扁拱结构的非线性振动分析

Analysis on Nonlinear Vibration of Shallow Arch Structure

建立了扁拱结构的非线性振动方程,并应用平均法求其一阶近似解;然后应用李雅普诺夫一次近似稳定理论研究了扁拱结构主参数共振时的动力稳定性;讨论了用平均法求解同时具有二次和三次非线性动力系统的局限性.研究表明:当扁拱结构系统非线性呈"渐软"弹簧特性时,二次非线性项在平均法求解过程不起作用,这时用平均法难以准确近似系统的真实响应,其方法是失效的;当扁拱结构系统非线性呈"渐硬"弹簧特性时,平均法求得的近似解较为准确,在此情况下平均法可以作为扁拱动力学特性的求解方法.

In this paper, nonlinear vibrations of a shallow arch subjected to action of a load distributed in the form of sinusoid function is investigated. A nonlinear vibration equation of a shallow arch structure is established first and the approximate solution of the equation is given with an averaging method. Then the analysis of stabiiity of a shallow arch structure is done with Lyapunov＇ s f＇~t-order approximate stability theory. Finally, the limitation of finding the solution of quadratic and cubic nonlinear dynamic systems with an averaging method is discussed. It is found that the square nonlinear term has no effect for finding the approximate solution by an averaging method in the soft-spring system. But the approximate solution by an averaging method is accurate in the hard-spring system, and therefore, an averaging method can be used as a solving method for dynamic characteristics of a shallow arch.