CHAPTER XII

TRAINS OF MECHANISM.

Value of ^a Train. — Directional Relation in Trains. — Clockwork. — Notation. — Method of designing Trains. — Approximate Numbers for Trains.

221. The required velocity ratio of two motions being given, it is always theoretically possible to obtain this ratio by the use of one of the elementary combinations described in the previous chapters. It often happens, however, that this ratio is so small, or so large, that, practically, the motion is better communicated by a train of such combinations ; each piece being at the same time the follower of the piece that drives it, and the driver of the piece that follows it.

For convenience, let us first consider the case in which all the pieces are circular wheels revolving about fixed axes. The usual arrangement in such cases is to secure two unequal wheels upon each axis, except the first and last, and to make the larger wheel on each axis gear with the smaller wheel on the next axis.

222. Value of ^a Train Let there be m such axes, and let us designate by ^ethe value of the train ; that is, the ratio of the angular velocities of the first and last axes, or, what amounts to the same thing, the ratio of their synchronal rotations.

Let $a_1, a_2, a_3, \ldots a_n$ be the angular velocities of the successive axes. Then we have

That is, the value of the train may be found by multiplying together the separate ratios of the angular velocities of the successive pairs of axes.

Again, let the synchronal rotations of the successive axes of the train be $L_1, L_2, L_3, \ldots, L_m$ . Then we have

That is, the value of the train may be found by multiplying together the separate ratios of the synchronal rotations of the successive pairs of axes. The value of $\epsilon$ is, of course, the same in both the above equations. This value will not be affected by the substitution, for any of the intermediate ratios, of any other two numbers that are in the same proportion; hence we may express the values of those ratios in the terms that may most easily be obtained from the train whose motions we wish to consider.

Letting a = angular velocity of one of two wheels in gear, R its radius, N its number of teeth, P its period, or time of one rotation, and L its number of rotations in a given time; and letting a’, R’, N’, P’, and L’ be the corresponding quantities for the other wheel, we have (Art. 35),

which equation will enable us to write the proper ratio in each case.

For instance, let iVj, JVj, JVj, . . . iV;“_i be the numbers of teeth of the drivers on the successive axes, and let n.^, n^, n^, . . . m” be the numbers of teeth of the corresponding followers. Then we may write, by making the proper substitutions for the intermediate ratios in Equation (1),

That is, the value of the train is equal to the quotient obtained by dividing the continued product of the numbers of teeth of all the drivers by the continued product of the numbers of teeth of all the followers.

It is obvious, that, in a train of this kind, the number of drivers, as well as the number of followers, is always one less than the whole number of axes.

223. Practical Example. —It is not necessary that all the ratios should be expressed in the same terms. As before stated, it is simply necessary to use, for each ratio, two numbers in the proper proportion.

For example, let there be a train of six axes, connected as above described.

Let the first axis revolve once per minute, and let the second axis revolve once in fifteen seconds. Hence

Let the second axis revolve three times while the third revolves five times. Hence

Let the third axis carry a wheel of sixty teeth, driving a wheel of twenty-four teeth on the fourth axis. Hence

Let the fourth axis carry a pulley of twenty-four inches diameter, driving, by means of a belt, a pulley of twelve inches diameter on the fifth axis. Hence

Let the fifth axis turn with an angular velocity two-thirds as great as that of the sixth axis. Hence

Substituting these ratios for the successive terms of Equation (1), we get

Hence, if the train consists of an odd number of axes, the first and last axes will revolve in the same direction ; if it consists of an even number of axes, they will revolve in opposite directions.

In this connection, it must be remembered that an annular wheel (Art. 36) revolves in the same direction as its pinion.

When the axes in ^a train are not parallel, the directional relation of the extreme axes can only be ascertained by tra cing the separate directional relations of each successive pair of axes in order.

Two separate wheels in a train may revolve concentrically about the same axis ; as, for example, the wheels to which are attached the hands of a clock. In this case, one of the wheels is fixed on the axis as usual, and the other is fixed on a tube, or cannon as it is sometimes called, which revolves freely on the first axis.

