Stability analysis of a whirling disk-spindle system supported by FDBs with rotating grooves

Transcription

1 Microsyst Technol (011) 17: DOI /s TECHNICAL PAPER Stability analysis of a whirling disk-spindle system supported by FDBs with rotating grooves Jihoon Lee Gunhee Jang Kyungmoon Jung Heonjeong Ha Received: 16 August 010 / Accepted: 7 December 010 / Published online: 3 January 011 Ó Springer-Verlag 011 Abstract This paper investigates the stability of a whirling disk-spindle system, supported by coupled journal and thrust bearings with rotating grooves. The stiffness and damping coefficients of the FDBs change periodically with the whirling motion of the disk-spindle system, which makes it difficult to define the stability problem in the inertia coordinate. However, with the introduction of the coordinate system which rotates with the disk-spindle system, the stiffness and damping coefficients are constant, which makes it possible to define the stability problem in the rotating coordinate system. The Reynolds equations and the perturbed equations of the coupled bearings were derived with respect to the rotating coordinate and were solved using FEM to calculate the stiffness and damping coefficients. The critical mass of the rotor-bearing system was determined by solving the linear equations of motion. As a result, the stability increases with an increase in the whirl radius and with a decrease in the rotating speed. It also decreases with an increase in the tilting angle under a small whirl radius while it increases with an increase in the tilting angle under a large whirl radius. 1 Introduction Fluid dynamic bearings (FDBs) are an important component of a computer hard disk drive (HDD) to support the disk-spindle system. Fluid lubricant in the micron-level clearance between the rotating and stationary parts of the FDBs provides both the stiffness and damping, as well as J. Lee G. Jang (&) K. Jung H. Ha Department of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul , Republic of Korea ghjang@hanyang.ac.kr prohibiting solid contact between the shaft and sleeve, which results in low noise and low non-repeatable vibration. The mechanical structures of the FDBs and the disk-spindle system of a HDD are both shown in Fig. 1. The FDBs of a HDD have a unique structure of coupled journal and thrust bearings to support both the axial and radial loads. The diskspindle system has whirling, tilting and flying motions due to the clearance between the rotating and stationary parts of the FDBs, resulting in the generation of large repeatable vibrations (Jang et al. 005). Many design parameters of FDBs are utilized to predict their performances, including pressure, load, friction torque, stiffness, and damping coefficients. As an important performance index, stability has been utilized to predict the stable and unstable operations of the disk-spindle system supported by FDBs because it incorporates the stiffness and damping coefficients with the design parameters of the disk-spindle system. Many researchers have performed stability analyses of a rotor supported by journal bearings mostly with respect to the x- and y-axes. Pai and Majumdar (1991) analyzed the stability of a plain journal bearing with a unidirectional constant load and a variable rotating load. Raghunandana and Majumdar (1999) investigated the stabilities of journal bearings due to the effect of non-newtonian lubrication under a unidirectional constant load. Kakoty and Majumdar (000) studied the stability of a journal bearing due to the inertia of the fluid effect. Jang and Yoon (003) investigated the stability of a rotor supported by a journal bearing with rotating grooves, which had time-varying stiffness and damping coefficients. Hwang and Ono (003) studied the stability and dynamic characteristics of an air foil journal bearing to be applied in a HDD, and Das et al. (005) performed a stability analysis of a journal bearing with micropolar lubrication with respect to the slenderness ratio of the shaft. Ene et al. (008) analyzed the stability of a

