The study of the equations of motion for a general space discretized system can be performed with reference to either the frequency domain or the time domain, yielding the natural frequencies, the mode shapes, and time free and forced responses. When the system is linear and the sets of equations are written in terms of vectors and matrices, the tools of linear algebra can be used to cope with the problems of motion (for example, see [19] or [18] for a general introduction to lumped parameters system, and [73] or [74] for the related theory of matrices, including numerical aspects).

In particular, for what the lateral dynamics of rotating systems is concerned, a further analytical tool is introduced and used throughout the present book, namely, the complex coordinates approach, in which complex numbers are used to represent generalized displacement vectors. This approach proved to be very expedient for modeling both single- and multi-degrees-of-freedom rotors, particularly when the whole system is axially symmetrical and the forcing functions acting in the rotation plane in the direction of the coordinate axes are in quadrature, i.e., with a phase delay of ?/2 with respect to each other, like unbalance forces.