CHAP. I. ON PURE MECHANISM
§ 22. Division of the Subject.—Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Kinematics, may be thus divided:
. Division i.—Motion of a point.
Division 2.—Motion of the surface of a fluid.
Division 3.—Motion of a rigid solid.
Division 4.—Motions of a pair of connected pieces, or of an " elementary combination " in mechanism.
Division 5.—Motions of trains of pieces of mechanism.
Division 6.—Motions of sets of more than two connected pieces, or of " aggregate combinations."
A point is the boundary of a line, which is the boundary of a surface, which is the boundary of a volume. Points, lines and surfaces have no independent existence, and consequently those divisions of this chapter which relate to their motions are only preliminary to the subsequent divisions, which relate to the motions of bodies.
Division 1. Motion of a Point.
§ 23. Comparative Motion.—The comparative motion of two points is the relation which exists between their motions, without having regard to their absolute amounts. It consists of two elements,—the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the directional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.
It is obvious that the motions of a pair of points may be varied in any manner, whether by direct or by lateral deviation, and yet that their comparative motion may remain constant, in consequence of the deviations taking place in the same proportions, in the same directions and at the same instants for both points.
Robert Willis (1800—1875) has the merit of having been the first to simplify considerably the theory of pule mechanism, by pointing out that that branch of mechanics relates wholly to comparative motions.
The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamilton's Quaternions,—the " tensor " expressing the velocity ratio, and the
versor " the directional relation.
Graphical methods of analysis founded on this way of representing velocity and acceleration were developed by R. H. Smith in'a paper communicated to the Royal Society of Edinburgh in 1885, and illustrations of the method will be found below.
Division 2. Motion of the Surface of a Fluid Mass.
§ 24. General Principle.—A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (called pistons or plungers) of the solid envelope or vessel in which the fluid is contained; and, when such transmission is the sole action, or the only appreciable action of the fluid mass, its volume is either absolutely constant, by reason of its temperature and pressure being maintained constant, or not sensibly varied.
Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward. Then the variation of the cubic contents of the vessel in a unit of time by reason of the motion of one piston is va. The condition that the volume of the fluid mass shall remain unchanged requires that there shall be more than one piston, and that the .velocities and areas of the pistons shall be connected by the equation
.va=o. (I)
§ 25. Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are al and a2, and their velocities vi and v2, their comparative motion is expressed by the equation
v2/vi = —aia/2 ; (2) that is to say, their velocities are opposite as to inwardness and outwardness and inversely proportional to their areas.
§ 26. Applications: Hydraulic Press: Pneumatic PowerTransmitter.—In the hydraulic press the vessel consists of two cylinders, viz. the pumpbarrel and the pressbarrel, each having its piston, and of a passage connecting them having a valve opening towards the pressbarrel. The action of the enclosed water in transmitting motion takes place during the inward stroke of the pumpplunger, when the abovementioned valve is open; and at that time the pressplunger moves outwards with a velocity which is less than the inward velocity of the pumpplunger, in the same ratio that the area of the pumpplunger is less than the area of the pressplunger. (See HYDRAULICS.)
In the pneumatic powertransmitter the motion of one piston is
transmitted to another at a distance by means of a mass of air contained in two cylinders and an intervening tube. When the pressure and temperature of the air can be maintained constant, this machine fulfils equation (2), like the hydraulic press. The amount and effect of the variations of pressure and temperature undergone by the air depend on the principles of the mechanical action of heat, or THERMODYNAMICS (q.v.), and are foreign to the subject of pure mechanism.
Division 3. Motion of a Rigid Solid.
§ 27. Motions Classed.—In problems of mechanism, each solid piece of the machine is supposed to be so stiff and strong as not to undergo any sensible change of figure or dimensions by the forces applied to it—a supposition which is realized in practice if the machine is skilfully designed.
This being the case, the various possible motions of a rigid solid body may all be classed under the following heads: (1) Shifting or Translation.; (2) Turning or Rotation; (3) Motions compounded of Shifting and Turning.
