The Wingsuit Theory, or How to fly like an eagle while barely moving on the couch

[yuri_base 1]

Yo!

Wingsuit flying lends itself to some simple, yet elegant and powerful math. By analyzing the equations of motion, we can try to find answers to questions like:

- how can L/D be calculated from the glide ratio in non-sustained flight modes?
- can a true L/D meter be built to give the pilot instant and accurate feedback?
- why on BASE and balloon jumps you seem to fly better?
- how to start flying as quickly as possible on base jumps?
- what is the altitude used up before you're flying at full glide?
- is bigger better?
- how does wingloading affect various characteristics of flight? (including "perceived" performance)
- how does balance affect the performance? (controversial, not completely researched, and partially wrong example here; another example of balance derived from windtunnel data here)
- how to achieve maximum horizontal speed possible?
- why sometimes you seem to be stuck and barely moving forward?

Let's see!

Yuri

Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

[yuri_base 1]

Consider forces acting on a flyer. One is gravity m*g, another is aerodynamic force that can be broken into two components: lift L perpendicular to current speed V, and drag D opposite to V. If the current glide angle to horizon is A, the horizontal components of lift and drag are L*sinA and -D*cosA, vertical -L*cosA and -D*sinA. Therefore, Newton's law gives us two equations:

Fx = m*ax = L*sinA - D*cosA
Fy = m*ay = m*g - L*cosA - D*sinA

where ax = dVx/dt, ay = dVy/dt are horizontal and vertical accelerations, Vx and Vy are horizontal and vertical speeds.

According to aerodynamics, lift and drag can be expressed as

L = (1/2)*Cl*ro*S*V^2
D = (1/2)*Cd*ro*S*V^2

where Cl and Cd are nondimensional coefficients of lift and drag, ro is density of air, S is effective area of wingsuit.

Since sinA = Vy/V, cosA = Vx/V, after some refactoring we arrive at these differential equations:

dVx/dt = k*V*(Cl*Vy - Cd*Vx)
dVy/dt = g - k*V*(Cl*Vx + Cd*Vy)

where k = (1/2)*ro*S/m, V = sqrt(Vx^2 + Vy^2).

Note that the lift and drag coefficients themselves can be changing with time. For simplicity, the surface area is presumed to be constant.

These equations can also be applied to tracking, canopy flight, etc. - any non-powered, not spinning out of control flying body.

Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

[yuri_base 1]

Without horizontal windtunnel, how do we know the coefficients of lift Cl and drag Cd? Fortunately, the answer is, we don't necessarily need to.

Suppose in some fixed body position your sustained horizontal and vertical speeds are Vxs and Vys. Since for sustained flight acceleration is zero by definition, the wingsuit equations read

k*Vs*(Cl*Vys - Cd*Vxs) = 0
k*Vs*(Cl*Vxs + Cd*Vys) = g

We have 2 equations for 3 unknowns (k and Cl, Cd). But note that if we combine k with the Cl and Cd into "adjusted" lift and drag coefficients (no longer nondimensional)

Kl = k/g*Cl
Kd = k/g*Cd

we have 2 equations for 2 unknowns:

Kl*Vys = Kd*Vxs
Vs*(Kl*Vxs + Kd*Vys) = 1

The solution of these is:

Kl = Vxs/Vs^3
Kd = Vys/Vs^3

With these adjusted coefficients, the wingsuit equations are

dVx/dt = g*V*(Kl*Vy - Kd*Vx)
dVy/dt = g*(1 - V*(Kl*Vx + Kd*Vy))

Now the unknown wingsuit parameters (wingloading mg/S and aerodynamic properties Cl, Cd) are "hidden" inside coefficients Kl and Kd, which can be easily calculated from sustained horizontal and vertical speeds.

Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

[yuri_base 1]

When speed has not yet reached the sustained value or when we change body position, the current glide ratio is not necessarily equal to L/D ratio. And it's L/D that determines the sustained glide ratio, and that's the value we strive to increase. Can we figure out L/D from the current non-equilibrium glide ratio?

The current glide ratio G is equal to Vx/Vy. Let's solve the wingsuit equations for Kl and Kd and then we can determine L/D = Cl/Cd = Kl/Kd.

