Horology Project  # 1


Prepared by R. H. Lebar.

Pendulum Q factor.

 1: A pendulum, as used in clocks, is a resonant structure. As such the same requirements and constraints apply as with  other resonant systems.

A factor that is often missed in clock construction is the need for a defined and solid termination. Christiaan Huygens, designer of the first practical pendulum timepieces had noted a curiousity.

When two similar pendulum clocks were placed on a mantle shelf they would settle into a synchronous relationship. With their pendulums swinging together, in anti-phase.

He first thought that air movements were locking them together, but clocks enclosed in cases exhibited the same effect. Friction of the pivots was finally decided on as the cause and formulae were devised to calculate the effect.

Friction does play a part, particularly with regards to the 'Q' of a pendulum, its quality as a resonator. Good pendulum systems have a higher Q than most balance wheel systems and are thus potentially more accurate.

Musical instrument designers are well aware of the need to terminate strings, reeds or tines correctly. Clock makers also know that, for chime tines to ring, they must be on  a heavy mounting block.

Yet most pendulums, more essential to the clock, are comparatively lightly mounted. Most of the weight of a grand piano is in its massive cast iron harp, providing its sonority, accurate tuning and sustain.

Yet our Kieninger wall clock, expensive, a fair time-keeper, has her pendulum pivot spring attached to the sheet brass action frame.

We enjoy her presence, as distinct from the radio clocks dotted round the house. An example of a bygone age, well made, attractive and with character. Nicer than moulded cases and plastic gears.

 Yet just a little more design effort could have made her an excellent time-keeper, rarely needing to be corrected. The heavy chimes block may provides a little reflected mass, but this is probably fortuitous.

Despite the massive harp, piano strings couple together in synchronism, mainly due to the sound board and bridge. That is why, where possible, three strings are placed together instead of two.

Two strings would vibrate in anti-phase, providing very little energy to the sound board. Three cannot all be in opposite phase, allowing a strong sound. In the bass, string thickness only alows two strings, which contributes to the higher upper harmonic content of bass tones.

With a pendulum, the same applies. Isaac Newton's third law of motion states 'for every action there is an equal and opposite reaction'. The motion of the pendulum is refracted through its mounting or termination.

Energy from the swing of the heavy bob is transfered through the pivot, even with the theoretical perfect frictionless pivot. That is what synchronises similar pendulum clocks.

It represents a loss, unless the mount is massive and hard enough to reflect it back. Otherwise that energy is lost, lower 'Q', less accuracy and more power required to keep the pendulum going.

To obtain a good 'Q', in the tens or hundreds of thousands is possible, requires a solid, hard and heavy mounting for the pendulum. Certain Web sites state that for good time-keeping a clock should be fixed to a solid wall.

This shows an understanding of the problem, but it may be too late. By the time it gets to the wall much of the errant energy will have dissipated in the clocks' structure and case. It may stop other clocks being synchronised but not improve the clock concerned.

To terminate a four Kilogramme pendulum requires a lot more than four Kilogrammes. An alternative technique is to use a double pendulum, each acting as termination for the other. This is similar to the principle of the tuning fork. One leg of the fork precisely terminates the other, resulting in good 'Q' and accuracy.

Such a clock is simpler in some ways and more complex in others. One of several potential advantages is that a heavy mass is not required. A number of such designs are featured on the Web, all have anti-phase pendulums.

1a; Pendulum pivots:

To obtain a high Q factor requires the pendulum pivots to provide secure termination, freedom of movement and low friction.

Movement in the desired orientation, usually left to right, requires the highest possible 'Q'. Movement in any other direction, needed only for initial self   settling, should have a lower 'Q'. Rotation about the pendulum rod's axis should be prevented.

Cheap timepieces, including some cuckoo clocks, have pendulums on simple spindles through the clock plates. High friction equals low 'Q', equals poor time-keeping.

A flat spring under tension is commonly used, instead of a pivot. Made from the thinnest gauge metal that can reliably hold the pendulum's weight. This shows only the fairly low inter-molecular friction of the metal, but has other issues.

The obvious one is a degree of interference with the gravitational restoring force. Adding a certain amount of spring tension, thus affecting the period of oscillation.

