Arrow modelling parameters

[DrRalph]

James, (Joe if you are reading too)

I'm setting up an adaptive step integrator to try some flight dynamics. I was wondering what parameters for drag coeffs, areas and so on you use for say triple 400s ,( eg 29.5", 100gr tips and spinnies, 340gr)? I'd like to do some verification of my code.

What are you using for the rotation inertia tensor? Or are you just using an average orientation and drag for each component throughout the flight?

Maybe we can set up some simple ballistic tests and test our codes independently. Say choose a 'standard arrow' set of coefficients and initial conditions and compare trajectories, ie max range, arrow height and speed at 10m ground intervals etc. What would you suggest?

Then I want to introduce arrow orientation/rotation dynamics and see what happens.

[James Park]

DrR,
I calculate the drag for each part of the arrow as follows:
Using: Drag=DragCoefficient*Area*VelocitySquared
____________________________
'Calculate the velocity pressure
VelocityPressure = 1271 * (Velocity ^ 2) / (2 * 9.8 )
'Calculate the Reynold's Numbers
Rshaft = 66521 * Velocity * ArrowLength / 1000
Rfletch = 66521 * Velocity * FletchLength / 1000
'Calculate the drag coefficient for each part of the arrow
Chead = 0.05
CNock = 0.2
CShaft = 0.074 / (Rshaft ^ 0.2) + 0.0016 * (ArrowLength / ArrowDiameter) / (Rshaft ^ 0.4)
CFletch = 1.328 / (Rfletch ^ 0.5)
'Calculate the drag for each arrow part in Grams
HeadDrag = VelocityPressure * Chead * HeadArea
NockDrag = VelocityPressure * CNock * NockArea
ShaftDrag = VelocityPressure * CShaft * ShaftArea
FletchDrag = VelocityPressure * CFletch * FletchArea
'Calculate the total drag in G's
ArrowDrag = (HeadDrag + NockDrag + ShaftDrag + FletchDrag) / ArrowWeight
_______________________________
Velocity in m/sec
diameters in mm
lengths in mm
I change the drag into "G's" by dividing by the arrow's mass (in grams).
I do allow for barrelled shafts of X10's and ACE's, and I do include a couple of small correction factors to match flight results (and refine them from time-to-time).

Note the need to recalculate the drag whenever the velocity changes.
I assume the arrow flying head-on without flexing.
I would be interested in your modelling.
As noted in other places, I have checked mine carefully in relation to predicted flight shot distances.
Note that in calculating trajectories and flight distances I have found it necessary to use quite small steps in the time increment (eg I use 1 millisecond in Accurate Sights for calculating trajectories).
As the above comes direct from Accurate Sights code, you can see the results of my calculations in the trajectory menu of AS.

[DrRalph]

Thanks, that gives me some comparison numbers to work with. My integrator will take variable steps to achieve a desired accuracy, say 10^-8. Some of the steps can indeed be very short. Makes sense, at 70m/s 1 cm is covered in 1.43e-4 of a second.

'Calculate the Reynold's Numbers
Rshaft = 66521 * Velocity * ArrowLength / 1000
Rfletch = 66521 * Velocity * FletchLength / 1000


Why use the arrow length for the characteristic Reynolds number? Wouldn't the arrow shaft diameter be the characteristic length scale for vortex generation? I guess not, but I'm curious as to why.

I take it that 66521 is the inverse of the kinematic viscosity of air. BTW here is a table I'm going to use to allow for air temperature:
T(C) nu 1/nu
0 1.32E-05 7.58E+04
20 1.50E-05 6.67E+04
40 1.69E-05 5.92E+04

Not a huge difference with 0.4 and 0.2 powers, but it might be interesting
to see what impact it'll have. I haven't data for humidity yet, but I'm looking.

Anyway, thanks, I'll see if I can work out a standard test shot now.

[Eberbachl]

OMG - :o

I gotta get out of here :D

This is far too complex for humans :D

....let me know when it's all done and it has a pretty little GUI to use!