The most common forms for the paths of the points of a piece of mechanism, whose motion is simple shifting, are the straight line and the circle.
Shifting in a straight line is regulated either by straight fixed guides, in contact with which the moving piece slides, or by combinations of linkwork, called parallel motions, which will be described in the sequel. Shifting in a straight line is usually reciprocating; that is to say, the piece, after shifting through a certain distance, returns to its original position by reversing its motion.
Circular shifting is regulated by attaching two or more points of the shifting piece to ends of equal and parallel rotating cranks, or by combinations of wheelwork to be afterwards described. As an example of circular shifting may be cited the motion of the coupling rod, by which the parallel and equal cranks upon two or more axles of a locomotive engine are connected and made to rotate simultaneously. The coupling rod remains always parallel to itself, and all its points describe equal and similar circles relatively to the frame of the engine, and move in parallel directions with equal velocities at the same instant.
§ 28. Rotation about a Fixed Axis: Lever, Wheel and Axle.—The fixed axis of a turning body is a line fixed relatively to the body and relatively to the fixed space in which the body turns. In mechanism it is usually the central line either of a rotating shaft or axle having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead centre round which a rotating bush turns; but it may sometimes be entirely beyond the limits of the turning body. For example, if a sliding piece moves in circular fixed guides, that piece rotates about an ideal fixed axis traversing the centre of those guides.
Let the angular velocity of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is o.r; and the path of that point is a circle of the radius r described about the axis.
This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.
In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of different radii described about their common fixed axis of turning, their velocityratio being that of their radii.
§ 29. Velocity Ratio of Components of Motion.—As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal. Hence it follows that, if A in fig. go be a point in a rigid body CD, rotating round the fixed axis F, the component of the velocity of A in any direction AP parallel to the plane of rotation is equal to the total velocity of the point m, found by letting fall Fm perpendicular to AP; that is to say, is equal to
a .Fm.
ponents of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fm and Fn.
§ 30. Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose axis of figure is B and angular velocity y, roll on a fixed cylinder aaa, whose axis of figure is A, either outside (as in fig. 91), when the rolling will be towards the same hand as the rotation, or inside (as in fig. 92), when the rolling will be towards the opposite hand; and at a given instant let T be the line of coptact of the two cylindrical surfaces, which is at their common intersection with the plane AB traversing the two axes of figure.
The line T on the surface bbb has for the instant no velocity ina 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 bbb, 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 perpendicular 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
vP =y . PT, (3) 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 circumference of bbb, that path becomes an epicycloid.
The velocity of any point in the axis of figure B is
va=y .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
v =a. AB, (5)
and consequently that a=y. TB/AB (6) For internal rolling, as in fig. 92, 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
13=7—a
whence d = y . TA/AB 5 ' (7) care being taken to attend to the sign of a, so that when that is negative the arithmetical values of y and a are to be added in order to give that of /S.
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.
§ 31. Instantaneous Axis of a Cone rolling on a Cone.—Let Oaa (fig. 93) be a fixed cone, OA its axis, Obb a cone rolling on it, OB
0
the axis of the rolling cone, OT the line of contact of the two cones at the instant under consideration. By reasoning similar to that of § 30, it appears that OT is the instantaneous axis of rotation of the rolling cone.
Let y denote the total angular velocity of the rotation of the cone B about the instantaneous axis, $ 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 : Q : y. (8) Or because of the proportionality of the sides of triangles to the sines of the opposite angles,
sin TOB : sin TOA : sin AOB : : a : : y, (8 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
Then CF=2X area ECO CE
and CG =2 X area ECO
OE
.•. CG : CF :: CE=OD : OE.
Let tic denote the linear velocity of the point C. Then vc=a.CF=y.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 vE be the linear velocity of the point E fixed in the plane of axes AOB. Then
v.= 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,
fi.EH=v$=a.EK;
therefore a : /3 :: 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 OPQ, and its velocity is
v,. = y . PQ. (9) The whole of 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 O.
