Pendulum Q factor.
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
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
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
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
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
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'
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
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
Cheap timepieces, including some cuckoo
clocks, have pendulums on simple spindles through the
clock plates. High friction equals low 'Q', equals poor
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
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
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
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
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.
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.
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.
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
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
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
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
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
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
2b1; Balancing of second order
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
Also shown for reference is a plot of a
single half second pendulum.
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
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
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
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.
second pendulum half swing along horizontal line, at
centre of oscillation (994 mm). Against angular and
Calculated using cosines to two or three decimal
places. Millimetres or degrees as appropriate.
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
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.
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
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
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
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
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
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.
What is normally called 'circular error' (CE) is a
reduction of a pendulum's swing rate with increased
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
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
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
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
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
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
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
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
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).