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Abstract
Following the recently revised presentation of the theory of small-perturbation flight dynamics by the authors, the present paper considers the newly corrected aircraft lateral-directional equations with trim angle of attack as a parameter. The dynamics is modeled by a set of second-order equations written in matrix form which makes it convenient to derive conditions for onset of instability. The static instability criterion is directly obtained and is seen to match entirely the condition for spiral mode instability. An approximate criterion for dynamic instability is obtained by considering the undamped lateral-directional dynamics, which happens to be a Hamiltonian system. The onset of dynamic instability corresponds to the occurrence of a Hamiltonian Hopf bifurcation which can be explicitly evaluated. This yields a new criterion for dynamic instability which is simpler and easier to use than the Routh discriminant. Also, the new criterion has the Cnβ, dyn parameter as a leading term which explains why this parameter may be used as an indicator for wing rock onset.
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References
- Ananthkrishnan, N. and Sinha, N.K., "A Simple, Correct Pedagogical Presentation of Airplane Longitudinal Dynamics", Journal of Aerospace Sciences and Technologies, accepted for publication.
- Ananthkrishnan, N. and Sinha, N.K., "A Simple, Correct Pedagogical Presentation of Airplane Lateral-Directional Dynamics", Journal of Aerospace Sciences and Technologies, Vol.65, No.3, August, 2013, pp.235-245.
- Ananthkrishnan, N., "Small-Perturbation Analysis of Airplane Flight Dynamics - A Reappraisal I: Longitudinal Modes", AIAA Atmospheric Flight Mechanics Conference, Providence RI, 16-19 August, 2004, pp.643-659.
- Raghavan, B. and Ananthkrishnan, N., "Small-Perturbation Analysis of Airplane Dynamics with Dynamic Stability Derivatives Redefined", Journal of Aerospace Sciences and Technologies, Vol.61, No.3, 2009, pp.365-380.
- Ananthkrishnan, N. and Unnikrishnan, S., "Literal Approximations to Aircraft Dynamic Modes", Journal of Guidance, Control, and Dynamics, Vol.24, No.6, 2001, pp.1196-1203.
- Ananthkrishnan, N. and Ramadevi, P., "Consistent Approximations to Aircraft Longitudinal Modes", Journal of Guidance, Control, and Dynamics, Vol.25, No.4, 2002, pp.820-824.
- Strogatz, S.H., "Nonlinear Dynamics and Chaos", Addison-Wesley, Reading, MA, 1994.
- Alligood, K.T., Sauer, T.D. and Yorke, J.A., "Chaos: An Introduction to Dynamical Systems", Springer-Verlag, New York, 1997.
- Sinha, N. K. and Ananthkrishnan, N., "Elementary Flight Dynamics with an Introduction to Bifurcation and Continuation Methods", CRC Press, Boca Raton FL, 2013 (in press).
- Abzug, M.J. and Larrabee, E.E., "Airplane Stability and Control: A History of the Technologies that Made Aviation Possible" Second Edition, Cambridge University Press, Cambridge, UK, 2002.
- Ananthkrishnan, N. and Sinha, N. K., "Level Flight Trim and Stability Analysis Using Extended Bifurcation and Continuation Procedure", Journal of Guidance, Control, and Dynamics, Vol.24, No.6, 2001, pp.1225-1228.
- Paranjape, A.A., Sinha, N.K. and Ananthkrishnan, N., "Use of Bifurcation and Continuation Methods for Aircraft Trim and Stability Analysis - A State-of-the-Art", Journal of Aerospace Sciences and Technologies, Vol.60, No.2, 2008, pp.85-100.
- Paranjape, A.A. and Ananthkrishnan, N., "Analytical Criterion for Aircraft Spin Susceptibility", Journal of Aircraft, Vol.47, No.5, 2010, pp.1804-1807.
- Chambers, J.R. and Anglin, E.L., "Analysis of Lateral-Directional Stability Characteristics of a Twin-jet Fighter Airplane at High Angles of Attack", NASA TN D-5361, August, 1969.
- Calico, R.A. Jr., "A New Look at Cnβ dyn ," Journal of Aircraft, Vol.16, No.12, 1979, pp.893-896.
- Lutze, F.H., Durham, W.C. and Mason, W.H., "Unified Development of Lateral-Directional Departure Criteria", Journal of Guidance, Control, and Dynamics, Vol.19, No.2, 1996, pp.489-493.
- Ananthkrishnan, N., Shah, P. and Unnikrishnan, S., "Approximate Analytical Criterion for Aircraft Wing Rock Onset", Journal of Guidance, Control, and Dynamics, Vol.27, No.2, 2004, pp.304-307.
- Duncan, W. J., "Control and Stability of Aircraft", Cambridge Univ. Press, Cambridge, England, U.K, 1952, pp.117-125.
