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A Second Article about How Two Amateurs Refined the Accuracy of a Pendulum Clock

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by C. L. Stong
August, 1960

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ALTHOUGH ELECTRONIC OSCILLATORS HAVE largely replaced pendulum clocks for the most accurate timekeeping, the mysteries of the pendulum continue to intrigue experimenters. Errors from at least a dozen sources prevent a pendulum from swinging at a constant rate. The sources include changes in temperature and in barometric pressure, mechanical irregularities in the suspension and in the system of drive, electrostatic forces, seismic motion and variations in gravity of a tidal nature. Ten years ago Vannevar Bush, formerly president of the Carnegie Institution of Washington, and John Early Jackson, director of the Office of Atomic, Biological and Chemical Warfare in the Department of Defense, undertook a collaboration as amateur horologists to determine just how accurately a pendulum clock could be made to keep time. In the course of their experiments they worked out a number of improvements in the mechanism for driving a pendulum and in conventional methods of compensating for changes in temperature and barometric pressure. They also solved the 300-year-old problem of "circular error": the change in rate that accompanies variations in the length of the pendulum's swing [see "The Amateur Scientist," July]. Still another kind of error, long known but largely neglected by clockmakers, arises from the motion communicated by the clock to the wall or to any other support on which it is mounted.


Figure 1: A twin-pendulum system for a precision clock

"There is no such thing as an immovable support," writes Jackson. "A pendulum with a 14-pound weight, or bob, swinging through an arc of 1.5 degrees at the rate of one beat per second produces a horizontal force on its support that varies from zero in the center of the swing to nearly a fifth of a pound at each end of the swing. If the circular error has been corrected, or if the pendulum swings through a very small arc, the force varies sinusoidally. There is a similar sinusoidal force in the vertical direction which varies with twice the pendulum frequency, but which is very much smaller in magnitude (about a hundredth as large). Vannevar Bush and I have devoted much effort over a period of several years to gaining a detailed understanding of the effects of such small forces on the rate of a pendulum and devising means of overcoming them. We call the effect 'support reaction.'

"Support reaction does not seem to have received much recognition in the past, other than the realization that a solid and massive support is desirable. We have found that the reaction is subject to experimental, quantitative investigation, and we have devised means for overcoming the effects, which turn out to be unexpectedly large.

"The experiments were made on a Shortt clock borrowed from the National Bureau of Standards. ( The clock had been superseded by quartz-crystal oscillators. ) The instrument was mounted on a cinder-block wall in the basement of my home, where it was fitted with an attachment that corrects circular error, and its original gravity-electric drive was replaced with a new kind of air drive. At that time it was felt that escapement errors were the chief cause of irregular behavior. Actually the air drive turned out to be difficult to adjust and generally not worth the trouble, but after modification the pendulum held to the time signals on the Bureau of Standards Station WWV within one millisecond for one period of 36 hours. Then, during the next eight-hour period, we observed a departure of several hundredths of a second from the signals of Station WWV.

"Everything about the clock seemed to be functioning perfectly during these last eight hours. We noted, however, that it had rained during the night. Subsequent observations showed that changes in humidity affected the rate of the clock, in spite of the fact that it was sealed in an airtight case! It became obvious that the mechanical properties of the cinderblock wall and/or the house foundation must be changing enough to influence the rate by several hundredths of a second a day.

"It can be shown mathematically that the horizontal force on the support of a theoretical pendulum swinging through a small angle (or, if the pendulum is corrected for circular error, through a large angle) varies sinusoidally. The force has a maximum value equal to the centrifugal force that would be produced if the mass of the pendulum were rotated uniformly about a vertical axis on a radius equal to half the amplitude of swing, at a rate of one revolution for each complete swing of the pendulum, that is, one revolution in two seconds for a' seconds' pendulum. It can also be shown that the sinusoidally varying vertical force has a maximum value equal to the centrifugal force that would be produced by rotating the mass of the pendulum about a horizontal axis on a radius equal to half its vertical rise during each half swing, at a rate of one revolution per half swing, that is, one revolution per second for a seconds pendulum.

