一种分段线性边界条件连续体振动响应求解方法

An Approach for Vibration Analysis of Continuous Systems with Piecewise-linear Boundary Conditions

基于振型转换的思想,提出了一种求解分段线性边界条件,以及可将其他非线性边界条件转化为分段线性边界条件的连续体振动问题的方法———相对振型转换法(Relative Mode Transfer Method,RMTM),用该法研究了端点带双阻挡悬臂梁的非线性振动。利用相对振型转换法处理了梁的接触振型与非接触振型的振动转换,与Moon于1983年所提方法进行了相互印证,通过时程图与幅频响应图,将两种方法得到的梁端点响应的结果进行对比,证实了相对振型转换法的正确性;并研究了一类边界条件梁的非线性振动,通过梁端点的幅值响应的分岔图,讨论了振型之间的耦合、模态阻尼及端点弹簧刚度对梁端点响应的影响。结果表明,弹簧刚度、阻尼、激励力等不同的参数组合可以导致梁的单周期运动、多周期运动以及混沌运动,得到了上述复杂非线性响应在激励力频域上存在的区域。

A new approach named the relative mode transfer method（RMTM） is developed in this study to solve the vibration problem of continuous systems with piecewise-linear boundary conditions or other nonlinear boundary conditions which can be transformed into piecewise-linear boundary conditions based on the mode transfer principle.The nonlinear vibration of a cantilever with double stops on the tip is investigated.The transfer between the contact states and non-contact states is dealt with using the new method,and its validity is proved by contrasting the results of a case studied both by the new approach and Moon＇s method which was proposed in 1983.The problem represented by the case is studied through the bifurcation response of the cantilever tip,and a discussion of the effects of the coupling between modes,modal damping,and stop stiffness.It is found that different combinations of the parameters of stop stiffness,damping ratio,driving force,etc.,can lead to period-one,period-n and chaotic vibration,and the area of complex nonlinear response in the range of the driving force is obtained.