Study on Nonlinear Vibration and Crack Fault of Rotor-bearing-seal Coupling System

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1 Sensors & Transducers 04 b IFSA Publishing S. L. Stud on Nonlinear Vibration and Crack Fault of Rotor-bearing-seal Coupling Sstem Yuegang LUO Songhe ZHANG Bin WU Wanlei WANG College of Electromechanical & Information Engineering Dalian Nationalities Universit Dalian 6600 China luog@dlnu.edu.cn Received: November 03 /Accepted: 8 Januar 04 /Published: 8 Februar 04 Abstract: The nonlinear dnamic model of rotor-bearing-seal sstem with crack in shaft is set up based on the coupling model of nonlinear oil-film force and Muszska s nonlinear seal fluid force. The dnamic vibration characteristics of the rotor-bearing-seal sstem and the effects of phsical and structural parameters of labrinth seal and crack fault on movement character of the rotor were analzed. The increases of seal length seal pressure differential seal radius and aial velocit are in favor of the stabilit of the sstem and it of seal gap and crack depth are not in favor of the stabilit of the sstem. Copright 04 IFSA Publishing S. L. Kewords: Rotor sstem Oil-film force Seal force Shaft crack Nonlinear vibration.. Introduction It has important influences of nonlinear oil-film force nonlinear seal force and shaft crack et al to the rotor sstem in safet running. Man researchers studied the nonlinear dnamic characteristics of the rotor sstem. Zhang Yanmei et al. [] analzed a bearing-rotor sstem b using a new analtic model of the unstead oil film force. The bifurcation and chaos behavior of the rotor sstem varing with the rotation angular frequenc were obtained through numerical simulations. Based on the analsis of the oil film force of a journal bearing four characteristic coefficients of the journal bearing are derived b Yang Jinfu et al. []. Moreover according to fluidstructure-interaction equilibrium equations the presented a stabilit criterion equation of the oil filmrotor sstem which can eplain the acting mechanism of the oil film force on the rotor. The dnamic destabilization characteristics of the journal bearing were given b testing on the work set. Chen Yushu [3] investigated the relation of the linearized stabilit of a single span rotor-seal sstem between its parameters using the Musznska s model of seal fluid dnamical forces. According to the theor of Hopf bifurcation the instabilit leads to self-ecited whirling motion of the rotor generated from its equilibrium position. The direction of Hopf bifurcation and the stabilit of the bifurcated periodic orbit are determined b Poore s algebraic criterion. The theoretical results are verified b the numerical results. Yang Xiaoming and Li Songtao et al. [4 5] analzed the nonlinear dnamic characteristics of labrinth seal-rotating wheel sstem in hdraulic turbines and investigated the nonlinear dnamic stabilit of hdraulic turbines with labrinth seal using the Musznska s model of seal fluid dnamic forces. The results show that there is occurrence of Hopf bifurcation after threshold speed is eceeded and the rotating wheel comes in to bifurcated periodic orbit. While the amplitude of bifurcated motion will rise according to improvement of rotor speed which ma cause the rotating wheel impact seal at certain speed. Lapunov s first approimate Article number P_867

