Machining Bevel Gears - Harder Than I Was Expecting

Welcome back to the middle of another project. Feels a bit weird to have skipped directly to about the halfway point, but in fairness making gears is something that I've done quite a few times before. In fact I could probably do it in my sleep, if I had a 4-axis CNC machine. But if you think about it, I've only actually made about three different types of gears. There's the normal spur gear, which we're all familiar with. Last year I did internal spur gears, and I've also tackled free-hobbing a worm gearbox. Now if you're wondering what I was making by the way, it's the basis for a follow focus attachment for my new camera. I recently switched over to using a small cine rig and whilst the lens is pretty good, it's let down by having a really short throw. And what that means is that it only requires a small amount of movement on the focus ring to move the focus a large distance. And that's not really what you want for close-up macro work. So what I'm planning on doing is adding a follow focus with a 4 to 1 step down so I can get a bit more control and resolution from the focus ring. And in addition to that they are really handy to have when you're doing handheld work. Now of course I can buy one of these off the shelf and they're really not that expensive but I've been making my own camera gear for the past 11 years and it's a tradition that I really want to continue. But outside of that this is really more about fact finding and see if I can actually make bevel gears using basically only the mill and the dividing head and not having to rely on buying them in or using a 3D printer. You know to be honest making bevel gears has been one of those things that has been on my list to do for the longest time. You know I could never really figure out how to do it but being able to change the axis of rotation on spitting parts is going to be really useful down the road I mean technically that is sort of what a worm gearbox does, and of course we made a few of those over the years, but they always suffer from having a ton of friction and efficiency loss, and they can't be backdriven. Although sometimes that is a good thing. But the biggest problem that I have with them is that they're usually made for a high speed reduction. I mean you can start playing around with multi-threaded worms, but at that point I don't think it's possible to free-hob the matching gears. But even then there's still a physical limit on the ratios which you can achieve. Anyway my point being is that worm gears, they are not suited for what I want here. What I really want to make is a bevel gear and by bevel gear I mean a matching set. And I really do mean that. Unlike a spur gear where you can mix and match as long as the teeth are the same size, bevel gears are always made as a matching set, two at a time because the geometry of one affects the geometry of the other. And that is one of many things that makes these gears a bit of a headache to calculate and also make. And that is why for most people, me included when I've done bevel gears in the past, you let the CAD software calculate the geometry for you, or you buy them off the shelf, or you simply 3D print them. I 3D printed these ones and they are more than adequate. In fact they work really well and they're really smooth, but there's obviously a limit to what you can get from a 3D print, and the best way to make them is from solid metal. So let's go ahead and do that, the hard way. Now the first thing worth saying is that unlike the one ring there is no one true type of bevel gear. I mean even if we ignore the spiral and the hypoid bevel gears even when you're talking about normal looking straight tooth bevel gears there are a ton of different variations. There's constant depth, tapered depth, tilted root line, constant width crest but really in the home shop the only one worth worrying about is the straight tooth with a constant depth. Which is exactly what these ones are. The maths for these should be more simple, and I can cut these with a normal set of spur gear cutters, and the resulting gears should have a smooth transmission of power. And speaking of maths, it is literally step one, because if you hadn't already noticed, bevel gear angles are not a clean 45 degrees. I mean despite the fact that they do come together at 90 degrees, the angle of the cone or the angle of the teeth is not 45 degrees. In fact the only time that they are 45 degrees is if they have the same number of teeth and they come together at a 90 degree intersection and in those cases we call them a mitre gear. Now for a set of bevel gears to determine the angle we simply visualize it as two intersecting cones and by using a bit of trigonometry we can find the cone angle of the gears. It's not particularly hard to do but it just means that every gear is going to be matched to a mating gear. It is worth saying of course that arctangent only works for gears that intersect at 90 degrees. That said bevel gears work at almost any angle it's just a different formula to work