Modern Machine-Shop Practice Part 2

revolutions of the 40 wheel, the discrepancy of 1/100 being due to the 6.66 leaving a remainder and not therefore being absolutely correct.

That the amount of power transmitted by gearing, whether compounded or not, is equal throughout every wheel in the train, may be shown as follows:--

[Ill.u.s.tration: Fig. 13.]

Referring again to Fig. 10, it has been shown that with a 50 lb. weight suspended from a 4 inch shaft E, there would be 30-33/100 lbs. at the perimeter of A. Now suppose a rotation be made, then the 50 lb. weight would fall a distance equal to the circ.u.mference of the shaft, which is (3.1416 4 = 12-56/100) 12-56/100 inches. Now the circ.u.mference of the wheel is (60 dia. 3.1416 = 188-49/100 cir.) 188-49/100 inches, which is the distance through which the 3-33/100 lbs. would move during one rotation of A. Now 3.33 lbs. moving through 188.49 inches represents the same amount of power as does 50 lbs. moving through a distance of 12.56 inches, as may be found by converting the two into inch lbs. (that is to say, into the number of inches moved by 1 lb.), bearing in mind that there will be a slight discrepancy due to the fact that the fractions .33 in the one case, and .56 in the other are not quite correct. Thus:

188.49 inches 3.33 lbs. = 627.67 inch lbs., and 12.56 " 50 " = 628 " "

Taking the next wheels in Fig. 12, it has been shown that the 3.33 lbs.

delivered from A to the perimeter of B, becomes 2.49 lbs. at the perimeter of C, and it has also been shown that C makes two revolutions to one of A, and its diameter being 40 inches, the distance this 2.49 lbs. will move through in one revolution of A will therefore be equal to twice its circ.u.mference, which is (40 dia. 3.1416 = 125.666 cir., and 125.666 2 = 251.332) 251.332 inches. Now 2.49 lbs. moving through 251.332 gives when brought to inch lbs. 627.67 inch lbs., thus 251.332 2.49 = 627.67. Hence the amount of power remains constant, but is altered in form, merely being converted from a heavy weight moving a short distance, into a lighter one moving a distance exactly as much greater as the weight or force is lessened or lighter.

[Ill.u.s.tration: Fig. 14.]

Gear-wheels therefore form a convenient method of either simply transmitting motion or power, as when the wheels are all of equal diameter, or of transmitting it and simultaneously varying its velocity of motion, as when the wheels are compounded either to reduce or increase the speed or velocity in feet per second of the prime mover or first driver of the train or pair, as the case may be.

[Ill.u.s.tration: Fig. 15.]

In considering the action of gear-teeth, however, it sometimes is more convenient to denote their motion by the number of degrees of angle they move through during a certain portion of a revolution, and to refer to their relative velocities in terms of the ratio or proportion existing between their velocities. The first of these is termed the angular velocity, or the number of degrees of angle the wheel moves through during a given period, while the second is termed the velocity ratio of the pair of wheels. Let it be supposed that two wheels of equal diameter have contact at their perimeters so that one drives the other by friction without any slip, then the velocity of a point on the perimeter of one will equal that of a point on the other. Thus in Fig. 15 let A and B represent the pitch circles of two wheels, and C an imaginary line joining the axes of the two wheels and termed the line of centres. Now the point of contact of the two wheels will be on the line of centres as at D, and if a point or dot be marked at D and motion be imparted from A to B, then when each wheel has made a quarter revolution the dot on A will have arrived at E while that on B will have arrived at F. As each wheel has moved through one quarter revolution, it has moved through 90 of angle, because in the whole circle there is 360, one quarter of which is 90, hence instead of saying that the wheels have each moved through one quarter of a revolution we may say they have moved through an angle of 90, or, in other words, their angular velocity has, during this period, been 90. And as both wheels have moved through an equal number of degrees of angle their velocity ratio or proportion of velocity has been equal.

Obviously then the angular velocity of a wheel represents a portion of a revolution irrespective of the diameter of the wheel, while the velocity ratio represents the diameter of one in proportion to that of the other irrespective of the actual diameter of either of them.

[Ill.u.s.tration: Fig. 16.]