§ 32, Screwlike or Helical Motion.—Since any displacement in a plane can berepresented 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 rotation 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 screwlike 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 screwlike figure, and moves in a guide of a corresponding figure. Helical motion and screws adapted to it are said to be right or lefthanded according to the appearance presented by the rotation to an observer looking towards the direction of the translation. Thus the screw G in fig. 94 is righthanded.
The translation of a body in helical motion is called its advance. Let v= 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; 2,r=6.2832 nearly, the circumference of a circle of the radius unity. Then
T = 2r/a (Io)
is the time of one turn at the rate a; and
p=v.T= 2rVx/a (II) 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 or helical projection.
Let r denote the perpendicular distance of a point in a body moving helically from the axis. Then
v, = ar (I2) is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity is
v=s/ Ivx2+v,.2}. (13)
The ratio of the two components of that velocity is
vx/v, = p/2rr = tan B. (14) where 0 denotes the angle made by the helical path of the point with a plane perpendicular to the axis.
Division 4. Elementary Combinations in Mechanism
§ 33. Definitions.—An elementary combination in mechanism consists of two pieces whose kinds of motion are determined by their connexion with the frame, and their comparative motion by their connexion with each other—that connexion being effected eitherby direct contact of the pieces, or by a connecting piece, which is not connected with the frame, and whose motion depends entirely on the motions of the pieces which it connects.
The piece whose motion is the cause is called the driver; the piece whose motion is the effect, the follower.
The connexion of each of those two pieces with the frame is in general such as to determine the path of every point in it. In the investigation, therefore, of the comparative motion of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connexion of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of the directional relation, so far only as it expresses the connexion between forward and backward movements of the driver and follower. When a continuous motion of the driver produces a continuous motion of the follower, forward or backward, and a reciprocating motion a motion reciprocating at the same instant, the directional relation is said to be constant. When a continuous motion produces a reciprocating motion, or vice versa, or when a reciprocating motion produces a motion not reciprocating at the same instant, the directional relation is said to be variable.
The line of action or of connexion of the driver and follower is a line traversing a pair of points in the driver and follower respectively, which are so connected that the component of their velocity relatively to each other, resolved along the line of connexion, is null. There may be several or an indefinite number of lines of connexion, or there may be but one; and a line of connexion may connect either the same pair of points or a succession of different pairs.
§ 34. General Principle.—From the definition of a line of connexion it follows that the components of the velocities of a pair of connected points along their line of connexion are equal. And from this, and from the property of a rigid body, already stated in § 29, it follows, that the components along a line of connexion of all the points traversed by that line, whether in the driver or in the follower, are equal; and consequently, that the velocities of any pair of points traversed by a line of connexion are to each other inversely as the cosines, or directly as the secants, of the angles made by the paths of those points with the line of connexion.
The general principle stated above in different forms serves to solve every problem in which—the mode of connexion of a pair of pieces being given—it is required to find their comparative motion at a given instant, or vice versa.
§ 35. Application to a Pair of Shifting Pieces.—In fig. 95, let PiP2 be the line of connexion of a pair of pieces, each of which has a motion of translation or shifting. Through any point T in that line draw TVi, TV2, respectively parallel to the simultaneous direction of motion of the pieces; through any other point A in the line of connexion draw a plane perpendicular to that line, cutting TVi, TV2 in Vi, V2; then, velocity of piece I: velocity of piece 2 :: TVi : TV2. Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V1V2 represents their velocity relatively to each other.
§ 36. Application to a Pair of Turning pieces.—Let al, a2 be the angular velocities of a pair of turning pieces; Bi, B2 the angles which their line of connexion makes with their respective planes of rotation; ri, r2 the common perpendiculars let fall from the line of connexion upon the respective axes of rotation of the pieces. Then the equal components, along the line of connexion, of the velocities of the points where those perpendiculars meet that line are
airs cos 0i = 0.27'2 COS 02 ;
consequently, the comparative motion of the pieces is given by the equation
a2 _ ri cos Bi (15) ai r2 cos Bi
§ 37. Application to a Shifting Piece and a Turning Piece.—Let a shifting piece be connected with a turning piece, and at a given instant let al be the angular velocity of the turning piece, ri the common perpendicular of its axis of rotation and the line of connexion, Bi the angle made by the line of connexion with the plane of rotation, B2 the angle made by the line of connexion with the direction of motion of the shifting piece, v2 the linear velocity of that piece. Then
,FIG. 95.
