Application of bifurcation methods to nonlinear flight dynamics problems
TL;DR: In this paper, applications of global stability and bifurcational analysis methods are presented for different nonlinear flight dynamics problems, such as roll-coupling, stall, spin, etc.
About: This article is published in Progress in Aerospace Sciences.The article was published on 1997-01-01. It has received 145 citations till now.
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TL;DR: In this article, the authors examined how these nonlinearities affect the ability to control the aircraft and how they may contribute to loss of control and how damage to control effectors impacts the capability to remain within an acceptable envelope and to maneuver within it.
Abstract: Loss of control is a major factor in fatal aircraft accidents. Although definitions of loss of control remain vague in analytical terms, it is generally associated with flight outside of the normal flight envelope, with nonlinear influences, and with a significantly diminished capability of the pilot to control the aircraft. Primary sources of nonlinearity are the intrinsic nonlinear dynamics of the aircraft and the state and control constraints within which the aircraft must operate. This paper examines how these nonlinearities affect the ability to control the aircraft and how they may contribute to loss of control. Specifically, the ability to regulate an aircraft around stall points is considered, as is the question of how damage to control effectors impacts the capability to remain within an acceptable envelope and to maneuver within it. It is shown that, even when a sufficient set of steady motions exist, the ability to regulate around them or transition between them can be difficult and nonintuitiv...
78 citations
Cites background from "Application of bifurcation methods ..."
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TL;DR: In this article, a systematic way of computing the set of all attainable steady states for a general class of helical trajectories is presented, and the proposed reconstruction of attainable equilibrium states and their local stability maps provides a comprehensive and consistent representation of the aircraft flight and maneuvering envelopes.
Abstract: An aircraft's performance and maneuvering capabilities in steady flight conditions are usually analyzed considering the steady states of the rigid-body equations of motion. A systematic way of computation of the set of all attainable steady states for a general class of helical trajectories is presented. The proposed reconstruction of attainable equilibrium states and their local stability maps provides a comprehensive and consistent representation of the aircraft flight and maneuvering envelopes. The numerical procedure is outlined and computational examples of attainable equilibrium sets in the form of two-dimensional cross sections of steady-state maneuver parameters are presented for three different aircraft models.
71 citations
TL;DR: In this paper, the problem of spin recovery of an aircraft was addressed as a nonlinear inverse dynamics problem of determining the control inputs that need to be applied to transfer the aircraft from a spin state to a level trim flight condition.
Abstract: The present paper addresses the problem of spin recovery of an aircraft as a nonlinear inverse dynamics problem of determining the control inputs that need to be applied to transfer the aircraft from a spin state to a level trim flight condition. A stable, oscillatory, flat, left spin state is first identified from a standard bifurcation analysis of the aircraft model considered, and this is chosen as the starting point for all recovery attempts. Three different symmetric, level-flight trim states, representative of high, moderate, and low-angle-of-attack trims for the chosen aircraft model, are computed by using an extended-bifurcation-analysis procedure. A standard form of the nonlinear dynamic inversion algorithm is implemented to recover the aircraft from the oscillatory spin state to each of the selected level trims. The required control inputs in each case, obtained by solving the inverse problem, are compared against each other and with the standard recovery procedure for a modern, low-aspect-ratio, fuselage heavy configuration. The spin recovery procedure is seen to be restricted because of limitations in control surface deflections and rates and because of loss of control effectiveness at high angles of attack. In particular, these restrictions adversely affect attempts at recovery directly from high-angle-of-attack oscillatory spins to low-angleof-attack trims using only aerodynamic controls. Further, two different control strategies are examined in an effort to overcome difficulties in spin recovery because of these restrictions. The first strategy uses an indirect, two-step recovery procedure in which the airplane is first recovered to a high- or moderate-angle-of-attack level-flight trim condition, followed by a second step where the airplane is then transitioned to the desired low-angle-of-attack trim. The second strategy involves the use of thrust-vectoring controls in addition to the standard aerodynamic control surfaces to directly recover the aircraft from high-angle-of-attack oscillatory spin to a low-angle-of-attack level-flight trim state. Our studies reveal that both strategies are successful, highlighting the importance of effective thrust management in conjunction with suitable use of all of the aerodynamic control surfaces for spin recovery strategies.
65 citations
TL;DR: Hollkamp et al. as mentioned in this paper proposed a self-tuning piezoelectric Vibration Absorber for passive vibration suppression in wing-like composite structures.
Abstract: 3Hollkamp, J. J., “Multimodal Passive Vibration Suppressionwith Piezoelectrics,” AIAA Paper 93-1683, 1993. 4Hollkamp, J. J., and Starchville, T. F., “A Self-Tuning Piezoelectric Vibration Absorber,” Journal of Intelligent Materials Systems and Structures, Vol. 5, No. 4, 1994, pp. 559–566. 5Rew, K.-H., Han, J. H., and Lee, I., “Adaptive Multimodal Vibration Control of Winglike Composite Structure Using Adaptive Positive Position Feedback,” AIAA Paper 2000-1422, 2000.
64 citations
15 Oct 1998-Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences
TL;DR: The nonlinear control law, which totally suppresses wing-rock motion, is derived, taking into account both local stability characteristics of aircraft equilibrium states and domains of attraction, along with the requirement that all other attractors be eliminated.
