Chapter 6

Mechanical point

Phase portrait of an oscillator

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

Let us consider the phase portrait of a damped harmonic oscillator : a mass having two forces exerted on : the elastic force of a spring (spring constant ), and a viscous fluid force ( ) , where is the velocity of the mass .

The mass is displayed of with respect to its equilibrium position O.

The study is done in the inertial frame of reference of the laboratory.

Phase portrait of an oscillator

Question

Find kind of oscillations are these ?

Hint

Solution

  • Underdamping : since there is friction, the phase curve is not closed.

  • It ends in an equilibrium point (here, O) called an attractive point.

Question

Graphically, find :

  • The initial position, .

  • The final position, .

  • The undamped period, .

  • The logarithmic decrement, .

Hint

Solution

Question

Deduce the undamped angular frequency , the -factor of the oscillator, the spring constant and the viscous friction coefficient .

Hint

Solution

We deduce :

, and

Note that gives the order of magnitude of the number of visible oscillations.

Then :

and

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Motions of charged particles inside electromagnetic fields