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Vibration predictions and verifications of

disk drive spindle system with ball

bearings

ARTICLE in COMPUTERS & STRUCTURES · JULY 2002

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Vibration predictions and verifications of disk drive

spindle system with ball bearings

J.P. Yang *, S.X. Chen

Data Storage Institute, DSI Building, 5 Engineering Drive 1, (off Kent Ridge Crescent, NUS), Singapore 117608, Singapore

Received 10 January 2001; accepted 4 May 2002

Abstract

With areal recording density of hard disk drives (HDD) historically growing at an average of 60% per year, it is

becoming increasingly more difficult to maintain the precise positioning required of the ever-smaller GMR heads to

read and write data. Any unexpected vibration will cause the data written to a wrong data track, even the vibration

amplitude is very small. Consequently, the dynamic behaviors of HDD spindle systems and their potential influence on

track misregistration rate must be clearly understood. This paper is to apply an approach based on efficient component

mode synthesis (CMS), incorporating multi-body system dynamics technology to predict dynamic characteristics of

HDD ball-bearing spindle systems. First, the discrete governing equations of motion for HDD spindle systems, which

consist of several flexible and rigid components, are derived through the use of Lagrangian equations. The elastic

component modal frequencies and modal shape vectors are then obtained using a finite-element analysis. For ball

bearing inherently defects, a mathematical model is used as a time-varying force, resulting in spindle vibrations. The

time-varying force and component modal shape vectors are incorporated into the governing equations of the whole

spindle systems. An implicit numerical integration method is used to obtain the forced vibration of the HDD spindle

system. Finally, the dynamic responses of two typical HDD spindle systems are investigated numerically to predict the

significant coupled vibration frequencies, mode shapes and resonance interactions. The results well agree with the

solutions predicted by other analytical methods and the experimental results, respectively.

2002 Published by Elsevier Science Ltd.

Keywords: Hard disk drive spindle system; vibration and component mode synthesis

1. Introduction

In recent years, a strong market demand has devel-oped for higher disk storage density (tracks per inch, or

TPI) and faster spindle operating speed (revolution per

minute, or rpm) in computer hard disk drives (HDD).

This represents a major technical challenge to design

engineers as the vibration characteristic of disk-spindle

system with increasing TPI and rpm becomes more

complex and sensitive to disk drive servo system. The

dynamic behaviors of the HDD spindle system and

its potential influence on misregistration between theread/write head and data track must be clearly under-

stood. However, the effects of rotary inertia, gyroscopic

and whirling motions on natural frequencies make the

problem be different from other mechanical system dy-

namics. Furthermore, nonrepeatable vibration of ball

bearings has become one of the major limiting factors to

the achievement of higher areal densities in the HDD

industry. Therefore, a better understanding of their dy-

namic behaviors is very important for HDD spindle

system design (Schirle and Lieu 1996). So far many

experimental techniques [1–4] appear to be effective

in identifying significant coupled vibration frequencies

Computers and Structures 80 (2002) 1409–1418

www.elsevier.com/locate/compstruc

* Corresponding author. Tel.: +65-874-8409; fax: +65-777-

1349.

E-mail address: [email protected] (J.P. Yang).

0045-7949/02/$ - see front matter 2002 Published by Elsevier Science Ltd.

PII: S0 0 4 5 -7 9 4 9 (0 2 )0 0 1 0 6 -2

http://mail%20to:%[email protected]/





and modes of HDD spindle systems. Meanwhile, many

researchers have investigated and predicted dynamic

characteristics of HDD spindle systems by using various

analytical methods [5–10].

The main purpose of this paper is to apply an ap-

proach based on efficient component mode synthesis

(CMS) [11,12], incorporating multi-body system dy-namics technology to predict dynamic characteristics of

HDD spindle systems. Applications of CMS-based

techniques to the dynamic analysis of high-speed rotor

bearing systems have been discussed for the past several

years [13,14]. However, little literature describes its ap-

plications in the dynamic analysis of HDD spindle

systems. Specially, effect of ball-bearing defect frequen-

cies on dynamic performance of the HDD spindle

system is limited. The essential ideal of CMS is to derive

the behavior of the overall system from its components.