This effect is temperature sensitive, the spring material is stiffer at lower temperatures, more pliable at higher.  Adding to expansion of the pendulum rod but it may not follow the same law, so is more difficult to compensate for.

A flat spring has little resistance to rotation of the pendulum rod, requiring a high degree of symmetry to prevent this. Many pendulum bobs are far from symmetrical front to back, having a decorative brass front cover, as with our Kieninger.

Thus the pendulum may rotate slightly about its axis, with every change in direction. This is both a loss of energy and a source of timing error. In addition, second order vibration of the spring is possible.

The cessation of motion, thus centrifugal force, at the end of each swing causes a reduction of tensile stress in the spring. This may trigger small compressive shock waves, resulting in additional perturbations in the smooth swing of the pendulum.

The design usually used in high end clocks, regulators etc. is a pair of knife edge pivots. This can be close to the ideal but problems may still exist. The knife edges are not ultimately sharp, they can only be honed down to a finite thickness, depending on the material used.

Ideally they should be machined to a small smooth radius. On a flat plate this will result in a rolling motion, rolling resistance is lower than sliding friction. Precious stone such as agate is ideal and was once commonly used.

Knife edges on a flat plate have no locating ability, to maintain correct orientation. A V groove will increase friction, motion will change from rolling to tangential sliding as with a spindle.

An axial semicircular groove, with a radius larger than that of the knife edge is best. Radiused at each end for axial location

Another potential problem is dust falling on the support plate, causing friction and wear. This may be more or less prevented by a radical rethink. Turn the whole thing upside down. Fixed knife edges facing up, capped by a precious stone strip with a groove facing down.

My method is a variation on this idea, Two small radius conical styli facing up, supporting larger radius cup jewels facing down. For a reasonably heavy bob the contact pressures will be high, so diamond is ideal for the styli.

Further protection from dust is achieved by mounting the assemblies within a cross shaft, housed in a bearing block. This has added benefits, the mass is greater, improving termination. Also by keeping clearances small the jewels are protected from damage if the clock receives an impact.

Plus of course a dust tight case.

1b; Pendulum drive:

A pendulum swinging always incurs frictional losses, so without a source of drive power the swing will decline and stop. A higher 'Q' will enable it to swing for longer, but not for ever.

Care is needed to ensure that applied drive has as small an effect on the pendulum's natural rate of swing as possible. A number of factors need to be taken into account and there are rules that should be followed.

Many clever people over the centuries have found, developed and refined these rules. We will do well to take account of their hard won knowledge.

Most of that knowledge was applied to mechanically coupled pendulums. using an escapement or similar device to provide drive and time the clock. But many of the factors involved also apply to free pendulums.

Some early efforts at electrical drive interfered more with the pendulum's swing than many mechanical systems. It is all too easy, when a new idea is born, to ignore what has gone before. More often than not this results in failure.

The following information, unless stated otherwise, comes from that store of knowledge, much comes from Christiaan Huygens. To advance we take a lead from those who went before, what they did makes what we try to do possible.

1c; Timing:

The best time in the swing of a pendulum to apply power is as it passes through its point of equilibrium, its rest point. This is the point of maximum angular velocity, where its mass is neither rising nor falling.

 1d; Amount:

The amount of power applied should be as small as possible, for as short a time as possible. Only just sufficient to maintain the desired swing amplitude.

Also it has been proved that adverse effects on timing are less if power is applied equally in both directions of swing. A little push should be applied in the direction of swing, any other angle is wasteful and more disruptive.

1e; Position:

The best position on the pendulum to apply the push is at the centre of oscillation (CoO). At this one point a drive pulse has no effect on the pivot and does not tend to vibrate the rod. Efficiency is also maximised, it is not the centre of gravity but usually a little lower, within the bob..

Christiaan Huygens calculated that a compound (real) pendulum can be hung upside down from this one point and its period will be the same. This provides a convenient way to determine the point, (see Kater's pendulum on the Web).

2; Dual Pendulums (Pendula):

A synchronous dual pendulum system has advantages if correctly designed.  It utilises the tendency for two resonant structures, with similar fundamental frequencies, to settle into synchronism when coupled sufficiently.