2 theor was used to analze nonlinear sstem stabilit and Runge-Kutta method was used to analze numerical simulation computation. Finall the influence of phsical and structural parameter of seal on rotor was investigated. Sensitivit analsis of nonlinear dnamic characteristics was performed b changing the phsical and structural parameters of seal-rotating wheel sstem. B comparing with eightparameter model it was tested that the variation tendenc of Musznska s model is identical. Luo Yuegang et al. [6] set up a dnamic model of twospan rotor-bearing sstem with crack fault. Using the continuation-shooting algorithm for periodic solution of nonlinear non-autonomous sstem the stabilit of the sstem periodic motion was studied b the Floquet theor. The periodical quasi-periodical and chaos motions were found in the sstem responses. The unstable form of the rotor sstem with crack fault was period-doubling bifurcation. There are unstable forms of period-doubling bifurcation and Hopf bifurcation in different rotate speed. There were man harmonic element s of /3 / /3 and so on within the sub-critical speed range. But the -harmonic element decreased within the supercritical speed range. Gao Chongren et al. [7] studied the seal behaviors taking cracked rotor sstem as object. The Musznska s model of the nonlinear seal force was applied to building the coupled dnamic equation and analzing the movement character of cracked rotor with airflow induced force. The effect of phsical and structural parameters of labrinth seal on movement character of cracked rotor was emphaticall discussed. The research results shown that the nonlinear dnamics character were ver plentiful in the cracked rotor seal sstem the airflow induced force can reduce the periodic response of the cracked rotor the main parameters of the sealed structure had important impact on the stabilit of cracked rotor sstem and the dnamic stabilit can be improved b adjusting seal parameters. Darpe [8] studied transient response of a simple Jeffcott rotor with two transverse surface cracks during its passage through first bending critical speed and the response of the rotor during its passage through /3rd and / subharmonic resonances was also investigated. In this paper the nonlinear dnamic model of rotor-bearingseal sstem with crack in shaft is set up based on the coupling model of nonlinear oil-film force and Muszska s nonlinear seal fluid force. The dnamic vibration characteristics of the rotor-bearing-seal sstem were analzed b numeral integrated method and the effects of phsical and structural parameters of labrinth seal and crack fault on movement character of the rotor were investigated. The results provide theoretical references for safe operation and fault diagnosis of the coupling sstems.. Dnamic Models of the Rotor Sstem and Motion Differential Equation Model of the rotor-bearing-seal sstem with crack fault is shown as Fig.. Shaft coupling connects the motor and rotor. The sstem mass is equivalentl concentrated on the center of ever disc and bearing support respectivel. The torsional vibration and gro moment are neglected and onl the lateral vibration of sstem is considered. Both ends of rotor are supported b journal bearings with smmetrical structures. O is geometric centers of bearing O is geometric centers of rotor O 3 is center of mass of rotor m is lumped masses of rotor at bearing m is equivalentl lumped masses at disc. The shaft with zero qualit connect disc with bearing k is the stiffness of elastic shaft. F and F are nonlinear oil film forces P and P are nonlinear seal fluid forces. There is a transversal crack with deepness of a in the middle of shaft. a t t Fig.. Models of rotor-bearing-seal coupling sstem and section of crack... Oil-film Force of Short Bearing Non-dimensional Renolds equation based on oilfilm of short bearing assumption is epressed as R L z ( h 3 p ) sin cos ( cos sin) z () where R is the radius of bearing L is the length of bearing. Non-dimensional nonlinear oil-film force can be obtained form above formula: L p D ( )sin ( ) cos (4z 3 ( cos sin ) ) () Non-dimensional nonlinear oil-film force can be finall epressed as f [( ) ( ) ] f 3 V ( ) sin G ( ) cos S ( ) 3 V ( ) cos G ( ) sin S ( ) (3)

3 where ( cos sin) G( ) V ( ) cos sin S( ) ( cos sin) (4) (5) where is the rotational angular velocit K D and m f are the represent equivalent stiffness equivalent damping and equivalent mass respectivel. f is the average velocit ratio of fluids in aiall. K D and f are the nonlinear functions of the displacements and which can be epressed as G( ) ( ) cos sin arctg ( ) (6) n K K0( e ) n D D0 ( e ) n 0.5 ~ 3 b ( ) 0 f 0 e b (9) arctg sign sign( ) (7).. Nonlinear Seal Fluid Force The Musznska s model is applied to investigate the nonlinear behaviors of seal fluid force it can be epressed as [9 0] P K m f f f D P f D K m f f D m f f m f 0 m f f D 0 m f (8) where e / cs is the relative eccentricit of rotor c s is the seal gap n b and 0 are the seal parameters general K 0 D 0 and m f can be calculated b Black-Childs formulas []..3. Motion differential equations The radial displacements of aes center of rotor sstem in left are assumed as ( ) respectivel the radial displacements of disc are ( ) respectivel then non-dimensional motion differential equation of sstem can be epressed as ( )[ F( )] F( )[( )cos t( )sin t] f( ) M ( )[ F( )] F( )[( )sin t( )cos t] f( ) -G M P ( )[ F( )] F( )[( )cos t( )sin t] cos N M P ( )[ F( )] F( )[( )sin t( )cos t] sin G N M (0) where ε and δ are the relative stiffness parameters about crack deepness a [6]; F ( ) is the open and cos close function of crack F ( ) t 0 arctan is the contained angle between crack direction and eccentricit; 0 is the c c initial phase place. c m ( m m f ) is the damping coefficient of bearing; c is the k damping coefficient of disc m kc s kc mc ( m mf ) cs k. The parameter c is the bearing ( m ) m f gap. mc M N ( m G m f ) mg ( m m f ) c s X cs c s Y M ( m m f ) cs m e g G c X Y c c. is the non-dimensional time F F t. f f are the non-dimensional nonlinear oil-film force components which are shown in formula (3). is the Sommerfeld RL R L correction coefficient is m g c R the viscidit of lubricant. P and P are the nondimensional nonlinear seal fluid forces i 3

