Du C., Xie L. Modeling and Control of Vibration in Mechanical Systems

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14 Modeling and Control of Vibration in Mechanical Systems

The Fourier transform of x(t) is

X(ω) = a(ω) − ib(ω). (1.29)

An equivalent form of the Fourier transform is given by

x(t) =

∞

−∞

X(ω)e

iωt

dω, (1.30)

and

X(ω) =

2π

∞

−∞

x(t)e

−iωt

dt. (1.31 )

The spectrum of a non periodic signal is the magnitude of its Fourier transform,

that is, X(ω). Equation (1.31) implies that the transform is sy mmetric about ω = 0,

that is

X(−ω) = X(ω). (1.32 )

The transform of a ti me-shifted signal x(t − d) is X(ω)e

−iωd

and its spectrum is

the same as the spectrum of x(t), while the time shift d affects only the phase angle

of the transform.

1.6.2 Relationship between the Fourier and Laplace transforms

The Fourier transform X(ω) of x (t) is related to the Laplace transform X(s) as

follows.

X(ω) =

2π

X(s)|

s=iω

. (1.33 )

1.6.3 Spectral analys is

The spectral density, or power spectral density, S

(ω), is one of the most useful

functions in vibration testing. I t is deﬁned as

(ω) =

2π

∞

−∞

(τ)e

−iωτ

dτ, (1.34)

where

(τ) = lim

T →∞

T /2

−T /2

x(t)x(t + τ)dt (1.35 )

is the autocorrelation function and T is the time duration.

Mechanical Systems and Vibration 15

(τ) can be written as the inverse Fourier transform of S

(ω):

(τ) =

∞

−∞

(ω)e

iωτ

dω. (1.36)

The cross-spectral density is written as

(ω) =

2π

∞

−∞

(τ)e

−iωτ

dτ, (1.37 )

where

(τ) = lim

T →∞

T /2

−T /2

x(t)y(t + τ )dt (1.38 )

is the cross-correlation function.

A very useful relati on is that

(0) =

∞

−∞

(ω)dω = E(x

), (1.39 )

which means that the mean-square value can be computed from the spectral density.

Modeling of Disk Drive System and It s

Vibration

2.1 Introduction

Modeli ng plays an important role in any control system analysis and design. A

physi cal system has to be modeled in order to design a contro l system. Physical

systems are more or less nonli near and may vary with time. Researchers have made

many attempts to deal with a physical system: approximating it with a linear model

at an operating point , ﬁnding a control strategy that is robust and adaptable to the

changes in the physical system.

The system modeling in this chapter is discussed from the ﬁrst principl e of actu-

ators. System dyn amics measurement in the frequency domain is used to determine

the parameters of a model in the form of transfer function. To have a more realist ic

model, uncertainties, especially due to high frequency unmodeled d ynamics, have to

be involved inherently. Moreover, nonlinearity induced by friction is measured under

the condition that the actuator is controlled to avoid unsteady signal measurement.

The hysteresis of friction versus actuator position is then ob tained f rom the measure-

ment in closed-loop. An operator based modeling approach is adopted to model the

hysteresis, and an optimal model is obtained by minimizing the energy gain between

the position and the modeling error.

In this chapter, vibr at ion modeling is based primarily on the spectral decomposi-

tion of the error signal measured in a closed-loop system. A decoupling procedure

is proposed which leads to approximate distu rbance and noise models. Particularly,

low-frequency disturbances are modeled as the output of an adaptive nonlinear mech-

anism with the error signal as the in put.

2.2 System description

A hard disk drive as shown in Figure 1.1 includes ﬁve major parts: baseplate and

cover, spindle and motor assembly, actuator assembly, disk, head/suspension assem-

bly, and electronics card.

18 Modeling and Control of Vibration in Mechanical Systems

The spindle and motor assembly includes disk clamps to clamp disks. The actuator

assembly contains an actuator driven by voice-coil motor (VCM) and mounted via

ball-bearing at each end of a pivot shaft, ﬂex cable carrying head and VCM leads, and

arms to support suspension/ head extension between the disks. In the head/suspension

assembly, an airbearing surface is created on surface next to the rotating disk, the

slider carrying heads ﬂies on top of the disk surface, and a gimbal attaches the slider

to the suspension. The electronics card involves drivers for the sp indle motor and

VCM, read/write (R /W) electronics, a servo demodulator, and micro processors for

servo control and control of interface to host computer.

