Stability And Unbalance Response Of Rotor Bearing System

Transcription

1 Stability And Unbalance Response Of Rotor Bearing Syste T.V.V..N. Rao and. Athre Mechanical Engineering Grop Departent of Mechanical Engineering Birla Institte of Technology & Science Indian Institte of Technology Delhi Pilani, Rajasthan 3333 New Delhi 6 Eail: nrao@bits-pilani.ac.in Eail: kathre@ech.iitd.ernet.in ABSTRACT This paper presents stability and nbalance response prediction for syetric rotor bearing syste sing siple bt realistic odels. The rotor syste is described with kinetic energy and potential energy fnctionals that flly accont for translational inertia, rotary intertia, and gyroscopic copling. Rayleigh Ritz ethod is sed to describe the flexral displaceents. The displaceent fnction chosen is the first ode shape of a classical free free bea in two orthogonal planes with constant cross section in bending. The rotor otion is derived sing agrange s eqations, and expressed as state space for that preserves the skew syetric gyroscopic atrix within the shaft and disks as well as cross copling effects of jornal bearings. Reslts are presented for threshold speed of instability, natral freqencies as a fnction of rotor speed (Capbell diagra), and nbalance response. Inflence of variation of jornal bearing paraeters (bearing length, radial clearance, oil viscosity) and distance between the flexible bearing spports on the rotor threshold speed of instability is evalated. INTRODUCTION A stdy of dynaic behavior of rotors onted on flid fil bearings is very iportant with increase in deand of power reqireents and decrease in weight of rotating achinery. The literatre on rotor dynaics is qite extensive. Becase of its fndaental iportance in the design of rotating achinery, rotor dynaics will persist as an active research area in ftre also. Several well established ethods are nowadays available for linear analysis of rotor bearing systes, sch as: transfer atrix [], finite eleents [], dynaic stiffness [3] and assed odes [4]. allane and Farris [5] derived the eqations of otion for siple rotor odels to show the basic phenoena occrring in rotordynaics. They have analyzed natral freqencies as a fnction of

2 speed of rotation, instability and responses to forces of excitation for syetric/asyetric rotor. The instability of rotors in flid fil bearings is generally attribted to the self excited vibrations known as oil whirl/whip of a rotor. Flid indced self excited vibrations (flid whirl and whip instability) in rotor bearing systes, theoretical odeling and experiental deterination are presented in detail by Mszynska [6] for rotors spported on a rigid (brass oilite) bearing at the inboard end and 36 oil lbricated jornal bearing at the otboard end. Based on nd s [7] linearized stability analysis, Rao [8] derived analytical expressions for Soerfeld nber, stiffness and daping coefficients as a fnction of steady state eccentricity ratio of a plain cylindrical jornal bearing. The present work focses on stability and nbalance response prediction for syetric rotor bearing syste sing assed odes or Rayleigh Ritz ethod, where the rotor displaceent field is represented as sperposition of basis fnctions of the space variable and generalized coordinates as fnctions of tie. The expressions for kinetic and strain energies of the rotor eleents and virtal work of flid fil bearings are calclated. angrange s eqations are applied to describe the dynaics of rotor syste. The Rayleigh Ritz ethod can be accrate with a sitable choice of assed odes and it has the advantage of representing the detail of a coplex rotor syste with relatively few degrees of freedo. THEORY Rotor-Bearing Model Figre [8] shows the coordinate syste of reference for the siplified analysis of shaft with three disks spported by bearings at both ends. The spport bearings are siilar and described by classical 8 linearized spring and daping coefficients. To deterine the gyroscopic inertial effects, the present analysis applies Stodola Green odel [9]. In a linearized theory, the shaft kinetic energy for the flexral displaceents and w relate to the orthogonal fixed transverse directions x and z respectively, and cross sectional rotations θ and ψ which are Elerian angles relative to those axes are: ρs ρi Ts = ( & + w& ) dy + ( θ& + ψ& ) dy + ρi ( + ψ& θ) dy () The kinetic energy of each attached disk d are: = ( + ) + ( θ& T & & + ψ& ) + ( + ψ& d M D w I Dx I Dy θ) () The strain energy of the shaft contains the inflence of bending deforation only withot taking into consideration of shear deforation [] and it is assed that copling of torsional and axial loadings with the flexral otion is niportant. The reslting potential energy fnctional is:

