Scale Model, Scale Force?

I vote for 1/2.

(The moment of inertia is 1/16 of before; the L³ factor is 1/8 of before.)

Wouldn't that be considered a shortbow?

OK, I'll say ¼. You only have half of half of the bending moment _{(I just made that up)}

Just to complicate it the projectile velocity is much higher than scale!
E.g Draw weight about 10 pounds (down from maybe a 70 -80 pound bow).
But the arrow velocity is over 80mph, whereas the fulsized one would be about 115mph.
I'll post a pic tonight.
Del

Interesting project.

Is the projectile scaled as well (diameter and length) ?

Yup, and it's fletched with feathers from a half scale Greylag Goose.
Del
_{(I just picked out some small feathers really)}

That changes the previous thoughts, the little arrow is only 1/8th the mass of the full size one. That explains somewhat. At least, I think it does. Does it fly as far as the big one, I ask because drag is proportional to surface and mass is proportional to volume, so drag is only a quarter of the full size but the energy is one eighth.

You better be careful though, you might be in danger of being re-named, Del (the cat) Littlebow.

I haven't shot it for distance due to the likelihood of loosing it as it would probably go over 100 yards at a guess.
It was made as a 'thank you' present for a guy in the USA who sent me some Oregon Yew, I made it from an offcut, so it will be going home, a very well travelled piece of timber!
Making it was fun as the process is exactly the same as making the full sized one.
Here's a close up of one of the horn nocks.

And the leather grip.

I had to replace the feather fletchings with paper ones for shipping to the US, feathers are banned, presumably because of worries about bird flu.
I emphasised it was a 'decorative' model, and neglected to mention it's fully functional.
Del

Hi Del

That is a very pretty 'nock' you have there. Imagine it 2m high and standing in a park or foyer. All well balanced, very pleasing to look at.

Occasionally I drive past this big stand of black bamboo and every time I am reminded of you. I think I mentioned it before. I am still planing to make a cross bow and use bamboo only. Well, some leather and modern glues. It might take me a few weeks of spare time but the imagined result seems very attractive.

As for the data you requested: Do a myth-busters thing. Just a jig to hold the bow and half every thing but the angle of attack. Measure distance traveled. Off the top of my head you would be good if you could achieve 40% of the full bow. Being an 'imaginator' it comes easy for me to visualize things.

Speaking of scale would it not be interesting to have a 6foot cats claw, with all it physical properties, and use that as a bow? They do have spring, don't they?

"Miniature bow from cats claw", I can see the YouTube headline already

May your arrows always be straight, Ky.

Hope I'm not breaching copy right

LOL
Del

Assuming you don't scale down the draw, and using straight beam calculations even though the bow is bent and the strings pull diagonally.

Draw=P₁xL³/48EI₁=P₂(L/2)³/48EI₂P1

(L/2)³=L³/8

I=bxh³/12 I₁=16I₂

P₂=P₁x8/16=P₁/2

It appears counter-intuitive because the deflection/draw is a proportional scale up.

Assuming you don't scale down the draw...

I'm assuming you mean draw length??
The draw length is scaled too.
The bow is 35" with a max draw length of 14 1/2".
You have to scale the draw as you couldn't draw a 35" bow the full sized 28" draw without smashing it.

I've lost sight of your conclusion. You'll have to spell it out simply for a cat.

The real question is, if the draw weight of the full sized bow is 80 pounds, what would you think the draw weight of the half scale mode is (at half scale draw length)
Del

Based on a quick glance at Passingtongreen's analysis, if the full scale bow had an 80# pull, the 1/2 scale bow would have a 40# pull.

However, if the half scale bow has smaller diameter in addition to the smaller length, my first guess is a 1/8 factor to the draw pull. I need to look at the analysis posted to see if he factored the smaller "diameter" in addition to the smaller length.

To clarify, as stated in O.P.
All linear dimensions 1/2 size.
E.G half thickness, half width, half length, half draw.
Del
(Time is left undilated and Orage juice undiluted, names have been changed to protect the guilty.)
Ah, here's a pic, I pulled it in off my blog.

Looks a little bit like the setup I had when I was 5! Of course mine was probably bought at Woolworth's for $2.98 and was basicly a wood slat with cotton string!

Now why would you assume that I/we would know that, I carried a Lee Enfield 303 when I was in the army, bows and arrows were long gone. Of course, the Lee Enfield is long gone from that service too, that is, if they actually brought the R.E.'s equipment up-to-date.

Given this information, I think the pull goes to 1/4 of the full size. But then, the force on the arrow would be a quarter of the full size and it would only be on it for half the distance reducing the speed to 1/8th of the big bow. It needs some thought.

.
I thought I'd clearly stated that all linear dimensions were halved.
It's a half scale bow.
Half height, half width, half thickness, half draw length.
Any parameter you can measure with a rule is halved.
(I shall go and play with my feathers and string unless ER has hidden them all).
The arrow mass is much lower of course as is the string mass, and limb mass and inertia, which helps the arrow speed.
There's more to a bow and arrow than meets the eye (as Harold said at the battle of Hastings)

Del

If it's 80 full size, it's 20 now.

