Static Equilibrium
Static equilibrium is the condition in which the applied forces and constraint forces on a mechanical system balance such that the system does not move. The principle of virtual work states that the virtual work of the applied forces is zero for all virtual movements of the system from static equilibrium, that is, δW=0 for any variation δr. This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is Fi=0.
Let the forces on the system be Fj, j=1, ..., m and let the virtual displacement of each point of application of these forces be δrj, j=1, ..., m, then the virtual work generated by a virtual displacement of these forces from the equilibrium position is given by
Now assume that each δrj depends on the generalized coordinates qi, i=1, ..., n, then
and
The n terms
are the generalized forces acting on the system. Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,
where vj is the velocity of the point of application of the force Fj.
In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is
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