This paper investigates the flexural vibration of a finite nonuniform Rayleigh beam resting on an elastic foundation and under travelling distributed loads. For the solution of this problem, in the first instance, the generalized Galerkin method was used. The resulting Galerkin’s equations were then simplified using the modified asymptotic method of Struble. The simplified second-order ordinary differential equation was then solved using the method of integral transformation. The closed form solution obtained was analyzed and results show that, an increase in the values of foundation moduli

This paper is sequel to an earlier one by Oni and Ayankop-Andi in [

In practice, cross-sections of elastic structures such as plates and beams are not usually uniform and the moving loads are commonly in distributed forms. To this end, in [

Emphatically speaking, in order to accurately model such physical situations in realistic manners, an accurate representation of the moving load is that it is distributed over a portion or over the entire length of the structure. In the light of this, recently, some researchers have addressed this shortcoming. Among these are Esmailzadez and Ghorashi in [

The nonuniform Rayleigh beam is a spatially one-dimensional elastic system. Thus, the problem of vibration of the finite nonuniform Rayleigh beam of length

The boundary conditions of the above problem are assumed to be general, while the initial conditions are

Substituting (

The differential equation describing the flexural vibrations of a finite nonuniform Rayleigh beam subjected to a moving distributed force may be obtained from (

If the inertia term is retained, then

Since

Substituting (

The variational equations are obtained by equating the coefficients of

Therefore, when the effects of the mass of the particle are considered, the first approximation to the homogeneous system is given as

Thus, to solve the nonhomogeneous equation (

In this section, practical examples of classical boundary conditions are used to illustrate the analyses presented in this paper.

The boundary conditions of a nonuniform Rayleigh beam clamped at both ends

Substituting (

As a second example, at end

Substituting (

The response amplitude of a dynamical system such as this may grow without bound. Conditions under which this happens are termed resonance conditions. For both illustrative examples, we observe that the nonuniform Rayleigh beam traversed by a moving distributed force at constant velocity reaches a state of resonance whenever

In order to illustrate the foregoing analysis, the nonuniform Rayleigh beam of length ^{−1}, ^{9} kg/m, and mass ratio ^{2}. The displacement response of the nonuniform Rayleigh beam is calculated and graphs are plotted for beam response against time for values of rotatory inertia correction factor

The transverse displacement response of the nonuniform clamped-clamped Rayleigh beam to distributed forces for various values of foundation moduli

Displacement response of a clamped-clamped nonuniform Rayleigh beam under distributed forces for various values of foundation moduli

Displacement response of a clamped-clamped nonuniform Rayleigh beam under distributed forces for various values of rotatory inertia correction factor

Displacement response of a clamped-clamped nonuniform Rayleigh beam to distributed masses for various values of foundation moduli

Displacement response of a clamped-clamped nonuniform Rayleigh beam under distributed masses for various values of rotatory inertia correction factor

Comparison of the deflection of moving distributed force and moving distributed mass for nonuniform clamped-clamped Rayleigh beam,

For the second illustrative example, the deflection profile of the nonuniform clamped-free Rayleigh beam to moving distributed forces for various values of foundation moduli

Displacement response of a nonuniform clamped-free Rayleigh beam under distributed forces for various values of foundation moduli

Displacement response of a nonuniform clamped-free Rayleigh beam under distributed forces for various values of rotatory inertia correction factor

Displacement response of a nonuniform clamped-free Rayleigh beam under distributed masses for various values of foundation moduli

Deflection profile of a clamped-free nonuniform Rayleigh beam under distributed masses for various values of rotatory inertia correction factor

Comparison of the deflection of moving distributed force and moving distributed mass cases of a nonuniform clamped-free Rayleigh beam,

This paper investigated the flexural vibration of a finite nonuniform Rayleigh beam under travelling distributed loads. Both gravity and inertia effects of the distributed loads are taken into consideration. The versatile technique due to Galerkin suitable for all variants of classical boundary conditions was employed to reduce the governing fourth-order partial differential equation with variable coefficients to a sequence of second-order ordinary differential equations. These series of equations were treated using a modification of the asymptotic method of Struble and integral transformations. It is shown that an increase in the values of foundation moduli and rotatory inertia correction factor reduces the response amplitudes of both the clamped-clamped nonuniform Rayleigh beam and the clamped-free nonuniform Rayleigh beam.

Analysis of the closed form solutions for both clamped-clamped and clamped-free boundary conditions showed that the critical speed for the moving distributed mass problem is smaller than that for the moving distributed force problem. Hence resonance is reached earlier in the former. Furthermore, for fixed

The authors declare that there is no conflict of interests regarding the publication of this paper.