Simultaneous Resonance Of Duffing Oscillator

Duffing oscillator,one of the most classic nonlinear oscillators,can describe many issues in engineering vibration,such as the vertical vibration of pantograph carbon strip suspension system and the nonlinear phenomena of rotor system,and the multiple-term frequency excitation is common case in these issues.For instance,the carbon belt suspension system of pantograph,it is excited by two kinds of harmonic excitations related to the distance between droppers and the span of catenary respectively.Moreover,there are many resonance phenomena in nonlinear dynamical system subjected to forced excitation,especially the excitation with multiple frequencies.Duffing oscillator subjected to the excitation with multiple frequencies may exhibit some complex resonance phenomena,such as the simultaneous resonance and combination resonance.The study of these resonance phenomena is significant both in theory and in engineering application,therefore,the simultaneous resonance of Duffing oscillator excited by two-term frequency is studied in this thesis,and it mainly including the following topics.(a)The background,significance and development about this thesis are elaborated,and the necessities of studying Duffing oscillators and introducing fractional calculus are explained.Moreover,the research methods,main works and highlights of this thesis are listed,respectively.(b)In the study on the primary and subharmonic simultaneous resonance of Duffing oscillator,the approximately analytical solution of the simultaneous resonance is obtained by the method of multiple scales,and the correctness and precision of the analytical solution are verified by numerical simulation.Then,Lyapunov’s first method is used to quantitatively calculate the stability condition of the steady-state response.Based on this,the stability,multi-value characteristic and frequency response of the simultaneous resonance are analyzed qualitatively.Furthermore,the effects of nonlinear factor on the steady-state response are analyzed by numerical simulation,and the results are discussed in depth.(c)In the study on the primary and super-harmonic simultaneous resonance of Duffing oscillator,the chaotic motion of the system is also investigated due to the fact that the approximate solution obtained by the singular perturbation method is not sufficient to describe the global characteristics of the system.The necessary condition for the chaos in the sense of Smale horseshoes is derived based on the Melnikov method,and the approach to chaos is studied by numerical simulation.Moreover,the relationship between the analytical and the numerical results of chaotic threshold is discussed,and it is found that the two results are qualitatively the same although there is quantitative difference.(d)In the study on the primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator,the difference and relationship between the conventional Duffing oscillator and the fractional-order one are discussed respectively.Moreover,the effects of fractional-order term on the steady-state response are also analyzed by numerical simulation,and the general effects mechanism of the fractional-order term on the mass-spring-damping(MSD)system is revealed.Finally,the main results of this thesis are summarized,and several issues worthy of further study are also put forward.