Fuzzy finite element method for vibration analysis of imprecisely defined bar

Abstract

This thesis investigates the vibration of a bar for computing its natural frequency with interval or fuzzy material properties in the finite element method. The problem is formulated first using the energy equation by converting the problem to a generalized eigenvalue problem. The generalized eigenvalue problem obtained contains the mass and stiffness matrix. In general these matrices contain the crisp values of the parameters and then it is easy to solve by various well known methods. But, in actual practice there are incomplete information about the variables being a result of errors in measurements, observations, applying different operating conditions or it may be maintenance induced error, etc. Rather than the particular value of the material properties we may have only the bounds of the values. These bounds may be given in term of interval or fuzzy. Thus we will have the finite element equations having the interval and fuzzy stiffness and mass matrices. So, in turn one has to solve by interval or fuzzy generalized eigenvalue problem. As such detail study related to interval and fuzzy computation has been done. First crisp values of material properties are considered. Then the problem has been undertaken taking the properties as interval and fuzzy. Initially, Young’s modulus and density as material properties have been considered as interval in two different cases, one for homogenous and other one for non-homogenous material properties. Then the problem has been analyzed using Young’s modulus and density properties as fuzzy. First Fuzzy material properties in terms of fuzzy number that is triangular fuzzy number is considered then trapezoidal fuzzy number is considered in the finite element method. The fuzzy material properties are solved using -cut to obtain the corresponding intervals. Then using interval computation natural frequencies are obtained and the fuzzy results are depicted in term of plots.