The Modern Clock / A Study of Time Keeping Mechanism; Its Construction, Regulation and Repair

by Ward L. Goodrich

Say we have lost a wheel; the pinion has 12 teeth and we know the arbor should go seven and one-half times to one of the missing wheel; we have our center distances established by the pivot holes which are not worn; what size should the wheel be and how many teeth should it have? 12 × 7.5 = 90, the number of teeth necessary to contain the teeth of the pinion 7.5 times. 7.5 + 1 = 8.5, the sum of the center distances; the pitch radius of the pinion can be closely measured; then 7.5 times that is the pitch radius of the missing wheel of 90 teeth. Other illustrations with other proportions could be added indefinitely but we have, we think, said enough to make this point clear.

Fig. 65. Spacing off center distances; c, center of wheel;
e, pitch circle; d, dedendum; b, addendum; a, center of pinion.

Conversion of Numbers.—There is one other point which sometimes troubles the student who attempts to follow the expositions of this subject by learned writers and that is the fact that a mathematician will take a totally different set of numbers for his examples, without explaining why. If you don’t know why you get confused and fail to follow him. It is done to avoid the use of cumbersome fractions. To use a homely illustration: Say we have one foot, six inches for our wheel radius and 4.5 inches for our pinion radius. If we turn the foot into inches we have 18 inches. 18 ÷ 4.5 = 4, which is simpler to work with. Now the same thing can be done with fractions. In the above instance we got rid of our larger unit (the foot) by turning it into smaller units (inches) so that we had only one kind of units to work with. The same thing can be done with fractions; for instance, in the previous example we can get rid of our mixed numbers by turning everything into fractions. Eighteen inches equals 36 halves and 4.5 equals 9 halves; then 36 ÷ 9 = 4. This is called the conversion of numbers and is done to simplify operations. For instance in watch work we may find it convenient to turn all our figures into thousands of a millimeter, if we are using a millimeter gauge. Say we have the proportions of 7.5 to 1 to maintain, then turning all into halves, 7½ × 2 = 15 and 1 × 2 = 2. 15 + 2 = 17 parts for our center distance, of which the pitch radius of the pinion takes 2 parts and that of the wheel 15.

The Shapes of the Teeth.—The second part of our problem, as stated above, is the shapes of the ends of our levers or the teeth of our wheels, and here the first consideration which strikes us is that the teeth of the wheels approach each other until they meet; roll or slide upon each other until they pass the line of centers and then are drawn apart. A moment’s consideration will show that as the teeth are longer than the distance between centers and are securely held from slipping at their centers, the outer ends must either roll or slide after they come in contact and that this action will be much more severe while they are being driven towards each other than when they are being drawn apart after passing the line of centers. This is why the engaging friction is more damaging than the disengaging friction and it is this butting action which uses up the power if our teeth are not properly shaped or the center distances not right. Generally speaking this butting causes serious loss of power and cutting of the teeth when the pivot holes are worn or the pivots cut, so that there is a side shake of half the diameter of the pivots, and bushing or closing the holes, or new and larger pivots are then necessary. This is for common work. For fine work the center distances should be restored long before the wear has reached this point.

If we take two circular pieces of any material of different diameters and arrange them so that each can revolve around its center with their edges in contact, then apply power to the larger of the two, we find that as it revolves its motion is imparted to the other, which revolves in the opposite direction, and, if there is no slipping between the two surfaces, with a velocity as much greater than that of the larger disc as its diameter is exceeded by that of the larger one. We have, then, an illustration of the action of a wheel and pinion as used in timepieces and other mechanisms. It would be impossible, however, to prevent slipping of these smooth surfaces on each other so that power (or motion) would be transmitted by them very irregularly. They simply represent the “pitch” circles or circles of contact of these two mobiles. If now we divide these two discs into teeth so spaced that the teeth of one will pass freely into the spaces of the other and add such an amount to the diameter of the larger that the points of its teeth extend inside the pitch circle of the smaller, a distance equal to about 1⅛ times the width of one of its teeth, and to the smaller so that its teeth extend inside the larger one-half the width of a tooth, the ends of the teeth being rounded so as not to catch on each other and the centers of revolution being kept the same distance apart, on applying power to the larger of the two it will be set in motion and this motion will be imparted to the smaller one. Both will continue to move with the same relative velocity as long as sufficient power is applied. Other pairs of mobiles may be added to these to infinity, each addition requiring the application of increased power to keep it in motion.

These pairs of mobiles as applied to the construction of timepieces are usually very unequal in size and the larger is designated as a “wheel” while the smaller, if having less than 20 teeth, is called a “pinion” and its teeth “leaves.” Now while we have established the principle of a train of wheels as used in various mechanisms, our gearing is very defective, for while continuous motion may be transmitted through such a train, we will find that to do so requires the application of an impelling force far in excess of what should be required to overcome the inertia of the mobiles, and the amount of friction unavoidable in a mechanism where some of the parts move in contact with others.

This excess of power is used in overcoming a friction caused by improperly shaped teeth, or when formed thus the teeth of the wheel come in contact with those of the pinion and begin driving at a point in front of what is known as the “line of centers,” i. e., a line drawn through the centers of revolution of both mobiles, and as their motion continues the driven tooth slides on the one impelling it toward the center of the wheel. When this line is reached the action is reversed and the point of the driving tooth begins sliding on the pinion leaf in a direction away from the center of the pinion, which action is continued until a point is reached where the straight face of the leaf is on a line tangential to the circumference of the wheel at the point of the tooth. It then slips off the tooth, and the driving is taken up on another leaf by the next succeeding tooth. The sliding action which takes place in front of the line of centers is called “engaging,” that after this line has been passed “disengaging” friction.

Now we know that in the construction of timepieces, friction and excessive motive power are two of the most potent factors in producing disturbances in the rate, and that, while some friction is unavoidable in any mechanism, that which we have just described may be almost entirely done away with. Let us examine carefully the action of a wheel and pinion, and we will see that only that part of the wheel tooth is used, which is outside the pitch circle, while the portion of the pinion leaf on which it acts is the straight face lying inside this circle, therefore it is to giving a correct shape to these parts we must devote our attention. If we form our pinion leaves so that the portion of the leaf inside the pitch circle is a straight line pointing to the center, and give that portion of the wheel tooth lying outside the pitch circle (called the addenda, or ogive of the tooth) such a degree of curvature that during its entire action the straight face of the leaf will form a tangent to that point of the curve which it touches, no sliding action whatever will take place after the line of centers is passed, and if our pinion has ten or more leaves, the “addenda” of the wheel is of proper height, and the leaves of the pinion are not too thick, there will be no contact in front of the line of centers. With such a depth the only friction would be from a slight adhesion of the surfaces in contact, a factor too small to be taken into consideration.