Author Topic: Launch Vehicles Trajectories Q&A  (Read 3943 times)

Offline pagheca

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Launch Vehicles Trajectories Q&A
« on: 06/08/2014 09:55 pm »
I would like to discuss here the trajectories of Launch Vehicles. In particular first stage ones.

As a start, let's assume that :

(1) thrust is constant all the way up
(2) trajectory maximizes payload mass
(3) atmospheric density is modelled as an exponential function (no atmosphere as a zero order approximation)
(4) Cx remains constant in sub, trans and supersonic velocity regimes
(5) no second order effects like aerodynamic lift, torsional forces, etc. etc.
(6) ADDED: payload is inserted into circular orbit (tangential flight for a flat earth approximation)

(a) given these constraints, is any analytical solution, even approximated, known to the  Lagrangian equations?

(b) in particular: for a SSTO LV in vacuum (imagine the LM lifting-off from the Moon) is an analytical solution known that maximize payload from lift-off to orbit?

Please note I know almost nothing about this but have a decent math background.
Even a reference to a good book would be fine.

Thanks in advance for any hint.
« Last Edit: 06/08/2014 10:39 pm by pagheca »

Offline Proponent

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Re: Launch Vehicles Trajectories Q&A
« Reply #1 on: 06/08/2014 10:30 pm »
In a vacuum and in a uniform gravitational field, it can be shown that a linear-tangent steering law is optimal -- see the attached paper for a calculus-of-variations derivation.  That doesn't give the optimal trajectory, though, just the optimal direction of the thrust vector as a function of time.

The assumptions underpinning the linear-tangent law are  too restrictive to apply to the entire trajectory of a launch vehicle from surface to orbit, but they're not bad for, say, the upper stage of a two-stage launch vehicle.

Offline pagheca

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Re: Launch Vehicles Trajectories Q&A
« Reply #2 on: 06/08/2014 11:35 pm »
In a vacuum and in a uniform gravitational field, it can be shown that a linear-tangent steering law is optimal

Many thanks. "linear-tangential" is a very good start. However, this is ok for the exo-atmospheric portion of ascent. Here I'm looking for a way to parametrize the optimal launch vehicle trajectory in "reasonable" yet simplified atmospheric models, spherical Earth conditions.

I understood that usually the two portions of the flight are set almost independently: a zero-lift trajectory after lift-off followed, after steering, by the linear-tangential trajectory as soon as the rocket is out of the dense atmosphere.

I was wondering if there is an analytical function, even as a series, that may describe and optimize both those sections of the flight.
« Last Edit: 06/09/2014 12:06 pm by pagheca »

Offline modemeagle

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Re: Launch Vehicles Trajectories Q&A
« Reply #3 on: 06/15/2014 10:06 pm »
What follows may not be realistic in the real world, but is used in my simulations.

My simulations runs all the way to orbit in 1/10 sec increments.  I wrote a somewhat complicated guidance system to reach orbit which is simplified below.

S-I pitches over to a preset pitch at 300 meters off the launch pad.  This pitch is the only manual input to the guidance and you use the graphs and final orbital parameters to adjust up or down for a more optimized trajectory.  This value can range from 89.99 to a low of ~84.00 depending on the rocket being simulated.

After pitch over the vehicle uses a zero AOA gravity turn until S-I MECO.  I have the vehicle hold it's pitch from 3 seconds before MECO to 3 seconds after S-II ignition to simulate a clean separation. 

The guidance on S-II is a complicated formula which uses "real-time" vehicle states and nominal orbital insertion parameters to calculate the pitch needed for the vertical velocity required.  It will also adjust pitch based on thrust, so it will also reach nominal MECO parameters even with throttle down at the end of the burn.

If the apogee is higher than the perigee then the guidance uses a zero pitch program until MECO.

I researched for years about different guidance systems and this was the best I could come up with for my simulations.  My research showed that most vehicles use low AOA during S-I atmospheric translation and then a closed loop during SII for accurate insertion.


Mods:  If this does not appear to add to the discussion please remove.

Offline Remes

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Re: Launch Vehicles Trajectories Q&A
« Reply #4 on: 06/16/2014 12:34 am »
I would like to discuss here the trajectories of Launch Vehicles. In particular first stage ones.

As a start, let's assume that :

(1) thrust is constant all the way up
(2) trajectory maximizes payload mass
(3) atmospheric density is modelled as an exponential function (no atmosphere as a zero order approximation)
(4) Cx remains constant in sub, trans and supersonic velocity regimes
(5) no second order effects like aerodynamic lift, torsional forces, etc. etc.
(6) ADDED: payload is inserted into circular orbit (tangential flight for a flat earth approximation)

(a) given these constraints, is any analytical solution, even approximated, known to the  Lagrangian equations?

(b) in particular: for a SSTO LV in vacuum (imagine the LM lifting-off from the Moon) is an analytical solution known that maximize payload from lift-off to orbit?

Please note I know almost nothing about this but have a decent math background.
Even a reference to a good book would be fine.

Thanks in advance for any hint.

I was searching for "Oberth’s Synergy Curve" and found this.

http://www.princeton.edu/~stengel/MAE342Lecture11.pdf

It's quite short, but gives some insight. Oberth wrote a book in 1923, where he explained it. IIRC he had quite some aspects in it solved.

I'm not quite sure about the circumstances you described and if there is an analytical solution, but in reality there are so many constraints, that an analytical solution most likely is not possible with todays math. You said no atmosphere, in reality there are of course Max-Q considerations, which are hard to describe and solve analytically. Furthermore most bigger rockets don't have a closed loop guidance as long as the rocket is heavy (which I understand as: the loops are closed as far as it is necessary to keep a desired direction and damp any oscillations or drift in yaw/pitch/roll, but deviations from the absolute trajectory are ignored until the rocket has burned of enough fuel and/or passed Max-Q).




Offline Remes

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Re: Launch Vehicles Trajectories Q&A
« Reply #5 on: 06/16/2014 12:37 am »
I researched for years about different guidance systems and this was the best I could come up with for my simulations.  My research showed that most vehicles use low AOA during S-I atmospheric translation and then a closed loop during SII for accurate insertion.

Did you ever compare it to real numbers?

http://www.ehartwell.com/afj/Reports/Apollo_17/Apollo-Saturn_V_Postflight_Trajectory_AS-512

(at the end, some csv files in different coordinate systems)

Offline pagheca

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Re: Launch Vehicles Trajectories Q&A
« Reply #6 on: 06/16/2014 01:02 am »
Thanks for this outburst of answers, guys...

I realized later that an analytical solution with the conditions I set would be worthless (and nonsense). In particular, conditions (1) and (4) are so factitious that any solution found - even if they exist (as an approximated series result of the integral) would be of zero value.

Thanks for the many links, ppt etc. Still studying them. Lots of material and interesting information. Question time later...
« Last Edit: 06/16/2014 01:09 am by pagheca »

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