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3 Computer Simulation of Dynamic Systems
Dividing the first of these equations by the second, we obtain a first-order nonlinear
equation
m
dυ
dr
=
−k(r )r
v
Separating the variables in the resulting equation
mvdυ = −k(r )r dr
And integrating the last equation twice, we find
υ
2
=
dr
dt
2
= −2
r
0
k
r
m
r
dr
+ C
dr
dt
= ±
C − 2
r
0
k(r )
m
r dr
t = ±
r
0
dr
C − 2
r
0
k(r )
m
r dr
−1
+ C 1
where in the implicitly written general solution, the constants C and C 1 can be
determined, knowing the initial data.
As we already know, in the linear case k(r ) = const, the phase trajectories of
the system are concentric ellipses with a center at the origin, the main semi-axes of
which are determined by the initial energy of the system and the “motion” along
which describes the oscillation process that is periodic in time.
We now consider a strongly nonlinear system in which the spring behaves as
“super soft,” for example, k(r ) = 1/
α + r
2
, α > 0. In the limiting case α = 0, the
nonlinear equation takes the form
m
dυ
dt
= −
1
r
υ =
dr
dt
And its solution is fundamentally different from the solution for a harmonic oscillator (Fig. 3.20a), in that the energy is not conserved; moreover, it grows unlimitedly
as r → ±0 (Fig. 3.20b). With the weakening of nonlinearity, the process becomes
normal.
Consider another type of nonlinearity. We assume that the elastic force is related
to the elongation r by the ratio: