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 delta wing experimental model with an external store has been designed and tested in the Duke University wind tunnel. The wing structure is modeled theoretically by using von Karman plate theory that allows 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 combination system. A 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 store pitch stiffness (attachment stiffness), the span location of store, and the store aerodynamics on the critical flutter velocity and limit cycle oscillations (LCO) are discussed. The correlations between the theory and experiment are good for both the critical flutter velocity and frequency but not good for the LCO amplitude, especially when the store is located near the wing tip. The theoretical structural model needs to be improved to determine LCO response, and improved results are shown in the companion paper as obtained with a higher-order structural model.

Flutter/Limit Cycle Oscillation Analysis and Experiment for Wing-Store Model

A computational investigation of the flutter onset and limit cycle oscillation behavior of various F-16 fighter weapons and stores configurations is presented. A nonlinear harmonic balance compressible Reynolds-averaged Navier–Stokescomputational fluiddynamic flowsolverisusedtomodeltheunsteadyaerodynamicsoftheF-16wing. Slender body/wing theory is used as an approximate method for accounting for the unsteady aerodynamic effects of wing-tip launchers and missiles. Details of the computational model are presented along with an examination of the sensitivity of computed aeroelastic behavior to characteristics and parameters of the structural and fluid dynamic model. Comparisons with flight-test data are also shown. I. Introduction T HE SEEK EAGLE Office at Eglin Air Force Base performs an essential task in clearing new aircraft/stores configurations through flight tests for safe and effective operation. Many of these flighttestsarefortheF-16aircraftwhichcontinuestobeaworkhorse for the U.S. Air Forcewith continually new stores (missiles, bombs, and fuel tanks) being considered for aircraft operations. Similar aeroelastic flight tests are expected for future fighter aircraft as they go into service in the coming years. The number of needed flight tests is projected to be well beyond the financial and staff resources available. Hence there is a pressing need to identify the most critical aircraft/store configurations for the limited flight-test resources available and also insofar as possibly reduce the number of flight tests needed. Virtual flight testing may be the answer. Using new improved computational capability that provides much more rapid solutions, computational simulation can help identify the most critical aircraft/ store configuration and also hasthe potential of reducingthe number ofneeded flighttestsifconfidencecanbeestablishedinthecapability of simulations to correlate with flight-test data. A new methodology has been developed to produce these computer simulations based upon the notion that because the response is periodic in time, the solution need only be obtained over a single period of oscillation in time. By avoiding the traditional time marching solution which computes the long transient before a steady-state periodic oscillation is reached, computational times are reduced by a factor of 10–100. This enables a sufficiently rapid solutiontomakesuchsimulationsapracticalrealityforthe flight-test engineer and support team. Future developments of this methodology hold the promise of further substantial reductions in computational cost and are being vigorously pursued. Also further refinements in the physical fidelity of the simulation models are being considered.

Theoretical Predictions of F-16 Fighter Limit Cycle Oscillations for Flight Flutter Testing

Modeling the LCO of a Delta Wing Using a High Fidelity Structural Model

Substantial advances have been made in improving the analytical methods available for prediction of interior noise and providing an experimental data base for their assessment and validation. A review of recent accomplishments, including representative results, is presented in the context of a Master Plan. It is suggested that modal representations of structural walls and acoustic cavities provide a useful conceptual and computational theoretical framework for including the mass, stiffness and damping of both the structure and acoustic cavity. Geometrical details and acoustic damping (absorption) are included in a rigorous manner. Future efforts, both near term and long term, are identified which are required to complete the Master Plan and provide the noise control practitioner with access to these improved methods.

Master Plan for Prediction of Vehicle Interior Noise

The Euler–Lagrange equations and the associated boundary conditions have been derived for an inextensible beam undergoing large deflections. The inextensibility constraint between axial and transverse deflection is considered via two alternative approaches based upon Hamilton's principle, which have been proved to yield equivalent results. In one approach, the constraint has been appended to the system Lagrangian via a Lagrange multiplier, while in the other approach the axial deflection has been expressed in terms of the transverse deflection, and the equation of motion for the transverse deflection has been determined directly. Boundary conditions for a cantilevered beam and a free–free beam have been considered and allow for explicit results for each system's equations of motion. Finally, the Lagrange multiplier approach has been extended to equations of motion of cantilevered and free–free plates.

Earl H. Dowell

Papers

Flutter/Limit Cycle Oscillation Analysis and Experiment for Wing-Store Model

Theoretical Predictions of F-16 Fighter Limit Cycle Oscillations for Flight Flutter Testing

Modeling the LCO of a Delta Wing Using a High Fidelity Structural Model

Master Plan for Prediction of Vehicle Interior Noise

Equations of Motion for an Inextensible Beam Undergoing Large Deflections