The individual component eigenvalues problems are

first solved by finite-element analysis (FEA) and thenthese components are assembled by posing the coupling

of the interfaces within multi-body dynamic equations

for the overall system. In this paper, a mathematical

model is also used and included in the governing equa-

tions as a time-varying force, which results in bearing

raceway deflections and causes spindle vibrations. Be-

cause the current work utilized the FEA-assisted com-

ponent dynamic condensation techniques, an efficient

computational effort can be achieved as compared with

other numerical methods. Furthermore, as for the

complicated nonlinear material and damped composite

laminated disk that uses alternative substrate mate-

rial [15], its finite-element modal frequencies and shapes

are also more easily computed and integrated into

the discrete governing equations than the analytical

method.

In following sections, first, the discrete governing

equations of motion for a HDD spindle system math-

ematical model, which consists of several components

(flexible and rigid), are derived through the use of

Lagrangian equations. Second, the individual flexible

component modal analyses are performed using FEA.

Third, CMS is applied to effectively approximate the

motion of the flexible components through the corre-

sponding component modal vectors. Next, a mathe-matical model for ball-bearing defects is described as

additional excitation sources, producing time-varying

forces that cause HDD spindle system vibrations, and

incorporated into the discrete governing equations.

Then, an implicit numerical integration method is used

to obtain the forced vibration of the HDD spindle

system. Finally, dynamic responses of the two typical

HDD spindle system subjected to an impact force

are investigated numerically. The results are compared

and verified with the solutions predicted by other an-

alytical methods and experimental method, respec-

tively.

2. Equations of motion for HDD spindle systems

Multi-body system dynamics has grown in the past

decade to be an important tool in the design and simu-

lation of complex mechanical dynamic systems. This is

especially true in the area of aerospace and vehicle dy-

namics and control, robotics, and bio-systems. Its ver-satility in analyzing a broad range of applications has

made it an attractive virtual prototype real-time simu-

lation tool prior to building a prototype. A multi-body

system, as considered here, that is, the HDD spindle

system, is composed of rigid components (e.g. spindle)

and flexible components (e.g. disks) that undergo large

gross motion accompanied by small elastic vibration.

Let us consider the general formulation for the po-

sition vector rG D of a point D, on the i th flexible disk of

the HDD spindle system as shown in Fig. 1, as follows.

rG

D

¼ xG þ s B

D

þ u B

D

ð1Þ

where xG ¼ ½ x; y ; z Tis the position vector from the ori-

gin of the ground reference frame G to the origin of the

local component reference frame B ; s B D is the position

vector of the undeformed position of point D with re-

spect to local component reference frame B ; and u B D is

the displacement of point D that is due to the defor-

mation of the disk. Also, we write the matrix form of the

position vector in the ground reference frame G as s

rG D ¼ xG þ AðsG D þ Uð DÞqÞ ð2Þ

where A is the transformation matrix from the local

component reference frame B with respect to ground G ;U is a matrix of space-dependent shape function; and q

is the time-dependent modal coordinate, which may be

obtained by applying the finite-element method. The

details are given below. Furthermore, if the position

vector rG H of a point H on the rigid spindle is considered,

Eq. (2) has a simpler form, as follows.

Fig. 1. Frames and vectors for HDD spindle system.

1410 J.P. Yang, S.X. Chen / Computers and Structures 80 (2002) 1409–1418



rG H ¼ xG þ AsG

H ð3Þ

We define the vector of generalized coordinates, p, as

follows.

p ¼

x

y

z

w

h

/

qi;i¼1;n

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

¼x

w

q

8<:

9=; ð4Þ

where w ¼ ½w; h;/T

are the body 3-1-3 Euler angles that

define the Euler transformation matrix, A. The velocity

expressions of points D and H in terms of _ p p are obtained

from the first time derivative of Eqs. (2) and (3)

v D ¼ ½I Að~ss D þ ~UUð DÞqÞC þ AUð DÞ _ p p ð5Þ

v H ¼ ½I A~ss H C _ p p ð6Þ

where I is the identity matrix, the tilde denotes the skew-

symmetrical matrix of a vector and C is described as

follows.

C ¼ X _ p p1 ¼x xx y x z

8<:

9=; _ p p1 ð7Þ

where X ¼ ½x x;x y ;x z T

is the angular velocity vector of

the local component reference frame with respect to

ground G .

