Nonlinear Dynamics and Chaos

Multi-stable systems, i.e, systems with multiple stable static equilibrium positions, exhibit a rich dynamical behavior. Such systems are nonlinear and undergo chaotic oscillations in the presence of a harmonic excitation of suitable amplitude and frequency. Examples of such systems in structural mechanics would be the forced vibrations of a buckled beam, the Duffing - Holmes magneto-elastic oscillator, etc.

The Duffing - Holmes magneto-elastic oscillator served as the first experimental evidence of chaos in structural mechanics. The oscillator consists of a cantilever beam vibrating in the field created by two external magnets. Depending on the distance between the magnets, the oscillator can have one (monostable), two (bistable) or three (tristable) stable static equilibrium positions.

The dynamics of the configuration with two stable states can be approximated by the well-known Duffing equation. Most of the studies reported so far study thus oscillator, either in terms of the Duffing equation or through experiments.

Recently, we have developed a mathematical model that relates the system dynamics to the parameters such as the distance between the magnets and the field strength of the magnets. The bifurcation diagrams obtained based on our model are shown below:

Experiments have also been performed to validate the model. The following videos show the chaotic oscillations obtained with the bistable and the tristable configurations.

Nonlinear dynamics has applications in various field, like structural analyses, thermal systems, space systems and Micro-electromechanical systems (MEMS). The dynamical behavior of these systems tends to exhibit periodic and chaotic patterns. It means that they have very sensitive dependence on initial state of the system. Due to that long term state prediction becomes impossible as the trajectories originated from almost same state diverge exponentially. In many applications it becomes necessary to control this diverging phenomenon and have more predictable behavior in near future for better design and application.

We have used well known OGY Method to control some of the chaotic systems be it discrete map or continuous time system. We have also developed a data base numerical scheme, which reconstruct system state and also calculate parameter of control variable.

As a reference problem we have done numerical experimentation on forced duffing oscillator and applied OGY control on it.

Blade disc assembly is one of the major rotating parts of turbines and compressors which work at very high rotating speed. Usually, these rotating parts of the system hold the cyclic symmetry features i.e., each bladed disc sector is designed to be identical. Therefore, this cyclic symmetry feature significantly simplifies the vibration analysis and is known as tuned system response. But, this symmetry of the structure will be destroyed because of differences in material properties, wear and tear, manufacturing tolerances, to name a few. This leads to the loss in periodicity of the rotor system and is known as a mistuned system. Consequently, mode localization phenomenon happens and makes rotor susceptible to early failure compare to the expected service life. In addition to that, if these mistuned systems will be subjected to rotation, the fundamental frequencies will be altered because of the centrifugal stiffening and gyroscopic effects. Therefore, it is very important to analyze the dynamic behavior of these rotating bladed disc systems at the design level with less computational complexities.