II. Theoritical study of the system

        II.1 Description of the system

        The dynamic system is composed of two pendulums linked with a spring. The two pendulums are identical. The spring is subjected to an exciting force depending on the time. The specifications of each element of the system are following one:

Pendulum :  lenght L=0,1m, mass M=25 kg
Spring : stiffness K =500 Nm-1
The value of the gravity is g = 9,81 ms-2

        The following picture shows us the mechanism:
 
 


 
 

PENDULUMS LINKED WITH A SPRING




x(t) represents the displacement of the point A
y(t) represents the displacement of the point B
represents the angle of the first pendulum with the vertical
' represents the angle of the second pendulum with the vertical

        To simplify the problem, one supposes two things:
-The friction forces are negligeable
-The angles are small enough, and one can suppose sin() =   at first order.
 
 

        II.2 The free oscillations

        In this following part, one will consider the previous system, without the exciting force. The system is then free. This study will allow us to know the value of the self-pulsations of the system.

        When the system is free, only the gravity and the spring are working: there is then a potential associated to the system. One has for the gravity force:

and one has for the elastic force:

If one consider that the angle stay weak, the potential of the system is:

        On the other hand, the kinetic energy of the system is:



        So, one can now find the self pulsation of this dynamic system. To that, it is necessary to solve [V-w2T]= 0 (the symbol [ ] represents the determinant of the matrix). One finds two solutions:

            

        These two values are the two self-pulsations of the system. There exist two self-modes associated to these pulsations. One will study this two modes in the part III.
        This allows us to know the resonance pulsation. Indeed, the self-pulsation and the resonance pulsation are linked with the relation 
As the system is not damped, the absorption factor is nil, so the resonance pulsation is equal to the self-pulsation.
 

        II.3 The forced oscillations

        The goal of the previous part was the determination of the self-pulsations of the system. Now that one knows the value of these self-pulsations, one will study how the system react under the action of an external force, which pulsation is equal to one of the self-pulsation of the system.
        In the studied case, the spring is subjected to an exciting force depending on the time. One apply the PFD to each pendulum and one obtains, after projection on horizontal and vertical axis the following system:


a, b and f depends on the characteristics of the problem and there expressions are:

        The previous system is a coupled  second order system. To solve it, one introduces two new variables:
 



which are solution of the following system:


        In what follows, one will only study the equation which deals with . Indeed, it is the same method to study the behaviour of   or (it is the same type of differential equation). Moreover, it is more interesting to study the physical phenomenon that describes the evolution of the gap between the two pendulums, especially when it is about the resonance.