_{1}

^{*}

The paper investigates the response of non-initially stressed Euler-Bernoulli beam to uniform partially distributed moving loads. The governing partial differential equations were analyzed for both moving force and moving mass problem in order to determine the behaviour of the system under consideration. The analytical method in terms of series solution and numerical method were used for the governing equation. The effect of various beam observed that the response amplitude due to the moving force is greater than that due to moving mass. It was also found that the response amplitude of the moving force problem with non-initial stress increase as mass of the mass of the load M increases.

In the recent years all branches of transport have experienced great advances characterized by increasing higher speeds and weight of vehicles. As a result, structures and media over or in which the vehicles move have been subjected to vibrations and dynamic stresses far larger than ever before. Many scholars have studied vibration of elastic and inelastic structures under the action of moving loads for many years, and effort are still being made to carry out investigation dealing with various aspect of the problem [1-15]. The structures on which these moving loads are usually modeled are by elastic beams, plates or shells. The problem of elastic beam under the action of the moving loads was considered by Willis [

This paper deals with the response of non-initially stresses Euler-Bernoulli beam with an attached mass to uniform partially distributed moving loads. The main objectives of this paper are

1) To present the analysis of the dynamic response of a non-initially stressed finite elastic Euler-Bernoulli beam with an attached mass at the end x = L, but arbitrary supported at the end x = 0, to uniform partially distributed moving load.

2) To present a very simple and practical analyticalnumerical technique for determine the response of beams with non-classical boundary conditions carrying mass.

With reference to

The governing equations describing the vibration behaviour of a uniform non-initially stressed Euler-Bernoulli with an attached mass at the end x = L but traversed by a concentrated moving loads are

where, E the is the modulus of elasticity, I is the second moment of area of the beam’s cross-sectional, m is the mass per unit length of the beam, is the damping constant, , Y is the deflection of the beam, x is the spatial coordinate, t is the time and f(x, t) is the applied force (i.e. the resultant concentrated force caused by the moving mass).

The applied force per unit length F(x, t) is the uniform partially distributed moving load which is defined as

M: is the mass of the load.

g: is the acceleration due to gravity.

: is the fixed length of load.

: is the length of the beam.

The differential operator is defined as

H(x) is function such that

Hence the governing equations describing the vibration behaviour of a uniform non-initially stressed EulerBernoulli beam with an attached mass at the end x = L becomes

Subject to the following boundary conditions

The corresponding initial conditions are

We assumed a solution in the form of a series

where are the known Eigen functions of the beam.

The Eigen functions satisfying the following equation

where are natural frequencies.

is the solution to Equation (10) and a, b, c, d are constants coefficients and are functions of time to be determined We further assumed

Substituting Equation (9) into Equation (5), we have

(14)

Multiply both sides of the R. H. S. of Equation (14) by and taking the definite integrals of both sides along the length of the beam with respect to x, we have

(15)

Evaluation the first definite integral in Equation (15) by carrying out integration by part with respect to using the following two properties of singularity function [ ]

Similar arguments to second, third to fifth definite integral in (15) hence evaluating the integrals using Taylor’s series expansion and applying orthogonality properties of the characteristics function the R. H. S. of (15), we finally obtain

Substituting Equation (18) into the R. H. S. of Equation (14) we have

Considering Equations (10) and (11), then Equation (19) becomes

The Equation (20) must be satisfied for arbitrary and this possible only when the expression in the curl bracket is equal to zero. Hence

The system of Equation (21) is a set of coupled ordinary second order differential equations and it is easily observed that a numerical approach is required to solve it.

The Eigen functions

We obtain the set of exact governing differential equation for the vibration of the beam by employing Equation (22) and evaluating the exact values of the integral in Equation (15) and we finally obtain

(23)

Note for the case i = j we replace the expression involving by

To solve Equation (23) recourse can be made to a numerical method, but two interesting cases are to be tackled.

A moving force problem is one in which the inertia effects of the moving load are neglected and only the force effects are retained. In other words by neglecting all the terms on the R. H. S. of Equation (23) except the first term.

This is the case in which both the inertia effect as well as the force effect are taken into consideration. The entire Equation (23) is the moving mass problem To obtain results given in this paper, an approximate central difference formulas have been made used of, for the derivatives in Equation (23) for both cases [5.1 and 5.2]. Thus, for N modal shapes, Equation (23) are transformed to a set of N linear algebraic equations, which are to be solved for each interval of time. Regarding the definition of approximation involved, in order to ensure the stability and convergence of the solution, sufficiently small time steps have been utilized.

Computer programe was developed and the following numerical data which are the same as those in reference [ ] were used for the purpose of comparison

Hence we have the graphs of results. See Figures 2-5.

The displacement profiles of the beam are display graphically to demonstrate the effect of the mass, the angular frequency and the viscous damping magnification factor.

From the response profile of the beam it was observed that the beam has more than one mode of vibration with

each mode having a different natural frequency. The amplitude increases with increase in viscous damping. Also it was observed from