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23.3 Vibration-Control Systems Design and Implementation

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23.3.1 Introduction

This section provides the basic fundamental concepts for vibration-control systems design and

implementation. These systems are classified into two categories: vibration absorbers and vibrationcontrol

systems. Some related practical developments in ARAs and piezoelectric vibration control of

flexible structures are also provided.

23.3.2 Vibration Absorbers

Undesirable vibrations of flexible structures have been effectively reduced using a variety of dynamic

vibration absorbers. The active absorption concept offers a wideband of vibration-attenuation

FIGURE 23.13 Schematic of the adjustable effective inertia vibration absorber. (Source: From Jalili, N. et al., Int. J.

Model. Simulat., 21, 148 – 154, 2001. With permission.)

23-12 Vibration and Shock Handbook

frequencies as well as real-time tunability as two

major advantages. It is clear that the active control

could be a destabilizing factor for the combined

system, and therefore, the stability of the combined

system (i.e., the primary and the absorber

subsystems) must be assessed.

An actively tuned vibration-absorber scheme

utilizing a resonator generation mechanism forms

the underlying concept here. For this, a stable

primary system (see Figure 23.7, top, for instance)

is forced into a marginally stable one through the

addition of a controlled force in the active unit (see

Figure 23.7, bottom). The conceptual design for

generating such resonance condition is demonstrated

in Figure 23.14, where the system’s

dominant characteristic roots (poles) are moved

and placed on the imaginary axis. The absorber

then becomes a resonator capable of mimicking the vibratory energy from the primary system at the

point of attachment. Although there seem to be many ways to generate such resonance, only two widely

accepted practical vibration-absorber resonators are discussed next.

23.3.2.1 Delayed-Resonator Vibration Absorbers

A recent active vibration-absorption strategy, the delayed resonator (DR), is considered to be the first

type of active vibration absorber when operated on a flexible beam (Olgac and Jalili, 1998; Olgac and

Jalili, 1999). The DR vibration absorber offers some attractive features in eliminating tonal vibrations

from the objects to which it is attached (Olgac and Holm-Hansen, 1994; Olgac, 1995; Renzulli et al.,

1999), some of which are real-time tunability, the stand-alone nature of the actively controlled absorber,

and the simplicity of the implementation. Additionally, this single-DoF absorber can also be tuned to

handle multiple frequencies of vibration (Olgac et al., 1996). It is particularly important that the

combined system, that is, the primary structure and the absorber together, is asymptotically stable when

the DR is implemented on a flexible beam.

23.3.2.1.1 An Overview of the Delayed-Resonator Concept

An overview of the DR is presented here to help the reader. The equation of motion governing the

absorber dynamics alone is

max€aðtÞ þ cax_aðtÞ þ kaxaðtÞ 2 uðtÞ ¼ 0; uðtÞ ¼ gx€aðt 2tÞ ð23:13Þ

where uðtÞ represents the delayed acceleration feedback. The Laplace domain transformation of this

equation yields the characteristics equation

mas2 þ cas þ ka 2 gs2 e2ts ¼ 0 ð23:14Þ

Without feedback ð g ¼ 0Þ; this structure is dissipative with two characteristic roots (poles) on the left

half of the complex plane. For g and t . 0; however, these two finite stable roots are supplemented by

infinitely many additional finite roots. Note that these characteristic roots (poles) of Equation 23.14 are

discretely located (say at s ¼ a þ jv), and the following relation holds:

g ¼

lmas2 þ cas þ kal

ls2l eta ð23:15Þ

where l·l denotes the magnitude of the argument.

Poles of passive absorber Poles of ARA

Real

Imag

S-plane

FIGURE 23.14 Schematic of the active resonator

absorber concept through placing the poles of the

characteristic equation on the imaginary axis.

Vibration Control 23-13

Using Equation 23.15, the following observation can be made:

* For g ¼ 0: there are two finite stable poles and all the remaining poles are at a ¼ 21:

* For g ¼ þ1: there are two poles at s ¼ 0; and the rest are at a ¼ þ1:

Considering these and taking into account the continuity of the root loci for a given time delay, t; and

as g varies from 0 to 1, it is obvious that the roots of Equation 23.14 move from the stable left half to the

unstable right half of the complex plane. For a certain critical gain, gc; one pair of poles reaches the

imaginary axis. At this operating point, the DR becomes a perfect resonator and the imaginary

characteristic roots are s ¼ ^jvc; where vc is the resonant frequency and j ¼

ffiffiffiffi

21 p : The subscript “c”

implies the crossing of the root loci on the imaginary axis. The control parameters of concern, gc and tc;

can be found by substituting the desired s ¼ ^jvc into Equation 23.14 as

gc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðcavcÞ2 þ ðmavc 2 kaÞ2

; tc ¼

tan21 cavc

mav2

c 2 ka

􀀒 􀀓

þ 2ð‘ 2 1Þp

􀀘 􀀙

; ‘ ¼ 1; 2; …

ð23:16Þ

When these gc and tc are used, the DR structure mimics a resonator at frequency vc: In turn, this

resonator forms an ideal absorber of the tonal vibration at vc: The objective of the control, therefore, is to

maintain the DR absorber at this marginally stable point. On the DR stability, further discussions can be

found in Olgac and Holm-Hansen (1994) and Olgac et al. (1997).

23.3.2.1.2 Vibration-Absorber Application on Flexible Beams

We consider a general beam as the primary system with absorber attached to it and subjected to a

harmonic force excitation, as shown in Figure 23.15. The point excitation is located at b; and the absorber

is placed at a: A uniform cross section is considered for the beam and Euler– Bernoulli assumptions are

made. The beam parameters are all assumed to be constant and uniform. The elastic deformation from

the undeformed natural axis of the beam is denoted by yðx; tÞ and, in the derivations that follow, the dot

(·) and prime (0) symbols indicate a partial derivative with respect to the time variable, t; and position

variable x; respectively.

Under these assumptions, the kinetic energy of the system can be written as

T ¼

ðL

›y

›t

􀀏 􀀐2

dx þ

maq_2

a þ

meq_2

e ð23:17Þ

The potential energy of this system using linear strain is given by

U ¼

ðL

›2y

›x2

􀁻 !2

dx þ

ka{yða; tÞ 2 qa}2 þ

ke{yðb; tÞ 2 qe}2 ð23:18Þ

c ka a

gqta(−τ)

ce f(t)

E, I, A, L

FIGURE 23.15 Beam – absorber – exciter system configuration. (Source: From Olgac, N. and Jalili, N., J. Sound Vib.,

218, 307 – 331, 1998. With permission.)

23-14 Vibration and Shock Handbook

The equations of motion may now be derived by applying Hamilton’s Principle. However, to facilitate the

stability analysis, we resort to an assumed-mode expansion and Lagrange’s equations. Specifically, y is

written as a finite sum “Galerkin approximation”:

yðx; tÞ ¼

i¼1

FiðxÞqbiðtÞ ð23:19Þ

The orthogonality conditions between these mode shapes can also be derived as (Meirovitch, 1986)

ðL

rFiðxÞFjðxÞdx ¼ Nidij;

ðL

EIF 00 i ðxÞF 00 j ðxÞdx ¼ Sidij ð23:20Þ

where i; j ¼ 1; 2; …; n; dij is the Kronecker delta, and Ni and Si are defined by setting i ¼ j in

Equation 23.20.

The feedback of the absorber, the actuator excitation force, and the damping dissipating forces in both

the absorber and the exciter are considered as non-conservative forces in Lagrange’s formulation.

Consequently, the equations of motion are derived.

Absorber dynamics is governed by

maq€aðtÞ þ ca q_aðtÞ 2

i¼1

FiðaÞq_biðtÞ

( )

þ ka qaðtÞ 2

i¼1

FiðaÞqbiðtÞ

( )

2 gq€aðt 2tÞ ¼ 0 ð23:21Þ

The exciter is given by

meq€eðtÞ þ ce q_eðtÞ 2

i¼1

FiðbÞq_biðtÞ

( )

þ ke qeðtÞ 2

i¼1

FiðbÞqbiðtÞ

( )

¼ 2f ðtÞ ð23:22Þ

Finally, the beam is represented by

Niq€biðtÞ þ SiqbiðtÞ þ ca

i¼1

FiðaÞq_biðtÞ 2 q_aðtÞ

( )

FiðaÞ þ ce

i¼1

FiðbÞq_biðtÞ 2 q_eðtÞ

( )

FiðbÞ

þ ka

i¼1

FiðaÞqbiðtÞ 2 qaðtÞ

( )

FiðaÞ þ ke

i¼1

FiðbÞqbiðtÞ 2 qeðtÞ

( )

FiðbÞ þ gFiðaÞq€aðt 2tÞ

¼ f ðtÞFiðbÞ; i ¼ 1; 2; …; n ð23:23Þ

Equation 23.21 to Equation 23.23 form a system of n þ 2 second-order coupled differential equations.