Kl*Vy - Kd*Vx = ax/g/V
Kl*Vx + Kd*Vy = (g - ay)/g/V

The solution is

Kl = (Vx*(g - ay) + Vy*ax)/g/V^3
Kd = (Vy*(g - ay) - Vx*ax)/g/V^3

thus

L/D = Kl/Kd = (Vx*(g - ay) + Vy*ax)/(Vy*(g - ay) - Vx*ax)

and finally

L/D = (G*(g - ay) + ax)/(g - ay - G*ax)

If we have accurate glide ratio and acceleration data, we can calculate the true L/D even for non-sustained flight (dive, starting to fly, planeout, etc.) by using the formula above.

Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

[yuri_base 1]

Imagine a weathervane attached to the flyer, with a 2-axis accelerometer installed on the vane, with one axis parallel to the vane and the other perpendicular.

What will the acceleromer read?

The vane is oriented at the glide angle A to horizon. In sustained (zero-acceleration) flight, the components of gravity parallel (g1s, the reading of the first accelerometer) and perpendicular (g2s, the reading of the second accelerometer) to the vane will be g1s = g*sinA and g2s = g*cosA , with cosA/sinA = G (glide ratio). Therefore, for sustained flight, L/D = g2s/g1s.

In non-sustained flight, the horizontal ax and vertical ay accelerations will skew the readings (lazy to draw a diagram, figure it out yourself):

g1 = g1s - ax*cosA - ay*sinA
g2 = g2s + ax*sinA - ay*cosA

Hence,

g2/g1 = (g*cosA + ax*sinA - ay*cosA)/(g*sinA - ax*cosA - ay*sinA)

g2/g1 = (G*(g - ay) + ax)/(g - ay - G*ax)

Compare to the formula in the post above. OMFG… The right hand side is equal to nothing but L/D!

L/D = g2/g1

Isn't it amazing?!
An accurate and fast true L/D meter can be built using a weathervane and a 2-axis accelerometer with its axes parallel and perpendicular to the vane. The true L/D is equal to the ratio of the readings of the parallel and perpendicular accelerometers.

Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

[yuri_base 1]

Ever get a balloon or base jump when it feels all too good... you max out the suit, and you effortlessly fly at insane glide ratio, you experience the quietness and smoothness you've never experienced before and can't figure out what the hell is right?

You may think that increased concentration and strength made you fly better. Well, now you can praise basic aerodynamics instead of praising yourself (or clown shoes)!

The wingsuit equations are solved in the attached spreadsheet using the simple Euler integration method for some sustained horizontal/vertical speeds (which, as we saw, determine the adjusted lift/drag coefficients). For simplicity, we assume constant Cl and Cd (that is, the wingsuit geometry and angle of attack are constant) and zero-speed exit. You can change the values of Vxs and Vys to match your speeds. In this particular example, Vxs = 86mph, Vys = 43mph, L/D = 2.0.

The graph PlaneoutTheoryVsExperiment.gif compares the calculations with one of the Phantom flights (the above parameters were chosen to best fit the experimental data). As you can see, from about 12 seconds to 24 seconds, the glide ratio is higher than 2.0 - we have a planeout with maximum glide ratio as much as 35% higher than the sustained glide ratio.

The graph PlaneoutVsWingloadingAndLD.gif shows the dependence of the planeout duration (time period when glide is better than L/D) vs. wingloading. The heavier flyers experience longer planeouts which start later. The graph of glide ratio increase vs. L/D shows that the planeout effect dramatically increases with the increasing L/D: the better you fly, the more you can be fooled into thinking that your insane glide ratio is your L/D. PlaneoutVsLD.gif shows the trajectories and glide ratio vs. time for different L/D.

Unlike intentinal dives and spirals and subsequent high-speed planeouts with quite high g-forces in skydiving, the smooth transition into full flight on a base or balloon jump hides the planeout in virtually unnoticeable ~0.1g decelerations that bleed your speed ever slowly, but do make your glide substantially better than your actual L/D.

In conclusion,

- when analyzing GPS data from a base or balloon jump, discard the first ~30s of the flight, even if it has a linear portion that looks like a sustained flight... it's not! (or better yet, correct the glide ratio for acceleration using the formula above)

- planeout effects can also manifest themselves when you change your body position and feel the decreased fallrate and improved glide - only to lose it in a few seconds. It could simply be a planeout! (again, accurate acceleration data can help you see if the improved glide was real or "fake")

- after the planeout, you will experience temporary "drop", a decrease of glide ratio even below your L/D. Plan your "do it or die" jump accordingly.

