Force-Couple system Reduction

The couple vector M or moment vector M of a force-couple system transformed from a force vector F is always prependicular to the force vector F.

 

Since the same resultant force vector on the rigid body must remain underchange, the equivalent force vector F should have the same magnitude and direction as the original force vector F. In order to create the equivalent moment effect at point B, the force F should move on a plane which passes through the force vector F or point B and is normal to the couple vector until the moment MAB of force F about B is equal to the moment of the couple vector Mc.

 

Therefore a force-couple system applying at an arbitary point B with a force vector F and a couple vector Mc  which is normal to the force vector F can be reduced to one single equivalent force vector F applying at point A on the rigid body.

 

Force-Couple system Conversion

However in general, a force-couple system usually consists force vector F and couple vector Mb which are which are not prependicular to each other. Because the force vector F is not prependicular to the couple vector Mb, the couple vector Mb cannot be replaced by the memont created by the force vector F.

 

But the force-couple system can be converted to a simpler one by couple decomposition. the couple vector Mb can be decomposed into couple vector M2 along the direction of force vector F and M3 normal to the force vector F ,on a plane which consists both the force vector F and couple vector Mb. .

 

Since couple vector M3 is normal to the force vector F, couple vector M3 can be replaced by moving the force vector F on a plane which passes through the force vector F or point B and is normal to the couple vector M3 until the moment MBD of force F about B is equal to the moment of the couple vector M3 as before.

 

Since the couple vector M2 is a free vector, it can also be moved to point D for convenience. And the equivalent system of a force-couple system is converted to a force vector  F and a couple vector  M2  along with the force vector direction named as wrench.