This paper presents the application of optimization method developed by Hashimoto to design oil lubricated thrust bearings for 2.5 inch form factor hard disk drives (HDD). The designing involves optimization of groove geometry and dimensions. Calculations are carried out to maximize the dynamic stiffness of the thrust bearing spindle motor. Static and dynamic characteristics of the modeled thrust bearing are calculated using the divergence formulation method. Results show that, by using the proposed optimization method, dynamic stiffness values can be well improved with the bearing geometries not being fixed to conventional grooves.
HDD has been used as the main storage multimedia for electronics devices. Currently, HDD widely depends on oil lubricated hydrodynamic bearings. Hydrodynamic bearing is mainly supported by thrust and journal bearings. A schematic view of bearings in a 2.5 inch HDD is shown in Figures
Bearings in spindle motor.
Initial spiral thrust bearing groove geometry and pressure distribution; (a) groove geometry, (b) pressure distribution.
Hydrodynamic bearing gives better performance characteristics compared to conventional ball bearings as it has high dynamic stiffness with additional much higher damping effects. These damping effect characteristics provide smaller nonrepeatable runout (NRRO). NRRO is the major contributor to the track misregistration in HDD readwrite mechanism. Even though the repeatable runout (RRO) of oil lubricated hydrodynamic bearing is higher than the ball bearing spindle motor, the RRO can be corrected by a readwrite servo. Therefore, to reduce NRRO and increase spindle performance is to improve the bearing performance.
Currently, groove geometries that are being widely used in HDD thrust bearings are mainly a spiral or a herringbone grooved geometries. Several investigators have conducted numerical analysis predictions of these grooves for HDD bearing performance [
This suggests that there is a probability of a further improvement if the groove geometry is not being fixed to any conventional grooves, either spiral or herringbone grooves. However, as far as authors know, there are no attempts at drastically improving the bearing characteristics changing the geometry and dimensions for oil lubricated 2.5 inch HDD spindle motor. Recently, HDD is being demanded to be thinner. If the characteristic of thrust bearing can be drastically improved, it is possible for HDD to be thinner. Therefore, in this paper, by adapting the optimization method initiated by Hashimoto, new optimum groove geometry and dimension to replace conventional grooves and increase the bearing performance of an oil lubricated 2.5 inch HDD had been calculated. The groove geometry and dimension of the thrust bearing for spindle motor are calculated using the hybrid method [
The optimization results showed that the dynamic stiffness of oil lubricated thrust bearing can be improved by introducing a new optimum geometry into the system. Vibration analyses were also presented to numerically verify the applicability of the improved bearings.
The geometrical optimization process is carried out using the hybrid method [
Figure
Initial 2.5 inch bearing parameters.
Parameters  Values 

Outer radius 

Inner radius 

Weight 
0.185 
Revolution speed 
5000~20,000 
Groove number 
16 
Seal ratio 
0.58 
Groove depth 

Groove width ratio

0.5 
Geometry parameter 
0.0 
Geometry parameter 
0.0 
Geometry parameter 
0.0 
Geometry parameter 
0.0 
Lubricant viscosity 

Design parameters for thrust bearing.
From there, the groove geometry design parameters are changed arbitrarily using the cubic spline interpolation function [
Figure
Modifications of groove geometry.
With the most inner point of the initial spiral groove curve fixed at point
In this paper, applying the boundaryfitted coordinate system to adjust the geometrical optimization method derives the calculation method of bearing characteristics. Moreover, in the process of analyzing the static and dynamic characteristics of the bearings, the perturbation method is applied to the Reynolds equivalent equation. The Reynolds equivalent equation can be solved using the NewtonRaphson iteration method, and the static component
The loadcarrying capacity
The minimum oil lubricating film thickness
The spindle motor’s friction torque on bearing surface is given by the following integration:
The spring coefficient
Finally, the bearing stiffness,
In this paper, the oil leak is not considered because the surround of thrust bearing of the spindle motor for 2.5 inch HDD, as shown in Figure
In formulating the optimum design problems for groove geometry, the design variable of vector
Another design variable considered in this optimization is the number of grooves. However, since design variables of groove numbers are of discrete values, the optimization is preliminarily calculated step by step for the groove number from
Furthermore, the constraint conditions based on a 2.5 inch HDD thrust bearing are expressed as
In (
Prescribed constraint values.
Parameters  Values 

