On the large amplitude free vibrations of tapered beams: an analytical approach S.H. Hoseini, T. Pirbodaghi, M.T. Ahmadian * , G.H. Farrahi School of Mechanical Engineering, Sharif University of Technology, 11365-9567 Tehran, Iran article info Article history: Received 6 March 2009 Received in revised form 4 August 2009 Available online 9 August 2009 Keywords: Non-linear vibration Analytical solution Tapered beam Homotopy analysis method Homotopy pade technique abstract In this study, the homotopy analysis method is used to obtain an accurate analytical solu- tion for fundamental non-linear natural frequency and corresponding displacement of tapered beams. Comparison between the obtained results and numerical solutions shows that the first-order approximation of present study leads to accurate solutions with a max- imum relative error less than 0.5%. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Tapered beams can model engineering structures which require a variable stiffness along the length, such as moving arms and turbine blades (Goorman, 1975; Timoshenko and Goodier, 1983; Hibbeler, 2001). In dimensionless form, the governing differential equation corresponding to fundamental vibration mode of a tapper beam is given by Goorman (1975): d 2 u dt 2 ! þ u þ e 1 u 2 d 2 u dt 2 ! þ u du dt  2  ! þ e 2 u 3 ¼ 0; ð1Þ where u is displacement and e 1 and e 2 are arbitrary constants. Subject to the following initial conditions: uð0Þ¼ a; duð0Þ dt ¼ 0; ð2Þ where a is an arbitrary constant. Under the transformation s ¼ xt, Eq. (1) can be rewritten as follows: x 2 d 2 u ds 2 ! þ u þ e 1 x 2 u 2 d 2 u ds 2 ! þ u du ds  2  ! þ e 2 u 3 ¼ 0; ð3Þ uð0Þ¼ a; duð0Þ ds ¼ 0; ð4Þ where x is the natural frequency. 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.08.003 * Corresponding author. Tel.: +98 21 66165503; fax: +98 21 66000021. E-mail address: ahmadian@sharif.edu (M.T. Ahmadian). Mechanics Research Communications 36 (2009) 892–897 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom