Modern Machine-Shop Practice Part 18

Second, 72/47.1 = 1.53 inch equal to the pitch.

This is nearly 1-1/2-inch pitch, and if possible the diameter would be reduced or the number of teeth increased so as to make the wheel exactly 1-1/2-inch pitch.

Rule 3.--Given ---- pitch and pitch diameter; to find ---- number of teeth.

First, ascertain from table the _pitch diameter_ for 1 _tooth_ of the given _pitch_.

Second, divide the _given pitch diameter_ by the _value_ found in table.

The quotient is the number required.

Example.--What is the number of teeth in a wheel whose pitch diameter is 42 inches, and pitch is 2-1/2 inches?

First, the pitch diameter, 1 tooth, 2-1/2-inch pitch, is 0.7958 inches.

42 Second. ------ = 52.8. Answer.

0.7958

This gives a fractional number of teeth, which is impossible; so the pitch diameter will have to be increased to correspond to 53 teeth, or the pitch changed so as to have the number of teeth come an even number.

Whenever two parallel shafts are connected together by gearing, the distance between centres being a fixed quant.i.ty, and the speeds of the shafts being of a fixed ratio, then the pitch is generally the best proportion to be changed, and necessarily may not be of standard size.

Suppose there are two shafts situated in this manner, so that the distance between their centres is 84 inches, and the speed of one is 2-1/2 times that of the other, what size wheels shall be used? In this case the pitch diameter and number of teeth of the wheel on the slow-running shaft have to be 2-1/2 times those of the wheel on the fast-running shaft; so that 84 inches must be divided into two parts, one of which is 2-1/2 times the other, and these quant.i.ties will be the pitch radii of the wheels; that is, 84 inches are to be divided into 3-1/2 equal parts, 1 of which is the radius of one wheel, and 2-1/2 of which the radius of the other, thus 84"/3-1/2 = 24 inches. So that 24 inches is the pitch radius of pinion, pitch diameter = 48 inches; and 2-1/2 24 inches = 60 inches is the pitch radius of the wheel, pitch diameter = 120 inches. The pitch used depends upon the power to be transmitted; suppose that 2-5/8 inches had been decided as about the pitch to be used, it is found by Rule 3 that the number of teeth are respectively 143.6, and 57.4 for wheel and pinion. As this is impossible, some whole number of teeth, nearest these in value, have to be taken, one of which is 2-1/2 times the other; thus 145 and 58 are the nearest, and the pitch for these values is found by Rule 2 to be 2.6 inches, being the best that can be done under the circ.u.mstances.

[Ill.u.s.tration: Fig. 183.]

[Ill.u.s.tration: Fig. 184.]

The forms of spur-gearing having their teeth at an angle to the axis, or formed in advancing steps shown in Figs. 183 and 184, were designed by Dr. Hooke, and "were intended," says the inventor, "first to make a piece of wheel work so that both the wheel and pinion, though of never so small a size, shall have as great a number of teeth as shall be desired, and yet neither weaken the wheels nor make the teeth so small as not to be practicable by any ordinary workman. Next that the motion shall be so equally communicated from the wheel to the pinion that the work being well made there can be no inequality of force or motion communicated.

"Thirdly, that the point of touching and bearing shall be always in the line that joins the two centres together.

"Fourthly, that it shall have _no manner of rubbing_, nor be more difficult to make than common wheel work."

The objections to this form of wheel lies in the difficulty of making the pattern and of moulding it in the foundry, and as a result it is rarely employed at the present day. For racks, however, two or more separate racks are cast and bolted together to form the full width of rack as shown in Fig. 185. This arrangement permits of the adjustment of the width of step so as to take up the lost motion due to the wear of the tooth curves.

Another objection to the sloping of the teeth, as in Fig. 183, is that it induces an end pressure tending to force the wheels apart _laterally_, and this causes _end_ wear on the journals and bearings.

[Ill.u.s.tration: Fig. 185.]

[Ill.u.s.tration: Fig. 186.]

To obviate this difficulty the form of gear shown in Fig. 186 is employed, the angles of the teeth from each side of the wheel to its centre being made equal so as to equalize the lateral pressure. It is obvious that the stepped gear, Fig. 184, is simply equivalent to a number of thin wheels bolted together to form a thick one, but possessing the advantage that with a sufficient number of steps, as in the figure, there is always contact on the line of centres, and that the condition of constant contact at the line of centres will be approached in proportion to the number of steps in the wheel, providing that the steps progress in one continuous direction across the wheel as in Fig.

184. The action of the wheels will, in this event, be smoother, because there will be less pressure tending to force the wheels apart.

But in the form of gearing shown in Fig. 183, the contact of the teeth will bear every instant at a single point, which, as the wheels revolve, will pa.s.s from one end to the other of the tooth, a fresh contact always beginning on the first side immediately before the preceding contact has ceased on the opposite side. The contact, moreover, being always in the plane of the centres of the pair, the action is reduced to that of rolling, and as there is no sliding motion there is consequently no rubbing friction between the teeth.

[Ill.u.s.tration: Fig. 187.]

[Ill.u.s.tration: Fig. 188.]

A further modification of Dr. Hooke's gearing has been somewhat extensively adopted, especially in cotton-spinning machines. This consists, when the direction of the motion is simply to be changed to an angle of 90, in forming the teeth upon the periphery of the pair at an angle of 45 to the respective axes of the wheels, as in Figs. 187 and 188; it will then be perceived that if the sloped teeth be presented to each other in such a way as to have exactly the same horizontal angle, the wheels will gear together, and motion being communicated to one axis the same will be transmitted to the other at a right angle to it, as in a common bevel pair. Thus if the wheel A upon a horizontal shaft have the teeth formed upon its circ.u.mference at an angle of 45 to the plane of its axis it can gear with a similar wheel B upon a vertical axis. Let it be upon the driving shaft and the motion will be changed in direction as if A and B were a pair of bevel-wheels of the ordinary kind, and, as with bevels generally, the direction of motion will be changed through an equal angle to the sum of the angles which the teeth of the wheels of the pair form with their respective axes. The objection in respect of lateral or end pressure, however, applies to this form equally with that shown in Fig. 183, but in the case of a vertical shaft the end pressure may be (by sloping the teeth in the necessary direction) made to tend to lift the shaft and not force it down into the step bearing. This would act to keep the wheels in close contact by reason of the weight of the vertical shaft and at the same time reduce the friction between the end of that shaft and its step bearing. This renders this form of gearing preferable to skew bevels when employed upon vertical shafts.

It is obvious that gears, such as shown in Figs. 187 and 188 may be turned up in the lathe, because the teeth are simply portions of spirals wound about the circ.u.mference of the wheel. For a pair of wheels of equal diameter a cylindrical piece equal in length to the required breadth of the two wheels is turned up in the lathe, and the teeth may be cut in the same manner as cutting a thread in the lathe, that is to say, by traversing the tool the requisite distance per lathe revolution.

In pitches above about 1/4 inch, it will be necessary to shape one side of the tooth at a time on account of the broadness of the cutting edges.

After the spiral (for the teeth are really spirals) is finished the piece may be cut in two in the lathe and each half will form a wheel.

To find the full diameter to which to turn a cylinder for a pair of these wheels we proceed as in the following example: Required to cut a spiral wheel 5 inches in diameter and to have 30 teeth. First find the diametral pitch, thus 30 (number of teeth) 5 (diameter of wheel at pitch circle) = 6; thus there are 6 teeth or 6 parts to every inch of the wheel's diameter at the pitch circle; adding 2 of these parts to the diameter of the wheel, at the pitch circle we have 5 and 2/6 of another inch, or 5-2/6 inches, which is the full diameter of the wheel, or the diameter of the addendum, as it is termed.

[Ill.u.s.tration: Fig. 189.]

[Ill.u.s.tration: Fig. 190.]