This is the effect that Chistiaan Huygens noted when he placed two clocks on the same shelf. That shelf provided the coupling. Each clock swayed slightly, in reaction to its pendulum, affecting the shelf and hence the other clock.

Such synchronism in a mechanical system has two main modes, in phase and in anti-phase. In-phase motion increases loss of energy to the environment and is usually unstable.

The two resonators tend to shift the phase of their movement until they are in anti-phase. This condition causes a minimum loss of energy, and is stable. Mr. Huygens' clocks exhibited this mode of movement.

Provided they are on a common support, any horizontal effect on the mounting of each pendulum is cancelled by the other, raising the 'Q' of both. If both pendulum bobs are the same mass, their swing amplitude will be the same.

If the bobs are of different mass, then the pendulums will naturally settle into a relationship where each transfers an equal and opposite amount of horizontal energy into the mounting. The lighter pendulum will swing wider.

This will increase its circular error, relative to the other, its natural frequency will slow slightly. The combination will swing at a frequency between the two natural frequencies, if these are slightly different.

This is an unsatisfactory situation, the frequency or swing rate will not be as stable as possible. For the combination to be advantageous, each pendulum's natural rate of swing should be identical, as near as possible.

Taking account of any differences in amplitude, so it is obviously easier if both pendula have similar mass.

With a piano, if one string of a set is detuned,  the result is a distinct 'twang'. Caused by an audible frequency shift, as the strings attempt to synchronise. Made use of for 'honky-tonk' instruments, not a desirable effect in a clock.

For horizontal forces to cancel, the two pendula should be in the same plane and at the same height. In other words, side by side. Then external horizontal forces, along the plane of movement, will have less effect.

External forces in an axial direction will not be cancelled, although such forces have less effect. Vertical movements, both caused by the pendula & external, are likewise not cancelled and do have a readily measurable effect.

The more common coaxial positioning results in a twisting moment, both of the mounting and the pendulums, with only partial horizontal cancellation.

2a; Setting up dual pendulums:

When setting up dual pendulums, each pendulum should be stopped in turn. The other's rate should then be adjusted until it is exactly correct, within the  measurement limits available.

Both pendulums should ideally be powered in use. Although the system will work if only one is powered, the other will impose a slight load.

 The sensor and drive systems can be independent for each pendulum. Then either can be the master, timing the clock. The possibility of electrical synchronism will be investigated, acheiving an exact phase relationship is a potential difficulty.

This section has no antecedents that I know of.

2b; The arguments for a third pendulum:

The possible advantages of a third pendulum are being empirically investigated. The questions to be answered include the obvious, are any potential gains in accuracy worth the additional complication?

2b1; Balancing of second order vibrations:

 Firstly a set of drawn curves showing the vertical and horizontal movements of dual one second pendulums swinging through a small angle. These show the reason why dual pendulums work and why they don't work better.

 Also shown for reference is a plot of a single half second pendulum.

This is a series of plots, depicting the motion of two, side by side, synchronised pendulums agains time. As can be seen, at small angular swings it is a close approximation to a sine wave.

 The top plot is the left hand pendulum, followed by the right. Above the datum line indicates a swing to the left of centre, below is to the right. The 180 degree phase relationship is clear.

The next two plots show the vertical movement of the same pendulums. First the left, then the right. The datum line is merely a drawing convenience here, all movement is above the centre rest point.

 As can be seen this movement is at twice the frequency, it is the second harmonic, two cycles per second.

 This phenomenom is also found with pairs of synchronised piano strings. It will be noticed that these motions are in phase, not out of phase.

 This frequency doubling and phase equality can be arithmetically explained. The square of both positive and negative number is always positive.

 The bob moves upwards as it leaves the centre to the left. or the right. Conversely it moves down when approaching the centre from either side.

 So the opposite horizontal phase relationship means that the pendulums effect on the mountings cancels, for first order vibrations. It also means that the effect of external vibrations in a horizontal plane (left to right) also cancel.

 The effect on the mounting of vertical movement is increased by using two pendulums, but this effect is smaller. So there is a net improvement in 'Q' and potentially in timekeeping accuracy.

 It follows that external vibrations in a vertical direction are not cancelled at all, although these have less effect on swing rate. This probably explains why precision dual pendulum clocks can register earth tremors origination many miles away.

