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Example of ment. It is not generally however so "well-conditioned” as
the trihedral hole, the V groove, and the horizontal plane contact, described above.
For investigation of the pressures on the contact surfaces of a geometrical slide or a geometrical clamp, see § 551, below.
There is much room for improvement by the introduction of geometrical slides and geometrical clamps, in the mechanism of mathematical, optical, geodetic, and astronomical instruments: which as made at present are remarkable for disregard of geometrical and dynamical principles in their slides, micrometer screws, and clamps. Good workmanship cannot compensate for bad design, whether in the safety-valve of an ironclad, or the movements and adjustments of a theodolite.
199. If one point be constrained to remain in a curve, there remain four degrees of freedom.
If two points be constrained to remain in given curves, there are four degrees of constraint, and we have left two degrees of freedom. One of these may be regarded as being a simple rotation about the line joining the constrained points, a motion which, it is clear, the body is free to receive. It may be shown that the other possible motion is of the most general character for one degree of freedom; that is to say, translation and rotation in any fixed proportions as of the nut of a screw.
If one line of a rigid system be constrained to remain parallel to itself, as, for instance, if the body be a three-legged stool standing on a perfectly smooth board fixed to a common window, sliding in its frame with perfect freedom, there remain three translations and one rotation.
But we need not further pursue this subject, as the number of combinations that might be considered is endless; and those already given suffice to show how simple is the determination of the degrees of freedom or constraint in any case that
may present itself. One degree 200. One degree of constraint, of the most general character,
is not producible by constraining one point of the body to a
curve surface; but it consists in stopping one line of the body character.
from longitudinal motion, except accompanied by rotation round this line, in fixed proportion to the longitudinal motion, and
of constraint of the most general
leaving unimpeded every other motion : that is to say, free rotation about any axis perpendicular to this line (two degrees of freedom); and translation in any direction perpendicular to the same line (two degrees of freedom). These four, with the one degree of freedom to screw, constitute the five degrees of freedom, which, with one degree of constraint, make up the six elements. Remark that it is only in case (6) below (S 201) that there is any point of the body which cannot move in every direction.
201. Let a screw be cut on one shaft, A, of a Hooke's joint, and Mechanical. let the other shaft, L, be joined to a fixed shaft, B, by a second Hooke's joint. A nut, N, turning on A, has the most general kind of motion admitted by one degree of constraint; or it is subjected to just one degree of constraint of the most general character. It has five degrees of freedom; for it may move, 1st, by screwing on A, the two Hooke's joints being at rest; 2d, it may rotate about either axis of the first Hooke's joint, or any axis in their plane (two more degrees of freedom: being freedom to rotate about two axes through one point); 3d, it may, by the two Hooke's joints, each bending, have irrotational translation in any direction perpendicular to the link, L, which connects the joints (two more degrees of freedom). But it cannot have a translation parallel to the line of the shafts and link without a definite proportion of rotation round this line; nor can it have rotation round this line without a definite proportion of translation parallel to it. The same statements apply to the motion of B if N is held fixed; but it is now a fixed axis, not as before a moveable one round which the screwing takes place.
No simpler mechanism can be easily imagined for producing one degree of constraint, of the most general kind.
Particular case (a).—Step of screw infinite (straight rifling), i.e., the nut may slide freely, but cannot turn. Thus the one degree of constraint is, that there shall be no rotation about a certain axis, a fixed axis if we take the case of N fixed and B moveable. This is the kind and degree of freedom enjoyed by the outer ring of a gyroscope with its fly-wheel revolving infinitely fast. The outer ring, supposed taken off its stand, and held in the hand, cannot revolve about an axis perpen
Mechanical dicular to the plane of the inner ring*, but it may revolve
freely about either of two axes at right angles to this, namely,
In this case the nut may run round freely, but cannot move along the axis of the shaft. Hence the constraint is simply that the body can have no translation parallel to the line of shafts, but may have every other motion. This is the same as if any point of the body in this line were held to a fixed surface. This constraint may be produced less frictionally by not using a guiding surface, but the link and second Hooke's joint of the present arrangement, the first Hooke's joint being removed, and by pivoting one point of the body in a cup on the end of the link. Otherwise, let the end of the link be a continuous surface, and let a continuous surface of the body press on it,
rolling or spinning when required, but not permitted to slide. One degree A single degree of constraint is expressed by a single equation
among the six co-ordinates specifying the position of one rigid expressed Analyti.
