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

A theoretical and experimental aeroelastic study of a typical airfoil section with control surface free play for nonzero angle of attack in low subsonic flow is presented. The study includes the flutter and limit cycle oscillation behavior and also the linear and nonlinear aeroelastic responses excited by periodic gust loads. The theoretical approach uses Peters's finite state airloads model. The experimental investigation has been carried out in the Duke University wind tunnel using a rotating slotted cylinder gust generator. The theoretical and experimental results show that the self-excited aeroelastic limit cycle oscillation is sensitive to the effect of initial pitch angle. For the gust response, the effect of initial pitch angle is smaller for the plunge and pitch responses and is larger for the flap response. The fair to good quantitative agreement between theory and experiment verifies that the present analytical approach has reasonable accuracy and good computational efficiency for nonlinear aeroelastic response analysis of such systems.

Aeroelastic Airfoil with Free Play at Angle of Attack with Gust Excitation

Navier–Stokes simulations are performed to determine the responses of a single-degree-of-freedom pitch airfoil system, as well as those of a two-degree-of-freedom pitch-and-heave system, in buffeting flows. The buffeting flow exhibits shock-wave oscillations of a unique frequency that occurs for some combinations of mean flow angle of attack and transonic Mach numbers. It was found that frequency lock-in may occur, when the frequency of the elastic system is close to the buffet frequency, and the amplitude of the elastic oscillation is above some threshold. Following frequency lock-in, the system response rapidly increases until reaching a limit-cycle oscillation. Synchronization of the transonic aerodynamic buffet phenomenon and the structural elastic modes provides a physical mechanism that leads to a limit-cycle oscillation, as observed in the current computational study.

Aeroelastic Responses of Elastically Suspended Airfoil Systems in Transonic Buffeting Flows

Theory and application of dynamic decoupling in structural analysis: Another view

We give a survey of recent results on flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are considered. The focus of the discussion here is on the interesting mathematical aspects of physical phenomena occurring in aeroelasticity, such as flutter and divergence. This leads to a partial differential equation treatment of issues such as well-posedness of finite energy solutions, and long-time (asymptotic) behavior. The latter includes theory of asymptotic stability, convergence to equilibria, and to global attracting sets. We complete the discussion with several well known observations and conjectures based on experimental/numerical studies.

Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures

Morphing-wing research has garnered much attention in the aerospace community over the last decade, and the folding wing is a promising concept that can improve aircraft performance over multiple types of missions. Several high-fidelity analyses of folding-wing aeroelastic stability have been published, but most analyses are specifi ct o certain wing planforms or a fixed number of wing segments. This paper presents a general aeroelastic model that predicts the flutter speed and flutter frequency of a folding wing with simplified geometry but with an arbitrary numberof wingsegments. Thebeam-theory structuralmodelandthe strip-theoryunsteadyaerodynamic modelare coupled using Lagrange’s equations. Three experimental models are constructed, and flutter tests are performed over a wide range of fold angles. The theoretical predictions for flutter speeds are within 10% of experimentally measured values for most configurations. In general, the results show that the essential physics of the problem is captured by the present first-principles model. Furthermore, data show that increasing the fold angle causes up to 30% increase in flutter speed, which has applications in extending the flutter boundaries of morphing aircraft.

Earl H. Dowell

Papers

Aeroelastic Airfoil with Free Play at Angle of Attack with Gust Excitation

Aeroelastic Responses of Elastically Suspended Airfoil Systems in Transonic Buffeting Flows

Theory and application of dynamic decoupling in structural analysis: Another view

Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures

Aeroelastic Model of Multisegmented Folding Wings: Theory and Experiment