DOI 10.1007/s00542-013-1844-6 TECHNICAL PAPER Optimal design of fluid dynamic bearings to develop a robust disk-spindle system in a hard disk drive utilizing modal analysis Jihoon Lee Gunhee Jang Kyungmoon Jung Received: 12 October 2012 / Accepted: 1 June 2013 Ó Springer-Verlag Berlin Heidelberg 2013 Abstract This research proposes an optimal design methodology for fluid dynamic bearings (FDBs) in a hard disk drive to improve the dynamic performance of the diskspindle system. We solved equations of motion for the rigid rotor supported by FDBs with five degrees of freedom. Five modal damping ratios were selected as multi-objective functions. The constraint equations were the friction torque of the FDBs and the stiffness and damping coefficients related to under-damped vibration modes. Ten major design variables of the FDBs were chosen for this optimization problem. The steady-state whirl radius and the shock response at half-speed whirl of the rotating rigid spindle-bearing system were evaluated as RRO and NRRO, respectively. The RRO and NRRO of the optimal design were compared with those of the conventional design. Our results show that the proposed method effectively reduces RRO and NRRO. 1 Introduction Fluid dynamic bearings (FDBs) have replaced ball bearings in hard disk drives (HDDs) since research in the early 1990s first showed that disk-spindle systems supported by FDBs generate smaller non repeatable runout (NRRO) than ball bearings, not only because FDBs prevent solid contact between stationary and rotating parts with fluid lubricant, but also because they provide damping to absorb vibration. However, disk-spindle systems supported by FDBs have J. Lee G. Jang (&) K. Jung PREM, Department of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea e-mail: ghjang@hanyang.ac.kr intrinsic instability, or half-speed whirl, in which zero pressure is generated at the frequency near the half of rotating speed. Disk-spindle systems are vulnerable to the excitation frequency with half of the rotating frequency, and most components of NRRO are populated near that frequency. Many researchers have investigated how to reduce the vibration of HDD disk-spindle systems. Jang and Yoon (2003) investigated the stability of a rotor supported by a journal bearing with rotating grooves that had time-varying stiffness and damping coefficients. Jang et al. (2005) studied whirling, tilting and flying motions of a HDD diskspindle system that existed due to clearance between the rotating and stationary parts of the FDBs. Yoon and Shen (2005) presented a numerical model predicting the shock response of a rotating-shaft spindle motor with nonlinear FDBs. Ono et al. (2005) studied the stability of journal bearings and their effects on translational motion. They analyzed the stability of a disk-spindle assembly supported by hydrodynamic plain journal bearings and a pivot bearing at the bottom of the flexible shaft, and studied the translation and tilting motion of a rotor with journal bearings in four degrees of freedom. Recently, some researchers have proposed designs to develop robust FDBs. Hashimoto and Matsumoto (2001) found multipeaks of objective function and suggested a hybrid method combining the direct search method and successive quadratic programming to find a global optimum solution for hydrodynamic journal bearings. Hirani and Suh (2005) investigated an optimal design for FDBs to minimize power loss, but their research was limited to the design parameters of journal bearings, and they did not consider the dynamic performance of the rotor. Hirayama et al. (2009) studied the optimization of grooved journal bearings to improve repeatable runout (RRO). Kim et al. (2011)
