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

Stellen Sie sich vier Pendel vor, die an eine Schnur gebunden und an einer starren Stütze aufgehängt sind. Seien die Längen von Pendel A und C identisch, während Pendel B und D unterschiedliche Längen haben.
Wenn das Pendel A verschoben wird, schwingt es frei mit einer Eigenfrequenz. Die Eigenfrequenz ist die Frequenz, bei der das Pendel ohne jegliche Dämpfung oder Antriebskraft auf natürliche Weise schwingt.
Die Energie des Pendels A wird auf die anderen Pendel übertragen, wodurch sie gezwungen werden, mit einer Antriebsfrequenz zu schwingen. Diese Schwingungen, die durch die äußere Antriebskraft des Pendels A entstehen, werden als erzwungene Schwingungen bezeichnet.
Die auf die Pendel wirkende Antriebskraft kann durch die periodisch variierende Kraft mit der Amplitude F0 und die treibende Kreisfrequenz dargestellt werden.
Durch Addieren der erhaltenen Antriebskraft zu den Kraftgleichungen des gedämpften harmonischen Oszillators erhält man die Bewegungsgleichung des erzwungenen Oszillators.

erzwungene schwingungen

Wenn ein Oszillator mit einer periodischen Antriebskraft gezwungen wird, kann die Bewegung chaotisch erscheinen. Die Bewegungen solcher Oszillatoren werden als Übergänge bezeichnet. Nachdem die Übergangsphasen abgeklungen sind, erreicht der Oszillator einen stabilen Zustand, in dem die Bewegung periodisch ist und die Verschiebung festgelegt wird.

<img alt="Equation1" src="/files/ftp_upload/12771/12771_Equation_1.png" style="text-align: center; height: 13px;" />

Unter Verwendung der Gleichungen für Kraft und Bewegung wird die Amplitude des angetriebenen Oszillators bestimmt. Diese hängt von der natürlichen und anregenden Frequenz ab.

<img alt="Equation2" src="/files/ftp_upload/12771/12771_Equation_2.png" style="text-align: center; height: 40px;" />

In der Gleichung für die Amplitude ist das Quadrat der Differenz der beiden Winkelgeschwindigkeiten positiv und groß, wenn die Anregungsfrequenz viel kleiner (oder viel größer) als die natürliche Frequenz ist. Dadurch ergibt sich eine kleine Amplitude für die Schwingungen der Masse. Wenn sich die Frequenz der Antriebskraft der natürlichen Frequenz des Systems nähert, nimmt die Amplitude der Schwingungen zu. Wenn die Frequenz der Antriebskraft der natürlichen Frequenz des Systems entspricht, ergibt sich dies in maximaler Amplitude.

Betrachten wir ein reales Beispiel. Die meisten von uns haben mit Spielzeugen gespielt, bei denen ein Objekt an einem Gummiband aufgehängt ist, wie beispielsweise ein Schlägerball, der von einem Finger gehalten wird. Anfangs, wenn der Finger ruhig gehalten wird, springt der Ball mit einer geringen Dämpfung auf und ab. Wenn der Finger langsam nach oben und unten bewegt wird, folgt der Ball ohne viel eigenes Springen. Mit zunehmender Frequenz, mit der der Finger nach oben und unten bewegt wird, reagiert der Ball mit einer zunehmenden Amplitude der Schwingungen. Wenn der Ball mit seiner natürlichen Frequenz angetrieben wird, nimmt die Amplitude der Schwingungen mit jeder Schwingung so lange zu, wie er angetrieben wird.

Das Phänomen, ein System mit einer Frequenz anzutreiben, die seiner natürlichen Frequenz entspricht, wird als Resonanz bezeichnet. Ein System, das mit seiner natürlichen Frequenz angetrieben wird, schwingt im Resonanzfall. Je höher die Anriebsfrequenz ist, desto kleiner wird die Amplitude der Schwingungen, bis die Schwingungen fast verschwinden und der Finger sich nur noch nach oben und unten bewegt, ohne den Ball zu beeinflussen.

Dieser Text wurde angepasst von <a href="https://openstax.org/books/college-physics/pages/16-8-forced-oscillations-and-resonance">Openstax, College Physics, Abschnitt 16.8: Forced Oscillations and Resonance</a> und <a href="https://openstax.org/books/university-physics-volume-1/pages/15-6-forced-oscillations">Openstax, University Physics Volume 1, Abschnitt 15.5: Forced Oscillations.</a>

Erzwungene Schwingungen

Wenn ein Oszillator mit einer periodischen Antriebskraft erzwungen wird, kann die Bewegung chaotisch erscheinen. Die Bewegungen solcher Oszillatoren werden als Transienten bezeichnet. Nach...

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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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Energy Conservation in Simple Harmonic Motion

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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...

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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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