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Finish sidereal, 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 collet.

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 assembled.


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 accurate reading.

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 spur. 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.

The finished 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 this before.


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 fine cigar...

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.


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Now Buchanan turns to the fabrication of the mockup for his reversible, third-order perpetual calendar.

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. This 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 reverse.

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 possible.  

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 and 2400 are leap years, while 1800, 1900, 2100, 2200, 2300 and 2500 are NOT leap years. When these criteria are accounted for the calculator is permanently perpetual; it is a third-order perpetual calendar calculator.

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.

What we have shown here in the first three photos are the basic components of the calendar module if it were to be a simple perpetual calendar. The first is the index wheel that has the days of the month represented by one through thirty one detents, but with 29 through 31 removed. This section is shown in detail in the next photo. In the third photo are the three movable detents that substitute for the three vacant spaces on the index wheel. These can be moved into position as needed and in the parlance of watch repeater mechanisms these would be called ‘surprise pieces’.


These levers are shown installed above the index wheel in the first photo. The three pieces do the following: 1. Account for the 30 and 31 regular monthly durations, 2. Account for the ‘not February’ rotation each month where February is the one month that has a shorter period than any other month, 28 days, and 3. The addition of the day in February, giving 29 days, for the leap year. The first photo shows these levers in place above the main index wheel. The next photo shows the cams that drive two of the surprise pieces, the the monthly deviations between 30 and 31 days and the ‘not February’ piece. The next photo is the leap year cam which runs on a twenty year cycle. This will later have a cam attached that in combination with this twenty year cycle will cycle once every 400 years for the next layer of complication to be discussed later.


These three photos show the normal month durations for non leap years. The first shows all three surprise pieces in their raised positions to give a 31 day period, next one piece lowered for a 30 day month and next all three pieces are lowered for the regular 28 day month of February.  


Next is the leap year where February has an extra day for 29 days with two of the three surprise pieces raised. The next photo shows one of the cams used to control the surprise pieces and next some of the gear work.

Now we add another layer of complexity to make the calendar perpetual for 100 years, second order.


Here we see a five lobed Geneva stepper cam attached to the cam. Since the larger cam to which the smaller cam is attached rotates every 20 years that smaller cam is stepped once every 20 years or will rotate one revolution in 100 years. Next is the piece that will step the Geneva cam.


The drive piece is installed and the 100 year cam is shown as if February was having 29 days, which is a leap year. Next the cam is shown not giving February 29 days so the leap year is eliminated. The third photo shows February having 29 days (arrow) .So now we have a system that will skip a leap once every 100 years.

The final layer of complexity, the 400 year correction, to make the calendar permanently perpetual is described below, third order.


The first photo shows the four stepped Geneva cam attached to the 400 year cam. Along with the associated gearing and bridge work needed to install this upon the existing calendar work. That cam is basically a smooth disc with one protruding arm. The next two photos show the four hundred year cam giving February 29 days once every 400 years, the next no intervention for the other 399 years.

One might ask how it can be that with this cam rotating only once every 400 years, the detent does not ride slowly up the one raised lobe thus confusing the insertion of the 29 day correction in the preceding and following years of the one correction year. Here is where the Geneva cam comes in. It ‘flips’ the cam every 100 years, so that cam is never actually continuously rotating but only jumps to the exact position each 100 years. The jump is made before the reader arm descends to the surface of the cam.


As with many other subsystems in this machine, a full scale working mockup had to be made to test for functionality before fabrication could begin. While other perpetual calendars have been made, there are no ready sources that we know of that would have given complete schematics. And even if this source existed, there has never been a perpetual calendar designed to operate in forward and reverse. The mechanism displays a wonderful layering effect. The 100 year cam and the bridge containing the 400 year cam works will also pirouette upon the 20 year cam, sadly a bit too slowly to be readily appreciated, but beautiful nonetheless!


The overall size of the finished mechanism will be about 2.5 inches (6 cm) and will contain about 30 parts. The entire mockup was completed in six days.

Shown here is an explanation of the various components of the perpetual calendar module.


Demonstration of all twelve months in a standard year.


Demonstration of standard 30 day month.


Demonstration of standard 31 day month.


Demonstration of a standard month of February, 28 days.


Demonstration of February in a leap year, 29 days.        



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