Super and Sub-Harmonic Resonance Conditions — Lesson 2

This lesson covers the concept of non-linear vibration, focusing on the Duffing equation and its responses. It delves into the stability of the obtained response and how to analyze a system subjected to hard excitation. The lesson also discusses the use of the Runge Kutta method for obtaining system responses and the Lindstedt Poincare technique for finding free vibration responses. It further explains the Lyapunov stability criteria and how to determine system stability using the Jacobian matrix. The lesson also provides an understanding of the method of multiple scales and how to analyze the free vibration response of a non-linear system with quadratic and cubic non-linearity.

Video Highlights

00:30 - Introduction to the Duffing equation
04:55 - System of equations in matrix form and eigenvalues
07:00 - System responses and their stability
29:25 - Frequency response of a system and affecting system parameters
37:11 - Stability of a system

Key Takeaways

- The Duffing equation is a non-linear differential equation used to model certain damped and driven oscillators.
- The Runge Kutta method and the Lindstedt Poincare technique are useful for obtaining system responses and finding free vibration responses respectively.
- The Lyapunov stability criteria is a method used to determine the stability of a system.
- The method of multiple scales is a perturbation technique used to find approximate solutions to non-linear problems.
- The Jacobian matrix is a crucial tool in determining system stability.