Chapter 6

# A few classical applications

### Fondamental : Damped 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. Underdamping
• Overdamping : (so )

If and if the initial velocity is equal to zero :   Overdamping
• 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

### Fondamental : Simple 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")

### Exemple : An impressive mechanical resonance : the Tacoma bridge resounding

Pont de Tacoma (Résonance mécanique)

Résonance

### Fondamental : Sinusoidally 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 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 : 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).