The last plot depicts a half second single pendulum, with no particular phase relationship to the others. Its similarity to the two plots above it suggests a way of reducing a dual pendulum clock's sensitivity to external vertical low frequency vibrations.

 2c; A third pendulum:

 Add a third half second pendulum, then synchronise it in anti-phase to the vertical movement of the other two. Couple it via a right angle lever (bell crank) to the dual pendulum's mounting bar. Its swing should then cancel their vertical effect on the mounting.

It is a matter of matching the degree of cancellation, by adjusting the lever ratio. Both first and second order vibrations will be cancelled. This also applies to external influences, will this create an earthquake proof clock?

At some point a drawing of the arrangement, or a photo, will be added. Together with the criteria to be taken into account.

 Of couse the vertical movements of the third pendulum remain uncorrected. But there is a law of diminishing returns, otherwise we could go on forever.


Pendulum Movement

 One second pendulum half swing along horizontal line, at centre of oscillation (994 mm). Against angular and vertical movement.

 Calculated using cosines to two or three decimal places. Millimetres or degrees as appropriate.

 Swing: Angle: Vertical: H/V ratio:

 2.5       .144      .003          833

5          .29        .013          385

7.5       .432      .03            250

10        .58        .05            200

15        .86        .11            136

20       1.15       .2              100

25       1.441     .314            80

30       1.73       .453            66

40       2.306     .805            50

 As can be seen, for small swings, the ratio of horizontal to vertical movement is approximately square law.. So a plot of vertical movement actually resembles a cycloidal form rather than a sine wave. 

This means that using the swing of a half second pendulum to correct the vertical movement of a one second pendulum is only an approximation. Only experience will show how well the arrangement performs in practice. 

As mentioned previously, the horizontal movement itself only approximates a sine wave for very small swings. It is also actually cycloidal, a matter of a vertical force (gravity) controlling a rotating movement. So it is balancing one approximation against another.

 What may work in our favour is that the bell crank translates vertical movement back into circular movement and vice versa.


Pendulum Control System.

Sensor and motor. Design information.

 Two versions of the sensor have been designed, inductive and capacitive. The inductive version is described here. Only an inductive version of the motor has been designed. A capacitive version is considered impractical.

 Inductive Sensor

 Horizontal steel rods are mounted on the left and right sides of the pendulum bob Adjusted to the height of the Centre of Oscillation, these two rods form both the pendulum motor armatures and the sensor actuators. Their outer ends  are each encircled by two stationary co-axial coils.

The inner left and right pair of coils form the position sensor, the other pair are the motor stator. The sensor coils are  wired in series, energised by a 'push-pull' signal, at a frequency of ten thousand cycles per second (10Kc/s).

When the pendulum is vertical, at rest, the two coils are of equal impedance (balanced), so no signal is present at their junction (sensor output).

If the pendulum is displaced from rest, the coils are unbalanced. One coil's rod enters it further, raising its inductance and impedance. The other coil's rod is partially withdrawn, lowering its inductance and impedance.

A 10Kc/s difference signal is produced at the sensor output. Its phase depends on which way the pendulum has moved. Movement to the left is arranged to give a signal in phase with the master oscillator. To the righ gives an out of phase signal. This difference is decoded as direction information ('DVI').

A one bit digital memory (flip/flop) holds 'DVI' for use by the motor. The signal is also rectified, giving a DC Voltage. The peak level of this Voltage indicates the peak swing (VS). This value is stored in a peak hold circuit for use by the motor comparator.

As the pendulum swings back the swing signal decreases. As it reaches zero the appropriate motor coil, (determined by 'DVI') receives a Voltage pulse. The peak hold circuit is discharged, ready to measure the next swing

The amplitude and duration of the motor pulse are determined by comparing VS with a corrected preset value (VC). This value is adjusted by the environmental sensors to maintain the correct swing rate.

Using full wave decoding, the theoretical resolution of the swing centre detection is fifty microseconds. One part in twenty thousand of the 15mm peak to peak swing, for the one second pendulums.

This is .75 of a micron, in practice, due to circuit noise, processing delays etc. this high resolution is unlikely to be achieved.

Rate correction:

 What is normally called 'circular error' (CE) is a reduction of a pendulum's swing rate with increased amplitude.

