The data storage capacity of disk drives continues to increase rapidly. A well-known trend in hard disk drive is the 60 percent areal density growth per annum.
FINITE ELEMENT MODELING AND MODAL TESTING OF VIBRATION CHARACTERISTICS OF DISK DRIVES
T. H. Yan and R. M. Lin Centre for Mechanics of Micro-Systems School of Mechanical & Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798
φX φA φ jr
ABSTRACT There are many factors that limit the performance of the servo system in current magnetic disk drives and prevent significant increases in track density. It is common in present hard disk drives that there exist some vibration modes between 2 kHz and 10 kHz, which involve coupled vibrations of slider, suspension, arm and bearing assembly. These modes are particularly troublesome for the servo system. So, a thorough understanding of the dynamic behavior of the head actuator is essential to control the vibration in future high-performance disk drives. The finite element model of a full head actuator is included. From the contributions of each mode and the DC gains, a system model can be established for control design. Experimental techniques are also proposed to evaluate the dynamic properties of VCM actuator-armsuspension–slider–air-bearing systems. Through the comparison and correlation of the experimental and FEM results, it has been established that the FE modeling method is quite accurate for drive systems.
Experimentally-measured mode shape Analytically-predicted mode shape j
th
element of r
th
mode shape/eigenvector
H jk (ω ) Individual FRF element between coords j and k.
HX HA
Experimentally-measured individual FRF Analytically-predicted individual FRF
1. INTRODUCTION Magnetic hard disk drives are fascinating mechatronic devices. They are high-precision and high-performance machines, which are produced in very high volumes and are sold at relatively low cost. Fig. 1 shows the major mechatronic components of a conventional HDD. Digital information is recorded in concentric tracks of the disks by means of miniaturized read/write (R/W) heads. R/W heads are mounted on self-lubricated sub-micron flying sliders. These sliders are connected to stainless-steel suspensions that are in turn connected to a carriage and a voice coil motor (VCM). This set up allows for cross-track seek and track-following operation of the R/W heads.
NOMENCLATURE
[M ] {F }
Structural mass matrix Structural stiffness matrix Structural damping matrix Input excitation force vector
[X ]
Frequency-dependent input excitation vector Time-dependent displacement vector
[K ] [C ]
[Fˆ ]
[Xˆ ] mr
ωr ξr
cr
Frequency-dependent displacement vector th Modal mass of r mode th
-1
Natural frequency of r mode (rad.s ) th Viscous damping ratio of r mode Modal/effective viscous damping of th r mode (proportional damping)
Fig. 1. Schematic view of a hard disk drive.
200
[M ]{X }+ [C ]{X }+ [K ]{X } = {F (t )}
The data storage capacity of disk drives continues to increase rapidly. A well-known trend in hard disk drive is the 60 percent areal density growth per annum. At this rate, an advanced storage device will have 100,000 to 200,000 tracks per inch (TPI) in the near future. This is projected to place stringent requirements on servo system to bring the R/W head on slider to the data track and to maintain the head over the track. The offset between the actual R/W head position and the data track centre is called the position error which, when exceeds certain limit, will lead to track misregistration (TMR). For future highperformance HDDs with over 100kTPI track density, the allowable TMR must be less than 0.16µm, which leads to a required bandwidth of greater than 1.4kHz [1]. A fundamental problem of servo control for magnetic disk drives is the physical flexibilities of the actuator, the voice coil motor (VCM) and the suspension which transfer the required motion to the R/W heads. It has been generally thought that the mechanical resonances in the suspension, the E-block and the pivot bearing limit in practice the actual achievable closed-loop bandwidth. From the transfer function, or the frequency response function of the slider in a typical head actuator [2], it is noted that there are strong resonances between 2kHz and 10kHz. Theoretically the bandwidth of the servo system of the HDDs must be at least two octaves lower than the resonant frequency of the lowest mechanical resonance. However, in realistic servo system in order to ensure the stability, the actual bandwidth is far less than the maximal bandwidth, say about 500 to 600 Hz. As we extend the track density into the 100,000 TPI regime, it is likely that these resonances caused by actuator assembly should be suppressed or pushed to higher frequency, thus the servo bandwidth could be increased above 1.4 kHz.
[ ][ ]
(1)
[ ]
Where M , C , and K are mass, damping, and stiffness matrices, respectively. These matrices are functions of the finite element model, such as the structure configuration, spring stiffnesses, elastic moduli, etc. These structural parameters can be updated using identification techniques.
{F (t )} represents the forcing input excitation.
There exist two types of analysis in finite element modeling: (a) natural mode analysis and, (b) frequency response analysis. In the first case, no input to system is
{ ( )}
assumed, that is F t = 0 , and damping is usually assumed to be zero, so in this case, (1) can be simplified as,
([K ] − ω [M ]){φ } ={0} (r = 1,2,) 2 r
r
(2)
Where ω r are the natural frequencies, and φ r are the corresponding modeshape vectors of the structure. In this case, structural parameters can be updated by correlating and φ r with measured natural frequencies and modeshapes. In the frequency response analysis, the excitations
are
{F (t )} = {Fˆ }e iωt ,
{}
the
corresponding
responses are {X (t )} = Xˆ e iωt , where {Fˆ } and {Xˆ } are the amplitude vector of the excitation input and the displacement response, respectively. Substituting them into (1), one can obtain,
A thorough understanding of the dynamics behavior of the head actuator through both numerical and experimental methods is essential to improve the dynamics properties of actuator and so as to increase the servo bandwidth. Several investigators have studied the dynamics of the components such as head and suspension separated from the actuator assemblies by using shaker [3]~[7]. Although these analyses of the individual components are an integral part of understanding the dynamics of the system, it is more important to analyze the dynamics of the entire head actuator as a whole. It appears to the author’s knowledge that little work has been done about it in previous publications. In this paper, the dynamics of the whole assembly was investigated both numerically and experimentally. The finite element model of a full head actuator is established and from which, plant matrix for servo control has been extracted through the contribution of each mode in the displacements from VCM excitation to the response of R/W heads. The corresponding experimental investigations under operating conditions are also included, and the detailed vibration modes of the disk drive have been investigated. Finally, the effectiveness of numerical analysis was illustrated by the experiments.
([K ] + iω [C ] − ω [M ]){Xˆ } = {Fˆ } 2
For a single forcing input with magnitude
(3)
Fˆk , the
frequency response function is H jk (ω ) = Xˆ j Fˆ . In this k case, the model parameters can be updated to correlate these analytical FRFs with those experimentally measured at the frequency range of interest.
3. Finite Element Modeling Rotary actuators from various manufacturers all have a very same mechanism despite differences in detailed structure configuration, size, and number of disks. As shown in Fig. 1, for a commercially available 3.5-in hard disk drive, there are four arms and four R/W heads, a coil is bonded onto a two-pronged fork, which is integrated with the hub. The head arms are integrated with a stainless steel hub containing a pivot, which has ball bearings. Opposite to the head arms is a flat coil of wire, which moves within the poles of a magnet structure to form the servo motor [8]. A finite element model of the head actuator is shown is Fig.2, which is created using the
2. Theoretical Background of Finite Element Analysis The governing equations of motion of a structure can be expressed in the matrix form
201
the coil, a force is produced which tends to accelerate actuator arm inward or outward, depending the direction of current. The force produced is a function of input current and a linearized nominal value. Without loss of any accuracy, we can use the force as excitation to extract the concerned transfer function. The displacements in each mode shape are extracted at both excitation and response nodes. Frequencies and modal masses are also extracted for each mode. Each mode is then modeled as a SDOF system. The eigenvalue/eigenvector results from finite element analysis will be used to define the equations of motion in principle coordinates. And to transform forces to principal coordinates. We will use Laplace transform to solve for the transfer functions in principal coordinates and back-transform to physical coordinates, where contributions of each individual mode become evident.