"This information enables us to construct a system of revolving weights that at every instant precisely balances and counteracts the dynamic forces exerted by a swinging pendulum on its support. The centrifugal force produced by a given mass revolving at a given radius is proportional to the product of the mass times the radius. So instead of using a rotating weight with a mass equal to that of the pendulum, we can use a weight with a 10th or 100th of this mass, mounted on a lever arm 10 or 100 times longer. The vertical motion of the bob of a seconds pendulum swinging through a total arc of 1.5 degrees is less than .004 inch, but if the bob weighs 10 pounds, we can use a weight of .01 pound rotating on a radius of slightly less than two inches to counteract the vertical force. This is obviously more convenient than trying to rotate a weight of 10 pounds on a radius of .002 inch.


Figure 2: Motor-driven weights for compensating support reaction in a single-pendulum clock

"To prevent motion in a plane at right angles to the plane of swing of the pendulum, two half-size weights on parallel shafts can be rotated in opposite directions to give a sinusoidal thrust in only one plane at right angles to the plane of the two shafts [see illustration]. The weights compensating for vertical motion can be conveniently mounted directly above or below the point of suspension of the pendulum so as to produce their thrust through this point, but the weights that generate horizontal thrust must be mounted so their centers of gravity are at the same level as that of the bob. As they move from side to side they will then not only cancel the horizontal forces but also obviate the torques exerted on the clock support by gravity.

"The problem of synchronizing these revolving weights so that they always rotate at exactly the right rate, and exactly 180 degrees out of phase with the pendulum, was a formidable one. It was finally solved with fair accuracy by driving the weights with a synchronous motor operating on ordinary 60-cycle alternating current, the frequency of which seldom varies more than five seconds from the correct time. The motor was coupled with the weight system through a differential gear. The 'cage' of the differential, in turn, was connected to a reversible direct-current motor that received impulses each second from a photoelectric-cell pickup on the pendulum. This signal was fed through a series of relays and cam-operated switches on the revolving shafts, arranged so that the direct-current motor added or subtracted a fraction of a revolution as needed to keep the weights 180 degrees out of phase with the pendulum.

"A stroboscopic-flash unit actuated by the weights as they passed their dead-center position illuminated a beat plate on the pendulum for initial adjustment of phase angle. Observation showed that this system remained synchronized and maintained phase angle to within less than a degree. The system is still not quite good enough, however, for a truly accurate clock.

"Such a system is capable of canceling all support reaction and movement only when the arc of the pendulum can be held to the value for which the mass of the weights and their lever arm was designed. Considerable difficulty has been experienced in maintaining the arc at constant amplitude. Added to this effect is the error caused by the slight variation in phase angle between the weights and the pendulum. With this arrangement, however, our clock has held the same rate as Station WWV to within a few milliseconds per day, with an accumulated error at the end of a week of only about three milliseconds.


Figure 3: Geometry of the support reaction

"In order to exaggerate the effect of support reaction, the whole clock case, together with its rigidly attached system of revolving weights, was mounted about seven inches out from the wall on four half-inch steel bolts. This provided a cantilever mounting that is sufficiently stiff to support the 100-pound clock, but is more resilient than the usual mounting, in which the clock is bolted directly to the wall.

"The horizontal resilience of this mounting, measured by the movement of the pendulum support when a known horizontal force was applied, was about 285 millionths of an inch per pound. The force of the swinging pendulum, weighing .118 pound, thus deflected it less than 60 millionths of an inch.

"Three stiff metal struts-two horizontal and one vertical-were also provided to reduce the resilience at will. (They could be anchored to the wall at their outer ends with lag screws.) The rate was adjusted with the struts removed and the compensating weights running. When the weights were stopped, the rate of the pendulum was 6.42 seconds per day slower. With the struts bolted to the wall (which made the mounting almost as stiff as a case mounted rigidly on the wall), there was still a loss of two seconds per day when the weights were not running, caused by movement of the wall and doubtless of the whole house. But with the weights operating, no difference could be detected when the struts were removed, showing that the weights were completely canceling the support movement.

"A most astonishing result emerged from these tests. No effect on the pendulum's rate could be traced to vertical support reaction! It makes no difference whether the vertical compensator is operated or not, or even if it is run in phase, instead of 180 degrees out of phase, so as to add its effect to that of the pendulum rather than providing a cancellation. This is true whether the struts are bolted to the wall or not.

"It now becomes evident that the simplest way to remove the effect of support reaction on rate is to use a double pendulum with two identical bobs swinging in opposite directions 180 degrees out of phase. The vertical forces developed by the pair on the support add up, but cause no change in rate; the horizontal forces cancel precisely at every instant. Hence, although it is not possible to find a perfectly rigid support for a pendulum, it is possible to reduce the harmful forces of support reaction to zero and thus to remove this source of error."