2 788 Microsyst Technol (011) 17: investigated using the transient motion of a rotor with respect to the inertia coordinates, as well as the rotating coordinates. It also investigated the stability of the disk-spindle system due to the whirl radius, rotating speed, and tilting angle. Method of analysis.1 Determination of dynamic coefficients Fig. 1 Mechanical structure of the disk-spindle system of a HDD three-wave journal bearing. Most of the prior research has been restricted to the stability analysis of a rotor in a twodimensional x y plane. However, the disk-spindle system supported by FDBs in a HDD is only allowed to rotate in the h z direction, which restrains the remaining five degrees of freedom motion in the x, y, z, h x and h y directions. Recently, Kim et al. (010) proposed a method of stability analysis for the five degrees of freedom of a general rotor-bearing system. However, their method did not consider the timevarying stiffness and damping coefficients of FDBs caused by the whirling motion of a disk-spindle system. This paper develops a method to determine the stability of a disk-spindle system with a conical whirl by introducing rotating coordinates. The Reynolds equations and the perturbed equations of the coupled bearings were derived with respect to the rotating coordinate system and were solved using FEM to calculate the stiffness and damping coefficients. The critical mass of the rotor-bearing system was determined by solving the linear equations of motion using the rotating coordinates. The validity of the proposed method was This study extended the method of Jang and Yoon (003)for FDBs with a rotating groove by including rotational degrees of freedom in the perturbed equations in order to investigate the moment coefficients and tilting effect. The coordinate system of the journal bearing and the thrust bearing with rotating grooves are shown in Fig.. The governing equations for the journal and thrust bearings were obtained by transforming the Reynolds equation into the hz and rh planes, respectively. o h 3 op þ o h 3 op RoH 1l RoH oz 1l oz o r h3 op þ o h 3 op ror 1l or roh 1l roh ¼ R_ h ¼ r _ h oh RoH þ oh ot oh roh þ oh ot ; ð1þ ðþ where R is the radius of the journal, _ h is the rotational speed of the shaft, h is the film thickness, p is the pressure, and l is the viscosity coefficient. The perturbations applied to the rotor at a quasiequilibrium position are shown in Fig. 3. Perturbation equations were derived by substituting into the Reynolds equation a first-order expansion of the film thickness and pressure with respect to small displacements and velocities. The film thickness, its time derivative, and the pressure may be expanded into the following forms: Fig. Inertia and rotating coordinates of the journal and thrust bearings

3 Microsyst Technol (011) 17: Fig. 3 Perturbations applied to the FDBs h ¼ h 0 þ X oh on Dn n ¼ x; y; z; h x; h y oh ot ¼ oh 0 ot þ X oh on D_ n þ X o ot p ¼ p 0 þ X op on Dn þ X op o _ n D_ n; oh on Dn ð3þ ð4þ ð5þ where h 0 and p 0 are the film thickness and the pressure in quasi-equilibrium, respectively. By substituting (3), (4) and (5) into (1) and () retaining only the first order terms, and separating the variables with respect to each perturbed displacement or velocity, the following perturbation equations are obtained for the journal and the thrust bearings, respectively: Table 1 Major design specifications of the FDBs of a.5-inch HDD spindle Design variable Journal bearing Thrust bearing Bearing width (mm) Upper journal 1.95 Upper thrust 0.5 Lower journal 1.35 Lower thrust 0.60 Radial clearance (lm).0 Axial total clearance (lm) 30.0 Groove pattern Herringbone Spiral Number of grooves 8 Upper thrust 10 Lower thrust 0 Groove depth (lm) 5 Upper thrust 8.00 Lower thrust Rotating speed (rpm) 5,400 Fig. 4 Finite element model and pressure distribution of the FDBs

