分数阶拟周期Mathieu方程的动力学分析

DYNAMIC ANALYSIS OF QUASI-PERIODIC MATHIEU EQUATION WITH FRACTIONAL-ORDER DERIVATIVE

分数阶微积分有着诸多优异的特点,目前在动力学领域主要用来提高非线性系统振动特性研究的准确性.本文在拟周期Mathieu方程的基础上,引入分数阶微积分理论,研究了分数阶微分项参数对方程稳定性的影响.首先,采用摄动法得到方程稳定区和非稳定区分界线(即过渡曲线)近似表达式,利用数值方法验证了解析结果的准确性,图像显示两者吻合较好.随后,通过归纳总结不同情况下的过渡曲线近似表达式,发现在系统中分数阶微分项以等效线性刚度和等效线性阻尼的方式存在.根据这一特点,得到了系统等效线性阻尼和等效线性刚度的一般形式,并且定义了非稳定区域厚度.最后,通过数值仿真直观地分析了分数阶微分项参数对方程稳定区域大小和过渡曲线位置的影响.结果发现,分数阶微分项不仅具有阻尼特性还具有刚度特性,并且以等效线性刚度和等效线性阻尼的方式影响着方程稳定区域大小和过渡曲线位置.合理选择分数阶微分项参数可以使其呈现不同程度的刚度特性或阻尼特性,方程稳定区域的大小和过渡曲线的位置也因此产生了不同程度的变化.

Fractional calculus has many excellent characteristics and is mainly used to improve the research accuracy for vibration characteristics of nonlinear systems in the field of dynamics.In this paper,the fractional-order derivative is introduced into the quasi-periodic Mathieu equation and the influences of fractional-order term on the stability of the equation are studied.Firstly,the conditions of the periodic solutions are obtained by the perturbation method,and the approximate expressions of the transition curves are also gotten.The accuracy of the approximate analytical solution is verified by comparing with the numerical solution,and they are in good agreement with each other.Moreover,approximate expressions of transition curves under different conditions are summarized.By analyzing their formal characteristics,it is found that the fractional-order term exists in the form of equivalent linear stiffness and equivalent linear damping in the equation,the general forms of equivalent linear damping and equivalent linear stiffness are obtained,and the thickness of unstable region is defined.Finally,the effects of fractional-order parameters on the size of stability region and the position of transition curves are analyzed intuitively by numerical method.It is found that the fractional-order term has both damping and stiffness characteristics,and the fractional coefficient and fractional order affect the transition curves of the equation in the form of equivalent linear stiffness and equivalent linear damping.Even in some cases,the effect of fractional-order term is almost equal to linear damping or linear stiffness.Reasonable selection of fractional-order parameters can make it show different degrees of stiffness or damping characteristics,and have different degrees of influence on the stability region of the equation and the position of the transition curve,thus affecting the value range of the stability parameters of the equation.These results are of great significance for the study of dynamic characteristics of such systems.