Skew Gears for Four Strokes (or, that blasted Feeney, Yet Again)
Now a bold experiment: It's said that for any printed work, the audience will drop by half for each equation that work contains. If this holds true, I can expect the hit counter for January to be about one quarter of that reached in December because I'm about to lay a couple on you. First a different sort of warning: I'm no kind of trained "professional" machinist. All the material following (in fact, all the material on this web site) is just my research, observation, and deduction. I could be, and frequently am, completely wrong! Follow any advice and recommendations here at your own risk...
Now I don't know about you, but I've often wondered why some commercial model 4 stroke designers, Saito for example, choose to drive the cam shaft with simple 2:1 spur gears, while others--like the eviscerated OS pictured here--choose a more complex skew gear arrangement. ET Westbury and Prof. Dennis Chaddock also appear to have favored helical gearing for this purpose (ref [MTO2] notes that the terms "spiral", "helical", and "skew" are synonymous when talking about this type of gearing).
For the amateur constructor, cutting spur gears is a very simple, if relatively exacting operation (this photo shows a trial gear I made some years back and the shop-made hob that cut it). Cutting spiral gears however, as this past month's research has disclosed, is far from simple. I've discovered that there are at least four ways to accomplish this operation, each requiring some rather specialized equipment and complicated set-up. Even for commercial manufacturers, this is going to increase the cost significantly over their spur gear competitors, so why do it? Admittedly, the use of skew gears will place the cam shaft at right angles to the crankshaft which, in turn, can lead to a better placement of pushrods and rocker arms, but there's got to be more to it than that.
The answer it turns out, lies not in the soil
, but in the math. First, some gear-related definitions:
Involute
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A curve that is most commonly used today as the optimal shape for the face of gear teeth for industrial purposes. The cycloidal form is still encountered in some applications, but mostly, gears and cutters are involute. See any text on gears to find out how to generate, or approximate the involute curve. The description in ref [WSP17] is particularly suited to amateur machinists.
Diametral Pitch
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Term applicable to imperial measurement, abbreviated to DP, is a ratio of the number of teeth to the pitch diameter. A designer can pack any number of teeth onto a given circumference--just vary the tooth size. But for two gears to mesh correctly, they must have the same DP. Involute gear cutters are made to (mostly) whole number DP values, although for any given DP, 8 different involute cutters are needed to cover the number of the teeth from maximum to minimum--the maximum being infinite, ie, no curviture, or a "rack" of teeth. What dictates the minimum is left as an exercise for the.... No! Stop that!
Pressure Angle
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The angle between a tooth profile and a radial line at the pitch point. Today, industry has largely standardized on 20 and 14.5 degrees with 20 being the preferred value as it results in a stronger tooth (the odd 14.5 figure was chosen I believe to simplify calculations in the days before hand held electronic calculators).
Pitch Point
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The point of tangency of two pitch circles. That is, the point at which the gears "mesh".
Pitch Circle
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A circle, centered on the gear axis, that passes through the pitch point.
Pitch Diameter
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Abbreviation PD, is the diameter of the gear at the pitch circle. Hence the distance center to center required to mesh two gears is one half the sum of pitch diameters.
The last 4 definitions are simplified for our purposes here, and like the Object Oriented programming terms class and instance, it's impossible to define one without mentioning the other and vice versa, only more so because there's four of them!
Now, all the foregoing was required to simply to explain that if we are going to cut teeth, we need a cutter and that cutter will be specified as having a specific Pressure Angle and Diametral Pitch (or Modulus, if we were using metric nomenclature). We started this whole business because we wanted our cam shaft to revolve at one half the engine crankshaft speed, and so required a 2:1 gearing ratio--which is to say one gear must have twice as many teeth as the other, and because they must mesh, they both must have the same DP.
The spur gear designer looks at how strong the gears need to be having regard to the speed and materials and how much space is available to pack the gears into, calculating the gear sizes from the formula:
PD = N / DP
where "N" is the number of teeth. Notice that no matter how you cut it, for the same DP, the diameter of the gear is going to be directly proportional to the number of teeth. So you end up with a little gear on the crankshaft driving a gear of twice that diameter on the cam shaft.
