Enhanced compound pendulums and systems

Enhanced compound pendulums provide thermal compensation and/or barometric compensation, such as for a mechanical clock system. The enhanced compound pendulums are simple to construct, and can be more easily compensated than conventional, single-bob pendulums. The enhanced compound pendulums typically comprise material that is added above the point of rotation. Thermal expansion factors for components of the enhanced compound pendulums may preferably be chosen to provide thermal compensation to the first order. In some embodiments of enhanced compound pendulums, volume is added above the pivot to provide barometric compensation, such as by equalizing the moments above and below the pivot, or by providing geometric symmetry above and below the pivot, with a lower density above the pivot.

BACKGROUND OF THE INVENTION

1. Technical Field

The invention relates to time keeping devices. More particularly, the invention relates to enhanced compound pendulums that can be thermally and/or barometrically compensated, such as for a mechanical clock.

2. Description of the Prior Art

Historically, the gravity pendulum has been the most successful device for accurately regulating the timing of a mechanical clock. The frequency of such a simple pendulum is approximately proportional to the square root of the ratio of earth's gravity to length of the pendulum (f=2π√{square root over (l/g)}). Because the force of gravity is reasonably constant, keeping the period constant is largely a matter of keeping the length constant, which can be accomplished by careful selection of the materials and geometry, while paying special attention to expansion due to changes in temperature.

While an idealized pendulum has all of its mass concentrated at a point, real pendulum are actually a compound pendulums, with a distributed mass. In general, a compound pendulum has a longer period than a corresponding idealized pendulum, because of the extra moment of inertia contributed by the distribution of the mass.

Another potential accuracy problem of a gravity pendulum is that the period of the swing actually depends slightly on the amplitude. The frequency formula mentioned above is based on the assumption that the restoring force created by gravity is proportional to the angle of the bob from vertical, which is only an approximation. Actually, the restoring force is proportional to the sine of that angle. This difference is small as long as the angle is small, but to hold the frequency constant, the average amplitude of the swing must also be held constant.

Friction creates most of the difficulties in holding constant amplitude. While the greatest source of friction is often the pendulum motion through the air, there is also friction in the unlocking of the escapement, as well as friction in the suspension. Each of these sources of friction is variable. Also, the existence of any type of friction requires that energy be put back into the pendulum, to keep the pendulum going. This impulsion of the pendulum can be a major source of variability, because it is difficult to deliver the exact same impulse on each tick.

Another source of the error in a pendulum is the variation of the density of air, which changes the buoyancy of the bob. Because some of the weight of the bob is supported by floating in the surrounding air, the restoring force of gravity varies with the density. Because the density of the air depends on the barometric pressure, variations in pressure contribute to variability in, the rate of the pendulum.

Eliminating air around the pendulum can reduce several sources of variability because, while air is not only the source of the variable density problem, air is also the source of much of the friction. For this reason, most accurate clock pendulums at this time are operated in a partial vacuum.

It would be advantageous to provide a pendulum that has some of the same advantages, but without the complexities of maintaining a partial vacuum.

It would also be advantageous to provide a pendulum that has thermal compensation and/or barometric compensation, that is simple to construct, and that is more easily compensated than a conventional, single-bob pendulum.

SUMMARY OF THE INVENTION

Enhanced compound pendulums provide thermal compensation and/or barometric compensation, such as for a mechanical clock system. The enhanced compound pendulums are simple to construct, and can be more easily compensated than conventional, single-bob pendulums. The enhanced compound pendulums typically comprise material that is added above the point of rotation. Thermal expansion factors for components of the enhanced compound pendulums may preferably be chosen to provide thermal compensation to the first order. In some embodiments of enhanced compound pendulums, volume is added above the pivot to provide barometric compensation, such as by equalizing the moments above and below the pivot, or by providing geometric symmetry above and below the pivot, with a lower density above the pivot.

DETAILED DESCRIPTION OF THE INVENTION

Some embodiments of the invention provide a new form of gravity pendulum, which is referred to as a low-displacement pendulum, because it displaces less air as it rotates. The low-displacement pendulum uses an unbalanced wheel instead of the more conventional suspended bob. Because the pendulum displaces less air as it rotates, it eliminates an important component of air drag that causes energy loss in a normal pendulum. The low-displacement also eliminates errors caused by variations in barometric pressure. In addition, it can be easily thermally compensated without the use of special materials, such as Invar.

