Introduction to Acoustics

910 Part G Structural Acoustics and Noise
Part G 22.2
Consequently, (22.57) becomes
〈Ψn, KΨn〉=ω 2 n . (22.62)
22.2.3 Admittances
In general, a given structure vibrates as a result of the
action of localized and distributed forces and moments.
For both numerical and experimental reasons, one often
has to work on a discrete representation of the structure.
In practice, the geometry of the structure is represented
by a mesh consisting of a number N of areas with dimensions
smaller than the wavelengths under consideration.
This results in having to consider the structure as an
N-DOF system. At each point of the mesh, the motion
is defined by three translation components plus three
rotation components. Similarly, the action of the external
medium on the structure can be reduced to three
force components plus three moment components, here
denoted by Fl. The translational and rotational velocity
components Vk at each point are thus related to the
forces and moments through a 6 × 6 admittance matrix,
such that
V = YF (22.63)
For each force component at a given point j on the structure,
denoted Fj|l, a motion at another point i, denoted
by Vi|k, can be generated. As a consequence one can
define the transfer admittance
Yij|kl = Vi|k
Fj|l
with 1 ≤ i, j ≤ N
and 1 ≤ k, l ≤ 6 .
In summary, the transfer admittance matrix is defined
by 6N ×6N coefficients such as Yij|kl.
Notation. In what follows, the indices (k, l) are omitted,
for the sake of simplicity. The transfer admittances are
written as Yij, which reduces to Yii (or, even to Yi) in
the case of a driving-point admittance. Attention here
is mostly focused on translation, though the formalism
remains valid for rotation.
The previous results obtained on modal decomposition
are now used to investigate the frequency
dependence of the admittances. Here, X(xi) ≡ Xi is one
component of displacement at point xi and F(x j) ≡ Fj
is one force component at point x j. The structure is
subjected to a forced motion at frequency ω.Wehave
� K − ω 2 M � Xi = Fj , (22.64)
where Xi = � Φn(xi)qn(t)and
� ω 2 n − ω 2� qn = fn
mn
tΦn(x j)Fj(ω)
=
. (22.65)
mn
For a structure discretized on N points, the displacement
at point xi is given by
Xi(ω) =
N� Φn(xi) tΦn(x j)
�
ω2 n − ω2� Fj(ω) . (22.66)
n=1
mn
The complete set of functions Xi(ω) givenin(22.66) is
the operating deflexion shape (ODS) at frequency ω.The
transfer admittance between points xi and x j is written
Yij(ω) = iω
N� Φn(xi) tΦn(x j)
�
ω2 n − ω2� . (22.67)
n=1
mn
The driving-point admittance at point xi becomes
Yi(ω) = iω
N�
mn
n=1
[Φn(xi)] 2
� ω 2 n − ω 2�
(22.68)
Remark. To include damping, (22.67) can be written
Yij(ω) = iω
N�
mn
n=1
Φn(xi) tΦn(x j)
�
ω2 n + 2iζnωnω − ω2� , (22.69)
where ζn is a nondimensional damping factor. The physical
origin of the damping terms will be presented in
more detail in Sect. 22.6.1.
Frequency Analysis and Approximations
For weak damping, and low modal density, the modulus
of Yij(ω) passes through maxima at frequencies close to
the eigenfrequencies ω = ωn (Fig. 22.4).
• For ω ≈ ωn, the main term in Yij(ω) is equal to
iω Φn(xi) tΦn(x j)
�
ω2 n − ω2� . (22.70)
mn
• For ω ≫ ωn, the admittance becomes
Yij ≈ iω � Φl(xi) tΦl(x j)
−mlω2 ≈ 1
iMω
l>n
with
1 � Φl(xi)
=
M
l>n
tΦl(x j)
ml
(22.71)
It can be seen that the modes with a rank larger than
a given mode n play the role of a mass M.