Nonlinear Vibrations of a Flexible L-shaped Beam Using Differential Quadrature Method

Samandari, Hamed
Ciğeroğlu, Ender
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.