I wanted to build the most accurate clock I could. Let me emphasize the "I" here. Not the most accurate clock ever built, but the best I could build. Thus it seemed a temperature compensated pendulum would be the first logical step. This quickly turned into a lot of formula seeking and materials science that I would not have dreamed of previously. Not wanting to build a typical grid iron pendulum and not wanting to deal with glass vials of mercury, I felt a simple pendulum of INVAR 36 and some compensation material would be easiest. Many examples exists that show the compensation tube inside the bob which shielded it from temperature changes. I decided that my entire bob would be the compensation "tube".

When I purchased the INVAR, I received the mill's spec sheet which stated the coefficient of linear thermal expansion was 2.2 to the 10[SUP]-6[/SUP]/°C. So how do you know how much material you will need to offset or overcome this expansion/contraction as the temperature changes? In order to solve that we need to know how long the rod will actually be. I spent a long time researching the formula for determining the period of a pendulum for any given length as shown here:

Period = 2PI * sqrt( Inertia / (Mass * gravity * d) )

d being the distance to center of mass.

The problem I had was I knew what period I wanted (one second), I just needed the formula to determine the length from the period. As far as I can tell no such formula exists for this. In the end I wrote a small computer program that ran in a loop to ever so slightly change the length, recalculate the center of mass, recalculate inertia, then calculate the period. Compare the results to 1 second and repeat until I got a length close to 1 second.

Of course to do this assumed that the pendulum bob's size and weight was already known. It wasn't and so I started with arbitrary values I thought were reasonable. The largest bob I could machine using the tools I have is 8 inches in diameter and that was the base value I used initially. Once I knew the length, I needed to determine how much the rod's length would change from one temperature to another. I chose 60°F to 80°F as my upper and lower temperature limits. Thus:

growth = Rod Length * (Coefficient * 0.000001) * (Upper Temp - Lower Temp)

(formula modified for Imperial units)

As a starting point I needed a bob that would expand the same amount. I chose 6061 aluminum because its coefficient of linear thermal expansion of 23.6 to the 10[SUP]-6[/SUP]/°C was enough to cancel out the rods change in length due to temperature. One problem with all of this is that changing the bob's diameter, changes the pendulums center of mass, which of course changes the period. Placing the pendulums rating nut at the bottom of the bob as is traditional would mean there would be no way to tune the temperature compensation. My pendulum has a tunable rating nut hidden inside the bob and a rough adjustment stem doubling as the pendulums pointer protruding from the bottom of the bob.

The below image (the aluminum was anodized) shows the interior of the bob. Seen are the pendulum rod and the tunable rating nut. The sections of the bob marked A & B cancel each other out as the bob expands and contracts with changing temperature. Section A grows downward, section B grows upward and vice-versa. The section marked C is that part of the bob that offsets the change in length of the pendulum rod as temperature changes. As you can see this is a relatively small section of aluminum to offset the entire length of the INVAR rod from the center of the suspension spring to the center of the tunable rating nut.

My plan to tune this is to adjust the pendulum to one second using the rough adjustment stem while the pendulum is at the lower temperature limit, then later when the pendulum is at the upper temperature limit re-adjust the pendulum to one second using the internal rating nut. The result will be a temperature compensated pendulum as best as I can make it.

About twelve inches below the top of the rod is two inches of 7/16 x 40 tpi thread to which a brass disk with 30 divisions threads onto. This is used for fine adjustment. This is handy when my Microset says the length needs to change about 1 or 2 ten-thousands of an inch.

So there you have it. This is my pendulum as it exists today.

Cheers