can be restored. The numerical values of the natural frequencies for corresponding eigenmodes obtained in accordance with the proposed projection algorithms at N0 = 4 are given in Table 8.4. Table 8.4: Eigenfrequencies for the beam with the isosceles cross section for ω < 0.4. i

2

3

0.0448

0.1193

0.2235

0.3497

0.1496

0.2992

−

−

ω(i) bl

0.0515

0.1359

0.2521

0.3909

ω(i) lb

0.3140

−

−

−

ω(i) bt ω(i) tb

1

4

The first mode of bending–longitudinal vibrations corresponding to the frequency ω(1) bl is shown in Figure 8.15. In contrast to the form of bending vibrations according to the Euler–Bernoulli model, the bending–longitudinal motions are characterized not only by transverse displacement w0 (x1 ) but also the longitudinal displacements u0 (x1 ). Here, w0 and u0 are the following integral parameters: w0 (x1 ) =

1 ∫ w (x) dS S S 3

and u0 (x1 ) =

1 ∫ w (x) dS . S S 1

The amplitudes w0 (x1 ) and u0 (x1 ) calculated at N0 = 4 can be compared by the ratio: β1 =

maxx1 ∈[0,l] |w0 (x1 )| maxx1 ∈[0,l] |u0 (x1 )|

= 68 350 .

At that, the deflections w0 are strongly dominant over u0 .

(8.46)

194 | 8 Spatial vibrations of elastic beams with convex cross-sections

Figure 8.15: Bending–longitudinal eigenforms: lateral (w0 (x1 ), solid curve) and longitudinal (u0 (x1 ), dashed curve) displacements for the frequency ω(1) bl .

Figure 8.16: Longitudinal–bending eigenforms: lateral (w0 (x1 ), solid curve) and longitudinal (u0 (x1 ), dashed curve) displacements for the frequency ω(1) lb .

The first longitudinal–bending mode of natural vibrations with the frequency ω(1) lb is shown in Figure 8.16. This form includes not only the component of the longitudinal displacements u0 , as follows from the classical concept, but also the lateral w0 . At that, the displacements u0 are dominant over w0 . The inverse amplitude ratio β2 =

maxx1 ∈[0,l] |u0 (x1 )|

maxx1 ∈[0,l] |w0 (x1 )|

= 48 540 ,

(8.47)

(1) at ω(1) bl is quite large conversely to the longitudinal–bending vibrations at ωlb . By taking into account equation (8.47), the influence of the longitudinal displacements can be neglected in the most cases. The values β1 and β2 confirm that the relationship between longitudinal and lateral vibrations is quite weak and must be ignored under certain assumptions for both eigenforms. The first bending–torsional and torsional–bending forms of natural beam vibrations are shown in Figure 8.17 and Figure 8.18, respectively. Amplitude relations change appreciably for the bending–torsional and torsional–bending modes. To com-

8.2 Natural vibrations of beams with triangular cross sections | 195

Figure 8.17: Bending–torsional eigenforms: bending (α3 (x1 ), solid curve) and torsional (α3 (x1 ), dashed curve) angles for the frequency ω(1) bt .

Figure 8.18: Torsional–bending eigenforms: bending (α3 (x1 ), solid curve) and torsional (α1 (x1 ), dashed curve) angles for the frequency ω(1) tb .

pare the bending and torsion, the functions α3 (x1 ) =

1 ∫ w (x)x2 dS Jz S 1

and α1 (x1 ) =

1 ∫ (w (x)x3 − w3 (x)x2 ) dS 2Jx S 2

with Jx = Jy + Jz

(solid and dashed curves in both Figure 8.17 and Figure 8.18) are used. Here, α3 (x1 ) is the integral rotation of the beam cross section with respect to the axis Ox3 , α1 (x1 ) is the average angle of cross-sectional rotation around the axis Ox1 . The principal moments of inertia Jy and Jz for the isosceles cross section are introduced in equation (8.32). The function α3 (x1 ) is dominant for the bending–torsion type of natural beam motions. However, the characteristic amplitude ratio for the frequency ω(1) bt β3 =

maxx1 ∈[0,l] |α3 (x1 )| maxx1 ∈[0,l] |α1 (x1 )|

≈ 131

is not so large as in the previous two examples (see equations (8.46) and (8.47)).

196 | 8 Spatial vibrations of elastic beams with convex cross-sections The inverse amplitude ratio β4 =

maxx1 ∈[0,l] |α1 (x1 )|

maxx1 ∈[0,l] |α3 (x1 )|

≈ 115

characterizes the relation between torsion and bending beam vibrations for the eigenfrequency ω(1) bt .

9 Double minimization in optimal control problems 9.1 Optimization of beam motions with polynomials In this chapter, a numerical approach to optimal control problems in solid dynamics is presented based on the MIDR. The early discretization is applied to reduce the original constrained minimization to successively resolve two algebraic systems relating respectively with quality and cost functionals. 9.1.1 Statement of an optimal control problem Study the controlled plane bending of an elastic rectilinear beam in the frame of the Euler–Bernoulli theory. One beam end is free of loads, whereas the other is rigidly attached to a carriage, which is allowed to move horizontally. The beam in the undeflected state settles vertically along the axis x (see Figure 9.1). The lateral displacements w(t, x) and the momentum density p(t, x) at the initial time instant t = 0 are known in the coordinate system Oxz connected with the carriage. The horizontal acceleration u(t) of the carriage is considered as a control input. The origin O of the moving system coincides with the lower beam end x = 0 and transfers in an inertial system O1 xz with the velocity vc ; the position of the point O in O1 xz is zc . It is natural that zċ = vc and vċ = u. The PDE system describing the beam motions with the boundary and initial conditions has the form: ρ(x)wtt + (κ(x)wxx )xx = −ρu(t)

for (t, x) ∈ Ω = (0, T) × (0, L) ,

w(t, 0) = wx (t, 0) = wxx (t, L) = wxxx (t, L) = 0 , w(0, x) = w0 (x) and wt (0, x) = ρ p0 (x) . −1

Figure 9.1: Elastic beam attached to a movable carriage. https://doi.org/10.1515/9783110516449-009

(9.1) (9.2) (9.3)

198 | 9 Double minimization in optimal control problems Here, L denotes the beam length, ρ is the linear density, κ is the bending stiffness, w0 and p0 are rather smooth functions of the coordinate x, T is the terminal time instant. Without loss of generality, the initial coordinate and velocity of the carriage are equal to zero: zc (0) = vc (0) = 0 .

(9.4)

It should be mentioned that the boundary and initial conditions (9.2), (9.3) must be consistent. For example, the function w0 (x) takes the values w0 (0) = w0′ (0) = 0. The control problem is formulated as follows. Problem 9.1. Find such a function u(t) in the set of admissible control laws 𝒰 that minimizes the cost functional J[u] → min , u∈𝒰

(9.5)

at the fixed instant T subject the equation of motions (9.1), the boundary and initial conditions (9.2), (9.3) as well as the terminal state zc (T) = zT ,

vc (T) = vT .

(9.6)

9.1.2 Discretization based on the MIDR Similarly, as in Section 4.1, the IBVP (9.1)–(9.3) is equivalently rewritten in the term of the displacements w, the momentum density p, and the bending moments s according to v(t, x) = 0 and q(t, x) = 0 with v ∶= wt − ρ−1 p and q ∶= wxx − κ−1 s ,

(9.7)

pt + sxx = −ρu(t)

(9.8)

for (t, x) ∈ Ω ,

w(t, 0) = wx (t, 0) = s(t, L) = sx (t, L) = 0 , w(0, x) = w0 (x) and

p(0, x) = p0 (x) .

(9.9) (9.10)

To solve Problem 9.1, the approach described in Section 4.1 is applied. In agreement with the MIDR, the local constitutive relations (9.7) are replaced by the integral equality: Φ = ∫ φ(t, x) dΩ = 0 Ω

1 with φ ∶= (ρv2 + κq2 ) . 2

(9.11)

Let us remind that the proposed integrodifferential statement (9.8)–(9.11) is reduced to the variational one: find such functions w∗ (t, x), p∗ (t, x), s∗ (t, x) that min-

9.1 Optimization of beam motions with polynomials | 199

imize the quality functional Φ[w∗ , p∗ , s∗ ] = min Φ[w, p, s] = 0 w, p, s

(9.12)

subject to the constraints (9.8)–(9.10). For numerical minimization of Φ, the positive integer numbers Mw and Mp are taken, and approximations of the optimal solution w∗ , p∗ , s∗ are found in the polynomial form: Mw

̃ x) = w0 (x) + tx2 ∑ wi a(i) w(t, w (t, x) , Mp

i=1

(9.13)

̃ x) = p0 (x) + t ∑ pi a(i) p(t, p (t, x) , i=1

̃ x) = − s(t,

L L u(t)(L − x)2 − ∫ ∫ pt (t, x1 ) dx1 dx2 , 2 x x2

(i) where wi , pi are unknown real coefficients. The basis functions a(i) w (t, x), ap (t, x) have to satisfy boundary conditions (9.9) and to be equal to zero at t = 0. Given the control signal u(t), the approximate solution w̃ ∗ (t, x), p̃ ∗ (t, x), s̃∗ (t, x) ̃ = Φ[w,̃ p,̃ s]̃ expressed in compliance can be found by minimizing the functional Φ with equation (9.11) through the parameters:

wi

for i = 1, … , Mw

and pi

for i = 1, … , Mp .

9.1.3 Parametric optimization of the beam motions The MIDR applied to the IBVP (9.1)–(9.6) provides various tools of numerical optimization, one of which is described below. At the first step, a specific set of approximations (9.13) are chosen, for example, polynomial, spline, etc. After that, a finite space of admissible control laws 𝒰 is defined: N+2

𝒰 = {u ∶ u(t) = ∑ ui a(i) u (t)} . i=1

(9.14)

It is necessary to note that the control u ∈ 𝒰 allows for the terminal conditions (9.6). Then the approximations w,̃ p,̃ s̃ from equation (9.13) are substituted into the quality functional Φ introduced in equation (9.11). As a result, the unknown coefficients wi , pi are all assembled into a vector of design parameters y ∈ ℝM , M = Mw + Mp , whereas the control parameters ui unresolved after implementation of the constraints (9.6) are combined in a vector u ∈ ℝN . Since the functional Φ is quadratic over y and u, the minimization (9.12) reduce to a linear

200 | 9 Double minimization in optimal control problems system of algebraic equations 𝜕Φ(y, u) = Ay − Bu − c = 0 𝜕y

(9.15)

with respect to the vector y. Here, A ∈ ℝM×M and B ∈ ℝM×N are given matrices, and the system vector c ∈ ℝM is derived from the initial and terminal conditions (9.5), (9.10). Based on the solution y ∗ = A−1 (Bu + c) of the system (9.15), the approximations w̃ ∗ (t, x, u) ,

p̃ ∗ (t, x, u) ,

s̃∗ (t, x, u)

are formed depending on the undefined control parameters ui . By using w̃ ∗ , p̃ ∗ , s̃∗ in the cost functional J, the minimum of the resulting function J(u) is searched in accordance with equation (9.5). If the objective function J is quadratic with respect to ui , then the optimal control problem is reduced to solving the following linear system: 𝜕J(u) = Du − e = 0 , 𝜕u where D ∈ ℝN×N and e ∈ ℝN are the matrix and vector of quadratic and linear terms, respectively. Finally, the value u∗ = D−1 e of the optimal vector is substituted into the control function u(t, u∗ ) ∈ 𝒰 from equation (9.14) and into the approximations: w̃ ∗ (t, x, u∗ ) ,

p̃ ∗ (t, x, u∗ ) ,

s̃∗ (t, x, u∗ ) .

9.1.4 Numerical examples of controlled motions One of possible optimization schemes for Problem 9.1 is based on the polynomial approximations of displacements and momentum density ̃ x) = w0 (x) + tx2 ∑ wi,j t i x j w(t, i+j≤M1

̃ x) = p0 (x) + t ∑ pi,j t i x j p(t, i+j≤M1

and

for i, j ∈ ℕ

as well as the polynomial control: N+1

u = ∑ ui t i . i=0

(9.16)

9.1 Optimization of beam motions with polynomials | 201

Example 9.1. To demonstrate the applicability of the MIDR to optimization in structural dynamics, motion of the elastic structure from the initial undisturbed state w0 (x) = p0 (x) = 0 to some terminal one (vT = 0) in the finite time T is studied. The goal of the control is to attain the minimal total energy of the system at the terminal state: J = W(T) ,

L

W(t) = ∫ ψ(t, x) dx ,

ψ(t, x) =

0

2 p2 κwxx + . 2ρ 2

(9.17)

The following dimensionless parameters have been taken in the simulation of system motions: the beam length L = 1, the bending stiffness κ = 1, the linear density ρ = 1. The lowest eigenfrequency for these constants is about ω1 ∼ 4 (see Section 6.1). To be specific, the desired carriage displacement is given as zT = 1. Asymptotic methods are advisable to be involved if the maximal period of beam natural vibrations T1 = 2πω−1 1 is much less than the control time T. In contrast, quasistatic approaches are preferable if T1 ≫ T. In this example, an intermediate case is under study with T = 4, which cannot be reliably studied and guided by the above mentioned methods. The optimization problem is solved over the polynomial control (9.16) for different numbers N = 0, … , 5. At N = 0, the linear control u = u0 + u1 t is uniquely reconstructed from the terminal conditions (9.6) with u0 = 6zT T −2

and u1 = −12zT T −3 .

