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

A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier–Stokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier–Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.

Stabilization of projection-based reduced order models of the Navier–Stokes

This study focuses on a novel harmonic balance formulation, the high-dimensional harmonic balance method. To investigate a non-linearity in the damping term, the system chosen for study is the Van der Pol's oscillator. Both unforced and forced oscillators are analyzed. The results from the analysis are compared with those obtained from the classical harmonic balance and the time marching (Runge–Kutta) methods.

A novel harmonic balance analysis for the Van Der Pol oscillator

The aeroelastic stability of an infinitely long panel of finite width with the upper surface exposed to an in viscid flow is examined. The panel behavior is accounted for by small deflection plate theory, whereas the aerodynamic forces acting on the panel are described by the classical linearized, potential theory. The panel is assumed to be simply supported along the side edges, although other edge conditions may be considered in a similar manner (e.g., clamped edges). Rather extensive numerical results have been obtained for flutter velocity (and dynamic pressure) indicating the effects of Mach number, air/panel mass ratio, and viscous structural damping. A comparison of the present results with those obtained by previous investigators for a long, narrow panel (using a more elaborate method that accounts for the finite length) shows some discrepancy between the two results.

Flutter of infinitely long plates and shells. I - Plate.

Theory and experiment for flutter of a rectangular plate with a fixed leading edge in three-dimensional axial flow

The purpose of this paper is to study the flutter instability and forced response of a nonrotating flexible rotor blade model with a geometrical structural nonlinearity and a freeplay structural nonlinearity. The ONERA stall aerodynamic model is used. External excitations are provided by base harmonic excitations in the pitch angle and the chordwise direction, respectively. Two cases are considered in this paper. Case A is for a nonlinear blade structure with an unstalled unsteady aerodynamic model. Case B is for the nonlinear blade structure with a large effective mean angle of attack. The effects of the structural and aerodynamic nonlinearities and initial disturbance on instability and forced response behavior are discussed. A wind-tunnel test has also been carried out in the Duke University low-speed wind tunnel. The wind-tunnel tests show generally good agreement between theory and experiment for both static and dynamic behavior. Although the mathematical and experimental model of the nonrotating blade is different from the operational rotating hingeless rotor blade, the results from the experimental-theoretical correlation study provide fundamental understanding of the nonlinear aeroelastic behavior for a flexural-flexural-torsional hingeless rotor blade.

Earl H. Dowell

Papers

Stabilization of projection-based reduced order models of the Navier–Stokes

A novel harmonic balance analysis for the Van Der Pol oscillator

Flutter of infinitely long plates and shells. I - Plate.

Theory and experiment for flutter of a rectangular plate with a fixed leading edge in three-dimensional axial flow

Experimental and Theoretical Study for Nonlinear Aeroelastic Behavior of a Flexible Rotor Blade