4 P K mf f fd P fd K mf f D m f f mf f D 3. Nonlinear Dnamic Characteristics Analsis of the Sstem () 3.. Dnamic Characteristics of the Sstem in Nonlinear Coupling Fluid Forces The parameters of the sstem are: m =4.0 kg m =3. kg R=5 mm L= mm c=0. mm c s =0.5 mm μ=0.08 P a s c =050 N s/m c =00 N s/m k= N/m e=0.05 mm. The phsical and structural parameters of seal are: n=.5 b= m The imported loss coefficient of seal is z=0. and the aial velocit is 5 m/s. The first critical rotate speed of the sstem (with no fault) is ω 0 =88.5 rad/s. Fig. is the bifurcation diagrams of rotorbearing-seal response with rotational speed. When the rotational speed is low the response of rotor sstem is periodic- motion. Along with the increase of the rotational speed at about ω=60rad/s the sstem loses stabilit of the periodic motion and bifurcation from periodic- to periodic-. The main motion in the region of critical rotational speed is periodic- and it in the supercritical rotational speed is periodic- periodic-4 and quasi-periodic motion. The main motion of the sstem in the ultrasupercritical rotational speed is quasi-periodic. For investigating the influences of phsical and structural parameter of seal on rotor sstem the seal length seal pressure differential seal radius aial velocit and seal gap are respectivel changed. The results are shown in Fig. 3. The instabilit critical speed of the sstem shows nonlinear improvement along with the increase of the seal length seal pressure differential seal radius and aial velocit. It testifies the increases of these parameters are in favor of the stabilit of the sstem. The instabilit critical speed of the sstem shows nonlinear decreased along with the increase of the seal gap which testifies the increase of the seal gap is not in favor of the stabilit of the sstem. It can be seen comparing references [4 5 7] that the influences of the phsical and structural parameters of seal on rotor sstem in the coupling actions of nonlinear oil-film force and nonlinear seal force are significantl different of it in action of nonlinear seal force onl. 3.. Influences of Crack to the Motion Character of Sstem Fig. 4(a)-(d) are the bifurcation diagrams of the rotor sstem with the rotating speed ω as the control parameters at different non-dimensional crack depths of a= When the crack is less (a=0.4) the effects of crack fault to the motion characteristics of rotor sstem is little. Along with the increases of crack depth the periodic-4 motion translates to chaotic motion around four regions in the region of critical rotational speed and there appear periodic-3 motions in the regions of supercritical rotational speed and ultra-supercritical rotational speed which can be the basis for judging the crack fault and degree. Fig. 3(f) shows when a 0. 4 the instabilit critical speed is about ω=60 rad/s. The instabilit critical speed of the sstem shows nonlinear decreased along with the increase of the crack depth which testifies the increase of the crack depth will reduce the stabilit of the sstem. Fig.. The bifurcation diagrams of rotor response with ω. 4

