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23.2 Vibration-Control Systems Concept

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23.2.1 Introduction

With a history of almost a century (Frahm, 1911), the dynamic vibration absorber has proven to be a

useful vibration-suppression device, widely used in hundreds of diverse applications. It is elastically

attached to the vibrating body to alleviate detrimental oscillations from its point of attachment (see

Figure 23.3). This section overviews the conceptual design and theoretical background of three types of

vibration-control systems, namely the passive, active and SA configurations, along with some related

practical implementations.

23.2.2 Passive Vibration Control

The underlying proposition in all vibration

control or absorber systems is to adjust properly

the absorber parameters such that the system

becomes absorbent of the vibratory energy within

the frequency interval of interest. In order to

explain the underlying concept, a single-degree-offreedom

(single-DoF) primary system with a

single-DoF absorber attachment is considered

(Figure 23.4). The governing dynamics is

expressed as

max€aðtÞ þ cax_aðtÞ þ kaxaðtÞ

¼ cax_pðtÞ þ kaxpðtÞ ð23:1Þ

cp kp

f(t)

ca ka

FIGURE 23.4 Application of a passive absorber to

single-DoF primary system.

23-4 Vibration and Shock Handbook

mpx€pðtÞ þ ðcp þ caÞx_pðtÞ þ ðkp þ kaÞxpðtÞ 2 cax_aðtÞ 2 kaxaðtÞ ¼ f ðtÞ ð23:2Þ

where xpðtÞ and xaðtÞ are the respective primary and absorber displacements, f ðtÞ is the external force, and

the rest of the parameters including absorber stiffness, ka; and damping, ca; are defined as per Figure 23.4.

The transfer function between the excitation force and primary system displacement in the Laplace

domain is then written as

TFðsÞ ¼

XpðsÞ

FðsÞ ¼

mas2 þ cas þ ka

HðsÞ

( )

ð23:3Þ

where

HðsÞ ¼ {mps2 þ ðcp þ caÞs þ kp þ ka}ðmas2 þ cas þ kaÞ 2 ðcas þ kaÞ2 ð23:4Þ

and XaðsÞ; XpðsÞ; and FðsÞ are the Laplace transformations of xaðtÞ; xpðtÞ; and f ðtÞ; respectively.

23.2.2.1 Harmonic Excitation

When excitation is tonal, the absorber is generally tuned at the disturbance frequency. For this case, the

steady-state displacement of the system due to harmonic excitation can be expressed as

XpðjvÞ

FðjvÞ

􀀈 􀀈 􀀈 􀀈 􀀈

ka 2 mav2 þ jcav

HðjvÞ

􀀈 􀀈 􀀈 􀀈 􀀈

ð23:5Þ

where v is the disturbance frequency and j ¼

ffiffiffiffi

21 p : An appropriate parameter tuning scheme can then

be selected to minimize the vibration of primary system subject to external disturbance, f ðtÞ:

For complete vibration attenuation, the steady state, lXpðjvÞl; must equal zero. Consequently, from

Equation 23.5, the ideal stiffness and damping of absorber are selected as

ka ¼ mav2; ca ¼ 0 ð23:6Þ

Notice that this tuned condition is only a function of absorber elements ðma; ka; and caÞ: That is, the

absorber tuning does not need information from the primary system and hence its design is stand alone.

For tonal application, theoretically, zero damping in the absorber subsection results in improved

performance. In practice, however, the damping is incorporated in order to maintain a reasonable tradeoff

between the absorber mass and its displacement. Hence, the design effort for this class of application is

focused on having precise tuning of the absorber to the disturbance frequency and controlling the

damping to an appropriate level. Referring to Snowdon (1968), it can be proven that the absorber, in the

presence of damping, can be most favorably tuned and damped if adjustable stiffness and damping are

selected as

kopt ¼

mam2pv2

ðma þ mpÞ2

; copt ¼ ma

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3kopt

2ðma þ mpÞ

ð23:7Þ

23.2.2.2 Broadband Excitation

In broadband vibration control, the absorber subsection is generally designed to add damping to

