Work of Forces Acting On A Rigid Body
The work of forces acting at various points on a single rigid body can be calculated from the work of a resultant force and torque. To see this, let the 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. This movement is given by the set of rotations and the trajectory d(t) of a reference point in the body. Let the coordinates xi i=1,...,n define these points in the moving rigid body's reference frame M, so that the trajectories traced in the fixed frame F are given by
The velocity of the points Xi along their trajectories are
where ω is the angular velocity vector obtained from the skew symmetric matrix
known as the angular velocity matrix.
The small amount of work by the forces over the small displacements δri can be determined by approximating the displacement by δr=vδt so
or
This formula can be rewritten to obtain
where F and T are the resultant force and torque applied at the reference point d of the moving frame M in the rigid body.
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