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.

- The system is stable if the eigenvalues remain within the unit circle and unstable if they lie outside it.
- For steady state, as time tends to infinity, the eigenvalues tend to infinity as well, leading to an unbounded response or system instability.
- The Floquet multiplier can be used to study the stability of a periodic system, with the system being stable if the multiplier is negative.

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