commercial finite element software, ANSYS55. 4-node shell elements (SHELL63) are used for the suspension. 3D 8-node solid elements with rotations (SOLID45) are used for the solids such as sliders and the arms. The pivot has two ball bearings biased against each other. Although the balls are of hardened steel, they are significantly springy due to very tiny areas of contact. A 3-D springdamper element (COMBIN14) was used. The boundary condition of the actuator is clamped at the top and bottom of the bearing shaft, and the two sliders are constrained to be free only in an in-plane translation. All mode shapes and frequencies for the actuator assembly under 10kHz are calculated. It should be pointed out that it is impossible to model the realistic pivot ball bearing accurately since it is a highly nonlinear component. As a result the ball bearing here is simulated as linear spring-damper element. The resonant frequencies obtained are shown in Table 1. The first mode is the rigid body motion rotating around the pivot, which translate the lateral motion from VCM to R/W heads. The other modes limit the control performance of the servo system. How each mode affects the dynamics of the actuator will be analyzed in the following sections.
Ref
+ -
0.01 1925.8 3042.4 4734.8 5215.6
828.55 2168.7 3729.6 4736.7 5290.1
1252.6 2229.4 3844.6 5182.2 5964.3
Ge (s )
G m (s )
+ +
PES
Other disturbance & Bias
Table 1 natural frequencies of actuator assembly (Hz) Mode 1-4 Mode 5-8 Mode 9-12 Mode 13-16 Mode 17-20
Actuator
Controller
1459.8 3040.8 4230.3 5186.1 6802.7
Fig.3 Block Diagram of Actuator Control The general equation of undamped and damped system can be expressed, in transfer function form, as the following:
()
N
H jk s = ∑ r =1
N
H jk (s ) = ∑ r =1
(s
2
φ jr φ kr mr
(s
+ ω r2 ) φ jr φ kr mr
(4)
2
+ 2ξ r ω r s + ω r2 )
(5)
In general, every transfer function is made of additive combinations of SDOF systems, with each system having its dc gain (transfer function evaluated with s=j0) determined by the appropriate modal constant,
Fig.2 Finite Element Model of Actuator Arm
4. Servo System Modeling Through Modal Superposition
r
A jk = φ jr φ kr mr
. Substituting
s = jω = j 0 = 0 , to
the rth mode frequency for the damped and frequency response at dc, the dc gain, which is the same for the undamped and damped cases is:
The track-following controller used in the drive is an embedded servo-system with a digital observer based controller. The block diagram of the plant-controller is shown in Fig.3. There are several inputs to the system: a bias force, often called windage; a force due to shock or vibration; runout from spindle rotation; noise, which corrupts the position measurement; and a target position. The effects of resonant mode of the actuator assembly are the main limitation on the bandwidth of the control system. Prior to controller design and implementation, we should analyze and extract the required model of actuator, Gm (s). Such transfer function needed can be established from previous FEM analysis. In operation, the Gm (s) is the transfer function between VCM current input and the response of R/W head. When a current is passed through
(
H jk r = φ jr φ kr mr ω r2
)
(6)
Where the dc gain is the modal constant divided by the square of the eigenvalue for ith mode. Obviously, at resonance, the peak gain amplitude of each mode is given by substituting
H jk r =
202
(− ω
s = jω r
into (5):
φ jr φ kr m r 2 r
+ 2 jξ r ω + ω 2 r
2 r
)
=
− j (dc gain ) 2ξ r (7)
employed. And the HP 35670 analyzer was used to generate signal and measure the response. The experimental sketch was shown in Fig.6. The HDD actuator assembly was measured under the constraint conditions the same as its operating ones, hence no additional effects were introduced. As a result, experimental results can be regarded as the true behavior of the disk drive.