Now suppose that in Fig. 16 A is a wheel of twice the diameter of B; that the two are free to revolve about their fixed centres, but that there is frictional contact between their perimeters at the line of centres sufficient to cause the motion of one to be imparted to the other without slip or lost motion, and that a point be marked on both wheels at the point of contact D. Now let motion be communicated to A until the mark that was made at D has moved one-eighth of a revolution and it will have moved through an eighth of a circle, or 45. But during this motion the mark on B will have moved a quarter of a revolution, or through an angle of 90 (which is one quarter of the 360 that there are in the whole circle). The angular velocities of the two are, therefore, in the same ratio as their diameters, or two to one, and the velocity ratio is also two to one. The angular velocity of each is therefore the number of degrees of angle that it moves through in a certain portion of a revolution, or during the period that the other wheel of the pair makes a certain portion of a revolution, while the velocity ratio is the proportion existing between the velocity of one wheel and that of the other; hence if the diameter of one only of the wheels be changed, its angular velocity will be changed and the velocity ratio of the pair will be changed. The velocity ratio may be obtained by dividing either the radius, pitch, diameter, or number of teeth of one wheel into that of the other.

Conversely, if a given velocity ratio is to be obtained, the radius, diameter, or number of teeth of the driver must bear the same relation to the radius, diameter, or number of teeth of the follower, as the velocity of the follower is desired to bear to that of the driver.

If a pair of wheels have an equal number of teeth, the same pairs of teeth will come into action at every revolution; but if of two wheels one is twice as large as the other, each tooth on the small wheel will come into action twice during each revolution of the large one, and will work during each successive revolution with the same two teeth on the large wheel; and an application of the principle of the hunting tooth is sometimes employed in clocks to prevent the overwinding of their springs, the device being shown in Fig. 17, which is from "Willis'

Principles of Mechanism."

For this purpose the winding arbor C has a pinion A of 19 teeth fixed to it close to the front plate. A pinion B of 18 teeth is mounted on a stud so as to be in gear with the former. A radial plate C D is fixed to the face of the upper wheel A, and a similar plate F E to the lower wheel B.

These plates terminate outward in semicircular noses D, E, so proportioned as to cause their extremities to abut against each other, as shown in the figure, when the motion given to the upper arbor by the winding has brought them into the position of contact. The clock being now wound up, the winding arbor and wheel A will begin to turn in the opposite direction. When its first complete rotation is effected the wheel B will have gained one tooth distance from the line of centres, so as to place the stop D in advance of E and thus avoid a contact with E, which would stop the motion. As each turn of the upper wheel increases the distance of the stops, it follows from the principle of the hunting cog, that after eighteen revolutions of A and nineteen of B the stops will come together again and the clock be prevented from running down too far. The winding key being applied, the upper wheel A will be rotated in the opposite direction, and the winding repeated as above.

[Ill.u.s.tration: Fig. 17.]

Thus the teeth on one wheel will wear to imbed one upon the other. On the other hand the teeth of the two wheels may be of such numbers that those on one wheel will not fall into gear with the same teeth on the other except at intervals, and thus an inequality on any one tooth is subjected to correction by all the teeth in the other wheel. When a tooth is added to the number of teeth on a wheel to effect this purpose it is termed a hunting cog, or hunting tooth, because if one wheel have a tooth less, then any two teeth which meet in the first revolution are distant, one tooth in the second, two teeth in the third, three in the fourth, and so on. The odd tooth is on this account termed a hunting tooth.

It is obvious then that the shape or form to be given to the teeth must, to obtain correct results, be such that the motion of the driver will be communicated to the follower with the velocity due to the relative diameters of the wheels at the pitch circles, and since the teeth move in the arc of a circle it is also obvious that the sides of the teeth, which are the only parts that come into contact, must be of same curve.

The nature of this curve must be such that the teeth shall possess the strength necessary to transmit the required amount of power, shall possess ample wearing surface, shall be as easily produced as possible for all the varying conditions, shall give as many teeth in constant contact as possible, and shall, as far as possible, exert a pressure in a direction to rotate the wheels without inducing undue wear upon the journals of the shafts upon which the wheels rotate. In cases, however, in which some of these requirements must be partly sacrificed to increase the value of the others, or of some of the others, to suit the special circ.u.mstances under which the wheels are to operate, the selection is left to the judgment of the designer, and the considerations which should influence his determinations will appear hereafter.

