Dynamical Behaviours Of The SD Oscillator With And Without Friction

SD oscillator is a geometrical nonlinear system admitting both smooth and discontinuous properties which can describe the large deformation and displacement in physics,mechanical engineering and aerospace. This oscillator has attracted the attention both in academic and engineering at home and abroad for a long period. With the rapid development of the science and technology, the geometrical nonlinear problems induced by friction are increasing prominent in the fields of engineering mechanics, mechanical dynamics, geomechanics and structural mechanics. However, the research on nonlinear friction dynamics remains at the hard starting stage because of limitation of the conventional nonlinear theory, the lack of systemic theory of friction, the unsteady of parameters in engineering, the neglect of weak nonlinear terms in modeling. We propose a geometrical nonlinear friction system of the SD oscillator with dry friction based upon the classical friction theory and SD oscillator with geometrical nonlinearity, which lays a important theoretical foundation for exploring the complex nonlinear behaviours and dynamical responses of the frictional systems with geometrical nonlinearity. This dissertation provides an important theoretical basis for engineering application by demonstrating a series of complex nonlinear dynamical phenomena of the SD oscillator with and without friction by employing analytical and numerical approaches. The main contents of this dissertation are listed as below.The periodic motions of the SD oscillator with and without a viscous damping and external excitation are discussed. The extended averaging method is employed by using the complete Jacobian elliptic integrals to obtain the primary resonance without any truncation for the perturbed SD oscillator, which are valid for both smooth and discontinuous stages. The stability of the periodic solutions of the system are analyzed by using the Lyapunov criteria. The distributions of the periodic solutions are investigated numerically which show a good agreement with the theoretical predictions.We investigate the stability of equilibrium sets of the SD oscillator with dry friction.The set of equilibrium states of the frictional system is analyzed together with Coulomb’s cone and differential inclusion. The theory of Lyapunov stability and Lasalle’s invariance principle are employed to derive the instability for the hyperbolic equilibrium set and the stability for the non-hyperbolic type. Analysis demonstrates the existence of a thick stable manifold of the hyperbolic equilibrium set due to the attractive sliding mode of the Filippov property, and shows that the unstable manifolds are that of the endpoints with their saddle property. Numerical simulations are carried out to present the bifurcations of the equilibrium set and the corresponding phase portraits.We propose a novel self-excited system with geometrical nonlinear friction called self-excited SD oscillator based upon the classical moving belt and a smooth and discontinuous(SD) oscillator. The moving belt friction is modeled as the Coulomb friction to formulate the mathematical model. The equilibrium states of the system are investigated, and bifurcations of the equilibrium are constructed to demonstrate the multiple stick regions, the hyperbolic structure transition, and the friction-induced asymmetry phenomena. The numerical simulations are carried out to present the multiple stick-slip period solution, multiple stick-slip chaotic solution and coexistence of the periodic and chaotic solutions of the system in the presence of the damping and external excitation.We investigate the stick-slip chaotic motion of the perturbed self-excited SD oscillator. The energy introduction or dissipation during the stick and slip modes in the original system are analyzed. The Melnikov’s method is employed to get the chaotic threshold of the perturbed self-excited SD oscillator, and two analytical criterions for chaos occurrence are obtained due to the asymmetry of the system, which are verified via some numerical simulations.The complex bifurcation behaviours of the self-excited SD oscillator with the Stribeck friction characteristic are presented. The collisions of tangent points are analytically explored to demonstrate the double tangency bifurcation by using the Filippov’s theory,and the bifurcation behaviours are presented in phase trajectory diagram. The sliding homoclinic bifurcations in the sense of Filippov occur as the change of the belt velocity. The subcritical Hopf-bifurcation of this system with damping is analyzed by making the examination of the equilibria and their stability, and the analytical criterions of the Hopf-bifurcation are obtained. Formal investigation in normal form of Hopf-bifurcation is discussed. Phase portraits are depicted to demonstrate the dynamical behaviours of subcritical Hopf-bifurcation and grazing bifurcation for the system.