out the cone angle. So for my setup I want a 2 to 1 reduction, so I'll be using a 40 tooth gear and a 20 tooth gear. And doing the maths it comes out to angles of 63.44 degrees and 26.57 degrees. Which angle you use depends on how you measure it, but the best rule of thumb is that the smaller gear always has a steeper angle, more of a cone shape, and the larger one is always more flat. And for this batch I'll be making them from aluminium. And despite what some people might say there is absolutely nothing wrong with aluminium gears. We just have to cut the aluminium first. Now before we can turn the blank to size, we also have to decide on the width of the tooth. Because unlike a spur gear, that will affect the diameter of our bevel gear. And the reason for that is that the inside diameter is our defining dimension, and then we go outwards. If that sort of makes sense. So for example, I'm going to be making a 40 tooth 0.8 module gear, with the module simply being the size of the tooth. Now normally for a gear of this size, I'd cut the blank to 33.6mm in diameter. That's a pitch diameter of 32mm plus an addendum of 0.8mm. But because this gear is at an angle, we only cut it to 32.72mm. The important thing to note though is that the pitch diameter is exactly the same. The exact same thing is true for the addendum or the tooth height. The only thing that we have to account for is that it's sloped at an angle, which is why the outside diameter is different. From there we then account for the width of the tooth, that in this case being 6mm, and also accounting for the slope of the tooth. Which we calculated earlier. That leaves us with an outside diameter of 42.32mm. I know that it is a lot of information to take in, but the important thing is that that inside ring has the correct specs for a 40 tooth gear. Or as I said before, the inside is what matters and then we go out. Of course I also need to cut that angle into the blank, so I'll rotate the compound to 26 degrees and then use an indicator to really dial it in. I'll then rotate the compound 90 degrees and then machine the inner cone. Its depth is not that critical, it only needs to go in as deep as the teeth that we need to cut. In this instance, it's 1.8mm. And to be on the safe side, I went a bit further in than that. And then lastly, I'll drill and ream a hole for mounting it later. And that is the blank now done. Technically, I did forget to add the back taper, or the back cone, but it really isn't that important for it to work. Now to do the gear cutting, I'll be using the dividing head. If you're not familiar with it, it's just a fourth axis spindle for the mill with a 40 to 1 gear reduction. 40 turns of the hand wheel spins the spindle once. And it's probably easy to guess why I chose a 40 tooth gear. And for reasons you'll see in a bit, I'm also going to be using the 49 hole dividing plate. And because I'm making a constant depth tooth gear, I want to align the travel of the table with the angle of the gear so that it's parallel. I'll then load the 55 to 134 tooth cutter into the arbor and then get it loaded into the mill. Technically it is the next size up from what you'd normally use but there are charts that give a call out for mounting a 40 tooth to a 20 tooth gear and it says to use the next size up. That being said I can't imagine the difference being that big. I'll then get it centered on the dividing head and then we can start cutting the gear teeth as we normally would. Same tooth depth as normal the only difference being is that we're doing this at an angle. Now at this point it is worth saying that there are dedicated bevel gear milling cutters that you can get and they look almost identical to the involute ones that I'm using here. Full transparency I don't know a whole lot about them but looking online I think they're ground a little bit thinner and they might produce a more optimal tooth profile. But at 120 dollars a set you'd really want to be making a lot of them to really justify buying them. And honestly for normal hobby shop work I think these normal cutters should work just fine. And with the first pass now done, we now have 40 teeth cut. And you know it does look like we have a bevel gear, but of course it isn't. Well at least not one that is usable. The problem of course is that we've cut a tooth with a valley that is a constant width. But of course the surrounding tooth around it is growing wider as the gear gets bigger. And to put it simply, it won't mesh. So what we need to do is angle it to make the cut widen as it goes out. And we can call it the tooth space taper. And this is where it starts to become a bit of a headache because what we want is to cut it so that the thickness of tooth A is the exact same width as the valley on gear B. Now of course that sounds simple on paper but actually doing that on the mill is a bit of a head scratcher because to do this we have to do two offsets. We have to rotate and offset the gear and also raise and lower the gear to cut each flank. And actually calculating both these offsets was the