 Open Access
 Total Downloads : 152
 Authors : B. Omolofe, O. K Ogunbamike
 Paper ID : IJERTV3IS10089
 Volume & Issue : Volume 03, Issue 01 (January 2014)
 Published (First Online): 16012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Influence of Some Vital Structural Parameters on the Dynamic Characteristics of Axially Prestressed Beam Under Moving Masses.
1 B. Omolofe and 2 O. K Ogunbamike
1Department of Mathematical Sciences, School of Sciences, The Federal University of Technology, PMB 704, Akure, Ondo state, Nigeria.
2 Department of Mathematics, Adeyemi College of Education, Ondo,Ondo State, Nigeria.
Abstract
This study concerns the problem of the longitudinal vibrations of beams resting on elastic foundation and under the actions of travelling exponentially varying load with a constant velocity type of motion. A mathematical formulation representing the transverse motions of the engineering structure which is valid for all variants of classical boundary conditions is set up. An analytical method of analysis is presented for investigating the effects of some vital structural parameters on the dynamic response characteristic of the structurally prestressed elastic beam with rotatory inertia correction factor. The theory proposed is applied to a beam having simple supports at both ends. However, the theory and application is not limited to the beam with simply supported boundary condition and without much mathematical difficulties, it can also be used to treat dynamic analysis of continuous beam structures including energy dissipation. Closed form solutions of the equation of motion describing the beamload interactions are obtained. Analysis is carried out and various results are presented in plotted curve.

Introduction
The vibration analysis (linear or nonlinear) of structural members has been and continues to be the subject of numerous studies in Engineering
Science and related fields; this is due largely to the fact that it embraces a wide class of problems with great relevance in many engineering applications. For example, the analysis and design of highway and railway bridges, cablerailways, cableways and overhead cranes to mention a few.
Thus, the effect of the nature of the complexity of the interactions between beams or beamlike structural elements and the load traversing them at various velocities on the dynamic characteristics of such structures have been investigated by many researchers [110].
Historically, three types of problems have been considered in the open literature. If the inertia effect of the moving subsystem is neglected, the problem reduces to the vibration of elastic structure under the actions of an external moving force and this is termed the moving force problem. When the inertia effect of the mass of the moving load is taken into account and assuming infinite stiffness of the coupling between the continuous system and the moving subsystem, we have the moving mass. The dynamical problem involving finite coupling stiffness leads to the moving oscillators problem. Depending on the type of model used and assumptions adopted, great numbers of publications have evolved during the past years [1126].
In spite of the enormous investigations and great amount of work that have been devoted to the dynamic response of beams under moving load, it unarguably remains a major topic for future scientific research because of the continuing advancements in design technology
and emergence of new materials with improved
the spatial coordinate taken along the axis of the beam and t is the time variable.
When the inertia of the moving load is taken into consideration, the transverse load can be expressed in the form
quality which enable the construction of lighter
1 d 2Qx,t
and more slender structures, vulnerable to fast traveling heavy loads [27].
P x,t
Pf x,t 1
g
dt 2
(2)
It is well known that a considerable amount of work dealing with the vibration of beams under the effect of moving load has been found in the open literature, however, to the
authors knowledge, the vibration analysis of
where the continuous moving force
Pf (x,t) acting on the beam model is given by
beams incorporating a rotatory inertia correction
P (x,t) et M g (x v t)
(3)
f i i
factor and subjected to exponentially varying load
is not common. Thus, this work concerns the d 2
dynamic response of axially prestressed elastic beam resting on elastic foundation and traversed by masses traveling at constant velocity. The
and the convective acceleration operator
defined as
dt 2 is
specific aims of this study is to classify the effects
d 2 2
2v 2
v 2
of some parameters namely prestressed,
i i
(4)
foundation subgrade, rotatory inertia correction factor, mass ratio etc on the flexural motions and critical velocity of elastic beams subjected to moving masses.
dt 2 t 2 xt x2
The boundary condition is arbitrary and without loss of generality, the initial conditions are of the form
2.0 Formulation of the problem
The governing differential equation for isotropic
Q(x,0) 0,
Q(x,0) 0
t
(5)
beam of length L on an elastic foundation and traversed by a moving load of mass M travelling with constant velocity v is given by
Substituting equations (2) to (4) in equation (1) one obtains
4 Q(x, t)
EI x 4 N
2 Q(x, t)
x 2
2 Q(x, t)
t 2

