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E-raamat: Non-Smooth Deterministic or Stochastic Discrete Dynamical Systems: Applications to Models with Friction or Impact

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 18-Mar-2013
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781118604328
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Non-Smooth Deterministic or Stochastic Discrete Dynamical Systems: Applications to Models with Friction or ImpactSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 18-Mar-2013
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781118604328
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This book contains theoretical and application-oriented methods to treat models of dynamical systems involving non-smooth nonlinearities. The theoretical approach that has been retained and underlined in this work is associated with differential inclusions of mainly finite dimensional dynamical systems and the introduction of maximal monotone operators (graphs) in order to describe models of impact or friction. The authors of this book master the mathematical, numerical and modeling tools in a particular way so that they can propose all aspects of the approach, in both a deterministic and stochastic context, in order to describe real stresses exerted on physical systems. Such tools are very powerful for providing reference numerical approximations of the models. Such an approach is still not very popular nevertheless, even though it could be very useful for many models of numerous fields (e.g. mechanics, vibrations, etc.). This book is especially suited for people both in research and industry interested in the modeling and numerical simulation of discrete mechanical systems with friction or impact phenomena occurring in the presence of classical (linear elastic) or non-classical constitutive laws (delay, memory effects, etc.). It aims to close the gap between highly specialized mathematical literature and engineering applications, as well as to also give tools in the framework of non-smooth stochastic differential systems: thus, applications involving stochastic excitations (earthquakes, road surfaces, wind models etc.) are considered.

Contents

1. Some Simple Examples. 2. Theoretical Deterministic Context. 3. Stochastic Theoretical Context. 4. Riemannian Theoretical Context. 5. Systems with Friction. 6. Impact Systems. 7. ApplicationsExtensions.





