96 Building acoustics
()
3
2
22
23 2,
12 1 1
0.81
1
(^25)
1
2
and 1.
12 (1 )
x
z
xz
Eh
B
H
L
EH h
B
H
L
Eh H
B
L
π
υ
π
υ
=
⎛⎞⎛⎞
−+⎜⎟⎜⎟
⎜⎟⎝⎠
⎝⎠
⎡ ⎤
⎢ ⎥
=−⎢ ⎥
⎢ ⎛⎞⎥
⎢ + ⎜⎟⎥
⎢⎣ ⎝⎠⎥⎦
⎡ ⎛⎞⎤
=+⎢ ⎜⎟⎥
+ ⎢⎣ ⎝⎠⎥⎦
(3.115)
Example We shall compare the natural frequencies of a flat square plate with the
corresponding ones for a wave corrugated plate. For a 1 mm thick steel plate with sides 1
metre long we get f1,1 ≈ 4.9 Hz, this by using Equation (3.109) with E equal to 2.1⋅ 1011
Pa, m equal to 7.8 kg/m^2 and υ equal to 0.3. Letting the height of the wave corrugated
plate be 20 mm (H equal to 10 mm) and the “wavelength” be equal to 100 mm, we get
f1,1 ≈ 25.5 Hz by using Equations (3.113) and (3.115). Proceeding to the (2,2) mode we
will have f2,2 ≈ 19.7 Hz for the flat plate and 102 Hz for the corrugated one. Selecting a
larger height and/or shorter wavelength for the corrugations will give even larger
differences. It should be observed that we have to take into account the fact that the mass
per unit area will increase when making the corrugations.
We shall later (see Chapter 6) demonstrate the effect of such corrugations on the
sound transmission as compared to a flat plate. However, we shall then employ the more
commonly used cladding type of plate, i.e. the type having trapezoidal corrugations.
Predicting the bending stiffness in this case, analogous expressions to the ones above
must be used. These are given in the literature; see e.g. Hansen (1993) or Buzzi et al.
(2003). The latter also cite expressions for L-shaped plates in additions to the trapezoidal
ones.^3
3.7.3.4 Response to force excitation
If the eigenfunctions for a given system are known we may, by analogy with the
calculations performed on the air-filled tube in section 3.6, calculate the response to a
given mechanical input. Again using a plate as an example we may calculate the transfer
function between a force (or moment) in a given point and a given response quantity
such as velocity or acceleration in the same or in another point. We shall have to solve
the wave Equation (3.107) but now modified by a term on the right-hand side
representing the excitation. The response and the relevant transfer function may then be
expressed by a sum of the eigenfunctions for the plate.
The measurement technique to determine such transfer functions is either by
attaching an electrodynamic vibration exciter to the structure or using a transient
excitation with a hammer blow or equivalent. The response quantity is measured at the
driving point or at other relevant positions. The latter option gives basic data to
determine the resonant vibration modes of the structure, a so-called modal analysis. We
shall present an example on such transfer functions, using bending waves on a plate. This
(^3) The quantity b in equation (38) lacks definition. It is the distance from the y-axis to the plate neutral axis.