4 790 Microsyst Technol (011) 17: Table Major design specifications of the disk-spindle system of a.5-inch HDD spindle Design variable Value Mass, m (kg) Moment of inertia about z, I z (kg m ) Moment of inertia about x and y, I x I y (kg m ) Location in the z direction (m) Top of upper thrust Mass center 0 Bottom of lower thrust (7). The global matrix equation of the finite element equation in the Reynolds equations was obtained in order to calculate the pressures of the coupled journal and thrust bearings in quasi-equilibrium, Ap 0 ¼ b 0 : ð1þ Once the pressure in the fluid film was determined, the global matrix equation of the finite element equation corresponding to each perturbation equation was determined. This allowed for the calculation of the perturbation pressure for the coupled journal and thrust bearings as follows: Ap n ¼ b n n ¼ x; y; z; h x ; h y ; _x; _y; _z; h _ x ; h _ y : ð13þ In Eq. (13), the matrix A is identical to that which was previously determined in the solution stage of the Reynolds equation, as shown in (1). Equations (8), (9), (10) and (11) show that the tilting displacements and velocities affect the perturbed displacements and velocities of both the journal and thrust bearings.. Stability analysis If the force of the thrust bearing is positive, a is?1; otherwise, it is 1. The terms u J, w J, u T and w T in (6) and (7) are defined as follows: u J ¼ z ð z 0 Þcos h x sin H ð8þ w J ¼ ðz z 0 Þcos h y cos H ð9þ u T ¼ ðz 0 zþsin h x r cos h x sin h ð10þ w T ¼ ðz 0 zþsin h y þ r cos h y cos h; ð11þ where z 0 is the axial coordinate of the center of mass. The FEM was used to solve the Reynolds equations in (1) and (), as well as the perturbation equations in (6) and The equation of motion of the rigid disk-spindle system supported by the coupled journal and thrust bearings in the fixed coordinate can be derived as follows (Kim et al. 010): M x þðcþgþ_x þ Kx ¼ 0 3 m a m a M ¼ 0 0 m a I x I y 3 c xx c xy c xz c xhx c xhy c yx c yy c yz c yhx c yhy C ¼ c zx c zy c zz c zhx c zhy 6 c hxx c hxy c hxz c hxh x c 7 4 hxh y 5 c hyx c hyy c hyz c hyh x c hyh y G ¼ hz _ 7 4 I z h _ z I z 0 3 k xx k xy k xz k xhx k xhy k yx k yy k yz k yhx k yhy K ¼ k zx k zy k zz k zhx k zhy 6 k hxx k hxy k hxz k hxh x k 7 4 hxh y 5 k hyx k hyy k hyz k hyh x k hyh y 8 9 Dx >< Dy >= x ¼ Dz Dh x >: >; Dh y ð14þ where m a, I x, I y, and I z are the mass and mass moments of inertia of the rotor, respectively. The radius of gyration is

5 Microsyst Technol (011) 17: Fig. 5 Stiffness coefficients in the inertia coordinate system introduced to express the mass moment of inertia with respect to the mass, I l ¼ Kl m; ð15þ where K l is the radius of gyration, and the subscript l denotes the x, y, and z axis. Therefore, Eq. (14) can be represented as a single variable, i.e., the mass of the rotor. The stiffness and damping coefficients of the FDBs change periodically as the disk-spindle system rotates, making it difficult to define the stability problem in the inertia coordinate system. Once the rotating coordinate system, which rotates with the disk-spindle system, is introduced, the stiffness and damping coefficients are constant with respect to the rotating coordinate, making it possible to define the stability problem with the rotating coordinates. The transformation matrix from the inertia coordinates to the rotating coordinates can be defined as follows: x cos h sin h u >< y >= sin h cos h >< v >= z ¼ w 6 7 h x cos h sin h 5 / x >: >; >: >; h y sin h cos h / y or x ¼ Tu: ð16þ The equations of motion in the rotating coordinate system can be written such that T T MT u þ T T ðc þ GÞT _u þ T T KTu ¼ 0 or M R u þðc R ð17þ þg R Þ _u þ K R u ¼ 0; where M R, C R, G R, and K R are the mass, damping, gyroscopic, and stiffness matrices with constant coefficients, and M R is equal to M.