proposed a robust optimal design utilizing the critical mass of rotor dynamics to decrease overall response but were not successful at decreasing all of the responses across five degrees of freedom in a rotating disk-spindle system. In addition, their research did not include all of the important design variables of FDBs. They performed a parametric study of several major design variables of FDBs. Lee et al. (2012) presented a design optimization for robust FDBs by utilizing the critical mass proposed by Kim et al. (2011). However, increasing the critical mass does not always guarantee a reduced steady state response or reduced shock responses in all five degrees of freedom even though it improves the overall dynamic performance of the rigid disk-spindle system supported by FDBs. This research proposes a method to develop robust FDBs in a HDD utilizing modal analysis and optimization to improve both the steady state response and the shock response of a disk-spindle system supported by FDBs in all five degrees of freedom. The design of the FDBs is optimized to maximize every modal damping ratio of the rotating rigid disk-spindle system supported by FDBs in five degrees of freedom. Our simulated results are verified by evaluating the steady-state and shock response of the rotating rigid disk-spindle system supported by FDBs. 2 Method of analysis 2.1 Modal analysis The equations of motion of the rigid disk-spindle system supported by the FDBs can be represented with five degrees of freedom as follows (Kim et al. 2010): MðtÞ x þ ðc þ GÞ_xðtÞþ KxðtÞ ¼ FðtÞ; ð1þ where M is a diagonal mass matrix, C is an asymmetric damping matrix, G is a skew symmetric gyroscopic matrix, and K is an asymmetric stiffness matrix. F(t) is the force vector which may be the centrifugal force due to unbalanced mass or external shock. Solving the eigenvalue problem of Eq. (1) provides natural frequencies and the modal matrix P. Solving the adjoint eigenvalue problem of Eq. (1) also provides the adjoint Eigen matrix Q, and Q T P ¼ I (Meirovitch 1967). To solve Eq. (1) by using mode superposition, the solution of Eq. (1) should be assumed as follows: xðtþ ¼SrðtÞ ¼ M 1=2 PrðtÞ ð2þ where S ¼ M 1=2 P (Inman 2001). Substituting Eq. (2) into (1) and pre-multiplying Q T M 1=2 ; Q T M 1=2 MM 1=2 P rðtþþq T M 1=2 ðc þ GÞM 1=2 P_rðtÞþQ T M 1=2 KM 1=2 PrðtÞ ¼ I rðtþþq T CP_rðtÞþQ ~ T ~KPrðtÞ ¼Q T M 1=2 FðtÞ ð3þ where C ~ ¼ M 1=2 ðc þ GÞM 1=2 and ~K ¼ M 1=2 KM 1=2. Q T ~KP is a diagonal matrix in which the i-th diagonal element is x 2 i. The off-diagonal elements of QT CP ~ are much smaller than the diagonal elements, so that it can reasonably be assumed to be a diagonal matrix in which the i-th diagonal element is 21 i x i. Equation (3) can be represented by n-decoupled equations in the modal domain. r i ðtþþ2f i x i _r i ðtþþx 2 i r iðtþ ¼f i ðtþ ð4þ The solutions of Eq. (4) with zero initial conditions are as follows: For 0\f i \1, r i ðtþ ¼ 1 Z t e f ix i t f t ðsþe f ix i s sin x di ðt sþds ð5þ mx di 0 and for f i [ 1, Z 1 t r i ðt) ¼ qffiffiffiffiffiffiffiffiffiffiffiffi e f ix i t f t ðsþe f ix i s 2mx i f 2 i 1 0 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ½e x i f 2 i 1ðt sþ e x i f 2 i 1ðt sþ Šds ð6þ qffiffiffiffiffiffiffiffiffiffiffiffi where x di ¼ x i 1 f 2 i is the damped natural frequency. Equations (5), (6) show that the modal displacement is a function of the modal damping ratio f i, and the amplitude of Eqs. (5), (6) are inversely proportional to the modal damping ratio. Therefore maximizing the modal damping ratio can reduce the amplitude of modal displacement. The norm of modal displacement rðtþ can be represented using Eq. (2) as follows. krðtþk ¼ S 1 xðtþ ¼ S 1 T xðtþ S 1 xðtþ ¼ ðxðtþþ T S 1 TS 1 xðtþ ¼ ðxðtþþ T M 1=2 P T P 1 M 1=2 xðtþ ¼ ðxðtþþ T MxðtÞ ð7þ where the mass matrix M is generally a diagonal matrix. Finally, Eq. (7) can be represented as follows: krðtþk ¼ Xn ðr i ðt) Þ 2 ¼ Xn i¼1 i¼1 m i ðx i ðt) Þ 2 ð8þ Hence, Eq. (8) shows that the reduction in the amplitude of the physical displacement xðtþ can be accomplished by reducing the amplitude of the modal displacement rðtþ