As with the Pythagorean comma in music, it is not truly an error, but a factor, to be allowed for or used.

The idea of using 'CE' was one of the first to be considered when the clock was planned.

The value 'VC' is initially adjusted to give a peak to peak swing, measured horizontally at the centre of oscillation, of 15 millimetres. This arbitrary value was chosen to give a small 'circular error', it may need to be changed.

Air temperature, pressure and humidity are measured by sensors. The values obtained, suitably scaled, are added to 'VC'. The purpose is to adjust swing amplitude to correct the rate.

A rise in temperature will extend the quartz rod, 5.5 parts in ten million per degree centigrade, slowing the pendulum by the square root of that change. The temperature signal is inverted and summed with 'VC', reducing its value.

The swing amplitude is reduced, speeding up the pendulum. The two changes cancel.

Likewise a fall in temperature will tend to speed up the pendulum. In this case a change in the temperature signal will increase 'VC'. The swing amplitude is increased, slowing the pendulum. Again the changes cancel.

The same method is used to correct for rate changes caused by barometric pressure and humidity. The idea is to eliminate the need for mechanical compensation. 'Time' will tell if it works satisfactorily.

 The critical parameters are the relative linearity’s of quartz's temperature coefficient, the various correcting signals and 'CE'. These can only be truly determined by careful measurement once the system is operational.


Potential errors with pendulum rate compensation systems.

The coefficient of thermal expansion of pendulum (CTE) rod materials is only approximately linear, over a limited range of temperatures.

The non-linearity may be different, over different ranges for various materials. Thus compensation may not be truly accurate when using, for example, bi-metallic compensation.

Fortunately commonly used materials have 'fairly' linear 'CTE' around comfortable room temperature. This includes fused quartz glass.

So thermal compensation, using two materials with different 'CTE', works 'fairly' well. It is not 'exact' compensation, over the full range of temperatures a clock may encounter.

Other factors, not amenable to compensation, affect a pendulum's rate. Such as variations in pivot friction due to age, wear etc. In the case of spring suspension the spring material will suffer gradual fatigue.

The best thermal compensation methods achieve accuracies sufficient that the uncontrollable errors are the major component. The Shorrt clock from 1921, when carefully set up, is accurate within 220 microseconds per day, 80.03 milliseconds per year.

That is one second in over twelve years, assuming error is cumulative, which is almost certainly not the case. So all is not lost, greater accuracy than this is hardly ever needed.

Electrical compensation using Circular Error (CE) is a different case. The relationship between 'CE' and swing amplitude is not linear.

Like many nonlinear factors, it can be assumed approximately linear over a small range. As any curve can be approximated by a number of straight lines.

The trick in obtaining sufficiently accurate thermal compensation here is to apply a compensating non-linearity to the correcting signal.

The aim at the outset was to produce a clock with a non-cumulative error of no greater than one millisecond per week. One second in over nineteen years, better than any production quartz clock.

 Quartz oscillators are, when carefully adjusted, very accurate at one temperature. The characteristics of a quartz crystal change, like any material, with temperature.

This thermal change, at room temperature, is much less in quartz than most metals, even INVAR. This is why a quartz clock can be more accurate than a more expensive mechanical clock.

A problem is that the common, lower frequency crystals used have a quadratic temperature coefficient of frequency. They are calibrated at their best temperature, usually around 25 to 28 degrees Centigrade.

Any change from this temperature, in either direction, causes a drop in frequency, following a square law.

For high accuracy, quartz oscillators are frequently placed in thermostatically controlled ovens, to eliminate temperature effects. Time standard clocks made after 1929 used this principle.

The best accuracy was around one part in ten million, just over 3 seconds a year. Regardless of room temperature. Not nearly as good as the Shorrt clock, which it never-the-less superseded. Newer is better?

Good makes of quartz watches use a similar principle, the metal case back conducts the wearer's stable body temperature to the crystal. This of course only works while the watch is worn.

Most current radio clocks use a crystal controlled mechanism which is periodically corrected (reset)  by radio signals from a national time standard. The accuracy between these resets depends on the quality of the clock.

 National time standards are synchronised by atomic clocks. These expensive and power hungry devices have accuracy beyond the capabilities of any mechanical system (at present).