Help those of us out who haven't earned their bowyer IQ yet...
Thats exactly the question that started me on the mass priniciple. 1st of all I am a backyard bow builder and not a geek by any stretch of the imagination. I do everything the hardway by actually building bows and then looking for patterns.
The first thing I had to do was develop some very basic common sense" bow logic" taking into consideration a few givens. The most important is that wood only has 1 correct thickness and it will be different on every bow we build. The challenge is to find and be able to express working techniques that will allow us to ge close to this perfect number on every bow we build. Obviously we are not going to do bend tests and calculations that are way above my head anyway.
The most common denominator I found that best predicts both the demensions and the proper tiller is mass. It is always good to think of thickness as how far a bow can bend and width as controlloing how far it will bend. If you have to reread that and think about it do so, it is an important basic piece of info that is good to drill into our heads.
When we bend something we will always create a radius, the radius may not be the same for the entire length of the limb and certainly the radius wont always be the same for different bows. working with the most elastic woods we have the inside radius needs to be less than 1% smaller than the outside radius, most woods are about 25% less than that. This is just something to keep in mind when you see a narrower section of the bow that is thicker but bending more than a wider thinner part of the bow. You know it just aint right. You don't have to figure anything here just keep it in mind.
Bow logic tells you that if you have paralell limbs it could only be for one reason and that is you want the outer limb to bend more than the inner limb giving you an eliptical tiller shape. Bow logic will alos tell you that if you have a tapered limb the bow should bend at least equally and the thickess should be about the same. Theoreticaly you could build a bow at one specific thickness and then tiller it completely from the sides for a nice round tiller.
Nearly all the woods we work with are going to fall into the specific gravity range of somehwhere between 50 and 100, the great majority of those will be between 60 and 80. The majority of the bow we build are going to be between about 45# and 65#. This is where I feel the mass principle is most accurate.
The mass principle can be very valuable in determining how to execute a particular design for instance. You may want to build a hickory backed ipe english longbow. You want it 6ft long but you only want it to be 50#. You don't want the bow so narrow that it is uncomfortable to shoot just so you can make mass, so what you do is modify the tiller shape until it is the right mass at the width you feel will be comfortable. Lets say for instance you have roughed the bow out and got it bending. it feels like a 100# still but is bending. You weigh the bow and find out it is 6 oz too heavy to hit your target mass and you know only about 2 oz more wood will come off to make your weight. You go the the program and start adding length to the handle and fade number. If you add 4 then you will use just a slightly elyptical tiller, if you add 8 you will use a full elyptical tiller, if you add 12" it will be whip tillered. They should perform pretty well if built like this and not have handshock.
Say you are working on an American longbow with parallel limbs most of the way down. You keep in mind the weight of the wood when you rough it out, for mid 60 bows for intance you might figure 1 3/8 for osage as a starting point 1/1/2 for locust and 2" for a lot of the white woods, + or minus depending on the density of the specimen. You simply rough out the bow, get it bending and then check the mass weight. Figure you have at least a couple more ounces comming off just to make weight so if you are within 2 or 3 ounces you just keep tillering the belly as you get closer you can adjust the width a little at a time or more to adjust for mass weight and fine tuning tiller.
The secret is knowing how to do the input, modify the length of handle and fade input to accomadate your tiller shape. If you want circular limbs use the exact measurement of your handle and fades if you want elyptical then add 2 or more inches depending on how extreme you are going.
If your tips are stiff say for six inches you may want to use a draw lenght figure 1 or 2 inches longer than you are actually going to draw it. If the tips are stiff for 10" you may want to use a figure 2 or 4" longer than your actual draw.
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