Investigation of the Free Vibrations of Radial Functionally Graded Circular Cylindrical Beams Based on Differential Quadrature Method

Abstract

In the current research, an effective differential quadrature method (DQM) has been developed to solve natural frequency and vibration modal functions of circular section beams along radial functional gradient. Based on the high-order theory of transverse vibration of circular cross-section beams, lateral displacement equation was reconstructed neglecting circumferential shear stress. Two equations coupled with deflection and rotation angles were derived based on elastic mechanics theory and further simplified into a constant coefficient differential equation with natural frequency as eigenvalue. Then, differential quadrature method was applied to transform the eigenvalue problem of the derived differential equation into a set of algebraic equation eigenvalue problems. Natural frequencies of the free vibrations of cylindrical beams with circular cross-sections were calculated at one time, and corresponding modal functions were solved together. The obtained numerical results indicated that the natural frequencies of functionally graded (FG) circular cylindrical beams obtained using differential quadrature method agreed with the results reported in related literatures. In addition, influences of varying gradient parameters on the modal shapes of circular cylindrical beams were found to be strongly consistent with previous studies. Numerical results further validated the feasibility and accuracy of the developed differential quadrature method in solving the transverse vibration of FG circular cross-section beams.