G. H. Jang e-mail: ghjang@hanyang.ac.kr J. W. Yoon PREM, Department of Mechanical Engineering, Hanyang University, Seoul, 133-791, Korea Nonlinear Dynamic Analysis of a Hydrodynamic Journal Bearing Considering the Effect of a Rotating or Stationary Herringbone Groove This research investigates the dynamic characteristics of a herringbone grooved journal bearing with plain sleeve (GJPS) and a plain journal bearing with herringbone grooved sleeve (PJGS) under static and dynamic load. FEM is used to solve the Reynolds equation in order to calculate the pressure distribution in a fluid film. Reaction forces and friction torque are obtained by integrating the pressure and shear stress along the fluid film, respectively. Dynamic behaviors of a journal, such as orbit or rotational speed, are determined by solving its nonlinear equations of motion with the Runge-Kutta method. Numerical results are validated by the experimental results of prior researchers. A GJPS produces less friction torque than a PJGS so that the GJPS consumes less input power than the PJGS. Under static load, the PJGS converges to the fixed equilibrium position, but the GJPS has a whirling motion due to the rotating groove even at the steady state, which produces the excitation frequencies corresponding to the integer multiple of the rotor speed multiplied by the number of grooves. The variation of rotational speed of a GJPS is always less than that of a PJGS due to less friction torque. Under the effect of mass unbalance, the excitation frequencies of the reaction force in a GJPS and a PJGS are the rotational frequency due to mass unbalance and its harmonics due to the nonlinear effect of fluid film. However, the GJPS has relatively big amplitude corresponding to the multiples of the number of grooves, in comparison with the amplitudes at the adjacent harmonics. DOI: 10.1115/1.1401019 1 Introduction Herringbone grooved journal bearings are being considered as one of the better alternatives to support the rotating disk-spindle system in a computer hard disk drive, replacing the conventional ball bearings, due to their outstanding low noise and vibration characteristics. Herringbone grooves have the advantage of selfsealing which causes the lubricant to be pumped inward, and therefore, reduces side leakage. They also prevent whirl instability that is commonly observed in the plain journal bearings at concentric operating conditions. Herringbone grooved journal bearings used in the spindle of a computer hard disk drive can be classified as a grooved journal bearing with plain sleeve GJPS and a plain journal bearing with grooved sleeve PJGS depending on their groove location as shown in Fig. 1. As shown in Fig. 1, the herringbone groove pattern of the GJPS has the opposite direction of that of the PJGS in order to make the lubricant pumped inward. The GJPS is easier and cheaper to manufacture than a PJGS, but the former may produce periodic hydrodynamic forces resulting from the rotating grooves that excites the spindle system. Many researchers have investigated the performance of herringbone grooved journal bearings. Bootsma 1 and Hirs 2 analyzed a herringbone grooved journal bearing using the narrow groove theory. Bonneau and Absi 3 showed that the narrow groove theory overestimates the load capacity of gas bearings by using FEM, and showed that GJPS produces periodic load whose frequency is equal to the product of the number of grooves and the rotating speed of the shaft. Zirkelback and San Andres 4 solved the modified Reynolds equation of the GJPS in order to perform the parametric study of the GJPS at the steady state using FEM. Jang and Kim 5 presented a method to calculate the dynamic coefficients in a hydrodynamic bearing considering five degrees of freedom. However, the above researchers did not include the dynamics of the journal in their analyses, and they only investigated the static characteristics and the dynamic coefficients of the GJPS in the steady state. Contributed by the Tribology Division of The American Society of Mechanical Engineers for presentation at the STLE/ASME Tribology Conference, San Francisco, CA, October 22 24, 2001. Manuscript received by the Tribology Division February 2, 2001; revised manuscript received July 3, 2001. Associate Editor: J. L. Streator. Fig. 1 Coordinate system and groove pattern: a PJGS; and b GJPS Journal of Tribology Copyright 2002 by ASME APRIL 2002, Vol. 124 Õ 297
