Finite element vibration analysis using bar and beam elements with a new method for lumping mass matrix

Abstract

The existing mass lumping schemes revolve around the concepts of mass and momentum conservation. Hence there is a gap for us to explore new schemes revolving around other conservation laws, such as the conservation of kinetic energy. By being able to develop a code for it, we would enable everyone to be able to solve complex problems regarding structures, as long as they have all the information required to solve the problem. This project seeks to derive new methods for lumping mass matrices. In order for the new method to be able to solve problems pertaining to higher order elements, this project also seeks to develop relevant computer program codes. The code will therefore be implemented and validated with existing problems to confirm the accuracy and reliability. If the code is deemed to be accurate, we can extend its usage even to people with no prior knowledge of the concepts. In this project, the development of a code will be done with the help of the MATLAB program. The code will then be put to the test to solve 2 node bar elements in order to determine natural frequencies and mode shapes. The project will then have to test the accuracy of the code by calculating the lowest natural frequencies and comparing it against the exact analytical solutions. In this case, we look to utilise the inverse iteration method which also acts as the backbone of the MATLAB code.