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

Second, the most desirable relative location of the doublet/averaging area is that which yields lifting line results when used subsonically with one patch per chord. When averaging over a parallelogram patch, the preferred doublet location is \/(e + 1) = 1 1 .92% of patch chord at the spanwise patch center, whereas Ref. 1 used the leading edge. This is discussed in Ref. 2 for the analogous downwash-point method. [Reference 2 also used unsteady/steady kernal ratios for supersonic solutions, as in Eqs. (19a) and (19b) of Ref. 1, but did not recognize the extra term in Eq. (19c), Ref. 1.] Figure 2 illustrates the use of this doublet location and of Eq. (1) with N=5, using the planforms of Ref. 3. Patch centers of pressure are not dictated by the doublet location used in forming the influence matrix, but may be assumed to be at 25% chord for M< 1, at 50% chord for supersonic patch leading edges, and interpolated for in-between Mach numbers.

Reply by Authors to K. L. Roger

Presented is a point collocation non-intrusive polynomial chaos approach for nonlinear aeroelastic uncertainty quantification analysis. A unique aspect of our approach is that we are using a nonlinear frequency-domain harmonic balance aeroelastic solution methodology, which allows us to solve directly for a limit cycle oscillation response amplitude versus reduced velocity response curve. Other researchers have typically framed their aeroelastic uncertainty quantification analysis methods around time-domain aeroelastic solvers, where an indirect measure of aeroelastic stability, such as aeroelastic damping, is the dependent variable that one performs an uncertainty quantification analysis for. Our method allows one to determine directly, uncertainty quantification for an aeroelastic limit cycle oscillation response curve, as well as unstable branches of an aeroelastic limit cycle oscillation response curve. We demonstrate the technique for uncertainty in freestream Mach number and airfoil static unbalance for a two-dimensional model of the NASA Benchmark Models Program B0012 NACA 0012 airfoil section aeroelastic wing configuration. 

Non-Intrusive Polynomial Chaos Approach for Nonlinear Aeroelastic Uncertainty Quantification

Three nonlinearities are considered. They are structural nonlinearities for plates and shells, fluid mechanical nonlinearities for bluff body oscillators, and aerodynamic nonlinearities for airfoils in transonic flow. For aeroelasticity of plates and shells, the equations of motion are well established. Results obtained by numerical time integration may be compared to those obtained by topological theories of dynamics and also from experiment. All of these suggest that the chaotic limit cycle oscillations may occur for this deterministic system. In a later chapter, chaotic oscillations in mechanical systems are discussed in a broader context. For bluff body oscillators, the equations of motion themselves are still in an emerging stage. Here a qualitative theory may be useful in establishing a generic model that describes the essential features of the physical phenomena. For aerodynamic nonlinearities in transonic flow effective methods of analysis are under development based on first principles of fluid and solid mechanics, and results from exploratory studies are now becoming available. An introductory discussion is contained in this chapter and a fuller treatment is provided in the following chapter.

Self-Excited Nonlinear Oscillations of Elastic Bodies in a Flow: An Introduction

There are many nonlinear systems used in engineering which require damage detection. For such systems, nonlinear damage detection methods may be useful. In this paper, a novel method is introduced that uses limit cycle oscillations that arise once the bifurcation (flutter) boundary is exceeded and shows greater sensitivity for damage detection versus linear damage detection techniques. Another advantage of this method is that it can be used for health monitoring of linear or nonlinear systems. Here a nonlinear aeroelastic panel is considered as a model to show the capabilities of the proposed damage detection technique. Also Proper Orthogonal Decomposition is used to find the number of independent damage locations in the panel.

Earl H. Dowell

Papers

Reply by Authors to K. L. Roger

Non-Intrusive Polynomial Chaos Approach for Nonlinear Aeroelastic Uncertainty Quantification

Self-Excited Nonlinear Oscillations of Elastic Bodies in a Flow: An Introduction

An Investigation on the Sensitivity of Limit Cycle Oscillations for Detecting Damage in an Aeroelastic Panel

An approach to modeling and estimation for uncertain systems