Chapter 6

Mechanical point

A few classical applications

FondamentalDamped harmonic oscillator

The motion of a damped harmonic oscillator follows the following (one dimensional) equation :

We note :

is the damping ratio and the quality factor :

Then :

We look for solutions like , with a complex number.

The characteristic equation is :

The discriminant is :

Different types of solution are observed : it depends on the value of (or ) :

  • Underdamping : (if )

    If and if the initial velocity is equal to zero :

    With :

    The undamped angular frequency.

    The -factor gives the order of magnitude of the number of oscillations we can experimentally measure.

  • Overdamping : (so )

    If and if the initial velocity is equal to zero :

  • Critical damping : (then )

    If and if the initial velocity is equal to zero :

Critical damping

Underdamping and Overdamping

Une vidéo sur les oscillations d'un flotteur (Référence : "Unisciel")

Oscillations d'un flotteur : régimes pseudo-périodique et apériodique

FondamentalSimple pendulum

If there is a viscous friction force :

The second law of motion projected on gives (the equivalent of the second law for angular momentum can be used as well) :

It follows :

If the angle is « small », the usual equation appears :

Simple pendulum
Résonance de pendules (Vidéo d'Alain Le Rille, enseignant en CPGE au Lycée Janson de Sailly, Paris)
Pendules couplés (animations)
Oscillations couplées de pinces à linge (Référence : "Unisciel")

ExempleAn impressive mechanical resonance : the Tacoma bridge resounding

Pont de Tacoma (Résonance mécanique)

ExempleUne animation sur la résonance (University of Colorado Boulder)


FondamentalSinusoidally driven mechanical oscillators

Why is this study useful ?

Harmonic analysis is the study of the output of a system when the input is a sinusoidal signal with variable driving angular frequency .

Chosen model : vertical mechanical oscillator with movable attached point.

The attached point is in A (see figure).

In the inertial reference frame of the Earth :

Equilibrium condition :

With the usual notations :

In what follows, we suppose :

This equation is equivalent to the one giving the tension of a capacitor in a RLC circuit (see electrokinetic course).

If is the tension of the capacitor : ( )

So :

Sinusoidally driven mechanical oscillators

Resolution with complex numbers (the same as in electricity) :

We note :

Then :

Output  :

Then :

Or :

The maximum value is the modulus of that expression :

is maximum for an angular frequency (which exists only if ) :

And the maximum value is :

By using the -factor instead of the damping ratio :

And :

Note :

If the damping is small ( "small" and "large"), then :

If , the amplitude of the resonance is times the one of the input signal : the resonance is « acute » and can cause the destruction of the oscillating system.

the output for different values of the Q-factor.

The previous figure gives the output for different values of the -factor.

We chose and .

The larger the -factor, the more acute the resonance becomes.

The oscillator is low-pass filter, with or without resonance.

Speed output :

If is real :

If is complex :  :

We note that . It follows :

Or :

The maximum amplitude is given by the modulus :

is maximum (velocity resonance) if the denominator is minimum, for an angular frequency :

The amplitude of the velocity is :

Speed output

The previous figure gives for different values of the -factor.

We chose and .

The larger the -factor, the more acute the resonance becomes.

The oscillator is a band-pass filter.

Bandwidth of this band-pass filter :

An angular frequency is in the bandwidth if the output gain is, by definition, superior to the maximal gain (which we obtain when is plugged in) divided by .

The bandwidth length is :

The smaller it is (acute resonance), the smaller the damping gets (and the larger the -factor becomes).

SimulationJAVA animations by Jean-Jacques Rousseau (Université du Mans)

  • Chute libre avec frottements : click HERE and HERE

  • Suspension d'un véhicule : click HERE

  • Oscillations amorties (frottements visqueux ou solide) : click HERE

  • Molécule biatomique (modèle harmonique) : click HERE

  • Potentiel de Morse : click HERE

  • Potentiel de Lennard-Jones : click HERE

  • Satellite géostationnaire : click HERE

  • Satellite artificiel : click HERE

  • Trajectoires célestes possibles : click HERE

  • Les satellites GPS : click HERE

Principles of Newtonian mechanics