vibrations of membranes 1

8/6/2019 Vibrations of Membranes 1

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Vibrations of Membranes


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Fundamentals of Membranes

A membrane is a plate subjected to tension and has small bendingresistance

Can be visualized as an assemblage of strings

Types of membranes: -

- Rectangular Membranes

- Circular Membranes

Based on the geometry of the membrane, suitable co-ordinate systems

are used for deriving the equations of motion.

- Rectangular Co-ordinates (x, y)

- Polar Co-ordinates (r, )


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Rectangular Membranes Consider a rectangular membrane lying in a plane in its equilibrium

position

Assumptions: -

- No stiffness

- Restoring force is supplied exclusively by uniform stress T in every orientation

Equation of motion of membrane can be obtained by using Newtons

Second law of motion

Net Vertical forces=

Net Horizontal forces

Inertia forces

Where T is tension in N/m

is surface density in N/m2

Fig.1 Elemental area of a membrane showing

the forces acting when the membrane is

displaced transversely


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Equating the sum of the vertical and horizontal forces to inertia force,

-------- 2D Wave Equation

Let

where

Or


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Method of Separation of variables, let (x, z) = X(x) . Z(z)

Let

- (1)

- (2)

- (3)

Solution of (1) and (2) is already known as given below:-

Complete Solutions is


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Boundary Conditions

1.

2.

For an example, consider a membrane that is fixed at the boundariesthen,

& ,

Since


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Thus, Standing waves on the membrane are given by

where A is the maximum displacement amplitude

Natural frequencies,

Schematic representation of four typical normal modes of rectangular

membrane with fixed rim


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Modes of Vibration of Rectangular membrane

(1,1)(1,2)

(2,1)(2,2)

http://paws.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html


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Circular Membrane

Consider a circular membrane lying in a plane in its equilibrium position Since the geometry is circular, write the wave equation in polar

co-ordinates

where and

Substituting above in (1), we get

Using method of separation of variables, let (r,) = R(r) (), we get


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By rearranging, we get

We will get two equations as shown below

Solution to (5), we know,

Solutions to (4) are Bessels functions of order m of the first kind Jm(kr) andsecond kind Ym(kr)

R(r)= A Jm(kr) +B Ym(kr)

Bessel functions are oscillatory functions ofkr whose amplitudes diminishroughly as 1/kr .

Ym(kr) become un bounded in the limit kr 0,

For finite displacement at r=0, B=0 R(r)= A Jm(kr)


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Boundary Conditions

R(a) =0 Jm(ka) = 0

Let the values argument of Jm that cause it to zero are denoted by Jmn

k kmn = Jmn /a

Final solution is

where


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Normal modes of a circular membrane with

fixed rim


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Modes of vibration of Circular membrane

(0,1) (0,3)

http://paws.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html


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(1,1) (2,1)


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vibrations of membranes 1

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