7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

1/10

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

2/10

Xa4,* Y,. Yb x and y translations at 1st and2nd supports

as,,(o) 2DOF system supportreceptance

aps,pw 2 DOF system predictedsupport receptanceb(w)1 (4x4) measured receptancematr ix0 Frequency radlsec

1. INTRODUCTION

The importance of accurately predicting rotor lateral critical

speeds and related amplification factors has never been

more evident. The imposition of ever tighter vibrationstandards means that it is in the equipment manufacturers

best interest to ensure that accurate, representative

rotordynamic models are available at the design stage.

Similarly, sophisticated rotordynamic-modeling techniques

are now being more widely employed by end users as amajor diagnost ic tool .

The recent dramatic increase in available desktop

computing power has provided the rotordynamicist with theability to analyze rotordynamic models of ever increasing

complexity. For instance, it is now not uncommon for asimulation to include real-life dynamic effects resulting

from, for example. rotor interaction with seals, fluid-film

bear ings and aerodynamic or hydraul ic inf luences.Even so, situations frequently arise where large disparities

exist between predicted vibration parameters and those

obtained f rom measurements on real rotor systems.

The most predictable part of the modeling process would

generally be considered the rotor structural dynamics. This

is because rotors are relatively simple structurally and theirgeometric and material properties are normally clearly

def ined.In recent decades great strides have been made in the

modeling of oil-film bearing dynamics although much effort

Nicholas [3] showed that the assumption of rigid supportscould result in large errors in predicted critical speeds andamplification factors in present day rotor systems. He also

proposed a simple support-model which could providesubstantial improvement in the predicted parameters.

Nicholas [4]. [5]. (61 also demonstrated the value ofincorporating measured support characteristics into the

theoretical model. Support levels of varying complexity

were presented and, although meaningfu l a n drepresentat ive results w e r e obtained, signi f icantdiscrepancies between measured and predicted responsewere still evident. It was suggested by Nicholas [4] thatone of the possible reasons for disagreement between

measured and predicted response was the fact that

measured FRF data was taken with the rotor mounted inthe machine case.

2. INFLUENCE OF INSTALLED ROTOR ON

IMPEDANCE TEST RESULTS.

It is quite common practice to obtain the dynamic

characteristics of machine casings by performing impact

tests [5]. In this way the relationship between measuredresponse and applied force is obtained in the form of a

receptance FRF. The procedure, although havinglimitations [7], is normally quick and simple to perform.Since the a im is to ident i fy the machine support

parameters, ideally the rotor should be removed from themachine casing. In reality this is normally not practical

making it necessary to perform testing with the rotorinstalled.

2.1 2 DOF ModelConsider the 2 DOF model shown in Fig.1 as a modal

representation of a rotor mounted on flexible suppolts. It isassumed that suppolt and rotor damping are negligiblesince, in practice, structural damping tends to be very low.Modal testing can be simulated by determining the support

response due to the applied dynamic force F The

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

3/10

The identified support receptance usup is now corrupt andif utilized in an analytical model would result in an invalid

predicted receptance function:-

The receptance function characterized by (2) provides

roots, or natural frequencies, different from those inferred

by (1). Therefore, one effect of using support datameasured with the rotor installed is that subsequent

analytical predictions would lead to shifting of the peak-response frequencies. This may explain the difficultyexperienced, in some cases [4,5], in attempting to matchpredicted and measured peak-response frequencies. For

this simple case it is easy to show that the predicted first

natural frequency [from (2)] would be lower than theoriginal, i.e. correct, system first frequency [from (I)].

Similarly, the predicted second natural-frequency given by

(2) would be higher than the actual frequency obtainedusing (1). Figure 2 is a plot of the receptance ratio, R,where:

R = Identified FRF a:,(@)True FRF = asup (0)

and the frequency ratio fsup/frot has been selectedarbitrarily as unity. The influence of support stiffness ratio,K, can be quite dramatic. It is seen that in the region of the

system frequencies the estimated receptance according to

(2) can be grossly in error when compared to the actualvalue described by (1). At frequencies well below or well

element with no mass. The stiffness was selected to give a

support stiffness ratio, K, of 10.0. Receptance was

computed numerical ly using a computer program based on

the direct stiffness-matrix formulation. The shaft was

simulated using twenty mass-less, elastic beam-elements

having lumped-mass located at the element ends.

