virtual displacement

The terms F are called the generalized forces associated with the virtual displacement δr.

Ideal constraints: those for which the work done by the constraint forces under a virtual displacement vanishes.

Since the virtual displacements are arbitrary, the preceding equality reduces to:

The work of a force on a particle along a virtual displacement is known as the virtual work.

The virtual work equation then becomes the principle of virtual displacements:

The equality between external and internal virtual work (due to virtual displacements) is:

A virtual displacement "is an assumed infinitesimal change of system coordinates occurring while time is held constant.

In modern terminology virtual displacement is a tangent vector to the manifold representing the constraints at a fixed time.

There are many forces in a mechanical system that do no work during a virtual displacement, which means that they need not be considered in this analysis.

Virtual displacements be zero on the part of the surface that has prescribed displacements, and thus the work done by the reactions is zero.

The only possible virtual displacement would be a displacement from the bead's position, to a new position (where could be positive or negative).

It is also worthwhile to note that virtual displacements are spatial displacements exclusively - time is fixed while they occur.

For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify:

In the application of the principle of virtual work it is often convenient to obtain virtual displacements from the velocities of the system.

In order to impose equilibrium on real stresses and forces, we use consistent virtual displacements and strains in the virtual work equation.

The principle of virtual displacements for the structural system expresses the mathematical identity of external and internal virtual work:

Virtual displacements consistent with virtual nodal displacements:

Mathematically the virtual work done δW on a particle of mass m through a virtual displacement δr (consistent with the constraints) is:

Since virtual displacements are automatically compatible when they are expressed in terms of continuous, single-valued functions, we often mention only the need for consistency between strains and displacements.

In analytical mechanics the concept of a virtual displacement, related to the concept of virtual work, is meaningful only when discussing a physical system subject to constraints on its motion.

Virtual work can now be described as the work done by the applied forces and the inertial forces of a mechanical system as it moves through a set of virtual displacements.

Lanczos presents this as the postulate: "The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints."

The static equilibrium of a mechanical system rigid bodies is defined by the condition that the virtual work of the applied forces is zero for any virtual displacement of the system.

An important benefit of the principle of virtual work is that only forces that do work as the system moves through a virtual displacement are needed to determine the mechanics of the system.

Among all of the possible displacements that a particle may follow, called virtual displacements, one will minimize the action, and, therefore, is the one followed by the particle by the principle of least action.