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

Equations of Motion

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

Rectangular Membrane• Using Separation of variables:

Rectangular Membrane• Solve the 3 ODEs:

)cos()sin()(0

tcHtcFtTTcT

yDyCyYYY

xBxAxXXX

Rectangular Membrane• Total Spatial Solution

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

)()(),(

yxAyxAyxAyxAyxW

yYxXyxW

Apply Boundary Conditions• Clamped Along all boundaries

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

0)cos()sin(

)()0(0),0(

yxAyxAyxW

AAyAyA

Apply Boundary Conditions• Clamped Along all boundaries

yAyAayYaX

:Similarly

solution trivial-nonfor

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

Natural Frequencies• Clamped Along all boundaries

ybmSinx

:ionsEigenfucnt

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

Full Solution• Clamped Along all boundaries

ctBctA

antyxw

cossin

sinsin,,

:Solution Full

Using orthogonality to evaluate the remaining constants from the initial conditions

Recall

nmmnmn

dxxmxn 2 ,0

,)sin()sin( 2

ctBctAab

dxdyybmx

antyxw

mnmnmnmn

cossin4

sinsin,,0 0

This leads to

dxdyybmx

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

Boundary Conditions

Clamped:

Simply Supported:

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

ntyxtyx

0 ),,(w

,0 ),,(w

Membranes and thin plates

Engineering

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