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Thread: Why crowned pulleys and bandsaw wheels?

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    [email protected] tonyfoale's Avatar
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    Why crowned pulleys and bandsaw wheels?

    This post is not about building a physical tool but I think that is a valid post here because it is about a mental tool, a tool of understanding that is an important part of many real hardware tools.

    As a teenager in the 1960s I had a Burgess 3 wheeled bandsaw, which like the milling machine of an earlier post, has followed me around the world and I still use it today. For some reason, which I don't recall, I built a 2 wheeled machine. Maybe I wanted full width at full height. Back then like Vyacheslav Nevolya I used to cast stuff out of scrap aluminium and that's how I made the new bandsaw wheels. Even though I was aware that pulleys for flat belt driven machinery and commercial band saws etc. had at least one crowned pulley, but the arrogance of youth lead me to believe that I knew better. It was obvious that a hollow pulley or bandsaw wheel would give better alignment. Arrogant I might have been back then but I was also quick to learn when I was wrong and I soon remachined the hollowed out wheel to a crown shape and had no further problems of flying bandsaw blades.

    Why crowned pulleys and bandsaw wheels?-bandsaw.jpg
    Here is a pic. of the little Burgess as it is today. I don't seem to have photo of the homemade bandsaw.

    As an inquizitive lad I searched books (yes, the internet has not always existed) in the library for an explanation of this seemingly counter intuitive behaviour, but found no answer. So I adopted my usual approach with such mysteries, that is I would lay awake in bed turning it over in my mind. One night it all clicked into place and I was a happy boy. However, back in October of this year during discussions with friends the subject resurfaced but I had forgotten the explanation. So back to a couple of sleepness nights to rediscover the mechanism. I hate to leave such matters hanging.

    So here is the flat belt on crowned pulleys explanation. I had to get it written down before I forget again and have to go through the mental gymnastics when the topic next resurfaces.
    The "trick" to it all, is to realise that the belt velocity is constant across its width but the pulley velocity is not. Then the relative velocities between pulley and belt work out in accordance with observation.
    It is easy to look at it in reverse and think that the part of the belt on the larger diameter is moving faster than the part of the belt on the smaller diam. That leads to the conclusion that the belts would fly off a crowned pulley, which I think is the intuitive reaction of most people. We know different though.

    To get an idea of the mechanisms at work we only need look at the crowned pulley not any parallel pulleys in the loop.
    Firstly, let us consider only a slice of half the width of the belt passing over just one side of the pulley as in Fig.1.

    Why crowned pulleys and bandsaw wheels?-pulley-1.jpg
    Fig.1.

    If we consider the linear velocities across the width of the belt, say V1, V2 and V3, shown near the bottom of Fig.1, we can see that they must be equal or the belt would be tearing itself apart. We known that is not happening. In other words the linear forward velocity of all points on the belt must be equal.
    Now consider the circumferencial velocity of the pulley surface at points P1 and P2. Due to different diameters at these points it is obvious that P! has a higher velocity than P2.
    The only way that we can reconcile this with a belt which has a constant velocity across its surface is to realize that over most of the contact area between belt and pulley there is slip of varying magnitude.
    Fig.2 shows us a view looking from "A" in Fig.1.

    Why crowned pulleys and bandsaw wheels?-pulley-2.jpg
    Fig.2.

    It is reasonable to assume that somewhere around half way between points 1 and 2 the belt and pulley surface will be travelling at matched velocities. So at point 2 the belt will be travelling faster relative to the pulley surface and at point 1 the belt will be slower than the pulley. This means that the belt will be trying to climb up the crown of the pulley, as in Fig.3. Think of a car in a curve, the outer wheels will be travelling faster that the inner wheels. The belt is like the car and the pulley surface is like the road surface.

    Why crowned pulleys and bandsaw wheels?-pulley-3.jpg
    Fig.3.

    Let us now put a new full width belt on the pulleys in place of our sliced belt as in Fig.4.

    Why crowned pulleys and bandsaw wheels?-pulley-4.jpg
    Fig.4.

