__III.1 Results
of the free oscillations case__

In the previous
part, it has been showed that the system had two self-modes, which correspond
to the two self-pulsations. To activate each of these two modes, it is
necessary to impose particular initial conditions. For the first mode,
one needs to impose = '
at the initial time, which means
= 0 (x(0)-y(0)= 0). Moreover, like the system is not impulsed at
the begining, there is a second initial condition:

To activate
the second mode, one needs to impose
= -' at the initial time, which
means that
is not nil. However, the system is still not impulsed at the begining,
so one still has:

After having
imposed the good initial conditions in the program (see part IV) we start
the simulation. For the two modes the simulation lasts 5 s and the time
step is 5 ms. The obtained results are represented on the following graphics:

Like one can see,
is always equal to zero for the first self-mode. This is normal because
in this mode, the two pendulums are in phase. For the second self-mode,
they are in opposition of phase, that is why one can see a sinusoid. These
two results can be find by an analytical resolution.

As it has
been previously seen, in the free oscillations case,
is solution of the differential equation:

and the solution of this equation is:

For each mode,
there are two initial conditions. Therefore, it is possible to know the
value of at every time. In
the case of the first self-mode,
= 0 and
so __A=B= 0__. At every time,
is nil.

In the case
of the second self-mode,
is equal to a constant C so __A+B = C__ and
so __A = B__. The solution is therefore

This allows us to check the validity of the previous graphic:
the value of the constant C should be the value of the amplitude, and the
value of the sinusoid's pulsation should be the square root of 2b-a.

__III.2 Results
of
the forced oscillations case__

In this case, the differential equation that describes the system is:

and like in the previous case, it is necessary to impose
two initial conditions to solve it. At time t = 0, the system is at its
equilibrium position, so one has x(0)= y(0)= 0, which means
= 0. But this time, the system is impulsed at the begining, so the value
of the derived at t = 0 is not nil. This value will be expressed later.

After having
imposed the good initial conditions, in the program (see part IV) we start
the simulation. All the simulation long, the initial conditions are the
same, but the frequency varies from 10 Hz to 0,5 Hz. For each frequency,
one mesure the maximum amplitude of the sinusoid and then one draws the
evolution of the amplitude in function of the pulsation. The time step
of each try is 15 ms and the final time is 50 s. One has obtained:

One can see there is a lot of different modes. Now, one draws the evolution of the amplitude in function of the pulsation:

the resonance pick is reached for w ~ 1,87 which
correspond to the theoritical value of w_{2} . The same study with
would have given the same results, but the resonance pick would have been
reached for a frequency equal to the theoritical value of the first self-pulsation,
w1.