R
o 4 Q(x, t)
x 2 t 2
4Q(x, t)
EI N
x 4
2Q(x, t)
x 2
2Q(x, t)
t 2
4Q(x, t)
o

R
x 2t 2
o

K ex Q(x, t) P(x, t)
2
x Q(x, t)
K e Q(x, t) M (x v t)

2v
2Q(x, t)
2Q(x, t)
v 2
(1)
where EI is the flexural rigidity of the beam,
Qx,t is the transverse deflection, is the mass
o i t 2
m
i i
et M g (x v t)
i 1
i xt i
(6)
x 2
per unit length of the beam, N is the constant axial which describes the flexural motions of axially
force, K is the elastic foundation,
Ro is the
prestressed beam resting on elastic foundation and
o
rotatory inertia,
Px,t
is the transverse load, x is
subjected to variable magnitude moving load. It is remarked at this juncture that equation (6) is valid for all variants of classical boundary conditions.
3.0 General Solution Procedures
4.0 Operational simplification
To obtain Wi (t) from equation (8), it is
It is evident that an exact closed form solution of the above partial differential equation (6) is not feasible. Consequently, an approximate solution is sought. Thus, the Galerkin technique described in
[24] is employed. By this technique the partialrequired that the expression on the left hand side of equation (8) be orthogonal to the functionU j (x) . Thus
differential equation (6) is reduced to a sequence
n

o
of ordinary differential equations in the first instance and the resulting set of second order
i1
EIH1 (i, j)
n
NH 2 (i, j)
L
Ko H 4 (i, j) Wi (t)
H 3 (i, j)
L
R H 2 (i, j) Wi (t)

M i Wi (t)
(x vi t)Ui (x)U j (x)dx 2viWi (t)
(x vi t)Ui (x)U j (x)dx
ordinary differential equations is further simplified
usng an asymptotic method of solution due to
i1 0
L
0
m

v2W (t) (x v t)U (x)U (x)dx et M gU (v t)
Struble and these set of equations are finally solved completely by the method of integral transformations. This versatile technique requires that the solution of equation (6) takes the form
i i
0
i i j
i j i
i1
(10)
n
Qn (x, t) Wi (t)Ui (x)
i1
(7)
we note that the dirac delta function as an even function can be expressed as
where
U (x)
is chosen such that pertinent
(x v t) 1 2
cos nvi t cos nx
i
boundary conditions are satisfied. Equation (7) when substituted into the equation (6) yields
L
L
L
n EIW (t)U iv (x) NW (t)U (x) W (t)U (x) RoW (t)U (x)
i
n1
L
(11)
i i
i1
i i i i i i
In view of equation (7), using equation (11) in
K exW (t)U (x) n M (x v t)W (t)U (x) 2v W (t)U (x)
equation (10), after some simplifications and
o i i
i
i1
i i i
i i i
rearrangement, one obtains
v2W (t)U (x) et n M g (x v t) 0
n
i i i
i i
i1
n
1 (i, j)
nvi t
(8)
Wi (t) (i, j) Wi (t)
(i, j) H a (i, j) 2
cos
L
H b (i, j, n)Wi (t)
An appropriate selection of functions for beam
i1
o i1 o
nv t
n1
th 2v H (i, j) 4v cos i H
(i, j, n)W (t)
d
i
problems are beam mode shapes. Thus, the m
normal mode of vibration of a uniform beam
i c i
n1 L
2
2 nvi t
m
et
vi H e (i, j) 2vi
cos
L
H f (i, j, n)Wi (t) (i, j)
M i gU j (vi t)
U (x) sin i x A cos i x B sinh i x C cosh i x
n1
o i1
i L i L i
L i L
(9)
where
(12)
is chosen such that the pertinent boundary
conditions are satisfied. In equation (9), is the L L
iv
mode frequency,
A , B , C
i
are constants which
H1 (i, j) Ui (x)U j (x)dx
0
H 2 (i, j) Ui (x)U j (x)dx
0
i j
i i i
can be determined by using the boundary conditions associated with the beam structure.
L
H 3 (i, j) U i (x)U j (x)dx
0
L
H 4 (i, j) e
0
xU (x)U (x)dx
L L nx n (i, j)
H a (i, j) Ui (x)U j (x)dx
Hb (i, j, n) cos
L Ui (x)U j (x)dx
W (t) 1 W (t)
i
i
n
0 0 i1
o (i, j)
H (i, j)
L
U (x)U (x)dx
H (i, j, n)
L nx
L
cos U (x)U (x)dx
H (i, j) 2
cos
nvi t
H b (i, j, n) Wi (t)

