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We consider the damping dynamical response of a finite string due to a uniformly distributed load and a point force moving with a constant velocity. The classical solution for the transverse displacement of a string has a form of a sum of two infinite series, one of which represents the forced vibrations (aperiodic vibrations) and the other one represents free vibrations of the beam. We show that the series which represents aperiodic (forced) vibrations of the string can be presented in a closed, analytical form.

The moving load problem (which is one of the most important problems in the dynamics of structures) has been studied by many authors for many years. This problem occurs in dynamics of bridges, roadways, railways, and runways as well as missiles and aircrafts. Different types of structures and girders like beams, plates, shells, and frames have been considered. Also different models of moving loads have been assumed [

We consider damping vibrations of a string of finite length

Scheme of string vibration due to moving load.

The boundary conditions have the form

After introducing dimensionless variables

Equation (

The boundary conditions (

The initial conditions have the form

The function

The closed solution of the function

Let us consider undamped vibration of a string due to a uniformly partially distributed load moving with a constant velocity. Assuming

Transverse displacement of the string due to moving load for

Let the vibrations of the string be caused by a moving load, uniformly distributed on

Interval uniformly distributed moving load.

The solution of the problem can be obtained by superposition of the solutions which are given in Sections

Phases of moving interval uniformly distributed load.

Let us consider the vibrations of the string under uniformly distributed moving mass

Scheme of string under moving uniformly distributed mass.

It is worth pointing out that when the velocity of the mass is bigger than the critical velocity, the string displacement is opposite to the direction of the gravity force which is consistent with our intuition.

Undamped vibration of the string due to the point force moving with a constant velocity has been considered in the paper [

Scheme of string moving force.

The solution for the moving point force, taking initial conditions (

The function

In Figures

Displacement of the string due to uniformly distributed load for

Displacement of the string due to a moving point force for

The dynamics response of a string loaded by a moving load uniformly distributed on an interval or a constant force moving with a constant velocity has been studied. The classical solutions for the transverse displacement of the string have the form of sums of infinite series. It has been shown that a part of the solution can be presented in a closed, analytical form. The closed, analytical solutions are derived from the fact that they are not only integrals of partial differential equations but also of some ordinary differential equations. The closed forms of the solutions take different forms whether the velocity of moving load is smaller, equal or larger than the wave velocity of the string. This follows from the fact that in a string wave phenomena may occur. The presented closed solutions have important meaning in the case when we consider the tension force in the string. The closed solutions allow analyzing the vibrations phenomena due to moving loads without performing numerical calculations; see Figure