Rotordynamic Fluid Force Moments on an Opentype Centrifugal Compressor Impeller in Precessing Motion

In recent years, increasing interest has been given to the rotordynamic fluid forces on impellers, from the view point of the shaft vibration analysis. Previous experimental and analytical results have shown that the fluid-induced forces and moments on closedtype pump impellers contribute substantially to the potential destabilization of subsynchronous shaft vibrations. However, to date few papers are known of the rotordynamic fluid forces on open-type centrifugal impellers. This paper reports about experimental investigations of the rotordynamic fluid force moments on an open-type centrifugal compressor impeller in precessing motion. For open-type impellers, the variations of the tip clearance and the clearance between the back shroud and casing due to the precessing motion contribute to the rotordynamic fluid force moments. Experiments were conducted to measure the rotordynamic fluid force moments directly using the 4-axis sensor, and the unsteady pressure on the front and back casing wall. In this paper, following results are obtained: (1) The fluid force moment becomes destabilizing in the region of negative precessing speed ratio (0.3 < f/c < 0), at the design flow rate; (2) At reduced flow rate, the destabilizing fluid force moments occurred at small positive precessing speed ratio (0.2 < f/co < 0.4); (3) From the comparison of direct measured fluid force moments with those estimated from the unsteady pressure measured on the front and back casing walls, it was found that the destabilizing moments in the backward precession are mainly caused by the fluid forces on the front surface of the present impeller, where there is large clearance between the back shroud and casing.


INTRODUCTION
For turbomachinery operating at supercritical shaft speed it is important to understand the characters of rotordynamic fluid forces and mo- ments on impeller which occur in response to shaft vibration.Figure l(a) shows the typical vibration of overhung impeller.The motion of impeller can be divided into two fundamental modes (Ohashi  et al., 1991).One is whirling motion, and the other is precessing motion.The vibration of a rotating impeller (rotational speed ) can be dissolved into two modes; whirling with a constant eccentricity, In the precession, the center of the impeller, "O", does not move but the direction of the shaft inclination rotates with constant speed f.For whirling motion, many experimental and analytical data have been obtained on the rotor- dynamic fluid forces on pump impellers, mainly at Caltech and the University of Tokyo.It is now widely recognized that for closed-type centrifugal impellers the fluid forces become destabilizing for the forward whirl generally at whirl speed ratio (/ a;) less than 0.5 (Jery et al., 1985, Bolleter et al.,  1987, and Ohashi et al., 1988).The destabilizing forces are caused by the unsteady interaction between the impeller and volute casing (Adkins  et al., 1988, and Tsujimoto et al., 1988A) or vaned diffuser (Tsujimoto et al., 1988B), or by unsteady leakage flow between impeller shroud and casing (Childs, 1989, and Guinzburg et al., 1994).The   results are summarized in the recent textbooks of Childs (1993) and Brennen (1994).
For precessing motion, Ohashi et al. (1991) have shown that, from direct force measurements on a closed-type pump impeller, the fluid force mo- ments become destabilizing for the forward pre- cessing speed ratio (f/) less than 0.5.This range of destabilizing speed ratio is quite similar to that of the whirling cases as mentioned above.Yoshida et al. (1997) and Tsujimoto et al. (1997) investi- gated the unsteady flow in the back shroud/casing clearance of a precessing pump impeller.They also observed that the fluid force moment on the back shroud is destabilizing at small positive precessing speed ratio.In addition, the frequency range of destabilizing moment increases as the increase in the leakage flow rate, caused by the increase in the steady circumferential velocity, Vo, in the back shroud/casing clearance.
Open-type impellers have been widely used for high-speed and high-pressure centrifugal compres- sors.For open-type impellers, the variation of the tip clearance due to the shaft vibration contributes to the rotordynamic fluid forces.Nevertheless, as far as the authors are aware, no rotordynamic data for these machines are available except the recent report of Yoshida et al. (1999).Yoshida et al.,   reported experimentally that for an open-type centrifugal compressor impeller in whirling motion the fluid force becomes destabilizing at small positive whirl speed ratio throughout all the flow range without the interaction of a volute or vaned diffuser.In addition, the fluid forces change dramatically into destabilizing near the whirling speed ratio f/=0.8 at the low flow rate.This peak of destabilizing fluid force is caused by the strong interaction of the whirling motion with the rotating flow instability, similar to "rotating stall", with propagating speed ratio fl/;=0.79.
This paper presents the results from an investi- gation of the rotordynamic fluid force moments on an open-type centrifugal compressor impeller in precessing motion.For the precessing motion, the variations of the tip clearance and the clearance between the back shroud and casing contribute to the rotordynamic fluid force moments.Fluid force moments were measured directly with a force balance device.Discussions on the unsteady pres- sure on the front and back casing walls, and blade to blade pressure distribution are also presented.

Description of Test Facility
Figure 2 shows the sketch of the mechanism used to generate the precessing motion.The inner sleeve supports the main shaft through two eccentric inner bearings set to produce a pure precessing motion.The main shaft is driven by an AC motor with the rotational speed (w) through a universal joint, and the outer sleeve is driven by a DC motor controlled to run at a prescribed speed (f).This in term is the precessing speed.The precessing ang- ular velocity, 9t, is defined as positive when it is in the same direction as the impeller rotation, w.The main shaft speed was maintained at 400 + rpm, and the precessing speed ratio (f/) was varied in the range from -1.4 to / 1.4.Uncertainty in the precessing speed ratio, 2/, is + 0.002.
Figure 3 shows the details around the impeller.centrifugal compressor with 12 blades (Zi), inlet blade angle 32 deg.and outlet 45 deg. at the tip (i.e., back swept blade), outer radius (r2) 149mm, exit width (b2) 23 mm; its non-dimensional type number is 1.3.The impeller is equipped with a vaneless diffuser with radius ratio r4/r2--1.61and a symmetrical collector to minimize the nonuni- formity of exiting flow.Although the test impeller was designed for gas, water was used as the working fluid to facilitate the measurement of the fluid forces.The Reynolds number (Re uzrz/u, uz=r2co) is 3.04.0x 106 for actual condition (air, u2=300400m/s), and 0.92 x 106 for this laboratory test condition (water, u2 6.2 m/s).The effect of compressibility of actual working fluid (gas) is neglected in the test condition (water).
Under the condition without a shaft inclination (a 0 deg.), the normal blade tip clearance (') is constant (mm) from the inlet to outlet.In the present tests, the center of precessing motion was placed at the impeller center "O", (see Fig. 3, placed on the center line of the impeller outlet height), and the apex angle a 0.19 deg.was used for the precessing motion.In this condition, the tip clearance varies in the angular direction in the range of 0.7mm 1.3mm at the impeller inlet, and 0.6mml.4mmat the outlet due to the precessing motion.The clearance between the back shroud and casing varies in the range of 19.5 mm 20.5 mm at the impeller outlet (r r2) with the precessing motion.
In present study, we focus on the fluid force moment caused by the forces on the front sur- face of the impeller.For fluid force moment on the impeller back shroud, perimentally that the destabilizing moment caused by the pressure on the back shroud increases as the increase in the leakage flow through the back shroud/casing clearance due to the in- crease in the steady circumferential velocity, Vo.
The magnitude of the moment increases as the decreases of the axial clearance between back shroud and casing, H, and of the radial clearance between impeller side plate and casing, Gap-A (see Fig. 3), caused by the increase in unsteady pressure fluctuation on the back shroud.From these results, the back casing of the present model compressor was designed with a wider clearance H (= 20mm), and a wider Gap-A (= 11 mm) than actual ones, and with no leakage flow in the clearance to minimize the fluid force moment on the back shroud.

Instrumentation and Data Acquisition System
The impeller is supported by the main shaft through a rotating force balance with a 4-axis force sensor, as shown in Figure 2. The force balance is composed of two couples of parallel plates and 4 strain gauges per plate to measure the 4-axis forces (2 forces and 2 force moments).The strain signals are taken out through a slipring.The output signals of the strain gauges are converted to two forces and two moments components using a transfer matrix determined from a dynamic calibration test.Output signals are ensemble- averaged over 32 precessing orbits based on a trigger signal that indicates the instance when both the direction of the shaft inclination and the impeller rotating angle come to a prescribed orien- tation.Force and force moment are measured twice, that is, in air and in water at the same rotational and precessing speed.The former meas- urement values merely inertia force and gyro- scopic moment of the impeller itself due to the precessing motion, therefore, the fluid-induced force and force moment can be obtained by sub- tracting the former from the latter.In the present paper we focus only on the fluid force moments, which determines the stability for the precessing motion.Figure 4 shows the coordi- nate system.The r-axis is set in the direction of maximum blade tip clearance and the t-axis per- pendicular to it, directed by 90 deg.from the r-axis in the direction of the impeller rotation.The fluid force moment M is represented with its radial (Mr) and tangential (Mr to the precessing motion (a: apex angle, : shaft rotational speed, 2: precessing speed, (r, t): radial and tangential to the precessing motion).
v/r/4 + b/12.In the present experiment, un- i= certainty in the dimensionless fluid force mo- ments rn and mt is -t-2.0.We should note here that the radial component of fluid force moment, mr, is destabilizing for the precessing motion when rnr x (f/) > O. (i.e., rnr and f/co are both positive or both negative.)In this case, rn x (f/) > 0, the radial moment feeds kinetic energy to the vibrating shaft and thus excitatory.P1 P6 on the front casing, and R1 R4 on the back casing in Figure 3 show the locations of pressure taps to measure the steady and unsteady pressure.P1 P3, P6, and R1 R4 were used to measure the steady pressure with a manometer.In addition to this, P1 P5 and R1 R4 were used to measure the unsteady pressure using pressure transducers to estimate the pressure force mo- ments on the impeller (described later).P1 P3 on the casing, wall are facing the impeller tip, and R1 R4 on the back casing wall are facing the impeller back shroud.The diameter of the pressure taps, P1 P3, is mm and silicone oil is filled in the cavity in front of the pressure transducers.The pressure transducers at P4, P5 and R1 R4 were installed flush with the casing wall.P4 and P5 are located at different circumferencial positions (separation angle 60 deg.) to facilitate the exam- ination of the circumferential propagation of a rotating flow instability.The resonance frequency of the measurement system is 2.2 kHz, while the blade passing frequency is 80 Hz (= Z x /27r.).