(In the case of a perfectly springy support Jackson has demonstrated by a simple algebraic analysis that the frequency of a pendulum is decreased by half the ratio of the horizontal movement of the bob. In the case of a non-resilient support with the same motion the demonstration shows that the loss is doubled. He has similarly shown that vertical support reaction has no effect on rate, as shown in the illustration in Figure 3. The reader can obtain a copy of Jackson's algebraic analysis by forwarding a stamped, self-addressed envelope to this department.)

"Bush has shown mathematically by an electrical analogy that increasing the mass of the clock case and its attachments tends to increase the loss in rate produced by support resilience, but that this effect is relatively small and can be neglected for all practical purposes.

"Most, if not all, actual clock supports introduce losses. Even if a perfectly elastic metal beam were used for the mounting, its support would ultimately be stone or earth, which would introduce some losses. In general, therefore, the loss in rate caused by support motion cannot be calculated by any formula, even if the resilience of the support is determined by deflecting it with a known force and measuring the deflection with a strain gauge. Furthermore, the losses and probably the resilience of the ultimate support are subject to unpredictable change with time and will cause the rate of the pendulum to vary accordingly.


Figure 4: Details of the twin-pendulum suspension

"Admittedly a cinder-block basement wall is not an ideal clock-mounting. The clock vault of the U. S. Naval Observatory in Washington has three specially constructed masonry piers, each carried down to a separate foundation and mechanically insulated from the rest of the building. Yet it has been found that if Shortt clocks are mounted on the two piers on opposite sides of the vault, they interfere so seriously that it is impossible to use them for precise timekeeping. If the clocks are on piers at right angles to each other, there is essentially no interference. Each pier still moves, however, and the clocks run slower than if they were mounted on an 'immovable' support. It seems reasonable to assume that hitherto unexplained changes in the rates of observatory clocks are, at least in part, caused by changes in the resilience or other physical characteristics of their mountings.

"The following rough design for a clock is submitted to the readers of Scientific American, without having been tested, in the hope that it may stimulate interest in amateur horology. A clock based on the design is now being built. The design includes the features and principles that have been studied over a period of several years, and that are believed essential to a really precise pendulum clock. We recognize that other designs may be even better.

"As has been indicated, a double pendulum will remove all support-reaction errors if the two pendulums are identical, and if they can be made to swing 180 degrees out of phase. The mechanical arrangement is important, because the horizontal forces must cancel without introducing twisting or other spurious forces that might move the support in any direction. However, a distinction can be made between that part of the support inside the clock enclosure, which is held at constant temperature, and the external support such as a wall or floor that is subject to non-uniform conditions. A constant loss in rate caused by movement of the internal frame can be corrected by shortening the pendulum, but variable effects cannot be corrected.

"The following design involves mounting at each end of a horizontal support bar two pendulums that swing in the plane of the bar, as shown in the accompanying illustration [above]. This requires two circular-error correctors, constructed as described in 'The Amateur Scientist' last month, with their springs mounted on temperature-compensated members carried by the same bar. The bar can be supported at the center, where there is no motion. The bar will be bent and alternately stretched and compressed slightly, of course, but even if it is supported at its ends, there will be no net motion of the base on which the clock rests; there will be only a slight bending of the metal inside the clock case. This will be uniform if temperature and other environmental conditions are held constant.

"An airtight clock case 45 or 50 inches tall is not convenient, and since beats of one-second duration are not necessary if the beats coincide with seconds at reasonably short intervals, a 2/3-second period was chosen. This frequency requires a pendulum only about 17.5 inches long and provides coincidence on every other second with radio time-signals. A half-second pendulum would coincide every second, but only when it was swinging in the same direction, making it difficult to tell when the clock was 'out of beat,' or 'ticking' evenly, as it would do if it was not perfectly level.

"The metal frame of the proposed arrangement can be made of aluminum alloy. It is proposed that the case be made of clear plastic about half an inch thick. All edges of the case would be cemented except for the bottom, which would be grooved to hold a ring-shaped rubber vacuum-seal. The case could be lifted from the bottom plate like a large bell jar, exposing the entire clock and frame for easy access. The metal frame would simply rest on the bottom plate, also made of clear plastic. The bottom plate would rest on a flat and reasonably firm bench or table that can be carefully leveled. The proposed design is depicted in the accompanying drawing [Figure 1].