which can be achieved by maximizing the modal damping ratios. 2.2 Formulation of design optimization The design optimization of the FDBs to improve both the steady state response and the shock response of the diskspindle system supported by FDBs in all five degrees of freedom was formulated as follows: Maximize; X5 f i i¼1 Subject to f i ðf i Þ 0 ; i ¼ 1to5 T f ðt f Þ 0 9 K xx ðk xx Þ 0 C xx ðc xx Þ 0 = K yy K yy C 0 yy C yy 0 ; K zz ðk zz Þ 0 C zz ðc zz Þ 0 ðx i Þ lower limit X i ðx i Þ upper limit ð9þ ð10þ ð11þ ð12þ ð13þ where X i are the design variables of the FDBs. To maximize the modal damping ratios decreases the modal and physical displacements as explained in Eqs. (5), (6) and (8). The friction torque of optimal FDBs is set to be smaller than that of the conventional one. The friction torque of FDBs was determined by Jang s method (Jang and Kim 1999). In a single degree of freedom system, the modal damping ratio can be represented as follows: f ¼ p c 2 ffiffiffiffiffiffi ð14þ mk Modal damping ratio is a function of the stiffness and damping coefficients. Even if the modal damping ratio increases, the stiffness or damping coefficient can be reduced. Therefore, K xx,k yy,k zz,c xx,c yy, and C zz of optimal FDBs are set to be greater than those of conventional FDBs. The stiffness and damping coefficients of the journal and thrust bearings were calculated by integrating the pressure change across the fluid film, which was determined by the perturbation equations of the Reynolds equation with respect to small displacements and velocities (Jang and Lee 2006). We chose ten design variables for the FDBs: the clearance, the bearing width and upper width to lower width ratio of herringbone groove in journal bearing, and the groove angle, groove depth, and groove to ridge ratio of the journal and thrust bearings. The upper width to lower width ratio of the herringbone groove of the journal bearing is an important design variable to determine the pumping direction of the journal bearings. 2.3 Procedure of design optimization Figure 1 shows the procedure of solving the proposed optimal design problem. This optimization procedure consists of FDB solver and optimization solver. In the FDB solver, the Reynolds equation is solved, and the flying height of the disk spindle system is determined for the given design variables of the FDBs and the given operating condition of the rotating disk-spindle system. The friction torque is calculated, and the dynamic coefficients are calculated by solving the perturbed Reynolds equations. The modal damping ratios are determined by modal analysis of the disk-spindle. Then, the output variables (modal damping ratios, friction torque, stiffness and damping coefficients) of the FDB solver are provided to the optimization Fig. 1 Solution procedure of the proposed optimal design
Fig. 2 Finite element model and pressure distribution of the FDBs solver as input variables. The optimization problem of FDBs is solved by applying the Progressive Quadratic Response Surface Modelling (PQRSM) algorithm. It takes approximately 24 h to converge to the optimal solution in a computer with a dual CPU of 3 GHz and RAM of 16 GB. 2.4 Evaluation of RRO and NRRO RRO and NRRO can be evaluated by solving the equations of motion of the rigid disk-spindle system supported by the FDBs in five degrees of freedom as follows: M xðtþ þðc þ GÞ_xðtÞ þkxðtþ ¼FðtÞ; ð15þ where FðtÞ is the force vector. At steady-state, the centrifugal force due to unbalanced mass is a major factor in determining the whirl radius or the RRO, and the centrifugal force can be represented as follows. F x ¼ m um ex 2 cosðxtþ; F y ¼ m um ex 2 sinðxtþ; ð16þ ð17þ where m um is the unbalanced mass, e is the distance of the unbalanced mass from the mass center, and x is rotating speed. NRRO is generated due to various sources such as manufacturing errors or instability of FDBs and external shocks in real operation. This research estimated NRRO due to half-speed whirl by applying the swept sine excitation in addition to the centrifugal force as follows: Table 1 Lower, upper and optimal values of design variables Design variable Lower bound Optimal value Upper bound Lower grooved