Several researchers studied the dynamic characteristics of plain journal bearings. Goenka 6 presented a numerical method to analyze the transient response of dynamically loaded plain journal bearings. Malik et al. 7 carried out the analysis of the transient response of a plain journal bearing during uniform acceleration and deceleration periods. Pai and Majumdar 8 analyzed the stability characteristics of submerged plain journal bearings under unidirectional constant load and variable rotating load. Choy et al. examined the nonlinear effect on the dynamic behavior and performance of a hydrodynamic journal bearing by comparing the linear analysis with the nonlinear analysis of the bearing. They concluded that high eccentricity may show the substantial increase in nonlinearity. However, these studies were only concerned with the translational motion of plain journal bearings without considering the rotational motion of the journal. The present work analyzes the dynamic behavior of both the GJPS and the PJGS under static and dynamic load conditions. FEM is used to solve the Reynolds equation for the pressure distribution in fluid film. Reaction forces and friction torque are obtained by integrating the pressure and shear stress along the fluid film, respectively. Dynamic behavior of a journal such as orbit, rotational speed and power consumption is investigated by solving the nonlinear equations of motion of a journal considering the rotational motion. 2 Method of Analysis 2.1 Governing Equation. Figure 1 shows the coordinate system of a herringbone-grooved journal bearing. The circumferential coordinate is determined from the fixed negative X-axis. The Reynolds equation can be written in the coordinate system (xr,z) fixed to the sleeve of a PJGS as shown in Fig. 1a. x h3 p 12 x z h3 p 12 z R h h (1) 2 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 (2) hce X cos e Y sin, (3) where c and c g are the clearance and groove depth, respectively. Since the coordinate system is fixed to the sleeve, the rotational motion of a journal does not affect the rate of change of the film thickness. The rate of change of film thickness can be expressed in the following equation: h t ė X cos ė Y sin. (4) However, in case of the GJPS Fig. 1b, film thickness changes as the grooved journal rotates and it makes the Reynolds equation of the GJPS very difficult to solve numerically. Introducing the assumption that the grooved journal is stationary and the sleeve is rotating in the opposite direction, the rate of change of the fluid film can easily be treated numerically. So the coordinate system (xr,z) is fixed to the journal in order to produce the following Reynolds equation for a GJPS 4: x h3 p 12 x z h3 p 12 z R h h 2 x t. (5) The circumferential coordinate of a GJPS can be expressed in terms of the rotational velocity and angular coordinate of the journal as t. (6) The rate of change of the film thickness in Eq. 5 can now be rewritten as h t ė X cos ė Y sin e X sin e Y cos. (7) Equation 6 makes it possible to consider the rate of change of the film thickness in the GJPS. 2.2 Finite Element Method. FEM is used to solve Reynolds equations for the PJGS and the GJPS, as shown in Eq. 1 and 5, respectively. The pressure p within each element can be approximated by its nodal values p i and shape functions N i p i1 n e N i p i, (8) where n e is the number of nodes in an element. The Reynolds equation is transformed to the following matrix equation by using the Galerkin method: q i K ij p j Q U i Q ḣ i. (9) Volume fluidity matrix K ij is expressed as follows: h 3 K ij A 12 N i N j da. (10) Volume flow q i, volume shear flow Q U i and volume squeeze flow Q ḣ i can be expressed in the following forms for a PJGS: h R p h2 2 12 nˆn ids (11) q i S R Q U i hn i da (12) A 2 Q i ḣ Aė X cos ė Y sin N i da, (13) where A and S represent the region and boundary of the fluid film, respectively, and nˆ is an outward normal vector along the boundary. They can also be expressed in the following forms for a GJPS: h R p h2 2 12 nˆn ids (14) q i S R Q U i hn i da (15) A 2 Q i ḣ Aė X cos ė Y sin e X sin e Y cos N i da. (16) 2.3 Reaction Force, Friction Torque and Power Loss. Once the pressure is determined in the fluid film, reaction force and load capacity can be calculated as follows: F X pcos dxdz (17) F Y psin dxdz (18) WF X 2 F Y 2. (19) Friction torque is calculated by integrating the shear stress xy along the fluid film T f R xy yh dxdz (20) 298 Õ Vol. 124, APRIL 2002 Transactions of the ASME
Table 1 Parameters of a herringbone grooved journal bearing investigated with the linear equations of motion. The nonlinear equations of motion of a rigid journal under load and centrifugal force are expressed as follows: xy yh h p R 2 x h. (21) Power loss can be evaluated from the following expression: P l T f F X ė X F Y ė Y. (22) 2.4 Nonlinear Equations of Motion of a Rigid Journal. A journal undergoes large whirling motion before it reaches the steady state, so that the transient analysis of the system cannot be Fig. 2 Comparison of load capacity with experimental data by Hirs 1965 Fig. 3 Speed profile of a rotating journal Fig. 4 Pressure distributions along the axial center of a journal bearing at the steady state zälõ2 : a PJGS N g Ä8 ; b GJPS N g Ä8 Journal of Tribology APRIL 2002, Vol. 124 Õ 299
where W X, W Y, and T are the loads in the X and Y directions and the input torque, respectively. 0 is the initial phase angle of the rigid journal. 3 Results and Discussion 3.1 Analysis Model and Validation. A computer program is developed to analyze both GJPS and PJGS. Table 1 shows the parameters of the herringbone grooved journal bearing used in this analysis. Fluid film is discretized by 6420 elements, and the boundary conditions are assumed to be the continuous pressure in the circumferential direction and the ambient pressure in both sides. The accuracy of the finite element code is validated by comparing the numerical results of the load capacity with the experimental data of a GJPS presented by Hirs 2. Load capacity is calculated by solving the finite element equation of the Reynolds equation only with the assumption that the herringbone grooved journal is rotating with the given eccentricity in the steady state. As shown in Fig. 2, the numerical results match well with the experimental data. The Hirs model has 20 grooves, which make the variation of the load capacity negligible when the rotor rotates. However, in the spindle system of a computer hard disk drive, the number of grooves is limited to less than 10 due to the small size of the shaft. 3.2 Effect of Static Load. The dynamic characteristics of the GJPS and the PJGS are investigated in the case where the static load of 0.21 N is applied to the journal in the X direction. First, the Reynolds equation is solved by using FEM to calculate the pressure distribution in the fluid film with conditions of zero eccentricity and velocity. Reaction force and friction torque are calculated by integrating the pressure and shear stress along the fluid film, respectively. Equations of motion in the X and Y directions are solved by using the fourth-order Runge-Kutta method with the time step of 10 6 sec to calculate the new position and translational velocities of the journal. Rotational speed of the journal is assumed to follow the speed profile defined in Fig. 3 without solving the equation of rotational motion in Eq. 25 directly, because the GJPS and the PJGS produce different speeds at the steady state even with the application of the same input torque. This numerical procedure is repeated until the journal reaches the steady state. Figure 4 shows the pressure distributions along the axial center of a journal bearing with 8 grooves at the steady state (zl/2). The peak pressure is produced where the thickness of the fluid film is abruptly decreased, i.e., at the transition boundary from groove to ridge for the PJGS, and at the transition boundary from ridge to groove for the GJPS. Peak pressure in the GJPS is slightly higher than that in the PJGS. Figure 5 shows the friction torque, input torque and power loss Fig. 5 Friction torque, input torque and power loss under static load: a friction torque; b input torque; and c power loss më X W X F X me u 2 cos t 0 (23) më Y W Y F Y me u 2 sin t 0 (24) I TT f, (25) Fig. 6 Trajectory under static load 300 Õ Vol. 124, APRIL 2002 Transactions of the ASME
Fig. 7 Load capacity due to variation of eccentricity in the cases where the GJPS and the PJGS have 4 and 8 grooves, respectively. As shown in Eq. 21, friction torque is mostly determined by the rotational speed and pressure gradient. The former has more dominant effect than the latter in the production of torque, but the former does not make any difference in the contribution of torque in both GJPS and PJGS at the steady state because they have the same rotational speed. As shown in Fig. 4, Fig. 9 Dynamic behavior due to shock: a impulsive force; b rotational speed; and c journal trajectory Fig. 8 Frequency spectra of reaction force and journal displacement of GJPS under static load: a reaction force; and b journal displacement the pressure gradients of the PJGS and the GJPS have almost the same magnitude, but they have the positive and negative effect on the production of the friction torque, respectively, so that the GJPS produces smaller friction torque than the PJGS, as shown in Fig. 5a. Furthermore, the absolute value of pressure gradient increases with the increase of the number of grooves, so that friction torque increases slightly with the increase of the number of grooves in case of the PJGS, or the decrease of the number of Journal of Tribology APRIL 2002, Vol. 124 Õ 301
