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 global transfer matrix. Therefore, it may be considered as an improved transfer matrix method with ap­ plication to plates. However, for rotor-bearing systems it seems to be unnecessary to obtain the element transfer matrix from the element stiff­ ness and mass matrices because the element transfer matrix may be derived directly from the governing equations in an exact manner (Lund and Orcutt, 1967; Lee et aI., 1991) . For rotor-bearing systems, the idea of combin­ ing the TMM and FEM to reduce the DOF was first proposed by Dimarogonas (1975) . A static deflection function between nodes were used in his article. This idea was used for stability analy­ sis of a rotating shaft. In this article an exact and direct modelling technique for rotor-bearing systems based on the combination of transfer and dynamic stiffness matrices is presented. In this technique, the en­ tire structure is first divided into several sub­ structures based on the required master DOF. Each substructure may consist of a large number of basic elements. The DOF for a substructure may be partitioned into two sets. One set is the internal DOF, and the other set is the boundary DOF. The transfer matrix of each substructure relates only the boundary DOF. The dynamic stiffness matrix of the substructure is obtained by rearranging the corresponding transfer matrix. In this way, the internal DOF are not used in the model. In other words, a substructure is reduced exactly to an equivalent element whose nodal coordinates are the boundary coordinates of the substructure. The boundary coordinates dy­ namic stiffness matrices of substructures become the basic matrices for assembling the global dy­ namic stiffness matrix for the original structure. The order of the system eigenvalue equation is equal to the number of physical boundary coordi­ nates between substructures and is frequency de­ pendent. Because the dynamic stiffness matrices of the basic elements such as shaft, lumped mass, and stiffness elements are exact, the number of modes predicted by the model is not limited by the number of master DOF used in the model. The consistent equations of motion of a rotat­ ing Timoshenko shaft subject to axial force de­ rived recently by Choi et al. (1992) are used as the governing equations. In the equations, the effects of rotary inertia, gyroscopic, and trans­ verse shear are taken into account. An exact transfer matrix for the shaft is derived directly from the governing equations following the pro­ cedure proposed by Lee et al. (1991) . The rotor­ bearing system is then reduced to a low dimen­ sion model with arbitrary selected DOF. All modal parameters of the rotor-bearing systems such as eigenfrequencies, mode shapes, modal masses, modal stiffness, and frequency response functions can be obtained from the established model. Two numerical examples are given to show the simplicity and high accuracy of the method. As a by-product, the exact dynamic stiffness matrix for a rotating shaft subject to ax­ ial force can be obtained by rearranging the cor­ responding exact transfer matrix using the tech­ nique described here. In the YZ plane: with the following quantities: E is the Young's modulus, G the shear modulus, K the shear fac­ tor, p the mass density, A the cross section area, and P the axial compression load. It is noted that not only the gyroscopic moments but also the axial load terms are consistently captured in the above equations. Transfer Matrix for a Shaft The steady-state solutions of Eq. (1) may be rep­ resented in the form DERIVATION OF EXACT TRANSFER MATRIX FOR A SHAFT Governing Equations Let us first examine a uniform shaft subject to constant compression axial load as shown in Fig. 1. For such a shaft, the consistent equations of motion have been derived recently using the finite strain beam theory (Choi et aI., 1992) . In the XZ plane: X(Z, t) = Xs(Z)sin f!t Y(Z, t) = Yc(Z)cos f!t (2) where f! is the whirling angular frequency. Substituting Eqs. (2) into (1) would result in two homogeneous equations as follows. In the XZ plane d4Xs dZ4 + (pf!2 E pf!2) d2Xs + KG dZ2 + (p2f!4 _ PAf!2) KGE EI Xs + 2pwf! d2yc -l- 2p 2wf!3 E dZ2 ' KGE Yc P ( P ) d2Xs pPf!2 p2 d2Xs pAf!2 (2P p2 p2 ) (3a) + EI 1 + KAG dZ2 - EIKG Xs - E2AI dZ2 - ---m - EA - EAKAG + (EA)2 X,\ = O. In the YZ plane P ( P ) d2yc pPf!2 p2 d2yc pAf!2 (2P p2 p2 ) + EI 1 + KAG dZ2 - EIKG Yc + E2AI dZ2 - ---m - EA - EAKAG + (EA)2 Y, = O. (3b) y p Z (4) (5) The solutions of Eqs. (3) take the form Xs = Use AZ Yc = Vc eAZ where Us, Vc are arbitrary real constants and }.. are the roots of the determinant equation II}..4 h+}..2f}.