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To determine a pendulum's axis of rotation, a small piece of paper is temporarily mounted on the front of the pendulum rod at the rod's top end, so that the paper extends up an inch or two past the suspension spring. With the pendulum stopped, two small dots (A and B) are marked on the paper about an inch directly above and another inch directly below the top end of the free unclamped part of the suspension spring. Although the one-inch dimensions are not critical, they accurately measure the actual distance between the two dots (A and B). The pendulum must be set swinging at its normal swing amplitude, and an accurate ruler (a six-inch machinist's scale calibrated in decimal inches is ideal) is used to measure the horizontal motion of each dot. The location of the axis of rotation changes slightly with the pendulum's swing amplitude.

In 1983, K. James published two useful but rather complicated equations whose purpose was to help design the suspension spring. The equations show the effect that different spring lengths, widths, and thicknesses will have on the pendulum. The two equations are quite helpful, as the suspension spring is without doubt the most complicated part of a pendulum, despite its apparent physical simplicity. The first equation calculates the maximum stress in the spring, which occurs at the spring's top end at the maximum angle of swing. The second calculates how much the pendulum will speed up due to the inherent torque exerted by the suspension spring on the pendulum rod. In this chapter, the second equation is used to show that the suspension spring exerts a temperature effect on the pendulum's timing that is roughly as big as the thermal expansion of the pendulum rod. Anywhere from 16% to 84% of a pendulum's total temperature sensitivity is due to the suspension spring, with the actual amount depending on the spring's dimensions, modulus of elasticity, and suspended weight.

A previous experiment showed that the walls of a pendulum clock case can slow down the pendulum via air drag by about 1 second per day. The pendulum has a 2-second period. This chapter describes an experiment designed to find out if the walls' drag on the pendulum could be reduced or made more constant by shaping the walls' inside surface for easier air flow. The concept basically involved rounding the square corners inside the clock case. The results show that a spherical bob had the smoothest airflow, while a large cylindrical bob disturbed the most air. There was very little, if any, air movement near the clock case walls. Any further attempt to affect the airflow should be aimed at the bob's surface-to-air interface, and not at the case walls.

An electromagnetically driven pendulum is more accurate than a mechanically (escapement) driven pendulum. This is because a pendulum is disturbed less by an electromagnetic drive pulse than by hitting and dragging a pallet across an escape wheel's tooth. This is empirically based on Q —the less disturbed a pendulum is, the more accurate it will be. A short drive pulse at the center of swing is superior to the continuous sine wave drive approach. This is due to the difficulty in avoiding spurious electrical drive currents at the ends of swing, where unwanted low level electrical currents in a continuous sine wave drive can cause significant time errors over long time intervals. In this clock, the pendulum is electronically driven by a short current pulse in each drive coil at the center of swing. This chapter describes some features of an electromagnetically driven pendulum clock. The clock's mechanical layout is presented.

This chapter deals with simple pendulums and with several things that can be done to improve their accuracy. Most of the items have only a minor effect on accuracy, but they add up. The pendulum should be enclosed in a case to protect it from the air currents of an open room, which will push the pendulum around and give erratic timing. A metal pendulum rod is recommended over a wooden one. If the pendulum is not temperature compensated, a low thermal expansion metal like iron must be chosen for the pendulum rod. If the pendulum is not temperature compensated, the bob must be supported at its bottom edge rather than at its middle or top edge. Other tips: use a low drag bob shape, walls dose to the pendulum cause a problem with relative humidity; slide the top end of the suspension spring up and down through a narrow slot; keep the number of piece parts and mechanical joints in a pendulum to a minimum.

Making a pendulum bigger or smaller involves more than just a linear scaling up or down of the pendulum's dimensions. If you want to make a larger or smaller pendulum than the one you have now, how should the dimensions change? It turns out that not all of the dimensions should change linearly with pendulum length. The free length of the suspension spring should be scaled directly proportional to the pendulum's length. James has shown that the vertical distance from the pendulum's axis of rotation up to the free top edge of the suspension spring is directly proportional to the spring's thickness. So the spring's thickness is then scaled directly proportional to the suspension spring's length (and incidentally, also directly proportional to the pendulum's length also). The suspension spring's width is adjusted to keep the static and bending stresses approximately constant for all pendulum lengths.

