The names given to the various lines of a tooth on a gear-wheel are as follows:
In Figure 233, A is the face and B the flank of a tooth, while C is the point, and D the root of the tooth; E is the height or depth, and F the breadth. P P is the pitch circle, and the space between the two teeth, as H, is termed a space.
It is obvious that the points of the teeth and the bottoms of the spaces, as well as the pitch circle, are concentric to the axis of the wheel bore. And to pencil in the teeth these circles must be fully drawn, as in Figure 234, in which P P is the pitch circle. This circle is divided into as many equal divisions as the wheel is to have teeth, these divisions being denoted by the radial lines, A, B, C, etc. Where these divisions intersect the pitch circle are the centres from which all the teeth curves may be drawn. The compasses are set to a radius equal to the pitch, less one-half the thickness of the tooth, and from a centre, as R, two face curves, as F G, may be marked; from the next centre, as at S, the curves D E may be marked, and so on for all the faces; that is, the tooth curves lying between the outer circle X and the pitch circle P. For the flank curves, that is, the curve from P to Y, the compasses are set to a radius equal to the pitch; and from the sides of the teeth the flank curves are drawn. Thus from J, as a centre flank, K is drawn; from V, as a centre flank, H is drawn, and so on.
The proportions of the teeth for cast gears generally accepted in this country are those given by Professor Willis, as average practice, and are as follows:
|Depth to pitch line,||3/10||of the||pitch.|
|Thickness of tooth,||5/11||"||"|
|Breadth of space,||6/11||"||"|
Instead, however, of calculating the dimensions these proportions give for any particular pitch, a diagram or scale may be made from which they may be taken for any pitch by a direct application of the compasses. A scale of this kind is given in Figure 235, in which the line A B is divided into inches and parts to represent the pitches; its total length representing the coarsest pitch within the capacity of the scale; and, the line B C (at a right-angle to A B) the whole depth of the tooth for the coarsest pitch, being 7/10 of the length of A B.
The other diagonal lines are for the proportion of the dimensions marked on the figure. Thus the depth of face, or distance from the pitch line to the extremity or tooth point for a 4 inch pitch, would be measured along the line B C, from the vertical line B to the first diagonal. The thickness of the tooth would be for a 4 inch pitch along line B C from B to the second diagonal, and so on. For a 3 inch pitch the measurement would be taken along the horizontal line, starting from the 3 on the line A B, and so on. On the left of the diagram or scale is marked the lbs. strain each pitch will safely transmit per inch width of wheel face, according to Professor Marks.
The application of the scale as follows: The pitch circles P P and P' P', Figure 236, for the respective wheels, are drawn, and the height of the teeth is obtained from the scale and marked beyond the pitch circles, when circles Q and Q' may be drawn. Similarly, the depths of the teeth within the pitch circles are obtained from the scale or diagram and marked within the respective pitch circles, and circles R and R' are marked in. The pitch circles are divided off into as many points of equal division, as at a, b, c, d, e, etc., as the respective wheels are to have teeth, and the thickness of tooth having been obtained from the scale, this thickness is marked from the points of division on the pitch circles, as at f in the figure, and the tooth curves may then be drawn in. It may be observed, however, that the tooth thicknesses will not be strictly correct, because the scale gives the same chord pitch for the teeth on both wheels which will give different arc pitches to the teeth on the two wheels; whereas, it is the arc pitches, and not the chord pitches, that should be correct. This error obviously increases as there is a greater amount of difference between the two wheels.
The curves given to the teeth in Figure 234 are not the proper ones to transmit uniform motion, but are curves merely used by draughtsmen to save the trouble of finding the true curves, which if it be required, may be drawn with a very near approach to accuracy, as follows, which is a construction given by Rankine:
Draw the rolling circle D, Figure 237, and draw A D, the line of centres. From the point of contact at C, mark on D, a point distant from C one-half the amount of the pitch, as at P, and draw the line P C of indefinite length beyond C. Draw the line P E passing through the line of centres at E, which is equidistant between C and A. Then increase the length of line P F to the right of C by an amount equal to the radius A C, and then diminish it to an amount equal to the radius E D, thus obtaining the point F and the latter will be the location of centre for compasses to strike the face curve.
