84
3 Buzzing Lips: Sound Generation in Brass Instruments
Fig. 3.22 One degree of
freedom model of a swinging
door lip
as indicated by the dashed lines in Fig. 3.21. We assume for simplicity that when
F io = 0, the lips lie in the vertical plane with a separation h 0 . The swinging motion
of the lips takes place in two dimensions, and a model with two or more dimensions
is necessary to describe all the effects of this type of lip valve. It is however possible
to incorporate the swinging door picture into the framework of the elementary 1DOF
model by making the further simplifying assumption that the lips rotate without
deformation (Adachi and Sato 1995). The displacement of the lip is then defined by
the single variable θ .
The lip mass m is taken to be concentrated at the end of a swinging rod of fixed
length l and negligible mass, as illustrated in Fig. 3.22. The direction of F io is always
perpendicular to the inner and outer lip surfaces and therefore rotates with the lip.
Assuming that F io acts at the centre of the lip surface, the torque about the pivot
point due to the pressure difference across the lips is
T io = F io l/2 = (p m − p)S io l/2.
(3.7)
The moment of inertia of the mass about the pivot point is I = ml 2 . The embouchure
muscles are modelled as a spring providing a restoring torque
T emb = −kl
2 θ = −ml
2 ω
2
l θ,
(3.8)
where ω l is the lip resonance frequency (see Eq. 3.2).
The equation of angular motion for the swinging door model can now be written
as
d 2 θ(t)
dt 2 +
ω l
Q l
dθ(t)
dt
+ ω
2
l θ =
S io
2ml
(p m − p(t)).
(3.9)
If the mouth pressure p m is below the threshold for self-sustained oscillation, a static
equilibrium state exists for a lip angle
3 Buzzing Lips: Sound Generation in Brass Instruments
Fig. 3.22 One degree of
freedom model of a swinging
door lip
as indicated by the dashed lines in Fig. 3.21. We assume for simplicity that when
F io = 0, the lips lie in the vertical plane with a separation h 0 . The swinging motion
of the lips takes place in two dimensions, and a model with two or more dimensions
is necessary to describe all the effects of this type of lip valve. It is however possible
to incorporate the swinging door picture into the framework of the elementary 1DOF
model by making the further simplifying assumption that the lips rotate without
deformation (Adachi and Sato 1995). The displacement of the lip is then defined by
the single variable θ .
The lip mass m is taken to be concentrated at the end of a swinging rod of fixed
length l and negligible mass, as illustrated in Fig. 3.22. The direction of F io is always
perpendicular to the inner and outer lip surfaces and therefore rotates with the lip.
Assuming that F io acts at the centre of the lip surface, the torque about the pivot
point due to the pressure difference across the lips is
T io = F io l/2 = (p m − p)S io l/2.
(3.7)
The moment of inertia of the mass about the pivot point is I = ml 2 . The embouchure
muscles are modelled as a spring providing a restoring torque
T emb = −kl
2 θ = −ml
2 ω
2
l θ,
(3.8)
where ω l is the lip resonance frequency (see Eq. 3.2).
The equation of angular motion for the swinging door model can now be written
as
d 2 θ(t)
dt 2 +
ω l
Q l
dθ(t)
dt
+ ω
2
l θ =
S io
2ml
(p m − p(t)).
(3.9)
If the mouth pressure p m is below the threshold for self-sustained oscillation, a static
equilibrium state exists for a lip angle