Part 22

677. There is thus no recoil, and the pendulum is allowed to reach the extremity of its swing to the right unretarded; but when the pendulum is returning, the crutch moves until the tooth passes from the circular arc D on to the pallet B: instantly the tooth slides down the pallet, giving the crutch an impulse, and escaping when the point has traversed B. The next tooth that comes into action falls upon the circular arc C, of which the centre is also at O; this tooth likewise remains at rest until the pendulum has finished its swing, and has commenced its return; then the tooth slides down A, and the process recommences as before.

678. The operations are so timed that the pendulum receives its impulse (which takes place when a tooth slides down a pallet) precisely when the oscillation is at the point of greatest velocity; the pendulum is then unacted upon till it reaches a similar position in the next vibration. This impulse at the middle of the swing does not affect the time of vibration.

679. There is still a small frictional force acting to retard the pendulum. This arises from the pressure of the teeth upon the circular arcs, for there is a certain amount of friction, no matter how carefully the surfaces may be polished. It is not however found practically to be a source of appreciable irregularity.

In a clock furnished with a dead-beat escapement and a mercurial pendulum, we have a superb time-keeper.

THE TRAIN OF WHEELS.

680. We have next to consider the manner in which the supply of energy is communicated to the escapement-wheel, and also the mode in which the vibrations of the pendulum are counted. A train of wheels for this purpose is shown in Fig. 99. The same remark may be made about this train that we have already made about the escapement,—namely, that it is more designed to explain the principle clearly than to show the actual construction of a clock.

681. The weight A which animates the whole machine is attached to a rope, which is wound around a barrel B; the process of winding up the clock consists in raising this weight. On the same axle as the barrel B is a large tooth-wheel C; this wheel contains 200 teeth. The wheel C works into a pinion D, containing 20 teeth; consequently, when the wheel C has turned round once, the pinion D has turned round ten times. The large wheel E is on the same axle with the pinion D, and turns with D; the wheel E contains 180 teeth, and works into the pinion F, containing 30 teeth: consequently when E has gone round once, F will have turned round six times; and therefore, when the wheel C and the barrel B have made one revolution, the pinion F will have gone round sixty times; but the wheel G is on the same shaft as the pinion F, and therefore, for every sixty revolutions of the escapement-wheel, the wheel C will have gone round once. We have already shown that the escapement-wheel goes round once a minute, and hence the wheel C must go round once in an hour. If therefore a hand be placed on the same axle with C, in front of a clock dial, the hand will go completely round once an hour; that is, it will be the minute-hand of the clock.

682. The train of wheels serves to transmit the power of the descending weight and thus supply energy to the pendulum. In the clock model you see before you, the weight sustaining the motion is 56 lbs. The diameter of the escapement-wheel is about double that of the barrel, and the wheel turns round sixty times as fast as the barrel; therefore for every inch the weight descends, the circumference of the escapement-wheel must move through 120 inches. From the principle of work it follows that the energy applied at one end of a machine equals that obtained from the other, friction being neglected. The force of 56 lbs. is therefore, reduced to the one hundred-and-twentieth part of its amount at the circumference of the escapement-wheel. And as the friction is considerable; the actual force with which each tooth acts upon the pallet is only a few ounces.

683. In a good clock an extremely minute force need only be supplied to the pendulum, so that, notwithstanding 86,400 vibrations have to be performed daily, one winding of the clock will supply sufficient energy to sustain the motion for a week.

THE HANDS.

684. We shall explain by the model shown in Fig. 101, how the hour-hand and the minute-hand are made to revolve with different velocities about the same dial.

G is a handle by which I can turn round the shaft which carries the wheel F, and the hand B. There are 20 teeth in F, and it gears into another wheel, E, containing 80 teeth; the shaft which is turned by E carries a third wheel D, containing 25 teeth, and D works with a fourth C, containing 75 teeth, C is capable of turning freely round the shaft, so that the motion of the shaft does not affect it, except through the intervention of the wheels E, F, and D. To C another hand A is attached, which therefore turns round simultaneously with C. Let us compare the motion of the two hands A and B. We suppose that the handle G is turned twelve times; then, of course, the hand B, since it is on the shaft, will turn twelve times. The wheel F also turns twelve times, but E has four times the number of teeth that A has, and therefore, when F has gone round four times, E will only have gone round once: hence, when F has revolved twelve times, E will have gone round three times. D turns with E, and therefore the twelve revolutions of the handle will have turned D round three times, but since C has 75 teeth and D 25 teeth, C will have only made one revolution, while D has made three revolutions; hence the hand A will have made only one revolution, while the hand B has made twelve revolutions.