i j
a
0

i j
0
i1 o (i, j)
n1 L

2v H (i, j) 4v cos nvi t
H (i, j)
L
U (x)U (x)dx
H (i, j, n)
L
cos
nx
U (x)U (x)dx
i c i
n1
H d (i, j, n) Wi (t)
L


i j
0

i j



L
0
2
2 nvi t
vi H e (i, j) 2vi cos L
H f (i, j, n) Wi (t)
(i, j) H (i, j) Ro H (i, j)
n1
o 3 2
EI N K
et m
jv t
1 (i, j)
H1 (i, j) H (i, j) H (i, j)
o
2 4
M g sin i
i
(13)
o (i, j) i1 L
5.0 Application and Illustrative examples
For the purpose of analysis, an elastic beam with simply supported boundary conditions, carrying fast traveling masses is considered. However, the analysis and formulation presented in this work are not limited to just simply supported boundary condition. The analysis in its general form may well be applied to beams with various boundary conditions. For beams with simple supports at both ends x = 0 and x = L, it can be shown that
(16)
Equation (16) represents the transformed equation of a uniform Rayleigh beam on a constant elastic foundation. Evidently, an exact closed form solution to this problem is not possible. Consequently, in what follows two cases of the coupled equation are considered.
6.0 Solution of the transform equation (6.1) The moving force problem
A B C
0 and
i
. To this effect,
If the inertia effect of the moving mass is
i i i i L
the transverse displacement response of beams having simple supports at both ends can be given taking into account (7) as
considered as negligible, we shall have the classical case of a moving force problem. Under this assumption 0 and after some simplifications and rearrangements and
n
Qn (x,t) Wi (t)sin
i1
ix L
(14)
considering only the ith particle of the system, equation (16) becomes
o
2
et
jvi t
and
U j
jx
(x) sin ,
L
U j (vi t) sin
jvi t L
(15)
Wi (t) mf Wi (t)
where
Mg sin
(i, j)
L
(17)
Substituting equation (14) into the transformed
2 1 (i, j)
(18)
governing equation (12) and after some simplifications and rearrangements one obtains
mf
o
(i, j)
It can be shown that the general solution of equation (17) can be written in the form
Wi (t) C1 cosmf t C2 sinmf t P1 (t)cosmf t P2 (t)sinmf t
(19)
where
Pmf 1 1
P (t) Pmf
et sin vi t sin
tdt
(20)
C1
2
mf
2 (a
mf
mf
)2 2 (a
)2
1
and
P2 (t)
mf
Pmf
et sin
L
vi t L
cos
mf
mf tdt
(21)
and
C2
Pmf
2mf
2
a mf
2
(a mf )
(25)
2
a mf
2
(a mf )
mf
C1 and C2 are constants to be determined and
P Mg .
(26)
substituting equations (25) and (26) into equation (24), simplifying and inverting yield
0
mf
i, j
n Mg
Qn (x,t)
cosmf t 2 2 2 2
Evaluating integrals (20) and (21) leads to
i1 2mf o (i, j)
(a mf ) (a mf )
(a mf )
(a mf ) t
(a mf ) sin(a mf )t

