Transition to Fluid-Induced Limit Cycle Self-Excited Vibrations of a Rotor and Instability Threshold " Hysteresis " " *

The transient process which starts at the instability threshold of a rotor rotating in a fluid environment, and ends up in the limit cycle of self-excited vibrations known as fluid whirl or fluid whip, is discussed in this paper. A one-lateral-mode, isotropic, nonlinear model of the rotor with fluid interaction allows for exact particular solutions and an estimation of the transient process. The fluid interacting with the rotor is contained in a small radial clearance area, such as in bearings, seals, or rotor-to-stator clearances, and its effects are represented by fluid film radial stiffness, damping, and fluid inertia rotating at a different angular velocities.


INTRODUCTION
The phenomena of fluid-induced instability of rotors called fluid whirl and fluid whip have been known since 1924, as reported by Newkirk.During the last decades, many physical descriptions and models of these phenomena have been investigated.Linear models of fluid forces acting at the rotor ("bearing and seal coefficients") are the most popular due to their ease of implementation into linear rotor model computerized analyses.It is known, however, that there exist substantial dis- crepancies between theoretical and experimental fluid force coefficients (e.g., Myllerup et al., 1992).
This poses a question of validity of the linearized modeling practices (Adams and Padovan, 1987).
Linearized fluid force coefficients, most often offered in numerical/tabular formats, can provide This paper was originally presented at ISROMAC-7.Tel." 702 782-3611.Fax: 702 782-9236.E-mail: agnes@brdrc.com.124   A. MUSZYNSKA a base for rotor instability prediction, but do not give any realistic description of the post-instability threshold behavior of the rotor.The coefficients can hardly be associated with any practical descrip- tion of the rotor/fluid interaction physical phe- nomena.Consequently, nonlinear models of the fluid force started gaining more attention.There is no agreement among the researchers, however, on what analytical expression for the fluid force should be adequate in specific cases.In the meantime, practical observations of the fluid-induced vibra- tions of machinery rotors in the field have provided rich documentation.However, in spite of the fre- quent occurrences of these fluid-induced vibrations, there still exists a cloud of misunderstanding of the basic physical phenomena of solid/fluid interaction in rotors (Crandall, 1983; Muszynska and Bently,  1996).This fact has considerably slowed down implementation of efficient fluid whirl/whip control mechanisms (Muszynska et al., 1988; Bently and   Muszynska, 1989).
This model is used in this paper for the rotor within its first lateral mode to analyze the transient process from instability threshold to limit cycle of self- excited vibrations.It is also used to investigate changes in the instability thresholds for the rotor transient process during startup and shutdown.The transition to limit cycle and instability threshold differences or "hysteresis" (the name introduced by Adams and Guo, 1996) represent new contributions to the knowledge on rotor/bearing dynamics.