We have already seen (Art. 681) how, by a train of wheels, one wheel can be made to revolve once in an hour. If that wheel be upon the shaft instead of the handle G, the hand B will be the minute-hand of the clock, and the hand A the hour-hand.

685. The adjustment of the numbers of teeth is important, and the choice of wheels which would answer is limited. For since the shafts are parallel, the distance between the centres of F and E must equal that between the centres of C and of D. But it is evident that the distance from the centre of F to the centre of E is equal to the sum of the radii of the wheels F and E. Hence the sum of the radii of the wheels F and E must be equal to the sum of the radii of C and D. But the circumferences of circles are proportional to their radii, and hence the sum of the circumferences of F and E must equal that of C and D; it follows that the sum of the teeth in E and F must be equal to the sum of the teeth in C and D. In the present case each of these sums is one hundred.

686. Other arrangements of wheels might have been devised, which would give the required motion; for example, if F were 20, as before, and E 240, and if C and D were each equal to 130, the sum of the teeth in each pair would be 260. E would only turn once for every twelve revolutions of F, and C and D would turn with the same velocity as E; hence the motion of the hand A would be one-twelfth that of B. This plan requires larger wheels than the train already proposed.

THE STRIKING PARTS.

687. We have examined the essential features of the going parts of the clock; to complete our sketch of this instrument we shall describe the beautiful mechanism by which the striking is arranged. The model which we represent in Fig. 102 is, as usual, rather intended to illustrate the principles of the contrivance than to be an exact counter-part of the arrangement found in clocks. Some of the details are not reproduced in the model; but enough is shown to explain the principle, and to enable the model to work.

688. When the hour-hand reaches certain points on the dial, the striking is to commence; and a certain number of strokes must be delivered. The striking apparatus has both to initiate the striking and to control the number of strokes; the latter is by far the more difficult duty. Two contrivances are in common use; we shall describe that which is used in the best clocks.

689. An essential feature of the striking mechanism in the repeating clock is the snail, which is shown at B. This piece must revolve once in twelve hours, and is, therefore, attached to an axle which performs its revolution in exactly the same time as the hour-hand of the clock. In the model, the striking gear is shown detached from the going parts, but it is easy to imagine how the snail can receive this motion. The margin of the snail is marked with twelve steps, numbered from one to twelve. The portions of the margin between each pair of steps is a part of the circumference of a circle, of which the axis of the snail is the centre. The correct figuring of the snail is of the utmost importance to the correct performance of the clock. Above the snail is a portion of a toothed wheel, F, called the rack; this contains about fourteen or fifteen teeth. When this wheel is free, it falls down until a pin comes in contact with the snail at B.

690. The distance through which the rack falls depends upon the position of the snail; if the pin come in contact with the part marked I., as it does in the figure, the rack will descend but a small distance, while, if the pin fall on the part marked VII., the rack will have a longer fall: hence as the snail changes its position with the successive hours, so the distance through which the rack falls changes also. The snail is so contrived that at each hour the rack falls on a lower step than it does in the preceding hour; for example, during the hour of three o’clock, the rack would, if allowed to fall, always drop upon the part of the snail marked III., but, when four o’clock has arrived, the rack would fall on the part marked IV.; it is to insure that this shall happen correctly that such attention must be paid to the form of the snail.

691. A is a small piece called the “gathering pallet”; it is so placed with reference to the rack that, at each revolution of A, the pallet raises the rack one tooth. Thus, after the rack has fallen, the gathering pallet gradually raises it.

