Domain-partition power series in vibration analysis of variable-cross-section rods

This paper presents a power series method with domain partition implemented in a matrix formulation, as an alternative to other power series techniques in vibration analysis. The proposed method solves linear differential equations efficiently up to a desired degree of accuracy and remedies two limitations of the conventional power series method. One limitation is related to the convergence domain of the series solution. If this domain does not include the region under analysis, the series expansion gives meaningless results. The other limitation is computational in nature; numerical difficulties arise when calculating natural frequencies, modes of vibration and dynamic stiffness of continuous models at high frequency. To compare some of the available implementations of the power series method in modal analysis, the longitudinal vibration of a rod with linearly varying area is studied. By means of this simple example, it is demonstrated that the power series method with domain partition provides more versatility than the power series approximation on complete domains.