(6)For internal rolling, as in fig. 92, AB is to be treated as negative, which will give a negative value to α, 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 axesAB, is obviously given by the equation—β = γ − αwhence β = γ · TA/AB(7)care being taken to attend to the sign of α, so that when that is negative the arithmetical values of γ and α are to be added in order to give that of β.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.Fig. 93.§ 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 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 γ denote the total angular velocity of the rotation of the cone B about the instantaneous axis, β its angular velocity about the axis OBrelativelyto the plane AOB, and α 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. ThenOD : OC : OE :: α : β : γ.(8)Or because of the proportionality of the sides of triangles to the sines of the opposite angles,sin TOB : sin TOA : sin AOB :: α : β : γ,(8A)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 OEThen CF = 2 ×area EC,CEand CG = 2 ×area ECO;OE∴ CG : CF :: CE = OD : OE.Let vcdenote the linear velocity of the point C. Thenvc= α · CF = γ · CG∴ γ : α :: 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 thatEK : EH :: OC : OD.Let vEbe the linear velocity of the point Efixed in the plane of axesAOB. ThenvK= α · 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,β · EH = vE= α · EK;thereforeα : β :: 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 isvP= γ · 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.Screw-like or Helical Motion.—Since 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 rotation is produced, occurs when there is a translationperpendicular to the plane and parallel to the axisof rotation.Fig. 94.Such a complex motion is calledscrew-likeorhelicalmotion; for each point in the body describes ahelixorscrewround 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. 94 is right-handed.The translation of a body in helical motion is called itsadvance. Let vxdenote the velocity of advance at a given instant, which of course is common to all the particles of the body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the circumference of a circle of the radius unity. ThenT = 2π/α(10)is the time of one turn at the rate α; andp = vxT = 2πvx/α(11)is thepitchoradvance per turn—a length which expresses thecomparative motionof the translation and the rotation.The pitch of a screw is the distance, measured parallel to its axis, between two successive turns of the samethreador helical projection.Let r denote the perpendicular distance of a point in a body moving helically from the axis. Thenvr= αr(12)is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity isv = √ {vx2+ vr2}.(13)The ratio of the two components of that velocity isvx/vr= p/2πr = tan θ.(14)where θ 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.—Anelementary combinationin 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 either by 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 thedriver; the piece whose motion is the effect, thefollower.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 betweenforwardandbackwardmovements 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 beconstant. 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 bevariable.Theline of actionorof connexionof 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 thatthe 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, thatthe 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.Fig. 95.§ 35.Application to a Pair of Shifting Pieces.—In fig. 95, let P1P2be 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 TV1, 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 TV1, TV2in V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1: TV2. Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V1V2represents their velocity relatively to each other.§ 36.Application to a Pair of Turning Pieces.—Let α1, α2be the angular velocities of a pair of turning pieces; θ1, θ2the angles which their line of connexion makes with their respective planes of rotation; r1, r2the 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—α1r1cos θ1= α2r2cos θ2;consequently, the comparative motion of the pieces is given by the equationα2=r1cos θ1.α1r2cos θ2(15)§ 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 α1be the angular velocity of the turning piece, r1the common perpendicular of its axis of rotation and the line of connexion, θ1the angle made by the line of connexion with the plane of rotation, θ2the angle made by the line of connexion with the direction of motion of the shifting piece, v2the linear velocity of that piece. Thenα1r1cos θ1= v2cos θ2;(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 Bétancourt, 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 comparativemotions of the pieces, and is, for that reason, defective, as Willis pointed out in his admirable treatiseOn 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:—DivisionA:Connexionbyrolling contact.”B:””sliding contact.”C:””wrapping connectors.”D:””link-work.”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 § 80, but in the present article the principle of Willis’s classification is followed mainly. The arrangement is, however, modified by taking themode of connexionas 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 twoshiftingpieces 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:—III. That the angular velocities of a pair of turning pieces in rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.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.Theline of contactis 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 calledsmoothortoothless wheels. Shifting pieces in rolling contact with turning pieces may be calledsmoothortoothless racks.VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion.§ 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 calledcentres; the point where the line of contact intersects it, thepoint of contact, orpitch-point; and the wheels are described ascircular,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 inoutside gearing; when beyond their centres, ininside 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 velocity-ratio of a pair of wheels is the inverse ratio of the distances of the point of contact from the centres respectively.Fig. 96.For outside gearing that ratio isnegative, because the wheels turn contrary ways; for inside gearing it ispositive, 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 ofrolling curves, subject to the condition that the distance between theirpoles(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 C1, C2(fig. 96) be the poles of a pair of rolling curves; T1, T2any pair of points of contact; U1, U2any 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= C1T1+ 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 radii-vectores shall be equal and contrary; or, denoting by r1, r2the radii-vectores 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 = −dr1/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 r1be the radius-vector of a point of contact on the wheel, x2the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Thendx2/ds = −dr1/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 (1 + eccentricity)/(1 − eccentricity) to (1 − eccentricity)/(1 + 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 hyperbolas, and the velocity ratio varying between (eccentricity + 1)/(eccentricity − 1) and unity; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix.Fig. 97.§ 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. 97 and 98).—Let O 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, OC2take 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 bases. In every case which occurs in practice, however, the velocity ratio is constant; the line of contact is constant in position, and the rolling surfaces of the wheels are regular circular cones (when they are calledbevel wheels); or one of a pair of wheels may have a flat diskfor its rolling surface, as W2in fig. 98, in which case it is adisk wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or disks, as shown by W1, W2in figs. 97 and 98.Fig. 98.Fig. 99.Fig. 100.§ 42.Sliding Contact (lateral): Skew-Bevel 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 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 component 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 have components along the line of contact which are unequal, and their difference constitutes alateral 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. 100, let B1C1, B2C2be the two axes; B1B2their common perpendicular. Through any point O in this common perpendicular draw OA1parallel to B1C1and OA2parallel to B2C2; make those lines proportional to the angular velocities about the axes to which they are respectively parallel; complete the parallelogram OA1EA2, and draw the diagonal OE; divide B1B2in D into two parts,inverselyproportional 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 calledskew-bevel wheels.In skew-bevel 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 cross-section; 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 skew-bevel 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 conical surfaces lie midway between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces are called thepitch surfacesof the wheels.The relative motion of the faces of contact of the ridges and grooves is arotatory slidingorgrindingmotion, about the line of contact of the pitch-surfaces 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 calledteeth; the hollows,spaces. The teeth of the driver push those of the follower before them, and in so doing sliding takes place between them in a direction across their lines of contact.Thepitch-surfacesof 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. Thepitch-circlesof a pair of circular toothed wheels are sections of their pitch-surfaces, made forspur-wheels(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 skew-bevel wheels the pitch-circles are a pair of contiguous rectangular sections of the pitch-surfaces. Thepitch-pointis the point of contact of the pitch-circles.The pitch-surface 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 pitch-surface is called theface; that part which lies within the pitch-surface, theflank.Teeth, when not otherwise specified, are understood to be made in one piece with the wheel, the material being generally cast-iron, brass or bronze. Separate teeth, fixed into mortises in the rim of the wheel, are calledcogs. Apinionis a small toothed wheel; atrundleis a pinion with cylindricalstavesfor teeth.The radius of the pitch-circle of a wheel is called thegeometrical radius; a circle touching the ends of the teeth is called theaddendum circle, and its radius thereal radius; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called theaddendum.The distance, measured along the pitch-circle, from the face of one tooth to the face of the next, is called thepitch. 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 necessarythat 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 calledsectors.II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essentialthat the pitch should be the same in each.III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii and inversely as the angular velocities.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 ofTrains 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 pitch-surfaces 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).Case 1.If n is a divisor of N,a = N; b = N/n; c = 1.(20)Case 2.If the greatest common divisor of N and n be d, a number less than n, so that n = md, N = Md; thena = mN = Mn = Mmd; b = M; c = m.(21)Case 3.If N and n be prime to each other,a = nN; b = N; c = n.(22)It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They therefore study that the numbers of teeth in each pair of wheels which work together shall either be prime to each other, or shall have their greatest common divisor as small as is consistent with a velocity ratio suited for the purposes of the machine.§ 45.Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point where they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack.In fig. 101, let C1, C2be the centres of a pair of spur-wheels; B1IB1′, B2IB2′ portions of their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower.Fig. 101.Let D1TB1A1, D2TB2A2be the positions, at a given instant, of the acting surfaces of a pair of teeth in the driver and follower respectively, touching each other at T; the line of connexion of those teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2be perpendiculars let fall from the centres of the wheels on the line of contact. Then, by § 36, the angular velocity-ratio isα2/α1= C1P1/C2P2.(23)The following principles regulate the forms of the teeth and their relative motions:—I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point.For, let P1P2cut the line of centres at I; then, by similar triangles,α1: α2:: C2P2: C1P1:: IC2:: IC1;(24)which is also the angular velocity ratio due to the rolling contact of the circles B1IB1′, B2IB2′.This principle determines theformsof all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE′, parallel to the direction of motion of the rack, and perpendicular to C1IC2, be substituted for a pitch-circle.II. The component of the velocity of the point of contact of the teeth T along the line of connexion isα1· C1P1= α2· C2P2.(25)III. The relative velocity perpendicular to P1P2of the teeth at their point of contact—that is, theirvelocity of slidingon each other—is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C1C2to rotate backwards round C1with the angular velocity α1, and the wheel 2 to rotate round C2as before, with the angular velocity α2relatively to the line of centres C1C2, so as to have the same motion as if its pitch-circlerolledon the pitch-circle of the first wheel. Thus therelativemotion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity α1+ α2. Hence thevelocity of slidingis that due to this rotation about I, with the radius IT; that is to say, its value is(α1+ α2) · IT;(26)so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with thepitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact.IV. Thepath of contactis the line traversing the various positions of the point T. If the line of connexion preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved.It is divided by the pitch-point I into two parts—thearcorline of approachdescribed by T in approaching the line of centres, and thearcorline of recessdescribed by T after having passed the line of centres.During theapproach, theflankD1B1of the driving tooth drives the face D2B2of the following tooth, and the teeth are slidingtowardseach other. During therecess(in which the position of the teeth is exemplified in the figure by curves marked with accented letters), thefaceB1′A1′ of the driving tooth drives theflankB2′A2′ of the following tooth, and the teeth are slidingfromeach other.The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact.V. Theobliquityof the action of the teeth is the angle EIT = IC1, P1= IC2P2.In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15°, nor the maximum value 30°.It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formulae is, that in equation (26) thedifferenceof the angular velocities should be substituted for their sum.§ 46.Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is obtained in the following manner (see fig. 102). Let C1, C2be the centres of two wheels, B1IB1′, B2IB2′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line P1IP2shall represent at once the constant line of connexion of teeth and the path of contact. Draw C1P1, C2P2perpendicular to P1IP2, and with those lines as radii describe about the centres of the wheels the circles D1D1′, D2D2′, calledbase-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also thatC1P1= IC1· cos obliquityC2P2= IC2· cos obliquity.
(6)
For internal rolling, as in fig. 92, AB is to be treated as negative, which will give a negative value to α, 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 axesAB, is obviously given by the equation—
(7)
care being taken to attend to the sign of α, so that when that is negative the arithmetical values of γ and α are to be added in order to give that of β.
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 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 γ denote the total angular velocity of the rotation of the cone B about the instantaneous axis, β its angular velocity about the axis OBrelativelyto the plane AOB, and α 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 :: α : β : γ.
(8)
Or because of the proportionality of the sides of triangles to the sines of the opposite angles,
sin TOB : sin TOA : sin AOB :: α : β : γ,
(8A)
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
∴ CG : CF :: CE = OD : OE.
Let vcdenote the linear velocity of the point C. Then
vc= α · CF = γ · CG∴ γ : α :: 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 vEbe the linear velocity of the point Efixed in the plane of axesAOB. Then
vK= α · 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,
β · EH = vE= α · EK;
therefore
α : β :: 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
vP= γ · 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.Screw-like or Helical Motion.—Since 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 rotation is produced, occurs when there is a translationperpendicular to the plane and parallel to the axisof rotation.
Such a complex motion is calledscrew-likeorhelicalmotion; for each point in the body describes ahelixorscrewround 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. 94 is right-handed.
The translation of a body in helical motion is called itsadvance. Let vxdenote the velocity of advance at a given instant, which of course is common to all the particles of the body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the circumference of a circle of the radius unity. Then
T = 2π/α
(10)
is the time of one turn at the rate α; and
p = vxT = 2πvx/α
(11)
is thepitchoradvance per turn—a length which expresses thecomparative motionof the translation and the rotation.
The pitch of a screw is the distance, measured parallel to its axis, between two successive turns of the samethreador helical projection.
Let r denote the perpendicular distance of a point in a body moving helically from the axis. Then
vr= αr
(12)
is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity is
v = √ {vx2+ vr2}.
