JZUS - Journal of Zhejiang University SCIENCE

Journal of Zhejiang University SCIENCE A 2017 Vol.18 No.7 P.497-510

Dynamics of a periodically driven chain of coupled nonlinear oscillators

Author(s): Paweł Fritzkowski, Roman Starosta, Grażyna Sypniewska-Kamińska, Jan Awrejcewicz
Affiliation(s): Institute of Applied Mechanics, Poznań University of Technology, Poznań 60-965, Poland; more
Corresponding email(s): pawel.fritzkowski@put.poznan.pl
Key Words: Nonlinear coupled oscillators, Synchronous motion, Averaging method

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Abstract: A 1D chain of coupled oscillators is considered, including the Duffing-type nonlinearity, viscous damping, and kinematic harmonic excitation. The equations of motion are presented in a non-dimensional form. The approximate equations for the vibrational amplitudes and phases are derived by means of the classical averaging method. A simple analysis of the resulting equations allows one to determine the conditions for the two basic synchronous steady-states of the system: the in-phase and anti-phase motions. The relations between the required excitation frequency and the natural frequencies of the abbreviated (linear) system are discussed. The validity of these predictions is examined by a series of numerical experiments. The effect of the model parameters on the rate of synchronization is analyzed. For the purpose of systematic numerical studies, the cross-correlation of time-series is used as a measure of the phase adjustment between particular oscillators. Finally, some essential issues that arise in case of the mechanical system with dry friction are indicated.

耦合非线性振子链的周期性驱动动力学研究

目的：考虑一个包含Duffing型非线性、粘性阻尼和运动谐波激励的一维耦合振子链，本文旨在推导描述振幅和相位的近似方程，得到并检验两类基本共振态（同相共振和反相共振）发生的条件，同时分析模型参数对同步率的影响。
创新点：1. 利用时间序列的互相关测量特定振子间的相位蝶，并通过一系列的数值实验来验证理论预测的结果； 2. 给出考虑干摩擦的系统共振的一些简要说明。
方法：1. 采用经典的平均值法进行理论分析；2. 采用MEBDFV求解器计算多自由度系统的数值解。
结论： 1. 利用平均值法确定了两种共振现象：同相状态（低频激励）和反相状态（高频激励）；2. 关于同相共振的预测非常琐碎但对多自由度的振子链非常适用；3. 对于反相共振的预测适用于短的振子链；4. 可以通过改变系统的物理参数来提高同步率。

关键词：耦合非线性振子；同步运动；平均值法

Reference

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耦合非线性振子链的周期性驱动动力学研究

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