Another method of finding the face curve, with compasses, is as follows: In Figure 238 let P P represent the pitch circle of the wheel to be marked, and B C the path of the centre of the generating or describing circle as it rolls outside of P P. Let the point B represent the centre of the generating circle when it is in contact with the pitch circle at A. Then from B mark off, on B C, any number of equidistant points, as D, E, F, G, H, and from A mark on the pitch circle, with the same radius, an equal number of points of division, as 1, 2, 3, 4, 5. With the compasses set to the radius of the generating circle, that is, A B, from B, as a centre, mark the arc I, from D, the arc J, from E, the arc K, from F, and so on, marking as many arcs as there are points of division on B C. With the compasses set to the radius of divisions 1, 2, etc., step off on arc M the five divisions, N, O, S, T, V, and at V will be a point on the epicycloidal curve. From point of division 4, step off on L four points of division, as a, b, c, d; and d will be another point on the epicycloidal curve. From point 3, set off three divisions, and so on, and through the points so obtained draw by hand, or with a scroll, the curve.
Hypocycloids for the flanks of the teeth maybe traced in a similar manner. Thus in Figure 239, P P is the pitch circle, and B C the line of motion of the centre of the generating circle to be rolled within P P. From 1 to 6 are points of equal division on the pitch circle, and D to I are arc locations for the centre of the generating circle. Starting from A, which represents the location for the centre of the generating circle, the point of contact between the generating and base circles will be at B. Then from 1 to 6 are points of equal division on the pitch circle, and from D to I are the corresponding locations for the centres of the generating circle. From these centres the arcs J, K, L, M, N, O, are struck. The six divisions on O, from a to f, give at f a point in the curve. Five divisions on N, four on M, and so on, give, respectively, points in the curve.
There is this, however, to be noted concerning the construction of the last two figures. Since the circle described by the centre of the generating circle is of a different arc or curve to that of the pitch circle, the length of an arc having an equal radius on each will be different. The amount is so small as to be practically correct. The direction of the error is to give to the curves a less curvature, as though they had been produced by a generating circle of larger diameter. Suppose, for example, that the difference between the arc a, b, and its chord is .1, and that the difference between the arc 4, 5, and its chord is .01, then the error in one step is .09, and, as the point f is formed in five steps, it will contain this error multiplied five times. Point d would contain it multiplied three times, because it has three steps, and so on.
The error will increase in proportion as the diameter of the generating is less than that of the pitch circle, and though in large wheels, working with large wheels, so that the difference between the radius of the generating circle and that of the smallest wheel is not excessive, it is so small as to be practically inappreciable, yet in small wheels, working with large ones, it may form a sensible error.
For showing the dimensions through the arms and hub, a sectional view of a section of the wheel may be given, as in Figure 240, which represents a section of a wheel, and a pinion, and on these two views all the necessary dimensions may be marked.
If it is desired to draw an edge view of a wheel (which the student will find excellent practice), the lines for the teeth may be projected from the teeth in the side view, as in Figure 240 a. Thus tooth E is projected by drawing lines from the corners A, B, C, in the side view across the face in the edge view, as at A, B, C in the latter view, and similar lines may be obtained in the same way for all the teeth.
When the teeth of wheels are to be cut to form in a gear-cutting machine, the thickness of the teeth is nearly equal to the thickness of the spaces, there being just sufficient difference to prevent the teeth of one wheel from becoming locked in the spaces of the other; but when the teeth are to be cast upon the wheel, the tooth thickness is made less than the width of the space to an amount that is usually a certain proportion of the pitch, and is termed the side clearance. In all wheels, whether with cut or cast teeth, there is given a certain amount of top and bottom clearance; that is to say, the points of the teeth of one wheel do not reach to the bottom of the spaces in the other. Thus in the Pratt and Whitney system the top and bottom clearance is one-eighth of the pitch, while in the Brown and Sharpe system for involute teeth the clearance is equal to one-tenth the thickness of the tooth.
