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...

/pdf/a-survey-of-projection-based-model-reduction-methods-for-3yjqbxk305.pdf

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.

/pdf/balanced-model-reduction-via-the-proper-orthogonal-5c77iwc45j.pdf

Balanced Model Reduction via the Proper Orthogonal Decomposition

On the use of higher-order finite-difference schemes on curvilinear and deforming meshes

Starting from a finite state model for a two-dimensional aerodynamic flow over an airfoil, the eigenmodes of the aerodynamic flow are determined. Using a small number of these aerodynamic eigenmodes, ie., a reduced-order model, the aeroelastic model is formed by coupling them to a typical section structural model with a trailing-edge flap. A free-play nonlinearity is modeled. Results are shown from the finite state model, the reduced-order model, and previous theoretical and experimental work. All results are in good agreement.

Reduced-Order Aerodynamic Model and Its Application to a Nonlinear Aeroelastic System

A method to predict unsteady aerodynamic forces on lifting surfaces in supersonic flow is presented. The wing is divided into small segments in which the lift force is expressed by a single-point doublet of the acceleration potential. This is the same concept as the doublet-point method developed by the authors for subsonic flows. In order to avoid sensitiveness to the Mach number, the upwash due to the point doublet is calculated by averaging over small areas. The integration is done analyticaly so that it requires no numerical quadrature. Pressure distributions are directly obtained as the unknowns of the algebraic equation. The results are compared with those obtained by other methods for various wing geometries, including the AGARD wing-tail configuration.

Doublet-Point Method for Supersonic Unsteady Lifting Surfaces.

In this paper the jump phenomena in quasilinear Duffing systems under sinusoidal and narrow band random excitations are examined by numerical simulations. The simulation results for sinusoidal excitations agree very well with analytical solutions obtained by the equivalent linearization method. The results showing the sensitivity of the periodic responses to the initial conditions are believed to be the first published in the literature. The simulation results for narrow band random excitations confirm that multi-level mean square responses can occur for mono-level excitations, but only for very narrow bandwidth excitations. The multi-level random responses are also sensitive to initial conditions. As the bandwidth of the excitation broadens, the multi-level responses merge into a single level one.

Numerical simulations of jump phenomena in stable Duffing systems

Limit cycle oscillations (LCOs) have been observed in flight for certain modern high-performance aircraft. The nonlinear physical mechanism responsible for the LCOs is still in doubt, even to the point of it not yet being determined whether the nonlinearity is principally in the flexible elastic structure of the aircraft or due to the fluid behavior in the surrounding aerodynamic flowfield. One observation from flight tests is that by changing the angle of attack of aircraft, the flight velocity at which LCOs begin may be raised or lowered and that the amplitude of the LCOs may be reduced. It has been suggested that this sensitivity to angle of attack indicates the nonlinearity is in the fluid rather than in the structure. We show that such effects of an angle of attack change can be the result of a structural nonlinearity. Specifically, an investigation to determine the effects of a steady angle of attack on nonlinear flutter and LCO of a delta wing-plate model in low subsonic flow has been made. A three-dimensional time domain vortex lattice aerodynamic model and a reduced order aerodynamic technique were used, and the structure is modeled using von Karman plate theory that allows for geometric strain-displacement nonlinearities in the delta wing structure. The results provide new insights into nonlinear aeroelastic phenomena not previously widely appreciated, i.e., LCOs for low aspect ratio wings that have a platelike nonlinear structural behavior. The effects of a steady angle of attack on both the flutter boundary and the LCOs are found to be significant. For a small steady angle of attack, α 0 ≤ 0.1 deg, the flutter onset velocity increases, whereas for larger α 0 , it decreases. Moreover, as α 0 increases, the maximum LCO amplitude decreases substantially. Such effects have been observed by Bunton and Denegri in flight flutter experiments. It is noted that the present theoretical results do not prove that the LCOs phenomena observed in flight are due to structural nonlinearities; however, the results of the present analysis are consistent with those observed in flight and do show that a structural nonlinearity can give rise to the observed effects of angle of attack on the LCOs.

Effects of Angle of Attack on Nonlinear Flutter of a Delta Wing

A novel nonlinear reduced-order-modeling technique for computational aerodynamics and aeroelasticity is presented. The method is based on a Taylor series expansion of a frequency-domain harmonic balance computational fluid dynamic solver residual. The first- and second-order gradient matrices and tensors of the Taylor series expansion are computed using automatic differentiation via FORTRAN 90/95 operator overloading. A Ritz-type expansion using proper orthogonal decomposition shapes is then used in the Taylor series expansion to create the nonlinear reduced-order model. The nonlinear reduced-order-modeling technique is applied to a viscous flow about an aeroelastic NLR 7301 airfoil model to determine limit cycle oscillations. Computational times are decreased from hours to seconds using the nonlinear reduced-order model.

Earl H. Dowell

Papers

Reduced-Order Aerodynamic Model and Its Application to a Nonlinear Aeroelastic System

Doublet-Point Method for Supersonic Unsteady Lifting Surfaces.

Numerical simulations of jump phenomena in stable Duffing systems

Effects of Angle of Attack on Nonlinear Flutter of a Delta Wing

Using Automatic Differentiation to Create a Nonlinear Reduced-Order-Model Aerodynamic Solver