    Imagine that we did not put the belt on properly and it is a little offset to the right. We have seen that the crowned pulley produces a force tending to force the belt towards the crown. The wider the section of the belt on one side of the pulley the greater the difference in velocities between the outer edge and the inner edge and hence the greater the force. So in Fig.4 we can see that the force "F" on the right hand side is greater than "f" on the left side. This force imbalance will make the belt move toward the left until f = F or in other words the belt becomes centralized over the pulley. That is it automatically corrects any errors of alignment.
    This is in accord with our experience.

    Intuition often leads many (including me in my naive and inexperienced youth) towards the idea that a hollow pulley would centralise a belt. However, anyone who has tried this will have learnt very quickly that it doesn't work. Looking at Fig.5 we can easily see why.

    Why crowned pulleys and bandsaw wheels?-pulley-5.jpg
    Fig.5.

    As in the preceding explanation we can see that if the belt is not perfectly centred there will be unbalanced lateral forces acting on each side of the belt. This time the forces will act outwards rather than inwards as before. So when F > f as shown the belt will be flung off, which is exactly what anyone knows who has tried such pulleys. It reinforces any intial alignment error rather than correcting it.

    QED.

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    Thanks Tony
    Brilliant explanation.
    As a lad I worked in an old brickworks with flat belts driving everything, I guess they sort of knew but without knowing exactly why.
    You mentioned sleepless nights, now if you could just explain how it can happen that one can fall asleep chewing on a problem only to wake up with the answer, it's happened to me often.
    I had one of those bandsaws once and swapped it for an old motorbike my cousin had in boxes, here it is now (best deal I ever made).

    [ATTACH=CONFIG]15708[/ATTACH
    Attached Thumbnails Attached Thumbnails Why crowned pulleys and bandsaw wheels?-imgp0006.jpg  

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    [email protected] tonyfoale's Avatar
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    Quote Originally Posted by olderdan View Post
    Thanks Tony
    Brilliant explanation.
    As a lad I worked in an old brickworks with flat belts driving everything, I guess they sort of knew but without knowing exactly why.
    Dan,
    Glad you like it. My problem has always been that I have never been satisfied knowing that stuff happens, I have to know why. It has cost me a lot of sleep over the decades.

    Quote Originally Posted by olderdan View Post
    I had one of those bandsaws once and swapped it for an old motorbike my cousin had in boxes, here it is now (best deal I ever made).
    Great deal obviously.

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    Thanks Tony for the brillant talk !
    I think it's the same for me, I fully understand you.
    BR
    Christophe
    Cheers !
    Christophe
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    I agree Tony just knowing that something works or does not work in not enough. Being born severely dyslexic, my early years in school were a nightmare, so much so that my teachers classed me as educationally stupid but then again in 1959 and 60/61 country school teachers were not well versed on dyslexia. I accidentally discovered a way to counter it in my 3rd grade with tinted cellophane then became voraciously hungry to learn things, so much so that I became addicted to knowledge even though I attended school I had become autodidactic. This served me well enough to eventually be able to retire as an industrial design engineer with a PHD in engineering philosophy.I remain so through this day to an extent even though I can no longer remember as much of my math as I once had and never achieved the level that mklotz and others have. So I really appreciate these explanations found here in these forums
    Never try to tell me it can't be done
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    [email protected] tonyfoale's Avatar
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    Quote Originally Posted by Frank S View Post
    .......... even though I can no longer remember as much of my math as I once had and never achieved the level that mklotz and others have. So I really appreciate these explanations found here in these forums
    Except for some stuff in quantum physics and cosmology, I believe that if you can't explain something in simple non-maths terms then you don't fully understand it. Many things that I have developed everyday explanations for would be much easier for me and much more concise if explained in terms of maths but that sometimes blurs a physical understanding. Gyroscopic effects are my favourite example of that. Pick up a physics book and it is bound to have a mathematical derivation of the standard relationship between torque and spin velocities. However, even most engineers and scientists that I have discussed this with don't have a good grasp of the relationship of the simple maths to the seemingly magic reactions you get when playing with bicycle wheels. I have developed a straightforward explanation of this and those who have seen it light up like a light bulb. They finally make the connection. It is equally enlightening to those with and without the maths understanding.
    I am far away from a maths-phobe. I have spend a lot of my working life developing mathematical models to simulate a wide variety of dynamic physical systems, mainly but not exclusively for land and air vehicles. Sometimes the results of such work help develop a physical understanding where none previously existed. Unless that comes I never feel that I understand anything, and I don't like that.

