4.7 Simplification of a Force and Couple System

A force has the effect of both translating and rotating about a body.

Remember the principle of transmissibility, which states that a force acting on a body is a sliding vector since it can be applied at any point along its line of action. Also, moments are free vectors and may affect at any point on the body.

A system of forces and couple moments may be replaced with an equivalent single resultant force and couple moments acting at a specified point O anywhere on the body.

Equivalent means same external effects of translation and rotation.

First scenario: a single force is applied at one point A and an equivalent system where the force is applied at another point O is desired. Note there are no couple moments.

1. Point 0 is on the line of action of a single force therefore moving F from point A to point O does not alter the body's tendency to rotate

2. Point O is not on the line of action of F. Therefore moving F from point A to point O changes the body's tendency to rotate. We must account for this change by adding in a couple moment.

Moving a single force F. from point A to point O.

1. draw in F at point Oa. this changes the resultant force on the body because there is no double the force acting

in the same direction2. add in negative F at point O to cancel out extra positive F force3. account for change in tendency to rotate resulting from force acting at a different point on body

by adding in a couple momenta. M=r X Fb. couple moment can be positioned anywhere because it is a free vectorc. often times will place it at point 0 because that is where the resultant force is located

Moving multiple forces to point O.

FR=∑ FiMRO=∑MOi

FR=F1+F2MRO=M 1+M 2

MRO=rO ¿A ¿× F1+rO ¿B ¿×F2

If there is an existing couple moment

FR=∑ FiMRO=∑MOi

FR=F1+F2MRO=MC1

+MC2+rO¿ A ¿×F1+rO¿B¿× F2

Procedure for Analysis when simplifying a force and couple moment system to an equivalent resultant force and couple system.

Establish the coordinate axes with the origin located at point 0 and the axess having a selected orientation.

Force summation

In two dimensions, resolve each force into the X. and Y. components In three dimensions, represent each force as a Cartesian vector before summing the forces

Moment summation

In two dimensions, use principle of moments i.e. determine the moments of the components of each force rather than the moment of the force itself.

In three dimensions, use the vector Cross product to determine the moment of each force about point O.

o Here the position vectors extend from O to any point on the line of action of each force.

4.8 Further Simplification of a Force and Couple System

Special cases: concurrent, coplanar, and parallel force systems can be further simplified.

Concurrent force system.

A concurrent force system is one in which the lines of action of all the forces intersect at a common point, thus causing no moment about this point.

As a result the equivalent system can be represented by a single resultant force equal to the sum of the forces acting at that point.

FR=∑ Fi

Coplanar force system

In a coplanar force system, the lines of action of all the forces lie in the same plane. Additionally, the moment that each of the forces about any point is directed perpendicular to this plane. Thus, the resultant moment and resultant force will be mutually perpendicular. The resultant moment can be replaced by moving the resultant force a perpendicular or moment arm distance D. away from point O such that the resultant force produces the same moment about point O.

FR=∑ FiMRO=FRd∨d=MRO

/FR

Parallel force system

FR=∑ FiMRO=FRd∨d=∑MRO

/FR