surface, prescribed; of freedom. Surface on In this case, then, there is but one independent variable-the both traces space passed over by the point of contact: and when the velocity one degree of the point of contact is given, the resultant angular velocity, and the direction of the instantaneous axis of the rolling body are determinate. We have thus a sufficiently clear view of the general character of the motion in question, but it is right that we consider it more closely, as it introduces us very naturally to an important question, the measurement of the twist of a rod, wire, or narrow plate, a quantity wholly distinct from the tortuosity of its axis (§ 7). Twist. Axis and transverse, 118. Suppose all of each surface cut away except an infinitely narrow strip, including the trace of the rolling. Then we have the rolling of one of these strips upon the other, each having at every point a definite curvature, tortuosity, and twist. 119. Suppose a flat bar of small section to have been bent (the requisite amount of stretching and contraction of its edges being admissible) so that its axis assumes the form of any plane or tortuous curve. If it be unbent without twisting, i.e., if the curvature of each element of the bar be removed by bending it through the requisite angle in the osculating plane, and it be found untwisted when thus rendered straight, it had no twist in its original form. This case is, of course, included in the general theory of twist, which is the subject of the following sections. 120. A bent or straight rod of circular or any other form of section being given, a line through the centres, or any other chosen points of its sections, may be called its axis. Mark a line on its side all along its length, such that it shall be a straight line parallel to the axis when the rod is unbent and untwisted. A line drawn from any point of the axis perpendicular to this side line of reference, is called the transverse of the rod at this point. straight rod is the angle The average twist is the The twist at any point The whole twist of any length of a between the transverses of its ends. integral twist divided by the length. is the average twist in an infinitely short length through this point; in other words, it is the rate of rotation of its transverse unit of length along it. per The twist of a curved, plane or tortuous, rod at any point is Twist. the rate of component rotation of its transverse round its tangent line, per unit of length along it. If t be the twist at any point, ftds over any length is the integral twist in this length. 121. Integral twist in a curved rod, although readily defined, as above, in the language of the integral calculus, cannot be exhibited as the angle between any two lines readily constructible. The following considerations show how it is to be reckoned, and lead to a geometrical construction exhibiting it in a spherical diagram, for a rod bent and twisted in any manner : of integral curve; 122. If the axis of the rod forms a plane curve lying in one Estimation plane, the integral twist is clearly the difference between the twist: inclinations of the transverse at its ends to its plane. For In a plane if it be simply unbent, without altering the twist in any part, the inclination of each transverse to the plane in which its. curvature lay will remain unchanged; and as the axis of the rod now has become a straight line in this plane, the mutual inclination of the transverses at any two points of it has become equal to the difference of their inclinations to the plane. 123. No simple application of this rule can be made to a tortuous curve, in consequence of the change of the plane of curvature from point to point along it; but, instead, we may proceed thus: Δ' D consisting portions in planes. First, Let us suppose the plane of curvature of the axis of In a curve the wire to remain constant through finite portions of the curve, of plane and to change abruptly by finite angles from one such portion different to the next (a supposition which involves no angular points, that is to say, no infinite curvature, in E the curve). Let planes parallel to the planes of curvature of three successive portions, PQ, QR, RS (not shown in the diagram), cut a spherical surface in the great circles GAG', ACA', CE. The radii of the sphere parallel to the tangents at the points Q and R of the curve where its curvature changes will cut its surface in A and C, the intersections of these circles. -H G B of integral twist: in a Let G be the point in which the radius of the sphere parallel to the tangent at P cuts the surface; and let GH, AB, CD (lines Estimation necessarily in tangent planes to the spherical surface), be parallels to the transverses of the bar drawn from the points P, Q, R of its axis. Then (§ 122) the twist from P to Q is equal to the difference of the angles HGA and BAG'; and the twist from Q to R is equal to the difference between BAC and DCA'. Hence the whole twist from P to R is equal to curve con sisting of plane portions in different planes. In a continuously tortuous curve. HGA BAG' + BAC-DCA', or, which is the same thing, A'CE+G'AC- (DCE— HGA). Continuing thus through any length of rod, made up of portions curved in different planes, we infer that the integral twist between any two points of it is equal to the sum of the exterior angles in the spherical diagram, wanting the excess of the inclination of the transverse at the second point to the plane of curvature at the second point above the inclination at the first point to the plane of curvature at the first point. The