Poincare Section, FFT and Lyapunov Exponent — Lesson 3

This lesson covers the study of non-linear vibration analysis, focusing on the numerical tools used in the course. It delves into the time response, FFT, Poincare section, and Lyapunov exponents, which are essential for the course. The lesson explains how to solve differential equations and plot the time response of a system, using a spring mass system as an example. It also discusses the use of Runge Kutta methods and ODE 45 in Matlab for plotting. The lesson further explores the concept of forced vibration, damped natural frequency, and external frequency. It also introduces the logistic map and cubic logistic map, explaining how to plot them and interpret the results.

Video Highlights

00:42 - Discussion on time response, FFT, Poincare section, and Lyapunov exponents
15:09 - Response plots
22:49 - Continuation techniques
24:43 - Fast Fourier Transform (FFT)
44:00 - Poincare section
51:45 - Lyapunov exponents

Key Takeaways

- Non-linear vibration systems can be studied using a variety of systems, including base-excited cantilever beams, magnetoelastic beams, and viscoelastic beams.
- The generalized Galerkin method is used to convert the spacio-temporal equation to its temporal form.
- The method of multiple scales is often used in the perturbation method to consider different resonance conditions.
- Poincare sections can be used to characterize fixed point, periodic, quasi-periodic, and chaotic responses.