:bday:

[Clare Barnes]

**** - apart from presumably made-up technical words I can't even find a spelling error! :o :roll: :D

[Joe]

DrRalph

My own flight models are (incomplete) theoretical ones aimed at understand arrow flight not trying to predict where an arrow will actually go so bear that in mind.

Drag calculation is done using the basic momentum drag equation (including the 0.5) as does James, only difference is that arrow movement is unconstrained so airflow never axial).

Drag coefficients:
Fletching surface = 2 (infinite flat plate; projected area along shaft axis of two fletchings at 60 degrees is used i.e. drag effect of fletching polar angle ignored )
Shaft = 1.2 (smooth cylinder laminar flow; don't take into account flow transition to turbulent at smaller arrow pitch/yaw angles)
Pile: pick a number. Problem is arrow vibration has a significant effect on drag. Intuitively feel that as long as the vibration frequency is high enough normal drag forces pretty much cancel out but you take a big hit on the axial drag force. Sensitivity analysis time. Vibrational drag is included in the pile drag coefficient - anything between 0.4 and 1.2 say.

Rotational Inertia:
Model iterates at 0.001 seconds. Centre of pressure calculated at each iteration and the moment of inertia calculated around the current arrow rotation axis. The calculation does not curently include the effect of arrow angular velocity on centre of pressure position. Tut tut! In order to understand bow tuning better this omission needs to be addressed - have started to upgrade the model to include angular velocity and vibration effects (arrow shaft broken down into 50 elements) but this seems to have been on the back burner for a while. :cry:

[DrRalph]

James, Does this square even approximately with AS? (I don't have AS for BSD Unix or MacOSX) Using your air density and viscosity, g = 9.80665 m/s/s.

Here is a ballistic check:
750mm shaft = l
5.5mm diameter = d = 2r
22.0g mass

Fletches = triangles, 2cm high, 5cm long x3 Area = 30cm^2 (both sides of fletch, air running past surface)
HeadArea = pi r^2 = 0.2375cm^2 (head on projected)
Nock Area = 1cm*pi*d = 1.727cm^2 (surface area approx)
Shaft Area = l*pi*d = 129.59 cm^2 (ditto, air sliding past)
Total Area = 161.556 cm^2

Launch from d = 0m, h = 1.35m
Speed = 70.0m/s
Angle = +6.25 degrees above horizontal

With Drag:
Initial Drag = -13.8634 m/s/s (~1.4 gravities)
End: t=1.47647s d=90m h=1.388m v=54.043m/s (10! if Target centre is also at 1.35m)

No Drag:
End: t=1.2934s d=90m h=3.004m v=69.768m/s (Miss)

[DrRalph]

Rotational Inertia:
Model iterates at 0.001 seconds. Centre of pressure calculated at each iteration and the moment of inertia calculated around the current arrow rotation axis. The calculation does not curently include the effect of arrow angular velocity on centre of pressure position..(snip)..(arrow shaft broken down into 50 elements)....

My integration starts with 1e-5s and adapts the step size to keep the relative error between a whole step and two equivalent half steps below a predefined limit. Then the step size is allowed to increase steadily until the limit is exceeded and reduced to compensate...

I really want to model fishtailing and shaft spin as well, and to start with I'm defining a rotation inertia tensor based on a simple long thin hollow rod geometry, but I know the arrow structure means I should actually integrate up moments. The arrows are hollow in parts and solid in others so the spinning will need this to be accurate. I'll do this once at the beginning for a static arrow geometry.

But maybe you are right, if it were recalculated on the fly the arrow could even bend.

[grantwomack]

I read in New Scientist about a small helicoptor that was being designed with very advanced rotors on it. The rotors were split into three sections and a variety of curvatures, thicknesses and lift were tested. From the data, the most efficient rotor blades were created. All this was done because the air on such a minute scale (less than two inches per blade) behaves with a much higher viscosity. Is this something that needs to be taken into account when canculating drag parameters? Will the fletches drag through the air as if it were honey?

[DrRalph]

Grant, somewhat yes. There is a hydrodynamical scaling factor called the Reynolds number:

Re = L x V/ kv

L is the length scale, V is the velocity and kv is the kinetic viscosity.

