非线性振动方程多重解求解方法

钱征文;程礼;陈卫;李应红

振动与冲击 ›› 2011, Vol. 30 ›› Issue (10) : 14-18.

PDF(1599 KB)
PDF(1599 KB)
振动与冲击 ›› 2011, Vol. 30 ›› Issue (10) : 14-18.
论文

非线性振动方程多重解求解方法

  • 钱征文; 程礼; 陈卫 ; 李应红
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A New Method to Solving the Multiple Solutions of Nonlinear Vibration Equations

  • Qian Zheng-wen; Cheng Li; Chen Wei; Li Ying-hong
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摘要

非线性振动方程多重解的求解过程中,迭代初值难以有效确定,不稳定周期解收敛域很小,利用通常的微分方程解法无法直接求解。针对这个问题,引入同伦算法,使得初始值的选取无任何限制;同时利用预测-校正算法对外激励参数变化下的解曲线进行追踪,得到系统的多重解。该方法不但可以计算稳定的周期解,而且不稳定的周期解也可以求出。采用Duffing振子运动方程对该方法进行了计算验证,通过与理论近似解以及龙格-库塔法计算结果的对比,验证了该方法的有效性。

Abstract

The initial iterative value is difficult to choose when there are multiple periodic solutions in solving the nonlinear vibration equations. Moreover, the simple iterative solution process of conventional approaches is inefficient to track the periodic solution with different outer excitation. In the view of these two problems, the homotopy method is employed so that the initial iterative value can be chosed easily. The solution curve variated with outer excitation is tracked by the predict-correct method. Both the stable and unstable periodic solutions can be calculated by using this method. The feasibility of the method is verified by calculating the Duffing oscillator equation. It is show that the simulation results using this method agrees well with the theoretic approximate results and the results using the Rung-Kuta method.

关键词

非线性 / 振动 / 多重解 / 谐波平衡法 / 预测-校正算法 / 同伦算法

Key words

nonlinear / vibration / multiple solution / harmonic balance method / predict-correct method / homotopy method

引用本文

导出引用
钱征文;程礼;陈卫;李应红. 非线性振动方程多重解求解方法[J]. 振动与冲击, 2011, 30(10): 14-18
Qian Zheng-wen;Cheng Li;Chen Wei;Li Ying-hong. A New Method to Solving the Multiple Solutions of Nonlinear Vibration Equations[J]. Journal of Vibration and Shock, 2011, 30(10): 14-18

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