The torsional natural frequencies and vibration mode shapes can be calculated using an eigenvector/eigenvalue matrix solution technique which directly solves the differential equations of motion for the lumped mathematical model of the torsional system.
Normally, the damping in torsional systems is relatively low and has little effect on the natural frequencies. Therefore, the torsional natural frequencies and mode shapes can often be calculated assuming the damping is zero. The torsional natural frequencies are determined from the eigenvalues and the mode shapes from the eigenvectors. Each set of eigenvectors is normalized with the maximum amplitude equal to one and plotted to illustrate the vibration mode shapes. The stiffness of non-linear couplings vary with load and speed, and may result in some slight changes of the mode shapes.
Torsional natural frequencies of some simple systems can be calculated using the formulas given below. Another method suitable for hand calculation is the Holzer method which is an iterative procedure for calculating torsional natural frequencies and mode shapes.
