ME4213 Lateral Vibration of Beams_04042012195822684

ME4213/4213E

ME4213/4213E Lateral Vibration of Beams

H.P. LEE Department of Mechanical Engineering

EA-05-20 Email: mpeleehp@nus.edu.sg

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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