The Sources of Sound in Birdsong - The Physics of Birdsong - page 49

that the restitution force is a function only of the displacement, and before
the walls were introduced, the friction was a function of the velocity only. As
we have introduced the walls, the dissipation of the system must be modeled
by a force that has a component which depends on both the velocity and the
position. The problem is no longer linear. This will bring about important
changes in the morphology of the oscillatory solutions of the problem.
The simplest conceivable model for our nonlinear dissipative force can
be written as F nl = Cx 2 x . Notice that for small values of x (i.e., close

to equilibrium) this term is negligible. It only becomes important when the
system is far from equilibrium. Other functional forms that are even in x can
be conceived of for F nl , but, since we are not interested in the detailed shape
of the solutions for now, we shall retain this form, which is the lowest-order
expresion in both x and y consistent with our discussion.
If the force proportional to the velocity F gain overcomes the friction
F friction , the system will begin to oscillate with progressively larger am-
plitude, but as soon as the walls are reached, the additional friction force
F nl will stop the system. The restitution force F spring will be responsible for
the return of the system, which will head towards the opposite wall, to be
stopped again. In this way, the oscillations around the equilibrium position
will continue, but with a shape quite different from the harmonic oscillations
that we would have obtained if all the forces were linear.
Taking into account all these forces, we can use Newton's laws as before
to write a differential equation for the midpoint position of a labium:

−

x = y,
y = −kx +( β − b ) y − cx 2 y, (4.5)

whose solutions we shall describe in the following subsection. The constants
k, b and c are the already defined K, B and C per unit M .

4.2.3 Nonlinear Forces and Nonlinear Oscillators

In order to illustrate the difference between the predictions of a model of a
linear oscillator and the model which we are building, we refer to Fig. 4.2. In
this figure, we show the time evolution of the variable x , which measures the
departure from equilibrium of the body under analysis. How did we generate
this figure? Simply with the help of Newton's equation (4.5), which prescribe
the acceleration of a body as a function of the forces acting on it. For given
initial values of the position x and the velocity y at some instant t 0 ,ifwe
know the net force we know the acceleration, and therefore we can calculate
the position x and velocity y of the body at an instant t 0 +∆ t later. In
this way, we can progressively build the trajectory of the body. We show the
result of this process in Fig. 4.2, for two different scenarios.
In the top part of Fig. 4.2, we show the result for the case in which
the body is subjected only to an elastic restitution force, which gives rise to