smart technologies for safety engineering

Smart Technologies in Vibroacoustics 311
(a) Plate resonance: 610 Hz (b) Coupled resonance: 1730 Hz (c) Plate resonance: 3060 Hz
max |p
Ωa
a | = 1 .00 Pa max |p
Ωa
a | = 717 Pa >> 1 Pa max |p
Ωa
a |=4.19Pa
Figure 8.11 Three resonances of the coupled system of the elastic plate and acoustic waveguide (subjected
to a time-harmonic pressure excitation of the amplitude of 1 Pa): (a,c) two plate resonances (the
plate deflections tend to increase infinitely), (b) the coupled resonance of the whole system (not only the
plate deflections but also the pressure p a in the acoustic medium a increase greatly).
shapes of the first five resonances. This is also confirmed by the frequency analysis (see curves
(b) and (c) in Figure 8.9). It was checked that, for much denser meshes but using quadratic
shape functions, this conformity is inferior. The comparison of the frequency analysis with the
eigenproblem solution illustrates the fact that in the case of uniformly distributed loading, the
resonances which cause antisymmetric modes of deformation are blocked (only symmetricshape
resonances, at 611 and 3057 Hz, appear in the relevant curve (b) of Figure 8.9). This is
an important observation since the case of uniform loading applies to the examined problem
of the small cell of the acoustic panel.
The acoustic medium waveguide (the air-gap) was modeled as a regular 5×5×5 brick
element mesh with quadratic shape functions used to approximate the acoustic pressure field.
The length of the waveguide is 50 mm and for the highest frequency considered (i.e. 4500 Hz)
the acoustic wavelength equals 76 mm. This is several times more than 10 mm, which is the
size of the waveguide finite element. Figure 8.11 shows the three resonances of the coupled
system of the elastic plate and acoustic waveguide: the two (pure) plate resonances are at
approximately 610 Hz (a) and 3060 Hz (c), and the coupled resonance of the elastic plate and
acoustic cavity at approximately 1730 Hz (b). It should be noted that, in the latter case, not
only the amplitude of maximal deflections of plate but also the amplitude of pressure in the
acoustic waveguide tend to increase significantly at this resonance frequency of the coupled
system.
8.10.3 Multilayer Analysis
Figure 8.12 shows the results obtained analytically for one-dimensional problems of wave
propagation in multilayered media. Two configurations were examined:
The two-layered medium has an acoustic layer of thickness of 50 mm coupled with a 0.8 mm
thick layer of aluminum (Figure 8.12(a)).
The three-layered medium has 38 mm of acoustic layer +12 mm of poroelastic layer
+0.8 mm of aluminum (Figure 8.12(b)).
It should be noted that the total thickness for both configurations is the same. The excitation
was a harmonic unit acoustic pressure applied on the acoustic layer and the results are the