This control law has been already considered in Section 5.1. The function u(t) for N > 0 includes unknown parameters that are used to minimize the cost function (9.17). With the help of the numerical algorithm described in the previous subsection, the minimum of the cost function (9.17) has been analytically found for the polynomial degrees M1 ≤ 15. At that, the total dimension of the algebraic system (9.15) is M = (M1 + 1)(M1 + 2). The results of optimal control approximation for the elastic beam attached to the carriage are presented in Figure 9.2. The solid (broken) line corresponds to the relative mechanical energy at the end of the optimal process J0 (N) =

TJ(u∗ ) Ψ(u∗ )

T

with Ψ = ∫ W(t, u∗ ) dx , 0

versus the degree N of the polynomial control. Here, W is the energy stored in the beam during the motion in accordance with equation (9.17). The relative error of discretization is given by Δ(N) =

Φ(u∗ ) , Ψ(u∗ )

202 | 9 Double minimization in optimal control problems

Figure 9.2: Relative errors J0 and Δ vs. control order N.

and this value is depicted in Figure 9.2 by a dashed line. As seen, the energy at the end of the motion decreases dramatically with increasing the number of control parameters. The calculation error at the approximation order M1 = 15 is lower than the relative terminal energy J0 but approaches to it at N = 6. The change of the discretization error (dashed line) and terminal energy (solid line) versus the approximation order M1 is shown in Figure 9.3 at the fixed dimension N = 6 of the control vector u. It can be seen that the quality of the obtained solution and control grows if the dimension M(M1 ) increases. Only for rather large orders M1 > 13, the error Δ turns out less than the normed terminal energy J0 . In this case, it is reasonable to talk about some reliable displacement, momentum, and moment fields.

Figure 9.3: Relative errors J0 and Δ vs. approximation order M1 .

It can be assumed from the above dependencies that the control error J0 exceeds the discretization one Δ when the parameter N is further increased. To avoid such a situation, it is necessary either to increase the dimension of the approximations, or to change, in a special way, the objective control function that is able to regulate the ratio of these errors. Such control strategies are discussed in Section 10.1.

9.1 Optimization of beam motions with polynomials | 203

The optimal carriage displacements zc (t) at both N = 5 and N = 6 are shown in Figure 9.4 respectively by dash-dot and dashed curves (M1 = 15). For this change of the polynomial degree, the control optimization leads to a slightly different solution. In the inertial coordinate system O1 xz (see Figure 9.1), the position of the free beam end wL (t) = zc (t) + w(t, L) is presented at N = 6 by the solid curve.

Figure 9.4: Optimal displacements of the beam ends zc (t) and wL (t).

In Figure 9.5, the elastic displacements w as functions of t and x are plotted at N = 6 and M1 = 15. The distribution of energy ψ(t, x) over space-time domain Ω is depicted in Figure 9.6. It is seen that the energy density is small enough at the end of the process due to the optimization performed.

Figure 9.5: Optimal displacements w(t, x) at N = 6.

The distribution of energy error φ defined in equation (9.11) at the same parameters of control and approximation is reflected in Figure 9.7. The maximum of φ appears at the beginning of the motion.

204 | 9 Double minimization in optimal control problems

Figure 9.6: Mechanical energy density ψ(t, x) at N = 6.

Figure 9.7: Error density φ at M1 = 15 and N = 6.

Example 9.2. An important class of control problems in elastodynamics for mechanical structures with distributed parameters is damping of initial vibrations in a finite time interval. The geometric and mechanical constants are chosen the same as in Example 9.1. For the sake of specificity, let the elastic system beam–carriage be placed initially at some static state (z(0) = v(0) = 0, p0 (x) ≡ 0). Suppose also that this static state is produced by the unit shear force applied at the beam top x = 1, that is, w0 (x) =

x3 x2 − . 6 2

The control purpose is minimization of the total mechanical energy in the system L

W0 (t) = ∫ ψ0 (t, x) dx 0

with ψ0 (t, x) =

2 (p + ρvc (t))2 κwxx + , 2ρ 2

(9.18)

W0 (0) = W(0) and W0 (T) = W(T) at the instant T = 4 by means of the polynomial law of carriage motions (9.16). The carriage must return to the initial position at the end of the process.

9.1 Optimization of beam motions with polynomials | 205

Numerical simulations show that the minimization of the cost function introduced in equation (9.18) leads to rather high amplitudes of the control signal u(t). In turn, this causes sufficiently large elastic deviations w(t, x) of the beam points from its midline during the motion. To decrease the amplitudes of u and w, the integral of vc2 (t) should be added to the cost function with some weighting coefficients. The new control problem is to minimized the following functional: J[u] → min u∈𝒰

T

with J = γ ∫ vc2 (t) dt + W(T) 0

subject to zc (T) = vc (T) = 0 . The optimal control u(t) is shown in Figure 9.8 by the solid curve for the given parameters γ = 10−5 , N = 6, and M1 = 15. The dashed curve demonstrates the time history of the carriage velocity vc (t).

Figure 9.8: Optimal acceleration u(t) (solid) and velocity vc (t) (dashed) of the carriage.

The place of the beam top in the phase plane (w(t, 1), p(t, 1)) is drawn in Figure 9.9 by the solid curve. For comparison, the dashed curve presents the approximate beam motion in the absence of the control signal (z(t) ≡ 0). Given the active damping, the vibrations of the elastic beam disappear rapidly. In Figure 9.10, the time history of total mechanical energy of the beam W0 (t) (solid) is presented. The mean energy W0 =

1 T ∫ W (t) dx , T 0 0

stored during the process is equal to W 0 = 0.0702. The initial and terminal energy are set to W0 (0) = 0.167 and W0 (T) = 2.68 ⋅ 10−5 , respectively. The integral error of discretization is much smaller that these values and equals to T −1 Φ = 8.88 ⋅ 10−6 .

206 | 9 Double minimization in optimal control problems

Figure 9.9: Phase-plane portrait of the optimal motions for the free beam end.

Figure 9.10: Total energy W0 (t) stored in the system.

The dashed line in Figure 9.10 shows the system energy W0 in the interval t ∈ (0, T) for the case when the carriage remains fixed (zc (t) ≡ 0). The approximated energy W0 decreases in the beginning of the process (W0 (0) − W0 (0.4) = 4.9 ⋅ 10−3 ) since the numerical error reaches then its maximal values. For t > 0.4, the energy changes very little (W0 (t) = 0.1618 ± 0.0001).

9.2 Polynomial control in dynamic problems of linear elasticity 9.2.1 Statement of an inverse dynamic problem Controlled motions of an elastic body occupying come volume x ∈ V ⊂ ℝ3 bounded by a piecewise smooth surface Γ = 𝜕V are studied in the time interval t ∈ (0, T). The displacement vector w(t, x) is defined on a fragment Γ1 of Γ, whereas the stress vector q(t, x) ∶= σ(t, x) ⋅ n(x)

9.2 Polynomial control in dynamic problems of linear elasticity | 207

is prescribed on the other part Γ1 . Here, σ denotes the stress tensor, and n is the outward normal. At that Γ1 ∩ Γ2 = ∅ and Γ1 ∪ Γ2 = Γ. The known volume force f (t, x) is also given. Assumed as in Section 3.1 the smallness of displacements, velocities, and deformations, the body motions are defined on the time–space domain Ω = (0, T) × V by the PDE system of linear elasticity v(t, x) = 0 and ξ (t, x) = 0 for (t, x) ∈ Ω , pt (t, x) = ∇ ⋅ σ(t, x) + f (t, x)

for (t, x) ∈ Ω ,

where the constitutive and kinematic functions v ∶= w t − ρ−1 p ,

ξ ∶= ε − C−1 (x) ∶ σ ,

1 ε ∶= (∇w + ∇w T ) 2

(9.19) (9.20)

(9.21)

are introduced. In eqution (9.21), p(t, x) denotes the vector of momentum density, ε(t, x) is the Cauchy strain tensor, ρ(x) is the volume density, C(x) is elastic modulus tensor. Where the normal exists, the boundary conditions are given by w(t, x) = u1 (t, x) q(t, x) = u2 (t, x)

for x ∈ Γ1 , for x ∈ Γ2

(9.22)

with the boundary displacements u1 and the external surface stresses u2 . At the beginning of the process, the displacements w and the momentum density p are set in the form: w(0, x) = w 0 (x)

and p(0, x) = p0 (x) .

(9.23)

The components of the boundary vectors u1 and u2 are either given functions of the time t and the coordinates x or unknown control inputs belonging to some sets 𝒰1 and 𝒰2 according to u1 (t, x) = w 0 (t, x)

for x ∈ Γ1 \Γ01 ,

u2 (t, x) = q0 (t, x)

for x ∈ Γ2 \Γ02 ,

u1 ∈ 𝒰1

u2 ∈ 𝒰2

for x ∈ Γ01 , for x ∈ Γ02 .

Problem 9.2. The optimal control problem is to find such admissible functions u∗i ∈ 𝒰i in x ∈ Γ0i for i = 1, 2 that lead the body from the initial state (9.23) to the terminal set w(T, x) ∈ 𝒲T

and p(T, x) ∈ 𝒫T

(9.24)

at the fixed time instant T. Simultaneously, this control law minimizes a cost function J[u∗1 , u∗2 ] → subject to the constraints (9.19)–(9.23).

min

u1 ∈𝒰1 ,u2 ∈𝒰2

J[u1 , u2 ]

(9.25)

208 | 9 Double minimization in optimal control problems 9.2.2 Time–space discretization based on the MIDR The numerical approach described in Section 5.4 is applied to discretize Problem 9.2. In accordance with the MIDR, the differential constitutive relations (9.19) are replaced by an integral equality: Φ = ∫ φ(t, x) dΩ = 0 Ω

1 with φ ∶= (ρv ⋅ v + ξ ∶ C ∶ ξ ) . 2

(9.26)

Such reformulation allows one to convert the original IBVP (9.19)–(9.23) to a variational problem on the constrained minimization Φ[w ∗ , p∗ , σ ∗ ] = min Φ[w, p, σ] = 0 w, p, σ

(9.27)

subject to Newton’s law (9.20) as well as the boundary and initial constraints (9.22), (9.23). ̃ x), σ(t, ̃ x), p(t, ̃ x) Define the number Ng for the finite-dimensional functions w(t, that can be represented in the vector-tensor form: Ng

(w,̃ p,̃ σ)̃ = ∑ zk (ak , bk , βk ) , k=1

(9.28)

where the functions (ak (t, x), bk (t, x), βk (t, x)) ∈ ℝ3+3+6

for k ∈ ℤ+

constitute a basis and zk are unknown coefficients. The basis functions are formed so that the approximation (9.28) meets all the essential constraints (9.20)–(9.23) for some properly defined constants zk . As a result, the finite-dimensional problem of unconstrained minimization for the functional Φ, quadratic with respect to unresolved coefficients zk , is reduced to a system of linear equations. The solution of this system for the known functions f , u1 , u2 ̃ p,̃ σ.̃ provides the necessary data to restore the kinematic and dynamic fields w,

9.2.3 Optimization of motion parameters The MIDR proposes a variety of numerical approaches to Problem 9.2, one of which relies on parametric optimization of the cost functional J. First of all, the basis and the number Ng in the approximation (9.28) are chosen in such a way that the PDE and initial constraints (9.20), (9.23) as well as the prescribed boundary conditions (9.22) on the surface (Γ1 \Γ01 ) ∪ (Γ2 \Γ02 ) can be satisfied for some combinations of the coefficients ̃ p̃ belong to the zk (see Section 5.4). It is also necessary that the state functions w, terminal sets 𝒲T and 𝒫T at t = T according to equation (9.24).

9.2 Polynomial control in dynamic problems of linear elasticity | 209

After that, the sets of admissible control functions 𝒰i for i = 1, 2 compatible with the specific approximations are fixed on the boundary part Γ01 ∪ Γ02 . It is follows from equation (9.28) that the control has to have the structure: Nu

(u1 , u2 ) = ∑ uj (u1 , u2 ) . (j)

(j)

j=1

Here, u1 and u2 are linearly independent vectors belonging respectively to the subspaces of w̃ and q̃ = σ̃ ∶ n traced on Γ01 ∪ Γ02 . N Let yj for j = 1, … , Ny be coefficients of the set {zk }1 g that remain unresolved after implementation of all imposed constraints. These coefficients are assembled into a vector of design parameters y ∈ ℝNy , whereas the control parameters form a vec̃ x, y, u), σ(t, ̃ x, y, u), p(t, ̃ x, y, u), which depend on tor u ∈ ℝNu . The approximations w(t, these two vectors in accordance with equation (9.28), are substituted in the functional Φ introduced in equation (9.26). Since Φ is quadratic with respect to y, the minimization (9.27) is reduced to solving of a linear system with the right-hand side, which is defined by the control vector u. The solution y ∗ of this system gives the approximations (j)

(j)

̃ x, y ∗ , u) , w̃ ∗ (t, x, u) = w(t, ̃ x, y ∗ , u) , p̃ ∗ (t, x, u) = p(t, ̃ x, y ∗ , u) , σ̃ ∗ (t, x, u) = σ(t, which are utilized for minimization of the cost function J[w̃ ∗ , p̃ ∗ , σ̃ ∗ , u] in agreement with equation (9.25). In the case considered below, the functional J is assumed to be a quadratic form of the control parameters ui . Its minimum can be found as a solution of another linear system with respect to the vector u (cf. Section 9.1).

