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Aeronautical and Astronautical Engineering

Dynamical Systems
^ Delay-Differential Equations
N. Sri Namachchivaya,* H. Van Roessel* (Univ. of Alberta)
University of Illinois; University of Alberta; Electric Power Research Institute, WO-8033-01

Unlike systems modeled by o.d.e.'s, mechanical systems modeled by delay-differential equations are, in some sense, infinite-dimensional. In this infinite-dimensional setting, the Hopf bifurcation theorem is not an elementary exercise, but requires some new ideas and some sophisticated mathematical machinery for its implementation. Researchers are studying the stability and bifurcations associated with delay-differential equations with multiple delays. In addition, a goal of this project is to examine the stability and bifurcations associated with stochastic time-delay systems. For a certain class of affine delay equations researchers obtain Lyapunov exponents. However, for general second-order systems with delay, researchers apply Lyapunov-like functions and the theory of differential inequalities in order to obtain stability in the pth mean.

^ Global Behavior of Nonlinear Aeroelastic Flat Panels in Supersonic Flow
N. Sri Namachchivaya,* S. Choi, A. Demir
U.S. Air Force Office of Scientific Research, 96-1-0265

The interaction of aerodynamic forces with a flexible structure such as a panel can create complicated vibrational effects that may adversely affect overall aircraft performance. Large-amplitude flutter, buckling, and fatigue failure are all possible results of the flow-induced dynamics. Researchers investigate the effect of nonlinearities on the global dynamics of flat panels in supersonic flow. The first part involves modeling the aerodynamic forces and moments that act on the flight vehicle and the nonlinearities that are inherent in the panel. The second part is concerned with the effects of boundary-layer turbulence on the panel dynamics.

^ Nonlinear Dynamics of Flexible Spinning Discs
N. Sri Namachchivaya,* N. Ramakrishnan
National Science Foundation, DMS-96-10456

Under consideration are the transverse dynamics of a high-speed spinning disc, clamped at its inner radius and rotating with a time-varying spin rate, such as found in turbine rotor systems and computer memory storage devices. Dynamical disturbances from time-varying spin rates, interactions with external dynamical systems, and inherent nonlinear effects may lead to disc instability. Such instabilities reduce the performance of the rotating system and can, in extreme cases, lead to failure. To examine these phenomena, the equations of motion of the system will be derived in detail. The effects of geometry, material characteristics, and aerodynamic interactions on disc displacement will be included. Local and global bifurcation analyses will then be implemented.

^ Nonlinear Dynamics of Parametrically Excited Gyroscopic Systems
N. Sri Namachchivaya,* R. McDonald, L. Vedula
U.S. Department of Energy, DE-FG02-97ER14795

Gyroscopic systems occur in many areas of engineering. Some examples are pipes conveying fluid, axially loaded rotating shafts, and systems subjected to Lorenz forces. Because of the widespread usage of gyroscopic systems, their stability characteristics are important in engineering application. Often the required stability analysis is complicated by parametric excitations. The effects of such excitations upon the stability of gyroscopic systems will be investigated through application of local and global stability analysis to discretized equations of motion incorporating the effects of symmetry breaking and linear and nonlinear dissipation. The results will be applied to several systems, analytically and numerically, and will be verified using a rotating shaft experimental apparatus.

^ Nonstandard Dimensional Reduction of Noisy Systems
N. Sri Namachchivaya,* R. B. Sowers* (Mathematics)
University of Illinois

This project is concerned with certain methods of dimensional reduction of nonlinear systems with symmetries and small noise. In the presence of a separation of scales, where the noise is asymptotically small, one exploits symmetries to use well-known methods to find an appropriate lower-dimensional description of the system. The interest of this project is when classical methods fail because the lower-dimensional description has singularities.

^ Sample Stability of Stochastic Dynamical Systems
N. Sri Namachchivaya,* H. Van Roessal* (Univ. of Alberta), N. Ramakrishnan, L. Vedula
National Science Foundation, DMS-96-10456; U.S. Department of Energy, DE-FG02-97ER14795

Lyapunov exponents are a generalization of the characteristic Floquet exponents so that more general nonperiodic orbits can be characterized. The spectrum of Lyapunov exponents provides the average exponential rates of divergence or convergence of nearby orbits in phase space. In addition, the Lyapunov exponents characterize the almost-sure stability of dynamical systems perturbed by noise. Furthermore, the moment Lyapunov exponent describes the moment stability of such systems. Asymptotic expansions of these exponents are constructed for stochastic dynamical systems when the noise (white or colored) is small. These results are used to understand stability and bifurcation characteristics of stochastic systems and are applied to various practical dynamical systems.

^ Stability of Tethered Spacecraft
N. Sri Namachchivaya,* R. McDonald
University of Illinois

Tethered spacecraft systems (TSS) have many applications ranging from the generation of electric power for a spacecraft to providing microgravity environments for experiments. The equations of motion governing the three dimensional motion of a two-body tethered system are highly nonlinear, yet most research on such systems has only considered the linear equations. Researchers propose to develop a low-dimensional nonlinear model for such systems to analytically study the stability, bifurcations, and chaos that may exist in these nonlinear systems. These analytic results will then be compared to numerical solutions to validate the low-dimensional model.

^ Stochastic Bifurcations
N. Sri Namachchivaya,* L. Arnold* (Univ. of Bremen), N. Ramakrishnan
National Science Foundation, DMS-96-10456

In this study, stochastic bifurcation implies either qualitative changes to the invariant measures that can be observed by examining the Fokker-Planck equation, or the appearance of a new invariant measure which is, at present, generated numerically through the forward and backward solutions of the stochastic differential equations. Researchers examine various concepts to describe stochastic bifurcations, namely the P-bifurcation and D-bifurcation. The analysis is carried out through studying a noisy Duffing-van der Pol oscillator that exhibits a variety of co-dimension one bifurcations along with certain global bifurcations in the absence of noise.


Summary of Engineering Research