It is now necessary to find what change wheels to put on the lathe to cut the teeth out the proper angle. Suppose then the axes of the shafts are at a right angle one to the other, and that the teeth therefore require to be at an angle of 45 to the axes of the respective wheels, then we have the following considerations. In Fig. 189 let the line A represent the circ.u.mference of the wheel, and B a line of equal length but at a right angle to it, then the line C, joining A, B, is at an angle of 45. It is obvious then that if the traverse of the lathe tool be equal at each lathe revolution to the circ.u.mference of the wheel at the pitch circle, the angle of the teeth will be 45 to the axis of the wheel.

Hence, the change wheels on the lathe must be such as will traverse the tool a distance equal to the circ.u.mference at pitch circle of the wheel, and the wheels may be found as for ordinary screw cutting.

If, however, the axes of the shafts are at any other angle we may find the distance the lathe tool must travel per lathe revolution to give teeth of the required angle (or in other words the pitch of the spiral) by direct proportion, thus: Let it be required to find the angle or pitch for wheels to connect shafts at an angle of 25, the wheels to have 20 teeth, and to be of 10 diametral pitch.

Here, 20 10 = 2 = diameter of wheel at the pitch circle. The circ.u.mference of 2 inches being 6.28 inches we have, as the degrees of angle of the axes of the shafts are to 45, so is 6.28 inches (the circ.u.mference of the wheels, to the pitch sought).

Here, 6.28 inches 45 25 = 11.3 inches, which is the required pitch for the spiral.

When the axes of the shafts are neither parallel nor meeting, motion from one shaft to another may be transmitted by means of a double gear.

Thus (taking rolling cones of the diameters of the respective pitch circles as representing the wheels) in Fig. 190, let A be the shaft of gear _h_, and B _b_ that of wheel _e_. Then a double gear-wheel having teeth on _f_, _g_ may be placed as shown, and the face _f_ will gear with _e_, while face _g_ will gear with _h_, the cone surfaces meeting in a point as at C and D respectively, hence the velocity will be equal.

When the axial line of the shafts for two gear-wheels are nearly in line one with the other, motion may be transmitted by gearing the wheels as in Fig. 191. This is a very strong method of gearing, because there are a large number of teeth in contact, hence the strain is distributed by a larger number of teeth and the wear is diminished.

[Ill.u.s.tration: Fig. 191.]

[Ill.u.s.tration: Fig. 192.]

Fig. 192 (from Willis's "Principles of Mechanism") is another method of constructing the same combination, which admits of a steady support for the shafts at their point of intersection, A being a spherical bearing, and B, C being cupped to fit to A.

Rotary motion variable at different parts of a rotation may be obtained by means of gear-wheels varied in form from the true circle.

[Ill.u.s.tration: Fig. 193.]

The commonest form of gearing for this purpose is elliptical gearing, the principles governing the construction of which are thus given by Professor McCord. "It is as well to begin at the foundation by defining the ellipse as a closed plane-curve, generated by the motion of a point subject to the condition that the sum of its distances from two fixed points within shall be constant: Thus, in Fig. 193, A and B are the two fixed points, called the _foci_; L, E, F, G, P are points in the curve; and A F + F B = A E + E B. Also, A L + L B = A P + P B = A G + G B. From this it follows that A G = L O, O being the centre of the curve, and G the extremity of the minor axis, whence the foci may be found if the axes be a.s.sumed, or, if the foci and one axis be given, the other axis may be determined. It is also apparent that if about either focus, as B, we describe an arc with a radius greater than B P and less than B L, for instance B E, and about A another arc with radius A E = L P-B E, the intersection, E, of these arcs will be on the ellipse; and in this manner any desired number of points may be found, and the curve drawn by the aid of sweeps.

