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

Flutter is an important topic to be considered during an aircraft development mainly due to its potentially catastrophic behavior. Typically a new aerial vehicle can require the addition of mass and limitations on its flight envelope to be flutter free. As an alternative, active control technologies can be used employing actuators and sources of power, and there are different techniques for computing the actuator’s force. In general, many approaches depend on describing the aeroelastic system through the state-space representation in the time domain. However, the state-space dynamic matrix typically contains nonphysical eigenvalues due to the unsteady aerodynamic forces approximations. In this context, this article introduces an algebraic arrangement for designing a modal controller to suppress the flutter phenomenon. The approach considers the modal state-space realization combined with the linear quadratic regulator to compute a control gain. The gain is converted from the new modal domain to the physical, and the control force introduces an effect of incremental damping for all flight envelope, although the controller is designed particularly for the flutter speed. To demonstrate the approach, numerical simulations are carried out considering the three-degree-of-freedom airfoil. The results show that this technique allows stabilizing the system.

A modal approach for designing controllers for active flutter suppression

By modeling an aero-refueling hose as a flexible pipe, an experimental research is performed on measuring the sectional bending stiffness of a flexible pipe withstanding a internal-pressurized fluid. A Revised Midpoint-Concentrate-Force-Loading Methodology (RMCFLM) with its valid range is developed to measure the sectional bending stiffness of a flexible pipe. The total system measurement error would be controlled within 7%. Three test specimens of a silica gel pipe, a polyvinyl chloride pipe and a rubber hose with fabric insert are conducted to measure their sectional-bending-stiffness in variable internal-pressurized fluid. The sectional-bending-stiffness increases with an incremental fluid pressure in a flexible pipe. Furthermore, the equivalent elastic modulus of a pipe is smaller and a incremental rate of the sectional-bending-stiffness is larger.

Measuring the Aero-Refueling Hose Model’s Sectional-Bending-Stiffness in the RMCFLM Experimental System

This work deals with the fundamentally similar dynamic behavior of two apparently different aircraft systems, namely, landing gears and all-movable control surfaces. Both systems typically contain stiffness and damping nonlinearities, leading to shimmy/flutter limit cycle oscillations (LCOs). In a series of earlier papers the authors have developed highly efficient computational methods, based on describing functions, to construct LCO responses for dynamic systems with concentrated nonlinearities in stiffness or damping. These methods are extended in the present paper, and the focus is on the physical insights that can be obtained by using them for systems with multiple nonlinearities. The model parameters are varied to obtain an array of interesting results. For example, each branch of the landing gear response is attributable to a particular nonlinearity. Based on this, a possible reason is advanced for the occurrence of shimmy in the F/A-18 aircraft. On the other hand, the entire flutter response of the all-movable surface can be viewed as being shaped by one nonlinearity and modified by the other. Finally, the well-known Wright Air Development Center wind tunnel flutter test results for an all-movable horizontal tail with freeplay are reinterpreted and explained in the light of the present findings.

Study of Nonlinear Stiffness and Damping Effects on Shimmy and Flutter

In this chapter is shown an application of the general formulation of the interaction problem presented in Chapter II and, in particular, of the condition on the contact surface. For this purpose the behavior of a cylindrical shell and curved plate with a transverse potential flow around them of an ideal incompressible liquid is considered.

Earl H. Dowell

Papers

A modal approach for designing controllers for active flutter suppression

Measuring the Aero-Refueling Hose Model’s Sectional-Bending-Stiffness in the RMCFLM Experimental System

Study of Nonlinear Stiffness and Damping Effects on Shimmy and Flutter

Bending of a Cylindrical Shell with a Transverse Flow Around It

A New Approach to Determine the Most Efficient Supersonic Mach Number