常数激励与简谐激励联合作用下Duffing系统的非线性振动

Nonlinear vibrations of Duffing system under the combination of constant excitation and harmonic excitation

以Duffing系统为研究对象,探讨常数激励与简谐激励联合作用下系统的骨架曲线及幅频响应特性,重点考察常数激励的影响。采用谐波平衡法求解该系统的振动方程,得到幅频响应关系,并给出了骨架曲线以及周期解的稳定性分析。采用幅频响应曲线和骨架曲线表征系统的基本动力学性质,讨论了常数激励和简谐激励幅值对系统幅频曲线性态和骨架曲线形态的影响。研究发现,系统振动响应中直流分量与谐波分量的振幅同步变化,但变化趋势相反。此外,两者骨架曲线的形态均是先向左微偏后转为向右弯曲,因此,在某些参数条件下,对应一个激励频率的周期解可能有5组,其中3组为稳定性,2组为不稳定解。进一步通过增大常数激励发现其能够对该系统造成的“刚度增强”效应,但同时也会伴随着愈加显著的“刚度渐软”特性。相应地,对于特定的简谐激励幅值,随着常数激励的增大,系统的幅频曲线能够由纯硬特性转变为软硬特性共存,甚至纯软特性。但是,在大尺度下观察,常数激励对系统骨架曲线的影响主要表现在骨架曲线根部的形态上,即随着简谐激励幅值的增大,常数激励对系统共振频率的影响变弱,不同常数激励下的骨架曲线趋于一致。

This paper focuses on the basic dynamical characteristics of a Duffing system under the combination of constant excitation and harmonic excitation.The Harmonic Balance method was employed to solve the motion equation of the system.The amplitude-frequency relationship was obtained,and the backbone curve and the stability of the obtained periodic solution were analyzed as well.The amplitude-frequency curves and the backbone curves were used to show the main dynamical characteristics of the system.These dynamical characteristics affected by the constant excitation and the amplitude of the harmonic excitation were discussed significantly.In the vibration response of the system,it was shown that when the excitation frequency increases the constant term changes synchronous with the amplitude of the harmonic component,but towards an opposite direction.Nevertheless,the backbone curves for them both bend slightly to the left at the first stage and bend rightward after that.As a result,in some parameter regions,the system may have at most five periodic solutions,three of them are stable and the other two are unstable.By increasing the constant excitation,an effect of“stiffness enhancement”is presenting in the system,but it is accompanied by a more significant“stiffness softening”characteristic.Consequently,for a certain harmonic excitation,the increasing of the constant excitation may change the amplitude-frequency curve from a pure soft spring characteristic to a coexistence of soft and hard spring characteristics,and even to a pure hard spring characteristic finally.In a larger scale,however,the influence of the constant excitation to the backbone curve is mainly reflected at the down part.In other word,with the increase of the harmonic excitation,the effect of the constant excitation on the excitation frequency becomes weak,the backbone curves for different values of constant excitation tend to be similar.