The big attraction of quartz as a pendulum material is its good dimensional stability over time. Stability over time is the biggest and most needed characteristic in an accurate pendulum. In contrast to invar, which was known to be unstable almost from its beginning, quartz has a long history of being a stable material. Dimensional stability is not the same as low thermal expansion. If a pendulum is temperature compensated, as all accurate pendulums are, then it does not matter much what the thermal expansion coefficient is, so long as the compensation has been done accurately. The accuracy of temperature compensation is limited by factors other than the thermal expansion coefficient. Because of their low density, quartz pendulum rods do have one drawback: they have a much higher sensitivity to barometric pressure changes than invar.

This chapter describes some transient temperature measurements made on a pendulum with a quartz pendulum rod. Time offset error occurs because different parts of the pendulum change temperature at different rates. Before and after a temperature change, the pendulum is the right length (hopefully) and runs at the right rate. But during the temperature change, the pendulum is the wrong length, due to its different parts changing temperature at different rates, and it runs at the wrong rate during the temperature change interval. Experiments were carried out to measure the pendulum temperatures using small thermistors. The pyrex sleeve provides about one-third of the temperature compensation, while two thin-walled pyrex tubes located on opposite sides of the quartz pendulum rod provide the other two-third. The temperature data provide an interesting look into the thermodynamics of a pendulum. The suspension spring assembly changes temperature relatively slowly, whereas the bob, with its large thermal mass, changes temperature the slowest of any part of the pendulum.

The sphere and the vertically oriented cylinder are more repeatable and predictable bob shapes than the more efficient, lower-drag prolate spheroid and football shapes. As a bob shape, prolate spheroids have low air drag for their volume, and give a high Q pendulum. Only football-shaped bobs, that is, bobs with pointed ends (120 degrees included angles) in the direction of travel give a higher Q. A prolate spheroid shape can be obtained by rotating an ellipse 360 degrees around its long axis. The improved repeatability and predictability of spheres and vertically oriented cylinders can be attributed to their not having the rotational alignment uncertainty that is present in the prolate spheroid and football bob shapes.

The usual coarse adjustment for trimming a pendulum's clock rate is a threaded nut beneath the bob, which moves the whole bob up and down the pendulum rod. The surface of the thread is somewhat rough, particularly if the material is invar, which machines poorly. The thread's roughness prevents any sort of smooth adjustment. Axially, 0.001 inch equals 1 second per day on a pendulum with a 2-second period. To get an adjustment sensitivity of 1 second per day, the thread must be lapped, or more correctly, rubbed smooth. And if the thread is not smooth, the whole bob weight will rest on the small raised points of the rough thread's two facing surfaces, creating high stress points and a potentially unstable joint (at the micro-inch level). Using a sleeve bushing or a thick washer of predetermined length underneath the bob as a coarse rate adjustment, and a small threaded nut at the top of the pendulum rod as a fine rate adjustment provide a better way to trim a pendulum's clock rate.

A crossed spring suspension provides an axis of rotation characterised by extremely low friction, something that a pendulum can use. The suspension typically consists of four flat strips of spring metal (two sets of two, with the two sets oriented at 90 degrees to each other) that are clamped at the ends. For small rotation angles, the axis of rotation is nominally located at the springs' crossover point in the middle of the springs. The suspension has three important characteristics. First, it has extremely low friction. Second, the rotational stiffness of the suspension springs (in units of torque per unit angle about the axis of rotation) varies with the total weight suspended from the springs, that is, with the weight of the pendulum. Third, the axis of rotation moves horizontally as the pendulum swings away from its vertical orientation in the center of swing. This chapter proposes an idea for barometric compensation of a pendulum using a crossed spring suspension.

Chops, located on the ends of a pendulum's suspension spring, are designed to get a more solid grip on the spring. With a degree of helpfulness varying from outstanding to useless, chops give a constant fixed length to the suspension spring and prevent the suspension spring from rocking back and forth on the top edge of the crosspin through the top end of the suspension spring. This chapter discusses the rocking of the suspension spring on one of its crosspins. A typical suspension spring has its ends pinned in narrow slots and contains no chops, with the slots slightly wider than the thickness of the spring. As the pendulum swings back and forth, the spring's ends bend or wiggle back and forth a little in the slots. The wiggling is more pronounced in the top slot than in the bottom slot. There would be no rocking motion if the spring's ends were a tight fit in the slots.

Over time, several different materials have been used for the pendulum rod such as steel, wood, and invar. The best material is quartz because of its proven stability and low thermal expansion. Steel is used for the pendulum rod in simple ordinary clocks because it is cheap and has relatively low thermal expansion. Wood is sometimes recommended because of its low linear thermal expansion coefficient along the grain, but it is an inherently unstable material. It warps, splits, and exhibits a high mechanical creep under load. Worst of all, wood expands and contracts considerably with relative humidity. Invar is a mixture of 36% nickel and 63% iron. It is magnetic and rusts in a humid environment. A new material of interest for the pendulum rod is carbon fibre, but it may not work too well as a pendulum rod as the epoxy absorbs moisture, changing the rod's length and weight.

Even though invar is the most common material used for the pendulum rod in a good clock, it is still a poor material for the purpose because of its relatively poor dimensional stability over time compared with other materials such as quartz or platinum. Invar is usually considered for its low thermal expansion coefficient (tempco) rather than its dimensional stability. Quartz, however, is an ideal material for a pendulum rod, if you can get around the glass breakage problem. There are three types of invar available: regular invar, regular invar free machining, and super invar. Each has a different tempco and is dependent on heat treatment and any coldworking or machining that the part has received. What never gets mentioned and is not widely known is how big the changes from heat treating and machining really are.

Almost since its invention in 1896, invar has been known to be a dimensionally unstable pendulum rod material. A few articles have been published over the years, trying to address the dimensional instability and eliminate it. Data published in 1927 showed a dimensional growth of 50 ppm over a 27-year interval. The growth was exponential, gradually slowing down with time. Invar's growth today still follows the same exponential pattern, although shrinkage is occasionally observed. In 1950, invar's instability was tied to the presence of impurities, especially carbon. The lower the level of impurities, the more stable the invar is. Invar's impurity level has been reduced over the years, so that today's invar, using the traditional furnace melt process, is more stable than it was 20 years ago. Today's regular invar has a dimensional stability of 2-27 ppm per year, at room temperature. There are three types of invar available today: regular invar, free machining invar, and super invar. Super invar's thermal expansion coefficient is three times smaller than that of regular invar.

Every clock's pendulum needs a fine trim to adjust its rate to the desired value. This is frequently done by adding small weights to a weight pan, which is usually located about one-third of the way down the pendulum rod. The clock literature says that the effect of adding a small weight to a pendulum will vary, depending on the weight's location along the pendulum rod. The literature also says that the weight will have maximum effect on the pendulum's rate if placed halfway between the bob and the suspension spring, and will have zero effect if placed at the center of the bob or at the suspension spring. This chapter describes an experiment that was carried out to measure the position sensitivity of a pendulum rod by clamping a 24-gram weight on the pendulum rod at a given location and calculating the change in clock rate.

A pendulum rod's air drag has a significant effect on the pendulum's Q. Many years ago, an experiment was carried out to determine the effect of bob shape on a pendulum's Q. The results showed that a football-shaped bob, pointed horizontally in the direction of swing, had the highest Q (least air drag) of any bob shape tested. The highest Q shape had a 2:1 length-to-diameter ratio. Spherical bobs had a little lower Q and right circular cylinders, with their cylindrical axis parallel to the pendulum rod's axis, had even lower Q. This chapter describes an experiment to measure the effect of the pendulum rod on Q. The pertinent rod variable is the rod's diameter. The data show that the Q decreases as the rod's diameter increases, and that the pendulum's Q with a spherical bob is 6-20% better than with a cylindrical bob. Better Q means proportionately better timekeeping.

This chapter describes how to calculate and remove the air pressure error from a clock's time error versus time chart. Depending on geographic location and the density of the pendulum materials, air pressure variations can cause roughly 2-18 seconds of error in a clock in a year's time interval. This is because the pendulum ‘floats’ in a sea of air, and variations in the air pressure make the pendulum slow down or speed up. In this chapter, the pressure error itself is described first, followed by an actual clock data run to show how the pressure error is calculated and then removed from a clock's time error versus time curve. Correcting for the pressure error corrects the clock's time to a constant average pressure at the clock site. The error is the product of the pressure difference times time. More specifically, it is the integral of the pressure difference with respect to time.