Vibration of a Complex Strings System under a Moving Force

Abstract

We consider vibrations of an elastically connected double-spring complex system subjected to moving concentrated forces. This system is treated as two strings connected by an elastic layer. The motion of the system is described by two partial differential equations. It is shown that some solutions of the system response can be obtained not only as series but also in closed forms. Formulation of the problemThe complex systems of strigs and beams occur, among others, in prestressed bridges of small and medium span, bridges reinforced by tensioning. We consider the transverse vibrations of an elastically connected double-springs complex under moving loads (Fig. 1). We look for the solution of the system in the classical forms and we show that we can find the closed forms of the deflection function. For the double-springs complex the equation of motion has the form of a system of differential equationswhere w j (x, t) are the deflection functions of the strings, S j are the tension forces, m j are the mass densities of the strings, EJ j are the axial stiffnesses of the strings, k is the stiffness parameters, c is the damping coefficient, whereas P (x, t) is the load process. We have boundary conditions w j (0, t) = w j (L, t) = 0 and initial conditions w j (x, 0) =ẇ j (x, 0) = 0. Solution of the problemIt is assumed that the system is loaded by concentrated force moving with velocity v = const and the tension forces are equal S 1 = S 2 = S and the mass densities are equal m 1 = m 2 = m, furthermore the damping coefficient is equal to zero. Let us introduce new dimensionless variables ξ = x/L and T = vt/L the range of which is [0, 1]. So we get the wave equationswhere δ(·) is the Dirac delta function andwhere v s is the velocity of the transverse wave in the strings.

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

“…One 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].…”

“…means the Heaviside or the unit step function. More examples of closed-form solutions can be found in [5][6][7][8].…”

Flexible Complex System of a Double-String Under Extreme Moving Loads

“…And Figure 6 shows graphs in a period of time i η ≤ T ≤ (i+1) η ≤ 1, where i = 1, 3, 5,…, 2n-1. More examples of closed-form solutions can be found [8,[18][19][20][21].…”

“…The analogies between a string and the beams have been considered in papers [5,6,11]. Various aspects of the dynamics response of a string under a moving load have been considered, among others, in the papers [2,4,8,10,[12][13][14][15][18][19][20]22]. The classical solution of the response of complex systems subjected to forces moving with a constant velocity has a form of an infinite series.…”

“…The presented solutions can be also used in axial and torsional vibration of the rod. Using this method, the closed solutions for undamped vibration of string and beam due to moving force have been obtained in the papers [18][19][20][21]. The solution for the dynamic response of the composite strings under moving force is important because it can be used also in order to find the solution for other types of moving loads.…”