Resonances of Compact Tapered Inhomogeneous Axially Loaded Shafts

An important technical area is the bending of shafts subjected to an axial load. These shafts could be tapered and made of materials with spatially varying properties (Functionally Graded Material – FGM). Previously the transverse vibrations of such shafts were investigated by the authors assuming the shafts had large slenderness ratios so that Euler-Bernoulli theory could be employed. Here compact shafts are treated necessitating the use of Timoshenko beam theory. For constant axial load case analysis of the effects of both FGMs and tapering on frequencies, the value of the compressive load is chosen to be 80% of the smallest critical (buckling) value for the shafts considered. The equations of motion give rise to two coupled differential equations with variable coefficients. These equations in general do not have analytic solutions and numerical methods must be employed (here using MAPLE®) to find the natural frequencies. MAPLE®’s built-in solver for two-point boundary value problems does not directly provide the eigenvalues. The strategy used is to solve a harmonically forced motion problem. On varying the excitation frequency and observing the mid-span deflection the resonance frequency can be found noting where a change in sign occurs. For example, results for FGM cylindrical and tapered shafts show that for a compact cylindrical beam the resonant frequency obtained differs from the Euler-Bernoulli prediction by 11%, and for a tapered beam by 12%, indicating that the effects of compactness can be significant. Since Timoshenko theory requires a value for the shear coefficient, which is not readily available for FGM beams, a sensitivity study is conducted in order to access the effect of the value on the results. Some effects of axial load variations on frequencies are also presented.