9.2.4 Structure of polynomial approximations An important type of discretization in Problem 9.2 relies on the representation of unknown functions w, p, σ by polynomials of the time t and the coordinates x [49] according to ̃ x) = ∑ w (i) t i0 x1i1 x2i2 x3i3 , w(t, |i| 102 . There is a region for 10−2 < α < 102 , where both functionals do not significantly vary and J2 ≪ J1 . It indicates that the obtained values of the energy J1 are rather accurate.

Figure 10.1: Terminal energy J1 (α) and energy error J2 (α).

The solution quality can be also verified by the relative integral error: Δ=

Φ , Ψ

T

Ψ = ∫ W(t) dt . 0

Let the coefficient α = 100 be fixed and the parameters of the approximation M or N be varied. The estimates of the solution quality Δ versus the differential order 2N1 = 2MN of the approximate system (10.9) are shown in Figure 10.2. The error is diminishing if either the number of finite elements is increased at the fixed degree of polynomials N = 2 (h-convergence, a dashed line with circles in the figure with 4 ≤ M ≤ 10), or the polynomial degree is raised at the given number of elements M = 4 (p-convergence, dashed line with squares for 2 ≤ N ≤ 5). Similar to the average energy of the system W = T −1 Ψ ≈ 0.28, the terminal energy J1 ≈ 5 ⋅ 10−8 slightly changes. Figure 10.3 shows displacements of the beam end points during the motion at M = N = 4, K = 9, α = 100. The solid curve corresponds to the optimal control function u∗ (t) = w(t, 0), and the dashed curve relates with the motion of the free end w(t, 1).

234 | 10 Semi-discrete approximations in inverse dynamic problems

Figure 10.2: Relative error Δ(N1 ) in the projection approach.

Figure 10.3: Displacements of the beam ends w(t, 0) = u(t) and w(t, 1).

̃ x). Figure 10.4: Optimal beam displacements w(t,

Such control accelerates the system quite smoothly at the beginning of the motion and slows gradually down at the ending. The distribution of the relative displacements for the beam points w̃ ∗ (t, x) is shown in Figure 10.4. For the chosen parameters, some elastic vibrations are excited

10.2 Optimal control of elastic body motions | 235

Figure 10.5: Linear energy density ψ(t, x).

Figure 10.6: Local error of the numerical solution φ(t, x).

in the system, but the control strategy is such that they almost disappear at the instant T. The vibration damping also confirms with the energy density ψ, as shown in Figure 10.5. Note that the distribution ψ(t, x), which defines the intensity of elastic vibrations, is close to symmetric function with respect to the line t = T/2. The local error φ(t, x) is presented in Figure 10.6. Its amplitude is small enough that confirms the good quality of the resulting approximations for the optimal control problem. In contrast to the variational approach, this error is distributed throughout the time interval without the outstanding peak at the beginning of the process. One can see spikes of the error at the interfaces between spatial elements.

10.2 Optimal control of elastic body motions 10.2.1 Variational formulation of a direct dynamic problem Consider an elastic body occupied some bounded volume V with the piecewise smooth boundary Γ in the 3D space. Introduce the kinematic variables w(t, x), ε(t, x) and the dynamic variables p(t, x), σ(t, x), which characterize the behavior of the elas-

236 | 10 Semi-discrete approximations in inverse dynamic problems tic system depending on the time t ∈ (0, T) and the coordinate vector x = (x1 , x2 , x3 ) ∈ V . Similar as in Chapter 3, w and p are respectively the vector-valued functions of displacements and momentum density, whereas σ and ε are the second-rank tensors that determine the distribution of elastic stresses and strains over the time–space domain Ω = (0, T) × V . In the linear theory, the constitutive equations in local form, relating the velocities of the body points w t with the function of the momentum density p as well as the strain tensor ε with the stress tensor σ, can be written as v(t, x) = 0

and ξ (t, x) = 0

for (t, x) ∈ Ω .

(10.12)

Here, the residual vector of the velocity and the residual tensor of deformations are introduced as v ∶= w t − ρ−1 p and ξ ∶= ε − C−1 ∶ σ .

(10.13)

The volume density ρ and the elastic modulus tensor C are given functions of coordinates x. The strain tensor ε is a linear function of the displacement vector according to 1 ε = (∇w + ∇w T ) . 2

(10.14)

By using equatiions (10.12) and (10.14), Newton’s second law is expressed in terms of the momentum density vector p and the stress tensor σ as pt (t, x) = ∇ ⋅ σ(t, x)

for (t, x) ∈ Ω .

(10.15)

It is supposed that the external volume forces are absent. Consider the case when the boundary conditions are defined by the displacements w and the surface stresses q = σ ⋅ n in the form: w(t, x) = w 0 (t, x) q(t, x) = q0 (t, x)

for x ∈ Γ1 ,

(10.16)

for x ∈ Γ2 .

Here, n is a unit vector of external normal to the body boundary Γ, w 0 and q0 are the given boundary vectors of displacements and stresses, respectively, Γ1 and Γ2 are non-intersecting parts of the boundary, so that Γ1 ∩ Γ2 = ∅ and Γ̄ 1 ∪ Γ̄ 2 = Γ. It is also necessary to determine the state of the body at the initial time by setting the initial distribution of the elastic displacements w 0 and the momentum density p0 according to w(0, x) = w 0 (x)

and p(0, x) = p0 (x)

for x ∈ V .

(10.17)

In Section 3.1, the following integrodifferential formulation of the IBVP (10.12)– (10.17) describing the motion of an elastic body was proposed to get such a field of

10.2 Optimal control of elastic body motions | 237

the unknown displacements w(t, x), momentum density p(t, x), and stresses σ(t, x), which satisfy the integral relation Φ[w, p, σ] = ∫ φ(t, x) dΩ = 0 , Ω

1 φ = (ρ(x)v ⋅ v + ξ ∶ C(x) ∶ ξ ) , 2

(10.18)

while the kinematic equation (10.14), the balance relation (10.15), the boundary and initial conditions (10.16), (10.17) are essential constraints. The non-negativity of the integral Φ allows us to reduce the problem (10.14)–(10.17), (10.18) to the minimization formulation: find such functions w, p, and σ that provide a minimum (zero) value of the functional Φ[w, p, σ] under the constraints (10.14)–(10.17). The integral quality of approximated functions w, p, and σ can be estimated by the value of the dimensionless ratio: Δ = ΦΨ−1 < δ ≪ 1 .

(10.19)

Here, δ is a selected positive number, and the time integral of the total mechanical energy Ψ is given by the formula: Ψ=

1 ∫ (ρ−1 (x)p ⋅ p + ε ∶ C(x) ∶ ε) dΩ . 2 Ω

(10.20)

The space and time distribution of the errors describing in the terms of w, p, and σ is characterized by the function φ(w, p, σ) defined in equation (10.18).

10.2.2 Projection formulation of the problem on body motions A variant of the Petrov–Galerkin approach [84] is described below, which uses the integral projections for the vector of the residual velocity v, and the tensor of residual strains ξ introduced in (10.12). In this case, the statement of a linear elasticity problem is: find the admissible displacements w, momentum densities p, and stresses σ, satisfying equations (10.15)–(10.17) such that the following integral equations: ∫ ρ(x)v t (t, x) ⋅ r(t, x) dΩ = 0 , Ω

∫ ξ (t, x) ∶ τ(t, x) dΩ = 0 , Ω

∀r ∈ L2 (Ω; ℝ3 ) ,

(10.21)

∀τ ∈ L2 (Ω; ℝ3×3 )

(10.22)

are valid. Here, r is a vector of virtual displacements, τ is a tensor of virtual stresses, and the vector v and the tensor ξ are defined in equation (10.13).

238 | 10 Semi-discrete approximations in inverse dynamic problems

Figure 10.7: Elastic cuboid body.

10.2.3 Statement of an optimal control problem Consider an elastic body having the shape of a cuboid with the length 2a1 and the cross-sectional dimensions 2a2 × 2a3 , and a1 ≫ a2 + a3 (see Figure 10.7). Introduce the Cartesian coordinate system Ox1 x2 x3 which origin is located in the midpoint of the body and the axis Oxk is parallel to the edges with the length 2ak for k = 1, 2, 3. The spatial volume occupied by the body is as follows: V = {x ∶ |xi | < ai , i = 1, 2, 3} . The case is under study when the long sides of the beam are free of stress, that is, σ(t, x) ⋅ en = 0

for xn = ±an

with n = 2, 3 .

(10.23)

One of the end cross sections is free of loads σ(t, x) ⋅ e1 = 0 for x1 = a1

(10.24)

and the other section remains undeformed and moves in accordance with a prescribed control law u(t): w(t, x) = (0, y1 (t), 0) with y1̈ (t) = u(t)

for x1 = −a1 .

(10.25)

Here, ek = (δ1,k , δ2,k , δ3,k ) for k = 1, 2, 3 are the unit vectors of the coordinate system Ox1 x2 x3 , normal to the different parts of the body boundary Γ, δj,k is a Kronecker symbol. Displacements of the end cross section y1 (t) along the axis Ox2 satisfy the following initial conditions: y1̇ (0) = y1 (0) = 0 .

(10.26)

Let us formulate the problem: find an acceleration u∗ (t), in other words, optimal control [65], which transfers the beam from the initial rest state w(0, x) = 0

and p(0, x) = 0

for x ∈ V

(10.27)

to the terminal one w(T, x) = (0, yT , 0)

and

p(T, x) = 0

for x ∈ V

(10.28)

10.2 Optimal control of elastic body motions | 239

and minimizes a quality functional J[u∗ ] = min J[u] . 2 u∈L (0,T)

(10.29)

The following quadratic functional is introduced: J=

1 T 2 ∫ u (t) dt + γΨ , 2 0

γ ≥ 0,

(10.30)

where γ is a weighting coefficient, Ψ is the time integral of the energy from equation (10.20).

10.2.4 Algorithm of discretization To apply the approach of semi-discretization described in Chapter 6, exclude from the consideration the vector-valued function of momentum density by integrating the equation (10.15) with respect to time and by taking into account the initial conditions (10.27) according to t

p(t, x) = ∫ ∇ ⋅ σ(t1 , x) dt1 . 0

(10.31)

By substituting equation (10.31) in the vector v introduced in equation (10.13), it is obtained that t

v = w t − ρ−1 ∫ ∇ ⋅ σ(t1 , x) dt1 . 0

(10.32)

After accounting the Cauchy tensor (10.14), the residual deformation tensor defined in (10.13) takes the form: 1 ξ = (∇w + ∇w T ) − C−1 ∶ σ . 2

(10.33)

Consider the approximations of unknown displacements N

w1 (t, x) = ∑ w1(k,l) (t, x1 )x̃ k2 x̃ l3 , k+l=0 N−1

w3 (t, x) = ∑ w3(k,l) (t, x1 )x̃ k2 x̃ l3 , k+l=0

N−1

w2 (t, x) = y1 (t) + ∑ w2(k,l) (t, x1 )x̃ k2 x̃ l3 , k+l=0

(10.34)

240 | 10 Semi-discrete approximations in inverse dynamic problems and stresses

N

(k,l) σ1,1 (t, x) = ∑ σ1,1 (t, x1 )x̃ k2 x̃ l3 , k+l=0

N

(k,l) σn,n (t, x) = gn ∑ σn,n (t, x1 )x̃ k2 x̃ l3 , k+l=0 N−1

(k,l) σ1,n (t, x) = gn ∑ σ1,n (t, x1 )x̃ k2 x̃ l3 ,

(10.35)

k+l=0 N−2

(k,l) σ2,3 (t, x) = g2 g3 ∑ σ2,3 (t, x1 )x̃ k2 x̃ l3 ,

gn = 1 − x̃ 2n ,

k+l=0

x̃ n = a−1 n xn ,

n = 2, 3 .

Here, N is a given positive integer that specifies the degree of the polynomial expansion of the unknown functions with respect to the dimensionless coordinates x̃ 2 and x̃ 3 . The approximations chosen in such a way automatically satisfy the homogeneous boundary conditions in stresses (10.23) at the long surfaces of the prism. By using the projection relations (10.21), (10.22), the approximations (10.34), (10.35) allow us to build a system of DAEs in partial derivatives of the time t and the coordinates x1 . At first, let us form a group of equations containing partial derivatives of x1 . To do so, write down the following projections: ∫ v t (t, x) ⋅ r(t, x) dΩ = 0 ,

∀r ,

Ω

(10.36)

and ∫ e1 ⋅ ξ (t, x) ⋅ s(t, x) dΩ = 0 ,

∀s .

