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The Physics of Aerobatics UnderstandingAirplanes.comThe Physics of Aerobatics UnderstandingAirplanes.com

The Physics of Aerobatics;What is your airplanephysically capable of?

Bernardo Malfitano – EAA 9026646

EAA Chapter 84 – June 13, 2017

© Bernardo Malfitano 1

The Physics of Aerobatics UnderstandingAirplanes.com

Important note• Always follow the POH, the placards on the panel, and other limitations/restrictions!

• This talk is about what is physically possible,(and how easily/safely), not what is legal/advisable!

• This material is intended for homebuilders who will write their own POH.

• I’m an engineer, NOTan FAA‐certified instructor!



Bernardo Malfitano• Academic:

• BS Mechanical Engineering, Stanford University

• MS Mechanical Engineering, Columbia University

• Aerodynamics research at wind tunnel,delaying separation to increase lift and reduce drag

• Propulsion coursework at engines lab: piston, jet, rocket

• Developed, implemented, tested autopilot for UAV

• Professional:

• 10 years at Boeing: Structural durability & damage tolerance researcher, fatigue analysis & maintenance planning instructor. Designedand ran fatigue tests for 787‐9, KC‐46, 737MAX, and now 777X.

• “On the side”:

• 15 years of airshow photojournalism

• Private pilot (+ tailwheel) :

• Creator/Teacher of“Understanding Airplanes”airplane design course for non‐engineers


2009: First solo.2012: Bought RV‐6 .2014: Flew to OSH


AgendaHow to quantify an airplane’s ability to (barely? safely?) perform…

•Aileron Rolls• Loops•Barrel Rolls• Slow Rolls•Knife‐edge• Spins & etc.



Rolls

• Slow Roll(‐1g)

•Aileron Roll(~0g)

•Barrel Roll(>0g)



Aileron Rolls



How much time at Zero G? Enough to roll 360⁰?

Verticalcomponentof velocity

Horizontal component of velocity

Angle of climb

Vertical component of velocity = Velocity X sine of angle of climb (vectors, trig)

. (definition of acceleration)

Time at Zero G = ( Vertical component of velocity / g ) x 2 (projectile/ballistic motion)

Pull up to 30⁰ (i.e. sine is half), Zero G time is ~5 sec. per 100 kt



Three key considerations• Roll rate = ?• Find out by banking 90⁰R to 90⁰L (i.e. roll 180⁰).• Rolling 360⁰ takes twice that long• Zero‐g time must be longer than 360⁰ roll time !!!

• Entry speed? Fast! (as long as no risk of hitting VNE).Go faster → More me at zero g & Faster roll rate.

• Gs during pull‐up = ? Circular motion equations, A=V

• A=V ( 9.81 m/s2 / 51.44m/s = 0.191 rad/s )For each 100 kts, each additional G (above 1)gets you about 11⁰ per second pitch‐up.• So at 2g and 100 kts, pulling up to 30⁰ takes 2.75 sec• And at 2g and 200 kts, pulling up to 30⁰ takes 5.5 sec

for aileron rolls



Sharpening the Pencil• Higher induced drag → some speed lost during pull‐up.

• Some is converted to potential energy. How much?• A=V2/R , assume ½mV2 = mgH (kinetic and potential energy), so…

• If V=100kt and A=3g ,R=V2/A=89.91m , H(30deg)=(1‐cos30⁰)R=12m , gH=118.2, ½V2=1323

Speed lost = 4.6 %

• If V=400kt and A=1.5g ,R=V2/A=2877m , H(30deg)=(1‐cos30⁰)R=385m , gH=3781, ½V2=21169

Speed lost = 9.4 %• So: zero‐g speed will lower than entry speed, up to 10%.So zero‐g time is lower by up to 10%: ~4.5 sec / 100 kt

• Worst case: Airplane end up with nose more than 30⁰ (or whatever) below the horizon. Not a problem if far from VNE. But if that’s the case, maybe precede the pull‐up with a dive to pick up higher entry speed.



In Summary: Aileron Rolls• How long to bank from 90⁰ left to 90⁰ right?

• Time to roll 360⁰ will be about twice that.

• If airplane flown at “fast but not dangerous” speed…• … then ~1.5g are pulled to 30⁰ nose‐up…• … then airplane is allowed to fall in zero‐g ballistic arc…• … zero‐g time will be about 4.5 seconds per 100kt.

• Roll is safe if zero‐g time is higher than 360⁰‐roll time.

