N. Sri Namachchivaya

Research Summary

The overall goal of our investigations has been to formulate and develop methods of analysis for complex interactions between noise and inherent nonlinearities in mechanical and structural systems, and to suppress undesirable vibrations which can lead to failure. Practical applications of these new fundamental results are beginning to appear across the entire spectrum of mechanics; for example, stability of aircraft at high angles of attack, wave propagation in random media, mixing and transport phenomena in fluid mechanics. Some of the results are even contrary to intuition, such as stabilization by noise and stochastic resonance.

Multi-scale phenomena, dimensional reduction, and stability and bifurcations of invariant measures that arise in the study of nonlinear and stochastic systems are the focus of several projects supported by funding from NSF, ONR, AFOSR, DOE, EPRI, and Industry.

Current Projects

• Reduction of Noisy Nonlinear Systems
  with Kristan Onu, Jun Park, and Lalit Vedula: Many physical systems have nonlinearities and symmetries. Often there are additional small random perturbations, and one would like to develop techniques of stochastic dimensional reduction to find a simpler model which captures relevant dynamics of the system. The goals of this project are the development of general techniques of stochastic averaging of randomly-perturbed typical and relevant four-dimensional gyroscopic systems (Lalit Vedula), autoparametric systems (Jun Park), and fluid dynamical system (Kristan Onu). (supported by NSF)

• Stability of Nonlinear Stochastic Systems
  with David Kok, Ludwig Arnold (University of Breman), Peter Imkeller (Humboldt-University of Berlin): The primary concern in the analysis of nonlinear dynamical systems, is the determination and prediction of steady-states or stationary motions or the invariant measures of the local random dynamical systems. The main goal of this research is to develop new mathematical techniques to determine their almost-sure (Lyapunov exponents) and moment (moment Lyapunov exponents) stability properties. The second objective of this project is to determine how these invariant measures can bifurcate due to various parameters. (supported by NSF and ONR)

• Noisy Nonlinear Non-Smooth Systems
  with Jun Park and Richard Sowers: Interactions of mechanical and structural systems with the boundaries are either of short duration, modeled as impacts, or sustained, necessitating contact descriptions, as in the presence of friction. Mathematically, such interactions result in non-smooth nonlinear effects which usually give raise to complex noise induced oscillations. Our goal is to develop a general collection of mathematical techniques which can be applied to understand stability properties. We shall achieve this through several specific physically-motivated problems such as dead band or free play, dry friction, and stiction. (supported by NSF)

• Nonlinear Functional Differential Equations with Periodic Delay
  with Lalit Vedula, W. F. Langford (University of Guelph), H. J. Van Roessel (University of Alberta): We investigate the effect of periodic time delay and noisy perturbations upon delay differential equations. These theoretical developments will be directly applicable to the study of machine tool chatter in the presence of noise and spindle speed variations. This work addresses the suppression of machine tool chatter.