In this post, Tim Robinson explains the form of the gear teeth needed to build the Analytical Engine. This work is part of ongoing research to demonstrate that Babbage's Analytical Engine was a practical design.
By Tim Robinson
To build an authentic version of the Analytical Engine we need to understand the form of the teeth which Babbage used on his gear wheels. While it will undoubtedly be the case that modern tools and methods will be used to actually fabricate parts, we should nevertheless aim to reproduce the original intent as closely as possible, rather than blindly substituting parts made directly to modern standards. Unfortunately, Babbage seems to have written nothing on this subject, so all we have to work from are the drawings. Despite being exquisitely executed in most cases, it is still not possible to determine precisely, from the drawings alone, the detailed geometries of gear teeth at the level necessary for manufacturing, so we need to search for other clues to inform our decisions. Indeed, most of the drawings do not show the gear teeth in detail at all, just the basic pitch circles for the gears, sometimes with additional circles to indicate the tips and roots of the teeth. Where teeth are drawn, it is often only in the elevation view, and while this allows the depth of the teeth to be determined with adequate precision, it tells us nothing about the actual tooth profile. In the few cases where teeth are shown in plan view, they may have been drawn using a simple template for drafting convenience, and not meant to be definitive for manufacturing purposes. A common practice at the time was to draw radial flanks and simple circular arcs, centered at the intersection of the pitch circle and the flank, for the faces. This appears to have been the case on most of the drawings. Babbage had been working with the same draughtsman, Jarvis, since the original Difference Engine No. 1 in the 1820s when it was certainly the case that wheelwork was a craft of the skilled artisan, so he probably just carried on with the same drafting techniques he had used back then, without regard for any recent progress in the gear art. Much might still have been left to the skill, knowledge, and experience of the craftsmen who would have been called upon to actually fabricate the wheels.
Babbage adopted a standardized notation on the drawings to distinguish various forms and functions of teeth (though without defining their detailed forms). He identified three classes: regular gear teeth, used to pass motion from one wheel to another as in standard gearing; locking teeth, used in conjunction with a locking lever of some form to precisely define the position of wheels when not in motion; and cams, special teeth formed or mutilated in particular ways to act on other parts in specific ways which vary on a case by case basis. He documented this in the Sketchbook.1 Teeth are marked in elevation using a cross with lines drawn diagonally between opposite corners. A straight cross indicates a regular gear tooth, a cross formed from concave lines with the crossing point towards the center of the wheel indicates a locking tooth, and a cross formed from convex lines with the crossing point away from the centre of the wheel indicates a cam. Figure 1, taken from the Sketchbook entry, illustrates this. Babbage provided an additional indicator for the number of teeth in the case that less than the full circle are to be provided, but he rarely, if ever, used this additional indicator on the actual drawings. He further states that, in the case of locking teeth and cams, the corresponding parts with which these interact should be similarly marked.
Here, we are only concerned with the regular gear teeth. Locking teeth, usually triangular in form, present no issues. Details of cams or teeth mutilated for special purposes (intermittent or interrupted motion, cam action of some form) must be deduced from the drawings together with an understanding of the specific motions intended to be produced when they act.
We should first consider the state of the art when Babbage commenced work on the Analytical Engine. The theory of gear tooth forms had been worked out by French and Swiss mathematicians in the previous century, particularly by Camus around 1733, Euler around 1745, and de la Hire, who as early as 1694 had recommended the use of the involute curve for gearing. However, that theory had not been put into practice in English workshops, where gear making was more a craft and an art. Dudley states: ‘The skilled craftsman could make gears that would fit and run smoothly without having even heard of an involute or a cycloid.’2 Babbage however, was well versed in continental mathematics, and so might have been familiar with the theoretical work of Euler.
Things began to change with the introduction of cast iron gears in millwork to replace earlier gears which had used wooden cogs fitted into a wood or iron frame. Wood can flex and absorb the shock loads created by imperfections in the teeth. But cast iron could not, and imperfections resulted in noise, excessive wear, and with prolonged use, fatigue failure. The important consideration in the design of gear teeth to eliminate these problems (aside from the obvious requirement of adequate accuracy in their production) is that they should communicate uniform motion. That is, if the driving wheel turns uniformly, so should the driven wheel.
