SOME INSIGHTS INTO THE SIMULTANEOUS FORWARD AND BACKWARD WHIRLING OF ROTORS

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1 SOME INSIGHTS INTO THE SIMULTANEOUS FORWARD AND BACKWARD WHIRLING OF ROTORS Milton Dias Jr., PhD Assistant Professor of Mechanical Engineering Department of Mechanical Design State University of Campinas Campinas - SP, Brazil Randall J. Allemang, PhD Professor of Mechanical Engineering Structural Dynamics Research Laboratory University of Cincinnati Cincinnati, OH ABSTRACT One of the greatest advantages of using complex coordinates in the study of rotating machinery is the gain of physical insight into the forward and backward precessional modes. Together with the complex modal analysis formulation, other interesting tools (like the Shape and Directivity Index Plot) were developed to characterize the shape and directivity of the whirl modes. These tools are helpful in understanding the bizarre phenomenon of mixed operational modes in rotors where some stations describe their precessional motion in the forward direction while others move in the backward direction. This study is carried out using a complex coordinate based finite element model of rigid as well flexible rotors supportad on anisotropic bearings. NOMENCLATURE [WJW~I 14 > ill (Yl 9 (4 h j=-, IT1 VI I [~,1~[~,1J~,1 {3 kl PfJb G/G R Parameter matrices, real coordinates Displacement and force vectors Displacement vectors in the y and z directions, respectively Angular displacement vectors in the y and z directions, respectively Number of nodes of the FE model Imaginary number Transformation matrix Identity and null matrices Parameter matrices, complex coordinates Complex displacement and force vectors Forward and backward components of the unbalance response Forward and backward components of the external force Rotational speed * Visiting Scholar at the Structural Dynamics Research Laboratory, University of Cincinnati, Cincinnati, USA. 1. INTRODUCTION It is well known that the unbalance force can excite the precessional backward modes only if the rotor is supported by bearings with different stiffness in two lateral orthogonal directions, as happens in many rotors of rotating machines [l-4]. While anisotropy in the bearings is often intentionally incorporated in rotating machine design, as it enhances the rotor stability, it may also introduce the undesirable unbalancerelated backward whirling motion, which can cause premature failure of the system due to stress reversal on the rotor [5]. Fortunately, the rotational speed regions where the backward precession occurs are relatively narrow and close to the critical speeds, which are avoided as operational speeds. Among other things, the amount of damping in the rotor/bearings system affects the existence, or not, of the backward precessional motion. Studies have shown that the more damping in the system, the narrower the rotational speed range where this motion can occur 151. The existence of the simultaneous forward and backward whirling on rotors has been proved experimentally by Rao et al [6,7l and Muszynska [5] but, in general, this phenomenon is briefly mentioned by the authors [l-3] and rarely discussed for flexible multi-discs rotors. Rao et a/ [S.fl studied a Jeffcott rotor supported on identical and dissimilar journal bearings. They showed the bearing clearance and the dissimilarities between the two bearings affect the existence of the mixed operational modes. Muszynska [5] studied a vertical, overhung unbalanced rotor with bent shaft and supported by flexible, anisotropic bearings. She showed that the combined effect of the mass unbalance together with the bow shaft unbalance could lead to simultaneous forward and backward precession motions at different stages of the rotor. The main goal of this work is to study the existence of the simultaneous forward and backward precessional motions on rotors through comr?x modal analysis. It is well known that the use of complex coordinates in the study of rotating machines can help in the interpretation and visualization of the whirling modes and of the precessional operational modes. These characteristics are used in this paper to study the simultaneous forward and backward motions of the many 1257