dc value
For any mode, if the force is excited at one of its nodal points, then the force applied cannot excite that mode, so the dc gain and peak gain will also be zero. If the mode cannot be excited, then it will have no effect on the frequency response and can hence be eliminated. For drive under discussion, the dc gains for each mode are shown in Fig.4, which shows that only few modes have big contributions to the lateral response from VCM current to the R/W head. The dc gains of individual modes can be used to rank the importance of each mode in the reduced system. Fig.5 shows the sorted dc gain values, the predicted FRF can be modeled by this reduced transfer function in 0~10KHz frequency range. Table 2 shows the important modes that have substantial contributions to the lateral response of the actuator in this range. Based the controller design and the properties of hardwares, the mode numbers for system modeling can be determined from Fig.5. At this stage, the plant model of actuator assembly, which contains the resonant modes, can be established by transfer function or state space form.
The accuracy of the finite element model can be evaluated by comparisons with measured natural frequencies and mode shapes, as well as frequency response function data. Modeshapes are typically compared using the wellknown Modal Assurance Criterion (MAC)[10] defined as: 2
{φ X }T {φ A } MAC ( A, X ) = ({φ X }T {φ X })({φ A }T {φ A })
dc 0gain of each mode contribution versus mode number 10 Head 1 Head 3 -5 10
The MAC takes on values from 0 to 1 (a MAC value of 1 means a perfect modeshape correlation) and can be evaluated for either the entire set or a partial set of modeshape vector data. Due to the dimensions, the number of nodes measured, and constraint conditions of hard disk drive, the points on some part cannot be measured under the operating conditions. The partial set of Frequency Response Assurance Criterion (FRAC)[9,10] had been used here to quantify the accuracy of FRFs predicted in the finite element model. In analogy to the MAC definition, the FRAC is defined as:
-10
10
-15
10
0
10
20 30 Mode number
40
50
sorted dc value
Fig.4 The dc gain vs mode number
2
dc 0gain of each mode versus number of modes included 10 Head 1 Head 3 -5 10 -10
10
0
10
20 30 Mode included
40
{H X (ω i )}T {H A (ω i )} FRAC (i ) = {H X (ω i )} 2 • {H A (ω i )} 2 (9) Table 2 Predicted vs. measured natural frequencies with non-negligible contributions (KHz) The predicted mode The measured mode
-15
10
(8)
50
0.01(Hz)
3.750
4.668
5.230
----
3.734
4.701
5.290
There are in total 12 points measured on the side of the actuator assembly from the tip to the VCM. The directions of the excitation and measured responses are both horizontal. Some typical FRFs are shown in Fig.8(a)-(d) together with their predicted counterparts. The FRAC values were evaluated as a function excitation frequency using (9) and are shown in Fig.9. The finite element model exhibits the highest average FRAC value (about 0.88) over the 0.1kHz-10kHz frequency range, so the accuracy of the model meet the requirement for predicting the dynamics properties of the system. From these comparisons with measured frequency response functions, we can conclude that the finite element modeling method used here for hard disk drive is very accurate. Some typical modeshapes of the hard disk drive were shown in Fig.10. Combined with
Fig.5 dc gain vs number of modes included in FRF
5. Experimental Results and Updated FEM Results Correlation The 3.5” hard disk assembly used in our experiment is shown in Fig.6 where the heads were numbered as head 1~4 from top to down. A Polytech LDV system, which measures motions of target points on the actuator assembly along the direction of laser beam, was
203
-80 Magnitude, dB
the FRFs of the heads, the first peak is produced by the butterfly mode of actuator assembly, which hinders the servo bandwidth of the controller.
