"Our collaboration in pursuing the mysteries of pendulums," writes Jackson, "got under way 10 years ago, when I received a letter from Dr. Bush suggesting that work on pendulum clocks would be 'a delightful hobby, largely because it is now a field where interest because of modern electrical devices, has to be academic.' If there is any practical use for a really accurate pendulum (one accurate to the order of a millisecond a day ), it would be in the field of gravity measurement, or in circumstances where a pendulum clock is more economical to build or to maintain than are electrical oscillators that have equal accuracy.

"During the past three centuries considerable progress has been made in the design of pendulums, but many improvements are still possible in the devices used to correct the well-recognized errors arising from changes in temperature and in barometric pressure, from friction in the drive mechanism and from the so-called circular error that makes a pendulum run faster when its amplitude of swing decreases. A much less known, but surprisingly large, error is caused by minute excursions of the pendulum's suspension induced by the motion of the pendulum's weight, or bob. We have investigated this error extensively and have called it support reaction. A number of other minor effects, including those arising from the influence of static electricity and of seismic motions of very small amplitude, prevents a pendulum from running at a uniform rate. Even if all these effects could be reduced to zero, changes in gravity of a tidal nature would produce periodic fluctuations of about .0002 second twice a day.

"Most of our recent experiments have been made with the aid of a clock designed by the British engineer W. H. Shortt and used by the National Bureau of Standards until it was retired from service in favor of a clock driven by a crystal-oscillator circuit. The instrument is of the 'master-slave' type and, until superseded by electronic techniques of timekeeping, was perhaps the most accurate mechanical clock available. The pendulum swings in an evacuated housing, has a rod made of an alloy that tends to retain its dimensions despite variations in temperature and is otherwise constructed to minimize the effects of environmental changes on its rate of swing. Our objective was to develop modifications for maximizing the performance of the Shortt clock and ultimately to build a clock of our own that would surpass it. As our standard of comparison we use the time signals broadcast continuously by WWV, the station of the Bureau of Standards. We have not yet built our clock, so the following discussion is in the nature of a progress report.

"Of the many variables that affect the performance of pendulums one of the most serious is temperature fluctuation. If the length of a pendulum rod changes by one part in 40,000, a clock driven by it will gain or lose about a second per day. Nearly all materials expand this much with temperature changes of only a few degrees, so it is essential to compensate in some way for the change in length. John Harrison, the eminent British horologist, made the first major contribution to the solution of the problem 200 years ago by inventing the 'gridiron' pendulum, in which brass rods in compression and expanding upward were alternated with steel rods in tension and expanding downward. The rods were arranged so that the length of the brass times its coefficient of expansion equaled the length of the steel times its coefficient. No net change in the length of the pendulum occurred when both metals expanded or contracted with temperature changes. It was an excellent arrangement, and it was widely used until it was outmoded by the invention of 'Invar' early in the present century. The small expansion-coefficient of this nickel-iron alloy permits a 40-inch pendulum rod to be compensated by only an inch or two of brass. In pendulums made of Invar the pendulum bob is bored oversize from the center down. A sleeve of brass is slipped over the lower end of the pendulum rod and rests on the adjustment nut on the end of the rod. The brass compensator supports the surrounding bob near its center of mass, as shown in the accompanying illustration.

"This is the worst possible location for the compensator, because the relatively large thermal capacity of the 10 to 15 pounds of metal in the bob prevents the brass from reaching a new temperature as fast as the rod, which is exposed for most of its length. Consequently if the ambient temperature varies, the rod's length varies, and the clock runs at a different rate until the heavy bob and the buried brass compensator reach the new temperature. This means that much greater care must be exercised to maintain a constant temperature than would be the case if the rod and its compensator had more nearly equal thermal masses and exposures, as did the original gridiron construction.

"An easy remedy is to add an Invar sleeve, similar to the brass compensator, to extend up into the bob, thus allowing the brass sleeve to be exposed below. It is still desirable to hold a fairly uniform temperature and particularly to guard against temperature stratification of the air. This can be taken care of relatively easily by constructing a closet of light insulating wallboard around the entire clock, by placing an electric fan inside the closet to provide continuous air movement and also by fitting the closet with a thermostatically controlled electric heater to hold the temperature a few degrees above the highest temperature expected in the room.

"For extreme accuracy it is desirable to compensate different parts of the rod separately. An ordinary brass or aluminum compensator can be placed inside the bob to take care of that portion of the rod which runs through the bob, while the exposed rod and suspension spring are compensated by brass or by aluminum members having thermal capacities similar to the parts they compensate.

