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.     

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.     

Complete determination of the dynamic coefficients of coupled journal and thrust bearings considering five degrees of freedom for a general rotor-bearing system

A complete method is presented for calculating the stiffness and damping coefficients of coupled journal and thrust bearings of a general rotor-bearing system considering five degrees of freedom. The Reynolds equations and their perturbation equations were derived by linearization of the bearing reaction with respect to the general five degrees of freedom, i.e., the tilting displacements and angular velocities as well as the translational displacements and velocities. The Reynolds equations and their perturbation equations were transformed into finite element equations by considering the continuity of pressure and flow at the interface between the journal and the thrust bearings. The Reynolds boundary condition was included in the numerical analysis so as to simulate the phenomenon of cavitation. The stiffness and damping coefficients of the proposed method were compared with those found from a numerical differentiation of the loads with respect to the finite displacements and velocities of the bearing center. It was shown that the proposed method may be used to calculate the dynamic coefficients of coupled journal and thrust bearings more accurately and efficiently than the differentiation method. The tilting motion was also been found to play an important role in the determination of force and moment coefficients.     

Robust optimal design of the FDBs in a HDD to reduce NRRO and RRO

This research proposes a robust optimal design methodology of the FDBs in a HDD to reduce RRO and NRRO. The critical mass, which determines the dynamic behavior of rotor-bearing system, was selected as an objective function, and the constraint equations were the friction torque of the FDBs, and the stiffness and damping coefficients related with under-damped vibration modes. Ten major design variables of the FDBs were chosen in 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. RRO and NRRO of the optimal design were compared with those of conventional one, and it showed that the proposed method could effectively reduce RRO and NRRO.     

Effects of thrust hydrodynamic bearing stiffness and damping on disk-spindle axial vibration in hard disk drives

This paper aims at investigating the effects of variations in thrust hydrodynamic bearing (HDB) parameters such as axial stiffness and damping coefficients on the axial vibration of disk-spindle systems in hard disk drives. For a parametric study, a closed-form axial frequency response function (FRF) of HDB spindle systems is derived as a function of the axial stiffness and damping coefficients of thrust HDBs. It is known that the axial vibration of the disk-spindle system is composed of two main parts: the vibration of the rigid hub in the axial direction and the disk deflection in the transverse direction. The results from this research clearly show that the vibration amplitudes at low frequency range is dominated by the axial vibration of the hub, and the amplitude of the unbalanced (0,0) mode is dominated by the disk deflection. The parametric study reveals that at low frequency range an increase in the bearing stiffness significantly reduces the hub axial vibration, and hence the axial vibration of the disk-spindle system. Surprisingly, a too much increase in the damping results in a higher amplitude of the unbalanced (0,0) mode. This is because a heavy damping constrains the hub vibration to nearly no motion, resulting in a direct transmission of vibration from the base to disk. To confirm the parametric study, a vibration test was performed on two HDB spindle motors with identical design but different fluid viscosity. The higher viscosity represents the higher axial stiffness and damping in the thrust bearing. The test result indicates that the spindle motor with higher viscosity has a larger unbalanced (0,0) mode amplitude when subjected to an axial base excitation.     

弹支-刚性转子系统过共振瞬态响应特性研究

针对结合弹支-刚性转子系统的动力学特点,利用Lagrange能量法建立了考虑变速特性的转子系统瞬态响应动力学方程,模型中区别考虑了非旋转阻尼和旋转阻尼的影响.采用精细积分算法计算获得过临界区的转子瞬态响应特性,进一步对比分析了角加速度和阻尼特性对转子系统瞬态振动响应幅频特性和相频特性的影响规律.研究结果表明：变转速引起系统的刚度矩阵变化并产生附件的激励力;瞬态过共振响应幅值明显小于稳态响应幅值,且过共振越快速、阻尼越大时系统瞬态振动响应幅值越小.针对过瞬态相频特性,在临界转速附近出现一个新的相位角(加速过共振小于90°),此相位不受角加速度值和旋转阻尼比的影响,但随着非旋转阻尼比的增大呈增大趋势.     

