Galerkin and Finite Difference Methods — Lesson 3

This lesson covers the Galerkin and Finite Difference Methods, two types of approximant methods used in vibration analysis. The Galerkin method requires a differential equation instead of an energy expression and uses an assumed shape function that satisfies boundary conditions. The Finite Difference Method converts a differential equation into a difference form, resulting in a set of algebraic homogeneous equations. The lesson also explains how to apply these methods to solve problems, using a simply supported beam as an example. It further discusses the concept of residual error and how it is orthogonal to the weight of the function in the domain of the structure.

Video Highlights

01:32 - Discussion on the development of the Gallarkin method from the virtual work principle.
26:30 - Finite difference method, an approximate method used in many problems in mechanics.
22:22 - Illustration of the Gallarkin method with an example of a simply supported beam of uniform cross section and material properties.
38:40 - Explanation of how to apply the finite difference method to vibration problems.
48:46 - Demonstration of a problem with a simply supported beam using the finite difference method.

Key Takeaways

- The Galerkin method requires a differential equation and an assumed shape function that satisfies boundary conditions.
- The Finite Difference Method converts a differential equation into a difference form, resulting in a set of algebraic homogeneous equations.
- Both methods generate a residual error, which is orthogonal to the weight of the function in the domain of the structure.
- The methods can be applied to solve problems in vibration analysis, with the lesson providing an example of a simply supported beam.
- Increasing the number of divisions in the Finite Difference Method can enhance the accuracy of the results.