Analysis of dynamic characteristics of a HDD spindle system supported by ball bearing due to temperature variation

Size: px

Start display at page:

Download "Analysis of dynamic characteristics of a HDD spindle system supported by ball bearing due to temperature variation"

Kelley Greer
5 years ago
Views:

1 Analysis of dynamic characteristics of a HDD spindle system supported by ball bearing due to temperature variation G. H. Jang, D. K. Kim, J. H. Han, C. S. Kim Microsystem Technologies 9 (2003) Ó Springer-Verlag 2003 DOI /s Abstract This paper presents a method to investigate the characteristics of a ball bearing and the dynamics of a HDD spindle system due to temperature variation. Finite element model is developed for the rotating and stationary parts of a HDD spindle system separately to determine their thermal deformations by using ANSYS, a finite element program. Then, the relative position of the rotating part with respect to the stationary part is determined by solving the equilibrium equation of the contact force between upper and lower ball bearings. The validity of the proposed method is verified by comparing the theoretical natural frequencies of a HDD spindle system with the experimental ones before and after temperature variation. The proposed method makes it possible to predict the characteristics of a ball bearing and the dynamics of a HDD spindle system due to temperature variation. It shows that the elevated temperature results in the increase of contact angle and the decrease of bearing deformation, contact force and bearing stiffness, which result in the decrease of the natural frequencies of a HDD spindle system. 1 Introduction Nonrepeatable runout (NRRO) is one of the major obstacles to achieve high track density in a computer hard disk drive (HDD). One of its sources is the waviness of a ball bearing in a HDD spindle system, and it generates bearing frequencies and the excessive vibration especially when one of the bearing frequencies matches with one of the natural frequencies of HDD spindle system [1 4]. In HDD industry, this resonance has been avoided by designing the natural frequencies of a HDD spindle system not to coincide with bearing frequencies. However, the operating temperature inside a HDD has thermal cycles, starting Received: 20 June 2002 / Accepted: 28 August 2002 G. H. Jang (&), D. K. Kim, J. H. Han PREM, Department of Mechanical Engineering, Hanyang University, 17, Haengdang-dong, Sungdong-ku, Seoul, , Korea ghjang@hanyang.ac.kr C. S. Kim Nano Storage Lab., Samsung Advanced Institute of Technology, Suwon, Korea Paper Presented at the 13th Annual Symposium on Information Storage and Processing Systems, Santa Clara, CA, USA, June, 2002 from room temperature up to 80 C. This may result in looseness of a ball bearing, natural frequency shifts of a HDD spindle vibration, coincidence with bearing frequencies, and eventually read/write errors. Several researchers have investigated the sources of natural frequency shifts of a HDD spindle system due to temperature rise, i.e. disk media and ball bearing. Jr-Yi Shen et al. (2000) studied both experimentally and theoretically how temperature variations affect natural frequencies of disk media [5]. They pointed out that natural frequency shifts of aluminum and glass disks at elevated temperatures result primarily from residual stresses and thermal membrane stresses of the disks, respectively. Chaw-Wu Tseng et al. studied how temperature variation affects natural frequencies of rocking vibration of a rotating disk and spindle system through mathematical modeling and experimental measurement [6]. They pointed out that frequency shifts of rocking modes at elevated temperature primarily result from relaxation of bearing stiffness. In their mathematical thermal model, they only included the radial displacement of the outer race with respect to the inner race at the elevated temperature without taking account of the axial displacement. In addition to radial displacement, axial displacement of the outer race with respect to the inner race at the elevated temperature affects significantly the characteristics of a ball bearing and natural frequency shifts of a HDD spindle system. This paper presents a method to investigate the characteristics of ball bearings and the dynamics of a HDD spindle system due to temperature variation. The characteristics of ball bearings at the elevated temperature are calculated by the bearing displacement due to temperature variation, which is obtained by the finite element analysis of thermal expansion and the force equilibrium equation between upper and lower bearings. The proposed method is verified by comparing the shifts of natural frequencies of a finite element model with those of the experimental modal testing of a HDD spindle system before and after temperature variation. It investigates rigorously how temperature variation affects the characteristics of a ball bearing and the dynamics of a HDD spindle system. 2 Method of analysis A HDD spindle system as shown Fig. 1 is composed of several components with different thermal expansion coefficients. In this model, base, flange and hub are made of aluminum, and shaft and ball bearings are made of steel. 243

Also, upper and lower ball bearings in a HDD spindle system do not undergo the same deformation due to the geometric asymmetry in the axial direction. This section consists of two parts.

2 244 Fig. 1. HDD spindle system Fig. 3. Change of contact angle before and after temperature variation Fig. 2. Bearing displacement due to temperature variation As the operating temperature changes, so does the clearance between ball and races. This affects the deformation and stiffness of a ball bearing. Also, upper and lower ball bearings in a HDD spindle system do not undergo the same deformation due to the geometric asymmetry in the axial direction. This section consists of two parts. In the first part, the bearing characteristics are determined in a single row ball bearing by using the kinematics and the Hertz contact theory with the assumption that the radial and axial displacements of a ball bearing due to temperature variation are known. In the second part, the radial and axial displacements of a ball bearing are determined in double row ball bearings by using a finite element analysis for the thermal deformation and the force equilibrium condition between the upper and lower ball bearings. 2.1 Determination of bearing characteristics in a single row ball bearing due to temperature variation Figure 2 shows the change of bearing displacement before and after temperature variation. In Fig. 2, Dr I ; Dh I ; Dr O ; Dh O A; B; A 0 and B 0 are the radial and axial displacements and the curvature centers of inner and outer races before and after temperature variation, respectively. Figure 3 shows the change of contact angles due to temperature variation, i.e. the relative displacement of the curvature center of the outer race with respect to the same curvature center of the inner race. The new contact angle after temperature variation (a 0 ) can be expressed with respect to the old contact angle before temperature variation (a) as follows. a 0 ¼ tan 1 AB sin a þ ðdh O Dh I Þ ð1þ AB cos a ðdr O Dr I Þ In Eq. (1), AB is the distance between the curvature centers of inner and outer race before temperature variation, and it can be expressed in terms of inner and outer race conformity (f i and f o ), ball diameter (d B ), and bearing deformation (d) as follows. AB ¼ d B ðf i þ f o 1Þþd ð2þ The bearing deformation after temperature variation (d 0 ) is the sum of the bearing deformation before temperature variation (d), the distance difference between the curvature centers of inner and outer race before and after temperature variation (A 0 B 0 AB), and the thermal expansion of ball (db 0 d B) after temperature variation as follows. d 0 ¼ d þ A 0 B 0 AB þ d 0 B d B ð3þ In Eq. (3), A 0 B 0 and db 0 are the distance between the curvature centers of inner and outer race, and ball diameter after temperature variation. They can be expressed in the following equations. A 0 B 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ 2 ¼ ABcosa ðdr O Dr I Þ ABsina þ ð DhO Dh I Þ d 0 B ¼ d B þ a TB DT d B ð4þ ð5þ In Eq. (5), a TB and DT are the thermal expansion coefficient of the ball and the temperature variation, respectively. According to the Hertz contact theory, the contact force between ball and race (F) can be represented with loaddeflection constant (K) and bearing deformation (d) as follows. F ¼ Kd 3=2 ð6þ

Then the stiffness of a single ball can be expressed by the partial derivative of the contact force with respect to the bearing deformation as follows. k single ball ¼ of od ¼ 3 2 Kd1=2 ð7þ In Eqs.

Once the temperature changes, it has to be recalculated because the geometry of ball and race changes, too. 2.