Looks pretty funny with you shooting it.

I go with your quarter:

the bending stiffness of a half scale thing is 1/8 th but because the force input is at half scale as well (the distance from your bow hand to the point the string is attached) the force lever is reduced to 1/2, meaning you have to pull twice as hard to create the same torque (which is what bends your bow). so you have an eigths stiffness multiplied by two =1/4

now me mustn't mix up energy and force. above we just figured the force needed to pull to be 1/4 . however, because the pull lengths is also 1/2 the energy given to the arrow will be 1/8th of the longbow's (at least if you limit yourself to pulling to scale. Which I doubt by the way, because the longbow's pull il likely limited by your arm's length, while the scaled one can be 'overpulled')

good luck with customs!

Forget this and read 28

If ALL materials involved in your project were homogeneous and uniformal (which they aren't, for example wood lines size to absolute dimension ratio counts, on metals crystal structure to overall size ratio counts etc) then, we could talk about 1/8 of force on sizing dimensions to half. In real world it will be a bit less than 1/8. S.M.

Are all the dimensions of the string 1/2 scale as well? Length and diameter?

You are just teasing me now.. yes of course the string is made to scale, and doing the whipping on it was a nightmare.
Del

Now that it's completed it would be interesting know what the actual draw force is compared to some of the calcs and theories. Can you get your hands on a spring scale?

Spring scale???

http://sites.google.com/site/delsbows/

Well alrighty then, I guess that was a stupid question!... Doh!

It's about 10 pounds, but it's difficult to measure accurately on my big scale. And I didn't want to risk taking it back to 14 1/2 on the tiller and blowing it.
There is an error in my measurement too as the rule needs adjusting down to measure from the top edge of the bow, the previous bow I'd had on there was a good 1" thicker (it was a 90 pound warbow).
So what we see there is about 10 pounds at 13", mind fiall finishing would have dropped some poundage hence my guestimate of 10# at 14.5"

I just went through your site and watched the videos. Fun stuff! I expecially like the cross bow video. It brought back some memories. I built a small pistol style cross bow about 27 yrs ago (I was 15) much to the disapproval of my parents! It took a while but I managed to figure out the trigger mecanism (no internet back then) and it's scary how close it resembled the sketch you have on your site. I used an old hockey stick for the stock (strong because it's laminated), spring steel from a small trailer, and shaped a pistol grip out of a 2x4 with a sander and coping saw. Not to mention the hours spent hand filling the steel trigger mechanism parts. Suffice it to say the thing was dangerous...but cool. It went well with my Ninja suit!

Thanks for the memories. Cheers.

Excellent.
Hockey stick stock! I like it
Del

Would leverage play any part in the draw weight? Leverage might not be the right term in this case but its 4 am and I haven't had my coffee yet. A longer bow would seem to give you a better leverage/angle to use to draw it back. Thus make it easier to draw a full sized bow as opposed to a half sized one.

Exactly, the leverage does come into it.... but...

My basic thinking was as follows.
Stiffness is proportional to cube of thickness so 1/2 the thickness is 1/8th the stiffness.
But...
You have half the leverage so twice the stiffness
but this is then cancelled out by having half the width so half the stiffness!
Taking us back to 1/8th, which seems to be empirically in the right ballpark.

(BTW. I mke no claim that my analysis/thinking is correct)
Del

Darn, I think you are right! the 1/8 is for a beam with half thickness and the width constant. I mistakenly thought that it was for all dimensions halfed. So my answer above is wrong :-(

That was my original basic thinking Del, but rethinking it after 3 cold beers, I came up with 1/4 minus something (instead of 1/8 minus something). It all comes down to that materials are characterized on acceptable pressure, i.e. force/surface or force/(length x length) so inverted square to (linear) size it is. S.M.

If you make NOT only the stiffness analysis but also the comparison of ENERGIES and consider that the energy and the velocity are related and that the arrow mass for scale 1:2 is 1/8 of the mass of arrow for scale 1:1 than the velocities ratio is V(1/2)/V(1/1)=1/ (2^0.5). If for scale 1:1 velocity was 115 then 115*2^-0.5=81.3 which corresponds perfectly to values mentioned.

The differences could come from the fact that I neglected the friction with air and other possible losses!

cool.
Del

Here's a graph of velocity vs mass for a crossbow bolt which nicely illustrates what you say in your post.
Blue line=velocity of bolt, red line=energy of bolt.
I did the testing using a small crossbow pistol as it was more repeatable than a bow.

BTW The G on the bolt as it is released is pretty high, can't remember the figure, but I did a back of the envelope calc' and was rather surprised.

Del

I made a short analysis of your values it results that the equivalent mass of the bow is in the range of 2 g. Which explains why for very low mass arrows the curves are different. When the arrow mas grows the more and more the equation M*W²= constant is confirmed. The equivalent mas of the bow is small because the bow makes small movements in comparison to the bolt. Some analysis I made a caouple of years ago show that the bow increases the displacement at the arrow with a foctor of 2.1 with respect to the bow deflection. At that opportunity I defined a dimensionless parameter for bows which can be used for similitude.

