This lesson covers the concept of transverse vibration in beams, a crucial aspect of structural dynamics. It delves into the different beam theories, including Euler Bernoulli, Release, Shear, and Timoshenko beam theories, and their respective assumptions. The lesson also explains how to derive the governing differential equation of motion for Euler Bernoulli beam model using Newton's Second Law and Hamilton's equation. It further discusses the formulation of a boundary value problem and how to obtain non-trivial solutions for determining infinite eigenvalues and eigenfunctions of the beam. For instance, the lesson uses the example of a cantilever beam to illustrate how different boundary conditions can lead to various solutions.

- Transverse vibration in beams is a significant mode of vibration in a continuous system.
- There are four main beam theories: Euler Bernoulli, Release, Shear, and Timoshenko, each with its assumptions and applications.
- The governing differential equation of motion for Euler Bernoulli beam model can be derived using Newton's Second Law and Hamilton's equation.
- A boundary value problem can be formulated to find the non-trivial solutions for determining infinite eigenvalues and eigenfunctions of the beam.
- Different boundary conditions can lead to various solutions, as illustrated using a cantilever beam example.

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