754 [APPLIED MECHANICS. a direction perpendicular to AB ; because for the instant it touches, without sliding, the line T on the fixed surface aaa. The line T on the surface bbb has also for the instant no velocity in the plane AB ; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface. The line of contact T, therefore, on the surface of the cylinder bib, is for the instant at rest, and is the "instantaneous axis" about which the cylinder bbb turns, together with any body rigidly attached to that cylinder. To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT ; the direction of motion of P will be perpendi cular to that plane, and towards the right or left hand according to the direction of the rotation of bbb ; and the velocity of P will be "PT 1 (^ PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind called epitrochoids. If P is in the circum ference of bbb, that path becomes an epicycloid. The velocity of any point in the axis of figure B is
- T*T> A
i B = 7 . TB 4) ; and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders. Let a denote the angular velocity with which the plane of axes AB rotates about the fixed axis A. Then it is evident that (5), _, ,,_, , area ECO Then CF = 2 x -- ^ TB and consequently that a = 7--r-j Ao For internal rolling, as in fig. 6, AB is to be treated as negative, which will give a negative value to a, indicating that in this case the rotation of AB round A is contrary to that of the cylinder bbb. The angular velocity of the rolling cylinder, relatively to the plane of axes AB, is obviously given by the equation 0-7- a ; TA whence 8 = 7 VT> (7) care being taken to attend to the sign of o, so that when that is negative the arithmetical values of y and a are to be added in order to give that of ft. The whole of the foregoing reasonings are applicable, not merely when aaa and bbb are actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration. 38. Composition and Resolution of Eolations about Parallel Axes. See above, p. 691, 73. 39. Instantaneous Axis of a Cone rolling on a Cone. Let Ctaa (fig. 7) be a fixed cone, OA its axis, Obb a cone rolling on it, OB the axis of the roll ing cone, OT the line of contact of the two cones at the instant under con sideration. By rea soning similar to that of sect. 37, it appears that OT is the instantaneous axis of rotation of Fig. 7. the rolling cone. Let y denote the total angular velocity of the rotation of the cone B about the instantaneous axis, /3 its angular velocity about the axis OB relatively to the plane AOB, and a the angular velocity with which the plane AOB turns round the axis OA. It is required to find the ratios of those angular velocities. Solution. In OT take any point E, from which draw EC parallel to OA, and ED parallel to OB, so as to construct the parallelogram OCED. Then OD : OC : OE
- = a : /3 : y
Or because of the proportionality of the sides of triangles to the sines of the opposite angles, sin^TOB : sin^TOA : sin^cAOB ) (8 A)
- : a : : )
that is to say, the angular velocity about each axis is proportional to the sine of the angle between the other two. Demonstration. From C draw CF perpendicular to OA, and CG perpendicular to OE. and CG = 2x area ECO OE . . CG : CF :: CE = OD : OE . Let r e denote the linear velocity of the point C. Then Vc = a . CF = 7 . CG . . y. a ::CF : CG : : OE : OD , which is one part of the solution above stated. From E draw EH perpendicular to OB, and EK to OA. Then it can be shown as before that EK : EH : : OC : OD . Let v z be the linear velocity of the point E fixed in the plane oj axes AOB. Then r, = a. EK. Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone. That is to say, /3. EH = v,, = a. EK ; therefore a : : : EH : EK : : OD : OC , which is the remainder of the solution. The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the instantaneous axis. Then the motion of P is perpendicular to the plane OFQ, and its velocity is The whole ot the foregoing reasonings are applicable, not merely when A and B are actual regular cones, but also when they are the osculating regular cones of a pair of irregular conical surfaces, having a common apex at 0. 40. Composition of notations about Two Axes meeting in a Point. See p. 691, 76. 41. Screw-like or Helical Motion. Since (see p. 690, 71, 72) any displacement in a plane can be represented in general by a rotation, it follows that the only combination of translation and rotation, in which a complex movement which is not a mere rota tion is produced, occurs when there is a translation perpendicular to the plane and parallel to the axis of rotation. Such a complex motion is called screw-like or helical motion; for each point in the body describes a helix or screw round the axis of rotation, fixed or instantaneous as the case may be. To cause a body to move in this manner it is usually made of a helical or screw-like figure, and moves in a guide of a corresponding figure. Helical motion and screws adapted to it are said to be right- or left - handed according to the appearance presented by the rotation to an observer looking towards the direction of the translation. Thus the screw G in fig. 8 is right-handed. The translation of a body in helical motion is called its advance. Let v x denote the velocity of advance at a given instant, which of course is common to all the particles of the body ; a the angular velocity of the rotation at the same instant ; 2ir=6 - 2832 nearly, the circumference of a circle of the radius unity. Then T = ?I (10) is the time of one turn at the rate o ; and (11) is the pitch or advance per turn, a length which expresses the comparative motion of the translation and the rotation. The pitch of a screw is the distance, measured parallel to its axis, between two successive turns of the same thread er helical pro jection. Let r denote the perpendicular distance of a point in a body moving helically from the axis. Then tv = or (12) is the component of the velocity of that point in a plane perpendi cular to the axis, and its total velocity is t , = V{r, 2 + rv 2 } (13). The ratio of the two components of that velocity is -- - <I4)! where denotes the angle made by the helical path of the point
with a plane perpendicular to the axis.