To make an airplane better at aileron rolls:

• Increase roll rate (e.g. larger ailerons – but watch out for twist/reversal !)…

• … and/or make it faster (i.e. higher VNE, e.g. stiffer structure to avoid flutter).



In Summary: Aileron Rolls


Max Speed

100 kt

200 kt

Time to roll 360⁰

5 sec 10 sec

roll possible

rollnotpossible


Aileron Rolls: In Practice…• Most airplanes are physically capable of aileron rolls.

• Rebecca Wallick says most Boeing test pilots have rolled their prototype airliners.



Loops• Very tall maneuver!Most kinetic energyat the bottom (at the start)becomes potential energyat the top, i.e. top gets slow.

• Less so if you pull tighter…but an airplane (and a pilot!) can only pull so many Gs…especially as it slows below VA near the top.

• Key parameter = Ability to fly at several times the stall speed• Yes, the ability to pull Gs helps too……but a high ratio of VS1 to VNE helps even more.

• Thrust does not help as much as you might think. (Unless you have a lot of it, it’s almost totally cancelled out by drag).



Kinetic and Potential Energy


• Say speed at bottom is Vbottomloop height is Hspeed at top is Vtop

• How much lower than Vtop is Vbottom?

• Kinetic energy at the bottom is ½ mVbottom2

Kinetic energy at the top is ½ mVtop2

• Potential energy is mgH

• So ½ mVtop2 + mgH = ½ mVbottom

2

i.e. Vtop2 + 2gH = Vbottom

2

This equation tells you how much speed is gained by anything that is going downwards due to gravity: a swing, a body in freefall, a bike going down a hill, etc.


Circular MotionWhat is the relationship between

• Speed, V• Centripetal acceleration (a.k.a. “Gs”), A• and the Radius of turn, R?

It’s A = V2/R(Also, A=V where = rate of turn, i.e.the change in direction angle per second)


This equation works for satellites, cars… anything on a curved path that is circular at least locally(i.e. that has some local radius‐of‐turn at each spot)


Three Ways to Model a LoopCircular Loop (constant radius)

• Very simple math and physics

• Almost uselessly conservative:You need to be able to pull 6g!!!

Constant or Increasing G Loop(Cornu spirals and clothoids)

• Very hard math and physics

• Pretty accurate, but still ignoresloss of G capability below VA

Discretized Loop(numerical analysis, circular segments)

• Very simple math and physics

• Arbitrarily accurate…

• … if slightly less elegant.


AZtop=0

Vtop2=1gR

Vbottom2=1gR+4gR

Vbottom2/R=5g (plus gravity)


What is “numerical analysis”?

A “closed‐form” or “exact solution” is:

• The most accurate way to solve a physics problem.

• You develop an equation and solve it exactly.

• But: Sometimes this is not possible.

A “numerical analysis” is:

• An approximate solution

• Not as accurate, but can get arbitrarily close

• You can get closer to the exact solution by…(1) brute computing power, and/or(2) tricks e.g. higher order methods

• Uses much simpler math, e.g. linear equations.

• Can be thought of as a “simulation”.

e.g. What is the shear stress in a beam, at each point along the height:xy = shear stressy = height of each pointV = shear forceI = moment of inertiah = height of beam



Numerical analysis for loop



The result:


Blue region:

Possible without ever dropping below VA

(more circular loop)


In practice…


MOSTAIRPLANES


Or rather: How “perfectly” must it be executed?


MOSTAIRPLANES

AEROBATICAIRPLANES

But… How safe is it?

So while the most important parameter to being able to perform a loop at all is VS1/VNE ratio, the most important parameter to being able to do it safely (i.e. going nowhere near the max structural G)is structural G capability!

This also allows for more circular loops.


In Summary: Loops• The higher the ratio between VNE and VS1, the fewer Gs are required to do a loop.

• The ability to pull more Gs makes it easier to do a loop, i.e. gives a wider margin of safety, and allows for a more circular loop (e.g. for aerobatic competitions).


To make an airplane better at doing loops (i.e. to make it easier to perform loops without having to hit the “perfect” speed and Gs):

• Decrease VS1 (larger wings, VGs, slots / auto‐slats)…

• … and/or increase VNE (i.e. stiffer structure to protect against flutter).


Immelmans, Split‐Ss, Cuban 8s• Made of parts of loops and rolls.

• So, similar requirements.

• However…


IMMELMAN

CUBAN 8


Immelmans, Split‐Ss, Cuban 8s• Made of parts of loops and rolls. So, similar requirements.

