Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves

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1 G. H. Jang J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 33-79, Korea Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves This paper presents an analytical method to investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves, even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill s infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone grooves increases with increasing eccentricity and with decreasing groove number, which play the major roles in increasing the average and variation of stiffness coefficients, respectively. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients. DOI: 0.5/ Introduction In recent years, computer hard disk drives HDDs require fast, quiet and reliable operation in addition to having high storage capacity. Traditionally, a spindle motor in a HDD is supported by ball bearings that generate a higher level of nonrepeatable runout NRRO and thus cannot ensure the reliability of reading and writing data, as the data tracks in a disk become dense. A spindle motor with a hydrodynamic herringbone grooved journal bearing is an ideal alternative to replace the ball bearing due to the damping effect of lubricant that results in low NRRO and low acoustic noise. However, a rotating grooved journal with a small number of grooves may induce parametric excitation in a spindle system employing this type of bearing. Many researchers have investigated the stability of hydrodynamic journal bearings. Pai and Majumdar analyzed the stability characteristics of submerged plain journal bearings under a unidirectional constant load and variable rotating load. Jonnadula et al. 3 analyzed the effects of surface roughness on the stability of submerged plain journal bearings. Raghunandana and Majumdar studied the effects of non-newtonian lubricant on the stability of oil film journal bearings under a unidirectional constant load. Kakoty and Majumdar 5 analyzed the stability of journal bearings under the effects of fluid inertia. These studies, however, were restricted to the stability of plain journal bearings. On the other hand, several researchers have studied the dynamic characteristics and the stability of herringbone grooved journal bearings. Zirkelback and San Andres 6 investigated the variation of the dynamic coefficients for a hydrodynamic journal bearing with rotating grooves at a fixed journal position. Jang and Yoon 7 investigated the dynamic behaviors of a hydrodynamic Contributed by the Tribology Division of the THE AMERICAN SOCIETY OF ME- CHANICAL ENGINEERS for presentation at the ASME/STLE Tribology Conference, Cancun, Mexico October 7 30, 00. Manuscript received by the Tribology Division February 9, 00; revised manuscript received July 0, 00. Associate Editor: J. L. Streator. journal bearing by solving the nonlinear equations of motion due to the effects of the rotating or stationary herringbone grooves. Kang et al. 8 presented a numerical analysis of static and dynamic characteristics and evaluated the mass parameter a measure of stability of oil-lubricated herringbone grooved journal bearings with 8 circular-profile grooves on the sleeve. Kang et al. were able to perform the stability analysis of the hydrodynamic herringbone grooved journal bearing by using the conventional mass parameter and whirl ratio because the herringbone grooves were inscribed in the a stationary sleeve. However, this cannot be appropriately applied to investigate the stability of the hydrodynamic journal bearing whose grooves are inscribed in a rotating journal because the dynamic coefficients change even in the steady state as the rotor rotates. This paper presents an analytical method to investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. It can be determined by solving Hill s infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. Method of Analysis. Determination of Dynamic Coefficients. Figure shows the coordinate system of the hydrodynamic journal bearing with rotating herringbone grooves. The circumferential coordinate Journal of Tribology Copyright 003 by ASME APRIL 003, Vol. 5 Õ 9