Groove number 

Minimum seal radius ratio 
0.4 
Maximum seal radius ratio 
0.8 
Minimum groove depth 

Maximum groove depth 

Minimum groove width ratio 
0.4 
Maximum groove width ratio 
0.9 
Minimum geometry parameter 
−180 
Maximum geometry parameter 
180 
Allowable film thickness 

The objective function being set to improve the performance characteristics of the spindle in hydrodynamic bearing is the dynamic stiffness because thrust bearing plays an important role in supporting the total axial direction load of the HDD spindle. The dynamic stiffness,
The optimization problem for HDD thrust bearing is then formulated as follows:
Once the optimization is being carried out, the vibration analysis is calculated to numerically verify the improvements. The vibration is conducted assuming that the oil film supporting the bearing is a viscously damped one degree free vibration model. The values of the spring and damping coefficient obtained by the optimum design are applied for the vibration response.
The equation of motion is given by
The external impulsive force being applied to the HDD disk platters is assumed as follows:
The nondimensional expressions for the gain
Figures
Groove geometry and pressure distribution of optimized oil lubricated bearings. (a) Optimized at 10,000 rpm,
As can be seen from the figures, all geometries of the optimized bearings changed from initial spiral groove geometry to a new geometry, a spiral groove with bends in the outer periphery. In this paper, such geometry is called the modified spiral geometry.
Generally, it is widely known that spiral groove geometry has the ability to elevate the film thickness with its pump in effects as the revolution increases due to pressure generation. This explanation is applicable for the inner periphery of the optimized groove geometry. As the bearing rotates, the oil lubricants flows inwards, starting from the outer bends periphery, and flows along the grooves and collides with the seal, hence generating pressure. From all of the pressure distribution figures, the point where the groove and seal meet has the highest point of pressure generated.
However, the phenomenon is different from the outer periphery for opposite spiral groove geometries. Instead of generating the pressure and increasing the film thickness, the oil lubricant flows away from the outer bends towards the most outer radius of the bearing surface. Hence, as can be seen from the pressure distribution figures, there are no pressure peaks generated in the vicinity of the most outer regions of the bearing surface. This is because the pressure in the outer periphery of the bearing is neutralized to the atmospheric pressure. The pressure equivalent to the atmospheric pressure caused by the outer geometry bends will act as a force to pull down the bearing and lower the film thickness. With the combination of spiral geometry in the inner periphery and the opposite spiral bends in the outer periphery, the film thickness decreased, thus maximizing the dynamic stiffness of the spindle.
Even though all groove geometries basically possess the modified spiral geometry, Figure
For the groove geometries with allowable film thickness of
If we focus on the inner periphery of the total geometry closely, that is, the end part of inner spiral just before the seal, we can see that another bend with the opposite direction of the outer bends started to appear. These inner bends however are smaller than the outer ones. This can be explained by the fact that in order to maintain the prescribed allowable film thickness, it is essential to increase the pump in effect of the inner spiral. It is insufficient with only the inner spiral geometry and outer bends. Therefore, another bend in the inner periphery, just beside the seal, appears to increase the film thickness of the bearing. The inner bends can be seen clearly for the case of allowable film thickness of
The details of groove geometry and dimension of the optimized bearings from Figure
Design variables of optimized bearings at 10,000 rpm.
Design variables  Optimized 
Optimized 
Optimized 

Angle 
7.94  4.46  2.26 
Angle 
−20.7  −11.2  −4.70 
Angle 
79.9  68.1  61.3 
Angle 
17.5  26.4  32.3 
Groove number 
6  11  13 
Seal ratio 
0.80  0.69  0.63 
Groove depth 



Groove width ratio 
0.4  0.45  0.48 
The other design parameters of groove dimensions such as seal radius ratio, groove depth, and groove width ratio also change. The seal radius ratio shows that the lower the allowable film thickness is being set, the more the groove geometry surface will be occupied with the seal area. This eventually made the groove geometry to be concentrated at the outer region of the total bearing surface. On the other hand, for the groove depth and groove width ratio values, results show that the lower the allowable film thickness is being set, the smaller those values turned into.
Figures
Bearing characteristics versus rotational speed; (a) dynamic stiffness
The objective function for this study is the improvement of the HDD spindle dynamic stiffness. As can be seen from Figure
Figure
For the friction torque values shown in Figure
To confirm the effectiveness of the optimum design presented here, the values of design variables and objective function for the optimized bearing with
Comparison of the optimized solution and random solutions under allowable film thickness of
Case  Obj. 