ROTOR/FLUID ENVIRONMENT MODEL
The mathematical model of a one-mode, isotropic rotor rotating and laterally vibrating within the fluid environment, contained in a relatively small clearance, is as follows (Muszynska, 1986): where x, y are two rotor orthogonal lateral displace- ments, M,K, Ds are rotor first lateral mode modal mass, stiffness, and damping respectively, KB, D, and Mf are fluid radial linear stiffness, damping, and fluid inertia effect, A and Af are fluid circum- ferential average velocity ratios of rotating damping and fluid inertia forces respectively, and f is the rotor rotative speed.The functionsf(lz]) and Dn(IZ]) represent the nonlinear stiffness and nonlinear damping of the fluid film, as functions of the rotor radial displacement ]z I.These functions can have any form, provided they are continuous within the range ]z < c, where c is the radial clearance.They cover an important class of nonlinearities; they are not, however, general.
The fluid model in Eq. (1) was identified through extensive modal perturbation testing (Muszynska, 1995;Grant et al., 1993).This model is based on the strength of the circumferential flow generated by rotor rotation.The products Af and Af represent angular velocities at which fluid damping and fluid inertia forces respectively rotate (for classical bear- ing or seal stiffness coefficients Af=(kxy-kyx)/ (Dxx + Dyy).Following research results by Bently   and Muszynska (1985), Grant et al. (1993), and  E1-Shafei (1993), the fluid force model included the linear fluid inertia effect, which often presents a non-negligible contribution in the fluid dynamic force.Rotation of the fluid inertia force with a different rate than that of the fluid damping force  (1995).Within the range ofrotative speeds limited to the first lateral mode, the isotropic rotor model ( 1) is adequate for the case of the rotor/ bearing system (with fluid whirl and whip tenden- cies), rotor/seal systems (with seal fluid whip tendency), fluid handling machines with blade-tip or rotor/stator periphery interactions (with fluid whip tendencies), and rotors with press fit rotating elements, exhibiting high internal/structural fric- tion (with internal friction whip tendencies; A 1, Mr= 0).
A justification of the use of the simple rotor model is based on the fact that the fluid-induced vibrations most often are associated with the rotor's lowest modes, either the rigid body mode (fluid whirl) or the first bending mode (fluid whip) (Muszynska and Bently, 1996).The advantage of a simple model is obvious: analytically explicit solutions allow for extended analysis and clear physical interpretations.
The rotor stability criterion is as follows: < st + M -+mf '72' where 7 At(1 + Ds/D)/A. (2) At the instability threshold, the real part of one of the rotor system eigenvalues becomes zero, and the natural frequency COnst (the corresponding imaginary part of the eigenvalue) is equal to: For the rotative speed f exceeding the instab- ility threshold (2), the real part (Re) of the cor- responding eigenvalue s becomes positive (the remaining three eigenvalues can also be analytically calculated): D+Ds Re(s) The positive value of the eigenvalue real part means that the rotor lateral vibrations are un- winding; the rotor orbit represents a spiral with increasing amplitude.The particular solution of the linearized Eq. ( 1) has the form: >.', (5) where Mf/f is the imaginary part of the eigenvalue, and C is a constant of integration.In the case when the rotor vibration amplitude Ce Re(*)' increases, the linear- ized model (1) ceases to be adequate, since the non- linear factors start playing a dominant role in the rotor response as the rotor displacement amplitude grows.This causes the rate of increase of the vibration amplitude during the transient motion to decrease, until a limit cycle of the self-excited vibrations is reached.
The limit cycle of the rotor self-excited vibrations can be obtained as a particular solution of Eq. (1): where B, co are amplitude and frequency of the rotor self-excited vibration respectively.While Eq. ( 5) of the transient process represents an unwinding spiral orbit, Eq. ( 6) describes a circular orbit of the rotor.
The amplitude B and frequency can be calculated if Eq. ( 6) is substituted into Eq.( 1): By splitting this equation into real and imaginary parts, two algebraic equations are generated which can be used to calculate the amplitude B and fre- quency co (Eqs.( 2) and (3) were used): D+Ds+Dn(B) + Ds/D + (Ds/(D qt_ Dn(B)) const const _qt_ 5, (8)   where the parameter 6 was introduced as: (5 COnst{(D/Ds) + D/[D + Dn(B)]}-'.( 9) For any given functions Dn(lz[) andY(ll), which now are functions of the self-excited vibration amplitude B, this amplitude can be calculated from Eq. ( 8).The limit cycle of the self-excited vibrations, known as fluid whirl or fluid whip, is, therefore, explicitly obtained.Note that the self-excited vib- ration frequency, co, in (8) differs only slightly from the rotor natural frequency at the instability thres- hold, const, in Eq. (4).
Using Eqs.(2), ( 3), (8), and (9) allows for the following simplification of the first Eq.( 8)" (52(M + Mr) 4-26const(M 4-Mr'y) -f(B) as the identity Mco2nst -+-Mf(const-,f(1 + Ds/ D) const//\) 2 K n t-KB can be eliminated.The func- tion ( 6) is a parabola, while the function (Eq.( 9)) is a hyperbola.Figure 2 presents the subsequent graphical solution obtained from the functions (B) and (a).It provides the amplitude of the self-excited vibration limit cycle when the nonlinear stiffness functionf(lzl) =f(B) is given.This amplitude B can be found at the intersection of the functions fiB) and [(B)].Since the function exists within the range [(const/(1 +D/Ds)) and (constD/D)], and the function f(B) exceeds these limits, the solution for B must exist, as two of these function plots cross (Fig. 2).

