A Modal Based Approach for the Lateral Dynamic Analysis of Rotating Shafts Supported by Elastic Substructures

Abstract

It is well known that the detailed modelling of the structures supporting modern, heavy-duty gas turbines is essential for accurately estimating the dynamic behaviour of the complete coupled system. In this paper, the equations of motion governing the lateral dynamics of a rotating shaft supported by an elastic substructure are assembled by making exclusive use of modal parameters — such as modal damping ratios, undamped natural frequencies and modal shapes — of both the supporting structure and the undamped, non-rotating and unconstrained rotor. Thanks to the particular approach of considering the rotor in its unconstrained configuration, the modal base used to derive the equations of motion becomes independent on the rotational speed. It therefore needs to be extracted only once. Gyroscopic effects can then be effortlessly introduced by projecting the gyroscopic matrix on the modal base described above. Harmonic, modal and transient analyses can be carried out by means of the proposed set of equations. Additionally, since the mathematical expression of a modal tracking parameter is reduced to a simple scalar product by the particular shape of the proposed system matrices, Campbell’s diagram analyses result to be simplified. An actual example of application to the modal and unbalance analyses of a heavy-duty gas turbine engine is presented. The proposed approach is used to produce the Campbell’s diagram of the complete rotor-structure system and extract bearing vibration velocities for a typical case of unbalance. Results are then compared on the one hand with those calculated by a well-known commercial tool and on the other hand with real measurement data. Since a key benefit of the present approach is that it is fully independent from the method used to extract the input modal parameters, it allows using results from models of the most appropriate complexity — 1D, 2D or 3D. Alternatively, and as shown in this paper, validated measurement data can also be used for fine-tuning of the input parameters in order to close potential gaps between numerical models and real machine dynamic behaviour.