"A pendulum rod is usually supported by a thin spring, though all pendulums used to measure gravity are supported by a knife-edge arrangement. Springs are probably better for continuous operation, because they are not subject to wear. Our experiments have shown that even the thinnest practical spring can exert a force equal to a thousandth of the force of gravity acting on the bob. This means that a spring-suspended pendulum is not a pure gravity-controlled device, and that extreme care is needed to keep the force of the spring constant. The stiffness of most spring materials changes with temperature. The coefficient of elasticity of steel changes about 200 parts per million per degree centigrade, and may change the rate of a pendulum clock a 10th of a second or more per day with a one-degree change in temperature. This can be taken care of in the compensation of the rod, but only if the compensator responds at the same rate as the spring. Hence the need for a well-exposed compensator of low thermal capacity, unless a material like 'Elinvar' or 'Ni-Span C' is used for the spring. These materials exhibit little or no change in elasticity with temperature, but their characteristics may change over long periods of time. Further experiments are desirable to determine the best material.

"The spring must also resist side-thrust produced by the inertia of the rod and bob as they oscillate about their common center of mass. A very light rod is desirable if a thin spring is used. In general a given spring will affect the rate of a short pendulum more than a long one, if each pendulum has a bob of equal weight.

"It has frequently been noted that clocks do not run at the same rate after they have been stopped and then started again. In many cases the pendulum rod is attached to the spring by means of a hook that hangs on a pin through the spring or its clamp. This facilitates assembly and permits motion at right angles to the plane of oscillation when the pendulum is adjusted. It is most important that the contacting surfaces of such a hook be V-shaped, rather than round, to assure that the length of the pendulum remains constant [see illustration in Figure 2].

"Attention should also be given to electrostatic forces that can, under certain circumstances, greatly affect the rate of a pendulum. A beat plate that moves very close to the bottom glass plate, as in the Shortt design, can be strongly attracted by stray static charges on the glass. A remedy for this is small pieces of polonium foil, such as is used in antistatic brushes for photographic work, which can be glued to the inside and outside of the glass near any moving parts of the pendulum system. These brushes can be purchased at most photographic-supply stores, and the polonium foil can be removed from them, though great care must be taken to prevent rupture of the thin protective coating over the polonium. The foil must not be allowed to come in contact with the bare skin. Alpha-particle emission from the polonium ionizes the air sufficiently to prevent any accumulation of electrical charge.

"That electrostatic forces can be a matter of serious concern to pendulum clock makers is illustrated by our experience with an air-drive escapement that will be described in a later article. It was necessary to provide a moving vane to switch the air from one jet to another to give an impulse in each direction as the pendulum swings from side to side. Various metallic stops were tried, to limit the movement of the vane on each side, but all stuck after a few hours of operation. Tiny glass stops proved satisfactory for several days of operation, but they also stuck eventually. A piece of polonium foil placed near them cured the trouble.

"Of course if the clock case is made entirely of metal, or is properly shielded on the inside, static cannot cause trouble unless there are insulated metal or nonmetallic parts on the moving system. Such nonmetallic parts should be covered with a conducting paint of the type used to ground the outside surface of a television picture-tube. (The paint should make contact with the metal parts of the pendulum.) "The effect of changes in barometric pressure, which varies the flotation of the bob, and hence its rate of swing, has been handled in all precise pendulum clocks by the rather clumsy expedient of sealing the pendulum in an airtight case. Aneroid units mounted on the pendulum to raise and lower weights have, in general, been unsatisfactory. Much can be done to improve the design of the case seals, so that the case can be easily opened and closed. We have found, for example, that a fine-threaded metal screw-cap, sealed with an ordinary rubber jar-gasket covered with stopcock grease, is satisfactory for frequent access to the weight pan or other parts during the adjustment period.

"In addition to flotation of the bob by the air there is a so-called inertia effect caused by the movement of the air as the bob swings. The magnitude of this effect varies with the design of the bob and the case, generally being greater if the case is small and less if the bob is streamlined. Obviously the effect will disappear completely if the pressure of the air is sufficiently reduced. Both the flotation and inertia effects were found to be directly proportional to pressure, producing a uniform increase in rate with reduction in pressure.

"The 'seconds' pendulum of a Shortt clock, fitted with a 14-pound cylindrical bob made of type metal and modified for an electromagnetic drive, runs about 12 seconds a day faster at a pressure of one centimeter of mercury than at 16 centimeters (normal atmospheric pressure), when the arc of swing is held constant at l.5 degrees. This is about .4 second per day per inch of change in the barometer. If the bob were made of denser material or better streamlined, the change in rate would be less.

