Nonlinear forced vibrations of rotating anisotropic beams
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O R I G I N A L PA P E R
Nonlinear forced vibrations of rotating anisotropic beams Ferhat Bekhoucha · Said Rechak · Laëtitia Duigou · Jean-Marc Cadou
Received: 15 November 2012 / Accepted: 16 August 2013 / Published online: 10 September 2013 © Springer Science+Business Media Dordrecht 2013
Abstract This work deals with forced vibration of nonlinear rotating anisotropic beams with uniform cross sections. Coupling the Galerkin method with the balance harmonic method, the nonlinear intrinsic and geometrically exact equations of motion for anisotropic beams subjected to large displacements, are converted into a static formulation. This latter is treated with two continuation methods. The first one is the asymptotic-numerical method, where power series expansions and Padé approximants are used to represent the generalized vector of displacement and the frequency. The second one is the pseudo-arclength continuation method. Numerical tests dealing with isotropic and anisotropic beams are considered. The natural frequencies obtained for prismatic beams are compared with the literature. Response curves are obtained and the nonlinearity is investigated for various geometrical conditions, excitation amplitudes and kinematical conditions. The nonlinearity related to the angular speed for prismatic isotropic beam is thus identified. The stability of the solution branch is exam-
F. Bekhoucha (B) · S. Rechak Ecole Nationale Polytechnique, Hassen Badi, El Harrach, Mechanical Engineering and Development Laboratory, 16200, Alger, Algeria e-mail: [email protected] L. Duigou · J.-M. Cadou Université de Bretagne Sud, Rue de Saint Maudé, Laboratoire d’Ingénierie des Matériaux de Bretagne, BP 92116, 56321 Lorient Cedex, France
ined, in the frequency domain using the Floquet theory. Keywords Nonlinear vibration of rotating beams · Galerkin method · Harmonic balance method · Asymptotic Numerical Method · Pseudo-arclength
1 Introduction Many studies have been performed for modeling the rotating flexible beams. The earlier linear analytical model to calculate the natural frequencies, based on the Rayleigh energy theorem, is attributed to Southwell and Gough [1] and has been extended by many studies based on different analytical methods, Putter and Manor [2], Wright et al. [3]. Later, complex models were developed to obtain accurate natural frequencies, introducing effects, such as the cross-sectional variation, Klein [4], and the pretwist Swaminathan and Rao [5]. Recently, other models have been developed for the vibration analysis of Timoshenko beams, including effects such as non-uniformity, pretwist and offset root [6, 7, 9]. In these models, the nonlinear terms are truncated and thus are valid for only linear vibration. A number of geometrically exact formulations for the nonlinear dynamics of beams were developed that can be used for nonlinear vibrations of beams; Borri and Mantegazza 1985 [10] Bauchau and Kang 1993
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[11]. Geometrically exact means that there is no approximation made to the geometry of the large d
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