body, relatively to another considered fixed. The effect of this cally.
on the body in any particular position is to prevent it from getting out of this position, except by means of component velocities (or infinitely small motions) fulfilling a certain linear equation among themselves.
Thus if w;w, wy, w, W., We, be the six co-ordinates, and F(@, ... ... ...) = 0 the condition; then dF
0 dw, is the linear equation which guides the motion through any particular position, the special values of w, a, a, etc., for the
dF dF particular position, being used in
dw, dw,' Now, whatever may be the co-ordinate system adopted, we may, if we please, reduce this equation to one between three velocities of translation u, v, w, and three angular velocities o, p, o.
* “ The plane of the inner ring" is the plane of the axis of the fly-wheel and of the axis of the inner ring by which it is pivoted on the outer ring.
of constraint expressed
ordinates are $, n, S.
Let this equation be
One degree Au + Bv + Cw + A'W + B'p + C'o = 0.
analytiThis is equivalent to the following :
A, B, C,
A', B, C",
A" + B’? + C"2
A + B? + C2
The angular velocities about the new axes,
W – PĚ + on=w'.
Au' + B6 + Cw' + (A' + BS-Cn) + etc. = 0.
, disappear, because this would require three conditions,
A(B-Cn) + B (CĚ - Af)+C (An- BĚ) = 0.
lu' + mo' + nu' + a (la + mp + no) = 0,
accompanied by a definite proportion of rotation about it.
Gencralized of either of the ordinary systems, rectangular or polar, required
to specify the position of a point, correspond to the three Of a point. degrees of freedom enjoyed by an unconstrained point. The
most general system of co-ordinates of a point consists of three sets of surfaces, on one of each of which it lies. When one of these surfaces only is given, the point may be anywhere on it, or, in the language we have been using above, it enjoys two degrees of freedom. If a second and a third surface, on each of which also it must lie, it has, as we have seen, no freedom left : in other words, its position is completely specified, being the point in which the three surfaces meet.: The analytical ambiguities, and their interpretation, in cases in which the specifying surfaces meet in more than one point, need not occupy us here.
To express this analytically, let y=, $=B, O=y, where 4, 0, 0 are functions of the position of the point, and a, b, a constants, be the equations of the three sets of surfaces, different values of each constant giving the different surfaces of the corresponding set. Any one value, for instance, of a, will determine one surface of the first set, and so for the others; and three particular values of the three constants specify a particular point, P, being the intersection of the three surfaces which they determine. Thus a, b, y are the “co-ordinates" of P; which may be referred to as “the point (a, b, y).” The form of the co-ordinate surfaces of the (4, 0, 0) system is defined in terms of co-ordinates (x, y, z) on any other system, plane rectangular co-ordinates for instance, if y, 0, 0 are given each as a function of (x, y, z).
203. Component velocities of a moving point, parallel to the three axes of co-ordinates of the ordinary plane rectangular system, are, as we have seen, the rates of augmentation of the corresponding co-ordinates. These, according to the Newtonian fluxional notation, are written x, y, ż; or, according
dx dy dz to Leibnitz's notation, which we have used above,
dt' dt' dt' Lagrange has combined the two notations with admirable skill and taste in the first edition* of his Mécanique Analytique, as we shall
* In later editions the Newtonian notation is very unhappily altered by the
Origin of the differential calculus.