Dynamic Stability Analysis of Euler-Bernoulli and Timoshenko Beams Composed of Bi-Directional Functionally Graded Materials

Abstract

In this paper, dynamic stability analysis of beams composed of bi-directional functionally graded materials rested on visco-Pasternak foundation under periodic axial force is investigated. Material properties of beam vary continuously in both the thickness and longitudinal directions based on the two types of analytical functions including exponential and power law distributions. Hamilton’s principle is employed to derive the equations of motion according to the Euler-Bernoulli and Timoshenko beam theories. Then, the generalized differential quadrature method in conjunction with the Bolotin method is used to solve the differential equations of motion under different boundary conditions. Various parametric investigations are performed for the effects of the gradient index, static load factor, length-to-thickness ratio and viscoelastic foundation coefficients on the dynamic stability regions of bi-directional functionally graded beam. The results show that the influence of gradient index of material properties along the thickness direction is greater than gradient index along the longitudinal direction on the dynamic stability of beam for both exponential and power law distributions. Also, the system become more stable and stiffer when beam is resting on visco-Pasternak foundation. Moreover, by increasing static load factor, the dynamic instability region moves to the smaller parametric resonance. The results of presented paper can be used to the optimal design and assessment of the structural failure and thermal rehabilitation of turbo-motor and turbo-compressor blades.