Effect of Tapered Thickness on the Logitudinal Free Vibrations of Cantilever Beam

Abstract

This paper deals with free longitudinal vibrations of nonuniform homogeneous cantilever beams. Cantilever of rectangular cross-section with constant width and tapered thickness variation are considered. Thickness at the clamped end is estimated while it changed with different values at free end at the ratio equal to the relation (thickness at free end hf / thickness at clamped end hc) where this ratio change from 0.05 to 0.9. The exact solution of differential equation in the linear case of free axial vibrations of nonuniform beam by using the analytic method by separation of variable in terms of Bessel function. Effect of thickness ratio between free end to clamped end (hf / hc) for different value of thickness of cantilever at clamped end and effect of different value of beam length on the characteristics of vibration ( natural frequency and mode shape) are studied. Some of results are compared with approximation method which called Raylieghs quotient. It is concluded that increasing the thickness of clamped end causes decrease in the natural frequency at any value of length of beam also increasing the thickness ratio and increasing the length of beam at assisted value of thickness at clamped end ( hc) causing decreased in the value of natural frequency. On the other hand it is found that the value of mode shape of cantilever beam decrease when increase the thickness ratio (hf / hc) at any value of thickness of clamped end and at the same value of length of beam also the mode shape decreased with increasing thickness of clamped end (hc). Finally at the same value of (hc) the value of mode shape decreased with increasing length of beam..