By proper selection of the feedback gain, the absorber can be tuned to the desired resonant frequency,

vc: This condition, in turn, forces the beam to be motionless at a; when the beam is excited by a tonal

force at frequency vc: This conclusion is reached by taking the Laplace transform of Equation 23.21 and

using feedback control law for the absorber. In short,

Y ða; sÞ ¼

i¼1

FiðaÞQbiðsÞ ¼ 0 ð23:24Þ

where Y ða; sÞ ¼ I{yða; tÞ}; QaðsÞ ¼ I{qaðtÞ} and QbiðsÞ ¼ I{qbiðtÞ}: Equation 23.24 can be rewritten in

time domain as

yða; tÞ ¼

i¼1

FiðaÞqbiðtÞ ¼ 0 ð23:25Þ

which indicates that the steady-state vibration of the point of attachment of the absorber is eliminated.

Hence, the absorber mimics a resonator at the frequency of excitation and absorbs all the vibratory

energy at the point of attachment.

Vibration Control 23-15

23.3.2.1.3 Stability of the Combined System

In the preceding section, we have derived the equations of motion for the beam – exciter – absorber system

in its most general form. As stated before, inclusion of the feedback control for active absorption is,

indeed, an invitation to instability. This topic is treated next.

The Laplace domain representation of the combined system takes the form (Olgac and Jalili, 1998)

AðsÞQðsÞ ¼ FðsÞ ð23:26Þ

where

QðsÞ ¼

QaðsÞ

QeðsÞ

Qb1ðsÞ

.. .

QbnðsÞ

8>>>>>>>>>><

>>>>>>>>>>:

9>>>>>>>>>>=

>>>>>>>>>>;

ðnþ2Þ£1

; FðsÞ ¼

2FðsÞ

.. .

8>>>>>>>>>><

>>>>>>>>>>:

9>>>>>>>>>>=

>>>>>>>>>>;

ðnþ2Þ£1

;

AðsÞ ¼

mas2 þcas þka 2 gs2 e2ts 0 2F1ðaÞðcas þkaÞ · · · 2FnðaÞðcas þkaÞ

0 mes2 þces þke 2F1ðbÞðces þkeÞ · · · 2FnðbÞðces þkeÞ

maF1ðaÞs2 meF1ðbÞs2 N1s2 þcs þS1ð1 þjdÞ · · · 0

.. .

. .

. .. .

maFnðaÞs2 meFnðbÞs2 0 · ·· Nns2 þcs þSnð1 þjdÞ

BBBBBBBBBBBBB@

CCCCCCCCCCCCCA

ð23:27Þ

In order to assess the combined system stability, the roots of the characteristic equation, detðAðsÞÞ ¼ 0 are

analyzed. The presence of feedback (transcendental delay term for this absorber) in the characteristic

equations complicates this effort. The root locus plot observation can be applied to the entire system. It is

typical that increasing feedback gain causes instability as the roots move from left to right in the complex

plane. This picture also yields the frequency range for stable operation of the combined system (Olgac

and Jalili, 1998).

23.3.2.1.4 Experimental Setting and Results

The experimental setup used to verify the findings is shown in Figure 23.16. The primary structure is a

3=80 in: £ 10 in: £ 120 in: steel beam (2) clamped at both ends to a granite bed (1). A piezoelectric actuator

with a reaction mass (3 and 4) is used to generate the periodic disturbance on the beam. A similar

actuator-mass setup constitutes the DR absorber (5 and 6). The two setups are located symmetrically at

one quarter of the length along the beam from the center. The feedback signal used to implement the DR

is obtained from the accelerometer (7) mounted on the reaction mass of the absorber structure. The

other accelerometer (8) attached to the beam is present only to monitor the vibrations of the beam and to

evaluate the performance of the DR absorber in suppressing them. The control is applied via a fast data

acquisition card using a sampling of 10 kHz.

The numerical values for this beam – absorber– exciter setup are taken as

* Beam: E ¼ 210 GPa, r ¼ 1:8895 kg/m

* Absorber: ma ¼ 0:183 kg, ka ¼ 10,130 kN/m, ca ¼ 62:25 N sec/m, a ¼ L=4

* Exciter: me ¼ 0:173 kg, ke ¼ 6426 kN/m, ce ¼ 3:2 N sec/m, b ¼ 3L=4

23-16 Vibration and Shock Handbook

23.3.2.1.5 Dynamic Simulation and Comparison with Experiments

For the experimental set up at hand, the natural frequencies are measured for the first two natural modes,

v1 and v2: These frequencies are obtained much more precisely than those of higher-order natural

modes. Table 23.1 offers a comparison between the experimental (real) and analytical (ideal) clamped –

clamped beam natural frequencies.

The discrepancies arrive from two sources: first, the experimental frequencies are structurally

damped natural frequencies, and second, they reflect the effect of partially clamped BCs. The

theoretical frequencies, on the other hand, are evaluated for an undamped ideal clamped – clamped

beam.

After observing the effect of the number of modes used on the beam deformation, a minimum of three

natural modes are taken into account. We then compare the simulated time response vs. the experimental

results of vibration suppression. Figure 23.17 shows a test with the excitation frequency vc ¼ 1270 Hz.

The corresponding theoretical control parameters are gc theory ¼ 0:0252 kg and tc theory ¼ 0:8269 msec.

The experimental control parameters for this frequency are found to be gc exp: ¼ 0:0273 kg and

tc exp: ¼ 0:82 msec. The exciter disturbs the beam for 5 msec, then the DR tuning is triggered.

The acceleration of the beam at the point of attachment decays exponentially. For all intents and

purposes, the suppression takes effect in approximately 200 ms. These results match very closely with

the experimental data, Figure 23.18. The only noticeable difference is in the frequency content of the

exponential decay. This property is dictated by the dominant poles of the combined system. The

imaginary part, however, is smaller in the analytical study. This difference is a nuance that does not affect

the earlier observations.

3/8"

14"

12"

7 6

(a)

(b)

FIGURE 23.16 (a) Experimental structure and (b) schematic depiction of the setup. (Source: From Olgac, N. and

Jalili, N., J. Sound Vib., 218, 307 – 331, 1998. With permission.)

Vibration Control 23-17

23.3.2.2 Active Resonator Vibration Absorbers

One novel implementation of the tuned vibration absorbers is the ARA (Knowles et al., 2001b). The

concept of the ARA is closely related to the concept of the delayed resonator (Olgac and Holm-Hansen,

1994; Olgac and Jalili, 1999). Using a simple position (or velocity or acceleration) feedback control within

the absorber section, it enforces the dominant characteristic roots of the absorber subsection to be on the

imaginary axis, and hence leading to resonance. Once the ARA becomes resonant, it creates perfect

vibration absorption at this frequency.

A very important component of any active vibration absorber is the actuator unit. Recent advances in

smart materials have led to the development of advanced actuators using piezoelectric ceramics, shape

memory alloys, and magnetostrictive materials (Garcia et al., 1992; Shaw, 1998). Over the past two

decades, piezoelectric ceramics have been utilized as potential replacements for conventional transducers.

TABLE 23.1 Comparison between Experimental and Theoretical Beam Natural Frequencies (Hz)

Natural Modes Peak Frequencies (Experimental) Natural Frequencies (Clamped – Clamped)

First mode 466.4 545.5

Second mode 1269.2 1506.3

FIGURE 23.17 Beam and absorber response to 1270 Hz disturbance, analytical. (Source: From Olgac, N. and Jalili,

N., J. Sound Vib., 218, 307 – 331, 1998. With permission.)

FIGURE 23.18 Beam and absorber response to 1270 Hz disturbance, experimental. (Source: From Olgac, N. and

Jalili, N., J. Sound Vib., 218, 307 – 331, 1998. With permission.)

