308
6 Shocks and Surprises: Refining the Elementary Model
6.4.3 Models with More Than One Degree of Freedom
A year after Adachi and Sato published their seminal paper describing a 1DOF lip
model (Adachi and Sato 1995), the authors extended the model by allowing the
pendulum arm on which the single mass swung to have finite stiffness (Adachi
and Sato 1996). The result is a one-mass model with two degrees of freedom,
one corresponding to the rotational mode and one corresponding to the vibrational
mode. Although this model, and a one mass 2DOF model including translation and
rotation (Strong and Dudley 1993), have been successfully used in brass instrument
simulations, they are capable of reproducing the alternation between converging and
diverging lip channels described in Sect. 3.4.
Numerical simulations by Neal et al. (2001) and Richards et al. (2003) have
demonstrated that a 2DOF model with two masses, in which a mode with outwardstriking characteristics is coupled to a mode with inward-striking characteristics,
allows the threshold frequency to lie above or below the acoustic resonance
frequency depending on the acoustic resonator conditions. Using linear stability
analysis of a simple two-mass 2DOF model, a similar effect to the experiments has
been obtained showing the continuous transition from playing above the acoustic
resonance frequency to playing below.
The 2DOF two-mass model is able to explain how a brass player can buzz the
lips in the absence of a mouthpiece or instrument (see, e.g. Boutin et al. (2015b))
without invoking the intervention of an upstream resonance (see Sect. 3.4). This
type of behaviour, in which sound is produced by self-sustained oscillations of a
mechanical system destabilised and then driven by air flow, is an example of the
phenomenon of ‘flow-induced vibration’, which occurs in many different contexts:
for an overview see Blevins (1986).
6.5 Playing Frequencies of Brass Instruments
Increasing the number of degrees of freedom in the lip model can improve its
ability to predict realistic playing behaviour, as explained in Sect. 6.4.3. Several
additional refinements, such as the inclusion of nonstationary flow terms (Elliott and
Bowsher 1982; Saneyoshi et al. 1987), can also be readily introduced. Increasing
the complexity of the lip model does however exacerbate the problem of selecting
appropriate values for the additional parameters (Velut et al. 2017a). A further
complication arises when we include the effects of coupling to resonances of
the player’s windway, described in Sect. 6.3, since these depend strongly on the
physiology and technique of the individual performer. To simulate the playing of a
note by a particular musician on a given instrument, it would be necessary to provide
the model with detailed information about the mechanical resonances of the lips, the
acoustic resonances of the player’s windway, the quasi-static pressure in the mouth
and input impedance of the instrument.
6 Shocks and Surprises: Refining the Elementary Model
6.4.3 Models with More Than One Degree of Freedom
A year after Adachi and Sato published their seminal paper describing a 1DOF lip
model (Adachi and Sato 1995), the authors extended the model by allowing the
pendulum arm on which the single mass swung to have finite stiffness (Adachi
and Sato 1996). The result is a one-mass model with two degrees of freedom,
one corresponding to the rotational mode and one corresponding to the vibrational
mode. Although this model, and a one mass 2DOF model including translation and
rotation (Strong and Dudley 1993), have been successfully used in brass instrument
simulations, they are capable of reproducing the alternation between converging and
diverging lip channels described in Sect. 3.4.
Numerical simulations by Neal et al. (2001) and Richards et al. (2003) have
demonstrated that a 2DOF model with two masses, in which a mode with outwardstriking characteristics is coupled to a mode with inward-striking characteristics,
allows the threshold frequency to lie above or below the acoustic resonance
frequency depending on the acoustic resonator conditions. Using linear stability
analysis of a simple two-mass 2DOF model, a similar effect to the experiments has
been obtained showing the continuous transition from playing above the acoustic
resonance frequency to playing below.
The 2DOF two-mass model is able to explain how a brass player can buzz the
lips in the absence of a mouthpiece or instrument (see, e.g. Boutin et al. (2015b))
without invoking the intervention of an upstream resonance (see Sect. 3.4). This
type of behaviour, in which sound is produced by self-sustained oscillations of a
mechanical system destabilised and then driven by air flow, is an example of the
phenomenon of ‘flow-induced vibration’, which occurs in many different contexts:
for an overview see Blevins (1986).
6.5 Playing Frequencies of Brass Instruments
Increasing the number of degrees of freedom in the lip model can improve its
ability to predict realistic playing behaviour, as explained in Sect. 6.4.3. Several
additional refinements, such as the inclusion of nonstationary flow terms (Elliott and
Bowsher 1982; Saneyoshi et al. 1987), can also be readily introduced. Increasing
the complexity of the lip model does however exacerbate the problem of selecting
appropriate values for the additional parameters (Velut et al. 2017a). A further
complication arises when we include the effects of coupling to resonances of
the player’s windway, described in Sect. 6.3, since these depend strongly on the
physiology and technique of the individual performer. To simulate the playing of a
note by a particular musician on a given instrument, it would be necessary to provide
the model with detailed information about the mechanical resonances of the lips, the
acoustic resonances of the player’s windway, the quasi-static pressure in the mouth
and input impedance of the instrument.