Non-Linear Vibration and Chaotic Responses — Lesson 3

This lesson covers the concept of non-linear vibration and chaotic responses in systems. It delves into the four types of responses in a system: fixed point response, periodic response, quasi-periodic response, and chaotic response. The lesson explains how these responses are determined and how they can be manipulated. It also discusses the concept of Lyapunov exponent and its role in determining the chaotic nature of a system. The lesson further explores the application of these concepts in real-life systems such as mechanical systems, biological systems, and electrical systems. For instance, a cantilever beam subjected to base excitation is used as an example to illustrate the concepts.

Video Highlights

00:37 - Explanation of the four types of responses in a system
01:27 - Detailed explanation of fixed point response, periodic response, and quasi-periodic response
15:49 - Period doubling response
21:11 - Explanation of crisis
29:28 - Duffing equation
44:13 - Lyapunov exponent

Key Takeaways

- Non-linear vibration and chaotic responses are crucial in understanding the behavior of various systems.
- The four types of responses in a system are fixed point response, periodic response, quasi-periodic response, and chaotic response.
- Chaotic responses are sensitive to initial conditions and can lead to different types of chaotic attractors.
- The Lyapunov exponent is a measure of the average rate of expansion and contraction of trajectories surrounding an attractor in a system.
- Controlling chaos involves changing the system parameter in such a way that it moves out of the chaotic domain.