基于改进Rayleigh-Ritz法的复杂形状平面薄板自振特性分析

Free vibration characteristics of plates with complicated shapes based on an improved Rayleigh-Ritz method

针对传统的Rayleigh-Ritz法难以求解复杂形状板的振动问题,提出了一种改进的Rayleigh-Ritz法。针对复杂形状板,通过将位移试函数的函数域拓展到曲边域外的矩形区域,并通过选择合适的位移试函数,能方便地对复杂形状板的位移场进行描述;利用不同刚度的弹簧来模拟曲边边界上的复杂边界条件,应用改进的傅里叶级数求解板的应变能、动能以及边界的弹性势能,并通过求取泛函极值获得板的固有频率。直边和曲边板的计算算例,与文献及有限元结果对比验证了该方法准确可靠,且该方法对于复杂形状和边界下薄板的振动求解具有良好的适应性。

An improved Rayleigh-Ritz method was proposed for solving vibration problems of plates with complicated shapes that were difficulty to be solved using the traditional Rayleigh-Ritz one.Aiming at plates with complicated shapes,through extending the function domain of their displacement trial function to rectangular domain outside curved edge domain and choosing appropriate displacement trial function,their displacement field could be easily described.Springs with different stiffness were used to simulate complicated boundary conditions at the curved boundary.Using the improved Fourier series,aplate’s strain energy,kinetic one and its boundary’s elastic potential one were obtained and natural frequencies of the plate were solved by searching the minimum value of the energy functional.Taking plates with straight edges and curved ones as calculation examples,the results obtained using the proposed method were compared with those using the finite element method or published in literature.Results showed that the proposed method is correct and reliable;it has good adaptability to solution of vibration problems of plates with complicated shapes.