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

By the use of a state-of-the-art computational e uid dynamic (CFD) method to model nonlinear steady and unsteady transonice owsin conjunction with a linearstructural model,an investigationismadeinto how nonlinear aerodynamics can effect the divergence, e utter, and limit-cycle oscillation (LCO) characteristics of a transonic airfoil cone guration. A single-degree-of-freedom (DOF) model is studied for divergence, and one- and two-DOF models are studied for e utter and LCO. A harmonicbalancemethod in conjunction with the CFD solver is used to determine the aerodynamics for e nite amplitude unsteady excitations of a prescribed frequency. A procedure for determining the LCO solution is also presented. For the cone guration investigated, nonlinear aerodynamic effects are found to produce a favorable transonic divergence trend and unstable and stable LCO solutions, respectively, for the one- and two-DOF e utter models. Nomenclature a = nondimensional location of airfoil elastic axis, e=b b, c = semichord and chord, respectively cl, cm = coefe cients of lift and moment about elastic axis, respectively e = location of airfoil elastic axis, measured positive aft of airfoil midchord h, ® = airfoil plunge and pitch degrees of freedom I® = second moment of inertia of airfoil about elastic axis

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Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations

On the aeroelastic instability of two-dimensional panels in uniform incompressible flow

An experimental high-aspect-ratio wing aeroelastic model with a slender body at the tip has been constructed, and the response due to flutter and limit-cycle oscillations (LCO) has been measured in a wind-tunnel test A theoretical model has been developed and calculations made to correlate with the experimental data Structural equations of motion based on nonlinear beam theory are combined with the ONERA aerodynamic stall model to study the effects of geometric structural nonlinearity and steady angle of attack on flutter and LCO of high-aspect-ratio wings Static deformations in the vertical and torsional directions caused by a steady angle of attack and gravity are measured, and results from theory and experiment are compared A dynamic perturbation analysis about a nonlinear static equilibrium is used to determine the small perturbation flutter boundary, which is compared to the experimentally determined flutter velocity and oscillation frequency Time simulation is used to compute the LCO response The results between the theory and experiment are in good agreement for static aeroelastic response, the onset of flutter, and dynamic LCO amplitude and frequency

Earl H. Dowell

Papers

Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations

On the aeroelastic instability of two-dimensional panels in uniform incompressible flow

Experimental and Theoretical Study on Aeroelastic Response of High-Aspect-Ratio Wings

Nonlinear Behavior of a Typical Airfoil Section with Control Surface Freeplay: a Numerical and Experimental Study

Power extraction from aeroelastic limit cycle oscillations