252
5 Blow That Horn: An Elementary Model of Brass Playing
the ‘trivial’ solution, both the mouthpiece pressure p(t) and the volume flow rate
u(t) are equal to zero.
Equilibrium states also exist for values of the mouth pressure p m > 0. Since
in these states there is a pressure gradient across the lip opening, there will be a
steady volume flow u eq . We assume that p(t) = 0, which is equivalent to setting
Z = 0 in the third constituent equation (Eq. 5.8). Setting all the terms involving time
derivatives equal to zero in the first constituent equation (as expressed in Eq. 5.5)
and substituting the result in the second constituent equation (Eq. 5.7) gives the
following expressions for h eq and u eq as functions of the control parameter p m :
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
h eq = h o +
p m
μω 2
l
u eq = w
h o +
p m
μω 2
l
2p m
ρ
.
(5.40)
These equilibrium states theoretically exist for all positive values of p m . They
can however only be observed in practice if p m is below the threshold p thr for
self-sustained oscillation. When p m is raised slowly through the threshold, the
equilibrium state becomes unstable, and the system bifurcates towards a stable
oscillating regime. It is the aim of linear stability analysis to derive these threshold
values from the equations of the global model.
The analysis of the stability of a state corresponding to a given value of the
control parameter p m follows the method outlined in Sect. 5.4.1 for the example
of the Van der Pol oscillator. The nonlinear dynamical system represented in the
state-space formalism by Eq. 5.31 or Eq. 5.38 is linearised around the equilibrium
position defined by Eq. 5.40. After linearisation the 2(N + 1) equations are coupled
by the linearised air flow equation (for details see Velut et al. (2017a) using the
‘complex mode’ representation). The resulting set of 2(N + 1) simultaneous firstorder linear ordinary differential equations can be written as the vector equation
dX
dt
= MX,
(5.41)
where M is the 2(N + 1) × 2(N + 1) Jacobian matrix.
Once the Jacobian matrix for the system is known, its stability is examined
by determining the eigenvalues of the matrix (Thomsen 1997). As explained in
Sect. 5.4.1, linear stability is preserved provided that the real parts of all eigenvalues
of the square matrix M are negative. If the control parameter p m is slowly increased
from zero, the equilibrium solution of the system becomes unstable when the real
part of one of the eigenvalues becomes positive. The threshold mouth pressure p thr
at which this transition occurs can thus be determined by studying the eigenvalue
behaviour as a function of p m . In a direct Hopf bifurcation, a stable oscillatory state
exists for p m > p thr (see Sect. 5.3.2). The threshold frequency f thr , defined as the
limiting value of the oscillation frequency as p m − p thr → 0, is determined by the
imaginary part of the eigenvalue λ:
5 Blow That Horn: An Elementary Model of Brass Playing
the ‘trivial’ solution, both the mouthpiece pressure p(t) and the volume flow rate
u(t) are equal to zero.
Equilibrium states also exist for values of the mouth pressure p m > 0. Since
in these states there is a pressure gradient across the lip opening, there will be a
steady volume flow u eq . We assume that p(t) = 0, which is equivalent to setting
Z = 0 in the third constituent equation (Eq. 5.8). Setting all the terms involving time
derivatives equal to zero in the first constituent equation (as expressed in Eq. 5.5)
and substituting the result in the second constituent equation (Eq. 5.7) gives the
following expressions for h eq and u eq as functions of the control parameter p m :
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
h eq = h o +
p m
μω 2
l
u eq = w
h o +
p m
μω 2
l
2p m
ρ
.
(5.40)
These equilibrium states theoretically exist for all positive values of p m . They
can however only be observed in practice if p m is below the threshold p thr for
self-sustained oscillation. When p m is raised slowly through the threshold, the
equilibrium state becomes unstable, and the system bifurcates towards a stable
oscillating regime. It is the aim of linear stability analysis to derive these threshold
values from the equations of the global model.
The analysis of the stability of a state corresponding to a given value of the
control parameter p m follows the method outlined in Sect. 5.4.1 for the example
of the Van der Pol oscillator. The nonlinear dynamical system represented in the
state-space formalism by Eq. 5.31 or Eq. 5.38 is linearised around the equilibrium
position defined by Eq. 5.40. After linearisation the 2(N + 1) equations are coupled
by the linearised air flow equation (for details see Velut et al. (2017a) using the
‘complex mode’ representation). The resulting set of 2(N + 1) simultaneous firstorder linear ordinary differential equations can be written as the vector equation
dX
dt
= MX,
(5.41)
where M is the 2(N + 1) × 2(N + 1) Jacobian matrix.
Once the Jacobian matrix for the system is known, its stability is examined
by determining the eigenvalues of the matrix (Thomsen 1997). As explained in
Sect. 5.4.1, linear stability is preserved provided that the real parts of all eigenvalues
of the square matrix M are negative. If the control parameter p m is slowly increased
from zero, the equilibrium solution of the system becomes unstable when the real
part of one of the eigenvalues becomes positive. The threshold mouth pressure p thr
at which this transition occurs can thus be determined by studying the eigenvalue
behaviour as a function of p m . In a direct Hopf bifurcation, a stable oscillatory state
exists for p m > p thr (see Sect. 5.3.2). The threshold frequency f thr , defined as the
limiting value of the oscillation frequency as p m − p thr → 0, is determined by the
imaginary part of the eigenvalue λ: