DYNAMIC MODELING OF A HARD DISK DRIVE
ACTUATOR USING SUBCOMPONENT FINITE
ELEMENT MODELS AND MODAL
SUPERPOSITION
By
JEFFREY DON ANDRESS
Bachelor of Science
Oklahoma State University
Stillwater, Oklahoma
1993
Submitted to the Faculty of the
Graduate School of the
Oklahoma State University
in partial fulfillment
ofthe requirements for
the Degree of
MASTER OF SCIENCE
May, 2000
DYNAMIC MODELING OF A HARD DISK DRIVE
ACTUATOR USING SUBCOMPONENT FINITE
ELEMENT MODELS AND MODAL
SUPERPOSITION
Thesis Approved:
ii

PREFACE
The purpose ofthis study was to develop a methodology whereby complex
structures, such as hard disk drive head stack assemblies, can be modeled accurately in a
compact form. The head stack assembly was divided into three subcomponents, the
yoke/coil, the actuator arm, and the suspension. Finite element models of each subcomponent
were created to provide the natural frequencies and corresponding mode
shapes ofthe individual subcomponents. Coupling terms were then derived that
describe how the subcomponents interact to form the overall system dynamics. The
resulting subcomponent fInite element analytical model was a 15 degree of freedom
model that could quickly be solved using commercially available matrix manipulation
software. The model proved to be accurate in predicting the offtrack motion ofthe head
stack assembly and helped provide understanding as to which resonances are the most
detrimental to drive performance. Initial simulations showed that some of the boundary
conditions and assumptions used in creating the subcomponent finite element models
were incorrect. However, by comparing the model results to measured data, the subcomponent
fInite element analytical model provided direction to improve the accuracy of
the subcomponent fInite element models.
I would like to thank Mr. Roy Wood and Mr. John Stricklin who provided
valuable insight and assistance study. Without their knowledge and advice, this research
would not have been possible.
jii
Chapter
TABLE OF CONTENTS
Page
1 Introduction 1
2 Background 8
3 Model Development 12
3.1 Background Theory 12
3.2 HSA Model Development 20
3.2.1 Development of the Mass and Stiffness Matrices
of the UndaIllped System 22
3.2.2 Development of the Force Input Vector 26
3.2.3 HSA Modeling Assumptions 30
3.2.4 HSA Model Discussion 32
3.2.5 State Space Representation of the HSA Model 35
4 FiniteElement Models 37
4.] FuJI HSA FiniteElement Model 37
4.2 SubComponent FE Models 39
4.2.1 Suspension FE Model 40
4.2.2 Actuator Arm FE Model 42
4.2.3 Yoke and Coil FE Model 44
4.3 Experimental Verification of the FE Models 46
5 HSA Model Simulation Results and Discussion 48
5.1 Reduced Order Model Simulation 48
5.2 FuJI Order Model Simulation 54
6 Conclusions and Future Work 61
6.1 Conclusions 61
6.2 Future Work 62
REFERENCES 64
APPENDIX A 66
iv

Chapter Page
APPENDIX B 69
APPENDIX C 72
APPENDIX D 75
v

LIST OF TABLES
Table Page
Table 41: Suspension FE Model vs. Measured Natural Frequencies 42
Table 42: Actuator Arm FE Model vs. Measured Natural Frequencies 44
Table 43: Yoke/Coil FE Model vs. Measured Natural Frequencies 45
vi
Figures
LIST OF FIGURES
Page
Figure 11: Head Stack Assembly (HSA) 3
Figure 12: Block Diagram of Control System 4
Figure 31: HSA Model 21
Figure 32: Forces due to Current Input to the Coil 28
Figure 41: Full HSA FE Model 37
Figure 42: HSA FE Model vs. Measured
Mechanical Transfer Function 39
Figure 43: Suspension SubComponent FE Model 41
Figure 44: Actuator Arm SubComponent FE Model 42
Figure 45: Yoke/Coil SubComponent FE Model 44
Figure 51: Rigid Body State Space Model
vs. Measured Frequency Response 49
Figure 52: Translational and Rotational Components
of the State Space Model 50
Figure 53: Corrected State Space Model
vs. Measured Frequency Response 51
Figure 54: NonCollocated Transfer Function
vs. Measured Frequency Response 53
Figure 55: Initial Simulation ofthe SubComponent
FE Analytical Model 55
Figure 56: SubComponent FE Model Simulation with
Corrected Sway Mode Coupling Factors 56
vii

Figures Page
Figure 57: Original Suspension 3rd Bending Mode 58
Figure 58: Modified Suspension 3rd Bending Mode 58
Figure 59: Modified SubComponent FE Analytical
Model Simulation 59
Figure AI: Suspension 1st Bending 67
Figure A2: Suspension 1st Torsion 67
Figure A3: Suspension 2nd Bending 67
Figure A4: Suspension 2nd Torsion 67
Figure A5: Suspension 3rd Bending 68
Figure A6: Suspension Sway 68
Figure B1: Ann 1st Bending 70
Figure B2: Ann Sway 70
Figure B3: Ann 2nd Bending 70
Figure B4: .Arm 1st Torsion 71
Figure CI: Yoke/Coil 1st Bending 73
Figure C2: Yoke/Coil 1st Torsion 73
Figure C3: Yoke/Coil Sway 74
viii
{A}
[A]
[B]
[C]
[Cdamp]
[D]
{F}
r
J
K
[K]
Kadjustmen,
Kbearlng
KBW
Ke
KTrans
NOMENCLATURE
Nonnalized mode shape vector
State space system matrix
State space input matrix
State space output matrix
Damping matrix
State space direct transmission matrix
Input force vector
Modal participation factor
Current (Amp)
Mass moment of inertia ofthe rigid actuator body (lbfins2
)
Total mass moment of inertia ofthe HSA about the pivot center
(lbfms2
)
Subcomponent fInite element model element stiffness
Stiffness Matrix
Adjusment gain required to correct the error in the state space model
Pivot bearing stiffness (lbf/in)
Gain factor associated with the drive electronics
DC gain of the rotational component transfer function
Voice coil motor torque constant (ozin/Amp)
DC gain of the translational component transfer function
ix

I Moment ann from pivot center to lumped nodal mass (in.)
[M] Mass matrix
m Subcomponent fInite element model lumped nodal mass
mB Mass ofthe rigid actuator body (lbfs2/in)
Mt! Effective mass
mk Modal stiffness
mm Modal mass
msr Total mass ofthe suspension subcomponent FE model (lbfs2/in)
MT Total mass ofthe HSA (lbfs2/in)
mT Total mass of the subcomponent FE model (lbfs2/in)
{ ¢} Mode shape vector normalized by the square root of the modal mass
q Modal coordinate
Rc Moment arm from pivot center to active length center (in.)
Rh Moment arm from pivot center to slider (in.)
rmel Rotational to modal coupling factor
co Natual frequency (Rad/s)
{y} State variable vector
~ Damping ratio
x

1 Introduction
As society continues to demand accurate and timely infonnation, the need to
quickly and reliably store and retrieve data is ever increasing. To date, rigid, magnetic
disk drives have become the primary data storage devices in today's computer systems.
Disk drives are now used in desktop computers, workstations, and servers as well as
portable devices such as notebook computers and digital cameras. As the applications in
which disk drives are used continue to expand, disk drive users constantly demand
increased storage capacity and improved drive perfonnance while insisting that drive
manufacturers maintain or decrease the physical size of the drive. To keep pace with the
demand for storage capacity, the current trend in the disk drive industry is a doubling of
capacity each year.
For drive dimensions to remain unchanged or even decrease, drive manufactures
must be able to write more data on a given surface to meet the growing capacity demand.
Increasing the storage capacity ofa drive therefore requires increasing the bits per square
inch (areal density) that are written on a disk surface. Areal density is the product ofbits
per inch (BPI) and tracks per inch (TPI). BPI is defined as the number of bits that can be
written along an inch of data track in the circumferential direction, while TPI is defined
as the number ofdata tracks that can be written on an inch ofthe disk surface in the
radial direction. The width of a data track is the reciprocal of TPI. For example a 100
microinch wide data track corresponds to 10,000 TPI. At 10,000 TPI approximately 40

data tracks could be written on the edge of a standard piece ofwhite paper (standard
white paper is approximately 0.004 inches thick). While BPI is limited mainly by the fly
height of the read/write head above the disk surface, TPI is limited by the track
following servo system. It is becoming increasing hard for drive manufactures to
increase the BPI, such that increased track densities are accounting for a larger
percentage of the annual areal density growth rate.
The limitations ofthe servo system are defined by the flexibility ofthe mechanical
components as well as the controller design. The flexibility of the mechanical
components can adversely affect drive performance in two ways. First, as TPI increases,
drives become increasingly susceptible to mechanical resonances that cause the
read/write head to move offtrack, since the displacement of the head becomes an
increasing percentage to the total track width. Second, to compensate for the TPI
increase, the drive bandwidth must increase to maintain the drives ability to track follow
accurately in the presence of external shock and vibration disturbances. However, as the
drive bandwidth increases, the stability of the servo loop is threatened due to the
mechanical resonances of the drive, since increasing the bandwidth of the drive can
decrease the gain margin of mechanical resonances. Resonances that were previously
ignored due to insufficient amplitude to be of interest can become stability issues if they
are not raised in frequency or reduced in amplitude.
One way to categorize drive resonances is by how they are excited and what
performance specifications they affect. Using this criterion, drive resonances can be
split into two categories, intheIoop and outoftheIoop. IntheIoop resonances are
resonances that modify the open loop or structural response and are excited by control
2

inputs to the actuator coiL IntheIoop resonances are typically head stack assembly
Actuator Body
Actuator Arm
Suspension
Slider
Figure 11: Head Stack Assembly (HSA)
(HSA) resonances, which include actuator ann, suspension, pivot bearing, and yoke/coil
resonances. Figure 11 shows a typical HSA. OutoftheIoop resonances are all other
resonances that do not directly modify the open loop or structural response since they are
not directly excited by the servo loop and the dynamic positioning of the HSA. Outofthe
loop resonances include disk pack, basedeck and topcover resonances that can be
excited by windage forces generated by the rotating disks, external shock and vibration
excitations, and spindle motor bearing defects. These types ofresonances can be
considered error excitations to the system.
As TPI continue to increase to meet the growing capacity demand and servo
bandwidths increase to help maintain drive performance margins, each generation of
drive becomes increasingly susceptible to drive resonances. IntheIoop HSA
resonances are the cause of significant servo stability and performance problems since
3

they modify the open loop response. The open loop response is a combination ofthe
servo system and mechanical system transfer functions as shown in Figure 12. The
mechanical
Noise
Head
Position
G
(Mechanics)
Coil
r, Current
H
(Compensator)
Position
Error 1
Position
Reference'
Head Position
Open Loop = GH = Position Error
Structw'al = G = Hea.d Position
COli Current
Coil Current
Compensator = H = P .. E
OSItIon ITor
Figure 12: Block Diagram of Control System
transfer function ofthe HSA relates coil current input to displacement of the slider. Any
significant dcfonnation of the HSA due to resonances ofthe yoke/coil, anns, suspension,
and pivot bearings can therefore adversely modify the open loop response. The ability to
accurately model the HSA in order to predict the effects ofHSA resonances on drive
perfonnance would be a valuable tool to both the servo and mechanical drive design
engineers. Such a model would give servo engineers the ability to design track
following loops to compensate for the mechanical resonances and would give
mechanical engineers the ability to optimize the actuator design to reduce the impact of
HSA resonances on the servo system in advance of a working drive.
4