The actuator servo channel consists of a demodulator producing position infor ma-

tion from the servo burst read from the disk during seeking and following; a servo

controller to control the po sition of the R/W head during reading, writing and seek-

ing; spindle control to keep the spindle ro tating at a speciﬁc speed wit h a minimum

speed ﬂuctuation and a power driver to drive the spindle motor and the VCM actu-

ator. The servo or position information is used to position the magnetic head on the

disk sur faces. Position measurement of the magnetic head is achieved by means of

analyzing the position error signal (PES) calculated from the read back signal.

The head-positioning servomechanism is a control syst em that p ositions the R/W

head from one track to another in minimum time, and repositions the R/W head over

a desired track with minimum statistical deviation from the track center. A settling

controller is used in between the above seeking and following modes.

The seek time is a measure of how fast the disk drive actuators can move the R/W

head to a desired location. The seek time is limited by the actuator behavior, acceler-

ation current level and the control algorith m. The major requirement in the seeking

process is fast and smooth seeking with small or even n o overshoot. Dual-st age ac-

tuation with a VCM as pri mary actuator and a mi croactuator as secondary actuator

works as one way to achieve fast seeking and settl ing due to higher bandwidth.

Once the actuator is regulating the position of the R/W head at the desired track,

the smaller the head position deviates from the desired track center, the closer the

tracks can be put together and the higher the track density becomes. In this stage,

the servo perfor mance is l imited by mechanical factors i n the actuator, d isk plat ter,

spind le motor, etc. An improved mechanical design is supposed to present less dis-

turbance, causing less off-track. On the other hand, a good closed-loop servo system

is expected to reject the disturbances. The error transfer function must be well de-

signed to yield a sufﬁciently small closed-loop non-repeatable runout. This typically

requires a satisfactory servo loop based on the disturbance spectrum. Generally, it

demands a high servo bandwidth, a high 0-dB crossover frequency and a low hump

of error rejection transfer function. A secondary microactuator activated to gether

with the VCM primary actuator is generally used to produce a higher bandwidth

closed-loo p system.

In a d isk drive, the positioning in formation or servo information (“servo bursts”)

is embedded in each disk surface. Servo bursts are conventionally written by costly

dedicated servo writing equipment external to t he disk drive, which uses a laser-

guided push-pin mechanism to position the write head on the disk su rface u ntil the

servo burst information is written on the d isk completely [53] [54]. The defects such

Modeling of Disk Drive System and Its Vibration 19

as non-circularity caused by the spindle motor vibration in the servo bursts will make

servo tracks more difﬁcult to follow in disk drives. The demand of bi g storage capac-

ity in hard disk drives requires high track density. More accurate placement of servo

bursts is thus required accordingly. As technologies such as servo mechanism and

head and media technology advance, the capability of writing and reading narrower

tracks is improved, result ing in an increased track density.

Conventional servo writers require a clean ro om enviro nment because the disk

and head will be exposed to th e environment to allow access to the external head and

actuator. A self-servo track writer regenerates timi ng and radial infor mation from

previously written t racks using the existing R/W head [135]. The external equip-

ment is no longer needed in servo pattern writing and thus no open access is required

for the head disk assembly and servo t rack writing does not have to be carried out

in a clean room environment. However, the self-servo track writing creates the ra-

dial error propagation prob lem, which will hinder the whole process of servo track

writing if not properly solved [140].

In an embedded servo, data and servo in formation are written on all tracks in an

interleaved manner. All tracks are divided into a ﬁxed number of radial sectors. Each

sector starts with a servo pattern followed by user d at a information. The pattern is

repeated for each sector. The ﬁelds in an embedded servo pattern include Pream-

ble, AGC ﬁeld, sector or index mark, track address in gray code and servo bursts.

The positio n feedback information for the disk drive servo mechanism consists of

two components: gray code and position error signal. The gray code track num-

ber provides coarse position information. It determines the absol ute posit ion o f the

read/write head during seeking and tracking. Position error sign al, or PES, is the

relative d isplacement of the R/W h ead from the track center. When PES and gray

code are combined together, the position of the read/write head i s obtained. Figure

2.1 shows one read back signal of the embedded servo. The A, B, C, D bursts give

ﬁne position error quant ifying the amount of track misregistration, i.e., the deviation

of the R /W head from the center of t he track.