and change the resonant characteristics of the primary structure in order to dissipate vibrational energy

maximally over a range of frequencies. The objective of the absorber design is, therefore, to adjust the

absorber parameters to minimize the peak magnitude of the frequency transfer function ðFTFðvÞ ¼ lTFðsÞls¼jv Þ over the absorber parameters vector p ¼ {ca ka}T: That is, we seek p to

min

{ max

vmin #v#vmax

{lFTFðvÞl}} ð23:8Þ

Alternatively, one may select the mean square displacement response (MSDR) of the primary system

for vibration-suppression performance. That is, the absorber parameters vector, p, is selected such that

Vibration Control 23-5

the MSDR

E{ðx􀀊pÞ2} ¼

ð1

{FTFðvÞ}2SðvÞdv ð23:9Þ

is minimized over a desired wideband frequency range. SðvÞ is the power spectral density of the

excitation force, f ðtÞ; and FTF was defined earlier.

This optimization is subjected to some constraints in p space, where only positive elements are

acceptable. Once the optimal absorber suspension properties, ca and ka; are determined, they can be

implemented using adjustment mechanisms on the spring and the damper elements. This is viewed

as a SA adjustment procedure as it adds no energy to the dynamic structure. The conceptual

devices for such adjustable suspension elements and SA treatment will be discussed later in

Section 23.2.5.

23.2.2.3 Example Case Study

To better recognize the effectiveness of the dynamic vibration absorber over the passive and optimum

passive absorber settings, a simple example case is presented. For the simple system shown in Figure 23.4,

the following nominal structural parameters (marked by an overscore) are taken:

m􀀊 p ¼ 5:77 kg; k􀀊p ¼ 251:132 £ 106 N=m; c􀀊p ¼ 197:92 kg=sec

m􀀊 a ¼ 0:227 kg; k􀀊a ¼ 9:81 £ 106 N=m; c􀀊a ¼ 355:6 kg=sec ð23:10Þ

These are from an actual test setting, which is optimal by design (Olgac and Jalili, 1999). That is, the peak

of the FTF is minimized (see thin lines in Figure 23.5). When the primary stiffness and damping

increase 5% (for instance during the operation), the FTF of the primary system deteriorates considerably

(the dashed line in Figure 23.5), and the absorber is no longer an optimum one for the present primary.

When the absorber is optimized based on optimization problem 8, the retuned setting is reached as

ka ¼ 10:29 £ 106 N=m; ca ¼ 364:2 kg=sec ð23:11Þ

which yields a much better frequency response (see dark line in Figure 23.5).

The vibration absorber effectiveness is better demonstrated at different frequencies by frequency sweep

test. For this, the excitation amplitude is kept fixed at unity and its frequency changes every 0.15 sec from

0.0

0.2

0.4

0.6

0.8

1.0

200 400 600 800 1000 1200 1400 1600 1800

Frequency, Hz.

FTF

nominal absorber de-tuned absorber re-tuned absorber

FIGURE 23.5 Frequency transfer functions (FTFs) for nominal absorber (thin-solid line), detuned absorber (thindotted

line), and retuned absorber (thick-solid line) settings. (Source: From Jalili, N. and Olgac, N., AIAA J. Guidance

Control Dyn., 23, 961 – 970, 2000a. With permission.)

23-6 Vibration and Shock Handbook

1860 to 1970 Hz. The primary responses with nominally tuned, with detuned, and with retuned absorber

settings are given in Figure 23.6a – c, respectively.

23.2.3 Active Vibration Control

As discussed, passive absorption utilizes resistive or reactive devices either to absorb vibrational

energy or load the transmission path of the disturbing vibration (Korenev and Reznikov, 1993; see

Figure 23.7, top). Even with optimum absorber parameters (Warburton and Ayorinde, 1980;

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Time, sec.

1.75

1.25

0.75

0.25

−0.25

−0.75

−1.25

−1.75

1.75

1.25

0.75

0.25

−0.25

−0.75

−1.25

−1.75

Dimensional disp. Dimensional disp.