+.2 k+ g }..4 h+}..f2}.+.2 k+ g "-- 0 with p02 f = E p02 P p2 p2 + KG + El + KAGEl - E2Al p204 g = KGE pA02 pP02 pA 2P02 ~ - EIKG + El EA pp202 + E 21AKG p2p02 E3Al h = 2pwO E ' k = 2p 2w03 KGE' Equation (5) is equivalent to the following equa­ tions: Case 2. Vbi - 4cI < bl • This is true for bl > 0, and c, > O. Physically, this case would corre­ spond to higher frequency vibrations. The solu­ tions of Eq. (5b) in this case are given by: The numerical test has shown that this case may happen only at very high frequency. This is not the case for most industrial rotors. Combining the above two cases, the roots of the characteristic function of Eq. (5) may take the form: for a constant value of w (see also Lee et al., 1991; Zu and Han, 1992) . In the same way as the displacements, the slopes, moments, and shear forces may be de­ fined as follows: a(Z, t) = as(Z)sin Ot (3(Z, t) = (3c(Z)cos Ot Mx(Z, t) = Mxs(Z)sin Ot My(Z, t) = Myc(Z)cos Ot QxCZ, t) = Qxs(Z)sin Ot Qy(Z, t) = QycCZ)cos Ot. The state vector is defined as follows: The state vector on the right-hand side of the shaft is related to the state vector on the left­ hand side by the following matrix equation: (6) (7) where [T] is the transfer matrix of a shaft ele­ ment with dimension 8 x 8 and of the substructure in the following way: where N is the total number of elements and N = Nb + Nk + N m + N d. It should be noted that although the shaft element length may be taken as long as possible in the case of uniform cross section because of the use of the exact transfer matrix, the total number of elements of a sub­ structure or the entire structure may still be large for very long rotor systems. In this case, numeri­ cal instability due to matrix multiplication may occur. This difficulty may be overcome by using symbolic computation, for example the Maple software. On the other hand, if N is very large, a lot of matrix multiplication is needed. To speed up the calculation, parallel algorithms may be used. Substructure Dynamic Stiffness Matrix The global transfer matrix [T] of a substructure relates the forces and displacements at both ends (9) with where Fl and Xl are the force and displacement vectors at the left end of the substructure, Fr and Xr are the same quantities at the right end. Equa­ tion (10) may be rewritten in the dynamic stiff­ ness matrix form: (10) (11) Dll = - TI2l Til Dl2 = TI2l D2l = -T2l + T22 T I}TIl D22 = - T22 TI-} where [D] is the global dynamic stiffness matrix of the substructure whose elements are fre­ quency dependent. Note that the global dynamic stiffness matrix of the substructure has the same dimension as the element transfer matrix. Equation (11) tells us that the substructure shown in Fig. 2 is reduced to an equivalent ele­ ment whose nodal coordinates are the boundary coordinates of the substructure. The internal coordinates of the substructure are not used in Eq. (11). Global Dynamic Stiffness Matrix The global dynamic stiffness matrix for the entire structure can be assembled using the above dy­ namic stiffness matrices of all substructures. Afi+4 i+6 A substructure. ter introducing the boundary conditions at both ends, the dimension of the global stiffness matrix can further be decreased. The restrained global dynamic stiffness matrix is denoted by [Dg]. The unwanted DOF such as rotational DOF can be removed in an exact manner as shown in the fol­ lowing numerical examples. It is of interest to note that the global dynamic stiffness matrix obtained in this way is very low in dimension. It uses only the boundary coordi­ nates of the substructures as the DOF. The inter­ nal DOF are not contained in the model. It should be pointed out that if the selected DOF happen to be the boundary coordinates of the entire structure, then only one substructure is needed and all the internal DOF are not con­ tained in the model. Global Frequency-Dependent Matrices The global frequency-dependent matrices, that is, mass matrix, [M], gyroscopic matrix [G], and stiffness matrix, [K], may be obtained from the global dynamic stiffness matrix [Dg] (Richards and Leung, 1977; Yang and Pilkey, 1992; Leung and Fergusson, 1993) : [M ( !1)] = _ d[Dg(w, 0)] g W, d(!12) d[Dg(w, 0)] d(!1) + [Dg(w, 0)]. [Kg(w, 0)] = !12[Mg(w, !1)] + i!1[G1/w, !1)] Modal Parameter Evaluation Once the global dynamic stiffness matrix [Dg] of the system is obtained, the natural frequencies are those values of!1 (for a given rotation speed w) for which [Dg(w, !1)]X = o. There are many ways to solve this nonlinear eigenvalue problem. Because [Dg] is a very low dimension matrix, Eq. (13) can be solved by a straightforward procedure of calculating det[Dg(w, !1)] at many closely spaced !1 values within the frequency band of interest, taking the two values that bracket each sign change, and then iteratively calculating each 0 crossing Wn value between the bracketed values. (12a) (12b) (12c) (13) 1P.. _-+ ·+~H-+---+---" - 1m .