Dynamic analysis models are customarily employed in turbomachinery design to predict critical whirling speeds and estimate dynamic response due to loads imposed by unbalance, misalignment, maneuvers, etc., Traditionally these models have been assembled from beam elements and been analyzed by transfer matrix methods. Recently there has been an upsurge of interest in the development of improved dynamic models making use of finite element analysis and/or component mode synthesis. We are currently developing a procedure for modelling and analyzing multi-rotor systems [1] which employs component mode synthesis applied to rotor and stator substructures. A novel feature of our procedure is a program for the automatic generation of the component modes for substructures modelled as Timoshenko beam elements connected to other substructures by bearings, couplings, and localized structural joints. The component modes for such substructures consist of constraint modes and internal modes. The former are static deflection shapes resulting from removing the constraints one at a time and imposing unit deflections at the constraint locations. The latter have traditionally been taken to be a subset of the natural modes of free vibration of the substructure with all constraints imposed. It has however been pointed out [2] that any independent set of geometrically admissible modes may be used. We take advantage of this and employ static deflections under systematically selected loading patterns as internal modes. All component modes are thus obtained as static deflections of a simplified beam model which has the same span and same constraints as the actual substructure but which has piecewise uniform dynamic properties. With the loading patterns we employ, all modes are represented by fourth order polynomials with piecewise constant coefficients. We have developed an algorithm for the automatic calculation of these coefficients based on exact integration of the Timoshenko beam equation using singularity functions. The procedure is illustrated by applying it to a simplified system with a single rotor structure and a single stator structure. The accuracy of the procedure is examined by comparing its results with an exact analytical solution and with a component mode synthesis using true eigenfunctions as internal modes.
Study of the Discrete Shear Gap Technique in Timoshenko Beam Elements