Research Article :
The dynamic response to moving distributed
masses of pre-stressed uniform Rayleigh beam resting on variable elastic
Pasternak foundation is examined. The equation governing this problem is a
fourth order partial differential equation with variable and singular co-efficients.
To solve this cumbersome equation, the method of Galerkin approach is adopted
to reduce the governing differential equation to a sequence of coupled second
order ordinary differential equation which is then simplified further with
modified asymptotic method of Struble. The more simplified equation is solved
using the Laplace transformation technique. The closed form solutions obtained
are analyzed in order to show the conditions of resonance, and to show that
resonance is attained earlier in moving mass system than in the moving force
system. The results in plotted graphs show that as the axial force, the
rotatory inertia, foundation modulus and shear modulus increase, the deflection
of the elastically supported non-uniform Rayleigh beam decreases in each case.
The transverse deflections of the beam on variable Pasternak elastic foundation
are higher under the action of moving masses than those when only the force
effects of the moving load are considered. This implies that resonance is
reached faster in moving mass problem than in moving force problem. Transport
structures such as railway or bridges are subjected to moving vehicles (loads)
which vary in both space and time. The branch of transport has experienced
great advances, characterized by increasing high speed and weights of vehicles.
These structures on which the vehicles move have been subjected to vibration
and dynamic stress more than ever before. Therefore, the moving load problem
has been a fundamental problem in several fields of applied mathematics,
mechanical engineering, applied physics and railway engineering. Rails and
bridges are examples of structured elements to be designed to support moving
masses. Most importantly, problems of this type are mathematically cumbersome
when the inertial effect of the load is taken into consideration. The
challenges of these designs have attracted the interest of many researchers in
the fields of applied mathematics, mechanical engineering, applied physics and
railway engineering. Some of these
researchers include Fryba [1], who studied the vibration of solids and
structures under moving loads. Gbadeyan and Dada [2] examined the influence of elastic
foundation on plate under a moving load without considering the influence of
rotatory inertia and shear deformation on the plate. The work of Stanistic et
al was taken up much later by Gbadeyan and Oni [3], who investigated the
dynamic analysis of an elastic plate continuously supported by an elastic
foundation and traversed by an arbitrary number of concentrated moving masses.
Yavari [4], studied the generalized solution of beams with jump discontinuities
on elastic foundation. Yin [5] also investigated the closed form solution for
reinforced Timoshenko beam on elastic foundation. In the same vein, Teodoru [6]
in his work, analyzed beam on elastic foundation by using finite difference
approach. Oni and Awodola [7]
investigated the vibrations under a moving load of a non-uniform Rayleigh beam
on variable elastic foundation. Oni and Awodola [8] also analyzed the dynamic
response under a moving load of an elastically supported non-prismatic
Bernoulli-Euler beam on variable elastic foundation. In the work of Oni and
Omolofe [9], the dynamic analysis of a pre-stressed elastic beam with general
boundary conditions under moving loads at varying velocities was investigated.
The study on exact series solution for the transverse vibrations of rectangular
plates with elastic boundary supports was carried out by Li [10]. Hsu [11],
studied the vibration analysis of non-uniform beams resting on elastic
foundation. The work of Ismail [12] on dynamic response of a beam due to an
accelerating moving mass using moving finite element approximation cannot be
ignored. Kargarmovin and Younesian [13] took further study on dynamic of
Timoshenko beams on Pasternak foundation under moving load. Recently, Adeoye and
Awodola [14] took a close studied on influence of rotatory inertial correction
factor on the vibration of elastically supported non-uniform Rayleigh beam
using Galerkin method and Struble technique.