1.75

1.25

0.75

0.25

−0.25

−0.75

−1.25

−1.75

Dimensional disp.

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Time, sec.

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Time, sec.

(a)

(b)

(c)

FIGURE 23.6 Frequency sweep each 0.15 with frequency change of 1860, 1880, 1900, 1920, 1930, 1950, and

1970 Hz: (a) nominally tuned absorber settings; (b) detuned absorber settings and (c) retuned absorber settings.

(Source: From Jalili, N. and Olgac, N., AIAA J. Guidance Control Dyn., 23, 961 – 970, 2000a. With permission.)

Vibration Control 23-7

Esmailzadeh and Jalili, 1998a), the passive absorption has significant limitations in structural

applications where broadband disturbances of highly uncertain nature are encountered.

In order to compensate for these limitations, active vibration-suppression schemes are utilized. With

an additional active force, uðtÞ (Figure 23.7, bottom), the absorber is controlled using different

algorithms to make it more responsive to primary disturbances (Sun et al., 1995; Margolis, 1998; Jalili

and Olgac, 1999). One novel implementation of the tuned vibration absorbers is the active resonator

absorber (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, 1995). Using a simple position (or velocity or

acceleration) feedback control within the absorber subsection, the delayed resonator enforces that the

dominant characteristic roots of the absorber subsection be on the imaginary axis, hence leading to

resonance. Once the ARA becomes resonant, it creates perfect vibration absorption at this frequency. The

conceptual design and implementation issues of such active vibration-control systems, along with their

practical applications, are discussed in Section 23.3.

23.2.4 Semiactive Vibration Control

Semiactive (SA) vibration-control systems can achieve the majority of the performance characteristics of

fully active systems, thus allowing for a wide class of applications. The idea of SA suspension is very

simple: to replace active force generators with continually adjustable elements which can vary and/or

shift the rate of the energy dissipation in response to instantaneous condition of motion (Jalili, 2002).

23.2.5 Adjustable Vibration-Control Elements

Adjustable vibration-control elements are typically comprised of variable rate damper and stiffness.

Significant efforts have been devoted to the development and implementation of such devices for a

variety of applications. Examples of such devices include electro-rheological (ER) (Petek, 1992;

Wang et al., 1994; Choi, 1999), magneto-rheological (MR) (Spencer et al., 1998; Kim and Jeon, 2000)

fluid dampers, and variable orifice dampers (Sun and Parker, 1993), controllable friction braces

(Dowell and Cherry, 1994), and variable stiffness and inertia devices (Walsh and Lamnacusa, 1992;

ca ka

Point of attachment

Absorber

Structure Primary

x1 Point of attachment

Absorber

Primary

u(t)

Compensator

Sensor (Acceleration, velocity, or

displacement measurement)

Structure

FIGURE 23.7 A general primary structure with passive (top) and active (bottom) absorber settings.

23-8 Vibration and Shock Handbook

Nemir et al., 1994; Franchek et al., 1995; Abe and Igusa, 1996). The conceptual devices for such

adjustable properties are briefly reviewed in this section.

23.2.5.1 Variable Rate Dampers

A common and very effective way to reduce transient and steady-state vibration is to change the amount

of damping in the SA vibration-control system. Considerable design work on SA damping was done in

the 1960s to the 1980s (Crosby and Karnopp, 1973; Karnopp et al., 1974) for vibration control of civil

structures such as buildings and bridges (Hrovat et al., 1983) and for reducing machine tool oscillations

(Tanaka and Kikushima, 1992). Since then, SA dampers have been utilized in diverse applications ranging

from trains (Stribersky et al., 1998) and other off-road vehicles (Horton and Crolla, 1986) to military

tanks (Miller and Nobles, 1988). During recent years, there has been considerable interest in the SA

concept in the industry for improvement and refinements of the concept (Karnopp, 1990; Emura et al.,

1994). Recent advances in smart materials have led to the development of new SA dampers, which are

widely used in different applications.

