3.3 The Mechanical Response of the Vibrating Lips
89
vations of low-frequency lip motion described in Sect. 3.1.5 and is in accord with
the musician’s experience that the lips bulge outward when buzzed. Nevertheless,
experimental measurements of the mechanical response of artificial and human lips
presented in Sect. 3.3 show evidence of resonances with both outward- and inwardstriking behaviour. We therefore retain at this stage the more general Eq. 3.15, which
can represent either outward- or inward-striking lip valves depending on the choice
of sign in the forcing term.
3.3 The Mechanical Response of the Vibrating Lips
The equation of motion for the 1DOF lip model with no external force is found
by setting the right-hand side of Eq. 3.3 to 0. There is a stationary solution to
this equation for h = h eq which represents the lips at rest with their equilibrium
separation. When a sinusoidally varying external force F (t) = A cos ωt is applied
to the lips, they are driven into oscillation at the forcing frequency f = ω/2π . The
strength of the response depends on the relationship between the forcing frequency
f and the lip resonance frequency f l = ω l /2π . If the two frequencies are far apart,
the response of the lips will be small, but when f f l , the amplitude of the lip
vibration can be many times larger.
The mechanical resonance behaviour of vibrating lips can be investigated experimentally by generating a small sinusoidally varying pressure difference
between the mouth and the mouthpiece and examining the resultant modulation of
the lip opening height h(ω). The mechanical response H mr at the angular frequency
ω is defined as
H mr (ω) =
h(ω)
p(ω)
.
(3.16)
If the lip valve did behave as a simple 1DOF oscillator, a plot of the mechanical
response as a function of frequency would have a single peak at the natural
resonance frequency of the lips. In reality the lips are complex structures with
multiple resonances, and this complexity is revealed by experimentally measured
mechanical response curves. It is nevertheless often possible to identify a peak
corresponding to the mechanical resonance principally involved in the lip valve
vibration, and analysis of the peak properties can provide information about the
parameters of the corresponding one-mass model.
3.3.1 Resonances of Artificial Lips
An experimental arrangement for measuring the mechanical response of a pair of
artificial lips is shown in Fig. 3.26. The artificial mouth illustrated in Fig. 3.15, which
89
vations of low-frequency lip motion described in Sect. 3.1.5 and is in accord with
the musician’s experience that the lips bulge outward when buzzed. Nevertheless,
experimental measurements of the mechanical response of artificial and human lips
presented in Sect. 3.3 show evidence of resonances with both outward- and inwardstriking behaviour. We therefore retain at this stage the more general Eq. 3.15, which
can represent either outward- or inward-striking lip valves depending on the choice
of sign in the forcing term.
3.3 The Mechanical Response of the Vibrating Lips
The equation of motion for the 1DOF lip model with no external force is found
by setting the right-hand side of Eq. 3.3 to 0. There is a stationary solution to
this equation for h = h eq which represents the lips at rest with their equilibrium
separation. When a sinusoidally varying external force F (t) = A cos ωt is applied
to the lips, they are driven into oscillation at the forcing frequency f = ω/2π . The
strength of the response depends on the relationship between the forcing frequency
f and the lip resonance frequency f l = ω l /2π . If the two frequencies are far apart,
the response of the lips will be small, but when f f l , the amplitude of the lip
vibration can be many times larger.
The mechanical resonance behaviour of vibrating lips can be investigated experimentally by generating a small sinusoidally varying pressure difference
between the mouth and the mouthpiece and examining the resultant modulation of
the lip opening height h(ω). The mechanical response H mr at the angular frequency
ω is defined as
H mr (ω) =
h(ω)
p(ω)
.
(3.16)
If the lip valve did behave as a simple 1DOF oscillator, a plot of the mechanical
response as a function of frequency would have a single peak at the natural
resonance frequency of the lips. In reality the lips are complex structures with
multiple resonances, and this complexity is revealed by experimentally measured
mechanical response curves. It is nevertheless often possible to identify a peak
corresponding to the mechanical resonance principally involved in the lip valve
vibration, and analysis of the peak properties can provide information about the
parameters of the corresponding one-mass model.
3.3.1 Resonances of Artificial Lips
An experimental arrangement for measuring the mechanical response of a pair of
artificial lips is shown in Fig. 3.26. The artificial mouth illustrated in Fig. 3.15, which