• However: Immelmans and Split‐Ss need enough speedat the top of the half‐loop to sustain 1g flight (i.e. more than VS1)

• So for an Immelman or a Split‐S, more Gs – or more speed –is needed at the start, when compared with just a loop.

• Here is the result of anothernumerical analysis, but theorange “barely” line is not“Speed barely above zero at90⁰ nose‐up”, it’s “Speed aboveVS1 at very top of loop”.

• This is basically the same asthe previous graph, butthe line is moved “oneto the right”, i.e. theairplane MUST enter themaneuver at over twiceits stall speed.


Max G

Max Speed / Stall Speed

6

5

4

3

2

1

0

0 1 2 3 4 5 6 7 8 9 10

Immelman& split‐Spossible


Barrel Rolls• When seen from behind, they’re like a loop.

• When seen from the side, they’re like an aileron roll.

• So both the loop equations and the aileron roll equations come in handy.



Barrel Rolls• Basically a combination of an aileron roll and a loop.

• What is the angle of the direction of flightrelative to the axis of the barrel?

• The smaller the angle, the more like an aileron roll.

• The larger the angle (closer to perpendicular), the more like a loop.

• Realistically, a barrel roll is an aileron roll where the pilot pulls up slightly“just to keep your butt in the seat” rather than fly through true zero G.


Small angle (aileron roll)

Perpendicular angle (loop)


Slow Rolls • Level flight,• knife‐edge one way,• ‐1g flight,• knife‐edge the other way,• Back to level flight:


What airplanes can sustain ‐1g flight?

How about knife‐edge?


Sustained Inverted Flight (‐1g)… requires three things:

• 1: Structural strength.• 2: Systems, e.g. inverted oil pump and fuel hose

• 3: Sufficient elevator authority. (This is easier at faster speeds)



“Elevator Authority”?• The first two are obvious. As for the third…• Recall that the horizontal stabilizer must push down for balance.



“Elevator Authority”?• So for sustained inverted flight, we need…


No way! Will fall!

Nope. Nose will come down.

That’s it!

Direction of flight

Direction of flight

Up

Up


“Elevator Authority”?• So the key questions are: Is the elevator big enough,and does it deflect through enough of an angle,to change the horizontal tail lift from “upwards” to “downwards” when the airplane is at a highly negative angle of attack?

• Note: The faster the speed, the less angle is needed. (Can the airplane do a‐1g push‐over? If so, then it can sustain ‐1g flight. And it can, at SOME speed).


Direction of flight

ProbablyCAN

sustain‐1g flight

ProbablyCANNOTsustain‐1g flight


Knife‐Edge Flight• Just as ‐1g capability depends on elevator authority,knife‐edge capability depends on rudder authority.

• Both the CG and the fuselage’s center of lift will be somewhere near the front. But the vertical stabilizer will make a lot of lift, wanting to bring the tail up and the nose down. Can the rudder overcome this? Not on most airplanes.

• (i.e. : In airplanes capable ofknife‐edge flight, a maximum slip –full rudder one way, then bankthe other way until the airplaneis not turning – will becomeknife‐edge).


Direction of flight

Up


Spins• I won’t get into what airplanes can or cannot safely be brought out from a spin.

• This is barely understood even by the big manufacturers. Airplanes as recent as the F/A‐18 and Global Hawk UAV had spin issues. New airplanes are tested with anti‐spin chutes.

• Subtle differences in aerodynamics and/or weight distribution can cause two nearly‐identical airplanes to spin quite differently / more or less “stubbornly”. There’s more to it than rudder effectiveness.

• Spins build up momentum and take some time to get out of.

• Some maneuvers, e.g. loops and tail‐slides, seem easy but come with a risk of inverted spins.

• Learn to recover from spins… then stay away from them (coordinated flight, not too close to the stall) except in airplanes shown to be able to reliably get out of spins.

• Then again, I’m biased, I’ve only ever spun an RV‐6A…



This just scratches the surface…• We did not discuss some relatively simple maneuvers such as hammerheads and snap rolls. How could they be modeled?

• What are the loop dynamics of an airplane with so much thrust, it barely loses any speed on the way up?

• What about high‐torque gyroscopic maneuvers like lomcevaks?

• How are things different in formation aerobatics?

• How about in airplanes that are neutrally stable or unstable,like a Sukhoi‐27 performing a Cobra maneuver?

• What possibilities are opened up by thrust‐vectoring jets and airplanes with large control surfaces placed in the prop‐wash?

• But hopefully tonight’s presentation was enough to help you think about the capabilities of a homebuilt to do basic aerobatics.



Questions?


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