Research On Nonlinear Vibration Characteristics Of Asymmetric Duffing Systems With Constant Excitation

In the vibration system,the asymmetry generally originates from the restoring force or the excitation force,and its existence will make the dynamic characteristics of the system more abundant.Therefore,it is of great significance to study the influence of asymmetric factors on the dynamic characteristics of the vibration system.In this paper,the main resonance and 1/2 subharmonic resonance of a strongly nonlinear Duffing equation and a Duffing rotor system under constant and harmonic excitations are studied,with focuses drawn on the nonlinear vibration characteristics of the system response caused by constant excitation.The details are as follows:First,the main resonance of the asymmetric Duffing equation is solved approximately analytically and its stability is analyzed.The phenomenon of vibration jump in the system response is discussed.The bifurcation set of the system about constant excitation and simple excitation frequency is calculated.The research shows that the bifurcation curve can be divided into five regions according to different constant excitation values,which correspond to the amplitude-frequency curves of five different behaviors.The increase of constant excitation makes the soft characteristics of the amplitude-frequency curve gradually dominate and mix up with the hard one.There are five steady-state solutions and complex vibration jumps at the intersection.Increasing the damping or decreasing the harmonic excitation amplitude eliminates multiple solutions and sudden jumps,making the amplitude-frequency curve appear approximately linear.Then,for the 1/2 subharmonic resonance of the asymmetrical Duffing equation,the approximately analytical solution and the stability analysis are given.The existence region of the subharmonic resonance solution in the plane of constant excitation-harmonic excitation amplitude is given,and the influences of the damping and the nonlinearity coefficients on the existence region are studied.Studies have shown that the presence of constant excitation is a necessary condition for 1/2 subharmonic resonance to occur.Increasing the damping or reducing the nonlinear coefficient can reduce the parameter region where the subharmonic resonance occurs;similar to the main resonance,with the constant excitationincreasing,the behavior of the amplitude-frequency curve under the sub-harmonic resonance gradually softens and mix with the hard characteristics,and the complex part of the multi-solution and vibration jump phenomenon appears at the intersection.Increasing the damping or reducing the harmonic excitation amplitude can attenuate the nonlinear characteristics exhibited by the amplitude-frequency curve.Afterwards,the periodic solution solution and the saddle-node bifurcation set analysis are performed for the main resonance of asymmetric Duffing rotor system.Studies have shown that the system's bifurcation curve for constant excitation and harmonic excitation frequency is based on the primary resonance of the single-degree-of-freedom Duffing equation,with a horizontal multi-feature and a hard-characteristic overall.It intersects with the vertical part;therefore,when the constant excitation is small,the two directions correspond to the main resonance area overlap,and the vertical direction amplitude-frequency curve adds a dislocation hard characteristic area;when it is in the middle range,the two resonance areas intersect.The vertical direction amplitude-frequency curve increases the overall hard characteristic and the middle dual-soft characteristic misalignment section.In this case,the multi-solution and vibration jump phenomenon is extremely complicated.When it is large,the two resonance regions are separated and the vertical-direction amplitude-frequency curve is similar to the case of single-degree-of-freedom Duffing equation,dislocations almost disappeared,with only soft characteristics remaining.As the damping increases or the amplitude of the harmonic excitation decreases,the softness characteristic of the amplitude-frequency curve gradually weakens or even disappears.In the end,only a narrow hard-feature region remains.Finally,numerical solutions and bifurcation characteristics analysis of 1/2subharmonic resonance of asymmetrical Duffing rotor system are performed.Studies have shown that when the energy consumed by damping and the energy provided by the excitation are in some kind of dynamic equilibrium,there are two types of quasi-periodic responses in the system with different evolutionary frequencies of the excitation frequency,and their irreducible frequency components are distributed over 1/2 times subharmonic frequency.