Identification Method in the Dynamics of Rotor- Bearing Systems Using Measurement Data of Unbalance Responses

Transcription

1 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 Identifiation Method in the Dynamis of otor- Bearing Systems Using Measurement Data of Unbalane esponses Kibong Han, Hyoungwoo Lee, Donghwan Lee, Department of Mehatronis Engineering, Jungwon University, South Korea Korea Institute of Mahinery & Materials, South Korea Abstrat Presented in this dissertation is a new method of identifying the ritial speed of rotor-bearing systems without atually reahing at the ritial speed itself. Using the method, it is possible to alulate the ritial speed by measuring a series of rotor responses at muh lower rotating speeds away from and without reahing at the ritial speed of the system. In the ourse of the proedures illustrated, not only the ritial speed but also the damping ratio and the eentriity of the system an be identified at the same time. Test rotor was tested on the otor Dynamis Test Faility at the Korea Institute of Mahinery & Materials, Korea, and the theory has been onfirmed experimentally. The method an be adopted to monitor hanges of the dynami harateristis of ritial rotating mahinery before and after overhauls, repairs, exhanges of various parts, or to detet trends or diretion of subtle hanges in the dynami harateristi parameters over a long periods of time. Keywords Critial speed, damping ratio, eentriity, identifiation, unbalane response I. INTODUCTION otating mahinery, mainly used in produts of the power plant industry, suh as gas turbines, ompressors, and high pressure turbine pumps, an experiene insignifiant problems whih result in the motion-stop of all dependent systems. Hene enormous prodution failure and eonomi loss frequently our. Therefore, the periodi overhaul, observation and diagnosis of the mahinery and the replaement of the produt inluding parts have been arried out in order to raise produtivity and stabilize prodution. However, the run-out ondition of this rotating mahinery is always varies. Moreover, hanges of rotor systems our in ases of long-term use, before and after overhaul, the replaement of produts or parts and the alteration of the operating ondition, as well as in ases of problems. But a simple method by whih the user an be aware of these harateristi hanges of suh mahine systems has never been developed or even known. Merely observing the ondition of mahinery in monitoring only vibration signals during operation and stopping and repairing the mahinery when signals indiating vibration in exess of the vibration set point is the method that has been used in the field until now. Among the harateristi hanges of mahine systems, the most remarkable and important to understand are how muh the operating point approahes the ritial speed after a hange of operating ondition, what the system damping ratio and unbalane eentriity in referene to the vibration level are whether any hange of ritial speed, damping ratio and unbalane eentriity is or is not owing to a harateristi hange, and whether that hange an affet the system or not. However, raising operation speed to the ritial speed of the mahine system is mostly impossible under field irumstanes suh as the safety of the field staff or the seurity of the rotating mahinery itself. Besides, a simple method of measuring or analyzing a system s damping ratio or unbalane eentriity has not yet been developed. A study on the dynami-harateristi parameters of rotor-bearing systems i.e. natural frequeny, vibration mode, damping oeffiient, and unbalane eentriity is as follows. To date, many researhers in the field of rotor dynamis have been absorbed in the development of various kinds of analytial, experimental or experimental/analytial omplex methods to identify the parameters of rotor-bearing systems from the Jeffott rotor system[-] to multi-degree-of-freedom rotor systems[4-5]. However, a pratial and simple method has not yet been devised. Methods[6-8] analytial identifying dynamiharateristi parameters inlude the method by modal analysis, but this is not effiient in diagnosing and alibrating the vibration problems of real rotor systems owing to the fat that it presents only qualitative results. Therefore quantifying the identifiation of dynamiharateristi rotor parameters has been mainly performed by modal test[9-] using experimental frequeny-response funtion measurements of real rotor systems. 70