Abstract: The use of nonlinear dynamics theory for the analysis of aircraft motion and the assessment of aircraft control systems is well known. In this paper the continuation and bifurcation methods are app...
61 citations
References
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01 Aug 1983TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.
12,669 citations
01 Jan 2015
TL;DR: In this paper, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.
12,485 citations
Book•
17 Aug 1976TL;DR: The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value as mentioned in this paper.
Abstract: The goal of these notes is to give a reasonably complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including stability calculations Historically, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930 Hopf's basic paper [1] appeared in 1942 Although the term "Poincare-Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it Hopf's crucial contribution was the extension from two dimensions to higher dimensions
The principal technique employed in the body of the text is that of invariant manifolds The method of Ruelle-Takens [1] is followed, with details, examples and proofs added Several parts of the exposition in the main text come from papers of P Chernoff, J Dorroh, O Lanford and F Weissler to whom we are grateful
The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations; see for example, Hale [1,2] and Hartman [1] Of course, other methods are also available In an attempt to keep the picture balanced, we have included samples of alternative approaches Specifically, we have included a translation (by L Howard and N Kopell) of Hopf's original (and generally unavailable) paper These original methods, using power series and scaling are used in fluid mechanics by, amongst many others, Joseph and Sattinger [1]; two sections on these ideas from papers of Iooss [1-6] and Kirchgassner and Kielhoffer [1] (contributed by G Childs and O Ruiz) are given
The contributions of S Smale, J Guckenheimer and G Oster indicate applications to the biological sciences and that of D Schmidt to Hamiltonian systems For other applications and related topics, we refer to the monographs of Andronov and Chaiken [1], Minorsky [1] and Thom [1]
The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value In Hopf's original approach, the determination of the stability of the resulting periodic orbits is, in concrete problems, an unpleasant calculation We have given explicit algorithms for this calculation which are easy to apply in examples (See Section 4, and Section 5A for comparison with Hopf's formulae) The method of averaging, exposed here by S Chow and J Mallet-Paret in Section 4C gives another method of determining this stability, and seems to be especially useful for the next bifurcation to invariant tori where the only recourse may be to numerical methods, since the periodic orbit is not normally known explicitly
In applications to partial differential equations, the key assumption is that the semi-flow defined by the equations be smooth in all variables for t > O This enables the invariant manifold machinery, and hence the bifurcation theorems to go through (Marsden [2]) To aid in determining smoothness in examples we have presented parts of the results of Dorroh-Marsden [1] Similar ideas for utilizing smoothness have been introduced independently by other authors, such as D Henry [1]
Some further directions of research and generalization are given in papers of Jost and Zehnder [1], Takens [1, 2], Crandall-Rabinowitz [1, 2], Arnold [2], and Kopell-Howard [1-6] to mention just a few that are noted but are not discussed in any detail here We have selected results of Chafee [1] and Ruelle [3] (the latter is exposed here by S Schecter) to indicate some generalizations that are possible
The subject is by no means closed Applications to instabilities in biology (see, eg Zeeman [2], Gurel [1-12] and Section 10, 11); engineering (for example, spontaneous "flutter" or oscillations in structural, electrical, nuclear or other engineering systems; cf Aronson [1], Ziegler [1] and Knops and Wilkes [1]), and oscillations in the atmosphere and the earth's magnetic field (cf Durand [1]) are appearing at a rapid rate Also, the qualitative theory proposed by Ruelle-Takens [1] to describe turbulence is not yet well understood (see Section 9) In this direction, the papers of Newhouse and Peixoto [1] and Alexander and Yorke [1] seem to be important Stable oscillations in nonlinear waves may be another fruitful area for application; cf Whitham [1] We hope these notes provide some guidance to the field and will be useful to those who wish to study or apply these fascinating methods
After we completed our stability calculations we were happy to learn that others had found similar difficulty in applying Hopf's result as it had existed in the literature to concrete examples in dimension ≥ 3 They have developed similar formulae to deal with the problem; cf Hsu and Kazarinoff [1, 2] and Poore [1]
The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type The new proof, relying on invariant manifold theory, is considerably simpler than existing proofs and should be useful in a variety of situations involving bifurcation theory for evolution equations
These notes originated in a seminar given at Berkeley in 1973-4 We wish to thank those who contributed to this volume and wish to apologize in advance for the many important contributions to the field which are not discussed here; those we are aware of are listed in the bibliography which is, admittedly, not exhaustive Many other references are contained in the lengthy bibliography in Cesari [1] We also thank those who have taken an interest in the notes and have contributed valuable comments These include R Abraham, D Aronson, A Chorin, M Crandall, R Cushman, C Desoer, A Fischer, L Glass, J M Greenberg, O Gurel, J Hale, B Hassard, S Hastings, M Hirsch, E Hopf, N D Kazarinoff, J P LaSalle, A Mees, C Pugh, D Ruelle, F Takens, Y Wan and A Weinstein Special thanks go to J A Yorke for informing us of the material in Section 3C and to both he and D Ruelle for pointing out the example of the Lorentz equations (See Example 4B8) Finally, we thank Barbara Komatsu and Jody Anderson for the beautiful job they did in typing the manuscript
Jerrold Marsden
Marjorie McCracken
1,878 citations