INTERNATIONAL JOURNAL OF ENERGY AND ENVIRONMENT Volume 9, Issue 6, 2018 pp.563-580 Journal homepage: www.IJEE.IEEFoundation.org ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2018 International Energy & Environment Foundation. All rights reserved. Semi-analytic solution for stability and free vibration of functionally graded (FG) material micro-pipe conveying fluid Talib EH. Elaikh, Nada M. Abed Department of Mechanical Engineering, College of Engineering, Thi-Qar University, Iraq. Received 2 Aug. 2018; Received in revised form 25 Sep. 2018; Accepted 27 Sep. 2018; Available online 1 Nov. 2018 Abstract A Micro-scale pipe conveying fluid and the functionally graded (FG) materials have many potential applications. In this article, an analytical solution is offered free vibration for a functionally graded (FG) material micro-pipe conveying fluid. On the basis of the Euler beam model and the modified coupled stress theory. The properties of the material are changed constantly across the micro-pipes thickness and depend on power law distribution. Utilized Hamilton’s principle to get an equation of motion for three end boundary conditions (Simply supported, clamped-clamped and cantilever micro-pipes).The differential transformation (DT) method is utilized to obtain the solution for motion's equation and concerned boundary conditions. The effect of fluid flow velocity, the gradient index and parameter of the material length scale on the vibration and stability of fluid conveying FGM micro scale pipes are discussed. The results show that critical velocities and natural frequencies are increased hastily with the increase in the gradient index p. Copyright © 2018 International Energy and Environment Foundation - All rights reserved. Keywords: Fluid- conveying FGM micro pipe; DTM; Functionally graded material; Natural frequency; Critical flow velocity. 1. Introduction Pipe Conveying fluid is very important components for most engineering structures, a nuclear reactor, heat exchanger, marine risers, oil pipelines, microfluidic, Nano fluidic devices and others. Free Vibration analysis of the fluid conveying pipe has been closely studied in past decenniumes. The vibration problems of the pipe conveying fluid were analyzed through different methods. By reviewing the literature in this field, it was observed that there are many numerical and analytical methods used in solving vibration problems of this structures both in nonlinear and linear dynamics, like finite element (FE) method by Zhang, Gorman, and Reese [1], Galerkin's method by Sarkar, and Païdoussis [2], DQM by Qian and Wang [3] simplistic method by Wang and Liu [4]. Paidoudssis and Issid [5] showed the linear dynamics of the pipe conveying fluid using the Galerkin's method. Yun-dong and Yi-ren [6] developed vibrational iteration (VI) method for analysis free vibration conveying fluid in the pipe, and they are obtained the critical velocity of flow and frequency for fluid conveying pipe with many end conditions. The (DT) Method was first suggested for solving linear and nonlinear elementary value problems at the analysis of the electrical circuit based on the expansion of Taylor’s series by Zhou [7]. This method is an