# Virtual Displacement

By Deepak Yadav|Updated : October 4th, 2022

Virtual Displacement is a change in coordinates that occurs instantly. Before going into specifics of the virtual displacement, let's first understand the displacement. The meaning of displacement is "change in the position of an object." It has a magnitude and a direction, making it a vector quantity. An arrow pointing from the starting to the finishing point serves as its symbol. For example, if an object moves from A to B, the object's location changes.

In mechanics, displacement refers to the distance traveled by a particle or body moving in one direction. Generally, point masses are used to describe particles and bodies. This means that, without sacrificing generality, bodies can be thought of as having all their mass concentrated in a single mathematical point. Let's now look further into the concepts of virtual displacement.

## What is Virtual Displacement?

Assume a rigid body is in equilibrium with a system of forces. The work done is 0 if there is no displacement on the body.

### Definition of Virtual Displacement

“The body in equilibrium is given an imaginary infinitesimal displacement, referred to as virtual displacement.”

### Virtual Displacement Symbol

The virtual displacement symbols for linear motion are x and y, while the virtual displacement sign for angular motion is δ. Virtual work is the resultant work performed by the forces/couples acting on the body in equilibrium due to virtual displacement.

### Virtual Displacement Formula

According to the virtual work principle, "the total work done by the system of external forces/couple acting on a rigid body in equilibrium is zero for any virtual displacement."

U = F11+ F22+ F33+….. +Fnn=0

## Virtual Displacement Principle or Principle of Virtual Work

"Total virtual work done by all the forces operating on a system in static equilibrium is zero for a collection of tiny virtual displacements from equilibrium," according to the principle of virtual work. This method was developed in 1717 by John Bernoulli. To understand the "concept of virtual work," we must be very clear about what work, virtual work, and virtual displacement are. When an external force moves an object over a distance, at least a fraction of that force must be exerted in the displacement's direction. In physics, this is known as "work."

A virtual force is any system of forces that is in equilibrium. Any displacement that is feasible without breaking any limitations is referred to as a virtual displacement of a system because it may be characterized as "an infinitesimal displacement of any mechanical system point that is compatible with the system's constraints and may or may not happen." In conjunction with this, virtual work is defined as "the work accomplished by a real force acting through a virtual displacement or a virtual force acting through a real displacement."

## Advantages of Principle of Virtual Work Method

Any tiny virtual displacement in configuration space consistent with the constraints takes no work, according to the principle of virtual work. Following are the advantages of the principle of virtual work:

• The systems do not necessarily need to be disassembled to establish connections between the active forces.
• Relationships between the active forces can be discovered directly without considering the reactive forces.
• This approach is beneficial for determining a system's equilibrium location under known loads.
• One notable benefit of the virtual work principle is that it only needs active forces to determine the system's mechanics when it experiences a virtual displacement.
 Important GATE Topics Determinate And Indeterminate Structures Astable Multivibrators Bistable Multivibrator Truss And Frame Network Layer Statically Determinate Anomalies In Dbms Eulers Equation Of Motion Dalembert's Principle Statically Indeterminate

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• The work done by a real force operating through a virtual displacement or a virtual force acting through a real displacement is referred to as virtual work. A virtual displacement is any displacement that is consistent with the structure's limitations, i.e., that meets the boundary requirements at the supports.

• It is important to remember that a virtual displacement just indicates how far we would move away from a given point on the manifold if we traveled along a chosen curve, which may not correspond to the system's actual motion as defined by time evolution.

• Moreover, a virtual displacement entails an instantaneous (time is maintained constant) change in coordinates, preserving the stress state (a real displacement would change the state of stress). So, if the system is in an equilibrium condition, all (real) forces acting on virtual displacements should have no effect.

• Virtual work is the total amount of work that a mechanical system's applied forces and inertial forces produce when it moves through a series of fictitious displacements. The concept of least action dictates that, when taking into account forces applied to a body in static equilibrium, their virtual work must be zero.

• Virtual displacement refers to a displacement that is only imagined because the body is not actually moved. Virtual work is the result of the force and the virtual displacement in the force's direction.

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