Name: resonance in driven oscillators
Uploaded: 2025-05-22T09:05:41.000Z
Duration: 1 min 6 s
Channel: JoVE

Considérons quatre pendules attachés à une ficelle et suspendus à un support rigide. Soit les longueurs des pendules A et C identiques, tandis que les pendules B et D ont des longueurs différentes.
Lorsque le pendule A est déplacé, il subit une oscillation libre avec une fréquence naturelle. La fréquence propre est la fréquence à laquelle le pendule oscille naturellement en l’absence de tout amortissement ou force motrice.
L’énergie du pendule A est transférée aux autres pendules, les forçant à osciller avec une fréquence d’entraînement. Ces oscillations résultantes dues à la force motrice externe exercée par le pendule A sont appelées oscillations forcées.
La force motrice agissant sur les pendules peut être représentée par la force variant périodiquement avec une amplitude F0 et la fréquence angulaire d’entraînement.
En ajoutant la force motrice obtenue aux équations de force de l'oscillateur harmonique amorti, l'équation du mouvement de l'oscillateur forcé est obtenue.

oscillations forcées

Lorsqu'un oscillateur est soumis à une force d'entraînement périodique, le mouvement peut sembler chaotique. Les mouvements de tels oscillateurs sont connus sous le nom de transitoires. Après la disparition des transitoires, l'oscillateur atteint un état stable, où le mouvement est périodique et le déplacement est déterminé.

<img alt="Equation1" src="/files/ftp_upload/12771/12771_Equation_1.png" style="text-align: center; height: 13px;" />

En utilisant les équations de la force et du mouvement, on obtient l'amplitude de l'oscillateur entraîné. Cela dépend de la fréquence naturelle et de la fréquence d'entraînement.

<img alt="Equation2" src="/files/ftp_upload/12771/12771_Equation_2.png" style="text-align: center; height: 40px;" />

Dans l'équation de l'amplitude, lorsque la fréquence d'entraînement est beaucoup plus petite (ou beaucoup plus grande) que la fréquence naturelle, le carré de la différence entre les deux fréquences angulaires est positif et grand, ce qui entraîne une faible amplitude pour les oscillations de la masse. À mesure que la fréquence de la force d'entraînement se rapproche de la fréquence naturelle du système, l'amplitude des oscillations augmente. Lorsque la fréquence de la force d'entraînement est égale à la fréquence naturelle du système, cela entraîne une amplitude maximale.

Prenons un exemple de la vie réelle. La plupart d'entre nous avons joué avec des jouets impliquant un objet soutenu par un élastique, comme une balle de paddle suspendue à un doigt. Au début, lorsque le doigt est tenu fermement, la balle rebondit en haut et en bas avec une petite quantité d'amortissement. Si le doigt est déplacé lentement vers le haut et vers le bas, la balle suit sans rebondir beaucoup par elle-même. À mesure que la fréquence à laquelle le doigt se déplace vers le haut et vers le bas augmente, la balle répond en oscillant avec une amplitude croissante. Lorsque la balle est entraînée à sa fréquence naturelle, les oscillations de la balle augmentent en amplitude à chaque oscillation tant qu'elle est entraînée.

Le phénomène de l'entraînement d'un système avec une fréquence égale à sa fréquence naturelle s'appelle la résonance. On dit qu'un système est en résonance lorsqu'il est entraîné à sa fréquence naturelle. À mesure que la fréquence d'entraînement devient progressivement plus élevée que la fréquence résonante ou naturelle, l'amplitude des oscillations devient plus petite jusqu'à ce que les oscillations disparaissent presque et que le doigt se déplace simplement vers le haut et vers le bas avec peu d'effet sur la balle.

This text is adapted from <a href="https://openstax.org/books/college-physics/pages/16-8-forced-oscillations-and-resonance">Openstax, College Physics, Section 16.8: Forced Oscillations and Resonance </a>et <a href="https://openstax.org/books/university-physics-volume-1/pages/15-6-forced-oscillations">Openstax, University Physics Volume 1, Section 15.5: Forced Oscillations.</a>

Oscillations forcées

Lorsqu’un oscillateur est forcé avec une force motrice périodique, le mouvement peut sembler chaotique. Les mouvements de ces oscillateurs sont connus sous le nom de transitoires. Après...

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Forced Oscillations

resonance in driven oscillators

Resonance in driven oscillators describes how a periodic force changes an object’s motion. A driven oscillator may first move in a messy way. These early motions are called transients. After the transients die out, the oscillator reaches a steady state. In this state, the motion becomes periodic and the displacement is determined.

The amplitude of a driven oscillator can be found from the equations for force and motion. That amplitude depends on the natural frequency of the system and the driving frequency. If the driving frequency is much lower or much higher than the natural frequency, the difference between the two angular frequencies is large. In that case, the amplitude is small. As the driving frequency gets closer to the natural frequency, the amplitude increases. When the two frequencies are equal, the amplitude reaches its maximum.

A common example is a paddle ball suspended from a finger by an elastic band. When the finger stays still, the ball bounces up and down with only a small amount of damping. If the finger moves up and down slowly, the ball follows along and does not bounce much on its own. As the finger moves faster, the ball responds with larger oscillations.

When the ball is driven at its natural frequency, its motion grows with each oscillation for as long as the force continues. This is resonance. A system driven at its natural frequency is said to resonate. If the driving frequency becomes higher than the resonant or natural frequency, the amplitude drops again. The oscillations then nearly disappear, and the finger moves up and down with little effect on the ball.

Resonance in Driven Oscillators

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Resonance in driven oscillators describes how a periodic force changes an object’s motion. A driven oscillator may first move in a messy way. These early motions are called transients. After the transients die out, the oscillator reaches a steady state. In this state, the motion becomes periodic and the displacement is determined. The amplitude of a driven oscillator can be found from the equations for force and motion. That amplitude depends on the natural frequency of the system and the...

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key terms and concepts

See how driven oscillators reach steady motion, why amplitude depends on driving and natural frequency, and how resonance makes a paddle ball swing higher.

Resonance in Driven Oscillators | JoVE K12

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Resonance in Driven Oscillators

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