"In general the lower the pressure the better the pendulum will perform, provided that a really good escapement, or drive mechanism, is used. Hermetically sealing the case provides a convenient method of fine rate-adjustment by permitting control of the slight remaining pressure in the case. A sensitive pressure-gauge, or short-tube mercury barometer, should be enclosed in the case to aid this adjustment and to indicate any air leakage. At atmospheric pressure, outside barometric changes producing an internal volume-change of one part in 15,000 can cause a one-millisecond-per-day change in rate, but at low internal pressure the rate of change is much less.

"Conventional pendulums swing in a circular arc, the restoring force of gravity varying with the trigonometric sine of the angle from the vertical. This causes the period to increase with the angle of swing, the effect known as circular error. Three hundred years ago Christiaan Huygens suggested a theoretically perfect method of making a pendulum bob swing in a cycloidal arc; he showed that this would result in simple harmonic motion and thus make the period independent of the length of the arc. The proposal required that a cam with a face in the shape of a cycloid be fixed on each side of the flat suspension spring. As the spring bent alternately around each cam, the path of the bob would become cycloidal. Unfortunately the cams cannot be made and installed to the required precision. All attempts to apply Huygens's theoretical solution to a real pendulum have introduced greater errors than the one it is designed to correct. So far as is known, no practical circular-error correction scheme has ever been used by the makers of precision pendulums, and none could be found in the available literature. Accordingly we have concentrated on this problem and have designed and tested what we believe to be the first practical arrangement for correcting circular error.

"The device consists of a fine wire connected from a point on the pendulum rod to a very weak fixed spring having a spring constant, K, defined by the equation in the accompanying illustration [below]. The spring should be made of Elinvar or a similar material having a negligible temperature-elasticity coefficient. Tungsten wire .002 inch in diameter provides excellent strength, elasticity and endurance. A cantilever spring, in the form of a horizontal flat strip anchored to the support at one end and fastened to the wire at the other, gives even better correction, and is easier to calibrate than a spiral spring. The spring should be mounted so the tension in the wire will not be affected by temperature changes. This calls for careful temperature compensation of its supporting rod or bracket, as well as of the wire itself.

"Temperature compensation for the wire alone can be provided by fastening the upper end of the wire to a relatively stiff spring on the pendulum rod, and stretching a similar wire of equal length from this spring to a point on the rod opposite the lower end of the first wire, with enough tension to stress the stiff upper spring. The lower end of the first wire will then move the same amount as the pendulum rod does at the point level with the attachment of the second wire. In addition the rod carrying the circular-error spring should be made the same size and of the same material as the pendulum rod, and be attached to the same support. The two expedients in combination give perfect temperature compensation.

"The system is sensitive to misalignment in the relative positions of the spring and pendulum rod and is therefore subject to malfunction if the mechanism is tilted from the vertical after adjustment. For very precise work a pendulous mounting of both the circular-error spring and the whole pendulum system is required. If the clock is mounted on a reasonably solid wall, however, the spring can be supported on a rod, similar to the pendulum rod, that is steadied at its lower end by attachment to the case. Such a mounting arrangement will considerably improve the performance of most pendulums.

"The spring-and-wire arrangement provides a restoring torque to the pendulum, in addition to that of gravity (but only about a 40,000th as great), which for small angles is very nearly equal to the difference between the angle of swing and the sine of that angle. This causes the pendulum to oscillate in simple harmonic motion, with the total restoring force proportional to displacement, thus freeing it from circular error and making the period of oscillation independent of arc length for angles of practical interest up to three or four degrees.

"Much ingenuity has been exhibited over a period of many years in the design of clock escapements (mechanisms that keep the pendulum going) and several excellent ones have been made. Most have been based of necessity on mechanical methods or simple electromechanical devices that involve the use of make-and-break electrical contacts, limiting the amount by which error can be minimized. It is now possible to apply electronic techniques to the problem of pendulum drive with a high degree of confidence in their long-term constancy and reliability.

"If a pendulum receives a driving impulse before it reaches dead center, the rate of swing is increased; if it receives an impulse after it has passed dead center, the rate is decreased. Conversely, a retarding force applied before the pendulum reaches dead center reduces the rate, while one applied after the pendulum has passed dead center increases it. Most escapement designs therefore attempt to apply a driving force that is symmetrical about the center of swing, so that any effect on rate before dead center is reached is canceled by an equa1 and opposite effect afterward. In the case of mechanical escapements this is extremely difficult to accomplish. Even with a perfect mechanical design there would be no easy way to adjust the mechanism so that the impulse would split the dead-center position exactly. Assuming that this could be accomplished, any slight shift of the case or support would disturb the adjustment and thereby introduce a change in rate. Any retarding force (such as is needed for unlocking escapements of certain designs) must also be perfectly constant and equal on both sides of the swing and must occur at the same position in the cycle of swing.

