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 general H∞ fault tolerant control and management for a linear system with actuator faults

A reduced order system ID approach to the modelling of nonlinear structural behavior in aeroelasticity

Nonlinear damping suspension is a promising method to be used in a rotor-bearing system for vibration isolation between the bearing and environment. However, the nonlinearity of the suspension may influence the stability of the rotor-bearing system. In this paper, the motions of a flexible rotor in short journal bearings with nonlinear damping suspension are studied. A computational method is used to solve the equations of motion, and the bifurcation diagrams, orbits, Poincare maps, and amplitude spectra are used to display the motions. The results show that the effect of the nonlinear damping suspension on the motions of the rotor-bearing system depends on the speed of rotor: (a) For low speeds, the rotor- bearing system presents the same motion pattern under the nonlinear damping (
 $$p=0.5, 2, 3$$
 ) suspension as for the linear damping (
 $$p=1$$
 ) suspension; (b) For high speeds, the effect of nonlinear damping depends on a combination of the damping exponent and damping coefficient. The square root damping model (
 $$p=0.5$$
 ) shows a wider stable speed range than the linear damping for large damping coefficients. The quadratic damping (
 $$p=2$$
 ) shows similar results to linear damping with some special damping coefficients. The cubic damping (
 $$p=3$$
 ) shows more stable response than the linear damping in general.

Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings

Presented is a technique for developing a nonlinear reduced order model of a computational uid dynamic solver. The method is based on a Taylor series expansion of a frequency domain harmonic balance computational uid dynamic solver residual. The computational routines used to compute the matrices and tensors of the Taylor series expansion are created using automatic dieren tiation. A Ritz type expansion using proper orthogonal decomposition shapes is then used in the Taylor series expansion to create the nonlinear reduced order model. Results are presented and compared to a linear reduced order model for an inviscid transonic airfoil conguration.

Using Automatic Differentiation to Create a Nonlinear Reduced Order Model of a Computational Fluid Dynamic Solver

The well known text by Lyon on Statistical Energy Analysis (SEA) [R. H. Lyon, Statistical Energy Analysis of Dynamical Systems: Theory and Applications (MIT Press, Cambridge, MA, 1975)] is the standard reference on the subject. As originally conceived SEA was based upon several plausible hypotheses. Various authors have considered the underlying basis for SEA and thereby advanced our understanding of it. In the present paper, the relationship between classical modal analysis (CMA) and SEA is studied. It is shown how the results of SEA may be obtained as an asymptotic limit of CMA. The present work only considers the response of a single general linear structure under a random or sinusoidal load. However future plans will consider (a) the structural acoustic response of an enclosure bounded by a flexible wall and excited by external forces and (b) two (or more) coupled structural systems. Here the emphasis is on (1) a new derivation of the SEA results from CMA for a single structure under a random load, (2) the generalization of the usual SEA result to show that, asymptotically, all points on the structure have the same response. In the literature this is sometimes invoked as an assumption. Here the result is derived as part of the asymptotic limit of CMA. (3) A numeriCal example which displays the manner in which the asymptotic limit is approached for random loading and (4) an extension of the usual SEA result for sinusoidal loading. In view of the above, the present results might be called more appropriately Asymptotic Modal Analysis (AMA) rather than SEA.

Earl H. Dowell

Papers

A general H∞ fault tolerant control and management for a linear system with actuator faults

A reduced order system ID approach to the modelling of nonlinear structural behavior in aeroelasticity

Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings

Using Automatic Differentiation to Create a Nonlinear Reduced Order Model of a Computational Fluid Dynamic Solver

Asymptotic modal analysis and statistical energy analysis of dynamical systems