Predicted FRF on head 2 measured FFR on head 2
-100 -120 -140 -160 -180 2 10
3
10 Frequency, Hz
4
10
(b)
Magnitude, dB
-80
Predicted FRF on head 3 measured FFR on head 3
-100 -120 -140 -160 -180 2 10
3
10 Frequency, Hz
4
10
(c)
Magnitude, dB
-80
Fig.6 The side and top view of the hard disk drive tested
Predicted FRF on head 4 measured FFR on head 4
-100 -120 -140 -160 -180 2 10
3
10 Frequency, Hz
4
10
(d)
Fig.8 Typical measured and predicted FRFs between VCM excitation and responses of actuator heads
FRAC (0~1)
1
0.5
0 2 10
Fig.7 Experimental measurement set-up
3
10 Frequency, Hz
4
10
Fig.9 Frequency Response Assurance Criterion (FRAC) for Hard Disk Drive
Magnitude, dB
-80 -100
Predicted FRF on head 1 measured FFR on head 1
6. CONCLUSIONS
-120 -140
Very close results have been obtained from finite element analyses and experimental investigations. Based on the finite element model developed in this paper for disk drive, accurate and reliable prediction results can be obtained. Based on both numerical and experimental results of the whole head actuator assembly and from the mechanics
-160 -180 2 10
3
10 Frequency, Hz
(a)
4
10
204
Normal Operating Conditions. Advances in Information Storage System. 1998,Vol.9, 47-61.
and dynamics viewpoint this paper supports the following conclusions. Due to the flexibility of the actuator assembly in lateral and vertical directions, the head actuator exhibits a series of resonant modes, but among them only few modes will actually contribute to the TMR. The head actuator vibration characteristics may be simplified as a series of single DOF mass-spring-damping systems. Under operating condition VCM excitation, only few resonances will eventually show up in the transfer function or the frequency response function curves in lateral direction. For the disk drive under consideration, the lateral butterfly mode, which is the first mode appeared in the FRF, should be improved or suppressed to avoid its hindrance on future increase of servo bandwidth of the system.
[8] C. Denis Mee, Eric D. Daniel, Magnetic Storage nd Edition, P. R. Donnelley Sons Handbook. The 2 Company, 1996. [9] NEFSKE, D. J. and SUNG, S. H., Correlation of a coarse-mesh FE model using structural system identification and a frequency response assurance criterion. Proceedings of IMAC, 1996, 14, 597-602. [10] Ewins, D.J., Modal Testing: Theory, Practice, and Application, Research Studies Press Ltd., Bladock, Hertfordshire, England, 2000.
ACKNOWLEDGEMENTS # 10: 3.73e+3Hz, Undeformed # 10:3.55e+3 Hz, Undeformed
The authors take great pleasure in acknowledging the financial support from the National Science and Technology Board and the Ministry of Education, Singapore.
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(a) The Butter Mode
[2] Guo, W., Weerasooriya, S. and Goh, T. B. Dual stage Actuators for High Density Rotating Memory Devices. IEEE Transactions on Mag.. 1998, 34(2), 450-455.
# 13:4.70e+3 Hz, Undeformed
# 13: 4.70e+3Hz, Undeformed
[3] Henze, D., Karam, R. and Jeans, A. Effects of Constrained layer Damping on the Dynamics of a Type 4 In-line Head Suspension. IEEE Trans. On Mag.. 1990, 26(5), 2439-2441. [4] Yoneoka, S., Owe, T. and Aruga, K. Dynamics of inline Flying-Head Assemblies. IEEE Trans. On Mag.. 1989, 25(5), 3716-3718.
(b) The Bending Mode
# 14:5.29e+3 Hz, Undeformed
# 18: 5.29e+3Hz, Undeformed
[5] Chious, S. S. and Miu, D. K. Tracking Dynamics of inline Suspensions in High-performance Rigid Disk Drives with Rotary Actuator. ASME J. of Vibration and Acoustics. 1992, 114, 67-73. [6] Jiang, L. X. Improved Performance of Hard Disk Drives with a Passive Vibration damper. Ph.D’s dissertation, University of New York. 1997.
(c) The Torsion Mode Fig.10 the typical mode shapes
[7] Jeong, T.G., Chun, J. I., Chung, C. C., Byun,Y.K., and Ro, K. C. Measurement Technique for Dynamic Characteristics of HDD Head-Suspension Assembly in
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