Dynamic stability of axially accelerating Timoshenko beam: Averaging method Xiao-Dong Yang a, * , You-Qi Tang a , Li-Qun Chen b, c , C.W. Lim d a Department of Engineering Mechanics, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, China b Department of Mechanics, Shanghai University, Shanghai 200436, China c Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China d Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong article info Article history: Received 21 February 2008 Accepted 17 July 2009 Available online 24 July 2009 Keywords: Axially moving beam of Timoshenko model Averaging method Subharmonic resonance Combination resonance Dynamic stability abstract This study investigates dynamic stability in transverse parametric vibrations of an axially accelerating tensioned beam of Timoshenko model on simple supports. The axial speed is assumed as a harmonic fluctuation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a finite set of ordinary differential equations. The method of averaging is applied to analyze the instability phenomena caused by subharmonic and combination resonance. Numerical examples demonstrate the effects of the mean axial speed, bending stiffness, rotary inertia and shear modulus on the instability boundaries. Ó 2009 Elsevier Masson SAS. All rights reserved. 1. Introduction Axially moving beams are involved in many engineering devices, such as band saws, serpentine belts, magnet tapes and paper in processing. The transverse vibrations associated with these devices have limited their applications (Wickert and Mote, 1988; Abrate, 1992). One major problem is the occurrence of large vibrations, termed as parametric vibrations, due to axial speed fluctuations, which often happen when the drive motors run at high speed (O ¨ z and Pakdemirli, 1999; O ¨ zkaya and Pakdemirli, 2000; O ¨ z, 2001; Chen et al., 2004a; Chen and Yang, 2005). For example, the vibration of the blade of band saws results in poor cutting quality. Therefore, understanding transverse vibrations of axially moving beams is important for the design of the devices (Lengoc and McCallion,1995). There are comprehensive studies on such systems. Most of them take the string and Euler beam as simplified model of the axially moving continuum. The parametric vibrations of axially moving strings have been investigated extensively. These researches include numerical simulations (Fung et al., 1997, 1998; Chen et al., 2004b), analytical expressions of steady-state responses (Zhang and Zu, 1999a,b), and chaotic behaviors (Chen et al., 2003). Accounting for the Euler model, Marynowski (2002) and Mar- ynowski and Kapitaniak (2002) used three-term Galerkin dis- cretization to investigate the nonlinear vibration response of an axially moving beam excited by a changing tension. Marynowski (2004) further studied numerically nonlinear dynamical behavior of an axially moving viscoelastic beam with time-dependent tension based on four-term Galerkin discretization. Yang and Chen (2005) studied numerically bifurcation and chaos of an axially accelerating nonlinear beam based on second-term Galerkin discretization. Perturbation methods, such as multiple scale method and the method of averaging are useful techniques in studying the nonlinear and parametric vibrations of axially moving continuum. O ¨ z et al. (1998) employed the method of multiple scales to study dynamic stability of an axially accelerating beam with small bending stiffness. O ¨ zkaya and Pakdemirli (2000) combined the method of multiple scales and the method of matched asymptotic expansions to construct non-resonant boundary layer solutions for multiple scales and the method of matched asymptotic expansions to construct non-resonant boundary layer solutions for an axially accelerating beam with small bending stiffness. O ¨ z and Pakdemirli (1999) and O ¨ z (2001) applied the method of multiple scales to calculate analytically the stability boundaries of an axially accelerating beam under pinned–pinned and clamped– clamped conditions respectively. Parker and Lin (2001) adopted a first-term Galerkin discretization and the perturbation method to study dynamic stability of an axially accelerating beam sub- jected to a tension fluctuation. Suweken and Van Horssen (2003) applied the method of multiple scales to a discretized system obtained by the Galerkin method to study the dynamic stability of an axially accelerating beam with pinned–pinned ends. Chen et al. * Corresponding author. Tel.: þ86 24 89723710; fax: þ86 24 89723727. E-mail address: jxdyang@163.com (X.-D. Yang). Contents lists available at ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol 0997-7538/$ – see front matter Ó 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.07.003 European Journal of Mechanics A/Solids 29 (2010) 81–90