airs cos Bi ='v2 cos B2 ; (16) which equation expresses the comparative motion of the two pieces. § 38. Classification of Elementary Combinations in Mechanism.—The first systematic classification of elementary combinations in mechanism was that founded by Monge, and fully developed by Lanz and Betancourt, which has been generally received, and has been adopted in most treatises on applied mechanics. But that classification is founded on the absolute instead of the comparative
motions of the pieces, and is, for that reason, defective, as Willis pointed out in his admirable treatise On the Principles of Mechanism.
Willis's classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connexion of the driver and follower. He divides the elementary combinations in mechanism into three classes, of which the characters are as follows:
Class A: Directional relation constant; velocity ratio constant.
Class B: Directional relation constant; velocity ratio varying.
Class C: Directional relation changing periodically; velocity ratio constant or varying.
Each of those classes is subdivided by Willis into five divisions, of which the characters are as follows:
Division A: Connexion by rolling contact.
B : „ ,, sliding contact.
C : „ „ wrapping connectors.
D: „ „ linkwork.
E : „ „ reduplication.
In the Reuleaux system of analysis of mechanisms the principle of comparative motion is generalized, and mechanisms apparently very diverse in character are shown to be founded on the same sequence of elementary combinations forming a kinematic chain. A short description of this system is given in § 8o, but in the present article the principle of Willis's classification is followed mainly. The arrangement is, however, modified by taking the mode of connexion as the basis of the primary classification, and by removing the subject of connexion by reduplication to the section of aggregate combinations. This modified arrangement is adopted as being better suited than the original arrangement to the limits of an article in an encyclopaedia; but it is not disputed that the original arrangement may be the best for a separate treatise.
§ 39• Rolling Contact: Smooth Wheels and Racks.—In order that two pieces may move in rolling contact, it is necessary that each pair of points in the two pieces which touch each other should at the instant of contact be moving in the same direction with the same velocity. In the case of two shifting pieces this would involve equal and parallel velocities for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces would move like one; hence, in the case of rolling contact, either one or both of the pieces must rotate.
The direction of motion of a point in a turning piece being perpendicular to a plane passing through its axis, the condition that each pair of points in contact with each other must move in the same direction leads to the following consequences:
I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane.
II. That. when one piece rotates, and the other shifts, the axis of the rotating piece, and all the points of contact, lie in a plane perpendicular to the direction of motion of the shifting piece.
The condition that the velocity'of each pair of points of contact must be equal leads to the following consequences:
IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.
The line of contact is that line in which the points of contact are all situated. Respecting this line, the above Principles III. and IV. lead to the following conclusions:
V. That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that the rolling surfaces are either plane or cylindrical (the term` cylindrical " including all surfaces generated by the motion of a straight line parallel to itself).
VI. That for a pair of turning pieces with intersecting axes the line of contact is also straight, and traverses the. point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term " conical " including all surfaces generated by the motion of a straight line which traverses a fixed point).
Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in rolling contact with turning pieces may be called smooth or toothless racks.
§ 40. Cylindrical Wheels and Smooth Racks.—In designing cylindrical wheels and smooth racks, and determining their comparative motion, it is sufficient to consider a section of the pair of pieces made by a plane perpendicular to the axis or axes.
The points where axes intersect the plane of section are called centres; the point where the line of contact intersects it, the point of contact, or pitchpoint; and the wheels are described as circular, elliptical, &c., according to the forms of their sections made by that plane.
When the point of contact of two wheels lies between their centres, they are said to be in outside gearing; when beyond theircentres, in inside gearing, because the rolling surface of the larger wheel must in this case be turned inward or towards its centre. From Principle III. of § 39 it appears that the angular velocityratio of a pair of wheels is the inverse ratio of the distances of the point of contact from the centres respectively.
