Introduction to Acoustics

notes the value of this eigenmode at one particular point
of the structure. For the structure vibrating in air, the
displacement becomes
ξ = φ10q1 + φ20q2 . (22.245)
From (22.226) one can derive a first-order approximation
for the bar displacement
�
� �
�
sC21
ξ = φ10
+ φ20 q20 + q10 .
q10 + sC12
q20
D1
D2
(22.246)
Operating Deflexion Shapes. Equation (22.246) can be
rewritten
�
ξ = q10 φ10 + sC21
�
φ20
D2
�
�
sC12
+ q20 φ20 + φ10
D1
with qi0 = φi0
F . (22.247)
mi Di
In experiments on structures, sinusoidal excitation is
often used. Imagine that we apply a sudden harmonic
force F(t) = H(t)sinωt at time t = 0tothestructure,
with excitation location and frequency such that q20 is
negligible compared to q10. H(t) is the Heaviside function.
In this case, the spatial pattern of the structure is
given by
sC21
φ1 = φ10 + φ20 . (22.248)
D2
Because of the time dependence of the second term
(through the Laplace variable s), the spatial shape
evolves with time. Here, we can use the Laplace limit
theorem, which states that the value of φ(t) as time tends
to infinity is given by the product of sφ(s) ass tends to
zero. Since s2C12/D2 tends to zero as s tends to zero,
the second term on the right-hand side of (22.248) vanishes
after some time. Calculating the inverse Laplace
transform shows that this decay time is of the order of
the magnitude of the decay time of the second structural
mode. After this transient regime, the spatial shape is
nearly equal to the spatial shape in vacuo φ10.Inthemore
general case, (22.247)showsthatFexcites both q10 and
q20. After a transient regime, the bar displacement then
finally converges to:
� �
φ10β1 φ20β2
ξ(ω, x) = + . (22.249)
D1(iω) D2(iω)
The quantity between in square brackets is called the
operating deflexion shape (ODS) ofthestructureat
Structural Acoustics and Vibrations 22.5 Structural–Acoustic Coupling 931
frequency ω. Since it is very difficult in practice to excite
a single qi0, theODS describes the multi-mode
shapes that are observed for sinusoidal excitation of
structures.
State-Space Formulation
The transfer function formulation is convenient if the
system is initially at rest, and for time-invariant systems.
For other applications, such as sound control, it is
useful to express the results in terms of state-space variables
[22.26–29]. In most cases, the mechanical state of
the system is given by the position and the velocity of
the DOFs. Since the unloaded structure is described by
its eigenmodes φn, all useful information about the state
of the system is contained in the modal participation factors
qn and in their first derivatives with time ˙qn. This
allows us to rewrite the equations for a 2-DOF coupled
system as follows
d
dt
where
⎛
⎜
⎝
X1
X2
X3
X4
⎞
⎟
⎠ =
⎛
0 0 1 0
⎜ 0 0 0 1
⎜
⎝−ω2
1 0 −2ζ1ω1 C12
0 −ω2 ⎞
⎟
⎠
2
C21 −2ζ2ω2
⎛ ⎞
X1
⎜ ⎟
⎜X2⎟
× ⎜ ⎟
⎝ ⎠ +
⎛ ⎞
0
⎜ ⎟
⎜ 0 ⎟
⎜ ⎟ F , (22.250)
⎝ ⎠
X3
X4
β1
β2
X1 = q1 ; X2 = q2 ; X3 = ˙q1 ; X4 = ˙q2 . (22.251)
Equation (22.250) can then be formulated equivalently
as
˙X = AX + BF , (22.252)
where F is the input and X is the state vector. The output
Y depends on the investigated mechanical problem. If
we decide, for example, to investigate the displacement,
then we can write for the output
� �t Y =
X = Ɣ X . (22.253)
φ1 φ2 00
Equations (22.252)and(22.253) are general expressions
for a linear system expressed in terms of state-space
variables.
Remark 1. The representation presented in (22.250) is
not unique. Selecting, for example
X1 = q1; X2 = ˙q1; X3 = q2; X4 = ˙q2 (22.254)
leads to different values for A and B.
Part G 22.5