2 Table Design parameters of the herringbone grooved journal bearing Fig. Coordinate system of the hydrodynamic journal bearing with rotating herringbone grooves is defined from the fixed negative X-axis. The Reynolds equation can be written in the coordinate system (xr,z) fixed to the journal as shown in Fig. 7. x h3 p x z h3 p z R h h () x t The thickness of the fluid film can be expressed in terms of in the groove and the ridge regions, respectively. hc g ce X cos e Y sin () hce X cos e Y sin (3) where c, c g, e X, and e Y are clearance, groove depth, and eccentricities in X and Y directions, respectively. The circumferential coordinate can be expressed in terms of the rotational velocity and angular coordinate of the journal as follows: t () Finally, the rate of change of the film thickness in Eq. can be written as follows: h t e X sin e Y cos (5) On the other hand, for small journal perturbations with respect to an equilibrium position, x and y, the pressure p and the film thickness h can be expressed as follows: pp 0 p p, x,y (6) hh 0 x cos y sin (7) where subscript 0 means the quasi-equilibrium position. Differentiation of film thickness can be written in the following form: h t h 0 t x sin y cos ẋ cos ẏ sin (8) Substituting Eqs. 6, 7, and 8 into Eq., five perturbation equations are obtained in the following forms: 3 h 0 p x x h 3 0 p z z R h 0 x e X sin e Y cos h 0 x cos p 0 h 0 x sin p 0 R sin x :ẋ R cos x :ẏ x x h 0 z h 0 z :0 cos p R 0 z sin p R 0 z cos x :x sin x :y (9) The solutions of Eq. 9 are easily obtained using the FEM 9. Load capacity, stiffness and damping coefficients can be determined by integrating the solutions in the bearing area. The relation between the hydrodynamic pressure and bearing reactions can be expressed in the following form: F X F Y F 0 K y x C ẏ ẋ (0) where F 0, K, and C are the load capacity, stiffness and damping matrix, respectively, and can be represented as follows: 9 Õ Vol. 5, APRIL 003 Transactions of the ASME

3 F 0 F X0 F Y0 K K XX K YX C C XX C YX A K XY K YY A C XY C YY A p 0 cos p 0 sin da () cos sin p x p y da () cos sin p ẋ p ẏ da (3). Stability Analysis of a Parametrically Excited System The equations of motion for a well-aligned rigid rotor with mass m supported by two identical hydrodynamic journal bearings can be written as follows: m 0 0 m ÿ ẍ C XX C XY C YX C YY ẏ ẋ K XX K XY K YY y x 0 0 K YX () For a hydrodynamic journal bearing with rotating grooves, the damping and stiffness coefficients have time-varying components even at the equilibrium position and they can be expressed in the following forms: C ij Cij Cij cost Cij (5) K ij Kij Kij cost Kij (6) where,,, and are the average value, the amplitude of the variation, the frequency of the parametric excitation and the phase angle of the dynamic coefficients, respectively. These equations are more complicated than the typical form of Mathieu s equation where time-varying component exists only in the displacement 0. However, after substituting Eqs. 5 and 6 into Eq., the following matrix with the average value of stiffness is utilized in order to transform the equations of motion into an eigenvalue problem later. 3 KXX K XX KXY KYX KYY (7) Multiplication of the inverse matrix of Eq. 7 into Eq. gives the following equation, in which the stiffness matrix with average value contains the same diagonal elements: Fig. Variation of the dynamic coefficients of the hydrodynamic journal bearing at 5,000 rpm Ä0.8 and N g Ä : a stiffness coefficient; and b damping coefficient. m m m 3 m ẍ ÿ E E E 3 E ẋ ẏ e e e 3 e ẏ ẋ cos t f f f 3 ẏ ẋ sin t K 0 0 K y x k k k 3 k y x cos t n n n 3 n y x sin t 0 (8) 0 In the above equation, each matrix can be expressed in the following form: f m m m/ m 3 m 3 3 (9) E E E 3 E 3 3 CXX CYX 3 CXY CXX CYX CXY CYY (0) e e e 3 e 3 3 cos CXX C XX cos CYX C YX 3 CXY cos CXY CYY cos CXX cos CXX CYX cos CYX CXY cos CXY CYY cos CYY () Journal of Tribology APRIL 003, Vol. 5 Õ 93

