THE MECHANICAL PRINCIPLES OF A CLOCK

Experimental Mechanics by Robert S. Ball is part of the HackerNoon Books Series. You can jump to any chapter in this book here. THE MECHANICAL PRINCIPLES OF A CLOCK

LECTURE XX. THE MECHANICAL PRINCIPLES OF A CLOCK.

Introduction.—The Compensating Pendulum.—The Escapement.—The Train of Wheels.—The Hands.—The Striking Parts.

INTRODUCTION.

659. We come now to the most important practical application of the pendulum. The vibrations being always isochronous, it follows that, if we count the number of vibrations in a certain time, we shall ascertain the duration of that time. It is simply the product of the number of vibrations with the period of a single one. Let us take a pendulum 39·139 inches long; which will vibrate exactly once a second in London, and is therefore called a seconds pendulum (See Art. 607). If I set one of these pendulums vibrating, and contrive mechanism by which the number of its vibrations shall be recorded, I have a means of measuring time. This is of course the principle of the common clock: the pendulum vibrates once a second and the number of vibrations made from one epoch to another epoch is shown by the hands of the clock. For example, when the clock tells me that 15 minutes have elapsed, what it really shows is that the pendulum has made 60 × 15 = 900 vibrations, each of which has occupied one second.

660. One duty of the clock is therefore to count and record the number of vibrations, but the wheels and works have another part to discharge, and that is to sustain the motion of the pendulum. The friction of the air and the resistance experienced at the point of suspension are forces tending to bring the pendulum to rest; and to counteract the effect of these forces, the machine must be continually invigorated with fresh energy. This supply is communicated by the works of the clock, which will be sufficiently described presently.

661. When the weight driving the clock is wound up, a store of energy is communicated which is doled out to the pendulum in a very small impulse, at every vibration. The clock-weight is just large enough to be able to counterbalance the retarding forces when the pendulum has a proper amplitude of vibration. In all machines there is some energy lost in maintaining the parts in motion in opposition to friction and other resistances; in clocks this represents the whole amount of the force, as there is no external work to be performed.

THE COMPENSATING PENDULUM.

662. The actual length of the pendulum used, depends upon the purposes for which the clock is intended, but it is essential for correct performance that the pendulum should vibrate at a constant rate; a small irregularity in this respect may appreciably affect the indications of the clock. If the pendulum vibrates in 1·001 seconds [Pg 320]instead of in one second, the clock loses one thousandth of a second at each beat; and, since there are 86,400 seconds in a day, it follows that the pendulum will make only 86,400 - 86·3 vibrations in a day, and therefore the clock will lose 86·3 seconds, or nearly a minute and a half daily.

663. For accurate time-keeping it is therefore essential that the time of vibration shall remain constant. Now the time of vibration depends upon the length, and therefore it is necessary that the length of the pendulum be absolutely unalterable. If the length of the pendulum be changed even by one-tenth of an inch, the clock will lose or gain nearly two minutes daily, according to whether the pendulum has been made longer or shorter. In general we may say that, if the alteration in the length amount to k thousandths of an inch, the number of seconds gained or lost per day is 1·103 × k with a seconds pendulum.

  664. This explains the practice of raising the bob of the pendulum when the clock is going too slow or lowering it when going too fast. If the thread of the screw used in doing this have twenty threads to the inch; then one complete revolution of the screw will raise the bob through 50 thousandths of an inch, and therefore the effect on the rate will be 1·103 × 50 = 55 nearly. Thus, the rate of the clock will be altered by about 55 seconds daily. Whatever be the screw, its effect can be calculated by the simple rule expressed as follows. Divide 1103 by the number of threads to the inch; the quotient is the number of seconds that the clock can be made to gain or lose daily by one revolution of the screw on the bob of the pendulum.

665. Let us suppose that the length of the pendulum has been properly adjusted so that the clock keeps accurate time. It is necessary that the pendulum should not alter in length. But there is an ever-present [Pg 321]cause tending to change it. That cause is the variation of temperature. We shall first illustrate by actual experiment the well known law that bodies expand under the action of heat; then we shall consider the irregularities thus introduced into the motion of the pendulum; and, finally, we shall point out means by which these irregularities may be effectually counteracted.

Fig. 97.