The Englishman Robert Willis, professor of natural and experimental philosophy at Cambridge University, was a pioneer in gear engineering. He wrote and taught extensively on the gear art, and in 1841 published the influential book: Principles of Mechanism, Designed for the Use of Students in the Universities, and for Engineering Students Generally. This book contains a lengthy chapter devoted to the forms of gear teeth which would deliver the required uniform motion, principally the cycloidal and involute forms, and Willis advocated for the involute form.3 Interestingly, it also contains a chapter titled “On Mechanical Notation” wherein he quotes at length, and sometimes verbatim, from Babbage's 1826 paper on the subject.4 Curiously, despite his work being substantially quoted here, and his stated desire that the Mechanical Notation should be taught in universities and engineering schools, Babbage does not appear to have owned a copy of this book, though surely he must have been aware of it.5
In 1835, the pre-eminent English machine toolmaker Joseph Whitworth was awarded a patent for the first gear cutting machine tool using the hobbing principle.6 This process uses a rotating cutter, in conjunction with a synchronized rotating blank, to generate a full set of perfectly formed teeth in a single continuous operation. For the cycloidal tooth form, a large number of distinct cutters would have been needed, because this tooth form depends not only on the number of teeth in the gear to be cut, but also on the number of teeth in the gear it is intended to mate with. In contrast, for the involute tooth form, a single cutter for a particular pitch of gear can be used to create wheels of any number of teeth. What is more, the form of this cutter is particularly simple as it requires only straight sided cutting teeth. This was a revolutionary step forward, because up to this point teeth had to be cut individually using form cutters made to match the space between teeth, which space itself varies with the number of teeth on the wheel to be cut. Unfortunately, limitations in creating the cutter at the time meant that Whitworth's machine could only cut spiral gears, not the straight toothed spur gears that the Analytical Engine requires. That limitation was only overcome much later in the century when in 1889 George Grant, an American pioneer of gear manufacturing, was awarded a patent for a machine using this method.7 Writing the following year, Grant noted when speaking of the Whitworth machine: ‘It was invented in 1835, and has not been materially improved since then, although there are numerous patents relating to it.’8 The London Science Museum holds one of the original Whitworth machines, dated to around 1854, in almost pristine condition, with a broad set of change wheels, as seen in Figure 2. By definition, change wheels, which needed to be selected according to the required relative motion of the hob and the gear blank, needed to be interchangeable, and so we presume Whitworth would have made them using the involute, rather than cycloidal tooth form. Modern mass produced gears are generally produced in the involute form by a hobbing process.
Whitworth had worked for Joseph Clement on Babbage's original Difference Engine and Babbage was in correspondence with Whitworth through most of his life. In a letter to a ‘Rev. Pearson’ in 1869 he said:
I have heard at different times from men I have employed in former years that amongst their own class it was said that:
Mr. Babbage made Clement
Clement made Whitworth
Whitworth made the tools.9
When later in life he was contemplating building a version of his Analytical Engine in the 1850s, Babbage ordered some machine tools from Whitworth.10 Thus it is reasonable to assume he kept abreast of Whitworth's latest developments and we might therefore expect he would have used Whitworth's state of the art machine tools had he been seriously interested in building the Engine of Plan 27.
However, despite the advantages of the involute tooth form, industrial practice continued to use almost exclusively the cycloidal form until late in the century, and that using mostly form cutters on machines of increasing sophistication and automation.11 So we conclude, despite Whitworth's invention, Babbage would most likely have adopted the cycloidal tooth form had the engine been built in the 1840s.
