Vibration of Strongly Nonlinear Discontinuous Systems

1 Operators of Linear Systems.- 1. Dynamic Compliance.- 1.1. Operator of Mechanical System.- 1.2 Fundamental Features of the Generalised Dirac ?-function.- 1.3. Green Functions for Systems with Lumped Parameters.- 1.4. Operator of Dynamic Compliance.- 1.5. The Eigenmode Decomposition of the Dynamic Compliance Operator.- 1.6. Linear System as a Low-pass Filter.- 1.7. Linear Single-Degree-of-Freedom System.- 1.8. Operators of Rod Systems.- 1.9. Expression of Forces Through Operator Functions.- 1.10. Some Generalisations.- 2. Periodic Green Functions.- 2.1. Periodic Generalised Functions.- 2.2. Periodic Green Functions.- 2.3. Features of Periodic Green Functions.- 2.4. Periodic Green Function on the Interval of Periodicity.- 2.5. Single-Degree-of-Freedom System.- 2.6. Eigenfunction Expansion of PGF.- 2.7. Steady-state Motion.- 2.8. Representation of PGF in the Form of Fast Convergent Fourier Series.- 3 Parametric Periodic Green Functions.- 3.1. Integral Equations of Periodic Vibration.- 3.2. Integral Fredholm Equations.- 3.3. Description of Parametric Periodic Green Functions.- 3.4. Excitation of Parametric Vibration by Impacts.- 2 Strongly Nonlinear Single-Degree-of Freedom Systems.- 4 Conservative Systems.- 4.1. Classification of Nonlinear Systems.- 4.2. Equations of Conservative Systems.- 4.3. Vibro-impact Systems.- 4.4. Singular Force of Impact.- 4.5. Motions of Vibro-impact Systems.- 4.6. Strongly Nonlinear Systems.- 4.7. Strongly Nonlinear Systems of Threshold Type.- 4.8. Singularisation.- 4.9. Improved Singularisation.- 4.10. Piecewise Linear Force of Threshold Type.- 4.11. Threshold-type Force Defined by the Power Function.- 4.12. Symmetric Threshold-type Forces.- 5 Forced Vibration.- 5.1. Problem Statement.- 5.2. Change of Variables.- 5.3. Resonant Processes.- 5.4. Averaging in Systems with Impact Interactions.- 5.5. Steady-state Vibration and Stability.- 5.6. Vibro-impact Systems Under Harmonic Excitation.- 5.7. Exact Laws of Motion of Vibro-impact Systems.- 5.8. Resonance Vibration of a System with Piecewise Restoring Force.- 5.9. Piecewise Power Restoring Force.- 5.10. Principle of Energy Balance.- 5.11. Conditions of Existence of Resonant Regimes under Harmonic Excitation.- 5.12. Bifurcation of Fundamental Resonant Regimes under Polyharmonic Excitation.- 5.13. Bifurcation of Solutions in Vibro-impact System.- 5.14. Analysis of Superperiodic and Combination Resonances.- 6 Vibration in Autonomous Systems.- 6.1. Preliminary Considerations.- 6.2. Analysis of Autonomous Systems using the Averaging Method.- 6.3. Chatter.- 6.4. Analysis of the Autoresonant System.- 6.5. Quasi-isochronous Approximation.- 6.6. Symmetric Systems.- 7 Parametric Vibration.- 7.1. Preliminary Considerations.- 7.2. Resonant Regimes Outside the Zones of Instability of a Linear System.- 7.3. Integral Equation of Parametric Vibration.- 7.4. Resonant Regimes within the Zone of Instability of Linear Systems.- 7.5. Parametric Systems with Force Excitation.- 7.6. Energy Condition of Instability.- 7.7. Mathieu Equation with Strong Nonlinearity.- 7.8. System with Symmetric Nonlinearity.- 7.9. Calculations for Systems under Combined Excitation.- 7.10. Bifurcation of Regimes in Parametric Systems.- 7.11. Explicit Solutions to a Specific Class of Model Problems.- 8 Random Vibration.- 8.1. Preliminary Considerations.- 8.2. Some Exact Solutions.- 8.3. Random Vibration in Self-sustained System with Small Clearance.- 8.4. Contact Damping.- 8.5. Deviations from Solutions of Averaged Systems.- 8.6. Quasi-resonant Regimes.- 8.7. Parametric Systems in a Quasi-isochronous Approximation.- 8.8. Perturbed Periodic Green Functions.- 8.9. Application of Perturbed Periodic Green Functions to the Analysis of a Vibro-impact system.- 8.10. Narrowband Excitation.- 3 Multiple-Degree-of-Freedom Systems.- 9 Forced Vibration in Multiple-Degree-of-Freedom Systems.- 9.1. Preliminary Considerations.- 9.2. Integro-differential Equation of Periodic Regimes in System of Two Strongly Interacting Linear Subsystems.- 9.3. Newtonian Interaction.- 9.4. Principle of Energy Balance.- 9.5. Singularisation.- 9.6. Interaction of Two Systems with Lumped Parameters.- 9.7. Interaction of Rod Systems.- 9.8. Resonant Regimes in Systems with Arbitrary Dynamic Compliance Operators.- 9.9. Quasi-resonant Regimes.- 9.10. Analysis of Multi-dimensional Systems using Markov Processes.- 9.11 Systems with Relaxation.- 9.12. Single-frequency Vibration in Systems Given by the Operator Equation.- 9.13. Analysis of Symmetric Systems.- 10 Parametric Vibration in the Multiple-Degree-of-Freedom Systems.- 10.1. Method of Analysis.- 10.2. Equation of Energy Balance.- 10.3. Auxiliary Analysis.- 10.4. The Second Approximation for Impact Impulse.- 10.5. Parametric Vibration of an Oscillator Suspended Inertially Inside a Container.- 10.6. Dynamics of Vibro-impact Mechanisms Mounted on a Vibrating Base.- Additional Bibliography.- Appendix I The Averaging Method in Systems with Impacts.- Appendix II On the Analysis of Resonant Vibration of Vibro-impact Systems Using the Averaging Technique.- Appendix III Structure-borne Vibroimpact Resonances and Periodic Green Functions.- Appendix IV Nonlinear Correction of a Vibration Protection System Containing Tuned Dynamic Absorber.