About the Authors

Jérôme Bastien is Assistant Professor at the University Lyon 1 (Centre de recherche et d'Innovation sur le sport) in France. Frédéric Bernardin is a Research Engineer at Département Laboratoire de Clermont-Ferrand (DLCF), Centre d'Etudes Techniques de l'Equipement (CETE), Lyon, France. Claude-Henri Lamarque is Head of Laboratoire Géomatériaux et Génie Civil (LGCB) and Professor at Ecole des Travaux Publics de l'Etat (ENTPE), Vaulx-en-Velin, France.
Introduction xi
Chapter 1 Some Simple Examples
1(26)
1.1 Introduction
1(1)
1.2 Frictions
1(15)
1.2.1 Coulomb's law
1(2)
1.2.2 Differential equation with univalued operator and usual sign
3(8)
1.2.3 Differential equation with multivalued term: differential inclusion
11(1)
1.2.4 Other friction laws
12(4)
1.3 Impact
16(6)
1.3.1 Difficulties with writing the differential equation
16(3)
1.3.2 Ill-posed problems
19(3)
1.4 Probabilistic context
22(5)
Chapter 2 Theoretical Deterministic Context
27(52)
2.1 Introduction
27(1)
2.2 Maximal monotone operators and first result on differential inclusions (in E)
27(18)
2.2.1 Graphs (operators) definitions
28(1)
2.2.2 Maximal monotone operators
29(4)
2.2.3 Convex function, subdifferentials and operators
33(5)
2.2.4 Resolvent and regularization
38(2)
2.2.5 Taking the limit
40(1)
2.2.6 First result of existence and uniqueness for a differential inclusion
40(5)
2.3 Extension to any Hilbert space
45(12)
2.4 Existence and uniqueness results in Hilbert space
57(2)
2.5 Numerical scheme in a Hilbert space
59(20)
2.5.1 The numerical scheme
59(1)
2.5.2 State of the art summary and results shown in this publication
60(1)
2.5.3 Convergence (general results and order 1/2)
61(6)
2.5.4 Convergence (order one)
67(5)
2.5.5 Change of scalar product
72(2)
2.5.6 Resolvent calculation
74(2)
2.5.7 More regular schemes
76(3)
Chapter 3 Stochastic Theoretical Context
79(50)
3.1 Introduction
79(1)
3.2 Stochastic integral
79(11)
3.2.1 The stochastic processes background
80(4)
3.2.2 Stochastic integral
84(6)
3.3 Stochastic differential equations
90(11)
3.3.1 Existence and uniqueness of strong solution
91(1)
3.3.2 Existence and uniqueness of weak solution
92(3)
3.3.3 Kolmogorov and Fokker-Planck equations
95(6)
3.4 Multivalued stochastic differential equations
101(3)
3.4.1 Problem statement
101(2)
3.4.2 Uniqueness and existence results
103(1)
3.5 Numerical scheme
104(25)
3.5.1 Which convergence: weak or strong?
106(2)
3.5.2 Strong convergence results
108(14)
3.5.3 Weak convergence results
122(7)
Chapter 4 Riemannian Theoretical Context
129(26)
4.1 Introduction
129(1)
4.2 First or second order
129(2)
4.3 Differential geometry
131(8)
4.3.1 Sphere case
131(1)
4.3.2 General case
132(7)
4.4 Dynamics of the mechanical systems
139(5)
4.4.1 Definition of mechanical system
139(2)
4.4.2 Equation of the dynamics
141(3)
4.5 Connection, covariant derivative, geodesics and parallel transport
144(4)
4.6 Maximal monotone term
148(1)
4.7 Stochastic term
149(2)
4.8 Results on the existence and uniqueness of a solution
151(4)
Chapter 5 Systems with Friction
155(170)
5.1 Introduction
155(1)
5.2 Examples of frictional systems with a finite number of degrees of freedom
155(60)
5.2.1 General framework
155(1)
5.2.2 Two elementary models
156(9)
5.2.3 Assembly and results in finite dimensions
165(28)
5.2.4 Conclusion
193(1)
5.2.5 Examples of numerical simulation
194(11)
5.2.6 Identification of the generalized Prandtl model (principles and simulation)
205(10)
5.3 Another example: the case of a pendulum with friction
215(16)
5.3.1 Formulation of the problem, existence and uniqueness
215(3)
5.3.2 Numerical scheme
218(1)
5.3.3 Numerical estimation of the order
219(2)
5.3.4 Example of numerical simulations
221(1)
5.3.5 Free oscillations
221(1)
5.3.6 Forced oscillations
221(1)
5.3.7 Transition matrix and calculation of the Lyapunov exponents
222(8)
5.3.8 Melnikov's method, transitory chaos and Lyapunov exponents
230(1)
5.4 Elastoplastic oscillator under a stochastic forcing
231(12)
5.4.1 Introduction
231(1)
5.4.2 Modeling
232(4)
5.4.3 Numerical scheme
236(2)
5.4.4 Numerical results
238(5)
5.5 Spherical pendulum under a stochastic external force
243(12)
5.5.1 Establishment of the model
243(5)
5.5.2 Numerical aspects
248(7)
5.6 Gephyroidal model
255(13)
5.6.1 Introduction
255(1)
5.6.2 Description and transformation of the model
256(7)
5.6.3 Quasi-static problems
263(2)
5.6.4 Numerical simulations
265(2)
5.6.5 Conclusion
267(1)
5.7 Chain
268(15)
5.7.1 Introduction
268(2)
5.7.2 Description of the model
270(1)
