Mechanical point
Phase portrait of an oscillator
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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.
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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 ?
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

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