(13)
The ratio of the two components of that velocity is
vx/vr= p/2πr = tan θ.
(14)
where θ 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.—Anelementary combinationin 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 either by 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 thedriver; the piece whose motion is the effect, thefollower.
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 betweenforwardandbackwardmovements 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 beconstant. 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 bevariable.
Theline of actionorof connexionof 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 thatthe 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, thatthe 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 P1P2be 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 TV1, 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 TV1, TV2in V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1: TV2. Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V1V2represents their velocity relatively to each other.
§ 36.Application to a Pair of Turning Pieces.—Let α1, α2be the angular velocities of a pair of turning pieces; θ1, θ2the angles which their line of connexion makes with their respective planes of rotation; r1, r2the 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—
α1r1cos θ1= α2r2cos θ2;
consequently, the comparative motion of the pieces is given by the equation
(15)
§ 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 α1be the angular velocity of the turning piece, r1the common perpendicular of its axis of rotation and the line of connexion, θ1the angle made by the line of connexion with the plane of rotation, θ2the angle made by the line of connexion with the direction of motion of the shifting piece, v2the linear velocity of that piece. Then
α1r1cos θ1= v2cos θ2;
(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 Bétancourt, 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 comparativemotions of the pieces, and is, for that reason, defective, as Willis pointed out in his admirable treatiseOn 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:—
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 § 80, but in the present article the principle of Willis’s classification is followed mainly. The arrangement is, however, modified by taking themode of connexionas 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 twoshiftingpieces 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:—
III. That the angular velocities of a pair of turning pieces in rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.
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.
Theline of contactis 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 calledsmoothortoothless wheels. Shifting pieces in rolling contact with turning pieces may be calledsmoothortoothless racks.
VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion.
§ 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 calledcentres; the point where the line of contact intersects it, thepoint of contact, orpitch-point; and the wheels are described ascircular,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 inoutside gearing; when beyond their centres, ininside 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 velocity-ratio 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 isnegative, because the wheels turn contrary ways; for inside gearing it ispositive, 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 ofrolling curves, subject to the condition that the distance between theirpoles(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 C1, C2(fig. 96) be the poles of a pair of rolling curves; T1, T2any pair of points of contact; U1, U2any 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= C1T1+ 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 radii-vectores shall be equal and contrary; or, denoting by r1, r2the radii-vectores 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 = −dr1/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 r1be the radius-vector of a point of contact on the wheel, x2the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then
dx2/ds = −dr1/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 (1 + eccentricity)/(1 − eccentricity) to (1 − eccentricity)/(1 + 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 hyperbolas, and the velocity ratio varying between (eccentricity + 1)/(eccentricity − 1) and 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. 97 and 98).—Let O 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, OC2take 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 bases. In every case which occurs in practice, however, the velocity ratio is constant; the line of contact is constant in position, and the rolling surfaces of the wheels are regular circular cones (when they are calledbevel wheels); or one of a pair of wheels may have a flat diskfor its rolling surface, as W2in fig. 98, in which case it is adisk wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or disks, as shown by W1, W2in figs. 97 and 98.
§ 42.Sliding Contact (lateral): Skew-Bevel 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 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 component 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 have components along the line of contact which are unequal, and their difference constitutes alateral 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. 100, let B1C1, B2C2be the two axes; B1B2their common perpendicular. Through any point O in this common perpendicular draw OA1parallel to B1C1and OA2parallel to B2C2; make those lines proportional to the angular velocities about the axes to which they are respectively parallel; complete the parallelogram OA1EA2, and draw the diagonal OE; divide B1B2in D into two parts,inverselyproportional 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 calledskew-bevel wheels.
In skew-bevel 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 cross-section; 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 skew-bevel 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 conical surfaces lie midway between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces are called thepitch surfacesof the wheels.
The relative motion of the faces of contact of the ridges and grooves is arotatory slidingorgrindingmotion, about the line of contact of the pitch-surfaces 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 calledteeth; the hollows,spaces. The teeth of the driver push those of the follower before them, and in so doing sliding takes place between them in a direction across their lines of contact.
Thepitch-surfacesof 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. Thepitch-circlesof a pair of circular toothed wheels are sections of their pitch-surfaces, made forspur-wheels(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 skew-bevel wheels the pitch-circles are a pair of contiguous rectangular sections of the pitch-surfaces. Thepitch-pointis the point of contact of the pitch-circles.
