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On the ‘inverse phugoid problem’ as an instance of non-linear stability in pitch

Published online by Cambridge University Press:  04 July 2016

L. M. B. C. Campos  and
A. J. N. M. Aguiar
L. M. B. C. Campos
Affiliation:
Instituto Superior Técnico, 1096 Lisboa Codex, Portugal
A. J. N. M. Aguiar
Affiliation:
Instituto Superior Técnico, 1096 Lisboa Codex, Portugal
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Summary

The motion of an aircraft along a straight flight path inclined to the vertical is considered from an arbitrary initial velocity which may be far from steady flight speeds. The velocity and incidence, as a function of distance or time, provide a solution to the ‘inverse phugoid problem’ of finding the control laws that keep an aircraft on a straight flight path by exactly cancelling the phugoid instability (the short period mode is omitted by neglecting the rotational inertia of the aircraft). This is a non-linear problem, namely finding the motion of an aircraft from an initial condition far removed from a stable steady state, or finding the large amplitude deviations from an unstable equilibrium. The stability and instability problems are solved analytically using the following assumptions:

  1. (i) the aircraft flies subsonically so that only form and induced drag need be considered;

  2. (ii) the thrust consists of a constant term, minus a term specifying a dependence on velocity similar to that of the total drag; and

  3. (iii) the aircraft incidence is away from stall so that lift is a linear function of incidence.

The response curves show that:

  1. (i) flight at velocities below the lower steady flight speed, which is unstable, is difficult to maintain due to fast divergence towards the stall; and

  2. (ii) for an initial velocity above the lower steady flight speed, there is convergence towards the higher steady flight speed.

The convergence, in the latter case (ii), may be slow along a straight flight path, although it can be hastened by suitable manoeuvres.

Type
Research Article
Information
The Aeronautical Journal , Volume 93 , Issue 927 , September 1989 , pp. 241 - 253
Copyright
Copyright © Royal Aeronautical Society 1989 

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References

1. Lanchester, F. W. Aerodonetics, Constahle, London 1908.Google Scholar
2. Von Mises, R. Theory of flight, McGraw-Hill, 1945.Google Scholar
3. Milne-Thomson, L. M. Theoretical aerodynamics. MacMillan, 1966.Google Scholar
4. Lecomte, P. Mécanique du Vol, Dunod, 1962.Google Scholar
5. Perkins, C. D. and Hage, R. E. Airplane performance, stability and control, Wiley, 1950.Google Scholar
6. Duncan, W. J. Principles of control and stability of aircraft, Cambridge University Press, 1952.Google Scholar
7. Etkin, B. Dynamics of flight stability and control, Wiley, 1974.Google Scholar
8. Brockhaus, R. Flugregelung, Holdenbourg, 1977.Google Scholar
9. Whittaker, E. T. Analytical Dynamics, Cambridge University Press, 4th ed, 1937.Google Scholar
10. Forsyth, A. R. Treatise of differential equations, MacMillan, 6th ed, 1927.Google Scholar
11. Painlevé, P. Cours de Mécanique, Gauthier-Villars, 1930-1936.Google Scholar
12. Painlevé, P. Étude sur le régime normal d’un aeroplane, Tech Aéronaut, 1910, 1, 3–11.Google Scholar
13. Routh, E. J. Dynamics of a system of rigid bodies, MacMillan, 1905.Google Scholar
14. Bryan, G. H. Stability in aviation, Macmillan, 1911.Google Scholar
15. Fuller, A. T. Stability of motion, Taylor & Francis, 1975.Google Scholar
16. Babister, A. W. Aircraft dynamic stability and response, Pergamon Press, 1980.Google Scholar
17. Lighthill, M. J. An informal introduction to fluid mechanics, Oxford University Press, 1986.Google Scholar
18. Prandtl, L. and Tietjens, O. Hydro- und Aeromechanik, Springer, 1929-1944.Google Scholar
19. Campos, L. M. B. C. Análise aplicada e campos de torça Calouste Gulbenkian Foundation, 1989.Google Scholar
20. Houghton, E. L. and Brock, A. E. Aerodynamics for engineering students, Edward Arnold, 1960.Google Scholar
21. Neumark, S. Problems of Longitudinal Stability Below the Minimum Drag Speed, and the Theory of Stability Under Constraint, ARC Report 2983, 1953.Google Scholar
22. Pinsker, W. J. G. Glide-Path Stability of an Aircraft Under Speed Constraint. ARC Report 3705, 1971.Google Scholar
23. Copson, E. T. Functions of a complex variable, Oxford University Press, 1935.Google Scholar
24. Guicheteau, P. Application de la théorie des bifurcations à l’étude des Pertes de Controle sur avion de combat, AGARD Conference Proceedings, 1981, CP-319, paper 17 (and Rech Aerosp, 1982, p. 61–73).Google Scholar
25. Churchill, R. V. Operational Mathematics, McGraw-Hill, 3rd ed, 1972.Google Scholar
26. Lepage, W. R. Complex variables and the Laplace transform for engineers, McGraw-Hill, 1961.Google Scholar
27. Sneddon, I. N. The use of integral transforms, McGraw-Hill, 1972.Google Scholar
28. Courant, T. and Hilbert, D. Methods of mathematical physics, Interscience, 1953.Google Scholar
29. Campos, L. M. B. C. On the fundamental acoustic mode in variable area, low Mach number nozzles, Prog Aerosp Sci, 1985, 22, 1–27.Google Scholar
30. Jeffreys, H. Operational Methods, Cambridge University Press, 2nd Ed. 1931 (3rd edition in Methods of mathematical physics, Cambridge University Press, 3rd ed. 1956).Google Scholar
31. Campos, L. M. B. C. On a pitch control law for constant glide slope through windshears Aeronaut J, October 1989, 93, (928).Google Scholar
32. Campos, L. M. B. C. On the influence of atmospheric disturbances on aircraft aerodynamics, Aeronaut J, June-July 1984, 88, (876), 257–264.Google Scholar
33. Campos, L. M. B. C. On aircraft flight performance in a perturbed atmosphere, Aeronaut J, October 1986, 90, (898), 302–312.Google Scholar