Adeoye and Akintomide [15] investigated dynamic behavior of
Bernoulli-Euler beam with elastically supported boundary conditions under
moving distributed masses on constant bi-parametric foundation using Galerkin
method and Struble technique. Akintomide
and Awodola [16] analyzed the dynamic response to variable-magnitude moving
distributed masses of Bernoulli-Euler beam on bi-parametric foundation and they
obtained the closed form solution using Runge-Kutta technique. In our recent research
work, Adeoye and Awodola [14] effort was made to investigate the influence of
rotatory inertial correction factor on the vibration of elastically supported
non-uniform Rayleigh beam on variable foundation. The objective of this paper
is to extend this research work to elastically supported uniform Rayleigh beam
on variable elastic bi-parametric foundation. This paper therefore investigates
dynamic response to moving distributed masses of pre-stressed uniform Rayleigh
beam resting on variable elastic Pasternak foundation. Substituting
equations (3) , (4)and (5) into
equation (1), one obtains The boundary condition
of the structure under consideration is first taken to be arbitrary and the
initial condition without any loss of generality is taken as Due
to complex nature of equation (1), no conventional method can be used to solve
the partial differential equation and till this moment, there is no exact
closed form solution to equation(1). Therefore, an approximate solution is
sought. The method of Galerkin is used to reduce equation (1) to second order
coupled ordinary differential equations, and this takes the form In
order to evaluate the integrals in E_17 (i,j),E_18 (i,j) and E_19 (i,j), one makes use of the Fourier series representation for the Heaviside
function in the form; Equation (29) is
the transformed equation governing the problem of supported beam on variable
bi-parametric elastic foundation. This coupled non-homogeneous second order
ordinary differential equation is assumed to have arbitrary boundary
conditions. Case I: Moving Force Problem In moving force
problem, only the load is being transferred to the structure. In this case, the
inertia effect is negligible. Setting ϖ=0 in the transformed equation (27), one
obtains Equation (33) represents the transverse
deflection of uniform Rayleigh beam under moving distributed force and resting
on variable Pasternak elastic foundation. Case II:
Moving Mass Problem In moving mass
problem, the moving load is assumed to be rigid, and the weight and as well as
inertia forces are transferred to the moving load. That is the inertia effect
is not negligible. Thus ϖ≠0 and so it is
required to solve the entire equation (27). Thus, equation (27) takes the form On
further rearrangements, one obtains Obviously, unlike the
moving force problem, an exact analytical solution to equation (35) is not
possible. In order to obtain approximate analytical solution, one makes use of
a modification of the asymptotic method of Struble. By this method, one seeks the
modified frequency corresponding to the frequency of the free system due to the
presence of the effect of the moving mass. An equivalent system operator
defined by the modified frequency then replaces equation (35). It
is therefore clear that for the same natural frequency, the critical speed for
the system consisting of elastically supported uniform Rayleigh beam resting on
variable elastic foundation and traverse by moving distributed force with
uniform speed is greater than that of moving distributed mass problem. Thus for
the same natural frequency, resonance is reached faster in the moving
distributed mass system than in the moving distributed force system. Using equations (75), (76) and (77), the frequency equation for
the dynamical problem is obtained as Substituting
equations (75), (76), (77) and (78) into equations (41) and (53) one obtains the displacement response
respectively to a moving force and a moving mass of Rayleigh beam elastically
supported at both ends and resting on a variable foundation. To illustrate the
analysis presented in this work, the uniform Rayleigh beam is taken to be of
length L = 12.192 m, the load velocity c = 8.128 m/s and modulus of elasticity In
this research work, the problem associating with the dynamic response to moving
distributed masses of pre-stressed uniform Rayleigh beam resting on variable
elastic Pasternak foundation has been studied. The closed form solutions of the
fourth order partial differential equations with variable and singular
co-efficients are obtained for both cases of moving force and moving mass. From
the closed form solutions obtained, the conditions of resonance are obtained
for both moving force and moving mass. Also from the closed form solutions, the
effects of beam parameters such as rotatory inertia, axial force, shear modulus
and foundation modulus on the beam for both cases of moving distributed force
and moving distributed mass were investigated. 1.
Fryba L. Vibration of solids and
structures under moving loads (1972)Noordhoff
International Publishing Groningen, Groningen, Netherlands. 2.
Gbadeyan JA and Dada MS. The dynamic
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Nigerian Association Math Physics 5: 185-200. 3.
Gbadeyan JA and Oni ST. Dynamic behavior
of beams and rectangular plates under moving loads (1995)Journal of Solid and Vibration 182: 677-695. https://doi.org/10.1006/jsvi.1995.0226 4.
Yavari A, Sarkani S and Reddy JN. Generalized
Solution of beams with jump discontinuities on elastic foundations (2001)
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Yin JH. Closed form solution for
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Oni ST and Omolofe B. Dynamic analysis of
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MH. Vibration analysis of non-uniform beams resting on elastic foundations
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E. Dynamic response of a beam due to an accelerating moving mass using moving
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MH and Younesian D. Dynamics of Timoshenko beams on Pasternak foundation under
moving load (2004) Mechanics Res Communications 31: 713 -723. https://doi.org/10.1016/j.mechrescom.2004.05.002 14. Adeoye
AS and Awodola TO. Influence of rotatory inertial correction factor on the
vibration of elastically supported non-uniform Rayleigh beam on variable
foundation (2017) Asian J Math 2 1-22. 10.9734/ARJOM/2017/31271 15. Awodola
TO and Akintomide A. Dynamic response to variable-magnitude moving distributed
masses of Bernoulli-Euler beam resting on Bi-Parametric elastic foundation
(2017) Asian J Math 5: 1-21. 10.9734/ARJOM/2017/33122 16. Adeoye
AS and Akintomide A. Dynamic behavior of Bernoulli-Euler beam with elastically
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Dynamic Response to Moving Distributed Masses of Pre-stressed Uniform Rayleigh Beam Resting on Variable Elastic Pasternak Foundation
Adeoye AS and Awodola TO
Abstract
Full-Text
Introduction
Analytical
Approximate Solution
Numerical
Results and Discussions
, the moment of inertiaFigures
Conclusion
For complete eqautions go through the pdf
References