The HSA is a complex structure made up of several subcomponents (yoke~ coil,
pivot bearing, actuator body, actuator arms, and suspensions) as shown in Figme 11.
Typically, the dynamic behavior of complex structures such as HSAs can be modeled
effectively using finite element (FE) models. In order to accurately predict the offtrack
displacement ofthe slider due to coil current input~ the entire HSA (all subcomponents)
must be modeled and correct boundary conditions must be applied. As a result, the full
HSA FE model can become very large, on the order of 120,000 degreesoffreedom
(DOF). FE models ofthis magnitude take numerous hours, even days, to develop and
solve, and can become cumbersome to work with. The goal of a HSA FE model is not
just to predict natural frequencies and mode shapes ofHSA resonances, but to provide a
tool whereby the actuator design can be optimized to meet performance specifications
and reduce the impact of resonances on drive performance. However, if a full HSA FE
model is used, the model development time, solve time, and size are not conducive to the
numerous iterations that might be necessary to arrive at an optimal design, thereby,
reducing the usefulness of the full FE model as a design tool.
Due to the complexity and size of the full HSA FE model, a smaller, more compact
alternative would be preferred. The goal of providing this alternative without the size
and complexity ofthe FE model would most likely be achieved through the development
of an analytical model. As opposed to the thousands ofDOF associated with the full
HSA FE model, the DOF of an analytical model would be limited to the number of
modeled resonances. There are numerous mechanical resonances ofthe HSA, however,
only resonances that cause offtrack displacements of the slider are of concern. Thus,
the analytical model could be reduced to approximately 15 DOF which would include
5

rigid body, yoke/coil, ann, suspension, and pivot bearing modes. Analytical models can
be expressed in a statespace fonn, which, can easily be solved by matrix operation
software such as Matlab in seconds as opposed to the hours required to solve the FE
model. Because ofthe compact size of the analytical model (15 OOF compared to
120,000 DOF for the analytical and FE models respectively), the analytical model can be
used to quickly and efficiently optimize the actuator design to provide improved HSA
dynamics. However, when analytical models are developed, the mechanical structure
being modeled is generally simplified such that the results of the model may not be
accurate or may not include higher order modes that are of interest. Since the HSA is a
complex structure, the assumptions and simplifications needed to create an analytical
model limit the accuracy ofthe model to approximately 6 kHz. The TPI, drive
bandwidths, samplerates, and system dynamics oftoday's drives require that the HSA
model be able to accurately predict the offtrack performance up to 15 kHz. There are
numerous HSA resonances above 6 kHz that significantly affect drive perfonnance and
servo stability that are difficult to model accurately by means ofan analytical model.
The disadvantages of both the FE and analytical HSA models limit the usefulness
of both methods in the prediction ofHSA dynamics on drive performance. The
limitations of these modeling methods also prevent both the FE and analytical models
from being useful design tools in the optimization of the actuator to meet performance
specifications and reduce the impact of drive resonances on servo stability. Therefore,
the need exists for a model that can accurately predict the dynamics of complex
structures yet provide the model in a compact form that can be quickly and easily solved.
The complexity of the HSA, however, requires the use of FE models to accurately
6

predict natural frequencies and mode shapes ofthe structure. The size, complexity,
development time, and solve time ofthe FE model can be reduced ifsubcomponents,
instead ofthe entire HSA, are modeled. If boundary conditions are modeled correctly,
subcomponent FE models of the HSA can provide the natural frequencies and mode
shapes with the same accuracy of the full HSA model. The modal information obtained
from the subcomponent FE model can be combined into a compact analytical model by
using mode summation techniques and by deriving the coupling equations that describe
how the subcomponents interact. The resultant model is a compact model (15 DOF like
the analytical model) that can accurately describe the complex dynamics of the HSA.
The coupling of the subcomponents describe how the flexibilities ofthe system interact.
The subcomponent FE analytical model can therefore provide directions for improving
the mechanics to reduce the resonance impact on drive performance.
The objective of the research described in this thesis is to develop the methodolgy
whereby complex structures, such as a hard disk drive HSA, can be modeled accurately
in a compact fonn. This study will detail the development of a compact HSA model that
can predict the dynamic perfonnance ofthe drive while track following. The details of
modeling and combining the servo system and HSA mechanical models are not
addressed in this paper since there are several papers in the open literature concerning
this subject matter.
7

2 Background
Little has been found in published literature concerning model development of
disk drive head stack assemblies. Currently, the majority of this research is being
performed by individual drive design companies or is funded by the disk drive industry
and, therefore, remains proprietary. However, there have been several papers published
recently that develop models to predict the track following and disturbance rejection
capabilities ofthe servo system. These papers develop and discuss methods for creating
full HSA models, as the mechanical resonances of the HSA significantly impact drive
performance and servo stability.
Radwan and Whaley [13] investigated how the mechanical flexibilities of a disk
drive interact with the servo system and were also able to demonstrate the capability to
predict both an open loop frequency response and the offtrack performance of the HSA
due to operating vibration excitations. TIrrough this research, Radwan and Whaley
developed a full FE model, which can be utilized to calculate the mechanical frequency
response ofthe HSA. Using this full FE model, a modal analysis was performed and
then the corresponding frequency response function due to a hannonic force input to the
coil was calculated. The plot ofthe frequency response was then copied to an ASCII file
and imported to MathCAD where a transfer function of the servo system was added,
resulting in the open loop transfer function ofthe system. While good correlation
between measured and modeled data was achieved, the process of obtaining
8
the mechanical transfer function was difficult. The data that was imported to MathCAD
was not a model ofthe HSA but was simply the magnitude and phase versus frequency
data points obtained from the FE hannonic analysis. Thus, if any structural
modifications were to be considered, the entire process starting with the full FE model of
the HSA would have to be repeated.
In Radwan et al. [11], the use ofFE models to predict a drive's track following
capability was further refined. Instead of simply exporting a transfer function graph
representing the HSA mechanical response, the full FE model was reduced to a state
space representation ofthe HSA. The statespace HSA model was then combined with a
model of the servo system and MATLAB was used to solve for the open loop response.
The state space HSA model was created by frrst performing a modal analysis on the full
FE model. Nodal displacements at both the excitation and response nodes (a node on the
coil and a node on the slider respectively) from each mode shape were extracted from
the FE solution. The modal mass and natural frequency for each mode was also
extracted. Each mode was then modeled as a single OOF system with the total system
response the superposition of the individual modes. Not all modes were used in the state
space model. Only modes that had significant contribution in the offtrack direction
were used in the modal sum. The benefits of this second model are that Radwan et al.
developed a method to reduce a detailed FE model of the HSA to a set of state space
equations. They also demonstrated that the offtrack response of the FE model can be
approximated by the superposition ofa reduced number ofmodes. However, like the
first work, the model was ultimately reliant on a full detailed FE model of the HSA.
9

While the two previous models discussed above dealt with the use ofdetailed FE
HSA models, a few examples of analytical HSA models can be found in the literature.
As previously mentioned, the complex nature of the HSA requires that the mechanical
structure be simplified in order to develop an analytical model. The simplifications
needed to create an analytical HSA model restrict the model's usefulness. While
assumptions and approximations can be made that allow first order modes to be
predicted, higher order modes of the system would be in error. Thus the use of
analytical models is limited. Radwan et al. [12] provides an example of a simplified
analytical model in order to predict track following performance under external shock
and vibration excitations. However, only the rigid body dynamics of the actuator were
modeled and mechanical resonances were not considered. Because mechanical
resonances were not considered, the frequency response of the model is valid for only a
few hundred hertz. The model presented by Radwan et al. is therefore useful only for
the investigation of the low frequency response of the actuator to external excitations.
The HSA model presented by Ono and Teramoto [10] expands the useful
frequency range ofthe model presented by Radwan et al. by including the flexibility of
the pivot bearings. The model presented by Ono and Teramoto was developed to
provide understanding of the interaction between the rigid body rotational mode ofthe
actuator and the flexible mode of the pivot bearings. While the model developed was
adequate for their study, the Ono and Teramoto model did not include flexible modes of
the yoke/coil, actuator anTIS, or suspensions. These modes must be included if the model
is to be used to understand the dynamic effects of the HSA on the servo system.
10

Aruga et al. [1] modeled the HSA as a 3 DOF system, including the rigid body
rotational mode, a pivot bearing mode, and one ann mode. Their study dealt with the
design of a new concept actuator meant to reduce the offtrack impacts of the
fundamental pivotbearing mode. The Aruga et al. model is still too simplified for
today's disk drives and servo systems. There are several detrimental suspension, ann,
and yoke modes that must be accounted for in order to predict drive perfonnance and
servo stability.
The models and methods detailed in the open literature are either too complicated
(full FE HSA models) or too simplified (analytical models), resulting in limited use in
the prediction ofHSA dynamics on drive perfonnance. The limitations of these
modeling methods also prevent the models from serving as useful design tools for drive
design engineers, who must optimize the actuator to meet performance goals while
reducing the impact ofresonances on servo stability. This study will detail the
development of a compact model that can be used to predict the dynamic performance of
complex structures such as HSAs.
11

3 Model Development
3.1 Background Theory
A multidegreeoffreedom system can be described by a set of n simultaneous
second order differential equations. In matrix fonn, the equations are expressed as
(31 )
The general solution to equation (31) is the stun of the complementary function and the
particular integral. The complementary function satisfies the homogeneous differential
equation (right side of the equation equals zero) which physically corresponds to the free
vibration problem ( {F(t)} = 0). Under free vibration, the system is not subjected to any
external excitation and its motion is governed only by the initial conditions. When the
excitation source is hannonic ( {F(t)} = {FoSincd} ), the solution to the particular
integral is the steady state oscillation ofthe system at the same frequency <0 as the
excitation. Seldom is it necessary to determine the motion of a system under conditions
of free vibration. However, the analysis of a system in free vibration provides two
important dynamic properties of the system: the natural frequencies of the system and
the corresponding mode shapes. For a multidegreeoffreedom system with n degreesof
freedom (DOF), there are n natural frequencies and mode shapes. Finite element
modeling software is often used to calculate the natural frequencies and mode shapes for
complicated systems with many degreesoffreedom.
12

If the effects ofdamping do not influence the natmal frequencies ofthe system,
when solving equation (31) for the free vibration case, the damping tenn is usually
omitted. For the undamped free vibration problem, equation (31) reduces to
(32 )
For free vibrations ofthe undamped structure, {y} is of the fonn
Yi = Ai sin(ax +VI) i = 1,2, ... , n
or in vector notation
{y} = {A }sin(M +VI) (33 )
where Ai is the displacement amplitude of the ith coordinate and n is the number of DOF.
Substituting equation (33) into equation (32) gives
(34 )
which is a set of n homogeneous linear algebraic equations in {A} with n unknown
displacements Ai and an unknown parameter aI. The nontrivial solution to equation
(34) requires that the detenrunant ofthe coefficient matrix ([[K]at[M]]) equal zero.
Equating the determinant to zero gives the characteristic equation
(35 )
from which the natural frequencies of the system are found. In general, equation (35)
results in a polynomial equation ofdegree n in aI which is satisfied by n values of at,
where
Wi i = 1, 2, ... , n
13

are the natural frequencies of the system and n is the number of DOF. Since equation
(34) is a homogeneous system of linear equations and the determinant of the coefficient
matrix is zero, the equations are linearly dependent such that there are an infinite number
of solutions for {A}. Thus, for each natural frequency (U; which satisfies equation (35),
there is not a unique solution for the corresponding displacement amplitude vector {A}.
Typically, one of the displacement amplitudes in the amplitude vector {A} is assigned a
unit value, although any number is sufficient. By substituting each of the natural
frequencies into equation (34), the corresponding amplitude vector {A} can be
calculated relative to the arbitrarily chosen element. These ratio values are known as the
mode shape or modal vector for a particular natural frequency. The mode shape is often
referred to as a normal mode since the elements in the vector {A} have been normalized
by an arbitrary value. Since there are n natural frequencies, there will be n
corresponding normalized mode shape vectors..
The normal modes may be conveniently arranged in the columns of a matrix
known as the modal matrix where each column represents a mode shape associated with
a particular natural frequency. For the general case of an n DOF system, the modal
matrix is written as
or
All AI2
[A]= ~I ~2 (36 )
14

