Reduction of a force system to its simplest form: principal vector, principal moment, wrench
Enter forces with their points of application and force couples. The calculator finds the principal vector and the principal moment about the centre of reduction, the invariants, decides the simplest form of the system (a resultant, a couple or a wrench) and the central axis.
Reduction of an arbitrary force system
Any system of forces can be replaced by an equivalent one: a single force equal to the principal vector R = ΣFᵢ applied at a chosen centre O, and a couple whose moment is the principal moment M_O = Σ(rᵢ × Fᵢ) + ΣMⱼ. The principal vector does not depend on the centre; when the centre moves from O to O', the principal moment changes by (O − O') × R.
Invariants
Two numbers do not change with the centre: I₁ = R² and I₂ = R·M_O. Therefore the projection of the principal moment on the direction of R is the same for all centres, and the smallest principal moment is M_min = (I₂/R²)·R.
Possible simplest forms
If R = 0 and M_O = 0 the system is in equilibrium. If R = 0 and M_O ≠ 0 it is equivalent to a couple. For R ≠ 0 and I₂ = 0 (always true for plane systems, concurrent and parallel forces) the system reduces to a resultant. For R ≠ 0 and I₂ ≠ 0 it cannot be simplified further: it is a wrench, a force R plus a couple parallel to R. The ratio p = I₂/R² is called the pitch of the wrench.
Central axis
The line on which the principal moment is parallel to R is the central axis. It passes through r₀ = (R × M_O)/R² parallel to R. For a resultant it is its line of action; for a wrench it is the axis along which the force and the couple act.
Example
A force of 5 along the x axis at the origin and a force of 5 along the y axis at the point (0; 0; 3). The principal vector is R = (5; 5; 0), M_O = (−15; 0; 0), I₂ = −75, the pitch is p = −1.5. The central axis passes through (0; 0; 1.5) parallel to R, and the moment on it is M_min = (−7.5; −7.5; 0). Check: r₀ × R + M_min = (−7.5; 7.5; 0) + (−7.5; −7.5; 0) = (−15; 0; 0) = M_O.
FAQ
What are the principal vector and the principal moment?
The principal vector R is the vector sum of all the forces and is the same for any centre of reduction. The principal moment M_O is the sum of the moments of the forces about the centre O plus the moments of the couples. When the centre changes it changes by R × (O − O'), so it is always quoted with its centre.
When does a system reduce to a resultant?
When R ≠ 0 and R·M_O = 0, that is, the principal moment is perpendicular to the principal vector. For a plane force system, concurrent forces and parallel forces this always holds, so there is never a wrench there.
What is a wrench (dynamic screw)?
It is a force R together with a couple whose plane is perpendicular to R. Such a system cannot be replaced by a single force. The pitch is p = I₂/R², the moment of the couple is M_min = p·R, and the sign of p tells a right-handed screw from a left-handed one.
How to find the equation of the central axis?
The point of the axis nearest to the centre O is r₀ = (R × M_O)/R², the direction is that of R. The parametric equation is r = r₀ + λ·R/∣R∣. For a plane system the equation of the line of action of the resultant, Ry·x − Rx·y = M_Oz, is more convenient.
How to enter a force couple?
By its moment vector, perpendicular to the plane of the couple: the direction follows the right-hand rule, the magnitude is force times arm. In a plane problem Mz is enough, positive for counter-clockwise rotation. A couple can also be entered as two equal opposite forces at different points.