For outside gearing that ratio is negative, because the wheels turn contrary ways; for inside gearing it is positive, because they turn the same way.
If the velocity ratio is to be constant, as in Willis's Class A, the wheels must be circular; and this is the most common form for wheels.
If the velocity ratio is to be variable, as in Willis's Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.
The following is the geometrical relation which must exist between such a pair of curves:
Let Cl, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2 any pair of points of contact; U1, U2 any other pair of points of contact.
Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled :
Sum of radii, C1U1+C2U2=CIT1+C2T2 = constant;
arc, T2U2 = T1U1. (17)
A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radiivectores shall be equal and contrary; or, denoting by r1, r2 the radiivectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact
dr2/ds = —dri/ds; (18) which is the differential equation of a pair of rolling curves whose poles are at a constant distance apart.
For full details as to rolling curves, see Willis's work, already mentioned, and Clerk Maxwell's paper on Rolling Curves, Trans. Roy. Soc. Edin., 1849.
A rack, to work with a circular wheel, must be straight. To work with a wheel of any other figure, its section must be a rolling curve, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant. Let r1 be the radiusvector of a point of contact on the wheel, x2 the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then
dx2/ds= —drl/ds (19)
is the differential equation of the pair of rolling curves.
To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis
of the ellipses, and the velocity ratio varying from {eccentricity
1—eccentricity
—eccentricity
to I +eccentricity' an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyper
rii
bolas, and the velocity ratio varying between eccent and eccentricity 1
— I
unity; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix.
§ 41. Conical or Bevel and Disk Wheels.—From Principles III. and VI. of § 39 it appears that the angular velocities of a pair of wheels whose axes meet in a point are to each other inversely as the sines of the angles which the axes of the wheels make with the line of contact. Hence we have the following construction (figs. 99 and g8).Let 0 be the apex or point of intersection of the two axes OC1, OC2. The angular velocity ratio
being given, it is required to find the line of contact. On OC1, OC2 take lengths OA1, OA2, respectively proportional to the angular velocities of the pieces on whose axes they are taken. Complete the parallelogram OA1EA2; the diagonal OET will be the line of contact required.
When the velocity ratio is variable, the line of contact will shift its position in the plane C1OC2, and the wheels will
be cones, with eccentric or irregular 'Ws bases. In every case which occurs in
practice, however, the velocity ratio is FIG. 97.
constant ; the line of contact is constant in position, and the rolling surfaces of the wheels are regular circular cones (when they are called bevel wheels) ; or one of a pair of wheels may have a flat disk
for its rolling surface, as W2 in fig. 98, in which case it is a disk wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or disks, as shown by W1, W2 in figs. 97 and 98.
§ 42. Sliding Contact (lateral): SkewBevel Wheels.—An hyperboloid
of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such
T hyperboloids E, F, equal or unequal, be placed in the closest possible contact, as in fig. 99, they will touch each other along one of the generating straight lines of each, which will form their line of
contact, and will be inclined to the axes AG, BH in opposite
directions. The axes will not be parallel, nor will they intersect
each other.
The motion of two such hyperboloids, turning in contact with
each other, has hitherto been classed amongst cases of rolling
contact; but that classification is not
strictly correct, for, although the com
ponent velocities of a pair of points of
contact in a direction at right angles
to the line of contact are equal, still,
as the axes are parallel neither to each
other nor to the line of contact, the
velocities of a pair of points of contact
contact which are unequal, and their difference constitutes a lateral sliding.
The directions and positions of the axes being given, and the required angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated:
In fig. too, let B1C1, B2C2 be the two axes; B1B2 their common perpendicular. Through any point 0 in this common perpendicular draw 0A1 parallel to BIC' and OA2 parallel to B2C2; make those lines proportional to the angular velocities about the axes to which they are respectively paiallel; complete the parallelogram 0A1 EA2, and draw the diagonal OE; divide B1B2 in D into two parts, inversely proportional to the angular velocities about the axes which they respectively adjoin; through D parallel to OE draw DT. This will be the line of contact.
