6.7 Going Further: Analytical Modelling of Vibroacoustic Coupling in Ducts
333
acoustic input impedance could be significantly modified by wall vibrations, a
design study was undertaken.
For slender tubes (L/a 1), the first eigenfrequencies of shells of finite length
are close to the cut on frequency f c
m of the flexural wave associated with the
circumferential index m ≥ 2 (m = 2 for ovalling modes) of infinite cylinders.
This frequency is given by
f
c
m =
m(m 2 − 1)
4π
√
3
√
m 2 + 1
h
a 2
E
ρ s (1 − ν 2 )
,
(6.48)
which can be used with m = 2 to estimate the eigenfrequency of the first ovalling
mode.
As experiments were to be carried out on a clarinet-like instrument, the sounding
length of the system was chosen to be about 50 cm and the internal radius
a = 7.5 mm. This gave the following series of acoustical eigenfrequencies for the
first five acoustic modes: 170, 510, 850, 1190 and 1530 Hz. Geometrical and
material parameters of the tube were determined so that the eigenfrequency of the
first ovalling mode was close to one of these values.
It was found to be possible to obtain the coincidence between the second acoustic
mode and the first ovalling mechanical mode, as illustrated in Fig. 6.41, by using a
thin plastic tube having the following characteristics: h = 0.2 mm, a = 7.5 mm,
E = 1.8 GPa, ρ s = 1350 kg/m 3 and ν = 0.3. The dimensioning is indicative:
the cut on frequency f c
m gives an order of magnitude of the ovalling mechanical
mode frequency, and the mechanical parameter values are not precisely known. The
plastic tube was connected to a rigid slide, making it possible to vary continuously
the acoustic resonance frequencies, thanks to a variable sounding length of the tube,
without changing the fixed mechanical resonance frequencies of the vibrating tube.
Fig. 6.41 Dispersion diagram representing the variations of the acoustic wave number (in red) and
of the flexural wave number associated with the circumferential number m = 2 (in blue) (Color
figure online). Adapted from Nief et al. (2006)
333
acoustic input impedance could be significantly modified by wall vibrations, a
design study was undertaken.
For slender tubes (L/a 1), the first eigenfrequencies of shells of finite length
are close to the cut on frequency f c
m of the flexural wave associated with the
circumferential index m ≥ 2 (m = 2 for ovalling modes) of infinite cylinders.
This frequency is given by
f
c
m =
m(m 2 − 1)
4π
√
3
√
m 2 + 1
h
a 2
E
ρ s (1 − ν 2 )
,
(6.48)
which can be used with m = 2 to estimate the eigenfrequency of the first ovalling
mode.
As experiments were to be carried out on a clarinet-like instrument, the sounding
length of the system was chosen to be about 50 cm and the internal radius
a = 7.5 mm. This gave the following series of acoustical eigenfrequencies for the
first five acoustic modes: 170, 510, 850, 1190 and 1530 Hz. Geometrical and
material parameters of the tube were determined so that the eigenfrequency of the
first ovalling mode was close to one of these values.
It was found to be possible to obtain the coincidence between the second acoustic
mode and the first ovalling mechanical mode, as illustrated in Fig. 6.41, by using a
thin plastic tube having the following characteristics: h = 0.2 mm, a = 7.5 mm,
E = 1.8 GPa, ρ s = 1350 kg/m 3 and ν = 0.3. The dimensioning is indicative:
the cut on frequency f c
m gives an order of magnitude of the ovalling mechanical
mode frequency, and the mechanical parameter values are not precisely known. The
plastic tube was connected to a rigid slide, making it possible to vary continuously
the acoustic resonance frequencies, thanks to a variable sounding length of the tube,
without changing the fixed mechanical resonance frequencies of the vibrating tube.
Fig. 6.41 Dispersion diagram representing the variations of the acoustic wave number (in red) and
of the flexural wave number associated with the circumferential number m = 2 (in blue) (Color
figure online). Adapted from Nief et al. (2006)