In drawing bevil gear wheels, the pitch line of each tooth on each wheel, and the surfaces of the points, as well as those at the bottom of the spaces, must all point to a centre, as E in Figure 241, which centre is where the axes of the shafts would meet. It is unnecessary to mark in the correct curves for the teeth, for reasons already stated, with reference to the curves for a spur wheel. But if it is required to do so, the construction to find the curves is as shown in Figure 242, in which let A A represent the axis of one shaft, and B that of the other of the pair of bevil wheels that are to work together, their axes meeting at W; draw the line E at a right angle to A A, and representing the pitch circle diameter of one wheel, and draw F at a right angle to B, and representing the pitch circle of the other wheel; draw the line G G, passing through the point W and the point T, where the pitch circles or lines E F meet, and G G will be the line of contact of the tooth of one wheel upon the tooth of the other wheel; or in other words, the pitch line of the tooth.
Draw lines, as H and I, representing the tooth breadth. From W, as a centre, draw on each side of G G dotted lines, as P, representing the height of the tooth above and below the pitch line G G. At a right angle to G G draw the line J K; and from where this line meets B, as at Q, mark the arc a, which will represent the pitch circle for the large diameter of the pinion D. [The smallest wheel of a pair of gears is termed the pinion.] Draw the arc b for the height, and circle c for the depth of the teeth, thus defining the height of the tooth at that end. Similarly from P, as a centre mark (for the large diameter of wheel C,) arcs g, h, and i, arc g representing the pitch circle, i the height, and h the depth of the tooth. On these arcs draw the proper tooth curves in the same manner as for spur wheels; that is, obtain the curves by the construction shown in Figures 237, or by those in Figures 238 and 239.
To obtain the arcs for the other end of the tooth, draw line M M parallel to line J K; set the compasses to the radius R L, and from P, as a centre, draw the pitch circle k. For the depth of the tooth draw the dotted line p, meeting the circle h and the point W. A similar line, from i to W, will give the height of the tooth at its inner end. Then the tooth curves may be drawn on these three arcs, k, l, m, in the same as if they were for a spur wheel.
Similarly for the pitch circle of the inner and small end of the pinion teeth, set the compasses to radius S L, and from Q as a centre mark the pitch circle d. Outside of d mark e for the height above pitch lines of the tooth, and inside of d mark the arc f for the depth below pitch line of the tooth at that end. The distance between the dotted lines as p, represents the full height of the tooth; hence h meets p, which is the root of the tooth on the large wheel. To give clearance and prevent the tops of the teeth on one wheel from bearing against the bottoms of the spaces in the other wheel, the point of the pinion teeth is marked below; thus arc b does not meet h or p, but is short to the amount of clearance. Having obtained the arcs d, e, f, the curves may be marked thereon as for a spur wheel. A tooth thus marked is shown at x, and from its curves between b and c, a template may be made for the large diameter or outer end of the pinion teeth. Similarly for the wheel C the outer end curves are marked on the arcs g, h, i, and those for the other end of the tooth are marked between the arcs l, m.
Figure 243 represents a drawing of one-half of a bevil gear, and an edge view projected from the same. The point E corresponds to point E in Figure 241, or W in 242. The line F shows that the top surface of the teeth points to E. Line G shows that the pitch line of each tooth points to E, and lines H show that the bottom of the surface of a space also points to E. Line 1 shows that the sides of each tooth point to E. And it follows that the outer end of a tooth is both higher or deeper and also thicker than its inner end; thus J is thicker and deeper than end K of the tooth. Lines F G, representing the top and bottom of a tooth in Figure 243, obviously correspond to dotted lines p in Figure 242. The outer and inner ends of the teeth in the edge view are projected from the outer and inner ends in the face view, as is shown by the dotted lines carried from tooth L in the face view, to tooth L in the edge view, and it is obvious from what has been said that in drawing the lines for the tooth in the edge view they will point to the centre E.