If these wheels are to move in opposite directions, a single bevel wheel may be used to connect them ; but if they are to turn in the same direction, as in a clock, they must be made in the form of spur-wheels, and connected by means of two other spur-wheels fixed to an axis parallel to the first.

225. Idle Wheel. — Let ^a spur-wheel be placed between and in gear with two other spur-wheels. Let the radii of the first, middle, and last wheels be R.^, R^, R^, and let their angular velocities be a^, a^, a^. Then we have, for the first and middle wheels,

That is, the velocity ratio of the two extreme wheels is precisely the same as though they were in immediate contact. The intermediate wheel is called an idle wheel; and, though it does not affect the velocity ratio, it does affect the directional relation. For, if the two extreme wheels were in direct contact, they would revolve in opposite directions; but, by the introduction of the idle wheel, they are eaused to revolve in the same direction.

226. Clockwork. — A familiar example of the employment of a train of wheels is afforded by a common clock.

In Fig. 169 is shown the arrangement of the wheels in a clock of the simplest kind. A is the barrel, and around it is wound a cord to the end of which is fastened the weight W. On the same axis with A is fixed the spur wheel B, which gears with the pinion b on a second axis. On the latter is

also fixed the spur wheel C, gearing with the pinion c on the third axis. This axis also carries an escapement wheel D(Art. 180), the verge or anchor d being fixed to the fourth axis, to which the pendulum is also hung at e. One tooth of the escape-wheel crosses the line of centres for every two vibrations of the pendulum. Let the time of one vibration of the pendulum be t seconds, and let the escape-wheel have $\Delta$ teeth; then the period or time of one complete rotation of this wheel is $2t\Delta$ seconds. To take a simple case, let the pendulum be a seconds pendulum; then t=1, and if $\Delta=$ 30, the swing-wheel will make one complete revolution in $2t\Delta = 2 \times 30 = 60$ seconds = 1 minute. Let B have 48 teeth; b, 6 teeth; C, 45 teeth; and c, 6 teeth. Then we have, for the value of the train connecting the barrel axis and the escapement axis,

That is, the escapement axis (or arbor, using the term employed by clockmakers) will make sixty revolutions while the barrel arbor makes one. Hence the barrel arbor will revolve once in sixty minutes, or one hour. The barrel $\Delta$ is not permanently secured to this arbor, but is connected to it, or to the wheel B, by means of a click and ratchet (Art. 201); so that, while it is free to move in one direction, its rotation in the other direction compels the wheel B to rotate with it. This arrangement permits the barrel to be rotated so as to wind up the cord without affecting the rest of the train. The number of times that the cord is wound round the barrel evidently depends on the length of time that the clock is to run without being wound. Generally not over sixteen coils of cord are so employed, which, in our clock, as the barrel arbor revolves once an hour, would be sufficient to make the clock run sixteen hours without re-winding.

227. The train of wheel-work just described is solely destined for the purpose of communicating the action of the weight to the pendulum in such a manner as to supply the loss of motion from friction and the resistance of the air. But besides this, the clock is required to indicate the hours and minutes by the rotation of two separate hands, and accordingly two other trains of wheel-work are employed for this purpose. The train just described is generally contained in a frame, consisting of two plates, shown edgewise at kl, mn, which are kept parallel and at the proper distance by means of three or four pillars not shown in the diagram. Opposite holes are drilled in these plates, which receive the pivots of the axes or arbors already described. But the axis which carries A and B projects through the plate, and other wheels E and F are fixed to it. Below this axis, and parallel to it, a stout pin or stud is fixed to the plate. On this stud revolves a tube, to one end of which is fixed the minute-hand M, and to the other the wheel e in gear with E. our present clock, the wheel E, being fixed to the barrel arbor, revolves once an hour; and as the minute-hand must also revolve once in that period, the wheel E and e must be equal. A second and shorter tube is fitted upon the tube of the minute-hand so as to revolve freely, and this carries at one end the hour-hand H, and at the other a wheel, f, which is driven by the pinion F. As f must revolve once in twelve hours, it must have twelve times as many teeth as F.
228. Notation. In discussing problems concerning trains of mechanism, we soon feel the need of some scheme of notation, whereby we may show, clearly and concisely, all the facts concerning the train which affect the transmission of motion. It is desirable to show, primarily, the order and nature of the several parts, and the manner in which the motion is transmitted; but such a scheme should also admit of the addition of dimensions and nomenclature, and should afford a ready means of calculating the velocity ratio.

Let the wheels be represented by their numbers of teeth, and write these numbers, beginning with the first driver, in horizontal lines; all the wheels that are on the same axis having their numbers written on the same horizontal line, and all the wheels that are in gear having the numbers of the followers written vertically below those of the respective drivers.

229. Example. — Thus, in the principal train of the clock (Fig. 169), if the letters represent the wheels, we should write the train thus:—

or, employing the numbers already selected,

Similarly we may represent the whole mechanism of our clock, adding to the numbers the names wherever it may be thought necessary. Thus—

The above shows clearly the three trains of mechanism from the barrel to the swing-wheel, the minute-hand, and the hour-hand respectively. It also distinctly classifies the pieces as drivers or followers, as the case may be, and shows the nature of their connection; that is, whether they are permanently fixed to the same axis, or connected by gearing. In case other connections are employed, such as links or bands, this must be written in the diagram, or expressed by a proper sign.

230. Method of Designing Trains. — We are now ready to undertake the solution of a problem of considerable importance in the contrivance of mechanism; namely, Given the velocity ratio of the extreme axes or pieces of a train, to determine the number of intermediate axes, and the proportions of the wheels, or numbers of their teeth. For simplicity, we will suppose the train to consist of toothed wheels only; for a mixed train, consisting of wheels, pulleys, linkwork, and sliding-pieces, can be calculated upon the same principles. Let the synchronal rotations of the first and last axes of the train be $L_1$ and $L_m$ respectively, and let $N_1$ , $N_2$ , . . . etc., be the numbers of teeth in the drivers, and $n_2$ , $n_3$ , etc., the number of teeth in the followers; then the value of the train is

both numerator and denominator of this fraction being composed of m-1 terms.

The value of $\epsilon$ being given in this shape, an equal fraction must be found, whose numerator and denominator shall each admit of being divided into m-1 factors of convenient magnitude for the number of teeth of a wheel.

The value of m, that is, the number of axes, is sometimes given with the other data of the problem, but more usually it is one of the quantities that are to be determined.

The order of succession of the drivers and followers is a matter of indifference, so far as the velocity ratio is con cerned ; for the value of the above fraction will evidently not vary with any variation in the order of the factors of either the numerator or denominator.

231. Least Number of Axes. — The number of axes will evidently depend upon the limits between which the numbers of teeth are to be allowed to vary.

For instance, let w be the greatest number of teeth that can be conveniently assigned to a wheel, and let p be the least that can be given to a pinion. Now, in any given case, let us suppose L,^ greater than i,, so that the wheels will be the drivers, and the pinions the followers. The least number of axes will then evidently be obtained by giving each wheel w teeth, and each pinion p teeth. The number of axes being m, we will have (Art. 222) m — ¹ wheels and m — ¹ pinions. Hence

whence loge = (m — 1) (logw — log^),

The least number of axes, under the assigned conditions of w and p^ is evidently the value of m thus found, if this value is a whole number ; or the whole number next larger than this value of m, if the latter is fractional. No general rule can be given for determining the values of w and p, which are governed by considerations that vary according to the nature of the proposed machine ; also, it will rarely happen

that the fraction will admit of being divided into factors so nearly equal as to limit the number of axes to the smallest value so assigned.

233. Practical Example of Clock Train We will now return to the consideration of the clock described in Art. 226, and show how the number of axes and the number of teeth of the wheels and pinions were determined. It was required that the first or barrel axis should revolve once per hour, and that the m”* or swing-wheel axis should carry a seconds hand, S. The swing-wheel axis must therefore revolve once per minute, or sixty times per hour.

Consequently

Let D be the numerator of this fraction, i.e., the continued product of ail the drivers, and let F be the denominator, i.e., the continued product of all the followers.

Then

$\epsilon = 60 = \frac{D}{F}$ $\therefore$ $D = 60 \times F$ ,

an indeterminate equation, for the solution of which any numbers may be employed that are within the assigned limits of w and p. Now, in ordinary clocks, w = 60, and p = Q, so that

From Equation (5) , we have

We can then determine the value of m by means of Equation (6) ; or, what is much simpler, determine, by inspection, the

value of m-1 from the above expression. The latter method is to be preferred, as the exact value of m-1, if it be fractional, is of no consequence, it being simply necessary to determine the next greater whole number.

Thus, in our example, it is evident, as 60 lies between $10^1$ and $10^2$ , that m-1 must lie between 1 and 2, consequently m must lie between 2 and 3; and, taking the next larger whole number, we fix on m=3, as the least number of axes. Consequently there will be two wheels and two pinions. Taking the pinions at six teeth each, we have

which is the product of the two wheels.

We are at liberty to divide this into any two suitable factors. The best mode of doing it is to begin by dividing the number into its prime factors, and writing it in this form,

For this enables us to see clearly the composition of the number, and it is easy to distribute these factors into two groups; as, for example,

$(2 \times 2 \times 2 \times 2 \times 3) \times (3 \times 3 \times 5) = 48 \times 45,$ or $(2 \times 2 \times 2 \times 5) \times (2 \times 3 \times 3 \times 3) = 40 \times 54,$ or

$(2 \times 2 \times 3 \times 3)$ $\times (2 \times 2 \times 3 \times 5) = 36 \times 60.$

The first group will give us the two wheels that are most

nearly equal, whick is a sufficient reason for selecting that pair for our train. We now have

So far we have only determined on the numbers of the teeth of the various wheels, without locating them as regards the different axes; and the above fractional expression is an excellent method of exhibiting the train under these conditions.

As before stated, the order in which the wheels come is a matter of indifference, so far as the velocity ratio is concerned; and, as no other considerations enter into this case, we will place driver 48 on the first axis, follower 6 and driver 45 on the second axis, follower 6 and swing-wheel 30 on the third axis, giving us, as in Art. 229, the train

233. Another Clock Train.—Six is, however, too small a number of leaves for the pinion, if perfect action is desired; for it is evident, from the table of Art. 134, that a pinion of 6 teeth cannot drive a wheel of less than 21 teeth, if the arc of recess equal two-thirds pitch; while, if this arc is increased to three-fourths pitch, a pinion of 6 cannot be made to work at all. In well-made clocks, p is generally taken between 8 and 12, while w ranges from 100 to 120.

Let us find a new train for our clock, having p = 12, and w = 105.

We have

from which we see, by inspection, that the value of m-1 is

fractional, and lies between 1 and 2; that the value of m lies between 2 and 3; and that the least number of axes will consequently be 3. Assuming the two pinions to be equal, and to have the smallest allowable number of teeth, we have

Proceeding as in the last example, we find the best values for the wheels to be $D=96\times 90$ . We then have

and, placing them on their axes, we have the train

Instead of assuming the pinions, we might have started with the wheels. Thus let us take

It is evidently impossible to divide 183.75 into two integer factors; and, as we cannot increase the assumed number of teeth for the wheels, we must diminish the number of one or both. Let us take one of the wheels as 104. This will give us

which can readily be factored, giving us $F=13\times 14$ , and the train

It very often happens, as just illustrated, that attempting to make the wheels and pinions with the limiting numbers of teeth gives rise to very awkward results, while an excellent train can, in such cases, be generally found by trying several numbers within the limits.

234. Clock with rapidly vibrating Pendulum.—If a clock has no seconds hand, the limitation as to the period of one revolution of the swing-wheel axis is removed. This is an advantage in clocks having short, and consequently rapidly vibrating, pendulums; for it would be impracticable to make the period of the swing-wheel axis one minute, as before, on account of the great number of teeth which would be required for the swing-wheel. If t =time of vibration of the pendulum in seconds, and $\Delta =$ number of teeth in the swing-wheel, then (as in Art. 226) $2t\Delta$ is the time required for one revolution of the swing-wheel.

But the vibrations of short pendulums are commonly expressed by stating the number of them in a minute. Let S be this number; then $\frac{2\Delta}{S}$ is the time of one revolution of

the swing-wheel in minutes; $\frac{S}{2\Delta}$ is the number of revolutions of the swing-wheel axis per minute; and, as the barrel arbor revolves once per hour, we have for the train between them,

For example, let the pendulum of a clock make 170 vibrations per minute; let there be 25 teeth on the swing-wheel; then

Taking w = 128, and p = 8, we have

and, as $204 = (16)^{m-1}$ , we see, by inspection, that the least number of axes is 3.

Assuming the pinions as each having 8 teeth, we have

Hence the train is

235. Eight-Day Clock.—All the trains so far explained were designed to establish the proper velocity ratio between the hour arbor and the swing-wheel axis. It was assumed in each case that the hour arbor also carried the weight-barrel; and, as we limited the number of coils of the cord to 16, it follows that the clocks so far considered will only run 16 hours without re-winding.

If we adhere to the limitation as to the number of coils of the cord, but still desire the clock to run longer than 16 hours, the barrel must be attached to a separate axis connected by wheel-work with the hour arbor, so that the barrel may revolve more slowly, consequently taking more time to uncoil all the cord.

For example, let the clock be required to go 8 days without re-winding; then, with 16 eoils of cord on the barrel, the latter must revolve once in $\frac{8 \times 24}{16} = 12$ hours. Then, assuming w = 100, and p = 8, we may use the train, —

It is often convenient to add to the notation the periods of the different arbors, as has been done in this ease.

236. Month Clock. — Let the clock be required to run 32 days without re-winding, and let there be 16 eoils on the barrel as before; then the latter must revolve once in $\frac{32 \times 24}{16} = 48$ hours. The train from the barrel to the

hour arbor is $\frac{D}{F}$ = 48, which will require an intermediate axis.

Letting w = 100, and p = 12, we may employ the following train:—

237. Now, in the clock (Fig. 169), the arbor of A is made to revolve in one hour, because the wheels E and e are equal. By making these wheels of different numbers, we get rid of the necessity of providing, in the principal train, an arbor that shall revolve in one hour; and we may thus, in many cases, distribute the wheels more equally. For example, in an eight-day clock let the swing-wheel revolve once

per minute, and let the train from the barrel-arbor to this minute-arbor be

in which case the barrel will revolve once in 810 minutes, or $13\frac{1}{2}$ hours.

The second wheel of this train, which, in Fig. 169, corresponds to D, will revolve in $\frac{12}{108} \times 810 = 90$ minutes, or $1\frac{1}{2}$ hours. On its arbor must be fixed, as in the figure, the wheels E and F for the minute and hour hands; and we may employ, for the two pairs of wheels,

$\frac{F}{f} = \frac{1\frac{1}{2}}{12} = \frac{1}{8} = \frac{10}{80};$ and $\frac{E}{e} = \frac{1\frac{1}{2}}{1} = \frac{3}{2} = \frac{54}{36}.$

So that our train will be as follows: -

Periods.

12-108 - - - - 54 - - 10 . . . . . . . . . . . . . . . . . .

238. The above examples have been confined to clockwork, because the action is more generally understood than that of other machines. The principles and methods are, however, universally applicable, or, at least, require very slight modifications to adapt them to particular cases.

For instance, in a screw-cutting lathe, there is usually one intermediate axis between the leading-screw and the head-stock spindle. Let the leading-screw be right-handed, and

have two threads to the inch; let $w=130,\ p=20$ ; and let it be required to cut a right-handed screw of 13 threads to the inch. Here

which is a good train for the purpose. The wheels for forming a series of such trains, calculated for the different numbers of threads to be produced, are known as a set of change-wheels; and tables for the use of such wheels are furnished by lathe-manufacturers with all screw-cutting lathes.

239. Frequency of Contact between Teeth.—It is sometimes a matter of interest to know how often any two given teeth will come into contact as the wheels rnn upon each other. We will take the case of a wheel of A teeth driving one of B teeth, where A is greater than B, and let $\frac{A}{B} = \frac{a}{b}$ when reduced to its lowest terms.

