On the existence and approximation of localized waves in a damped Euler–Bernoulli beam on a nonlinear elastic foundation

Abstract

Localized waves in an Euler–Bernoulli beam on a weakly nonlinear elastic foundation, which has a specific weakly nonlinear structural damping, and is under the action of a moving (constant speed) concentrated load, are studied. A structural nonlinear damping, and a damping in the elastic foundation are taking into account. New approximation formulas, which describe in one expression both linear and nonlinear damping characteristics depending on the model parameters, are found. The localized approximations of the solutions of the weakly nonlinear problem have been obtained by using the Ritz method and perturbation techniques. Relatively simple formulas obtained for the asymptotic localized solution, show that the nonlinearity in the elastic foundation leads to smaller amplitudes in the displacements than in case of a linear foundation. The nonlinearity also leads to a smaller value of the critical velocity in the considered system compared to the linear case. The coefficient introduced in the weakly nonlinear elastic foundation also leads to a larger cut-off frequency in the system as compared to the linear case. Another property and effect of the nonlinearity in the elastic foundation is that at a “velocity resonance” the amplitudes of displacement remain finite even when damping (viscosity) is absent. The critical velocity for the weakly nonlinear problem has also been obtained, and the stability of the localized solution, which depends on the system parameters, is investigated.