# Aerospace Engineering Research Works

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Item The Phase Space Portrait of an Integrable Dynamical System(Science, 1990-02-16) Coffey, Shannon; Deprit, André; Deprit, Etienne; Healy, LiamFor an integrable dynamical system with one degree of freedom, "painting" the integral over the phase space proves to be very effective for uncovering the global flow down to minute details. Applied to the main problem in artificial satellite theory, for instance, the technique reveals an intricate configuration of equilibria and bifurcations when the polar component of the angular momentum approaches zero.Item A Toolbox for Nonlinear Dynamics(Springer-Verlag, 1991) Coffey, Shannon; Deprit, André; Deprit, Eitenne; Healy, Liam; Miller, BruceUsing the main problem of artificial satellite theory as an illustration, we review several developments which have had a significant impact on research in nonlinear dynamics. On the mathematical front, we point to the theory of Lie transformations; in the area of computational software, we explain how massively data parallel machines open the way for symbolic solution of large problems. Finally, we show how color graphics assist in the qualitative analysis of dynamical systems.Item Paint by number: Uncovering phase flows of an integrable dynamical system(American Institute of Physics, 1991-09) Healy, Liam; Deprit, EtienneGiven an integrable dynamical system with one degree of freedom, "painting" the integral over phase space proves to be a powerful technique for uncovering both global and local behavior. This graphical technique avoids numerical integration, employing instead a nonlinear method of assigning contrasting colors to the energy values to distinguish subtle details of the flow.Item Parallel Computing for Space Surveillance(MIT Lincoln Laboratory, 1992) Healy, Liam; Coffey, ShannonThis paper reports on an application of massively parallel processors to multiple satellite propagation and the calculation of miss distances between objects (COMBO). Unlike serial computations, we do not pre-filter the data but rather sort the data set in a way that dramatically cuts the number of comparisons required in order to be assured of a complete catalog-to-catalog comparison. The same general algorithm allows two logical sets to be compare to each other. Run time for this demonstration code on an 8K Connection Machine is about one second per time step, including propagation, complete catalog-to-catalog calculation of miss distances, plotting satellite positions, and recording of the miss distances to a file. Propagation of the objects is performed with an analytic propagator, using J2 only at present, though the code may easily be extended to other propagators. We demonstrate a second application of parallel computing to the problem of debris propagation resulting from a satellite breakup. The spread of such debris into n pieces is simulated by replicating the element set for the original satellite n times, then altering each to represent a distribution of velocities to the center of mass.Item Computation of error effects in nonlinear Hamiltonian systems using Lie algebraic methods(American Institute of Physics, 1992-06) Healy, Liam; Dragt, Alex; Gjaja, IvanThere exist Lie algebraic methods for obtaining transfer maps around any given trajectory of a Hamiltonian system. This paper describes an iterative procedure for finding transfer maps around the same trajectory when the Hamiltonian is perturbed by small linear terms. Such terms often result when an actual system deviates from an ideal one due to errors. Two examples from accelerator physics are worked out. Comparisons with numerical computations, and in simple cases exact analytical calculations, demonstrate the validity of the procedure.Item Deterministic Studies of Debris Hazards with Parallel Processors(European Space Agency, 1993-04-05) Healy, Liam; Coffey, ShannonA new generation of parallel processing computers makes possible the ability to propagate all objects in the space surveillance catalog with simulated objects, and detect close approaches. With this capability, it is possible to test deterministically debris scenarios, without resorting to statistical models. To compare the positions of objects we have developed two methods, an all-to-all comparison and a one-to-all comparison. For the former, a seive significantly reduces computation time; for the latter, direct comparison is possible in parallel. We show results from several simulations, including simulated multiple sources of debris, hazard to the space station, and close contacts amongst the catalog itself, to show potential for debris studies. The techniques described here have potential application the general problem of catalog maintenance.Item Close Conjunction Detection on Parallel Computer(American Institute of Aeronautics and Astronautics, 1995-07) Healy, LiamClose conjunction detection is the task of finding which satellites will come within a given distance of other satellites. The algorithms described here are implemented on the Connection Machine (CM) in a program called CM-COMBO. It will find close conjunctions of satellites over a time range for one, a few, or all satellites against the original or another catalog and works with an arbitrary propagator. The problem of comparing an entire catalog against itself is beyond the computing power of current serial machines. This program does not prefilter any orbits and does not make assumptions about the type of orbit (that it be nearly circular, for instance). This paper describes the algorithm for this computation, the implementation on the CM, and resuls of several studies using this program.Item Symbolic and Parallel Computation in Celestial Mechanics(Society for Industrial and Applied Mathematics, 1996) Healy, LiamOne aspect of celestial mechanics is the computation of the long-term orbits of celestial bodies. This type of computation is