of forces which are statically equivalent, according to the principles
of §§ 4, 8, will (to the first order of small quantities) do the
same amount of work in any infinitely small displacement of a
rigid body to which they may be applied. It is also evident that
the total work done in two or more successive infinitely small
displacements is equal to the work done in the resultant displacement.
The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed. Let the couple consist of two forces P, P (fig. 47) in the plane of the paper, and let J be the point where this plane is met by the axis of rotation. Draw JBA perpendicular to the lines of action, and let ε be the angle of rotation. The work of the couple is
if G be the moment of the couple.
The analytical calculation of the work done by a system of forces in any infinitesimal displacement is as follows. For a two-dimensional system we have, in the notation of §§ 3, 4,
| Σ(Xδx + Yδy) | = Σ{X(λ − yε) + Y(μ + xε)} |
| = Σ(X)·λ + Σ(Y)·μ + Σ(xY − yX) ε | |
| = Xλ + Yμ + Nε. |
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. |
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
= Rε {(p + p′) cos θ − h sin θ},
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.
A screw is determined by its axis and its pitch, and therefore involves five independent elements. These may be, for instance, the five ratios ξ : η : ζ : λ : μ : ν of the six quantities which specify an infinitesimal twist about the screw. If the twist is a pure rotation, these quantities are subject to the relation
In the analytical investigations of line geometry, these six quantities, supposed subject to the relation (4), are used to specify a line, and are called the six “co-ordinates” of the line; they are of course equivalent to only four independent quantities. If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its co-ordinates are accordingly subject to the further relation
where the coefficients are constant. This is the equation of a “linear complex” (cf. § 8).
Two screws are reciprocal when a wrench about one does no work on a body which twists about the other. The condition for this is
if the screws be defined by the ratios ξ : η : ζ : λ : μ : ν and ξ′ : η′ : ζ′ : λ′ : μ′ : ν′, respectively. The theory of the screw-systems which are reciprocal to one, two, three, four given screws respectively has been investigated by Sir R. S. Ball.
Considering a rigid body in any given position, we may contemplate the whole group of infinitesimal displacements which might be given to it. If the extraneous forces are in equilibrium the total work which they would perform in any such displacement would be zero, since they reduce to a zero force and a zero couple. This is (in part) the celebrated principle of virtual velocities, now often described as the principle of virtual work, enunciated by John Bernoulli (1667–1748). The word “virtual” is used because the displacements in question are not regarded as actually taking place, the body being in fact at rest. The “velocities” referred to are the velocities of the various points of the body in any imagined motion of the body through the position in question; they obviously bear to one another the same ratios as the corresponding infinitesimal displacements. Conversely, we can show that if the virtual work of the extraneous forces be zero for every infinitesimal displacement of the body as rigid, these forces must be in equilibrium. For by giving the body (in imagination) a displacement of translation we learn that the sum of the resolved parts of the forces in any assigned direction is zero, and by giving it a displacement of pure rotation we learn that the sum of the moments about any assigned axis is zero. The same thing follows of course from the analytical expression (2) for the virtual work. If this vanishes for all values of λ, μ, ν, ξ, η, ζ we must have X, Y, Z, L, M, N = 0, which are the conditions of equilibrium.
The principle can of course be extended to any system of particles or rigid bodies, connected together in any way, provided we take into account the internal stresses, or reactions, between the various parts. Each such reaction consists of two equal and opposite forces, both of which may contribute to the equation of virtual work.
The proper significance of the principle of virtual work, and of its converse, will appear more clearly when we come to kinetics (§ 16); for the present it may be regarded merely as a compact and (for many purposes) highly convenient summary of the laws of equilibrium. Its special value lies in this, that by a suitable adjustment of the hypothetical displacements we are often enabled to eliminate unknown reactions. For example, in the case of a particle lying on a smooth curve, or on a smooth surface, if it be displaced along the curve, or on the surface, the virtual work of the normal component of the pressure may be ignored, since it is of the second order. Again, if two bodies are connected by a string or rod, and if the hypothetical displacements be adjusted so that the distance between the points of attachment is unaltered, the corresponding stress may be ignored. This is evident from fig. 45; if AB, A′B′ represent the two positions of a string, and T be the tension, the virtual work of the two forces ±T at A, B is T(Aα − Bβ), which was shown to be of the second order. Again, the normal pressure between two surfaces disappears from the equation, provided the displacements be such that one of these surfaces merely slides relatively to the other. It is evident, in the first place, that in any displacement common to the two surfaces, the work of the two equal and opposite normal pressures will cancel; moreover if, one of the surfaces being fixed, an infinitely small displacement shifts the point of contact from A to B, and if A′ be the new position of that point of the sliding body which was at A, the projection of AA′ on the normal at A is of the second order. It is to be noticed, in this case, that the tangential reaction (if any) between the two surfaces is not eliminated. Again, if the displacements be such that one curved surface rolls without sliding on another, the reaction, whether normal or tangential, at the point of contact may be ignored. For the virtual work of two equal and opposite forces will cancel in any displacement which is common to the two surfaces; whilst, if one surface be fixed, the displacement of that point of the rolling surface which was in contact with the other is of the second order. We are thus able to imagine a great variety of mechanical systems to which the principle of virtual work can be applied without any regard to