Kinematic investigation of tripod sliding universal joints based on coordinate transformation

In order to understand the kinematics of the tripod sliding universal joint, its kinematic equations were established based on coordinate transformation and then the effects of the joint angle, the rotating radius of the slide-rods, the length of the output shaft, and the frequency of the input shaft on its kinematics were investigated. The increase of the joint angle or the rotating radius of the slide-rods or the frequency of the input shaft enhances not only the relative motion of the slide-rods, but also the periodic oscillation of the output shaft along the rotating direction. However, increasing length of the output shaft weakens its periodic oscillation and hardly affects the relative motion of the slide-rods. In addition, the increase of the joint angle or the rotating radius of the slide-rods increases the fluctuation of the joint angle, but increasing length of the output shaft reduces it.     

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.     

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     

磁悬浮飞轮储能系统机电耦合非线性动力学研究

针对磁悬浮飞轮储能系统的"磁悬浮飞轮-发电机"机电耦合非线性动力学特性进行研究.通过推导磁悬浮飞轮储能系统在偏心条件下的动能、势能、发电机系统的磁场能以及系统的耗散函数,由Lagrange-Maxwell方程建立磁悬浮飞轮系统和两相四极永磁发电机系统的机电耦合动力学方程.采用数值法对0.6MW磁悬浮飞轮储能系统进行了仿真分析,研究结果表明,系统机电耦合非线性方程存在稳定的与转速同频的基频和三倍频周期运动解,且基频振动幅值比三倍频振动幅值大.对于稳定的磁悬浮储能飞轮机电耦合系统,飞轮转速增大,或磁轴承系统刚度减小或阻尼增大,或磁场能(电枢反应磁场能或永磁励磁磁场能)减小,可使系统的非线性振动幅值减小.而增大磁轴承系统的刚度,或减小磁轴承系统的阻尼,或增大系统的磁场能有可能破坏机电耦合系统的稳定性,使飞轮失稳.     

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.     

Dynamic characteristics of a coupled journal and thrust hydrodynamic bearing in a HDD spindle system due to its groove location

 This research numerically analyzes the dynamic characteristics of a coupled journal and thrust hydrodynamic bearing due to its groove location which has the static load due to the weight of a rotor in the axial direction and the dynamic load due to its mass unbalance in the radial direction. The Reynolds equation is transformed to solve a plain member rotating type of journal bearing (PMRJ), a grooved member rotating type of journal bearing (GMRJ), a plain member rotating type of thrust bearing (PMRT), and a grooved member rotating type of thrust bearing (GMRT). FEM is used to solve the Reynolds equations 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, such as whirl radius or axial displacement of a rotor, are determined by solving its nonlinear equations of motion with the Runge–Kutta method. This research shows that the groove location affects the pressure distribution in the fluid film and consequently the dynamic performance of a HDD spindle system. Received: 5 July 2001/Accepted: 17 October 2001     

Stability and Control of a Parametrically Excited Rotating System. Part I: Stability Analysis

This paper analyses the stability of a parametrically excited double pendulum rotating in the horizontal plane. The equations of motion for such a system contain time varying periodic coefficients. Floquet theory and the method of Hill's determinant are used to evaluate the stability of the linearized system. Stability charts are obtained for various sets of damping, parametric excitation, and rotation parameters. Several resonance conditions are found, and it is shown that the system stability can be significantly altered due to the rotation. Such systems can be used as preliminary models for studying the lag dynamics and control of helicopter blades and other gyroscopic systems.     

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.     