[Ill.u.s.tration: Fig. 18.]

Modern practice has accepted the curve known in general terms as the cycloid, as that best filling all the requirements of wheel teeth, and this curve is employed to produce two distinct forms of teeth, epicycloidal and involute. In epicycloidal teeth the curve forming the face of the tooth is designated an epicycloid, and that forming the flank an hypocycloid. An epicycloid may be traced or generated, as it is termed, by a point in the circ.u.mference of a circle that rolls without slip upon the circ.u.mference of another circle. Thus, in Fig. 18, A and B represent two wooden wheels, A having a pencil at P, to serve as a tracing or marking point. Now, if the wheels are laid upon a sheet of paper and while holding B in a fixed position, roll A in contact with B and let the tracing point touch the paper, the point P will trace the curve C C. Suppose now the diameter of the base circle B to be infinitely large, a portion of its circ.u.mference may be represented by a straight line, and the curve traced by a point on the circ.u.mference of the generating circle as it rolls along the base line B is termed a cycloid. Thus, in Fig. 19, B is the base line, A the rolling wheel or generating circle, and C C the cycloidal curve traced or marked by the point D when A is rolled along B. If now we suppose the base line B to represent the pitch line of a rack, it will be obvious that part of the cycloid at one end is suitable for the face on one side of the tooth, and a part at the other end is suitable for the face of the other side of the tooth.

[Ill.u.s.tration: Fig. 19.]

A hypocycloid is a curve traced or generated by a point on the circ.u.mference of a circle rolling within and in contact (without slip) with another circle. Thus, in Fig. 20, A represents a wheel in contact with the internal circ.u.mference of B, and a point on its circ.u.mference will trace the two curves, C C, both curves starting from the same point, the upper having been traced by rolling the generating circle or wheel A in one direction and the lower curve by rolling it in the opposite direction.

[Ill.u.s.tration: Fig. 20.]

To demonstrate that by the epicycloidal and hypocycloidal curves, forming the faces and flanks of what are known as epicycloidal teeth, motion may be communicated from one wheel to another with as much uniformity as by frictional contact of their circ.u.mferential surfaces, let A, B, in Fig. 21, represent two plain wheel disks at liberty to revolve about their fixed centres, and let C C represent a margin of stiff white paper attached to the face of B so as to revolve with it.

Now suppose that A and B are in close contact at their perimeters at the point G, and that there is no slip, and that rotary motion commenced when the point E (where as tracing point a pencil is attached), in conjunction with the point F, formed the point of contact of the two wheels, and continued until the points E and F had arrived at their respective positions as shown in the figure; the pencil at E will have traced upon the margin of white paper the portion of an epicycloid denoted by the curve E F; and as the movement of the two wheels A, B, took place by reason of the contact of their circ.u.mferences, it is evident that the length of the arc E G must be equal to that of the arc G F, and that the motion of A (supposing it to be the driver) would be communicated uniformly to B.

[Ill.u.s.tration: Fig. 21.]

Now suppose that the wheels had been rotated in the opposite direction and the same form of curve would be produced, but it would run in the opposite direction, and these two curves may be utilized to form teeth, as in Fig. 22, the points on the wheel A working against the curved sides of the teeth on B.

To render such a pair of wheels useful in practice, all that is necessary is to diminish the teeth on B without altering the nature of the curves, and increase the diameter of the points on A, making them into rungs or pins, thus forming the wheels into what is termed a wheel and lantern, which are ill.u.s.trated in Fig. 23.

[Ill.u.s.tration: Fig. 22.]

A represents the pinion (or lantern), and B the wheel, and C, C, the primitive teeth reduced in thickness to receive the pins on A. This reduction we may make by setting a pair of compa.s.ses to the radius of the rung and describing half-circles at the bottom of the s.p.a.ces in B.

We may then set a pair of compa.s.ses to the curve of C, and mark off the faces of the teeth of B to meet the half-circles at the pitch line, and reduce the teeth heights so as to leave the points of the proper thickness; having in this operation maintained the same epicycloidal curves, but brought them closer together and made them shorter. It is obvious, however, that such a method of communicating rotary motion is unsuited to the transmission of much power; because of the weakness of, and small amount of wearing surface on, the points or rungs in A.