sin t
2
2 2 cos mf t 2 2
2
Pmf a mf t
t
(a mf )
(a mf ) (a mf )
mf mf

e
P1 (t)
2 2 (a )2 e
sin(a mf )t
2 (a )2 e
mf
cos(a mf )t
a
cos(a mf )t (a mf ) sin(a mf )t (a mf ) cos(a mf )t
mf
mf
mf

mf et sin(a
mf
2 (a )2 mf
)t
mf
2 (a )2
et cos(a
)t
mf
2 (a )2 2 (a )2
2 (a )2
2
2
sin(a mf )t
(a mf ) cos(a mf ) sin(a mf )t
(22)

sin mf t
2
2
(a mf )
(a mf )
2
2
(a mf )
(a mf ) cos(a mf )t
ix
and
2
2
(a mf )
sin
L
2
2 2 (a )2
P (t) Pmf
mf mf
et sin(a
a
)t mf et cos(a )t
mf
mf 2 (a )2 mf
mf
a
Equation (27) represents the transverse displacement response to a moving force of a uniform Rayleigh beam resting on an elastic
mf
mf
2 (a )2
et sin(a
)t
mf
mf
2 (a )2
et cos(a
(23)
)t
foundation.
substituting equations (22) and (23) into equation
(19) yields
(6.2) The moving mass problem
In this case, 0 that is, the mass of the
t (a mf ) sin(a mf )t
moving load is commensurable with that of the
Wi (t)
C1 cos mf t C2 sin mf t e
cos mf t 2 2
(a mf )
structure, the inertia effect of the moving mass is

cos(a mf )t
mf
2 (a ) 2

(a mf ) sin(a mf )t
mf
2 (a ) 2

(a mf ) cos(a mf )t
mf
2 (a ) 2
taken into consideration. This is termed the moving mass problem. To this end equation (16)
sin(a )t

sin t mf


(a mf ) cos(a mf )t
mf
sin(a mf )t
is rearranged to take the form
mf
mf
2 (a ) 2
2 (a ) 2
2 (a ) 2
nv t
W (t) 2 W (t) H (i, j)W (t) 2cos i H
(i, j, n)W (t)
B
i
(a mf ) cos(a mf )t
i mf i
A i
n1 L
mf
2 (a ) 2
L

