Arrow speed: theory vs chrono

[diliviu]

Hello
For quite long time I have this dilemma: why maximum possible speeds for 10 gpp at 28 look lower on the paper than on reality (chrono).
So v*v in m/s will be F in newton * x% (max 84-86%?) because we have the "average" force actually * d in meters (0.712-BH-handle thickness) * elastic inner return (max 0.93?) all dividend to arrow mass în kg * multiplication coefficient like 1.1 or so (because of the limbs and string mass). Using this I hardly get 170 fps. Maybe average force coefficient and mass coefficient are wrong estimated? But hard to believe to adjust the max force with something more than 0.86-0.90... of course there are other marginal factors (string moving a little over BH at release, so a small gain; handle width - a small loss)
So where am I wrong in the math?
Thank you.

[Del the cat]

I Can't follow your maths, if you want it do be followed, it's better to write it out on paper in the formal manner, and post a picture.
It looks like it's full of fiddle factors and approximations so, no wonder it doesn't work out right.
You must remember maths and physics is a way of trying to understand and predict the real world (rather than the real world having to follow the maths and physics, which is only valid until proved otherwise).
The maths and physics used for models of bows always includes plenty of simplification or caveats.
 e.g Beam deflection equations are always "for small deflections". A bow isn't a small deflection.
Remember classical aerodynamic theory says a bumblebee can't fly... it also struggles to analyse a boomerangs flight.
Del

[Hrothgar]

I think I need another cup of coffee.

[Badger]

  If you are talking about straight bows 170 fps is pretty much about it for the most part.

[Digital Caveman]

I agree with Del, to many assumptions.  I'm not so sure about the 'average force', it depends on what you mean by it.

[PatM]

Chronograph error or you are pushing the bow on the loose and helping out. 

[willie]

Could you also point to the source for your equasion?  a simple "rule of thumb" calc seems useful.

[bradsmith2010]

lots of variables I think the math does not see

[Digital Caveman]

Engineer in me says the math will see as many variables as you choose to include.  Also says close enough is good enough. 

[Selfbowman]

I’m just reading. Arvin

[bradsmith2010]

im not engineer, so can it measure variation in draw speed,, or pressure on each finger and such, just asking,,

[Digital Caveman]

Sure, but it'd be expensive.    You wouldn't necessarily need the pressure on each finger though, the angle, length, and tension of each half of the string should work fine.

[bradsmith2010]

but the pressure of your fingers will change the speed,, through a chrono, and also how tight you hold the bow,,
just just wondering,, Im happy just with the speed the choron says, but have notice small changes in my form effect the speed, thanks for the input,,

[diliviu]

Thank you all for your input.
Del, you are right, I should put on a paper in a formal and clear manner.
Willie, I could not find the source any more, but here it is something close: https://math.stackexchange.com/questions/3486775/how-do-you-calculate-the-trajectory-of-an-arrow
But in the original source it was like f/2×d×elastic return = (m+X%×M)×v×v. Where f is draw weight in newtons, d draw length minus braceheight minus handle thickness all in meters, elastic return something like 0.90 for yew they said, m arrow mass in kg, M limbs and string mass, X% from M something close to 0 - depending on tillering and bow design like molle or pyramid, v speed of arrow in meters per second.
But in real life we don't have f/2 because the bow is pre-stressed - we have f × something between 0.5 and 1.
I agree, too many assuptions - I mean the one about force-draw curve (the one giving that number between 0.5 and 1) and the one regarding limbs and string mass and their adjustment coefficient.
Badger, what I got around 170 I thought it was like  for a recurve bow with high overall reflex and also very efficient. But, again, we have at least 3 factors that we estimate way too roughly.
Thank you.

[Digital Caveman]

That formula assumes a straight FD curve, which is only a rough estimate. 

I've been wondering for a long time how to calculate the kinetic energy of a returning limb, but I'm getting there.