Torsional Vibrations of Rotor Systems by the Continuous System and Finite Element Methods

In Chapter 6 , we considered torsional vibrations of rotor systems using Newton’s second law of motion and the transfer matrix method. The transfer matrix method especially proved its applicability for analyzing even large rotor systems in a more systematic (algorithmic) fashion. The advantage of the method is that the size of matrices does not increase with the degrees of freedom (DOFs) of the system. The only disadvantage of the method is that natural frequencies have to be obtained for large systems using root searching numerical techniques. Moreover, it has some other shortcomings that we observe, especially when we analyze transverse vibrations in Chapter 8 . There, the method requires special treatment when rotor systems have multiple supports and when the shaft is treated as continuous (i.e., when the mass and the stiffness of the shaft are distributed throughout its span). In the present chapter, we first analyze torsional vibrations using the analytical approach by treating the shaft as continuous, which has infinite DOFs. For this case, the governing equation becomes a partial differential equation (i.e., identical to the wave equation ). For simple boundary conditions, governing equations can be solved in closed form. However, for complicated rotor-support systems, the continuous rotor system is very difficult to analyze by analytical methods, and often approximate methods are used.