Question

A square membrane of side lengths L, which is initially at rest on the xy-plane has its edges fixed on the xy-plane. A periodic force per unit area, given by A⋅cos⁡(ω⋅t) for t > 0,and A=constant, acts at every point in the membrane. Apply appropriate finite Fourier transforms to show that the displacements in the membrane.

Answer 1

A basic problem in free vibration of the membrane is to solve Eq. (16.1) subject to proper boundary conditions. To this end, express the membrane displacement as

(16.5)w(x,y,t)=w(x,y)cosωt

which, when substituted into Eq. (16.1), yields

(16.6)(∂2∂x2+∂2∂y2)w(x,y)+ρω2Tw(x,y)=0

Equation (16.6), along with the boundary conditions, defines an eigenvalue problem of the membrane, in which ω is an eigenvalue or natural frequency, and W(x, y) is an eigenfunction or mode shape function.

The solution of Eq. (16.6), by separation of variables, is written as

(16.7)W(x,y)=Φ(x)Ψ(y)

Substitute Eq. (16.7) into Eq. (16.6) to obtainthe differential equations

(16.8a)d2Φdx2+α2Φ=0

and

(16.8b)d2Ψdy2+β2Ψ=0

where parameters α and β are related by

(16.9)α2+β2=ρω2T

Equations (16.8) indicate that functions Φ(x) and Ψ(y) are of sinusoidal form. It follows that the solution of Eq. (16.6) is

(16.10)W(x,y)=A1sinαxsinβy+A2sinαxcosβy+A3cosαxsinβy+A4cosαxcosβy

where Ak are constants to be determined.

Equation (16.10), when used with the boundary conditions, leads to the eigensolutions (natural frequencies and mode shapes). As an example, consider a membrane that is clamped at edges x = 0, x= a, and y = b, and is free at edge y = 0. The boundary conditions of the membrane are

(i)

At edge x = 0  W(0,y) = 0

(ii)

At edge y = 0  ∂W(x,y)∂y|y=0=0

(iii)

At edge x = a  W(a,y) = 0

(iv)

At edge y = b  W(x,b) = 0

Application of conditions (i) and (ii) to Eq. (16.10) gives A1 = A3 = A4 = 0, and

W(x,y)=A2sinαxcosβy

which, by conditions (iii) and (iv), leads to the characteristic equations

sinαa=0,undefinedcosβb=0.

The characteristic roots are

αm=mπa,m=1,2,…βn=(n−1/2)πbn=1,2,…

With the relation (16.9), the natural frequencies of the membrane are

ωmn=π(ma)2+(n−1/2b)2Tρ,undefinedm,n=1,2,…

and the associate mode shapes are

Wmn(x,y)=sinαmxcosβny.