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    Thank you Tony for your intuitive explanation for why pulleys with a center crown work so well in keeping the belts centered.

    By the way, after reading your crowned pulley posting, I went on to see one of my favorite YouTube channels Keith Appleton, an expert in restoring model steam engines. Near the end of Keith's recent video "REBUILDING A VINTAGE MODEL OPEN STEAM LAUNCH - PART #3" he shows a rare tour of his shop. On his workbench is the same style Burgess bandsaw. I think both of you have had this type of bandsaw forever and great to see tools owned and maintained for over 50 years by the same owner.

    Regards, Paul

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    [email protected] tonyfoale's Avatar
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    Quote Originally Posted by Paul Jones View Post
    Thank you Tony for your intuitive explanation for why pulleys with a center crown work so well in keeping the belts centered........... he shows a rare tour of his shop. On his workbench is the same style Burgess bandsaw. I think both of you have had this type of bandsaw forever and great to see tools owned and maintained for over 50 years by the same owner.
    Paul,
    It is amazing how good those little bandsaws were/are. The only thing that I have modified are the blade guides. I often think that a motor upgrade would help but it has never been a big enough problem to motivate me to do it.
    I believe that Black and Decker bought the company or maybe just the rights to make the saw. I tried one once and it was nothing like as good as the original, it had been cheapened down.
    Talking of keeping tools for a long time, I still have a number of hand tools that were my father's. He wasn't involved in hobbies which required tools like I am but he had a few tools for jobs around the house. he never had a single power tool. I use almost daily a small ball pane hammer and several pliers etc. that were his. Probably 80/90 years old.
    When I started working at 15, I worked with an 85 year old, he had two sons who had no interest in dirty hands stuff and one day he invited me to his home, told me that he would not be using several tools again and insisted that I take them. I still have and use several that must be well over 100 years old. The throw-away idea didn't exist when these tools were made and I expect that in due course my son or grandson will continue with them, but they'll have to wait until I'm 6' under before I'll be giving them up.

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    Tony,

    Wonderful to know you have such an appreciation for keeping, using and preserving the older hand tools from family and friends, plus the ones you acquired along the way. I too have kept all of my fathers hand tools he used for repairing and improving things around our homes. My dad always included me as his helper for this work around the house until I was in college and he acknowledged my craftsman skills far exceeded his craftsman skills.

    My dad was a captain in the merchant marine and at sea for long periods so he left me in charge of making repairs and improvements to our houses by providing a long list of things to build or repair. He also paid me well rather than hiring general contractors and I used the money to buy more tools. By sixteen, I built my first metal lathe but I think the most important tool that allowed me to be very creative was a Rockwell-Delta table saw and jointer that my dad gave me on my 18th birthday in the 1960's. My parents had just purchased a new waterfront home on Bainbridge Island and asked me to do the finish construction for built-in cabinets in hallways, dining room, bedrooms and closets, fireplace surrounds, and to completely finish the basement, and add fencing and custom gates and doors. The table saw and jointer was in use everyday well past sun down and I learned at lot about wood joinery. For me this was all fun and not work, and I still have the table saw and jointer, and now use it for our own custom home. Give your kids good tools and computers, and let their imaginations take over.

    Thank you for the discussions and ideas here on HMT.

    Paul

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    Hi Tony
    I to love my old tools, I have all my fathers spanners and gadgets that he made, my uncles bench vice and things that retiring toolmakers would hand to workmates, micrometers etc. I feel a connection when I use them and I think its all about respect.

    I have been wondering lately about drum brakes after your post of your grinding machine, I have made a Tls conversion for my Triumph (spare backplate), the shoes are floating and the action seems disappointingly spongy, maybe not such a good idea. I do always skim the shoes on a fixture to -.020” drum dia.
    The subject has got me wondering about heat expansion of drums that are laced into the wheel and how it effects spoke tension and wheel stability. Is this why Vincents chose to have bolt on drums instead?.
    I have a feeling you will know the answer.

    Why crowned pulleys and bandsaw wheels?-photo0058.jpg

    Why crowned pulleys and bandsaw wheels?-imgp0016.jpg

    Regards
    Olderdan

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