Name: stable equilibrium through potential energy
Uploaded: 2023-09-22T11:49:37.000Z
Duration: 1 min 16 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

Energy

HS-PS3-5

potential-energy-criterion-for-equilibrium

Potential-Energy Criterion for Equilibrium

位置が 1 つの独立変数によって定義される摩擦のないばね質量システムについて考えます。ここで、ポテンシャルエネルギーは重力ポテンシャルエネルギーと弾性ポテンシャルエネルギーの合計であり、その変化の負はシステム上で行われる仕事に等しくなります。
システムが平衡状態にあり、仮想変位が発生した場合、行われた作業は仮想作業に置き換えることができます。
仮想作業の原理から、すべての仮想変位に対して作業はゼロであるため、位置エネルギーの変化もゼロです。小さな仮想変位の場合、位置エネルギーは位置座標を使用した1次導関数で表されます。
仮想変位はゼロではないため、位置エネルギーの一次導関数はゼロでなければなりません。したがって、システムは、その位置エネルギーの1次導関数がゼロであるときに平衡状態にあります。
この基準をばね質量システムに適用すると、その平衡位置が得られます。
ポテンシャルエネルギーがいくつかの独立変数に依存する場合、各座標の位置エネルギーの偏微分は平衡のためにゼロでなければなりません。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス:静力学。第14版、ニュージャージー州:ピアソン。600-602ページ。</li>
	<li>メリアム、JL、クレイジ、LGおよびボルテン、JN(2016)。エンジニアリングメカニクス:静力学とダイナミクス。第8版、シンガポール:John Wiley & Sons。415-416ページ。</li>
</ol>

平衡のためのポテンシャルエネルギー基準

ポテンシャルエネルギーまたはポテンシャル関数は、機械系の安定性を決定する上で重要な役割を果たします。もし系が重力力と弾性力の両方にさらされている場合、系のポテンシャル関数は重力ポテンシャルエネルギーと弾性ポテンシャルエネルギーの代数的な和として表すことができます。もし系が平衡状態であり、わずかな量だけ変位された場合、系にかかる仕事は初期位置から最終位置までの系のポテンシャルエネルギーの変化の負に等しいです。もし系が実際の変位ではなくバーチャル変位を経験する場合、バーチャルワーク関係によれば、全てのバーチャル変位に対して系のポテンシャルエネルギーの変化はゼロでなければなりません。つまり、機械系の平衡配置は、系の全ポテンシャルエネルギーが一定の値を持つ状態です。ポテンシャルエネルギーとその導関数が配置を表す単一変数xの連続関数である1自由度系の場合、平衡条件は以下のように書かれます。

<img alt="方程式1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;" />

上記の方程式は、機械系がそのポテンシャルエネルギーの導関数がゼロであるときに平衡していることを述べています。複数の自由度を持つ系の場合、各座標に対するポテンシャルエネルギーの偏微分は、平衡状態でゼロでなければなりません。

平衡配置での安定性は、そのポテンシャルエネルギーがその配置で最小である限り維持されます。最小のポテンシャルエネルギーを確保するためには、変位座標に対するポテンシャル関数の2階導関数が正でなければなりません。これが系の安定平衡の第2の重要な条件です。

ポテンシャルエネルギーまたはポテンシャル機能は、機械システムの安定性を決定する上で重要な役割を果たします。システムが両方にさらされている場合...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Virtual Work

仮想仕事

KEY TERMS AND CONCEPTS

主要用語と概念

stable equilibrium through potential energy

Potential energy helps determine when a mechanical system is stable at equilibrium. When a system is acted on by gravitational and elastic forces, its total potential function is the algebraic sum of gravitational potential energy and elastic potential energy.

If the system is already in equilibrium and is given a small displacement, the work done on the system equals the negative of the change in potential energy between the starting and ending positions. For a virtual displacement, which is an imagined small shift rather than an actual motion, the virtual work relation shows that the change in potential energy must be zero for every virtual displacement.

This means the equilibrium configuration is a state where the total potential energy has a stationary value. For a system with one degree of freedom, where the configuration is described by a single variable x and the potential energy is a continuous function of x, equilibrium occurs when the derivative of total potential energy with respect to x is zero. For systems with several degrees of freedom, the partial derivative of potential energy with respect to each coordinate must also be zero.

Stability depends on whether the potential energy is at a minimum at the equilibrium position. To make sure the equilibrium is stable, the second derivative of the potential function with respect to the displacement coordinates must be positive. This positive second-derivative condition is the second requirement for stable equilibrium.

