246
5 Blow That Horn: An Elementary Model of Brass Playing
to be promising tools in giving a broad overview of the dynamical behaviour of brass
instruments, not only above but sometimes also below the mouth pressure threshold.
5.4.2 State-Space Representations of the Elementary Brass
Playing Model
As explained in the previous subsection dealing with Van der Pol self-sustained
oscillator, it is useful to reformulate the equations of the brass instrument model
(Sect. 5.1) as a set of first-order ODE equations in order to use LSA and continuation
methods. To do this it is necessary to rewrite the input impedance equation (Eq. 5.8)
in terms of a sum of individual acoustic resonance modes in the frequency domain
and then to transform them into the time domain. The acoustical response of the
instrument tube can be represented as a sum of either real modes, as in Eq. 5.23
(see, e.g. Debut and Kergomard (2004)), or complex modes, as in Eq. 5.33 (see, e.g.
Silva et al. (2014)). These two ways of approximating the input impedance in the
frequency domain lead to two different sets of first-order vector equations
dX
dt
= F (X).
We begin with the ‘real mode’ representation of the input impedance Z.
The input impedance fitted with N resonance modes is written as
Z(ω) =
N
n=1
Z n
j
ω n
q n
ω
ω 2
n + j
ω n
q n
ω − ω 2
.
(5.23)
where the nth resonance is defined by three real constants, the amplitude Z n , the
dimensionless damping coefficient q n and the angular frequency ω n .
Transformation of Eq. 5.23 into the time domain, and decomposition of p(t) into
its real modal components p n defined by
p(t) =
N
n=1
p n (t),
(5.24)
results in a second-order ODE for each p n :
d 2 p n
dt 2 +
ω n
q n
dp n
dt
+ ω
2
n p n (t) = Z n
ω n
q n
du(t)
dt
,
(5.25)
with u(t) being the total volume flow rate.
5 Blow That Horn: An Elementary Model of Brass Playing
to be promising tools in giving a broad overview of the dynamical behaviour of brass
instruments, not only above but sometimes also below the mouth pressure threshold.
5.4.2 State-Space Representations of the Elementary Brass
Playing Model
As explained in the previous subsection dealing with Van der Pol self-sustained
oscillator, it is useful to reformulate the equations of the brass instrument model
(Sect. 5.1) as a set of first-order ODE equations in order to use LSA and continuation
methods. To do this it is necessary to rewrite the input impedance equation (Eq. 5.8)
in terms of a sum of individual acoustic resonance modes in the frequency domain
and then to transform them into the time domain. The acoustical response of the
instrument tube can be represented as a sum of either real modes, as in Eq. 5.23
(see, e.g. Debut and Kergomard (2004)), or complex modes, as in Eq. 5.33 (see, e.g.
Silva et al. (2014)). These two ways of approximating the input impedance in the
frequency domain lead to two different sets of first-order vector equations
dX
dt
= F (X).
We begin with the ‘real mode’ representation of the input impedance Z.
The input impedance fitted with N resonance modes is written as
Z(ω) =
N
n=1
Z n
j
ω n
q n
ω
ω 2
n + j
ω n
q n
ω − ω 2
.
(5.23)
where the nth resonance is defined by three real constants, the amplitude Z n , the
dimensionless damping coefficient q n and the angular frequency ω n .
Transformation of Eq. 5.23 into the time domain, and decomposition of p(t) into
its real modal components p n defined by
p(t) =
N
n=1
p n (t),
(5.24)
results in a second-order ODE for each p n :
d 2 p n
dt 2 +
ω n
q n
dp n
dt
+ ω
2
n p n (t) = Z n
ω n
q n
du(t)
dt
,
(5.25)
with u(t) being the total volume flow rate.