# Fun with laminated limb physics - 2.25 times more energy!

I built a pyramidal bow out of ash back in the 80's. I designed it for 30# at 29", and even today, it delivers 28#. That's pretty wimpy... so what if I made a composite bow out of two laminations of ash? Often people laminate different materials, but what can I do with just two laminations of the same stuff?

It's cool. Instead of a 1/2" constant limb thickness like my ash bow (thus leading to a nearly pyramidal shape), take two stick bows like mine that are 1/4" thick and laminate them together. Without pre-stressing them in any way, you just get back the bow I built. Don't do that. Instead, warp both laminations in a circular arc to the point of almost breaking, and glue them together. When the glue sets, and you unclamp everything, the bow will snap back by 1/4th of the arc that you used to stress it. When you force the laminated limb straight, you will have nearly every fiber of the wood maximally stressed, with one of the laminations uniformly in tension everywhere, and the other uniformly in compression throughout it's entire volume. This is very unlike the stresses seen in my original bow, which are maximum on the surface and taper off linearly to zero in the middle. The new laminated limb should pull without breaking with 50% more force than the original. It will also bend 50% farther, storing 2.25 times as much energy!

Now we have a bow that can't be strung because it reaches the breaking point when the limbs are pulled straight. We have to pull the limbs backward to string the bow. So, introduce a riser, and have the circularly recurved limbs start off pointing back towards the archer in a reflex/deflex configuration. Also, use constant thickness in the limbs and a true linear width taper all the way to the nocks. Angle the limbs back so that at full draw the string makes a 90 degree angle with the now straightened limbs. This minimizes the limb mass required to hold the bow string at full draw, which should be good for arrow speed. This is basic tapered cantilever stuff, which applies exactly to my straight limbs at full draw.

Here's a diagram showing the idea for the new bow:

For physics geeks out there, here's how you can figure this stuff out in your head. When the two laminations are bent to nearly breaking, the glue intersection is interesting. One surface is under maximum tension and the other is under maximum compression. When the glue dries, assuming both laminations are identical and behave linearly in both compression and tension, then this boundary will remain in this state of maximum stress permanently. As you straighten the bow, the rest of the volume of one limb becomes entirely compressed, while the other is entirely under tension. This stores 3 times more energy than the original limb, but the rest state of the laminated limb stores 75% as much as the original limb, so the total usable increase is 2.25X. Remember that when we take the limb out of the mold, it unbends by 25%, which is why the limb does not still contain 100% of the maximum original energy used to bend the laminations. When we straighten the bow, we push the laminated limb, which is 8X more stiff, through 3/4ths of the original arc. The original limb could only be pushed through 1/2 of this arc before reaching the breaking point. So, we push the limb through 50% more arc and reach 50% more force, so the new energy is 1.5*1.5 = 2.25 times more than the original, which is the same answer as before.

Naturally, most "new" ideas are reinventions of old ones. Here's a great article discussing the history of archery in America. The most important inventor was probably Clarence Hickman. Check out his bow design in the paper. It's exactly the same as mine! In his patent, all his math works out exactly like mine as well, confirming the 2.25X energy improvement.

Hickman took the idea even further, showing that with more than 2 laminations, we can have beneficial prestress with whatever bow limb shape we like. Oddly, this idea is not used in modern commercial bows, but for accuracy, modern limbs optimize stability rather than efficiency. However, for "flight" bows, modern designers are on top of this. "avcase" on an archery forum says:

Beneficial pre-stress doesn't have to be so strange looking in a bow. All you need is two laminations that are Naturally deflexed, but glued together in reflex so that the resulting unstrung profile looks like a normal straight bow. I used to glue two sets of three thin laminations in deflex then glue the sets again into slight reflex to produce a similar effect. The resulting "super stave" was very strong bending in one direction but extremely weak when bent the opposite direction. It took awhile to perfect, but I made a few up and for a trick would hand it to someone in such a way that they would try to break it by bending it in the strong direction. When they failed at the attempt and handed it back to me, I sneakily flipped it around and bent it the opposite weak direction. For me, it would snap like a twig. Fun stuff. Hahaha! Alan

Here's a picture of Hickman's original bow, which blew away all competitors in flight distance competitions. Here's a picture of the two limbs I crafted. They each pull 24.5# when drawn flat, and combined, including the riser attachment portion, they weigh 9 oz. I'd guess the active portion is about 3.5 oz each. They're 19" from riser attachment (first node) to node at the end, where made the nocks.

My craftmanship stinks, so it's surprising they came out close to what I wanted. I'm only angling them back from vertical by 36 degrees, rather than the 60 degrees I'd started with, or the 45 degrees Hickman used. It makes it look almost look like a conventional highly recurved bow. Also, I tapered the limb thickness slightly rather than keeping it constant, which is why the limbs curve a bit more at the tips. This requires high-end wood glue such as Unibond 800. I delaminated a couple of limbs I built glued with Tightbond III.

