A Floating-Frame-of-Reference Formulation for Deformable Rotors Using the Properties of Free Elastic Vibration Modes

Formulations in rotordynamics are usually based on the assumption that the displacements of the bearings of the rotor are small, such that, besides the axial rotation, no large rigid-body motions have to be taken into account. This results in linear equations of motion with gyroscopic terms. When the axial angular speed of a rotor is increased, however, as well as for rapidly changing transient conditions, a non-linear coupling between the large axial rotation and the small rigid body motion induced by the compliance of the bearings and the small elastic deformation of the rotor body itself is to be expected. It is the scope of the present contribution to present a rational strategy for dealing with this situation. First, we present a problem-oriented version of the floating-frame-of-reference formulation (FFRF). We use a co-rotating rigid rotor as reference configuration, which allows using linear modes of the non-rotating elastic rotor as Ritz approximations. The position vector of the origin of a body-fixed coordinate system and three suitable Bryant angles are used as rigid body coordinates, and free elastic modes of the rotor are considered as elastic Ritz approximations. The properties of the latter and their consequences upon simplifying the necessary spatial integrals in the FFRF are addressed in some detail. The free modes are obtained from a Finite Elements pre-processing of the elastic rotor body. The non-linear equations of motion of the rotor are obtained afterwards by means of symbolic computation This formulation leads to a set of relations, in which the rigid-body degrees of freedom need not to be small, and which is integrated using an implicit scheme. Results for a rotor with unbalance forces, accelerated by external forces and having linear visco-elastic bearings are successfully compared to a commercial multi-body dynamics code.