sum of those exterior angles is what is defined below as the "change of direction in the spherical surface" from the first to the last side of the polygon of great circles. When the polygon is closed, and the sum includes all its exterior angles, it is (§ 134) equal to 2π wanting the area enclosed if the radius of the spherical surface be unity. The construction we have made obviously holds in the limiting case, when the lengths of the plane portions are infinitely small, and is therefore applicable to a wire forming a tortuous curve with continuously varying plane of curvature, for which it gives the following conclusion :— Let a point move uniformly along the axis of the bar: and, parallel to the tangent at every instant, draw a radius of a sphere cutting the spherical surface in a curve, the hodograph of the moving point. From points of this hodograph draw parallels to the transverses of the corresponding points of the bar. The excess of the change of direction (§ 135) from any point to another of the hodograph, above the increase of its inclination to the transverse, is equal to the twist in the corresponding part of the bar. of integral continu tortuous The annexed diagram, showing the hodograph and the Estimation parallels to the transverses, illustrates this rule. Thus, for in- twist: in a stance, the excess of the change of direction in the spherical ously surface along the hodograph from A to C, above DCS-BAT, curve. is equal to the twist in the bar between the points of it to which A and C correspond. Or, again, if we consider a portion of the bar from any point of it, to another point at which the tangent to its axis is parallel to the tangent at its first point, we shall have D B Ti H G F a closed curve as the spherical hodograph; and if A be the point of the hodograph corresponding to them, and AB and AB the parallels to the transverses, the whole twist in the included part of the bar will be equal to the change of direction all round the hodograph, wanting the excess of the exterior angle B'AT above the angle BAT; that is to say, the whole twist will be equal to the excess of the angle BAB' above the area enclosed by the hodograph. The principles of twist thus developed are of vital importance in the theory of rope-making, especially the construction and the dynamics of wire ropes and submarine cables, elastic bars, and spiral springs. of twist in For example: take a piece of steel pianoforte-wire carefully Dynamics straightened, so that when free from stress it is straight: bend kinks. it into a circle and join the ends securely so that there can be no turning of one relatively to the other. Do this first without torsion: then twist the ring into a figure of 8, and tie the two parts together at the crossing. The area of the spherical hodograph is zero, and therefore there is one full turn (27) of twist; which (§ 600 below) is uniformly distributed throughout the length of the wire. The form of the wire, (which is not in a plane,) will be investigated in § 610. Meantime we can see that the torsional couples" in the normal sections farthest from the crossing give rise to forces by which the tie at the crossing is pulled in opposite directions perpendicular to the plane of the crossing. Thus if the tie is cut the wire springs back into the circular form. Now do the same thing again, 7 VOL. I. of twist in kinks. Dynamics beginning with a straight wire, but giving it one full turn (2) of twist before bending it into the circle. The wire will stay in the 8 form without any pull on the tie. Whether the circular or the 8 form is stable or unstable depends on the relations between torsional and flexural rigidity. If the torsional rigidity is small in comparison with the flexural rigidity [as (§§ 703, 704, 705, 709) would be the case if, instead of round wire, a rod of + shaped section were used], the circular form would be stable, the 8 unstable. Surface rolling on sur traces given. Lastly, suppose any degree of twist, either more or less. than 2π, to be given before bending into the circle. The circular form, which is always a figure of free equilibrium, may be stable or unstable, according as the ratio of torsional to flexural rigidity is more or less than a certain value depending on the actual degree of twist. The tortuous 8 form is not (except in the case of whole twist 27, when it becomes the plane elastic lemniscate of Fig. 4, § 610,) a continuous figure of free equilibrium, but involves a positive pressure of the two crossing parts on one another when the twist > 2π, and a negative pressure (or a pull on the tie) between them when twist <2: and with this force it is a figure of stable equilibrium. = 124. Returning to the motion of one surface rolling and face; both spinning on another, the trace on each being given, we may consider that, of each, the curvature (§ 6), the tortuosity (§ 7), and the twist reckoned according to transverses in the tangent plane of the surface, are known; and the subject is fully specified in § 117 above. 1 and be the curvatures of the traces on the rolling ρ and fixed surfaces respectively; a' and a the inclinations of their planes of curvature to the normal to the tangent plane, reckoned as in 116; ' and their tortuosities; t' and t their twists; and q the velocity of the point of contact. All these being known, it is required to find: w the angular velocity of rotation about the transverse of the traces; that is to say, the line in the tangent plane perpendicular to their tangent line, σ the angular velocity of rotation about the tangent line, and of spinning. |