To save work in drawing bevil gear wheels, they are sometimes drawn in section or in outline only; thus in Figure 244 is shown a pair of bevil wheels shown in section, and in Figure 245 is a drawing of a part of an Ames lathe feed motion. B C D and E are spur gears, while G H and I are bevil gears, the cone surface on which the teeth lie being left blank, save at the edges where a tooth is in each case drawn in. Wheel D is shown in section so as to show the means by which it may be moved out of gear with C and E. Small bevil gears may also be represented by simple line shading; thus in Figure 247 the two bodies A and C would readily be understood to be a bevil gear and pinion. Similarly small spur wheels
may be represented by simple circles in a side view and by line shading in an edge view; thus it would answer every practical purpose if such small wheels as in Figures 246 and 247 at D, F, G, K, P, H, I and J, were drawn as shown. The pitch circles, however, are usually drawn in red ink to distinguish them.
In Figure 248 is an example in which part of the gear is shown with teeth in, and the remainder is illustrated by circles.
In Figure 250 is a drawing of part of the feed motions of a Niles Tool Works horizontal boring mill, Figure 251 being an end view of the same, f is a friction disk, and g a friction pinion, g' is a rack, F is a feed-screw, p is a bevil pinion, and q a bevil wheel; i, m, o, are gear wheels, and J a worm operating a worm-pinion and the gears shown.
Figure 249 represents three bevil gears, the upper of which is line shaded, forming an excellent example for the student to copy.
The construction of oval gearing is shown in Figures 252, 253, 254, 255, and 256. The pitch-circle is drawn by the construction for drawing an ellipse that was given with reference to Figure 81, but as that construction is by means of arcs of circles, and therefore not strictly correct, Professor McCord, in an article on elliptical gearing, says, concerning it and the construction of oval gearing generally, as follows:
But these circular arcs may be rectified and subdivided with great facility and accuracy by a very simple process, which we take from Prof. Rankine's "Machinery and Mill Work," and is illustrated in Figure 252. Let O B be tangent at O to the arc O D, of which C is the centre. Draw the chord D O, bisect it in E, and produce it to A, making O A=O E; with centre A and radius A D describe an arc cutting the tangent in B; then O B will be very nearly equal in length to the arc O D, which, however, should not exceed about 60 degrees; if it be 60 degrees, the error is theoretically about 1/900 of the length of the arc, O B being so much too short; but this error varies with the fourth power of the angle subtended by the arc, so that for 30 degrees it is reduced to 1/16 of that amount, that is, to 1/14400. Conversely, let O B be a tangent of given length; make O F=1/4 O B; then with centre F and radius F B describe an arc cutting the circle O D G (tangent to O B at O) in the point D; then O D will be approximately equal to O B, the error being the same as in the other construction and following the same law.
The extreme simplicity of these two constructions and the facility with which they may be made with ordinary drawing instruments make them exceedingly convenient, and they should be more widely known than they are. Their application to the present problem is shown in Figure 253, which represents a quadrant of an ellipse, the approximate arcs C D, E, E F, F A having been determined by trial and error. In order to space this off, for the positions of the teeth, a tangent is drawn at D, upon which is constructed the rectification of D C, which is D G, and also that of D E in the opposite direction, that is, D H, by the process just explained. Then, drawing the tangent at F, we set off in the same manner F I = F E, and F K = F A, and then measuring H L = I K, we have finally G L, equal to the whole quadrant of the ellipse.