692. On the same axle as the gathering pallet, and turning with it, is another piece C, the object of which is to arrest the motion when the rack has been raised, sufficiently. On the rack is a projecting pin; the piece C passes free of this pin until the rack has been lifted to its original height, when C is caught by the pin, and the mechanism is stopped. The magnitude of the teeth in the rack is so arranged with reference to the snail, that the number of lifts which the pallet must make in raising the rack is equal to the number marked upon the step of the snail upon which the rack had fallen; hence the snail has the effect of controlling the number of revolutions which the gathering pallet can make. The rack is retained by a detent F, after being raised each tooth.

693. The gathering pallet is turned by a small pinion of 27 teeth, and the pinion is worked by the wheel C, of 180 teeth. This wheel carries a barrel, to which a movement of rotation is given by a weight, the arrangement of which is evident: a second pinion of 27 teeth on the same axle with D is also turned by the large wheel C. Since these pinions are equal, they revolve with equal velocities. Over D the bell I is placed; its hammer E is so arranged that a pin attached to D strikes the bell once in every revolution of D. The action will now be easily understood. When the hour-hand reaches the hour, a simple arrangement raises the detent F; the rack then drops; the moment the rack drops, the gathering pallet commences to revolve and raises up the rack; as each tooth is raised a stroke is given to the bell, and thus the bell strikes until the piece C is brought to rest against the pin.

694. The object of the fan H is to control the rapidity of the motion: when its blades are placed more or less obliquely, the velocity is lessened or increased.

APPENDIX I.

The formulæ in the tables on p. 73 and after can be deduced by two methods,—one that of graphical construction, the other that of least squares. The first method is the more simple and requires but little calculation; though neatness and care are necessary in constructing the diagrams. The second method will be described for the benefit of those who possess the requisite mathematical knowledge. The formulæ used in the preparation of the tables have been generally deduced from the method of least squares, as the results are to a slight, though insignificant, extent more accurate than those of the method of graphical construction. This remark will explain why the figures in some of the formulæ are carried to a greater number of places of decimals than could be obtained by the other method.

We shall confine the numerical examples to Tables III. and IV., and show how the formulæ of these tables have been deduced by the two different methods.

Tables V., XIV., XVI., XXI., are to be found in the same manner as Table III.; and Tables VI., IX., X., XI., XV., XVII., XVIII., XIX., XX., XXI., XXII., in the same manner as Table IV.

_THE METHOD OF GRAPHICAL CONSTRUCTION._

TABLE III.

A horizontal line APS, shown on a diminished scale in Fig. 103, is to be neatly drawn upon a piece of cardboard about 14" × 6". A scale which reads to the hundredth of an inch is to be used in the construction of the figure. A pocket lens will be found convenient in reading the small divisions. By means of a pair of compasses and the scale, points are to be marked upon the line APS, at distances 1"·4, 2"·8, 4"·2, 5"·6, 7"·0, 8"·4, 9"·8, 11"·2 from the origin A. These distances correspond to the magnitudes of the loads placed upon the slide on the scale of 0"·1 to 1 lb. Perpendiculars to APS are to be erected at the points marked, and distances F₁, F₂, F₃, &c. set off upon these perpendiculars. These distances are to be equal, on the adopted scale, to the frictions for the corresponding loads. For example, we see from Table III., Experiment 3, that when the load upon the slide is 42 lbs., the friction is 12·2 lbs.; hence the point F₃ is found by measuring a distance 4"·2 from A, and erecting a perpendicular 1"22. Thus, for each of the loads a point is determined. The positions of these points should be indicated by making each of them the centre of a small circle 0"·1 diameter. These circles, besides neatly defining the points, will be useful in a subsequent part of the process.

It will be found that the points F₁, F₂, &c. are very nearly in a straight line. We assume that, if the apparatus and observations were perfect, the points would lie exactly in a straight line. The object of the construction is to determine the straight line, which on the whole is most close to all the points. If it be true that the friction is proportional to the pressure, this line should pass through the origin A, for then the perpendicular which represents the friction is proportional to the line cut off from A, which represents the load. It will be found that a line AT can be drawn through the origin A, so that all the points are in the immediate vicinity of this line, if not actually upon it. A string of fine black silk about 15" long, stretched by a bow of wire or whalebone, is a convenient straight-edge for finding the required line. The circles described about the points F₁, F₂, &c. will facilitate the placing of the silk line as nearly as possible through all the points. It will not be found possible to draw a line through A, which shall intersect all the circles; the best line passes below but very near to the circles round F₁, F₂, F₃, F₄, touches the circle about F₅, intersects the circles about F₆ and F₇, and passes above the circle round F₈. The line should be so placed that its depth below the point which is most above it, is equal to the height at which it passes above the point which is most below it.