"Because the coils of the electronic drive mechanism deliver less than a microwatt of energy to each pendulum, and because the thrust acts through the entire swing in each direction, there is no need to mount the pendulums so that their thrust is exactly at the center of percussion, and no need for a stiff pendulum rod. One-eighth-inch Invar rods, or even thin wires, are ample for bobs in the form of right cylinders three inches in diameter by three inches long, each weighing about 7 1/2 pounds.

"A light, polished aluminum vane with a vertical slit at its center is fastened opposite the center of each bob so that the vanes overlap and the two slits coincide when the pendulums are at rest Being nonmagnetic, the vanes are not attracted to each other by magnetism induced by the earth's field, or by leakage from the magnet systems on the bobs, even though they swing very close together. Good metallic bonding to the pendulums prevents any electrostatic attraction between the vanes.

"The narrow vertical slits in the vanes allow a pulse of light to reach a photocell when the slits coincide near the center of each swing. A collimated beam of light somewhat larger in diameter than the length of the slits is directed on the vanes from front to back. An oversize beam eliminates the need for fussy adjustments and allows the same amount of light to reach the photocell even if slit coincidence does not occur exactly at dead center. A lens beyond the slits focuses the light on the photocell. The electrical impulse from the cell is displayed on an oscilloscope, along with radio time-signals, for accurate comparison. If the slits are quite narrow, a photomultiplier tube may be required. The pulse may be made as sharp as desired to give extremely accurate time comparison, and can, of course, operate a chronograph if sufficiently amplified.


Figure 5: Details of electrical driving-mechanism for the twin pendulums

"The advantage of this arrangement lies in the fact that the slits coincide with each swing at the instant representing the mean time of the two pendulums. If for any reason the phase of one pendulum is slightly more than 180 degrees ahead of the other, it will pass dead center a bit too soon; however, the other pendulum, approaching from the opposite direction, will be a bit late. The slits will pass at the same instant that coincidence would have occurred if the phase displacement had remained exactly 180 degrees and they had passed dead center at the same instant.

"Since the two pendulums are normally swinging in opposite directions at any given time, any small seismic or other disturbance that speeds one up is apt to slow the other down by the same amount, but the mean time of the two remains the same. Moreover, if such a disturbance occurs, the system is corrected automatically. The two voltage-pickup coils are connected in series, so that the net voltage fed to the amplifier, and the resulting current in both drive coils, are in phase with the correct mean time of the two pendulums. This makes the driving force lead the slow pendulum and lag the fast one, causing them to pull back into synchronization.

"Furthermore, if the phase angle is not exactly 180 degrees, a slight horizontal movement of the support occurs, which also produces a synchronizing force. Two pendulums hanging from a half-inch steel bolt in a stone wall were found to lock into step in opposite phase and to swing at their average rate, even if their individual periods differed as much as five or six seconds per day. The vanes from front to back. An oversize beam eliminates the need for fussy adjustments and allows the same amount of light to reach the photocell even if slit coincidence does not occur exactly at dead center. A lens beyond the slits focuses the light on the photocell. The electrical impulse from the cell is displayed on an oscilloscope, along with radio time-signals, for accurate comparison. If the slits are quite narrow, a photomultiplier tube may be required. The pulse may be made as sharp as desired to give extremely accurate time comparison, and can, of course, operate a chronograph if sufficiently amplified.

"The advantage of this arrangement lies in the fact that the slits coincide with each swing at the instant representing the mean time of the two pendulums. If for any reason the phase of one pendulum is slightly more than 180 degrees ahead of the other, it will pass dead center a bit too soon; however, the other pendulum, approaching from the opposite direction, will be a bit late. The slits will pass at the same instant that coincidence would have occurred if the phase displacement had remained exactly 180 degrees and they had passed dead center at the same instant.

"Since the two pendulums are normally swinging in opposite directions at any given time, any small seismic or other disturbance that speeds one up is apt to slow the other down by the same amount, but the mean time of the two remains the same. Moreover, if such a disturbance occurs, the system is corrected automatically. The two voltage-pickup coils are connected in series, so that the net voltage fed to the amplifier, and the resulting current in both drive coils, are in phase with the correct mean time of the two pendulums. This makes the driving force lead the slow pendulum and lag the fast one, causing them to pull back into synchronization.