Thus, the kinetic energy of the HDD spindle systemis written as:

T ¼ T H þ T D

¼ 1

2ðX I H X þ M H v H v H Þ

þ 1

2

Xni¼1

q

Z V i

v Di v Di dV i ð8Þ

where q is the density of the disk; n is the number of

disks; M H and I H are the mass and the mass moment of

inertia of the spindle. The dissipation function C and

potential energy V for the HDD spindle system are

C ¼ 1

2

X2

b¼1

fcbxð_r r bx X r by Þ2 þ cby ð_r r by þ X r bxÞ

2 þ cbz _r r 2bz g

ð9Þ

V ¼ V B þ V D ¼ 1

2

X2

b¼1

ðk bxr 2bx þ k by r

2by þ k bz r

2bz Þ þ

Xni¼1

V i

ð10Þ

where r bx, r by and r bz , _r r bx, _r r by and _r r bz are the displacements

and velocities of spindle at the bth ball-bearing supports,

respectively; k bx;

k by and k bz , cbx, cby and cbz are the stiff-

ness and viscous damping coefficients of the bth ball

bearing; V i is the strain energy for the i th rotating disk,

which may be represented as

V i ¼ 1

2 Xl

i¼1 Xl

j¼1

K ijqiðt Þq jðt Þ ð11Þ

where K ij is based on the stiffness properties and mode

shape of the disk; l is the total degrees of freedom (DOF)

for the disk.

Finally, the general equation of motion for the HDD

spindle system can be derived using Lagrangian equa-

tions

d

dt

oT

o _ p p

oT

o p þ

oC

o _ p p þ

oV

o p ¼ 0 ð12Þ

The equation of motion expressed in matrix form is

M € p pn o

þ G _ p pn o

þ D _ p pn o

þ K pf g ¼ F ð13Þ

where M is a mass matrix, G the skew-symmetric gy-

roscopic matrix, D the modal damping matrix, K the

modal stiffness matrix, all of dimension b b; € p p, _ p p, and p

are b 1 column matrices of general coordinates having

elements € p p i, _ p p i, and p i, respectively, in the i th row; F is a

b 1 column force matrices.

3. Component mode synthesis

Eq. (13) includes not only rigid components but alsoflexible components that need a large number of DOF to

describe the related flexibility. The finite-element method

is usually a commonly used tool to model flexibility of a

component with arbitrary geometry. But, its accuracy

depends on a reasonable fine mesh. Thus, a few of

flexible components require finite-element models with

the large number of nodal variables and result in a very

large set of the dynamic equations. To achieve a lower

order dynamic model, the FEA-based CMS method is

utilized to represent motion of the flexible disks in the

HDD spindle system equations. This approach involves

using a relatively small number of component modalvectors U together with the mass M f , stiffness K f , and

inertia invariants I f for each disk. The i th flexible disk

finite element can be written in a partitioned form:

M BBi 0

0 M IIi

€uuB

i

€uuIi

" #þ

K BBi K BI

i

K IBi K II

i

uB

i

uIi

¼

F BiF Ii

ð14Þ

where uBi and uI

i represent the boundary and interior

coordinates of the disk, respectively, they may include

both elastic deformations and small rigid-body motions.

In this study, the i th disk modal vector Ui includes the

J.P. Yang, S.X. Chen / Computers and Structures 80 (2002) 1409–1418 1411



normal modal matrix UNi and the constraint modal

matrix UCi .

The normal modes are defined as the eigenvectors of

the disk with all boundaries uBi restrained. Thus, UN

i is

obtained as uBi ¼ 0 and normalized with respect to M II

i ,

i.e.,

ðK IIi M II

i XÞ UNi

¼ 0 ð15Þ

UNi

TM II

i UN

i

¼ I ; UN

i

TK II

i UN

i

¼ X ð16Þ

where X ¼ diagðx2i Þ, xi are the i th modal frequency and

I is an identity matrix. The normal mode matrix UNi is

usually truncated to a matrix UNim

of the first m normal

modes (m l) so that a lower order system model is

reached.

The constraint modes are defined as the mode shapes

for interior uIi due to a unit deflection in each of the

boundary uBi

successively with all other boundary uBi

fixed. The i th disk equilibrium equation is:

K BBi K BI

i

K IBi K II

i

uB

i

uIi

¼

F BiF Ii

ð17Þ

We set the force FIi to zero, Eq. (17) yields

K IBi uB

i þ K IIi uI

i ¼ 0 ð18Þ

or

uIi ¼ ðK II

i Þ1

K IBi

uBi ¼ UC

i uBi ð19Þ

The final form of UCi

is given by UCi ¼ ðK II

i Þ1

K IBi .