"Having completed this ellipse, prolong its major axis, and draw a similar and equal one, with its foci, C, D, upon that prolongation, and tangent to the first one at P; then B D = L P. About B describe an arc with any radius, cutting the first ellipse at Y and the line L at Z; about D describe an arc with radius D Z, cutting the second ellipse in X; draw A Y, B Y, C X, and D X. Then A Y = D X, and B Y = C X, and because the ellipses are alike, the arcs P Y and P X are equal. If then B and D are taken as fixed centres, and the ellipses turn about them as shown by the arrows, X and Y will come together at Z on the line of centres; and the same is true of any points equally distant from P on the two curves. But this is the condition of rolling contact. We see, then, that in order that two ellipses may roll together, and serve as the pitch-lines of wheels, they must be equal and similar, the fixed centres must be at corresponding foci, and the distance between these centres must be equal to the major axis. Were they to be toothless wheels, if would evidently be essential that the outlines should be truly elliptical; but the changes of curvature in the ellipse are gradual, and circular arcs may be drawn so nearly coinciding with it, that when teeth are employed, the errors resulting from the subst.i.tution are quite inappreciable. Nevertheless, the rapidity of these changes varies so much in ellipses of different proportions, that we believe it to be practically better to draw the curve accurately first, and to find the radii of the approximating arcs by trial and error, than to trust to any definite rule for determining them; and for this reason we give a second and more convenient method of finding points, in connection with the ellipse whose centre is R, Fig. 193. About the centre describe two circles, as shown, whose diameters are the major and minor axes; draw any radius, as R T, cutting the first circle in T, and the second in S; through T draw a parallel to one axis, through S a parallel to the other, and the intersection, V, will lie on the curve. In the left hand ellipse, the line bisecting the angle A F B is normal to the curve at F, and the perpendicular to it is tangent at the same point, and bisects the angles adjacent to A F B, formed by prolonging A F, B F.

[Ill.u.s.tration: Fig. 194.]

"To mark the pitch line we proceed as follows:--

"In Fig. 194, A A and B B are centre lines pa.s.sing through the major and minor axes of the ellipse, of which _a_ is the axis or centre, _b_ _c_ is the major and _a_ _e_ half of the minor axis. Draw the rectangle _b_ _f_ _g_ _c_, and then the diagonal line _b_ _e_; at a right angle to _b_ _e_ draw line _f_ _h_ cutting B B at _i_. With radius _a_ _e_ and from _a_ as a centre draw the dotted arc _e_ _j_, giving the point _j_ on the line B B. From centre _k_, which is on line B B, and central between _b_ and _j_, draw the semicircle _b_ _m_ _j_, cutting A A at _l_. Draw the radius of the semicircle _b_ _m_ _j_ cutting _f_ _g_ at _n_. With radius _m_ _n_ mark on A A, at and from _a_ as a centre, the point _o_. With radius _h_ _o_ and from centre _h_ draw the arc _p_ _o_ _q_. With radius _a_ _l_ and from _b_ and _c_ as centres draw arcs cutting _p_ _o_ _q_ at the points _p_ _q_. Draw the lines _h_ _p_ _r_ and _h_ _q_ _s_, and also the lines _p_ _i_ _t_ and _q_ _v_ _w_. From _h_ as centre draw that part of the ellipse lying between _r_ and _s_. With radius _p_ _r_ and from _p_ as a centre draw that part of the ellipse lying between _r_ and _t_.

With radius _q_ _s_ and from _q_ draw the ellipse from _s_ to _w_. With radius _i_ _t_ and from _i_ as a centre draw the ellipse from _t_ to _b_. With radius _v_ _w_ and from _v_ as a centre draw the ellipse from _w_ to _c_, and one half the ellipse will be drawn. It will be seen that the whole construction has been performed to find the centres _h_ _p_ _q_ _i_ and _v_, and that while _v_ and _i_ may be used to carry the curve around the other side or half of the ellipse, new centres must be provided for _h_ _p_ and _q_; these new centres correspond in position to _h_ _p_ _q_.

"If it were possible to subdivide the ellipse into equal parts it would be unnecessary to resort to these processes of approximately representing the two curves by arcs of circles; but unless this be done, the s.p.a.cing of the teeth can only be effected by the laborious process of stepping off the perimeter into such small subdivisions that the chords may be regarded as equal to the arcs, which after all is but an approximation; unless, indeed, we adopt the mechanical expedient of cutting out the ellipse in metal or other substance, measuring and subdividing it with a strip of paper or a steel tape, and wrapping back the divided measure in order to find the points of division on the curve.