6 79 Microsyst Technol (011) 17: Fig. 6 Damping coefficients in the inertia coordinate system The homogeneous solution of Eq. (17) can be assumed to be an exponential function such that 8 9 u h >< v h >= u ¼ u h expðxtþ; u h ¼ w h : ð18þ >: / xh / yh Substituting Eqs. (15) and (18) into Eq. (14), the following equation can be obtained, X T T MT þ XT T ðc þ GÞT þ T T KT uh expðxtþ ¼0: ð19þ The solution of Eq. (19) can be generally expressed as X =-X real? ix img. The motion of the rotor is stable if X real is greater than zero, and it is unstable if X real is less than zero. Therefore, the critical condition can be determined when X real is equal to zero. The solutions >; of the characteristic determinant of Eq. (19) are the critical mass, (m a ) c, and the corresponding frequency, X img. The behavior of the disk-spindle system supported by coupled journal and thrust bearings can be regarded as stable if the mass, m a, is less than the critical mass, (m a ) c ; however, it is considered unstable if m a is greater than (m a ) c. In addition, the critical mass determined from the equations of motion for the rotating coordinates in Eq. (17) is the same as that determined from the equations of motion for the inertia coordinates in Eq. (14) because the displacement x in Eq. (16) is dependent upon u. 3 Simulation model Table 1 shows the major design specifications of the FDBs in a.5-inch HDD. The FDBs consist of two

7 Microsyst Technol (011) 17: Fig. 7 Stiffness coefficients in the rotating coordinate system rotating grooved journal bearings, four plain journal bearings, two rotating grooved thrust bearings, and one plain thrust bearing. Figure 4 shows the finite element model and the pressure distribution of the coupled journal and thrust bearings. The fluid film was discretized using 5,640, four-node, isoparametric bilinear elements, and the Reynolds boundary condition was applied to guarantee the continuity of pressure and the pressure gradient. In this research, the disk-spindle system rotating at 5,400 rpm was assumed to have an equilibrium flying height of 7.85 lm, where the axial load generated by the FDBs was equal to the weight of a rotor. Table shows the major design specifications of a.5-inch HDD spindle system used to analyze the stability and dynamic behavior. The disk-spindle system was assumed to have a whirling motion at a tilting angle due to the mass unbalance of the rotating disk-spindle system. 4 Results and discussion 4.1 Dynamic coefficients Figures 5 and 6 show the respective stiffness and damping coefficients of the FDBs with rotating grooves due to the change in the whirl radius of the fixed coordinates (Kim et al. 010). The tilting angle of the shaft was assumed to be zero. The stiffness and damping coefficients changed periodically with a period of 180, and they increased with an increase in the whirl radius. Figures 7 and 8 show the respective stiffness and damping coefficients of the FDBs with rotating grooves due to the change in the whirl radius of the rotating coordinates. They were first calculated using the proposed method in Sect..1, and they were also calculated using the transformation matrices in Eqs. (16) and (17). Both results were exactly the same. The stiffness and

8 794 Microsyst Technol (011) 17: Fig. 8 Damping coefficients in the rotating coordinate system damping coefficients of the FDBs with rotating grooves under whirling motion were constant in the rotating coordinate system and they also increased with an increase in the whirl radius. 4. Stability analysis Figure 9 shows the stability, or critical mass, of the diskspindle system with a tilting angle of 0.01 due to the change in the whirl radius, illustrating that the tilting angle increased with an increase in whirl radius. It shows that the disk-spindle system of this research is always stable under this operating condition because the mass of disk-spindle system is smaller than critical mass. Figures 10 and 11 show the displacements x, y, z, h x and h y and the trajectory of the mass center of the disk-spindle system with a whirl radius of 1. lm and a mass of 100 kg, corresponding to the stable position (a) in Fig. 9. The displacements and the trajectory converged in both the rotating and fixed coordinate systems. Figures 1 and 13 show the displacements x, y, z, h x and h y, as well as the trajectory of the mass center of the disk-spindle system with a whirl radius of 0.8 lm and a mass of 100 kg, corresponding to the unstable position (b) in Fig. 9. The displacements and the trajectory diverged in both the rotating and the fixed coordinate systems. Figure 14 shows the stability of the disk-spindle system due to the rotating speed and whirl radius when the tilting angle was assumed to be 0.01 and the z-axis was the critical mass. If the mass of the disk-spindle system was heavier than the critical mass on the surface, the system was unstable; otherwise, it was stable. The critical mass,

