Generalized Coordinates - Generalized Coordinates and Virtual Work

Generalized Coordinates and Virtual Work

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. When formulated in terms of generalized coordinates, this is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is F_i=0.

Let the forces on the system be F_j, j=1, ..., m be applied to points with Cartesian coordinates r_j, j=1,..., m, then the virtual work generated by a virtual displacement from the equilibrium position is given by

where δr_j, j=1, ..., m denote the virtual displacements of each point in the body.

Now assume that each δr_j depends on the generalized coordinates q_i, 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 v_j is the velocity of the point of application of the force F_j.

In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is