This paper describes an accurate mathematical model that can predict forced vibration of a rotating spindle system with a flexible stationary part. In particular, we demonstrate this new formulation on a hard disk drive (HDD) spindle to predict its position error signal (PES). This improved method is a nontrivial extension of the mathematical model by Shen and his fellow researchers, as the improved method allows the flexible stationary part to comprise multiple substructures. When applied to HDD vibration, the improved model consists not only a rotating hub, multiple rotating disks, a stationary base, and bearings (as in Shen’s model) but also an independent flexible carriage part. Moreover, the carriage part is connected to the stationary base with pivot bearings and to the disks with air bearings at the head sliders mounted on the far end of the carriage. To build the improved mathematical model, we use finite element analysis (FEA) to model the complicated geometry of the rotating hub, the stationary base and the flexible carriage. With the mode shapes, natural frequencies, and modal damping ratios obtained from FEA, we use the principle of virtual work and component-mode synthesis to derive an equation of motion. Naturally, the stiffness and damping matrices of the equation of motion depend on properties of the pivot and air bearings as well as the natural frequencies and mode shapes of the flexible base, the flexible carriage, the hub, and the disks. Under this formulation, we define PES resulting from spindle vibration as the product of the relative displacement between the head element and the disk surface and the error rejection transfer function. To verify the improved model, we measured the frequency response functions using impact hammer tests for a real HDD that had a fluid-dynamic bearing spindle, two disks, and three heads. The experimental results agreed very well with the simulation results not only in natural frequencies but also in gain and phase.

1.
Satomitsu
,
I.
, and
Hidehiko
,
N.
, 1999, “
Estimation Method of Head-Positioning Error Due to Disk Flutter From the Flutter Amplitude
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
0387-5024,
65
(
636
), pp.
3075
3081
.
2.
Mote
,
C. D.
, Jr.
, 1970, “
Stability of Circular Plates Subjected to Moving Loads
,”
J. Franklin Inst.
0016-0032,
290
(
4
), pp.
329
344
.
3.
Iwan
,
W. D.
, and
Stahl
,
K. J.
, 1973, “
The Response of an Elastic Disk With a Moving Mass System
,”
ASME J. Appl. Mech.
0021-8936,
40
, pp.
445
451
.
4.
Chen
,
J.-S.
, and
Bogy
,
D. B.
, 1992, “
Effects of Load Parameters on the Natural Frequencies and Stability of a Flexible Spinning Disk With a Stationary Load System
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
S231
S235
.
5.
Shen
,
I. Y.
, and
Ku
,
C.-P. R.
, 1997, “
A Nonclassical Vibration Analysis of a Multiple Rotating Disk and Spindle Assembly
,”
ASME J. Appl. Mech.
0021-8936,
64
, pp.
165
174
.
6.
Lee
,
C.-W.
, and
Chun
,
S.-B.
, 1998, “
Vibration Analysis of a Rotor With Multiple Flexible Disks Using Assumed Modes Method
,”
ASME J. Vibr. Acoust.
0739-3717,
120
, pp.
87
94
.
7.
Jia
,
H. S.
, 1999, “
On the Bending Coupled Natural Frequencies of a Spinning, Multispan Timoshenko Shaft Carrying Elastic Disks
,”
J. Sound Vib.
0022-460X,
221
, pp.
623
649
.
8.
Shen
,
J.-Y.
,
Tseng
,
C.-W.
, and
Shen
,
I. Y.
, 2004, “
Vibration of Rotating Disk/Spindle Systems With Flexible Housing/Stator Assemblies
,”
J. Sound Vib.
0022-460X,
271
, pp.
725
756
.
9.
Tseng
,
C.-W.
,
Shen
,
J.-Y.
, and
Shen
,
I. Y.
, 2003, “
Vibration of Rotating-Shaft HDD Spindle Motors With Flexible Stationary Parts
,”
IEEE Trans. Magn.
0018-9464,
39
(
2
), pp.
794
799
.
10.
Tseng
,
C. W.
, 2002, “
Vibration of Rotating-Shaft Design Spindles With Flexible Bases
,” Ph.D. dissertation, Univ. Washington.
11.
Hou
,
S. N.
, 1969, “
Review of Modal Synthesis Technique and a New Approach
,”
Shock Vib. Bull.
,
40
(
4
), Naval Research Laboratory, Washington, D.C., pp.
25
39
.
12.
Craig
,
R. R.
, Jr.
, 1987, “
A Review of Time-Domain and Frequency-Domain Component Mode Synthesis Methods
,”
Int. J. Anal. Exp. Modal Anal.
0886-9367,
2
, pp.
6
72
.
13.
Edwards
,
J. R.
, 1999, “
Finite Element Analysis of the Shock Response and Head Slap Behavior of a Hard Disk Drive
,”
IEEE Trans. Magn.
0018-9464,
35
(
2
), pp.
863
867
.