A key aspect of a low-displacement pendulum is the use of an unbalanced wheel instead of a suspended bob. The unbalance can be accomplished by increasing the amount or density of material in the lower part of the wheel or by decreasing density in the upper part of the wheel. Because a wheel is symmetric about its center of rotation it does not displace air as it rotates. This eliminates a major component of aerodynamic drag, referred to as form drag, leaving only the skin drag caused by sheer in the boundary layer. The lack of displacement also eliminates rotational forces that are caused by buoyancy and which vary with barometric pressure.

The wheel of a low-displacement pendulum may be a full disk, but the inertia per weight can be increased by thinning the center of the pendulum, putting most of the mass in a thin ring around the edge. The mass farther from the center contributes most to the inertia of the wheel. Thus, this lightens the mass required to achieve a given inertia. The skin drag can be reduced by replacing the center hub with one or more spokes between the ring and the center support suspension. If these spokes are placed symmetrically across from each other, the low-displacement property of the pendulum is preserved. In any case, the displacement of the spoke or spokes is small.

The use of spokes creates some additional air drag. This can be minimized by streamlining the shape of the spoke by keeping the cross-section small, and by keeping the number of spokes to a minimum. If a single spoke is used, it can either be above the center, in compression, or below the center, in tension. Which configuration is best depends upon specific details, such as the choice of materials and type of suspension. Materials, such as glass and ceramics, are often much stronger in compression, favoring the spoke-up configuration. Flexible materials, such as metal, may tend to buckle, favoring the spoke-down configuration. The spoke-down configuration is also particularly simple because it uses a flexure bearing, whereas a spoke-up configuration is very simple because it uses a knife-edge suspension.

A Slower Swing

The low-displacement pendulum is a type of compound pendulum. Accordingly, it swings more slowly than a conventional pendulum of the same length. This can be used to advantage in further reducing air drag, which depends on the velocity of the pendulum. In a simple pendulum, the period is determined by the length because both the inertia and the restoring force scale together with the mass, but in a compound pendulum these two factors need to be controlled independently. In the low-displacement pendulum, the degree of imbalance, which controls the restoring force, can be made arbitrarily small. This allows the period to be increased without changing the length.

For a thin ring, if the distance from the center of rotation to the center of gravity is h, and the radius of the ring is r, then period of the pendulum is:

period≈2⁢π⁢r2gh(1)
where g is the acceleration due to gravity, or expressed in terms of angular frequency:

Skin drag on an oscillating ring is proportional to area of the surface, the velocity, and to the square root of the viscosity and density of the medium and the frequency of the oscillation. The drag on a low-displacement pendulum of radius r, half-angle φ, and angular frequency ω is:

drag=area⁢⁢r⁢⁢ϕω⁢μρω2(3)
where μ is the viscosity and ρ is the density of the medium, in this case air. Note that benefits to reducing the frequency of oscillation are better than linear. This means that the quality factor Q of the oscillator actually increases as the pendulum is made slower. This may seem counterintuitive because Q is sometimes expressed as frequency divided by damping factor, but in this case the damping factor goes down faster than the frequency. For a low-displacement pendulum:

The above calculations neglect the drag due to the spokes, but they are small and streamlined, and are small compared to the drag of the ring.

There are also additional advantages of having the pendulum swing more slowly. One is that the gear train is simplified because there is less reduction required. Another is that less energy is required to keep the pendulum swinging. The impulse variability per unit time can be reduced if the impulses are delivered less often.

Temperature Compensation for Low Displacement Pendulums. If the pendulum undergoes thermal expansion, the period changes as specified by Equation (1) above. Changes in both r and h effect the period, but these two effects work in opposite directions. The period changes in proportion to r and in inverse proposition to the square root of h. This is because the increase in r increases the inertia of the pendulum, whereas increasing h increases the restoring force. If suitable materials are chosen for the spoke and the ring, the two effects can be made to cancel.

To achieve first-order temperature compensation in a single-spoke-system with a downward spoke, in tension, the coefficient of thermal expansion of the spoke must be slightly greater than that of the ring. Specifically,

This can be achieved by making a portion of the spoke out of material having a higher coefficient of expansion than the material of the ring. For example, if the ring is made of nickel or nickel copper alloy such as Monel™ it may be sufficient to make the suspension spring out of stainless steel or phosphor bronze, and the rest of the spoke from the same material as the ring. The exact proportion of spoke length that is made of the high-expansion material is determined by ratio of h to r.

