System of Forces Transformation

Sometimes a system of forces acting on a rigid body is needed to be transformed into an equivalent system at the point of interest.

Force Couple

A couple of force F is applied at points A, B on a rigid body.

 

The equivalent system is a free couple vector M acting on any location of the rigid body.

 

Consider the moment MBA at point B,  the tendency of rotation above B is caused by the applied force F at A. The applied force -F at B can be considered as the reaction supporting force. Both the magnitude and direction of the moment vector MBA  is same as the couple vector M.

 

Force Vector

Suppose a force vector acting at point A on the rigid body is to be considered at point B.

 

When only a single force F is applied at point A on the rigid body, the moment of the force at point A about point B is also equal to  MBA .

 

To simulate the force couple condition, two forces, F and  -F with magnitude equal to the applied force F are attached at point B. After this transformation, a force couple of moment MBA is formed with an additional force -F acting at point B.

 

Force F acting at point A is now moved to point B. The formed force couple of moment MBA can be replaced by a couple vector M. The couple vector M is a free vector and it can be located at point B for convenience.

 

Therefore the equivalent system of a force vector F acting at point A on a rigid body can be transformed to an arbitrary point B by moving the force vector to pont B and adding a couple vector M with moment equals to the moment MBA of the force vector F about point B. At which, the force vector F tend to provide the same linear motion and the couple vector M tend to provide the same rotational motion as the force vector F at point A. The equivalent system of combining a force vector and a couple vector is named as force-couple system. And the couple vector is