RESULTS AND DISCUSSIONS
Compressor Pressure Performance Figure 5 shows the static pressure coefficient (b) at the collector and the steady pressure at various locations on (a) front casing, and (b) back casing wall plotted against the flow coefficient (b), under the condition without the shaft inclination.The design flow coefficient is ba= 0.424.From the flow visualization through the transparent casing with air bubble, backflow onset at the impeller inlet was observed at b=0.32, where the pressure rise reaches a local peak.The performance curve has a positive slope in a range of b 0.30 0.32.For the measurements of fluid force moments, the flow rate was varied from b=0.285 to b=0.508.The steady circumferential velocity on the back shroud, Vo, was estimated from the pressure distribu- tions in Figure 5(b).Assuming that the pressure gradient is balanced by the centrifugal force (i.e., dp/dr-pV/r-pK22r), it was found that K= Vo/r is in the range of 0.460.51 at the lo- cations R1 R4 throughout all the flow range.This result shows that the swirl on the back shroud has nearly forced vortex pattern, since there is no leakage flow in the back shroud/casing clearance.
Fluid Force Moments on Impeller Measured with Force Balance measured directly by the force balance versus the precessing speed ratio, f2/co, for various flow rates.
At the design flow rate, q5 qSd=0.424, it can be seen that the radial fluid force moment, mr, on the open-type centrifugal impeller are destabilizing (i.e., mr < 0) for the backward precession in the range -0.3 < f/w < 0. This destabilizing range of the fluid force moment extends to the negative direction of the precessing speed ratio as the decrease in flow rate.In addition, at lower flow rate, the radial fluid force moments become destabilizing (i.e., mr > 0) at positive precessing speed ratio in the range 0.2 < f/a < 0.4.Figure 7 shows with negative small precessing speed ratio, while the closed-type impeller has the destabilizing range (mr > 0) with positive small speed ratio.
Although the fluid force moments on the open- type impeller are smaller than those of the closedtype impeller, the radial component is roughly linear and the tangential component is quadratic with precessing speed ratio for both cases.As mentioned in the previous section, the clearance between the back shroud and casing of the present impeller, H-20 mm, is wider than that of the closed-type impeller, q 7 ram.This wider clear- ance H and wider Gap-A may be one of the reasons of the smaller fluid force moments on the present open-type impeller than those on the closed-type impeller.
The peak of the force moment occurred near [2/ co=0.20.3 at low flow rate 05=0.330.For the whirling motion of the present impeller, Yoshida et al. (1999) observed that the fluid force increases dramatically near the whirling speed ratio fUco 0.8 at the low flow rate (q5 < 0.32), caused by the interaction of the whirling motion with the rotating flow instability at the impeller inlet, similar to "rotating stall", with propagating speed ratio fZ'/w=0.79.Childs (1989) predicted in his calculation that the peak of the force on the front shroud occurred near fUco=0.3 0.5 due to the fluid structure interaction under the whirling motion.On the other hand, Tsujimoto et al.
-1,5 -1 -0.5 0 0.5 1.5 FIGURE 7 Comparison of the dimensionless fluid force moments on the impeller, radial mr and tangential mt components, with those on the closed-type pump impeller, from Ohashi et al. (1991), at design flow rates respectively.
we couldn't observe the rotating stall in the vane- less diffuser nor the rotating stall in the impeller inlet at this flow rate q5=0.330from the meas- urements of the pressure fluctuation at the dif- fuser inlet, P6, and the impeller inlet, P4 and P5.Unfortunately, full explanation of this peak force moment near f/c 0.2 0.3 at the low flow rate has not been obtained.
Fluid Force Moments on the Back Shroud and Front Surface of Impeller The fluid force moments on the back shroud, and the front surface of the impeller are calculated using the direct measurements of the fluid force moment and the unsteady pressure measure- ments on the back casing wall to obtain a better understanding of their origin.
Unsteady pressure on the back casing wall (Ap at locations R1, R2, R3, and R4) consists mainly of the precessing frequency component.Figure 8(a) shows a typical example of unsteady pressure, Ap, measured on the back casing wall at the location of R1, for qS=qd=0.424,and f/c=0.61.In this figure, the horizontal axis represents the phase of the precessing during a period, in which the clearance gap is widest at r, and smallest at 0 and 27r.The pressure fluctuation with precessing fre- quency, denoted as Apa, was obtained from the Fourier analysis of Ap, as shown in the right of Figure 8(a).The fluid force moments on the back shroud were obtained from the integration of the pressure distribution between the radii r 149 mm and 60mm using Ap a at locations R1, 2, 3, and R4.
Figure 9 shows the comparison of the dimen- sionless fluid force moments on the back shroud, m and mr,' obtained by integrating the pressure distribution, Apa, with the total moments on the impeller, mr and mr, measured directly by the force balance, at =qSd=0.424.In this figure, theoretical moments on the back shroud (from Tsujimoto et al., 1997) are also shown.It is calculated under the conditions of no leakage flow, K= 0.5, and no resistance at Gap-A.Under these conditions, the calculated radial com- ponent is 0 for all values of f/c, and the tan- gential component is 0 at f/a=-0.5 and + 1.5   with the maximum at f/co= +0.5.It can be found that the tangential fluid force moment, shows almost the same character as the cal- mt, culated moment with precessing speed ratio.
