Vibration of Membranes and thin Plates

1. Periodic Structure
Maged Mostafa
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Vibration of Continuous
Structures

2. Periodic Structure
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Vibration of Membranes &Thin Plates

3. Periodic Structure
Maged Mostafa
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Course Contents
SDOF
M-DOF
Cables/String
 Bars
 Shafts
 Vibration Attenuation
 Beams
 Membranes & Thin Plates
o FEM for Vibration
o Aeroelasticity

4. Periodic Structure
Maged Mostafa
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Vibration of Membranes

5. Periodic Structure
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Objectives
• What is Membrane?
• The equation of motion for membranes
• Estimate the Natural Frequencies
• Understand the concept of mode shapes
• Apply BC’s and IC’s to obtain structure response

6. Periodic Structure
Maged Mostafa
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Objectives
• What is Membrane?
• The equation of motion for membranes
• Estimate the Natural Frequencies
• Understand the concept of mode shapes
• Apply BC’s and IC’s to obtain structure response

7. Periodic Structure
Maged Mostafa
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What is Membrane?
• 2-D material (Structure) does not resist
bending
• Analogous to cables
• Examples:
• Tents
• Human Tissues
• Light Aircraft

8. Periodic Structure
Maged Mostafa
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Objectives
• What is Membrane?
• Derive the equation of motion for
membranes
• Estimate the Natural Frequencies
• Understand the concept of mode shapes
• Apply BC’s and IC’s to obtain structure response

9. Periodic Structure
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Equations of Motion
• The structure itself does not resist
bending, so the restoring force is due
only to the tension
w: Out of plane displacement
: tension per unit length
: mass per unit area
Laplace operator in Cartesian coordinates:
),,(w),,(w2
tyxtyx tt 
2
2
2
2
2
yx 






10. Periodic Structure
Maged Mostafa
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Equations of Motion
Boundary Conditions are defined along the shape
of the boundaries not just a point!!











c
t
w
cy
w
x
w
2
2
22
2
2
2
1

11. Periodic Structure
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Rectangular Membrane
• Using Separation of variables:
22222
2
2
2
2
2
,,
1
1
















Y
Y
X
X
Y
Y
X
X
T
T
c
T
T
cY
Y
X
X



12. Periodic Structure
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Rectangular Membrane
• Solve the 3 ODEs:
)cos()sin()(
0
)cos()sin()(
0
)cos()sin()(
0
22
2
2
tcHtcFtT
TcT
yDyCyY
YY
xBxAxX
XX














13. Periodic Structure
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Rectangular Membrane
• Total Spatial Solution
)cos()cos()sin()cos(
)cos()sin()sin()sin(),(
)()(),(
43
21
yxAyxA
yxAyxAyxW
yYxXyxW






14. Periodic Structure
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Apply Boundary Conditions
• Clamped Along all boundaries
)cos()sin()sin()sin(),(
0
)cos()sin(
)()0(
0),0(
21
43
43
yxAyxAyxW
AA
yAyA
yYX
yW








15. Periodic Structure
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Apply Boundary Conditions
• Clamped Along all boundaries
 
b
m
a
n
yAyAa
yYaX
yaW










:Similarly
solutiontrivial-nonfor
)cos()sin()sin(
)()(
0),(
21

16. Periodic Structure
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Natural Frequencies
• Clamped Along all boundaries


























y
b
m
Sinx
a
n
Sin
nm
b
m
a
n
mnmn



:ionsEigenfucnt
,...,3,2,1,,
22

17. Periodic Structure
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Full Solution
• Clamped Along all boundaries
 
 
 


















 



 ctB
ctA
y
b
m
x
a
n
tyxw
mnmn
mnmn
m n 

cos
sin
sinsin,,
:SolutionFull
1 1

18. Periodic Structure
Maged Mostafa
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Using orthogonality to evaluate the remaining
constants from the initial conditions
Recall
nm
mn
mn
dxx
m
x
n


2,0
,
)sin()sin(
2
0











19. Periodic Structure
Maged Mostafa
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Using orthogonality to evaluate the remaining
constants from the initial conditions
 
    ctBctA
ab
dxdyy
b
m
x
a
n
tyxw
mnmnmnmn
b a


cossin
4
sinsin,,
0 0














This leads to

20. Periodic Structure
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Using orthogonality to evaluate the remaining
constants from the initial conditions
 
  dxdyy
b
m
x
a
n
yxw
cab
B
dxdyy
b
m
x
a
n
yxw
ab
A
b a
mn
mn
b a
mn




























0 0
0 0
sinsin0,,
4
sinsin0,,
4





21. Periodic Structure
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Homework
• Draw the first 3 mode shapes for
square membrane with side
length =1 and all sides are
clamped

22. Periodic Structure
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Vibration of Plates

23. Periodic Structure
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Objectives
• What is Plate?
• The equation of motion for Plates
• Estimate the Natural Frequencies
• Understand the concept of mode shapes
• Apply BC’s and IC’s to obtain structure response

24. Periodic Structure
Maged Mostafa
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Objectives
• What is Plate?
• The equation of motion for Plates
• Estimate the Natural Frequencies
• Understand the concept of mode shapes
• Apply BC’s and IC’s to obtain structure response

25. Periodic Structure
Maged Mostafa
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What is Plate?
• 2-D material (Structure) that can resist
bending
• Analogous to beam
• Examples

26. Periodic Structure
Maged Mostafa
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Objectives
• What is Plate?
• The equation of motion for Plates
• Estimate the Natural Frequencies
• Understand the concept of mode shapes
• Apply BC’s and IC’s to obtain structure response

27. Periodic Structure
Maged Mostafa
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Equations of Motion
• Assume small deflections with respect to the
thickness h.
• Normal stresses transverse to the plate is
negligible
Where:
),,(w),,(w4
tyxtyxD ttE 
4
4
22
4
4
4
4
2
yyxx 








 2
3
112 

Eh
DE

28. Periodic Structure
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Boundary Conditions
Clamped:
Simply Supported:
0
),,(w
,0),,(w




n
tyx
tyx
0
),,(w
,0),,(w
2
2




n
tyx
tyx