3 Then the stiffness of a single ball can be expressed by the partial derivative of the contact force with respect to the bearing deformation as follows. k single ball ¼ of od ¼ 3 2 Kd1=2 ð7þ In Eqs. (6) and (7), the load-deflection constant is a function of the geometry and the material properties of ball and race [7, 8]. Once the temperature changes, it has to be recalculated because the geometry of ball and race changes, too. 2.2 Determination of bearing displacement in double row ball bearings due to temperature variation Finite element analysis in thermal expansion and the force equilibrium equation between upper and lower bearings are used to determine the radial and axial displacements of inner and outer race due to temperature variation. In the finite element analysis, thermal expansion due to temperature variation is calculated for two parts, i.e. the stationary and rotating parts, separately. Figure 4 shows the finite element models developed using ANSYS. In Fig. 4, the stationary part consists of base, flange, shaft and inner race, and the rotating part consists of outer race and hub. The radial displacements of inner and outer races and the axial displacements of inner races of upper and lower ball bearings due to temperature variation (DrI U, DrU O, DrL I, DrL O, Dh U I and Dh L I ) can be easily determined by subtracting the initial positions of inner and outer races before temperature variation from the final positions after temperature variation, because a HDD spindle system has axisymmetric geometry with respect to the shaft. But the same method cannot be applied to determine the axial displacements of outer races of upper and lower ball bearings (Dh U O and Dh L O ) because outer races of upper and lower ball bearings in a HDD spindle system undergo the thermal expansion as well as the axial movement of hub in such a way that the bearing forces between the upper and lower ball bearings should satisfy the force equilibrium condition. A HDD spindle system is not symmetric with respect to the span center of upper and lower bearings. This asymmetry results in the different thermal expansion of upper and lower bearings. It also makes the axial forces of the two bearings different, which moves the hub in the axial direction until the axial forces of upper and lower bearings are the same. Eq. (8) is the equilibrium equation of the axial forces between upper and lower ball bearings, and it is used to determine the axial movement of the hub. X F 0 A ¼ n F 0L sin a 0L þ n F 0U sin a 0U ¼ 0 ð8þ Figure 5 shows the numerical procedures to determine the axial displacements of the outer races of upper and lower 245 Fig. 4a, b. Finite element models of a HDD spindle system for thermal analysis. a Stationary part. b Rotating part Fig. 5. Numerical procedure to determine the axial displacements of outer races of upper and lower ball bearings

246 ball bearings by using the Secant method. First, the finite element model of outer race and hub as shown in Fig.

Second, an initial guess of an axial hub movement is made, and the axial displacements of outer races of upper and lower ball bearings (Dh U O ; DhL O ) are obtained by adding the axial movement of a

temperature variation to Eq. (6). Then, the Secant method is applied to Eq. (8) in order to determine the final axial displacements of outer races of upper and lower ball bearings.