23-18 Vibration and Shock Handbook

These materials are compounds of lead zirconate – titanate (PZT). The PZT properties can be optimized

to suit specific applications by appropriate adjustment of the zirconate – titanate ratio. Specifically, a

piezoelectric inertial actuator is an efficient and inexpensive solution for active structural vibration

control. As shown in Figure 23.19, it applies a point force to the structure to which it is attached.

23.3.2.2.1 An Overview of PZT Inertial Actuators

PZT inertial actuators are most commonly made out of two parallel piezoelectric plates. If voltage is

applied, one of the plates expands as the other one contracts, hence producing displacement that is

proportional to the input voltage. The resonance of such an actuator can be adjusted by the size of the

inertial mass (see Figure 23.19). Increasing the size of the inertial mass will lower the resonant frequency

and decreasing the mass will increase it. The resonant frequency, fr; can be expressed as

fr ¼

ffiffiffiffiffi

ð23:28Þ

where ka is the effective stiffness of the actuator, ma; is defined as

ma ¼ mePZT þ minertial þ macc ð23:29Þ

The PZT effective mass is mePZT

; minertial is the inertial mass, and macc is the accelerometer mass. Using a

simple single-DoF system (see Figure 23.19), the parameters of the PZT inertial actuators can be

experimentally determined (Knowles et al., 2001a). This “parameter identification” problem is an inverse

problem. We refer interested readers to Banks and Ito (1988) and Banks and Kunisch (1989) for a general

introduction to parameter estimation or inverse problems governed by differential equations.

23.3.2.2.2 Active Resonator Absorber Concept

The concept of ARA is closely related to that of the DR (Olgac, 1995; Olgac and Holm-Hansen, 1995;

Olgac and Jalili, 1998). Instead of a compensator, the DR uses a simple delayed position (or velocity, or

acceleration) feedback control within the absorber subsection for the mentioned “sensitization.”

In contrast to that of the DR absorber, the characteristic equation of the proposed control scheme is

rational in nature and is hence easier to implement when closed-loop stability of the system is concerned.

Similar to the DR absorber, the proposed ARA requires only one signal from the absorber mass, absolute

or relative to the point of attachment (see Figure 23.7 bottom). After the signal is processed through a

compensator, an additional force is produced, for instance, by a PZT inertial actuator. If the compensator

parameters are properly set, the absorber should behave as an ideal resonator at one or even more

frequencies. As a result, the resonator will absorb vibratory energy from the primary mass at given

frequencies. The frequency to be absorbed can be tuned in real time. Moreover, if the controller or the

actuator fails, the ARA will still function as a passive absorber, and thus it is inherently fail-safe. A similar

vibration absorption methodology is given by Filipovic´ and Schro¨der (1999) for linear systems. The ARA,

however, is not confined to the linear regime.

Inertial mass

PCB Series 712 Actuator

Structure Structure

ka ca u(t)

FIGURE 23.19 A PCB series 712 PZT inertial actuator (left), schematic of operation (middle), and a simple single-

DoF mathematical model (right). (Active Vibration Control Instrumentation, A Division of PCB Piezotronics, Inc.,

www.pcb.com.)

Vibration Control 23-19

For the case of linear assumption for the PZT actuator, the dynamics of the ARA (Figure 23.7, bottom)

can be expressed as

max€aðtÞ þ cax_aðtÞ þ kaxaðtÞ 2 uðtÞ ¼ cax_1ðtÞ þ kax1ðtÞ ð23:30Þ

where x1ðtÞ and xaðtÞ are the respective primary (at the absorber point of attachment) and absorber mass

displacements. The mass, ma; is given by Equation 23.29 and the control, uðtÞ; is designed to produce

designated resonance frequencies within the ARA.

The objective of the feedback control, uðtÞ; is to convert the dissipative structure (Figure 23.7, top) into

a conservative or marginally stable one (Figure 23.7, bottom) with a designated resonant frequency, vc:

In other words, the control aims the placement of dominant poles at ^jvc for the combined system,

where j ¼

ffiffiffiffi

21 p (see Figure 23.14). As a result, the ARA becomes marginally stable at particular

frequencies in the determined frequency range. Using simple position (or velocity or acceleration)

feedback within the absorber section (i.e., UðsÞ ¼ U􀀊 ðsÞXaðsÞ), the corresponding dynamics of the ARA,

given by Equation 23.30, in the Laplace domain become

ðmas2 þ cas þ kaÞXaðsÞ 2 U􀀊 ðsÞXaðsÞ ¼ CðsÞXaðsÞ ¼ ðcas þ kaÞX1ðsÞ ð23:31Þ

The compensator transfer function, U􀀊 ðsÞ; is then selected such that the primary system displacement at

the absorber point of attachment is forced to be zero; that is

CðsÞ ¼ ðmas2 þ cas þ kaÞ 2 U􀀊 ðsÞ ¼ 0 ð23:32Þ

The parameters of the compensator are determined through introducing resonance conditions to the

absorber characteristic equation, CðsÞ; that is, the equations Re{CðjviÞ} ¼ 0 and Im{CðjviÞ} ¼ 0 are

simultaneously solved, where i ¼ 1; 2; …; l and l is the number of frequencies to be absorbed. Using

additional compensator parameters, the stable frequency range or other properties can be adjusted in

real time.

Consider the case where U ðsÞ is taken as a proportional compensator with a single time constant based

on the acceleration of the ARA, given by

U ðsÞ ¼ U􀀊 ðsÞXaðsÞ; where U􀀊 ðsÞ ¼

gs2

1 þ Ts ð23:33Þ

Then, in the time domain, the control force, uðtÞ; can be obtained from

uðtÞ ¼

ðt

e2ðt2tÞ=T x€aðtÞdt ð23:34Þ

To achieve ideal resonator behavior, two dominant roots of Equation 23.32 are placed on the imaginary

axis at the desired crossing frequency, vc: Substituting s ¼ ^jvc into Equation 23.32 and solving for the

control parameters, gc and Tc; one can obtain

gc ¼ ma

a v2 2

􀀏 􀀐 2

mav2 þ 1

BBB@

CCCA

; Tc ¼

ffiffiffiffiffiffiffi

ka=ma

ffiffiffiffiffiffi

maka

v2 2

􀀏 􀀐; for v ¼ vc ð23:35Þ

The control parameters, gc and Tc; are based on the physical properties of the ARA (i.e., ca; ka; and maÞ as

well as the frequency of the disturbance, v; illustrating that the ARA does not require any information

from the primary system to which it is attached. However, when the physical properties of the ARA are

not known within a high degree of certainty, a method to autotune the control parameters must be

considered. The stability assurance of such autotuning proposition will bring primary system parameters

into the derivations, and hence the primary system cannot be totally decoupled. This issue will be

discussed later in the chapter.

23-20 Vibration and Shock Handbook

23.3.2.2.3 Application of ARA to Structural-Vibration Control

In order to demonstrate the effectiveness of the proposed ARA, a simple single-DoF primary system

subjected to tonal force excitations is considered. As shown in Figure 23.20, two PZT inertial actuators

are used for both the primary (model 712-A01) and the absorber (model 712-A02) subsections. Each

system consists of passive elements (spring stiffness and damping properties of the PZT materials) and

active compartment with the physical parameters listed in Table 23.2. The top actuator acts as the ARA

with the controlled force, uðtÞ; while the bottom one represents the primary system subjected to the force

excitation, f ðtÞ:

The governing dynamics for the combined system can be expressed as

max€aðtÞ þ cax_aðtÞ þ kaxaðtÞ 2 uðtÞ ¼ cax_1ðtÞ þ kax1ðtÞ ð23:36Þ

m1x€1ðtÞ þ ðc1 þ caÞx_1ðtÞ þ ðk1 þ kaÞx1ðtÞ 2 {cax_aðtÞ þ kaxaðtÞ 2 uðtÞ} ¼ f ðtÞ ð23:37Þ

where x1ðtÞ and xaðtÞ are the respective primary and absorber displacements.

23.3.2.2.4 Stability Analysis and Parameter Sensitivity

The sufficient and necessary condition for asymptotic stability is that all roots of the characteristic

equation have negative real parts. For the linear system, Equation 23.36 and Equation 23.37, when

utilizing controller (Equation 23.34), the characteristic equation of the combined system (Figure 23.20,

right) can be determined and the stability region for compensator parameters, g and T; can be obtained

using the Routh – Hurwitz method.

