On nonlinear vibrations of Timoshenko FG porous micropipes in thermal environment: analysis and optimization

Abstract

This study focuses on an optimization analysis of the nonlinear free vibration of a functionally graded porous micropipe conveying fluid in uniform steady thermal environment using the Timoshenko beam theory. The nonlinear equations of motions are derived based on the modified strain gradient elasticity theory and Von–Kármán’s strain relations. By means of the Galerkin method, the nonlinear partial differential equations of motion are transferred into an ordinary 4^th-order nonlinear ordinary differential equation. An analytical closed-form solution for this nonlinear differential equation has been presented using homotopy analysis method. As a consequent, closed–form expressions for the nonlinear critical flow velocity, time history and n^th nonlinear frequency are obtained. The exact solution for the critical flow velocity of the micropipe resting on elastic foundation has been used to find the optimum pipe length. The results illustrate as the micropipe’s length increases, the nonlinear frequency significantly drops for short micropipes but it decreases slightly for longer ones. Additionally, in high temperatures, the nonlinear frequency is less affected by the variation of the power-law exponent. Furthermore, in the absence of elastic substrate, the critical fluid velocity decreases with increasing the microtube length. However, when the microtube is placed on an elastic substrate, the optimum value of the microtube length is observed in higher mode shapes.