thrust bearing Groove angle ( ) 10 13.5 45 Groove depth (lm) 5 7.4 20 Groove to ridge ratio (-) 0.25 0.29 0.67 Upper grooved journal bearing Groove angle ( ) 10 27 45 Groove depth (lm) 3.5 3.5 5 Groove to ridge ratio (-) 0.25 0.35 0.67 Upper part width of journal bearing (mm) 0.7 0.88 1.0 Lower part width of journal bearing (mm) 0.5 0.80 0.8 Lower grooved journal bearing Groove angle ( ) 10 23.5 45 Groove depth (lm) 3.5 3.5 5 Groove to ridge ratio (-) 0.25 0.32 0.67 Upper part width of journal bearing (mm) 0.25 0.54 0.6 Lower part width of journal bearing (mm) 0.35 0.60 0.7 Radial clearance (lm) 1.7 1.75 2
Table 2 Comparison of the damping ratios and natural frequencies between conventional and optimal models Mode Modal damping ratio (-) Natural frequency (Hz) Conventional model Optimal model Difference (%) Conventional model Optimal model Mode 1 1.00 1.00 0 Mode 2 6.44 21.88 239.8 Mode 3 0.75 0.77 1.9 48.9 46.8 Mode 4 0.89 0.90 1.1 41.7 39.5 Mode 5 0.98 0.99 0.2 129.6 132.1 F ¼ p 0 sin a 2 t2 þ x s t ; ð18þ a ¼ x e x s ; T ð19þ where a, T, x s and x e are sweep rate, sweep period, starting and ending frequencies. 3 Simulation model Figure 2 shows a finite element model of the FDBs of a 2.5 00 HDD with a rotating speed of 5,400 rpm and calculated pressure distribution of the FDBs. This finite element model consists of two grooved journal bearings, four plain journal bearings, two grooved thrust bearings, and one plain thrust bearing. Fluid film was discretized by 7,240 isoparametric bilinear elements with four nodes. The Reynolds boundary condition was applied to guarantee continuity of pressure and pressure gradient. The accuracy of the developed program was verified by comparing the calculated flying height of the coupled journal and thrust bearings in equilibrium (where the axial load generated by the FDBs was equal to the weight of a rotor) with the measured flying height at various rotating speeds (Jang et al. 2006). 4 Results and discussion 4.1 Results of optimal design Table 1 shows the optimal value of design variables, and manufacturable upper and lower bounds of the proposed optimal design problem. Table 2 compares the damping ratios and natural frequencies between conventional and optimal models. The rotating rigid disk-spindle system in Fig. 3 has two over-damped modes (a pure radial translational mode (mode 1) and a pure axial translational mode (mode 2)), three under-damped vibration modes (two tilting modes (modes 3 and 5) and a coupled radial and tilting mode (mode 4)) in which four modes (modes 1, 3, 4 and 5) except the axial mode are half-speed whirl modes. Four modal damping ratios of the optimal model are increased while the modal damping ratio of mode 1 is maintained at the same value. Table 3 compares friction torque, direct stiffness, and damping coefficients between conventional and optimal models. The friction torque of the optimal model is almost the same as that of the conventional model, and the direct stiffness and damping coefficients of the optimal model are greater than those of the conventional model. Groove depth of the optimal journal bearing is smaller than that of the conventional journal bearing. It decreases average clearance, and increases pressure, friction torque, stiffness and damping coefficients. The grooved journal bearings of the optimal model have smaller bearing areas than the conventional model. However, optimal journal bearings result Table 3 Comparison of friction torque, stiffness and damping coefficients between conventional and optimal models Conventional model Optimal model Difference (%) Fig. 3 Mechanical structure of the disk-spindle system of a HDD T f (mnm) 0.220 0.219-0.1 K xx (N/m) 4.00E?06 4.07E?06 1.6 K yy (N/m) 3.99 E?06 4.03 E?06 1.1 K zz (N/m) 2.67 E?05 4.86 E?05 81.7 C xx (Ns/m) 1.06 E?04 1.09 E?04 3.1 C yy (Ns/m) 1.06 E?04 1.09 E?04 3.0 C zz (Ns/m) 9.65 E?02 4.42 E?03 357.9