Optimum solution 

4.46  −11.2  68.1  26.4  11  0.69 

0.45 
Random 1 

−60.0  −105.7  −31.4  82.8  6  0.79 

0.87 
Random 2 

−54.3  −37.1  159.9  162.8  9  0.44 

0.79 
Random 3 

−131.4  −42.8  −125.7  57.1  11  0.4 

0.66 
Random 4 

17.1  −5.7  −128.5  88.5  7  0.65 

0.49 
Random 5 

−168.5  142.8  −97.1  −25.7  10  0.44 

0.73 
Random 6 

20.0  17.1  −119.9  −105.7  11  0.68 

0.5 
Random 7 

40.0  −60.0  74.2  −91.4  6  0.58 

0.58 
Random 8 

77.1  62.8  −117.1  −14.3  12  0.64 

0.72 
Random 9 

102.8  −42.8  −165.6  −122.8  6  0.71 

0.41 
Random 10 

131.4  142.8  −142.8  −157.1  8  0.45 

0.85 
Figures
Vibration characteristics of optimized at 10,000 rpm bearings and conventional spiral bearing; (a) displacement with impulsive force, (b) gain versus frequency, and (c) phase angle versus frequency.
Figure
Figures
In this paper, the optimum design method for 2.5 inch HDD was carried out to improve the dynamic stiffness of a thrust bearing spindle. The results can be concluded as follows.
Bearing characteristics using the proposed hybrid method showed that, compared with conventional spiral groove bearings, improvements of dynamic stiffnesses can be obtained when a new geometry of modified spiral groove geometry is introduced.
To maximize the dynamic stiffness is to set the maximum seal radius ratio, that is, by concentrating the geometry at the most outer periphery of the bearing.
Vibration analysis of optimized bearings showed better characteristics than the initial spiral bearings. Therefore, optimized bearings are expected to improve vibration characteristic of HDD.
In this paper, we confirmed that dynamic stiffness of thrust bearing for 2.5 inch HDD was drastically improved numerically using the proposed hybrid method. However, it is necessary to verify the calculated data by experiments. We have prepared the experiments to verify the calculated data by using original experimental test rigs as shown in Figure
Experimental test rigs for verification of calculated data.
Vibration test rig
Impulse test rig
When optimizing the groove geometry, it is necessary to perform a sequential analysis of characteristics of a bearing with a groove geometry modified in succession by the finite difference method, but because of the
Bearing geometry transformation based on the boundaryfitted coordinate system; (a) Original bearing geometry, (b) Transformed bearing geometry.
The following Reynolds equivalent equation can be obtained from the equilibrium between the mass flow rates of oil inflowing into and outflowing from the control volume due to the shaft rotation and the squeezing motion:
Definition of control volume.
Then, in (
Assuming that variations of the bearing clearance are microscopic, the minimum oil lubricating film thickness
In the equations stated above,
The substitution of (
Solving Equations (
Groove width (m)
Land width (m)
Damping coefficient of oil lubricated film (N·s/m)
Objective function
Constraint functions;
Impulse external force (N)
Gain
Allowable film thickness (m)
Groove depth (m)
Minimum oil lubricating film thickness (gap between shaft and lower groove surface) (m)
Spring coefficients of oil lubricated film (N/m)
Dynamic stiffness of oil lubricated film (N/m)
Mass of the rotor (kg)
Groove number
Rotational speed of spindle (rpm)
Oil lubricated film pressure (Pa)
Atmospheric pressure (Pa)
Static component of oil lubricated film pressure (Pa)
Dynamic component of oil lubricated film pressure (Pa/m)
Coordinate of radius direction (m)
Bearing outer radius (m)
Bearing inner radius (m)
Seal radius (m)
Seal ratio;
Times (s)
Friction torque on bearing surface (N·m)
Loadcarrying capacity (N)
Vector of design variables
Equipartition space of
Ratio of groove width;
Viscosity of lubricant (Pa·s)
Coordinate of circumferential direction (rad)
Coordinate of change based on boundaryfitted coordinate system (m)
Coordinate of change based on boundaryfitted coordinate system (rad)
Phase angle (deg)
Groove geometry design parameter; (displacement for
Natural damped frequency (rad/s)
Squeeze frequency of the shaft revolution (rad/s)
Angular velocity of the shaft (rad/s).
Maximum value of state variables
Minimum value of state variables.