Minimum stiffness and optimal position of intermediate elastic support to maximize the fundamental frequency of a vibrating Timoshenko beam

Abstract

The optimal position and minimum support stiffness of a vibrating Timoshenko beam are investigated to maximize the fundamental frequency. The Finite element method is employed. According to the maximum-minimum theorem of Courant, the optimum position is at the zero of the second mode shape function. The intermediate support's position and minimal stiffness for a wide variety of slenderness proportions were achieved. It was observed that the ideal position of intermediate support and its minimum stiffness are sensitive to the slenderness ratio. Also, for thick cantilever beams with intermediate support at the optimal location, the minimum support stiffness is less than 266.9, which was reported in the literature for the Euler-Bernoulli beam. The minimum stiffness of familiar end conditions of an optimally located beam is presented for a wide range of slenderness ratios. Since, in many practical applications, it is impossible to locate support at the optimal position, the minimum support stiffness for a beam in which its intermediate support is not located at the optimal position is obtained for various boundary conditions and slenderness ratios. Furthermore, empirical evaluations were carried out, and the findings were contrasted with hypothetical estimates of the initial two natural frequencies.