When this number is small, the flow is generally smooth, and when large it is turbulent.

When smooth, drag goes with velocity, when turbulent is increases with v^2. Arrows generally operate with v^2 drag.

The stuff James put up shows that the drag coefficients in some parts are a function of the size and the Reynolds numbers ( Rshaft and Rfletch)

Shaft:
Rshaft = 66521 * Velocity * ArrowLength / 1000
CShaft = 0.074 / (Rshaft ^ 0.2) + 0.0016 * (ArrowLength / ArrowDiameter) / (Rshaft ^ 0.4)

Fletches:
Rfletch = 66521 * Velocity * FletchLength / 1000
CFletch = 1.328 / (Rfletch ^ 0.5)

I am still curious why it is the shaft length not diameter, but I think it is because James has the arrow always perfectly lined up with the airflow, so it is along the shaft, not across it.

So drag affects things somewhat depending on it's size compared to it's speed through a medium. (which is also why model boats on water never look like large boats in movies, the speeds are wrong, and they haven't changed the liquid to scale down the viscosity).

[Joe]

to start with I'm defining a rotation inertia tensor based on a simple long thin hollow rod geometry, but I know the arrow structure means I should actually integrate up moments.

Won't work as you need the turning moment as well as the MOI to get to the angular acceleration. A hollow tube (zero FOC) has zero turning moment (Munk effect apart).

Email me If you want a source code copy of the basic calculation engine for the 2 dimensional flight simulator. (It's in C++)

Joe

[James Park]

DrR,
I will run it though Accurate Sights and see how your results compare.
By my calculations, the Reynold's Number is sufficiently high that we will always have turbulent flow, hence the V^2. Note that if we had laminar flow the drag would be much higher, and that does not fit thenexperimental results, so I am confident that it is turbulent.
I have used the shaft area in calculating the shaft drag because the arrow is assumied to be moving along its axis (as will be the case with a well tuned bow, arrows with fletches, and after the arrow has had time to stabilise in its flight.
One good test of my calculations is: "do the calculated sight settings match the real settings"? They do, and very well indeed. For example, if I use 15M and 70M as the input data for Accurate Sights, and I have the drag calculation correct, the 40M sight setting should be accurate. It is. If the drag was in error that intermediate distance setting would be in error.

For your example, at 70 m/s I calculate a lower drag (of about 0.9G).

[grantwomack]

Great explanation, DrRalph! I am sort of understanding it now.

So are you saying also that James' formulas don't take into account things like a weaker spine than required? Wouldn't it be possible to use both shaft diameter and shaft length for the drag? I'm sure there is a reason but physics ended for me in first year... :)

[James Park]

sh*t - apart from presumably made-up technical words I can't even find a spelling error! :o :roll: :D
If it had a spelling error the program would crash!! What we need is for DrR to invent a programming language called "DWIMNWIT". That is: "Do What I Meant, Not What I Typed". Because DrR has not yet invented DWIMNWIT, engineers need to be profoundly excellent at spelling.

[DrRalph]

I will run it though Accurate Sights and see how your results compare.
By my calculations, the Reynold's Number is sufficiently high that we will always have turbulent flow, hence the V^2. Note that if we had laminar flow the drag would be much higher, and that does not fit thenexperimental results, so I am confident that it is turbulent.
.
I agree.

...(as will be the case with a well tuned bow, arrows with fletches, and after the arrow has had time to stabilise in its flight.

For fast arrows in steady shallow arcs, particularly well shot arrows from compound bows, I suspect this is a good approximation, particularly since it is borne out by experiment.

I'm interested in more wobbly, finger release or miss tuned or strongly arced flight, from low power bows. I'd like to model the tuning-from-scratch process and how we interpret tuning methods from a flight point of view. I'd like to look at the spin aspects in particular.


For example, if I use 15M and 70M as the input data for Accurate Sights, and I have the drag calculation correct, the 40M sight setting should be accurate. It is. If the drag was in error that intermediate distance setting would be in error.

A very powerful argument that the steady flight limit is useful in the real world.

