The Rectangular Membrane Notes | EduRev

: The Rectangular Membrane Notes | EduRev

 Page 1


Vibrations of Structures
Module IV: Vibrations of Membranes
Lesson 32: The Rectangular Membrane
Contents:
1. Modal Analysis
2. Modal Degeneracy
Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy
Page 2


Vibrations of Structures
Module IV: Vibrations of Membranes
Lesson 32: The Rectangular Membrane
Contents:
1. Modal Analysis
2. Modal Degeneracy
Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy
The Rectangular Membrane
1 Modal Analysis
x
y
(1; 0)
T
(1; 0)
T
(0; 1)
T
(0;1)
T
a
b
Figure 1: A rectangular membrane in cartesian coordinates
Consider a rectangular membrane with all edges xed as shown in Fig. 1,
and governed by
w
;tt
 T(w
;xx
+ w
;yy
) = 0; (1)
w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2)
Assume a solution of the form
w(x; y; t) = W(x; y)e
i!t
; (3)
where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency.
Substituting this solution into the equation of motion leads to the eigenvalue
2
Page 3


Vibrations of Structures
Module IV: Vibrations of Membranes
Lesson 32: The Rectangular Membrane
Contents:
1. Modal Analysis
2. Modal Degeneracy
Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy
The Rectangular Membrane
1 Modal Analysis
x
y
(1; 0)
T
(1; 0)
T
(0; 1)
T
(0;1)
T
a
b
Figure 1: A rectangular membrane in cartesian coordinates
Consider a rectangular membrane with all edges xed as shown in Fig. 1,
and governed by
w
;tt
 T(w
;xx
+ w
;yy
) = 0; (1)
w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2)
Assume a solution of the form
w(x; y; t) = W(x; y)e
i!t
; (3)
where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency.
Substituting this solution into the equation of motion leads to the eigenvalue
2
problem of the Helmholz equation
r
2
W +
!
2
c
2
W = 0 (4)
W(0; y) = 0; W(a; y) = 0; W(x; 0) = 0; and W(x; b) = 0 (5)
whererW = W
;xx
+ W
;yy
. Assume a solution of (4) of the form
W(x; y) = Be
i(k
x
x+k
y
y)
; (6)
where B is a complex constant. Substituting (6) in (4) yields the dispersion
relation of the membrane as
k
2
x
 k
2
y
+
!
2
c
2
= 0: (7)
The solutions of (7) are of the form k
x
= and k
y
= such that 
2
+
2
=
!
2
=c
2
. The general solution of (4) can then be written as
W(x; y) = (B
1
e
ix
+ B
2
e
ix
)(B
3
e
iy
+ B
4
e
iy
); (8)
or in the form
W(x; y) = A
1
cos x cosy + A
2
cos x siny +
A
3
sin x cosy + A
4
sin x siny: (9)
Using (9) in the rst and third boundary conditions in (5), one obtains
A
1
cos y + A
2
sin y = 0 (10)
and A
1
cos x + A
3
sin x = 0: (11)
3
Page 4


Vibrations of Structures
Module IV: Vibrations of Membranes
Lesson 32: The Rectangular Membrane
Contents:
1. Modal Analysis
2. Modal Degeneracy
Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy
The Rectangular Membrane
1 Modal Analysis
x
y
(1; 0)
T
(1; 0)
T
(0; 1)
T
(0;1)
T
a
b
Figure 1: A rectangular membrane in cartesian coordinates
Consider a rectangular membrane with all edges xed as shown in Fig. 1,
and governed by
w
;tt
 T(w
;xx
+ w
;yy
) = 0; (1)
w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2)
Assume a solution of the form
w(x; y; t) = W(x; y)e
i!t
; (3)
where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency.
Substituting this solution into the equation of motion leads to the eigenvalue
2
problem of the Helmholz equation
r
2
W +
!
2
c
2
W = 0 (4)
W(0; y) = 0; W(a; y) = 0; W(x; 0) = 0; and W(x; b) = 0 (5)
whererW = W
;xx
+ W
;yy
. Assume a solution of (4) of the form
W(x; y) = Be
i(k
x
x+k
y
y)
; (6)
where B is a complex constant. Substituting (6) in (4) yields the dispersion
relation of the membrane as
k
2
x
 k
2
y
+
!
2
c
2
= 0: (7)
The solutions of (7) are of the form k
x
= and k
y
= such that 
2
+
2
=
!
2
=c
2
. The general solution of (4) can then be written as
W(x; y) = (B
1
e
ix
+ B
2
e
ix
)(B
3
e
iy
+ B
4
e
iy
); (8)
or in the form
W(x; y) = A
1
cos x cosy + A
2
cos x siny +
A
3
sin x cosy + A
4
sin x siny: (9)
Using (9) in the rst and third boundary conditions in (5), one obtains
A
1
cos y + A
2
sin y = 0 (10)
and A
1
cos x + A
3
sin x = 0: (11)
3
This implies A
1
= 0, A
2
= 0 and A
3
= 0, and hence
W(x; y) = A
4
sinx sin y: (12)
The second and fourth boundary conditions in (5) now lead to
sin a sin y = 0 and sinx sin b = 0 (13)
which requires
 =
m
a
; and  =
n
b
; m; n = 1; 2; : : :;1: (14)
The eigenfunctions are then obtained from (12) as
W
(m;n)
= sin
mx
a
sin
nx
b
; m; n = 1; 2; : : :;1; (15)
where W
(m;n)
represents the eigenfunction of the (m; n) mode. These eigen-
functions satisfy the orthogonality condition
hW
(m;n)
(x; y); W
(r;s)
(x; y)i =
Z
a
0
Z
b
0
W
(m;n)
(x; y)W
(r;s)
(x; y) dx dy
=
ab
4