The helical gear designer still has to arrive at 2:1 gearing, so still requires twice as many teeth on the driven gear as on the driver, but now the formula must take the angle of the spiral into account:
PD = N / (DP * Cos A)
where again, "N" is the number of teeth, but this time, the PD (and hence the overall diameter of the gear, which will be a little larger) is increased by dividing by some value less than one, this being the cosine of the angle "A", which is the angle of the helix relative to the gear axis. If we want the cam shaft to be at right angles to the crankshaft axis (and we do), the two helix angles must sum to 90 degrees. The easy way out is to choose 45 degrees as the helix for both gears and end up with a big one and a little one--with pitch circle diameters also in the ratio 2:1, just like their skew counterparts, only slightly larger (the Sin and Cos of 45 degrees both being, as we all remember, "one on root two", or 0.707 thereabouts--an easy number for air-minded folk to memorize).
But wait! Because there is that fourth variable, we can solve the equations for a set of 2:1 gears using the same PD, but with different helix angles:
PD = N / (DP * Cos A) = 2N / (DP * Cos A')
or
Cos A' = 2 Cos A
where
A + A' = 90
Now with a little jiggling, we can derive the magic angle and this comes out at about 26 degrees 34 minutes for the 20 tooth driven gear, and 63 degrees 26 minutes for the 10 tooth drive gear. Given these spiral angles, the PD's will remain the same for any number of teeth and DP that produce the required 2:1 gearing.
A pragmatic designer is probably going to round those to the nearest ten degrees in order to simplify manufacture and live with the fact that one gear will be about 13% larger than the other--that's still a lot better than 100% which is where we'd be if we'd gone with spur gears. This fact is probably drummed into mechanical designers during their formative years, so as to be second nature. To me it was shock, surprise, revelation!.
That's possibly why OS, Feeney and Westbury chose skew gearing and the odd angles. Even though the skew gears are going to be bigger than their spur counterparts with the same number of teeth, the relative sizes of the gears can be managed using the skew angle, allowing the driver gear to be made larger and hence stronger. To achieve the same tooth size, a spur gear driver would need twice as many teeth, meaning the driven gear would need twice as many teeth, and the diameters (on the PC) overall would double. By using skew gearing, we achive a more compact design for a given strength. The photo here shows a pair of Feeney gears and the helical hob used to produce them. The tooth profile is "distorted" on the face by the helix angle, but it's plain to see that the gears are of roughly similar diameters, yet one has twice the teeth of the other.
It appears that the 10 tooth "driver" gear has larger, stronger teeth, but this is an illusion brought about by the helix angle slice. The profile of the teeth taken at some point normal to the spiral would show the teeth on both gears to be the same (otherwise they would not mesh).
Saito frequently uses spur gears mounted at the front. This places the crankshaft and camshaft axies in line (you can see by the location of the tappets and push rods here that both cam lobes must be behind the large spur gear).
Hence the driver gear must be integral with the main crankshaft and be of equal or grater diameter to maintain strength of the shaft. This means that the driven gear must be twice that diameter (as measured on the PC), hence the "hump" on the shaft journal. If we wanted the crankshaft and camshaft axis to be in-line with skew gears, this could be simply done by using one left-hand and one right-hand gear, with equal helix angles. But because the angles must be the same, we can't play games with the angles like we did before to manipulate the PC diameters, so there is no advantage that I can see in using skew gears arranged like this. In fact, the extra friction generated by the end-thrust inherant in skew gears would be a distinct disadvantage.
For the Feeney and Westbury designs, with cam assemblies and valve gear at the rear (a safe place during an unexpected encounter with the ground), a similar argument may apply. Then again, maybe they just plain really, really wanted a transverse cam shaft and hang the expense!
Back to the Feeney Gear Problem
Skew gears with a 45 degree helix angle are a stock item in the HPC (UK) catalog. They can supply in steel, brass, delrin, etc and the cost is very low, even in one-off quantity. The big question is can the Feeney casting accommodate the change in gear diameters? With 40 DP, 45 degree skew gears, the change in center to center distance is nominal (-0.008"), but the driven gear is larger (a 45 degree helix angle yields 0.757" OD compared to 0.577" for the 30 degree helix). The two pictures here show a photo of the relevant Feeney drawing sheet (I was too lazy to scan it), loaded into TurboCAD and scaled to 100%, with the outline of the two gear options superimposed. On the left, in blue, is the 30/60 gear pair as called out by the drawings. On the right, in the red corner, is the HPC 45/45 set. If these were used, the case would be very thin at the aft end (assuming the drawing accurately reflects the case as cast), and the amount of metal supporting the tappet guides would be reduced a little. There are other issues in making the change, but they are minor and I won't bother with them here--suffice to say, the substitution is not completely impractical.
