双向变厚度矩形薄板的自由振动分析

Free vibration analysis of rectangular plates with varying thickness in two directions

针对传统级数法只能求解特定边界条件下矩形板的振动问题,通过采用改进Fourier级数的方法,将双向变厚度薄板的振动位移函数表示为标准的二维Fourier余弦级数和辅助Fourier级数的线性组合,从而使级数法能适用于任意的弹性边界条件.利用Rayleigh-Ritz法建立了与变厚度薄板控制方程等价的矩阵表达示,通过特征值分解求得板的固有频率和振型.数值算例表明了该方法具有很高的求解精度和很好的收敛性.

To use the traditional series method, only vibrations of rectangular plates with specific boundary condi- tions can be analyzed, but an improved Fourier series method has been employed so that the vibration displace- ments of rectangular plates with varying thickness in two directions could be expressed as the linear combination of a standard two-dimensional Fourier cosine series and auxiliary series functions, and this method can be universally applied to general elastic boundary conditions. The Rayleigh-Ritz method can give the matrix＇s eigenvalue equation which is equivalent to governing differential equations of the tapered plate, and all the eigenvalues and eigenvectors can be obtained by solving the matrix equation. The accuracy and good convergence characteristics of this method are demonstrated through numerical examples.