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

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xx Modeling and Control of Vibration in Mechanical Systems

12.8 |Φ| versus frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 239

12.9 3σ of PES NRRO. . . . . . . . . . . . . . . . . . . . . . . . . . . 240

12.10 Frequency response of the H

controller. . . . . . . . . . . . . . . 241

12.11 Frequency response of the open-l oop system with the H

controller. 241

12.12 Comparison of sensitivity functions. . . . . . . . . . . . . . . . . . 242

12.13 σ value o f PES NRRO versus track number. . . . . . . . . . . . . . 243

12.14 2-D controller for SSTW servo loop. . . . . . . . . . . . . . . . . 245

12.15 σ of PES NRRO versus track number. . . . . . . . . . . . . . . . . 246

12.16 Open-loop frequency response with stabil ization controller. . . . . 24 6

12.17 Sensitivi ty function with stabilization controll er (A

, B

, C

, D

). 247

12.18 Frequency response of controller (A

, B

, C

, D

). . . . . . . . . 248

12.19 Open-loop frequency response with controller (A

, B

, C

, D

). . 24 9

12.20 Sensitivi ty function with controll er (A

, B

, C

, D

). . . . . . . 249

12.21 Step respon se (Channel 1/2 /3: Reference/Output /Control signal). . 250

13.1 Friction compensation fo r the actuation system. . . . . . . . . . . . 252

13.2 Input u versus di splacement x with and wit hout compensation. . . . 253

13.3 Actuat or frequency responses with and without friction compensa-

tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

13.4 Actuat or frequency responses with fricti on compensation fo r differ-

ent displacements in voltage with 0. 5µm/V (Straight smooth lines:

the pur e double integrator). . . . . . . . . . . . . . . . . . . . . . . 25 4

13.5 Actuat or frequency responses with fricti on compensation fo r differ-

ent disp lacements in voltage wi th 0.5 µm/V (Straight smoot h lines:

the pur e double integrator). . . . . . . . . . . . . . . . . . . . . . . 25 4

13.6 Sensitivity functions with and without friction compensation. . . . . 255

13.7 Control structure of a plant P (s) with Youla parametrization ap-

proach and adaptive nonlinear compensation. . . . . . . . . . . . . 256

13.8 Comparison of error rejection frequency response without and with

of different p and Γ. . . . . . . . . . . . . . . . . . . . . . . . . 258

13.9 NR RO power spectrum with KYP lemma-based linear contro l and

nonlinear compensation (80% reduct ion before 400 Hz). . . . . . . 259

14.1 Experi mental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . 263

14.2 Frequency response of the VCM actuator measured by injecting a

swept sine wave of 5mV amplitude. . . . . . . . . . . . . . . . . . 263

14.3 The servo loop i n experiment. . . . . . . . . . . . . . . . . . . . . 2 64

14.4 Frequency response of the controller C(z). . . . . . . . . . . . . . 264

14.5 Frequency response of the sensitivity function S(z) with different

reference levels (i.e., actuator moving ranges are different). . . . . . 265

14.6 Frequency response of the quantizer before and after compensation

(bit number n = 6). . . . . . . . . . . . . . . . . . . . . . . . . . . 267

14.7 Frequency response of the quantizer with compensation (bit number

n = 8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

List o f Figures xxi

14.8 Frequency response of the quantizer with compensation (bit number

n = 1 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

14.9 Measured sensitivity function S

(z) with different bits n. . . . . . . 268

14.10 Sensitivi ty function |S

(f)| with f = 10 Hz versus bit n. . . . . . 269

14.11 Compensati on scheme of quantization effect. . . . . . . . . . . . . 270

14.12 Choosing the threshold δ and the scaling factor a. . . . . . . . . . . 270

14.13 Sensitivi ty function with quantization compensation. . . . . . . . . 271

15.1 Block diagram of FXLMS algorith m. . . . . . . . . . . . . . . . . 276

15.2 Filtered-X LMS adaptive feedback algorithm. . . . . . . . . . . . . 279

15.3 Adaptive inverse control scheme. . . . . . . . . . . . . . . . . . . 279

15.4 Block diagram of 2 × 2 adaptive feedback algorithm. . . . . . . . . 282

15.5 Block diagram of 6 × 1 FXLMS adaptive feedback control system. 283

15.6 General l ayo ut of the experimental setup. . . . . . . . . . . . . . . 286

15.7 60 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 287

15.8 210 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 287

15.9 240 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 288

15.10 270 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 288

15.11 Simulink diagram of automatic gain control. . . . . . . . . . . . . 291

15.12 180 Hz error signal without automatic gain control. . . . . . . . . . 2 92

15.13 180 Hz error signal with automatic gain control. . . . . . . . . . . 292

Symbols and Acronyms

: n-dimensio nal real Euclidean space

n×m

: set of n × m real matrices

: n × n identity matrix

(A, B, C, D): state-space representation of a system









: compact representation of system:

x(k + 1) = Ax(k) + B

w(k) + B

u(k)

z(k) = C

x(k) + D

w(k) + D

u(k)

y(k) = C

x(k) + D

w(k) + D

u(k)

diag{A

, A

, ···, A

}: block diagon al matrix with A

( not necessarily

square), j = 1, 2, ···, n, on the diagonal

: transpose of matrix X

∗

: complex conju gate transpose of matri x X

P ≥ 0: symmetric positive semideﬁnite matrix P ∈ R

n×n

P > 0: symmetric positive deﬁnite matrix P ∈ R

n×n

P ≥ Q: P − Q ≥ 0 for symmetric P, Q ∈ R

n×n

P > Q: P − Q > 0 for symmetric P, Q ∈ R

n×n

¯σ(X): largest singular value of X

Trace(X): trace of X

k · k: Euclidean vector norm

kwk

: ℓ

-norm of a si gnal {w(k)}, i.e.,

∞

k=0

kw(k))k

xxiii

xxiv Modeling and Control of Vibration in Mechanical Systems

ℓ

{[0, ∞)}: space of square summable sequences on {[0, ∞)}.

The signal {w(k)} is said to be from ℓ

{[0, ∞)}or simply ℓ

if kwk

< ∞.

kGk

: H

norm of transfer function G

kGk

∞

: H

∞

norm of transfer function G

Re( ): the real part of a complex number

Im( ): the imagin ary part of a complex number

ρ( ): spectral radius

AGC : automatic gain con trol

AV C : active vib ration contro l

deg: degree

det: determinant

DSA: Dy namic Signal Analyzer

F F T : fast Fourier transform

F XLMS: ﬁltered-X LMS

HDD: hard disk drive

LDV : Laser Dopp ler Vibrometer

LF T : Linear fractional transformation

LMI: linear matrix inequality

LMS: least mean square

LQG: linear quadratic Gaussian

LT R: loop transfer recovery

MEM S: micro electro-mechanical system

MSE: mean square error

NRRO: nonrepeatable runout

P ES: position error signal

Symbols and Acronyms xxv

P ID : proportional-integral- derivative

P LP F : phase lead peak ﬁlter

P ZT : lead zirconate titanate/piezoelectric

RBF : radial basis function

RM S: root mean square

RP M : rotations per min ute

RRO: repeatable runout

SST W : self-servo track writing

ST W : servo t rack writing

T M R: track misregistration

V CM : voice coil motor

Mechanical Systems and Vibration

When studying mechanical systems, we have t o include the subject of dyn amics and

vibration. Dynamics is a branch of mechanics th at deals with motion and its effect on

a bo dy. Unlike statics, wh ich deals with bodies at rest, dynamics takes into account

the effect of velocities and accelerations on the forces acting on bodies. Vibratio n is

regarded as a branch of dynamics, since forces and masses are taken into account in

vibration analysis. Thus it is natural that both dynamics and vibrations of mechanical

systems are studied in this book.

