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This chapter deals with the design and performance of various types of spring suspensions for an accurate pendulum clock. It also covers the three basic oscillation modes of a pendulum, the interactions between them, and the effects they have on the clock's timing accuracy. These oscillation modes affect the design of the suspension. To simplify the discussion, left-right pendulum motion is used to refer to the normal left-right motion of the pendulum bob, as viewed from in front of the clock case. Front-to-back pendulum motion refers to the front-to-back motion of the pendulum bob, again as viewed from in front of the clock case. The pendulum rod is made of invar. The lengths given for suspension springs are the free flexure lengths, and do not include the clamped portions at the ends of the springs. The springs in this chapter are made of thin flat stock, with their ends thickened afterward by clamping or soldering ‘chops’ (thicker end pieces) onto the spring's two ends.

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 describes an experiment designed to determine the effect of the walls of a pendulum clock case on a pendulum. The basic experimental approach was to measure the pendulum's clock rate and drive force, with the walls set at different distances from the pendulum. All measurements were made at atmospheric pressure. The walls affect the pendulum via the air between the walls and the pendulum. The bob is the biggest part of a pendulum, so its size and possibly its shape are important in determining the walls' effect on the pendulum. The results show that the walls made the pendulum run slower and increased the drive force. The walls' slowdown effect on the pendulum is independent of the angle of swing. Relative humidity causes changes in clock rate that are rather large for an accurate clock.

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.

The rate of a pendulum clock is affected by air pressure. As the air pressure increases, the clock slows down, and vice versa. The basic cause is that the pendulum floats in a sea of air, and when the density of the air changes, the effective weight of the pendulum changes by a small but significant amount. A pendulum's sensitivity to air pressure depends on bob shape and density, and is in the range of 0.2-0.4 second per day per inch of mercury. A pendulum clock is normally set to run true over some length of time, meaning a nominally zero time error is obtained at the average air pressure during that time period. This chapter shows that a clock's time error varies considerably with location. In addition, the predominant effect of air pressure is long-term time error, not short term, as any effects of one year or more in duration are considered long term.

The dimensional stability of a pendulum directly affects its accuracy. If the bob sags downward or warps upward, the clock slows down or speeds up. If the temperature compensator shrinks or expands in length, the same thing happens. If the pendulum rod gets longer or shorter, again the clock slows down or speeds up. It does not take much to disturb an accurate dock. Unless a pendulum is kept in a constant temperature room, it is apparent that heat treatment will improve the pendulum's stability by 2-8 times, the exact amount of improvement depending on the material. The combination of annealing plus temperature cycling is very effective in reducing the thermal hysteresis of pendulum materials, making a pendulum clock more stable and more accurate in the typical home environment where the temperature is not perfectly constant.

Q is a clock oscillator's quality factor, a measure of how low its energy losses are with respect to the total energy stored in the motion of the oscillator. This chapter provides some new information on Q and looks at arguments both for and against it. The value of Q rests on two facts: that increasing a pendulum's Q means lower drive forces, which means less pendulum disturbance, which means a more accurate clock; and that historically (but not scientifically), a higher Q generally means a higher long-term accuracy. On a global scale of wristwatches to atomic frequency standards, Q is historically often proportional to long-term clock accuracy. This makes Q significant as an indicator of clock accuracy on a global clock scale. On the more limited scale of pendulum clocks by themselves, both temperature compensation and aging are more important issues than Q since their errors are bigger than the increased accuracy available from an increased Q.

Photoelectronics make good sensors for pendulum clocks, because they add no power losses to the pendulum. A swinging pendulum interrupts a light beam, and a light detector provides an electrical signal to compare to a time standard or to incrementally drive the second hand on a clock face. Most photoelectronic circuitry is aimed at very simple applications, such as counting slow-moving cans or boxes on a production line, or detecting the passage of a slow-moving pendulum. For the pendulum application, where the light source and light detector are about a half-inch or so apart, the most suitable light source is an infrared light emitting diode. There are four basic things that can be done to improve the dimensional and time resolutions of a pendulum: narrow the light beam down to just a slit width, use a voltage comparator on the light detector's output signal, better stray light reduction, and use a faster light detector.

The traditional plot of a clock's time error versus time is far superior to the Allan variance for showing a pendulum clock's time performance. The Allan variance is admittedly a universal and statistically more accurate measure of a clock or oscillator's random frequency and time variations (that is, variance), because it is averaged over multiples of each time interval. However, it is a whole curve on a graph instead of being just a single memorable number, and its value is drastically reduced by the short time span it is able to cover. The variance, however, can be used to generate an oscillator's ‘root mean square (rms) time error versus time’ curve, a curve that is much easier to understand. But the rms time error's equally short time span also drastically limits its value to the clockmaker.

This chapter covers the temperature compensation of pendulum clocks. Temperature compensators can be divided into four groups: mercury for medium expansion pendulum rods (iron), gridiron for medium expansion pendulum rods (iron), sleeve for low expansion pendulum rods (invar, quartz), and miscellaneous low accuracy schemes (bimetal, length of the suspension spring, barometer on pendulum, and gears and levers). The story starts in the early 1700s, when people noticed that different metals expanded at different rates. Several clockrnakers independently measured the relative thermal expansions of different metals, with each using his own arbitrary scale of expansion.

This chapter discusses how to make solid one-piece type of pendulum suspension springs. There are three types of pendulum suspension springs: the mechanically clamped type, the soldered assembly type, and the solid one-piece type. The mechanically clamped type is the easiest to make, but its clock rate is variable and dependent on the clamping forces and clamping surfaces. A 10-microinch change in the length of a 1-second beat pendulum amounts to an error of 3.7 seconds per year. To meet the microinch length tolerance needed for an accurate pendulum clock, a mechanically bolted assembly must have microinch tolerances on the clamped surfaces, that is, an optical finish. Mechanically machined surfaces are not good enough, and introduce uncertainty as to where the spring ends and the end clamp actually begins. The uncertainty causes the pendulum's length and its timing to vary with temperature and the suspension spring's clamping pressure.

Chapter 3 explores changes in conceptions of time by examining the work of Tokugawa-period astronomers. It begins in the seventeenth century with the astronomical methods of Shibukawa Shunkai, follows the introduction of Jesuit astronomical texts from China in the eighteenth century, and describes the rise of the Asada Gōryū school of astronomy, which culminated in the Kansei calendrical reform led by Takahashi Yoshitoki and Hazama Shigetomi. Focusing on the development in astronomers’ practices, this chapter shows that astronomers’ conceptions of time were neither value-neutral nor obvious. Rather, individual astronomers constructed their notions of time based on a multitude of associations they gained from practices dependent on time-measurement, such as observation and mathematical calculation. As astronomers’ methods changed, so did the cognitive basis for their perceptions of time. The use of spherical trigonometry, sextants and pendulums, and calculation using diagrams led to an understanding of measured time in terms of celestial arcs and angles. Furthermore, as astronomers’ practices came to include elements of Western astronomy, the notions of time they formed came to resonate with those seen in European astronomy — despite the fact that Japanese astronomers did not read European languages and could not converse with European colleagues.