4 Fig. 3 Variation of the direct stiffness coefficients for increasing rotational speed: a Ä0. N g Ä ; b Ä0. N g Ä8 ; c Ä0.8 N g Ä ; and d Ä0.8 N g Ä8. f f f 3 f 3 3 sin CXX C XX sin CYX C YX 3 CXY sin CXY CYY sin CXX sin CXX CYX sin CYX CXY sin CXY CYY sin CYY () K 0 0 K KXX 0 0 KXX (3) k k k 3 k 3 3 cos KXX K XX cos KYX K YX 3 KXY cos KXY KYY cos KXX cos KXX KYX cos KYX KXY cos KXY KYY cos KYY () n n n 3 n 3 3 sin KXX K XX sin KYX K YX 3 KXY sin KXY KYY sin KXX sin KXX KYX sin KYX KXY sin KXY KYY sin KYY (5) Because the frequency of the parametric excitation in Eq. 8 is, the solution has the period of T/. Therefore, the solution x(t) has the property that xttxt (6) where is a constant. If there is a solution for which, the response x(t) will grow as time increases and the solution will be unstable. If, the response will decrease with time and the solution will be stable. If, the response will continue indefinitely without a change of amplitude and the solution will also be critically stable, in the sense that the response will be bounded. Also, the boundaries between the stable and unstable regions are determined when. If, x(t) must have period T and if, the response must have period T. However, when 9 Õ Vol. 5, APRIL 003 Transactions of the ASME

5 Fig. Variation of the cross-coupled stiffness coefficients for increasing rotational speed: a Ä0. N g Ä ; b Ä0. N g Ä8 ; c Ä0.8 N g Ä ; and d Ä0.8 N g Ä8., the characteristic equation includes all the eigenvalues as when 0. So, in this research, the characteristic equation is only derived in the case of. For a periodic solution x(t) with period T, Eq. 6 can be rewritten as follows: xttxt (7) In this case, the responses x(t) and y(t) can be expressed as Fourier expansions in the following forms: xa 0 n yc 0 n a n cos nt b n sin nt (8) c n cos nt d n sin nt (9) where a n, b n, c n, and d n are Fourier coefficients. The velocity and acceleration can be obtained as follows: ẋ n ẏ n ẍ n ÿ n n nt a n sin b n n nt c n sin d n n a n cos nt b n n c n cos nt d n n n n n cos nt (30) cos nt (3) sin nt (3) sin nt (33) Substituting Eqs into the equations of motion in Eq. 8 and equating the coefficients of the constant, sine and cosine terms to zero, linear algebraic equations can be obtained in terms of a n, b n, c n, and d n. The following are the linear algebraic equations obtained from the first equation in Eq. 8: constant n0: e b e d Ka a 0k k c f a f c n b n d 0 (3) Journal of Tribology APRIL 003, Vol. 5 Õ 95

6 Fig. 5 Variation of the damping coefficients for increasing rotational speed: a Ä0. N g Ä ; b Ä0. N g Ä8 ; c Ä0.8 N g Ä ; and d Ä0.8 N g Ä8. cos t/ n: m a m c E b e 3 b 3 e b E d e 3 d 3 e d a 3 Ka k k a k c 3 k c f a f 3 a 3 f c f 3 c 3 b n n b 3 n d n d 3 0 (35) cos t n: m a m c E b e b E d e d Ka k a k a 0 cos nt/n3: m a n n k c k c 0 f a f c n b n m n c n d 0 (36) n E b n e n b n n b n n E d n e n d n n d nka n k a n a n k c n c n f n a n n a n f n c n n c n n b n b n n d n d n 0 (37) 96 Õ Vol. 5, APRIL 003 Transactions of the ASME

7 Fig. 6 Stability chart of the hydrodynamic journal bearing with rotating grooves: a Ä0. N g Ä ; b Ä0. N g Ä8 ; c Ä0.8 N g Ä ; and d Ä0.8 N g Ä8. sin t/ n: m b m d E a e 3 a 3 e a E c e 3 c 3 e c Kb b 3 k k b k d 3 k d f b f 3 b 3 f d 3 f d 3 n a n a 3 n c n c 3 0 (38) b sin t n: m b m d E a e a E c e c Kb k k d a f b f d n n c 0 (39) sin nt/n3: m b n n e n f n m d n n c n n b n n E n a n e n a n n n a ne c n c nkb n k b n b n k d n d n b n f n d n n d n n a n a n n c n c n 0 (0) Journal of Tribology APRIL 003, Vol. 5 Õ 97

