一类非等截面杆的纵向自由振动

Free Axial Vibrations of a Non-uniform Rod

研究一类变截面杆,其横截面积呈指数函数变化。经适当变换后,杆的纵向自由振动方程转换为退化的超几何方程,其解可以用Kummer函数来表示。得到了三种简单边界条件下的频率方程和振型函数。频率方程一般是超越方程,需要数值求解其固有频率。在特殊情形下,可以求得各阶固有频率。

In this paper, the free axial vibrations of a rod with variable cross section area of the cross section of the rod varies exponentially along the axes of the rod. In terms are investigated. The of the proposed transformation, the governing equations of the free vibration are transformed as degenerated hyper-geometry equations, the solutions of which can be expressed by Kummer functions. The frequency equations and mode functions are obtained under three kinds of boundary conditions. Since the frequency equations are transcendental equations, the natural frequencies are solved with a numeric method. For some special cases, the explicit expressions of natural frequencies can be obtained.