(1)
Again, for a three-dimensional system, in the notation of §§ 7, 8,
| Σ(Xδx + Yδy + Zδz) |
| = Σ{(X(λ + ηz − ζy) + Y(μ + ζx − ξx) + Z(ν + ξy − ηx)} |
| = Σ(X)·λ + Σ(Y)·μ + Σ(Z)·ν + Σ(yZ − zY)·ξ + Σ(zX − xZ)·η + Σ(xY − yX)·ζ |
| = Xλ + Yμ + Zν + Lξ + Mη + Nζ. |
(2)
This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axes. The first three terms express the work done by the components of a force (X, Y, Z) acting at O, and the remaining three terms express the work of a couple (L, M, N).
| Fig. 48. |
The work done by a wrench about a given screw, when the body twists about a second given screw, may be calculated directly as follows. In fig. 48 let R, G be the force and couple of the wrench, ε,τ the rotation and translation in the twist. Let the axes of the wrench and the twist be inclined at an angle θ, and let h be the shortest distance between them. The displacement of the point H in the figure, resolved in the direction of R, is τ cos θ − εh sin θ. The work is therefore
R (τ cos θ − εh sin θ) + G cos θ
= Rε {(p + p′) cos θ − h sin θ},
(3)
if G = pR, τ = p′ε, i.e. p, p′ are the pitches of the two screws. The factor (p + p′) cos θ − h sin θ is called the virtual coefficient of the two screws which define the types of the wrench and twist, respectively.