Natural Frequencies for Various End Conditions — Lesson 2

This lesson covers the concept of transverse vibration of beams. It starts with the derivation of the differential equation that includes damping properties and external excitation. The lesson then moves on to the solution of the free vibration problem, demonstrating how the solution is obtained in terms of hyperbolic and trigonometric functions. The lesson further discusses how to find the natural frequencies and mode shapes of the beam with various end conditions. It also explains the significance of the free vibration solution in determining these properties. The lesson concludes with a discussion on the orthogonality of mode shapes and a numerical example to illustrate the concepts.

Video Highlights

02:07 - Different boundary conditions for beams and their applications.
04:10 - Formulation of natural frequencies and mode shapes, and the use of numerical techniques to solve transcendental equations.
04:47 - Derivation of the partial differential equation of motion of a beam.
06:02 - Mode shape function and how to find the natural frequencies.
58:25 - Importance of nodal points in the vibration of a beam.

Key Takeaways

- The differential equation for transverse vibration includes damping properties and external excitation.
- The solution of the free vibration problem is obtained in terms of hyperbolic and trigonometric functions.
- The free vibration solution is significant in determining the natural frequencies and mode shapes of the beam.
- The orthogonality of mode shapes is an important concept in understanding beam vibrations.
- Practical examples and numerical problems help in understanding and applying the concepts.