For an n OOF system, each natural frequency has its own mode shape and
behaves essentially as a single DOF system. The total motion of the system, the solution
to equation (32), is the superposition of the normal modes ofthe system.
where the C's and VS are arbitrary constants of integration that are detennined by the
initial conditions ofthe system.
An interesting property of the nonnal modes of a system is that they are
orthogonal with respect to the mass and stiffness matrices. The orthogonal nature ofthe
normal modes is defined as follows
{A}~[MRAL = 0 for i *j
{Ar[KHAL =0 for i *j
and
{A};[M]{A}, =mm, i = 1,2, ... , n
{A};[KHA}, =mk, i = 1,2, ... , n
(38 )
(39 )
(310 )
(311 )
where mmj and mk, are defined as the modal mass and modal stiffness ofthe ith mode.
The resulting modal mass and modal stiffness matrices are diagonal.
As previously mentioned, the amplitudes of vibration for normal modes are only
relative values which may be scaled or normalized to an arbitrary value. It is often
convenient to nonnalize the mode shapes by the square root of the modal mass.
15
or
~.
'" _ U
'f'ij  r:::::::
"mm,
(312 )

where tAj is the modal mass normalized jth component of the jth modal vector. The modal
matrix can be rewritten as
¢JII ¢12
[¢J] = ¢J21 ¢22 (313 )
Substituting {¢} i for {A} i in equation (38) through (311) results in
{¢J}~[M]{¢L =0 for i :;It j (314 )
{¢J}~[K]{¢L =0 for i :;It j (315 )
{¢J};[MM¢Jt =1 i = 1,2, ... , n (316 )
{¢J}~ [K M¢J t =OJ/2 i = 1,2, ... , n (317 )
The advantage of this normalization method will be shown below in equation (323).
The equations ofmotion represented in equation (32) are generally coupled
through the mass and/or the stiffness matrices. Dynamic coupling exists if the mass
matrix is nondiagonal where as static coupling exists if the stiffness matrix is nondiagonal.
If the equations are uncoupled by the proper choice of coordinates, each
equation can be solved independently of the others and each mode can then be examined
as an independent singleDOF system. Although it is always possible to decouple the
equations of motion for the undamped system, it is not always possible to decouple
damped systems. Due to the orthogonal properties of the modal matrix, the modal
matrix, equation (36) or equation (313), can be used to decouple the mass and
stiffness matrices in equation (32). To decouple the mass and stiffness matrices, a
modal coordinate q, is defmed such that
16

{y}= I{;}/qi
iI
Substituting equation (318) into equation (32) and premultiplying by [¢.IT gives
(318 )
(319 )
However, from equations (38), (39), (316) and (317), equation (319) reduces to a set
of n decoupled equations of the fonn
(320 )
So far, damping tenns have been ignored. As previously mentioned, when
damping is considered, it is not always possible to decouple the equations ofmotion. If,
however, the damping matrix [C] is proportional to the mass [M] and/or stiffness [K]
matrix, the following hold true
(321 )
and
(322 )
so that equation (320) becomes
(323 )
In equation (323), each mode is expressed as a singleDOF system where the
total system response is the superposition of the contributing modes (equation (318».
The advantage of equation (323) is that an nDOF system can be represented in tenns of
the modal parameters of the system without the knowledge of the mass and stiffness
17

matrices. As previously mentioned, the modal parameters, such as the natural
frequencies and mode shapes, of complex systems can be determined from FE models.
Equations (323) and (318) allow a FE model of a complex system to be reduced to a
set of n singleDOF equations, where n is the nwnber of flexible modes considered, by
first solving the FE model for the natural frequencies and mode shapes of the system.
The mass and stiffness matrices generated by the FE software can be extremely large
(the mass and stiffness matrices will have as many rows and colwnns as there are DOF
of the system), but are not needed to calculate the system response.
For lightly damped systems, such as HSAs, the damping terms in equation (323)
serve mainly to limit the amplitude response at resonance. Appropriate damping terms
for each mode can be determined experimentally by direct measurement or can be
estimated by matching the resonant amplitudes from modeled and measured frequency
response functions. For the HSA model that is being developed, the damping in the
system is predominately due to structural damping such that ~ is generally asswned to be
less than one percent.
As in free vibration, it is also possible to express the response of a system to
forced vibration as the superposition of the nonnal modes. When the normal modes are
used to decouple the set of system equations, the modal superposition method reduces
the problem of finding the response of a multiDOF system to the determination ofthe
response of n singleDOF systems. In forced vibration, the equations ofmotion are decoupled
by first solving for the natural frequencies (l4) and nonnalized mode shapes (¢)
from the free vibration case. For the forced vibration condition, equation (323)
becomes
18

(324 )
where
For systems with a large number of degrees of freedom, every mode may not
significantly contribute or be of interest in the response of the system to forced
excitation. Modal superposition allows the response of a system to be approximated by
the sum of a limited number of normal modes thereby decreasing the number ofdegrees
of freedom ofthe system.
The response of a system to ground or base motion is a specialized case of forced
excitation. For an nDOF system excited by base or ground motion, equation (324) can
be rewritten as
where the term
n
Lm/pj ,
r. =..:;..)' . I n
Imj¢J],
jI
( 325)
(326 )
is called the modal participation factor and ji(t) is the motion of the ground. The modal
participation factor relates how the ground motion excites a given mode. In equation
(326), mj is the actual mass value from the mass matrix [M], associated with the jth row
and column. As previously mentioned, the differential equations of motion for a multi
DOF system are generally coupled through the mass and/or stiffuess matrix. Equation
(326) assumes that system of equations only exhibit static coupling such that the mass
19
matrix is diagonal and the equations are coupled through the stiffness matrix. When
deriving the equations of motion for nDOF system, the choice ofthe system coordinates
will define the type of coupling. Just as a coordinate system can be found that decouples
the system equations, a coordinate system can also be found that ensures only
static coupling. From equation (316), the denominator of equation (326) is equal to
one so that the modal participation factor becomes
n
r; =Lmj¢j;
j=1
3.2 HSA Model Development
The goal ofthis thesis is to develop a compact model that can describe the
dynamics of complex structures such as HSAs. The model that will be developed
combines both FE and analytical modeling methods. In order to minimize the
(327 )
complexity, development time, and solve time that is often associated with FE models of
complex structures, the HSA is divided into subcomponents. The subcomponent FE
models can be developed and solved in less time compared to a full FE model of the
HSA. The subcomponent FE models provide the natural frequencies and mode shapes
of the individual subcomponents. In order to describe the complete system, coupling
terms must be derived that describe how the subcomponents interact to form the overall
system dynamics.
Figure (31) shows a representation of the HSA used for the model development.
In the model, only the offtrack displacements of the various components are considered
since it is the offtrack motion that adversely effects drive perfonnance. In the above
20

figure, the offtrack direction is indicated by the yaxis. The HSA model consists of five
Lumped Nodal Mass (ms;, mAil mYi)
Element St~(Kg;, K". KyO
Rigid
Actuator
Body
..0....0..
IYi
Suspension SubComponent FE Model (Illsi, KsJ
Arm SubComponent FE Model (mAi. KAJ
Yoke/Coil SubComponent FE Model (mYi. KyJ
lSi
IAi
~
Figure 31: HSA Model
main parts: the yoke and coil, an actuator ann, a suspension, the actuator body, and the
pivot bearings. The yoke/coil, actuator ann, and suspension are subcomponent FE
models. The actuator body is assumed to be rigid and the pivot bearings are modeled as
a linear spring. Each subcomponent FE model is represented by a series of masses and
springs. In Figure (31) and the model development that follows, the subscripts S, A, and
Yare used to identify parameters associated with the suspension, ann, and yoke/coil subcomponent
FE models respectively. The mass terms associated with the subcomponent
FE models (mSi, rnA;, and mYi) represent the lumped mass values at individual nodes and
the spring terms (Ksi, KAi, Kyi) represent the stiffness values of individual elements. The
total displacement of each lumped mass is expressed as the swn of the rigid body
21
t'
displacement, both translation and rotation, and the displacement due to the defonnation
ofthe subcomponents.
{y},OIOJ ={y}rigid +{y}de/onnorI01l (328)
The deformation in the offtrack direction of each nodal lumPed mass is obtained from
the results of the corresponding subcomponent FE model. Using modal superposition,
the displacement due to deformation can be expressed in terms ofthe modal coordinates
as the sum of the nonnal modes as in equation (318). Substituting equation (318) into
equation (328) yields
11
{y },O,OJ ={y }rigld +L {¢}; ql
3.2.1 Development of the Mass and Stiffness Matrices of the Undamped
System
The equations of motion are derived through the use of Lagrange's equation
(329)
(330)
where T is the total kinetic energy of the system, U is the total potential energy of the
system, q is a generalized coordinate and Qis a generalized force. In order to simplify
the derivation, damping tenns are omitted since the effects of damping do not influence
the natural frequencies ofthe system. Damping terms are, however, important and will
be addressed later in the model development. From Figure (31) and equation (329) the
total kinetic and potential energy of the system are expressed as
22
T 1 . ~ =mBJ' +1J BO· 2 +1m ' 1 O· At. • At.. \2 A1 \y+ Al +'f'AlIqAI +'f'AI2qA2 +'''J 2 2 2
+!mA2(y+IA2B+¢J'i2\l.1AI +¢JA2lJA2 + ...y2
+...
+!msl (y + ISIB +¢JAnl4 AI + ¢JAn24A2 +... +¢Js1I4sl +¢JS\24s2 + ...)
2
+!mS2 (y +ls2B+ ¢JAnI~LI +¢An24A2 +... + ¢JS214s1 +¢JS224s2 +..f 2
+...
+!my ,(y+IYlB+¢YIl4n +¢Y1A12 + ...y2
+!m12 (y+Z12B+¢JY214Y1 +¢Y224Y2 +..f 2
+...
U =1KBrgy2+1K ( )2 AI ¢AllqAI +¢J,412qA2 + ...
2 2
+ ~ K A2 ((¢.m ¢J,4l1)qAI + (¢A22 ¢JAIJ1A2 +...y
+ ...
+ ~ K An ((;Ani  ¢A("I)1 hAl + (¢An2  ¢A(III)2kA2 +...y
+ ~ KS1 (¢SlIqSI + ;S12qS2 +...y
+ ~ KS2 ((;.'1'21  ¢SII )qSI + (;S22  ¢S12 )qS2 +.. .y
+...
+ ~ KSn ((;SIII  ;S(nI)I hSI + (;Sn2 ¢JS(III}2 #S2 + .. ·r
+!KYI(¢Yllqn +;Y12QY2 +...y
2
+ ~ K 12 ((¢121 ;m )qYI + (;122  ¢YJ2)q12 +...y
+...
+ ~ KY,,{(;YnJ  ;Y(nI)IhY1 + (;Yn2  ;Y(nI)2 #12 + ...r
23
(331)
(332)
Substituting the kinetic and potential energy equations into Lagrange's equation and
solving yields a set of n equations where n is number ofDOF of the system. The
number of flexible modes and rigid body modes that are considered determines the
number ofDOF of the system. Putting the equations in matrix form, as in equation
(32), yields the following mass and stiffness matrices.
col.17
mr 0 r YI r Y2 r AI +msrt/JAlII r A2 + msr t/JAII 2
0 J r 0 0 0 0
r Yl 0 1 0 0 0
r n 0 0 1 0 0
[M]= rAJ +msrt/JAnl 0 0 0 1+ msrt/J~I msrt/JAn2t/JAni
r A2 +msrt/JAn2 0 0 0 msr t/JAnI¢AII2 1+ msrt/J~2
r A3 +msrt/JAll3 0 0 0 msrt/JAlII t/JAll3 msrt/JAn2¢All3
o
o
oo
o
o
col.812
r A3 + msrt/JAn3
o
o
o
msrt/JAll3t/JAnI
msrt/JAn3t/JAn2
1+msr¢~3
r SI r S2
0 0
0 0
0 0
rSI¢An, r S2¢AnI
(3·33)
r S\¢An2 r S2¢An2
r sl¢An3 r S2t/JAn3
1 0
0 1
24
KHariIIg 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 2 (UY) 0 0 0 0 0 0 0 0 0
0 0 0 {U2n 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 2 0 0 0 0 0 0 [K]= {UAI
0 0 0 0 0 0 2 0 0 0 0 0
(334)
{UA2
0 0 0 0 0 0 0 {UA23 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 2 0 0 {Us I
0 0 0 0 0 0 0 0 0 0 W S22 0
0 0 0 0 0 0 0 0 0 0 0
where
y
()
qyl
qy2
{y} = qAI (335)
qA2
q A3
qSI
QS2
In the derivation of the system equations, the identities established below were utilized,
in conjunction with the identities established in equations (314), (315), (316), (317),
and (327), to simplify the mass and stiffness matrices..
n mST =LmSi
1=1
25
(336)
The total mass of the suspension subcomponent FE model is represented by mIT, and
each lumped nodal mass is represented by mSi.
(337)
The total mass ofthe HSA is represented by mr. The mass of the rigid body (as shown
in Figure (31» is represented by mB. As in equation (336), the mass ofthe yoke/coil,
arm, and suspension subcomponent FE models are represented by the Im terms.
(338)
The total mass moment of inertia of the HSA about the pivot center is represented by Jr.
The mass moment of inertia of the rigid body about the pivot center is represented by J B.
The mass moment of inertia of the yoke/coil, arm, and suspension subcomponent FE
models about the pivot center are represented by the Im/2 terms.
3.2.2 Development of the Force Input Vector
From Figure (31) and equation (324) the input force vector can be shown to be
26
Fo
F;,Rc
tPYi1Fo
tPr;2Fo
{F}= 0
0
(339)
0
0
0
where Rc represents the moment ann from the HSA pivot center to the force applied by
the coil. The HSA is moved and positioned by a voice coil motor (VCM) which consists
ofa coil ofwire (as shown in Figure (11)) "sandwiched" between two magnets.
Applying a current I through the coil induces a torque T on the HSA which causes an
angular acceleration. The magnitude of the torque is proportional to the product of the
current f and torque constant K, ofthe VCM.
T =K,J
The units ofKt are usually expressed in ozin/Amp. Since the forces that move and
(340)
position the HSA are generated by applying current to the VCM, the input forces in the y
and Bdirections of equation (339) can be rewritten in terms of the torque constant and
input current as
..J..
~
and
F = K'f
o R
c
27
(341)
(342)
The force generated by current flow through a constant magnetic field is the vector cross
product ofthe vector representing the direction ofthe current flow and the vector
representing the direction of the magnetic flux density. Due to the orientation ofthe coil
in the magnetic field, only the sides, or the active lengths, of the coil are useful for
generating torque to move the HSA (as shown in Figure (32)). The force vectors
Active Lengths
o Magnetic Flux out of the Page
e Magnetic Flux into the Page
Figure 32: Forces due to Current Input to the Coil
generated by the ends of the coil, or end turns, are directed either towards or away from
the pivot center and provide little or no torque. Typically, only the active lengths of the
coil are exposed to the magnetic field. Each active length provides onehalfofthe force
and therefore onehalf of the torque generated when current is passed through the coil.
The force is assumed to be unifonnly distributed along the active length of the coil and
can be approximated as a single force vector acting at the center of the active length, as
28

