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

Рассмотрим четыре маятника, привязанных к веревке и подвешенных на жесткой опоре. Пусть длины маятника A и C одинаковы, в то время как маятник B и D имеют разную длину.
Когда маятник А смещается, он претерпевает свободные колебания с собственной частотой. Собственная частота — это частота, на которой маятник колеблется естественным образом при отсутствии какого-либо демпфирования или движущей силы.
Энергия маятника А передается другим маятникам, заставляя их колебаться с движущей частотой. Эти результирующие колебания из-за внешней движущей силы, оказываемой маятником А, известны как вынужденные колебания.
Движущая сила, действующая на маятники, может быть представлена периодически изменяющейся силой с амплитудой F0 и движущей угловой частотой.
Путем сложения полученной движущей силы к силовым уравнениям затухающего гармонического осциллятора получается уравнение движения вынужденного осциллятора.

Принудительные колебания

Когда осциллятор принуждается периодическим силовым воздействием, движение может показаться хаотичным. Движения таких осцилляторов известны как переходные процессы. После затухания переходных процессов осциллятор достигает установившегося состояния, где движение периодическое, и смещение определяется.

<img alt="Equation1" src="/files/ftp_upload/12771/12771_Equation_1.png" style="text-align: center; height: 13px;" />

Используя уравнения силы и движения, получают амплитуду принуждаемого осциллятора. Она зависит от собственной и вынуждающей частоты.

<img alt="Equation2" src="/files/ftp_upload/12771/12771_Equation_2.png" style="text-align: center; height: 40px;" />

В уравнении для амплитуды, когда частота принуждения намного меньше (или намного больше) собственной частоты, квадрат разности двух угловых частот положителен и большой, что приводит к малой амплитуде колебаний массы. По мере приближения частоты силы принуждения к собственной частоте системы, амплитуда колебаний увеличивается. Когда частота силы принуждения равна собственной частоте системы, это приводит к максимальной амплитуде.

Давайте рассмотрим пример из реальной жизни. Большинство из нас играли с игрушками, включающими объект, удерживаемый на упругой ленте, например, мяч на палочке, подвешенный к пальцу. Сначала, когда палец находится в неподвижном состоянии, мяч подпрыгивает вверх и вниз с небольшим затуханием. Если палец медленно двигается вверх и вниз, мяч следует за ним, не отскакивая самостоятельно. С увеличением частоты движения пальца вверх и вниз, мяч начинает колебаться с увеличивающейся амплитудой. Когда мяч принуждается его собственной частотой, амплитуда его колебаний увеличивается с каждым колебанием до тех пор, пока он принуждается.

Феномен принуждения системы с частотой, равной её собственной частоте, называется резонансом. Система, принуждаемая собственной частотой, называется резонирующей. По мере увеличения частоты силы принуждения выше резонансной или собственной частоты, амплитуда колебаний становится меньше, пока колебания почти не исчезают, и палец просто двигается вверх и вниз с небольшим влиянием на мяч.

Этот текст адаптирован из <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> и <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>

Когда осциллятор подвергается воздействию периодической движущей силы, движение может казаться хаотичным. Движения таких осцилляторов известны как переходные процессы. После...

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

Oscillations

Колебания

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

Period and Frequency in Simple Harmonic Motion

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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characteristics of simple harmonic motion

characteristics-of-simple-harmonic-motion

Simple Harmonic Motion: Period and Phase

Simple harmonic motion depends on how displacement, force, and acceleration work together. In this motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and points in the opposite direction. That restoring force pulls the object back toward equilibrium. The period and frequency of a simple harmonic oscillator do not depend on amplitude. A diving board shows this idea well. A stiff, thick diving board has a large force constant, so it has a...

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oscillations about an equilibrium position

oscillations-about-an-equilibrium-position

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...

energy in simple harmonic motion

Simple Harmonic Motion: Energy and Speed

Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

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frequency of spring-mass system

How Mass and Spring Stiffness Set Oscillation

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 and uniform circular motion

simple-harmonic-motion-and-uniform-circular-motion

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...

problem solving: energy in simple harmonic motion

problem-solving-energy-in-simple-harmonic-motion

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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simple pendulum

Pendulum Motion and Period

A simple pendulum is a small ball, or bob, hanging from a string that has negligible mass and does not stretch. Pendulums appear in everyday life, including clocks, swing sets, and a sinker on a fishing line. Their motion is a useful model for studying repeating movement. The period of a simple pendulum depends on two factors: its length and the acceleration due to gravity. The period does not depend on the bob’s mass or on the maximum displacement. For small angles, the pendulum behaves like...

torsional pendulum

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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physical pendulum

Physical Pendulum Motion and Inertia

A physical pendulum is a rigid body that hangs from a fixed pivot point and swings when it is displaced. Its motion is similar to a simple pendulum. To describe that motion, physicists use the small-angle approximation, where sin θ ≈ θ. This approximation is valid up to about 14°. The period and angular frequency of a physical pendulum can be found by comparing it with a spring-mass system. The mass moment of inertia is the key quantity that describes how the body’s mass is spread around the...

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measuring acceleration due to gravity

Using Pendulums to Find Gravity

Pendulums can be used to measure the acceleration due to gravity. A simple pendulum is a point mass attached to a string. A physical pendulum is an object that swings in a similar way, but its mass is spread out, so it cannot be treated as a point mass on a string. A physical pendulum is described using rotational motion and moment of inertia. In both types of pendulum, gravity provides the restoring force that brings the object back toward the middle of its swing. For a simple pendulum,...

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damped oscillations

Energy Loss in Oscillations

Oscillations in the real world often lose energy as they move. A system that keeps the same amplitude without any loss is called undamped. That kind of motion is rare, because friction and other non-conservative forces usually reduce the motion over time. Common examples show this clearly. A guitar string stops vibrating a few seconds after it is plucked. A swing also needs a person to keep pushing it if the motion is to continue. In some devices, such as car shock absorbers, damping is useful...

types of damping

How Damping Changes Oscillations

Damping changes how a system oscillates by reducing its back-and-forth motion. As damping increases, the period and frequency begin to change because damping opposes motion and lowers the net force in both directions. When damping is very large, the system stops oscillating altogether. It then moves slowly toward equilibrium. An overdamped system reaches equilibrium slowly. An underdamped system reaches equilibrium faster, but it oscillates around the equilibrium point while doing so. A...

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

concept-of-resonance-and-its-characteristics

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

physics

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Resonance in Driven Oscillators

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