2 station of rigid and flexible rotors. This paper also introduces the Shape and Directivity Index (SDI) Plot, a helpful tool that clearly shows the direction (forward or backward) and shape (circular, elliptical or rectilinear) of any station of the rotor at any rotational speed. 2. UNBALANCE RESPONSE - COMPLEX COORDINATES In this section, the procedure for transforming the lateral, orthogonal displacements of each station of the rotor into their corresponding complex coordinates is briefly presented. The frequency equation needed for the computation of the unbalance response is also developed. A more detailed development can be found in Lee [8] and Nelson [9]. Let one first consider the well-known equation of motion of a 4N degrees of freedom rotating system: [~lc4+[~lw +rmw = 10 I (1) where [m is the symmetric, positive definite mass matrix. [D] and [/(I are rotational speed dependent matrices and, in general, are neither symmetric nor positive definite. Matrix [D] represents the damping (internal, bearings and surrounding environment) and gyroscopic terms and matrix [Kj includes the stiffness and circulatory (internal damping, bearings and surrounding environment) terms. A very convenient way to assemble the displacement vector (x) in a finite element code is the following: where (y) and (z) are the displacement vectors on each of the two directions perpendicular to the rotation axis of the rotor, and &] and (&} are the angular displacement vectors of all the nodes of the model in the directions y and 2, respectively. The expressions for the transformation of the displacements of node k into their corresponding complex coordinates are: p,,(t) = Yk (r) + izr (9 and ~dj) = &Jf) - &,,(O Considering all nodes of the finite element model, one has: where [MC]. [&I and [KC] are the complex mass, damping/gyroswpic, stiffness/circulatory matrices [IO]. (g) is the complex vector force. The complex displacement (p) as well as the complex external force (g) can be decomposed into its forward, (P ) ((G/J ), and backward, {P&J( (GbJ), components as: (p(t)} = {P,}e & +(P,)e- a (g(t)} = {G,,dcd +(G,)e- Considering harmonic excitation and response and substituting Equation (3) in (2) one concludes that: (-w2[hfc,]+ jm[dc]+[kc.,,{~~:). = {%;I. (4) The unbalance force is pure forward and dependent on the rotational speed. It can be expressed by (3) k(j)) = IG,)e@ (5) The amplitude vector (G,) is a function of the unbalance rn~, of the initial phase of the unbalance mass relative to the real axis (direction y), and of the rotational speed of the rotor. The backward component of the unbalance force is null. To obtain the unbalance response of the rotating system, one needs to solve the following set of algebraic equations: The relative amplitude between the forward, {P,), and the backward, (Pr,), components of the unbalance response define whether the precessional motion of a specific station of the rotor will be forward or backward, circular, elliptical or rectilinear. These two components can be combined in one parameter, called Shape and Directivity Index, or SDI, defined by [3.1 I]: where the bar denotes complex conjugate. Applying the same transformation [A on the force vector and substituting these results in Equation (1) yields: (7) The relations among the values of the SDI, the shape of the orbit of a station of the rotor and the direction of the precessional motion are: 1258

3 . SDI=-1 - Circular backward precessional motion. -lcsd/<o - Elliptical backward precessional motion. SD/=0 - Rectilinear motion l OcSDI<l - Elliptical forward precessional motion. SDI=l - Circular forward precessional motion It is easy to conclude that the sign of the SDI defines the direction of the precessional motion while the shape of the orbit is defined by its absolute value. 3. NUMERICAL RESULTS AND DISCUSSION In order to illustrate the potential of the complex formulation, especially the SDI Plot, on the analysis of the precessional operational modes of rotating systems and to verify the existence, or not, of simultaneous forward and backward whirling motions on rotors, two systems are studied: the first one, from now on denoted Rotor l, is a rigid Jeffwtt rotor supported on anisotropic bearings. Three cases are considered on the analysis of the Rotor 1: 1. undamped system with identical bearings at the two ends; 2. undamped system with dissimilar bearings; and 3. damped system with two identical bearings. The second system, Rotor 2, is a simple flexible rotor with two discs and three bearings. For this system, the influence of several different parameters on the existence of the mixed operational modes is studied. These parameters are the amplitude of the unbalance force on each disc, the relative angular position of the unbalance in the two discs, the damping in the bearings and the position of the discs along the shaft. presents Campbell diagram (RPM Spectral Map), the unbalance response at node 1, and the SDI Plot. According to the theory, this system should whirl in the backward direction between the two split resonances and in the forward direction elsewhere. Figure 3 shows that the amplitude of the backward component (dotted line) is bigger than the forward component between the two first critical frequencies. The plots at the other nodes (not shown) indicate that the unbalance response of all nodes are approximately the same. It means that, in this rotational speed range, the whole rotor whirls in the backward direction. k f II < ; L.. j..a z -TV--.,-_ i c i-: h 5 1 /IT R,c Figure 2 -Campbell diagram Rotor 1: Jeffwtt rotor supported by anisotropic bearings Figure 1 shows the finite element model of the rotor considered in this section. The length of the shaft is 051m and its diameter is 0.022m. The disc, placed at midspan. has a diameter of m and its thickness is equal to 0.044m. The unbalance mu = 1.104kg.m applied at the disc (node 5). FIWITE ELEMEWT YODEL x- PLANE Nader Figure 3 - Unbalance response at node I. Figure 1 - Finite element model of Rotor 1. - Identical Undamped Bearings The first case considers a rotor supported by two identical undamped anisotropic bearings with the following stiffness: kw = 8000Nlm and kll = 6000N/m. Figures 2, 3 and 4 Figure 4 - SDI Plot. 1259

