Resultant of Non-coplanar Non-concurrent Forces:
As indicated earlier, there are various engineering problems which needs analysis of non-coplanar, non-concurrent system of forces that is three dimensional in nature.
Such type of analysis calls for representation of the system through a pictorial drawing or via two or more orthographic projections.
The resultant of a non-parallel, non-concurrent, non-coplanar force system may be a single force or a couple, but generally it is a force and a couple.
While all the forces of the system are parallel, the resultant shall be a single force (parallel to the given forces) or in parallel plane or a couple in the plane of the system or the resultant can be zero. The resultant is a force while ΣF is different from zero and a couple while ΣF is equal to zero unless ΣMo is also zero wherein case the system is in equilibrium and the resultant is zero.
The resultant of any general force system can be obtained by resolving each force into a parallel force through some common point and a couple in the manner defined in Sub-section. Thus the general system is decreased to a set of concurrent forces & a set of couples. The resultant of concurrent forces is attained as the vector sum of the component forces, R = ΣFi, as discussed in previous section. The couples are combined vectorially to attain a resultant couple M. The vector addition of the moment of the resultant of the set of the couples may be attained if the moment of each couple is resolved out in the rectangular components. Three rectangular components of the resultant couple are ΣMx, ΣMy, and ΣMz that are the algebraic sums of the moments of the couples of the system around x, y and z axes, respectively.
Now, the resultant couple is specified by
and the direction of the resultant couple can be specified by its direction cosines, that
cos α x = ∑ Mx/M, cos αy = ∑My /M and cos α z = ∑M z/ M
Here, the angles θx, θy and θz are the angles, the vector representing the couple M makes along with x, y and z axes, respectively.