For your example, at 70 m/s I calculate a lower drag (of about 0.9G).

This is good, it is so hard to check you have it right. Assuming I have the correct areas (?) I'll go back and try to verify the coefficients. This is debugging at this point and I need to get drag correct before adding dynamics!!

Thanks, I'll go off and test some more and get back later.

[Joe]

DrRalph

Completely misunderstood what you meant by rotation inertia.
To make amends: :P

Spin moment of inertia = ad^2(pwt/8 + (awt-pwt)/4)
ad = arrow diameter
pwt = pile weight
awt = total arrow weight

Triangular fletching; centroidal plane 1/3 of fletching height above shaft surface.

Skin friction is negligible and you never get a high enough RPM for turbulence effects between your spinwings.

Incidentally the relationship between drag force and velocity depends on the drag mechanism not whether the flow is laminar or turbulent. Agree with James that the typical arrow's Reynolds numbers mean the drag is dominated by inertial effects hence the v squared but this has nothing to do with laminar/turbulent flow.

[Clare Barnes]

I have used the shaft area in calculating the shaft drag because the arrow is assumied to be moving along its axis (as will be the case with a well tuned bow, arrows with fletches, and after the arrow has had time to stabilise in its flight.

:D :D

It's okay guys - he's human! :wink:

[2Dogs]

:rofl: ...ok the last post I understood

[James Park]

DrR,
I can see value in pushing it to where we can model the flexing of the arrow as it leaves the bow. Then we really could get to grips with the basics of bow tuning!!
This is where I have been heading with modelling arrow resonant frequency and vertical/horizontal nock travel as the arrow is accelerated out of the bow.
That is: understand how the arrow can flex as well as the forcing function and we can perhaps then model the actual flex and match that to the bow.
It is a mighty complicated area, I know, but an area where I think we could extract a lot of value. I am not aware of anyone having rigorously analysed it as yet (and also an area where there is a lot of unsubstantiated hype).
It is also of interest to note that many of the tests used (for example in bow tuning) are not able to isolate single variables, and hence are difficult to apply for the majority of archers.

[DrRalph]

Spin moment of inertia = ad^2(pwt/8 + (awt-pwt)/4)
ad = arrow diameter
pwt = pile weight
awt = total arrow weight

Triangular fletching; centroidal plane 1/3 of fletching height above shaft surface.

Skin friction is negligible and you never get a high enough RPM for turbulence effects between your spinwings.

Great, that is the sort of initial geometry and inertias I was working in. Ta.

Incidentally the relationship between drag force and velocity depends on the drag mechanism not whether the flow is laminar or turbulent. Agree with James that the typical arrow's Reynolds numbers mean the drag is dominated by inertial effects hence the v squared but this has nothing to do with laminar/turbulent flow.

I've misunderstood then. I thought that with perfect laminar flow around a sphere for example you get stokes equation. When large vortex shedding occurs you move to v^2 drag and hide all the boundary layer physics, geometry and turbulent cascades in a semi-empirical 'coefficient'.

BTW, Has anyone put tiny 'dimples' on an arrow shaft surface yet?

[James Park]

BTW, Has anyone put tiny 'dimples' on an arrow shaft surface yet?
No, not useful, because the Reynold's Number is sufficiently high that the flow will be turbulent in any case (different to a golf ball where it is in the transition zone and you use the dimples to force it to be turbulent as long as possible to minimize the drag).

[DrRalph]

Nock Area = 1cm*pi*d = 1.727cm^2 (surface area approx)
With Drag:
Initial Total Drag = -13.8634 m/s/s (~1.4 gravities)


with
CNock = 0.2
NockDrag = VelocityPressure * CNock * NockArea


I think I got the NockArea wrong. I assumed it was simply an extension of the shaft cylinder, but the very large coefficient of 0.2 implies a different geometry.

Should I use the projected area of the end of the Nock, ~pi*r^2 = ~0.2376cm^2 ?

Then the drag contribution of the Nock region is still important, but now my initial decceleration is 9.645 m/s/s or 0.98 gs, dominated by shaft (7.64) then fletches (1.17) then nock(0.67) and finally head (0.16 m/s/s). Before I had shaft (7.64), nock(4.89), fletches (1.17) and head (0.16 m/s/s).