I had a ton of fun with hard to characterize bamboo. These two limbs are made from a raw pole I bought from Brightside Bamboo, about 3 miles from my house, where they harvest locally grown bamboo. I split the pole in quarters, and got 4 staves, which I use to make 4 almost identical blanks, so if I mess up, I've got spares. I went against experienced advice on bow building message boards, and heat treated my bamboo, first at 180 degrees for 3 hours, and then 12 minutes at 350 degrees. I did it in our oven, which was possible because the limbs are so short. That's a slightly toned down version of what is used to make bamboo fly fishing rods. This will reduce the max tension my bamboo can take, but reduce the set a ton, which is what I need to take full advantage of beneficial pre-stress, which is sometimes called "Perry reflex". I would have taken off the enamel, but my son loves it. Here's a picture of the scrap bamboo I threw out while trying to get this right.

Here's the riser I built. I laminated some cedar with scraps of IPE with TB3 glue. I needed a combination of light weight and strength, because my son complains that his fiberglass recurve is too heavy, and this riser is quite a bit bigger. There's IPE on the back where the arrow might bang against the riser, and IPE on both sides of the handle to give the riser strength at it's most stressed point. The complete bow weighs 1.25#.

My son loves it!

To help the kids understand a lot of physics about energy storage, levers, gear ratios, coefficients of friction, and such, I helped my daughter build this bow-limb powered "car". She won her school's racing competion with it.

Continuing the fun, Here's a weird recurve design that can have nearly constant weight pull for a significant portion of the draw.

Here's a weird recurve design that can have nearly constant weight pull for a significant portion of the draw. I am considering building one to see how well it works.

First, take Clarence Hickman's bow, with circularly recurved limbs, and curve them even more. This would work best, I suspect, with solid fiberglass limbs. Angle the limbs back from a very short riser at 45 degrees, just like Hickman did. This bow has a good force-draw profile already for a flight bow, though it is unstable for target accuracy. I'm just trying to improve the arrow speed, not stability.

Next, add two stiff forked limbs angled 45 degrees back, between the archer and the limbs, solidly attached to the riser. The forked limbs have a slot in them for the string to pass through. The point of these limbs is to support portions of the flattened curved limb as it's drawn. Instead of a tapered width, as in Hickman's bow, use a constant width for most of the curved limb, and only taper near the tips. Use a very short brace height.

Once the string is pulled to 45 degrees, which happens early because of the short brace height and short riser, the circularly curved constant width limbs will uncurl with constant string tension, and the string will stay at 45 degrees as it's drawn further. At some point, the tip loses contact with the string, and the leverage we have between the nock and the point where the limb is resting on the fixed forks begins to increase. That's why we need to taper the width near the tips.

In reality, constant force would be bad in a design like this, since the arrow nock could move up or down freely during the draw. There is a trade-off where the draw has more stability and predictability vs how flat we want the draw-force curve.

Here's a diagram showing the rough idea.

Here's what it looks like mid-draw when the draw-force remains constant:

If I were shooting very heavy arrows... say a javelin... then this would work well. As the limbs of any bow I've seen are heavier than the arrows, it's amazing that we can get in the 80% efficiency range. That can only be explained by the way the string acts as a brake as the limbs pull the string close to the brace position. This design fails to capitalize on this braking effect. I think Hickman's bow has a similar issue, though probably not as bad.

So, can this design be modified to provide braking? How about something like this:

The idea here is to have a bow shaped more like an efficient flight bow that will brake limb movement near the end of a shot. At some point during the draw, the limbs contact the fixed support structure, and the draw force becomes close to constant until full draw. I think the initial acceleration of the arrow will be higher with this design, because initially only a small portion of the limbs near the tips move at all, allowing most of the energy to be imparted to the arrow rather than limb movement. However, as the limbs roll along the fixed support structure, the moving part increases in weight, and a higher portion of the energy is imparted to the limbs rather than the arrow. At some point, it becomes more important to start braking the limbs to get the energy out of them than to continue accelerating them, and that's why we should leave contact with the supports and go into the traditional braking mode near the end of the shot.

Now this design will have heavier tips, since there will be a portion where the limbs don't taper. That will lower efficiency. On the other hand, we get more energy storage for the same draw weight. My gut tells me that there is an overall benefit in arrow speed for reasonable arrow weights if this is designed properly. I doubt this would improve arrow speed for the very light arrows used with flight bows, but I'm thinking it might help propel arrows in the 10 grains/lbs a bit faster.