In the case of an upward pointing spoke, the coefficient of thermal expansion of the spoke must be slightly lower than that of the ring. Specifically,

This can be achieved by making a portion of the spoke out of material having a lower coefficient of expansion than the material of the ring, e.g. such as but not limited to quartz.

Presently Preferred Embodiments of Low Displacement Pendulums.FIG. 1is a side view showing a spoke up configuration16of a low displacement pendulum having a knife-edge bearing14, where a portion12of the pendulum ring10is made of a higher density material.FIG. 2is a side view showing a spoke up configuration26of a low displacement pendulum having a suspension spring24, where voids22are formed in the pendulum ring20to create imbalance.

FIG. 3is a side view showing a spoke up configuration of a low displacement pendulum having upward compensation, where a portion34of the pendulum spoke36is made of a lower expansion material, and where an optional false spoke32is provided to preserve symmetry of the pendulum ring30.

FIG. 4is a side view showing a spoke down configuration of a low displacement pendulum having a suspension spring44, where a pendulum spoke46provides weight to create pendulum imbalance in the ring40.

FIG. 5is a side view showing a symmetrical spoke configuration56of a low displacement pendulum having a knife-edge bearing54, where a lower portion53of the pendulum ring52is thicker to create imbalance in the ring50.

FIG. 6is a side view showing a curved spoke configuration of a low displacement pendulum having downward compensation66in the ring60.

The reduction of frictional and barometric errors, combined with simple thermal compensation, make the low-displacement pendulum an alternative to the conventional bob pendulum operated in air. The reduced power requirements and high Q suggest that the slower low-displacement pendulum has greater stability than a conventional pendulum. Whether or not it is more accurate, the reduced input power, reduced wear and simplification of the gear train associated with a longer period, make this an attractive pendulum. Initial experiments look promising. An 18-inch test pendulum, having periods of about three seconds, has a Q of several thousand, which compares favorably with conventional pendulums of the same size and weight.

Enhanced Compound Pendulums. Several embodiments of enhanced compound pendulums100, e.g.100a(FIG. 7) are also disclosed herein, which can be easily compensated, such as to provide thermal compensation and/or barometric compensation, without the use of exotic materials such as Invar™ or mercury. These enhanced compound pendulums100are simple to construct, and can be far more easily compensated than conventional, single-bob pendulums.

As noted above, compound pendulums generally have a longer period than a corresponding idealized pendulum, due to the extra moment of inertia contributed by the distribution of the mass. The enhanced compound pendulums100disclosed herein deliberately take advantage of this effect, by putting material above the point of rotation.

FIG. 7is a schematic front view120of a first exemplary embodiment of an enhanced compound pendulum100a.FIG. 8shows exemplary rotational movement140of a first exemplary embodiment of an enhanced compound pendulum100a.

As seen inFIG. 7andFIG. 8, two point masses, e.g. comprising an upper point mass A134and a lower point mass B128, are located above and below the point of rotation122, at distances a136and b130, respectively.

The model applies to any compound pendulum with radius of gyration r and distance from center of rotation to center of gravity h, where

The angle of the pendulum θ changes with time according to Newton's equation

For small amplitudes, the solution to this equation is a sine wave with period

Or in terms of the distance to the center of gravity and the radius of gyration

Notice that if h=r (when upper point mass A=0) this reduces to the more familiar

The period p remains constant as long as the ratio of force to inertia does not change. If the upper spoke132and lower spoke126are comprised of materials with coefficients of thermal expansion α and β, respectively, then a temperature change of ΔT will change the period to

This may be expanded as a Taylor series around ΔT=0 to

By setting this ΔT term in the expansion to zero, the coefficient of thermal expansion α is given as:

Or in the case where a=b=r

Therefore, the expansion factors for components of enhanced compound pendulums100may preferably be chosen to provide thermal compensation to the first order.

FIG. 9is a schematic front view160of a second exemplary embodiment of an enhanced compound pendulum100b, having a ring and spoke structure, such as to provide the further advantages of a low displacement pendulum, as described above. As seen inFIG. 9, a lower arm126extends from the center of rotation122to a lower point mass128, while an upper point mass134is mounted to the assembly100bthrough a ring162that extends to the lower point mass128, wherein the upper mass134is connected to the center of rotation through the lower mass128, instead of directly to the point of rotation122.