Thus, the i th disk mode set are approximated with a

modal summation Ui

ui ¼ uB

i

uIi

¼

I

UCi U

Nim

qC

i

qNim

¼ Uifqig ð20Þ

Through ui ¼ Uifqig, the corresponding generalized

mass matrix M f , stiffness matrix K f and inertial invar-

iants I f are obtained in terms of modal coordinates, q i.

Finally, Eq. (13) can be cast in the form

M € p pn o

þ G _ p pn o

þ D_ p p

n oþ K pf g ¼ F ð21Þ

where M , G , D, K and F are the mass matrix, skew-

symmetric gyroscopic matrix, modal damping matrix,

modal stiffness matrix and force matrix associated with

Ui (i ¼ 1; . . . ; n), respectively.

4. Ball-bearing waviness model

Within the HDD spindle system, ball bearings

are more complex components that are different from

other components such as the spindle and disks. They

are considered the inherent excitation source of HDD

spindle self-induced vibrations. A ball bearing generally

consists of four types of parts: an inner ring (also called

the inner raceway), an outer ring (the outer raceway), a

set of ball elements, and a cage as shown in Fig. 2. These

parts interact with each other in a dynamic way. Theload distribution within the bearing varies cyclically and

results in the bearing deflections. From a kinematics

point of view [16], the fundamental bearing frequencies

resulting from ball-bearing defects can be determined by

the following equations, for cage frequency, inner and

outer raceway frequencies to cage, and ball spin fre-

quency, respectively.

f 1 ¼ x

2 1

d b

D cos h

ð22Þ

f 2 ¼

x

2 1

d b

D cos h

ð23

Þ

f 3 ¼ x

2 1

þ

d b

D cos h

ð24Þ

f 4 ¼ D x

2d b1

d 2b D2

cos2 h

ð25Þ

Fig. 2. Structure of ball bearing.




where D is pitch diameter; d b is the ball diameter, x is

the rotational frequency of the spindle, h is the bearing

contact angle. These equations assume a single defect,

rolling contact, and a rotating outer raceway with the

fixed inner raceway.

When the outer raceway is rotating (defect index ¼

1), the bearing defect frequencies are given as follows,for outer raceway, inner raceway, and ball diameter

wavinesses, respectively.

f bo ¼ Zf 2 x ð26Þ

f bi ¼ Zf 3 ð27Þ

f bb ¼ 2 f 4 f 3 ð28Þ

where Z is the number of ball elements. These ball-

bearing defect frequencies are linear with respect to the

spindle speed x. They are transmitted well to the outer

raceway or the spindle hub and cause deflections at thesedefect frequencies. For illustrative purposes, the fol-

lowing sinusoidal waviness model is utilized to approx-

imate the outer raceway:

r o ¼ Q sinð2p f it Þ ð29Þ

and

€r r o ¼ Qð2p f iÞ2

sinð2p f it Þ ð30Þ

where frequency f i is the rate at which balls pass over a

complete wave cycle; r o and €r r o are the radius of the outer

raceway and its acceleration in radial direction; Q is

the estimated waviness amplitude. These simplifying as-

sumptions are sufficient to demonstrate the importance

of relatively small irregularities on ball-bearing raceway.

Thus, the force F b ¼ m€r r o from the ball-bearing defectfrequencies can be added into Eq. (21) as the inherent

excitation source, producing time-varying forces that

cause mechanical vibrations of HDD spindle systems.

M € p pn o

þ G _ p pn o

þ D_ p p

n oþ K pf g ¼ F þ F b ð31Þ

5. Solution method

Eq. (31) represents a sparse and uncoupled set of

mixed, nonlinear differential-algebraic equations (DAE)

governing the behavior of the HDD spindle system. It is

time-dependent. An implicit numerical integration with

backward differentiation formulas (BDF) [17,18] is used

to solve the DAE. In this algorithm, the equations

are linearized by employing the GEAR variable-order,

variable-step stiff integration algorithm [19] and effec-

tively solved using a predict-corrector process [18]. The

whole solution process of the proposed CMS-based

approach for HDD spindle system dynamics is depicted

in Fig. 3.

Fig. 3. Solution process of CMS-based HDD spindle system dynamics.




6. Numerical examples

To demonstrate the proposed approach, dynamic

responses of two typical HDD spindle system models

supported by ball bearings were presented as examples.