Hopefully this is OK now and I can start modifying the simulator further.

[James Park]

I would use the projected area of the nock along the axis of the arrow. Yes, I think the contribution from the nock is quite significant.

[CMB50]

:o OK, this is doing my head in....i'm outta here!!

[Clare Barnes]

Does anyone else wonder if Jim, Ralph and Joe are just making it all up as they go along, waiting for someone to point out it's total rubbish? :o :-? :wink:

[Shirt]

BTW, Has anyone put tiny 'dimples' on an arrow shaft surface yet?
No, not useful, because the Reynold's Number is sufficiently high that the flow will be turbulent in any case (different to a golf ball where it is in the transition zone and you use the dimples to force it to be turbulent as long as possible to minimize the drag).

Hey, deja vu! :D

[Joe]

Clare

Of course were all making it up as we go along and essentially it is rubbish :P . There's nothing wrong with an empirical approach if the results are good enough.

'Dimples' were used on a javelin design a few years ago - their use was made illegal. On an arrow dimples would maybe reduce wind drift but imo the effect would be fairly insignificant because of the small diameter.

I thought that with perfect laminar flow around a sphere for example you get stokes equation.

No - Reynolds no is ratio of inertial (v^2) to viscous (v something horrible) forces. See above where James is using Re to vary the drag coefficient to cater for laminar/turbulent flow transition NOT changing from inertial to viscous drag description. On sphere drag relates to v^2 if Re is high enough - inertial forces dominate.

[James Park]

Clare

Of course were all making it up as we go along and essentially it is rubbish :P .
Absolutely!! (But it is fun).
Where is Mike? I thought he would be commenting on this topic by now! (Must be studying for his exams).

[grantwomack]

Hey, deja vu! :D
I was about to say the same thing... :D
Except it all makes more sense in context now... :)

[mike]

I don't think I have anything to contribute!! This is all a bit hard whilst I am studying for a geophysics exam ... :cry:

[recurve boy]

sh*t - apart from presumably made-up technical words I can't even find a spelling error! :o :roll: :D
If it had a spelling error the program would crash!! What we need is for DrR to invent a programming language called "DWIMNWIT". That is: "Do What I Meant, Not What I Typed". Because DrR has not yet invented DWIMNWIT, engineers need to be profoundly excellent at spelling.

No James, everything needs to be spelt the same, not correctly ;) I sure as hell can't spell but all my programs work! Or just abbreviate the word or use one letter variables if you're the only one who has to read code.

Can anyone suggest a textbook that deals with the necessary topics to understand and model arrow flight? I spent some time last night talking to some mechanical engineers and will give arrow modeling a go during the holidays. If I think I can understand the physics.

I knew there was a reason why I hate mechanics ....

[DrRalph]

Recurvebuy,

1st year undergraduate physics texts cover about enough mechanics (except for some of the materials technology), and then add some 2nd year numerical analysis to make a good integrator and you are there.

Kreyszig 'advance engineering mathematics' covers more than enough maths, and any physics handbook, the sort that summarises physics and includes sections on Newton's laws, drag, ballistics, angular momentum etc will give the basics. You might even find useful stuff trawling university websites for physics 101 courses online. For numerical techniques, Acton's book is the original, but the 'Numerical Recipes' is the modern reference.

Good luck.

R

[compound boy]

actually, i'd stay clear of the 1st year physics undergrad textbooks.

look for books with, "modelling for static and dynamic systems"

u want the 2nd year mechanical engineering stuff.....

remember all your v for final and u for init vel? forget it all and start with a different perspective.... have fun........ i guess if your engineering math is sound, you will.

but dr ralph is correct........ the physics books will give you a clearer understanding of the concepts involved..........just not the eng math needed

[Harald]

Think dimples was turned down in this thread
http://www.archery-forum.com/viewtopic.php?t=85290
where the new cloud golfball/arrow hybrid was invented :D :D :D

Guess shooting with pimples doesn't improve anything neither... 8)