4 246 ball bearings by using the Secant method. First, the finite element model of outer race and hub as shown in Fig. 4b is used to determine the axial displacements of the outer races of upper and lower ball bearings due to temperature variation only (Dh U OT ; DhL OT ). Second, an initial guess of an axial hub movement is made, and the axial displacements of outer races of upper and lower ball bearings (Dh U O ; DhL O ) are obtained by adding the axial movement of a hub. Third, the bearing contact forces of upper and lower bearings after temperature variation (F 0U ; F 0L ) are determined by applying the changed geometric parameters of ball and race after temperature variation to Eq. (6). Then, the Secant method is applied to Eq. (8) in order to determine the final axial displacements of outer races of upper and lower ball bearings. 3 Verification The proposed method is verified by comparing the natural frequencies of the finite element model with those of the experimental modal testing. Table 1 shows the major design specifications of a ball bearing used in this analysis. This research does not include a disk in a HDD spindle system in order to exclude the frequency shifts due to the residual stress of the disk [5]. Once the displacements of inner and outer races of upper and lower ball bearings in a HDD spindle motor due to temperature variation are determined by using the finite element analysis for thermal expansion and the force equilibrium condition, the stiffness of a ball bearing is calculated by using Eq. (7). ANSYS is used to develop a finite element model of a HDD spindle system as shown in Fig. 6, and the natural frequencies of a HDD spindle system are calculated. Figure 7 shows the experimental setup to measure the natural frequencies of a HDD spindle system before and after temperature variation. Inside the glass thermal chamber, a hot air blower is used to increase the temperature, which is measured by thermocouples. Impact hammer excites the hub and base and the laser doppler vibrometer measures the frequency response function of the HDD spindle system at the hub and base at 20 and 60 C, respectively, as shown Fig. 8. Table 2 shows the natural frequency shift of a HDD spindle system due to the temperature variation from 20 to 60 C in the experimental modal test and the finite element analysis, respectively. It shows that the experimental results match closely with the analytical ones. Natural frequency shifts of Fig. 7. Experimental setup to measure the natural frequencies of a HDD spindle system Table 1. Major design specifications of a ball bearing Number of balls 10 Ball diameter (mm) Pitch diameter (mm) 9.1 Contact angle ( ) Preload (N) 16 Inner race conformity Outer race conformity Radial clearance (mm) Fig. 6. Finite element model of a HDD spindle system Fig. 8a, b. Frequency response function of a HDD spindle system. a Frequency response function measured at hub. b Frequency response function measured at base

5 Table 2. Experimental and analytical results of natural frequency shifts of a HDD spindle system due to temperature variation Index Vibration mode Major source of flexibility Experiment Analysis 20 C 60 C Shift 20 C 60 C Shift 1 Axial mode Base )1 2 Axial mode Base ) )3 3 Rocking mode Ball bearing ) )24 4 Rocking mode Ball bearing ) )21 5 Axial mode Ball bearing ) )14 a HDD spindle system due to temperature variation will be discussed in the next section in detail Results and discussion 4.1 Bearing characteristics due to temperature variation Bearing characteristics in a HDD spindle system are calculated by using the proposed method whenever the operating temperature increases 10 C in the range from 20 to 60 C. Figure 9 shows the change of contact angle due to temperature variation. It shows that contact angle increases as temperature increases. The thermal expansion rate of the rotating part is larger than that of the stationary part, because the former is made of aluminum and the latter steel. Therefore, the curvature center of the outer race is moved in the outward direction with respect to that of the inner race, increasing the contact angle. Figure 9 also shows that the increase rate of the contact angle of the upper bearing is larger than that of the lower bearing. One of the reasons is the yoke and permanent magnet located in the lower part of the hub. They have a smaller thermal expansion coefficient than aluminum, which decreases the thermal expansion of the lower bearing. Another source is the upward axial movement of a hub to satisfy the force equilibrium, which increases and decreases the contact angle of upper and lower bearings, respectively. Figure 10 shows the change of bearing deformation, contact force and stiffness of a single ball due to temperature variation. They decrease as temperature increases. It also shows that the bearing deformation, contact force and stiffness of the single ball of lower bearing are larger than those of the upper bearing due to the small thermal expansion coefficients of permanent magnet and yoke. Figure 11 shows the change of preload due to temperature variation. It is equal to the axial force in upper or lower bearing. It decreases as temperature increases, and it decreases almost to half when the temperature increases up to 60 C. Figure 12 shows the change of equivalent radial and axial stiffness due to temperature variation. They are calculated with the following equations [9]. k RR ¼ K n 2 k AA ¼ K n! d 3=2 d þ AB þ d1=2 d þ 3AB cos 2 a 2 d þ AB! d 3=2 d þ AB þ d1=2 d þ 3AB sin 2 a 2 d þ AB ð9þ ð10þ Fig. 9. Contact angle due to temperature variation of a HDD spindle system In Eqs. (9) and (10), k RR, k AA and n are the equivalent radial and axial stiffness and the number of balls, respectively. The stiffnesses decrease as temperature increases, but the equivalent radial stiffness decreases more rapidly than the equivalent axial stiffness. 4.2 Dynamics of a HDD spindle system due to temperature variation The decrease of bearing stiffness due to temperature rise results in the decrease of the natural frequencies of a HDD spindle system. As shown in Fig. 8 and Table 2, the first two vibration modes are the axial modes mostly governed by the stiffness of the base, so that their frequency shifts are very small because the stiffness change of base due to temperature variation is negligible. The next three vibration modes are two rocking and axial modes mostly governed by the stiffness of the ball bearing. These frequencies decrease significantly because the stiffness of the ball bearing decreases due to temperature rise. Also, the frequencies of rocking vibration modes decrease more rapidly than that of the axial vibration mode because the equivalent radial stiffness decreases more rapidly than the equivalent axial stiffness as shown in Fig Conclusions (a) In the case where the thermal expansion coefficient of the stationary part of a HDD spindle system is smaller than that of the rotating part, elevated temperature