The radial fluid force moment, mr, on the back shroud is nearly 0 in the range -0.3 < f/a < 0, however the radial fluid force moment, mr, on the impeller directly measured by the force balance becomes negative (i.e., destabilizing).Further- more, the influence of the flow rate upon the fluid force moment on the back shroud was very small.
To obtain the understanding of the fluid force moments on the "front surface" of the impeller, " and " the fluid force moments on the back mr mt and shroud, m me, were subtracted from the direct measured moments on the impeller, mr and mr, at the same test conditions.Here, we call the sur- face of the impeller except the ""back shroud" as the front surface".Figure 10 shows the dimen- " and sionless fluid force moments, m mr, on the front surface of the impeller estimated by the above method, versus the precessing speed ratio for various flow rates.A solid line, mtgyro in Figure 10(c) shows the gyro moment of the fluid in the impeller due to the precession.This mtgyro is agree fairly well with the tangential moment, m so that for the design flow rate some fraction of the fluid force moment on the front surface of the impeller is caused by the inertia of the fluid in the impeller.The radial component, m on the front surface of the impeller becomes destabilizing (m < 0) for the backward precession with a de- crease in flow rate.This tendency is similar to the radial component, mr, on the impeller.From these results it can be concluded that the destabi- lizing moment (mr < 0) in backward precession is mainly caused by the forces on the front surface of the impeller due to the variation of the tip clearance, or the blade channel height.with moments on the impeller, mr and mr, for design flow rate b bd= 0.424 (uncertainty in m, mr--4-2.0, in f/w= + 0.002).-" 0 :--.,,.
Force Moments on Front Surface of Impeller Estimated from Unsteady Pressure Two simple estimations of the force moment on the front surface of the impeller are employed.The first is the integration of the pressure distribution on the front casing wall.In this case the forces resulted from the pressure distribution and the momentum transfer at the impeller inlet and outlet, and the rate of change of fluid momentum in the impeller are neglected.The second is the integration of the blade forces evaluated from the pressure difference across the blade measured on the front casing wall.This corresponds to the non- uniform blade loading model in axial flow turbine proposed by Thomas (1958)  and Alford  (1965).
During one period of the precession, the number of blades passing by the pressure transducer on the front casing is Zi ]/1.For the condition shown in Figure 8(b), twenty waves (Z; ]/] 19.7) due to the blade passing are clearly observed.The amplitudes of the component with blade pass- ing frequency, denoted by Ap and used for the estimation of the blade loading, were obtained from the reading of peak-to-peak values for each blade passing, as shown in the lower right of For the casing pressure force moments evalu- ation, the fluid force moments (mp, m")tp are obtained by integrating the pressure distributions Ap a from the blade leading edge to trailing edge on the casing wall.The pressure distributions are interpolated and extrapolated from Lkp a at P1, P2 and P3.For the blade load evaluation, it is assumed that Ap represents the pressure differ- ence across the blade at the tip, and the pressure difference is proportional to the square of its radius from the tip to hub.The force moments (m'r m") tb are estimated by integrating the assumed pressure differences on the three segments of the blade using/kp at P1, P2 and P3 taking account of the back swept blade.
Figure 11 shows the comparison of the esti- mated force moments (a) (m, " " mtp ), and (b) (mrb, m't ) with the moments on the front surface of the impeller (rn m"), for the design flow rate (0 4d-0.424).The estimated moments (mp, m) and (rn, rn) are not in good agreement with (m, m).Thomas (1958) and Alford (1965)   explained the destabilizing mechanism in axial flow turbines from the blade loading nonunifor- mity due to the change in tip clearance.Yoshida et al. (1999) reported that for the whirling open- type centrifugal impeller some fraction of the fluid force on the impeller is caused by the nonuniform pressure distribution on the casing wall.However, the present results suggest that the fluid force moment on the front surface of the tested impeller cannot be approximated by the nonuniform blade loading nor nonuniform pressure distribution on the casing wall.Therefore, a 15 10 "10 "15 1.5 -1 -0.5 0 0.5 0 m", mr" 0 mtp', lilt" , o -0 5 1.5 -1.5 -I -0.5 0 0.5 1.5 (a) mrp" mtp" 01=d=0.424) 5 0 5 (b) mrb', mt" (l=d---0;424)   and tangential FIGURE 11 Comparison of the dimensionless fluid force moments on the front surface of the impeller, radial m m components with (a) moments, mp and mtp, estimated from component of unsteady pressure on the front casing, and (b) moments, mrb" and mtb", estimated from the pressure tifference across the blades, for the design flow = 0.424 (uncertainty in -+-0.01, in / + o.oo2).

