# Principle of Virtual Work: Problem Solving

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Mechanical Engineering
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JoVE Core Mechanical Engineering
Principle of Virtual Work: Problem Solving

### Nächstes Video11.6: Mechanical Efficiency of Real Machines

Consider a scissors linkage system connected to a spring, which is unstretched at a certain angle.

In the presence of an external force, the spring stretches and the system undergoes a virtual displacement. In this condition, only the spring force and the external force contribute to work.

To determine the deflection angle to maintain the system's equilibrium, the position coordinates of the two forces with respect to a fixed point are determined.

Then, the amount by which the spring is compressed and the corresponding spring force are evaluated.

Differentiating position coordinates gives the virtual displacements corresponding to the two forces.

The external force acts in the direction of virtual displacement, so it does positive work. Meanwhile, the spring force acts opposite to the virtual displacement and does negative work.

Next, the principle of virtual work is used to write down the virtual-work equation for the system.

Solving the virtual-work equation for the given force, the angle needed to maintain the equilibrium of the scissors linkage system can be obtained.

## Principle of Virtual Work: Problem Solving

The principle of virtual work is an essential concept in the field of mechanics and engineering. This is used to solve problems related to the equilibrium of a structure or system. It is based on the assumption that if a system is in equilibrium, the work done by all the forces during a virtual displacement is zero. This principle is applied by considering virtual displacements of the system and the corresponding work done by internal and external forces.

To apply the principle of virtual work, it is necessary to express the position coordinates of the internal and external forces in terms of a common coordinate parameter. This parameter is typically an angle, distance, or displacement that is relevant to the problem at hand.

Once the coordinates have been expressed, the virtual displacements corresponding to each of the forces are found by differentiating the coordinate expressions. The virtual displacements represent hypothetical displacements that the system could undergo, and they allow us to calculate the virtual work done by each force acting on the system.

The external force acting on the system does positive work when it is moving in the direction of its virtual displacement, while the internal forces do negative work when they are opposing their virtual displacements.

To solve a problem using the principle of virtual work, first, identify the unknown forces and moments that exist within the system. Then evaluate the work done by these forces and moments during a hypothetical virtual displacement of the system. The virtual displacement is very small and represents a hypothetical displacement that the system undergoes. This allows the determination of each force and moment acting on the system and the work done by them during this virtual displacement.

Next, write down the virtual-work equation for the system using the principle of virtual work. This equation considers the virtual displacements and corresponding work done by each force and moment acting on the system. Finally, solving the virtual work equation gives the values of the unknown forces and moments in the system.

The principle of virtual work is beneficial for solving problems related to complex systems involving many interconnected parts. It is widely used in engineering and mechanics to calculate a structure or system's unknown forces and moments.