On The Torsional Vibration of Branched Systems Using Extended Transfer Matrix Method

Abstract

A novel method for the analysis of free vibration of branched torsional systems is presented. The method is radically different from the traditional methods in that an extended transfer matrix relation is formulated for each branch. For this, the calculations are propagated from the junction and proceed simultaneously in all branches toward their respective ends. Then by substituting the compatibility and equilibrium conditions, a frequency dependent characteristic matrix is formulated. This procedure automatically eliminates the need of any additional operation such as matrix inversion and the solution of a system of equations for the formulation of the characteristic matrix and also reduces the size of the matrix. Finally, the boundary conditions are applied to the matrix relation and the natural frequencies are determined from the roots of a frequency determinant derived from the characteristic matrix. For this purpose, the paper introduces a method based on the Newton-Raphson iterative technique which systematically finds the roots of the frequency determinant using both the value of the determinant and its derivative with respect to square of the natural frequency. The paper also presents a procedure for calculating these derivatives directly from the formulation of the extended transfer matrices. Numerical examples are given to illustrate the simplicity and straightforwardness of the proposed method in finding the natural frequencies of complex branched torsional systems. Results indicate that the method is accurate and allows a greater degree of error in the selection of trial frequencies.