二向受力不等的平面薄膜自由振动问题解

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Solution for Free Vibration Problem of the Membrane with Unequal Tension in Two Directions

摘要: 本文中求解了双向受力不等的矩形、圆形、椭圆形平面薄膜的自振频率与振型,还给出了任意外形边界的平面薄膜的近似解.矩形薄膜,先经过坐标变换将方程变换成常见的薄膜振动方程,因此很容易求得解.圆形薄膜.先将坐标作与上述同样的变换,再把它变换成椭圆坐标,将方程化为马丢(Mathieu)方程,这样利用马丢函数的性质,不难求得其解.椭圆形薄膜解法与圆形薄膜相似.文末还给出了例题.

Abstract: In this paper, we obtain the analytic solution of free vibration frequency and mode shapes of rectangle, circle and elliptic membranes. The approximate solution of membrane with arbitrary boundary is also obtained. All of these membranes are acted on by unequal tension in two directions.For the rectangle membrane, in this paper we transform its vibration equation into one of usual membranes by trnasforming the coordinate, thus it is easy to get the solution. For the circle membrane, first we transform the coordinate in the same way we deal with the rectangle membrane. Next we transform the vibration equation into the Mathieu equation, then we get a formula of frequency of that membrane with some Mathieu function's property. In the solution the elliptic membrane is similar to that of the circle membrane.Finally, some examples are given.

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