Ω

(10.37)

The vectors of virtual momenta r and stresses s = τ ⋅ e1 are introduced here in the form: N

r1 (t, x) = ∑ r1(k,l) (t, x1 )x̃ k2 x̃ l3 , k+l=0 N

k l s1 (t, x) = ∑ s(k,l) 1 (t, x1 )x̃ 2 x̃ 3 , k+l=0 N−1

rn (t, x) = ∑

k+l=0 N−1

(10.38)

rn(kl) (t, x1 )x̃ k2 x̃ l3 ,

k l sn (t, x) = ∑ s(k,l) n (t, x1 )x̃ 2 x̃ 3 k+l=0

for n = 2, 3. Equations (10.36) and (10.37) can be explicitly resolved with respect to the first (k,l) (k,l) derivatives of the functions 𝜕wm /𝜕x1 , 𝜕σ1,m /𝜕x1 for m = 1, 2, 3. In accordance with

10.2 Optimal control of elastic body motions | 241

equations (10.34), (10.38), the total number of these functions coincides with the (k,l) number of the virtual functions rm , s(k,l) and is equal to 2Nd , where Nd = (N + m 1)(3N + 2)/2. By substituting the expression for these derivatives into the functional Φ of equation (10.18), the algebraic relations can be obtained that are necessary to find the stress (k,l) (k,l) (k,l) functions σ2,2 , σ3,3 , σ2,3 . The total number of such functions is Na = 32 N 2 + 52 N + 2. The corresponding relations are derived with vanishing of the first variation δσ2,2 Φ + δσ2,3 Φ + δσ3,3 Φ = 0

(10.39)

while values of the other functions of stresses and displacements keep frozen. The system of integral equations (10.36), (10.37), (10.39) is equivalent to the system (k,l) (k,l) of 2Nd + Na linear equations of the variables wm , σm,n at an arbitrary choice of test (k,l) (k,l) (k,l) (k,l) (k,l) functions rm , sm and variations δσ2,2 , δσ3,3 , δσ2,3 . The differential order of the system both with respect to time t and the coordinate x1 is equal to 2Nd . The Nd boundary conditions in stresses σ1,1 (t, a1 ) = 0

for i + j ≤ N ,

(k,l) (k,l) (t, a1 ) = 0 σ1,2 (t, a1 ) = σ13

for k + l ≤ N − 1 ,

(i,j)

(10.40)

follow from the boundary condition (10.24), whereas the Nd conditions in displacements w1 (t, −a1 ) = 0

for i + j ≤ N ,

w2(k,l) (t, −a1 ) = w3(k,l) (t, −a1 ) = 0

for k + l ≤ N − 1

(i,j)

(10.41)

are obtained from the condition (10.25) and the approximations (10.34). The initial conditions for the displacements are derived directly from the relations (10.27) according to w1 (0, x1 ) = 0

for i + j ≤ N ,

w2(k,l) (0, x1 ) = w3(k,l) (0, x1 ) = 0

for k + l ≤ N − 1 ,

w1,t (0, x1 ) = 0

for i + j ≤ N ,

(k,l) (k,l) w2,t (0, x1 ) = w3,t (0, x1 ) = 0

for k + l ≤ N − 1 .

(i,j)

(i,j)

(10.42)

To deduce a consistent system, it is necessary to add to equations (10.36), (10.37), (10.39)–(10.42) the differential equation (10.25) together with the initial conditions (10.26) with respect to the displacement of the end cross section y1 (t).

242 | 10 Semi-discrete approximations in inverse dynamic problems 10.2.5 Spectral boundary value problem To solve the formulated finite dimensional problem in partial derivatives, the unknown functions (10.34) are represented [96] as M−1

w1 (t, x) = ∑ w̃ 1,i (x)yi+1 (t) , i=1 M−1

w3 (t, x) = ∑ w̃ 3,i (x)yi+1 (t) , i=1

M−1

(10.43)

w2 (t, x) = y1 (t) + ∑ w̃ 2,i (x)yi+1 (t) , M−1

i=1

σ = ∑ σ̃ i (x)yi+1 (t) , i=1

where M is a positive number. By substituting the expression yi (t) = exp(iωt) with i = 1, … , M in equation (10.43), the problem (10.36), (10.37), (10.39)–(10.42) can be reduced to a system of ODEs with respect to x1 under homogeneous boundary conditions given at the ends of the beam. In other words, the problem is to find the eigenfrequencies ω. As in Chapter 8, the solution of the eigenvalue problem with using the notation introduced in (10.34) can be represented in the form [55]: N

k l w̃ 1 = ∑ w̃ (k,l) 1 (x1 )x̃ 2 x̃ 3 , k+l=0 N

N−1

k l w̃ n = ∑ w̃ (kl) n (x1 )x̃ 2 x̃ 3 , k+l=0 N

k l σ̃ 1,1 = ∑ σ̃ (k,l) 1,1 (x1 )x̃ 2 x̃ 3 ,

k l σ̃ n,n = gn ∑ σ̃ (k,l) n,n (x1 )x̃ 2 x̃ 3 ,

k l σ̃ 1,n = gn ∑ σ̃ (k,l) 1,n (x1 )x̃ 2 x̃ 3 ,

k l σ̃ 2,3 = g2 g3 ∑ σ̃ (k,l) 2,3 (x1 )x̃ 2 x̃ 3 .

k+l=0 N−1

k+l=0

k+l=0 N−2

(10.44)

k+l=0

Here, w̃ i(k,l) (x1 ), σ̃ (k,l) i,j (x1 ) are the components of the corresponding eigenvectors. Further, the special case of vibrations for a homogeneous isotropic beam with a square cross section (a2 = a3 ) is investigated. By introducing the characteristic length x̃ = a2 and time t ̃ = a2 √ρ/E, where E is Young’s modulus, all equations of linear elasticity can be transformed to the dimensionless form. The resulting system has two parameters: the relative length of the beams a = a1 /a2 and Poisson’s ratio ν. The uñ To simplify the notation, the known quantity is the dimensionless frequency ω̃ = tω. sign tilde is omitted. Further, the sample values ν = 0.3, a = 20 are used in the calculations. To improve the efficiency of the algorithm, the properties of symmetry can be taken into account (see Section 8.1). Due to the fact that the cross section of the beam

10.2 Optimal control of elastic body motions | 243

has two axes of symmetry, the system of equations (10.36), (10.37), and (10.39) is split into four independent subsystems. According to the polynomial parity of the basic functions in equation (10.44), these subsystems approximately describe the tension– compression, bending about axes Ox2 and Ox3 , as well as the torsion of the beam. The maximal degrees of variables x2 and x3 , which are present in the approximations of displacements and stresses in equation (10.44), are given in Table 8.1. It is worth reminding that the integer n = k/2 ≥ 0 in the table characterizes the differential order of the corresponding boundary value problem. The number of differential equations is: (n + 1)(3n + 2) for the tension–compression problem; (N + 1)(3n + 4) for the bending problems; (N + 1)(3n + 6) for the torsion problem. The minimum possible dimensions of approximations in equation (10.44) are equal to 2, 4, and 6, respectively. In accordance with equation (10.25), the end of the beam at x1 = −a1 moves in the direction of the Ox2 axis. Thus, the bending vibrations around the axis Ox3 are only occurring in the system. As an example, consider the eigenvalue problem at n = 0. In this case, the functions (10.44) have the form: w̃ 1 = w̃ (1,0) (x1 )x2 , 1

w̃ 2 = w̃ (0,0) (x1 ) , 2 2 σ̃ 1,2 = σ̃ (0,0) 1,2 (x1 )(1 − x2 ) ,

σ̃ 1,1 = σ̃ (1,0) 1,1 (x1 )x2 , 2 σ̃ 2,2 = σ̃ (1,0) 2,2 (x1 )x2 (1 − x2 ) ,

(10.45)

w̃ 3 = σ̃ 1,3 = σ̃ 2,3 = σ̃ 3,3 = 0 . As a result, the DAE system (10.36), (10.37), (10.39) can be written down as 4 dσ̃ 1,1 3 dx1

8 4 − σ̃ (0,0) + ω2 w̃ (1,0) = 0, 1 3 1,2 3

8 dσ̃ 1,2 3 dx1

+ 4ω2 w̃ (0,0) = 0, 2

(1,0)

(0,0)

16 dσ̃ 1,2 45 dx1

(0,0)

+

16 (1,0) σ̃ = 0, 15 2,2

(10.46)

4 dw̃ (1,0) 4 4 1 − σ̃ (10) + σ̃ (1,0) = 0, 3 dx1 3 1,1 25 2,2 4 dw̃ (0,0) 4 208 (0,0) 2 + w̃ (1,0) − σ̃ = 0, 3 dx1 3 1 75 1,2 with the boundary conditions (0,0) w̃ (1,0) (−a1 ) = w̃ (0,0) (−a1 ) = σ̃ (1,0) 1 1,1 (a1 ) = σ̃ 1,2 (a1 ) = 0 . 2

(10.47)

244 | 10 Semi-discrete approximations in inverse dynamic problems The eigenfrequencies ω are found from the system (10.46) as the roots of the characteristic equation 4 4 406 2 2 104 4 λ + ω λ − 4ω2 + ω = 0, 3 75 25

(10.48)

where λ is the corresponding wave number. For comparison, the characteristic equation for the Timoshenko beam with the same structural parameters is given [97] by 4 4 412 2 2 104 4 λ + ω λ − 4ω2 + ω =0. 3 75 25

(10.49)

As can be seen, equations (10.48) and (10.49) differ only in coefficients at λ2 . At that, this difference is less than 2%. For a more accurate calculation of natural beam vibrations, it is necessary to apply a higher degree of polynomial approximations n > 0. In the first three rows of Table 10.1, the values of the four lower eigenfrequencies for a cantilever beam with the square cross section for the model of Euler–Bernoulli, as well as for the model described here at n = 0 and n = 1 are shown. The differences between the frequencies obtained at these approximations are put in the fourth and fifth rows, respectively. The differences between these frequencies and the frequencies calculated based on the model of Euler–Bernoulli are greater and reaches to 5.8 % already for the fourth mode. The last line reflects the relative accuracy Δ1i in calculation of the i-th eigenform at n = 1 according to the integral criterion (10.19). Table 10.1: Eigenfrequencies ωi for the cantilever beam. i

1

ωci

1.269 ⋅ 10 1.266 ⋅ 10−3 1.269 ⋅ 10−3 0.2% 0.21% 0.30%

n = 0, ω0i n = 1, ω1i (ωci − ω0i )/ω0i (ω1i − ω0i )/ω0i Δ1i

2 −3

3

7.951 ⋅ 10 7.843 ⋅ 10−3 7.861 ⋅ 10−3 1.4% 0.22% 0.31% −3

4

2.226 ⋅ 10 2.157 ⋅ 10−2 2.162 ⋅ 10−2 3.2% 0.25% 0.32% −2

4.363 ⋅ 10−2 4.123 ⋅ 10−2 4.135 ⋅ 10−2 5.8% 0.29% 0.33%

10.2.6 System of ODEs with respect to time In order to compose an ODE system with respect to the time t, replace the eigenforms (10.44), corresponding to the frequencies ωi with i = 1, … , M − 1, on the displacements w̃ i (x) and the stresses σ̃ i (x) in the approximations (10.43). These approximations are substituted, in turn, into the integral equations (10.36), (10.37), and (10.39).

10.2 Optimal control of elastic body motions | 245

In equations (10.36) and (10.37), the vector v t and the tensor ξ are projected onto the basis vector-valued functions r i (x) and si (x) with the components N

N+1

̃ = ∑ r (k,l) ̃ (x1 )x̃ k2 x̃ l3 , r 1,i 1,i

̃ = ∑ r (kl) ̃ (x1 )x̃ k2 x̃ l3 , r n,i n,i

̃ (x1 )x̃ k2 x̃ l3 , s(k,l) 1,i

̃ (x1 )x̃ k2 x̃ l3 . s(k,l) n,i

k+l=0 N

̃ = ∑ s1,i

k+l=0

k+l=0 N−1

̃ = ∑ sn,i

k+l=0

(10.50)

̃ (x1 ) and s̃(k,l) The unknown functions r (k,l) j,i j,i (x1 ) for j = 1, 2, 3 are found as a solution of the adjoint boundary eigenvalue problem [26]. The choice of such projections leads the resulting integral equations (10.36), (10.37), and (10.39) to the diagonal form: yj̈ = −ω2j−1 yj + bM+j u(t) ,

j = 2, … , M ,

(10.51)

where ωi for i = 1, … , M − 1 are the approximate eigenfrequencies obtained at the chosen degree n, and bk for k = M + 1, … , 2M are the control coefficients. 10.2.7 Finite-dimensional control problem By taking into account the initial conditions (10.26), (10.42), the ODE system (10.51) can be completed by adding the equation (10.25): yj̇ = yM+j { { { ̇ yM+j = −ω2j−1 yj + bM+j u(t) { { { { yk (0) = 0

for j = 1, … , M , for j = 1, … , M ,

(10.52)

for k = 1, … , 2M .