Fig. 10 Dynamic behavior due to mass unbalance: a journal trajectory N g Ä8 ; b journal trajectory N g Ä4 ; c rotational speed; and d power loss. grooves in case of the GJPS. Input torque in Fig. 5b is obtained by inversely solving the equation of rotational motion in Eq. 25. Peak values of input torque are almost the same at the transient state, but the GJPS requires less input torque than the PJGS at the steady state due to less friction torque. Power loss has almost the same variation as friction torque Fig. 5a and c, which means that power loss mostly depends on friction torque rather than reaction forces even at the transient state. Figure 6 shows the trajectories of a journal. The journal moves to the positive X-direction abruptly due to the static load, and then it converges to the equilibrium position. The PJGS converges to the fixed equilibrium position, but the GJPS has a whirling motion resulting from the rotating groove even at the steady state. As the number of grooves increases, the rotational period of each groove pattern decreases, that results in small variation of pressure distribution, reaction forces, and whirl radius consequently. Whirl radii of GJPS with 4 and 8 grooves are 410 3 and 310 5, respectively. Figure 7 shows the static analysis of load capacity resulting from the variation of eccentricity. Under the application of 0.21 N Fig. 7, eccentricity is in the increasing order of the PJGS with 8 grooves, the PJGS with 4 grooves, the GJPS with 8 grooves and the GJPS with 4 grooves. From the initial position, each journal under static load converges to the same equilibrium positions predicted by static analysis. Figure 8 shows the frequency spectra of reaction force and journal displacement of the GJPS. The GJPS changes the thickness of fluid film even at the steady state so that the excitation frequencies of reaction force are determined by the integer multiple of the rotor speed multiplied by the number of grooves. Their contribution increases with the decrease of the number of grooves. They also excite the journal of GJPS, which results in the same frequency contents of the journal displacement. However, PJGS does not produce any excitation frequency components in the reaction force because of its concentric motion at the steady state. 3.3 Effect of Shock and Mass Unbalance. Shock resistance is one of important design considerations in the spindle system of a computer hard disk drive. It is investigated for dynamic characteristics due to the application of additional impulses. As shown in Fig. 9a, impulsive force is assumed to be half sine wave with a period of 10 msec and peak value of 1.26 N, which is six times that of static load. Figure 9b shows the speed variation under impulsive excitation. Three equations of motion in Eqs. 23, 24, and 25 are solved once the reaction force and the friction torque are determined from the solution of the Reynolds equation. Input torques calculated under the static load in Fig. 5b are applied to these models, respectively. The GJPS has a smaller speed variation than the PJGS, because the GJPS produces less friction torque than the PJGS, as shown in Fig. 5a. Speed variation slightly increases due to the increase in the number of grooves in the GJPS and the PJGS. Figure 9c shows the trajectories of the journal under impulsive excitation. The GJPS has a 302 Õ Vol. 124, APRIL 2002 Transactions of the ASME
Fig. 11 Frequency spectra of reaction force due to mass unbalance: a PJGS N g Ä8 ; b PJGS N g Ä4 ; c GJPS N g Ä8 ; and d GJPS N g Ä4. smaller return trajectory than the PJGS with the same number of grooves, but several small whirls are observed in the case of the GJPS with 4 grooves along the overall whirling motion before it reaches to the steady state. Mass unbalance always exists even in the highest precision spindle, and it plays the role of dynamic load in the form of centrifugal force. Figure 10 shows the dynamic characteristics resulting from the application of mass unbalance in addition to the static load. Mass unbalance, me u, is assumed to be 0.01 percent of total mass. Even at the steady state, mass