In the first example, the ball-bearing defects are not

considered as the excitation sources. In the second ex-ample, effect of the ball-bearing defect frequencies on the

HDD spindle system is investigated. Each HDD spindle

system model includes: (1) a few identical 3.500 flexible

disks (with inner radius Ri, outer radius Ro, and thick-

ness T ); (2) a spindle motor (including one rigid rotating

hub and one rigid stationary stator and shaft); (3) a pair

of ball bearings; and (4) a rigid base plate. The hub and

shaft are connected by ball bearings that are modeled

using generic spring elements (stiffness coefficient k and

damping coefficient c). The inner rims of the identical

disks are clamped on the hub. The stator and shaft are

mounted on the base plate. The base plate is fixed onground. Fig. 4 describes the schematic model of ball-

bearing HDD spindle system. Fig. 5 shows its mathe-

matical model with the ball-bearing modeling. Table 1

lists the material and geometrical properties for the

HDD spindle system.

6.1. Example 1: without ball-bearing defect assumption

The four disks are attached to the hub. The finite

element model of every disk platter has a total of 1770shell elements. Based on the boundary conditions as-

sumed, the first 8 component natural frequencies and

modal shape vectors were obtained using ANSYS. They

include disk modes ð p ; qÞ for p ¼ 0 and q ¼ 0, 1, 2, 3.

The disk mode ð p ; qÞ implies that the disk has p nodal

circles and q diameters. A few selected modes are shown

in Fig. 6. These modal frequencies xi and modal shape

vectors U i together with the mass M f , stiffness K f , and

inertia invariants I f were then incorporated into the

HDD spindle system dynamics Eq. (21), to represent

motion of the four flexible disks. During dynamic sim-

ulation, the hub and disks were rotating about the hubself-axial axis with a constant speed x. The model was

excited at the outer rim of the top disk by an impact

force ( F z ¼ 0:1 N) within a short time. Finally, the in-

Fig. 4. Schematic model of HDD spindle system.




terior coordinate uIi

as shown in Eq. (20), that is, the

axial displacement responses at the outer rim of the top

disk during each time step, were approximately calcu-lated using the implicit numerical integration with BDF

under certain spindle rotating speeds. The responses

from t ¼ 0 s to t ¼ 0:1 s form a frequency response

function at the specific spindle speed.

Fig. 7 displays the spin-up waterfall plots of the

HDD spindle system for the top disk axial displacement

responses from 0 rpm to 11.7 krpm. In the figure, many

valleys at the natural frequencies were observed. From

the results, it can be noticed that most significant cou-

pled vibration frequencies and modes can be effectively

identified using the proposed approach. A pair of thepeaks (A1, A2) denotes the ð0; 1Þ unbalanced modes [7].

They are caused by four disk ð0; 1Þ modes (with the same

deformation direction) coupled with spindle rocking

mode. Peaks (B1, B2) represent the ð0; 0Þ unbalanced

and balanced mode frequencies, which are caused by

four disk ð0; 0Þ axisymmetric modes. Peaks (C1, C2)

show the ð0; 1Þ balanced mode frequencies that are

caused through combinations of four disk ð0; 1Þ modes

(with different deformation directions) coupled with

spindle rocking mode. Peaks (D1, D2) indicate the ð0; 2Þbackward mode and forward mode, respectively. They

are produced by disk ð0;

2Þ mode coupled with spindlerocking mode. Similarly, peaks (E1, E2) display the

ð0; 3Þ backward mode and forward mode. It is noted

that the HDD spindle vibration mode splits except ð0; 0Þincrease with the spindle motor spinning speed as

Fig. 6. Component natural frequencies and modal shapes for disk.

Table 1

Material and geometric properties for HDD spindle system

Disk E d ¼ 6:7 1010 kg/mm2, M d ¼ 1:353 102 kg, T ¼ 0:8 mm

qd ¼ 2:7 106 kg/mm3, R i ¼ 17:0 mm

md ¼ 0:3, Ro ¼ 47:0 mm

Hub I xh ¼ 2

:

38 kgmm2

, I oh ¼ 3

:

30 kgmm2

, M h ¼ 2:

29 102

kg

Stator and shaft I xs ¼ 1:95 kgmm2, I os ¼ 3:72 kgmm2, M s ¼ 9:70 103 kg

Ball bearings k x ¼ k y ¼ 3:0 104 N/mm, k z ¼ 2:5 104 N/mm, S ¼ 17:66 mm

c x ¼ c y ¼ 0:01 N s/mm, c z ¼ 0:02 Ns/mm

d b ¼ 1:584 mm, D ¼ 8:9 mm, h ¼ 23:65, Z ¼ 10

Fig. 5. Mathematical model for simulation.




shown. These predicated natural frequency-splitting

amounts due to the gyroscopic effects are about twice

rotating speed. These results agree reasonably well with

the analytical solutions [7], as shown in Fig. 8.