It is evident that the same points of the two pitch circles would be in contact after a revolutions of B, or b revolutions of A. Hence, the smaller the numbers which express the velocity ratio of the two axes, the more frequently will the contact of the same teeth occur.

1. Let it be required to bring the same teeth into contact as often as possible.

Since this contact occurs after b revolutions of A, or a revolutions of B, we shall effect our object by making a and b as small as possible; this is, by providing that A and B shall have a large common divisor.

For example, assume that the comparative angular velocity of the two axes is intended to be as nearly as possible as 5 to 2. Now make A = 80, B = 32; then

or, the same pair of teeth will come in contact after 5 revolutions of B, or 2 of A.

2. Let it be required to bring the same teeth into contact as seldom as possible.

Now change A to 81, and we shall have $\frac{A}{B} = \frac{81}{32} = \frac{5}{2}$ very nearly; or, the angular velocity of A relatively to B will be scarcely distinguishable from what it was originally. But the alteration will effect what we require, for now $\frac{a}{b} = \frac{81}{32}$ . There will, therefore, be a contact of the same pair of teeth

There will, therefore, be a contact of the same pair of teeth only after 81 revolutions of B, or 32 revolutions of A.

The insertion of a tooth in this manner was an old contrivance of millwrights to prevent the same pair of teeth from meeting too often, and was supposed to insure greater regularity in the wear of the wheels. The tooth inserted was called a hunting cog, because a pair of teeth, after being once in contact, would gradually separate, and then approach each other by one tooth in each revolution, and thus appear to hunt each other as they went round.

Clockmakers, on the contrary, appear to have adopted the opposite principle; though it has probably been partly forced on them, as the velocity ratio of the clock arbors must necessarily be exact.

240. Approximate Numbers for Trains.—If $\frac{L_m}{L_1}$ = k, when k is a prime number, or one whose prime factors are too large to be conveniently employed in wheel-work, an approximation may be resorted to. For example, assume $\frac{L_m}{L_1} = k \pm h$. This will introduce an error of $\pm h$ revolutions of the last axis during one of the first, and the

revolutions of the last axis during one of the first, and the nature of the machinery in question can alone determine whether such a variation is permissible.

For example, let $\epsilon = \frac{L_m}{L_1} = 269$, which is a prime number. Take $\epsilon = 269 + 1 = 270$, which can readily be factored into $6 \times 5 \times 9$; and we may employ the train $\frac{D}{F} = \frac{72 \times 60 \times 90}{12 \times 12 \times 10}$. This train will cause an error of one revolution of the last axis for every revolution of the first axis, the altered value of $\epsilon$ varying less than two-fifths of one per cent from the correct value.

241. But we may obtain a better approximation than this, without unnecessarily increasing the number of axes in the train; for, determine, in the manner already explained, the least number m of axes that would be necessary if k were decomposable, and the number of teeth that the nature of the machine makes it practicable to give to the pinions, and let F be the product of the pinions so determined; hence

$\frac{L_m}{L_1}$ or $\frac{D}{F} = k = \frac{Fk}{F}$ ,

supposing the wheels to drive.

Assume

where h must be taken as small as possible, but so as to obtain for $Fk \pm h$ a numerical value decomposable into factors. There will be, in this case, an error of $\pm h$ revolutions in the last axis during F of the first, or an error of $\frac{\pm h}{F}$ during one of the first. If the pinions are to be the drivers, then, in the same manner, assume

and there will then be an error of $\frac{\pm h}{D}$ revolutions in the first

axis during one revolution of the last axis. Let us take, as in the previous example, $\epsilon = 269$. Let w = 90, and p = 10; then

whence we find the least number of axes to be four.

Let us assume that pinions of 10 will be employed; then

Now add 1 to the numerator, and we have

This will give a good train with an error of only 1 revolution in 269000.

As another example, let it be required to find a train that shall connect the twelve-hour wheel of a clock with a wheel revolving in a lunation (viz., 29 days, 12 hours, 44 minutes nearly), for the purpose of showing the moon’s age on a dial.

Reducing the periods to minutes, we have

of which the numerator contains a large prime; viz., 10631; but

giving a good train, with an error of one minute in a lunation.