- heavier jumpers will experience the planeout effect longer. Vampire will exhibit stronger planeout glide imrovement than Prodigy. The higher the performance, the better the planeout - and the worse the "drop" after it.

Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

[yuri_base 1]

What is the most efficient technique to start flying as quickly as possible from still air exit? Is it an intentionally extended superterminal dive? Is it totally flat? Or somewhere in between?

The wingsuit equations can help us understand what works and what not. Let's use the lift/drag coefficients from this post. Although they may not represent any wingsuit, the aerodynamics of the wings at high angle of attack (see, for example, this article) suggests that the lift coefficient is maximum at 45 degrees and is almost independent of the geometry of the wing.

Let's model the initial 10s of the flight for the angles of attack 6 degrees (L/D=2.0, Cl=0.26, Cd=0.13, Vxs=132mph, Vys=65mph for some wingloading), 12 degrees (best L/D=2.8; Cl=0.55, Cd=0.2, Vxs=98mph, Vys=35mph), 45 degrees (L/D=0.85, Cl=0.92, Cd=1.09, Vxs=43mph, Vys=51mph), and 70 degrees (L/D=0.35, Cl=0.56, Cd=1.62, Vxs=20mph, Vys=58mph). See StartingToFly.gif.

45 degree technique wins hands down. Due to its high lift coefficient, it propels you forward much more efficiently than even the best L/D AoA. Below 1000ft down, best L/D catches up and soon beats the 45 degree. At these altitudes, the glide ratio is already 1.0 and increasing, so it's time to switch from 45 degrees to full maxed out flight. The headdown and "flat from the start" techniques are the least efficient.

StartingToFlyVsWingloading.gif shows how wingloading affects starting to fly. The distance you fly in the first few seconds is inversely proportional to weight. (This explains Deadmanwalking's amazing discovery. ) The bigger the suit, the faster it starts to fly, even if it is not the best L/D suit around.

Conclusions:

- to start flying as quick as possible, maintain the 45 degree angle of attack in the first ~6s into the flight. Maintain the balance using the leg wing and decrease the pitch angle in sync with the increasing glide until you reach your maxed out pitch.

- light flyers start flying faster

- the bigger the suit, the easier it starts to fly. Phantom will start flying faster than Prodigy, Vampire faster than Phantom, squarrel suits will fly faster than Vampires.

Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

[TomAiello 26]

All,

I've removed a bunch of off topic one liners from this thread. Please remember that the wingsuit forum is a topical forum. If you feel a need to discuss the mechanics of ponies and lube, that's what the Bonfire is for.

The nature of the wingsuit community is going to mean that there will always be a certain amount of banter here, but let's try to keep it from wandering too far, and creating a giant string of one liners in this forum.

Thanks!

-- Tom Aiello

Tom@SnakeRiverBASE.com
SnakeRiverBASE.com

[SkymonkeyONE 4]

That's an outstanding series of posts, Yuri! Very well thought out. To everyone else: this is a perfect example of a great thread that does not need to be junked-up with nonsense.

Chuck

[Waldschrat 0]

  Quote

- the bigger the suit, the easier it starts to fly. Phantom will start flying faster than Prodigy, Vampire faster than Phantom, squarrel suits will fly faster than Vampires.

What means this?

Very interesting posts yuri.

[mccordia 74]

  Quote

  Quote

- the bigger the suit, the easier it starts to fly. Phantom will start flying faster than Prodigy, Vampire faster than Phantom, squarrel suits will fly faster than Vampires.

What means this?

Very interesting posts yuri.

I think he means mono-wing suits (S-Fly/Matter etc)

JC
FlyLikeBrick
I'm an Athlete?

[robibird 3]

Do not agree with this.
Smaller and well balanced wingsuit is way easier to get it off sable than just whatever bigger WS.
One of the primary reasons why is recommended to start with small WS from the cliff is stability. If stability and transition in to the flight is screwed start will took long time.
Size has important role in '' start flying faster'' but it is not the primary one!
Average jumper w Phantom or Prodigy will have better result than when using really big one.