2 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 At this time, exitation methods inlude a diret method using the run-up and oast-down of the rotating speed of the rotor and its own residual unbalane as the exitation soure, and a method designing and using an extraordinary tool apable of the exitation of the rotating rotor. However, the former method requires preise balaning in order to surpass the ritial speed, so it annot be applied to a real rotor. Moreover, it is very diffiult to exite the rotor during rotation using the latter method, unlike a ommon struture, and it is also diffiult to distinguish the exitation signal from the measured vibration signal. Therefore, the design and appliation of an extraordinary tool or an extraordinary data proessing method is required. For instane, H. Li et al. designed and produed the torsional hammer for torsional exitation of rotating mahinery[], and W. D. Marsher introdued a time-averaged modal exitation method as the proposed method to find the natural frequeny of the shaft and the dynamis onstant of the bearing[4]. This method improves the S/N ratio of the measured signal using the time-average funtion of a signal analyzer after repeatedly and suffiiently exiting the neighborhood of the bearing housing with an impat hammer. However, suh methods require the aomplishment of a preise experiment by a veteran engineer, disturb the working environment, and are dependent on the working environment itself. Hene these methods annot be used effiiently in the field. In this study, using the response-harateristi equation of the Jeffott rotor model, an algorithm whih an experimentally identify the dynami-harateristi parameters of the rotor-bearing system i.e. ritial speed, damping ratio, and unbalane eentriity has been developed. Furthermore, an experimental method whih determines rotor-bearing system parameters using measured data on unbalane response before ritial speed, has been proposed. For the appliation of the proposed method to a real rotor, a rotor-bearing system was assembled, the system parameters were identified from the measured vibration data, and the validity of the proposed method was verified through experimentation. II. ALGOITHM DEVELOPMENT FO IDENTIFICATION OF SYSTEM PAAMETES A. Vibration response of Jeffott otor The first-degree ritial speed of rotating mahinery has the approximate harateristis of the Jeffott rotor whether it is unneessary of a omplex struture or simple rotating mahinery. Beside, onsidering the operation of most rotating mahinery within the first-degree ritial speed, the importane of the Jeffott rotor model annot be overlooked. 70 Fig. A model of Jeffott rotor As shown in Fig., the equation of motion of the Jeffott rotor having a disk at its enter is expressed as follows, : d M dt d M dt dz dt z os t kz 0 dy dt y sin t ky 0 Where, M,,, k are the effetive mass and unbalane eentriity of the disk, and the damping oeffiient and stiffness oeffiient of the rotor system respetively. ewriting above equation, M z z k z M os t M y y k y M sin t 4 Defining whirl radius r to equation 5 as follows, a ombination of the aforementioned equation and 4 gives equation 6 as follows. r z i y 5 M r i t r k r M e 6 In the above equations, the unique exitative fore is the entrifugal fore of the whirling of the disk rotating on the elasti shaft aording to effetive mass M and unbalane eentriity. The general solution of equation 6 is as follows: i t r e 7 where, M 8 tan k M k M 9

3 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 70 B. Theoretial equation for system parameter identifiation It an be definitely known that the measured vibration signal of the rotor ontains the information of the dynamiharateristi parameters of the system from the responseharateristi equation of the Jeffott rotor model. Therefore, finding three pairs of rotating speed and vibration response by measurement, the dynamiharateristi system parameters suh as ritial speed, damping ratio, and unbalane eentriity an be found using the Jeffott rotor response equation. The arrangement of the Jeffott rotor equation of motion 6 divided into the oeffiient of the highest degree M in both lateral is as follows: t i e r r r 0 In equation 0, the unknowns are,, and, and the solution of the above-mentioned three equations is found in order to find these unknowns using the required three equations. To transform and define the form of,, from system parameters M,, k and, the rearrangement of equation 8 is as follows: Where, M k, Mk r To find an independent solution of the three unknowns, and from equation, assuming eah different rotating speed and vibration quantity,,,, and, three simultaneous equations are found as follows: 4 5 ] From equations, 4 and 5, eentriity existing as a ommon variable is found in the form inluding two unknowns,. 6 Arranging the first and seond equalities of equation 6, 7 earranging the first and third equalities of equation 6 is as follows: 8 To find from equations 7 and 8, first performing equation 7 about, 4 9 where, 0 Performing equation 8 about similarly, 4 where, from equations 9 and,

4 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August From equation, two simultaneous equations in whih the parameter is determined by are found. Squaring the first and seond equalities of equation, the next equation is ompleted earranging equation 4, ] [ 5 where, 6 Arranging equation 5 with respet to, 0 ] [ 7 Finding equation7 with respet to, A 8 A From equation 8, sine the relationship of 0 is ompleted, is simply expressed as follows. 9 From equation 9, is uniquely determined by the known quantities of,, and,,. Namely, when is uniquely determined, is uniquely determined by equation 6, then, the unique is found by equation 6 again. Hene, only measuring the rotation speeds of, and and displaement quantities of, and, the system parameters natural frequeny, damping ratio and eentriity an be found. III. IDENTIFICATION AND VEIFICATION OF SYSTEM PAAMETES A. Experimental devie for identifiation of system parameters A preise experiment bed is designed and onstruted for the experiment identifying the rotor bearing system, as shown in Fig.. In order to inrease the degree of experimental repetition, the upper surfae of the steel bed omposed of the rotor bearing was ground down to maintain the horizontal degree of m. Fig. An overview of rotor dynamis test faility The preise proposed test bed is installed on a suffiiently heavy onrete base of , and the Jeffott rotor shaft with disk is plaed in the test bed. In order to drive the shaft, an AC spindle motor and rotation speed ontroller were used. Besides, the motor driving part and the rotor driven part, the middle shaft, are mutually divided, installed, and power is transmitted by belt as shown in Fig.. Hene, neither vibrations are transmitted. Moreover, this experiment bed is applied to the journal bearing whih an flexibly self-align the horizontal vibration displaement of the shaft, the bearing housing, the lubriation devie of the bearing, and the safety devie of the shaft

