A parametric study on rocking vibration of rotating disk/spindle systems with hydrodynamic bearings: rotating-shaft design

 The system studied in this paper is a rotating disk/spindle assembly supported by hydrodynamic bearings with a rotating shaft design. Based on an experimentally verified mathematical model [1, 2], this paper presents how various spindle parameters affect critical vibration modes of the system, such as half-speed whirls and (0, 1) unbalanced modes (i.e., rocking modes). The parameters studied include number of disks, hub/shaft interface stiffness, shaft rigidity, thrust bearing location, radial bearing stiffness, radial bearing damping, and radial bearing locations. To simulate operational tests, the numerical study focuses on frequency response functions (FRF) of rotating disk/spindle systems subjected to linear base excitations. Simulation results show that 1-disk configuration has smaller FRF amplitude than the 4-disk configuration. In addition, the amplitude of half-speed whirl is primarily controlled by the radial bearing stiffness. In contrast, the amplitude of (0, 1) unbalanced modes is dominated by hub/shaft interface stiffness. Finally, radial bearing locations significantly affect the amplitude of half-speed whirls and (0, 1) unbalanced modes simultaneously. Received: 16 October 2001/Accepted: 31 December 2001     

A parametric study on rocking vibration of hard disk drive spindle motors with fluid-dynamic bearings and rotating-shaft design

This paper presents a thorough parametric study to identify critical parameters controlling rocking vibration of hard disk drive (HDD) spindle motors. The spindle motors studied are of rotating-shaft design with fluid-dynamic bearings (FDB). The rocking vibration of interest results primarily from first three pairs of (0,1) unbalanced modes (also known as rocking modes, pitch modes, or gyro modes) and half-speed whirls. The parametric study shows that the transverse mass moment of inertia of the rotating part is the most critical parameter affecting the rocking amplitude. Also, FDB in-line stiffness dominates the amplitude of the half-speed whirls. Surprisingly, FDB in-line damping coefficient can considerably affect the amplitude of the second (0,1) unbalanced mode. Finally, bearing locations are very critical parameters for rocking amplitude of FDB spindles. Unfortunately, there is not a set of optimal bearing positions that will minimize vibration for all (0,1) unbalanced modes.The work was completed when the first author was a graduate student at the University of Washington, Seattle, Washington, USA.

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

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.     

Finite element modal analysis of an HDD considering the flexibility of spinning disk–spindle, head–suspension–actuator and supporting structure

This paper presents a finite element method to analyze the free vibration of a flexible HDD (hard disk drive) composed of the spinning disk–spindle system with fluid dynamic bearings (FDBs), the head–suspension–actuator with pivot bearings, and the base plate with complicated geometry. Finite element equations of each component of an HDD are consistently derived with the satisfaction of the geometric compatibility in the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Base plate, arm, E-block and fantail are modeled by tetrahedral elements. Pivot bearing of an actuator and air bearing between spinning disk and head are modeled by stiffness elements. The restarted Arnoldi iteration method is applied to solve the large asymmetric eigenvalue problem to determine the natural frequencies and mode shapes of the finite element model. Experimental modal testing shows that the proposed method well predicts the vibration characteristics of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode are transferred to a head–suspension–actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock.     

Transient dynamic analysis of ferro-fluid bearing spindle motor

 With areal recording density of hard disk drives (HDD) historically growing at an average of 60% per year and fast spindle speed to continue to reduce access time, it is becoming increasingly more difficult to maintain the precise positioning required of the GMR heads to read and write data. Any unexpected vibration will cause the data written to a wrong data track. Consequently, the dynamic behaviors of HDD spindle systems and their potential influences on track misregistration are key issues in disk drive design. With rapid advances in the emerging consumer device market, the fluid bearing spindle motors, which have low NRRO, low acoustic noise and high damping, are being developed as next generation spindles. This paper is to study transient dynamic performance of HDD ferro-fluid bearing spindle systems. The FEA based component mode synthesis method is used to reduce the overall spindle system dimensions. The effect of the unbalanced magnetic pulls (UMP) due to two different types of motor configurations (balanced and unbalanced configurations) on the dynamic behaviors of spindle system was investigated. The simulated results show that the motor with balanced configuration provides better spindle dynamic performance due to absence of UMP. The UMP derived from the unbalanced configuration can result in some frequency resonance interactions and adversely affect the HDD servo-tracking system. Received: 5 July 2001/Accepted: 17 October 2001     

Robust shaft design to compensate deformation in the hub press-fitting and disk clamping process of 2.5″ HDDs

We investigated deformation of the outer diameter of a shaft due to the hub press-fitting and disk clamping processes associated with a 2.5″ hard disk drive. We propose a new robust shaft design to minimize the effect of deformation on the outer diameter of the shaft. We numerically show the effect of deformation on the shaft due to the pressure, stiffness, and damping coefficients of fluid dynamic bearings (FDBs), and the critical mass and excitation response of the rotor-bearing system. We also experimentally measured the axial non-repeatable runout and the amplitude at the half speed whirl frequency of FDBs with both conventional and proposed designs. Through these tests we confirm that the proposed design improves the static and dynamic performance of the FDBs and rotor-bearing system.     

