Start First Page interview
Freeabsorbedmaintainedparametricspecial

The free oscillator : the simple pendulum without any friction ( non-dissipativ system).

  The picture below show the different components of this system.

 

The fundamental principle of dynamic permit to write the equation:

For a small angle, this equation can be simplified:

This is a periodical motion, which solution is

T is the period and w the pulsation.

Two values are sufficient to describe the whole instantaneous movement : the angle (q ) and the angular speed (dq /dt).

 

Furthermore it is not necessary to integrate because of the representation that can be used to describe the movement. You can easily express the angular speed (dq /dt) in function of the angle (q ). The geometrical representation is called "portrait des phases".

 

In this case you can use E(q ,dq /dt):

 

This is proportional to the energy of the pendulum.

 

So the curves solution of the equation are isoenergetical curves in the phase-space ("espace des phases"). Cf. the picture below.