变截面杆纵振动问题的试探函数方法

TRIAL FUNCTION METHOD FOR THE LONGITUDINAL VIBRATION OF RODS WITH VARIABLE CROSS-SECTIONS

给出了一种试探函数法,并研究了变截面杆的纵振动问题.先给出振动控制方程的特殊函数形式的试探解,然后要求此解满足控制方程,反过来确定了控制方程各种可能的系数函数(即截面变化函数)并得到了控制方程的精确解.作为例子,给出了一种变截面杆在3种边界条件下的频率方程,计算出了固有频率.研究表明,试探函数法简单、直接,适合于研究变截面杆的纵振动问题.对于杆扭转振动、薄膜振动以及管中波传播等问题,该方法同样有推广应用价值.

A trial function method is proposed to study the longitudinal vibration of rods with variable crosssections. A trial solution of the governing equation in a form of special functions is suggested, satisfying the governing equation, various possible coefficient functions （namely cross-section functions of rods） are determined and the exact analytical solutions of the governing equation is obtained. As an example, the frequency equations and the natural frequencies of a kind of variable cross-section rod are given, under three boundary conditions. It is shown that the trial function method is simple and convenient, and suitable for the study of the longitudinal vibration of rods with variable cross-sections. The method can also be applied to the study of the torsional vibration of rods, the vibration of membranes as well as the wave propagation in ducts.