This paper assumes a requirement for an unmanned multiunit satellite to be assembled in orbit. The requirement to be met is to bring the satellites together so tha t they do not collide but actually rendezvous. The equations of motion of the rendezvous satellite in a relative coordinate system are derived and used to compute a final injection velocity which would effect collision after a time r. The velocity is corrected periodically by a command guidance system and just before impact retrothrust is applied. A terminal infrared homing sj^stem is required to actually accomplish physical contact and joining of the satellites. The first satellite placed in orbit is the "control satellite" and controls all the satellites to be assembled and contains the ccmputer, command guidance equipment, precision orientation equipment, and other features necessary to effect rendezvous. The succeeding satellites contain a propulsion system, a rough at t i tude control system, and a command receiver plus whatever scientific equipment they carry to perform their basic mission. This paper presents the following:

Terminal Guidance System for Satellite Rendezvous

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A new method for performing a balanced reduction of a high-order linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshotsisused to obtainlow-rank,reduced-rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain.Theapproximationsarethenusedtoobtainabalancedreducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized high-order system that models unsteady motion of a two-dimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.

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Balanced Model Reduction via the Proper Orthogonal Decomposition

On the use of higher-order finite-difference schemes on curvilinear and deforming meshes

For fans of a variable-cycle engine, a good aerodynamic performance over a wide range of rotating speeds is essential. However, when supersonic cascades designed for a high upstream Mach number operate under a low upstream Mach number, a starting problem may occur, which significantly decreases the aerodynamic performance of fan blades. As a result, the operating range of the supersonic cascades is severely limited. To solve the starting problem caused by the mismatch between geometry of supersonic cascades and upstream Mach number, a morphing supersonic cascade is developed. Thus, this study mainly focuses on the effect of the static deformation of supersonic cascades driven by smart materials on aerodynamic characteristics under a low upstream Mach number. To investigate this issue, numerous simulations are conducted on an S-type cascade by finite element method and computational fluid dynamics. As demonstrated by flow structure analysis, the morphing cascades are started under certain morphing configurations while the original cascade operates at nonstarted state. The results show that the deformation driven by smart materials alters the shock wave structures under a low upstream Mach number by adjusting key cascade geometric parameters. Specifically, the morphing cascades achieve 80% reduction of detached shock loss compared with the original cascades.

Aerodynamic Characteristics of Morphing Supersonic Cascade Under Low-Upstream-Mach-Number Condition

The representation of unsteady aerodynamic flow fields in terms of global aerodynamic modes has proven to be a useful method for reducing the size of the aerodynamic model over those representations that use local variables at discrete grid points in the flow field. Eigenmodes and Proper Orthogonal Decomposition (POD) modes have been used for this purpose with good effect. This suggests that system identification models may also be used to represent the aerodynamic flow field. Implicit in the use of a systems identification technique is the notion that a relative small state space model can be useful in describing a dynamical system. The POD model is first used to show that indeed a reduced order model can be obtained from a much larger numerical aerodynamical model (the vortex lattice method is used for illustrative purposes) and the results from the POD and the system identification methods are then compared. For the example considered, the two methods are shown to give comparable results in terms of accuracy and reduced model size. The advantages and limitations of each approach are briefly discussed. Both appear promising and complementary in their characteristics.

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System Identification and Pod Method Applied to Unsteady Aerodynamics

Unsteady transonic aerodynamics and aeroelasticity.

Effects of Steady Angle of Attack on Nonlinear Flutter of a Delta Wing Plate Model

In this study, analysis and results of linear and nonlinear aeroelastic of a cantilever beam subjected to the airflow as a model of a high aspect ratio wing are presented. A third-order nonlinear beam model is used as structural model to take into account the effects of geometric structural nonlinearities. In order to model aerodynamic loads, Wagner state-space model has been used. Galerkin method is implemented to solve dynamic perturbation equations about a nonlinear static equilibrium state. The small perturbation flutter boundary is determined by these perturbation equations. The effect of geometric structural nonlinearity of the beam model on the flutter behavior is significant. As it is observed the system’s response to upper speed of flutter goes to limit cycle oscillations and also the oscillations lose periodicity and become chaotic.Copyright © 2017 by ASME

Earl H. Dowell

Papers

Aerodynamic Characteristics of Morphing Supersonic Cascade Under Low-Upstream-Mach-Number Condition

System Identification and Pod Method Applied to Unsteady Aerodynamics

Unsteady transonic aerodynamics and aeroelasticity.

Effects of Steady Angle of Attack on Nonlinear Flutter of a Delta Wing Plate Model

Linear and Nonlinear Aeroelastic Analysis of a High Aspect Ratio Wing