Let us return now to the problem at hand, defining the precise form of the teeth for the gears of Plan 27. We should be up front in saying that, at least in the data-path of the Analytical Engine, the gears only ever move slowly and are not called upon to transmit significant power. In effect, the wheels are being used to count teeth, not transmit power. Further, as the machine is inherently digital in nature, using the extensive system of locks to ensure correct alignment at the end of any motion, the exact nature of the teeth is probably of little practical importance. Any tooth form which meshes freely, even if it does not deliver the uniform motion so important in power transmission applications, will probably work satisfactorily. Indeed, the author's own Meccano models of some of Babbage's machines and mechanisms used gear teeth formed from nothing more than bolt heads and yet calculated correctly. Nevertheless, with our eye on authenticity, let us consider the issues in more detail, keeping open for now the choice between involute and cycloidal tooth forms.
In most of the drawings up to and including those of Plan 27, the gears are characterized by short, stubby teeth, with wide flat tips. However, the wheels of the subsequent Plan 28, which interestingly are drawn in much more detail and showing most of the teeth, have very long and fully pointed teeth. We might wonder why the sudden change—had Babbage stumbled on some new insight into gear design? No, what changed here was that, in order to reduce height, Babbage eliminated the dedicated triangular locking teeth, and applied the locking lever directly to the regular gear teeth instead. With the older, short stubby teeth, applying locks to those teeth would have significantly reduced the amount of correction a lock could provide, because any large error in the position of the wheel would result in the lock colliding with the flat top of the tooth and jamming. However, the fully pointed teeth of Plan 28 allow the lock to provide the same degree of correction (fully half a tooth) as previously achieved with the dedicated triangular locking teeth. There is in fact one place where Babbage had already used this technique in Plan 27, for the outer teeth of the long stepping pinions. After the change to the reduced unit of space following Plan 13, there was no longer room to use the clever internal lock Babbage had used in that Plan, and no room for dedicated locking teeth, so he made the lock act directly on the gear teeth. See drawing BAB/A/099 for details.
Surprisingly though, if we look at the teeth of the wheels in Difference Engine No. 2, which was designed after Plan 28, we see a return to the shorter stubby teeth of Plan 27. When the builders of that Engine in modern times confronted this same issue, they avoided the rathole and quickly decided to adopt a modern involute tooth profile even though it does not accurately match the original drawing.12 The resulting engine functions perfectly nevertheless.
The first decision we must make is the choice between cycloidal and involute tooth forms. Both have the desirable property of delivering uniform motion. The cycloidal form also has the property that it produces little if any lateral force between the mating wheels, because when the teeth meet at the pitch line the force is tangential. This may be important for long columns of wheels which might tend to deflect. It is also possible to make cycloidal gears with a very low number of teeth without running into the problem of undercutting. In Plan 27 there are regular gear wheels with as few as 18 teeth, and even fewer in certain specialized circumstances. However, there are several significant disadvantages. Specifically, the correct form of the tooth on a driving wheel in general depends not only on the number of teeth on that wheel, but also on the number of teeth on the driven wheel. This means that if we have two wheels each with n teeth, but they drive two wheels with different numbers of teeth, they are themselves different, and not interchangeable. Thus we need as many different types of wheel with n teeth as there are distinct wheels we need them to drive. Given that the wheels would have been machined using form cutters which remove material to create the spaces between successive teeth, the shape of the cutter had to precisely match the shape of the desired space. A potentially huge number of form cutters would have been needed. A further disadvantage is that perfect meshing only occurs when the gears are placed at exactly the correct centre distance for which they were designed. Any error, say caused by wear, or inaccurate placement of the axes, means that is no longer the case, and the transmitted motion will not be uniform.
In contrast, the involute form (which is used almost exclusively today) has the distinct advantage that any two wheels cut with the same hob will interwork correctly regardless of their respective numbers of teeth. This is an important consideration in the Analytical Engine, because there are many places where a wheel must drive more than one other, and they typically do not have the same number of teeth. Further, any error in the centre to centre spacing of a pair of gears does not affect the uniform motion property. It is also possible, via a process termed profile shifting, to make modified gears using the same hob which operate with slightly altered centre to centre distances, and this turns out to be desirable in a few places in Plan 27, in order to obtain perfect meshing of some groups of gears which form a loop.13 Profile shifting does result in a tiny modification of the effective pressure angle for the mated pair, but this is of no practical significance. Finally, though hardly relevant to our application, another major advantage is that the involute teeth are wider at the base and so considerably stronger than the corresponding cycloidal form, an important consideration in power transmission applications. However, involute gears with a small number of teeth do run into the problem of undercutting, which weakens the teeth and is generally to be avoided.
