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

Рассмотрим тело в статическом равновесии, претерпевающее бесконечно малое виртуальное смещение или вращение.
Выполняемая виртуальная работа является продуктом виртуальной силы и перемещения. Точно так же и при виртуальном вращении выполняемая работа является произведением момента и виртуального углового перемещения.
Принцип виртуальной работы гласит, что алгебраическая сумма виртуальной работы, выполненной всеми силами и моментами, равна нулю для любого виртуального смещения или вращения.
Когда мяч претерпевает виртуальное смещение вниз, его вес выполняет положительную виртуальную работу, в то время как нормальная сила выполняет отрицательную виртуальную работу.
Для условий равновесия сумма всей выполненной виртуальной работы должна быть равна нулю.
Точно так же, когда опорная балка подвергается виртуальному вращению, работу выполняют только две силы.
Компоненты силы реакции на опоре не способствуют виртуальной работе.
Рассматривая виртуальные смещения вдоль y-направления и применяя принцип виртуальной работы, выводится уравнение виртуальной работы.
Член в скобках указывает на состояние вращательного равновесия.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика: Статика. Четырнадцатое издание, Нью-Джерси: Пирсон. Раздел 11.1, стр. 583-584.</li>
	<li>Мериам, Д.Л., Крейдж, Л.Г. и Болтон, Д.Н. (2016). Инженерная механика: статика и динамика. Восьмое издание, Сингапур: John Wiley & Sons. Раздел 7.2 С. 394-395.</li>
	<li>Бир, Ф.., Джонстон, Э.Р., Мазурек, Д.Ф., Корнуэлл,.Д. и Селф, Б.. (2016). Векторная механика для инженеров. Одиннадцатое издание, Нью-Йорк: McGraw-Hill Education. С. 577-578.</li>
</ol>

Виртуальная работа

Принцип виртуальной работы утверждает, что если тело находится в статическом и динамическом равновесии, то сумма всех виртуальных работ, выполняемых всеми внешними силами и моментами сил для любого заданного виртуального перемещения, должна быть равна нулю.

В статическом равновесии тело может испытывать воображаемое или виртуальное движение, такое как перемещение или вращение. Виртуальная работа, выполненная силой, равна скалярному произведению силы и виртуального перемещения в направлении силы. Когда речь идет о виртуальном вращении момента силы, применяется тот же принцип. Виртуальная работа, выполненная при виртуальном вращении, определяется умножением момента силы на соответствующее виртуальное вращение.

Чтобы проиллюстрировать эту концепцию на примере, предположим, что на верхушке плоской поверхности находится мяч. Изображение его полного тела покажет, что при внизовиртуальном перемещении вес будет выполнять положительную виртуальную работу, в то время как нормальная сила будет выполнять отрицательную виртуальную работу. Для достижения равновесия сумма всех этих сил должна быть равна нулю, и поэтому можно составить соответствующее уравнение, представляющее эту условность.

Концепция виртуальной работы используется для решения проблем, связанных как с частицами, так и с жесткими телами. При работе с жесткими телами, подвергающимися копланарным силам, требуются три раздельных уравнения, так как они относятся к разным типам перемещений - трансляции в направлениях x и y и вращения вокруг оси, перпендикулярной плоскости x-y.

В заключение, виртуальная работа является фундаментальной концепцией в механике, которая позволяет инженерам и ученым предсказывать поведение структур и машин без физического тестирования. Это мощный инструмент, который может быть использован для анализа поведения статических и динамических систем, и у него широкий спектр применений в инженерии и науке.

Принцип виртуальной работы гласит, что если тело находится в статическом и динамическом равновесии, то сумма всей виртуальной работы, выполненной всеми внешними силами...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Virtual Work

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Virtual Work in Force Equilibrium

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