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

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