equation of time, calendar drives. Create mockup for third-order perpetual calendar
calculator module. - October 2014
A decision was made to make a slight change in the
design outlined last month
for the equation cam and sidereal time dial
drives. We have substituted contrate wheels for
bevel wheels in the lead offs to the cam worm drives. In keeping with trying to induce surprise to the viewer
I opted for contrate wheels since this type of wheel design has yet to be used in
this project. In other areas where this could have applied we chose not to
use this type of gear because the speeds were fairly high, such as the
remontoire and strike governors. There a bevel delivers a far smoother
meshing result than a contrate wheel. But here the rotation is one per year in the
case of the equation kidney cam and twice per year in the case of the
sidereal time dial so this is not a factor here.
Also the design of
having both worms driven from similar sized wheels with the worms having a
single and two start thread giving the sidereal time twice the speed of the
equation cam (which rotates in step with mean solar or regular time) is
being changed. The wheels that mesh with the worms are at different
positions in the horizontal plane which would lead to the worm arbors being out of
parallel to each other if driven from the same diameter wheels. To rectify
this we use two different diameter wheels to allow the arbors to remain
parallel to each other. This also obviates the need for one of the worm
gears being a two-start and both will be conventional one-start worms since
the different diameter drive wheels will take care of the need to double
the drive ratio of the sidereal function. While I think this quirky
non-parallel look would be visually attractive, a worm gear that is not
parallel to the face of its mating wheel is not a good mechanical design.
Here is Buchanan's
solution to the positioning of the worm drives. Instead of two identically
sized drive wheels as depicted in the drawing last month (09/10/14), he uses
two different diameter wheels which then position the attached arbors at the
correct depths to have the worms sit parallel to their mating wheels. Just
an aside here. Look at the construction of the contrate wheel, it is made
from one piece of material. The
easy way would be to make the flat drive wheel and then attach a separate
contrate wheel to it. The reason that was not done here are the clearance
issues from the wheel directly above it, yellow arrow, second photo. Had the
new design been noted from the beginning the conventional way could have
been done. This is a clever way around a potential problem and will make for
an interesting spoking project.
Here the mating wheel
for the worm drive is fly cut. The teeth are canted at 4.00 to
match the worm tooth profile. Note how different the tooth profile is to
those of a conventional wheel offering another visual note.
Here the equation
components continue to be made. A small wheel is being spoked out. The
second photo shows Buchanan trimming off excess length from the screws
securing the equation cam plate to its collet, next a close up of that
The various parts of
the equation cam clutch are shown in the first photo. The second has the
dished clutch washer that is mounted under the knurled securing nut. Next the parts
The first photo shows the cam and its drive mounted to the movement. Next
the compound drive and contrate wheel is shown after spoking. Compare this
to the first photo in this installment.
The last photo shows this wheel within the compliment of
its surrounding drive wheels. This plate as well as its mating rear plate is
shown fully skeletonized later.
The two helical wheels are now shown within the context of the machine as a
whole. The upper wheel drives the sidereal dial counterclockwise and the
lower the kidney cam for the equation of time function.
Next the angles of the
worm drive arbors need to be accurately measured and the digital bevel tool
in the second photo does this. Buchanan temporarily has two brass cylinders
mounted on the arbor so as to allow the tool to span the worm gear and take an
Both drives will originate from the two
wheels of differing diameters. Those pivots are located in the cylindrical
part shown in the these photos and their locations were determined from the
data derived from the bevel tool.
This part will later undergo
extensive machining and hand finishing to achieve its final complex shape.
The cylinder is next shown mounted in
place, partially cut away and jeweled.
The first two photos show the lower end of the smaller sidereal worm
arbor, first without and then with it drive wheel mounted to the cylinder.
Next the larger equation worm drive wheel is shown located below the
sidereal assembly and mounted to the same cylinder.
Next the upper worm
arbor pivot is fabricated for the sidereal time. In the next photo the lower
pivot for the sidereal time worm arbor is positioned using some putty to
keep it in place. Note the jewel is also in place.
The lower pivot must
wrap around the equation kidney cam.
This set of photos show
a few the complex series of machining operations that Buchanan has to do in
order to make the potence that will hold the lower end of the equation worm
drive arbor. Because it must wrap around the equation cam its shape is more
complex than the one for the sidereal arbor. There were over 20 separate
machining operations in addition to the hand filing needed to achieve the
final curved profiles.
Here is the final part.
It is obvious from the second photo how this is designed to curve around the
equation cam obstruction. Even here there is a small decorative organic ivy
Detailing is retained in even the smaller parts. Overall length
is 0.75" or 2 cm.
In the meantime the cylinder responsible for holding the other end of
the arbor is being cut out into into final shape with a fine jewel's saw.