2v H (i, j)W (t) 4v cos nvi t H
(i, j)W (t) v2 H (i, j, n)W (t)
(24)
i C i
i D
n1
i i E i
2
nvi t
et
jvi t
When use is made of the initial conditions (5) in
2vi
cos
L
H F (i, j, n)Wi (t) (i, j) M i g sin L
conjunction with equation (24), one obtains
n1 o
where
H (i, j) H a (i, j)
o
A (i, j)
H (i, j, n) Hb (i, j, n)
o
B (i, j)
O( 2 )
(32)
H (i, j)
H (i, j, n)
and
H (i, j) c H (i, j, n) d 1
C
D
o (i, j) o (i, j)
nvi t
H (i, j) He (i, j)
E
H (i, j) H f (i, j, n)
F
1 H A (i, j) 2cos L H B (i, j, n)
o (i, j)
o (i, j)
(29)
n1
Equation (28) after some simplifications and rearrangements take the form
nvi t
1 H A (i, j) 2
cos
L
H B (i, j, n)
4v
cos nvi t H
i, j, n
n1
* 2vi H c i, j i L D
i
W t n1 W t
(33)
i
nvi t
1 * H A i, j 2
cos
L
H B i, j, n
where
2 2
n1
2v 2
cos nvi t H
i, j, n
mf
* vi H E i, j
i L F
nv t
n1 W t
i
H
(i, j) 2cos i H
(i, j, n) 1
nv t
A L B
1 * H
A i, j 2
cos i H
L B
i, j, n
n1
et
n1
MgU v t
(34)
o i, j
i
2
i
L B
cos nvi t H
i, j, n
Substituting equations (32) and (33) into equation
1 * H A i, j
n1
(30)
(30), one obtains
In equation (30), unlike in the case of moving
W (t) 2v H
(i, j) 4v cos nvi t
force problem an exact analytical solution is not
i i C
i
n1
H D (i, j, n)Wi (t)
L
feasible. Though the equation yields readily to
2 2
2 nvit
numerical technique, an analytical approximate
mf vi H E (i, j) 2vi
cos
L
H F (i, j, n) Wi (t)
method is desirable as solutions so obtained often
n1
nv t
shed light on vital information about the vibrating system.
2 H
mf
et
(i, j) 2 cos i H
A
B
n1 L
i
M g sin jvi t
(i, j, n)
Here we seek the modified frequency corresponding to the frequency of the free system due to the presence of the effect of the mass of the system. An equivalent free system operator defined by the modified frequency then replaces
o (i, j) L
to O() only.
(35)
equation (30). To this end, we set the right hand
When
0
in equation (35), a case
side of (30) to zero and consider a parameter
1 for any arbitrary mass ratio * define as
corresponding to the case when the inertia effect of the mass of the system is neglected, then the solution of (35) can be written as
1
(31)
W (x,t) Amf cos(mf t m )
(36)
So that
where Amf and m are constants.
Furthermore as 1strubles technique required
solving equations (40) and (41) respectively gives
that the asymptotic solutions of the homogeneous part of the equation (28) can be written as
mf
W (x,t) C(m,t)cos t (m,t) (1,t) O( 2 )
(37)
C(m,t) A eZt
m
and
H
(42)
(i, j)
(m,t)
v2 E
i
H (i, j)t
m
where
C(m,t)
and
(m,t)
are slowly varying
2
mf
mf A
function of time or equivalent, where implies is of
To obtain the modified frequency, equation (37)
where m
is a constant.
(43)
and its derivatives are substituted into the homogeneous part of equation (35). We extract only the variational part of the equation describing the behaviour of C(m,t) and (m,t) during the
Therefore, when the inertia effect of the moving mass is considered, the first approximation to the homogeneous system is
W (t) A eZt cos t
motion of the mass. Thus, making this i m
substitution and taken into account the following trigonometric identities
kf m
(44)
sin
t (m,t)cos nvit 1 sin
t (m,t) nvit sin t (m,t) nvit
where
mf L 2 mf
L mf L
cos
t (m,t)cos nvit 1 cos
t (m,t) nvit cos
t (m,t) nvit
H
(i, j) v2 H E (i, j) 45)
mf L 2 mf
L mf
L
(38)
mm mf 1 A
2
i 2
mf
and neglecting terms which do not contribute to the variational equations we obtain
represents the modified natural frequency due to the presence of the moving mass. It is observed that when 0 , we recover the frequency of the
mf
mf
2C (m,t)mf sin t (m,t) 2C(m,t)(m,t)
cos t (m,t)
movig force problem when the inertia effect of
2v H (i, j)C(m,t) sin
t (m,t)C(m,t)v2 H (i, j) cos
t (m,t)
i C mf mf
mf
A
mf
2 H (i, j) cos t (m,t) 0
Retaining terms to O() only.
mf
i E mf
the moving mass is neglected. Thus, to solve the nonhomogeneous equation (28), the differential
operator which acts on Qm,t and Qk,t is
mf
The variational equations are obtained by equating the coefficients of sin t (m,t) and
mf
cos t (m,t)and setting them to zero, thus
replaced by the equivalent free system operator defined by the modified frequency kf . Using equation (45), the homogeneous part of equation
(30) can be written as
d 2W (t)
2C (m,t) 2v H
(i, j)C(m,t) 0
i 2
W (t) 0
(46)
i C mf
(40)
dt 2
mm i
and
Thus, the entire equation (30) becomes
2Cm,tm,t
Cm,tv2 H
i, j2 H i, j 0
d 2W (t)
i 2
W (t)
et
Mg sin jvi t
mf i E
mf A
(41)
dt 2
mm i
o (i, j)
L
(47)
retaining O() only.
Equation (47) is analogous to equation (17). Thus, using similar argument as in moving force problem, Wi (t) can be obtained which on
concentrated masses moving at constant speed as shown in figure 4.
The deflection profile for various values of applied axial force N for both cases of moving
inversion gives
n Lg
force and moving mass problems of the uniform beam are displayed in figures 2 and 5 respectively. It is observed that as the applied axial force N increases the transverse displacement of the beam
Qm (x, t) 2
(i, j) cos mmt 2 (a
)2 2 (a )2
decreases.
i 1
mm o mm
mm
(a mm )
(a mm )
t
(a mm ) sin(a mm )t
sin mmt 2 (a
)2 2 (a
)2 e cos mmt
2 (a )2
In Figures 3 and 6, the dynamic response
mm
mm mm
cos(a )t (a ) sin(a )t (a ) cos(a )t
of BernoulliEuler beam simply supported at both
mm