弹性箔片轴承-盘式永磁电机转子系统轴向冲击响应

轴向磁通永磁(Axial Flux Permanent Magnet,AFPM)电机在给转子输出周向电磁转矩的同时,还作为一种“磁轴承”,与弹性箔片轴承(Gas Foil Bearings,GFBs)并联工作.当GFBs支承的单定子单转子AFPM电机转速发生突变时(如加、减速),叶轮或者透平会产生一个瞬时轴向力.为揭示此轴向冲击对转子系统振动的影响机理,针对采用2个径向GFBs和1个轴向GFB支承的单定子单转子AFPM电机,建立了系统的刚性转子动力学方程,计入了永磁体的轴向吸力、径向回复力以及轴向GFB的推力对转子振动的影响.计算显示：轴向冲击对横向振动的影响非常小,箔片结构刚度的非线性效应会对转子振荡幅值和时间均产生影响,永磁体的负刚度绝对值越大,轴向振幅越大.在设计时,要注意永磁轴承刚度、箔片结构刚度和结构阻尼的合理匹配,以保证转子在轴向冲击下的轴向振荡时间和振幅均不超过允许值.     

Whirling, tilting and axial motions of a HDD spindle system due to the manufacturing errors of FDBs

This paper investigates the whirling, tilting and axial motions of a hard disk drive (HDD) spindle system due to manufacturing errors of fluid dynamic bearings (FDBs). HDD spindle whirls around the sleeve with tilting angle due to the centrifugal force of unbalanced mass and the gyroscopic moment of rotating spindle in addition to axial motion. The whirling, tilting and axial motions may be increased by the manufacturing errors of FDBs such as imperfect cylindricity of sleeve bore, or imperfect perpendicularity between shaft and thrust plate. They increase the disk run-out to limit memory capacity and they may result in the instability of the HDD spindle system. This paper proposes the modified Reynolds equations for the coupled journal and thrust FDBs to include the variable film thickness due to the cylindricity of sleeve bore and the perpendicularity between shaft and thrust plate. Finite element method is used to solve the modified Reynolds equation to calculate the pressure distribution. Reaction forces and friction torque are obtained by integrating the pressure and shear stress, respectively. The whirling, tilting and axial motions of the HDD spindle system are determined by solving the equations of a motion of a HDD spindle system in six degrees of freedom with the Runge-Kutta method. It shows that the imperfect cylindricity and perpendicularity increase the whirl radius, axial runout and tilting angle of the HDD spindle system. However, the degradation of dynamic performance due to the imperfect perpendicularity between shaft and thrust plate can be improved by allowing the other manufacturing error of the cylindricity of sleeve bore in such a way to compensate the bad effect of the imperfect perpendicularity.     

Magnetically induced vibration of a flexible rotating disk-spindle system due to the internal magnetic force arising from the spindle motor of a HDD

This paper investigates the magnetically induced vibration of a flexible rotating disk-spindle system and stationary stator-base due to the internal excitation of the local magnetic force arising from the spindle motor of a HDD. A three-dimensional magnetic finite element model of the spindle motor is developed, and the Maxwell stress tensor method is applied to calculate the local magnetic force acting on the stationary teeth and rotating permanent magnet of the spindle motor. Also, a three-dimensional structural finite element model is developed and local magnetic force is applied to teeth and permanent magnet. The simulated forced vibration of the base plate matched well with the measured one. The dominant frequency component of local magnetic force is the 12th harmonic corresponding to the number of poles, but the dominant frequency component of vibration is the 36th harmonic corresponding to the least common multiple of the number of poles and slots because the 12 and 24th harmonics in local force are canceled out when they are summed up along the air gap. The 12th, 24th and 36th harmonics of the axial vibration are mostly affected by the axial magnetic force, and the amplitudes of those harmonics are increased with the increase of stator eccentricity.     

