Effect of Functionally Graded Materials on Resonances of Bending Shafts Under Time-Dependent Axial Loading

The effect of functionally graded materials (FGMs) on resonances of bending shafts under time-dependent axial loading is investigated. The axial load is taken to be a sinusoidal function of time and the shaft is modeled via an Euler–Bernoulli beam approach (pin-pin boundary conditions). The axial load enters the formulation via a “buckling load type” approach. For generality, two distinct particulate models for the FGM are considered, namely, one involving power law variations and another based on a volume fraction approach, for both Young’s modulus and material density. The spatial dependence in the partial differential equation of motion is suppressed by utilizing Galerkin’s method with homogeneous beam mode shapes. To check the accuracy of this approximation, numerical solutions for the boundary value problem represented by the original partial differential equation are obtained using MAPLE^®’s PDE solver. Good agreement (within 5%) was found between the PDE results and the one-mode approximation. The approximation leads to ordinary differential equations that have time-dependent coefficients and are prone to parametric and forced motions instabilities. Hill’s infinite determinant approach is used to study stability. The main focus is on the primary parametric resonance. It was found that in most cases the FGM shafts increase the parametric resonance frequencies substantially, while decreasing the zone thicknesses, both desirable trends. For instance, for an axial load about one-third of the buckling value, an aluminum/silicon carbide shaft, when compared to a pure aluminum shaft, increases the primary parametric resonance by 21% and decreases instabilities by 23%. For one model of FGM, the sensitivity of the results to volume fraction variations is examined and it was found that increasing the volume fraction is not uniformly beneficial. Results for other parametric zones are also presented. Forced resonances are also briefly treated.