Fig. 4 Whirling motion of a HDD spindle system Fig. 5 Swept sine excitation in time and its frequency domain in the same level of friction torque due to low clearance. The groove depth and groove to ridge ratio of the optimal lower grooved thrust bearing are smaller than that of the conventional one. It decreases the flying height of the optimal model, and it increases axial direct stiffness and damping coefficients of the optimal model. 4.2 Verification of optimal design Figure 4 shows the 3-D loci of the conventional and optimal models at steady-state. The mass unbalance of the disk spindle system is assumed to be 0.2 g mm. The whirl radii of the mass centers of the conventional and optimal models Fig. 6 Frequency spectrum of the radial displacement of the mass center applied by the x-directional swept sine excitation were 67.2 and 54.0 nm, respectively. This implies that the radial RRO of the optimal model is smaller than that of the conventional model by 19.6 %. Figure 5 shows the swept sine excitation with amplitude of 5 G from 0 to 300 Hz over 5 s. It is applied to the mass center of the HDD spindle in the positive radial, axial and angular directions, respectively, when the disk-spindle system is at steady state. Figures 6, 7 and 8 show the frequency transforms of the radial displacement, the axial displacement and the angular displacement of the mass
Fig. 7 Frequency spectrum of the axial displacement of the mass center applied by the x-directional swept sine excitation Fig. 9 Frequency spectrum of the radial displacement of the mass center applied by the z-directional swept sine excitation Fig. 8 Frequency spectrum of the angular displacement of the mass center applied by the x-directional swept sine excitation center applied by the radial swept sine excitation. In the x-displacement as shown in Fig. 6a, the amplitudes of halfspeed whirl frequency of the conventional and optimal designs are 23.0 and 19.1 nm, and the amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 59.9 and 48.0 nm, respectively. Figure 6b shows the y-displacement, and the amplitude of the fundamental rotating frequency of the optimal model is 46.8 nm which is smaller than the conventional model by 11 nm. In the axial displacement as shown in Fig. 7, the amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 0.6 and 0.1 nm, respectively. Figure 8a shows the x-directional angular displacement, and the amplitude of the fundamental rotating frequency of conventional and optimal designs are 15.6 and 13.0 lrad. In the y-directional displacement as shown in Fig. 8b, the amplitudes of half-speed whirl frequency of the conventional and optimal designs are 6.3 and 5.4 lrad, and the amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 18.1 and 14.9 lrad, respectively. Figures 6, 7 and 8 show that the optimal FDBs have smaller radial and axial displacements, and smaller angular displacement than the conventional one with the application of radial-directional swept sine excitation. Figures 9, 10 and 11 show the frequency transforms of the radial displacement, the axial displacement and the angular displacement of the mass center applied by the axial-directional swept sine excitation. In the radial displacement as
Fig. 10 Frequency spectrum of the axial displacement of the mass center applied by the z-directional swept sine excitation Fig. 12 Frequency spectrum of the radial displacement of the mass center applied by the x-directional angular swept sine excitation Fig. 11 Frequency spectrum of the angular displacement of the mass center applied by the z-directional swept sine excitation shown in Fig. 9, the amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 67.3 and 54.2 nm, respectively. In the axial displacement as shown in Fig. 10, the amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 43.7 and 10.4 nm, respectively. Figure 11 shows the angular displacements with the application of axial swept sine excitation, and the amplitudes of the fundamental rotating frequency of conventional and optimal FDBs are 18.3 and 15.2 lrad. Half-speed whirl frequency does not show in Figs. 9, 10 and 