TRANSIENT PROCESS STARTING AT THE INSTABILITY THRESHOLD
The rate at which the unstable linear vibrations start unwinding, when the rotative speed reaches the instability threshold (2), can be calculated as a derivative of Eq. ( 4), ORe(s)/Of at ft---st, multi- plied by f8t: an increase of M, Mr, and A. It increases with K+ KB and Af.The role of the damping in s* is better seen if the fluid inertia in Eq. ( 11) is neglected: This function has a maximum when D + Ds 2v/M(K + KB) which resembles the critical damp- ing.For subcritical damping, s* increases with a damping increase; for supercritical damping, it decreases.The above statements are also true for Eq. ( 11) in a qualitative sense, as some quantitative changes take place when the fluid inertia is incorporated: the maximum occurs at a smaller damping value.
Splitting Eq. ( 15) into real and imaginary parts provides: The characteristic equation for Eqs. ( 16), from which the variational eigenvalues Sv can be calcu- lated, is as follows: ( D+ Ds+Dn(B)) One of the roots "Sv" is zero, and it has been ex- tracted from Eq. ( 21).The analysis of the poly- nomial (17) shows that there exists only one real, negative root -s** which is approximately equal to" Using Eqs.
(2), (3), and (8) some terms in Eq. ( 18) can be transformed: As can be seen from the first Eq.( 19), the second term in the numerator of Eq. ( 18) is smaller than the first one, as it directly depends on damping D, and can practically be neglected.
The behavior of s** as a function of the system parameters is very similar to that of s*: s** decreases with increasing M, Mr, and A, and it increases with and Ac and f(B).The effect of K+ Ku is opposite than for s*, since nowf(B) is dominating the system stiffness.
The particular solution of Eqs. ( 16) is as follows: u(t) Cue -'** ', /(t)-C/e -'***', (20) where Cu, C; are constants of integration.The solution describing the rotor motion around the limit cycle of the self-excited vibrations (6) will, therefore, be as follows: z(t) (B + Cue-**')e j('+c;-'**').( 21) By comparing it with the solution (12), it can be seen that at the instability threshold the rotor amplitude exponent, which starts from the positive value + s*, ends up, during the transition time, as the negative value -s**, when it reaches the limit cycle.This transition to the limit cycle of the self-excited vibration is qualitatively illustrated in Fig. 3.  Freqmmy (m) FIGURE 4 Full spectrum cascade plot of rotor response dur- ing startup with angular acceleration 3.5 rad/s (full spectrum uses data from two orthogonal transducers; elliptical orbits at each frequency component are split into circular forward and reverse components).Instability threshold occurs at 6279 rpm.

