转动弹性支承矩形板自由振动的一般解析解法

GENERAL ANALYTICAL METHOD FOR FREE VIBRATION OF RECTANGULAR PLATES ELASTICALLY RESTRAINED AGAINST ROTATION

转动弹性支承边与简支边,平夹边均有不同,一方面板边均能防止上下移动,即其挠度均为零。而转动弹性支承边,由于在边界装有均匀分布的转动弹簧使边界弯矩受到斜度的制约而与板边的斜度成正比。采用矩形薄板自由振动横向位移函数的微分方程建立了一般性的解析解,该一般解包括三角函数和双曲线函数组成的解,它能满足4个边为任意边界条件的问题。解中的待定常数可由4边的边界条件来确定,由此得出的齐次线性代数方程系数矩阵行列式等于零可以精确地求得各阶固有频率及其振型。由于矩形板对中间轴具有对称性,利用对称和反对称条件可使求解大大简化。对于正方形板还可利用对角线的对称性而毫无遗漏地找出最低的各阶频率及其振型。以四边均为转动弹性支承方板为例进行计算和讨论。

Different from simple supports and clamped supports,the supports with elastic restrains against rotation restrain not only the vertical translation,but the moment at the supported edge,which linearly related to the slope of the edge.Using the differential equation for transverse displacement function of rectangular thin plates in free vibration,a general analytical solution is established.This general solution includes both trigonometric function solutions and hyperbolic function solutions,and can satisfy arbitrary boundary conditions.The unknown constants can be determined by applying boundary conditions of four edges.Each natural frequency and vibration mode can be exactly solved by the condition that the determinate of coefficient matrix from the homogeneous linear algebraic equations equals to zero.As the rectangular plate is symmetric in plane,it can facilitate the analysis to take advantages of symmetric and anti-symmetric conditions.Finally,a square plate with four edges elastically restrained against rotation is analyzed and discussed.