Overview

Another of the assumptions on which the analysis seen in Part 1 was based is that of linearity. If this assumption is dropped, no general solution of the equations of motion can be achieved and the alternatives are the numerical solution by step-by-step integration in the time domain or the search for approximate solutions, using the classic approximation techniques typical on nonlinear dynamics.

The difference between the behavior of axi-symmetrical and nonisotropic systems is even larger in the case of nonlinear rotors than that already seen for linear ones. If the system is axially symmetrical, circular whirling is an exact solution for the unbalance response, although the nonlinearity of the system makes it possible for other solutions to exist. This differentiates the behavior of a nonlinear rotating system from that of a nonlinear oscillator, where no closed-form solutions can usually be found. The closed-form solutions (in some cases, a single solution, but in other cases, the existence of multiple solutions has been confirmed) constitute attractors, whose basins of attraction share the phase space with other possible solutions of different type, if and when they exist at all. The impossibility of demonstrating that a solution exists and that it is unique leads to the possible existence, even in those cases, of multiple solutions, with the jump phenomenon typical of nonlinear dynamics, polyharmonic, and even chaotic solutions.

On the contrary, if no axial symmetry exists, particularly if the nonrotating parts of the machine are nonisotropic, a situation similar to that characterizing...