Robert Pecnik
robert@phoenix-fly.com
www.phoenix-fly.com

[robibird 3]

Hello,
Nice stuff here Yuri.
Every second of time and effort involved in reading those posts is absolutely worth it .
Most common mistake on WS arena ( and in life in general ) is to try to learn something by using shortcuts, for example, like flying good only by picking the tips and tricks without trying to really understand the minimum of theory....why we fly or how we fly.

All physics and mathematics here are accurate, but also require above average knowledge. Therefore my only suggestion here which goes to Yuri would be... to explain all this but without numbers, equations, etc...It is hard but it will be way better accepted. ( similar approach I had on articles on PF web site - Aerodynamics I and II )

If we give up from going to the theory we will soon be at the beginning, claming that the bigger is better, slower is better, etc.... If parachute manufactures had followed the same ignorant approach to theory or physics we would be still jumping and hanging under round canopies, not having knowledge of the square canopies and best canopy would be the one which hangs out in the air for the longest time.

It is frustrating to feel lack of progress just because of lack of knowledge. Aerodynamics is very demanding science. WS flying is part of it and part of flying,and we are dealing with so many different variables and problems that is almost impossible to predict the results accurately enough.... Therefore guys, try to listen, read and understand what Yuri is saying, because this source ( theory ) can tell you a lot.

My dream is to fly fast and for long time with the best possible glide ratio!! I want to have a hangglider with wingspan equal to the span of my arms. If we want to achieve that , one of the most crucial part is physics, or in particular, Theory of aerodynamics. It is a long journey, it is not easy and every help is appreciated.

Robert Pecnik
robert@phoenix-fly.com
www.phoenix-fly.com

[KrisFlyZ 0]

This is a good site as well

[KrisFlyZ 0]

  Quote

If we give up from going to the theory we will soon be at the beginning, claming that the bigger is better, slower is better, etc....

Granted just making a big suit will not make it the best gliding suit.....

But bigger is better . The best suit(V2) you make is bigger than any other suits that you make(Prodigy, Phantom).

Now slower........

Kris.

[tr027 0]

  Quote

  Quote

Granted just making a big suit will not make it the best gliding suit.....
Kris.

The M1 should put that theory to good test (7:1 remember), once it's at more than just 1 dropzone in 1 country.

"The evil of the world is made possible by nothing but the sanction you give it. " -John Galt from Atlas Shrugged, 1957

• Quote

[VectorBoy 0]

Granted just making a big suit will not make it the best gliding suit.....

But bigger is better . The best suit(V2) you make is bigger than any other suits that you make(Prodigy, Phantom).

Now slower........

Kris.

[KrisFlyZ 0]

Not comparing the different planforms Glen.

But as a side note.

There should be no doubts that the V suits have larger area then the S-3 or the S6 in addition to having a whole lot less drag and a better wing distribution. See Chucks V1 vs S6 comparision.

One of Robi's own articles on aerodynamics goes on for three paragraphs about the problems with increasing area of the wingsuits. Increase in area is a good thing but not just anywhere.

One striking feature of the V series arm wings is the concavity from wingtip gripper to body attachment point. See the pic of my V2(previously owned).

Try holding a flat object under a tap and see where most of the water goes, then hold a spoon under the water and see what happens. Its the simple things in life
Creating a channel for the air to flow under the suit and cutting back on the span of the arm wings gave me the best results and this is counterintuitive. But. I do not have enough jumps on the V series of suits to say for sure.


Kris.

[mccordia 74]

  Quote

in addition to having a whole lot less drag and a better wing distribution

I tend to think the larger grippers at the end of the wing tend to ad a lot of drag. Like holding a small flag in each hand.
It causes larger arm pressure, and makes for few people that can fly the suit maxed out for an entire skydive.

Looking at aircraft wings, the tip of the wing does quite little and seems to actualy ad drag, generated by the vortexes that rolll/bleed off the side.
Having seen a few pics of weird experiments in wingsuits with winglets at the tip of the wing (probably scary as sh*t at pulltime).
How much of the long grippers is actualy affective/working wing area?

But wingtips etc. aside...isnt there still anyone jumping the M-1 that actualy has some true data on what the suit can do?

I'd gladly buy into the whole 7:1 thing, but if those numbers can be posted here, surely there must be GPS plots that support those numbers..?

I gladly believe its an incredible suit, with amazing performance. But so far the claims (2.5 times better performance then other top of the line suits from other manifactures) seem more like marketing hype then anything supported by facts or data?

