Vibration analysis of a beam with uncertain-but-bounded parameters using interval finite element method

Abstract

This thesis investigates the vibration of beam for computing its natural frequency with uncertain-but-bounded parameters i.e. interval 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. Thus we will have the finite element equations having the interval stiffness and mass matrices. So, in turn one has to solve by the problem by interval generalized eigenvalue problem. This requires the complex interval arithmetic and so detail study of interval computation related to the present problem has been done. First homogeneous beam with crisp values of material properties are considered. Then the problem has been undertaken taking the material properties as interval. Initially, Young’s modulus and density have been considered as interval separately, and then the problem has been analyzed using both Young’s modulus and density properties as interval. Next, similar investigations for non-homogeneous beam have also been done. Although the non-homogeneity makes the problem more complex but this may be the actual representation of a general beam. The considered interval material properties are in term of , where is called the uncertainty factor. Using interval computation the interval generalized eigenvalue problem has been solved by a new proposed method. Solution of the interval eigenvalue problem gives the interval eigenvalues which are the natural frequencies in each cases of the beam as above. The computed results are shown in terms of table and plots.