illustrated in Figure (32). The moment ann, Rc, is therefore the distance from the pivot
center to the center of an active length.
Since the input force to the HSA model can be approximated as two equal forces,
each force is assumed to act at a single node at the midpoint of each active length in the
yoke/coil subcomponent FE model. From equation (324), for each yoke mode
considered, the mode shape values «(i) for the two nodes where the forces are assumed to
act, must be know, and can be obtained from the yoke/coil subcomponent FE model.
Thus, the input force from equation (339) can be rewritten as
A. F  fA. + A. )! K,
\l'YII 0  ~'T')JI"I 'f'Yk In 2 R
c
(343)
Substituting equations (341), (342), and (343) into equation (339) yields the force
input vector in tenns ofthe input current.
K1
Rc
K1
(¢y) 1"1 + ¢Ylrln ) ~ ~I
C
(¢)J2.. 1 + ¢Y/C2n ) ~ ~I .
c
{F}= o
o
o
oo
29
I (344)
I.'.

3.2.3 HSA Modeling Assumptions
During the development of the HSA model, several assumptions were made in
order to simplify the equation derivations and the resulting mass and stiffuess matrices.
First, as previously mentioned, damping was omitted. The resonances ofthe HSA are
typically lightly damped and minimally coupled to adjacent modes so that damping does
not effect the calculation of the natural frequencies ofthe individual resonances. The
primary influence of damping in the HSA model is that oflimiting the amplitude
response at resonance. Since damping limits the amplitude at resonance, damping will
be accounted for later in the model development in order for the HSA model to
accurately predict offtrack motion during track following. When damping is added,
damping terms will be treated as viscous, or proportional, damping.
Second, the HSA model is limited to track following. By limiting the model to
track following as opposed to seeking, the HSA can be assumed to behave as a linear
system. During seeking, the input current to the coil, and therefore the forces acting on
the HSA, is significantly greater compared to the input current and resulting forces that
act on the HSA during track following. The increased forces applied on the HSA and
the frequency content of a seek profile, can excite nonlinear resonances that otherwise
would not be excited. Also during seeking, modes that were previously considered
linear may begin to exhibit nonlinear behavior.
During track following, there is very little rotation of the HSA. By limiting the
model to track folJowing, the angular accelerations, velocities, and displacements of the
HSA are limited to small values such that the coupling between the rotational DOF (f1)
and the modal (qYi, qAt, qSt) DOF can be asswned negligible. This is an important
30
assumption since the coupling factors between the rotational and modal OOF are
difficult to calculate. The modal participation factors (I) and mode shape values ((I)
used to couple the translational (y) and modal (qy;, qA.l, qSi) OOF in the mass matrix are
obtained from the subcomponent FE normal mode solution. The modal participation
factors and mode shape values are typically made available as outputs of the FE
software. The rotational to modal coupling factors that are obtained through the
derivation ofthe system equations are of the form
fl
rmcfj = Lmi¢Ji),.
/.1
where rmcjj is the rotational to modal coupling factor for the jib mode, m/ is the ith
(345)
lumped nodal mass, ~j is the correspond mode shape value, and /1 is the corresponding
moment arm from the pivot center to the ith nodal mass. In order to calculate the
rotational to modal coupling factors, a custom solver would have to be developed and
used to solve the subcomponent FE models. The validity ofthe assumption that the
rotational to modal coupling factor is zcro may be tested by placing terms in the mass
matrix that couple the rotational and modal DOF (the artificial coupling terms must be of
the same order of magnitude as other terms in the matrix). Resolving the system
equations shows that the addition ofthe coupling terms does not alter the frequency
response of the system. The conclusion can therefore be drawn that the assumption
previously discussed is valid.
Lastly, the HSA is assumed to be perfectly balanced so that the center of gravity
is at the pivot center. This assumption removes any coupling terms between the
translational (y) and rotational (8) OOF. Imbalance ofthe HSA is primarily a concern for
31
external shock and vibration disturbance inputs. In the case of external disturbances, any
amount of imbalance in the HSA can be detrimental to the overall drive performance.
Thus drive design engineers strive to minimize imbalance in the HSA. Realistically,
because of manufacturing tolerances, the center of gravity is always offset from the pivot
center so that every HSA has some amount ofimbalance. Even though imbalance terms
are not included, the effects of imbalance can easily be studied with the HSA model that
has been derived. Imbalance terms can be calculated and placed in the mass matrix to
couple the translational and rotational DOF. However, just as the rotational to nodal
coupling terms do not effect the modeled frequency response, the rotational to
translational coupling terms will have almost no effect on the track following frequency
response of the model.
3.2.4 HSA Model Discussion
In the model development, the mechanical resonances ofthe HSA have been
divided into flexible and rigid body modes. The flexible resonances, as calculated
utilizing the subcomponent FE models, result from the material properties and boundary
conditions of each subcomponent system. The rigid body modes result from the
moving mass and rotational inertia ofthe HSA. Specifically, the two rigid body modes
considered in the model are the modes relating to and resulting from the translation and
rotation of the HSA in the y and () coordinate systems.
Since the HSA is free to rotate about the pivot bearings, the rigid body rotational
mode is an unconstrained mode at 0 Hz. The rigid body translational mode, or bearing
translational mode, is actually, in all technical accuracy, not a true rigid body mode since
32
)..
~
°0 ..
..
10
:;
~.:.
it is dependent on the deformation of the pivot bearings. Also, at the frequency that the
rigid body translational mode occurs, the subcomponents may defonn. However, it is
referred to as a rigid mode in this paper as it primarily results from the moving mass of
the HSA and the spring constants of the pivot bearings_ The natural frequency ofthe
rigid body translation mode can be approximated by
[K::;
(J)TranslaJion =V~ (346 )
The resulting HSA model is an nDOF model where n is the total number of
modes considered. The model consists of two rigid body modes, as previously
mentioned, and n2 flexible modes ofthe yoke/coil, suspension, and arm. Each flexible
mode is considered as a singleDOF system in a modal.coordinate system qj_ The
system equations are dynamically coupled through the mass matrix. The coupling tenns
in the mass matrix are comprised ofthe modal participation factors (r) and specific
mode shape values (rf) obtained from the ann subcomponent FE model. The~e coupling
factors indicate how the flexible subcomponent and rigid body modes interact to form
the overall system dynamics. In the presence of support or ground motion, the modal
participation factor describes to what extent a given mode is excited. For the forced
vibration case, the modal participation factor describes to what extent the excited mode
transmits the input force to the ground or support structure.
The HSA model does not require that all mode shape values associated with each
node in the subcomponent FE model be extracted. For each subcomponent mode
considered, only the mode shape values at nodes where displacements of the FE model
must be known (such as the arm tip and slider) and nodes where forces are assumed to
33
':~
.'.I
..
:..~. •
..
,..'"
act are extracted from the subcomponent model. This limits the number mode shape
values that must be obtained from the FE model to one or two values per mode for each
subcomponent.
For each yoke mode considered, only the two mode shape values associated with
the nodes were the input force is asswned to act are required. These mode shape values
are used in the force input equation (equation (344». For each ann mode considered,
only one mode shape value at the tip of the ann where the suspension is attached is
required. The ann mode shape value describes the displacement due to deformation of
the arm tip. This arm mode shape value is the only modal value (¢JAni) that shows up in
the mass matrix (equation (333». It is part of the coupling factor between the ann and
suspension since the total displacement of the base of the suspension is the swn of the
rigid body motion and the defonnation displacement of the ann tip. Likewise, for the
suspension FE model, only one mode shape value per mode is required. The mode
shape that is required from the suspension FE model is a modal value on the slider since
it is ultimately the displacement of the slider that is detrimental to the drive perfonnance.
The mode shape value extracted from the suspension FE model is used only in the output
equation shown below (equation (353).
The FE software used in this thesis was Structural Dynamics Research
Corporation (SDRC) Ideas Master Series 7. SDRC calculates the modal participation
factor as part of an effective mass term. SDRC defines the effective mass as
,j '.
,'.
".,,
(347 )
34
where Me is the effective mass, Fis the modal participation factor, mm is the modal
mass, mT is the total mass of the FE modal and i denotes the mode nwnber. The modal
participation factor can be calculated by rearranging terms.
(348 )
3.2.5 State Space Representation of the HSA Model
For simulation purposes, the equations are recast in a state space representation
where the system equations are described by a set of first order differential equations. In
state space [onn, the state of the system at any given time can be described by the
displacement and velocity of the system, which are called state variables. The state
space matrix equations take the general fonn given below
{i} = [A ~x}+ [B]u (349 )
where [A] is the system matrix, [B] is the input matrix, and {x} is the state variable
vector. The output equation can also be expressed in matrix form as
•·
{Y}= [cXx}+ [D]u (350 ) ·· ,
where [C] is the output matrix and [D] is the direct transmission matrix. Expressing
equation (31) in state space [ann yields the following
(351 )
35
where [M] and [K] are the mass and stiffness matrices and {F} is the input force vector.
For the HSA model, the input term u represents the coil input current I from equation
(344).
In equation (351), the term [M]"l[Cdamp] represents the damping ofthe system.
Damping terms were previously ignored in the model development. For the HSA model,
proportional damping is assumed so that the resulting damping matrix is diagonal.
(352 )
For the HSA model, the output and direct transmission matrices [q and [D] are
[C] =[1  Rh 0 0 ... 0 0 0 . .. "'Snl A. ...)
'" "','),02
and (353 )
[n]= 0
where ~ is the distance from the pivot center to the slider. The output matrix [C] is the
modal superposition of the rigid body and flexible modes that directly contribute to the
offtrack displacement of the slider.
36
1

