含分数阶微分的线性单自由度振子的动力学分析(Ⅱ)

Dynamical analysis of linear SDOF oscillator with fractional-order derivative(Ⅱ)

研究了含两类分数阶微分项的线性单自由度振子,通过平均法得到了系统的近似解析解.在近似解中,两类分数阶微分项的系数和阶次均以等效线性阻尼和等效线性刚度的形式影响着系统的动力学特性,这一点和现有文献中大多数直接将分数阶微分项归类为阻尼进行处理是完全不同的.对近似解析解和数值解进行了比较,二者符合精度很高,证明了该结果的准确性.然后分析了两类分数阶微分项的系数和阶次对系统响应特性的影响,发现两类分数阶微分项的系数和阶次都既可以影响系统的共振振幅,又可以影响系统的共振频率.最后研究了第二类分数阶微分项对共振频率的影响,指出了在振动控制工程中如何通过选取合适的第二类分数阶微分项的系数达到满意的控制效果.

A linear single degree-of-freedom（SDOF） oscillator with two kinds of fractional-order derivatives is investigated by the averaging method,and the approximately analytical solution is obtained.The effects of the parameters on the dynamical properties,including the fractional coefficients and the fractional orders in the two kinds of fractional-order derivatives,are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness,and the results is entirely different from the results given in the existing literature. A comparison of the analytical solution with the numerical results is made,and their satisfactory agreement verifies the correctness of the approximately analytical results.The following analysis of the effects of the fractional parameters on the amplitude-frequency is presented,and it is found that the fractional coefficients and the fractional orders can affect not only the resonance amplitude through the equivalent linear damping coefficient,but also the resonance frequency by the equivalent linear stiffness.Finally,the effects of the fractional coefficient in the second fractional-order derivative on resonance frequency are analyzed,and the design rule for the fractional coefficient in the second fractional-order derivative to meet the satisfactory vibration control performance is pointed out.