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ME4213/4213E
ME4213/4213E Lateral Vibration of Beams
H.P. LEE Department of Mechanical Engineering
EA-05-20 Email: [email protected]
Semester 2 2011/2012
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ME4213/4213E 2
Lateral Vibration of beams
You have done the experiment on the beam
vibration (a clamped steel ruler).
Recall that the various transverse or lateral
modes of vibration are as follows.
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ME4213/4213E 3
Illustration of a vibrating beam
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ME4213/4213E 4
It can be something very small
A micro beam
A read write head
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ME4213/4213E 5
As well as something very big
Rama IX Bridge, Bangkok, Thailand OPAC, AES, and Kinemetrics were engaged in 2000 by the Expressway and Rapid Transit Authority of Thailand to inspect, instrument, and evaluate the Rama IX Bridge, a 450m span cable stayed bridge. Excessive vibration of the bridge has led to concerns about its fatigue life.
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ME4213/4213E 6
Free body diagram A free body diagram of an elementary length dx of the
beam is shown.
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ME4213/4213E 7
Free body diagram
Note that V and M are shear and bending
moments, respectively, and p(x) represents the
loading per unit length of the beam.
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ME4213/4213E 8
Equation of motion
Summing the forces in the y-direction
dV - pdx = 0
Summing moments about any point on the right
face of element
0))(( 2
21 dxxpVdxdM
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ME4213/4213E 9
Equation of motion
The above equations result in the following
important relationships
)(xpx
V
Vx
M
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ME4213/4213E 10
Equation of motion
The first equation states that the rate of change
of shear force along the length of the beam is
equal to the loading per unit length, and
The second equation states that the rate of
change of moment along the beam is equal to
the shear
From the two equations, we have
)(2
2
xpx
V
x
M
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ME4213/4213E 11
Equation of motion
The bending moment is related to the curvature
of the flexure equation (from your solid
mechanics course)
Therefore
2
2
x
yEIM
)(2
2
2
2
xpx
yEI
x
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ME4213/4213E 12
Free lateral vibration of a beam
For a beam vibrating under its own weight, the
equation of motion (assuming harmonic motion)
is (y = Yeiωt)
In the case where the flexural rigidity EI is
constant, the above equation becomes
))((0 22
2
2
2
2
YxpyYdx
YdEI
dx
d
02
4
4
Ydx
YdEI
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ME4213/4213E 13
Free lateral vibration of a beam
Substituting
we obtain the fourth-order differential equation
for the vibration of a uniform beam.
EI
24
04
4
4
Ydx
Yd
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ME4213/4213E 14
General solution
The general solution is
The solution can be derived by assuming the
solution to be of the form
xDxCxBxAY sincossinhcosh
xeY
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ME4213/4213E 15
General solution
which will satisfy the differential equation when
= , and = i
Since
the form of solution can be established.
xxe
xxe
x
x
sincos
sinhcosh
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ME4213/4213E 16
Natural frequencies
The natural frequencies of vibration are given by
EInn
2
4
2
l
EIlnn
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ME4213/4213E 17
Natural frequencies
where the number n depends on the boundary
conditions of the problem
The mode shapes of a uniform beam for
different end conditions are as shown
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ME4213/4213E 18
Natural frequencies and mode shapes
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ME4213/4213E 19
Summary of equations for uniform beam under various end conditions
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ME4213/4213E 20
The roots l of the frequency equation for a uniform beam
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ME4213/4213E 21
Example 1
Determine the natural frequencies of vibration of
a uniform beam clamped at one end and free at
the other.
The boundary conditions are
00
00
0
0
0
3
3
2
2
dx
YdorV
dx
YdorM
lxAt
dx
dY
Y
xAt
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ME4213/4213E 22
Example 1
Sub these boundary conditions into the general
solution
(y)x=0 = A + C = 0, A = - C
0cossincoshsinh0
0
x
x
xDxCxBxAdx
dY
DBDB ,0
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ME4213/4213E 23
Example 1
0sincossinhcosh2
2
2
lDlClBlAdx
Yd
lx
0sinsinhcoscosh llBllA
0cossincoshsinh3
3
3
lDlClBlAdx
Yd
lx
0coscoshsinsinh llBllA
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ME4213/4213E 24
Example 1
From the last two equations, we obtain
ll
ll
ll
ll
coscosh
sinsinh
sinsinh
coscosh
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ME4213/4213E 25
Example 1
which reduces to
There are a number of values of l which can
satisfy the above equation, corresponding to
each normal mode of oscillation. The first and
second modes are given by 1.875 and 4.695,
respectively. The natural frequency given by the
first mode is
0coscosh ll
EI
l
EI
l22
2
1
515.3875.1
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ME4213/4213E 26
Animation – mode 1
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ME4213/4213E 27
Animation – mode 2
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ME4213/4213E 28
Animation – mode 3
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ME4213/4213E 29
Animation – mode 4
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ME4213/4213E 30
Animation – mode 5
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ME4213/4213E 31
Animation – mode 6
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ME4213/4213E 32