5.2 Crossing the Threshold: Small Amplitude Oscillating Solutions
229
either known or can be estimated. It is also necessary to know the form of the
input impedance Z of the instrument; in Chap. 4 methods were described for either
calculating or measuring this function.
In reality finding the solution of the brass model equations is far from straightforward. It is possible to measure typical values of mouth pressure p m during
performance, and realistic estimates of the lip opening width w and equilibrium
opening height h eq can be extracted from video recordings of players using
transparent mouthpieces. However the parameters γ , ω r and μ depend on the
effective mass, stiffness, surface area and internal damping of the vibrating section
of lip; these quantities are not susceptible to direct measurement.
A more fundamental difficulty in finding solutions to the model equations arises
from the nonlinear nature of the lip valve dynamics. In Sect. 5.2.1 the view of the
player-instrument system as a feedback loop was discussed. This approach helps to
explain the system behaviour very near its oscillation threshold, which is a useful if
limited step on the way to a fuller understanding of brass playing. An alternative way
of investigating near-threshold behaviour treats the player-instrument combination
as a dissipative nonlinear dynamical system, which can be analysed using the
mathematical technique of linear stability analysis (see, e.g. Cullen et al. (2000) and
Velut et al. (2017a)). The principle of linear stability analysis (LSA) is described
in the present section, and an example is given of its ability to explain some basic
features of brass playing. A full mathematical treatment is reserved for the Going
Further Sect. 5.4.
The linearisation of the model equations can be achieved by replacing each
system variable q i (t) by q i,eq + ˜
q i (t), where q i,eq is the threshold equilibrium value
of q i . Terms which are more than first order in one of the new variables ˜
q i (t) are then
eliminated. The resulting linearised equations describe infinitesimal oscillations
of the system variables about their equilibrium values. As a simple example, we
consider the case in which the instrument has only one acoustic mode. The equations
can then be rewritten in terms of the four variables ˜
h,
d ˜
h
dt
, ˜
p, and
d ˜
p
dt
, which can be
combined into a 4 × 1 state vector
X =
˜
h
d ˜
h
dt
˜
p
d ˜
p
dt
T
.
(5.9)
The linearised model equations can be written as a single matrix equation
dX
dt
= MX
(5.10)
where M is a 4 × 4 matrix. This equation has solutions of the form
X = We
λt ,
(5.11)
229
either known or can be estimated. It is also necessary to know the form of the
input impedance Z of the instrument; in Chap. 4 methods were described for either
calculating or measuring this function.
In reality finding the solution of the brass model equations is far from straightforward. It is possible to measure typical values of mouth pressure p m during
performance, and realistic estimates of the lip opening width w and equilibrium
opening height h eq can be extracted from video recordings of players using
transparent mouthpieces. However the parameters γ , ω r and μ depend on the
effective mass, stiffness, surface area and internal damping of the vibrating section
of lip; these quantities are not susceptible to direct measurement.
A more fundamental difficulty in finding solutions to the model equations arises
from the nonlinear nature of the lip valve dynamics. In Sect. 5.2.1 the view of the
player-instrument system as a feedback loop was discussed. This approach helps to
explain the system behaviour very near its oscillation threshold, which is a useful if
limited step on the way to a fuller understanding of brass playing. An alternative way
of investigating near-threshold behaviour treats the player-instrument combination
as a dissipative nonlinear dynamical system, which can be analysed using the
mathematical technique of linear stability analysis (see, e.g. Cullen et al. (2000) and
Velut et al. (2017a)). The principle of linear stability analysis (LSA) is described
in the present section, and an example is given of its ability to explain some basic
features of brass playing. A full mathematical treatment is reserved for the Going
Further Sect. 5.4.
The linearisation of the model equations can be achieved by replacing each
system variable q i (t) by q i,eq + ˜
q i (t), where q i,eq is the threshold equilibrium value
of q i . Terms which are more than first order in one of the new variables ˜
q i (t) are then
eliminated. The resulting linearised equations describe infinitesimal oscillations
of the system variables about their equilibrium values. As a simple example, we
consider the case in which the instrument has only one acoustic mode. The equations
can then be rewritten in terms of the four variables ˜
h,
d ˜
h
dt
, ˜
p, and
d ˜
p
dt
, which can be
combined into a 4 × 1 state vector
X =
˜
h
d ˜
h
dt
˜
p
d ˜
p
dt
T
.
(5.9)
The linearised model equations can be written as a single matrix equation
dX
dt
= MX
(5.10)
where M is a 4 × 4 matrix. This equation has solutions of the form
X = We
λt ,
(5.11)