Monday, October 15, 2012

Mechanical Analysis of Baumgartner's Dive (Part II)

(This is the second and final article of the Felix Baumgartner dive saga - click here for part 1)

By now you have no doubt heard that Felix Baumgartner has shattered several records with his successful sky dive on October 14, 2012.  Fearless Felix stepped off of his perch, fell freely for 4 min 18 sec, and then pulled his parachute, coasting safely to the surface about five minutes later.

The lead up to the historic event was similar to that of a rocket launch, complete with weather delays.  This jump was originally set for October 8, but on several occasions, it got bumped.  You know you are involved in something risky when a little too much wind is cause for serious worry.

Imagine you are Felix, and you wake up on October 8, having probably not slept much the night before, ready for the leap of your life.  You down a few red bulls, get your adrenalin up, and then some guy in a lab coat gives you the news that the jump must be postponed.  Repeat this a few more times, and you just might go mad.  I do not know this for certain, but I would imagine that a psychologist was on site with Baumgartner to help him maintain his mental well-being through this go/no-go roller coaster that lasted more than a week.

Many videos of the dive have circulated on YouTube, though most have been yanked by the sponsor (Red Bull).  Here is their 90 second summary of the event.

One can only imagine what it must have been like to look down from 128,000 ft (8,000 ft more than originally planned), and to behold the planet.  From that altitude, one can begin to get a sense of the Earth's curvature.  With a final salute (to his family, and mankind I suppose), Baumgartner stepped off from his pod and quickly vanished from view.

Based on some of the information given in the video, as well as some educated guesses, I have constructed approximate graphs of Baumgartner's speed and altitude as a function of time for the free-fall portion of his descent.

(Note that it is possible to generate theoretical results by solving the governing equation numerically, but as I do not have access to the particular parameters associated with his specially designed space suit, such as mass and drag coefficient, I elected to plot these 'experimentally')

From the speed graph, we see that the increase is linear at first, since he is very far from the terminal velocity at that altitude, which is around 350 m/s.  As expected, his initial acceleration is around 9.8 m/s/s (the initial slope of the speed graph).  This acceleration begins to decrease significantly once his speed begins to approach 350 m/s.  Then, because the density of air increases the lower he descends, his speed decreases, roughly matching (though slightly overshooting) the terminal velocity associated with the given altitude.  After 4 min 18 sec (258 seconds), he pulls his chute, having decelerated to about 55 m/s.

From the altitude graph, we see that he jumps from about 39 km and pulls his chute at around 3 km.  The beginning of this graph looks just like an ideal free-fall graph (one that omits air drag), with negative concavity.  The curve adjusts once he hits terminal velocity, causing a much more gradual descent.  If there were no atmosphere on Earth, Baumgartner would have struck the surface after just 90 seconds (it should be mentioned that opening a parachute would not affect his kinematics in such a case - note to self, never sky dive on the moon).

There is so much more that can be discerned by watching the video of the dramatic event...

The most dangerous portion of the dive - the part where Baumgartner's expertise as a sky diver was essential - was the time-frame between 60 and 90 seconds.  As can be seen in the speed graph, there is a fairly dramatic deceleration that occurs in this span of time.  When he encounters denser air at such a high speed, even the slightest asymmetry in his form causes a torque about his center of mass that sends him spinning.  Once this spinning begins, Baumgartner must adjust his form continuously so as to restore a stable shape and orientation.  With unimaginable focus, he succeeds in doing so, and those watching from the ground erupt in cheers.

The one record set in 1960 by Joe Kittinger during his 31 km plunge that was not broken by Baumgartner was the time of free-fall.  Kittinger fell for 4 min 36 sec (18 seconds more than Baumgartner).  This seems counter-intuitive given that Baumgartner jumped from 8 km higher, but it means that his terminal velocity parameters were not the same.  In particular, Baumgartner's suit made the total mass greater, and his coefficient of drag smaller.

One final question I wish to address is on the topic of going supersonic, which Baumgartner succeeded in doing, becoming the first person to do so without the aid of an aviation vehicle, ie, some kind of on-board thrust.  Accelerated by gravity alone, his speed eventually exceeded that of sound in the region of air he traveled through.  By my estimation, he was supersonic for about thirty seconds.

The speed of sound on Earth varies a bit with temperature, but a typical value is 340 m/s.  In the low to mid stratosphere (the region of space where supersonic flight took place), the speed of sound is around 300 m/s, owing to the significantly lower temperature.  So, it can be said that Baumgartner reached about Mach 1.2.

When a supersonic jet reaches such speeds, a white cone forms around it, and a sonic boom can be heard by stationary spectators.  It is not clear from the video whether such a cone formed around Baumgartner, or whether a sonic boom was emitted.  If this particular supersonic flight created a 'weak' sonic boom, it may be due to the low density air that the event took place in.

In any case, the situation does pose an interesting question that I would like to leave you with (feel free to post answers in the comments section):

If you travel faster than the speed of sound, do you hear a sonic boom?


Anonymous said...

Good article! News stories say he reached a peak Mach number of around 1.24 so your "experimental" analysis looks fairly good. One thing needed to take into account if you did a numerical simulation is that approaching Mach 1 the drag coefficient increases dramatically, by a factor of 2 or 3 what it is below Mach 0.8.

As to the question of whether he heard a sonic boom, I say no. A sonic boom is formed from the pressure discontinuity of the shock wave as it passes by you. Since Felix was always behind the shock wave (since he was creating it) it didn't pass over him. One other factor is that the air is so thin at that altitude the absolute magnitude of the pressure discontinuity is not that great.

