A SUBSTRUCTURE METHOD FOR THE TRANSIENT ANALYSIS OF NON-LINEAR ROTORDYNAMIC SYSTEMS USING MODAL ANALYSIS

摘要

Rotordynamic systems are characterized by unsymmetric matrices due to gyroscopic and circulatory effects. Additionally, non-linearities are present resulting e.g. from the bearing behaviour. There is a need for efficient numerical routines to treat the case of catastrophic loading of such systems in the time domain. In order to get a minimum number of non-linear equations, a partitioning of the system into two substructures is performed by separating the linear parts of the system from the non-linearities. The motion of the linear rotor, which forms substructure 1, is described in the state space using the configuration-space modes and modal impulse response functions of the linear symmetric parts of the system matrices. This avoids the more costly analysis with complex numbers due to gyroscopic effects and relates the state-space formulation to modal analysis with real eigenvectors of the configuration space. Substructure 2 carries the circulatory terms and the non-linearities. It is formulated analogous to substructure 1, where the non-linearities, the circulatory terms and the coupling forces are treated again as pseudo-forces of the linear system with symmetric matrices. The time evolution of the response in both substructures thus can be formally represented using a Duhamel-type integration procedure. Inserting the coupling conditions and approximating the pseudo-forces, a set of non-linear equations is reached, where the number of equations corresponds to the number of non-trivial components of the non-linear restoring force in the configuration space. Using a time-stepping procedure, this system of incremental equations can be solved by means of any appropriate non-linear solution procedure. In the more simple case of non-linearities depending only on the coordinates of substructure 2, the state-space representation breaks down to a formulation in the configuration space. Further reductions of the computational effort can be achieved using modal analysis with a reduced base of eigenvectors. As a numerical application, the sudden occurrence of an eccentricity due to a broken part of the rotor is treated. The efficiency of the present method is demonstrated with respect to the Newmark integration method.