thing that stumped me for the longest time. Unfortunately, I couldn't find one chart on doing it, and it seems to be different for every size of gear. And on top of that, there are several methods on how to calculate it, and all the answers seem to be slightly different. I know everyone praises the Machinery's Handbook, but honestly, I didn't find its method particularly helpful. There is a chart and formula for calculating the cutter offset, but for some reason there's no formula for calculating the angular offset. Instead, it relies on test cuts and measurements to get the actual figure. And whilst that might work for larger gears, I think I would struggle trying to do this for a smaller 0.8 module gear. Now a lot of my engineering books had formulas for calculating the angles for bevel gears, but unsurprisingly it didn't extend to calculating the angular offset when it came to manufacturing them on a dividing head. And frustratingly, several of my machining books either outright skip, or are only very brief when it comes to bevel gears. And to an extent, I can sort of understand it, because most people either buy them, or simply use CAD to auto-generate the properties. But even so, it is a bit frustrating. Now eventually, I did find a method called the offset and roll method, which does all the calculations from first principles. Which does work, although I have to admit, it took me ages to figure out how to do this, and actually make it work. A lot of work, a lot of wrong answers, but eventually I was able to verify it in CAD that I got an answer that worked. Not a method that I would recommend, but it is the one that I used. And it was only when I was racking my brain on how I would explain this that I finally stumbled on a much more simplified formula by a creator called Tom's Rabbit Hole. I'll leave a link in the description, but essentially all we have to do is offset the tool by half the tool width. So for my setup, I'm using a 0.8 module, module simply being pi or 3.14. 3.14 times 0.8 is 2.51, divided by 2 is 1.225, and that is our tool offset in each direction. And for the angular offset, it is the normal dividing number divided by 4. So for a 40 tooth gear that is 340 degrees divided by 40 which comes out to 9 degrees, 9 divided by 4 is 2.25 or one quarter of a rotation of the handle. So for a cut like this we've ended up with 2.25 degrees of offset and 1.225 mils of cutter offset. And according to the CAD it should yield almost exactly the same results as the units which I used. That being 0.015 mil of offset and 0.353 degrees of rotation. Either way it doesn't really matter which method we use, we now have our offset values which should produce the correct tooth shape. So I'll lower the spindle by 0.15mm and then rotate the dividing head clockwise by 0.353 degrees. Except unfortunately I can't do that, at least on a manual dividing head. Or at least not without making custom indexing plates. Thankfully indexing two holes on a 49 plate comes out to being 0.367 degrees, which is easily close enough. My main point being is that before you choose the gear you may want to check just to make sure you can approximate the desired offset at least with the plates that you have. And depending on the gear that you're making you may have to use different plates to cut the gear and then do the offset. I mean it's either that or you use a CNC attachment. At this point it's now time to cut all the teeth again maintaining that offset. And the important thing to remember is that we don't want to be machining away material on that inner ring. We only want to modify the outward taper. And with the first pass now done, we now need to raise the cutter by 0.3mm and also offset it back by 4 holes. And then we can do our second pass to cut the other flank. Alright and after all that cutting, we now have a bevel gear. I mean we'll have no clue how well it's done until we make the other one, but looking close up at it, I mean I know there is material removed and I know that it is different, but honestly it sort of looks mostly the same. Oh well, we'll wait and see. Nothing left to do but make the second one. Now the second one we'll make will be the 20 tooth pinion. Now again normally if I was cutting a normal gear I'd make a gear of length that's 17.6mm but once you factor in the face width and everything else the OD comes to 22.18mm and the inside cone is 17.43mm. So I'll cut down some 25mm round bar and then get it in the lathe and get it cleaned up. And just to visually check you can see just how well those two line up at 90 degrees and you can see the face widths do match I'll then come in and drill and ream the hole which should also make it easier to cut that inside taper. It did rub a little bit on the inside but that should be okay. I can then get the blank in the mill and then thankfully unlike the lathe I don't have to reset the dividing head I just have to move the saddle not the table. And once again I do have to change the cutters this one here being number five per the handbook which again is different to what