23.3.2.2.5 Autotuning Proposition

When using the proposed ARA configuration in real applications where the physical properties are not

known or vary over time, the compensator parameters, g and T; only provide partial vibration

suppression. In order to remedy this, a need exists for an autotuning method to adjust the compensator

c1 k1

f (t)

u(t)

FIGURE 23.20 Implementation of the ARA concept using two PZT actuators (left) and its mathematical

model (right).

TABLE 23.2 Experimentally Determined Parameters of PCB Series 712 PZT

Inertial Actuators

PZT System Parameters PCB Model 712-A01 PCB Model 712-A02

Effective mass, mePZT (gr) 7.20 12.14

Inertial mass, minertial (gr) 100.00 200.00

Stiffness, ka (kN/m) 3814.9 401.5

Damping, ca (Ns/m) 79.49 11.48

Vibration Control 23-21

parameters, g and T; by some quantities, Dg and DT; respectively (Jalili and Olgac, 1998b; Jalili and

Olgac, 2000a). For the case of the linear compensator with a single time constant, given by Equation

23.33, the transfer function between primary displacement, X1ðsÞ; and absorber displacement, XaðsÞ; can

be obtained as

GðsÞ ¼

X1ðsÞ

XaðsÞ ¼

mas2 þ cas þ ka 2

gs2

1 þ Ts

cas þ ka ð23:38Þ

The transfer function can be rewritten in the frequency domain for s ¼ jv as

GðjvÞ ¼

X1ðjvÞ

XaðjvÞ ¼

2mav2 þ cavj þ ka þ

gv2

1 þ Tvj

cavj þ ka ð23:39Þ

where GðjvÞ can be obtained in real time by convolution of accelerometer readings (Renzulli et al., 1999)

or other methods (Jalili and Olgac, 2000a). Following a similar procedure as is utilized in Renzulli et al.

(1999), the numerator of the transfer function (Equation 23.39) must approach zero in order to suppress

primary system vibration. This is accomplished by setting

GðjvÞ þ DGðjvÞ ¼ 0 ð23:40Þ

where GðjvÞ is the real-time transfer function and DGðjvÞ can be written as a variational form of

Equation 23.39 as

DGðviÞ ¼

›G

›g

Dg þ

›G

›T

DT þ higher order terms ð23:41Þ

Since the estimated physical parameters of the absorber (i.e., ca; ka; and ma) are within the vicinity of the

actual parameters, Dg and DT should be small quantities and the higher-order terms of Equation 23.41

can be neglected. Using Equation 23.40 and Equation 23.41 and neglecting higher-order terms, we have

Dg ¼ Re½GðjvÞ􀀉

2Tcav2 2 ka þ kaT2v2

" #

þ Im½GðjvÞ􀀉

ca 2 T2v2ca þ 2kaT

" #

;

DT ¼ Re½GðjvÞ􀀉

ca 2 T2v2ca þ 2kaT

gv2

" #

þ Im½GðjvÞ􀀉

ka 2 2Tcav2 2 kaT2v2

gv3

" #

ð23:42Þ

In the above expressions, g and T are the current compensator parameters given by Equation 23.35, ca; ka;

and ma are the estimated absorber parameters, v is the absorber base excitation frequency, and GðjvÞ is

the transfer function obtained in real time. That is, the retuned control parameters, g and T; are

determined as follows

gnew ¼ gcurrent þ Dg and Tnew ¼ Tcurrent þ DT ð23:43Þ

where Dg and DT are those given by Equation 23.42. After compensator parameters, g and T; are adjusted

by Equation 23.43, the process can be repeated until lGð jvÞl falls within the desired level of tolerance.

GðjvÞ can be determined in real time as shown in Liu et al. (1997) by

Gð jvÞ ¼ lGðjvÞleð jfðjvÞÞ ð23:44Þ

where the magnitude and phase are determined assuming that the absorber and primary displacements

are harmonic functions of time given by

xaðtÞ ¼ Xa sinðvt þ faÞ; x1ðtÞ ¼ X1 sinðvt þ f1Þ ð23:45Þ

23-22 Vibration and Shock Handbook

With the magnitudes and phase angles of Equation 23.44, the transfer function can be determined from

Equation 23.44 and the following:

lGð jvÞl ¼

; fðjvÞ ¼ f1 2 fa

23.3.2.2.6 Numerical Simulations and Discussions

To illustrate the feasibility of the proposed absorption methodology, an example case study is presented.

The ARA control law is the proportional compensator with a single time constant as given in Equation

23.34. The primary system is subjected to a harmonic excitation with unit amplitude and a frequency of

800 Hz. The ARA and primary system parameters are taken as those given in Table 23.2. The simulation

was done using Matlab/Simulinkw and the results for the primary system and the absorber displacements

are given in Figure 23.21.

As seen, vibrations are completely suppressed in the primary subsection after approximately 0.05 sec,

at which the absorber acts as a marginally stable resonator. For this case, all physical parameters are

assumed to be known exactly. However, in practice these parameters are not known exactly and may vary

with time, so the case with estimated system parameters must be considered.

To demonstrate the feasibility of the proposed autotuning method, the nominal system parameters

ðma; m1; ka; k1; ca; c1Þ were fictitiously perturbed by 10% (i.e., representing the actual values) in the

simulation. However, the nominal values of ma; m1; ka; k1; ca, and c1 were used for calculation of the

compensator parameters, g and T: The results of the simulation using nominal parameters are given in

Figure 23.22. From Figure 23.22, top, the effect of parameter variation is shown as steady-state

oscillations of the primary structure. This undesirable response will undoubtedly be encountered when

the experiment is implemented. Thus, an autotuning procedure is needed.

The result of the first autotuning iteration is given in Figure 23.22, middle, where the control

parameters, g and T; are adjusted based on Equation 23.43. One can see tremendous improvement in the

primary system response with only one iteration (see Figure 23.22, middle). A second iteration is

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

− 8

− 6

− 4

− 2

Time [sec]

× 10−7 [m]

Absorber

Primary

FIGURE 23.21 Primary system and absorber displacements subjected to 800 Hz harmonic disturbance.

Vibration Control 23-23

performed, as shown in Figure 23.22, bottom. The response closely resembles that from Figure 23.21,

where all system parameters are assumed to be known exactly.

23.3.3 Vibration-Control Systems

As discussed, in vibration-control schemes, the control inputs to the systems are altered in order to

regulate or track a desired trajectory while simultaneously suppressing the vibrational transients in the

system. This control problem is rather challenging since it must achieve the motion-tracking objectives

while stabilizing the transient vibrations in the system. This section provides two recent control methods

developed for the regulation and tracking of flexible beams. The experimental implementations are

also discussed. The first control method is a single-input vibration-control system discussed in

0 0.01 0.02 0.03 0.04 0.05

Time, sec.

0.06 0.07 0.08 0.09 0.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

−1

− 0.5

0.5

−10

−5

−10

−5

× 10−6

× 10−7

Absorber

Primary

Absorber

Primary

Absorber

Primary

FIGURE 23.22 System responses (displacement, m) for (a) nominal absorber parameters; (b) after first autotuning

procedure and (c) after second autotuning procedure.

23-24 Vibration and Shock Handbook

Section 23.3.3.1, while the second application utilizes a secondary input control in addition to the

primary control input to improve the vibrational characteristics of the system (see Section 23.3.3.2).

23.3.3.1 Variable Structure Control of Rotating Flexible Beams

The vibration-control problem of a flexible manipulator consists in achieving the motion-tracking

objectives while stabilizing the transient vibrations in the arm. Several control methods have been

developed for flexible arms (Skaar and Tucker, 1986; Bayo, 1987; Yuh, 1987; Sinha, 1988; Lou, 1993; Ge

et al., 1997; de Querioz et al., 1999). Most of these methods concentrate on a model-based controller

design, and some of these may not be easy to implement due to the uncertainties in the design model, large

variations of the loads, ignored high frequency dynamics, and high order of the designed controllers. In

view of these methods, VSC is particularly attractive due to its simplicity of implementation and its

robustness to parameter uncertainties (Yeung and Chen, 1989; Singh et al., 1994; Jalili et al., 1997).

23.3.3.1.1 Mathematical Modeling

As shown in Figure 23.23, one end of the arm is

free and the other end is rigidly attached to a

vertical gear shaft, driven by a DC motor. A

uniform cross section is considered for the arm,

and we make the Euler– Bernoulli assumptions.

