Matrix Techniques for Determination of Fundamental Mode Shape and Frequency Parameter of Torsional Oscillations in Engines

Abstract

Matrix techniques for the solution of natural frequencies of a torsional system are proposed for their simplicity and easy manipulation. The tedious calculations in obtaining a dynamic matrix are alleviated by evolving a general expression for dynamic matrix for fundamental mode of a multi-degrees of freedom straight torsional system.

Further, the dynamic matrices for engine systems having four to eight cylinders with equal moments of inertia per cylinder line and equal shaft stiffnesses between the cylinder centres are deduced from a general expression. They are iterated upon five times to obtain expressions for the fundamental frequency parameter, which are further used to obtain the relative amplitudes and the frequency parameter of different engine systems. The results obtained by five times iteration are in close agreement with the exact values.