2.3 System modeling

2.3.1 Modeling of a VCM actuator

A linear VCM actuator moves in and out along a disk radius in one direction. It

contains a coil which is rigidly attached to the structure to b e moved and suspended

in a magnetic ﬁeld created by permanent magnets. When a current passes through the

coil, a force is prod uced which accelerates the actuator radi ally inward or outward,

depending on the direction of the current. The produced force is a function of the

current i

. Approximately,

= k

, (2.1)

20 Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.1

A read b ack signal of embedded servo.

where k

is a linearized nominal value called torque constant.

The resonance of the actuator is mainly due to the ﬂexibility of the pivot bearing,

arm, suspension, etc. When the bandwidth of a control loop is very low and the

resonance may not be a l imiting factor to the control design, the actuator model

can be considered as the simpliﬁed and rigid one which is a double integrator with

transfer function k/s

, i.e.,

y =

u, (2.2)

˙y = k

v, (2.3)

˙v = k

u, (2.4)

where u is th e i nput to the actuator, y and v are the displacement and t he velocity of

the read/write head, k

is the position measurement gain, and k

= k

/m with the

actuator mass m.

With higher bandwidth, the actuato r resonances have to b e considered in the con-

trol design, since the ﬂexible resonance modes will r educe the sy stem stability and

affect control performance if ignored. Then the actuator model becomes

y =

(s)u, (2.5)

which in cl udes the resonance model P

(s). Let ω

= 2πf

correspond to a sing le

resonance frequency f

, and ξ

be the associated damping coefﬁcient. A second

order transfer function can be used to represent the resonance, i.e.,

(s) =

+ 2ξ

s + ω

. (2.6)

Modeling of Disk Drive System and Its Vibration 21

Different ξ

gives different frequency responses, shown in Figure 2.2. The peak

of magnitude is higher when ξ

decreases.

Magnitude(dB)

Frequency(Hz)

Phase(deg)

FIGURE 2.2

Frequency responses of a second o rder transfer fun ction.

Other forms of P

(s) in clude

(s) =

s + b

+ 2ξ

s + ω

(2.7)

and

(s) =

+ b

s + b

+ 2ξ

s + ω

(2.8)

with zeros included to facilitate a phase lift which is usually associated with reso-

nance modes.

For lightly damped resonance, 0 .005 ≤ ξ

≤ 0.05 is typical.

22 Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.3

Measured VCM Bode plots (straight lines: pure d ouble integrator k/s

Modeling of Disk Drive System and Its Vibration 23

Figur e 2.3 shows the measured frequency response of a VCM actuator system. By

curve ﬁtti ng each resonant mode, one can obtain the parameters of the t ransfer func-

tion. The deviation fr om the double integrator model in low frequencies in Figure

2.3 is due t o the pivot f riction and other nonlinearities. The nonlinearity modeling

will be discussed in detail in the next section.

Consider the resonance model in (2.7) ((2.6) and ( 2.8) can be handled similarly).

˙y = k

v, (2.9)

˙v = k

s + b

+ 2ξ

s + ω

u. (2.10)

Deﬁne

+ 2ξ

s + ω

u, x

sω

+ 2ξ

s + ω

u, (2.11 )

i.e.,

˙x

= ω

. (2.12 )

Then, we have a state-space description







˙y

˙v

˙x













0 k

0 0

0 0 k

0 0 0 ω

0 0 −ω

−2ξ































u. (2.13 )

More resonance modes can be modeled similarly.

Assume that nonlinearities such as friction and bias force and resonances are prop-

erly compensated, the actuator model is a double integrator model. It is wr itten as



˙y

˙v





0 k

0 0









u. (2.14 )

On the other hand, to make the mathematical models more realistic, we need to

consider uncertainties inh erent in the plant model. Note that it is difﬁcult to involve

all high-frequency dynamics i n a model -based servo control design. In this case, an

uncertainty is introduced in the plant model. For instance, a model with multiplica-

tive uncertainty can be given by P = (1+∆)P

. Fig ure 2.4 shows the multi plicative

uncertainty ∆ of a VCM. To meet the robustness requirement against this u nmodeled

high frequency dynamics, we need to properly handle the uncertain system so that it

can perform under strict er margins.

2.3.2 Modeling of friction

In this section, we focus on friction modeling. There are basically two kinds of

methodologies for friction control in the literature: Model-based fricti on compen-

sation and non-model based f riction control. Friction models for the former can be