complicated by the interaction of the many bodies that need to be considered to derive accurate long-term behavior. For reasons explained in this chapter, it is necessary to do this symbolically rather than numerically. Symbolic computations performed on a Lisp machine are described. The visualization of the solution is accomplished in a massively parallel SIMD machine.Item High-Frequency Nonlinear Vibrational Control(IEEE, 1997-01) Shapiro, Benjamin; Zinn, B. T.This paper discusses the feasibility of high-frequency nonlinear vibrational control. Such control has the advantage that it does not require state measurement and processing capabilities that are required in conventional feedback control. Bellman et al. [1] investigated nonlinear systems controlled by linear vibrational controllers and proved that vibrational control is not feasible if the Jacobian matrix has a positive trace. This paper extends previous work to include nonlinear vibrational controllers. A stability criteria is derived for nonlinear systems with nonlinear controllers, and it is shown that a nonlinear vibrational controller can stabilize a system even if the Jacobian matrix has a positive trace.Item Automatic Rendering of Astrodynamics Expressions for Efficient Evaluation(American Astronautical Society, 1998) Healy, Liam M.; Travisano, Jeffrey J.In this paper, we describe the automatic rendering of expressions computed using symbolic manipulation. Computations from astrodynamics frequently can be put in a fixed hierarchy of polynomials and Fourier series. Once in this form, FORTRAN subprograms can be generated automatically in a form that lends itself to numerical evaluation. The goal of the current work is to present an approach for using symbolic manipulation techniques to produce a Fortran representation of the normalized Hamiltonian and other supporting equations representing as many of the actual physical effects on satellites as possible.Item Symmetry Approach to Extension of Flutter Boundaries via Mistuning(American Institute of Aeronautics and Astronautics, 1998-05) Shapiro, BenjaminA general framework is presented for analyzing and optimizing stability increases resulting from mistuning. The framework given is model independent and is based primarily on symmetry arguments. Difficult practical issues are transformed to tractable mathematical questions. It is shown that mistuning analysis reduces to a block circular matrix eigenvalue/vector problem that can be solved efficiently even for large problems. Similarly, the optimization becomes a standard linear constraint quadratic programming problem and can be solved numerically. Because the methods given are model-independent, they can be applied to various models and allow the researcher to easily conclude which models accurately capture mistuning and which do not. A simple quasisteady model for flutter in a cascade is used to illustrate and validate results in this paper.Item Solving for Mistuned Forced Response by Symmetry(American Institute of Aeronautics and Astronautics, 1999-03) Shapiro, BenjaminThe introduction of mistuning in jet-engine bladed disks can lead to large changes in stability and forced response. Even small random mistuning (within the bounds of manufacturing tolerance) can lead to unacceptable response and high-cycle fatigue. Meanwhile, intentional mistuning may improve stability and forced response under manufacturing uncertainty. This paper presents a general framework for predicting forced response as a function of mistuning. Because the forced response problem is an almost singular linear problem, its solution is highly nonlinear in the mistuning parameters. Our methods exploit symmetry arguments and eigenstructure perturbation to provide a method valid for any model. It is shown that, by perturbing eigenvectors in the numerator and the inverse of eigenvalues in the denominator (exploiting symmetry in both computations), we can accurately approximate the forced response as a function of mistuning. Results are demonstrated for a simple lightly damped model, and the consequent sharp nonlinear behavior is captured almost perfectly. We also show that intentional mistuning may guarantee improved stability and forced response under fixed manufacturing tolerances. Thus, intentional mistuning should be considered as a practical means of increasing safety and enhancing engine performance.Item The Main Problem in Satellite Theory Revisited(Springer Science+Business Media B.V., 2000) Healy, Liam M.Abstract. Using the elimination of the parallax followed by the Delaunay normalization, we present a procedure for calculating a normal form of the main problem (J2 perturbation only) in satellite theory. This procedure is outlined in such a way that an object-oriented automatic symbolic manipulator based on a hierarchy of algebras can perform this computation. The Hamiltonian after the Delaunay normalization is presented to order six explicitly in closed form, that is, in which there is no expansion in the eccentricity. The corresponding generating function and transformation of coordinates, too lengthy to present here to the same order; the generator is given through order four.Item The generalized Sundman transformation for propagation of high-eccentricity elliptical orbits(Univelt, Inc., 2002-01) Berry, Matthew M.; Healy, Liam M.A generalized Sundman transformation dt = cr^n ds for exponent n >= 1 may be used to accelerate the numerical computation of high-eccentricity orbits, by transforming time t to a new independent variable s. Once transformed, the integration in uniform steps of s effectively gives analytic step variation in t with larger time steps at apogee than at perigee, making errors at each point roughly comparable. In this paper, we develop techniques for assessing accuracy of s-integration in the presence of perturbations, and analyze the effectiveness of regularizing the transformed equations. A computational speed comparison is provided.Item