From A measure AS, a length of 10", and erect the perpendicular ST. We find by measurement that ST is 2"·7. If, then, we suppose that the friction for any load is really represented by the distance cut off by the line AT upon the perpendicular, it follows that

_F_ : _R_ :: 2"·7 : 10". or _F_ = 0·27 _R_.

This is the formula from which Table III. has been constructed.

TABLE IV.

By a careful application of the silk bow-string, X Y Q can be drawn, which, itself in close proximity to A, passes more nearly through F₁, F₂, &c. than is possible for any line which passes exactly through A. X Y Q will be found not only to intersect all the small circles, but to cut off a considerable arc from each. Measure off X P a distance of 10", and erect the perpendicular P Q; then, if _R_ be the load, and _F_ the corresponding friction, we must have from similar triangles—

_AY_ _F_ - ———— × 1 lb. 0"·1 _PQ_ ————————————————— = ——————. _R_ _PX_

By measurement it is found that _AY_ = 0"·14, and _PQ_ = 2"·53.

We have, therefore,

_F_ = 1·4 + 0·253 _R_.

This is practically the same formula as

_F_ = 1·44 + 0·252 _R_,

from which the table has been constructed. In fact, the column of calculated values of the friction might have been computed from the former, without appreciably differing from what is found in the table.

_THE METHOD OF LEAST SQUARES._

TABLE III.

Let _k_ be the coefficient of friction. It is impossible to find any value for _k_ which will satisfy the equation,

_F_ - _k R_ = 0,

for all the observed pairs of values of _F_ and _R_. We have then to find the value for _k_ which, upon the whole, best represents the experiments. _F_ - _k R_ is to be as near zero as possible for each pair of values of _F_ and _R_.

In accordance with the principle of least squares, it is well known to mathematicians, the best value of _k_ is that which makes

(_F₁_ - _k R₁_)² + (_F₂_ - _k R₂_)² + &c. + (_Fₘ_ - _k Rₘ_)²

a minimum where F₁ and R₁, F₂ and R₂ &c. are the simultaneous values of F and R in the several experiments.

In fact, it is easy to see that, if this quantity be small, each of the essentially positive elements,

(_F₁_ - _k R₁_)², &c.

of which it is composed, must be small also, and that therefore

_F_ - _k R_

must always be nearly zero.

Differentiating the sum of squares and equating the differential coefficient to zero, we have according to the usual notation,

Σ _R₁_ (_F₁_ - _k R₁_) = 0;

Σ _R₁ F₁_ whence _k_ = ————————————————. Σ _R₁_²

The calculation of _k_ becomes simplified when (as is generally the case in the tables) the loads _R₁_, _R₂_, &c., _Rₘ_ are of the form,

_N_, 2_N_, 3_N_ &c., _m N_.

In this case,

Σ _R₁_ _F₁_ = _N_ (_F₁_ + 2_F₂_ + 3_F₃_ + &c. + _mFₘ_).

Σ _R₁_² = _N_² (1² + 2² + &c. + _m_²)

_m_(_m_ + 1) (2_m_ +1) = _N_² ——————————————————————— 6

(_F₁_ + 2_F₂_ + &c. + _mFₘ_) ∴ _k_ = 6 ————————————————————————————— . _Nm_(_m_ + 1) (2_m_ + 1)

In the case of Table III.

_m_ = 8, _N_ = 14,

_F₁_ + 2_F_ + 3_F₃_ + _mFₘ_ = 770·9;

whence _k_ = 0·27.

Thus the formula _F_ = 0·27 _R_ is deduced both by the method of least squares, and by the method of graphical construction.

TABLE IV.

The formula for this table is to be deduced from the following considerations.

No values exist for _x_ and _y_, so that the equation

_F_ = _x_ + _y R_

shall be satisfied for all pairs of values of _F_ and _R_, but the best values for _x_ and _y_ are those which make

(_F₁_ - _x_ - _y R₁_)² + (_F₂_ - _x_ - _y R₂_)² + &c.