From our modern viewpoint at least, we see a compelling argument for adopting the involute form, but our objective is to be faithful to Babbage's actual design, which by casual inspection of the drawings, almost certainly did not use this form. However, because of greater familiarity, we will explore this involute option first, then compare the cycloidal. For either case, there are several parameters which need to be defined, each of which affects the form of the finished teeth, and we wish to choose these parameters to get the best match possible to Babbage's drawings.
The first of these parameters is the pressure angle. This is the angle the line of action (the locus of the points of contact of the driving and driven teeth) makes to the common tangent to the pitch circles. Historically, a value of 14.5° was common, while modern gears mostly use a value of 20°, though tooling for 14.5° is still readily available. The actual value can be chosen somewhat arbitrarily, but its value determines the slope of the teeth on the generating hob (and also the slope of the teeth on a mating rack). The considerations are that a smaller value leads to somewhat weaker teeth, whereas a larger value will result in a larger lateral force between the gears. Too small a value also increases the minimum number of teeth which can be used on a pinion before they become seriously undercut, which further weakens them. Whatever value is chosen for the pressure angle, it must be the same for all gears in an interchangeable set, a fact which is automatically guaranteed by the use of the hobbing process. In 1841, Willis had suggested 15° as the ideal value ‘to give the best form to the teeth,’ but also said that 14.5° was a good choice because of the fact that sin(14.5°) is almost exactly ¼, simplifying certain calculations, which at that time would have had to be done by hand.14 Babbage's drawings are much more consistent with 14.5° than 20°, though even that leads to teeth that are rather more tapered than the drawings indicate. It requires a pressure angle of around 8° to obtain a closer match, which would require custom tooling, and would need further research to ensure there are no unexpected consequences of such a non-standard choice.
The next parameter we need to set is the addendum. This is a measure of the height of the teeth above the pitch circle. In Babbage's time the pitch of gears was usually specified by the circular pitch, i.e., the distance around the pitch circle between the centres of adjacent teeth. Alternatively, this is the circumference of the pitch circle divided by the number of teeth. Later, the term diametrical pitch (Pd) was preferred. (Originally this was called Manchester pitch.) This is the number of teeth on a gear of unit diameter. Thus a gear of 1 inch diameter with 10 teeth would be described as 10DP (Pd = 10). The larger the Pd, the finer the teeth. More modern practice is to use the term module, essentially the inverse of the diametrical pitch (but now generally specified using metric units in millimetres). Thus, a 10DP gear would have a module of 0.1 in, or 2.54 mm. Babbage's gears are for the most part 10DP, though the larger portion of the stepping pinions and some gears in the carriage have larger teeth of 8DP.
As with the pressure angle, the choice of the addendum is somewhat arbitrary. Increasing the value makes the teeth more pointed, decreasing it makes them more stubby. When pitch was measured as circular pitch, it was common to choose the addendum to be 0.3 × pitch, whereas with gears specified by diametrical pitch the convention is 1/Pd. We should emphasize, these choices are arbitrary; pure convention. Decreasing the addendum has a small effect on what is called the time of action, the duration over which a particular pair of teeth remain in contact. In general, it is desirable to always have at least two pairs of teeth in contact at any given time, and an extreme reduction in the addendum might compromise this. The clearest measure of the addendum chosen by Babbage is perhaps obtained from drawing BAB/A/086, where the elevation shows a side view of the tooth, together with a line representing the pitch circle. We measure it to be 0.8/Pd, rather smaller than modern practice.
