Exact and Direct Modeling Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom

Shock and Vibration, Jul 2018

An exact and direct modeling technique is proposed for modeling of rotor-bearing systems with arbitrary selected degrees-of-freedom. This technique is based on the combination of the transfer and dynamic stiffness matrices. The technique differs from the usual combination methods in that the global dynamic stiffness matrix for the system or the subsystem is obtained directly by rearranging the corresponding global transfer matrix. Therefore, the dimension of the global dynamic stiffness matrix is independent of the number of the elements or the substructures. In order to show the simplicity and efficiency of the method, two numerical examples are given.

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Exact and Direct Modeling Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom

Received February Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom 0 Shilin Chen Michel Geradin Aerospace Laboratory (LTAS) University An exact and direct modeling technique is proposed for modeling of rotor-bearing systems with arbitrary selected degrees-of-freedom. This technique is based on the combination of the transfer and dynamic stiffness matrices. The technique differs from the usual combination methods in that the global dynamic stiffness matrix for the system or the subsystem is obtained directly by rearranging the corresponding global tramfer matrix. Therefore, the dimension of the global dynamic stiffness matrix is independent of the number of the elements or the substructures. In order to show the simplicity and efficiency of the method, two numerical examples are given. © 1994 John Wiley & Sons, Inc. INTRODUCTION The subject of rotor-bearing dynamics has re­ ceived considerable attention over the last few decades. The reason for this interest in rotor­ bearing dynamics is almost certainly the demand placed on manufacturers to continuously im­ prove both the power rating and the reliability of rotating machinery (Goodwin, 1992) . For dy­ namic analysis, modeling of the rotor-bearing systems is the first step. Various methods for modeling rotor-bearing systems have been devel­ oped and widely used during the past few dec­ ades. Among these techniques, the transfer ma­ trix method (TMM) and the finite element method (FEM) may be most commonly used for multi-degree-of-freedom (MDOF) rotor-bearing systems. The outstanding advantage of the TMM is that it requires calculations using matrices of fixed size, irrespective of the number of DOF in the problem. This means that the computational complexity is low even when dealing with sys­ tems with hundreds of DOF. However, only the natural frequencies, mode shapes, and harmonic response are available so far by this method. The other important modal parameters such as modal mass, modal stiffness, and transfer function of the system cannot be obtained. Furthermore, the stiffness and mass matrices that are important for modeling the systems are not available when the TMM is used. At the current state of rotor dynamic technol­ ogy, the FEM has proved to be powerful and versatile. It is also the only validated tool avail­ able at the present time for nonlinear systems and for transient dynamic analysis. However, in this method, the number of DOF is very high for large rotor-bearing systems. Many unwanted DOF such as rotational and internal DOF are used in the model. This makes it difficult for the FEM model to compare with the experimental model due to the large difference in the numbers of coordinates used in both models. If such comparison is necessary, the FEM model has to be reduced, which may be uneconomical in some cases. Another alternative is the dynamic stiffness matrix method (DSM). It may be considered as an improved FEM. Being different from the FEM, the DSM uses the analytical solutions of the governing equations as shape functions. Therefore, the obtained DSM is exact in the sense ofthe exact governing equation. These ma­ trices are in general parametric in terms of the vibration frequency and the load factor. One of the most important advantages of the DSM is that the DOF needed to model a struc­ ture is significantly reduced compare to the FEM. This is due to the fact that a uniform shaft, for example, can be taken as long as needed in the DSM. There is a significant number of refer­ ences describing the method. Fergusson and Pilkey (1991 a,b, 1993a,b) have reviewed the rele­ vant literature up to 1992. Most of the literature up to the end of 1992 may be found in their re­ views. However, there are some drawbacks associ­ ated with the DSM. First, the number of DOF is still high for large structures. Similarly to the FEM, many unwanted DOF such as rotational degrees remain in the model. Second, the calcu­ lation of modal parameters requires the solution of a highly nonlinear (trascendental) eigen­ problem: [D(w)]X = o. To this purpose, some special algorithms should be used, for example, the algorithms described by Williams and Wittrick (1970) , Wittrick and Williams (1971) , and Richards and Leung (1977) . The former algorithm requires calculation of nat­ ural frequencies for each individual beam ele­ ment with its end fixed. This increases the com­ putational time significantly. In order to minimize the dynamic DOF with­ out any loss of accuracy, a combination method was introduced by Dokainish (1972) for plate vi­ bration problems in which the element transfer matrix was obtained directly from the element stiffness and mass matrices. In recent years, this combination method has been improved by other researchers (Chiatti and Sestieri, 1979; Ohga et aI., 1983; Degen et aI., 1985) for different ap­ plications. In this method, the eigenfrequencies and mode shapes are calculated from the g (...truncated)


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Shilin Chen, Michel Géradin. Exact and Direct Modeling Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom, Shock and Vibration, 1, DOI: 10.3233/SAV-1994-1601