4 FiniteElement Models
4.1 Full HSA FiniteElement Model
To illustrate the difficulties in dealing with large FE models of complex structures, a
full FE model ofthe HSA will fIrst be presented. The results ofthe full HSA model
were not used in the development ofthe subcomponent FE analytical model. Figure 41
Figure 41: Full HSA FE Model
shows the full FE model of a typical HSA. The FE software used for all FE models
presented in this paper is SDRC Ideas Master Series 7 running on a Windows NT
platform. The HSA FE model consists of the actuator, coil, pivot bearings, and six
suspensions. The actuator and coil are composed of solid tetrahedral elements with
37
midside nodes. Each suspension is comprised mainly of thin shell elements with the
exception of the sliders, which were meshed using solid brick elements. The pivot
bearings were modeled as sixteen linear springs connected between the actuator and
bearing shaft. The total number of elements for the full HSA model is approximately
22,000 and the total number of nodes is approximately 39,000. The boundary conditions
ofthe HSA model are such that the actuator is clamped at the top and bottom ofthe
bearing shaft and the six sliders are constrained to move only in the ofTtrack direction.
The model development time for the full HSA FE model was approximately 80 hours.
All mode shapes and frequencies for the HSA under 17 kHz, approximately 80 modes,
were calculated. Due to the size of the model, the solve time was over 12 hours for a
single processor Pentium II 350 system with 256 MB of memory. The calculated natural
frequencies ofthe HSA are generally within 10% ofexperimental data, although the
mode shape amplitudes do not correspond quite as well. The discrepancies between the
FE HSA model and measured data can be seen in Figure 42. Figure 42 compares the
mechanical frequency response ofthe FE model due to a harmonic input at the coil and a
displacement output at the slider to the measured frequency response from a hard disk
drive. Th.e discrepancies between the measured and modeled transfer function are due to
errors in the FE model. Possible sources of error include improperly modeled pivot
bearings, improperly modeled coil (including inaccurate assumptions regarding coil
material properties), improperly modeled interface between the actuator and bearings,
improper boundary conditions (either too rigid or too soft), and the fact that the center of
gravity of the FE model is offset from the pivot center. All ofthe errors in the FE model
that contribute to the discrepancies between the measured and modeled transfer
38
'~
HSA FE Model vs. Measured FrElQ.Jency Response
40 r,..,.;.,...,...,.."'"l
,, ,,
Measured Bearing
Translational Mode
 FE Yoke Sway Mode _._~,_._._~~
_._.~_._ ··, ,. • • ~t ~I J, ___ .JI _
I I I , •
• I I I I
• I • I ,
I I .,I ,I,
,,,
,,
I , I I I I
_.y~_._._.,,~,    A •
...... FE Bearing Translational Mode
40
.,,
• I • I I' • I ._ .. ,,._,, rr rea, ~ I • I I' I'
1 I • • I I I I
, • • I • I I I
I I • I I I I I
f', I I I I I 'I ~ ii~~r~i
FE Model : : : :: ::
Measured : : : :: ::
60 ''~~~~'"~"''"'
103
10
20
30
EO 0
~
<D g 10 .......
'c
~ 20
30
Frequency (/Iz)
Figure 42: HSA FE Model vs. Measured Mechanical Transfer Function
functions can be corrected so that the modeled results agree more closely with the
measured data. However, due to the size of the model and the required solve time,
determining the exact cause ofthe errors and correcting the FE model would be
extremely time conswning. Thus the full FE model should not be used to produce
theoretical transfer fimctions.
4.2 SubComponent FE Models
.,
"I
.J
I
.1..~I
Because of the complexity of the full HSA FE model, the HSA can be broken into
three subcomponent FE models. The three smaller FE models consist of the coil and
yoke, a single ann with two suspensions, and a single suspension. These three models
are much smaller than the full model and take only minutes, compared to hours, to solve.
39

The smaller subcomponent models can be solved in system memory where as the full
model requires the use ofvirtual memory. Virtual memory requires dumping the system
memory to the hard drive and reading it back at a later time when the information is
needed. Any time a solution requires the use of virtual memory, the solve time increases
dramatically.
The boundary conditions ofthe subcomponent models are critical if these models
are to agree with measured data. To verify the boundary condition ac;sumptions as well
as the accuracy of each subcomponent FE model, the calculated natural frequencies and
mode shapes are compared against measured data. Typically, the calculated natural
frequency for each mode shape is within 7% of experimental data. Even though each FE
model represents only a subcomponent ofthe total HSA, all experimental data is
measured from full HSAs in working drives. The subcomponent FE models are
compared to full HSAs since the purpose ofthe subcomponent model is to reduce the
complexity associated with creating and solving FE models of complex structures while
retaining accurate prediction capabilities for the natural frequencies and mode shapes of
the various HSA components. Thus, even though the subcomponent FE models
represent only a portion of the entire HSA, the associated FE model should be able to
accurately predict the natural frequencies and corresponding mode shapes as if each subcomponent
FE model were part of a complete HSA.
4.2.1 Suspension FE Model
Figure 43 shows a typical model of a suspension. The model consists of
approximately 360 elements and approximately 490 nodes. The suspension is modeled
using predominately thin shell elements, the exception being the slider, which is meshed
40

Figure 43: Suspension SubComponent FE Model
using solid brick elements. The nodes of the suspension armmd the swage hole, or base
ofthe suspension, are constrained in all six degrees of freedom and the four comers of
the slider are constrained with zero displacement in the vertical or zaxis. The slider is
free to move side to side or front to back. This constraint on the slider accurately models
the air bearing, the thin layer of air that separates the slider from the disk under operating
conditions, which has an axial stiffness ofgreater than 30 kHz. The justification for the
rigidly clamped boundary conditions around the swage hole is based on the assumption
that the flexibility of the ann does not significantly affect the mode shape of the
suspension. This is a valid assumption since drive design engineers strive to minimize
coupling not only between the arm and suspensions but also between any components in
the disk drive. If coupling exists, the track following performance ofthe disk drive is
degraded since coupled or adjacent modes without sufficient frequency separation can
cause greater displacements ofthe slider than individual modes. The model
development time for the suspension model was approximately 5 hours and the solve
41

time on a Pentium II 350 system with 256 MB of memory was less than 1 min. Table
41 shows the results of the single suspension FE model compared with measured data.
The suspension mode shapes calculated from the subcomponent FE model are shown in
AppendixA.
Modeled Measured
Mode Shape Frequency (kHz) Frequency (kHz) % Difference
1' Bending 1.87 1.97 5.1
11 Torsion 4.05 3.68 10.1
2"U Bending 5.82 5.61 3.7
2"" Torsion 10.66 9.88 7.9
3'" Bending 11.77 11.80 0.3
Table 41: Suspension FE Model vs. Measured Natural Frequencies
4.2.2 Actuator Arm FE Model
The second subcomponent model is ofthe actuator arm as seen in figure 44. The
Figure 44: Actuator Arm SubComponent FE Model
42
model consists of approximately 2150 elements and approximately 4000 nodes. The
ann model consists of a single actuator arm and two suspensions and is meshed with
solid tetrahedral elements with midside nodes (the suspensions are as described above).
For the arm, the nodes at the base of the arm that fonn the body ofthe actuator are
constrained in all six degrees of freedom to model the arm being rigidly clamped. The
two suspensions are joined to the ann model by merging coincident nodes on both the
ann and suspensions so that the suspensions are rigidly attached to the arm. The
boundary conditions for the sliders are the same as for the single suspension model. For
the FE model of the subcomponents to be accurate, the boundary conditions must be
correct. Thus it is necessary to include the two suspensions in the arm model. The
suspensions provide mass and inertia that dramatically affect the mode shapes and
frequencies ofthe arm. It appears redundant to solve FE models of the arm with two
suspensions as well as a separate model of a single suspension since the arm model
solution will give the natural frequencies and mode shapes for the both the arm and
suspensions. However, there are two advantages in treating the suspension and arm as
separate subcomponents in the model development. First, by treating the two as
separate subcomponents, the coupling factors between the ann and suspension modes
can be calculated, lending understanding as to how the suspension is excited by the arm
and how the arm is excited by the suspension. Second, once the model is developed,
new suspension designs can be evaluated quickly without the creation of a new ann
model. The suspension model must be created first anyway. The model development
time for the arm model was approximately 1 hour (the suspension model is assumed to
already exist) and the solve time on a Pentium II 350 system with 256 MB ofmemory
43
.1
'1
~
I.,'
'o.J
It
~.f1
was less than 3 min. Table 42 shows the results of the ann FE model compared with
measured data. The arm mode shapes calculated from the subcomponent FE model are
shown in Appendix B.
Modeled Measured
Mode Shape Frequency (kHz) Frequency (kHz) °/. Difference
1'" Bending 1.08 1.14 5.3
2'"' Bending 7.51 7.69 2.3
1'" Torsion 7.73 7.79 0.8
Table 42: Actuator Arm FE Model vs. Measured Natural Frequencies
4.2.3 Yoke and Coil FE Model
The third subcomponent model is ofthe yoke and coil as seen in Figure 45. The model
Figure 45: Yoke/Coil SubComponent FE Model
consists of approximately 2600 elements and 5100 nodes. The yoke/coil model consists
of three parts: the yoke, the coil, and the epoxy that bonds the coil to the yoke. All
44
".,
'.•1
"