=1 The modal mass for mode n may be obtained by (Richards and Leung, 1977) : (14) The response to harmonic excitation at any fre­ quency !1 can be easily obtained in this stage. In fact, [Dg(O)]-1 is the exact frequency response function (FRF) matrix for the given DOF. NUMERICAL EXAMPLES Example 1: Simply Supported Rotor with Disk The simply supported rotor with a disk subject to axial force shown in Fig. 3(a) is studied as the first example. The geometric parameters of the system are also given in the figure. The material of the rotor is steel with Poisson's ratio /.L = 0.3, Young's modulus E = 2.1 X 101 1 N/m and mass density p = 7800 kg/m3. The purpose is to model this rotor to 2DOF with the translational dis­ placements X 2 and Y2 at the disk as the general­ ized coordinates. The six basic elements (five shaft elements and one disk element) may be represented by two substructures as shown in Fig. 3(b). The transfer matrices [TIl, [T2] for each substructure are eas­ ily obtained. Their corresponding dynamic stiff­ ness matrices [D I], [D2] are obtained by rearrang­ ing the matrices [Til, [T2] according to Eq. (11). The global dynamic stiffness matrix [D] for the entire rotor system may be obtained by applying the standard assembly procedure of the finite ele­ ment method. It is noted that the obtained unre Substructure 1 Substructure 2 strained global dynamic stiffness matrix is 12 x 12 in our case. After introducing the boundary conditions at both ends, one finally obtains the dynamic equilibrium equation: where [Dg] is the restrained global dynamic stiff­ ness matrix. The unwanted DOF (ho 81y , 82x , 82y , 83n 83y may be eliminated by solving the linear set of Eqs. (15) and letting their corresponding excit­ ing moments MIx, M ly , M2x , M2y , M 3x , M 3y be O. Finally, we obtain: ( F2x) = [D2(w, fl)] (X2). F 2y Y2 It is important to note that the system shown in Fig. 3(a) is exactly reduced to 2DOF with dis­ placements X2 and Y2 at the disk as generalized coordinates for a given frequency and rotation speed. It is also noted that the reduced dynamic stiffness matrix [D 2(w, fl)] of the system is both frequency and rotation speed dependent due to the gyroscopic effect. The FRF matrix H(w, fl) = [D 2(w, 0)]-1 completely determines the dy­ namic behaviors of the system. The modal pa­ rameters can be obtained with ease at this stage. 301 I ~25r j ~ 1- § ~ ~ : : :::1 i ::r~--=-=------:-=",-rpm,-,,-:-::-,l o 5000 10000 15000 a) First mode ~360t 1 i340 00 ~0~ ~ 320r ~ 0 c o l ~-3ool 0 0 2801o rpm150100 5000 10000 c) Third mode To validate the correctness of the established model, Fig. 4 depicts the first four natural fre­ quencies predicted by the model as a function of the rotation speed (referred as TDM). For com­ parison, the finite element (Samcef Software, 20 shaft elements) results (FEM) are also presented in the figure. The influence of the axial force on the natural frequencies is given in Table 1. It is seen that the natural frequencies decrease with the increase of the axial force. Example 2: Three-Disk and Two-Bearing Rotor A three-disk and two-bearing rotor system, fixed at one end (Fig. 5) is studied as the second exam­ ple. The radius ofthe shaft is 0.025 m. The radius and thickness of the disks are 0.1 and 0.02 m, respectively. The material of the rotor and disks are steel with Poisson's ratio /.t = 0.3, Young's modulus E = 2.1 X 1011 N/m and mass density p = 7800 kg/m3. The two bearings are assumed to be iden­ tical and the stiffnesses are: Kxx = 1 X 106 N/m, Kxy = Kyx = O. Kyy = 3 X 106 N/m, The rotation speed of the rotor ranges from 0 to 30,000 rpm. The objective is to reduce the system to 6DOF with the translational displacements at disk 1, disk 2, and disk 3 as its generalized coor­ dinates. The 10 basic elements (five shaft ele­ ments, two bearing elements, and three disk elements) may