Couples: Scalar and Vector Formulation

Consider equal and parallel forces applied to rotate a ship's steering wheel. They are oppositely directed and create a couple moment that rotates the wheel in an anticlockwise direction.

The right-hand rule can be used to determine the direction of the couple moment. When the fingers are curled along the rotation direction, the thumb indicates the moment direction. It always acts perpendicular to the plane of the two forces.

In scalar formulation, the couple moment equals the force magnitude multiplied by the perpendicular distance between the forces.

In vector formulation, the couple moment of one of the forces about any point on its line of action equals zero. The moment of another force about this point equals the cross product of the position vector from the line of action of one force to the other and the applied force.

Consider two couple moments acting on a speed-reducer unit. Since each is a free vector, their resultant can be estimated by vector addition.

If there are more than two couple moments, this expression can be generalized.