5 (a) Seal length (b) Seal pressure differential (c) Seal radius (d) Aial velocit (e) Seal gap (f) Crack depth Fig. 3. Effects of parameters on critical unstable rotating speed. Fig. 4 (a). The bifurcation diagrams of rotor response with ω at different crack depth a=0.4. 5

6 Fig. 4 (b). The bifurcation diagrams of rotor response with ω at different crack depth a=0.7. Fig. 4 (c). The bifurcation diagrams of rotor response with ω at different crack depth a=.0. Fig. 4 (d). The bifurcation diagrams of rotor response with ω at different crack depth a=.3. 6

7 4. Conclusions ) The nonlinear dnamic model of rotor-bearingseal sstem with crack in shaft is set up based on the coupling model of nonlinear oil-film force and Muszska s nonlinear seal fluid force. The dnamic vibration characteristics of the rotor-bearing-seal sstem were analzed b numeral integrated method and the effects of phsical and structural parameters of labrinth seal and crack fault on movement character of the rotor were investigated. ) The increases of seal length seal pressure differential seal radius and aial velocit are in favor of the stabilit of the sstem and it of the seal gap is not in favor of the stabilit of the sstem. The influences of the phsical and structural parameters of seal on rotor sstem in the coupling actions of nonlinear oil-film force and nonlinear seal force are significantl different of it in action of nonlinear seal force onl. 3) The increase of the crack dep will reduce the stabilit of the sstem. The periodic-3 motions appearing in the regions of supercritical rotational speed and ultra-supercritical rotational speed can be the basis for judging the crack fault and degree. Acknowledgements This research is supported b the Fundamental Research Funds for the Central Universities (No. DC000) supported b the Natural Science Foundation of Liaoning Province China (No. 0004) and supported Funds for the State Ethnic Affairs Commission of China (No. DLZ006). References []. Zhang Yanmei Lu Qishao. Bifurcation and stabilit analsis on vibrations of rotor sstems with unstead oil film force Acta Aeronautica Et Astronautica Sinica Vol. 4 Issue 003 pp []. Yang Jinfu Liu Zhansheng Yu Daren et al. Research on nonlinear oil film force and its stabilit of journal bearing Power Engineering Vol. 4 Issue pp [3]. Chen Yushu Ding Qian Hou Shujun. A stud on the stabilit and hopf bifurcation of nonlinear rotor-seal sstem Journal of Vibration Engineering Vol. 0 Issue pp [4]. Yang Xiaoming Ma Zhenue Zhang Zhenguo. Stud of nonlinear dnamic stabilit of labrinth seal-rotating wheel sstem in hdraulic turbines Journal of Dalian Universit of Technolog Vol. 47 Issue 007 pp [5] Li Songtao Xu Qingu Wan Fangi. A stud on nonlinear dnamic stabilit of labrinth seal-rotor sstem Chinese Journal of Applied Mechanics Vol. 9 Issue 00 pp [6]. Luo Yuegang Zhang Songhe Liu Xiaodong et al. Stabilit of a two - span rotor - bearing sstem with crack fault Transactions of the Chinese Societ for Agricultural Machiner Vol. 38 Issue pp [7]. Gao Chongren Yin Yufeng Yao Dechen et al. Stud on coupled vibration of nonlinear cracked rotor seal sstem Machine Design and Research Vol. 5 Issue 009 pp [8]. A. K. Darpe. Dnamic response of an accelerating two - crack Jeffcott rotor Advances in Vibration Engineering Vol. 9 Issue 00 pp [9]. A. Musznska. Model testing of Rotor/Bearing sstems The International Journal of Analtical and Eperimental Model Analsis Vol. Issue pp [0]. A. Musznska D. E. Bentl. Frequenc swept rotating in put perturbation techniques and identification of the fluid models in rotor/bearing/seal sstems and fluid handing machines Journal of Sound and Vibration Vol. 43 Issue 990 pp []. Zhang Wen Theoretical basis of rotor dnamics Science Press of China Copright International Frequenc Sensor Association (IFSA) Publishing S. L. All rights reserved. ( 7