Name: how damping changes oscillations
Uploaded: 2023-04-30T11:09:03.000Z
Duration: 1 min 20 s
Channel: JoVE

水面に浮かぶ木の丸太を考えてみましょう。下向きに押し下げられ、上下に揺れると、平均位置を中心に振動します。時間の経過とともに、散逸力は振動の振幅を減少させ、それは減衰調和運動の方程式で表すことができます。
解くと、虚数部がゼロでない場合、時間依存の指数減衰エンベロープ内の位置と角周波数に対する微分方程式の一般解が得られます。
角周波数に応じて、システムはさまざまなタイプの減衰条件を示します。減衰力が小さい場合、角周波数は固有振動数に似ています。したがって、減衰が不十分な状態では、システムはいくつかの可能なサイクルを行った後、減衰振幅で振動します。
臨界減衰条件では、減衰力と復元力が等しくなり、角周波数がゼロになり、システムの振動が指数関数的にフェードアウトします。
一方、減衰が過剰に進んだ状態では、減衰力が比較的大きくなるため、システムはゆっくりと平衡状態に達します。

減衰の種類

系内の減衰の量が徐々に増加すると、減衰が前後の動きに対抗し、したがって前後の動きを遅くするため、周期と周波数が影響を受け始めます (正味の力は両方向で小さくなります)。 減衰量が非常に大きい場合、系は振動さえしません。 代わりに、ゆっくりと平衡に向かって進みます。 簡単に言うと、過減衰系は平衡に向けてゆっくりと移動しますが、過小減衰系は平衡に急速に移動しますが、その際に平衡点を中心に振動します。 対照的に、臨界減衰系は、平衡点の周りで振動することなく、平衡に向かってできるだけ早く移動します。

一般に、このような系は平衡状態に迅速に戻るだけでなく、平衡状態を維持するため、臨界減衰が望まれることがよくあります。 さらに、臨界減衰系に一定の力が加えられると、オーバーシュートや新しい位置を中心に振動することなく、可能な限り短時間で系が新しい平衡位置に移動します。 たとえば、針ゲージが付いている体重計の上に人が立つと、針は振動することなく平衡位置に移動します。 針が安定する前に、新しい平衡位置の周りで長時間振動すると、非常に不便になります。 減衰力の特性は大きく異なります。 たとえば、摩擦は速度に依存しない場合があります。 ただし、多くの減衰力は速度に依存します。その場合は複雑な場合もあれば、単純に速度に比例する場合もあります。

このテキストは <a href="https://openstax.org/books/college-physics/pages/16-7-damped-harmonic-motion"> から改変されたものです。 Openstax, College Physics, Section 16.7: Damped Harmonic Motion </a>および<a href="https://openstax.org/books/university-physics-volume-1/ page/15-5-damped-oscillations"> Openstax, University Physics Volume 1, Section 15.4: Damped Oscillations.</a>

システム内のダンピングの量を徐々に増やすと、ダンピングが反対に、したがって遅くなるため、周期と周波数が影響を受け始めます...

Education

JoVE Core

Physics

物理学

Unit 1

ユニット1

Oscillations

振動

KEY TERMS AND CONCEPTS

主要用語と概念

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 critically damped system is different. It moves to equilibrium as quickly as possible without overshooting or oscillating. This is often the desired behavior because the system returns to equilibrium rapidly and stays there.

Critical damping is also useful when a constant force creates a new equilibrium position. In that case, the system moves to the new position in the shortest time possible without overshooting. A bathroom scale with a needle gauge is a common example. The needle moves to its equilibrium position and does not keep bouncing around it, which would be inconvenient.

Damping forces can also have different forms. Friction is sometimes independent of velocity. In many other cases, damping depends on velocity, either in complex ways or simply in proportion to velocity.

How Damping Changes Oscillations

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

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

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

Learn how damping affects oscillations, including underdamped, overdamped, and critically damped motion, plus why critical damping is often useful.

How Damping Changes Oscillations | JoVE K12

physics

JoVE Core 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

How Damping Changes Oscillations

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