The pitch-surface 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 pitch-surface is called theface; that part which lies within the pitch-surface, theflank.
Teeth, when not otherwise specified, are understood to be made in one piece with the wheel, the material being generally cast-iron, brass or bronze. Separate teeth, fixed into mortises in the rim of the wheel, are calledcogs. Apinionis a small toothed wheel; atrundleis a pinion with cylindricalstavesfor teeth.
The radius of the pitch-circle of a wheel is called thegeometrical radius; a circle touching the ends of the teeth is called theaddendum circle, and its radius thereal radius; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called theaddendum.
The distance, measured along the pitch-circle, from the face of one tooth to the face of the next, is called thepitch. 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 necessarythat 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 calledsectors.
II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essentialthat the pitch should be the same in each.
III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii and inversely as the angular velocities.
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 ofTrains 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 pitch-surfaces 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).
Case 1.If n is a divisor of N,
a = N; b = N/n; c = 1.
(20)
Case 2.If the greatest common divisor of N and n be d, a number less than n, so that n = md, N = Md; then
a = mN = Mn = Mmd; b = M; c = m.
(21)
Case 3.If N and n be prime to each other,
a = nN; b = N; c = n.
(22)
It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They therefore study that the numbers of teeth in each pair of wheels which work together shall either be prime to each other, or shall have their greatest common divisor as small as is consistent with a velocity ratio suited for the purposes of the machine.
§ 45.Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point where they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack.
In fig. 101, let C1, C2be the centres of a pair of spur-wheels; B1IB1′, B2IB2′ portions of their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower.
Let D1TB1A1, D2TB2A2be the positions, at a given instant, of the acting surfaces of a pair of teeth in the driver and follower respectively, touching each other at T; the line of connexion of those teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2be perpendiculars let fall from the centres of the wheels on the line of contact. Then, by § 36, the angular velocity-ratio is
α2/α1= C1P1/C2P2.
(23)
The following principles regulate the forms of the teeth and their relative motions:—
I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point.
For, let P1P2cut the line of centres at I; then, by similar triangles,
α1: α2:: C2P2: C1P1:: IC2:: IC1;
(24)
which is also the angular velocity ratio due to the rolling contact of the circles B1IB1′, B2IB2′.
This principle determines theformsof all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE′, parallel to the direction of motion of the rack, and perpendicular to C1IC2, be substituted for a pitch-circle.
II. The component of the velocity of the point of contact of the teeth T along the line of connexion is
α1· C1P1= α2· C2P2.
(25)
III. The relative velocity perpendicular to P1P2of the teeth at their point of contact—that is, theirvelocity of slidingon each other—is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C1C2to rotate backwards round C1with the angular velocity α1, and the wheel 2 to rotate round C2as before, with the angular velocity α2relatively to the line of centres C1C2, so as to have the same motion as if its pitch-circlerolledon the pitch-circle of the first wheel. Thus therelativemotion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity α1+ α2. Hence thevelocity of slidingis that due to this rotation about I, with the radius IT; that is to say, its value is
(α1+ α2) · IT;
(26)
so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with thepitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact.
IV. Thepath of contactis the line traversing the various positions of the point T. If the line of connexion preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved.
It is divided by the pitch-point I into two parts—thearcorline of approachdescribed by T in approaching the line of centres, and thearcorline of recessdescribed by T after having passed the line of centres.
During theapproach, theflankD1B1of the driving tooth drives the face D2B2of the following tooth, and the teeth are slidingtowardseach other. During therecess(in which the position of the teeth is exemplified in the figure by curves marked with accented letters), thefaceB1′A1′ of the driving tooth drives theflankB2′A2′ of the following tooth, and the teeth are slidingfromeach other.
The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact.
V. Theobliquityof the action of the teeth is the angle EIT = IC1, P1= IC2P2.
In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15°, nor the maximum value 30°.
It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formulae is, that in equation (26) thedifferenceof the angular velocities should be substituted for their sum.
§ 46.Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is obtained in the following manner (see fig. 102). Let C1, C2be the centres of two wheels, B1IB1′, B2IB2′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line P1IP2shall represent at once the constant line of connexion of teeth and the path of contact. Draw C1P1, C2P2perpendicular to P1IP2, and with those lines as radii describe about the centres of the wheels the circles D1D1′, D2D2′, calledbase-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that
C1P1= IC1· cos obliquityC2P2= IC2· cos obliquity.