Eigen Value Problems in Vibrations — Lesson 1

This lesson covers the concept of Eigen value problems in the context of vibrations. It delves into the intricacies of boundary conditions, geometric boundary conditions, and natural or force boundary conditions. The lesson further explains the types of boundary conditions and their implications in different problems. It also discusses the concept of admissible functions, comparison functions, and Eigen functions. The lesson then moves on to the topic of self-adjoined systems and positive definite systems, explaining their significance in obtaining solutions. The lesson concludes with the orthogonality of Eigen functions. An example of a beam with a tip mass is used to illustrate these concepts.

Video Highlights

05:33 - Explanation of the trivial solution and the aim to find non-zero solutions.
8:27 - Discussion on the conditions for the type of problems and the concept of Sturm Liouville theorem.
18:16 - Explanation of the concept of admissible functions, comparison functions, and Eigen functions.
26:12 - Discussion on the concept of self adjoined system and positive definite system.
43:38 - Explanation of the orthogonality of Eigen functions.
65:23 - Explanation of the orthogonality conditions for Eigen functions.

Key Takeaways

- Boundary conditions play a crucial role in solving Eigen value problems in vibrations. They can be geometric or natural/force boundary conditions.
- Admissible functions, comparison functions, and Eigen functions are different types of functions used in these problems. Each has its own characteristics and requirements.
- Self-adjoined systems and positive definite systems are important concepts in obtaining solutions. They ensure that all Eigen values are real and positive.
- Eigen functions are orthogonal, which is a key property used in solving these problems.
- Real-world examples, like a beam with a tip mass, help in understanding these abstract concepts.