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 title problem is investigated, emphasizing two effects that are usually ignored in the literature. These are (i) nonlinear plate stiffness and (ii) mutual interaction between the plate and external and/or internal (cavity) air flow. The method of analysis employs modal expansions for the plate and cavity (air) motions, with the solution effected in the time (rather than the frequency) domain. The external flow‐pressure perturbations due to the plate motion are assumed to be described adequately by classical acoustic theory including convection effects. A discussion is given as to when the two effects will be important and a practical, numerical example is worked out to illustrate the method.

Transmission of Noise from a Turbulent Boundary Layer through a Flexible Plate into a Closed Cavity

Comparison of Theory and Experiment for Non-Linear Flutter and Stall Response of a Helicopter Blade

An experimental delta-wing/store model with freeplay has been designed and tested in the Duke wind tunnel. The wing structure is modeled theoretically using von Karman plate theory that accounts for geometric strain- displacement nonlinearities in the plate wing structure. A component modal analysis is used to derive the full structural equations of motion for the wing/store system. A linear three-dimensional time-domain vortex lattice aerodynamic model including a reduced-order model aerodynamic technique and a slender-body aerodynamic theory for the store are also used to investigate the nonlinear aeroelastic system. The effects of the freeplay gap, the span location of the store, and the initial conditions on the limit-cycle oscillations (LCO) are discussed. The correlations between the theory and experiment are good for the smaller LCO amplitudes, that is, for flow velocities slightly higher than the flutter velocity, but are not good for the larger LCO amplitudes, that is, higher flow velocities. The theoretical model needs to be improved to determine LCO response for larger-amplitude motions.

Flutter and Limit-Cycle Oscillations for a Wing-Store Model with Freeplay

The nonlinear effects of transonic aerodynamic forces on the flutter boundary of a typical section airfoil are studied. The flutter speed dependence on amplitude is obtained by utilizing a novel variation of the describing function method which takes into account the first fundamental harmonic of the nonlinear oscillatory motion. By using an aerodynamic describing function, traditional flutter analysis methods may still be used while including the effects of aerodynamic nonlinearities. Results from such a flutter analysis are compared with those of brute force time-marching solutions. The aerodynamic forces are computed by the LTRAN2 aerodynamic code for an NACA 64A006 airfoil at Mx = 0.86.

Earl H. Dowell

Papers

Transmission of Noise from a Turbulent Boundary Layer through a Flexible Plate into a Closed Cavity

Comparison of Theory and Experiment for Non-Linear Flutter and Stall Response of a Helicopter Blade

Flutter and Limit-Cycle Oscillations for a Wing-Store Model with Freeplay

Flutter Analysis Using Nonlinear Aerodynamic Forces

Forced response of a cantilever beam with a dry friction damper attached, part I: Theory