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研究了含分数阶导数阻尼的一类线性系统在不同周期信号激励下系统的响应问题.首先在简谐信号的激励下,利用谐波平衡法得到了系统响应的近似解,这一结果和已有文献(申永军,杨绍普,邢海军2012物理学报61110505)的结果完全相同,但本文的求解过程大为简化,而且本文进一步扩展了分数阶导数阻尼微分阶数的取值范围.接着,利用傅里叶级数展开法和线性系统的叠加原理,求得了一般周期信号激励下系统响应的近似解,并以周期方波信号和周期全波正弦信号为例进行了说明.本文的结果表明,分数阶导数阻尼的微分阶数影响系统响应中各阶谐波的共振频率和共振振幅.系统响应的幅值与分数阶导数阻尼的微分阶数之间的单调关系主要受外激信号频率的影响.除解析分析外,本文还用数值模拟对相关结论进行了验证,两种结果符合良好,表明本文的分析方法是可行的.
The response of a class of linear system with fractional-order derivative damping under different periodic signal excitation is studied. First, the harmonic response is used to obtain the approximate solution of the system response under the excitation of harmonic signal. The results of the literature (Shen Yongjun, Yang Shaopu, Xing Haijun 2012 Physical Science 61110505) are completely the same, but the solving process of this paper is greatly simplified, and this article further extends the range of the fractional order derivative damping differential order .Furthermore, Leaf series expansion method and linear system superposition principle, the approximate solution of the system response under the general periodic signal excitation is obtained, and the periodic square wave signal and periodic full-wave sinusoidal signal are taken as an example to illustrate.The results of this paper show that the fractional order The derivative order of the derivative damping affects the resonance frequency and the resonance amplitude of each harmonic in the system response.The monotonic relationship between the amplitude of the system response and the derivative order of the fractional derivative damping is mainly affected by the frequency of the exogenous signal Analytical analysis, the paper also numerical simulation of the relevant conclusions were validated, the two results in good agreement, indicating that the analysis method is feasible.