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

Stellen Sie sich einen Körper im statischen Gleichgewicht vor, der einer verschwindend kleinen virtuellen Verschiebung oder einer Drehung unterliegt.
Die virtuelle Arbeit, die geleistet wird, ist das Produkt der virtuellen Kraft und Verdrängung. Ähnlich wie bei einer virtuellen Drehung ist die geleistete Arbeit das Produkt aus dem Moment und der virtuellen Winkelverschiebung.
Das Prinzip der virtuellen Arbeit besagt, dass die algebraische Summe der virtuellen Arbeit, die von allen Kräften und Momenten geleistet wird, für jede virtuelle Verschiebung oder Rotation Null ist.
Wenn ein Ball eine virtuelle Abwärtsverschiebung erfährt, leistet sein Gewicht positive virtuelle Arbeit, während die Normalkraft negative virtuelle Arbeit leistet.
Für Gleichgewichtsbedingungen muss die Summe aller virtuell geleisteten Arbeiten Null sein.
Ähnlich verhält es sich, wenn ein gelagerter Balken eine virtuelle Drehung durchläuft, nur zwei Kräfte die Arbeit erledigen.
Die Komponenten der Eingreiftruppe an der Unterstützung tragen nicht zur virtuellen Arbeit bei.
Unter Berücksichtigung der virtuellen Verschiebungen entlang der y-Richtung und unter Anwendung des Prinzips der virtuellen Arbeit wird die Gleichung für die virtuelle Arbeit abgeleitet.
Der Term in den Klammern gibt den Zustand des Rotationsgleichgewichts an.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik: Statik. Vierzehnte Auflage, New Jersey: Pearson. Abschnitt 11.1, S. 583-584.</li>
	<li>Meriam, J.L., Kraige, L.G. und Bolton, J.N. (2016). Technische Mechanik: Statik und Dynamik. Achte Auflage, Singapur: John Wiley & Sons. Abschnitt 7.2, S. 394-395.</li>
	<li>Bier, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. und Self, B.P. (2016). Vektormechanik für Ingenieure. Elfte Auflage, New York: McGraw-Hill Education. S. 577-578.</li>
</ol>

virtuelle arbeit

Das Prinzip der virtuellen Arbeit besagt, dass die Summe der virtuellen Arbeit aller externen Kräfte und Momente für jede gegebene virtuelle Verschiebung null ist, wenn ein Körper im statischen und dynamischen Gleichgewicht ist. Im statischen Gleichgewicht kann ein Körper eine imaginäre oder virtuelle Bewegung, wie Verschiebung oder Rotation, erfahren. Die virtuelle Arbeit, die von einer Kraft verrichtet wird, entspricht dem Skalarprodukt von Kraft und virtueller Verschiebung in Richtung der Kraft. Wenn es um die virtuelle Rotation eines Momentenpaares geht, gilt dasselbe Prinzip. Die virtuelle Rotationsarbeit wird bestimmt durch das Multiplizieren des Momentenpaares mit seiner entsprechenden virtuellen Rotation. Um dieses Konzept anhand eines Beispiels zu veranschaulichen, nehmen wir an, es gibt einen Ball, der auf einer flachen Oberfläche liegt. Das Zeichnen seines Vollkörperdiagramms wird zeigen, dass bei einer nach unten gerichteten virtuellen Verschiebung die Gewichtskraft positive virtuelle Arbeit verrichtet, während die Normalkraft negative virtuelle Arbeit verrichtet. Um das Gleichgewicht zu erreichen, muss die Summe all dieser Kräfte null sein, und somit kann entsprechend eine Gleichung formuliert werden, die diesen Zustand darstellt. Das Konzept der virtuellen Arbeit wird verwendet, um Probleme sowohl mit Partikeln als auch mit starren Körpern zu lösen. Wenn es um starre Körper unter Einwirkung von koplanaren Kräften geht, sind drei separate Gleichungen erforderlich, da sie sich auf verschiedene Arten von Verschiebungen— beziehen Translation in den x- und y-Richtungen und Rotation um eine Achse senkrecht zur x-y-Ebene. Zusammenfassend ist virtuelle Arbeit ein grundlegendes Konzept in der Mechanik, das Ingenieuren und Wissenschaftlern ermöglicht, das Verhalten von Strukturen und Maschinen ohne deren physische Prüfung vorherzusagen. Es ist ein leistungsstolles Werkzeug, das verwendet werden kann, um das Verhalten von statischen und dynamischen Systemen zu analysieren, und es hat vielfältige Anwendungen in der Ingenieur- und Naturwissenschaft.

Virtuelle Arbeit

Das Prinzip der virtuellen Arbeit besagt, dass, wenn sich ein Körper im statischen und dynamischen Gleichgewicht befindet, die Summe aller virtuellen Arbeit, die von allen äußeren Kräften geleistet wird, ...

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Mechanical Engineering

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Unit 1

Einheit 1

Virtual Work

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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

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Work in Force and Motion

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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...

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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

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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,...

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Conservative Forces and Path-Independent Work

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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...

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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...

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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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