The control torque, t; acting on the output shaft, is

normal to the plane of motion. Viscous frictions

and the ever-present unmodeled dynamics of the

motor compartment are to be compensated via a

perturbation estimation process, as explained later

in the text. Since the dynamic system considered

here has been utilized in literature quite often, we

present only the resulting partial differential

equation (PDE) of the system and refer interested

readers to Junkins and Kim (1993) and Luo et al.

(1999) for detailed derivations.

The system is governed by

u€ðtÞ þr

ðL

xz€ðx; tÞdx ¼ t ð23:46Þ

rz€ðx; tÞ þ EIz 0000ðx; tÞ ¼ 0 ð23:47Þ

with the corresponding boundary conditions

zð0; tÞ ¼ 0; z 0ð0; tÞ ¼ uðtÞ; z 00ðL; tÞ ¼ 0; z 000ðL; tÞ ¼ 0 ð23:48Þ

where r is the arm’s linear mass density, L is the arm length, E is Young’s modulus of elasticity, I is the

cross-sectional moment of inertia, Ih is the equivalent mass moment of inertia at the root end of the arm,

It ¼ Ih þ rL3=3 is the total inertia, and the global variable z is defined as

zðx; tÞ ¼ xuðtÞ þ yðx; tÞ ð23:49Þ

Clearly, the arm vibration equation (Equation 23.47) is a homogeneous PDE but the boundary

conditions (Equation 23.48) are nonhomogeneous. Therefore, the closed form solution is very tedious to

obtain, if not impossible. Using the application of VSC, these equations and their associated boundary

conditions can be converted to a homogeneous boundary value problem, as discussed next.

23.3.3.1.2 Variable Structure Controller

The controller objective is to track the arm angular displacement from an initial angle, ud ¼ uð0Þ; to zero

position, uðt ! 1Þ ¼ 0; while minimizing the flexible arm oscillations. To achieve the control

x y(x,t)

X′

Y ′

q (t)

Sec. A-A

t (t)

FIGURE 23.23 Flexible arm in the horizontal plane

and kinematics of deformation. (Source: From Jalili, N.,

ASME J. Dyn. Syst. Meas. Control, 123, 712 – 719, 2001.

With permission.)

Vibration Control 23-25

insensitivity against modeling uncertainties, the nonlinear control routine of sliding mode control with

an additional perturbation estimation (SMCPE) compartment is adopted here (Elmali and Olgac, 1992;

Jalili and Olgac, 1998a). The SMCPE method, presented in Elmali and Olgac (1992), has many attractive

features, but it suffers from the disadvantages associated with the truncated-model-base controllers.

On the other hand, the infinite-dimensional distributed (IDD)-base controller design, proposed in Zhu

et al. (1997), has practical limitations due to its measurement requirements in addition to the complex

control law.

Initiating from the idea of the IDD-base controller, we present a new controller design approach in

which an online perturbation estimation mechanism is introduced and integrated with the controller to

relax the measurement requirements and simplify the control implementation. As utilized in Zhu et al.

(1997), for the tip-vibration suppression, it is further required that the sliding surface enable the

transformation of nonhomogeneous boundary conditions (Equation 23.48) to homogeneous ones. To

satisfy vibration suppression and robustness requirements simultaneously, the sliding hyperplane is

selected as a combination of tracking (regulation) error and arm flexible vibration as

s ¼ w_ þs w ð23:50Þ

where s . 0 is a control parameter and

w ¼ uðtÞ þ

zðL; tÞ ð23:51Þ

with the scalar, m; being selected later. When m ¼ 0; controller (Equation 23.50) reduces to a sliding

variable for rigid-link manipulators (Jalili and Olgac, 1998a; Yeung and Chen, 1988). The motivation for

such sliding a variable is to provide a suitable boundary condition for solving the beam Equation 23.47,

as will be discussed next and is detailed in Jalili (2001).

For the system described by Equation 23.46 to Equation 23.48, if the variable structure controller is

given by

t ¼ cest þ

1 þ m

2k sgnðsÞ 2 Ps 2

y€ ðL; tÞ 2s ð1 þmÞ u_ 2

y_ ðL; tÞ

􀀏 􀀐

ð23:52Þ

where cest is an estimate of the beam flexibility effect

c ¼ r

ðL

xy€ðx; tÞdx ð23:53Þ

k and P are positive scalars k $ 1 þ m=It lc 2 cestl, 21:2 , m , 20:45; m – 21 and sgnð Þ represents

the standard signum function, then, the system’s motion will first reach the sliding mode s ¼ 0 in a finite

time, and consequently converge to the equilibrium position wðx; tÞ ¼ 0 exponentially with a timeconstant

1=s (Jalili, 2001).

23.3.3.1.3 Controller Implementation

In the preceding section, it was shown that by properly selecting control variable, m, the motion

exponentially converges to w ¼ 0 with a time-constant 1=s; while the arm stops in a finite time. Although

the discontinuous nature of the controller introduces a robustifying mechanism, we have made the

scheme more insensitive to parametric variations and unmodeled dynamics by reducing the required

measurements and hence easier control implementation. The remaining measurements and ever-present

modeling imperfection effects have all been estimated through an online estimation process. As stated

before, in order to simplify the control implementation and reduce the measurement effort, the effect of

all uncertainties, including flexibility effect ð

ÐL

0 xy€ðx; tÞdxÞ and the ever-present unmodeled dynamics, is

gathered into a single quantity named perturbation, c; as given by Equation 23.53. Noting Equation

23.46, the perturbation term can be expressed as

c ¼ t 2 It

u€ðtÞ ð23:54Þ

23-26 Vibration and Shock Handbook

which requires the yet-unknown control feedback t: In order to resolve this dilemma of causality, the

current value of control torque, t; is replaced by the most recent control, t ðt 2 dÞ; where d is the small

time-step used for the loop closure. This replacement is justifiable in practice since such an algorithm is

implemented on a digital computer and the sampling speed is high enough to claim this. Also, in the

absence of measurement noise, u€ðtÞ ø u€calðtÞ ¼ ½ u_ðtÞ 2u_ðt 2dÞ􀀉=d:

In practice and in the presence of measurement noise, appropriate filtering may be considered and

combined with these approximate derivatives. This technique is referred to as “switched derivatives”. This

backward differences are shown to be effective when d is selected to be small enough and the controller is

run on a fast DSP (Cannon and Schmitz, 1984). Also, y€ðL; tÞ can be obtained by attaching an

accelerometer at arm-tip position. All the required signals are, therefore, measurable by currently

available sensor facilities and the controller is thus realizable in practice. Although these signals may be

quite inaccurate, it should be pointed out that the signals, either measurements or estimations, need not

be known very accurately, since robust sliding control can be achieved if k is chosen to be large enough to

cover the error existing in the measurement/signal estimation (Yeung and Chen, 1989).

23.3.3.1.4 Numerical Simulations

In order to show the effectiveness of the proposed controller, a lightweight flexible arm is considered

(h .. b in Figure 23.23). For numerical results, we consider ud ¼ uð0Þ ¼ p=2 for the initial arm base

angle, with zero initial conditions for the rest of the state variables. The system parameters are listed in

Table 23.3. Utilizing assumed mode model (AMM), the arm vibration, Equation 23.47, is truncated to

three modes and used in the simulations. It should be noted that the controller law, Equation 23.52, is based

on the original infinite dimensional equation, and this truncation is utilized only for simulation purposes.

We take the controller parameter m ¼ 20:66; P ¼ 7:0; k ¼ 5; 1 ¼ 0:01 and s ¼ 0:8: In practice, s is

selected for maximum tracking accuracy taking into account unmodeled dynamics and actuator

hardware limitations (Moura et al., 1997). Although such restrictions do not exist in simulations (i.e.,

with ideal actuators, high sampling frequencies and perfect measurements), this selection of s was

decided based on actual experiment conditions.

