Vibrations of a beam‐string complex system under a moving force

Abstract

In this paper, the dynamic response of an Euler-Bernoulli beam and string system traversed by a constant moving force is considered. The force is moving with a constant velocity on the top beam. The complex system is finite, simply supported, parallel one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a beam-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case the aperiodic part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations. The dynamic influence lines of complex systems may be used for the analysis the complex models of moving load. Formulation of the problemOne of the most important issues in the dynamics of structures is moving load problems witch modeling is very difficult in its complications and generates many mathematical problems [1][2][3][4][5][6][7][8]. Even simply models give very complex and unpredicted solutions of structural in dynamical and stability meaning [1,3]. We consider the transverse vibrations of an elastically connected beam-string complex system, under moving loads (Fig. 1). We search a solution of the system in the classical-forms and we show that we can find the closed-forms of the deflection function. For the beam-string complex the equation of motion has a form of a system of partial differential equationswhere w j (x, t) are the deflection functions of the beam and the string, wherein j = [b, s] denote the first beam and the second string, S and N are the tension forces, m j are mass spread over the length, EI is flexural rigidity of the beam, E Young's modulus of elasticity, I moment of inertia of the cross-section area, EA are the axial stiffnesses of the string, A the cross-section area, k stiffness modulus of a Winkler elastic element, whereas P (x, t) is the load process. We have boundary conditions