Cantilever Beam and Combination Resonance — Lesson 2

This lesson covers the study of non-linear dynamics of a flexible beam, focusing on different types of beams such as elastic, viscoelastic, and magneto-elastic beams. The lesson delves into the equation of motion derived using Newton's second law and the extended Hamilton principle. It further explores the system with principle parametric resonance condition and combination parametric resonance condition. The lesson also discusses the concept of detuning parameters, the method of multiple scale, and the transformation of equations. It concludes with an in-depth analysis of stability, bifurcation points, and the response plot of the system.

Video Highlights

01:28 - Combination resonance condition
03:10 - Mathematical modeling
18:45 - Intermittency route to chaos
25:00 - Example - flexible beam with payload
40:00 - Example - nonlinear beam with pulsating axial load

Key Takeaways

- The non-linear dynamics of a flexible beam can be studied using different types of beams such as elastic, viscoelastic, and magneto-elastic beams.
- The equation of motion is derived using Newton's second law and the extended Hamilton principle.
- The system with principle parametric resonance condition and combination parametric resonance condition is explored.
- Detuning parameters are used to consider the nearness of the external frequency to the frequency Omega one plus Omega two.
- The method of multiple scale is used to get a set of first-order differential equations.
- The stability of the system is determined by the eigenvalues of the Jacobian matrix.
- Bifurcation points are identified in the response plot of the system.