4 The same conclusion can be obtained from the SDI Plot, This plot is built by computing the SDI for all nodes and for each rotational speed. Specific colors are assigned to each SDI value in order to be possible to distinguish between forward and backward precessional motion of the stations of the rotor. Qualitative information about the shape of the orbit is also available. Qg& - Dissimilar Undamped Bearings This case considers the same rotor described in the previous study but supported by two dissimilar undamped anisotropic bearings with the following characteristics: lefl bearing (element 10 on Figure 1) - k, = 80OON/m and kn = 6000N/m; right bearing (element 11) - k, = 700ON/m andk,=12000n/m. Comparing the Campbell diagrams shown in Figures 2 and 5 one, can note that the difference between the first two critical speeds is lower in 2 than in 1. Other than that, the overall characteristic of the system looks very similar. However, it is worth noting that the Campbell diagram only gives information about the eigenvalues of the system, without considering the eigenvectors. Figures 5 and 6 present the unbalance response of the rotor at nodes 1 (left bearing) and 5 (disc position), respectively. The behavior of this system, until after the second critical speed, is basically the same as that observed in 1, with the whole rotor whirling in backward motion between the spit resonances, and forward elsewhere. The major difference between the two systems is evident when the rotational spaed approaches the third critical speed. Figure 6 shows that the amplitude of the backward component (dotted line) is bigger than the amplitude of the forward component, meaning that this node describes its precessional motion in backward direction. On the other side, Figure 7 shows that the whirling motion of the disc will be in the backward direction in a very small rotational speed range. Instead of watching the frequency response of all stations of the rotor (which is a lot of work for more complex systems) to understand the global behavior of the machine, it is easier to analyze the SDI Plot. Figure 8 shows the SDI Plot for 2 and the corresponding operational deflection shapes in the frequency range where the simultaneous forward and backward whirling motion occurs. Through this plot one can easily see that for some values of rotational speed the system do precession in backward motion in some parts of the rotor and in forward motion in other parts of the structure. Basically, as the rotational speed increases, the ends of the rotor start whirling in the backward direction while the disc describes its orbit in the forward direction. Gradually, the backward whirl motion extends to the portion of the rotor closer to the center of the system. When the rotational speed coincides with the third critical speed, the whole system whirls in the backward motion. It can be observed from the operational modes illustrated on Figure 8 that, at the point of transition from forward to backward whirling, the whirl orbit becomes a straight line [WI. The amount of damping in the bearings plays an important role on the excitation of the backward modes of the system and, consequently, on the existence of the mixed operational modes. To avoid the simultaneous forward and backward whirl motion, the minimum damping value on both bearings must be C, = Cn = 6Nslm. The corresponding damping ratios are: 1.59%, I.35%, 4.93%, and 4.71% for modes 1 to 4, respectively. The minimum damping value on both bearings to completely avoid backward whirl orbits must be C, = C, = 45Nslm. 15,+... ;....j...:. -...: : i...:_ i... i.. i... 1 R : : :. 0LL i / I I ii ,000 llod 11m 1600 lso adlatso al SPBSd lb, Figure 5 -Campbell diagram / I...i i BOO ,dOO 1600 lsd Fro~"oncy,w"l Figure 6 - Unbalance response at node 1. Figure 7 - Unbalance response at node

5 DOETIP um.ei o,,*t,)~ O/.~I~c.m.,,l~,.:/..- ; + \ OosTr~~ /omm,6 0.u rll~ D,rp,*rrm~nl,.m, r~,lllln~lip~~d=li~ll~m -..,. ;.:, \,.. 1, s*rna iln i rmadr.lt msr mof i Figure 8 - SDI Plot and the operational deflection shapes in the frequency range where the simultaneous forward and backward whirling motion occurs Rotor 2: Flexible Rotor Supported on Anisotropic 18 is an isotropic bearing with stiffness k, = k, = 1.106N/m; Bearings the right bearing is anisotropic with stiffness kw = 1.10s N/m and ku = N/m. For all the simulations presented in this Figure 9 shows the finite element model of the flexible rotor paper these stiffness values remain the same. considered in this section. The length of the shah is 0.61m and its diameter is 0.01 m. The diameter of the two identical - Influence of the unbalance phase relationship discs, placed initially at nodes 7 and 13 (all the elements have the same length), is 0.08m and its thickness is equal to In order to quantify the influence of the relative angular m. The unbalance force will always be applied on the position of the unbalance (unbalance phase relationship) in discs. m.z is initially set to l.io-5kg.m. The rotor has three the two discs, the unbalance responses, and the SDI, were supports: the left one (element 20) represents the elastic calculated for four different values of phase: 0, 90, 180. coupling between the electric motor and the shaft; element and 270. The results for 90 and 270 were exactly the same and are not shown here. The results for the other two phase relationship are presented on Figures 10 and 11. FiNlTEELEMENT X-Y PLANE MODEL Figure 10 indicates that this rotor exhibits simultaneous whirl motions in a wide rotational speed range. Its dynamic behavior is far more complicated than the rigid rotor previously studied. Figure 10 also presents the operational modes for five different rotational speeds. Also in this case, it can be observed that the whirl orbit becomes a straight line at the point of transition from forward to backward whirling. From Figure 11 one can conclude that, if the phase relationship between the two unbalances is 180, the rotor does not exhibit simultaneous whirl orbits at any rotational Figure 9 - Finite element model of Rotor 2. speed! 1261