- Van der Meer, J. C., "Hamiltonian Hopf Bifurcation", Lecture Notes in Mathematics, Vol.1160, Springer-Verlag, Berlin, 1985, p.15.
- Nguyen, L.T., Ogburn, M.E., Gilbert, W.P., Kibler, K.S., Brown, P.W. and Deal, P.L., "Simulator Study of Stall/Post-Stall Characteristics of a Fighter Air-plane With Relaxed Longitudinal Static Stability", NASA TP-1538, December, 1979.
References
Ananthkrishnan, N. and Sinha, N.K., "A Simple, Correct Pedagogical Presentation of Airplane Longitudinal Dynamics", Journal of Aerospace Sciences and Technologies, accepted for publication.
Ananthkrishnan, N. and Sinha, N.K., "A Simple, Correct Pedagogical Presentation of Airplane Lateral-Directional Dynamics", Journal of Aerospace Sciences and Technologies, Vol.65, No.3, August, 2013, pp.235-245.
Ananthkrishnan, N., "Small-Perturbation Analysis of Airplane Flight Dynamics - A Reappraisal I: Longitudinal Modes", AIAA Atmospheric Flight Mechanics Conference, Providence RI, 16-19 August, 2004, pp.643-659.
Raghavan, B. and Ananthkrishnan, N., "Small-Perturbation Analysis of Airplane Dynamics with Dynamic Stability Derivatives Redefined", Journal of Aerospace Sciences and Technologies, Vol.61, No.3, 2009, pp.365-380.
Ananthkrishnan, N. and Unnikrishnan, S., "Literal Approximations to Aircraft Dynamic Modes", Journal of Guidance, Control, and Dynamics, Vol.24, No.6, 2001, pp.1196-1203.
Ananthkrishnan, N. and Ramadevi, P., "Consistent Approximations to Aircraft Longitudinal Modes", Journal of Guidance, Control, and Dynamics, Vol.25, No.4, 2002, pp.820-824.
Strogatz, S.H., "Nonlinear Dynamics and Chaos", Addison-Wesley, Reading, MA, 1994.
Alligood, K.T., Sauer, T.D. and Yorke, J.A., "Chaos: An Introduction to Dynamical Systems", Springer-Verlag, New York, 1997.
Sinha, N. K. and Ananthkrishnan, N., "Elementary Flight Dynamics with an Introduction to Bifurcation and Continuation Methods", CRC Press, Boca Raton FL, 2013 (in press).
Abzug, M.J. and Larrabee, E.E., "Airplane Stability and Control: A History of the Technologies that Made Aviation Possible" Second Edition, Cambridge University Press, Cambridge, UK, 2002.
Ananthkrishnan, N. and Sinha, N. K., "Level Flight Trim and Stability Analysis Using Extended Bifurcation and Continuation Procedure", Journal of Guidance, Control, and Dynamics, Vol.24, No.6, 2001, pp.1225-1228.
Paranjape, A.A., Sinha, N.K. and Ananthkrishnan, N., "Use of Bifurcation and Continuation Methods for Aircraft Trim and Stability Analysis - A State-of-the-Art", Journal of Aerospace Sciences and Technologies, Vol.60, No.2, 2008, pp.85-100.
Paranjape, A.A. and Ananthkrishnan, N., "Analytical Criterion for Aircraft Spin Susceptibility", Journal of Aircraft, Vol.47, No.5, 2010, pp.1804-1807.
Chambers, J.R. and Anglin, E.L., "Analysis of Lateral-Directional Stability Characteristics of a Twin-jet Fighter Airplane at High Angles of Attack", NASA TN D-5361, August, 1969.
Calico, R.A. Jr., "A New Look at Cnβ dyn ," Journal of Aircraft, Vol.16, No.12, 1979, pp.893-896.
Lutze, F.H., Durham, W.C. and Mason, W.H., "Unified Development of Lateral-Directional Departure Criteria", Journal of Guidance, Control, and Dynamics, Vol.19, No.2, 1996, pp.489-493.
Ananthkrishnan, N., Shah, P. and Unnikrishnan, S., "Approximate Analytical Criterion for Aircraft Wing Rock Onset", Journal of Guidance, Control, and Dynamics, Vol.27, No.2, 2004, pp.304-307.
Duncan, W. J., "Control and Stability of Aircraft", Cambridge Univ. Press, Cambridge, England, U.K, 1952, pp.117-125.
Van der Meer, J. C., "Hamiltonian Hopf Bifurcation", Lecture Notes in Mathematics, Vol.1160, Springer-Verlag, Berlin, 1985, p.15.
Nguyen, L.T., Ogburn, M.E., Gilbert, W.P., Kibler, K.S., Brown, P.W. and Deal, P.L., "Simulator Study of Stall/Post-Stall Characteristics of a Fighter Air-plane With Relaxed Longitudinal Static Stability", NASA TP-1538, December, 1979.