be represented by three substruc­ tures as shown in Fig. 6. The dynamic stiffness matrix for each sub­ structure may be obtained from the correspond­ ing transfer matrix. The global dynamic stiffness matrix [D] for the entire rotor system may then be obtained by applying the standard assembly procedure of the FEM. It is noted that the ob­ tained unrestrained global dynamic stiffness ma­ trix is 16 x 16 in this case. After introducing the boundary conditions at both ends, one finally ob­ tains the dynamic equilibrium equation: (17) M 3x M 3y F4x F4y M 4x M 4y So far, we have finally established a 6DOF model with the translational displacements at disk 1, disk 2, and disk 3 as its generalized coordinates. It is noted that the elements of matrix [D6] are both frequency and rotation speed dependent. The FRF matrix can be easily obtained by in­ verting the dynamic stiffness matrix [D6] given in Eq. (18). The modal parameters can easily be obtained using the same procedure as the dy­ namic stiffness matrix method (Leung, 1980; Leung, 1983) at this stage. To validate the correctness of the established model, Table 2 gives the natural frequencies pre­ dicted by the transfer function matrix H(w, n) = [D6(w, ,0,)]-1. For comparison, the finite element (Samcef Software, 13 shaft elements) results are also listed. Figure 7 depicts the transfer function (at disk 3) predicted by the established model (TDM model). Modeling of rotor-bearing systems by the FEM or the DEM usually results in large stiffness and mass matrices. In order to compare with the ex­ perimental model, it is necessary to reduce the n I I I 100 100 , 150 nn IIII 150 ,Fre., Hz I 200 ,Fre., Hz I 200 ~ j where [Dg] is the restrained global dynamic stiff­ ness matrix. The unwanted DOF, Bix , Biy , i = 2,3,4 may be removed by solving the linear set of Eqs. (17) and setting their corresponding exciting moments Mix, M iy , i = 2, 3, 4 to O. Finally, we obtain: = [D6(w, n)]6x6 (18) FEM or the DEM model to lower dimension in terms of selected DOF. In this article, an exact and direct modeling technique is presented in which the global dynamic stiffness matrix is ob­ tained directly from the global transfer matrix whose dimension is independent of the number of elements or substructures. Using this method, the rotor-bearing systems can be modeled di­ rectly with arbitrary selected DOF. Because the transfer matrix is exact, the global stiffness ma­ trix is also exact. Hence, the results predicted by the model have a higher accuracy than the finite element results. The method presented is different from the Dokainish combination method. The first differ­ ence is that an exact transfer matrix derived di­ rectly from the governing equations is used; in Dokainish's method, the transfer matrix was de­ rived from the element stiffness and mass matri­ ces, which are not exact. The second difference, which is the most important difference, is that the Dokainish's method was actually an im­ proved TMM because the eigenproperties were obtained from the global transfer matrix. Only eigenfrequencies, mode shapes, and harmonic responses are available in Dokainish's method. In the present method, the global transfer matrix is transformed to the global dynamic stiffness matrix in terms of which all the modal parame­ ters such as eigenfreqencies, mode shapes, modal masses, modal stiffness, and transfer function can be obtained. Furthermore, any ro­ tor-bearing system can be directly modeled with arbitrary selected DOF. Therefore, the presented method may be considered as an improved dy­ namic stiffness matrix method. The authors would like to thank DBS company (Belgium) for the financial support of the project and for the permission to publish this article. APPENDIX: Matrices [A] and [N] in Eq. (8) The nonzero elements of matrix [A] are: 1 All = 1; A 23 = 1; A27 = KGA; pil2 A31 = KG; 1 A35 = EI; A43 = b; 1 A6S = KGA; pil2 An = KG; 1 A76 = EI; A S3 -- -2pEwD AS4 = b; Ass = a; with 1 pil2 (1 1) a = EI - KGA E - KG; the other elements being equal to O. Matrix [N] is obtained as follows: The matrix [Hb] takes the form H2) -H2 pil2 b = KG pil2 E ; with the submatrices [HIl, [H2] As a by-product, the exact dynamic stiffness matrix [D] for a rotating shaft subject to axial force can be obtained by rearranging the transfer matrix [T] using Eq. (11) in the text. 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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, DOI: 10.3233/SAV-1994-1601