Vibration of Strongly Nonlinear Discontinuous Systems

Among the wide diversity of nonlinear mechanical systems, it is possible to distinguish a representative class of the systems which may be characterised by the presence of threshold nonlinear positional forces. Under particular configurations, such systems demonstrate a sudden change in the behaviour of elastic and dissipative forces. Mathematical study of such systems involves an analysis of equations of motion containing large-factored nonlinear terms which are associated with the above threshold nonlinearity. Due to this, we distinguish such discontinuous systems from the much wider class of essentially nonlinear systems, and define them as strongly nonlinear systems'. The vibration occurring in strongly nonlinear systems may be characterised by a sudden and abrupt change of the velocity at particular time instants. Such a vibration is said to be non-smooth. The systems most studied from this class are those with relaxation (Van Der Pol, Andronov, Vitt, Khaikhin, Teodorchik, etc. [5,65,70,71,98,171,181]), where the non-smooth vibration usually appears due to the presence of large nonconservative nonlinear forces. Equations of motion describing the vibration with relaxation may be written in such a manner that the highest derivative is accompanied by a small parameter. The methods of integration of these equations have been developed by Vasilieva and Butuzov [182], Volosov and Morgunov [190], Dorodnitsin [38], Zheleztsov [201], Mischenko and Rozov [115], Pontriagin [137], Tichonov [174,175], etc. In a system with threshold nonlinearity, the non-smooth vibration occurs due to the action of large conservative forces. This is distinct from a system with relaxation.

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.