The sampling rate for the simulations is d ¼ 0:0005 sec, while data are recorded at the rate of only

0.002 sec for plotting purposes. The system responses to the proposed control scheme are shown in

Figure 23.24. The arm-base angular position reaches the desired position, u ¼ 0, in approximately 4 to

5 sec, which is in agreement with the approximate settling time of ts ¼ 4=s (Figure 23.24a). As soon as

the system reaches the sliding mode layer, lsl , 1 (Figure 23.24d), the tip vibrations stop (Figure 23.24b),

which demonstrates the feasibility of the proposed control technique. The control torque exhibits some

residual vibration, as shown in Figure 23.24c. This residual oscillation is expected since the system

TABLE 23.3 System Parameters Used in Numerical Simulations

and Experimental Setup for Rotating Arm

Properties Symbol Value Unit

Arm Young’s modulus E 207 £ 109 N/m2

Arm thickness b 0.0008 m

Arm height h 0.02 m

Arm length L 0.45 m

Arm linear mass density r 0:06=L kg/m

Total arm base inertia Ih 0.002 kg m2

Gearbox ratio N 14:1 —

Light source mass — 0.05 kg

Position sensor sensitivity — 0.39 V/cm

Motor back EMF constant Kb 0.0077 V/rad/sec

Motor torque constant Kt 0.0077 N m/A

Armature resistance Ra 2.6 V

Armature inductance La 0.18 mH

Encoder resolution — 0.087 Deg/count

Vibration Control 23-27

motion is not forced to stay on s ¼ 0 surface (instead it is forced to stay on lsl , 1) when saturation

function is used. The sliding variable s is also depicted in Figure 23.24d. To demonstrate better the feature

of the controller, the system responses are displayed when m ¼ 0 (Figure 23.25). As discussed, m ¼ 0

corresponds to the sliding variable for the rigid link. The undesirable oscillations at the arm tip are

evident (see Figure 23.25b and c).

23.3.3.1.5 Control Experiments

In order to demonstrate better the effectiveness of the controller, an experimental setup is constructed

and used to verify the numerical results and concepts discussed in the preceding sections. The

experimental setup is shown in Figure 23.26. The arm is a slender beam made of stainless steel, with the

same dimensions as used in the simulations. The experimental setup parameters are listed in Table 23.3.

One end of the arm is clamped to a solid clamping fixture, which is driven by a high-quality DC

servomotor. The motor drives a built-in gearbox ðN ¼ 14:1Þ whose output drives an antibacklash gear.

The antibacklash gear, which is equipped with a precision encoder, is utilized to measure the arm base

angle as well as to eliminate the backlash. For tip deflection, a light source is attached to the tip of the

arm, which is detected by a camera mounted on the rotating base.

The DC motor can be modeled as a standard armature circuit; that is, the applied voltage, v; to the DC

motor is

v ¼ Raia þ La dia=dt þ Kb

u_m ð23:55Þ

FIGURE 23.24 Analytical system responses to controller with inclusion of arm flexibility, that is, m ¼ 20:66:

(a) arm angular position; (b) arm-tip deflection; (c) control torque and (d) sliding variable sec. (Source: From

Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 – 719, 2001. With permission.)

23-28 Vibration and Shock Handbook

FIGURE 23.25 Analytical system responses to controller without inclusion of arm flexibility, that is, m ¼ 0:

(a) arm angular position; (b) arm-tip deflection; (c) control torque and (d) sliding variable sec. (Source: From

Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 – 719, 2001. With permission.)

FIGURE 23.26 The experimental device and setup configuration. (Source: From Jalili, N., ASME J. Dyn. Syst. Meas.

Control, 123, 712 – 719, 2001. With permission.)

Vibration Control 23-29

where Ra is the armature resistance, La is the armature inductance, ia is the armature current, Kb is the

back-EMF (electro-motive-force) constant, and um is the motor shaft position. The motor torque, tm

from the motor shaft with the torque constant, Kt; can be written as

tm ¼ Ktia ð23:56Þ

The motor dynamics thus become

u€m þ Cv

u_m þ ta ¼ tm ¼ Ktia ð23:57Þ

where Cv is the equivalent damping constant of the motor, and Ie ¼ Im þ IL=N2 is the equivalent inertia

load including motor inertia, Im; and gearbox, clamping frame and camera inertia, IL: The available

torque from the motor shaft for the arm is ta:

Utilizing the gearbox from the motor shaft to the output shaft and ignoring the motor electric time

constant, ðLa=RaÞ; one can relate the servomotor input voltage to the applied torque (acting on

the arm) as

t ¼

NKt

v 2 Cv þ

KtKb

􀀏 􀀐

N2u_ 2 Ih

u€ ð23:58Þ

where Ih ¼ N2Ie is the equivalent inertia of the arm base used in the derivation of governing

equations. By substituting this torque into the control law, the reference input voltage, V ; is

obtained for experiment.

The control torque is applied via a digital signal processor (DSP) with sampling rate of 10 kHz, while

data are recorded at the rate 500 Hz (for plotting purposes only). The DSP runs the control routine in a

single-input – single-output mode as a free standing CPU. Most of the computations and hardware

commands are done on the DSP card. For this setup, a dedicated 500 MHz Pentium III serves as the host

PC, and a state-of-the-art dSPACEw DS1103 PPC controller board equipped with a Motorola Power PC

604e at 333 MHz, 16 channels ADC, 12 channels DAC, as microprocessor.

The experimental system responses are shown in Figure 23.27 and Figure 23.28 for similar cases

discussed in the numerical simulation section. Figure 23.27 represents the system responses when

controller (Equation 23.52) utilizes the flexible arm (i.e., m ¼ 20:66). As seen, the arm base reaches

the desired position (Figure 23.27a), while tip deflection is simultaneously stopped (Figure 23.27b).

The good correspondence between analytical results (Figure 23.24) and experimental findings

(Figure 23.27) is noticeable from a vibration suppression characteristics point of view. It should be

noted that the controller is based on the original governing equations, with arm-base angular

position and tip deflection measurements only. The unmodeled dynamics, such as payload effect

(owing to the light source at the tip, see Table 23.3) and viscous friction (at the root end of the

arm), are being compensated through the proposed online perturbation estimation routine. This, in

turn, demonstrates the capability of the proposed control scheme when considerable deviations

between model and plant are encountered. The only noticeable difference is the fast decaying

response as shown in Figure 27b and c. This clearly indicates the high friction at the motor, which

was not considered in the simulations (Figure 24b and c). Similar responses are obtained when the

controller is designed based on the rigid link only, that is, m ¼ 0: The system responses are

displayed in Figure 23.28. Similarly, the undesirable arm-tip oscillations are obvious. The overall

agreement between simulations (Figure 24 and Figure 25) and the experiment (Figure 27 and Figure 28)

is one of the critical contributions of this work.

23.3.3.2 Observer-Based Piezoelectric Vibration Control of Translating Flexible Beams

Many industrial robots, especially those widely used in automatic manufacturing assembly lines, are

Cartesian types (Ge et al., 1998). A flexible Cartesian robot can be modeled as a flexible cantilever

beam with a translational base support. Traditionally, a PD control strategy is used to regulate the

movement of the robot arm. In lightweight robots, the base movement will cause undesirable

vibrations at the arm tip because of the flexibility distributed along the arm. In order to eliminate

23-30 Vibration and Shock Handbook

such vibrations, the PD controller must be upgraded with additional compensating terms.

In order to improve further the vibration suppression performance, which is a requirement for

the high-precision manufacturing market, a second controller, such as a piezoelectric (PZT) patch

actuator attached on the surface of the arm, can be utilized (Oueini et al., 1998; Ge et al., 1999; Jalili et al.,

2002).

In this section, an observer-based control strategy is presented for regulating the arm motion (Liu et al.,

2002). The base motion is controlled utilizing an electrodynamic shaker, while a piezoelectric (PZT)

patch actuator is bonded on the surface of the flexible beam for suppressing residual arm vibrations. The

control objective here is to regulate the arm base movement, while simultaneously suppressing the

vibration transients in the arm. To achieve this, a simple PD control strategy is selected for the regulation

of the movement of the base, and a Lyapunov-based controller is selected for the PZT voltage signal. The

selection of the proposed energy-based Lyapunov function naturally results in velocity-related signals,

which are not physically measurable (Dadfarnia et al., 2003). To remedy this, a reduced-order observer is

designed to estimate the velocity related signals. For this, the control structure is designed based on the

truncated two-mode beam model.

23.3.3.2.1 Mathematical Modeling

For the purpose of model development, we consider a uniform flexible cantilever beam with a PZT

actuator bonded on its top surface. As shown in Figure 23.29, one end of the beam is clamped into

FIGURE 23.27 Experimental system responses to controller with inclusion of arm flexibility, that is, m ¼ 20:66:

(a) arm angular position; (b) arm-tip deflection; (c) control voltage applied to DC servomotor. (Source: From

Jalili, N., ASME J. Dyn. Syst. Meas. Control, 123, 712 – 719, 2001. With permission.)