6 Figure 10 - SDI Plot for phase = 0 and the operational deflection shapes in the frequency range where the simultaneous forward and backward whirling motion occurs. - Influence of the amplitude of the unbalance forces In this case, the unbalance nr~ of the disc 2 (element 17) was set to kg.m while the unbalance of the disc 1 (element 16) was kept the same. The results of the unbalance response calculation for two different phase relationships are presented in Figures 12 and 13. It can be seen that, in both cases, the rotor exhibits simultaneous whirl motions in a wide rotational speed range. Figure 11 - SDI Plot Phase = Influence of damping in the bearings The influence of the damping in the bearings on the existence of synchronous whirl orbit on this rotor was studied. It was verified that the minimum damping value on the right bearing to avoid the existence of mixed operational modes should be C, = C, = 12NsJm; and the minimum damping value to completely avoid backward whirl orbits should be C, = Cn = 40NsJm. In this last case, the damping ratios are as high as 70% Figure 12 - SDI Plot. Phase = 0.

7 CONCLUSION The simultaneous forward and backward whirling motions in rigid and flexible rotors supported on anisotropic bearings have been studied numerically in this paper. It was verified that the existence and extension of this phenomenon is strongly affected by the amount of damping in the bearings, by the spatial distribution of the unbalance forces, by the mass distribution along the shaft, by the level of anisotropy of the bearings and by the dissimilarities among the bearings. The SDI Plot was also introduced and shown to be a helpful and easy-to-use tool in the analysis of the whirling motion of rotors. Figure 13 - SDI Plot. Phase = Influence of the position of the discs on the shaft Two different configurations were tested: a. disc 1 at node 14 and disc 2 at node 13 and b. disc 1 at node 7 and disc 2 at node 8. The computed unbalance responses for each one of these configurations are presented in Figures 13 and 14. For both cases the phase relationship between the unbalance masses was 0. From these figures it is relatively easy to conclude that the mass distribution along the shaft is esskntial to avoid, or minimize, the existence of simultaneous whirling in rotating machines. Figure 13 - SDI Plot. Disc placed 16 at node 14. Figure 14 - SDI Plot. Disc 17 placed at node 8. ACKNOWLEDGEMENT The authors would like to thank the FAPESP - Funda@o de Amparo a Pesquisa do Estado de SBo Paulo - for the financial support of this project. REFERENCES [11 [31 [41 [51 VI Vance, J. M., Rotordynamics of Tutiomachinery, New York: John Willey & Sons, Ehrich, F. F. (Editor), Handbook of Rofordynamics, New York: McGraw-Hill, Inc., Ktimer, E., Dynamics of Rotors and Foundations, Berlin: Springer-Verlag, Lalanne, M. and Ferraris, G., Rotordynamics Predictions in Engineering, 2 d edition, Wiley, New York, Muszynska, A. - Forward and Backward Precession of a Vertical Anisotropically Supported Rotor, Journal of Sound and Vibration, v192, nl, pp Rao, C., Bhat, R. B. and Xistris, G. D., Experimental Verification of Simuiianeous Forward and Backward Whiding at Different Points of a Jeffcoff Rotor Supported on identical Journal Bearings, Journal of Sound and Vibration, v198, n3, pp , Rao, C., Bhat, R. B. and Xistris, G. D. -Simultaneous Forward and Backward Whirling in a Jeffcoft Rotor Supported on Dissimilar Hydrodynamic Bearings, Journal of Sound and Vibration, ~203. n4, pp Lee, C. W., A Complex Modal Testing Theory for Rotating Machinery, Mechanical Systems and Signal Processing, v5, n2. pp , Nelson, H. D. and McVaugh, J. M., The Dynamics of Rotor-bearing Systems Using Finite Elemenfs, Journal of Engineering for Industry, pp , [IO] Joh, Y. D. and Lee, C. W., Excifation Methods and Modal Parameter Idenfififation in Complex Modal Testing of Rofating Machinery, The International Journal of Analytical and Experimental Modal Analysis, v8, n3, pp , [II] Lee, C. W. and Han, Y. S., Use of Directional Wigner Distribution for identification of the lnsfanfaneous Whirling Ort~if in Rotating Machinery, Proceedings of the International Symposium on Transport Phenomena and Dynamics of Rotating machinery, Hawaii, USA,