me4213 lateral vibration of beams_04042012195822684
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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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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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Illustration of a vibrating beam
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It can be something very small
A micro beam
A read write head
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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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Free body diagram A free body diagram of an elementary length dx of the
beam is shown.
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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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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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Equation of motion
The above equations result in the following
important relationships
)(xpx
V
Vx
M
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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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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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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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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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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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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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Natural frequencies
The natural frequencies of vibration are given by
EInn
2
4
2
l
EIlnn
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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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Natural frequencies and mode shapes
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Summary of equations for uniform beam under various end conditions
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The roots l of the frequency equation for a uniform beam
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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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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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Example 1
0sincossinhcosh2
2
2
lDlClBlAdx
Yd
lx
0sinsinhcoscosh llBllA
0cossincoshsinh3
3
3
lDlClBlAdx
Yd
lx
0coscoshsinsinh llBllA
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Example 1
From the last two equations, we obtain
ll
ll
ll
ll
coscosh
sinsinh
sinsinh
coscosh
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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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Animation – mode 1
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Animation – mode 2
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Animation – mode 3
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Animation – mode 4
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Animation – mode 5
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Animation – mode 6
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