Stability analysis of a disk-spindle supported by a plain journal bearing and pivot bearing

A hydrodynamic bearing is widely used for hard disk drives, and it is better than a ball bearing in terms of vibration suppression, noise reduction and shock resistance. However, its cost to performance ratio should be further improved. In this study we analyzed the stability of a disk-spindle assembly supported by a hydrodynamic plain journal bearing and a pivot bearing at the bottom of the shaft. As a result, we found that a half-frequency whirl of a vertical spindle caused by the plain journal bearing becomes stable if the gyro factor of the rotor is larger than 0.5. We examined the effects of the bearing design parameters on the stability of the disk-spindle assembly, including the flexibility of the shaft. We also compared the stability of a disk-spindle assembly supported by two plain journal bearings and found that the vertical spindle is always unstable. Thus, the bottom end of the shaft should be supported as a fulcrum.     

Robust design of a HDD spindle system supported by fluid dynamic bearings utilizing the stability analysis of five degrees of freedom of a general rotor-bearing system

This paper proposes a method to improve the robustness of a hard disk drive (HDD) spindle supported by fluid dynamic bearings (FDBs) by utilizing the stability analysis of the five degrees of freedom of a general rotor-bearing system. The Reynolds equations and the perturbed equations of the coupled journal and thrust bearings were solved by FEM to calculate the dynamic coefficients. The paper introduces the radius of gyration to the equations of motion in order to consistently define the stability problem with respect to a single variable, i.e., the mass. The critical mass, which is the threshold between the stability and instability of the HDD spindle, is determined by solving the linear equations of motion. The proposed method was applied to improve the robustness of a HDD spindle supported by FDBs by varying the groove parameters. It shows that the optimized groove design obtained using the proposed method increases both the stability and the modal damping ratio of the half-speed whirl mode. This research also determines the motions of the rotating disk-spindle system by solving its nonlinear equations of motion with the Runge?CKutta method. It shows that the groove design optimized using the proposed method has a small whirl radius in the steady state. It also shows that it has very little displacement due to the shock excitation, and that it quickly recovers to the equilibrium state.     

Optimal design of fluid dynamic bearings to develop a robust disk-spindle system in a hard disk drive utilizing modal analysis

This research proposes an optimal design methodology for fluid dynamic bearings (FDBs) in a hard disk drive to improve the dynamic performance of the disk-spindle 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.     

Improvement of slider off-track vibration through design parameter modifications of a compatible FDB spindle system

Recently, the hard disk drive (HDD) industry has tried to use a compatible spindle system regardless of the number of disks because of the resulting cost reduction and standardization of components. The center of gravity (CG) location predominantly affects the disk and slider off-track vibration, which is why the rocking mode of a spindle system is affected by the CG. Any changes to the CG affect the operational vibration of the spindle system. In a compatible fluid dynamic bearing (FDB) spindle system, changing the number of disks may alter the CG. Nevertheless, research into the compatibility of FDB designs has not been undertaken. In this study, FDB design parameters were selected to reduce the slider off-track vibration with variations in the CG considering a compatible spindle system. First, a verified finite element (FE) model of a spindle system was constructed. The amplitude and frequency of the rocking mode were compared between a one-disk spindle system and a two-disk spindle system using the FE model, considering the relationship between the CG location, which is changed by the number of disks, and the location of the upper and lower journal bearings. HDD prototypes were then manufactured using the improved design. Based on the manufactured spindle system, the variations in the rocking mode characteristics and slider off-track vibration were measured and operational vibration tests were performed to verify the effect of the number of disks on the slider off-track vibration. An improved FDB spindle design was developed with a reduced rocking mode, and a compatible spindle system was proposed.     

On Taylor vortices and Ekman layers in flow-induced vibration of hard disk drives

This paper is about flow-induced vibration (FIV) of disks in hard disk drives (HDD) influenced by two classical flow structures in fluid dynamics, Taylor Couette vortices (TCV) and Ekman layers. FIV is computed with a fully coupled commercial aerodynamics/structural code. The emphasis is on FIV of disks and geometries under conditions typical for high speed, server HDDs. In typical server drives computational fluid dynamic (CFD) analysis predicts the occurrence of TCVs in the disk to shroud clearance. TCVs typically do not occur in mobile and desktop drives. The main controlling non-dimensional parameters are the Reynolds number, the Taylor number and the aspect ratio of the disk to shroud clearance. The existence of Ekman layers on the disk surfaces is persistent. The Ekman layers and their radial return flow interact in a complex manner with the flow in the disk to shroud clearance. The turbulent viscosity between shrouded disks results from “bursting” phenomena that are typical for the flow field near the disk rims and shroud. The details of a turbulent burst are presented together with its momentary disk excitation effect. The benchmark case used is a fully shrouded set of two disks with a disk to shroud clearance and a disk thickness to shroud aspect ratio such that TCVs occur in the disk to shroud clearance. The TCVs interact with the Ekman layers such that the outer TCVs are continuously destroyed and recreated. An example is presented of fully coupled FIV of a two-disk axi-symmetric benchmark case. The two co-rotating shrouded disks attract aerodynamically: they deflect statically inward. The results also show the dynamic disk deformation dominated by the disk (0,0) “umbrella” mode. In addition, there is random disk deflection caused by the turbulent bursting. At server drive conditions and a 70 mm diameter disk the peak to peak deflection is approximately 20% of the mean deflection. Three dimensional effects are also presented such as wavy TCVs. In another benchmark with a cavity the flow near unshrouded disk edges is shown. In that case the pressure fluctuations can be an order of magnitude greater than in shrouded regions.     

