Or a Basic Physics Lesson on Oscillatory Motion
Dr. Ed Fasanella
The following presentation was made on February 13, 2000 at the Old Dominion Chapter 34 Workshop which was held at the Ft. Magruder (Radisson) Conference Center.
The invention of the "balance wheel made possible a portable timekeeper. A pendulum is excellent for a stationary timepiece, but it certainly is not practical to carry around a clock. This reminds me of a joke - that you probably have heard before.
A man had purchased a fine old grandfather clock and was carrying it on his shoulder to his van when a drunk meandering down the sidewalk stumbled right into him, knocking him down and demolishing the fine clock.
The furious man said to the drunk, "You fool, do you see what you have done to my fine antique clock.
The drunk responded in a slurred voice, " Wah --wah --, why don't 'cha ware a watch like e'ry body else!"
Actually, a pendulum and a balance wheel are just two mechanical devices that can be used to measure the passing of time because they exhibit periodic or oscillatory motion. A "cheap" pendulum clock can be made very accurate. However, a balance wheel is difficult to make accurate - especially before special metals were formulated that do not expand or contract or change elastic properties with temperature. In the first half of the last century (gosh that seems strange to say), it took a lot of work and effort to get a watch to run accurately. That is why a genuine railroad watch cost so much. It was not the 21 jewels that made it expensive. Much of the cost was in the labor required to meticulously adjust the balance wheel so that it would keep good time. With a proper balance wheel and balance jewels, a 7 jewel watch can be made to keep time as well as a 21 jewel watch. The seven jewels required are used as follows: one for the balance wheel roller jewel, two jewels for the pallet (or lever), two jewels to restrain the balance wheel, and two cap jewels for the balance wheel staff. Cap jewels limit the end shake of the balance wheel, and allow a conical pivot to be used for the balance staff, which minimizes friction and MAXIMIZES the strength of the staff as it eliminates a sharp corner. For a watch to keep accurate time, the balance wheel has to be adjusted very accurately to orientations (position) and it has to be adjusted to keep good time whether warm or cold. (Aside - it is relatively easy to get an old watch to run - it is not easy to get an old watch to keep good time - it may have NEVER kept good time!!)
Of course, if all the wheel holes in a movement are jeweled, the watch will not likely wear as much if properly serviced, and friction will be minimized. Thus 17 jewels is the optimum number for a watch. A 17 jewel watch will last longer than a 7 jewel watch. However, consider that many marine chronometers do not have the main wheels jeweled, except for the escape wheel. The extra four cap jewels in the 21 jewel watch, two caps for the escape wheel and two for the lever, are for show and are not utilitarian.
A balance wheel must have a hairspring to provide a restoring force to the motion. The hairspring is a type of spring that operates by deforming "elastically." Elastic simply means that it does not deform permanently. Most materials behave elastically - but if deformed too much they either break, or deform plastically; that is, they take a permanent set.
Suppose we take a mechanical watch to the International Space Station. Will it work? Suppose we take a pendulum clock to the International Space Station. Would it work. Could we make a clock work in the space station. Even a 400 day rotary clock would not likely work too well in the Space Station without a few modifications. Note that while a balance wheel uses a spring to supply the restoring force, a pendulum uses gravity. If there were no gravity, a pendulum would not come back to the equilibrium position. However, with a little ingenuity by adding a spring to act like gravity, we might be able to make a clock work in the Space Station.
The balance wheel can be used to keep time because of the physics of the periodic motion of a rotary device attached to a spring. We are all familiar with vibrations and vibratory behavior. For example, wire under tension at a given length vibrates at a give rate or frequency. That is why a string instrument always sounds the same when you pluck it. The vibrating wire vibrates the air and we hear the sound. Also, a weight suspended on a spring will oscillate at a given periodic rate depending on the strength of the spring and the "weight" (or technically the mass). There are many types of "balance wheels." A 400 day clock "torsional pendulum" is really basically a balance wheel However, it would not fit well into a watch. But if we modify the arbor of the 400 day "pendulum" by turning pivots and then mount the 400 day pendulum arbor between plates with jewel holes, then we really have a big balance wheel. If it were small enough, it could theoretically fit in a watch!
Actually, the balance wheel would not have to be a wheel. We could use a bar. However, a wheel has advantages. It is easier to balance a wheel than a bar. And gravity and balance cause us lots of grief in designing a balance wheel. The rotary inertia "I" of a mass "M"on a bar is just the mass times the distance from the center of rotation squared. For a thin wheel, the formula for rotary inertia looks about the same. The equation of motion for our device with a rotary spring is simple. The spring gives us a force proportional to the
angle of twist of the wheel.