9 Microsyst Technol (011) 17: the stability of the disk-spindle system, increased with an increase in the whirl radius and with a decrease in the rotation speed. Figure 15 shows the stability of the diskspindle system due to the whirl radius and tilting angle when the rotating speed of the disk-spindle system was assumed to be 5,400 rpm. Under a small whirl radius, the increasing rates of cross-coupled stiffness and damping coefficients in radial and tilting directions are larger than those of direct stiffness and damping coefficients with an increase in the tilting angle. It decreases the stability with an increase in the tilting angle under the small whirl radius. However, under a large whirl radius, the magnitudes of the cross-coupled stiffness and damping coefficients in radial and tilting directions decrease with an increase in the tilting angle. It increases the stability with an increase in the tilting angle under the large whirl radius. 5 Conclusions Fig. 9 Stability of the tilted disk-spindle system at 0.01, rotating at 5,400 rpm due to the whirl radius This paper proposes the dynamic analysis and stability of a disk-spindle system with a conical whirl supported by coupled journal and thrust bearings with rotating grooves and considering the five degrees of freedom of a general rotor-bearing system. The stiffness and damping coefficients of the FDBs are constant with respect to the rotating Fig. 10 Displacements and angular displacements of the mass center of the disk-spindle system (whirl radius 1. lm, mass 100 kg, rotation speed 5,400 rpm)

10 796 Microsyst Technol (011) 17: Fig. 11 Trajectory of the mass center of the disk-spindle system (whirl radius 1. lm, mass 100 kg, rotation speed 5,400 rpm) Fig. 1 Displacements and angular displacements of the mass center of the disk-spindle system (whirl radius 0.8 lm, mass 100 kg, rotation speed 5,400 rpm) coordinates, making it possible to define the stability problem in the rotating coordinate system. The Reynolds equations and their perturbed equations of the coupled bearings were derived with respect to the rotating coordinates and were solved using FEM to calculate the stiffness and damping coefficients. The critical mass of the rotorbearing system was determined by solving the linear equations of motion, and this research was validated through the transient analysis of the equations of motion. As a result, the stability increased with an increase in the

11 Microsyst Technol (011) 17: Fig. 13 Trajectory of the mass center of the disk-spindle system (whirl radius 0.8 lm, mass 100 kg, rotation speed 5,400 rpm) whirl radius and with a decrease in the rotating velocity. The stability also decreased with an increase in the tilting angle under a small whirl radius and increased with an increase in the tilting angle under a large whirl radius. Subsequently, the results of this research can be used to develop a robust rotor-bearing system. References Fig. 14 Stability of the tilted disk-spindle system at 0.01 due to the whirl radius and rotation speed Das S, Guba SK, Chattopadhyay AK (005) Linear stability analysis of hydrodynamic journal bearings under micropolar lubrication. Tribol Int 38: Ene NM, Dimofte F, Keith TG Jr (008) A stability analysis for a hydrodynamic three-wave journal bearing. Tribol Int 41: Hwang T, Ono K (003) Analysis and design of hydrodynamic journal air bearings for high performance HDD spindle. Microsyst Technol 9: Jang GH, Yoon JW (003) Stability analysis of a hydrodynamic journal bearing with rotating herringbone grooves. J Tribol 15: Jang GH, Oh SH, Lee SH (005) Experimental study on whirling, flying and tilting motions of a 3.5 in. FDB spindle system. Tribol Int 38: Kakoty SK, Majumdar BC (000) Effect of fluid inertia on stability of oil journal bearings. J Tribol 1: Kim MG, Jang GH, Kim HW (010) Stability analysis of a diskspindle system supported by coupled journal and thrust bearings considering five degrees of freedom. Tribol Int 43: Pai R, Majumdar BC (1991) Stability of submerged oil journal bearings under dynamic load. Wear 146: Raghunandana K, Majumdar BC (1999) Stability of journal bearing systems using non-newtonian lubrications: a non-linear transient analysis. Tribol Int 3: Fig. 15 Stability of the disk-spindle system rotating at 5,400 rpm due to the whirl radius and tilting angle