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

Consideriamo quattro pendoli legati a una corda e sospesi a un supporto rigido. Siano le lunghezze del pendolo A e C identiche mentre il pendolo B e D hanno lunghezze diverse.
Quando il pendolo A viene spostato, subisce un'oscillazione libera con una frequenza naturale. La frequenza naturale è la frequenza alla quale il pendolo oscilla naturalmente in assenza di qualsiasi smorzamento o forza motrice.
L'energia del pendolo A viene trasferita agli altri pendoli costringendoli ad oscillare con una frequenza di pilotaggio. Queste oscillazioni risultanti dovute alla forza motrice esterna esercitata dal pendolo A sono note come oscillazioni forzate.
La forza motrice che agisce sui pendoli può essere rappresentata dalla forza che varia periodicamente con ampiezza F0 e dalla frequenza angolare di guida.
Sommando la forza motrice ottenuta alle equazioni di forza dell'oscillatore armonico smorzato, si ottiene l'equazione del moto dell'oscillatore forzato.

oscillazioni forzate

Quando un oscillatore è forzato con una forza propulsiva periodica, il movimento può sembrare caotico. I movimenti di tali oscillatori sono conosciuti come transitori. Una volta che i transitori svaniscono, l'oscillatore raggiunge uno stato stazionario, dove il movimento è periodico, e lo spostamento è determinato.

<img alt="Equation1" src="/files/ftp_upload/12771/12771_Equation_1.png" style="text-align: center; height: 13px;" />

Usando le equazioni della forza e del movimento, si ottiene l'ampiezza dell'oscillatore forzato. Essa dipende dalla frequenza naturale e da quella di guida.

<img alt="Equation2" src="/files/ftp_upload/12771/12771_Equation_2.png" style="text-align: center; height: 40px;" />

Nell'equazione per l'ampiezza, quando la frequenza propulsiva è molto più piccola (o molto più grande) della frequenza naturale, il quadrato della differenza delle due frequenze angolari è positivo e ampio, con una conseguente piccola ampiezza per le oscillazioni della massa. Visto che la frequenza della forza propulsiva approccia la frequenza naturale del sistema, l'ampiezza delle oscillazioni aumenta. Quando la frequenza della forza propulsiva è uguale a quella naturale del sistema, il risultato si traduce in un'ampiezza massima.

Consideriamo un esempio di vita reale. La maggior parte di noi ha giocato con giocattoli composti da un oggetto supportato da una fascia elastica, come una pallina sospesa da un dito. All'inizio, quando il dito è fermo, la pallina rimbalza su e giù con una piccola quantità di smorzamento. Se il dito è lentamente mosso su e giù, la palla segue senza rimbalzare più di tanto. Quando la frequenza del dito mosso su e giù aumenta, la palla risponde aumentando l'ampiezza dell'oscillazione. Quando la palla è portata alla sue frequenza naturale, l'ampiezza della sua oscillazione aumenta ad ogni oscillazione durante la quale è guidata.

Il fenomeno del guidare un sistema con una frequenza uguale alla sua frequenza naturale è chiamato risonanza. Un sistema che viene guidato alla sua frequenza naturale è detto risuonato. Come la frequenza di guida diventa progressivamente maggiore di quella risuonante o di quella naturale, l'ampiezza delle oscillazioni inizia a diminuire finché le oscillazioni quasi scompaiono e il dito si muove semplicemente su e giù con poco effetto sulla palla.

Questo testo è adattato da <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> e <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>

Oscillazioni forzate

Quando un oscillatore viene forzato con una forza motrice periodica, il movimento può sembrare caotico. I movimenti di tali oscillatori sono noti come transitori. Dopo...

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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

simple harmonic motion

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Simple harmonic motion describes oscillatory motion when the net force follows Hooke's law. In this type of motion, the force acts toward the equilibrium position. If there is no damping from friction or other non-conservative forces, the oscillator moves with equal displacement on both sides of equilibrium. The period of a simple harmonic oscillator is found from the equation of motion. Period is the time for one complete oscillation. Frequency is the inverse of period, so the two values can...

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Stable Equilibrium and Atomic Oscillation

Stable equilibrium is a key idea in oscillation. When an object sits at a stable equilibrium point, a small push moves it away, but it then tends to oscillate around that point. If the equilibrium point is unstable, a slight disturbance will not bring the object back. The transcript also describes a half-stable equilibrium point. It is still considered unstable, but the name comes from the fact that the force on only one side points toward the equilibrium point. That makes it different from a...

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A spring-mass system shows simple harmonic motion, and its oscillation rate depends on the mass and the spring force constant. In this model, the angular frequency, period, and frequency are set by those two values. They do not depend on the motion’s amplitude or on the initial conditions. To find these relationships, the system can be analyzed with Newton’s second law. Imagine a block on a spring on a frictionless surface. Three forces act on the mass: weight, the normal force, and the spring...

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Simple Harmonic Motion from Circular Motion

Simple harmonic motion and uniform circular motion are closely linked. Simple harmonic motion is oscillatory motion in a system where the net force follows Hooke’s law. Uniform circular motion is motion along a circular path at constant speed. A simple way to create simple harmonic motion is to use uniform circular motion. One example is a ball attached to a vertical turntable that spins at a steady rate. If you watch the ball’s shadow on the floor, the shadow, or projection, moves in simple...

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Energy Conservation in Simple Harmonic Motion

Energy conservation in simple harmonic motion explains how a spring and mass system shares energy as it moves back and forth around equilibrium. Simple harmonic motion, or SHM, is periodic motion in time and position. The object oscillates with a constant amplitude and frequency. In SHM, kinetic energy and potential energy change into each other continuously. A good example is the spring in a car shock absorber as the car moves on a bumpy road. The wheel attached to the spring oscillates in...

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Angular Oscillation in a Torsion String

Angular oscillation in a torsion string is shown by a rigid body suspended from a string that twists as it moves. The twisting of the string provides the restoring force. The string is ideally massless, and in practice its mass is much smaller than the rigid body&amp;#39;s mass, so it is neglected. When the angular displacement is small, the motion can be modeled as linear angular oscillation. The amplitude is measured as an angle. In this system, the role of mass is played by the rigid...

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Energy Loss in Oscillations

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How Damping Changes Oscillations

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forced oscillations

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...

concept of resonance and its characteristics

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Resonance in Music, Cars, and Radios

Resonance is a key feature of driven oscillators, and it depends strongly on damping. Damping is the process that reduces motion by removing energy from the system. When a system must resonate at one specific frequency, very light damping is needed. Piano strings and many other musical instruments are common examples of this kind of response. Heavy damping works better when the goal is small-amplitude oscillations. A car suspension system uses this approach so that bumps do not produce large...

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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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