6 248 Fig. 11. Preload due to temperature variation Fig. 12. Equivalent radial and axial stiffness due to temperature variation Fig. 10. a Bearing deformation, b contact force and c stiffness of single ball due to temperature variation results in the increases of contact angle and the decrease of the bearing deformation, contact force and bearing stiffness. (b) The rate of increase of the contact angle of the upper bearing due to temperature rise is larger than that of the lower bearing because of the small thermal expansion coefficients of the yoke and permanent magnet located in the lower part of the hub and the upward axial movement of a hub to satisfy the force equilibrium. (c) The bearing deformation, contact force and stiffness of the single ball of lower bearing after temperature rise are larger than those of the upper bearing because of the small thermal expansion coefficients of the permanent magnet and yoke located in the lower part of the hub. (d) The decrease of bearing stiffness due to temperature rise results in the decrease of the natural frequencies of a HDD spindle system only affected by bearing stiffness. (e) The frequencies of rocking vibration modes decrease more rapidly than the axial vibration mode because the equivalent radial stiffness decreases more rapidly than the equivalent axial stiffness. References 1. Richter WO; Talke FE (1988) Nonrepeatable radial and axial runout of 5 1/4 00 disk drive spindles. IEEE Trans Magn 24: Ono K; Saiki N; Sanada Y; Kumano A (1991) Analysis of nonrepeatable radial vibration of magnetic disk spindles. Trans ASME J Vibration Acoust 133: Jang GH; Kim DK; Han JH (2001) Characterization of NRRO in a HDD spindle system due to ball bearing excitation. IEEE Trans Magn 37: Jang GH; Jeong SW (2002) Nonlinear excitation model of ball bearing waviness in a rigid rotor supported by two or more ball bearings considering five degrees of freedom. ASME J Tribology 124: 82 90

7 5. Shen Jr- Yi; Tseng Chaw-Wu; Shen IY; Ku C-PR (2000) Vibration of disk media at elevated temperatures. J Info Storage Proc Syst 2: Tseng Chaw Wu; Shen Jr- Yi; Ku C PR; Shen IY Effects of elevated temperatures on rocking vibration of rotating disk and spindle systems. ASME J Tribology 7. Harris TA (1991) Rolling bearing analysis. 3rd edn. John Wiley & Sons 8. Hamrock BJ; Dowson D (1981) Ball Bearing Lubrication. John Wiley & Sons 9. Jones AB (1960) A general theory for elastically constrained ball and roller bearing under arbitrary load and speed conditions. ASME J Basic Eng June:

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

DOI 10.1007/s00542-016-2850-2 TECHNICAL PAPER Robust shaft design to compensate deformation in the hub press fitting and disk clamping process of 2.5 HDDs Bumcho Kim 1,2 Minho Lee 3 Gunhee Jang 3 Received: