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

Weźmy pod uwagę cztery wahadła przywiązane do sznurka i zawieszone na sztywnym wsporniku. Niech długości wahadła A i C będą identyczne, podczas gdy wahadło B i D mają różne długości.
Kiedy wahadło A jest przemieszczone, ulega swobodnej oscylacji z częstotliwością drgań własnych. Częstotliwość drgań własnych to częstotliwość, z jaką wahadło oscyluje naturalnie przy braku jakiejkolwiek siły tłumiącej lub napędowej.
Energia z wahadła A jest przekazywana do pozostałych wahadeł, zmuszając je do oscylacji z częstotliwością sterującą. Te wynikające z tego oscylacje spowodowane zewnętrzną siłą napędową wywieraną przez wahadło A są znane jako oscylacje wymuszone.
Siła napędowa działająca na wahadła może być reprezentowana przez okresowo zmieniającą się siłę o amplitudzie F0 i napędzającą częstotliwość kątową.
Dodając uzyskaną siłę napędową do równań siły tłumionego oscylatora harmonicznego, otrzymuje się równanie ruchu oscylatora wymuszonego.

wymuszone oscylacje

Kiedy na oscylator działa okresowa siła napędowa, ruch może wydawać się chaotyczny. Ruchy takich oscylatorów nazywane są stanami przejściowymi. Po wygaśnięciu stanów nieustalonych oscylator osiąga stan ustalony, w którym ruch jest okresowy i określane jest przemieszczenie.

<img alt="Równanie1" src="/files/ftp_upload/12771/12771_Equation_1.png" style="text-align: center; height: 13px;" />

Korzystając z równań siły i ruchu, uzyskuje się amplitudę napędzanego oscylatora. Zależy to od częstotliwości naturalnej i częstotliwości jazdy.

<img alt="Równanie2" src="/files/ftp_upload/12771/12771_Equation_2.png" style="text-align: center; height: 40px;" />

W równaniu na amplitudę, gdy częstotliwość wymuszająca jest znacznie mniejsza (lub znacznie większa) niż częstotliwość drgań własnych, kwadrat różnicy dwóch częstotliwości kątowych jest dodatni i duży, co skutkuje małą amplitudą oscylacji Msza. W miarę jak częstotliwość siły napędowej zbliża się do częstotliwości drgań własnych układu, amplituda oscylacji wzrasta. Gdy częstotliwość siły napędowej jest równa częstotliwości drgań własnych układu, daje to maksymalną amplitudę.

Rozważmy przykład z życia wzięty. Większość z nas bawiła się zabawkami, w których używano przedmiotu zawieszonego na elastycznej gumce, na przykład piłki wiosłowej zawieszonej na palcu. Na początku, gdy palec jest trzymany nieruchomo, piłka odbija się w górę i w dół z niewielkim tłumieniem. Jeśli palec porusza się powoli w górę i w dół, piłka podąża za nią, sama nie odbijając się zbytnio. Wraz ze wzrostem częstotliwości ruchu palca w górę i w dół piłka reaguje oscylacją ze wzrastającą amplitudą. Kiedy piłka jest napędzana z częstotliwością naturalną, amplituda drgań piłki zwiększa się z każdą oscylacją tak długo, jak długo jest napędzana.

Zjawisko napędzania układu częstotliwością równą jego częstotliwości własnej nazywa się rezonansem. Mówi się, że system, który jest napędzany swoją częstotliwością naturalną, rezonuje. W miarę jak częstotliwość sterująca staje się stopniowo wyższa niż częstotliwość rezonansowa lub częstotliwość drgań własnych, amplituda oscylacji staje się mniejsza, aż oscylacje prawie zanikają, a palec po prostu porusza się w górę i w dół, z niewielkim wpływem na piłkę.

Ten tekst jest adaptacją  <a href="https://openstax.org/books/college-physics/pages/16-8-forced-oscillations-and-resonance">Openstax, College Physics, sekcja 16.8: Wymuszone oscylacje i rezonans </a>oraz <a href="https://openstax.org/books/university-physics-volume -1/pages/15-6-forced-oscillations">Openstax, University Physics, tom 1, sekcja 15.5: Oscylacje wymuszone.</a>

Wymuszone oscylacje

Kiedy oscylator jest wymuszany okresową siłą napędową, ruch może wydawać się chaotyczny. Ruchy takich oscylatorów są znane jako stany nieustalone. Po...

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KEY TERMS AND CONCEPTS

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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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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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Simple Harmonic Motion from Circular Motion

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problem solving: energy in simple harmonic motion

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Energy Conservation in Simple Harmonic Motion

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damped 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

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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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