Dynamics of Anisotropically Supported Rotors

The paper discusses dynamic effects occurring in machinery rotors supported in bearings and 
pedestals with laterally different characteristics. In the considered rotor model the anisotropy 
of radial stiffness and tangential (“cross”) stiffness components are included. Within certain 
ranges of the rotative speed the support anisotropy leads to the specific, excited-by-unbalance 
rotor lateral synchronous vibrations in a form of backward (reverse) precession. In 
addition, one section of the rotor may precess backward, while the other section simultaneously 
precesses forward. Experimental results illustrate this phenomenon. The analytical 
model of the system is based on multimode modal approach. It is also shown in this paper 
that greatly enhanced information for machine malfunction diagnostics can be obtained by 
simulated rotation of the XY transducer system observing rotor lateral vibration. This simulated 
rotation can be accomplished by the machine diagnostic data acquistion and processing 
system. The data processing also includes extraction of forward and backward components 
of elliptical orbits filtered to one frequency, and the filtered orbit major axis magnitude and 
its angular orientation.

1. INTRODUCTION Most rotating machine support structures are charac- terized by lateral anisotropy.The anisotropy of the ro- tor system can originate in bearing support pedestals, foundations, and/or asymmetric piping attachments to the machine casing.It can also originate in fluid-lu- bricated bearings or seals, and process flow asymme- tries.The anisotropy can affect mass, damping, and stiffness matrices.In effect, the rotor mode character- istics become anisotropic in two lateral orthogonal di- rections.This results in closely spaced, coupled "pairs" of rotor lateral modes revealed, for instance, in rotor (synchronous) response polar and Bode plots as "split resonances."Due to the anisotropy, the orbits excited in response to simple unbalance are el- liptical with various degrees of ellipticity.It has been known that in certain rotative speed regions the rotor unbalance response orbits are backward (reverse).*Corresponding author.Tel." (702)  782-3611.Fax: (702) 782-9236.E-mail: agnes@brdrc.com.This classical effect is discussed in papers and books on rotordynamics, such as Gunter et al [1993], Vance [1993], and Handbook of Rotordynamics [1993].With a specific unbalance distribution along the rotor axis, it may also happen that a portion of the rotor would precess forward, while another one precesses back- ward.This fact, briefly mentioned by Vance [1993], is discussed in this paper.
Vibration monitoring systems installed on rotating machines include a number of pairs of rotor displacement measuring transducers mounted at or nearby bearings in the orthogonal XY configuration.A spe- cific transducer angular orientation seldom coincides with the support structure major or minor axis of stiffness anisotropy.In addition, these axes are usu- ally nonorthogonal.Independently from the trans- ducer lateral location, following the "oscilloscope conVention," the vibrational information from both XY transducers is used to correctly recreate the rotor orbits, the magnified images of the rotor centerline motion.The transducer information is also used to obtain rotor filtered single frequency response vec- tors, such as 1 or 2 in the Bode and polar plot formats.For the purpose of these plots, the informa- tion from only one lateral transducer is required, thus the Bode and polar plot display data is characteristic for the specific transducer location.The anisotropy affects the response vectors, which observed from a different angular location, would be different.The questions arise about how to properly identify the unbalance ("heavy spot") angular location, especially at low speed, and how to evaluate the Synchronous Amplification Factors for anisotropic rotors.The fact is that the response phase and amplitude magnitudes vary significantly with observation angle.
The problems mentioned above are discussed in this paper using, as an example, a mathematical model of a two-mode rotor, based on the multimode modal approach, discussed by Muszynska [1994].This model includes stiffness and tangential force anisotropy.