Let it now be required to lay out twenty-four teeth upon this ellipse; that is, six in each quadrant; and for symmetry's sake we will suppose that the centre of one tooth is to be at A, and that of another at C, Figure 253. We, therefore, divide L G into six equal parts at the points 1, 2, 3, etc., which will be the centres of the teeth upon the rectified ellipse. It is practically necessary to make the spaces a little greater than the teeth; but if the greatest attainable exactness in the operation of the wheels is aimed at, it is important to observe that backlash, in elliptical gearing, has an effect quite different from that resulting in the case of circular wheels. When the pitch-curves are circles, they are always in contact; and we may, if we choose, make the tooth only half the breadth of the space, so long as its outline is correct. When the motion of the driver is reversed, the follower will stand still until the backlash is taken up, when the motion will go on with a perfectly constant velocity ratio as before. But in the case of two elliptical wheels, if the follower stand
still while the driver moves, which must happen when the motion is reversed if backlash exists, the pitch-curves are thrown out of contact, and, although the continuity of the motion will not be interrupted, the velocity ratio will be affected. If the motion is never to be reversed, the perfect law of the velocity ratio due to the elliptical pitch-curve may be preserved by reducing the thickness of the tooth, not equally on each side, as is done in circular wheels, but wholly on the side not in action. But if the machine must be capable of acting indifferently in both directions, the reduction must be made on both sides of the tooth: evidently the action will be slightly impaired, for which reason the backlash should be reduced to a minimum. Precisely what is the minimum is not so easy to say, as it evidently depends much upon the excellence of the tools and the skill of the workman. In many treatises on constructive mechanism it is variously stated that the backlash should be from one-fifteenth to one-eleventh of the pitch, which would seem to be an ample allowance in reasonably good castings not intended to be finished, and quite excessive if the teeth are to be cut; nor is it very obvious that its amount should depend upon the pitch any more than upon the precession of the equinoxes. On paper, at any rate, we may reduce it to zero, and make the teeth and spaces equal in breadth, as shown in the figure, the teeth being indicated by the double lines. Those upon the portion L H are then laid off upon K I, after which these divisions are transferred to the ellipse by the second of Prof. Rankine's constructions, and we are then ready to draw the teeth.
The outlines of these, as of any other teeth upon pitch-curves which roll together in the same plane, depend upon the general law that they must be such as can be marked out upon the planes of the curves, as they roll by a tracing-point, which is rigidly connected with and carried by a third line, moving in rolling contact with both the pitch-curves. And since under that condition the motion of this third line, relatively to each of the others, is the same as though it rolled along each of them separately while they remained fixed, the process of constructing the generated curves becomes comparatively simple. For the describing line we naturally select a circle, which, in order to fulfil the condition, must be small enough to roll within the pitch ellipse; its diameter is determined by the consideration that if it be equal to A P, the radius of the arc A F, the flanks of the teeth in that region will be radial. We have, therefore, chosen a circle whose diameter, A B, is three-fourths of A P, as shown, so that the teeth, even at the ends of the wheels, will be broader at the base than on the pitch line. This circle ought strictly to roll upon the true elliptical curve; and assuming, as usual, the tracing-point upon the circumference, the generated curves would vary slightly from true epicycloids, and no two of those used in the same quadrant of the ellipse would be exactly alike. Were it possible to divide the ellipse accurately, there would be no difficulty in laying out these curves; but having substituted the circular arcs, we must now roll the generating circle upon these as bases, thus forming true epicycloidal teeth, of which those lying upon the same approximating arc will be exactly alike. Should the junction of two of these arcs fall within the breadth of a tooth, as at D, evidently both the face and the flank on one side of that tooth will be different from those on the other side; should the junction coincide with the edge of a tooth, which is very nearly the case at F, then the face on that side will be the epicycloid belonging to one of the arcs, its flank a hypocycloid belonging to the other; and it is possible that either the face or the flank on one side should be generated by the rolling of the describing circle partly on one arc, partly on the one adjacent, which, upon a large scale, and where the best results are aimed at, may make a sensible change in the form of the curve.
The convenience of the constructions given in Figure 252 is nowhere more apparent than in the drawing of the epicycloids, when, as in the case in hand the base and generating circles may be of incommensurable diameters; for which reason we have, in Figure 254, shown its application in connection with the most rapid and accurate mode yet known of describing those curves. Let C be the centre of the base circle; B, that of the rolling one; A, the point of contact. Divide the semi-circumference of B into six equal parts at 1, 2, 3, etc.; draw the common tangent at A, upon which rectify the arc A 2 by process No. 1; then by process No. 2 set out an equal arc A 2 on the base circle, and stepping it off three times to the right and left, bisect these spaces, thus making subdivisions on the base circle equal in length to those on the rolling one. Take in succession as radii the chords A 1, A 2, A 3, etc., of the describing circle, and with centres 1, 2, 3, etc., on the base circle, strike arcs either externally or internally, as shown respectively on the right and left; the curve tangent to the external arcs is the epicycloid, that tangent to the internal ones the hypocycloid, forming the face and flank of a tooth for the base circle.