Once the addendum is selected, the next parameter to consider is the full depth, or the depth of the spaces between the teeth below the pitch circle. (The full depth is made up of the dedendum, or flank of the tooth, plus additional clearance.) Clearly, the full depth must be greater than or equal to the addendum, or otherwise the teeth of one wheel would jam against the bottom of the spaces in the mating wheel. However, we also have to allow for manufacturing tolerances, and the fact that the center to center distance may have some error, so additional clearance has to be provided. Modern practice would set the full depth to be 1.25/Pd. This value also allows room for a fillet at the root of the teeth which improves strength. However, Babbage usually did not indicate the use of fillets on his teeth, and since the gears are not going to be under heavy load, the reduction in strength from not having them is unimportant. Modern practice usually also puts a small amount of relief on the tips of the teeth, but there is no indication Babbage did that either. We measure Babbage's full depth from drawing BAB/A/086 to be 1.15/Pd.
The final consideration is backlash, which measures the amount of clearance between teeth measured along the pitch circle. Perfectly made gears, perfectly spaced, would mesh with zero clearance between the teeth, but practical gears have manufacturing tolerances, expand and contract with temperature changes, and may not have their axes placed exactly at the nominal positions. For all these reasons the teeth must be made narrower, implying a corresponding increase in the spaces between the teeth, so as to allow for these factors. We know that Babbage was concerned about backlash, or as he called it ‘lost motion,’ because in a train of gears the final member of the train will not turn quite as far as it nominally should when the source gear moves because some motion is lost to taking up the slack between the teeth. Babbage's system of lockings, every two gearings, applied sequentially from the source to the destination, was specifically intended to deal with the lost motion, ensuring the destination wheel ends up at the desired position. If anything, the number of lockings he included is probably overkill, which means that a reasonable amount of backlash should be of no consequence. A larger value reduces the required precision in manufacturing the gears, which in turn lowers cost. It is impossible by measurement of the drawings to determine how much backlash Babbage intended to allow, so we will need to pick a value based on current industry practice, to avoid unnecessary precision.
In the Analytical Engine it is necessary to put gears in and out of mesh. This is accomplished by raising or lowering columns of pinions which interpose between gears which it is desired to connect. Wheels are nominally aligned by the system of locks to make this possible, but even the slightest error in the position of one of the wheels would result in a clash of the teeth, preventing clean engagement. To obviate this, Babbage arranged to put chamfers on the sides and tips of the teeth on any wheels that must slide into mesh in this way. The chamfers ensure that as the teeth approach axially, any slight error in position will be corrected as they come into contact, ensuring a smooth mesh. The chamfers are clearly indicated by Babbage in any place where they are required. The dimensions of the chamfers are not particularly critical, so values read from the drawings should be perfectly adequate. What is less clear is how these chamfers would be applied in the manufacturing process, because they do not feature on standard gears, and this is not something that can be done as part of the hobbing process. It will require a post-processing step.
We have used a CAD system to generate profiles for involute gears of varying parameters in order to compare with the drawings to see how closely they match. Figure 3 shows an overlay of a generated outline on one of the drawings (BAB/A/051), which shows a four-inch figure wheel. Here we have used default modern parameters, 20° pressure angle, addendum coefficient of 1, and full depth of 1.25. The teeth are longer, wider at the root and with much more sloping flanks.
In Figure 4 we have changed to a pressure angle of 14.5°, and reduced the addendum coefficient to 0.8 with a full depth of 1.2 to better match Babbage's shorter teeth.
Finally, in Figure 5 we try the non-standard pressure angle of only 8°. Here we come very close. However, if we were to choose this option, custom tooling would be required. In addition, further work must be done to compare this tooth form for wheels of other sizes, and Babbage gives us few clear drawings of wheels with lower numbers of teeth for comparison.
Turning now to the cycloidal option, the primary variable we have here is the ratio of the size of the generating circle to the pitch circle, for which there is no conventional symbol. When the generating circle is rolled around the interior of the pitch circle, a fixed point on it traces out the hypocycloidal form of the tooth dedendum. We express this ratio as a fraction of the pitch circle, and a commonly used value is 0.3, which results in concave flanks which hug the convex surface of the mating tooth face giving a large contact area, spreading the load and reducing wear. The addendum of the tooth is created in epicycloidal form by rolling the generating circle of the mating gear around the outside of the pitch circle. This is the reason that gears of different sizes cannot in general be made interchangeable in this system, because the shape of the addendum in one gear depends on the number of teeth in the gear it is intended to mesh with.