Here, the new variables yM+j (t), denoted the time derivatives with respect to unknown yj (t), as well as the frequency ω0 = 0 and the control coefficient bM+1 = 1 are introduced. System (10.52) can be written in the vector form ̇ = f (y, u) = Ay(t) + bu(t) , y(t)

(10.53)

with the homogeneous initial conditions y(0) = 0 ,

(10.54)

y(T) = (yT , 0, … , 0)

(10.55)

and the terminal conditions

246 | 10 Semi-discrete approximations in inverse dynamic problems following from equation (10.28). Here, y = (y1 (t), … , y2M (t)) is the vector of phase variables, A ∈ ℝ2M×2M and b2M are constant matrix and vector, respectively. For the finite-dimensional dynamic systems (10.53), which defines the approximate lateral motions for the considered elastic body, let us formulate an optimal control problem corresponding to equations (10.27)–(10.29): find a control function u∗ (t), which transfers the linear system (10.53) from the initial zero state (10.54) to the terminal rest state (10.55) at a fixed time T and minimizes the quality functional: ̃ . ̃ ∗ ] = min J[u] J[u 2 u∈L (0,T)

(10.56)

Here, the quadratic integral T

J ̃ = ∫ f0 (t) dt 0

1 γ for f0 = u2 (t) + y(t) ⋅ W ⋅ y(t) , 2 2

(10.57)

is obtained by discretizing the functional J in equation (10.30). By introducing the vector of conjugate variables z(t) ∈ ℝ2M , the Hamiltonian of the system is defined as ℋ[y, z, u] = −f0 + f ⋅ z

(10.58)

in accordance with the Pontryagin maximum principle [74]. By using the Hamiltonian (10.58), the conjugate system of equations can be obtained according to ̇ =− z(t)

𝜕ℋ = γWy(t) − AT z(t) . 𝜕y

(10.59)

The optimal control u = b ⋅ z(t) ,

(10.60)

is a linear function of conjugate variables. By substituting this function into equation (10.53), the optimal motions are found by solving the BVP (10.53)–(10.55), (10.59), (10.60). To illustrate the efficiency of the proposed algorithm, the following dimensions for approximations n = 1, M = 4, 5, and the control parameters yT = 20, T = 5 000, γ = 5 ⋅ 10−7 are selected. The control horizon T is slightly higher than the period of beam vibrations for the first mode T1 ≃ 4 952; the displacement yT is a half of the dimensionless beam length a. Figure 10.8 presents the optimum control law u(t) at the approximation parameter M = 5 (only the first 4 modes of vibrations are considered). This function is slightly different from the control at M = 4. The difference between these two laws Δu(t) is demonstrated in Figure 10.9. As follows from these two figures, a high-frequency signal is added to the control function with raising the system dimension.

10.2 Optimal control of elastic body motions | 247

Figure 10.8: Optimal control law u(t).

Figure 10.9: Variation of the control Δu(t) with raising the problem dimension.

Figure 10.10 presents in the phase plane the displacements w2 (t, x ± ) of two elastic body points with coordinates x ± = (±a, 0, 0). The dashed curve shows the position and velocity of the midpoint of the boundary cross section at x1 = −a, which moves as a rigid body. The solid curve denotes the phase trajectory of the midpoint of the boundary cross section at x1 = a, which is free of external loads. It can be seen that these curves significantly deviate from each other. This means that considerable elastic deformations are excited in the system when the optimal control law is applied. The distribution of energy among vibration modes is depicted in Figure 10.11. The solid curve presents the change of the kinetic energy E1 (t) which corresponds to the translational motion of the beam. This energy is proportional to the square of the variable y1̇ (t). The dashed curve corresponds to the kinetic energy of the first mode E2 (t) and dotted line reflects the second mode energy E3 (t). These values are proportional to the y22̇ (t) and y32̇ (t), respectively. The other modes are not shown on the chart due to their smallness. The first mode can be referred to as the main elastic motion, as the values of E2 (t) are comparable with the energy of the translational motion. To refine the optimal control law u∗ (t) for the elastic body motion and to find the corresponding displacements, stresses, and momenta, one needs to increase the order of the polynomial expansion n and the number of involved modes M. It should be noted that increasing the dimensionality of the model leads to the appearance of high-

248 | 10 Semi-discrete approximations in inverse dynamic problems

Figure 10.10: Displacements in the phase plane for the midpoints x ± of the boundary cross sections.

Figure 10.11: Changing the kinetic energy Ei (t) for lowest modes.

frequency oscillations in the control circuits, which can significantly complicate the implementation of the proposed strategy in technical applications.

10.3 Variational approach to optimization of parabolic systems 10.3.1 Statement of a control problem Consider a homogeneous thin thermally conductive rod, which is heated at one end. The equation describing the variation of temperature w(t, x) in different points of the rod has the form: c1 (x)wt (t, x) − c2 (x)wxx (t, x) + c3 (x)w(t, x) = f (t, x)

(10.61)

10.3 Variational approach to optimization of parabolic systems | 249

over the space–time domain (t, x) ∈ Ω = (0, T) × (0, L), where ci for i = 1, 2, 3 are given coefficients and the function f (t, x) describes the distributed external disturbances acting on the rod. Subscripts t and x denote partial derivatives with respect to time and coordinate. To find a unique solution of equation (10.61), it is necessary to define boundary conditions at the ends of the rod and consistent initial conditions. Let the points for which the boundary conditions are specified have the coordinates x = 0 and x = L. In order that the proposed system describes more realistically the behavior of a laboratory setup, which will be discussed in this section, give the following boundary conditions: c2 (0)wx (t, 0) = q0 (t)

and w(t, L) = u(t) .

(10.62)

The temperature initial condition can be written as w(0, x) = w0 (x) .

(10.63)

In equations (10.62) and (10.63), q0 (t) and w0 (x) are specified functions of time and position, and u is a control. Since the temperature w(t, x) should be a continuous function with respect to the variables t and x, the following condition of compatibility is given: u(0) = w0 (L) .

(10.64)

It is considered that the feedforward control for the system (10.61)–(10.63), i.e., the time function u(t) is constructed. It is assumed that the resulting temperature of the control rod at the point x = xd with 0 < xd < L is an output signal. The objective of the control is to choose an input signal u = u∗ (t) so that the output signal w(t, xd ) is rather accurately coordinated with respect to the desired temperature profile wd (t), which is supposedly smooth enough. A more strict formulation of the control problem will be given further. To illustrate the proposed approach to modeling and optimization of the heat transfer dynamic system (10.61)–(10.64), consider the experimental setup schematically shown in Figure 10.12. It was developed at the University of Rostock, Germany [75]. A control of the heating rod can be performed by Peltier elements combined with cooling devices. The rod is divided into four equal segments. It is assumed that the input temperature u(t) is supplied by the element located under the fourth heating or cooling segment of the rod and the output temperature is measured in the first segment. The coefficients introduced in equation (10.61) are defined by geometrical and physical parameters of the system in the form: c1 = 1 ,

c2 =

λ , ρcp

c3 =

α . hρcp

(10.65)

250 | 10 Semi-discrete approximations in inverse dynamic problems

Figure 10.12: Scheme of the experimental setup.

Distributed disturbance is determined only by convection heat transfer with the environment (ambient air) according to f (t, x) = c3 wa (t) . Here, ρ is the volume density of the material, cp is the specific heat capacity, λ is the thermal conductivity, α is the heat-transfer coefficient, wa is the air temperature, and h is the height of the rod, which has the shape of a cuboid. It is assumed that due to the adiabatic insulation on one end of the rod the heat flux across the border is equal to zero, that is, q0 (t) ≡ 0 in (10.62). The input temperature u(t) = w(t, L) is given at the middle of segment 1 (x = L = Lr − Ls /2), where Lr is the full length of the rod, Ls = Lr /4 is the length of each of the four segments shown in Figure 10.12. The parameters of this model have been identified in a series of experiments. Further, it is assumed that the initial temperature of the rod is uniformly distributed along the length and is equal to the external temperature, w0 (x) = wa (0). 10.3.2 Fourier method in heat transfer problems The solution of the IBVP (10.61)–(10.63) can be obtained on the basis of the modified Fourier method [22]. The main idea of this modification is that the original problem with inhomogeneous boundary conditions (10.62) reduces to the problem with homô x) so geneous ones. To do this, let us introduce a new function of the temperature w(t, that ̂ x) + u(t) , w(t, x) = w(t,

(10.66)

where u(t) is the control function. After substituting the expression (10.66) for w(t, x) in equation (10.61)–(10.63) and taking into account the representation of the system parameters (10.65), a new IBVP is formulated: Find an unknown distribution of the ̂ x) that satisfies the relations: temperature w(t, ŵ t −

λ α α α ̇ − ŵ + ŵ = w (t) − u(t) u(t) , ρcp xx hρcp hρcp a hρcp

ŵ x (t, 0) = 0 t=0∶

̂ L) = 0 , and w(t,

̂ x) = 0 w(0,

and

u(0) = wa (0) .

(10.67)

10.3 Variational approach to optimization of parabolic systems | 251

After that, the temperature ŵ is represented as an infinite series ∞

̂ x) = ∑ yk (t)bk (x) , w(t,

(10.68)

k=0

where bk are normalized eigenfunctions of the boundary value problem 2 b″ k (x) + κk bk (x) = 0 ,

b′k (0) = bk (L) = 0 .

These functions are found in an explicit way as 2 bk = √ cos(κk x) . L

The corresponding eigenvalues κk are the roots of the characteristic equation cos(κk L) = 0, i.e., 1 π κk = ( + kπ) . L 2 It can be shown that the eigenfunctions bk (x) constitute an orthonormal basis: L

∫ bk (x)bl (x) dx = δk,l , 0

‖bk ‖22

L

=∫

0

b2k (x) dx

=1

for {k, l} ∈ ℕ .

(10.69)

After multiplying the first equality of equation (10.67) by the eigenfunction bk (x) for k ∈ ℕ, let the obtained product integrate over the segment x ∈ [0, L]. By taking into account expressions (10.68) and (10.69), a countable system of first-order ODEs with respect to the Fourier coefficients yk (t) is given in the form: yk̇ + νk yk = Fk (t) , Fk (t) =

νk =

λκk2 α + , ρcp hρcp

(−1)k 2 α α √ ( ̇ − w (t) − u(t) u(t)) , κk L hρcp a hρcp

yk (0) = 0

(10.70)

for k ∈ ℕ .

The solution of the system (10.70) can be expressed in quadratures as t

yk (t) = ∫ Fk (τ) exp(νk (τ − t)) dτ 0

for k ∈ ℕ .

To give a numerical representation of the temperature field, consider m-mode approximation: m

w̃ f (t, x) = ∑ bk (x)yk (t) + u(t) . k=0

This temperature approximation is used to verify the quality of the numerical solutions obtained, as described below, based on special procedures of the MIDR developed for an optimal control problem of rod heating.

252 | 10 Semi-discrete approximations in inverse dynamic problems 10.3.3 Optimal control problem of rod heating Recall first the physical laws underlying heat transfer processes. Equation (10.61) follows from the Fourier law of heat conductivity in conjunction with the first law of thermodynamics (conservation of energy). The law of the thermal conductivity relates linearly the heat flux density q(t, x) to a temperature gradient: s(t, x) = 0 for (t, x) ∈ Ω

with s ∶= q + λwx .

(10.71)

In turn, the first principle of thermodynamics leads to the equation: ρcp wt (t, x) + qx (t, x) + h−1 α(w(t, x) − wa (t)) = 0

for (t, x) ∈ Ω .

(10.72)

The boundary conditions (10.62) expressed through the temperature w and the heat flux q are represented as q(t, 0) = 0

and

w(t, L) = u(t) ,

(10.73)

where the insulation on one end of the rod is taken into account. In accordance with the assumptions made in the preceding subsection, the initial condition takes the form: w(0, x) = wa (0) .

(10.74)

Now let us formulate the optimal control problem: Find a function u∗ (t) ∈ 𝒰, which transfers the system (10.71)–(10.73) from the initial state (10.74) to a terminal state in the fixed time T and minimizes the quadratic functional J[u] → min ,

(10.75)

J = ∫ (w(t, xd ) − wd (t))2 dt .

(10.76)

u∈𝒰

where

T

0

Here, wd is the desired temperature trajectory, 𝒰 represents a set of admissible controls. To find the numerical solution of the problem (10.71)–(10.76), restrict ourselves in this subsection with the class of polynomial control signal n

𝒰 = {u(t) ∶ u = wa (0) + ∑ uk t k }, k=1

(10.77)

where uk are the unknown real coefficients. Polynomial functions of 𝒰 exactly satisfy the condition (10.64). The control parameters uk are assembled into a vector: u = (u1 , … , un ).

