Results

Our study is about the reaction of the pendulum when we modify the frequency of the density variations.

In the first case, the frequency is the double of the system frequency so, when the two variations are in phase, the movement is the same than a swing. So, we must observe an entertained movement for the little values of the gravity coefficient.

In the second case, we make tests with a frequency of the gravity variations equal to the frequency of the system.

In all the simulations, the average gravity is taken equal to 9, the mass is 1Kg and the rope is one meter long, so the pulsation of the system is 3rd/s.

1) The swing mouvement.

In this first case, the pulsation of the gravity (6rd/s) is twice the pulsation of the system (3rd/s) and gravity coefficient is 0,3.

After a short period, the system converge to a stable solution.

variation of the angle with the time variation of the angle with the velocity This first case is the movement expected.

If we increase the coefficient of gravity ( 0.9 ), we obtaine the results following.  There seem to be different attractors, but finaly the system converge to the same kind of solution than in the first case.

Now, we take a coefficient of gravity equal to 1,1. So in this case the gravity is oscillating between -0,1g0 and 2,1g0.

We get the results following :  In this case, the system diverges.

If we continue to increase the coefficient of gravity, we get curious results. In the following case the coefficient is equal to 10, it has no physic sens.  The solution we get seems to be chaotique.

2) Equality of the pulsations.

Now, the two pulsations are the same.

If we take a coefficient of gravity equal to 0.2, we get the results following :  That result represents a system amorted.

If we continue to increase it to 0.6, we get the results following :  The system is converging.

If we still increase the coefficient (k=1.5), we get :  It seems to be chaotic with no attractor.

Now, we take k=2  The attractor is quite strange.