MATHEMATICAL MODEL OF A TWO-MODE ANISOTROPIC ROTOR
Consider two modes of a laterally symmetric rotor supported in anisotropic susceptible pedestals.The rotor model which includes anisotropic tangential force is as follows: M: + D,Jc + K,x + D(Jc + X.,.l'2y)-F cos(tot My + D,.) + g,,y + D@-A,,Qx)-F sin(tot d/dt where x(t), y(t) are rotor orthogonal lateral deflec- tions, M, D,. are rotor modal mass and damping re- spectively, D is surrounding fluid radial damping, X,, are fluid circumferential average velocity ratios [Muszynska, 1994], Q is rotative speed, K,-, K,, are rotor/supporting structure stiffnesses in x and y direc- tions.For positive X.,. and )t,. the expressions and -DA,,Qx represent nonsymmetric components of a forward (acting in the direction of rotation) tangential force.This force is due to circumferential flow of the rotor surrounding fluid (process and/or lubricating fluid).As discussed, for instance, in the Handbook of Rotordynamics [1993], the tangential force may also originate from other sources.The parameters F, to and i denote external exciting, nonsynchronously ro- tating force amplitude, frequency, and angular orien- tation, respectively.Muszynska [1989] showed that equations (1) can be solved analytically.There.exist two cases: (a) weak coupling, and (b) strong coupling, for which the eigenvalues and modal functions are slightly differ- ent.In case (b) instability may occur.The results are summarized in Table I.

VIBRATION DATA PROCESSING FOR MODE DECOUPLING" TRANSDUCER ROTATION SIMULATION
In machine monitoring systems the displacement transducers observing the rotor are mounted in XY configuration which usually does not coincide with the major or minor rotor/support stiffness axis direc- tions.The vibrational data obtained from the trans- ducers most often indicate some level of the system anisotropy: 1 orbits are elliptical in a broad rota- rive speed range, 1 Bode and polar plots display "split resonances."An improvement for easier inter- pretation of such data can be achieved if the 1 response vectors obtained from X and Y transducers are post processed, in particular, rotated by an angle (R) (Fig. 3).The new orthogonal response vectors A,ei'', Awe i<' will have the following amplitudes and phases:

FIGURE
Synchronous (1 ) response vectors of the rotor (co 11) in Bode and polar plot format, calculated from Eqs. (1).The phase crossing in the range of 1070 to 1160 rpm indicates back- ward orbiting.Data flom the Y probe on the polar plot is rotated by 90 to coincide better with data from the X probe (in case of iso- tropic rotor the polar circles are identical).
A. MUSZYNSKA et al. 3000 rl)m in Bode and polar plot format, calculated from Eqs.
(1).Note a difference in amplitude in comparison with Fig, 1.
cos" O + A sin O + A,A sin 20 cos(R,. If the rotation angle O corresponds to one of the main stiffness axes, and, in a particular case, is equal to either 6) arctan b or e): arctan b2 where qb 2 are rotor eigenfunctions (see Table I), then one coordinate (u or w in the rotated system) becomes either uncoupled from the other (case (a)), or partially decoupled with minimum coupling effect (case (b), Table I).If () and 0 2 are orthogonal, which occurs in a very particular case when X.,.+ X,. 0, lull decoupling is possible in case (a). Figure 4 presents the same data as in Figure 2, rotated by the corre- sponding decoupling angle calculated as arctan Another decoupling angle in this case is -68.08.The response vectors rotated the way that there is a minimum coupling effect serve better for diagnostic purposes.It is illustrated using the machine field data, following Hatch et al [1995].Figure 5 presents gas turbine synchronous responses and Figure 6 presents the rotated data with minimum coupling.
(1) into the forward and backward mode variables z..t z, can certainly be presented in the classical format with one amplitude and one phase for each variable.The ex- pressions (6) emphasize the correlation between so- lutions for original x, y and transformed variables Z b The backward component response amplitude and phase directly depend on anisoti'opic parameters, and for an isotropic system, they vanish: A, -F /(K K,,):.+ [D(X x,,)a] 2, D(R,-% -+ ot + arctan (7)  The forward response amplitude and phase are as follows: F At= 2A 'V/(Kr -ff K 2M(.o2) -1--[2(D + D.,.)m + DD,(R.,.+ X,.)] 2, at. 8o + arctan 2(D + D.,.)o + DD(R.,.+ K,-+ Kr-2Mm2 The response orbit major axis magnitude S and its angular orientation o-measured from the horizontal axis can directly be obtained as S .At + A/,, o- 0.5 arctan [2A.,A,, cos(% o,,)/(A a)], or using the original parameters of the system, the latter is: The same initial data as in Fig. 1.The backward component amplitude is larger than the for- ward one in the speed range 107{} to 116{} rpm.In this range the orbits are backward.Note that the low speed forward synchronous response points toward the heavy spot (here at -30).Amplitudes nondimensionalized by multiplying by M/mr.
Figure 8 illustrates the major axis magnitude S and angular orientation (r of the rotor synchronous orbits.Three orbits at frequencies close to resonances ac- company the Bode and polar plots.
Figures 9 and 10 present the gas turbine data in the forward/reverse format, and in the format of the orbit major axis magnitude and angular orientation.Both these formats represent new tools in diagnostics of machine malfunction, and their usefulness will be es- tablished as soon as they are used.