Vibration Control 23-31

a moving base with the mass of mb; and a tip mass, mt; is attached to the free end of the beam.

The beam has total thickness tb; and length L; while the piezoelectric film possesses thickness and

length tb and ðl2 2 l1Þ; respectively. We assume that the PZT and the beam have the same width, b:

The PZT actuator is perfectly bonded on the beam at distance l1 measured from the beam support.

The force, f ðtÞ; acting on the base and the input voltage, vðtÞ; applied to the PZT actuator are the

only external effects.

To establish a coordinate system for the beam, the x-axis is taken in the longitudinal direction and the

z-axis is specified in the transverse direction of the beam with midplane of the beam to be z ¼ 0; as shown

in Figure 23.30. This coordinate is fixed to the base.

The fundamental relations for the piezoelectric materials are given as (Ikeda, 1990)

F ¼ cS 2 hD ð23:59Þ

E ¼ 2hTS þ bD ð23:60Þ

where F [ R6 is the stress vector, S [ R6 is the strain vector, c [ R6£6 is the symmetric matrix of elastic

stiffness coefficients, h [ R6£3 is the coupling coefficients matrix, D [ R3 is the electrical displacement

vector, E [ R3 is the electrical field vector, and b [ R3£3 is the symmetric matrix of impermittivity

coefficients.

(a)

−25

100

(b)

−10

−8

−6

−4

−2

(c)

−6

−4

−2

q (t), deg

v, volts

y(L,t), mm

Time, sec.

Time, sec. Time, sec.

0 1 2 3 4 5 6 7 8

FIGURE 23.28 Experimental system responses to controller without inclusion of arm flexibility, that is, m ¼ 0:

(a) arm angular position; (b) arm-tip deflection; (c) control voltage applied to DC servomotor. (Source: From Jalili,

N., ASME J. Dyn. Syst. Meas. Control, 123, 712 – 719, 2001. With permission.)

23-32 Vibration and Shock Handbook

An energy method is used to derive the equations

of motion. Neglecting the electrical kinetic energy,

the total kinetic energy of the system is expressed as

(Liu et al., 2002; Dadfarnia et al., 2004)

Ek ¼

mbsðtÞ2 þ

ðl1

rbtbðsðtÞ þ w_ ðx; tÞÞ2dx

ðl2

l1 ðrbtb þrptpÞðsðtÞ þ w_ ðx; tÞÞ2dx

ðL

rbtbðsðtÞ þ w_ ðx; tÞÞ2dx

mtðsðtÞ þ w_ ðL; tÞÞ2

mbsðtÞ2 þ

ðL

rðxÞ

􀀄

sðtÞ þ w_ ðx; tÞÞ2dx

mtðsðtÞ þ w_ ðL; tÞÞ2 ð23:61Þ

where

rðxÞ ¼ ½rbtb þ GðxÞrptp􀀉b

GðxÞ ¼ Hðx 2 l1Þ 2 Hðx 2 l2Þ

ð23:62Þ

and HðxÞ is the Heaviside function, rb and rp are the respective beam and PZT volumetric densities.

Neglecting the effect of gravity due to planar motion and the higher-order terms of quadratic in w 0

(Esmailzadeh and Jalili, 1998b), the total potential energy of the system can be expressed as

Ep ¼

ðl1

ðtb =2

2tb =2

FTS dy dx þ

ðl2

ðtb =2

2tb =2

FTS dy dx þ

ðl2

ððtb =2Þþtp

tb =2 ½FTS þ ETD􀀉dy dx

ðL

ðtb =2

2tb =2

FTS dy dx

ðL

cðxÞ

›2wðx; tÞ

›x2

" #2

dx þ hlDy ðtÞ

ðl2

›2wðx; tÞ

›x2 dx þ

blðl2 2 l1ÞDy ðtÞ2 ð23:63Þ

(l1 − l2)

z PZT patch

geometric beam

center of the

beam

neutral axis

FIGURE 23.30 Coordinate system.

s(t)

w(x,t)

f (t)

FIGURE 23.29 Schematic of the SCARA/Cartesian

robot (last link).

Vibration Control 23-33

where

cðxÞ ¼

11t3

􀁻 !

þ GðxÞ 3cb

11tbz2

n þ c p

11 t3

p þ 3tp

2 zn

􀀏 􀀐2

þ3t2

2 zn

( 􀀘 􀀏 􀀏 􀀐􀀐􀀙)

hl ¼ h12tpbðtp þ tb 2 2znÞ=2; bl ¼ b22btp

ð23:64Þ

and

zn ¼

c p

11tpðtp þ tbÞ

11tb þ c p

11tp

The beam and PZT stiffnesses are c b

11 and c p

11; respectively.

Using the AMM for the beam vibration analysis, the beam deflection can be written as

wðx; tÞ ¼

i¼1

fiðxÞqiðtÞ; Pðx; tÞ ¼ sðtÞ þ wðx; tÞ ð23:65Þ

The equations of motion can now be obtained using the Lagrangian approach

mb þ mt þ

ðL

rðxÞdx

􀀒 􀀓

€sðtÞ þ

j¼1

mjq€jðtÞ ¼ f ðtÞ ð23:66aÞ

mi€sðtÞ þ mdiq€iðtÞ þv2i

mdiqiðtÞ þ hlðf0i

ðl2Þ 2 f0i

ðl1ÞÞDy ðtÞ ¼ 0 ð23:66bÞ

j¼1

{ðf0j

ðl2Þ 2 f0j

ðl1ÞÞqjðtÞ} þ blðl2 2 l1ÞDy ðtÞ ¼ bðl2 2 l1ÞvðtÞ ð23:66cÞ

where

mdj ¼

ðL

rðxÞf2j

ðxÞdx þ mtf2j

ðLÞ; mj ¼

ðL

rðxÞfjðxÞdx þ mtfjðLÞ ð23:67Þ

Calculating Dy ðtÞ from Equation 23.66b and substituting into Equation 23.66c results in

mi€sðtÞ þ mdiq€iðtÞ þv2i

mdiqiðtÞ 2

h2l

ðf0i

ðl2Þ 2 f0i

ðl1ÞÞ

blðl2 2 l1Þ

j¼1

{ðf0j

ðl2Þ 2 f0j

ðl1ÞÞqjðtÞ}

¼ 2

hlbðf0i

ðl2Þ 2 f0i

ðl1ÞÞ

vðtÞ; i ¼ 1; 2; … ð23:68Þ

which will be used to derive the controller, as discussed next.

23.3.3.2.2 Derivation of the Controller

Utilizing Equation 23.66a and Equation 23.68, the truncated two-mode beam with PZT model reduces to

mb þ mt þ

ðL

rðxÞdx

􀀒 􀀓

€sðtÞ þ m1q€1ðtÞ þ m2q€2ðtÞ ¼ f ðtÞ ð23:69aÞ

m1€sðtÞ þ md1q€1ðtÞ þv21

md1q1ðtÞ 2

h2l

ðf01

ðl2Þ 2 f01

ðl1ÞÞ

blðl2 2 l1Þ

􀀐

􀀣

ðf01

ðl2Þ 2 f01

ðl1ÞÞq1ðtÞ þ ðf02

ðl2Þ 2 f02

ðl1ÞÞq2ðtÞ

􀀤

¼ 2

hlbðf01

ðl2Þ 2 f01

ðl1ÞÞ

vðtÞ

ð23:69bÞ

23-34 Vibration and Shock Handbook

m2€sðtÞ þ md2q€2ðtÞ þv22

md2q2ðtÞ 2

h2l

ðf02

ðl2Þ 2 f02

ðl1ÞÞ

blðl2 2 l1Þ

􀀐

􀀣

ðf01

ðl2Þ 2 f01

ðl1ÞÞq1ðtÞ þ ðf02

ðl2Þ 2 f02

ðl1ÞÞq2ðtÞ

􀀤

¼ 2

hlbðf02

ðl2Þ 2 f02

ðl1ÞÞ

vðtÞ

ð23:69cÞ

The equations in Equation 23.69 can be written in the following more compact form

MD€ þ KD ¼ Fe ð23:70Þ

where

M ¼

c m1 m2

m1 md1 0

m2 0 md2

664

775

; K ¼

0 0 0

0 k11 k12

0 k12 k22

664

775

; Fe ¼

f ðtÞ

e 1vðtÞ

e 2vðtÞ

8>><

>>:

9>>=

>>;