Stable Equilibrium Through Potential Energy

stable-equilibrium-through-potential-energy

work

Work in Force and Motion

Work connects force, motion, and energy in mechanics. It happens when a force acts on an object and the object moves along the line of action of that force. Work can also be described as energy transfer caused by applying a force, which leads to a change in the object’s motion or position. Work is calculated as the force multiplied by the part of the displacement in the same direction as the force. It can also be written as the dot product of the applied force and the object’s displacement. In...

work of a couple moment

Couple Moment Work in Rotation

Mechanical engineering studies motion, energy, and force, and it uses the idea of a couple moment to describe how two equal and opposite forces act on a rigid body. A couple moment is created when those forces act at two points on the body that are separated by a distance. When a rigid body moves because of a couple, its displacement can be split into two parts. First, the two points move by the same amount to their final positions. Then the body rotates about one of those points. During the...

virtual work

Virtual Work in Force Equilibrium

Virtual work is a mechanics principle used to check equilibrium with imaginary motion. If a body is in static or dynamic equilibrium, the total virtual work from external forces and couple moments must be zero for any virtual displacement. A virtual displacement is an imagined small movement. It can be a shift or a rotation. The virtual work done by a force equals the force dotted with the virtual displacement in the force’s direction. For a couple moment, virtual rotational work is found by...

virtual work for a system of connected rigid bodies

virtual-work-for-a-system-of-connected-rigid-bodies

Finding Forces with Virtual Work in Linked Bodies

Virtual work helps find forces in a system of connected rigid bodies. It is a useful method when several links move together and the system is in equilibrium. In that case, the total virtual work is zero. To analyze the system, start with a free-body diagram. Use one independent coordinate to describe the link configuration. Then mark the deflected position that comes from a positive virtual displacement. This makes the small change in position clear before any calculations begin. Next,...

principle of virtual work: problem solving

principle-of-virtual-work-problem-solving

Virtual Work for Finding Unknown Forces

The principle of virtual work helps find unknown forces and moments in equilibrium problems. It is an important idea in mechanics and engineering. The method uses a small, hypothetical motion called a virtual displacement. If the system is in equilibrium, the work done by the forces during that displacement is zero. To apply this principle, the positions of the internal and external forces must be written in terms of one common coordinate parameter. That parameter can be an angle, a distance,...

mechanical efficiency of real machines

Friction’s Role in Machine Efficiency

Mechanical efficiency shows how well a machine turns input work into output work. It is calculated as the ratio of output work to input work. An ideal machine has an efficiency of one, which means the input work equals the output work. Real machines are never perfectly ideal because some energy is lost. Friction is one of the main causes of that loss. In a toggle vise used to compress wooden blocks, the sliding block rubs against the horizontal plane and creates friction. To find the...

conservative forces

Conservative Forces and Path-Independent Work

Conservative forces are forces whose work depends only on an object’s starting and ending positions. The path between those positions does not change the amount of work done. This idea is important in mechanical engineering because it helps describe how energy moves through a system. A conservative force also conserves energy in the sense that its work is path independent. Gravity is one example. Electrostatic forces and springs modeled by Hooke’s law are also examples. A ball rolling down a...

potential energy

Gravitational and Elastic Energy Stored

Gravitational and elastic potential energy are forms of stored energy from conservative forces. A conservative force, such as gravity or an elastic force, gives a body the capacity to do work. This capacity is measured as potential energy and depends on the body’s position relative to a fixed reference point, or datum. For gravitational potential energy, the datum is the level where the energy is taken as zero. If a body is above this reference plane, its weight has positive gravitational...

potential-energy criterion for equilibrium

Potential energy helps determine when a mechanical system is stable at equilibrium. When a system is acted on by gravitational and elastic forces, its total potential function is the algebraic sum of gravitational potential energy and elastic potential energy. If the system is already in equilibrium and is given a small displacement, the work done on the system equals the negative of the change in potential energy between the starting and ending positions. For a virtual displacement, which is...

stability of equilibrium configuration

Types of Equilibrium in Mechanical Systems

Types of equilibrium in mechanical systems are stable, neutral, and unstable. These three cases help explain how a structure or object responds when it is pushed slightly away from balance. They are important for predicting whether a system will return to its original position, stay where it is, or move farther away. Stable equilibrium occurs when a system returns toward its original position after a small displacement. In this case, the potential energy is at a minimum. A cantilever beam...

one-degree-of-freedom system

Spring-Mass Motion and Energy Stability

A one-degree-of-freedom system uses a single independent variable to describe its state and behavior. In mechanical engineering, this kind of model helps explain how parts in a larger system respond to force. It also gives engineers a way to predict motion before a design is built in real life. A simple harmonic oscillator is one example of a one-degree-of-freedom system. A mass attached to a spring moves along one axis, and that position determines the system’s state. The spring force makes...

stability of equilibrium configuration: problem solving

stability-of-equilibrium-configuration-problem-solving

Equilibrium Stability Through Potential Energy

Stability of equilibrium in physics and engineering can be studied by using a system’s potential energy. An equilibrium configuration is a state where the system can remain at rest. Stability describes whether the system tends to return to that state after a small disturbance. Problem solving begins by finding the equilibrium points of the system. This is done by setting the derivative of the potential energy function equal to zero and solving for the independent variable values that satisfy...

key terms and concepts

Learn how gravitational and elastic potential energy determine equilibrium, why the first derivative is zero, and how a positive second derivative shows stability.

Stable Equilibrium Through Potential Energy | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

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HS-PS3-1

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Stable Equilibrium Through Potential Energy

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