Introduction to Acoustics

from which we can derive through (22.174) the eigenfrequencies
ωn ≈ nπc � n
1 + ε 2π2 �
. (22.176)
L
2
The eigenfrequencies are raised by the stiffness. The
difference increases as n2 . The inharmonicity coefficient
can be defined as
i = ε n2π 2
. (22.177)
2
22.4.3 Flexural Vibrations of Thin Plates
The thin plate, or Kirchhoff–Love plate, described below
is a generalization of the Euler–Bernoulli beam
in 2-D [22.13]. The general case of orthotropic plates
is treated here. The problem is solved in Cartesian
coordinates, where w(x, y, t) denotes the transverse displacement.
It is assumed that the coordinates coincide
with the symmetry axes of the material. The assumption
of orthotropy leads to the following relations between
plane stresses σij and plane strains εkl
⎛
⎜
⎝
σxx
σyy
σxy
⎛
Ex νyx Ex
⎞ ⎜
⎜1
− νxyνyx 1 − νxyνyx
⎜
⎟ ⎜
⎠ = ⎜ νyx Ex Ey
⎜
⎜1
− νxyνyx 1 − νxyνyx
⎝
0
0
⎞
⎟
⎠
⎛ ⎞
εxx
⎜ ⎟
× ⎝ ⎠ .
0 0 2Gxy
(22.178)
εyy
εxy
These relations involve two elastic moduli Ex and
Ey, two Poisson’s ratios νxy and one torsional
modulus Gxy. In addition, we have the property
Exνyx = Eyνxy [22.14]. The bending and twisting moments
are obtained by integration of the elementary
moments over the thickness h of the plate:
Mx =
My =
�h/2
−h/2
�h/2
−h/2
Mxy = Myx =
zσxx dz ;
zσyy dz ;
�h/2
−h/2
zσxydz (22.179)
Structural Acoustics and Vibrations 22.4 Bars, Plates and Shells 923
which leads to the matrix relation between the moments
and curvatures
⎛
∂
⎛ ⎞ ⎛
Mx D1
⎜ ⎟ ⎜
⎝ My ⎠ =−⎝D2/2
Mxy 0
D2/2
D3
0
⎞ ⎜
0 ⎜
⎟ ⎜
0 ⎠ × ⎜
D4/2 ⎜
⎝
2w ∂x2 ∂2w ∂y2 ∂2 ⎞
⎟ ,
⎟
w ⎠
∂x∂y
(22.180)
where
D1 =
D2 =
D3 =
Exh 3
12(1 − νxyνyx) ;
Exνyxh3 Eyνxyh3 = , (22.181)
6(1 − νxyνyx) 6(1 − νxyνyx)
Eyh 3
12(1 − νxyνyx) ;
Gxyh3 D4 = . (22.182)
3
Again, the equation of motion can be obtained using
Hamilton’s principle [22.12]. Finally, given an external
force density term f (x, y, t), we obtain the equation of
motion for the plate:
ρph ∂2w ∂t2 = ∂2Mx ∂x2 + ∂2My ∂y2 + 2 ∂2Mxy + f (x, y, t) .
∂x∂y
(22.183)
Boundary Conditions. As for the beam, the boundary
conditions follow from integration by parts of the elastic
energy. The number of possible conditions depends on
the selected geometry. For rectangular plates, for example,
Leissa lists 21 possible boundary conditions [22.15].
Along the principal directions, for example, the conditions
of greatest practical interest are:
1. Clamped edge: displacement w = 0 and rotation
∂w
∂x = 0;
2. Simply supported edge: displacement w = 0and
bending moment Mx = 0;
3. Free edge: bending moment Mx = 0 and shear force
∂Mx ∂Mxy
∂x + 2 ∂y = 0.
Part G 22.4