Investigations on characteristics of micro/meso scale gas foil journal bearings for 100–200 W class micro power systems using first order slip velocity boundary conditions and the effective viscosity model

In this paper, a modified compressible Reynolds equation for micro/meso scale gas foil journal bearings considering first order slip and effective viscosity under rarefied flow conditions is presented. The influence of rarefaction effect on the load carrying capacity, attitude angle, speed and frequency dependent stiffness and damping coefficients, modal impedance, natural frequencies and unbalance response is studied. From numerical analysis, it has been found that there is significant change in all the static and dynamic characteristics predicted by the no-slip model and model with effective viscosity. There is also a considerable difference between the values predicted by a model with effective viscosity and a model without effective viscosity. For a given eccentricity ratio, the influence of effective viscosity on load carrying capacity and attitude angle is more significant for the typical operating speed range of micro/meso scale gas turbines. The influence of effective viscosity decreases with increase in compliance of bearing structure. Similarly, the influence of effective viscosity on frequency dependent stiffness and damping coefficients increases with excitation frequency ratio. Significant change in natural frequency, modal impedance and unbalance response for model with no slip and slip with effective viscosity is observed. The influence of effective viscosity is found to be significant with increase in Knudsen number.     

Dynamic analysis of a HDD spindle system with FDBs due to the bearing width and asymmetric grooves of journal bearing

This paper investigates the dynamic behavior of a HDD spindle system with fluid dynamic bearings (FDBs) by solving the Reynolds equation and the equations of a motion of a HDD spindle system in five degrees of freedom. FEM is used to solve the Reynolds equation in order to calculate 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 behaviors of a HDD spindle system, such as the whirling and the tilting motion, are determined by solving its nonlinear equations of motion with the Runge-Kutta method. This paper also proposes two design methods to improve the dynamic characteristics of a HDD spindle system without increasing friction torque, i.e., optimization of the width of the lower and the upper journal bearings and the journal bearings with asymmetric grooves.     

Geometric Stiffness for Real‐time Constrained Multibody Dynamics

This paper focuses on the stable and efficient simulation of articulated rigid body systems for real‐time applications. Specifically, we focus on the use of geometric stiffness which can dramatically increase simulation stability. We examine several numerical problems with the inclusion of geometric stiffness in the equations of motion, as proposed by previous work, and address these issues by introducing a novel method for efficiently building the linear system. This offers improved tractability and numerical efficiency. Furthermore, geometric stiffness tends to significantly dissipate kinetic energy. We propose an adaptive damping scheme, inspired by the geometric stiffness, that uses a stability criterion based on the numerical integrator to determine the amount of non‐constitutive damping required to stabilize the simulation. With this approach, not only is the dynamical behavior better preserved, but the simulation remains stable for mass ratios of 1,000,000‐to‐1 at time steps up to 0.1 s. We present a number of challenging scenarios to demonstrate that our method improves efficiency, and that it increases stability by orders of magnitude compared to previous work.     

Investigation of the electromechanical variables of the spindle motor and the actuator of a HDD due to positioning and free fall

This research investigates the electromechanical variables of a spindle motor and an actuator of an operating hard disk drive (HDD) due to the positioning and the free-fall of a HDD. Magnetic fields of a brushless DC motor and a voice coil motor are determined by the time-stepping finite element equation of the Maxwell equation and the driving circuit equation. The pressure of the fluid dynamic bearings (FDBs) is determined by solving the finite element equation of the Reynolds equation to calculate the reaction force and the friction torque. Dynamic equations of the rotating disk-spindle, actuator, and stationary bodies of a HDD are derived from the Newton–Euler’s equation. The speed control of the rotating disk-spindle and the servo control of the actuator are included to describe the head positioning between the rotating disk and the head. The simulation is performed to investigate the electromechanical variables of the spindle motor and the actuator due to the positioning and the free-fall of a HDD. This research shows that the positioning and the free-fall of a HDD change the electromechanical variables of the spindle motor and the actuator of an operating HDD, and that monitoring their electromechanical variables may identify the positioning and the free-fall of a HDD without using extra sensors.     

Simulation of a magneto-mechanical damping machine: analysis,discretization, results

This paper presents and analyzes a method for the simulation of the dynamical behavior of a coupled magneto-mechanical system such as a damping machine. We consider a two-dimensional model based on the transverse magnetic formulation of the eddy currents problem for the electromagnetic part and on the motion equation of a rotating rigid body for the mechanical part.The magnetic system is discretized in space by means of Lagrangian finite elements and the sliding mesh mortar method is used to account for the rotation. In time, a one step Euler method is used, implicit for the magnetic and velocity equations. The coupled differential system is solved with an explicit procedure.