摘要
工程学中的许多参数激励振动都会产生带有非线性项的Mathieu方程.本文以底鼓触发器为实物模型,将底鼓槌子的击打过程等效为一个类Kapitza单摆系统,运用泰勒展开得到含有五次方非线性项的Mathieu-Duffing方程.并采用摄动法全面分析了首个参数共振舌域的过渡曲线及附近的稳定性变化,确定了系统具有多个平衡点共存.借助雅可比矩阵的特征值,详细揭示了平衡点处超临界叉式分岔与亚临界叉式分岔的发生机制,深入探讨了分岔与非线性参数之间的内在关联.最后运用数值模拟得到了在三维参数状态空间中的首个参数共振舌域的分岔图,并验证了理论推导的正确性.
In engineering systems,many parametrically excited vibrations can be modeled by the Mathieu equation with nonlinear terms.This study establishes a physical model using a bass drum trigger,where the striking process of the drum beater is equivalently represented as a Kapitza-like pendulum system.Through Taylor expansion,we derive a Mathieu-Duffing equation incorporating a quintic nonlinear term.A perturbation method is systematically employed to analyze the transition curves and stability variations near the first parametric resonance tongue,revealing the coexistence of multiple equilibrium points in the system.By investigating the eigenvalues of the Jacobian matrix,we explicitly demonstrate the emergence mechanisms of both supercritical and subcritical pitchfork bifurcations at equilibrium points,with particular emphasis on their intrinsic relationships with nonlinear parameters.Numerical simulations are conducted to reconstruct the bifurcation diagram within the three-dimensional parameter state space for the first resonance tongue,thereby validating the theoretical predictions.The results comprehensively illustrate the intricate interplay between parametric excitation,nonlinear stiffness,and bifurcation dynamics in such hybrid oscillator systems.
作者
张风
李险峰
魏周超
Zhang Feng;Li Xianfeng;Wei Zhouchao(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China;School of Mathematics and Physics,China University of Geosciences(Wuhan),Wuhan 430074,China)
出处
《动力学与控制学报》
2025年第8期29-36,共8页
Journal of Dynamics and Control
基金
国家自然科学基金资助项目(12462002)
甘肃省自然科学基金资助项目(22JR5RA348)。