; D ¼

sðtÞ

q1ðtÞ

q2ðtÞ

8>><

>>:

9>>=

>>;

ð23:71Þ

and

c ¼ mb þ mt þ

ðL

rðxÞdx; e 1 ¼ 2

hlb

bl ðf01

ðl2Þ 2 f01

ðl1ÞÞ; e 2 ¼ 2

hlb

bl ðf02

ðl2Þ 2 f02

ðl1ÞÞ;

k11 ¼ v21

md1 2

h2l

blðl2 2 l1Þ ðf01

ðl2Þ 2 f01

ðl1ÞÞ2;

k12 ¼ 2

h2l

blðl2 2 l1Þ ðf01

ðl2Þ 2 f01

ðl1ÞÞðf02

ðl2Þ 2 f02

ðl1ÞÞ;

k22 ¼ v22

md2 2

h2l

blðl2 2 l1Þ ðf02

ðl2Þ 2 f02

ðl1ÞÞ2

ð23:72Þ

For the system described by Equation 23.70, if the control laws for the arm base force and PZT voltage

generated moment are selected as

f ðtÞ ¼ 2kpDs 2 kdsðtÞ ð23:73Þ

vðtÞ ¼ 2kvðe 1q_1ðtÞ þe 2q_2ðtÞÞ ð23:74Þ

where kp and kd are positive control gains, Ds ¼ sðtÞ 2 sd; sd is the desired set-point position, and kv . 0

is the voltage control gain, then the closed-loop system will be stable, and in addition

lim

t!1

{q1ðtÞ; q2ðtÞ; Ds} ¼ 0

See Dadfarnia et al. (2004) for a detailed proof.

23.3.3.2.3 Controller Implementation

The control input, vðtÞ; requires the information from the velocity-related signals, q_1ðtÞ and q_2ðtÞ; which

are usually not measurable. Sun and Mills (1999) solved the problem by integrating the acceleration

signals measured by the accelerometers. However, such controller structure may result in unstable closedloop

system in some cases. In this paper, a reduced-order observer is designed to estimate the velocity

signals, q_1 and q_2: For this, we utilize three available signals: base displacement, sðtÞ; arm-tip deflection,

PðL; tÞ; and beam root strain, e ð0; tÞ; that is

y1 ¼ sðtÞ ¼ x1 ð23:75aÞ

y2 ¼ PðL; tÞ ¼ x1 þ f1ðLÞx2 þ f2ðLÞx3 ð23:75bÞ

y3 ¼ e ð0; tÞ ¼

2 ðf00 1ð0Þx2 þ f00 2ð0Þx3Þ ð23:75cÞ

Vibration Control 23-35

It can be seen that the first three states can be obtained by

8>><

>>:

9>>=

>>;

¼ C21

1 y ð23:76Þ

Since this system is observable, we can design a reduced-order observer to estimate the velocity-related

state signals. Defining X1 ¼ ½ x1 x2 x3 􀀉T and X2 ¼ ½ x4 x5 x6 􀀉T; the estimated value for X2 can be

designed as

X^ 2 ¼ Lry þ z^ ð23:77Þ

z ¼ Fz^ þ Gy þ Hu ð23:78Þ

where Lr [ R3£3; F [ R3£3; G [ R3£3; and H [ R3£2 will be determined by the observer pole placement.

Defining the estimation error as

e2 ¼ X2 2 X^ 2 ð23:79Þ

the derivative of the estimation error becomes

e2 ¼ X_ 2 2 _^

X2 ð23:80Þ

Substituting the state-space equations of the system (Equation 23.77 and Equation 23.78) into Equation

23.80 and simplifying, we obtain

e2¼Fe2þðA212LrC1A112GC1þFLrC1ÞX1þðA222LrC1A122FÞX2þðB22LrC1B12HÞu ð23:81Þ

In order to force the estimation error, e2; to go to zero, matrix F should be selected to be Hurwitz and the

following relations must be satisfied (Liu et al., 2002):

F ¼A22 2 LrC1A12 ð23:82Þ

H ¼B2 2 LrC1B1 ð23:83Þ

G ¼ðA21 2 LrC1A11 þFLrC1ÞC21

1 ð23:84Þ

The matrix F can be chosen by the desired observer pole placement requirement. Once F is known, Lr; H;

and G can be determined utilizing Equation 23.82, to Equation 23.84, respectively. The velocity variables,

X^ 2; can now be estimated by Equation 23.77 and Equation 23.78.

23.3.3.2.4 Numerical Simulations

In order to show the effectiveness of the controller, the flexible beam structure in Figure 23.29 is

considered with the PZT actuator attached on the beam surface. The system parameters are listed in

Table 23.4.

First, we consider the beam without PZT control. We take the PD control gains to be kp ¼ 120 and

kd ¼ 20: Figure 23.31 shows the results for the beam without PZT control (i.e., with only PD force

control for the base movement). To investigate the effect of PZT controller on the beam vibration, we

consider the voltage control gain to be kv ¼ 2 £ 107: The system responses to the proposed controller

with a piezoelectric actuator based on the two-mode model are shown in Figure 23.32. The comparison

between the tip displacement, from Figure 31 and Figure 32, shows that the beam vibration can be

suppressed significantly utilizing the PZT actuator.

23.3.3.2.5 Control Experiments

In order to demonstrate better the effectiveness of the controller, an experimental setup is

constructed and used to verify the numerical results. The experimental apparatus consists of a

flexible beam with a PZT actuator and strain sensor attachments, as well as data acquisition,

23-36 Vibration and Shock Handbook

amplifier, signal conditioner and the control software. As shown in Figure 23.33, the plant consists

of a flexible aluminum beam with a strain sensor and a PZT patch actuator bound on each side of

the beam surface. One end of the beam is clamped to the base with a solid clamping fixture, which

is driven by a shaker. The shaker is connected to the arm base by a connecting rod. The

experimental setup parameters are listed in Table 23.4.

Figure 23.34 shows the high-level control block diagram of the experiment, where the shaker provides

the input control force to the base and the PZT applies a controlled moment on the beam. Two laser

sensors measure the position of the base and the beam-tip displacement. A strain-gauge sensor, which is

attached near the base of the beam, is utilized for the dynamic strain measurement. These three signals

TABLE 23.4 System Parameters Used in Numerical Simulations and

Experimental Setup for Translational Beam

Properties Symbol Value Unit

Beam Young’s modulus cb

11 69 £ 109 N/m2

Beam thickness tb 0.8125 mm

Beam and PZT width b 20 mm

Beam length L 300 mm

Beam volumetric density rb 3960.0 kg/m3

PZT Young’s modulus cp

11 66.47 £ 109 N/m2

PZT coupling parameter h12 5 £ 108 V/m

PZT impermittivity b22 4.55 £ 107 m/F

PZT thickness tp 0.2032 mm

PZT length l2 2 l1 33.655 mm

PZT position on beam l1 44.64 mm

PZT volumetric density rp 7750.0 kg/m3

Base mass mb 0.455 kg

Tip mass mt 0 kg

0 1 2 3 4

s(t), mm

0 1 2 3 4

−1

P(L,t), mm

0 1 2 3 4

−0.2

0.2

0.4

0.6

Time, sec.

f (t), N

0 1 2 3 4

−50

−25

Time, sec.

v(t), volt

(a) (b)

FIGURE 23.31 Numerical simulations for the case without PZT control: (a) base motion; (b) tip displacement;

Vibration Control 23-37

are fed back to the computer through the ISA MultiQ data acquisition card. The remaining required

signals for the controller (Equation 23.66) are determined as explained in the preceding section. The data

acquisition and control algorithms are implemented on an AMD Athlon 1100 MHz PC running under

the RT-Linux operating system. The Matlab/Simulink environment and Real Time Linux Target are used

to implement the controller.

The experimental results for both cases (i.e., without PZT and with PZT control) are depicted in

Figure 35 and Figure 36, respectively. The results demonstrate that with PZT control, the arm vibration

is eliminated in less than 1 sec, while the arm vibration lasts for more than 6 sec when PZT control is

not used. The experimental results are in agreement with the simulation results except for some

differences at the beginning of the motion. The slight overshoot and discrepancies at the beginning of

the motion are due to the limitations of the experiment (e.g., the shaker saturation limitation) and

unmodeled dynamics in the modeling (e.g., the friction modeling). However, it is still apparent that the

PZT voltage control can substantially suppress the arm vibration despite such limitations and modeling

imperfections.

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