The Engineer said...

Thanks for the great insight 'anonymous'.

Mitch said...

I did a numerical simulation of his jump a couple of years ago when his intentions were announced. At the time, they mentioned that he had been practicing in a skydiving chamber (one of those facilities where they blast air upwards at you fast enough so that you basically hover), and that his terminal velocity in this situation was 130 MPH. I took a guess at his weight (100kg/220lb), assuming he was an average sized fellow wearing a 50-pound spacesuit. That was enough to estimate an "equivalent drag area" for the purposes of my numerical simulation.

The sim was an Excel spreadsheet. One page in my spreadsheet held a model of the atmosphere, providing pressures, densities and temperatures at 100-foot increments for altitudes up to 500K feet (you can construct this model by starting at sea level with STP conditions and incrementing your way upward using the ideal gas law and known atmospheric temperature profiles). Each line in the freefall sim increments from the previous line by 0.5 seconds. The sim starts at an altitude of 128K feet, and a velocity of zero. At each time step, for his current altitude, you grab local air density/pressure/temperature from the atmosphere model, calculate drag force as a function of velocity, calculate acceleration as a function of drag+gravity, and then calculate the increment in velocity (acc * timestep) to be added to the velocity on the next line. Along the way I was able to calculate the adiabatic temperature rise he experienced due to compressive heating of the air piling up around him. My plots are here:

and the good news (for both of us) is that they look a lot like yours! I overshot on his peak velocity by a few percent, and also underestimated his total freefall duration by a bit. I was watching during his ascent, and the local pressures they were reporting from 80K feet and higher were about a 1/4-psi higher than in my atmosphere model. Not sure whether my model was inaccurate, or if this was just local barometric variations, but it might explain why he fell a bit more slowly than my model predicts.

The Engineer said...

That is an impressive analysis that you have done Mitch. Out of curiosity, are you an engineering student? Either way, kudos on a job well done.

As a previous commenter mentioned, one reason why one might overshoot his speed through such a theoretical analysis would be if you took the coefficient of drag to be constant for all velocities (it goes up a fair bit beyond Mach 0.8).

I was tempted to do such an an analysis, but did not, in part because the drag coefficient data was not available to me.

Mitch said...

>are you an engineering student?

Was, many moons ago. I got my Ph.D. in mech engineering back in '98, and sometimes find it entertaining to run the numbers for every-day phenomenon like this. For the past several days I've been searching the web, trying to find real data from his jump to assess the accuracy of my simulation. Your is one of a handful of pages I found. The other two are here:

Not too surprisingly, we're all in the same ballpark. :-)

I did have a course in compressible flow phenomenon in grad school, but we didn't really discuss transonic drag behavior at all; I just now cracked open my (dusty) textbook, and they didn't mention it in there, either, which strikes me as a bit odd. It is a real thing - Mach ~1 was the only time (other than takeoff) when the Concorde used its afterburners - but I'll have to scour the web to find info about it.

BTW, the condensation clouds your referred to in your initial post are called Prandtl-Glauert singularities:

They tend to happen in warm, humid air; under these conditions the air is already close to the dewpoint, and there's enough moisture present that the resulting condensation cloud will be dense/opaque enough to see. Not sure what the relative humidity is at 88,000 feet, but even if it's close to 100%, the absolute moisture content (grams per cubic meter) of such cold air is awfully low; even if you were able to condense all of it, the resulting cloud around Baumgartner might have been too thin to be visible.

Incidentally, the web has been having a lot of fun with his jump:

I was particularly amused by this pic, where someone pasted a PG singularity around Baumgartner's waist:

Anonymous said...

Hi! I did numerical solution of this fall taking into account air density change (ISA model) with altitude and g change at every altitude. I've got max speed ~390 m/s for effective cross-section of his body of about 0.7m, drag coeff about Cd=0.8. yes small variations of these (size and Cd) gives a bit different results.

solid line on the graph is speed of sound vs altitude.


The Engineer said...

Nice work Alex. It is nice to overlay a plot of terminal velocity. Due to inertia, Felix's speed overshoots this curve by a bit, and then gradually approaches it.

Anonymous said...

Yes agree, added the line there

Anonymous said...

Thank you for this great work.
Is the video still available ? The link is no longer working or maybe I am unable to watch it because of my location (France).

The Engineer said...

Hello anonymous,

I have updated the link on the article... It now shows the sponsor's 90 second summary clip. Glad you enjoyed the post.

enzuber said...

Thanks for a great post. Would you be able to post a link to the data points you used to generate the graphs? I would like to give them to my students so they can work out the motion characteristics themselves. (V hard to find the raw data on the descent and I was hoping since you have already collected it from the video ... :-)


The Engineer said...

Hi enzuber,

As mentioned in the article, some of these data points were "fudged" by me via educated guesses based on kinematic relations of displacement and velocity. I had a few data points for velocity, and a couple for displacement, and then did some interpolation and imposed kinematic constraints. All this to say, I would not want the data points I used to be assumed as "true values".

If I were to use these graphs in an exercise, I would focus on a few aspects of the graph...(1) Initial slope of velocity graph (approx -9.8 m/s/s), (2) max speed corresponds to max slope of altitude. Since the values are approximate, they may as well be read approximately off of the graph. I feel strange publishing unofficial data points.

Good luck, I hope it goes well.

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