[Ill.u.s.tration: Fig. 23.]

[Ill.u.s.tration: Fig. 24.]

In place of points or rungs we may have radial lines, these lines, representing the surfaces of ribs, set equidistant on the radial face of the pinion, as in Fig. 24. To determine the epicycloidal curves for the faces of teeth to work with these radial lines, we may take a generating circle C, of half the diameter of A, and cause it to roll in contact with the internal circ.u.mference of A, and a tracing point fixed in the circ.u.mference of C will draw the radial lines shown upon A. The circ.u.mstances will not be altered if we suppose the three circles, A, B, C, to be movable about their fixed centres, and let their centres be in a straight line; and if, under these circ.u.mstances, we suppose rotation to be imparted to the three circles, through frictional contact of their perimeters, a tracing point on the circ.u.mference of C would trace the epicycloids shown upon B and the radial lines shown upon A, evidencing the capability of one to impart uniform rotary motion to the other.

[Ill.u.s.tration: Fig. 25.]

To render the radial lines capable of use we must let them be the surfaces of lugs or projections on the face of the wheel, as shown in Fig. 25 at D, E, &c., or the faces of notches cut in the wheel as at F, G, H, &c., the metal between F and G forming a tooth J, having flanks only. The wheel B has the curves of each tooth brought closer together to give room for the reception of the teeth upon A. We have here a pair of gears that possess sufficient strength and are capable of working correctly in either direction.

[Ill.u.s.tration: Fig. 26.]

But the form of tooth on one wheel is conformed simply to suit those on the other, hence, neither two of the wheels A, nor would two of B, work correctly together.

They may be qualified to do so, however, by simply adding to the tops of the teeth on A, teeth of the form of those on B, and adding to those on B, and within the pitch circle, teeth corresponding to those on A, as in Fig. 26, where at K' and J' teeth are provided on B corresponding to J and K on A, while on A there are added teeth O', N', corresponding to O, N, on B, with the result that two wheels such as A or two such as B would work correctly together, either being the driver or either the follower, and rotation may occur in either direction. In this operation we have simply added faces to the teeth on A, and flanks to those on B, the curves being generated or obtained by rolling the generating, or curve marking, circle C upon the pitch circles P and P'. Thus, for the flanks of the teeth of A, C is rolled upon, and within the pitch circle P of A; while for the face curves of the same teeth C is rolled upon, but without or outside of P. Similarly for the teeth of wheel B the generating circle C is rolled within P' for the flanks and without for the faces. With the curves rolled or produced with the same diameter of generating circle the wheels will work correctly together, no matter what their relative diameter may be, as will be shown hereafter.

[Ill.u.s.tration: Fig. 27.]

In this demonstration, however, the curves for the faces of the teeth being produced by an operation distinct from that employed to produce the flank curves, it is not clearly seen that the curves for the flanks of one wheel are the proper curves to insure a uniform velocity to the other. This, however, may be made clear as follows:--

In Fig. 27 let _a_ _a_ and _b_ _b_ represent the pitch circles of two wheels of equal diameters, and therefore having the same number of teeth. On the left, the wheels are shown with the teeth in, while on the right-hand side of the line of centres A B, the wheels are shown blank; _a_ _a_ is the pitch line of one wheel, and _b_ _b_ that for the other.

Now suppose that both wheels are capable of being rotated on their shafts, whose centres will of course be on the line A B, and suppose a third disk, Q, be also capable of rotation upon its centre, C, which is also on the line A B. Let these three wheels have sufficient contact at their perimeters at the point _n_, that if one be rotated it will rotate both the others (by friction) without any slip or lost motion, and of course all three will rotate at an equal velocity. Suppose that there is fixed to wheel Q a pencil whose point is at _n_. If then rotation be given to _a_ _a_ in the direction of the arrow _s_, all three wheels will rotate in that direction as denoted by their respective arrows _s_.

a.s.sume, then, that rotation of the three has occurred until the pencil point at _n_ has arrived at the point _m_, and during this period of rotation the point _n_ will recede from the line of centres A B, and will also recede from the arcs or lines of the two pitch circles _a_ _a_, _b_ _b_. The pencil point being capable of marking its path, it will be found on reaching _m_ to have marked inside the pitch circle _b_ _b_ the curve denoted by the full line _m_ _x_, and simultaneously with this curve it has marked another curve outside of _a_ _a_, as denoted by the dotted line _y_ _m_. These two curves being marked by the pencil point at the same time and extending from _y_ to _m_, and _x_ also to _m_. They are prolonged respectively to _p_ and to K for clearness of ill.u.s.tration only.