A pair of thin frusta of a pair of hyperboloids are used in practice to communicate motion between a pair of axes neither parallel nor intersecting, and are called skewbevel wheels.
In skewbevel wheels the properties of a line of connexion are not possessed by every line traversing the line of contact, but only by every line traversing the line of contact at right angles.
If the velocity ratio to be communicated were variable, the point D would alter its position, and the line DT its direction, at different periods of the motion, and the wheels would be hyperboloids of an eccentric or irregular crosssection; but forms of this kind are not used in practice.
§ 43. Sliding Contact (circular): Grooved Wheels.—As the adhesion or friction between a pair of smooth wheels is seldom sufficient to prevent their slipping on each other, contrivances are used to increase their mutual hold. One of those consists in forming the rim of each wheel into a series of alternate ridges and grooves parallel to the plane of rotation; it is applicable to cylindrical and bevel wheels, but not to skewbevel wheels. The comparative motion of a pair of wheels so ridged and grooved is the same as that of a pair of smooth wheels in rolling contact, whose cylindrical or conicai surfaces lie midway between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces are called the pitch surfaces of the wheels.
The relative motion of the faces of contact of the ridges and grooves is a rotatory sliding or grinding motion, about the line of contact of the pitchsurfaces as an instantaneous axis.
Grooved wheels have hitherto been but little used.
§ 44. Sliding Contact (direct): Teeth of Wheels, their Number and Pitch.—The ordinary method of connecting a pair of wheels, or a wheel and a rack, and the only method which ensures the exact maintenance of a given numerical velocity ratio, is by means of a series of alternate ridges and hollows parallel or nearly parallel to the successive lines of contact of the ideal smooth wheels whose velocity ratio would be the same with that of the toothed wheels. The ridges are called teeth; the hollows, spaces. The teeth of thedriver push those of the follower before them, and in so doing sliding takes place between them in a direction across their lines of contact.
The pitchsurfaces of a pair of toothed wheels are the ideal smooth surfaces which would have the same comparative motion by rolling contact that the actual wheels have by the sliding contact of their teeth. The pitchcircles of a pair of circular toothed wheels are sections of their pitchsurfaces, made for spurwheels (that is, for wheels whose axes are parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described about the common apex. For a pair of skewbevel wheels the pitchcircles are a pair of contiguous rectangular sections of the pitchsurfaces. The pitchpoint is the point of contact of the pitchcircles.
The pitchsurface of a wheel lies intermediate between the points of the teeth and the bottoms of the hollows between them. That part of the acting surface of a tooth which projects beyond the pitchsurface is called the face; that part which lies within the pitchsurface, the flank.
Teeth, when not otherwise specified, are understood to be made in one piece with the wheel, the material being generally castiron, brass or bronze. Separate teeth, fixed into mortises in the rim of the wheel, are called cogs. A pinion is a small toothed wheel ; a trundle is a pinion with cylindrical staves for teeth.
The radius of the pitchcircle of a wheel is called the geometrical radius; a circle touching the ends of the teeth is called the addendum circle, and its radius the real radius; the difference between these radii, being the projection of the teeth beyond the pitchsurface, is called the addendum.
The distance, measured along the pitchcircle, from the face of one tooth to the face of the next, is called the pitch. The pitch and the number of teeth in wheels are regulated by the following principles:
I. In wheels which rotate continuously for one revolution or more, it is obviously necessary that the pitch should be an aliquot part of the circumference.
In wheels which reciprocate without performing a complete revolution this condition is not necessary. Such wheels are called sectors.
II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each.
IV. Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.
From this principle arise problems of a kind which will be referred to in treating of Trains of Mechanism.
V. Let n, N be the respective numbers of teeth in a pair of wheels, N being the greater. Let t, T be a pair of teeth in the smaller and larger wheel respectively, which at a particular instant work together. It is required to find, first, how many pairs of teeth must pass the line of contact of the pitchsurfaces before t and T work together again (let this number be called a); and, secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called b) ; thirdly, with how many different teeth of the smaller wheel the tooth T will work at different times (let this be called c)
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