(10.78)

10.3 Variational approach to optimization of parabolic systems | 253

10.3.4 Variational formulation of the IBVP For numerical solution of the problem (10.71)–(10.74), use the MIDR, as discussed in Section 2.4. In accordance with this method, differential constitutive relations (in this case, Fourier’s law (10.71)) are replaced by the corresponding integral equality Φ = ∫ φ(t, x) dΩ = 0 Ω

with φ ∶= s2 ,

(10.79)

which is involved constitutive function s(t, x) introduced in (10.71). Thus, the original IBVP can be reformulated. It is necessary to find such temperature distribution w∗ (t, x) and heat flux q∗ (t, x) that satisfy the first law of thermodynamics (10.72), the boundary and initial conditions (10.73), (10.74), and also obey the integral equality (10.79). It is worth emphasizing that the functional Φ is non-negative and reaches its absolute minimum on the solution, due to the non-negativity of the integrand φ for arbitrary values of w and q given in equation (10.79) (see Chapter 2). Therefore, the integrodifferential problem (10.72)–(10.74), (10.79) can be reduced to the variation one: Find unknown functions w∗ and q∗ that minimize the functional Φ according to Φ[w∗ , q∗ ] = min Φ[w, q] = 0

(10.80)

w,q

subject to the constraints (10.72)–(10.74). 10.3.5 Spatial discretization with polynomials Introduce the following approximation for the temperature and the heat flux: m−1

w̃ = ∑ yk (t) k=0

xk k!

and

m

q̃ = ∑ qk (t) k=0

xk k!

(10.81)

Here, yk and qk are unknown coefficients depending on the time t, n is given integer. After that, the original 2D PDE system of heat transfer (10.71), (10.72) can be rewritten as a finite system of ODEs with respect to the time variable t. To implement this, explicitly resolve equation (10.72) according to α q1 = −ρcp y0̇ − (y0 − wa ) , h α ̇ − yk−1 for k = 2, … , m . qk = −ρcp yk−1 h

(10.82)

Then it is possible to satisfy the boundary conditions (10.73) as follows: q0 = 0 ,

ym−1 =

m−2

(m − 1)! Lk (u − y ). ∑ k Lm−1 k! k=0

(10.83)

254 | 10 Semi-discrete approximations in inverse dynamic problems By using the approximations (10.81) as well as equations (10.82) and (10.83), the constitutive function s̃ = s(w,̃ q)̃ from equation (10.71) is expressed by m

s ̃ = ∑ sk k=0

xk , k!

α s1 = λy2 − ρcp y0̇ − (y0 − wa ) , h α ̇ − yk−1 for k = 2, … , M − 2 , sk = λyk+1 − ρcp yk−1 h α ̇ − yl−1 for l = m − 1, m . sl = −ρcp yl−1 h s0 = λy1 ,

(10.84)

It is follows from equation (10.84) that the integral equation (10.79) exactly holds if all the coefficients of sk for k = 0, … , m are strictly equal to zero for t ∈ (0, T). In general, the system sk = 0 for k = 0, … , M is overdetermined because the number of unknown functions yl for l = 0, … , m − 2 is less than the number of equations. Substitute the expressions (10.81)–(10.84) and polynomial control (10.77), (10.78) in the functional Φ defined in equation (10.80). After that, it is possible to formulate the following minimization problem with respect to the unknown vector y(t) = (y0 (t), … , ym−2 (t)) ∶ Find the solution y ∗ (t, u), which guarantees the minimum of the functional Φ̃ ∶= Φ[w,̃ q]̃ according to ̃ ∗ (t, u), u] = min Φ[y, ̃ u] Φ[y

(10.85)

y(0) = (wa (0), 0, … , 0) .

(10.86)

y

under the constraints

The functional Φ̃ in equation (10.85) has the form: T

Φ̃ = ∫ ζ dt 0

L

with ζ = ∫ φ̃ dx 0

and

φ̃ = s̃2 (t, x, y, y,̇ u) ,

where u is the arbitrary control vector introduced in equation (10.78). The necessary condition of stationarity δy Φ̃ = 0 is equivalent to the system of the Euler–Lagrange equation and the terminal condition of transversality: d 𝜕ζ 𝜕ζ − =0 dt 𝜕ẏ 𝜕y

for t ∈ (0, T)

with

𝜕ζ | = 0. 𝜕ẏ t=T

(10.87)

The ODE system (10.86) and (10.87) with respect to time are consistent. Its solution ̃ u]. The components yl∗ of the vector y ∗ y ∗ (t, u) is the extremal of the functional Φ[y, for l = 0, … , m − 2 are linearly dependent on the control coefficients uk for k = 1, … , n.

10.3 Variational approach to optimization of parabolic systems | 255

The parameters uk can be used to design the optimal control law that minimizes the ∗ functional (10.76). The functions ym (t, u) define the approximate temperature fields ∗ ∗ w̃ (t, x, u) and the heat flux q̃ (t, x, u) in accordance with equations (10.81)–(10.83). Note that the functional J introduced in (10.76) does not depend explicitly on the heat flux q. After substituting the approximate solution w̃ ∗ (t, xd , u) instead of w(t, xd ) in equation (10.76) and taking into account the polynomial parameterization of the control function u(t, u) given in (10.77) and (10.78), the optimization problem (10.75) is reduced to unconstrained minimization of a quadratic functional with respect to the unknown vector u. The optimal control vector u∗ is found by solving the system of algebraic equations T 𝜕J ̃ = 0 with J ̃ = ∫ (w̃ ∗ (t, xd , u) − wd (t))2 dt . (10.88) 𝜕u 0 The functions w̃ ∗ (t, x, u∗ ) and q̃ ∗ (t, x, u∗ ), obtained after optimization, are an approximate solution of the original optimal control problem (10.71)–(10.78). 10.3.6 Discretization error in the variational approach The proposed MIDR formulation of the IBVP in thermodynamics gives us the possibility to estimate a posteriori the quality of any numerical solution. Let w∗ (t, x) and q∗ (t, x) be the exact solution of the control problem (10.72)–(10.78), (10.80). Then the integral error of the approximations w̃ ∗ (t, x, u∗ ), q̃ ∗ (t, x, u∗ ) can be defined by the functional: ∗ Φ̃ ≥ Φ[w∗ , q∗ ] = 0

with Φ̃ ∶= Φ[w̃ ∗ , q̃ ∗ ] . ∗

The dimensionless ratio Δ=

∗ Φ̃ ∗ Ψ̃

with Ψ̃ = ∫ (q̃ ∗ (t, x, u∗ ))2 dΩ ∗

Ω

(10.89)

serves as the relative error of the approximate solution w̃ ∗ , q̃ ∗ . The function φ̃ ∗ = (s̃∗ )2

with s̃∗ ∶= s(w̃ ∗ (t, x, u∗ ), q̃ ∗ (t, x, u∗ ))

represents the distribution of the local computational error. 10.3.7 Numerical results of heat control The following system parameters defined on the basis of experimental data (see Figure 10.12) have been used in the numerical simulation: ρ = 7 800 kg/m3 , 2

α = 50 W/m /K ,

cp = 420 J/kg/K , h = 12 mm ,

λ = 110 W/m/K ,

Lr = 320 mm ,

wa (t) = 296.15 K.

256 | 10 Semi-discrete approximations in inverse dynamic problems The aim of control is to provide the input temperature u in the system that makes the output temperature w(t, xd ) with xd = Ls /2 follows along the prescribed temperature profile wd (t) = w0 +

(wT − w0 ) T kT (1 + tanh(α(t − )) coth( )) , 2 2 2

(10.90)

with the least square deviation (10.88). The control parameters are given by w0 = wa (0) ,

wT = w0 + 10 K ,

T = 3 600 s ,

α = 1.5 ⋅ 10−3 s−1 .

The optimal control strategy proposed in the previous subsection has been tested with computer simulations. Figure 10.13 shows the trajectories of the optimal control u∗ (t) = w(t, L) minimizing the functional (10.76) (upper curve), and the input temperature w(t, xd ) (bottom curve). This solution is found based on the MIDR at the fixed parameters of the approximation m = 10 and control n = 14.

Figure 10.13: Input and output temperature trajectories.

The deviation Δw(t) = w(t, xd )−wd (t) of the output temperature at the point xd from the desired trajectory defined in equation (10.90) is presented in Figure 10.14. It is worth noting that the maximum deviation from the prescribed temperature for the polynomial control law u∗ (t) is not more than 0.02 K. The distribution of the local error φ(t, x) introduced in equation (10.79) is shown in Figure 10.15. For optimal heat transfer control, the function φ is close to zero almost everywhere except for a narrow region at the beginning of the process t = 0. The error reaches its maximum at the input point x = L. For the approximate solution w̃ ∗ and q̃ ∗ obtained under the optimal control u∗ (t), the relative integral error Δ defined in equation (10.89) is equal to Δ = 1.2 ⋅ 10−7 . To verify the numerical results obtained by the proposed variational approach, the Fourier method is used. The values of the first five eigenvalues νk∗ given in equation (10.70) are listed in Table 10.2. The difference between the exact eigenvalues νk∗

10.3 Variational approach to optimization of parabolic systems | 257

Figure 10.14: Deviation Δw(t) from the prescribed temperature.

Figure 10.15: Distribution of the local error φ(t, x).

and their approximations ν̃ k found as eigenvalues of the BVP (10.87) is shown in the third row. It should be mentioned that the values ν̃ k can serve as an upper bound of the corresponding eigenvalue of the heat transfer system. In Figure 10.16, the difference ̃ xd ) Δwf (t) = wf (t, xd ) − w(t, between the output temperature wf (t, xd ), which is obtained by the Fourier method ̃ xd ), which is found with the for the 21-mode approximation, and the temperature w(t, help of the MIDR at m = 10, is shown under the identical optimal control law u∗ . The maximum of temperature deviation at the system output is less than 5 ⋅ 10−4 K. The perTable 10.2: Exact and approximate eigenvalues of the heat transfer system. k νk∗ νk∗

− ν̃ k

0

1

2

3

4

−0.00232 2 ⋅ 10−25

−0.0108 1.0 ⋅ 10−14

−0.0277 5.6 ⋅ 10−10

−0.0531 4.8 ⋅ 10−7

0.0869 4.8 ⋅ 10−5

258 | 10 Semi-discrete approximations in inverse dynamic problems

Figure 10.16: Discrepancy of the output temperature trajectories.

formed simulations verify that increasing the number of modes in the Fourier analysis does not noticeably influence the output temperature. As it has been suggested, the feedforward control signal is designed without taking the current temperature distribution w(z, t) into account. In other words, the optimal temperature control law is calculated in advance for the entire time horizon t ∈ [0, T], in which the desired temperature profile wd (t) is given. The feedback control can be applied to compensate modeling error, uncertainty in parameters, and external disturbances [76]. In this case, it is necessary to measure the temperature at output points of the system and to utilize these values to correct the input signal u∗ (t) obtained for the approximate system of heat transfer in the rod.

11 Modeling and control in mechatronics 11.1 Optimal rotations of an electromechanical manipulator 11.1.1 Motion of a flexible link by a drive Consider a electromechanical system, whose main part is a thin elastic rod rotating in the plane OX1 X2 (see Figure 11.1). One end of the rod is fixed in the inertial space OX1 X2 X3 , but a rigid body A is located at the other end. It is supposed that the thickness of the body is small in comparison with the length of the rod L. The axis of rotation is perpendicular to the plane OX1 X2 . The electric drive is situated at the origin O. It contains a separately excited DC motor and an absolutely rigid reducer.

Figure 11.1: Elastic link loaded by a point mass.

To describe the motion of the system, the coordinate system Ox1 x2 x3 is introduced rotating synchronously with the rod. It is supposed that the planes of the OX1 X2 and Ox1 x2 coincide. The axis x1 is collinear with the midline of the rod at the point O. It is also assumed that the elastic deformations of the system are described in the framework of the linear theory of thin rectilinear rods. The relative elastic displacements w = w(t, x) are small and perpendicular to the line Ox1 . The rotational speed of the rod as a whole is small in comparison with the eigenfrequency corresponding to the lowest mode of elastic vibrations. The equations described this model are given in [41]. The constitutive equations and the boundary conditions described planar rotations of the rod have the form: ̈ for (t, x) ∈ Ω = (0, T) × (0, L) ρ(x)wtt + (κ(x)wxx )xx = −ρ(x)xα(t)

(11.1)

with w(t, 0) = wx (t, 0) = wxx (t, L) = 0 , ̈ (κ(x)wxx (t, L))x − m0 (wtt (t, L) + Lα(t)) = 0. https://doi.org/10.1515/9783110516449-011

(11.2)

260 | 11 Modeling and control in mechatronics Here, α is the angle between Ox1 and OX1 , m0 is the mass of the point body at the end of the rod. The linear density of the rod ρ and its bending stiffness κ are functions of x. The initial conditions are given as follows: w(0, x) = w0 (x),

wt (0, x) = v0 (x) ,

(11.3)

where w0 (x), v0 (x) are sufficiently smooth functions, which are consistent with the conditions (11.2). The changing of the angular momentum for the system with respect to the rotation axis Ox3 is described by the integral–differential equation (the equation of moments) and the initial conditions: L

∫ ρ∗ (xα̈ + wtt )x dx = μ1 , 0

α(0) = α0 ,

̇ = ω0 . α(0)

(11.4)

Here, μ1 is the moment produced by the electric drive around the axis Ox3 . The point mass A at the end of the rod is taken into account by adding the Dirac delta function with intensity m0 to the linear density ρ(x). It is supposed thus that ρ∗ (x) = ρ(x) + m0 δ(L − x) ,

x ∈ [0, l] .

