摘要
利用解析和数值方法,以弹簧摆为对象讨论了线性的时滞位移反馈控制对一类平方非线性系统动力学行为的影响。根据多尺度法得到了1:2内共振情况下一次近似解的慢变方程,基于此讨论了反馈控制参数对零解的稳定性和周期解振幅的影响。结果表明:耦合的反馈项在平均方程中并不出现。根据罗斯-霍尔维茨判据发现,没有反馈控制时该系统的零解总是不稳定的,而通过调整反馈增益或反馈时滞就可以很容易地使零解稳定。反馈时滞对周期解振幅的影响呈现周期性,反馈增益或时滞发生变化时,周期解振幅的变化会表现出鞍结分岔现象;同时基于MATLAB软件的数值计算结果验证了该理论分析的正确性。
Based on the assumption that quadratic nonlinear terms and 1:2 internal resonance are considered,the influence of linear feedback control with time delay on the dynamics of spring-pendulum systems is investigated analytically and numerically.By means of the method of multiple scales,the slow-flow equations for the first order approximation,which can be used to discuss the stability of the trivial solution and the variation of amplitude of periodic solutions with control parameters,are obtained.It is shown that there are no the coupled feedback terms in the slow-flow equations.According to Routh-Hurwitz criterion,the trivial solution can be easily stabilized by feedback control although it is always unstable when no feedback control is applied to it.The influence of time delay in feedback control on amplitude of periodic solutions is periodic,and the saddle-node bifurcation may occur when control parameters are varied.Some simulations by MATLAB are presented to verify those analytical results.
出处
《应用力学学报》
CAS
CSCD
北大核心
2012年第4期421-425,486,共5页
Chinese Journal of Applied Mechanics
基金
国家自然科学基金(10872063)
关键词
弹簧摆
内共振
反馈控制
时滞
多尺度法
spring pendulums,internal resonance,feedback control,time delay,method of multiple scales.
