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SpaceX’s Third Attempt

The first stage worked flawlessly but the staging malfunctioned. I wasn’t watching, but went to bed at 4 am local time after waiting through some launch delays, thinking that it’d take forever anyway, and of course the launch was right after.

What’s always been a mystery to me, if SpaceX is selling their Falcon 1 rocket at just about eight million dollars, why don’t they do a lot more test flights with dummy payloads? It seems that it would accelerate development a lot, since the hundreds of workers can’t be cheap to just keep working – a delay of a few months in selling some rocket flights will surely be costly.

During the webcast run-up to the launch, they showed Elon Musk doing a tour of their factory, and everything in their production processes seemed well automated and thought out, since they have to do so many engines for Falcon 9 anyway. Just to mention the totally automated copper milling for the chamber and nozzle root liner, as well as the automatic pipe bending machine for the nozzle. 80% of the hardware of Falcon 1 is produced in house from bare metal.

On the other hand, the Merlin 1 main engine has gone through many changes, with power increases (don’t remember if the turbomachinery or injectors have changed) and the regenerative nozzle at least. That means early flight testing could not have been very representative of the design or the build or integration processes. Now they seem further along in that, with the Falcon 9 first stage recently having done a full-up hold down firing of its nine engines.

Design vs Test

There’s something fundamental about the whole issue of designing vs testing. It’s not a totally simple picture, with the current advanced computation and simulation capabilities making the boundary fuzzy. And there has always been partial hardware non-destructive testing too, like structural test models. So, can expensive destructive test flights be seen as just an extension of finding a workable design, as well as production, integration and operation processes? In that sense they all can be pooled into one, as just means of getting to some combination of capability, cost and time goals.

Even if there are no rigid mental boundaries between development and testing, one still has to make more careful judgements before doing a very expensive destructive flight test vs running a few minute configuration simulation. Of course you have to be careful with time allocation in design too, conceptual design is one tool for that, to avoid spending huge amounts of time and thus money for elaborate dead-end designs and configurations. The previous post is about that, where NASA spent a lot of design time for launchers and components that ended up too small anyway. But actually they started from very far and little assumptions in ESAS, eliminating lots of fundamental concepts in a tree analysis, pictured below.

ESAS conceptual launch vehicle design

Conceptual design is from the top down, but real hardware testing is from the bottom up. Both are needed to work in a real capability. Armadillo Aerospace’s John Carmack has mentioned innumerable times how building working hardware always discards too ambitious and overcomplicated designs (I have to shamefully admit, I have very little experience in designing built hardware, though I am in the process of changing that). If you only do conceptual design without basing anything in real hardware, you are moving on very thin ice. The NASP program was quite a good example of that.

Armadillo Aerospace's Module in tethered hover testing

In a sense, Falcon 1 is the hardware and process test platform for the real rocket, Falcon 9. Now that SpaceX seems to have their production line ready, I hope they would just do Falcon 1 test launches in rapid succession to iron out their bugs. (Of course, first do a lot of nondestructive ground testing, like for the pyrobolts this time.) Hopefully without charging money from the payload customers.

Again, when looking back to Wernher von Braun, after V-2 he proposed in USA the development of some new rockets and humongous amounts of test launches were included in the plan. Hardware got more reliable and rockets bigger and more complicated and thus expensive, which eliminated this approach, but it is an interesting historical mindset and viewpoint.

And again, too, of course, the fundamental property of expendable rockets that every flight ends in destruction, prevents economic partial testing. You can’t do careful envelope expansion or survive flight anomalies. Reusable flight vehicles on the other hand allow a lot of flight testing.

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Powered landing, practical considerations (1-dim)

There was some excellent commentary in the last post, and that has partly been one inspiration in writing this post.

In the last part, the absolute minimum efforts by the launch vehicle making a vertical landing were determined. On top of these, there are lots of slightly more practical ones. This is still a one-dimensional vertical analysis.