The requirement for the existence of the partial derivatives for the function w(t, x) of x up to the fourth order is, as shown in [88], very strong since equation (11.1) is derived from the integrodifferential equation that requires the existence of the derivatives for this function up to the second order (see [96]). The system of equations for the electric drive has the form: L0

dj dβ + Rj + F = u(t) , dt dt

J0

d2 β μ =μ− 1 , dt 2 ν

j(t0 ) = j0

(11.5)

μ = Fj ,

(11.6)

and α=

β , ν

where L0 is the inductance, F is the magnetic flux, R is the ohmic resistance, j is the current in the motor coil, J0 is the moment of inertia of the armature, β is the rotor rotation angle, ν is the gear ratio, and u(t) is the voltage. Equation (11.5) describes the voltage balance in the motor armature circuit, and the ODE in (11.6) relates to the armature rotation. By eliminating the equations (11.1), (11.4)–(11.6), the variables μ (electromagnetic moment), μ1 , j, β, it is possible to obtain the integrodifferential equation: 2 d3 α dα 2d α + RJ ν + F 2 ν2 0 3 2 dt dt dt L d3 α d2 α ∗ + ∫ ρ (x)[L0 (x 3 + wttt ) + R(x 2 + wtt )]x dx = Fνu(t) . dt dt 0

L0 J0 ν2

(11.7)

11.1 Optimal rotations of an electromechanical manipulator |

261

By using equations (11.1), (11.4)–(11.6), the initial value for the α̈ can be found: ̈ 0) = α(t

κ(0)wxx (0, 0) + Fνj0 , J0 ν2

wxx (0, 0) =

d2 w0 (0) . dx2

(11.8)

Further, the system of equations (11.1), (11.7), boundary conditions (11.2), and initial conditions (11.3), (11.4), (11.8) are considered. The BVP with respect to x and the Cauchy problem in terms of t can be studied by methods of mathematical physics [96] at the given control signal u(t). For certainty, the solution of the control problem is built at the zero initial conditions corresponding to the rest of the rod and the drive, as well as the case when the rod is considered as a homogeneous (ρ = const, κ = const). For the system (11.1), (11.7), the problem is formulated as follows. Find a voltage control u(t) that turns the rod, which is initially in a rest state, at a desired angle with braking and damping of vibrations at the end of the process t = T. At the same time, this control provides the minimum of a cost functional J[u] in the class of admissible controls 𝒰, and the solution of the problem exists: J[u] → min , u∈𝒰

α(0) = α0

and α(T) = αT ,

(11.9)

̇ = α(0) ̈ = 0, α(0) w(0, x) = wt (0, x) = 0 . 11.1.2 Optimal angular rotation of the link

In this subsection, the solution of the PDE (11.1) with the initial and boundary conditions (11.2), (11.3) is constructed by the method of separation of variables. By using the dimensionless variables x∗ =

x , L

u∗ =

u , L

t∗ =

t , τ

τ = L2 √

ρ κ

(11.10)

and by omitting the asterisk subscript in equations (11.1)–(11.3), the PDE system ̈ wtt (t, x) + wxxxx (t, x) = −xα(t) for (t, x) ∈ Ω , w(t, 0) = wx (t, 0) = wxx (t, 1) = 0 , wxxx (t, 1) = m(wtt (t, 1) + α)̈

m with m = 0 , ρL

(11.11)

is obtained, where m is the dimensionless mass of the point A. The elastic displacements can be presented as ∞

w(t, x) = ∑ yk (t)bk (x) . k=1

(11.12)

262 | 11 Modeling and control in mechatronics Here, bk (x) are the eigenfunctions defined in x ∈ (0, 1), which are a solution to the problem: 4 b(4) k = λk bk ,

bk (0) = b′k (0) = b″ k (1) = 0 ,

4 b‴ k (1) = −mλk bn (1) .

(11.13)

The sequence {λk }∞ 1 consists of the eigenvalues λk (m), which are the roots of the transcendental characteristic equation: 1 + cosh λ cos λ + mλ(sinh λ cos λ − cosh λ sin λ) = 0 .

(11.14)

The eigenfunctions bk (x) of the boundary problem (11.13) have the form: bk (x) = cosh λk x − cos λk x −

cosh λk − cos λk (sinh λk x − sin λk x) . sinh λk − sin λk

(11.15)

Note that the positive eigenvalues λk (k ∈ ℤ+ ) and the corresponding eigenfunctions bk (x) should be only considered due to the symmetry of equations (11.14), (11.15). The functions bk (x) are orthogonal in the space L2 (0, 1) with the inner product 1

⟨bi , bj ⟩ = ∫ bi (x)bj (x)ρ∗ (x) dx = δi,j ‖bi ‖2 ,

(11.16)

0

where ρ∗ (x) is the weighting coefficient and δi,j is Kronecker symbol. After multiplying equation (11.11) by Xn (x)ρ∗ (x), integrating on x, and dividing by ‖bk ‖2 , the countable system of equations for the Fourier coefficients yk (t) is derived. By taking into account equations (11.12) and (11.16), this system has the form: yk̈ + λk4 yk = Ck α̈ Ck = −

for k ∈ ℤ+ ,

1

2 1 , ∫ b (x)ρ∗ (x)x dx = − 2 ‖bk ‖2 0 k λk ‖bn ‖2

‖bk ‖2 = 1 + m(

(11.17) 2

sinh λk cos λk − cosh λk sin λk ) . sinh λk + sin λk

̈ is a known function of time, the Fourier coefficients yk (t) from equaIf α(t) tion (11.11) can be found as yk (t) =

Ck t ̈ dτ . ∫ sin λk2 (t − τ)α(τ) λk 0

(11.18)

Let us integrate equation (11.7), take into account the previous simplifications, and return to the old time t = τt∗ from equation (11.10). After that, substitute the values of the second and third derivatives with respect to time for the displacement function w(t, x) into equation (11.7). After integration by parts, the following PDE is obtained: A

d3 α d2 α dα G H − w (t, 0) − 2 wxx (t, 0) = u∗ (t) , +B 2 +D 3 dt dt dt τ2 txx τ

A = L0 J0 ν2 ,

B = RJ0 ν2 ,

D = F 2 ν2 ,

G = ρL4 L0 ,

H = ρL4 R ,

u∗ (t) = Fνu(t) .

(11.19)

11.1 Optimal rotations of an electromechanical manipulator |

263

In accordance with equations (11.12), (11.18), the value α̈ are included in the w(t, x) (as well as wtxx (t, 0) and wxx (t, 0)). According to (11.12), equation (11.19) can also be considered as a differential, containing the sums of yṅ and yk . The infinite dimensional system of ODEs with constant coefficients is obtained together with the system (11.17). Indeed, by substituting the values wtxx (t, 0) and wxx (t, 0) of equation (11.12) into equation (11.19) and taking into account equations (11.15), (11.17), the countable system of equations with initial and terminal conditions (according to equation (11.9)) can be represented as J[u∗ ] → min , ∗ ∗ A

3

u ∈𝒰 2

dα dα d α − τ−2 ∑ (Gyk̇ + Hyk ) = u∗ (t) , +B 2 +D dt 3 dt dt k=1 ∞

yn̈ + τ−2 λn4 Θn = Cn α̈ α(0) = α0 ,

for n ∈ ℤ+ ,

(11.20)

α(T) = αT ,

̈ = α(0) ̇ = yn (0) = yṅ (0) = 0 . α(0) 11.1.3 Projection approach to the problem on link motions Consider the approach based on the MIDR, which has been described in Section 6.3. Restrict ourselves to the case when the transients in the drive do not have a significant influence on the motion of the elastic link, and assume that the inductance of the motor coil is negligible (L0 = 0). Introduce new dynamic variables, namely, the linear density of momentum p(t, x) and the bending moment in the cross section of the rod s(t, x) according to ̈ v ∶= ρ(x)(wt + xα(t)) −p=0

and

q ∶= κ(x)wxx − s = 0 ,

(11.21)

where v and q are the constitutive functions. The equation of the rod rotation in the new variables takes the form: pt + sxx = 0 .

(11.22)

This equality is satisfied automatically if the new unknown functions, kinematic ̃ x), are introduced so that ̃ x) and dynamic r(t, w(t, ̃ x) , w = w0 (x) + w(t, ̃ (t, x) , p = ρ(x)v0 (x) + r xx

s = m0 (x − L)pt (t, L) − r t̃ (t, x) .

(11.23)

Here, w0 and v0 are the initial distribution of the elastic displacements and velocities of the rod points from equation (11.3), respectively.

264 | 11 Modeling and control in mechatronics The boundary conditions (11.2) can be expressed in terms of the functions w̃ and r ̃ in the form: ̃ L) = r x̃ (t, L) = 0 . ̃ 0) = w̃ x (t, 0) = r(t, w(t,

(11.24)

The initial conditions (11.3) are given by (11.25)

̃ x) = 0 . ̃ x) = r(0, w(0,

For L0 = 0, the ODE (11.19) with respect to the angle of the link rotation α can be written in the new variables as ̃ , J 0̃ (α̈ + τd−1 α)̇ − s(t, 0) = u(t) ũ = R−1 Fνu(t) ,

τd = RJ0 F −2 ν−2 ,

J 0̃ = J0 ν2 ,

(11.26)

where ũ is the new control function with the dimension of the moment, τd is the characteristic time of electric drive transients, J 0̃ is the effective moment of drive inertia. The tilde sign is further omitted. The orientation of the inertial coordinate system OX1 X2 X3 (see Figure 11.1) can be always chosen so that the initial angle of the link rotation is equal to zero. Then the initial conditions of equation (11.4) take the form: α(0) = 0

and

̇ = ω0 . α(0)

(11.27)

An approximate solution of the problem (11.21), (11.23)–(11.27) is found based on the MIDR and the FEM as it has been discussed in Sections 6.2 and 6.3. To compose a joint system of ODEs, the integral projections of constitutive functions are equated to zero according to L

∫ v(t, x)g(x)μ(x) dx = 0 , 0

μ=1+

m0 δ(x − 1) . ρ(1)

L

∫ q(t, x)g(x) dx = 0 , 0

∀g ∈ 𝒮g ,

By accounting for a point mass m0 attached to the end of the rod, the first integral projection is taken with the dimensionless weighting factor μ, where δ is the delta function. The fields r,̃ w,̃ and g are defined as piecewise polynomials with respect to x: ̃ x) ∶ w̃ = w̃ ∈ 𝒮w = {w(t,

∑

j+k=N+1

wi,j (t)bj,k (zi (x)) ,

̃ 0) = w̃ x (t, 0) = 0}, x ∈ Ii , i = 1, … , M ; w̃ ∈ C 1 , w(t, ̃ x) ∶ r ̃ = r ̃ ∈ 𝒮p = {r(t,

∑

j+k=N+1

ri,j (t)bj,k (zi (x)) ,

̃ 1) = r x̃ (t, 1) = 0}, x ∈ Ii , i = 1, … , M ; r ̃ ∈ C 1 , r(t, g ∈ 𝒮g = {g(x) ∶ g =

∑

j+k=N−1

gi,j bj,k (zi (x)) , x ∈ Ii , i = 1, … , M},

11.1 Optimal rotations of an electromechanical manipulator |

where Ii = [xi−1 , xi ] with x0 = 0

and xi =

iL , M

zi =

265

x − xi−1 . xi − xi−1

The scaled control voltage u(t) introduced in equation (11.26) is given in the class of polynomial functions: K

u = u0 + ∑ uk t k k=1

with u = (u1 , … , uK ) ,

(11.28)

where u ∈ ℝK is the vector of control parameters and the coefficient u0 is used to satisfy the terminal condition (see below). The aim is to find a control u(t) that turns the electromechanical link from the initial position to the final one in a finite time T, i.e., α(T) = αT and minimizes the following energy functional: WT [u] → min , u

WT = W1 + γW2 ,

γ ≥ 0,

L

W(t) = ∫ ψ(t, x) dx + 0

Φ = ∫ φ(t, x) dΩ , Ω

W1 = W(T) ,

J0 α2̇ (t) , 2

W2 = T −1 Φ ,

1 μp2 2 ψ= ( + κwxx ), 2 ρ

1 μv2 q2 + ). φ= ( 2 ρ κ

Here, W1 is the terminal energy of the link, which includes the total mechanical energy of an elastic link as well as the kinetic energy of the point mass and the rotor of the electric motor, W2 is the integral error of the approximated system, expressed in energy units, γ is the weighting coefficient that allows us to regulate the computational error, Ψ is the time integral of rod energy density with adjoined mass m0 , and φ is the distribution of discretization error. In this statement, the total mechanical energy of the system at the end of the process is minimized by taking into account the function φ. In accordance with the algorithm described in Section 10.1, the optimal vector of control parameters u∗ and the control function u∗ (t) = u(t, u∗ ) corresponding to this vector are constructed. After that, the following approximations are found: the lateral displacements w̃ ∗ (t, x) = w(t, x, u∗ ), the momentum density p̃ ∗ (t, x) = p(t, x, u∗ ), and bending moments s̃∗ (t, x) = s(t, x, u∗ ). To represent the elastic motions of a specific electromechanical link, choose the dimensionless parameters of the system: L = 1, m0 =

x 4 κ = (1 − ) , 8

1 L ∫ ρ dx , 10 0

x 2 ρ = (1 − ) , 8 1 L 2 J0 = ∫ ρx dx , 10 0

τd = 0.1 .