5 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 Fig. Belt drive system showing motor, belt, drive shaft, rubber oupling, and rotor shaft In order to identify the rotor-bearing system parameters, and, the experiment devie is onstruted as shown in Fig. 4. In Fig. 4, the AC spindle motor with an output power of 7.5 kw rotates the middle shaft by a flat belt, and the middle shaft and experimental rotor are onneted by a flexible of light rubber. The bearing is supported by a spherial plain journal bearing of mm width, and silion oil of 0 Cst is supplied to the upper part of the oil tank by oil pump with respet to the lubriation of the bearing, and the oil supply itself is of the gravitational type. In Fig. 4, the displaement is measured by installing nonontat-type displaement sensors X-Probe, Y-Probe and the key phasor is transmitted to PC through a digital vetor filter DVF, two osillosopes, an FFT analyzer and an AD onverter, and analyzed. The digital vetor filter DVF transforms the rotor unbalane vibration signal to the filtered X signal omponent and indiates displaements in the x diretion and y diretion as well as the rotating speed and phase angle. Two osillosopes monitor the unfiltered signal orbit of the original x- diretion and y-diretion displaements filtered as well as that of the x-diretion and y-diretion displaement X omponents. The FFT analyzer indiates the frequeny spetrum of the original vibration signal and monitors frequeny harateristis suh as the magnitude of the degree omponent per rotation speed. The PC, as the analysis system, inputs the measured vibration signal as an algorithm by the three-equations method, and outputs the parameter data of the rotor system. Moreover, it is used to find the dynami-harateristi parameters of the real system through the statistial proess of data analysis. Fig. 4 Shemati diagram of measurement and analysis system for rotor. A. System parameter identifiation The y-diretional displaement and phase per rotation speed of the rotor, used in the identifiation of system parameters after aomplishing the response measurement experiment, are illustrated in Table and Fig. 5. Those measured data are alibrated values of runout, whih are used in the algorithm of the three equations in order to the system parameters, and respetively. The algorithm analysis results of the measured data are shown in Figs. 6, 7 and 8. Figure 6 shows both the horizontal and vertial axes as the same axes of the ritial speed of the rotor. Figure 7 shows the distribution of the damping ratio versus the ritial speed. Furthermore, Fig. 8 shows the distribution of the unbalane eentriity versus the ritial speed of the rotor. Table Measurement data of Y- axis rotor vibration Amplitude Unit: mil= 5.4 m, peak-to-peak Y otor Speedrpm Amplitude Phase Angle mil deg

6 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 Fig. 5 Plot of response vs. shaft speed Analysing the experimental and analytial results of Figs. 6, 7 and 8, most values of ritial speed, damping ratio and unbalane eentriity are distributed neighboring,94 rpm, 0.07 mil and 4. mil, respetively. The analysis and investigation of the probability distribution of the ritial speed of the system is shown in Fig. 9. Figure 9 shows the same result as Fig. 6. Aordingly, dynamiharateristi parameters aording to the algorithm of the three equations of the experimental and analytial results are summarized as illustrated in Table. Table Dynami harateristis of otor Fig. 7 Plot of damping ratio vs. ritial Speed Critial Speed rpm Damping atio Eentriity mil Fig. 8 Plot of eentriity vs. ritial speed Fig. 9 Probabilities of ritial speed Fig. 6 Plot of ritial speed vs. ritial speed 706

7 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 A. Experimental verifiation of system parameters. Experimental verifiation of the ritial speed To verify the ritial speed of the proposed method, the measurement of the ritial speed was tested. After the rotor was rotated through the ritial speed to,600 rpm, the ritial speed was investigated by lowering the rotation speed. Sine the burden that the rotor should overome the load an be diminished when raising the rotation speed, it is helpful to investigate the ritial speed preisely. From the experiment, x-axis-diretional Bode plots suh as those in Figs. 0 and are found. These are shown as polar plots suh as those in Fig.. Moreover, y-axis-diretional Bode plots suh as those in Figs. and 4 are found. These are shown as polar plots suh as those in Fig. 5. From these dynami-harateristi rotor plots, the ritial speed of the rotor of,960 rpm an be known by the x- and y-axes diretional plots of,960 rpm respetively. Sine the ritial speed is indiated by the diretion of both axes, the experimental devie performs well aording to the harateristis of the Jeffott rotor. A omparison of the ritial speeds of the proposed method and the experiment is illustrated in Table. The proposed method has an error rate within %. Fig. Expanded Bode plot of X-response Fig. Polar plot of X-response Fig. 0 Bode plot of X-response Fig. Bode Plot of Y- esponse 707