Effects of a Time Delay

Let us see what a delay of t_delay before starting the main deceleration would do to a landing vehicle. The craft would travel $t_{delay} v_{ter}=y_m$ extra distance before starting deceleration. That would mean that the actual reaching of zero speed would be y_m behind the original deceleration target. If we assume the original target was on ground level, then the new target would be underground. The difference between the targets is $y_m=0.5 a t_m^2$. (We used the trick of counting backwards in time from the underground target.) We can solve the imaginary underground time duration, $t_m=sqrt(2 y_m/a)$. And the speed at ground level thus is $v_m = t_m a = sqrt(2 y_m a) = sqrt (2 a v_{ter} t_{delay})$

Example Vehicle

Our example vehicle, with v_ter=100 m/s and a=20 m/s^2 would stop in 5 seconds and 250 meters normally. If it traveled without any margins and would miss the engine ignition by one second (t_d=1 s) and 100 meters (y_m=100 m), it would impact the ground at over 60 m/s (130 mph, 220 km/h), or roughly two thirds of its original terminal velocity of 100 m/s. That would destroy it completely.

If we used a gentler and more lossy 10 m/s^2 deceleration (and started the engines higher accordingly, but still had the unfortunate one second delay), then the impact speed would be about 40 m/s. Still far too much.

Even a realistic 0.1 s of delay would cause a 20 to 14 m/s impact, depending on planned acceleration. Complete crash, remotely possibly survivable.

The results might look unintuitive. That’s because in a constant deceleration, the most time is spent on the end, low speed distances, while most distance is covered with high speeds, early on. So if the deceleration is cut short even by a short percentage in distance, the time is cut short by a much bigger percentage, so the vehicle does not have enogugh time to brake much. Running out of a little distance you run out of a lot of time. This in total makes the vertical landing a very time sensitive problem!

It is also the reason why in the classical driving school example, even if you lock-brake and the car overshoots the pedestrian crossing by just a few meters, it was still going quite fast when you crossed it. The car’s speed doesn’t change much per meter when you start braking, but in the end half it changes a lot because it’s going slower and there are many more seconds per meter and thus the force has more time to effect. The drag is not dependent on velocity so in a sense it functions like a rocket engine. Accelerating with the car’s engine is the opposite as it generates constant power, not constant acceleration.

So, if we anyway design the vehicle to survive some vertical touchdown speed v_touch , then we can calculate the margin for the fast deceleration directly to pad in seconds. $t_d = v_{touch} / sqrt (2 a v_{ter})$. With our example vehicle it could be 4 m/s, equal to a drop from 0.8 meters height. Then the acceptable time delay would be 0.063 s or 63 milliseconds. That’s probably too small of a margin.

More Realistic Approaches

So, in reality, one can not have a direct high deceleration to pad. One could have multiple other approaches and ways to think about it. These are the most obvious to me:

1. Put the the deceleration aim target (hover) some distance above the ground. Layne Cook told that was an approach the DC-X used.
2. Stop the high deceleration at some determined velocity and switch to a low deceleration. For example at 10 m/s.
3. Use some more elaborate forecast of the vehicle position and throttle accordingly to always have a plan to reach very slow speed at touchdown. I think Armadillo is using something like this approach.

These are handled below.

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Powered landing, analytical solution

There is a long standing dispute whether winged landing or vertical rocket powered landing is more efficient and overall better for reusable rockets on earth. 🙂
Powered landing can be looked at in a broad overview. The post documents what I doodled in my notebook some day in the summer.

Summary

Deceleration from terminal velocity to hover usually requires about 5-10% of total landed mass as propellants. The amount is directly proportional to the vehicle’s terminal velocity. Also, the higher acceleration, the better. The powered landing penalty fraction is $N_{penalty}=\frac{2g}{3a}$, so landing at 2 gee acceleration (3 gee felt) gives a penalty of 0.3. The required delta vee is $\Delta v = v_{ter} (1+N_{penalty} )$ where v_ter is terminal velocity. And the required impulse $I=m \Delta v$ and propellant mass $m_{prop}=I/v_{ex}$.

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