The rotation of the three wheels being continued, when the pencil point has arrived at O it will have continued the same curves as shown at O _f_, and O _g_, curve O _f_ being the same as _m_ _x_ placed in a new position, and O _g_ being the same as _m_ _y_, but placed in a new position. Now since both these curves (O _f_ and O _g_) were marked by the one pencil point, and at the same time, it follows that at every point in its course that point must have touched both curves at once.

Now the pencil point having moved around the arc of the circle Q from _n_ to _m_, it is obvious that the two curves must always be in contact, or coincide with each other, at some point in the path of the pencil or describing point, or, in other words, the curves will always touch each other at some point on the curve of Q, and between _n_ and O. Thus when the pencil has arrived at _m_, curve _m_ _y_ touches curve K _x_ at the point _m_, while when the pencil had arrived at point O, the curves O _f_ and O _g_ will touch at O. Now the pitch circles _a_ _a_ and _b_ _b_, and the describing circle Q, having had constant and uniform velocity while the traced curves had constant contact at some point in their lengths, it is evident that if instead of being mere lines, _m_ _y_ was the face of a tooth on _a_ _a_, and _m_ _x_ was the flank of a tooth on _b_ _b_, the same uniform motion may be transmitted from _a_ _a_, to _b_ _b_, by pressing the tooth face _m_ _y_ against the tooth flank _m_ _x_. Let it now be noted that the curve _y_ _m_ corresponds to the face of a tooth, as say the face E of a tooth on _a_ _a_, and that curve _x_ _m_ corresponds to the flank of a tooth on _b_ _b_, as say to the flank F, short portions only of the curves being used for those flanks. If the direction of rotation of the three wheels was reversed, the same shape of curves would be produced, but they would lie in an opposite direction, and would, therefore, be suitable for the other sides of the teeth. In this case, the contact of tooth upon tooth will be on the other side of the line of centres, as at some point between _n_ and Q.

[Ill.u.s.tration: Fig. 28.]

In this ill.u.s.tration the diameter of the rolling or describing circle Q, being less than the radius of the wheels _a_ _a_ or _b_ _b_, the flanks of the teeth are curves, and the two wheels being of the same diameter, the teeth on the two are of the same shape. But the principles governing the proper formation of the curve remain the same whatever be the conditions. Thus in Fig. 28 are segments of a pair of wheels of equal diameter, but the describing, rolling, or curve-generating circle is equal in diameter to the radius of the wheels. Motion is supposed to have occurred in the direction of the arrows, and the tracing point to have moved from _n_ to _m_. During this motion it will have marked a curve _y_ _m_, a portion of the _y_ end serving for the face of a tooth on one wheel, and also the line _k_ _x_, a continuation of which serves for the flank of a tooth on the other wheel. In Fig. 29 the pitch circles only of the wheels are marked, _a_ _a_ being twice the diameter of _b_ _b_, and the curve-generating circle being equal in diameter to the radius of wheel _b_ _b_. Motion is a.s.sumed to have occurred until the pencil point, starting from _n_, had arrived at _o_, marking curves suitable for the face of the teeth on one wheel and for the flanks of the other as before, and the contact of tooth upon tooth still, at every point in the path of the teeth, occurring at some point of the arc _n_ _o_. Thus when the point had proceeded as far as point _m_ it will have marked the curve _y_ and the radial line _x_, and when the point had arrived at _o_, it will have prolonged _m_ _y_ into _o_ _g_ and _x_ into _o_ _f_, while in either position the point is marking both lines. The velocities of the wheels remain the same notwithstanding their different diameters, for the arc _n_ _g_ must obviously (if the wheels rotate without slip by friction of their surfaces while the curves are traced) be equal in length to the arc _n_ _f_ or the arc _n_ _o_.

[Ill.u.s.tration: Fig. 29.]