含分数阶微分项和参数激励的Duffing⁃van der Pol振子的动力学分析  

Dynamic analysis of Duffing⁃van der Pol oscillators with fractional⁃order derivative and parametric excitation

作　　者：崔腾达 申永军 CUI Tengda;SHEN Yongjun(School of Mechanical Engineering,Shijiazhuang Tiedao University,Shijiazhuang 050043,China;State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures,Shijiazhuang Tiedao University,Shijiazhuang 050043,China)

机构地区：[1]石家庄铁道大学机械工程学院,河北石家庄050043 [2]石家庄铁道大学省部共建交通工程结构力学行为与系统安全国家重点实验室,河北石家庄050043

出　　处：《振动工程学报》2025年第4期715-721,共7页Journal of Vibration Engineering

基　　金：国家自然科学基金资助项目(U1934201,12272242)。

摘　　要：对含有分数阶微分项和参数激励的Duffing⁃van der Pol振子的动力学行为进行了研究,分析了在黏惯性(1≤p≤2)和参数激励共同作用下系统的各项参数对系统幅频曲线的影响。采用平均法分析此系统,用等效线性阻尼和等效质量的概念处理分数阶微分项,得到系统的近似解析解。将所得近似解析解与数值解进行比较,二者具有较高的吻合度,证明了解析解的正确性。分析了系统参数对幅频响应曲线的影响,发现共振峰值、共振频率、共振区域、多值解的范围和解的数量都会受到系统参数的影响。经过分析发现,外激励幅值和分数阶微分项系数在一定程度上会抑制参数激励的效果。The dynamic behavior of the Duffing-van der Pol oscillator with fractional-order derivative and parametric excitation is studied in this paper.The effects of various parameters on the amplitude-frequency curves of the system under the combined action of viscous inertia(1≤p≤2)and parametric excitation are analyzed.The system is analyzed by the averaging method,and the fractional-order derivative is treated by the concepts of equivalent linear damping and equivalent mass.The approximate analytical solution of the system is obtained and compared with the numerical solution.The curves of the two solutions agree well with each other to a large extent,which proves the correctness of the analytical solution.The influences of system parameters on the amplitude-frequency curve are analyzed.It is found that the resonance peak value,resonance frequency,resonance region,the range and the number of multivalued solutions are all affected by the system parameters.Through analysis,it is found that the external excitation amplitude and the coefficient of fractional-order derivative can suppress the effect of parametric excitation to some extent.

关 键 词：非线性振动 分数阶微分项 黏惯性 参数激励 平均法 

分 类 号：O322[理学—一般力学与力学基础] O313[理学—力学]