*edit for bad zpelling*

JC
FlyLikeBrick
I'm an Athlete?

[KrisFlyZ 0]

I knew that mentioning suit names was a mistake. Anyway...

Angle of Attack and Pitch.
==========================

http://www.aerospaceweb.org/question/aerodynamics/q0165.shtml

http://en.wikipedia.org/wiki/Angle_of_attack

It seems like AoA can never be negative for a wingsuit flight(pitch can be but not AoA). A negetive AoA would mean that the airstream is hitting the top surface of the wing. This would mean that lift is negative. Playing with the FoilSim applet on the NASA aerodynamics website shows this.

Camber
=======
http://en.wikipedia.org/wiki/Camber_%28aerodynamics%29

This is a good thing. We can create Camber by pressing the legs down and sucking the gut(kinda hard and needs a lot of ab strength...shoulders on level with feet as some of the suit nmanuals describe it) and(or) by bending and rolling the shoulders but keeping the legs straight...this seems to reflect the way most real world wings are...the bulge in the wing shape is nearer the leading edge.

Angle of Incidence
==================

http://en.wikipedia.org/wiki/Angle_of_incidence


On some suits this is not zero. To the best of my recollection, the manufacturers reason for this was that it aids in reducing the angle of attack.

Kris.
Edited to make clicky's

[robibird 3]

_____________________________________________________________
I tend to think the larger grippers at the end of the wing tend to ad a lot of drag. Like holding a small flag in each hand.
It causes larger arm pressure, and makes for few people that can fly the suit maxed out for an entire skydive.
_____________________________________________________________

If they feel this , they are not holding proper position for maxing out in terms of either FF time nor distance!

Robert Pecnik
robert@phoenix-fly.com
www.phoenix-fly.com

[VectorBoy 0]

It causes larger arm pressure, and makes for few people that can fly the suit maxed out for an entire skydive.
_____________________________________________________________

If they feel this , they are not holding proper position for maxing out in terms of either FF time nor distance!

[tr027 0]

I agree with Robi and Vector, when using the proper technique there's little pressure from the gripper. But I do remember my first flights when I didnt know anything about flying technique for the suit that I did have a moderate amount.

"The evil of the world is made possible by nothing but the sanction you give it. " -John Galt from Atlas Shrugged, 1957

[KrisFlyZ 0]

Your post confuses AoA and pitch.

  Quote

45 degree technique wins hands down. Due to its high lift coefficient, it propels you forward much more efficiently than even the best L/D AoA. Below 1000ft down, best L/D catches up and soon beats the 45 degree. At these altitudes, the glide ratio is already 1.0 and increasing, so it's time to switch from 45 degrees to full maxed out flight.

The headdown and "flat from the start" techniques are the least efficient.

In the last sentence do you mean pitch or AoA? Because headdown and flat from start refer to pitch.
What equations did you use and what ax and ay did you use? Because these are non zero.

Are you talking about 45 degree AoA? Because at zero velocity 45 degree AoA = what pitch and what L/D?

The graphs are more than a bit off in regards to real world observations.

Take a look at Terje's exit in this movie. The flight starts way before 200'(if that of fall) and I cannot tell if the AoA is 45 degrees.

By the time a wingsuit flyer falls 300m, well over 1:1 GR can be reached.

You have seen James's bridge day video? Depending on opening altitude, that final GR is probably over 1:1 at that point and the altitude lost is not even 600'?

Kris.

[mccordia 74]

  Quote

If they feel this , they are not holding proper position for maxing out in terms of either FF time nor distance!

Proper arm position maybe the reason?
Several people flying V1/V2s complained about it around here, thats why I asked...

Ive seen people use some really akward arm and body positions trying to fly their suits. We're always trying to assist in heloing them see what they could change to improve flying wise. So any tips on this subject would be great..

We noticed that in your video's, the arm wing is always perfect. One smooth shape.

While with a lot of flyers, there is a clear line. Cut across the wing. like a fold. Is that whats causing their armwing to be 'heavy to fly'?
And maybe also tipping them headlow a bit to much (having trouble staying up with slower formations)

Its seen a little in this picture
But its more pronounced with some of the people I've seen.
Does it require more pushing down on the wing, or pulling grippers outward to create a smooth wingshape there?

It would be great of (body position-wise) you'd have some tips and pointers on that subject?

JC
FlyLikeBrick
I'm an Athlete?