Work of Forces Acting On A Rigid Body
Consider the set of forces F1, F2 ... Fn act on the points X1, X2 ... Xn in a rigid body. The trajectories of Xi, i=1,...,n are defined by the movement of the rigid body with rotation and the translation d(t) of a reference point in the body, given by
where xi are coordinates in the moving body.
The velocity of each point Xi is
where ω is the angular velocity vector and v is the derivative of d(t).
The work by the forces over the displacement δri=viδt of each point is given by
Define the velocities of each point in terms of the twist of the moving body to obtain
Expand this equation and collect coefficients of ω and v to obtain
Introduce the twist of the moving body and the wrench acting on it given by
then work takes the form
The 6x6 matrix is used to simply the calculation of work using screws, so that
where
and is the 3x3 identity matrix.
Read more about this topic: Screw Theory
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