# Conclusions

When analyzing this model, we must remember that it is supposed to
represent a real-life situation of a shock absorber being driven by a
driving force. In our analysis, we discovered that the origin was
stable if lambda < 0 and c1 > 0. This makes physical sense since
in this situation, the forcing term decays with time, and the shock
absorber becomes a damped unforced oscillator which decays to 0. When
lambda < 0 and c1 < 0 then the forcing term decays, but the
damping constant is negative and negative damping is similar in effect
to a forcing term. Thus, the origin is unstable unless, w(0) and
z(0), the two frocing terms, are equal to 0. Similarly, if lambda >
0 and c1 > 0, we have a forced, damped oscillator which will settle
into a limit cycle unless there is no initial damping, in which case
the solution acts as a damped, unforced oscillator. Finally, if
lambda > 0 and c1 < 0, we have both a positive forcing term and
negative damping. The solution does not go to infinity as normal
undamped forced oscillators do because of the quadratic damping term
which is always positive. Instead, the solution always approaches a
limit cycle regardless of the initial conditions and the soution never
decays to 0 unless it starts at (0,0,0,0). Again, this makes sense
since this situation describes an oscillator that starts at rest from
its equilibrium condition and with no damping force. Thus, we should
expect the oscillator to remain at rest.
One of the major faults of this model is that we cannot predict the
behavior when c1 = 0 or lambda = 0. These are cases where there is no
linear damping or no forcing repectively so we should be able to
analyze these possible situations. A better model should allow us to
predict the behavior at all possible values of lambda and c1.

Created by Jason Sardell (hyena@wpi.edu) & Thomas Szymkiewicz
(mrtom@wpi.edu)