elements used in the yoke/coil model are solid tetrahedral elements with midside nodes.
Similar to the ann model, the nodes at the base of the yoke that form the body of the
actuator are constrained in all six degrees of freedom to model the yoke being clamped
rigidly. Since the coil is not solid, the material properties (Young's modulus and mass
density) are difficult to determine. The density value that is used in the FE model is
adjusted so that FE model of the coil has the correct weight for the modeled coil volume.
The correct Young's modulus is found by iterating until the frequencies ofthe first few
modes agree with measured data. Since the model of the yoke/coil is small, the iteration
process only takes a few minutes. Once the Young's modulus is found, this value can be
used in future yoke models that contain similar coil designs. The epoxy that bonds the
coil to the yoke provides no structural stiffness to the system. The Young's modulus
that is used for the epoxy is simply several orders ofmagnitude less than that of
aluminum, the material ofthe yoke. Often the coil is modeled as an orthotropic material,
but this step is not necessary to accurately predict the frequency and mode shapes for the
first few modes. The model development time for the yoke/coil model was
approximately 2 hours and the solve time on a Pentium II 350 system with 256 MB of
memory was less than 5min. Table 43 shows the results of the yoke/coil FE model
compared with measured data. The yoke/coil mode shapes calculated from the subcomponent
FE model are shown in Appendix c.
Modeled Measured
Mode Shape Frequency (kHz) Frequency (kHz) % Difference
1'" Bending 1.12 1.19 5.5
1R Torsion 1.82 1.90 4.2
Table 43: Yoke/Coil FE Model vs. Measured Natural Frequencies
45
",
"
.'
"
11
4.3 Experimental Verification of FE Models
Measurements of the natural frequencies and corresponding mode shapes of the
suspension, actuator arm, and yoke/coil were made with a Scanning Laser Doppler
Vibrometer (LDV). The LDV is an optical instrument that provides a noncontact
means to measure the velocity and displacement vibrations of a surface. It is important
that a noncontact technique be used since the components that are being measured are
relatively small with little mass. Devices such as accelerometers can mass load the
component and significantly alter the measurements. Mass loading occurs when the
mass ofthe measurement device is a significant portion ofthe effective mass of a
particular mode.
The scanning LDV automatically measures a set of user predefined points on the
structure and calculates the frequency response function for each measured point. The
frequency response function at each measurement point is calculated by dividing the
measurement output response by the input excitation for the system. For the scanning
LDV measurements of the HSA, the measurement output is the LDV vibration
measurement (typically velocity) and the input excitation is a random noise current input
to the coil. Once all of the points are scarmed, animations of the structure can be viewed
for any frequency in the measurement range since the magnitude and phase of each
measurement point are know from the frequency response functions. The animations
provided by the scanning LDV software are actually operating deflection shapes (ODS)
not mode shapes, since no modal parameters are calculated. However, ODS are
typically an accurate approximation of the mode shapes and tmdarnped natural
frequencies of a system provided adjacent modes are not closely coupled. The scanning
46
LDV eliminates the tediousness and saves hours over manual data collection techniques
associated with modal analysis methods.
In order to animate an ODS, numerous points on the surface of the structure to be
analyzed must be measured. Since the components of the HSA are small in size,
measurements with the scanning LDV are typically limited to axial (perpendicular to the
disk plane) measurements to ensure a sufficient surface area for proper animation and
mode shape identification. As a result, only modes that have displacement components
perpendicular to the disk plane can be measured. This includes bending and torsional
modes ofthe HSA subcomponents. Modes that have defonnation predominately in the
plane of the disk, such as arm sway modes, are very difficult to accurately measure with
the scanning LDV. Thus, in the previous tables which compare calculated and measured
natural frequencies of the subcomponents, sway modes are excluded. However, since
the bending and torsion modes are typically within 7% of measured values, it will be
assumed that the natural frequencies ofthe yoke, ann, and suspension sway modes are
also be within 7% of the actual natural frequency.
47
5 HSA Model Simulation Results and Discussion
5.1 Reduced Order Model Simulation
The results of the frequency response simulation ofa reduced order model will
fIrst be discussed. lbis model simplifies the overall dynamics of the HSA by removing
the flexible modes of the subcomponents. The purpose ofthis section is to discuss a
deviation between the statespace model and measured data and show how the full order
model will be compensated to correct for modeling inaccuracies. Considering only the
rigid body rotational mode and the bearing translational mode reduces the model to a set
of two uncoupled system equations
( 51 )
where the total displacement ofthe slider is
(52 )
In Figure 51 below, the frequency response of the reduced order model is compared
with measured data from a hard disk drive. From Figure 51, it can be seen that the
modeled system transfer function deviates from the measured data. The modeled
frequency response starts to diverge from the measured frequency response at
approximately 2 kHz and the modeled bearing translational mode at 5.4 kHz has a
48
Mooeled YS Measured FreqJeflcy Respoose
2O.....~.....________.__r__"""T"___r___>
Bearing Translational Mode
en
:g a
Q)
u
..:.:.:.J..
.~ 20
ro IJ....
2
·,··
· , ~_._~_.
I ,
·,I ,•,.,
.I
· ·." . _____ • IL ~I _. •L ~I I IL
I • • I I I
I , t I I I
I • , I I •
I • • I I •
I • • I I •
I • • I I I
I • • I I I
I t I I I I
r~~~.,~ r J I I I I I ,
, I I I I I I
, I , I f I ,
, I • I • , I
• I I I I I
I I I I I I
• I I I I I 1500 L ~....,;.'.,;~'.:.'~..:....'__':'J
1~ 1~
~ 500
~
(J)
Ul ro if 1000
Frequency (Hz)
Figure 51: Rigid Body State Space Model vs. Measured Frequency Response
different amplitude and phase compared to the measured data. Since the output of the
state space model is the sum of the translational and rotational displacements (equation
(52)), the relationship ofthe two equations is critical to the overall shape ofthe total
displacement frequency response. This is illustrated in Figure 52 where the individual
tranlsational and rotational components are shown. The differences between the
modeled and measured frequency response functions are due the frequency at which the
mass line of the rotational transfer function and the stiffness line ofthe translational
transfer function intersect. The DC gain of each transfer function determines where the
two curves intersect. For the rotational component, the DC gain is
49
K =KrRh K
8 J BW
T
and for the translational component, the DC gain is
(53 )
(54 )
where KBW is the gain factor associated with the drive electronics and compensator. The
,,,
~~~ , I ,,
Stiffness Line
~ ,,
.,,,
I
State Space Model Frequency Response
20 r,,..,.....,.....,.....~ , ,
~ / Mass, Line ~
 I t I I , • ~ ~ J L ~ L __ ~ __
It. I I • •
I • I t I I
I I I I
I I • I
I I • I
I ,I ,• ,I
... _""""",, ...... ~ .........1......
I ........ ""'' ,,
.40 L ' '_~'_____~__..,;",_~.......;...._:..._..:::l
103 4 10
Or,...,...,.....,,..,.,,
total disp
translational disp
rotational disp
..................................................... _1,_ _ ~_ 4 ~ .. ,,
·300
m1oo
Q)
~~
.200 t.::,.'.:".::":'.'.:".:::.".:".::.::'.':".:::'.'.:".::":'..':".:::_:.t".::."'.:".:::..~.:::.~_:_.::;.::::. ~.~.~.~.~;~_.:;:;:s..
.."c:' I I • I I •
a.. :, :, :,
Frequency (Hz)
Figure 52: Translational and Rotational Components of the State Space Model
terms that comprise the rotational and translational DC gain terms are measurable
properties of the system (torque constant (K,), rotational inertia (Jr), bearing stiffness
(Kbearlng), moment arm from input force to pivot center (Rc), and moment arm from slider
to pivot center(Rh». In order for the sum ofthe translational and rotational equations to
intersect and form a smooth transition, either the DC gain ofthe rotational component
must be lowered or the DC gain of the bearing translational mode must be increased to
50
achieve the correct frequency response function. However, since the measured and
modeled frequency responses do not diverge until 2 kHz, it appears that the gain for the
rotational component is correct and the gain for the bearing tranlsational mode is
incorrect. Since the DC gain for the transational mode is proportional to the input force
(equation (54)), the input force is increased until the state space model matches the
measured data. The resulting gain adjustment needed to correct the error in the state
space model is
(55 )
where MT is the total mass ofthe actuator, Rh is the moment ann from the pivot center to
Modeled YS Measured FreQ,Jency Response
20....,,....,..,..,
_.__._~ ~..~ ,,,
.,,
,, . ,,,
I • I , I ,
,,_.,~'r
I , I ,
f I I I
corrected state space model : : : :
40 L measured ...l..:'_:_';'_:'I.L..~__J
1~ 1~
OJ
:g. 0
([)
"'0
:::J
+'
.~ 20
(tJ ~..l....L.__,
2
O....~~~.:_:____:I
,,
, ,
t I I I ,
I , I I I I • "6) 500 ~ ~ ~ ~ ~ ~ ~ ~ ~ 
Q) 'I I I I I I ,
~ ::: : : : : : I I , , I I I I
<I> '" I I I I
~ fl' I I I I
I I I I I I I 00 I • I I I I I ~ 10 ~._.~~ ..  .. ._~.~.~.;. ; ,
0... I I I • , I I I
I I • I • I I I
I I , • I I I I
I I I • • I I •
I I I 1 , I I ,
I I I , • I I I
1500 L .:I... ~I _~I _ __'_I __I ..I ...I~~I J~
1~ 10"
Frequency (Hz)
Figure 53: Corrected State Space Model vs. Measured Frequency Response
51
the slider, and Jr is the total mass moment of inertia ofthe HSA about the pivot center.
The physical reason why the adjustment gain is needed is not fully understood at this
time. However the gain adjustment tenn derived above is valid for all cases, not just the
model presented in this paper. Applying the gain adjustment factor to the force input for
the translational DOF corrects the offset problem as is seen in Figure (53).
In Figures 51 and 53 there is a slight difference between the calculated and
measured phase below 4 kHz. From 1 kHz to 4 kHz, the measured phase decreases as
frequency increases, where as the modeled phase remains constant. This phase
divergence is a result of the measurement technique that was used to obtain the
frequency response function from the drive. Referring to Figure 12, the structural
response ofthe drive is defmed as the head position divided by coil current. Both the
head position and input current to the coil can be measured directly from the drive.
However, the disk drive is a digitally sampled system with inherent computational delay
in the control electronics. Both the computational delay and the delay due to sampling
result in the phase loss shown in the measured transfer functions. The HSA model
presented in this paper is a continuous model and does not account for any delay terms.
Another method to achieve the correct frequency response for the two rigid body
modes, is to express the 2DOF HSA model as a fourth order transfer function. The
HSA is a good example of a noncollocated system, where there is a flexible mechanical
member (pivot bearings) between the actuator (coil) and the sensor (read/write head).
Noncollocated systems accounting for one mechanical resonance can be modeled as a
fourth order system by the following normalized transfer function
52

(56 )
where
(j) =
"
Kbearmg and K = K,RIt K
J BW r
(57 )
From Figure 54 below, it can be seen that the noncollocated transfer function of
Modeled \IS Measured Frequency Response
2Or....,.........,~.~~ ,
~ 0 r~.~.~.~~.~~~~~_IJ!"_.~~~'!!.""'....""'~it",,iii"iii"lI~~;0.;I"~:
"0 '
.:.:.::.J. ,, ,•
'~20 stele spa:e model ~~..+. .. ~.i"'l:lIl!'lILrj+'M
:2 <ro transfer function model 1 1 1 1 ~
measured :::::
40 ''''''.....~''' '~...O...J
103
Or....,.........~~.~..._,
, . ... ~~ .... ,
,
,I •. , ,..,
'._
: : : :~