In the diagram, Figure 253, we have shown a part of an ellipse whose length is ten inches, and breadth six, the figure being half size. In order to give an idea of the actual appearance of the combination when complete, we show in Figure 255 the pair in gear, on a scale of three inches to the foot. The excessive eccentricity was selected merely for the purpose of illustration. Figure 255 will serve also to call attention to another serious circumstance, which is, that although the ellipses are alike, the wheels are not; nor can they be made so if there be an even number of teeth, for the obvious reason that a tooth upon one wheel must fit into a space on the other; and since in the first wheel, Figure 255, we chose to place a tooth at the extremity of each axis, we must in the second one place there a space instead; because at one time the major axes must coincide; at another, the minor axes, as in Figure 255. If, then, we use even numbers, the distribution, and even the forms of the teeth, are not the same in the two wheels of the pair. But this complication may be avoided by using an odd number of teeth, since, placing a tooth at one extremity of the major axes, a space will come at the other.
It is not, however, always necessary to cut teeth all round these wheels, as will be seen by an examination of Figure 256, C and D being the fixed centres of the two ellipses in contact at P. Now P must be on the line C D, whence, considering the free foci, we see that P B is equal to P C, and P A to P D; and the common tangent at P makes equal angles with C P and P A, as is also with P B and P D; therefore, C D being a straight line, A B is also a straight line and equal to C D. If then the wheels be overhung, that is, fixed on the ends of the shafts outside the bearings, leaving the outer faces free, the moving foci may be connected by a rigid link A B, as shown.
This link will then communicate the same motion that would result from the use of the complete elliptical wheels, and we may therefore dispense with the most of the teeth, retaining only those near the extremities of the major axes, which are necessary in order to assist and control the motion of the link at and near the dead-points. The arc of the pitch-curves through which the teeth must extend will vary with their eccentricity; but in many cases it would not be greater than that which in the approximation may be struck about one centre; so that, in fact, it would not be necessary to go through the process of rectifying and subdividing the quarter of the ellipse at all, as in this case it can make no possible difference whether the spacing adopted for the teeth to be cut would "come out even" or not, if carried around the curve. By this expedient, then, we may save not only the trouble of drawing, but a great deal of labor in making, the teeth round the whole ellipse. We might even omit the intermediate portions of the pitch ellipses themselves; but as they move in rolling contact their retention can do no harm, and in one part of the movement will be beneficial, as they will do part of the work; for if, when turning, as shown by the arrows, we consider the wheel whose axis is D as the driver, it will be noted that its radius of contact, C P, is on the increase; and so long as this is the case the other wheel will be compelled to move by contact of the pitch lines, although the link be omitted. And even if teeth be cut all round the wheels, this link is a comparatively inexpensive and a useful addition to the combination, especially if the eccentricity be considerable. Of course the wheels shown in Figure 255 might also have been made alike, by placing a tooth at one end of the major axis and a space at the other, as above suggested. In regard to the variation in the velocity ratio, it will be seen, by reference to Figure 256, that if D be the axis of the driver, the follower will in the position there shown move faster, the ratio of the angular velocities being P × D/P × B; if the driver turn uniformly, the velocity of the follower will diminish, until at the end of half a revolution, the velocity ratio will be P × B/P × D; in the other half of the revolution these changes will occur in a reverse order. But P D = L B; if then the centres B D are given in position, we know L P, the major axis; and in order to produce any assumed maximum or minimum velocity ratio, we have only to divide L P into segments whose ratio is equal to that assumed value, which will give the foci of the ellipse, whence the minor axis may be found and the curve described. For instance, in Figure 255 the velocity ratio being nine to one at the maximum, the major axis is divided into two parts, of which one is nine times as long as the other; in Figure 256 the ratio is as one to three, so that the major axis being divided into four parts, the distance A C between the foci is equal to two of them, and the distance of either focus from the nearest extremity of the major axis is equal to one, and from the more remote extremity is equal to three of these parts.