Natural frequencies of a beam with varying section properties

Abstract

Non-uniform beams with cross-section varying in a continuous or non-continuous manner along their lengths are used in many structural applications in an effort to achieve an optimum distribution of strength and weight. The authors have investigated the problem of calculating the natural frequencies for beams with inertia, area and mass varying in a general manner. The procedure proposed allows the effect of each of these variables to be evaluated separately and leads to the concept of effective inertia, effective area and effective mass. It also enables one to determine the natural frequencies of a beam with varying section properties using a simple formula similar to the one used in practice for computing the natural frequencies for a uniform beam. It is also shown that the effect of shear deformation upon the natural frequencies of a beam with classical boundary conditions can be estimated within the limits of practical accuracy by using a simple formula similar to the one used for simply supported beams. An important feature of the results presented here is the simplicity and relative ease with which these expressions can be applied to beams with various types of boundary conditions. The authors believe that a practicing engineer will find these results easy to apply in computing the natural frequencies of non-uniform beams within practical limits.