Goddard's 85 Years Optimal Ascent Problem Finally Solved
- Paper number
IAC-06-E4.3.05
- Author
Dr. Radu Rugescu, Politechnic University of Bucharest, Romania
- Year
2006
- Abstract
It was in 1919 when Robert H. Goddard had first published his observations on possible optimal atmospheric ascent of rockets towards high altitudes (R.H. Goddard, A Method of Reaching Extreme Altitudes, Miscellaneous Collections, Smithsonian Inst., 1919), without giving any clue. Four years later only, Hermann Oberth is publishing a discussion on possible optimal paths in the atmospheric ascent (H. Oberth, Wege zur Raumschiffahrt, Berlin, 1923) and gives in 1929 the first, partial solution for this problem (H. Oberth, Die Rakete zu den Planeten-Raumen, Von Oldenburg Verlag, Berlin, 1929). The well-known professor in mathematics Georg Hamel is formulating in the meantime (G. Hamel, Eine mit dem Rakete zusamenhängende Aufgabe der Variationsrechnung, ZFW December 1927) the Goddard problem in strong variational terms that will state the basis for all eventual developments on the subject, but still with no explicit solution. Despite a number of earlier trials, Tsien and Evans are the first to find a partial, variational solution of the Goddard problem in 1951, in a very beautiful work. A flow of developments in atmospheric ascent optimization is then emerging with hundreds of high quality papers, including the most famous ones by Gudju, Leitmann, Miele and many others. By continuous variational methods the solution is found that consists of an impulsive start, followed by an accommodated sustainer flight and the injection to coast up to pick altitude, but in all these solutions the basic discontinuity in the equations of motion at burn-off is fortuitously neglected. Salutary are the impressive solutions of Bulirsch team in Munich, which emphasize on the challenge of discontinuities and give some answers. Despite these considerable attempts, pure numerical solutions only are currently adopted to optimize, in part, the vehicle ascent, as the classical calculus of variations proves inadequate for this discontinuous integrand problems. In fact, the entire amount of variational work does not respond to the real problem of rocket ascent and we badly realize that up to recent time no complete and documented answer to the 85-years old Goddard problem seemed found. We try to follow in our paper the history of those tremendous findings, which had found very recently only an unexpected answer.
- Abstract document
- Manuscript document
IAC-06-E4.3.05.pdf (🔒 authorized access only).
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