As with the pressure angle of involute gears, the ratio of the generating circle to the pitch circle is an arbitrary choice which we can adjust to best match Babbage's gears. Two interesting special cases arise. If the generating circle is made equal in size to the pitch circle, then it cannot be rolled at all. There is no flank in the dedendum, and no contact occurs below the pitch circle, but we still need clearance for the tooth of the mating gear. Since the situation is symmetrical between the two members of the pair (assuming the same ratio of generating circle to pitch circle), we deduce that contact only occurs on the line of the pitch circle. For this reason, this system is referred to as point contact gearing. While it provides very low friction, it is rarely used (except in some precision instruments), because under load, severe wear occurs at the pitch line on account of the infinitesimal contact area. The other special case is when the generating circle is one half the diameter of the pitch circle, in which case the flanks of the teeth are straight radial lines. This form is often referred to as clock gearing, because at a time when clock makers had to laboriously cut and file the teeth by hand, the straight flanks were much easier to produce than the curved flanks necessary for other ratios.
While it is often thought that cycloidal gears are not interchangeable (unlike involute gears), that is not necessarily so. In 1837, Willis had introduced a system of interchangeable cycloidal gearing, and it is described in his 1841 treatise.15 The key is to use the same size generating circle for both the addendum and dedendum for all gears in the set. He chose the diameter of this circle to be equal to one half of the pitch diameter of a pinion with 12 teeth. This means that a 12 tooth pinion will have the straight flanks as in clock gearing, but all larger sizes will have convex flanks. It is thus possible to create any gear in this system of 12 or more teeth using a form cutter. It is not clear if this system was widely adopted in practice at the time, but later, around 1865, the American machine tool company Brown and Sharpe introduced a similar system, this time based on a 15 tooth pinion as the base.16 They manufactured sets of 24 form cutters for any particular pitch, allowing any gear of 15 or more teeth to be cut with acceptable accuracy. We note that for the involute tooth form, similar accuracy is possible using a set of just 8 form cutters, another fact favouring the involute system.
The same considerations of addendum length, full depth, and backlash apply to cycloidal gears as to involutes, so again we just need to make choices to best match Babbage's drawings.
We again used a CAD system to generate profiles for cycloidal gears of varying parameters in order to compare with the drawings to determine the best choices. Figure 6 shows an overlay of the generated outlines on the same drawing (BAB/A/051). In this first version, we used the ratio 0.3 for the generating to pitch circles, and we assumed the mating gear to be of the same size. As in the involute cases we have set the addendum coefficient to 0.8 and clearance to 0.2 to get the best match to Babbage's short teeth.
In Figure 7, we have set the generating circle equal to the pitch circle, producing the point contact form. In Figure 8, the generating circle is one half of the pitch circle, the clock gearing form. Finally, in Figure 9, we have a gear created in Willis's interchangeable system in which the generating circle is fixed for all gears to one half the diameter of the pitch circle of a 12 tooth pinion.
The best match for the cycloidal case appears to be the clock gearing form of Figure 8. But as we know, this is not an interchangeable system, so meshing one wheel with multiple others of varying diameters is potentially problematical. As with the involute case, further comparison needs to be made against wheels of different sizes, to the extent that Babbage gives us detailed examples.
Referring back to the issue of the stepping pinions which require the long teeth for purposes of locking: the plan view in drawing BAB/A/099 shows the form of these teeth. Figure 10 overlays a CAD generated cycloidal gear of the clock variety in which the addendum coefficient has been increased to 1.5 with the full depth 1.7. In this case, Babbage did add root fillets, so we have done the same. The match is very close. It is not possible to come anywhere close to this tooth form using an involute profile.
Thus we conclude with high certainty that Babbage intended to use cycloidal teeth of the clock gearing type. From the limited number of cases where the teeth are drawn in detail it appears he used a generating circle for the addendum of the same diameter as that used for the dedendum of the same gear, i.e., one half of the pitch circle. This means that gears of differing sizes will not fulfill the uniform motion condition. As we have noted already, this is probably of very little consequence for the slow moving wheels in the Analytical Engine.