Buchanan now begins to
skeletonize the brass frames shown in the first photo. The wheels depicted
against the blue background are a photo of the equation differential work
already completed in July of 2010. The goal here is to continue the
curvilinear design of the differential work through to the plates that
contain the wheels that will connect this to the demonstration drives. The
second photo shows how this is done with the frame supporting the
differential on the left blending in with and continuing onward to the right
into the frames currently designed.
This design is for the
front and rear frames of the section containing the wheels that connect the
demonstration drives to the sidereal and equation complications.
Next the plates for
the wheel set are fretted out and the two frames (formerly plates) are now
shown devoid of wheels in the second photo.
The two arrows point
out a unique way that Buchanan has made the frames curling from opposite
directions on each frame to hold a wheel.
The first photo has a
good view of the complex pair of pivots that were carved out of the solid
cylinder shown earlier in this installment. This has yet to be refined, but
it looks as if Buchanan had somehow simply taken a pair of pliers and
twisted them into the desired shape. Next both worm arbors and their drive
wheels are shown mounted into that assembly.
assembly with the wheels mounted within. The uppermost wheel is now
supported by the two curved extensions that approach the wheel pivot in
opposite directions in mid-air in the first and second photo below. When was
the last time one had seen this done in a clock movement? Normally such
structures are mirrored on both sides of the wheel. Email me if you've seen
The first photo is a close up of the wheel suspended by
the opposite facing pivots. The second photo shows the frames mounted
into the clock. It is reminiscent of the curl of smoke being blown from a
A demonstration of the celestial train two speed transmission. This is
second gear, a 12:1 step up in speed.
See how the curvilinear
frames of the equation drive flow into the new frames, just as depicted on
the drawings. Too bad a great deal of this is hidden behind the time dial,
second photo, only the lower smoke curls are visible. But there are so many
wonderful things to look at that it’s inevitable that something of visual
value will be hidden behind dial or other work. And if one looks hard from
the rear it will be visible. This is why it will take a visitor so much time
simply to take in all the detail. I strove to have as few dials as possible
to convey the information needed and to place them as out of the way of the
main movement as I could. Hence the two main dial sets being placed in
the upper right and left hand quadrants of the clock.
Now Buchanan turns
to the fabrication of the mockup for his reversible, third-order perpetual
The need for the calendar to be able to operate in both forward
and reverse and keep all its data in tact makes this perpetual calendar
design unique. The way we achieve this is to do away with the conventional
manner in which the dates are advanced using a stepper in the form of a star
wheel for the dial indications. Instead everything is directly geared
together and is advanced each day at midnight using a remontoire. The
perpetual calculator mechanism has a special provision to allow it to step
backward using an index wheel with all of the other calculating components
being geared together facilitating the ability to run in reverse.
is a third-order calculator in that it encompasses all three criterion
needed to keep the calendar in synch with the tropical year.
It is likely that the handful of clocks which exist that have
a perpetual Easter calculator has this third-order calendar calculator
included, but there are none, to the best of my knowledge, that can run in
Why do we need such a
reversible perpetual calendar in this machine? The reason is that we will
use the calendar indications of the day, date, month and year to give an
exact temporal reference to the demonstration of the celestial functions. In
other words, when the machine is in demonstration mode for all of the rest
of the celestial functions, the calendar will advance or go backwards in
synch with that demonstration. In this way one can see exactly how certain
celestial events will look like or occur on any given date. This makes the
prediction or verification of events such as a solar or lunar eclipse
In the Gregorian calendar three criteria must be taken into account to identify
leap years: A year
will be a leap year if it is divisible by 4 but NOT by 100.
If a year is divisible by 4 and by 100, it is NOT a leap
year unless it is also divisible by 400.
This means that 2000 and2400 are leap years, while 1800, 1900, 2100, 2200,
2500 are NOT leap years. When these criteria are accounted for the
calculator is permanently perpetual; it is a third-order perpetual calendar
Why do we need Leap Years?
Leap Years are needed to keep our
modern day Gregorian Calendar in
alignment with the Earth's revolutions around the sun. It takes the Earth
approximately 365.242199 days – or 365 days, 5 hours, 48 minutes, and 46
seconds – to circle once around the Sun. This is called a tropical year.
However, the Gregorian calendar has only 365 days in a year, so if we didn't
add a day on February 29 nearly every 4 years, we would lose almost six
hours off our calendar every year. After only 100 years, our calendar would
be off by approximately 24 days.
A basic first-order perpetual calendar accounts for
the quadrennial leap years. That is, a four year perpetual calendar. The
next complication is the 100 year calendar where at the end of that 100 year
period the leap year is skipped. To keep the calendar in synch that skipped
leap year must be continued for the next three 100 year intervals. That is for
the next 300 years, every 100 years a leap year is skipped. The 400 year
perpetual calendar then allows the leap year to be re-inserted once every
400 years; making it a perpetual calculator.
Of course it goes without saying that the calendar properly accounts for the
regular sequence of the months
ending in 30 or 31 days.
Below we begin with the first layer of complication to make
the calendar a simple perpetual calendar, first order.