EXPERIMENTAL RESULTS
in an oil-lubricated bearing; the second bearing is relatively rigid bronze bushing) during its startup and shutdown.Details of the experimental rig can be found in references by Muszynska (1995), Muszynska and Bently (1996), and Grant et al.  (1993).Due to the action of the fluid force in the bearing, the instability thresholds occur at 6279 and 6240 rpm respectively.The rotor orbit in Fig. 6 illustrates the transition to the limit cycle whirl vibrations with frequency 0.48f.Figures 4 and 5 illustrate the well-known phenomenon of "hysteresis": differences in the instability thresholds for increasing and decreasing speed.This effect was discussed and the name, "hysteresis," was introduced in by Adams and Guo (1996), but it seems that the sophisticated analysis used there is not necessary.The threshold differ- ences increase with an increasing angular accelera- tion of the rotor, and can be explained using the fluid force model applied in this paper: the fluid circumferential average velocity ratio (especially that of the fluid damping force), which is driven by the rotor rotation, at startup is lagging (is smaller) than that during shutdown, when it is leading, in comparison to the quasi-stationary case.Another lagging/leading effect can be contributed by the fluid inertia.It was, however, less obvious in the experiments.
In order to confirm the physical observations on the "hysteresis," a specific new perturbation test was performed.An external forward rotating unbalance force at a constant frequency CUp WaS applied through an auxiliary system to the rotor during its fast startup and shutdown at 26 rad/s 2 angular acceleration.The response of the well- balanced rotor was filtered to the constant pertur- bation frequency CUp, and presented by the Bode/ polar plots (Fig. 7).The data was then transformed into the dynamic stiffness format (Muszynska, 1995) (Fig. 8).The differences in the dynamic stiffness components for the rotor fast startup and shutdown are visible.Using the linear model (1) (f=Dn=0), these direct (DDS) and quadrature (QDS) components can be expressed as follows: (DDS) -(M + Mf)cu2p + 2Mfcup/\ff Mf,, 2 -+-K q--KB, (QDS) (D + Ds)cup D,.
Note in Fig. 8 that the direct stiffness does not have a constant value for CUp--const.This indicates that DDS depends on the fluid inertia which involves the rotative speed f.The differences, A, in the dynamic stiffness components for the startup minus shut- down are: A(DDS) -Mfu(CU-fufl) 2 q-Mfd(CUp ,fdfi) 2, ( 23) where indexes u and d denote startup and shutdown respectively.From Fig. 8 and Eq. ( 24), the fluid damping force average circumferential velocity ratios can be identified: Au-0.484, /d=0-471.The ratio Au for startup is, therefore, about 2.6% lower than that for shutdown for this example of the rotor high angular acceleration.
Note that the difference between Au and d can by itself explain the instability threshold (2)  8 Rotor startup and shutdown direct (DDS) and quadrature (QDS) dynamic stiffness components versus rota- tire speed.Humps in DDS at fluid whirl resonance speed are due to stiffness nonlinearity (Muszynska, 1995).Qst and the fluid inertia ratio Ar has a peak value when At= A/(1 + Ds/D).Since a direct identifica- tion of A is not possible, qualitative analysis of the effect of A on the startup/shutdown characteristics is made taking into account the instability thresh- old and direct dynamic stiffness information.In order to concentrate attention on the fluid inertia effect, an assumption is made below that in the instability thresholds for the startup and shutdown (Eq.(2)) Au--Ad A and Ds =0.Using notation (3), it results, therefore, in a4 d/a f. > af.)/(a-afd)] (2S) On the other hand, Eq. ( 23) and perturbation results (Fig. 8) provide another inequality: (26) The right-side expressions of ( 25) and ( 26) are concave, crossing parabolas versus Afu, which define the lower border of the area of possible Mfd/Mfu ratios (Fig. 9).If Aru=Afa, then Mfd > Mfu which seems unlikely.If Mfu= Mfd, then Afu is limited: [max(2A-Afd 2COp/CO--Afd < Afu<Afd] (provided that Afd is high enough, Afd > max(A, cdp/f)).Note that for the fluid reso- nance conditions (ap Aft) both inequalities (25)   and ( 26) are the same.So far the evidence is still FIGURE 9 Area of possible ratios of the shutdown and startup fluid inertia ratio, Mrd/Mru, versus fluid inertia circum- ferential average velocity ratio at startup, Aru.
insufficient to conclude about the variability of the fluid inertia during the startup and shutdown of the rotor.The research on this subject continues.

SUMMARY AND CONCLUSIONS
As the first subject, the post-instability threshold behavior of rotors rotating in fluid environment enclosed in small radial clearances was discussed in this paper.The nonlinear fluid force model identi- fied by Muszynska (1988b, 1995), Muszynska and  Bently (1996), and Grant et al. (1993), using the modal perturbation testing during the past 15 years, was implemented into the first lateral mode of an isotropic rotor.The equations provide analytical values for the instability threshold, and the limit cycle self-excited vibration amplitude and fre- quency.The transient process starting at the instabil- ity threshold, and ending at the limit cycle, was evaluated analytically and illustrated experimentally.As the second subject, the differences in the instability thresholds for rotor startups and shut- downs, known as a "hysteresis" (Adams and Guo,  1996), were presented and discussed (the "hyster- esis" here does not have any correlation to energy loss).The sole differences in the rates of the fluid damping rotation rate (A) explain the basic portion of this hysteresis.These differences are very well intuitively understood, and can easily be quantified by the novel perturbation testing: the external perturbation rotating force with a constant ampli- tude and frequency is applied to the rotor during its startup and shutdown transition.The rotor response filtered to the perturbation frequency is then presented in the dynamic stiffness format, which makes parameter identification easy.The effect of the fluid inertia on the instability "hysteresis" was also evaluated, but quantitatively it remains inconclusive.The discussed two subjects represent new contributions.The applied fluid force model, based on the strength of circumferential flow, once again proved to be useful and adequate in describing the physical phenomena occurring in rotors in a fluid environment.
Figure presents the graphical solution for 5 as a function of B when the nonlinear damping Dn(lZl)= Dn(B) is a given function.

Figures 4
Figures 4 and 5 present the full spectrum cascade plots of lateral response of a rotor (centered at rest

FIGURE 6
FIGURE 6 Rotor orbit between the instability threshold and limit cycle.Each bright dot represents one rotation.Pic- ture from oscilloscope screen.

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