The direct receptance at the right hand support location

was computed for a range of frequencies up to seven times

the first rigid-support shaft mode (Fig. 4a). The receptanceis presented in a normal ized fashion whereby the computed

receptance is multiplied by the support stiffness Ksup.Since the supports exhibit no dynamic effects a normalized

receptance of unity would correspond to a support stiffness

of Ksup and would also indicate that the shaft dynamicsplay no role in the system - a desirable feature since theaim is to identify the support characteristics only. Any

deviation from a normalized receptance of 1 O correspondsto unwanted parameter identification errors. Referring to

Figure 4a we can see the three resonance peaks resulting

from the shaft dynamics. Since the supports are relativelystiff, Kz10.0, the first shaft frequency is quite close to thatfor the pinned-pinned case. The important question iswhat happens when we feed this ident i f ied receptance into

an analytical model.

Normally the identified data would be combined with an

available rotor model and dynamic-response computed. Asimilar procedure is followed here where the direct

receptance at the right hand support is predicted. Figure

4b shows the receptance data of Fig. 4a superimposed on

the shaft critical speed map. The figure is informative inshowing how use of the identified support data would lead

to the introduction of further false-resonances.

In cont rast , the constant support -s t i f fness line

corresponding to K=l.O correctly intercepts the natural-frequency plots at frequencies corresponding to those

resonances shown in Fig 4a Using the identif ied data and

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

4/10

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

5/10

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

6/10

where [Arot] contains the appropriate coefficients from Inter-Support Stiffness.[KrJ as outlined in Equation (6). kxbxa 4.36E6 N / m

The ident i f icat ion procedure can be summarized as fol lows.

At each measurement frequency: - Inter-Suppolt Damping,Cybxa 2.07E4 Ns I mCompute the dynamic stiffness matrix [D(o)] - onlyincluding the (static) rotor effects.Re-arrange the coeff icients i f necessary to

Invert the measured receptance matrix [u(a)].Compute the RHS of Equation (13) to determine [Asup]which contains the support coefficients.

It should be noted that [As,,] contains the coefficients of

[Ksup] shown in (7).The mathematical approach is very similar to that employed

for static condensation and shows that provided a valid

rotor model exists then measurement of a small number of

selective system FRFs should provide sufficient informationto enable effective identification of the support structure.

Although the identified support data could be furtherreduced and fitted to suitable structural models this aspect

is considered beyond the scope of this paper.

3.2 Validation

The foregoing procedure has been incorporated into a

Fortran program for se on a desktop PC. Both theprocedure and software were validated by analyzing anumber of test cases having known support parameters.

Figure 6 shows some of the results of an analysis of onesuch test case. The model employed for the analysis is

shown in Fig. 5b and includes inter-support stiffness and

damping cross-effects in addition to planar coupling

Although showing only informat ion for three supportparameters, the figure typifies the level of error involved in

the identification of all 32 support parameters. T h e

maximum error observed was approximately 1 .O%.

It should be noted that the method does not explicitly

identify the support mass terms but produces only the

direct Dynamic-stiffness values

The model, which utilized a 23.station rotor, was analyzedusing a 33MHz/466 PC and took approximately 20 minutesto identify the support parameters at 400 frequency points.

The run-time is approximately linearly proportional to the

number of frequency points to be analyzed and roughlyproportional to N3.Single-precision computations have been found to providesufficient accuracy for the cases analyzed to date. It is

conceivable that significant numerical errors could result if

rotor-modeling was such that the magnitude of those rotor

stiffness coefficients used in (13) was vastly different to that

of the corresponding support stiffness coefficients.Increasing the computational precision (e.g. going to

double-precision) would alleviate the problem in most

circumstances. However, it is felt that application of sound

model ing techniques should ensure the el iminat ion of such

problems.

4. ELECTRIC MOTOR EXAMPLE

The company employs a number of 4000 HP, Z-pole

induction motors for gas compression purposes in plantsthroughout Saudi Arabia. A schematic of the machine

configuration is presented in Fig. 7. The 1830 kg rotor has

a bearing span of 2.1 m and is supported on 125 mm

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

7/10

Results from tests on two of these motors will be presented

here. The motors are subsequently identified as motors A

and B.

Each motor was mounted on test-bed and an instrumented

force-hammer employed to excite the motor casing.Casing response was measured using an accelerometer.

Four measurement points were utilized - i.e. the horizontaland veltical locations at each bearing housing. The 16lnertance FRFs were obtained using a portable analyzerand subsequently transferred to PC. Each FRF wasdefined using 400 data points over the frequency-ranges O-12000 cpm and O-6000 cpm for motors A and B

respectively. The identification algorithm presented earlier

was utilized to estimate the support parameters, by making

use of the aforementioned validated rotor model. Asanother check on the procedure the identified parameterswere entered into a forced-response program and

simulated FFiFs produced by considering unit loading ateach of the measurement points in turn. Figure 9 shows

the measured and simulated drive-end horizontal direct

inertance for motor A. The illustration is typical in that themeasured and simulated plots are identical, for all practical

purposes, confirming the validity of the combined rotor-support system model and algor i thm.

