A differential quadrature procedure for linear and nonlinear steady state vibrations of infinite beams traversed by a moving point load

Abstract

A differential quadrature procedure is proposed to study the steady state linear and nonlinear vibrations of an infinite beam resting on an elastic Winkler foundation and subjected to a moving point load. The governing nonlinear partial differential equation of motion of the beam is first expressed with respect to a moving coordinate system. This step reduces the governing nonlinear partial differential equation of motion of the beam to an ordinary nonlinear differential equation. This equation is then converted to a set of nonlinear algebraic equations by application of the differential quadrature method. The Newton–Raphson method is used to solve the resultant system of nonlinear algebraic equations. Issues related to implementation of infinite boundary conditions and modeling the point load are addressed. To accurately predict the dynamic behavior of the beam at high speeds of the moving point load, an efficient and robust absorbing boundary condition is also introduced. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison study with available analytical solutions in the literature is performed. Numerical results reveal that the proposed procedure can be used as an effective tool for handling nonlinear moving load problems on infinite domains.