"The so-called free pendulum of the Shortt clock gets around the unlocking troubles by using a cleverly synchronized 'slave' pendulum to do this work, together with an electrically tripped and reset weight, or gravity arm, which drops on the rim of a small wheel on the moving pendulum rod to produce the driving impulse. This is perhaps the best escapement heretofore devised, and some Shortt clocks have established records of astonishing accuracy. The indirectly tripped gravity-arm escapement also has the advantage of applying an extremely constant driving-force to the pendulum and hence of maintaining the length of the arc at an impressively constant value. Our experience has shown, however, that changes of only a few seconds of arc can produce unacceptable changes in rate unless the Shortt pendulum is compensated for circular error. Even if the driving force and the friction losses in the drive mechanism were to remain absolutely constant, which they seldom do for long, slight changes in support losses (energy spent in moving the structure to which the pendulum is attached, which must be supplied by the clock drive ) would result in changes in the length of the arc.

"An escapement that produced a genuinely constant driving-force, or one perfectly symmetrical about the center throughout the entire swing, would have no effect on the rate of a pendulum, because any increase in rate produced during the swing to dead center would be exactly canceled by an equal decrease in rate after dead center. Even if the driving force were not symmetrical about the center, but were duplicated exactly in each direction, any error produced in the full swing one way would be canceled on the return swing.

"An escapement that meets these requirements is now relatively easy to produce by means of an electromagnetic drive, although we found none described in the literature. We have made one and believe that it is unique. It consists of a center-tapped coil of fine wire carried by the pendulum and arranged to swing in a reasonably constant magnetic field. This coil generates a voltage that would be a perfect sine wave if the magnetic field were uniform. But irrespective of its wave form the voltage goes through zero and changes sign precisely at the end of each swing, and the two halves of the full cycle are symmetrical.

"This voltage is fed to the grids of a push-pull amplifier, the output of which goes to another coil mounted on the pendulum. This coil moves in a similar, but not necessarily identical, magnetic field. The reaction between the second coil and its magnetic field drives the pendulum. The magnitude of the driving current can be controlled by a series resistor that regulates the driving force, and hence the length of swing.

"The power output of such an arrangement is unstable. If the arc does change slightly, the magnitude of the voltage generated and the resulting driving current change proportionally, thus maintaining the new arc. This defect can be corrected by connecting a device in the circuit (a so-called nonlinear element) to prevent the output current from changing in direct proportion to the input voltage. A simple diode clipper-circuit, for example, and a couple of small nickel-cadmium storage batteries accomplish this quite satisfactorily. With the proper design, Zener diodes, which require no batteries, might be used. Incidentally, these special diodes, which begin to conduct suddenly when the voltage reaches a certain fixed value, are now replacing standard-voltage cells in electrical measuring instruments.

"The magnetic-field-and-coil arrangement and the electronic circuit that we have used to drive the Shortt clock are shown in the accompanying illustration [Figure 4]. Each half of the voltage pickup coil consists of 4,500 turns of No. 40 enameled wire wound on a spool that moves along a 3/16-inch iron bolt between two Alnico magnets, each about 3/4 inch by 1/2 inch by 1/4 inch. The drive coil has 6,000 turns of No. 40 wire and moves in a similar magnetic field. Both coils are mounted below the bob, and the metal parts of the pendulum are magnetically shielded with stationary sheets of highly permeable material to prevent magnetic attraction or eddy-current effects from changing the rate of the pendulum.

"At full atmospheric pressure the Shortt pendulum requires only about 1.5 microwatts of driving energy; at a pressure of one or two centimeters of mercury it needs only about a fifth of this power. The electromechanical efficiency of the driving system can therefore be quite low. Substantially the full output of the push-pull amplifier is used to keep the batteries charged, to replace the losses due to the series resistor and to operate a relay for counting seconds. Power for the amplifier is provided through a voltage-regulating transformer. The entire electronic circuit is mounted inside the constant-temperature enclosure built around the clock.

"The relay, inserted in series with the clipper, serves to actuate the dial mechanism of the Shortt clock, but the accuracy of its closure is not better than about 10 milliseconds-at least an order of magnitude worse than the pendulum itself. For this reason we installed a photoelectric cell, which is energized by a light beam reflected from a mirror on the pendulum, to generate time pulses. These are compared with WWV signals by means of an oscilloscope. The pulses are also used to operate a relay or chronograph, as desired."

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