8 Similar linear algebraic equations can be derived from the second equation in Eq. 8 in the same manner. For the n terms of the Fourier expansion of the solution, the equations of motion can be written in the matrix form as follows: PU0 () where the upper and lower half of the matrix P come from the first and second equations in Eq. 8, respectively. The elements of vector U are composed of the Fourier coefficients as follows: (/ )/N g 6. They can be reasonably assumed to vary harmonically with the fundamental parametric-excitation frequency of ( N g )/, as explained in Eq. 5 and 6. Figure 3 shows the variation of the direct stiffness coefficients with respect to the rotational speed for two cases of eccentricity (0. and 0.8 and groove number (N g and 8, respectively. The bars denote the maximum and minimum values of the dynamic coefficients. K XX is bigger than K YY in this analysis model. Ua 0,a,...,a n,b,...,b n,c 0,c,..., c n,c n,d,...,d n T () The appendix shows the elements of the matrix P with the dimensionless variables as follows: K m (3) k i i m, i,..., () i E i, i,..., (5) m i e i, i,..., (6) m i f i, i,..., (7) m n i i m, i,..., (8) In order that Eq. has a non-trivial solution, the determinant of matrix P should be zero, which results in the characteristic equation corresponding to the equations of motion in Eq. 8. Since the diagonal elements of matrix P only contain, must be the eigenvalues of the matrix P, and it can be rewritten as follows: P 0 I0 (9) where P 0 is the matrix excluding from the matrix P, and I is the identity matrix. Consequently, Eq. 9 takes the form of Hill s infinite determinant and its solution corresponds to the boundary between the stable and unstable regions 0,,. 3 Results and Discussion 3. Analysis of Dynamic Coefficients. A computer program developed by Jang and Yoon 7 is used to calculate the dynamic coefficients of the hydrodynamic journal bearing with rotating herringbone grooves. Table shows the design parameters of the herringbone grooved journal bearing used in this analysis. In a finite element modeling, fluid film is discretized by 60 elements with -node isoparametric elements, and the boundary conditions are assumed to be the continuous pressure in the circumferential direction and the ambient pressure in both sides. Figure shows the variation of the stiffness and damping coefficients of the hydrodynamic journal bearing with rotating grooves at 5,000 rpm for the case of 0.8 and N g. The dynamic coefficients vary in time with the period equal to Fig. 7 Time response of the whirl radius in the stable, critically stable and unstable positions: a stable; b critical; and c unstable. 98 Õ Vol. 5, APRIL 003 Transactions of the ASME