A vibr ation may be a signal, force, or temperature variation that affects the re-

sponse of a system in an unacceptable manner. If our analysis of sy stem response to

a vibration sh ows th at it regularly affects the system performance in an unacceptable

manner, we need to alter or control the vibratio n response and bring the response

within acceptable levels by addin g appropriate forces, called control forces which

are functions of system response such as displacement. This may require a design

by usi ng the value of the system response to generate add itional forces according to

certain rules or laws such that the modiﬁed response behaves according to desired

performance and within certain bounds. This results in a closed-loop system that

incorporates feedback controls. The purpose of d esigning a system with feedback

force is to minimize unwanted behavior. Examples include magnetic recording sys-

tems, Stewart platforms, positioning stages [27], the atomic force microscope (AFM)

[28], industry robot s [ 29], as well as some automotive systems [34]. The magnetic

recording system and Stewart platform are examples to be presented next to help

in ﬁxing these ideas more ﬁrmly. With such a grou nding, more advanced problems

become accessible.

1.1 Magnetic recording system

Figur e 1.1 shows a servo control loop of a hard disk drive (HDD) with a voice coil

motor (VCM) and a piezoelectric (PZT) actuated servo system. It consists of a stack

of ﬂat rotating disks with position ing information or servo information embedded in

their surfaces. The servo information is used to position the magnetic heads on the

disk surfaces. Position measurement of the magnetic heads is achieved by means o f

analyzing the po sition error signal (PES) calculated from the read back signal. To

2 Modeling and Control of Vibration in Mechanical Systems

have a disk d rive with a high storage capacity, the head positioning error with re-

spect to the target track center needs to be as small as possible. The error is mainly

due to, (1) t orque disturbances from spindle motor, (2) actuator pivot friction, (3)

airﬂow-induced non-repeatable di sk, suspension and slider vibrations, (4) mechani-

cal resonance vibr at ion, and (5) head sensing and electronic noises, media noise, and

quantization noises. Hence how to deal with the variety of the disturbances is crit-

ical to the head positioning accuracy, and subsequently the t rack density for a high

capacity disk drive.

FIGURE 1.1

The servo control loo p of a hard disk drive.

1.2 Stewart platform

The six- leg parallel linkage mechanism known as the “Stewart platform” was dis-

covered as early as in 1965 [30]. I t is desig ned according to a cubic conﬁguration,

consisting of two tri ang ular parallel plates connected to each other by six active legs

orthogonal to each other. Each leg is equipped with a voice coil actuator, a force

sensor and two ﬂexible joints. The closed k inematical linkage structure of a Stewart

platform has major advantages over any serial link robots: great rigidity, hig h force

to weight ratio, six d egrees of freedom (DOF), etc. [31] The Stewart platfor m i s

Mechanical Systems and Vibration 3

widely used as space born structures, as well as a high precision pointing device and

vibration isol at or.

Stewart platforms can be divided into two main classes according to the stiffness

of the legs: stiff type and soft type. For the soft design, each leg essentially acts as

an axial sp ringing parallel with a voice coil actuator, while the stiff design involves

piezoelectric or magneto restrictive legs whose extensions can b e controlled [32].

The Stewart platform has been widely used in active vibration control. It has an

important property for vibration control application: forces transmitted between the

mobile plate and th e b ase plate are totally axial forces of actuators. Thi s implies that

if the axial forces can be measured and eliminated, the vi bration created by these

forces can thus be eliminated. Thus the Stewart platform has become one of the

most popu lar approaches for 6-DOF active vibration control in pr ecision systems

due to its attractive properties.

FIGURE 1.2

Hexapod from Micromega Dynamics.

The Stewart platform (Hexapod) from Micromega-Dynamics used as a vibration

isolat ion device is shown in Fi gure 1.2. The hexapod has a cubic archi tecture and

consists of two parallel plates connected to each other by six active legs. The plates

are made of aluminum with a thickness of 20 mm and diameter of 250 mm, with

the weight of the mobile plate at 1 kg. Each leg of the active interface consists of

a linear piezoelectric actuator, a collocated fo rce sensor, and ﬂexible tips to connect

the two end plates. Flexible tips are u sed in order to avoid the problem of friction and

backlash, which comes with the use of spherical joints. The hexapod can be used to