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Model Forum / General / Rockets / January 2009LEO Rocket Trajectory Optimization
My name is Blake Chambers and I am new to this group. I am very interested in rocketry and am working on a Rocket Physics and Design project for my Independent Study class at my high school. I am attempting to use physics to describe the flight of a 10kg rocket with a payload of 1kg, assuming 9kg of fuel, from the ground to low earth orbit. The point of the project is to learn the physics required in large scale rocket flight by pursuing the problem. So far I have been able to calculate all of my variables using calculus or other physics equations, and I have imaginary, yet reasonable defined forces. The trajectory optimization to define my flight path is the only part of my flight I have not been able to define. I have read a couple articles online and in this group and have not been able to fin the answers I need. I understand that the Earth is rotating causing the rocket to move in an arc relative to its launch position and opposite to the motion of the Earth. However, is there an optimal ascension arc or trajectory which will minimize the aerodynamic effects and maximize my engine efficiency: and if so, how do I go about doing the calculations? Blake It seems that your project has more fantasy than reality. On Dec 14, 3:52 pm, blakejwchamb...@gmail.com wrote: > My name is Blake Chambers and I am new to this group. I am very > interested in rocketry and am working on a Rocket Physics and Design > project for my Independent Study class at my high school. > > I am attempting to use physics to describe the flight of a 10kg rocket > with a payload of 1kg, assuming 9kg of fuel, How do you build a real rocket with no other masses except for fuel and payload? > from the ground to low earth orbit. What kind of fuel actually has that high a specific impulse? The Independent Study class is a way for students to learn about something they enjoy outside of a classroom. In essence the project is a theoretical exercise. I probably should have worded it as a 1kg rocket with 9kg of fuel. I am using a 10kg rocket to satisfy the physics equations; although in real life, the actual values of specific impulse (somewhere around 1160) and other values are unrealistic. One piece of the flight process that I don't understand is the trajectory and its effects on aerodynamic forces. > The Independent Study class is a way for students to learn about > something they enjoy outside of a classroom. In essence the project [quoted text clipped - 8 lines] > One piece of the flight process that I don't understand is the > trajectory and its effects on aerodynamic forces. Well, the aerodynamic forces affect the trajectory more than the other way round. The forces are the total of all the atmospheric factors, including temperature, pressure, humidity, wind speed and direction, etc. As weather is non-deterimalistic and non-linear in location and time, as well as in effect, calculating a true optimum is impossible. You would need to assume an 'ideal' atmosphere - no wind, constant temerature, linear pressure, etc. Then too, you would need to assume the rocket will stay under mach 0.8 for the whole flight or ignore trans-sonic effects altogether. Models of the world are neat and regular only if you build them out of Lego. Perhaps you be a little more specific with what it is that you are trying to find? >The Independent Study class is a way for students to learn about >something they enjoy outside of a classroom. In essence the project [quoted text clipped - 5 lines] >real life, the actual values of specific impulse (somewhere around >1160) and other values are unrealistic. It would help if we knew your physics equtations that are satisfied with a 10kg rocket. It would also help to know which parameters you are free to make up, which are fixed, and which should be physicaly realistic. I doubt that you couold actualy get your 10kg rocket into orbit with an ISP of 1160, and even 1160 is optimistic by a factor of over two, for chemical propulsion. >One piece of the flight process that I don't understand is the >trajectory and its effects on aerodynamic forces. I'm assuming that that you are free to chose any trajectory. The aerodynamic forces along that trajectory are mainly a function of aatlitude and vehicle speed. Even with an imaginary propulsion system, the trajectory and aerodynamic forces are interdependant. Alan > However, is there an optimal ascension arc or trajectory which will > minimize the aerodynamic effects and maximize my engine efficiency: > and if so, how do I go about doing the calculations? Wikipedia has some good information to get started with -- the atmospheric pressure according to the barometric formula http://en.wikipedia.org/wiki/Barometric_formula and a page on drag http://en.wikipedia.org/wiki/Atmospheric_drag >My name is Blake Chambers and I am new to this group. I am very >interested in rocketry and am working on a Rocket Physics and Design [quoted text clipped - 19 lines] > >Blake No. You can't optimixe to maximize and or minimixe two or more different things. You need to develop a single objective functions that combines all the desireable objectives in a suitable way, and then optimixe the single function over over your design variables. For example, you could simply minimize fuel used to achieve a specified orbit with consraints on max q (aerocynamic forces) and peak heating rates. You can't really do this. Typicaly you need an education to about the masters degree level. What you might be able to do do is find a find a simulation program that does all the work for your, and it could even incorporate some simple optimizations for you. You might learn a lot just by putting in various inputs and seeing what happemns. You might want to just pick a simple steering law, such as bilinear or inverse tangent, and then just play around with thrust untill you get something that is acceptable. IF you are up for some research, you could find a good engineering library and look for papers on similar projects. YOu could look in AIAA and AAS jornals and confrence procedings. You could also survey real launch vehicles (Don't espect to find any real SSTO vehicles.) and learn from them. You might like "International Reference Guide to Sspace Launch Systems" by Sssteven J. Isakowitz, AIAA, 1991 edition. Small rockets are particularly difficult to get into orbit because because aerodymaics is more signivicant. Aerodynamic drag is mostly proportional to frontal area, while weight and mass are more proportional to volume. let's assume you keep rocket lenth-to-diameter fixed for structural reasons, say at 12. If you double the length, weight will increase by a factor of 8 while drag only increases by a factor of 4. As size increases, drag becomes less significant compared to other factors. A SSTO vehicle is going to be huge, and have marinal payload peformance compared to multi stage vehicles. Alan Thank you for your help Alan. I decided to use the Rocketsim program by Apogee. I made a few assumptions, but was able to find an equation to calculate my position with respect to all my variables. I also looked into the AIAA and found some information on other rocket flights. My project is working out very well. Thank you again for your help. Blake > Thank you for your help Alan. I decided to use the Rocketsim program > by Apogee. I made a few assumptions, but was able to find an equation [quoted text clipped - 4 lines] > > Blake To answer your original question for explicitly, one of the ways that optimal rocket trajectories are calculated are through variational calculus. There's plenty of papers out on rocket trajectory optimization, but you'll be hard-pressed to find on that isn't at the post-graduate level. Alan is correct in that you can't optimize fuel efficiency AND reduce aerodynamic loads to a minimum at the same time. Minimizing aerodynamic loads would mean your rocket is hovering and isn't moving (zero velocity = zero drag), which is at odds of maximizing fuel efficiency. You could set a maximum acceptable aerodynamic load and optimize fuel around that. It seems to me that the simplest model you could make would be launching your rocket from the equator (or assuming the earth isn't rotating), therefore you could make your model 2D. The variables you could modify (as a function of time) to achieve some trajectory would be thrust and pitch (i.e. elevation as an angle). Assuming a constant drag coefficient and a constant ISP (which in reality changes as a function of exit pressure at the nozzle) would also make your problem a lot easier to solve. Hope that helps, Dave |
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