The exemplary enhanced compound pendulum100bshown inFIG. 9makes the thermal expansion factors for providing thermal compensation convenient. For example, assuming that the ring162hangs from the point of rotation122by a massless downward pointing spoke126of radius r and distance from the center of rotation to the center of gravity h, and that both the ring162and the spoke126have a coefficient of thermal expansion β, then the change in period accounting for temperature change is

Both the change due to temperature and period differ from the idealized pendulum by a factor of

However, if the spoke126that has a coefficient of thermal expansion β is replaced by a spoke126that has a coefficient of thermal expansion α, then there is an additional shift of r(α−β)ΔT, wherein the period p is shown as

The (r(α−β)ΔT)2term in the numerator is the extra inertia due to the parallel axis shift. Using the same Taylor series method used above, the enhanced compound pendulum100bis thermally compensated, i.e. the ΔT term is zero, when

If h is small compared to r in this embodiment of an enhanced compound pendulum100b, the two coefficients of thermal expansion β and α are similar in magnitude.

FIG. 10is a schematic front view180of an alternate exemplary embodiment of an enhanced compound pendulum100chaving a ring and spoke structure, wherein a central member182, e.g. a ring structure182, extends locally around a central compensator structure190. As seen inFIG. 10, the central ring182does not extend around the entire length of either the upper spoke132or the lower spoke126.

In practice, the coefficient of the spokes126,132for some embodiments of the enhanced compound pendulum100can be tuned, by using a second material for only a portion of the length for one or both spokes126,132, e.g. a portion of length b of the spoke126, and/or a portion of the length a of the upper spoke132. For example, a steel pendulum100ccan be thermally compensated with a short length of brass or aluminum replacing a portion of the steel upper spoke132.

As seen inFIG. 10, the ring182does not extend around the entire spoke, but just around the compensator190. Furthermore, the compensator190may preferably be comprised of a material that has either a higher or a lower thermal expansion than the rest of the pendulum100, e.g.100c. This is accomplished by using an upward pointing spoke in the case where the compensator190has lower thermal expansion, also shown inFIG. 10. In this case, the compensation occurs when

FIG. 11is a detailed schematic view200of an exemplary embodiment of an enhanced compound pendulum100dhaving a ring and spoke structure, wherein a peripheral ring202extends from the lower spoke126. The lower arm126shown inFIG. 11extends from a first end206a, at the center of rotation122, to a second lower end206b.

In the exemplary embodiment100dseen inFIG. 11, the outer peripheral ring202comprises a 20-inch diameter ring202of 0.25-inch diameter steel rod. The peripheral ring202is suspended from the pivot122, by the downward pointing spoke126. A portion204aof the lower spoke126comprises 1-inch diameter medium carbon steel rod.

In the exemplary embodiment shown inFIG. 11, the mass of the ring and spoke themselves serve as the weights, wherein no additional bobs, e.g. a lower mass128and/or an upper mass134, are used.

The required length of the compensator207may be determined by parameter h and r, which can be calculated or measured directly. For a ring202and bar portion204acomprising medium carbon steel having a density of 0.284 lb/in3(7.86 g/cm3), the total weight of the enhanced compound pendulum100dis calculated to be approximately about 5.7 pounds.

The distance h, from the center of rotation122at the pivot124, to the center of gravity h is about 2 inches. The period for the exemplary enhanced compound pendulum100dis about 2 seconds. The radius of gyration r calculated from either the measured period or from the geometry is about 8.5 inches. Using the formulas above with β=15μ/° C., which is the thermal expansion coefficient of medium carbon steel, the required thermal expansion coefficient α of the lower spoke126is calculated as:

Since yellow brass has a thermal expansion coefficient of about 20μ/° C., a lower spoke126with the desired temperature coefficient of 18.5μ/° C. may preferably be provided, such as by replacing a section204babout 7 inches for the 10-inch spoke126with brass. Since the density of brass is similar to that of steel, the parameters h and r change only slightly. In practice, the expansion coefficients are often not exact, such that the length of the compensator may preferably be tuned by experiment.

Therefore, some embodiments of enhanced compound pendulums may preferably provide thermal compensation for a pendulum by replacing at least a portion of a spoke126with one of higher thermal expansion. Expansion of the radius of gyration slows the pendulum, unless it is compensated by a greater force through the lengthening of h. In embodiments of enhanced compound pendulums100, these two parameters can be controlled independently.

