Chapter 12 Nonlinear Vibrations of a Flexible L-shaped Beam Using Differential Quadrature Method Hamed Samandari and Ender Cigeroglu Abstract Flexible L-shaped beams are integrated sub-components of several navy and space structures where overall response of the system is affected by these structures. Hence, an understanding of the dynamical properties of these structural systems is required for their design and control. Recent studies show that the dynamic response of beam like structures undergoing large deformation is nonlinear in nature where phenomenon such as jump and chaotic response can be detected. In this study, nonlinear free vibrations of L-shaped beams are studied using a continuous beam model with a focus on the internal resonance of these structures. Nonlinearity considered is due to large deflection of the beams (geometric nonlinearity). Hamilton principle and Euler Bernoulli beam theory are used to obtain the nonlinear equations of motion. The differential quadrature method (DQM) is utilized to discretize the partial differential equations of motion in spatial domain, which resulted in a nonlinear set of ordinary differential equations of motion in time domain. Harmonic balance method is used to convert the ordinary differential equations of motion into a set of nonlinear algebraic equations which is solved numerically. Numerical simulations, based on the mathematical model, are presented to analyze the nonlinear responses of the L-shape beam structure. Keywords Nonlinear vibrations • L-shaped beam • Differential quadrature method • Geometric nonlinearity • Euler-Bernoulli beam theory 12.1 Introduction Flexible structures composed of beams are one of the most important components of engineering structures [1]. Recent studies confirmed that response of these structures are affected significantly by their geometric nonlinearity as they go through large deflections. Understanding the nonlinear dynamics of flexible structures becomes more essential with recent advances in science and deployment of these structures in advance engineering structures such as space stations and energy harvesters [2, 3], Among flexible beam structures, L-shaped beam structures are the most common ones and the dynamics of them have been studied by several researchers in the past decades. Haddow et al. [4] were one of the first who studied the planar dynamic responses of a flexible L-shaped beam-mass structure analytically and experimentally. Later, similar structures are studied by Nayfeh and Zavodney [5], Nayfeh and Balachandran [6–8], and several other researchers [9–14]. The internal resonance of an L-shaped beam structure is studied in [15, 16] where motion of the structure is limited by using several stops. In a detailed study, Nayfeh and Pai (2004) [1] have investigated the nonlinear dynamics of a similar structure to primary-resonant excitations, experimentally and theoretically. These studies confirmed that periodic and chaotic solutions, saddle-node and Hopf bifurcations and saturation phenomena can exist for the case of L-shaped beam structures. However, in all of these studies a reduced order model, two degrees of freedom model, is used to study the dynamics of the system. The reduced order model is obtained by utilizing a single trial function, which is the exact eigenfunction of the relevant linear system, in Galerkin type methods. However for nonlinear systems, the resulting nonlinear eigenfunctions can be significantly different than the eigenfunctions of the linear system, and depending on the nonlinearity, it may not be possible to capture the nonlinear characteristics correctly by using a single trial function [17–19]. Therefore, to determine the dynamics of L-shaped beams correctly, methods that can model and approximate the nonlinear eigenfunctions of the structure is required. In the present study, nonlinear free vibrations of a flexible L-shaped beam is investigated using Differential quadrature method (DQM). DQM is a well-developed numerical method for quick solutions of linear and nonlinear partial differential equations. Furthermore, application of DQM eliminates the need for finding suitable trial functions that satisfies all the H. Samandari • E. Cigeroglu ( ) Department of Mechanical Engineering, Middle East Technical University, Ankara 06800, Turkey e-mail: ender@metu.edu.tr © The Society for Experimental Mechanics, Inc. 2016 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15221-9_12 145
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