分数阶达芬振子的超谐与亚谐联合共振

SUPER-HARMONIC AND SUB-HARMONIC SIMULTANEOUS RESONANCES OF FRACTIONAL-ORDER DUFFING OSCILLATOR

研究了含分数阶微分项的达芬(Duffing)振子的超谐与亚谐联合共振.采用平均法得到了系统的一阶近似解析解,提出了超、亚谐联合共振时等效线性阻尼和等效线性刚度的概念.建立了联合共振定常解幅频曲线的解析表达式,并对联合共振幅频响应的近似解析解和数值解进行了比较,二者吻合良好,证明了求解过程及近似解析解的正确性.然后,将等效线性阻尼和等效线性刚度的概念与传统整数阶系统进行比较,证明分数阶微分项不仅起着阻尼的作用同时还起着刚度的作用.最后,通过数值仿真研究了不同的分数阶微分项系数和阶次对联合共振幅频曲线多值性和跳跃现象的影响,并与单一频率下超谐共振或亚谐共振进行了对比.研究发现,分数阶微分项系数与阶次不仅影响着系统的响应幅值、共振频率,同时还对系统的周期解个数、发生区域面积、发生先后等有重要影响.并且,在不同的基本参数下该系统分别表现出单独超谐共振、单独亚谐共振以及超谐共振和亚谐共振同时存在的现象.这些结果对系统动力学特性的研究具有重要意义.

The super-harmonic and sub-harmonic simultaneous resonance of Duffing oscillator with fractional-orderderivative is studied in this paper.The first-order approximate analytical solution is obtained by averaging method.Thedefinitions of equivalent linear damping coefficient and equivalent linear stiffness efficient for super-harmonic and subharmonicsimultaneous resonance are presented.The analytical amplitude-frequency equation for steady-state solution ofsimultaneous resonance is established.A comparison of the analytical solution with the numerical results is made,andtheir satisfactory agreement verifies the correctness and higher-order precision of the approximately analytical results.Then,a further comparison between the fractional-order and traditional integer-order Duffing oscillator is fulfilled throughthe definitions of equivalent linear damping coefficient and equivalent linear stiffness coefficient,and the results provethat the fractional-order parameters has the effects of both damping and stiffness,which is similar in other fractionalordersystem.At last the numerical simulation is used to analyze the effects of different fractional-order parameters onmulti-value characteristics and jumping phenomena of the amplitude-frequency curve under simultaneous resonance,andthe differences between the super-harmonic and sub-harmonic resonances under single-frequency excitation are analyzed in detail.It could be found that the fractional-order parameters not only affect the response amplitude and resonantfrequency of the system,but also has significant influence on the number,existing area and the occurrence order ofperiodic solutions.Moreover,single super-harmonic resonance,single sub-harmonic resonance and both existing of thesetwo resonances could be respectively found under different basic parameters,which is important to study the dynamiccharacteristics of the similar system.