mr

ns
: (16)
The rst few mode-shapes of the membrane are shown in Fig. 2. Using (14)
in the condition 
2
+ 
2
= !
2
=c
2
yields the frequency equation
!
(m;n)
= c
r
m
2
a
2
+
n
2
b
2
; (17)
where !
(m;n)
represents the circular eigenfrequency of the (m; n) mode. Note
that the eigenfrequencies of the membrane are not, in general, integral mul-
tiples of the fundamental frequency (as in the case of a string).
4
Page 5


Vibrations of Structures
Module IV: Vibrations of Membranes
Lesson 32: The Rectangular Membrane
Contents:
1. Modal Analysis
2. Modal Degeneracy
Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy
The Rectangular Membrane
1 Modal Analysis
x
y
(1; 0)
T
(1; 0)
T
(0; 1)
T
(0;1)
T
a
b
Figure 1: A rectangular membrane in cartesian coordinates
Consider a rectangular membrane with all edges xed as shown in Fig. 1,
and governed by
w
;tt
 T(w
;xx
+ w
;yy
) = 0; (1)
w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2)
Assume a solution of the form
w(x; y; t) = W(x; y)e
i!t
; (3)
where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency.
Substituting this solution into the equation of motion leads to the eigenvalue
2
problem of the Helmholz equation
r
2
W +
!
2
c
2
W = 0 (4)
W(0; y) = 0; W(a; y) = 0; W(x; 0) = 0; and W(x; b) = 0 (5)
whererW = W
;xx
+ W
;yy
. Assume a solution of (4) of the form
W(x; y) = Be
i(k
x
x+k
y
y)
; (6)
where B is a complex constant. Substituting (6) in (4) yields the dispersion
relation of the membrane as
k
2
x
 k
2
y
+
!
2
c
2
= 0: (7)
The solutions of (7) are of the form k
x
= and k
y
= such that 
2
+
2
=
!
2
=c
2
. The general solution of (4) can then be written as
W(x; y) = (B
1
e
ix
+ B
2
e
ix
)(B
3
e
iy
+ B
4
e
iy
); (8)
or in the form
W(x; y) = A
1
cos x cosy + A
2
cos x siny +
A
3
sin x cosy + A
4
sin x siny: (9)
Using (9) in the rst and third boundary conditions in (5), one obtains
A
1
cos y + A
2
sin y = 0 (10)
and A
1
cos x + A
3
sin x = 0: (11)
3
This implies A
1
= 0, A
2
= 0 and A
3
= 0, and hence
W(x; y) = A
4
sinx sin y: (12)
The second and fourth boundary conditions in (5) now lead to
sin a sin y = 0 and sinx sin b = 0 (13)
which requires
 =
m
a
; and  =
n
b
; m; n = 1; 2; : : :;1: (14)
The eigenfunctions are then obtained from (12) as
W
(m;n)
= sin
mx
a
sin
nx
b
; m; n = 1; 2; : : :;1; (15)
where W
(m;n)
represents the eigenfunction of the (m; n) mode. These eigen-
functions satisfy the orthogonality condition
hW
(m;n)
(x; y); W
(r;s)
(x; y)i =
Z
a
0
Z
b
0
W
(m;n)
(x; y)W
(r;s)
(x; y) dx dy
=
ab
4

mr

ns
: (16)
The rst few mode-shapes of the membrane are shown in Fig. 2. Using (14)
in the condition 
2
+ 
2
= !
2
=c
2
yields the frequency equation
!
(m;n)
= c
r
m
2
a
2
+
n
2
b
2
; (17)
where !
(m;n)
represents the circular eigenfrequency of the (m; n) mode. Note
that the eigenfrequencies of the membrane are not, in general, integral mul-
tiples of the fundamental frequency (as in the case of a string).
4
m = 1; n = 1 m = 1; n = 2
m = 2; n = 1 m = 2; n = 2
m = 3; n = 1 m = 3; n = 2
Figure 2: First few mode shapes of a rectangular membrane with xed boundaries
5
Read More
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!