Contents
1. Introduction 1
2. Research.........................................................................................................................................3
3. Tabla - an insight. 4
3.1 Recommended listening. 7
4. Aims of the project. 8
5. General Methodology 8
6. Mathematical modelling of systems in state space. 9
6.1 Derivation of state equations. 10
6.1.1 Method 1. Explicit.......................................................................................................................11
6.1.2 Method 2. The daring approach. 13
6.2 Multiple spring-mass System 14
6.3 A three mass system 15
6.4 The choice of output variables. 16
6.5 Initial conditions to the system 18
6.6 Implementation on matlab. 18
6.7 The ‘hits’ Implemented 20
6.8 Constants used in the file tabla.m 21
6.9 The full 2 Dimensional model 22
6.9.1 Boundary conditions 25
6.10 The choice of output variables for 2d model 26
7. Verification of quality of models 27
8. Results of State-space models 28
8.1 The one dimensional model 28
8.2 The two-dimensional model 28
9. What are the eigenmodes and eigenvalues:- 31
9.1 Experiment to find the eigenvalues and eigenvectors. 32
9.1.1 Results of the Experiment 34
9.2 Some thoughts and ideas on implementation 36
9.3 Finding the Eigenvalues and Eigenvectors of the 2D state space model 37
9.3.1 Degenerate modes. 40
10. General theory of Finite Elements 41
10.1 A simple one dimensional element : the pin jointed bar 43
10.2 A one dimensional finite element example : a string. 45
10.3 Further finite element theory. 50
10.3.1 The Finite Element eigenvalue expression 50
10.4 Derivation of the higher order element stiffness matrices. 52
10.4.1 Natural co-ordinates. 52
10.4.2 Derivation of a stiffness matrix for a quadratic triangular element 53
10.4.3 Derivation of a stiffness matrix for a quadratic triangular element 55
10.4.4 Gauss Quadrature 56
10.4.5 Membrane forces 57
11. Introduction to the implementation of the finite element method.59
11.1 Break down of the finite element input file 62
11.2 Results of Finite element analysis. 77
12. Conclusion. 88
13. Appendix 89
13.1 Tabla.m 89
13.2 drum.m 92
13.3 sscon.m 94
13.4 Finite element example of a ‘string’ implemented on Matlab. 95
13.5 The Finite element input file - drum2.inp 96
14. Bibliography. 99