~ 500 ~ .._..  .... ...  ........  ....  ~ ........  ..  .... .. ~_ ..........  ~_ ..... ~ .. _.. ~ ..  .... ~
"'0 ~:: : : :
 • I • I • I
Q) ::: : : ;
Ul ::: : : : ~ 1000  ~ ~ ~ ~ ~ r" r""
CL ::: : : : :
I I • I t I I
I I I I I I I
I I I I I I I
, I I I I I I
t I I I I • I 1500 ' ~' ...'._____'_'__..'.._''_~''''
103
Frequency (Itz)
Figure 54: NonCollocated Transfer Function vs. Measured Frequency Response
equation (56) matches the measured data and the corrected state space model. The
transfer function can be easily transformed into an equivalent state space model using
MatJab so that the rigid body model can be used with the HSA model previously
53
developed. However, for the full order model simulation results presented below, the
gain adjusted state space model is used.
5.2 Full Order Model Simulation
The full order model includes subcomponent flexible modes and is a 15 DOF
model. The model consists of 2 rigid body modes, 3 yoke modes, 4 ann modes, and 6
suspension modes. The flexible modes that are included in the model are those
resonances whose mode shapes have displacement in the offtrack direction or have the
potential to cause displacement in the offtrack direction. The Matlab script file used in
the simulation is shown in Appendix D.
HSA modes can be identified as either inplane or outofplane resonances. Inplane
resonances are resonances in which the motion ofthe structure is primarily in the
plane of the disk. The resulting motion ofthe slider due to inplane resonances is radial
offtrack motion. From an analytical point of view, these modes can be modeled as a
complex pole pair. Examples of inplane resonances that will be considered in the HSA
model are the bearing translational mode and the sway modes ofthe arm, suspension,
and yoke. Outof plane resonances are resonances in which the motion ofthe structure
is primarily perpendicular to the plane of the disk. In order for these modes to modify
the open loop or structural response, there must be some amount ofradial or offtrack
motion associated with the mode shape. Ideally, outofplane resonances would not add
to the offtrack motion of the slider. However, unless the slider is at a nodal point of the
mode shape, outofplane modes will usually exhibit some amount offtrack motion.
Outofplane modes can typically be modeled as complex pole/zero pairs. Examples of
54
outofplane resonances that will be considered in the HSA model are bending and
torsion modes of the arm, suspension, and yoke.
The zeros associated with outofplane resonanaces make it difficult to accurately
model the offtrack displacements. The relationship ofthe zeros and the poles determine
the magnitude and phase response of outofplane resonances. Slight differences in arm
or suspension geometry or time and temperature effects that can change the mode shape
and make the offtrack displacements due to outofplane modes vary greatly from drive
to drive.
The results of the initial simulation of the subcomponent FE analytical model can
Modeled vs Measured Mecha1ica Transfer Function
Suspension Sway
.~_.. .. _. ·····
______ ... : Bearing Tranlation
m 0
~
<D g 20 ..6...>..
~ 40 ~Mode1.........,···
Measured
2Or,.,.,....~~__,
or"""""""=::::;:====:::::=::=,===~s.:::;~~~:T""""I
,..
. , . . I I I I ., I
····~f·~·~~~ .~~~..
, I I' f I I I
I I • t. ••
, , ..., I
l • ,... •
t , f'" I , I I""
I I I' I I I
_.__.__._~._._~_.~~~~;~ ~.  I • , I I , I
I • I , I I
I • I , I I
I I I I I I
I • , I I I
• • I I I I
I , • I I •
I , • I I • 1500 L.....:~'~~ ............~...................4....""'
103 10
<D
[6
t£ 1000
g; 500
"0
Frequency (Hz)
Figure 55: Initial Simulation of the SubComponent FE Analytical Model
be seen in Figure 55 compared with data from a measured drive. As can be seen in
Figure 55, there are several differences between the measured and modeled data
55
resulting primarily from the sway modes of the yoke, arm and suspension. The modal
participation factors (1) and modal displacements ((i) for these three modes are
approximately two times the desired magnitude. The appropriate values for the modal
participation factors and modal displacements are determined by iterating on these tenns
in the subcomponent FE model until the model converges with the measured data. By
Modeled ~ Measured MechCllicai Transfer Function
2Grr.,.....rrr.r.
Yoke Swa Ann Sway m 0 .~~~__....ry
~ ,
(l) , "3 2G ··t· Bearing TransJatio
.~ :
~ 40 t......L..,
Model
Measured
oI==~,F==::::::====:::;:::~~~~::,
,,,,
, ,'
• • • • II I I I • I I I • ~ __ • ~ ._J .L L•• J L __ J ._
I I • • • I l , ,
I I • • I I , • •
I I I • I I I I I
I , I I I I I f
I I I I f I "
I I I • f I I I
I I • I I I I I
• I I • I I I f
____________________ •I •• 1I 1I I I I I I rr.·r 1 
I , • I ,, ., ,, ,, ,, ,. , , . , , , 1500 L '~__~.__~ ' _~~_~'"__.L.__~___J
1lT
~ 500 :s
(l)
Ul ro tf. 1000
Frequency (Hz)
Figure 56: SubComponent FE Model Simulation with Corrected Sway Mode
Couplin2 Factors
reducing the modal displacements and coupling factors of the three sway modes by a
factor of two, the modeled frequency response agrees much more closely with the
measured data as seen in Figure 56. The remaining differences between the modeled
and measured transfer functions in Figure 56 result from two primary sources. First, the
natural frequencies ofthe subcomponent FE models can differ from measured data by
56
approximately 7% as previously discussed. Second, outofplane modes (bending and
torsion modes) do not exhibit sufficient offtrack displacements in the subcomponent
FE models.
The differences between the modeled and measured frequency response functions
arise from errors in the subcomponent FE models. Errors in the FE model results can
be due to incorrect boundary conditions and incorrect assumptions and simplifications
regarding the modeled part. For example, when parts are modeled using FE models, the
part is typically assumed to be a nominal part and tolerances are not considered.
However, all parts deviate from nominal manufacturing values. Due to the increasing
TPls and the corresponding decrease in track widths, displacement ofthe slider of only a
few microinches can be detrimental to drive perfonnance. As a result, part tolerances
can have a significant impact on the dynamic perfonnance of the HSA. Slight bends,
twists, or asymmetries that arise due to part tolerances can change the mode shape so
that the resulting offtrack motion for a given mode is reduced or increased. The HSA
model that has been developed is only as accurate as the subcomponent FE models.
One of the greatest advantages ofthe HSA model developed in this paper is that it
that it can be used to provide direction for improving the subcomponent FE models.
The subcomponent FE analytical model that has been developed is a compact model
that can be used to determine the appropriate modal participation and displacement
values so that the model matches measured data. The new values obtained from the
HSA model can be used to improve the accuracy and reliability ofthe FE models. As an
example, the appropriate modal parameters (rand (J) were determined using the subcomponent
FE model for the 3rd bending mode of the suspension such that the modeled
57
frequency response matched the measured frequency response for the mode of concern.
Using the new rand ¢modal parameters as target values, the suspension subcomponent
FE model was modified until the appropriate modal participation factor and mode shape
value for the 3rd bending mode matched the values dictated by the subcomponent FE
analytical model. The necessary changes to the suspension FE model included twisting
the suspension two degrees and adding the copper runs that carry the read/write signal to
and from the head. The added twist to the suspension approximates the handling and
assembly tolerances that allow for a slightly imperfect suspension and the copper runs
add an asymmetrical mass loading to the suspension. The differences in the unmodified
and modified 3rd bending mode ofthe suspension can be seen in Figures 57 and 58. In
Figure 57: Original Suspension 3rd Bending Mode
OffTrack Displacement
Figure 58: Modified Suspension 3rd Bending Mode
Figure 57, the nodal lines are perpendicular to the center line ofthe suspension,
resulting in little or no offtrack displacement of the slider due to the 3rd bending mode.
58
However, in Figure 58, the nodal lines are an angle to the suspension center line as a
result of the modifications that were made, so that the slider will exhibit offtrack motion
associated with the 3rd bending mode. The resulting changes that were required for the
3rd bending mode, show that the asswnptions and simplifications used to create original
suspension FE were incorrect.
By decreasing the sway mode coupling factors, adjusting the natural frequencies
to account for modeling/measurement differences, and correcting the subcomponent FE
models so that outofplane modes exhibit sufficient offtrack displacements, the subcomponent
FE analytical model shows good agreement with measured data as shown in
Figure 59. The only significant discrepancies that remain between the modeled and
Modeled vs Measured Mecha1ical Transfer Function
2O,...rr,rrrr,
.    ..  ..    ..  ; ..  ~ ..  ~  ~ \t~~~'~
iD 0  •••  ••• . •• ~T~••~~~.~~,.~.
:g.
(I)
"....g... ·20
"5, UnModeled Dynamics .
~ 40\.............  . 1_  _1 __ "~ ..  to ..  ~4
Model
60 ' Measured'__~_~_~~___'__~~~...........'_____J
103
~ 500
'0(I)
~ ct 1000
,,
.,:
UnModeled Dynamics: .
, , . ~.~.,,rr~r ,. 
1500 l ~~_~_~_~~_o...__L__ALJ
1d
Frequency (Hz)
Figure 59: Modified SubComponent FE Analytical Model Simulation
59
measured data are due to unmodeled dynamics between 8 kHz and 9 kHz. The
yoke/coil, arm, and suspension subcomponent FE models do not predict flexible
mechanical resonances between 8 kHz and 9 kHz. These modes are possibly system
modes where all of the subcomponents contribute significantly to the overall mode
shape or coupled modes where multiple arms and suspensions deform together as a
group. In the case ofcoupled modes, the arms or suspensions of the HSA appear to be
coupled together since they deform simultaneously in similar mode shapes. Typically
patterns are established such that one coupled mode might involve every other ann of
the HSA at a particular frequency, while at a different frequency, the inner arms might
deform as a group outofphase with the outer arms. However, since the subcomponent
FE analytical model only considers one arm and suspension, and not the entire HSA, the
model will not predict system or coupled modes.
60
6 Conclusions and Future Work
6.1 Conclusions
The resulting subcomponent finite element analytical model was a 15 degree of
freedom model that could quickly be solved using commercially available matrix
manipulation software. The model proved to be accurate in predicting the offtrack
motion ofthe head stack assembly and helped provide understanding as to which
resonances are the most detrimental to drive performance. Initial simulations showed
that some of the boundary conditions and assumptions used in creating the subcomponent
finite element models were incorrect. Due to modeling errors in the subcomponent
FE models, the offtrack amplitude and coupling factors of the inplane sway
modes were approximately two times the desired magnitude. Conversely, the amplitude
and coupling factors of the outofplane modes were not of sufficient magnitude.
However, by comparing the model results to measured data, the subcomponent finite
element analytical model provided direction to help improve the accuracy of the
individual subcomponent models, as was demonstrated with the 3rd bending mode of the
suspension. Also, since coupling terms were derived that describe how the subcomponents
interact to form the overall system dynamics, the subcomponent finite
element analytical model can provide direction to help optimize the head stack assembly
design in order to reduce the impacts of mechanical resonances.
61
6.2 Future Work
One of the main advantages of the HSA model developed in this paper, is that it
provides a means to predict track following perfonnance in advance of a working drive.
However, without the servo control loop, only the mechanical interactions between the
yoke, pivot bearings, arm, and suspension can be studied with the current HSA model. It
