Abstract
This paper presents an analytical study of the longitudinal dynamics of a thrusting, lifting, orbital vehicle in a nearly circular orbit. The translational motion is composed of a nonlinear oscillation, or phugoid, and a spiral mode which results in either decay or dilatation of the orbit depending on the perturbed initial conditions. The nonlinear effects on the phugoid period and damping are small in the altitude range considered. Elements of the orbit such as radial distance, velocity, and flight path angle were obtained explicitly as functions of time. The behavior of the variations of these elements is correctly predicted. Explicit expressions for period and damping of the angle-of-attack mode were derived. It is shown that a critical altitude may exist at which the phugoid mode and the angle-of-attack mode have nearly equal periods. Near this resonance altitude linearized solutions are no longer valid and a study of the nonlinear equations shows that there is a strong interactions between the translational and the rotational modes resulting in a switching of the two frequencies of oscillations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beletskii, V. V.: 1966,Motion of an Artificial Satellite About Its Center of Mass, NASA Translation, NASA TTF-429.
Bellman, R.: 1964,Perturbation Techniques in Mathematics, Physics, and Engineering, Holt, Rinehart and Winston, Inc., New York, pp. 5–7.
Bogoliubov, N. N. and Mitropolsky, Y. A.: 1961,Asymptotic Methods in the Theory of Non-Linear Oscillations, Hindustan Publishing Corp., India.
Lancaster, O.E. (ed.): 1961,High Speed Problems of Aircraft and Experimental Methods, Vol. VIII in the series ‘High Speed Aerodynamics and Jet Propulsion’, Princeton University Press, Princeton, New Jersey, pp. 149–152.
Etkin, B.: 1960, ‘Longitudinal Dynamics of a Lifting Vehicle in a Circular Orbit’, UTIA Report No. 65, AFOSR TN-60-191.
Etkin, B.: 1961, ‘Longitudinal Dynamics of a Lifting Vehicle in Orbital Flight’,J. Aerospace Sci. 28, 779–88.
Laitone, E. V. and Chou, Y. S.: 1965, ‘Phugoid Oscillations at Hypersonic Speeds’,AIAA J. 3, 732–5.
Morth, R. and Speyer, J. L.: 1961, ‘Divergence from Equilibrium Glide Path at Supersatellite Velocities’,Am. Rocket Soc. 31, 448–50.
Norman, W. S. and Meier, T. C.: 1963, ‘Approximate Longitudinal Dynamics of a Lifting Orbital Vehicle’,AIAA Journal 1, 1662–3.
Porter, R. F.: 1961, ‘The Linearized Long-Period Longitudinal Modes of Aerospace Vehicles in Equilibrium Flight’, Air Force Flight Test Center, AFFTC-TN-61-2.
Rangi, R. S.: 1960, ‘Non-Linear Effects in the Longitudinal Dynamics of a Lifting Vehicle in Orbital Flight’, UTIA TN-40.
Roberson, R. E.: 1958, ‘Gravitational Torque on a Satellite Vehicle’,J. Franklin Inst. 265, 13–22.
U.S. Standard Atmosphere: 1962, U.S. Government Printing Office, Washington, D.C.
U.S. Standard Atmosphere Supplements: 1966, U.S. Government Printing Office, Washington, D.C.
Author information
Authors and Affiliations
Additional information
This work was supported by NASA Contract No. NASr 54(06).
Rights and permissions
About this article
Cite this article
Vinh, N.X., Dobrzelecki, A.J. Nonlinear longitudinal dynamics of an orbital lifting vehicle. Celestial Mechanics 3, 427–460 (1971). https://doi.org/10.1007/BF01227791
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01227791