L is the bow half length (bending only)

f is the bow extremity displacement under load

ΔL is the "shortening" of the bow under load

T is the bolt stroke from the non loaded position and ψ is the string half angle (unloaded it is 90°)

This graph is NOT highly precise but it gives a lot of qualitative information.

The parameter is F*L²/(E*Jeq) with F the load in axial direction, E the Young modulus for the material and Jeq is the equivalent inertia moment considering the bow profile. For tapered beams the Jeq= Jroot* q where q<1 and depends on the way the beam is tapered (one side or both).

Excellent, I'm not sure I'm up to understanding it all .
In practical terms the optimum arrow mass depends what you want the arrow to do, fly the max distance, shoot as smooth and accurate as possible, or penetrate plate armour.
I have an ambivalent relationship with the maths /physics/mechanics... it's a love hate thing, I feel I should be able to get my head round it but it slips from my grasp like a hand full of sand. (I failed my 2nd year applied physics degree).
There is quite a bit of talk of the 'mass princial' and the effective mass of a bow limb amongs some bowyers.
I tend to go for feel, I try to get the curve of the bow to be an arc of a circle at full draw and reduce as much unworking mass as possible (eg tip mass). Because I tend to work with self wood bows (e.g a single piece of timber from the tree) there is more art and feel than when using machined and laminated wood staves which are much more even.
I s'pose growing older and wiser is about becoming at peace with ones limitations and maximising what talents we do have.
Blimey getting a bit philosophical, must be time for a cuppa.
Del

Del; All this talk has given me a headache. Let's stick to "nuts" and "bolts" (Couldn't help the the pun).

You are totally right, I had the intention to give free the method how a bow behavior could be predicted that means the pulling force, the arrow speed and so on but since I do not want to disturb the majority of participants I will not do it.

Thanks to give me the right feeling about the true interest.

I think 1/8 is correct if the draw is proportional, i.e., 1/2 as far back for the scaled version. The bow is bent at a certain angle (the same for both). The force generated is the strain times the lever arm times the width. Since the strain is half as far away from the neutral axis and half as much, the force would be 1/4 as much for a given width. When the width is cut in half, the force is then 1/8. Since the geometry of both bows is the same, the force vectors should add in the same way, so the pull should be 1/8 for the scaled down version.

Great thread Del! It's the kind of question we used to ask of the "scientists" I worked with years ago during our lunch...it usually made for a loooong lunch break! So much fun!

What is the difference in maximum distance between the two?

If you figure the energy imparted to the arrow, it would be the force integrated over the distance. If the force is reduced to 1/8 and the distance is reduced to 1/2, the energy is reduced to 1/16. Since the mass of the arrow is reduced to 1/8, v² = 2E/m, the initial velocity of the scaled model is sqrt(1/2)=.707 times the velocity of the full size arrow. The ballistic coefficient of the reduced size arrow will be about 1/2 the full size arrow, and therefore it will be more subject to air drag, reducing its range.

Have you ever seen the series "Decisive Weapons" on BBC4? They are repeating it and last week I recorded the one about the Longbow.

Historians have found that the original Longbows were between 100 and 180 Lbs of full pull weight. Far stronger than previously thought. A quantity of original Bows were found on the Mary Rose in perfect condition....Your 80 Lbs Bow would be considered a boy's weapon!!!

They show how the bow must be cut from a particular part of the European Yew tree, with just the right amount of heart wood to new wood (in one single piece of course, no gluing!), they also showed un-worked blanks as well, showing the amount of each wood.....

I can probably make a Video copy (I believe its still on the hard disk) for you on DVD if you wish and have not seen it......contact me over the CR4 email. The program was made in 1996 originally.

They talk mostly about Agincourt and a little about Crecy as well.....I found the program fascinating....the Europeans took a long time to learn about Yeoman - the hard way....

Robert Hardy the actor has written a book called "The Longbow" and he is also on the video.....

I've handled the Mary Rose bows and I have a copy of 'Longbow' signed by Robert Hardy when he saw 3 of my bows at a shoot and cast an aproving eye over them.
I've pretty much seen and read all the longbow stuff, but thanks for the offer.
heaviest weight bow I've got to full draw and shot was 90 pounds. Here for the video.
I expect in my youth I could have trained up to handle over 100 no prob.
Del

I'm not sure I understand the parameters. You say all linear dimensions. Does that mean the cross dimensions stay the same? If so, the shorter you make it, the harder the draw. ??? But what the hey, There's at least a couple other things I don't understand.

As you point out I say all... That is any dimension which you measure with a rule.
If you look at the pics it should be self evident.
Linear as oposed to area, volume, mass, colour, time, etc.

If you wade through the whole thread I'm sure it's all been explained somewhere.

I expressed linear because I once took a label which had I'd draw at a scale of 2:1 to a label company to be reduced to 'half size'... they reduced it to half the area not half the linear dimensions
Del

I suspect some folks selling binoculars pull that trick also.