would be very beneficial to include the control loop so that the effects ofHSA
resonances on servo stability could be investigated as well. By adding the servo control
loop, the HSA model could be used to study drive performance impacts for new actuator
and suspension designs in the presence of increasing TPI as well as provide a platform
for theoretical compensator design work.
Another area of work that needs to be furthered is an investigation as to the correct
boundary conditions and assumptions to be used for the subcomponent FE models.
Specifically, the cause of the excessive displacement amplitudes associated with the
sway modes and the insufficient displacement amplitudes associated with the outofplane
modes needs to be understood so that accurate subcomponent FE models can be
generated. As previously mentioned, the subcomponent FE analytical model is only as
accurate as the subcomponent models.
Lastly, no drives are identical. Each drive has a unique frequency response
function due to differences in each HSA caused by manufacturing and assembly
tolerances. These differences result in drive to drive variations in the natural frequencies
and amplitudes of the intheIoop resonances. The offtrack displacement amplitudes of
outofplane resonances are particularly susceptible to variations in the HSA due to the
zeros that are associated with these modes. In order represent the variations in the
62

frequency response that might be expected over a large population of drives, the HSA
model can be used to generate probabilisitic distribution of transfer functions by using a
statistical sampling method. One such method is often referred to as a Monte Carlo
analysis where each variable in the model is asswned to have a statistical distribution.
For the HSA model, the offtrack amplitude and natural frequency distribution for each
mode would best be obtained by measuring a large sample of drives. Using a Monte
Carlo analysis, numerous HSA transfer functions could be generated by randomly
selecting natural frequency, modal participation, and mode shape values from the
corresponding distribution functions. Once all the transfer functions have been
calculated, a minimum and maximum "envelope" transfer function can be detennined
that would represent a range of values that a population of drives should fall within.
63

REFERENCES
1. Aruga, K., Kuroba, Y., Koganezawa, S., Yamada, T., Nagasawa, Y., Komura, Y.,
"HighSpeed Orthogonal Power Effect Actuator for Recording at Over 10,000 TPI,"
IEEE Transactions on Magnetics, Vol. 32, No.3, May 1996, pp. 17561761.
2. Chiou, S. S., Miu, D. K., "Tracking Dynamics ofInline Suspension in HighPerformance
Rigid Disk Drives with Rotary Actuators," ASME Journal ofVibration
andAcoustics, Vol. 114, January 1992, pp. 6773.
3. Evans, R, Carlson, P., Messner, W., "TwoStage Microactuator Keeps Disk Drive
on Track, " Data Storage, Apri11998, pp. 4344.
4. Franklin, G. F., Powell, J. D., Workman, M. L., Digital Control of Dynamic
Systems, 3Td ed., AddisonWesley, Inc., Reading, MA, 1998, pp. 649687.
5. HewlettPackard, The Fundamentals ofModal Testing Application Note 2433,
HewlettPackard Co., Palo Alto, California, 1986.
6. Jeans, A. H., "Analysis of the Dynamics of a Type 4 Suspension, " ASME Journal of
Vibration and Acoustics, Vol. 114, January 1992, pp. 7478.
7. Miu, D. K., "Physical Interpretation of Transfer Function Zeros for Simple Control
Systems with Mechanical Flexibilities," Robotics Research, Vol. 26, 1990, pp. 67·
73.
8. Miu, D. K., Frees, G. M., Gompertz, R. S., "Tracking dynamics of Read/Write Head
Suspension in HighPerfonnance Small Fonn Factor Rigid Disk Drives," ASME
Journal ofVibration and Acoustics, Vol. 112, January 1990, pp.3339.
9. Miu, D. K., Yang, B., "On Transfer Function Zeros of General Colocated Control
Systems with Mechanical Flexibilities," ASMEJournal ofDynamic Systems,
Measurement, and Control, Vol. 116, March 1994, pp. 151154.
10. Ono, K., Teramoto, T., "Design Methodology to Stabilize the Natural modes of
Vibration ofa SwingAnn Positioning Mechanism," Advances in Information
Storage Systems, Vol. 4, 1992, pp. 343359.
64

11. Radwan, H. R., Huang, F., Serrano, 1., Oettinger, E., "ControlStructure Interaction
in Disk Drives Using Modal Superposition and FiniteElement Analysis," Journal of
Information Storage and Processing Systems, Vol 1, 1999, pp. 8794.
12. Radwan, H. R., Phan, Do To, Cao, K., "Effect of disk Drive Actuator Unbalance on
Track Following Response to External Vibration and Shock," IEEE Transactions on
Magnetics, Vol. 32, No.3, May 1996, pp. 17491755.
13. Radwan, H. R., Whaley, R., "ServoStructure Interaction in Disk Drives Using Finite
Element Analysis," ASME Advances in In/ormation Storage Systems, Vol. 5, 1993,
pp.lOl118.
14. Spanos, J. To, "ControlStructure Interaction in Precision Pointing Servo Loops,"
Journal o/Guidance, Vol. 12, No.2, MarchApril 1989, pp. 256263.
15. Wilson, C. J., Bogy, D. B., "Experimental Modal Analysis ofa Suspension
Assembly Loaded on a Rotating Disk," ASMEJournal o/Vibration andAcoustics,
Vol. 116, January 1994, pp. 8592.
65
APPENDIXA
66
Figure AI: Suspension 1st Bending
Figure A2: Suspension 1st Torsion
Figure A3: Suspension 2Dd Bending
Figure A4: Suspension 2nd Torsion
67

Figure A5: Suspension 3rd Bending
Figure A6: Suspension Sway
68
APPENDIX B
69

Figure B1: Arm lit Bending
Figure B2: Ann Sway
Figure B3: Arm 2Dd Bending
70

Figure B4: Arm lit Torsion
71

APPENDIX C
72

Figure CI: Yoke/Coil lit Bending
Figure C2: Yoke/Coil ttt Torsion
73

Figure C3: Yoke/Coil Sway
74

APPENDIX D
75
SubComponent FE Analytical HSA Model
Written by: Jeff Andress
1/17/99
'"""
function [mag,phase,freq]=actmod
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Model Parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Rigid Body Parameters %%%%%%%%%%%%%%%%%%%%%%%
Mt=13.5/4541386;
Kbrg=90000;
J=O. 00037/16;
Rhead=1.80;
Rcoil=.85;
Kt=9.6/16;
TPI=18145;
vpt=5;
Kbw=.0455;
%tota1 actuator mass (lbfs A 2/in)
%bearing stiffness (lbf/in)
%actuator inertia about pivot (lbfins A 2)
%pivot to gap distance (in)
%coil e.g. to pivot distance (in)
%torque constant (lbfin/Amp)
%tracks/inch
%volts/track
%bandwidth adjustment factor to compensate
%for gain of drive electronics
p=normal mode (normalized by modal mass)
mrn=modal mass
g=modal participation factor
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Model Parameters from FEM Analysis
%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% FEM Yoke Parameters: %%%%%%%%%%%%%%%%%%%%%%%%
my=2.57e5; %total mass of the yoke/coil FE model
%%% Mode 1: Yoke 1st Bending %%%
wyl=1l24 *2*pi;
mmy1=3.52826e3;
py1=(9.8118e3+4.0265e3)/sqrt(mmy1);
gy1=sqrt(0.0*my) ;
%%% Mode 2: Yoke Torsion %%%
wy2=182l*2*pi;
mmy2=2.95827e3;
py2=(3.0722e2+7.0812e3)/sqrt(mmy2) ;
gy2=sqrt(0.0*my);
%%% Mode 3: Yoke Sway %%%
wy3=5068*2*pi;
76
...
mrny3=4.93233e3;
py3=(1.46ge1+1.3564el)/sqrt(mrny3)/2;
gy3=sqrt(O.0087117*my)/2;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% FEM Arm Parameters: %%%%%%%%%%%%%%%%%%%%%%%%%
mas=6.133e6; %tota1 mass of the arm FE model
%%% Mode 1: Arm 1st Bending %%%
wal=1079*2*pi;
mrnal=l. 21434e8;
pa1=4.9603e6/sqrt(mrna1);
ga1=sqrt(O*mas);
%%% Mode 2: Arm 2nd Bending %%%
wa2=7508*2*pi;
rnma2=4.411692e10;
pa2=4.8836eS/sqrt(mma2);
ga2=sqrt(O.OOOOOl*rnas);
%%% Mode 3: Arm Sway %%%
wa3=7240*2*pi;
rnma3=1.10235e7;
pa3=3.1594el/sqrt(mma3)/3;
ga3=sqrt(O.216352*mas);
%%% Mode 4: Arm 1st Torsion %%%
wa4=7600*2*pi;
mrna4=6.50653elO;
pa4=1.3948e4/sqrt(rnma4);
ga4=sqrt(O.05*mas};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% FEM Suspension Parameters: %%%%%%%%%%%%%%%%%%
mst=2.699ge7; %total mass of suspension FE Model
%%% Mode 1: Suspension 1st Bending %%%
wsl=1970*2*pii
mrnsl=2.66717elO;
psl=5.0782e4/sqrt(mmsl);
gsl=sqrt(O.000042*rnst);
%%% Mode 2: Suspension 1st Torsion %%%
ws2=4200*2*pii
mrns2=3.8027gell;
ps2=3.2672e3/sqrt(rnms2);
gs2=sqrt(O.OOOll6*mst) ;
%%% Mode 3: Suspension 2nd Bending %%%
ws3=S820*2*pi;
mrns3=4.6880gell;
77

ps3=1.012ge3/sqrt(rnms3)*2;
gs3=sqrt(O.0215*mst);
%%% Mode 4: Suspension 2nd Torsion %%%
ws4=9800*2*pi;
mms4=9.73791e12;
ps4=3.1100e3/sqrt(mms4);
gs4=sqrt(O.00695*mst);
%%% Mode 5: Suspension 3rd Bending %%%
ws5=11770*2*pi;
rnms5=1.43310e11;
ps5=6.232ge3/sqrt(mms51:
gs5=sqrt(O.06*mst);
%%% Mode 6: Suspension Sway %%%
ws6=12830*2*pi;
rnms6=1.01493e12;
ps6=4.1307e3/sqrt(mms6)/2;
gs6=sqrt(0.156*mst);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Form Mass Matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
m=[Mt 0 gyl gy2 gy3 gal+mst*pal ga2+mst*pa2 ga3+mst*pa3 ga4+mst*pa4 gs1
gs2 gs3 gs4 gs5 gs6;
o J 0 0 0 0 0 a 0 0 a a 0 0 0;
gy1 0 1 0 0 0 0 0 0 a 0 0 0 0 0;
gy2 0 0 1 a 0 0 0 0 0 0 0 a 0 0;
gy3 0 a 0 1 a 0 0 0 0 0 0 0 0 0;
gal+mst*pal 0 0 0 0 1+mst*pa1*pal mst*pal*pa2 mst*pa1*pa3 mst*pa1*pa4
gsl*pal gs2*pal gs3*pa1 gs4*pal gs5*pal gs6*pal;
ga2+mst*pa2 0 0 0 0 mst*pal*pa2 1+mst*pa2*pa2 mst*pa2*pa3 mst*pa2*pa4
gsl*pa2 gs2*pa2 gs3*pa2 gs4*pa2 gs5*pa2 gs6*pa2;
ga3+mst*pa3 0 0 0 0 mst*pa1*pa3 mst*pa3*pa2 1+mst*pa3*pa3 mst*pa3*pa4
gsl*pa3 gs3*pa3 gs4*pa3 gs4*pa3 gs5*pa3 gs6*pa3;
ga4+mst*pa4 a 0 0 0 mst*pa1*pa4 mst*pa4*pa2 mst*pa4*pa3 1+mst*pa4*pa4
gsl*pa4 gs3*pa4 gs4*pa4 gs4*pa4 gs5*pa4 gs6*pa4;
gsl 0 0 0 0 gsl*pal gsl*pa2 gsl*pa3 gsl*pa4 1 0 0 0 0 0;
gs2 0 0 0 0 gs2*pal gs2*pa2 gs2*pa3 gs2*pa4 0 1 0 0 0 0:
gs3 0 a 0 0 gs3*pal gs3*pa2 gs3*pa3 gs3*pa4 0 0 1 0 0 0:
gs4 0 0 0 0 gs4*pal gs4*pa2 gs4*pa3 gs4*pa4 0 0 0 1 0 0;
gs5 0 a 0 0 gs5*pal gs5*pa2 gs5*pa3 gs5*pa4 0 0 0 0 1 0;
gs6 0 a 0 0 gs6*pal gs6*pa2 gs6*pa3 gs6*pa4 0 000 0 1];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Form Stiffness Matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
temp=[Kbrg 0 wyl~2 wy2~2 wy3~2 wa1~2 wa2~2 wa3~2 wa4~2 wsl~2 ws2~2
ws3~2 ws4~2 ws5~2 ws6~2]:
k=diag(temp,O);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Form Input Array
78

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f=Kt/Rcoil*[Mt*Rhead/J Rcoil pyl/2 py2/2 py3/2 0 0 0 0 0 0 0 0 0 0] I;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Form State Space
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
za=0.005;
zs=O.OI;
zy=O.OI;
%damping ratio for the arm modes
%damping ratio for the suspension modes
%damping ratio for the yoke modes
a=zeros(30,30);
temp(1:15)=I;;
a(1:15,16:30)=diag(temp,O);
a(16:30,1:15)=(inv(m)*k);
temp=[2*0.03*sqrt(Kbrg/Mt) 0 2*zy*wyl 2*zy*wy2 2*.05*wy3 2*za*wal
2*za*wa2 2*.Ol*wa3 2*za*wa4 2*zs*wsl 2*zs*ws2 2*zs*ws3 2*
O.OI*ws4 2*zs*ws5 2*zs*ws6];
a(16:30,16:30)=diag(temp,0);
b(I:15,1)=0;
b(16:30)=inv(m)*fi
c=[l Rhead 0 0 0 0 0 0 0 psI ps2 ps3 ps4 ps5 ps6];
c(16:30)=O;
d=O;
input=100*2*pi:5*2*pi:15000*2*pi;
if nargout==O
%frequency range for bode plot
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% load measured data for comparison
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
load c:\matlab\toolbox\user\measmag.txti
load c:\matlab\toolbox\user\measph.txti
load c:\matlab\toolbox\user\freq.txti
[mag2,phase2,freq2]=bode(a,b,c,d,l,input);
mag2=mag2*TPI*vpt*Kbw;
subplot (2, 1, 1)
handlel=semilogx(freq2/(2*pi),20*logI0(mag2), 'k',freq,measmag, 'k');
grid;
maxy=max(20*logI0(mag2))+lO;
miny=min(20*loglO(mag2)lO;
axis([1000 15000 60 20]);
ylabel('Magnitude (dB) ');
title('Modeled vs Measured Mechanical Transfer Function');
subplot(2,1,2)
handle2=semilogx(freq2/(2*pi),phase2, 'k',freq,measph, 'k');
grid;
79

axis«(1000 15000 1500 10));
ylabel('Phase (deg) ');
xlabel('Frequency (Hz) ');
set (handlel (1) , 'LineWidth' ,2) ;
set(handle2(1), 'LineWidth',2);
legend('Model', 'Measured');
else
[mag,phase,freq)=bode(a,b,c,d,l,input) ;
freq=freq/(2*pi);
mag=mag*TPI*vpt*Kbw;
end
80

VITA
j .. ,
Jeffrey Don Andress
Candidate for the Degree of
Master of Science
Thesis: DYNAMIC MODELING OF A HARD DISK DRIVE ACTUATOR USING
SUBCOMPONENT FINITE ELEMENT MODELS AND MODAL
SUPERPOSITION
Major Field: Mechanical Engineering
Biographical:
Education: Graduated from Bartlesville High School, Bartlesville, Oklahoma in
May 1989. Received Bachelor of Science Degree in Mechanical
Engineering from Oklahoma State University, Stillwater, Oklahoma in
December 1994. Completed requirements for the Master of Science
Degree in Mechanical Engineering at Oklahoma State University in May
2000.
Professional Experience: Worked as a Mechanical Design Engineer for Seagate
Technology, Oklahoma City, Oklahoma 1995 to present.