3 EI w U s = + dy y y (3) Bearing forces are taken to be arbitrary linear fnctions of the transverse displaceents and velocity coponents at the bearing locations. The virtal work done by the forces at the bearing locations b is: δwb = ( xx + xzw + B & xx + B w& xz ) δ (4) ( + w + B & + B w& ) δw zx zz zx zz z y x l. l... l Fig. ikchi rotor [8] Application of angrange s eqations to the expressions for the kinetic energy of shaft disk assebly with nbalance ass, strain energy of shaft and virtal work de to bearing reactions on the shaft leads to a set of copled second order linear differential eqations of the for: q& + B q& + B q + q + q = f ( l ) d sin t (5) & B q& + B q& q& + + q + q = f ( l ) d cost (6) = d where l l [ M f () l + I g () l ] + ρs f ( y) dy + ρi g ( y) D Dx M D M D ( R R ) ρ ; I = I = ( 3R + 3R h ) ; I = ( R R ) M D = π h Dx Dz + D y + πr S = πr ; I = 4 The fnction f ( y) that describes spatial coordinates of the orthogonal displaceents y, t) f y q t and ( y, t) = f y q t is derived by applying the ( = ( ) ( ) ( ) ( ) w dy 3

4 bondary conditions for this rotor bearing odel given by classical Eler Bernolli first ode fnctions [] for a non rotating free-free bea. The fnctions g ( y), h( y) are the first and second derivatives respectively of f ( y). For an aligned jornal bearing, classical 8 spring and daping coefficients are coonly sed to odel the dynaic radial force interactions between the jornal and bearing. In this work 8 spring and daping coefficients are deterined sing the analytical expressions based on the short bearing approxiation [8]. In Eqs. (5-6), the ass atrix is syetric, while the daping and stiffness atrices are not syetric. The non-syetries arise fro jornal bearings and gyroscopic oents. If the net energy iparted to the rotor per cycle of haronic otion by the non conservative force is negative, then the rotor bearing syste is stable for the specified freqency, otherwise it is nstable []. Stability Analysis The instability deterination is based on the stdy of the rotor bearing syste in free otion. The characteristic eqation for the rotor bearing syste is: 4 3 s + a s + a s + a s + a = (7) a a 3 where ( B + B ) = ; 3 4 a = ( ) + ( B B B B ) ( B + B B B ) ( ) = ; B B + 4 a = Unbalance Response The soltion of eqations of otion (Eqs. 5-6) with nbalance force of excitation is to be obtained in the for [5] as: q = A sin t + B cost (8) q = A sin t + B cost (9) The sbstittion of Eqs. (8) (9) in Eqs. (5) (6), yields two eqations to the coefficients of sin t, cost. The atrix fors of reslting linear set of eqation are: B B B B B B A B = A B f ( l f ( l ) d ) d () 4

5 ( ) ( ) ( ) ( ) The soltion of Eq. () gives vales of A, B, A, B fnctions of rotational speed. The ajor and inor axis of the elliptical whirl orbits are [8]: A, B = ( A + B + A + B ) ± ( A B + A B ) + 4( A B + A ) B RESUTS AND DISCUSSION which are The geoetry of the rotor bearing syste for the syetric ikchi rotor is given in Table. The instability threshold speed for the disks-shaft-bearings configration given in Table is 88 rad/s. The roots of the characteristic eqation (Eq. 7) are coplex nbers [] and the iaginary parts of the roots give the freqencies of the rotor bearing syste. When the syste becoes nstable, any one of the real part of coplex roots becoe positive and this indicates growing whirl orbit. The inflence of variation of jornal bearing design variables on the instability threshold speed is given in Table. Bearing length, oil viscosity and radial clearance are chosen as the variables as the jornal diaeter and load on the bearing are inflenced by the rotor design. Instability threshold speed for b /D ratio of.5 ( b =. ) is higher copared to b /D ratio of. and.5 ( b =.4 and b =.6 ). Redcing the b /D ratio for a given load on bearing, increases the operating eccentricity ratio and hence increase in instability threshold. Decrease in oil viscosity fro.6 Pas to. Pas also increases the operating eccentricity ratio and hence instability threshold speed. Instability threshold speed has decreased with increase in radial clearance fro 5x -6 to 5x -6. Increase in clearance obviosly increases operating eccentricity ratio (as Soerfeld nber decreases), however increase in clearance also redces the bearing spring and daping coefficients for a given load and speed condition. The inflence of distance between the bearing spports on the instability threshold is given in Table 3. The flexral displaceent of the bea for the end conditions (free-free) is zero when l o =.38, and hence choice of bearing spports at this location reslts in a lowest vale of instability threshold speed (.3 rad/s) which indicates that, the instability threshold is lowest when the flexral displaceent of rotor are axi. () Table Geoetry for the Rotor bearing syste Disk Shaft Bearing R..6 D.4 R.9 ρ 78 kg/ 3 b.4 H.4 E x N/ C r 5x -6 ρ 78 kg/ 3 rad/s µ.6 Pas 5