Comparison Of Accuracy Assessment Techniques For Numerical Integration(Univelt, Inc., 2003-02) Berry, Matt; Healy, LiamKnowledge of accuracy of numerical integration is important for composing an overall numerical error budget; in orbit determination and propagation for space surveillance, there is frequently a computation time-accuracy tradeoff that must be balanced. There are several techniques to assess the accuracy of a numerical integrator. In this paper we compare some of those techniques: comparison with two-body results, with step-size halving, with a higher-order integrator, using a reverse test, and with a nearby exactly integrable solution (Zadunaisky's technique). Selection of different kinds of orbits for testing is important, and an RMS error ratio may be constructed to condense results into a compact form. Our results show that step- size halving and higher-order testing give consistent results, that the reverse test does not, and that Zadunaisky's technique performs well with a single-step integrator, but that more work is needed to implement it with a multi-step integrator.Item Comparison of MSIS and Jacchia atmospheric density models for orbit determination and propagation(Univelt, Inc., 2003-02) Akins, Keith A.; Healy, Liam M.; Coffey, Shannon L.; Picone, J. MichaelTwo atmospheric density model families that are commonly chosen for orbit determination and propagation, Jacchia and MSIS, are compared for accuracy. The Jacchia 70 model, the MSISE-90 model, and the NRLMSISE-00 model may each be used to determine orbits over fitspans of several days and then to propagate forward. With observations kept over the propagation period, residuals may be computed and the accuracy of each model evaluated. We have performed this analysis for over 4000 cataloged satellites with perigee below 1000km for September-October 1999, and the 60 HASDM calibration satellites with a large observation set for February 2001. The purpose of this study is to form a picture of the relative merits of the drag models in a comprehensive view, using all satellites in a manner consistent with the operational practice of US space surveillance centers. A further goal is to refine this knowledge to understand the orbital parameter regions where one of the models may be consistently superior.Item Orbit propagation with Lie transfer maps in the perturbed Kepler problem(Springer Science+Business Media B.V., 2003-02) Healy, Liam M.The Lie transfer map method may be applied to orbit propagation problems in celestial mechanics. This method, described in another paper, is a perturbation method applicable to Hamiltonian systems. In this paper, it is used to calculate orbits for zonal perturbations to the Kepler (two-body) problem, in both expansion in the eccentricity and closed form. In contrast with a normal form method like that of Deprit, the Lie transformations here are used to effect a propagation of phase space in time, and not to transform one Hamiltonian into another.Item Analyzing Mistuning of Bladed Disks by Symmetry and Reduced-Order Areodynamic Modeling(American Institute of Aeronautics and Astronautics, 2003-03) Shapiro, Benjamin; Willcox, KarenThe mistuned behavior of bladed disks is analyzed and optimized using an unsteady, transonic, computational fluid dynamic model (CFD). This result is enabled by the integration of two frameworks: the first is based on symmetry arguments and an eigenvalue/vector perturbation scheme, while the second is a reduction technique based on the proper orthogonal decomposition (POD). The first framework reduces the complexity of the problem, reveals engineering trade offs and suggests the existence of an intentional robust mistuning which improves both stability and forced response with respect to random variations in blade parameters. The second framework permits the reduction of state-of-the-art computational fluid dynamic codes to reduced-order models, which capture the accuracy of the original simulation but fit within the mistuning analysis framework. Together, these methodologies allow the analysis of a transonic, bladed disk with stiffness mistuning (see Fig. 1).Moreover, because of the low order of the aeroelasticmodel, a robust control¹ uncertainty analysis can be used to prove that the intentional mistuning suggested by the symmetry analysis framework is indeed robust. Hence this paper contains the first rigorous demonstration that intentional mistuning can robustly improve both the stability and forced response for a model that includes sophisticated aerodynamic effects.Item Accuracy and Speed Effects of Variable Step Integration for Orbit Determination and Propagation(Univelt, Inc., 2003-08) Berry, Matthew M.; Healy, Liam M.In this paper the fixed step Gauss-Jackson method is compared to two variable step integrators. The first is the variable step, variable order Shampine-Gordon method. The second is s-integration, which may be considered an analytical step regulation. Speed tests are performed for orbit propagation with the integrators set to give equivalent accuracy. The integrators are also tested for orbit determination, to determine the speed benefit of the variable step methods. The tests give an indication of the types of orbits where variable step methods are more efficient than fixed step methods.Item Student Projects for Space Navigation and Guidance(Univelt, Inc., 2003-08) Healy, Liam M."Space Navigation and Guidance," taught every fall at the University of Maryland, is required of all space track undergraduate aerospace engineering majors. Every student is required to participate in a group project where real observations are used in the solution of a navigation problem with estimation from observations. In this paper, I discuss two such projects, an observatory project in which the students use a telescope to track a satellite and determine its orbit, and a GPS project in which they analyze GPS receiver data to determine the receiver's position.