666. We have here a brass bar a yard long; it is at present at the temperature of the room. If we heat the bar over a lamp, it becomes longer; but upon cooling, it returns to its original dimensions. These alterations of length are very small, indeed too small to be perceived except by careful measurement; but we shall be able to demonstrate in a simple way that elongation is the consequence of increased temperature. I place the bar a d in the supports shown in Fig. 97. It is firmly secured at b by means of a binding screw, and passes [Pg 322]quite freely through c; if the bar elongate when it is heated by the lamp, the point d must approach nearer to e. At h is an electric battery, and g is a bell rung by an electric current. One wire of the battery connects h and g, another connects g with e, and a third connects h with the end of the brass rod b. Until the electric current becomes completed, the bell remains dumb, the current is not closed until the point touches e: when this is the case, the current rushes from the battery along the bar, then from d to e, from that through the bell, and so back to the battery. At present the point is not touching e, though extremely close thereto. Indeed if I press e towards the point, you hear the bell, showing that the circuit is complete; removing my finger, the bell again becomes silent, because e springs back, and the current is interrupted.

667. I place the lamp under the bar: which begins to heat and to elongate; and as it is firmly held at b, the point gradually approaches e: it has now touched e; the circuit is complete, and the bell rings. If I withdraw the lamp, the bar cools. I can accelerate the cooling by touching the bar with a damp sponge; the bar contracts, breaks the circuit, and the bell stops: heating the bar again with the lamp, the bell again rings, to be again stopped by an application of the sponge. Though you have not been able to see the process, your ears have informed you that heat must have elongated the bar, and that cold has produced contraction.

668. What we have proved with respect to a bar of brass, is true for a bar of any material; and thus, whatever be the substance of which a pendulum is made, a simple uncompensated rod must be longer in hot weather than in cold weather: hence a clock will generally have a tendency to go faster in winter than in summer.[Pg 323]

669. The amount of change thus produced is, it is true, very small. For a pendulum with a steel rod, the difference of temperature between summer and winter would cause a variation in the rate of five seconds daily, or about half a minute in the week. The amount of error thus introduced is of no great consequence in clocks which are only intended for ordinary use; but in astronomical clocks, where seconds or even portions of a second are of importance, inaccuracies of this magnitude would be quite inadmissible.

Fig. 98.

670. There are, it is true, some substances—for example, ordinary timber—in which the rate of expansion is less than that of steel; consequently, the irregularities introduced by employing a pendulum with a wooden rod are less than those of the steel pendulum we have mentioned; but no substance is known which would not originate greater variations than are admissible in the performance of an astronomical clock.

We must, therefore, devise some means by which the effect of temperature on the length of a pendulum can be avoided. Various means have been proposed, and we shall describe one of the best and simplest.

671. The mercurial pendulum (Fig. 98) is frequently used in clocks intended to serve as standard time-keepers. The rod by which the pendulum is suspended is made of steel; and the bob consists of a glass jar of mercury. The distance of the centre of gravity of the mercury from the point of suspension may practically be considered as the length of the pendulum. The rate of expansion of mercury is about sixteen times that of steel: hence, if the bob be formed of a column of mercury one-eighth part of the length of the steel rod, the compensation would be complete. For, suppose the temperature of the [Pg 324]pendulum be raised, the steel rod would be lengthened, and therefore the vase of mercury would be lowered; on the other hand, the column of mercury would expand by an amount double that of the steel rod: thus the centre of the column of mercury would be elevated as much as the steel was elongated; hence the centre of the mercury is raised by its own expansion as much as it is lowered by the expansion of the steel, and therefore the effective length of the pendulum remains unaltered. By this contrivance the time of oscillation of the pendulum is rendered independent of the temperature. The bob of the mercurial pendulum is shown in Fig. 98. The screw is for the purpose of raising or lowering the entire vessel of mercury in order to make the rate correct in the first instance.

THE ESCAPEMENT.

  672. Practical skill as well as some theoretical investigation has been expended upon that part of a clock which is called the escapement, the excellence of which is essential to the correct performance of a timepiece. The pendulum must have its motion sustained by receiving an impulse at every vibration: at the same time it is desirable that the vibration should be hampered as little as possible by mechanical connection. The isochronism on which the time-keeping depends is in strictness only a characteristic of oscillations performed with a total freedom from constraint of every description; hence we must endeavour [Pg 325]to approximate the clock pendulum as nearly as possible to one which is swinging quite freely. To effect this, and at the same time to maintain the arc of vibration tolerably constant, is the property of a good escapement.

Fig. 99.

