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.     

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.     

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.     

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.     

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.     

A bi-directional rotating fluid bearing system

 Most fluid bearing systems with grooves on the journal/thrust bearing surfaces were designed to rotate in a specified direction and cannot be reversed. This feature of such fluid bearings limits their application range and hence, a bi-directional rotating fluid bearing system is proposed. The results of numerical simulation on the dynamic characteristics of such bearing system are presented and compared with those of one-directional rotating fluid-bearing system. It shows that for the same load capacity and stiffness requirement, the bi-directional rotating fluid bearing system has a higher power consumption than that of the one-directional counterpart. However, the bi- directional rotating fluid bearing system provides the freedom of rotating spindle motor in either direction and widens the application range of fluid bearing spindle motors. Received: 5 July 2001/Accepted: 17 October 2001     

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     

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.     

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.     

A mixed shear flexible finite element for the analysis of laminated plates

A mixed shear flexible finite element based on the Hencky-Mindlin type shear deformation theory of laminated plates is presented and their behavior in bending is investigated. The element consists of three displacements, two rotations, and three moments as the generalized degrees of freedom per node. The numerical convergence and accuracy characteristics of the element are investigated by comparing the finite element solutions with the exact solutions. The present study shows that reduced-order integration of the stiffness coefficients due to shear is necessary to obtain accurate results for thin plates.     

Prediction of dynamic properties of rotor supported by hydrodynamic bearings using the finite element method

The purpose of this paper is to present general programs for rotor bearing analysis using the Finite Element Method. A consistent representation of both mass and stiffness is used for the rotor shaft, while hydrodynamic bearings are calculated by solving Reynolds equation. Dynamic characteristics of these bearings are then obtained with a perturbation method.These programs are first compared with both numerical and experimental results from literature. We also study the influence of bearing characteristics on stability threshold and unbalance response of a rotor.     

Numerical solution of finite modified Reynolds equation for couple stress squeeze film lubrication of porous journal bearings

In this paper, the rheological effect of couple stress fluids on the static and dynamic characteristics of squeeze film lubrication in finite porous journal bearings is studied. The finite modified Reynolds equation is derived from the Stokes constitutive equations for couple stress fluids and is solved numerically by using the finite difference technique. The applied load is considered as a sinusoidal function of time to simulate the bearings operating under cyclic loads. Under a cyclic load, the effect of couple stress is to reduce the velocity of the journal centre and to increase the minimum permissible height of the squeeze film.     

基于6DOF模型及动网格的动静压轴承刚度阻尼数值计算

针对传统雷诺方程求解三维油膜流场特性的局限性,提出基于6DOF模型及动网格的动静压轴承刚度阻尼计算方法.以具有典型结构形式的液体动静压轴承为例,通过加载6DOF自定义程序,采用非线性迭代方法计算外载荷作用下轴心轨迹的瞬态变化过程,得到轴颈在外载荷作用下的静平衡位置;通过嵌入UDF宏程序以动网格更新方法实现对轴颈在静平衡位置的扰动,求解Navier-Stokes方程得到轴颈扰动前后位置变化后的瞬态油膜力,利用差分法求得动静压轴承油膜刚度和阻尼,并分析了不同转速下轴承刚度和阻尼的变化规律.     

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     

PZT微驱动器磁头的动特性研究

采用摄动法对描述超薄气体润滑理论的修正雷诺方程进行处理,建立了气体润滑的静动态方程,求解得到磁头气膜的无量纲刚度系数和阻尼系数,并讨论了磁头线速度对空气膜刚度系数和阻尼系数的影响.采用的磁头模型是一种PZT微驱动器磁头,其主体部分飞高为20 nm,读写头处飞高可达5 nm.模拟结果表明,磁头末端下降15 nm后,其动态特性和稳定性明显提高;磁头线速度在8.0 m/s～11.2 m/s变化时,气膜刚度系数线性增加,阻尼系数却线性减小.     

Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.     

具有轴承不对中的多跨柔性转子系统非线性动力学研究

主要研究了具有不对中轴承支承的柔性多转子耦合系统的动力学建模和非线性动力学行为.首先在短轴承假设、小轴承的不对中量和圆盘不平衡量等几个基本假设条件下,考虑了转子的柔度、不对中轴承的非线性油膜力和圆盘的不平衡等因素后,建立了一个具有轴承不对中的10自由度多跨转子系统非线性动力学模型；最后采用数值方法研究了系统的非线性动力...     

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.     

Dynamic characteristics of micro air bearings for microsystems

This paper reports on the dynamic characteristics of micro air bearings that include thrust and journal air bearings for microsystems. The dynamic thrust air bearing employs a spiral groove configuration. Analysis shows that the motion stability and load capacity of a thrust air bearing imposes a contradictive requirement on the groove pattern of the air bearing. The dynamic journal air bearing is realized by using a plain circular trench with a narrow radial clearance of C, and a very small aspect ratio of bearing length versus diameter (L/D). Analysis on the shock tolerance of the journal air bearing shows that the shock tolerance increases with the bearing number, the bearing aspect ratio of L/D and its initial equilibrium eccentricity ratio ε ₀. The optimum values of the bearing parameters are explored and recommended. A prototype of turbine device has been designed based on the recommended bearing configurations and realized via microfabrication.