FIG. 12is a schematic partial cutaway view220of an enhanced compound pendulum100ehaving an upward pointing spoke232, wherein a large portion of the structure100eis a pendulum body222comprised of brass. In an exemplary embodiment of the enhanced compound pendulum100e, the upward pointing spoke232is comprised of steel, and serves as both a thermal compensator and a pivot122.

In one specific embodiment of an enhanced compound pendulum100e, his about 1.4 inches, and is r about 7.3 inches, yielding a period of about 2 seconds. The exact period can be adjusted by changing the position of the spoke232. The length236of the compensator232may preferably be adjusted by changing the position of the clamping mechanism234, e.g. a clamping screw, which forms the contact with the compensator232. In one embodiment, the clamping mechanism further comprises one or more washers that are pressable against the compensator bar232. In some embodiments, the thickness of the material is the same for all parts, e.g. between central elements224. In one specific embodiment, the overall pendulum height228is 20 inches, with central elements having a width230of 2.625 inches and a height238of 4 inches, front and rear face members226a,226beach having a width239of ¼ inches, and an upper spoke compensator232having a 0.5 inch diameter and an initial height236of about 8.5 inches. For some embodiments of the exemplary enhanced compound pendulum100eshown inFIG. 12, a barometric compensator element240may preferably be added to the top of the pendulum100e, such as directly attached to the upper spoke232, which may further comprise a hollow region242defined therein.

If the enhanced compound pendulum100eis suspended by flexure, the variation in restoring force due to the change in stiffness of the flexure may preferably be taken into account. This is particularly important in a compound pendulum, because the restoring force is lower in proportion to the mass.

The speedup s can be calculated from the speedup equation shown in K. James. “The Design of Pendulum Springs for Pendulum Clocks”,Timecraft, June-August 1983, which is incorporated herein. While the speedup s is typically less than one percent, its change with temperature can be a significant source of error.

For example, a compound pendulum that is suspended by a spring would have a temperature stiffness dependency of γ (typically negative), wherein the value γ takes into account both the change in dimensions of the spring and the (generally larger) change in Young's modulus, as described by A. Rawlings,The Science of Clock and Watches, Pitman Publishing, p. 144. For example, the value γ is about 200 parts per million per degree C. for spring steel, or about 300 ppm/C for Ti-6-4 titanium alloy, as noted in R. Boyer, G. Walsch, Materials Properties Handbook Titanium Alloys, ASM International, p. 493. For small s the period changes by sγΔT, as shown:

This is compensated when

When both s and β are small this is approximately

If the suspension spring has a coefficient of thermal expansion δ, when the thermal lengthening and effective length L of the spring (the length below the center of rotation) are also taken into account, then by a similar calculation

Or in the case of an upward spoke

The effective length L is typically ¾ of the actual length of the material, as noted in P. Woodward, “Some Thoughts on Suspension Springs”,Horological Journal,1998-5, pp. 3-6.

Finally, the variation of force due to barometric error (due to change in buoyancy) is dependent on the first moments of the volumes of the masses above and below the center, Va and Vb,

Some barometric compensators use an aneroid pressure-sensing element to adjust the distance b with changing air pressure, as noted in P. Woodward,My Own Right Time, Oxford University Press, p. 102. In a pendulum with mass above the point of rotation, the barometric error can be eliminated by balancing the first moments of the volume about the pivot. This can be accomplished by making the material above the pivot less dense, either by using a low-density material, or by building a sealed hollow chamber into the upper bob. For example, the pendulum will be barometrically compensated if the pendulum is geometrically symmetric about the pivot. It would be possible to build a similar compensator for a conventional pendulum, by attaching a bob of low-density material to the rod above the pivot point.

For instance in the example pendulum described above, barometric compensation can be accomplished by adding a symmetrical upward pointing spoke of a low-density material, such as a light wood or plastic. A thin-walled tube with sealed end caps would also serve the purpose. This 1-inch diameter compensator can be attached to the ring or to the spoke, but a small gap208may preferably be defined in this “false spoke” to allow the ring202to expand freely, as shown inFIG. 11.

By using the methods described, thermally and barometrically compensated pendulums can be easily constructed without the use of special low-expansion materials or devices such as aneroids.