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

Nonlinear aeroelastic response of panels

Radiation from panels as a source of airframe noise

A partial control surface is constructed near the midspan of a NACA 0012 airfoil model with and without freeplay gaps. Static and dynamic control surface pitch motion measurements were performed in wind-tunnel tests at higher angles of attack including buffet or von Karman vortex street flows. Two typical angles of attack, α=20 and 40 deg, and four freplay gaps in the pitch stiffness, δ=0.25, 0.5, 0.75, and 1.0 deg, are considered in the tests. This study has measured the static and dynamic responses of the flap pitch motion (with and without freeplay gaps) induced by the buffeting flow. Also, the buffeting frequency has been determined from the measured flap pitch responses. The buffet flow occurs at large angles of attack, and this leads to a larger preload from the aerodynamic forces and a larger static pitch equilibrium position. In this case, there is no limit cycle oscillations of the flap pitch motion induced by the freeplay gap to be found. Because the flap pitch motions are around the upper bound...

Experimental Aeroelastic Response for a Freeplay Control Surface in Buffeting Flow

An investigation of modeling unsteady transonic buffeting flow for the NACA 0012 airfoil when using the NASA OVERFLOW 2 flow solver code is presented. For a fixed freestream Mach number and Reynolds number, buffeting flow is modeled for various angles-of-attack. The flow model is based on the unsteady Reynolds averaged Navier-Stokes equations, and three different turbulence models are considered as well as four different numerical flux functions and four different numerical limiter options. It is shown that the computed buffet response is very sensitive to details of the computational model.

Airfoil Transonic Flow Buffet Calculations Using the OVERFLOW 2 Flow Solver

It is well known that the effectiveness of a trailing-edge control surface can be substantially diminished due to the elastic twist of an airfoil or rolling wing. This aeroelastic phenomenon is known as control surface reversal when the lift or rolling rate vanishes at a sufe ciently large ratio of e ow dynamic pressure to airfoil or wing stiffness. However, a leading-edge control surface can be used to counteract control surface reversal, and, indeed, in principle, a leading-edge control surface can entirely cancel the tendency of the trailing-edge control surface to undergo reversal. Moreover, analysis shows that by using a simple adaptive control strategy, one can use a combination positive and negative control surface rotations to maximize lift and rolling effectiveness or minimize control surface rotations. In the present work, a theoretical ‐experimental study of the effectiveness of trailingand leading-edge control surfaces has been made for a rolling wing ‐fuselage model. An experimental model and wind-tunnel test are used to assess the theoretical results. The theoretical model includes the inherently nonlinear dry friction damping moment between the spindle support and the experimental aeroelastic wing model for the rollingdegreeoffreedom.Athree-dimensionalvortexlatticeaerodynamictheoryisemployed.Newinsightsintothe behavior and design of an adaptiveaeroelastic wing using trailing- and leading-edge control surfacesare provided. Nomenclature c = wing chord length K® = torsional stiffness km, kn = numbers of vortex elements on the wing in x and y directions, respectively kmm = total number of vortices on both the wing and wake in the x direction L = total lift on the wing Llocal = local lift along the span

Earl H. Dowell

Papers

Nonlinear aeroelastic response of panels

Radiation from panels as a source of airframe noise

Experimental Aeroelastic Response for a Freeplay Control Surface in Buffeting Flow

Airfoil Transonic Flow Buffet Calculations Using the OVERFLOW 2 Flow Solver

Experimental and Theoretical Study on Rolling Effectiveness of Multiple Control Surfaces