Virtual work

Virtual work is a term of analytical mechanics and engineering mechanics and describes the work performed by a force on a system with a virtual displacement. Under a virtual displacement is defined as a form or position change of the system, which is compatible and " instantaneously ", but otherwise arbitrary and might also infinitesimally small with the bonds (eg warehouse). The principle of virtual work is used to calculate the equilibrium in statics and for setting up equations of motion ( d' Alembert's principle).

Description

Virtual displacement, virtual work

The following is an N- particle system is considered, which is restricted by constraints.

A virtual displacement is a fictional infinitesimal displacement of the - th particle, which is compatible with the constraints. The function of the time is not considered. [Footnotes 1]

The holonomic constraints, are met by the use of so-called generalized coordinates:

( The holonomic constraints are thus explicitly eliminated by selection and corresponding reduction in the generalized coordinates. )

To fulfill also the constraint conditions anholonomen the [Footnotes 2] subject to additional terms, such as differential non- integrable equations:

The virtual work that would do the force with virtual displacement at the -th particle is:

System is in equilibrium

If the - particle system in equilibrium, then, for each particle the acceleration equal to zero:

Therefore, the resultant force on each particle must be equal to zero:

If the system is in equilibrium, the virtual work of the force on displacement equal to zero because the force itself vanishes:

Thus, the sum of the work done by all forces at work virtual displacements equal to zero:

The resulting forces can be composed of embossed forces and constraining forces:

Substituted into the above equation:

Principle of virtual work

In most cases the constraining force is perpendicular to the virtual displacement, so that applies. This is for example the case when the movement is limited to curves or surfaces.

However, there are systems in which individual constraint forces do work.

The principle of virtual work is now demanding that the sum of all work done by the constraint forces virtual work vanishes for a system in equilibrium:

For the active forces, the principle of virtual work means:

Note that the principle of virtual work is only an equilibrium principle of statics. The extension to the dynamics provides the D' Alembert's principle.

Principle of virtual work in conservative systems

In conservative systems, all active forces are derivable from a potential:

In this case, the principle of virtual work

In the form of

Represent. Here, the symbol is to be interpreted as sign of variation in terms of variational calculus. so that means the first variation of the Potential Energy.

Example

To an angle lever which is freely rotatably mounted on an axle, access 2 applied forces F1 and F2. The virtual work of the force application points are? X 1 and δx2. The virtual work of the impressed forces is thus

Because of the angle lever is considered to be rigid, the sizes? X 1 and δx2 are not independent. Their dependence can be expressed by the variation δΦ of the generalized coordinate Φ:

This is the virtual work:

Because of the randomness of δΦ the left side of this equation may disappear only when the parenthetical expression disappears, resulting ultimately follows:

Then the system is in equilibrium, ie it tilts neither to the right nor to the left.

Principle of virtual work for dynamic systems

The virtual work of the constraint forces or moments is in dynamic systems is equal to zero. Pressing the virtual displacements in the generalized coordinates, can be set up with the principle of virtual work equations of motion for large multibody systems.