9 The average and variation of direct stiffness coefficients increase with increasing rotational speed. They also increase with increasing eccentricity and with decreasing groove number. Figure shows the variation of the cross-coupled stiffness coefficients with respect to the rotational speed for two cases of eccentricity ( 0. and 0.8 and groove number (N g and 8, respectively. K XY is bigger than K YX in this analysis model. The average and variation of cross-coupled stiffness coefficients increase with increasing rotational speed, and they also increase with increasing eccentricity and with decreasing groove number. Figure 5 shows the variation of the damping coefficients with respect to the rotational speed for two cases of eccentricity (0. and 0.8 and groove number (N g and 8, respectively. The average and variation of damping coefficients are not dependent on rotational speed, but on eccentricity and groove number. They increase with increasing eccentricity and with decreasing groove number. 3. Stability Analysis. Stability analysis of the hydrodynamic journal bearing with rotating grooves is performed using the dynamic coefficients calculated in section 3.. The mass of the rotor is assumed to be. g, which corresponds to the mass of the rotating part of the spindle with one disk in a computer hard disk drive. Stability analysis in section. is repeated with the terms of the Fourier coefficients being increased until the stable and unstable regions in the stability chart are unchanged. This research includes 0 terms of Fourier coefficients. Figure 6 shows the stability chart of the hydrodynamic bearing with rotating herringbone grooves with variations of eccentricity and groove number. All the lines are drawn in such a way that eigenvalues are calculated with the variation of. Integrating the equations of motion and observing the response, determine the stability of each area. The unstable region in Fig. 6 increases with increasing eccentricity and with decreasing groove number, which increase the average and variation of the stiffness and damping coefficients as shown in Fig. 3,, and 5. This might signify that the increased values of the average and variation of the stiffness coefficients play the major role in increasing the unstable region of the herringbone grooved journal bearing in this analysis, because the increased damping coefficients generally increase the stable region. In this analysis, the herringbone grooved journal bearing at 5,000 rpm operates in the stable region well above the unstable region. However, high rotational speed increases the stiffness coefficients without changing the damping coefficients as shown in Fig. 3,, and 5, so that it may result in the instability of the hydrodynamic journal bearing with rotating grooves. Figure 7 shows the time response of the whirl radius for the stable, critically stable and unstable positions in Fig. 6c. The equations of motion in Eq. 8 are solved by using the fourthorder Runge-Kutta method with a time step of 0 6 sec and an initial whirl radius of 0.5 m. Figure 7 shows that the whirl radius in the stable, critically stable and unstable regions, converges to the equilibrium position, repeats with bounded amplitude and diverges, respectively. These time responses of whirl radius validate the stability analysis proposed by this research. Conclusion The hydrodynamic journal bearing with rotating herringbone grooves can be regarded as a parametrically excited system due to the time-varying dynamic coefficients. This paper presents an analytical method to investigate its stability by transforming the equations of motion to the eigenvalue problem with Hill s infinite determinant. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone groove increases with increasing eccentricity and with decreasing groove number, which play the major role in increasing the average and variation of the stiffness coefficients. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients. Appendix The elements of the matrix P including 3 terms of the Fourier coefficients are as follows: Journal of Tribology APRIL 003, Vol. 5 Õ 99

10 References Bouchard, G., Lau, L., and Talke, F. E., 987, An Investigation of Nonrepeatable Spindle Runout, IEEE Trans. Magn., 35, pp Pai, R., and Majumdar, B. C., 99, Stability of Submerged Oil Journal Bearings under Dynamic Load, Wear, 6, pp Jonnadula, R., Majumdar, B. C., and Rao, N. S., 997, Stability Analysis of Flexibly Supported Rough Submerged Oil Journal Bearings, Tribol. Trans., 03, pp. 37. Raghunandana, K., and Majumdar, B. C., 999, Stability of Journal Bearing Systems Using Non-Newtonian Lubricants: A Non-Linear Transient Analysis, Tribol. Int., 3, pp Kakoty, S. K., and Majumdar, B. C., 000, Effect of Fluid Inertia on Stability of Oil Journal Bearings, ASME J. Tribol.,, pp Zirkelback, N., and San Andres, L., 998, Finite Element Analysis of Herringbone Groove Journal Bearings: A Parametric Study, ASME J. Tribol., 0, pp Jang, G. H., and Yoon, J. W., 00, Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Due to the Effect of a Rotating or Stationary Herringbone Groove, ASME J. Tribol.,, pp Kang, K., Rhim, Y., and Sung, K., 996, A Study of the Oil-Lubricated Herringbone-Grooved Journal Bearing-Part : Numerical Analysis, ASME J. Tribol., 8, pp Jang, G. H., and Kim, Y. J., 999, Calculation of Dynamic Coefficients in a Hydrodynamic Bearing Considering Five Degrees of Freedom for a General Rotor-Bearing System, ASME J. Tribol.,, pp Newland, D. E., 989, Mechanical Vibration Analysis and Computation, Longman Scientific and Technical. Nayfeh, A. H., and Mook, D. T., 979, Nonlinear Oscillations, John Wiley & Sons, Inc. Hayashi, C., 985, Nonlinear Oscillations in Physical Systems, Princeton University Press, Princeton, New Jersey. 300 Õ Vol. 5, APRIL 003 Transactions of the ASME

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