The Nonlinear Systems Research Laboratory, directed by Professor Subhash C. Sinha, consists of SUN workstations and PC's which are used to study various problems encountered in nonlinear dynamics and control. A number of software including IMSL, MATLAB, Mathematica, AUTO, Dynamical and Phaser are available to the users.

---

Laboratory Personnel:

      Sangram Redkar - Research Assistant, Ph.D. student

        Daniel Bell - Research Fellow, MS student

        Gerald Milek - Research Assistant, MS student

        Susan Wooden - Research/Teaching Assistant, MS student

        Emmanuel Gourdon - Visiting Scholar from ENTPE, France

        Michelle Farinash - UG Research Assistant

        Natalia Senkova - Administrative Assistant

                                  RESEARCH PROJECTS

         STABILITY AND CONTROL OF NONLINEAR TIME - PERIODIC SYSTEMS

Modeling of many mechanical systems such as asymmetric rotor bearing systems, helicopter blades, turbines, gear trains, spinning satellites, connecting rods in internal combustion engines, and structural elements subjected to in-plane periodic loads leads to a set of nonlinear differential equations with periodic coefficients. The principle aim of this proposed study is to develop a set of practical and efficient techniques that can be applied to a wide class of analysis and control problems encountered in nonlinear systems with periodically varying parameters.

a) Development of control techniques via ‘Lyapunov-Floquet (L-F) transformation’ and time dependent normal form theory. Illustrative examples include mechanical elements subjected to periodic axial loads and rotor-bearing system.

b) Bifurcation and control of critical systems – a general procedure for bifurcation analysis of nonlinear systems with ‘strong’ parametric excitations is being developed. This is achieved through an application of L-F transformation and ‘time-dependent center manifold theory’. Then the idea is extended to bifurcation control where systems may be linearly uncontrollable.

c) Application of ‘backstepping’ and  Lyapunov’s  second method to time-periodic nonlinear systems – algorithms are being developed to design feedback controllers for time-periodic linear and nonlinear dynamical systems.

d) Design of control systems via symbolic computation –‘mathematica’ based symbolic algorithms are being developed to design controllers for parametrically excited linear and nonlinear systems. First, the monodromy matrix is computed in symbolic form in terms of the unknown control gains. Then through Shur-Cohn or Juri’s criteria the controller gains are determined to guarantee asymptotic stability.

e) Control of chaotic or irregular motion to a desired periodic motion – control system design to drive a chaotic (or any irregular) motion to a periodic orbit or a fixed point is being pursued.  Applications include Duffing’ oscillator, Rossler’s system and cardiac models. It will be shown that an effective controller can be designed to change an irregular heat beat to a desired periodic rhythm.

f) Stability, robustness and Lyapunov redesign of time-periodic linear systems subjected to time-varying nonlinear perturbations are investigated.

g) ‘Near-identity transformation’ and ‘normal form theory’ are being developed for nonlinear systems with quasi-periodic coefficients. Illustrative examples include free and forced response of Mathieu type equations with periodic/quasi-periodic coefficients.
 

                                ORDER REDUCTION OF LARGE - SCALE SYSTEMS AND STRUCTURES

This project aims at the development of unique methods for non-liner order reduction of general large-scale systems. These dynamical systems could contain constant coefficients (i.e., Time-invariant) or periodic coefficients (i.e., Time-varying). The methods under development are based on the concept of Non-linear Normal Modes (NNMs). NNMs are special non-linear periodic motions, which are a non-linear analog of traditional linear normal modes.

a) Order Reduction of Time Invariant Systems:
  For time-invariant non-linear systems, the existing methods are simply variations of Guyan Reduction (a popular linear order reduction technique). These techniques do not account for nonlinearity. But NNM based order reduction method will capture non linear characteristics of system better, which are valid over a wide range of amplitudes, with fewer non linear modes.

b) Order Reduction of Time Varying Systems:
  An important class of problem is dynamical systems with time periodic coefficients e.g., helicopter blades, asymmetric rotor bearing systems, etc. For order reduction of these systems, first Liapunov-Floquet Transformation is used to transform the linear time-periodic parts to time-invariant forms. The subsequent use of time-dependent normal forms yields nonlinear time-invariant systems for which nonlinear normal modes may be extracted and order reduction algorithms can be applied.

This research brings together several recent concepts in vibration and nonlinear dynamics and tackles an important class of unsolved problem. The truncated models obtained by these techniques would help industries like automobile, aircraft, military to save millions of dollars on model testing programs.
 

 

 

---