Disk resonance due to electro-magnetic excitation

Disk vibration is a substantial contributor to the Track-Mis-Registration error budget for the high rpm disk drive applications (10k, 15k rpm) because of high air turbulence. Measures are taken to reduce disk vibration such as to add stator between disks, build shroud around the disk stack to direct the airflow, or use thicker disks etc. In addition to air turbulence excitation, disk is also subjective to excitation from electro-magnetic forces, occurring only when the disk mode frequency lined-up with excitation frequency. In this paper, the focus is not on how to reduce disk vibration, but on how to avoid disk resonance from a spindle motor design point of view. Basic Campbell plot is presented as a reference useful to all spindle motor designers. Case studies are discussed including root cause analysis and solutions.     

受轴向激励弹性支承梁的稳定性分析

对于广泛存在的弹性支撑梁,首次呈现支承弹簧刚度对轴向激励下梁横向振动稳定性的影响.应用Hamilton原理,建立了两端由线性弹簧支撑的受轴向激励梁的动力学控制方程.通过解析方法计算了受轴向压力梁的固有频率,得到了支撑弹簧刚度与系统固有频率和临界轴力的关系.Galerkin截断后,通过多尺度法和Runge-Kutta法,计算得到了梁参激振动稳态响应的半解析与数值解.讨论了激励幅值、支撑弹簧刚度、平均轴力对系统非线性响应幅值及软硬特性的影响.利用Routh-Hurwitz稳定性判据,求得系统的参激稳定边界,着重讨论了支撑弹簧刚度、阻尼系数的影响.研究发现,边界支撑弹簧的刚度可以显著改变受轴向激励梁的参激稳定边界.因此,研究结果将为广泛存在受到轴向激励结构的设计提供指导.     

Dynamic analysis of torsional resonance mode of atomic force microscopy and its application to in-plane surface property extraction

In the torsional resonance (TR) mode of atomic force microscopy (AFM), the changes in the torsion-related dynamic characteristics of a tip–cantilever system due to in-plane (lateral) tip–sample interaction are used for surface property imaging. This paper investigates the fundamental dynamics of a tip–cantilever system when it is operated in TR mode, with or without tip–sample interaction. With the actual location of the tip on the cantilever taken into consideration, modal analysis is carried out to obtain the cantilever TR frequencies/mode shapes under linear elastic tip–sample interaction. The relations of lateral contact stiffness and viscosity to torsional amplitude/phase shift are established. A comprehensive understanding on the effects of lateral contact stiffness and viscosity, driving frequency, and tip location on the cantilever torsional amplitude/phase shift is achieved by parametric analysis. The basic methodology to extract in-plane surface properties in TR mode is described. This work will help in advancing the development and applications of the techniques and instruments using TR mode of AFM.     

Attenuation of rotating disk vibrations with sector-shaped plates in disk storage devices

A parametric study has been conducted to understand the mass loading contribution to rotating disk vibrations in disk storage devices. With sector-shaped plates retaining air between the two plates and a rotating disk, results have shown frequency crossing of disk flutter modes with non-zero nodal diameters where significant modal pattern switching is observed. Extending the range of the analysis to sector geometries provides insight into the disk vibrations on the degree of frequency separation and mode switching as a span angle of the sector-shaped plates and disk rotating speed. Design guidelines are formulated to minimize this mode crossing phenomenon and thus minimize the number of spindle disturbances present during track follow or the servotrack writing process.     

Dynamic behaviour of curved piles

The paper presents an analytical model and associated computer program which is developed to investigate the dynamic behaviour of curved piles embedded in a homogeneous elastic half-space and subjected to forced harmonic vertical vibration, wherein the movement of piles in the axial and lateral directions are considered. The analysis accounts for a soil-pile interaction in a simplified way and predicts the response of such foundations. Equivalent stiffness and damping parameters of pile in longitudinal and horizontal directions have been studied and typical results have been presented. A parametric study of the dimensionless amplitudes in longitudinal and lateral directions is also presented. The directions are found to depend on the frequency of vibrations, slenderness ratio and curvature of pile. Analytical results are compared with experimental ones.