VIRTUAL WORK

Virtual work is a concept in dynamics that is necessary for understanding some energy methods for deriving equations of motion such as Lagrange's method and Hamilton’s principle. It can also be used in statics to derive equilibrium equations for systems.

Consider a particle with mass m_i whose position at some instant in time is defined by the vector σ_i. Now consider an infinitesimal change in the particle’s position, consistent with the constraints of the system, denoted by δσ_i. This infinitesimal change in position is not real; that is, the particle does not actually move according to this change, so it is called a virtual displacement. These displacements are imagined constructs for the purpose of representing the system in a slightly different state, so they do not require any time to develop or change the real forces acting on the system.

Now, assume each of the particles in the system is acted upon by some outside force F_i. The virtual work done by the force F_i is defined as

where the line over δW_i indicates that this quantity is not necessarily the variation of any functional W_i, but is instead the infinitesimal work done by the force F_i to effect the virtual displacement σ_i. As with the virtual displacements, the virtual work is not real work done on the system, and it does not require any change in time to be performed. In many cases, it is more useful to define the total virtual work instead of that acting on a particular particle, or

where n is the number of particles in the system. In energy methods, the concept of virtual work is used to determine the generalized forces acting on the system.

For a system at equilibrium in which constraint forces are normal to the virtual displacements, the principle of virtual work dictates that the virtual work performed by applied forces external to the system is zero, or

where F_i is defined as an external force applied to particle i. The principle of virtual work can be used to determine the equilibrium equation for a system.

For more details on virtual work and the principle of virtual work, see Meirovitch [1].

[1] Leonard Meirovitch, Principles and Techniques of Vibrations, 1997: Prentice Hall.