"Furthermore, if the phase angle is not exactly 180 degrees, a slight horizontal movement of the support occurs, which also produces a synchronizing force. Two pendulums hanging from a half-inch steel bolt in a stone wall were found to lock into step in opposite phase and to swing at their average rate, even if their individual periods differed as much as five or six seconds per day. The proposed mounting has considerably higher resilience than such a bolt, making the synchronizing force even greater. During experiments with the single Shortt pendulum ( using synchronized revolving weights as described above) it was found that the pendulum could be driven by the motion produced by the weights alone. The necessary lead angle of the weights was only about five degrees greater than 180 degrees when the air in the clock case was at a pressure of a few centimeters of mercury.

"Bush has made a complete mathematical analysis of the synchronizing force produced by slight support movement and has applied this same principle to the synchronization of the two opposed pistons of a free-piston engine. There energy is transferred primarily by means of gas bled through small pipes from one end of the cylinder to the other, but the basic principle is the same. This makes it possible to keep the two pistons in opposite phase without mechanical linkages.

"A. L. Rawlings, the noted horologist, suggested that instead of using moving coils, with connections made through fine springs near the point of support of the pendulum (as was done with the electromagnetic drive described last month), the arrangement be reversed: fixed coils react with magnets that move with the pendulum. He cited the advantage that such a magnet system would :, maintain constant weight and dimensions over long periods of time, whereas coils of insulated wire may gain or lose weight and thus change the rate.

"Recently it has been possible to work out a practical adaptation of this suggestion. The great difficulty is that any magnetic system with an appreciable external field would not only attract the magnets on the second pendulum but also would induce eddy currents in it and in any nearby stationary metal, thus absorbing energy from the pendulums and influencing their rates.

"The proposed design, however, provides a magnet system so completely self-shielded that essentially no lines of magnetic force escape to interact with nearby metal. These structures can therefore be mounted directly on pendulums. Of course strong nearby magnetic fields will cause trouble, but they should be avoided anyway if accurate time is expected from any clock or watch.

"The steel shell of the self-shielded magnet system is made from a piece of one-inch electrical conduit about two inches long. The ends are plugged with steel disks a quarter of an inch thick. This unit houses a center rod made of two Alnico magnets a quarter of an inch in diameter and a half inch long, with a matching steel rod between them. The magnets butt against the outer plugs, with like poles in contact with the plugs, as shown in the accompanying drawing [Figure 5]. The magnet assembly is supported by and moves with the pendulum. Its Alnico magnets and steel rod oscillate through the air core of a fixed coil. The coil is supported by a nonmetallic rod that passes through a longitudinal slot in the side of the pipe. The leads to the coil also enter the shielded enclosure through this slot. The entire coil structure and its supporting rod should be coated with conducting paint to eliminate electrostatic forces.

"The lines of magnetic force extend radially from the center rod to the outer shell, as indicated, but the entire outer surface has the same magnetic polarity. Any substantial flux leakage is therefore inward, even near the slit where the coil support enters. The electrical drive is similar to the one described last month If a 12AT7 tube, or its commercial version, is used, each voltage and drive coil should be wound on a plastic spool with about 2,000 turns of No. 38 magnet wire to produce between eight and 10 volts of amplified peak output for application to the clipper circuit. This will give good arc regulation and provide a few hundred microamperes for relay operation. A winding space of 118 inch by 3/8 inch is needed, which allows the coils to fit nicely in the magnet system. Heavy Formvar insulation should be used.

"Instead of employing grid resistors in the amplifier circuit, the midpoint of the two voltage coils should be connected directly to the end of the bias resistor as shown in the circuit diagram accompanying last month's article. The drive coils, one for each pendulum, are connected to the amplifier output in series. The shape and mass of the two pendulums should be the same, but to equalize the arcs (in case of a slight difference, particularly in the strength of the magnets ) a high resistance may be shunted across one drive coil. The energy represented by the loss in this resistor is supplied from the amplifier, which produces a voltage across the coil and resistor at all times sufficiently greater than the back electromotive force generated in the drive coil to produce a positive driving force on the pendulum.. There is never any absorption of energy from the pendulum itself, beyond that necessary to charge the grids of the amplifier."

 

Bibliography

CLOCK REPAIRING AND MAKING. F. J. Garrard. C. Lockwood and Son, 1920.

ELECTRICAL TIMEKEEPING. F. Hope Jones. N. A. G. Press Ltd., 1940.

 

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