Chapter 6

# 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.

## Question

Find kind of oscillations are these ?

### 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, .

### Solution

• By reading the diagram : and • Also : • The logarithmic decrement is : (see the lesson "A few classical applications") ## Question

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

### Solution

We deduce : , and Note that gives the order of magnitude of the number of visible oscillations.

Then : and Next
Motions of charged particles inside electromagnetic fields