EXPERIMENTAL RESULTS DEMONSTRATING SIMULTANEOUS FORWARD AND BACKWARD ORBITING OF TWO SECTIONS OF THE ROTOR
An experimental vertical rotor with an overhung un- balanced disk was driven through an elastic coupling by an electric motor mounted at the top.At inboard side the rotor was supported by a relatively rigid, laterally pivoting rolling element bearing.The rotor support anisotropy was achieved by sets of "horizon- tal," x, and "vertical," y, springs mounted to the rotor through rolling element bearings at two different ax- ial locations.The rotor shaft was slightly bent, and also carried an unbalance.The  -1 -0 -5 0 FIGURE 8 Rotor synchronous response orbit major axis magni- tude and angular orientation in Bode (a) and polar plot format (b); orbits at 900, 1000, and 1100 rpm (c).The same initial data as in Fig. 1.Note that the orbits at 900 and 1100 rpm are fl)rward and the orbit at 100()rpm is backward.the rotor were observed by three sets of XY noncon- tacting proximity probes (inboard, midspan, and out- board).The vibrational data were processed by a computerized acquisition system.Figure 11 illustrates midspan and outboard rotor full spectrum cascades and IX orbits at selected ro- tative speeds.The full spectrum contains forward and backward components of elliptical orbits at separate frequencies decomposed by Fourier transformation.The main portion of the rotor response is IX with two distinct ("split") resonances (occurring at about 950 rpm and 1250 rpm) due to the anisotropic support.Classically, between these two resonance ."150" 90 80 pm PP Full Scale RPM FIGURE !0 Gas turbine data from Figure 5 in IX orbit major axis magnitude and angular orientation Bode and polar plot format.In comparison to Fig. 8, at higher speeds this data indicates a presence of the next mode.speeds, the outboard disk orbits are backward.The inboard data, which is not displayed here, looked qualitatively similar to the midspan data; this portion of the shaft responded in phase.Due to the existence of shaft bow, all lX amplitude plots exhibited a sig- nificant response at low frequency (slow roll).The midspan bow was about twice larger than the out- board bow.The rotor average centerline plot versus rotative speed did not show any significant changes.Maximum centerline displacement was less than 4 mils, which indicated that there was no specific ac- tivity affecting the rotor centerline.
The sequence of orbits in Figure 11 reveals a phenomenon documented by the Muszynska in 1996: At certain rotative speeds the shaft midspan orbits are forward, while the outboard orbits are reverse (see the orbits at 1160 rpm in Fig. 112).This phenomenon originally raised the question: how is the shaft able to move counterclockwise at one section, and clockwise at another?Further analysis confirmed and quantified this behavioral feature.The next question concerned the deformation and stress patterns of the shaft fibers in the situation of different precession directions of two sections of the shaft.In order to assess the shaft stress, the midspan and outboard orbits at 1060 rpm (both backward) and at 1160 rpm were plotted again, respectively, on one figure (Figs.12a and 13a).The numbers on the orbits corresponded to the same tim- ing; the vectors connecting these timing points represent the outboard-to-midspan relative displacements.When drawn separately, these relative displacements reveal very similar orbits for both speeds 1060 and 1160 rpm.Both relative orbits are forward, with some amplitude differences, but very little phase change (Figs.12b and 13b).These relative orbit graphs show that nothing unusual occurs in the shaft rotating in the 1060 to 1160 rpm speed range.The relative orbits can be interpreted in terms of a "relative" mode, when the midspan location of the shaft is "frozen" (Fig. 14b).The full analysis of the system is presented by Muszynska 19961.