Figure 10 compares receptance functions obtained from

motor B using the method presented here with those

derived following the procedure adopted by Nicholas [4],where the rotor influence is neglected. It is clear that

substantial differences occur between the identified FFiFswithin certain frequency ranges. More specifically, thefrequencies at which the peak-responses occur differ

significantly depending upon which identification technique

is applied.

Synchronous unbalance response was computed using a

model similar to that presented in section 3.1 with the

inclusion of linearized oil-film coefficients. The identified

Close examination of the data suggested the presence of a

mechanical rub invalidating any type of linear analysis in

this region.

A final assessment of the need to account for the rotordynamics during impedance testing was undertaken by

examining the computed unbalance response for motor B.Figure 12 presents shaft and casing synchronous response

plots using support data acquired with and without

correction for the shaft dynamics. The response plots are

given for an unbalance of 6.6 kg mm at the non-drive en d

cooling fan.

It is very noticeable that the peak-response speeds differ

significantly, in line with the theory presented earlier. It is

clear that even the inclusion of oil-film effects, at least in

this case, is not sufficient to diminish the error resultingfrom the neglect of shaft dynamics.

5. CONCLUSIONS

It has been shown theoretical ly that in certain

circumstances substantial errors may occur when

employing measured support-data which has been

acquired from impedance tests with the rotor installed.

A method has been presented to enable identification of the

machine-support parameters even when the modal-test

data includes dynamic effects from the installed rotor. The

approach is similar to static condensation and requiresmeasurement of a small number of selective system FRFsin conjunction with the utilization of a valid rotor model.The validity of the approach was demonstrated by

analyzing a numerical model having known support

parameters. All 32 suppolt damping and stiffnesscoefficients were identified to within 1% of their true values.

The analysis procedure was applied to a practical caseinvolving a 4000 HP electric motor. Measured shaft

synchronous response was compared to that predicted

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

8/10

6. ACKNOWLEDGMENTS I I

The author wishes to thank K.M. Al-Hussain and MB.Al-Aidarous for their effort in performing the rotor modal tests.The assistance of B.B. Sanchez Jr. in preparing the

manuscript is gratefully acknowledged.

7. REFERENCES

[l] Morton, P.G.Recent Advances in the Study of Oil Lubricated

Journal Bearings,Proc. of the 4th Intl. Conf. on Rotor Dynamics,IFToMM. Sept. 7-9, 1994.

[Z ] Childs, D.Turbomachinery Rotordynamics :Phenomena,Modeling, and Analysis, J. Wiley, 1993

[3] Nicholas, J.C. and Barrett, LX.The Effect of Bearing Suppolt Flexibility on CriticalSpeed Prediction.

ASLE Transactions, Vol. 29, No. 3, July 1986.[4] Barrett, L.E., Nicholas, J.C. and Dhar. D.

The Dynamic Analysis of Rotor Bearing Systems Using

Experimental Bearing Support Compliance Data. Proc.of the 4th Intl. Modal Analysis Conference, 1966.

[5] Nicholas, J.C., Whalen, J.K. and Franklin, S.D.Improving Critical Speed Calculations Using Flexible

Bearing Support FRF Compliance Data.Proc. of the 15th Turbomachinery SymposiumTexas A&M University, Nov. 1966

[6] Nicholas, J.C.

N/P///Fig.1. Rotor - Support Model

Operating Turbomachinery On or Near the SecondCritical Speed in Accordance with API Specifications. 0 06 1 15 I ss 8Proc. of the 16th Turbomachinerv Svmposium, 1969 Reqary*,f

(71 Halvorse, W.G. and Brow, D.i. 1Impulse Technique for Structural Frequency Response

Testing.Sound and Vibration, Nov. 1977.

[8] Vernon, J.B.

Fig.2 Plot of Receptance Ratio for2 DOF Dystem

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

9/10

Fig.4.(a) Shaft System Receptance Plot

Fig.4.(b) Critical Speed Map ShowingFalse Resonances

Fig. 4.(c) Receptance Plot Using

Identified data for theSystem Shown in Fig.%

Fig.j.(a) General Rotor System

-%F--

Fig.5.(b) Rotor System Model

Fig.6 Comparison of Actual andIdentified Parameters forNumerical Model.

Fig.7. Schematic of 4000HP Induction Motor

Fig 8. Comparison of a) Measured andb) Computed Rotor Free-Free

Mode Shapes

7/30/2019 Sem.org IMAC XIII 13th 13-10-6 Practical Rotordynamics Modeling Using Combined Measured Theoretical

10/10

I,.,