A procedure is presented for dynamic modeling of rotor-bearing systems which consist of rigid disks, distributed parameter finite rotor elements, and discrete bearings. The formulation is presented in both a fixed and rotating frame of reference. A finite element model including the effects of rotatory inertia, gyroscopic moments, and axial load is developed using the consistent matrix approach. A reduction of coordinates procedure is utilized to model elements with variable cross-section properties. The bearings may be nonlinear, however, only the linear stiffness and viscous damping case is considered. The natural whirl speeds and unbalance response of a typical overhung system is presented for two sets of bearing parameters: (i) undamped isotropic, (ii) undamped orthotropic. A comparison of results is made with an independent lumped mass analysis.
The use of finite elements for simulation of rotor systems has received considerable attention within the last few years. The published works have included the study of the effects of rotatory inertia, gyroscopic moments, axial load, and internal damping; but have not included shear deformation or axial torque effects. This paper generalizes the previous works by utilizing Timoshenko beam theory for establishing the shape functions and, thereby including transverse shear effects. Internal damping is not included but the extension is straight forward. Comparison is made of the finite element analysis with classical dosed form Timoshenko beam theory analysis for nonrotating and rotating shafts.
The implementation of finite element simulations for the study of rotor dynamic systems has been the subject of recent publications. Since the finite element offers obvious modeling advantages, particularly in modeling large-scale systems, this study extends the linear finite element concept to provide a detailed evaluation of damped rotor stability. In this work the effects of both internal viscous and hysteretic damping have been incorporated into the finite element model. Both produce circulatory terms in the generalized equations of motion which encourages the destabilization of this nonconservative system. Results are presented for both hysteretic and viscous forms of damping. Both forms of internal damping destabilize the rotor system and induce nonsynchronous forward precession. The stabilizing effects of anisotropic bearing stiffness and external damping are also demonstrated.
A method of component mode synthesis is presented for the analysis of multishaft rotor-bearings systems. The motion of each component of the system is described as the superposition of constraint modes associated with boundary coordinates and constrained precessional modes associated with internal coordinates. The constrained precessional modes for each component are truncated and the reduced component equations are assembled to yield a set of system equations. The nonsymmetric nature of the general problem requires the utilization of biorthogonality relations between right and left vector sets in order to decouple the component precessional modes. The method is developed for damped whirl speed/stability analysis and comparative results are presented for various levels of mode truncation for two example systems.
A new quantitative method of estimating steady state periodic behavior in nonlinear systems, based on the trigonometric collocation method, is outlined. A procedure is developed to analyze large rotor dynamic systems with nonlinear supports by the use of the above method in conjunction with Component Mode Synthesis. The algorithm discussed is seen to reduce the original problem to solving nonlinear algebraic equations in terms of only the coordinates associated with the nonlinear supports and is a big improvement over commonly used integration methods. The feasibility and advantages of the procedure so developed are illustrated with the help of an example of a typical rotor dynamic system with an uncentered squeeze film damper. Future work on the investigation of the stability of the periodic response so obtained is outlined.
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