6 .374* -3 kg.3 W 5.9*9.8 N d. Table Inflence of jornal bearing on the instability threshold speeds b, Instability threshold speed, rad/s µ,pas Instability threshold speed, rad/s C r, Instability threshold speed, rad/s x x x Table 3 Inflence of bearing spport span on the instability threshold speeds l o, Instability threshold speed, rad/s Figre shows the Capbell diagra for syetric rotor. Coplex conjgates roots are obtained when the syste becoes nstable. Figre 3 shows the response de to nbalance excitation for the rotor bearing systes. The nbalance ass is considered for the first rotor (disk) located at.33 fro the origin. The ajor and inor axes of elliptical orbit (Eq. ) are obtained solving [8] the syste of linear eqations (Eq. ). The ajor axis of the elliptical orbit is indicated as the aplitde of response in Fig. 3. The nbalance response aplitde for the syetric rotor increases in agnitde ntil the threshold speed of instability and the aplitde of response is nearly constant in the nstable regie. The nbalance response exhibits forward whirl ode in the range of rotor speed considered in this stdy. CONCUSION Rayleigh Ritz ethod is applied to the rotating shaft with thin rigid disks and flexible jornal bearing spports. The rotor is analyzed by classical bea end conditions (free-free) in two orthogonal planes. inetic energy and strain energy of the rotor eleents and virtal work of flexible bearing spports are calclated. agrange s eqations are sed to derive dynaic eqations of rotor otion. Based on the analysis of disk-shaft-bearing geoetry for the syetric rotor configration, the following conclsions are obtained. Jornal bearing design variables sch as bearing length, oil viscosity and radial clearance inflence the instability threshold. Decreasing the following variables independently can increase rotor instability threshold speed a. bearing length b. oil viscosity c. radial clearance. 6

7 . Rotor flexral displaceents significantly inflence the instability threshold speed. The rotor exhibits ini instability threshold, when the rotor flexre is axi. Whirl speed, rad/s ikchi rotor Rotational speed, rad/s Unbalance response,.5e-6.e-6.5e-6.e-6 5.E-7.E+ ikchi rotor Rotational speed, rad/s Fig. Capbell diagra Fig. 3 Unbalance response NOMENCATURE C r radial clearance () D jornal diaeter () d nbalance ass location in the disk () E Modls of elasticity (Pa) g gravitational acceleration (/s ) H thickness of the disk () I second oent of area of circlar shaft, ( 4 ) I Dx, I Dy, I Dz inertia of the disk in x, y, z directions (Fig. ) (kg ) length of the shaft () l location of the bearings (Fig. ) () l location of the first disk fro the origin of xyz coordinate syste, () b length of the bearing () M D ass of the rotor (kg) nbalance ass of the rotor (kg) R inner radis of the disk () R oter radis of the disk () S area of shaft ( ) W load on the flexible bearing spports 7

8 x,y,z coordinate syste of reference shown in Fig. () µ viscosity of lbricant (Pa s) ρ density of lbricant (kg/ 3 ) anglar velocity of shaft (rad/s) REFERENCES. nd, J. W., Stability and Daped Critical Speeds of a Flexible Rotor in Flid Fil Bearings, ASME Jornal of Engg. for Indstry, pp , Nelson, H. D., and McVagh, J. M., The Dynaics of Rotor-Bearing Systes sing Finite Eleents, ASME Jornal of Engg. for Indstry, 98, pp , Rieger, N. F., Thoas, C. B., and Walter, W. W., Dynaic Stiffness Matrix Approach for Rotor Bearing Syste Analysis, Proc. of IMechE conference on Vibration in Rotating Machinery, C 87/76, pp. 87 9, Morton, P.G., Analysis of Rotors Spported on Many Bearings, J. Mech. Engg. Sci., 4, pp. 5 33, allane, M., and Ferraris, G., Rotor dynaics Prediction in Engineering, John Wiley and Sons, Mszynska, A., Flid Indced Instabilities of Rotors: Whirl and Whip, Bently Rotor Dynaics Research Corporation Report No. 3., nd, J. W., Review of the concept of Dynaic Coefficients for Flid Fil Jornal Bearings, ASME Jornal of Tribology, 9, pp. 37-4, Rao, J. S., Rotor Dynaics, New Age International (P) td, Childs, D., Trboachinery Rotordynaics, John Wiley and Sons, New York, Zirkelback, N.., Ginsberg, J. H., Ritz Series Analysis of Rotating Shaft Syste: Validation, Convergence, Mode Fnctions, and Unbalance Response, ASME Jornal of Vibration and Acostics, pp. 49 5,.. Rao, S. S, Mechanical Vibrations, Addison-Wesley Pblishing Copany, 99.. Adas, M.., and Padovan, J., Insights into inearized Rotor Dynaics, Jornal of Sond and Vibration, 76 (), pp. 9 4, 98. 8