mm mm mm mm mm
mm
mm
2 (a )2
2 (a )2
2 (a )2
ends for various values of rotatory inertia
Ro are
sin(a mm )t
(a mm ) cos(a mm )t
sin(a mm )t
mm
sin mmt 2 (a )2
mm
2 (a )2
mm
2 (a )2
showcased for both cases of moving force and
moving mass problems. These figures depict that
(a mm ) cos(a mm )t
ix
mm
2 (a )2
sin
L
(48)
as the rotatory inertia correction factor Ro increases, the response amplitudes of vibration of the elastic beam decreases.
which represents the transversedisplacement
response to a moving mass at a constant velocity of a uniform Rayleigh beam resting on elastic foundation.
7.0 Results and Discussions
In order to illustrate the analytical results, the uniform BernoulliEuler beam is taken to be of the length L=15.192. Other values used are M 8407.27Kg ,
The comparison of the transverse displacement of moving force and moving mass cases for the simply supported beam transverse by a moving load travelling at constant velocity for K 40,000N / m3 and
K=0 

K=40000 

K=400000 

K=4000000 

0 
0.5 
1 
1.5 
2 
2.5 
3 
3.5 
4 
4. 
0.6
E 2.10924109
andV 3.128m / s . The
K=0 K=40000 K=400000
K=4000000
0.4
transverse deflections of the beam are calculated and plotted against time for various values of foundation constant (moduli), axial force and the rotatory inertia factor. Values of K between
0.2
V(L/2, t)m
0
5
0.2
0N / m3 and 4.0106 N / m3 were used while the
0.4
values of N were varied between
N 0N and N 2.0106 N .
Figure 1 illustrates the displacement response of the simply supported Bernoulli
Euler beam for a moving force problem for fixed values of N and various values of K. Clearly, the results show that as the foundation modulus increases, the transverse displacement of the beam decreases. Similar results are obtained when the simply supported beam is transverse by a
0.6
0.8
Time (t secs)
Fig 1: Transverse displacement of a simply supported moving force for various values of foundation modulus and fixed values of axial force N 20000N and
rotatory inertia R0 0.8
K=0 

K=40000 

K=400000 

K=4000000 

0 
0.5 
1 
1.5 
2 
2.5 
3 
3.5 
4 
4. 
0.4
0…6
0…4
0…2
V(L/2.t)m
0
0…2
0…4
0…6
0…8
N=20000 N=200000
N=2000000
N=20000000
N=20000 N=200000 N=2000000
N=20000000
N=20000
N=200000
4…
4
3…5
3
2…5
2
1…5
1
0…5
0
5
Tiiime (secs)
0.2
0
V(L/2, t)m
0.2
0.4
0.6
0.8
5
K=0 K=40000 K=400000
K=4000000
Time (t secs)
Fig 2: Deflection profile of simply supported moving force for various values of axial force and fixed values of foundation modulus
Fig 4: Transverse displacement of a simply supported moving mass for various values of foundation modulus and fixed axial force
K 40000N / m3 and rotatory inertia R0 0.8
N 20000N and rotatory inertia R0 0.8 .
R0=1.5 R0=3.5 R0=5.5
R0=7.5
R0=1.5 R0=3.5 R0=5.5
R0=7.5
R0=1.5 R0=3.5 R0=5.5
R0=7.5
0.4
0.2
0.4
N=20000 N=200000 N=2000000
N=20000000
N=20000 N=200000 N=2000000
N=20000000
0.2
V(L/2, t)m
0
0.2
0.4
0.6
0.8
5
4.
4
3.5
3
2.5
2
1.5
1
0.5
0
Time (t secs)
0
V(L/2,t)m
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
time(secs)
Fig 3: The deflection of simply supported moving force for various values of rotatory inertia and fixed values of foundation modulus
K 40000N / m3 and axial force N 20000N
Fig 5: Deflection profile of simply supported moving mass for various values of axial force and fixed values of foundation odulus K 40000N / m3 and
rotatory inertia R0 0.8
0.4
0.2
0
V(L/2,t)m
0.2
0.4
0.6
0.8
R0=1.5
R0=3.5 R0=5.5
R0=7.5
R0=1.5 R0=3.5 R0=5.5
R0=7.5
4
4
3.5
3
2.5
1.5 2
1
0.5
0
.5
time (secs)
Fig 6: The deflection of simply supported moving mass for various values of rotatory inertia and fixedvalues of foundation
modulus K 40000N / m3 and axial force
N 20000N .
Moving Force Moving Mass 