266 | 11 Modeling and control in mechatronics Let us fix the parameters of approximation M = N = 4 and control T = 3, K = 8, αT = 1, γ = 10. This means that the inhomogeneous rod (tapering to the loaded end) is turned at one radian in the controlled process. The differential order of the approximation for ODE system is equal to MN + 2 = 34. By using a polynomial control of equation (11.28), one succeeds to reach a rather small quantity of the terminal energy W1 = 1.8 ⋅ 10−10 as compared with the average energy distributed in the elastic rod, W̄ = 0.743. The relative error obtained for the −1 optimal process does not exceed Δ = W̄ W2 = 1.1 ⋅ 10−10 . In Figure 11.2, the solid curve shows the voltage u∗ (t) supplied to the motor winding. The optimal control signal has two alterations of the sign and vanishes gradually at the end of this motion. The rotational angle of the rod α(t) is presented by the dashed ̇ is given by the dot-dash curve. As it is seen, the curve, and the angular velocity α(t) link has the reversing displacements at the beginning of the process. The elastic displacements of the rod w(t, x) are shown in Figure 11.3, whereas the distribution of the linear density of momentum corresponding to these displacements are presented in Figure 11.4. The elastic vibrations arising in the optimal control process as well as the maximum momentums are rather large in the central part of the

Figure 11.2: Optimal control u(t), angle of rotation ̇ α(t), and angular velocity α(t).

Figure 11.3: Elastic displacements w(t, x) of the rod points.

11.1 Optimal rotations of an electromechanical manipulator |

267

Figure 11.4: Distribution of the linear momentum density p(t, x).

chosen time interval. Near the terminal instant t = T, the vibrations vanish what corresponds to the low level of the terminal energy W1 discussed above. The energy distribution in the rod ψ(t, x) is shown in Figure 11.5. Note that the link begins and ends its motion in a smooth way. Such rotation does not cause noticeable high-frequency vibrations in the system. This fact is reflected in the value of local computation error φ(t, x) presented in Figure 11.6. Here, the function of error has high

Figure 11.5: Linear density of the rod energy ψ(t, x).

Figure 11.6: Distribution of the local error φ(t, x).

268 | 11 Modeling and control in mechatronics ridges at the ends of the rod and at the nodes of the mesh. But this circumstance does not significantly affect the overall high quality of the solution.

11.2 Control of a flexible structure with viscoelastic links Motions of a flexible electromechanical structure consisting of two joined beams clumped on a mobile carriage controlled with an electric drive are discussed in this section. An optimal control problem is considered to transfer the structure from its initial state to terminal one in a fixed time and to minimize the mean mechanical energy of the beams. The numerical solution of the inverse dynamic problem, which is based on the method of integrodifferential relations and a model reduction technique, is presented. In the current section, the MIDR is combined with the finite element technique developed in [87] for hybrid dynamic systems with distributed and lumped parameters. This technique is extended to modeling and optimal control of rack feeder systems with distributed viscoelastic and inertial parameters, which have already been considered by using an alternative system representation in [5].

11.2.1 Controlled mechanism with flexible links The experimental setup built up at the Chair of Mechatronics of the University of Rostock represents the structure of a typical high bay rack feeder as pictured in Figure 11.7. The flexible structure consists of two identical beams clamped vertically to a horizontally movable carriage. Plane motions of the system are described in the non-inertial Cartesian coordinates with the vertical axis Oy2 and the horizontal axis Oy1 . The origin O is located equidistantly from the points where the beams are attached to the carriage. Both beams are rigidly connected at their tips by a pulley block, which is necessary for the vertical positioning of a cage. The cage is a payload sliding along beam 1. In addition, the beams are hingedly coupled by means of two rigid rods as shown in the figure. Motions of the structure’s elements are considered in the frame of the Euler– Bernoulli beam theory. The following parameters of the structure are used in the subsequent modeling: the length of the beams L = 1.07 m, their cross section area A = 3.1 ⋅ 10−4 m2 , their moment of area I = 2.14 ⋅ 10−9 m4 , the distance 2b = 2.45 ⋅ 10−4 m between the beams, Young’s modulus E = 70 GPa, the volume density ρ = 2700 kg m−3 of the beam material, the viscosity coefficient μ = 65 MPa s, the masses of the pulley block mp = 0.95 kg and the cage mc = 0.91 kg, their moments of inertia Ip = 2.0 ⋅ 10−3 kg m2 and Ic = 1.0 ⋅ 10−3 kg m2 , the vertical position L2 = L/2 of the cage, as well as the heights L1 = L/3 and L3 = 2L/3 of the hinging. The drive force acted on the carriage is supposed

11.2 Control of a flexible structure with viscoelastic links | 269

Figure 11.7: Beam structure of a high bay rack feeder.

to be a scalar control input in this dynamic system. The transient time of the drive is τ = 0.019 s. 11.2.2 Optimal control problem of structure motion The projection approach based on the MIDR [56] is applied to the IBVP of structure motions with the generalized constitutive relations given in the form: ∫ (q(t, y) ⋅ v(t, y) + g(t, y) ⋅ r(t, y)) dΩ = 0 , Ω

pt (t, y) = 𝒟s(t, y) + f (t, y) 0

w(0, y) = w (y),

∀v, r ∈ L2 (Ω; ℝn ) ;

for (t, y) ∈ Ω = (0, T) × (0, 1) ;

p(0, y) = p0 (y) ;

𝒜a w(t, a) + ℬa s(t, a) = u(t)ca + f a (t) ,

(11.29)

a ∈ {0, 1} .

Here, t denotes the time and y is the common spatial coordinate unified by the corresponding scaling for all structure finite elements (segments of beam 1 or 2). The following constitutive vector-valued functions of t and y: q ∶= p − M ⋅ w t

and g ∶= s − K ⋅ 𝒟w − R ⋅ 𝒟w t

(11.30)

are introduced in equation (11.29). The components of the operators 𝒟 and 𝒟 appearing in equations (11.29) and (11.30) are represented as [𝒟]i,j = δi,j

𝜕α 𝜕yα

and [𝒟]i,j = (−1)α+1 δi,j

𝜕α 𝜕yα

(11.31)

with α = 2 for i ≤ m (lateral motions) and α = 1 for i > m (longitudinal motions), where m is the number of elements. The unknown trial variables are the generalized dis-

270 | 11 Modeling and control in mechatronics placements w(t, y) ∈ ℝn of midline points for each structural element, the generalized momentum density p(t, y) ∈ ℝn , and the generalized forces s(t, y) ∈ ℝn (bending moments or normal loads). The dimension n = 2m appears for the plane motions. The test functions v(t, y) ∈ ℝn and r(t, y) ∈ ℝn are virtual velocities and displacements. The mass, stiffness, and damping matrices M(y), K(y), R(y), respectively, depend on the structural parameters mentioned in Subsection 11.2.1. The functions w 0 (y) and p0 (y) define the initial state of the structure. The forms of the linear operators 𝒜a and ℬa are chosen in accordance with the boundary and interelement conditions. The function u(t) ∈ ℝ is the control signal, ca ∈ ℝ3m is the input vector. External disturbances are introduce through the functions f (t, y) ∈ ℝn and f a (t) ∈ ℝ3m . The control objective under study is to move the flexible structure to a terminal state at the fixed time T and to minimize the quadratic cost function: ms T ∫ (u(t) − v(t))2 dt + γU , 2 0 1 p(T, y) = p1 (y) , U[w] = ∫ (𝒟w ⋅ K ⋅ 𝒟w) dΩ , 2T Ω

J[u∗ ] = min J[u] with J = 2 u∈L (0,T)

w(T, y) = w 1 (y) ,

(11.32)

where U is the mean elastic energy of the structure, u∗ (t) is the optimal control funċ = u(t) − v(t) has to be taken into account, tion. The equation of the electric drive τv(t) where v(t) is the carriage velocity. The weight γ ≥ 0 in equation (11.32) is dimensionless, since the first term in the cost function is multiplied by the characteristic mass ms = mp + mc + 2LAρ. The functions w 1 (x) and p1 (x) define the terminal conditions. 11.2.3 Finite element algorithm The piecewise polynomial approximations w̃ i , s̃i , p̃ i , ṽ i , r ĩ to the corresponding components wi , si , pi , vi , ri , of both trial functions w, p, s and test functions v, r are given according to w̃ i (t, y) = wi0 (y) + ∑ bj,k (y)wi,j (t) , j+k=α

sĩ (t, y) = ∑ bj,k (y)si,j (t) j+k=α

with α = l + 1 for i ≤ m and p̃ i (t, y) = p0i (y) +

α = l for m < i ≤ n ,

∑ bj,k (y)pi,j (t) ,

(11.33)

j+k=l−1

ṽi (t, y) = ∑ bj,k (y)vi,j (t) , j+k=l−1

r ĩ (t, y) = ∑ bj,k (y)ri,j (t) . j+k=l−1

A consistent semi-discretization scheme is worked out [5]. In equation (11.33), wi,j (t), pi,j (t), si,j (t) are unknown functions of time, vi,j (t), ri,j (t) are arbitrary test functions.

11.2 Control of a flexible structure with viscoelastic links | 271

j

The Bernstein polynomials are given by bj,k (y) = Ci+k yj (1 − y)k ; the number l defines the polynomial degree. An explicit energy criterion is used to estimate the solution quality: Δ=

Φ ≪ 1, Ψ

Φ = ∫ (q̃ ⋅ M−1 ⋅ q̃ + g̃ ⋅ K−1 ⋅ g)̃ dΩ , Ω

̃ dΩ Ψ = ∫ (w̃ t ⋅ M ⋅ w̃ t + 𝒟w̃ ⋅ K ⋅ 𝒟w)

(11.34)

Ω

with q̃ = p̃ − M ⋅ w̃ t

and g̃ = s̃ − K ⋅ 𝒟w̃ − R ⋅ 𝒟w̃ t .

The ratio Δ of the functional Φ and energy integral Ψ means the relative error of nũ y) ∈ ℝn in equation (11.34) denotes ̃ y) ∈ ℝn and g(t, merical solutions. The notation q(t, approximations of the constitutive functions introduced in equation (11.30). The resulting system of linear DAEs can be represented as ̇ + B ⋅ z(t) = cu(t) + d(t) A ⋅ z(t) na

z(t), d(t), c ∈ ℝ , n

2m

α

for A ⋅ z(0) = 0 ;

na = 3(l + 2)m , l−1

{zi }1 a = ⋃(⋃{wi,j , si,j } ⋃{pi,j }) , α=l+1

i=1 j=0

j=0

for i ≤ m and α = l

(11.35)

for m < i ≤ 2m ,

where z(t) denotes the state vector, u(t) ∈ ℝ is the scalar control, c is the input vector, d(t) is disturbances. It is possible to reduce the system (11.35) after identical transformation to the diagonal form: ̇ = L ⋅ x(t) + u(t)b + e(t) x(t) z(t) = V ⋅ x(t) ∈ ℝna , [L]i,j = δi,j λi ,

for x(0) = 0 ;

x(t) ∈ ℝnd ,

Re λi ≥ Re λj

nd = rank A ≤ 4lm ,

(11.36)

for i < j .

Here, V is the eigenvector matrix, λi denote the eigenvalues, b is a new input vector, e(t) is a new disturbance vector.

11.2.4 Numerical simulation of structure dynamics Results of numerical simulations of the system (11.36) are obtained by using the following approximation parameters: m = 7, l = 3, na = 76, nd = 168. Some spectral data of the considered structure are presented in Table 11.1. In agreement with the Kelvin–Voigt viscoelastic model, only 10 modes are oscillatory for the given coefficient of viscosity, all the others are aperiodic. The errors Δλi are found as the discrepancy between the

272 | 11 Modeling and control in mechatronics Table 11.1: Eigenvalues λi of the structure. i 1 2, 3 4, 5 6 7, 8 9, 10 11, 12 >12

Re λi , s−1

|Im λi |, s−1

0 −0.727 −18.15 −52.63 −223.7 −349.6 −920.5