Introduction to Acoustics

924 Part G Structural Acoustics and Noise
Part G 22.4
Example: Homogeneous Rectangular Simply Supported
Plate. From (22.180) and(22.183), we obtain
for a homogeneous orthotropic plate
ρph ∂2w ∂
+ D1
∂t2 4w ∂x4 + (D2 + D4)×
∂4w ∂x2 + D3
∂y2 ∂2w = 0 . (22.184)
∂y4 For a rectangular plate of length a and width b, simply
supported at its edges, the boundary conditions are
⎧
W(0, y, t) = W(a, y, t) = W(x, 0, t)
⎪⎨
= W(x, b, t) = 0 ,
Mx(0, y, t) = Mx(a, y, t) = My(x, 0, t)
⎪⎩
= My(x, b, t) = 0 .
(22.185)
The eigenfunctions satisfying (22.185) are of the form
Φmn(x, y) = sin mπx nπ y
sin (22.186)
a b
and the associated eigenfrequencies are given by:
ωmn = π 2
�
1
ρph
�
m
× D1
4 n
+ D3
a4 4
b4 + (D2 + D4) m2n 2
a2 .
b2 (22.187)
In both expressions, m and n are positive integers.
As for strings and beams, the eigenfunctions Φmn are
orthogonal with respect to mass and stiffness, so that
�a
0
�
0
b
ρphΦmn(x, y)Φm ′ n ′(x, y)dx dy =
⎧
⎨
0 ifm�= m
⎩
′ or n �= n ′ ,
Mmn if m = m ′ and n = n ′ ;
(22.188)
where Mmn is the modal mass for the mode (m, n).
The eigenfrequencies for the orthotropic plate are distributed
between two limiting curves in the (k, f )-plane
(Fig. 22.12).
Particular Case: Isotropic Plate
In the isotropic case, the rigidity coefficients become
D1 = D3 = D =
Eh 3
12(1 − ν 2 )
(22.189)
Frequency (kHz)
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80 90 100
Wavenumber
Fig. 22.12 Dispersion curves for an orthotropic plate made
of carbon fibers with D1 = 8437 MPa; D2 = 463 MPa;
D3 = 852 MPa; D4 = 2267 MPa; h = 2mm; ρp =
1540 kg/m 3 ; a = 0.4m;b = 0.2m
and
D4 = Eh3
6(1 + ν) ,
D2 = 2D1 − D4 = Eνh3
6(1 − ν2 . (22.190)
)
Here, the elastic behavior of the material is fully determined
by two constants: the Young’s modulus E and the
Poisson’s ratio ν. The eigenfunctions Φmn(x, y) arethe
same as in (22.186). However, the eigenfrequencies now
reduce to
ωmn = π 2
� �
D m2 n2
+
ρph a2 b2 �
. (22.191)
Prestressed Isotropic Plate. If a tension (or compression)
Tx in the x-direction and, simultaneously, a tension
(or compression) Ty in the y-direction are applied in the
plate plane, then the flexural equation is modified as
follows [22.1]:
ρph ∂2 �
w ∂4w + D
∂t2 ∂x4 + 2 ∂4w ∂x2∂y 2 + ∂2w ∂y4 �
∂
− Tx
2w ∂
− Ty
∂x2 2w = 0 . (22.192)
∂y2