11, because the axial shock cannot excite the half-speed whirl of radial and angular motion, which are orthogonal to the axial direction. Figures 9, 10 and 11 show that the radial, axial and angular displacements of the optimal FDBs with the application of the axial swept sine excitation are smaller than those of the conventional FDBs. Figures 12, 13 and 14 show the frequency transforms of the radial displacement, the axial displacement and the angular displacement of the mass center applied by the x-directional angular swept sine excitation. Figure 12a shows the radial x-displacement in which the amplitude of the fundamental rotating frequency of the optimal model is smaller than the conventional model by 0.6 lm. In the radial y-displacement as shown in Fig. 12b, the amplitudes of the half-speed whirl frequency of the conventional and optimal designs are 6.4 and 5.4 lm, and the amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 5.6 and 4.7 lm, respectively. In the
amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 1.9 and 1.7 mrad, respectively. Figure 14b shows the y-directional angular displacement in which the amplitude of the fundamental rotating frequencies of conventional and optimal designs are 0.9 and 0.8 mrad, respectively. Figures 12, 13 and 14 show that the optimal FDBs have smaller radial and axial displacements, and smaller angular displacements than the conventional FDBs due to the x-directional angular swept sine excitation. Fig. 13 Frequency spectrum of the axial displacement of the mass center applied by the x-directional angular swept sine excitation 5 Conclusions We propose a method to develop a robust optimal design for FDBs in a HDD to improve the NRRO and RRO by utilizing modal analysis considering the five degrees of freedom of a general rotor-bearing system. The proposed optimal design methodology is verified by comparing the steady-state and the swept sine excitation responses of the disk-spindle system supported by conventional FDBs with those supported by optimal FDBs. At steady state with the application of centrifugal force, the whirl radius of the optimal model is smaller than that of the conventional model. In the application of radial, axial and angular shock, all the peak values corresponding to half speed whirl and fundamental frequencies of the optimal model are smaller than those of the conventional model. It shows that the proposed optimal model is more robust than the conventional model. The proposed method can be utilized to develop robust design of FDBs in a HDD disk-spindle system. Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0021919). References Fig. 14 Frequency spectrum of the angular displacement of the mass center applied by the x-directional angular swept sine excitation axial displacement as shown in Fig. 13, the amplitudes of the fundamental rotating frequency of the conventional and optimal designs are 75.4 and 12.7 nm, respectively. In the x-directional angular displacement as shown in Fig. 14a, the amplitudes of half-speed whirl frequency of the conventional and optimal designs are 2.2 and 2.0 mrad, and the Hashimoto H, Matsumoto K (2001) Improvement of operating characteristics of high-speed hydrodynamic journal bearings by optimum design: part I-formulation of methodology and its application to elliptical bearing design. J Tribol :305 312 Hirani H, Suh NP (2005) Journal bearing design using multiobjective genetic algorithm and axiomatic design approaches. Tribol Int 38:481 491 Hirayama T, Yamaguchi N, Sakai S, Hishida N, Matsuoka T, Yabe H (2009) Optimization of groove dimension in herringbonegrooved journal bearings for improved repeatable run-out characteristics. Tribol Int 42:675 681 Inman DJ (2001) Engineering vibration. Prentice-Hall, Inc., New Jersey Jang GH, Kim YJ (1999) Calculation of dynamic coefficients in a hydrodynamic bearing considering five degrees of freedom for a general rotor-bearing system. ASME J Tribol 121:499 505
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