This document discusses vibration of membranes and plates. It covers the objectives, equations of motion, boundary conditions, natural frequencies, and mode shapes for rectangular membranes and plates. The key points are: - Membranes are 2D structures that do not resist bending, while plates can resist bending. - The equation of motion for membranes involves in-plane tension and the Laplace operator, while plates use the bending stiffness and biharmonic operator. - Boundary conditions for clamped, simply supported, and free edges are defined. - Natural frequencies and mode shapes are determined by solving the equations of motion subject to boundary conditions using separation of variables.

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Vibration of Continuous
Structures

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Vibration of Membranes &Thin Plates

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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

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Vibration of Membranes

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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

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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

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

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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

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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 






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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

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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



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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














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

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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








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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

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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

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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

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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











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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

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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





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

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Vibration of Plates

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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

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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

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

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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

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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

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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