• Cover (1)
• Foundations of Engineering Mechanics (2)
  • Tractable Models of Solid Mechanics (4)
    • ISBN 9783642153716 (5)
    • Preface (6)
    • Abbreviations (10)
    • Contents (12)
• 1 Introduction: Historical Development of Tractable Models (16)
  • References (26)
• 2 Discrete Finite Systems (28)
  • 2.1 Linear Oscillators (28)
    • 2.1.1 Linear Conservative Oscillator (29)
    • 2.1.2 Linear Oscillator with Viscous Damping (30)
      • 2.1.2.1 Strong Energy Dissipation (ε >>1) (31)
      • 2.1.2.2 Weak Energy Dissipation (ε<<1) (34)
      • 2.1.2.3 Exercise (36)
    • 2.1.3 Linear Oscillator with Viscous Damping and Periodic (Harmonic) Forcing (36)
      • 2.1.3.1 Exercise (38)
    • 2.1.4 Two Coupled Oscillators (39)
      • 2.1.4.1 Weakly Coupled Oscillators with Strongly Different Frequencies (39)
      • 2.1.4.2 Exercise (42)
      • 2.1.4.3 Weakly Coupled Oscillators with Close Frequencies (42)
      • 2.1.4.4 Exercise (46)
      • 2.1.4.5 Strongly Coupled Oscillators with Essentially Different Frequencies (46)
      • 2.1.4.6 Exercise (48)
      • 2.1.4.7 Strongly Coupled Oscillators with Close Frequencies (49)
      • 2.1.4.8 Exercise (49)
  • 2.2 Single-DOF Nonlinear Oscillator (49)
    • 2.2.1 Quasilinear Oscillator (51)
    • 2.2.2 Strongly Nonlinear Oscillator and Vibro-Impact Approximation (54)
      • 2.2.2.1 Special Solutions for Vibro-Impact Motions (54)
      • 2.2.2.2 Vibro-Impact Systems Treatable by Analytic Functions (57)
    • 2.2.3 Oscillator with Multiple Equilibriums (61)
  • 2.3 Forced Nonlinear Oscillator (62)
    • 2.3.1 General Remarks (62)
    • 2.3.2 Governing Equations (63)
    • 2.3.3 The Dynamics of the Oscillator without Dissipation and the LPT (65)
    • 2.3.4 The Transient Dynamics of a Weakly Damped Oscillator (68)
      • 2.3.4.1 Non-smooth Temporal Transformations (69)
      • 2.3.4.2 The Construction of a Generating Solution (70)
      • 2.3.4.3 Calculation of A0(t0) and ω0(t0) (73)
    • 2.3.5 Quasi-Linear Oscillations (76)
  • 2.4 Entrainment, Synchronization and Resonance Capture (79)
    • 2.4.1 Pendulum with Constant External Torque (80)
    • 2.4.2 Entrainment of the Van der Pol Oscillator by External Harmonic Force (83)
    • 2.4.3 Synchronization of Oscillators and Related Models (90)
    • 2.4.4 Resonance Capture (92)
      • 2.4.4.1 Exercises (95)
    • 2.4.5 Forced Oscillator with Multiple States of Equilibrium (96)
  • 2.5 Symmetric Systems of Coupled Nonlinear Oscillators Beating Phenomena (99)
    • 2.5.1 Exercise (108)
  • 2.6 2DOF Systems of Nonlinear Oscillators with Essential Asymmetry Targeted Energy Transfer (TET) (109)
    • 2.6.1 Targeted Energy Transfer in an Unforced 2DOF System (109)
    • 2.6.2 Targeted Energy Transfer in Forced 2DOF System (120)
  • 2.7 Coupled Nonlinear Oscillators with Time Delays (148)
    • 2.7.1 Analytic Model (148)
    • 2.7.2 Numeric Verification -- Straight Modes (156)
    • 2.7.3 Numeric Verification -- "Oval"-- Modes and Phase -- Locked Solutions (159)
  • 2.8 Low-DOF Discrete Nonlinear Systems (162)
  • 2.9 Concluding Remarks (175)
  • References (176)
• 3 Infinite Discrete Systems (182)
  • 3.1 Dynamics of Infinite Nonlinear Chains (182)
    • 3.1.1 Long-Wavelength Approximation. Equation of Supersonic Extension Solitons in an Infinite FPU Chain (182)
    • 3.1.2 Zigzag Chain and Long-Wave Solitons (184)
    • 3.1.3 Envelope Solitons (186)
    • 3.1.4 Optical Breathers in a Zigzag Chain (191)
    • 3.1.5 Torsional Solitons (201)
    • 3.1.6 Approximation of Immobile Neighbour Chains (203)
  • 3.2 Dynamics of Essentially Nonlinear and Vibro-Impact Chains (207)
    • 3.2.1 Oscillatory Chain with Rigid Barriers (208)
    • 3.2.2 Discrete Breathers in a Vibro-Impact Chain (213)
      • 3.2.2.1 System of Klein-Gordon (KG) Type (214)
      • 3.2.2.2 System of Fermi-Pasta-Ulam (FPU) Type (218)
  • 3.3 The Problem of Heat Conduction in Dielectrics (222)
  • 3.4 Solitons in Energetically Nondegenerate Quasi-One-Dimensional Models (232)
    • 3.4.1 Quasi-One-Dimensional Model of a Molecular Crystal: Soliton Modes of Motion in a Bistable Nonlinear System (234)
  • 3.5 Dynamics of Ensembles of Interacting Nonlinear Chains (240)
  • 3.6 Concluding Remarks (248)
  • References (248)
• 4 Continuous Systems (252)
  • 4.1 One-Dimensional Models (252)
    • 4.1.1 Bolotin Model (252)
      • 4.1.1.1 Exercises (256)
    • 4.1.2 Simplification of the Timoshenko Beam (256)
  • 4.2 The Planar Dynamical Problem and Tractable One-Dimensional Models of an Elastic Solid (263)
    • 4.2.1 Exercises (268)
  • 4.3 The Two-Dimensional Orthotropic Model and Its Application to a Complex Contact Problem (268)
    • 4.3.1 Basic Asymptotic Decomposition of the Orthotropic Plate Problem (268)
    • 4.3.2 The Contact Problem for a Planar Orthotropic Strip (274)
  • 4.4 Models of Elastic Foundation (279)
    • 4.4.1 General Equations and Asymptotic Analysis (279)
    • 4.4.2 Example -- Dynamical Problem (283)
    • 4.4.3 Example -- An Axisymmetric Stamp (285)
  • 4.5 On the Concept of Solids (296)
  • 4.6 Models of Non-Fourier Heat Conduction (299)
  • 4.7 Concluding Remarks (306)
  • References (306)
• Afterword (310)
• Index (312)