Name: virtual work in force equilibrium
Uploaded: 2023-09-22T11:49:36.000Z
Duration: 1 min 20 s
Channel: JoVE

静止平衡状態にある物体が、極小の小さな仮想変位または回転を受けているとします。
行われる仮想作業は、仮想の力と変位の産物です。同様に、仮想回転の場合、行われる作業はモーメントと仮想角度変位の積です。
仮想作業の原理は、すべての力とモーメントによって行われる仮想作業の代数的合計は、仮想変位または回転に対してゼロであると述べています。
ボールが仮想的な下向きの変位を受けると、その重量は正の仮想的な働きをし、垂直な力は負の仮想的な働きをします。
平衡条件の場合、行われたすべての仮想作業の合計はゼロでなければなりません。
同様に、支持されたビームが仮想回転を受けると、2つの力のみが機能します。
サポートの反力のコンポーネントは、仮想作業には寄与しません。
y方向に沿った仮想変位を考慮し、仮想仕事の原理を適用することにより、仮想仕事方程式が導出されます。
括弧内の項は、回転平衡の状態を示します。

<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。セクション11.1、583-584ページ。</li>
	<li>メリアム、JL、クレイジ、LGおよびボルトン、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。セクション7.2、394-395ページ。</li>
	<li>ビール、FP、ジョンストン、ER、マズレック、DF、コーンウェル、PJおよびセルフ、BP(2016)。エンジニアのためのベクトル力学。第11版、ニューヨーク:マグロウヒル教育。577-578ページ。</li>
</ol>

仮想作業

仮想作業の原理は、物体が静的および動的な平衡状態にある場合、与えられた仮想変位に対して外力およびカップルモーメントによって行われる仮想作業の総和はゼロでなければならないと述べています。

静的平衡では、物体は仮想的な動き、例えば変位や回転を経験することができます。力によって行われる仮想作業は、力と仮想変位の力方向におけるドット積と等しいです。カップルモーメントを仮想的に回転させる場合も、同じ原理が適用されます。仮想的な回転作業は、カップルモーメントとそれに対応する仮想回転の積によって決定されます。

この概念を例として説明するために、平らな表面の上に球が乗っているとしましょう。全体の体図を描くと、下方向の仮想変位がある場合、重量は正の仮想作業を行い、法線力は負の仮想作業を行います。平衡を達成するために、これらの力の総和はゼロでなければならず、それに応じた条件を表す方程式が構築されます。

仮想作業の概念は、粒子と剛体に関連する問題を解決するために使用されます。コプレナーな力にさらされた剛体を扱う場合、異なるタイプの変位- xおよびy方向への変位とx-y平面に垂直な軸周りの回転-に関連する3つの別々の方程式が必要です。

結論として、仮想作業は、構造や機械の振る舞いを物理的なテストなしで予測することを許す力学の基本的な概念です。これは、静的および動的なシステムの振る舞いを分析するために使用される強力なツールであり、エンジニアリングや科学に広範な応用があります。

仮想作業の原理は、物体が静的および動的平衡状態にある場合、すべての外部力によって行われるすべての仮想作業の合計が...

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

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 multiplying the moment by the virtual rotation.

A simple example is a ball on a flat surface. If the ball has a downward virtual displacement, its weight does positive virtual work. The normal force does negative virtual work. The equilibrium condition then gives an equation in which the sum of these virtual works is zero.

Virtual work can be used for both particles and rigid bodies. For rigid bodies under coplanar forces, three equations are needed. They correspond to translation in the x direction, translation in the y direction, and rotation about an axis perpendicular to the x-y plane. Engineers and scientists use this method to study structures and machines without physically testing them.

Virtual Work in Force Equilibrium

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 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-criterion-for-equilibrium

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...

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 virtual work helps test equilibrium using imagined displacement, rotation, and force balance in particles and rigid bodies.

Virtual Work in Force Equilibrium | 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。

mechanical engineering

Virtual Work in Force Equilibrium

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