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

The vibration of clamped and simply supported elastic panels due to subsonic and supersonic turbulent boundary-layer flows is investigated by a Monte Carlo technique. The resulting generalized random forces are simulated numerically from boundary-layer turbulence spectra, and the response analysis is performed in the time domain. The mutual interaction between panel motion and external and/or internal airflow is included. Response studies are performed with respect to rms response, probability structure, peak distribution, threshold crossing and spectral density. The effect on the response statistics of in-plane loading, static pressure differential, and cavity pressure is investigated.

Nonlinear Panel Response by a Monte Carlo Approach

On Asymptotic Approximations to Beam Model Shapes

Fluid eigenmodes can be used to construct extremely efficient reduced order unsteady aerodynamic models of the Euler equations, which can also be used to perform highly efficient aeroelastic analyses. This is done by transforming the large set of primitive variable (density, velocity, etc.) equations into a much smaller set of decoupled (modal) equations. The technique of constructing a reduced order aerodynamic model is quite general and applies to most any CFD solver, regardless of the size of the primitive variable computational model. However, in practice the construction of reduced order representations of the CFD model is limited by the ability to accurately calculate the fluid eigenmodes. The difficulty of this calculation increases dramatically with the number of degrees of freedom in the primitive variable model. In this paper, two techniques are presented for calculating the eigenvalues and the corresponding eigenvectors of the large nonsymmetric matrices typically encountered with the discretized Euler equations: a Lanczos reduction and a Modified WYD reduction. The Lanczos reduction procedure is limited to small computational meshes (O(10) DOF). The Modified WYD reduction is developed here to extend the reduced order modelling technique to primitive variable systems with 0(10) DOF. Results presented in this paper demonstrate that a reduced order aeroelastic model created from a primitive variable system of this size predicts the onset of flutter within 0.5% of the conventional, refined mesh CFD based analysis, but at 10~ the cpu cost per analysis point. Nomenclature \9P] [vr.\

Reduced order Euler equations for unsteady aerodynamic flows - Numerical techniques

A theoretical aeroelastic study for a flexibly suspended high-aspect ratio wing aeroelastic model excited by gust loads is presented along with a companion wind-tunnel test. A flexible support system at the wing root has been constructed to simulate a rigid body mode. Structural equations of motion based on a nonlinear beam theory are combined with the ONERA aerodynamic model. Also, a dynamic perturbation analysis about a nonlinear static equilibrium is used to determine the small perturbation flutter boundary and structural natural frequencies. The effects of the flexible support system (bi-beam system) on the wing structural dynamics, the static aeroelastic displacement at root, nonlinear flutter, and gust response to a harmonic or a frequency sweep excitation are discussed. Also the effect of gust distribution along the span on the response is described based upon computations. The fair to good quantitative agreement between theory and experiment demonstrates that the present analysis method has reasonable accuracy.

Gust Response for Flexibly Suspended High-Aspect Ratio Wings

Chaotic oscillations have now been observed in non-linear mechanical systems by analytical, numerical and experimental methods. Nevertheless a more fundamental understanding of why and when such oscillations occur is of great importance. This goal will be pursued here by considering the relationship between chaos induced by forced oscillations versus self-excited oscillations, the relationship of indeterminancy of the final equilibrium state in the initial value problem to chaos in the sustained oscillation problem, comparison of theory to physical experiment, necessary and sufficient conditions for chaos to occur, and the question of convergence of systems of modal ordinary differential equations which derive from partial differential equations.

Earl H. Dowell

Papers

Nonlinear Panel Response by a Monte Carlo Approach

On Asymptotic Approximations to Beam Model Shapes

Reduced order Euler equations for unsteady aerodynamic flows - Numerical techniques

Gust Response for Flexibly Suspended High-Aspect Ratio Wings

Chaotic oscillations in mechanical systems