ABSTRACT
In Chapter 5, gyroscopic effects on a rotor with a single disc and massless elastic shaft were discussed in great detail by using quasi-static and dynamic analyses with the help of the analytical approach. Multi–degree-of-freedom (DOF) systems with the gyroscopic effect of thin discs were described while discussing the transfer matrix method in Chapter 8. There we did not consider gyroscopic effects due to thick flexible shafts (i.e. a shaft with distributed mass and stiffness properties). In the previous chapter, we dealt with analysis of a simple rotor with the analytical (i.e. the continuous) and numerical (finite-element) approaches without considering the important higher effects, such as shear deformation, rotary inertia, and gyroscopic effects. In the present chapter, the finite-element analysis will be extended to initially a single disc of a simple rotor (i.e. the Euler–Bernoulli beam with distributed properties). Then it will be extended for more general rotor systems (i.e. the Timoshenko beam model) with the help of a powerful analysis tool of the finite-element method. The Timoshenko beam theory will be used for the development of a governing equation of the continuous system analysis. The finite-element formulation will be developed for the spinning Timoshenko beam, which includes higher effects like rotary inertia, shear deformation, and gyroscopic effects. The eigenvalue problem will be developed through the state-space form of the governing equation. With the help of examples, the extraction of modal parameters from the special eigenvalue problem will be explained. The standard Campbell diagram for various cases is obtained, which shows the natural whirl frequency variation with the spin speed of the shaft for asynchronous whirl. This diagram can be used to obtain critical speeds of any complex rotor systems.