FINAL REMARKS
While anisotropy in bearing supports is often specifically incorporated in rotating machine design, as it is known to enhance the stability of the rotor (see, for instance, Handbook of Rotordynalnics, 1993), it also introduces, in some rotative speed ranges, the inevi- table, unbalance-related, backward precessional mo- tion, resultine in damaein-rotor reversal stresses The brighter side of this is related to the fact that rotative speed bands where the backward precession takes place are relatively narrow, and they occur near closely spaced rotor natural frequencies which are normally avoided as operational speeds.The  (Forward) FIGURE 13 (a) Rotor midspan and outboard orbits at 1160 rpm.
The midspan orbit is forward, the outboard orbit is backward. (b) Differential orbit is again lbrward with no phase change compared to Fig.   however, always exists during startups and shut- downs of the machine.Careful balancing and shaft straiehtenin-,.. considerably improve rotor transient condition operations.Also, as shown by Corley [19861, a sufficient amount of damping in the rotor/ support system may effectively suppress the back- ward precessional motion.It has been shown in the paper that vibration data processing using filtered forward/backward orbit coinponents and orbit major axis amplitude and its orientation may be useful for machine malfunction diagnostics purposes.As new tools, the usefulness of these new forlnats will be proven following acculnu- lated experience when specific machine lnalfunctions will be associated with a growth (or decrease) of a specific response component.NOMENCLATURE AI; AI,, i o;, Amplitudes and phases of forward and backward circular components of rotor elliptical orbits respectively A.,., A,., o.,., c,.Rotor response amplitudes and phases in .vand v direction respec- tively A,,, A,., %,, o,,.Rotor response amplitudes and phases in u and

FIGURE 2
FIGURE 2 Nonsynchronous response vectors of the rotor for {}, cos

FIGURE 4
FIGURE 4 Nonsynchronous response of the rotor for [ 3000 rpm rotated by angle e) arctan4 -2.92.The same data as

Figure 7 FIGURE 7
Figure7presents forward and reverse components of the synchronous response vectors of the rotor cor- responding to the data illustrated in Figure1.At low speed the forward component phase points correctly toward the unbalance angular location.

FIGURE 9
FIGURE 9 Gas turbine data from Fig.5in the forward and re- verse Ix component Bode and polar plot format.The low speed forward response points toward the heavy spot angular.rientation.Compare with Figs.4 and 5.
FIGURE 11 Full spectrum cascade of the rotor midspan (a) and outboard (b) "vertical" (north-south) responses accompanied by midspan and outboard filtered 1 (synchronous) orbits at selected speeds.
FIGURE13 (a) Rotor midspan and outboard orbits at 1160 rpm.The midspan orbit is forward, the outboard orbit is backward. (b) Differential orbit is again lbrward with no phase change compared to Fig.12.
FIGURE13 (a) Rotor midspan and outboard orbits at 1160 rpm.The midspan orbit is forward, the outboard orbit is backward. (b) Differential orbit is again lbrward with no phase change compared to Fig.12.

Fluid
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The same data as FIGURE 3 Coordinate systems, in Figure2.The response marked "x" is decoupled. danger,