I'd normally use if I was machining a normal spur gear. I'll also swap to a 20 hole pattern plate again to achieve the desired offset. I'll then cut the 20 tooth gear as I normally would. I think if I was to do this again I could probably skip this step and then do it all in two passes. I'll then do the offsets the same way that I did last time, although that being said it is a different offset. The angular offset is now 0.88 degrees, which I rounded up to 0.9 which is two holes on a 20 tooth plate. And with the cuts now done we now have produced a bevel geared looking pinion. Somehow again I also managed to forget to add the back cone and the pinion looks a little bit odd without it. So I'll get it back in the lathe and then machine in the back taper and that is looking a lot more like a pinion. Although one thing I didn't count on were the burrs that it was going to kick up. Probably should have added the cone before cutting the teeth. The important thing is that now that it's all deburred it feels like a proper bevel gear. You know just judging it by feel it's smooth, it's not binding and it naturally wants to follow the curve of the gear that it's meshing with Honestly I'm kind of shocked that it works so well on the first go. I mean obviously it's not 100% there yet, we still need to get it held in and then we'll know how well I machined it and how well my maths holds up but so far I am really impressed. So in order to hold it the first thing I'll do is find the center on the dividing head and then I'll drill and tap it for a grub screw so I can hold it onto a rod and then I can get them in the lathe and get them parted off. And once they're loaded into the 3D printed holder they are looking pretty amazing. You know bevel gears are pretty sensitive to alignment issues and all the mating dimensions had to be predetermined in CAD. So looking at it I think it came out really well. I think I may have cut in a little bit too far with the pinion cutter. You know those crests are looking a little bit too sharp for my liking and that is probably what is causing that bit of backlash. But outside of that small mistake which is on me, I think that is a pretty good bevel gear setup. It's really smooth it's really quiet and it runs really nicely and of course most importantly it's transferring the power at 90 degrees and it's doing it at a one to two or a two to one ratio depending how you look at it. So yeah bevel gears took me a long time to figure it out but we finally got there. Now as for the follow focus that I was making, I mean it still is a prototype but let's try and finish it anyway. The next part that I need to make is the ring that will connect the follow focus ring onto the lens. I have this leftover offcut which I'll put in the lathe and then widen the bore. Now unfortunately due to its size and the size of the gear cutter, I can't actually hold it in the chuck like I normally would because the cutter would mill into the chuck. So instead I'll use a smaller chuck to hold onto the ring and then I'll hold that chuck in the larger one. It's probably a mess in terms of run out but that's not hugely critical here. It also helps that I only need to cut teeth about a third of the way around it. And I can get away with that simply because I'm making this specifically for a lens that has 90 degrees of focus throw. And I gotta say if there were runout issues they certainly didn't show up in the gear teeth. I can then get it in the vice and then bore downwards using an end mill. This will create a pocket for a screw head. I can then drill through and then tap a thread for a clamping screw. And finally I'll cut out a slot so I can expand and contract the ring. And once that's on that fits nicely onto the camera lens. I've also added that 60 tooth gear which we made at the beginning. So as it is at the moment it's a half step reduction through the bevel gears and another half step reduction through that 60 tooth gear meshing with well one third of a 120 tooth gear. And that gives us our four to one reduction that I was looking for. The last thing that I need is a handle which I'll cut from the billet that I've been using. So I'll first turn a step down at the front. And I know I've used the dividing head a lot in this project, but I wouldn't mind using it one last time to add some rounded cutouts on the knob. Or I guess not. In any event, I prefer these V-notches better anyway. Alright, and that's the follow focus prototype now together and connected to the camera. I still need to remake that body from aluminium, but even so, I am really liking it as it is at the moment. I mean, it has been so long since I've used one of these, and I am glad that I have this one now. I haven't had that much time to use it but I think this is going to be a real game changer at least when I'm doing handheld work with this lens. But of course the more important thing is that I now have the ability to make bevel gears which opens the door to so many new projects. And with that, thank you very much for watching and I'll see you next time.