8 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 Namely, by finding the residual unbalane based on the influene oeffiient method ICM the results of the measuring test attahing gradual trial masses and measuring vibration quantity are shown in Fig. 6 as the distribution of the residual unbalane quantity applying trial masses of.76 g and.64 g. From this polar plot, the unbalane eentriities of.77 mil and.79 mil are found, respetively. A omparison of the unbalane eentriities of the proposed method and the experiment is illustrated in Table 4. This shows an error rate of % for the proposed method 4. mil. Fig. 4 Expanded Bode plot of Y-response NO Fig. 5 Polar plot of Y-response Table Estimated and verified ritial speed. Method Present Method Critial speed rpm Error % to Experiment ISO Standard Verifiation of unbalane eentriity by ISO Standard Calulating the residual unbalane eentriity based on the ISO standard[5], the result is as follows. It is applied by aquiring the amplitude and phase information of the ISO standard. Fig. 6 esidual unbalane alulated by influene oeffiient method. NO Table 4 Estimated and verified eentriity. Method Present Method Eentriity mil Error % to Experiment 4. + ISO Standard.8 - IV. CONCLUSION Using data measuring the vibration quantity before the ritial speed based on the response-harateristi equation of the Jeffott rotor model, an experimental method identifying the ritial speed, unbalane eentriity and damping ratio of a rotor-bearing system was proposed, and an experimental model was produed. From the experiment, the results were as follows. 708

9 International Journal of Emerging Tehnology and Advaned Engineering Website: ISSN , ISO 900:008 Certified Journal, Volume, Issue 8, August 0 From the vibration-response equation with respet to the Jeffott rotor model, the algorithm of three equations whih an determine the dynamiharateristi parameters of a rotor-bearing system suh as ritial speed, damping ratio, and unbalane eentriity was developed. Using the Bode plots of the experimental model, the ritial speed was measured to,960 rpm. It was impossible to overome the ritial speed due to exessive vibration during the measurement. Hene, several preise balaning of the uni-setion were aomplished to overome the vibration safely. After alulating the ritial speed and unbalane eentriity using the algorithm for the identifiation of the developed parameters, the ritial speed and unbalane eentriity were found to be,94 rpm and 4. mil, whih were preise within error rates of % and %, respetively. EFEENCES [] Ehih, F. F., Handbook of otor dynamis, MGraw-Hill, In., 99, pp.4-4. [] Jeffott, H. H., The Lateral Vibration of Loaded Shafts in the Neighbourhood of a Whirling Speed The Effet of Balane, Philosophial Magazine and Journal of Siene, Vol. 7, Series 6, pp. 04 4, 99. [] ieger, N. F., Vibrations of otating Mahinery, Part I: otor- Bearing Dynamis, Vibration Institute, Clarendon Hills, Ill., 98. [4] Eshleman,.L., Flexible otor-bearing System Dynamis- I. Critial Speeds and esponse of Flexible otor System, The Flexible otor Systems Subommittee, The Engineering Division, ASME. [5] Shapiro, W., and umbarger, J.H., Flexible otor-bearing System Dynamis-II. Bearing Influene and epresentation in otor Dynami Analysis, The Flexible otor Systems Subommittee, The Engineering Division, ASME. [6] Lee, C.-W., and Jei, Y.-G., Modal Analysis of Continuous otor- Bearing Systems, J. Sound and Vibration, 6, 988, pp45-6. [7] Zenetta, G.A.., Identifiation Methods in the Dynamis of Turbo Generator otor, Proeedings of the Institute of Mehanial Engineering, July, 99, pp 7-8. [8] Birembaut, Y., Peigney, J., Predition of Dynami Properties of otor Supported by Hydrodynami Bearings Using the Finite Element Method, Computer & Strutures, Vol., 980, pp [9] Subbiah,., Bhat,.B., Sankar, T.S., Determination of Modal Parameters of otors Supported on Hydrodynami Bearing Through Experimental Modal Analysis, pp [0] Ewins, D. J., Modal Testing: Theory and Pratie, esearh Studies Press Wiley, 984 [] Marsher, W. D., Analysis and Test of Multistage Pump Wet Critial Speeds, STLE Preprint No. 89-TC-6E-, 989. [] Marsher, W. D., Determination of Pump otor Critial Speeds During Operation Through Use of Modal Analysis, Pro. ASME, 986 WAM, pp. 7-78, De [] Li, H., Entwistle,.D., Stone, B.J., Impat Exitation of Torsional Vibration, The International Journal of Analytial and Experimental Modal Analysis, Vol. 9. No., July 994, pp [4] Jen, C.-W., Marsher, W.D. Using Time-Averaged Modal Exitation to Determine the otor Dynami Bearing Coeffiients of Centrifugal Pumps. [5] ISO Standard, Mehanial Vibration-Balaning Quality equirements of igid otor, ef. No. ISO 940/-986E, pp