0 
0.5 
1 
1.5 
2 
2.5 
3 
3.5 
4 
4. 
0.6
Moving Force Moving Mass
0.4
0.2
Fig 8: The graph of modified natural frequency against mass ratio for fixed value of foundation modulus K 40000N / m3 and axial force N 20000N .
V(L/2,t)m
0
0.2
0.4
0.6
0.8
5
Time (secs)
Fig 7: Comparison of the displacement response of moving force and moving mass cases of a simply supported fixed values
N 20000N, K 40000N / m3 and R0 0.8
Fig 9: The graph of modified natural frequency against velocity for fixed value of foundation modulus K 40000N / m3 and axial force N 20000N .
Fig 10: The graph of modified natural frequency against rotatory inertia for fixed value of foundation modulus K 40000N / m3 and axial force
N 20000N .
Fig 11: The graph of critical velocity against mass ratio for fixed value of foundation
Fig 12: The graph of critical velocity against foundation modulus for fixed value of axial force N 20000N .
Fig 13: The graph of critical velocity against axial force for fixed value of foundation
modulus K 40000N / m3 .
modulus K 40000N / m3 and axial force
N 20,000N
is displayed in figure 7. It is
N 20000N .
observed from this figure that relying on moving force solution as a safe approximation to moving mass problem is quite misleading.
In figure 8, the relationship between the natural frequency and the mass ratio is displayed. It is shown from the figure that as the mass ratio
increases the natural frequency of the system also increases. Similar result is obtained in figure 9 which depicts that as the velocity of the traversing load increases the natural frequency of the system also increases. For fixed values of foundation modulus K and axial force N figure 10 clearly shows that as the values of rotatory inertia correction factor increases, the natural frequency of the system decreases.
In figure 11, it is clearly shown that as the mass ratio increases the critical velocity of the dynamical system decreases. While figure 12 depicts that as the values of foundation modulus K increases for fixed values of other parameters the critical velocity of the system also increases. Similar result is obtained in figure 13 which show that for fixed value of foundation modulus K and rotatory inertia correction factor R0 the critical velocity of the system increases as the value of axial force N increases. These interesting results confirm that the presence of these structural parameters in appropriate measures in the design of engineering structures will enhance safety and reliability in the design of such structures.

Conclusion
The dynamic behaviour of a beam simply supported at both ends and carrying moving concentrated varying magnitude loads has been analyzed. An approximate method of solution has been employed to treat the governing differential equations of motion describing the dynamic interactions of the continuous system and the moving subsystems. The system response in series form has been obtained with the inclusion of the inertial effect of the moving mass in the governing differential equations of motion. It was observed that the moving mass inertial effect is very significant. Results further show that structural parameters such as the foundation modulus, axial force, mass ratio and the rotatory inertia correction factor have significant effects on the flexural motions and critical velocity of elastic structures carrying moving load. Thus, in the design of engineering structures such as railway bridges, overhead cranes, cableways and tunnels effects of these important parameters should put
into considerations to guarantee the safety and reliability of the design.
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