673. A common form of escapement is shown in Fig. 99. The arrangement is no doubt different from that actually found in a clock; but I have [Pg 326]constructed the machine in this way in order to show clearly the action of the different parts. g is called the escapement-wheel: it is surrounded by thirty teeth, and turns round once when the pendulum has performed sixty vibrations,—that is, once a minute. i represents the escapement; it vibrates about an axis and carries a fork at k which projects behind, and the rod of the pendulum hangs between its prongs. The pendulum is itself suspended from a point o. At n, h are a pair of polished surfaces called the pallets: these fulfil a very important function.

674. The escapement-wheel is constantly urged to turn round by the action of the weight and train of wheels, of which we shall speak presently; but the action of the pallets regulates the rate at which the wheel can revolve. When a tooth of the wheel falls upon the pallet n, the latter is gently pressed away: this pressure is transmitted by the fork to the pendulum; as n moves away from the wheel, the other pallet h approaches the wheel; and by the time n has receded so far that the tooth slips from it, h has advanced sufficiently far to catch the tooth which immediately drops upon h. In fact, the moment the tooth is free from n, the wheel begins to revolve in consequence of the driving weight; but it is quickly stopped by another tooth falling on h: and the noise of this collision is the well known tick of the clock. The pendulum is still swinging to the left when the tooth falls on h. The pressure of the tooth then tends to push h outwards, but the inertia of the pendulum in forcing h inwards is at first sufficient to overcome the outward pressure arising from the wheel; the consequence is that, after the tooth has dropped, the escapement-wheel moves back a little, or “recoils,” as it is called. If you look at any ordinary clock, which [Pg 327]has a second-hand, you will notice that after each second is completed the hand recoils before starting for the next second. The reason of this is, that the second-hand is turned directly by the escapement-wheel, and that the inertia of the pendulum causes the escapement-wheel to recoil. But the constant pressure of the tooth soon overcomes the inertia of the pendulum, and h is gradually pushed out until the tooth is able to “escape”; the moment it does so the wheel begins to turn round, but is quickly brought up by another tooth falling on n, which has moved sufficiently inwards.

The process we have just described then recurs over again. Each tooth escapes at each pallet, and the escapements take place once a second; hence the escapement-wheel with thirty teeth will turn round once in a minute.

675. When the tooth is pushing n, the pendulum is being urged to the left; the instant this tooth escapes, another tooth falls on h, and the pendulum, ere it has accomplished its swing to the left, has a force exerted upon it to bring it to the right. When this force and gravity combined have stopped the pendulum, and caused it to move to the right, the tooth soon escapes at h, and another tooth falls on n, then retarding the pendulum. Hence, except during the very minute portion of time that the wheel turns after one escapement, and before the next tick, the pendulum is never free; it is urged forwards when its velocity is great, but before it comes to the end of its vibration it is urged backwards; this escapement does not therefore possess the characteristics which we pointed out (Art. 672) as necessary for a really good instrument. But for ordinary purposes of time-keeping, the recoil escapement works sufficiently well, as the force which acts upon the pendulum is in reality extremely small. For [Pg 328]the refined applications of the astronomical clock, the performance of a recoil escapement is inadequate.

The obvious defect in the recoil is that the pendulum is retarded during a portion of its vibration; the impulse forward is of course necessary, but the retarding force is useless and injurious.

676. The “dead-beat” escapement was devised by the celebrated clockmaker Graham, in order to avoid this difficulty. If you observe the second-hand of a clock, controlled by this escapement, you will understand why it is called the dead beat: there is no recoil; the second-hand moves quickly over each second, and remains there fixed until it starts for the next second.

The wheel and escapement by which this effect is produced is shown in Fig. 100. a and b are the pallets, by the action of the teeth on which the motion is given to the crutch, which turns about the centre o; from the axis through this centre the fork descends, so that as the crutch is made to vibrate to and fro by the wheel, the fork is also made to vibrate, and thus sustain the motion of the pendulum. The essential feature in which the dead-beat escapement differs from the recoil escapement is that when the tooth escapes from the pallet a, the wheel turns: but the tooth which in the recoil escapement would have fallen on the other pallet, now falls on a surface d, and not on the pallet b. d is part of a circle with its centre at o, the centre of motion; consequently, the tooth remains almost entirely inert so long as it remains on the circular arc d.

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: [Pg 329]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.

Fig. 100.

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.[Pg 330]

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, [Pg 331]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.

Fig. 101.

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; [Pg 333]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.

[Pg 334]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.

Fig. 102.

[Pg 336]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.[Pg 337]

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.

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