+ (_Fₘ_ - _x_ - _y Rₘ_)²

a minimum.

Differentiating with respect to _x_ and _y_, and equating the differential coefficients to zero, we have

Σ (_F₁_ - _x_ - _y R₁_) = 0,

Σ _R₁_(_F₁_ - _x_ - _y R₁_) = 0.

This gives two equations for the determination of _x_ and _y_.

Suppose, as is usually the case, the loads be of the form,

_N_, 2_N_, 3_N_, 4_N_ &c. _mN_,

and making

_A_ = _F₁_ + _F₂_ + _F₃_ + &c. + _Fₘ_

_B_ = _F₁_ +2_F₂_ + 3_F₃_ + &c. + _mFₘ_,

we have the equations

_m_(_m_ + 1) _A_ - _mx_ - ————————————— _N y_ = 0, 2

_m_(_m_ + 1) _m_(_m_ + 1) (2_m_ + 1) _B_ - ————————————— _x_ - ——————————————————————— _N y_ = 0. 2 6

Solving these, we find

2 + 4_m_ 6 _x_ = ——————————— _A_ - ————————— _B_, _m_² - _m_ _m_² - _m_

12 _B_ 6 _A_ _y_ = ————————— ————— - -————————— ————— . _m_³ - _m N_ _m_² - _m N_

In the present case,

_m_ = 8, _N_ = 14, _A_ = 138·4, _B_ = 770·9; whence _x_ = 1·44 _y_ = 0·252,

and we have the formula,

_F_ = 1·44 + 0·252 _R_.

APPENDIX II.

DETAILS OF THE WILLIS APPARATUS USED IN ILLUSTRATING THE FOREGOING LECTURES.

The ultimate parts of the various contrivances figured in this volume are mainly those invented by the late Professor Willis of Cambridge. They are minutely described and illustrated in a work written by him for the purpose under the title _System of Apparatus for the use of Lecturers and Experimenters in Mechanical Philosophy_, London, Weale & Co., 1851. This work has long been out of print. It may therefore be convenient if I give here a brief account of those parts of this admirable apparatus that I have found especially useful The illustrations have been copied from the plates in Professor Willis’ book.[2]

[2] I ought to acknowledge the kindness with which Mr. J. Willis Clark, of Cambridge, the literary executor of Professor Willis, has responded to my queries, while I am also under obligations to the courtesy of Messrs. Crosby, Lockwood, & Co.

The Willis system provides the means for putting versatile framework together with or without revolving gear for the purpose of mechanical illustration. Many parts which enter into the construction of the machine used at the lecture to-day will reappear to-morrow as essential parts of some totally different contrivance. The parts are sufficiently substantial to work thoroughly well. The scantlings and dimensions generally have been so chosen as to produce models readily visible to a large class.

It will of course be understood that every model contains some one or more _special_ parts such as the punch and die in Fig. 73, or the spring balance in Fig. 17, or the pulley-block in Fig. 33. But for the due exhibition of the operation of the machine a further quantity of ordinary framework and of moving mechanism is usually necessary. This material, which may be regarded as of a _general_ type, it is the function of the Willis system to provide.

THE BOLTS.—The system mainly owes its versatility and its steadiness to the use of the iron screw bolt for all attachments. The bolts used are ⅜ diameter; the shape of the head is hemispherical and the shank must be square for a short distance from the head so that the bolt cannot turn round when passed through the slits of the _brackets_ or _rectangles_. When the head of the bolt bears on a slit in one of the wooden pieces a circular iron washer 2" in diameter, or a square washer 2" on each side, is necessary to protect the wood from crushing. There is to be a square hole, in the washer to receive the square shank of the bolt and the thickness of the washers should be ⅛". The nut is square or hexagonal, and should _always_ have a washer underneath when screwed home with a spanner or screw-wrench. The most useful lengths are 2", 4", 6". The proper kind are known commercially as _coach-bolts_, and they should be chosen with easy screws, for facility in erecting or modifying apparatus. At least two dozen of the intermediate size and a dozen of each of the others are required. For elaborate contrivances many more will be necessary.