Mathematical modeling of a beam using finite elements, with application to an Mbira reed

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Unversity of Dar es Salaam
The main purpose of this study was to develop the general mass and stiffness matrices for standard beam equation and eventually obtain the finite element equations for the mbira reed. So far few studies have been done on mathematical modelling of a beam using finite elements, with applications to an mbira reed. However none of these studies had considered the aspect of a tapering mbira reed. By construction, the mbira reed is not of uniform cross-section. It is a tapering beam along its length. The study intends to examine the relationships that exist between the natural frequencies and the varying dimensions and properties of mbira reed. The finite element method was used as a numerical method to solve the mathematical model of the mbira reed. Two reeds were selected from the mbira and some breadth and depth measurements were taken. These measurements were used in the development of a mathematical model of the mbira reed. Each selected reed was plucked and through the microphone, its frequency was recorded (the measured frequency). The model was solved by the aid of a MATLAB code to obtain the model frequency. The two frequencies were compared. There were little differences between the measured and modelled frequencies. However, it was found that the mathematical model predicted the natural frequencies of the two reeds reasonably accurately. To improve or extend this study, one may carry a study on; The effect of curvature on the mbira reed. A finite element model of the wooden base to investigate its resonances.
Available in print form, East Africana Collection, Dr. Wilbert Chagula Library, Class mark (THS EAF TA347.F5M63)
Finite element method, Beam, Mbira reed