CONCLUSIONS
From the experimental results and discussions, the following conclusions can be drawn: (1) For an open-type centrifugal compressor impeller, the radial fluid force moment be- comes destabilizing for the backward precessing motion in the range -0.3 < f/cv < 0 at the design flow rate (b=0.424).This destabiliz- ing range of the fluid force moment extends to the negative direction of the precessing speed ratio as the decrease in flow rate (q5 0.285 0.330).
(2) At lower flow rate (b=0.2850.330), the radial fluid force moment becomes destabilizing also for the forward precessing motion in the range 0.2 < f2/a < 0.4.
(3) From the direct measured moments and the unsteady pressure measurements on the back casing, it was shown that the destabiliz- ing fluid force moments on the impeller at negative precessing speed ratio (-0.3 < f/ c < 0) are mainly caused by the forces on the front surface of the present impeller, where there is a large clearance in the back casing wall.
(4) The force moments on the front surface of the impeller estimated from the unsteady pressure on the front casing wall, and the pressure dif- ference across the blade don't agree with the direct measured force moments.
(5) More detailed experiments and sophisticated analyses are needed to explain the fluid force moments on open-type centrifugal impellers in precessing motion, and get more general conclusion.

NOMENCLATURE b2 f
Gap-A H mr, mt impeller axial width at outlet 23 mm (see Fig.  3) coefficient of unsteady pressure Ap, normalized by p(r2) 2 coefficient of unsteady pressure Apa, normalized by p(rza) 2 coefficient of unsteady pressure normalized by p(r2cv) frequency (Hz) clearance between impeller side plate and casing 11 mm (see Fig. 3)   clearance between back shroud and casing (normal clearance 20mm) moment of inertia of a disk with ra- dius r2, thickness, b2 pTrrb2i2, where i= r/4 + b22/12 velocity ratio (= Vo/rcv) components of fluid force moment; radial (r) and tangential (t) component (see Fig.  Ap , radial (r) and tangential (t) com- ponents, normalized by Mo gyro moment of the fluid in the impeller due to the precession pressure total pressure at inlet unsteady pressure amplitude of unsteady pressure on casing wall at precessing frequency peak-to-peak amplitude of unsteady pressure on casing wall at blade passing frequency radius radial and tangential axis (see Fig. 4)   impeller outlet radius 149mm (see Fig. 3) blade tip clearance (normal clearance S-lmm) time mean circumferential velocity in the clearance between back shroud and casing number of impeller blades 12 apex angle of precessing motion 0.19 deg.radius of circular whirl orbit fluid density flow coefficient-flow rate/(27rrb2) pressure coefficient (p-Ptl)/p(rzco) 2 precessing angular velocity angular velocity of rotating stall angular velocity of impeller precessing speed ratio or whirling speed ratio propagating speed ratio of rotating stall

FIGURE
FIGUREMotion of a overhung impeller on a vibrating shaft, and two fundamental vibration modes of an impeller.

FIGURE 4
FIGURE 4 Scheme showing the rotordynamic fluid force moments.Mr and Mt are the radial and tangential component

Figure 6 FIGURE 5
Figure6shows the dimensionless radial, m and tangential, mr, fluid force moments on the impeller the comparison of the dimensionless fluid force moments of the present open-type com- pressor impeller with those of the closed-type pump impeller from Ohashi et al. (1991), at the design flow rates respectively.The present open- type impeller has the destabilizing range (mr < O)

V=d=0FIGURE 8 FIGURE 9
FIGURE 8 Typical wave from of unsteady pressure ACp, (a) on back casing wall at R1 and (b) on front casing wall at P1, for b ba 0.424, Ct/w 0.61.AC shows the component of frequency f, and AC shows the plot of the amplitude of the blade passing fluctuation (uncertainty in ACp + 0.005, in b + 0.01, in f/w= -t-0.002).

Figure 8
Figure 8(b).On the other hand, the pressure fluctuation at the precessing frequency, denoted by Ap and used for the evaluation of the casing pressure force moment, was obtained from the Fourier analysis of Ap shown in the upper right of Figure 8(b).
different flow model is needed to explain the rotordynamic fluid force moment on open-type centrifugal impellers.More detailed experiments and sophisticated analyses are necessary to esti- mate the fluid force moment on open-type centri- fugal impellers in the future.
moment due to pressure distribution on casing wall: Ap , radial (r) and tangential(t)    components, normalized by Mo dimensionless fluid force moment due to pressure difference across the blades: Tsujimoto et al. (1997) reported analytically and Yoshida et al. (1997) ex-