张思进 刘喻 吉德三

摘   要：采用摄动法和Poincaré映射方法推导出了具有立方非线性项和外部激励项的二自由度碰振系统周期解的扩展Melnikov函数，并运用该Melnikov函数研究了二自由度碰振系统的双碰周期解特性，确定了系统稳定双碰周期2运动的存在条件，即在参数域内的一条临界曲线.通过数值模拟验证，结果表明：该临界曲线下方区域参数是双碰周期2运动，上方区域参数是非双碰周期2运动;当保持其他参数不变，仅增加系统激励幅值f时，系统的运动状态会从多碰多周期运动逐步向双碰周期2运动转变;当保持其他参数不变，仅增加系统恢复系数η0时，系统的运动状态会从双碰周期2运动逐步向多碰多周期运动转变.

关键词：碰振系统;Melnikov方法;双碰周期2运动;Poincaré映射;扩展Melnikov函数

中图分类号：O322                                   文献标志码：A

Melnikov′s Method of Periodic Solutions with Double Impacts for

a 2-DOF Vibro-impact Quasi-Hamiltonian System

ZHANG Sijin1，LIU Yu1，JI Desan2

（1. College of Mechanical and Vehicle Engineering， Hunan University，Changsha 410082，China;

2. College of Science，Wuhan University of Science and Technology，Wuhan 430065，China）

Abstract： Perturbation method and Poincaré mapping method were used to derive the generalized Melnikov function of the periodic solution for a two-degree-of-freedom vibro-impact system with cubic non-linearity and external excitations. By using the Melnikov′s method， the characteristics of periodic motions with double-impact of the 2-dof system were studied， and the existence condition of period-2 motions with double-impact was determined as a critical curve in the parameter domain. The results of numerical simulations show that the regions below the critical curve are the period-2 motions with double-impact， the upper regions of the critical curve are not period-2 motions with double-impact;Meanwhile，increasing the force amplitude and keeping the other parameters unchanged， the motion state of the system changes from multi-period motions with multi-impact to period-2 motions with double-impact， while increasing the system restitution coefficient and keeping the other parameters unchanged， the motion state of the system changes from period-2 motions with double-impact to multi-period motions with multi-impact.

Key words： vibro-impact system;  generalized Melnikov′s method;period-2 motion; Poincaré maping; generalized Melnikov′s function

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