Free longitudinal vibrations of functionally graded tapered axial bars by pseudospectral method

Abstract

In this work, the problem of free longitudinal vibration of rods with variable cross-sectional area and material properties is investigated using the pseudospectral method. With the gradation of material properties like modulus of elasticity and mass density in the axial direction, the results corresponding to a functionally graded axial bar are obtained using the proposed pseudospectral formulation. The pseudospectral formulation used is relatively easy to implement and powerful in analyzing vibration problems. With the help of several numerical examples, the non-dimensional natural frequencies of rods obtained using the pseudospectral method are compared with those obtained by the analytical solution, generalized finite element method, the discrete singular convolution method and differential transformation method. The numerical results obtained show that the proposed technique allows boundary conditions to be incorporated easily and yields results with good accuracy and faster convergence rates than other methods.

Analysis of the Axial Vibration of Non-Uniform and Functionally Graded Rods via an Analytical-Based Numerical Approach

“…As shown in Table 20, the results obtained from this study are in very good agreement with [28], even with 20 subdomains. However, for further research, additional computations were conducted, and the results shown in Tables 21-23 were obtained according to different combinations of boundary conditions.…”

“…Pillutla et al [28] studied the longitudinal vibrations of functionally graded rods with variable cross-sectional areas and material properties using the pseudospectral method.…”

“…This Study [17] This Study [17] This Study [17] This Shahba et al [18], Shahba and Rajasekaran [19], and Pillutla et al [28] studied the longitudinal free vibration of axially functionally graded rods using various computational approaches. They considered a tapered bar with a rectangular cross-section for which its breath taper ratio is c b and height taper ratio is c h , and its cross-section is given by A(x) = (1 − c b x)(1 − c h x), where x = x/L.…”

“…However, to determine the first six vibration frequencies, 100 subdomains were used in the calculations. In [18,19,28], the first three vibration frequencies of fixed-fixed and fixed-free rods were considered.…”

“…These additional results are given in Tables 18 and 19 below. Shahba et al [20] and Pillutla et al [28] studied the longitudinal free vibration of axially functionally graded rods with a cross-section given by A(x) = (1 − cx), where x = x/L. The modulus of elasticity and mass density varied according to E = E 0 e x and ρ = ρ 0 e x .…”