Thrust face
One last point about skew gears. Like threads, skew gears can be cut as left hand, or right hand (an easy way to tell which is which, and this applies to threads too, is hold it with the axis horizontal; if the slant of a crest points down towards the right, it is right-handed; down to the left means left-handed). The importance of this is that working skew gears generate axial thrust, the direction of which depends on the "handedness" of the gear, direction of rotation, and which is the driven gear (ref [MH25]). In Reference [ME45], John Hellewell, writing about the Westbury Kinglet gears, states that while spiral gears provide a smoother drive than spur gears due to the longer, gradual engagement of the teeth, they also suffer have one outstanding disadvantage: end-thrust.
To understand this, imagine one gear being held stationary as the other turns. The effect is like a bolt and nut. The rotating gear will move in a direction governed by the direction of rotation and thread "hand" exactly as if it were a screw thread. In operation, both gears will experience this effect, which will be imposed on the shafts to which the gears are fixed. So taking this illustration, if gear 'B' were fixed and the right-hand spiral gear 'A' rotated as shown by the arrow, it would have to move toward the top of the diagram. Even with everything free, this force remains on the driven gear. Changing the direction of rotation, or direction of the spiral will reverse the direction of the thrust, as will swapping which is the driver, and which is the driven gear (ref [MH25]). The greater the spiral angle, the greater the thrust load, so with our 30/60 arrangement, the 60 degree helix driver affixed to the well supported crankshaft takes the lion's share, while the driven 30 degree cam shaft, sitting in relatively light bearings, has a lesser end-thrust applied. This is good, but probably not massively significant.
The Feeney design calls for left-handed gears. This means that for normal crankshaft rotation, the force on the driver gear will be pulling the crankshaft rearwards. As this can't happen, the net effect, unless I'm completely wrong, will be that the gear will be trying to pull itself off the shaft it is pressed onto. And, given the way the case is bored to insert the cam assembly, the force from the driven gear will be trying to push the cover plate off the crankcase! This seems completely backwards to me. If right handed gears were used, the driver would be being forced against it's sholder on the crankshaft, and there would be no load on the cam cover plate at all. Can Feeney have got it that wrong? Look at the illustration in the previous paragraph which depicts right hand spiral gears. Imagine the crankshaft attached to gear 'A', with the front of the engine at the top of the figure--in other words, the viewpoint is underneath the crankshaft, looking up. The direction of rotation shown for the driver gear 'A' is correct for normal rotation. Clearly, the driver gear is now being pushed against its seat on the end of the crankshaft, and the driven gear is being pushed into the crankcase cavity (as shown in the sections A-O-B-C and D-E of the GA here). To me, this looks "correct", even though the spiral direction is opposite the that called out by Feeney. Must check one of Westbury's skew designs (The Wyverne, Kinglet, or the Channel Island Special). More mystery; more inventigation required; more next month...
References:
Thanks go to a reader/supporter in the USA who took the time and trouble to copy and mail me the two MW articles, to Eric Offen who dug into his copius ME archive for the Westbury related material, Bert Streigler for the OS skew gear shot, and all the Motor Boys for trying valiently to make sense of my sudden skew gear obsession. Much appreciated!
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[MH25] |
Green, RE (ed): Machinery's Handbook, Twenty-Fifth Edition, Industrial Press Inc, New York 1996, pps 1984-1998.
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[MTO2] |
Burghardt, HD et al: Machine Tool Operation Part II, McGraw Hill, fourth edition 1960, chapter 10.
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[ME45] |
Hellewell, J: Spiral Gears for Petrol Engines, Model Engineer, June 21, 1945, p 581.
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[MW1] |
Cooper, J: Gear Cutting Adventures: Helical Gears, The Machinist's Workshop, June 1999, p23.
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[MW2] |
Sexton, T: Experimental Helical Milling Attachment, The Machinist's Workshop, June 1999 , p12.
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[WSP17] |
Law, I: Gears and Gear Cutting (Workshop Practice Serien Number 17), Nexus Special Interests Ltd, 1995 , Chapters 2, 6.
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