Introduction to Acoustics

940 Part G Structural Acoustics and Noise
Part G 22.6
Lx
Ly
z
y
0 x
Fig. 22.18 Geometry of the thin isotropic baffled radiating
plate
The radiation impedance matrix Ra is defined as
Ra = ωρ
⎛
��
Φ∗ (kx, ky)Φ
Re ⎝
8π2 t (kx, ky)
�
k2 − k2 x − k2 ⎞
dkx dky⎠
y
(22.331)
which generalizes the results obtained in (22.265)to2-D
systems. Each element (Ra)ij of the matrix Ra quantifies
the mutual radiation resistance resulting from the
interference between the fields of modes (m, n) and
(m ′ , n ′ ), respectively. If (m, n) = (m ′ , n ′ ), we obtain the
self-radiation resistances, which are the diagonal terms
of the matrix Ra. These terms can be written explicitly
as
(Ra)ij = (Ra)mn,m ′ n ′
= ωρ
Re
8π2 22.6 Damping
��
Φ∗ m (kx) Φ∗ n (ky)Φm ′ (kx) Φn ′ (ky)
�
k2−k2 x−k2 dkx dky .
y
(22.332)
In this section, we start by summarizing briefly the
conditions for modal decoupling in discrete damped
systems, and the concept of proportional damping is
introduced. The modal approach is convenient for treating
the case of localized damping (Sect. 22.6.1). The
following example of a string with a dissipative end illustrates
the limit of the modal approach, and shows
that such a system cannot exhibit stationary solutions.
A physical interpretation is presented in terms of damped
propagating waves. Other authors use a state-space
modal approach to address the analysis of damped
structures [22.41, 42].
The section continues with the presentation of
three damping mechanisms in plates: thermoelasticity,
viscoelasticity and radiation, with emphasis on
the time-domain formulation. The section ends with
For a baffled simply supported plate, the radiation resistances
become
(Ra)mn,m ′ n ′ = mm′ nn ′ ωρπ 2
8L 2 x L2 y
� ��
×Re ( fmm ′(kx Lx) fnn ′(ky L y)dkx dky)
� � � k 2 x − (mπ/Lx) 2�� k 2 x − (m′ π/Lx) 2�
� k 2 y − (nπ/L y) 2�� k 2 y − (n′ π/L y) 2���
, (22.333)
where the functions of the form fmm ′(kx Lx) aregiven
by
fmm ′(kx Lx) =
⎧
2(1 − cos kx Lx) for m even, m
⎪⎨
⎪⎩
′ even
2(1 + cos kx Lx) for m odd, m ′ odd
2i sin kx Lx for m odd, m ′ even
−2i sin kx Lx for m even, m ′ ⎫
⎪⎬
.
⎪⎭
odd
(22.334)
For the application of the radiation modal expansion to
active structural control of sound, see, for example, the
work by Gibbs et al.[22.38]. In a recent work, Kim investigated
structural–acoustic coupling for active control
purpose, using an impedance/mobility approach [22.39].
Alternative techniques make use of multipole expansion
of the radiated sound pressure [22.40].
a brief presentation of friction, stick–slip vibrations
and hysteretic damping, which are often encountered
in structural damping models.
22.6.1 Modal Projection in Damped Systems
Discrete Systems
For a linear system with multiple degrees of freedom,
with a dissipation energy of the form
ED = 1 t ˙XC ˙X , (22.335)
2
where C is a symmetric damping matrix with positive
elements, the equations of motion can be written as
M ¨X + C ˙X + KX = F . (22.336)