3.2 An Equation of Motion for the Lips
79
3.2 An Equation of Motion for the Lips
The lips of a brass player are flexible, continuous structures subjected to aerodynamic forces which depend on the pressure field around them. The vibrating area
of each lip is limited by the rim of the mouthpiece against which it is pressed.
Many of the material properties of the lips are difficult to measure experimentally,
and the internal forces exerted by the muscles of the player’s embouchure are in
general unknown. A complete mechanical model of this vibrating system with many
degrees of freedom would be extremely complex and would include a large number
of parameters whose values could not be accurately determined. A more fruitful
approach to understanding the essential physics of the vibrating lip is to begin with
a highly simplified model, as outlined in Sect. 2.2.
In Sect. 3.2.1 a model is developed in which the complex multidimensional lip
motion revealed by the experimental studies described in Sect. 3.1 is represented
by the time dependence of a single spatial variable. A ‘sliding door’ model, in
which the lip vibration is assumed to take place only in the vertical plane, is
described in Sect. 3.2.2. A ‘swinging door’ model, which allows for the observed
two-dimensional motion of the lips (Sect. 3.1.5), is discussed in Sect. 3.2.3.
3.2.1 A One-Mass Model of the Lips
The simple lip model with only one degree of freedom (1DOF) is frequently
described as a one-mass model, since in it the distributed mass of the moving
section of the lip is concentrated into a single mass m (Fig. 3.18). To begin with,
we assume that the single degree of freedom corresponds to motion along the
Fig. 3.18 The one-mass
model of a single degree of
freedom oscillator with mass
m and stiffness k
k
m
79
3.2 An Equation of Motion for the Lips
The lips of a brass player are flexible, continuous structures subjected to aerodynamic forces which depend on the pressure field around them. The vibrating area
of each lip is limited by the rim of the mouthpiece against which it is pressed.
Many of the material properties of the lips are difficult to measure experimentally,
and the internal forces exerted by the muscles of the player’s embouchure are in
general unknown. A complete mechanical model of this vibrating system with many
degrees of freedom would be extremely complex and would include a large number
of parameters whose values could not be accurately determined. A more fruitful
approach to understanding the essential physics of the vibrating lip is to begin with
a highly simplified model, as outlined in Sect. 2.2.
In Sect. 3.2.1 a model is developed in which the complex multidimensional lip
motion revealed by the experimental studies described in Sect. 3.1 is represented
by the time dependence of a single spatial variable. A ‘sliding door’ model, in
which the lip vibration is assumed to take place only in the vertical plane, is
described in Sect. 3.2.2. A ‘swinging door’ model, which allows for the observed
two-dimensional motion of the lips (Sect. 3.1.5), is discussed in Sect. 3.2.3.
3.2.1 A One-Mass Model of the Lips
The simple lip model with only one degree of freedom (1DOF) is frequently
described as a one-mass model, since in it the distributed mass of the moving
section of the lip is concentrated into a single mass m (Fig. 3.18). To begin with,
we assume that the single degree of freedom corresponds to motion along the
Fig. 3.18 The one-mass
model of a single degree of
freedom oscillator with mass
m and stiffness k
k
m