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

A trial function method is proposed to study the longitudinal vibration of rods with variable cross-sections. 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.