Hooke formulated the law called "Hooke's Law" several centuries ago. Huygens, who takes credit for the pendulum, and Hooke both claimed credit for the watch hairspring. Hooke hypothesized that the force exerted by a spring is proportional to the stretch of the spring. This is written, F=-kx. - For rotations of a balance wheel, the force couple is proportional to the angle the balance is rotated. Now if the force (or more accurately couple or torque) exerted by twisting a balance wheel through an arc were exactly proportional to the arc; and if heat, friction. and gravity could be neglected, a balance wheel would be a near perfect timekeeper. It would be isochronous, it would complete an oscillation in the same amount of time, no matter what the amplitude (small arcs or large arcs would take the same amount of time). We know that theoretically, a pendulum does not keep the same time for large arcs as for small arcs. Clocks are designed for small pendulum arc. However, we know that even the most accurate balance wheel clock, the marine chronometer, is no match for the best pendulum clocks. Why is this??
If you like formulas, the time of oscillation for a mass on a spring is:
T=2x3.1415 sqrt( m/k)
Or in words, the period of oscillation is twice "pi" (which is 3.1415) times the square root of the mass divided by the stiffness of the spring. A similar formula applies to a balance wheel except we use moment of inertia instead of mass. If the formula worked exactly, we would have a perfect timekeeper as the period would be the same independent of the amplitude of the balance wheel or the amplitude the mass on a spring. What causes a balance wheel to be inaccurate. Well, there are several types of errors we have alluded to that we must now examine:
1. temperature error - a big one
2. pivot friction
3. hairspring lack of isochronism (small arcs completed in different time than large arcs)
4. poise (balance) errors where gravity causes problem
5. escapement errors
Temperature Considerations. Even clock pendulum rods can be effected by temperature as they typically expand with temperature. That is why wood is often used for the rod. Theoretically, a pendulum can be considered as a point mass on a very light rod. Then the period of the pendulum for small amplitudes depends only on its length. But, if the rod expands, the pendulum will be longer and the clock slows down. For a pendulum, this is a small effect. Various devices, such as the gridiron pendulum have been invented for temperature compensation for clocks.
For watches, the temperature effect is quite large. Temperature causes the balance and hairspring to expand. In addition, a steel hairspring loses strength (softens) with temperature. The net effect is that an uncompensated watch will lose time with an increase in temperature. A typical watch may lose about 10 seconds a day for only a two degree Fahrenheit temperature increase. This is a lot! Thus, much research was aimed at temperature compensation. The bimetallic balance wheel was the result of much effort and is quite effective. The bimetallic balance wheel is made of two strips of metal, brass on the outside and steel on the inside. The wheel is cut so that we effectively have two arms. Since brass expands faster than steel, the arms move in with a rise in temperature. This speeds up the watch to compensate for the loss due to the softening or loss of elasticity of the steel hairspring. Screws are next placed on the balance wheel to provide for precise adjustment. There are typically twice as many holes as screws. By moving the screws nearer the cut which is the end of the arm, more compensation is achieved. The screws on the balance spokes are used for balancing the wheel, and for slight time adjusting. They are not effective for temperature compensation. I have some examples showing the construction of the bi-metallic balance wheel.
1. A solid steel disk is chosen. A center hole is drilled and an outer hole to grasp in lathe.
2. Brass is cast around the steel disk, and then turned on the lathe to clean it up.
3. The inside of the steel disk is turned out .
4. The excess steel is filed away to leave the spokes of the balance.
5. The balance is drilled, and tapped for screws.
6. The balance is cut to form bimetallic arms
Properly adjusted, the balance will keep good time at two temperatures; however, there will still be middle temperature errors.
The best solution was to develop metals whose elasticity was not effected by temperature. A material named elinvar (invariable elasticity)was developed for hairsprings with these properties. Later other rust resistant nickel-steel materials for hairsprings were developed including Nivarox and Conel (Waltham) that were more robust and less delicate than elinvar. Also, materials were developed for the balance wheel so that it did not expand with temperature. With these improved hairsprings, a solid balance was used.
Poise or Balance - Last of all, consider poise - or balance. Suppose a balance wheel has a heavy spot. Then we have the heavy spot wanting to fall to the lowest point because of gravity. This essentially gives us an additional force acting on the balance in addition to the hairspring. To eliminate this problem, a poising tool with ruby jaws is used to balance the wheel similar to the method that tires are statically balanced. Either metal is cut from the underside of the heavy side screw, or a washer is used to add weight to the light side.
Well, this is about all I have time to discuss today. If you would like to do some further reading on developing accurate timekeepers, I suggest Dana Sobol's book on John Harrison, who won the prize (after many many years of petitioning) for developing a clock accurate enough to determine the longitude at sea.