An analytical solution for nonlinear vibration of functionally graded porous micropipes conveying fluid in damping medium

Abstract

This paper presents an analytical solution for nonlinear free vibration of functionally graded porous micropipes conveying fluid embedded in viscous damping medium. By applying Von-Kármán's nonlinear strain relations, strain gradient theory and Hamilton's principle, the nonlinear governing equations of motion are derived in the presence of viscous damping medium. The nonlinear governing equations of motion are discretized and converted to a set of nonlinear ordinary differential equations by using Galerkin's method. For the first time, closed-form expressions are obtained for nonlinear frequency, critical fluid velocity and damped time response of the functionally graded porous micropipe conveying fluid by employing the homotopy analysis method. In addition, the convergence control parameter is considered to diminish the error of the damped vibration response. The effects of the geometrical properties of the micropipe and the viscous damping medium on the vibration of the functionally graded porous micropipe conveying fluid are investigated. The results show that the optimum value of the convergence control parameter dramatically decreases the error of the homotopy solution. Also, the vibration period is increased by the increment of the fluid velocity, damping coefficient, porosity volume fraction, power-law exponent and the length of the functionally graded porous micropipe conveying fluid.