unbalance produces the whirling motion in the PJGS as well as in the GJPS, as shown in Fig. 10a and b. And several small whirls are observed in case of the GJPS with 4 grooves along the overall whirling motion at the steady state. Because of the large friction torque of the PJGS, it has a higher variation of rotational speed, and consequently, higher power loss than the GJPS Fig. 10c and d. Figure 11 shows the frequency spectra of the reaction force of the GJPS and the PJGS. Excitation frequencies are the rotational frequency due to mass unbalance and its harmonics due to the nonlinear effect of fluid film. In the PJGS, the amplitude of high harmonics decreases monotonically. However, the GJPS has relatively higher amplitude at the harmonics of the number of grooves than those at the adjacent harmonics. The small number of grooves in the GJPS may be one of the sources that excite the spindle system. 4 Conclusion This research investigates the dynamic characteristics of the GJPS and the PJGS under static and dynamic load. FEM is used to solve the Reynolds equation in order to calculate the pressure distribution in fluid film. Reaction force and friction torque are determined by integrating the pressure and shear stress along the fluid film. The dynamic behaviors of a journal, such as orbit or rotational speed, are determined by solving its nonlinear equations of motion considering the rotational motion. The groove location of a journal bearing affects the pressure distribution in the fluid film and consequently its dynamic performance. The GJPS produces less friction torque than the PJGS, so that the former produces less power loss and has a smaller speed variation than the latter under the effect of mass unbalance in addition to static load. The GJPS also shows better shock resistance than the PJGS. Although the GJPS has the above-mentioned advantage compared to the PJGS, the small number of grooves in the GJPS produces the periodic reaction forces so that it may be one of the sources that excite the spindle system. Nomenclature A region c, c g clearance m, groove depth m e u mass eccentricity Journal of Tribology APRIL 2002, Vol. 124 Õ 303
e X, e Y eccentricity in X and Y direction m F X, F Y fluid film force in X and Y direction N G mass center of journal h film thickness m I mass moment of inertia kgm 2 K ij volume fluidity matrix L length of journal bearing m m mass of journal kg N g number of groove N i shape function n e total number of nodes per element O geometric center P l power loss W p pressure N/m 2 p a ambient pressure N/m 2 Q ḣ i Q U i q i volume squeeze flow m 3 /s volume shear flow m 3 /s volume flow m 3 /s R radius of journal bearing m S boundary T input torque Nm T f friction torque Nm t time sec W load capacity N W X, W Y load in X and Y direction N X, Y inertial coordinate system xr,z coordinate system fixed to sleeve xr,z coordinate system fixed to grooved journal groove angle ratio of the ridge width to the sum of the groove and ridge widths eccentricity ratio, e/c 0 initial phase angle of journal fluid viscosity Pa s t circumferential coordinate fixed to sleeve circumferential coordinate rotating with journal rotational speed of journal rad/s Subscripts J journal S sleeve References 1 Bootsma, J., 1975, Liquid-Lubricated Spiral-Groove Bearings, Phillips Research Reports-Supplements, No. 7, The Netherlands. 2 Hirs, G. G., 1965, The Load Capacity and Stability Characteristics of Hydrodynamic Grooved Journal Bearings, ASLE Trans., 8, pp. 296 305. 3 Bonneau, D., and Absi, J., 1994, Analysis of Aerodynamic Journal Bearings With Small Number of Herringbone Grooves by Finite Element Method, ASME J. Tribol., 116, pp. 698 704. 4 Zirkelback, N., and San Andres, L., 1998, Finite Element Analysis of Herringbone Groove Journal Bearings: A Parametric Study, ASME J. Tribol., 120, pp. 234 240. 5 Jang, G. H., and Kim, Y. J., 1999, Calculation of Dynamic Coefficients in a Hydrodynamic Bearing Considering Five Degrees of Freedom for a General Rotor-Bearing System, ASME J. Tribol., 121, pp. 499 505. 6 Goenka, P. K., 1984, Dynamically Loaded Journal Bearings: Finite Element Method Analysis, ASME J. Tribol., 106, pp. 429 439. 7 Malik, M., Bhargava, S. K., and Sinhasan, R., 1989, The Transient Response of a Journal in Plane Hydrodynamic Bearing During Acceleration and Deceleration Periods, Tribology Transactions, 32, pp. 61 69. 8 Pai, R., and Majumdar, B. C., 1991, Stability of Submerged Oil Journal Bearings under Dynamic Load, Wear, 146, pp. 125 135. 304 Õ Vol. 124, APRIL 2002 Transactions of the ASME