Effect of Initial Geometric Imperfections on Nonlinear Vibration of Thin Plate by an Asymptotic Numerical Method
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ORIGINAL ARTICLE
Effect of Initial Geometric Imperfections on Nonlinear Vibration of Thin Plate by an Asymptotic Numerical Method Lahcen Benchouaf1 · El Hassan Boutyour2 Received: 6 May 2020 / Accepted: 12 September 2020 © Indian National Academy of Engineering 2020
Abstract In this work, the influence of initial geometric imperfections on geometrically nonlinear free vibrations of thin elastic plates has been investigated by an asymptotic numerical method. The nonlinear strain displacement relationship of von Karman theory is adopted to calculate the elastic strain energy. The harmonic balance approach and Hamilton’s principle are used to convert the equation of motion into an operational formulation. The nonlinear problem is transformed into a sequence of linear ones having the same stiffness matrix, which can be solved by a classical finite-element method. To improve the validity range of the power series, Padé approximants are incorporated and a continuation technique is also used to get the whole solution. Numerical results are discussed and compared to those available in the literature and convergence of the solution is shown for various amplitudes of imperfection. Keywords Nonlinear vibrations · Geometric imperfections · Thin plates · Asymptotic numerical method · Harmonic balance method
Introduction Thin structures such as plates are widely used in various industrial fields such as civil engineering, mechanics, and aeronautics. They are subjected to dynamic loads that can cause vibration amplitude of the order of the thickness of the structure and giving rise to significant nonlinear phenomena. Consequently, the study on nonlinear vibrations of thin plates has assumed considerable importance in recent years. Thin plates are generally object of geometric imperfections that can significantly influence its vibrational behavior. Therefore, attempts to find the vibratory behavior incorporating sensitivity to imperfection have never ceased. In the literature, several researchers have studied the nonlinear vibrations of plates in the presence of geometric imperfections. Ilanko and Dickinson (1991) studied the effect of geometric imperfection on the vibratory behavior * Lahcen Benchouaf [email protected] 1
Hassan First University of Settat, Higher School of Education and Training, Berrechid, Morocco
Hassan First University of Settat, Faculty of Sciences and Technologies, Department of Applied Physics, Settat, Morocco
2
of simply supported plates. Nonlinear vibrations of imperfect plates with different initial conditions are also studied by Liu and Yeh (1993). Ostiguy et al. (1998) used a direct integration method to solve the nonlinear behavior of the initial geometrically imperfect plate. Lin and Chen (1989) presented a set of nonlinear equations to describe the behavior of imperfect isotropic plates. The authors have shown that the nonlinear frequency of the plate is related to the size of the imperfections and that an increase in the size of the imperfection could change the behavior of the n
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