5.7.3 Transformation of the equations
271(12)
5.7.4 Conclusion
283(1)
5.8 An infinity of internal variables: continuous generalized Prandtl model
283(18)
5.8.1 Introduction
283(1)
5.8.2 Description of the continuous model
284(3)
5.8.3 Existence, uniqueness and regularity results
287(2)
5.8.4 Application to the discrete case, and convergence of the discrete model to the continuous model
289(2)
5.8.5 Numerical scheme
291(2)
5.8.6 Study of hysteresis loops
293(8)
5.8.7 Numerical simulations
301(1)
5.9 Locally Lipschitz continuous spring
301(24)
5.9.1 Introduction
301(1)
5.9.2 The studied model
301(2)
5.9.3 Results for the existence and uniqueness of the solutions
303(8)
5.9.4 Convergence results for the numerical schemes
311(2)
5.9.5 The locally Lipschitz continuous case
313(1)
5.9.6 Identification of the parameters from the hysteresis loops
314(6)
5.9.7 Numerical simulations
320(5)
Chapter 6 Impact Systems
325(30)
6.1 Existence and uniqueness for simple problems (one degree of freedom)
326(22)
6.1.1 The work of Schatzman-Paoli
326(1)
6.1.2 Simple case with one degree of freedom, forcing and impact: piecewise analytical solutions
327(2)
6.1.3 Adaptation of some classical methods
329(4)
6.1.4 Movement with the accumulation of impacts and a sticking phase
333(4)
6.1.5 Behavior of the numerical methods
337(1)
6.1.6 Convergence and order of one-step numerical methods applied to non-smooth differential systems
338(5)
6.1.7 Results of numerical experiments
343(5)
6.2 A particular behavior: grazing bifurcation
348(7)
6.2.1 Approximation of the map in the general case
349(1)
6.2.2 Particular case
350(1)
6.2.3 Stability of the non-differentiable fixed point
351(2)
6.2.4 Numerical example
353(2)
Chapter 7 Applications-Extensions
355(76)
7.1 Oscillators with piecewise linear coupling and passive control
355(23)
7.1.1 Description of the model
356(1)
7.1.2 Free oscillations of the system
356(6)
7.1.3 Order 1
362(4)
7.1.4 Case of periodic forcing
366(11)
7.1.5 Conclusion
377(1)
7.2 Friction and passive control
378(8)
7.2.1 Introduction
378(1)
7.2.2 Introduction to the models: smooth and non-smooth systems
379(7)
7.3 The billiard ball
386(4)
7.3.1 Maximal monotone framework
386(3)
7.3.2 More realistic but non-maximal monotone framework
389(1)
7.4 An industrial application: the case of a belt tensioner
390(6)
7.4.1 The theory
390(2)
7.4.2 The tensioner used
392(1)
7.4.3 Identification of the parameters
392(1)
7.4.4 Validation
393(3)
7.5 Problems with delay and memory
396(4)
7.5.1 Theory
396(3)
7.5.2 Applications
399(1)
7.6 Other friction forces
400(23)
7.6.1 More general forms (variable dynamical coefficient)
401(18)
7.6.2 With a variable static coefficient
419(2)
7.6.3 With variable static and dynamical coefficients
421(2)
7.7 With the viscous dissipation term
423(1)
7.8 III-posed problems
424(7)
7.8.1 First model: limit of a well-posed friction law
426(1)
7.8.2 Second model: a differential inclusion without uniqueness
427(2)
7.8.3 Conclusion
429(2)
Appendix 1 Mathematical Reminders
431(12)
A1.1 Two Gronwall's lemmas
431(1)
A1.2 Norms, scalar products, normed vector space, Banach and Hilbert space
432(1)
A1.2.1 Scalar products, norms
432(1)
A1.2.2 Banach and Hilbert space, separable space
433(2)
A1.3 Symmetric positive definite matrices
435(1)
A1.4 Differentiable function
435(1)
A1.5 Weak limit
436(1)
A1.6 Continuous function spaces
436(1)
A1.7 Lp space of integrable functions
437(1)
A1.7.1 Lp(Ω) space
437(1)
A1.7.2 Lp(Ω Rq) space
438(1)
A1.7.3 Lp(Ω H) spaces
438(1)
A1.8 Distributions
439(1)
A1.8.1 Real values distributions
439(1)
A1.8.2 Distributions with values in Rq
440(1)
A1.8.3 Distributions with values in Hilbert space
440(1)
A1.9 Sobolev space definition
441(1)
A1.9.1 Functions with real values
441(1)
A1.9.2 Functions with values in Hilbert space
441(2)
Appendix 2 Convex Functions
443(4)
A2.1 Functions defined on R
443(3)
A2.2 Functions defined on Hilbert space
446(1)
A2.2.1 Any Hilbert space
446(1)
A2.2.2 Particular case of the finite dimension
446(1)
Appendix 3 Proof of Theorem 2.20
447(8)
Appendix 4 Proof of Theorem 3.18
455(12)
Appendix 5 Research of Convex Potential
467(10)
A5.1 Method used
467(1)
A5.2 Lemma 5.1
468(5)
A5.3 Lemma 5.4
473(3)
A5.4 Lemma 7.1
476(1)
Bibliography 477(18)
Index 495
Claude-Henri LAMARQUE, Head of LGCB, Professor ENTPE, Vaulx-en-Velin, France.

Jérôme BASTINE, Assistant Professor at the University Lyon 1, Lyon, France.

Frédéric BERNARDIN, Research Engineer at CETE de Lyon (DLCF), Lyon, France.
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