6.2. Example 2: with ball-bearing defect frequency

In the model, the HDD spindle system carries threerotating 3.500 disks. The excited forces from ball-bearing

defect frequencies, as given in the Eqs. (22)–(24), are

included in the HDD spindle dynamic Eq. (31). Other

assumptions are the same as those in the previous ex-

ample.

Fig. 9 displays the spin-up waterfall plots of the

spindle system from 6400 to 7800 rpm. The spindle

rocking modes, the disk modes and ball-bearing defect

frequencies can be identified. Peaks f d1 and f d3 denote

the unbalanced ð0; 1Þ mode frequencies of the spindle

disk assembly. Peak f d2 represents the disk axial mode

frequency. Peaks f d4 and f d5 are referred to the balanced

ð0; 1Þ mode frequencies of the spindle disk assembly. The

dashed lines indicate the ball-bearing defect frequencies.

For example, when the spindle is rotating at 6400

rpm, the ball-bearing defect frequencies are as follows:

(1) backward-going outer raceway waviness, 340 Hz;

(2) forward-going outer raceway waviness, 553 Hz; (3)

backward-going ball diameter waviness, 521 Hz; and (4)

forward-going ball diameter waviness, 645 Hz. From the

figure, the interactions between the spindle disk assem-

Fig. 8. Vibration predictions by the analytical method [7]

(Example 1).

Fig. 9. Numerical results for waterfall plot of HDD spindle

system (Example 2).

Fig. 7. Waterfall plot of HDD spindle system for axial displacement (Example 1).




bly vibration modes and ball-bearing defect frequencies

can be observed. When the ball-bearing frequencies

interact with the unbalanced ð0; 1Þ mode, a ‘‘distur-

bance’’ amplitude is being excited. However, when they

interact with the disk axial mode and the balanced ð0; 1Þmodes, the magnitudes of the peaks are changed a little.

Similar phenomena are observed in the experimental

results as shown in Fig. 10. This is a measurement of the

axial deflection of the outer rim of the top disk measured

with the laser Doppler vibrometer as the rotation speed

is increase from 6400 to 7800 rpm. The deflection output

of the laser Doppler vibrometer and the signal of thevibrator were fed into the dynamic signal analyzer, and

frequency response functions were calculated at 15-rpm

increments. It is noted that some spindle harmonics also

appear in the figure. These results show that the pro-

posed CMS-based approach is able to better predict

dynamic resonance interactions induced by ball-bearing

defect frequencies. This helps designers to estimate

HDD spindle safe operating window at the prototype

design stage. The safe operating window means that

when the spindle is operated within the window, the

spindle system vibration modes will not interact with the

ball-bearing defect frequencies. Thus, the resonance in-teractions can be avoided in operation. From the figures,

it is noticed that the HDD spindle safe operating win-

dow is from 6800 to 7600 rpm.

7. Conclusions

The dynamic responses of the two typical HDD

spindle system using the proposed CMS-based approach

were studied and verified experimentally. The results

indicate that the CMS-based approach incorporating

multi-body system dynamics is directly applicable to

dynamic analysis of high-speed HDD spindle systems

and is able to better predict the spindle safe operating

window. The various HDD spindle vibration frequen-

cies and their resonance interactions can be effectively

and accurately identified using the approach. In addi-

tion, since the component mode synthesis technique was

used to reduce the size of the dynamic equations for the

overall system, the computational efficiency is improved

in solving the HDD spindle system dynamic equations.

Also, this approach is expected to have more generic

applicability to the HDD spindle system with more

complex material properties in future. Apparently, thisstudy provides an alternative generic simulation ap-

proach for investigating the dynamic characteristics of

HDD spindle system during high rotating speed condi-

tions. It can be utilized to systematically evaluate the

effects of various parameters (e.g. damping coefficients,

mass imbalance, flexible disk modes and bearing prop-

erties) on dynamic performance of HDD spindle system.

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