Membranes and thin plates

Periodic StructureMaged Mostafa

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Vibration of Continuous Structures



Vibration of Membranes &Thin Plates



Course ContentsSDOF M-DOFCables/String Bars Shafts Vibration Attenuation Beams Membranes & Thin Plateso FEM for Vibrationo Aeroelasticity



Vibration of Membranes



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



Objectives

• What is Membrane?• The equation of motion for membranes




What is Membrane?• 2-D material (Structure) does not resist

bending• Analogous to cables• Examples:

• Tents• Human Tissues• Light Aircraft



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



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

22

yx



Equations of Motion

Boundary Conditions are defined along the shape of the boundaries not just a point!!

c

tw

cyw

xw

2

2

22

2

2

2 1



Rectangular Membrane• Using Separation of variables:

22222

2

22

22

, ,

1

1

YY

XX

YY

XXTT

c

TT

cYY

XX



Rectangular Membrane• Solve the 3 ODEs:

)cos()sin()(0

)cos()sin()(0

)cos()sin()(0

22

2

2

tcHtcFtTTcT

yDyCyYYY

xBxAxXXX



Rectangular Membrane• Total Spatial Solution

)cos()cos()sin()cos()cos()sin()sin()sin(),(

)()(),(

43

21

yxAyxAyxAyxAyxW

yYxXyxW



Apply Boundary Conditions• Clamped Along all boundaries

)cos()sin()sin()sin(),(

0)cos()sin(

)()0(0),0(

21

43

43

yxAyxAyxW

AAyAyA

yYXyW



Apply Boundary Conditions• Clamped Along all boundaries

bm

an

yAyAayYaX

yaW

:Similarly

solution trivial-nonfor

)cos()sin()sin()()(

0),(

21



Natural Frequencies• Clamped Along all boundaries

ybmSinx

anSin

nmbm

an

mnmn

:ionsEigenfucnt

,...,3,2,1, ,22



Full Solution• Clamped Along all boundaries

ctBctA

ybmx

antyxw

mnmn

mnmn

m n

cossin

sinsin,,

:Solution Full

1 1



Using orthogonality to evaluate the remaining constants from the initial conditions

Recall

nmmnmn

dxxmxn 2 ,0

,)sin()sin( 2

0




ctBctAab

dxdyybmx

antyxw

mnmnmnmn

b a

cossin4

sinsin,,0 0

This leads to




dxdyybmx

anyxw

cabB

dxdyybmx

anyxw

abA

b a

mnmn

b a

mn

0 0

0 0

sinsin0,,4

sinsin0,,4



Homework• Draw the first 3 mode shapes for

square membrane with side length =1 and all sides are clamped



Vibration of Plates



Objectives

• What is Plate?• The equation of motion for Plates




Objectives





What is Plate?• 2-D material (Structure) that can resist

bending• Analogous to beam• Examples



Objectives





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

44 2

yyxx

2

3

112

EhDE



Boundary Conditions

Clamped:

Simply Supported:

0 ),,(w ,0 ),,(w

ntyxtyx

0 ),,(w

,0 ),,(w

2

2

ntyx

tyx

Engineering

Membranes and thin plates