Periodic, Quasi-Periodic and Chaotic systems — Lesson 1

This lesson covers the concept of non-linear vibration, specifically focusing on parametrically excited systems. The lesson begins with an explanation of what a parametrically excited system is, using examples such as a cantilever beam subjected to an excitation force and a spring mass damper system. The lesson then delves into the concept of Euler buckling load, explaining how a column or beam will start to buckle when a constant force exceeds the Euler buckling load. The lesson also discusses the concept of parametrically excited systems, where a force is applied in one direction and displacement occurs in a perpendicular direction. The lesson concludes with a detailed explanation of how to use Floquet theory to determine the stability of a system, and how to find the parametric instability region.

Video Highlights

00:54 - Characterization of quasi-periodic and chaotic responses
01:46 - Generating periodic, quasi-periodic, and chaotic responses
05:00 - Numerical methods to find the response of systems
25:43 - Study of responses for a base excited cantilever beam example

Key Takeaways

- Understanding of non-linear vibration and its different responses: periodic, quasi-periodic, and chaotic.
- Knowledge of the stability and bifurcation of the periodic response.
- Ability to distinguish between periodic, quasi-periodic, and chaotic responses.
- Understanding of the time response phase portrait, Poincare section, and Lyapunov exponent to characterize the chaotic response.
- Familiarity with different types of systems like the Duffing oscillator and the van der Pol oscillator.