In view of these SA dampers, ER and MR fluids probably serve as the best potential hardware

alternatives for the more conventional variable-orifice hydraulic dampers (Sturk et al., 1995). From a

practical standpoint, the MR concept appears more promising for suspension, since it can operate, for

instance, on vehicle battery voltage, whereas the ER damper is based on high-voltage electric fields.

Owing to their importance in today’s SA damper technology, we briefly review the operation and

fundamental principles of SA dampers here.

23.2.5.1.1 Electro-Rheological Fluid Dampers

ER fluids are materials that undergo significant

instantaneous reversible changes in material

characteristics when subjected to electric potentials

(Figure 23.8). The most significant change is associated

with complex shear moduli of the material,

and hence ER fluids can be usefully exploited in SA

absorbers where variable-rate dampers are utilized.

Originally, the idea of applying an ER damper to

vibration control was initiated in automobile

suspensions, followed by other applications

(Austin, 1993; Petek et al., 1995).

The flow motions of an ER fluid-based damper

can be classified by shear mode, flow mode,

and squeeze mode. However, the rheological

property of ER fluid is evaluated in the shear

mode (Choi, 1999). As a result, the ER fluid

damper provides an adaptive viscous and frictional

damping for use in SA system (Dimarogonas-

Andrew and Kollias, 1993; Wang et al., 1994).

23.2.5.1.2 Magneto-Rheological Fluid Dampers

MR fluids are the magnetic analogies of ER fluids and typically consist of micron-sized, magnetically

polarizable particles dispersed in a carrier medium such as mineral or silicon oil. When a magnetic field is

applied, particle chains form and the fluid becomes a semisolid, exhibiting plastic behavior similar to

that of ER fluids (Figure 23.9). Transition to rheological equilibrium can be achieved in a few

milliseconds, providing devices with high bandwidth (Spencer et al., 1998; Kim and Jeon, 2000).

Moving cylinder

Fixed

cup

Aluminum

foil

ER Fluid r h

Ld Lo

y. y

FIGURE 23.8 A schematic configuration of an ER

damper. (Source: From Choi, S.B., ASME J. Dyn. Syst.

Meas. Control, 121, 134 – 138, 1999. With permission.)

Vibration Control 23-9

23.2.5.2 Variable-Rate Spring Elements

In contrast to variable dampers, studies of SA springs or time-varying stiffness have also been geared

for vibration-isolation applications (Hubard and Marolis, 1976), for structural controls and for

vibration attenuation (Sun et al., 1995 and references therein). The variable stiffness is a promising

practical complement to SA damping, since, based on the discussion in Section 23.2, both the absorber

damping and stiffness should change to adapt optimally to different conditions. Clearly, the absorber

stiffness has a significant influence on optimum operation (and even more compared to the damping

element; Jalili and Olgac, 2000b).

Unlike the variable rate damper, changing the effective stiffness requires high energy (Walsh and

Lamnacusa, 1992). Semiactive or low-power implementation of variable stiffness techniques suffers

from limited frequency range, complex implementation, high cost, and so on. (Nemir et al., 1994;

Franchek et al., 1995). Therefore, in practice, both absorber damping and stiffness are concurrently

adjusted to reduce the required energy.

23.2.5.2.1 Variable-Rate Stiffness (Direct Methods)

The primary objective is to directly change the spring stiffness to optimize a vibration-suppression

characteristic such as the one given in Equation 23.8 or Equation 23.9. Different techniques can be

utilized ranging from traditional variable leaf spring to smart spring utilizing magnetostrictive materials.

A tunable stiffness vibration absorber was utilized for a four-DoF building (Figure 23.10), where a spring

is threaded through a collar plate and attached to the absorber mass from one side and to the driving gear

from the other side (Franchek et al., 1995). Thus, the effective number of coils, N; can be changed

resulting in a variable spring stiffness, ka:

ka ¼

d4G

8D3N ð23:12Þ

where d is the spring wire diameter, D is the spring diameter, and G is modulus of shear rigidity.

23.2.5.2.2 Variable-Rate Effective Stiffness (Indirect Methods)

In most SA applications, directly changing the stiffness might not be always possible or may require large

amount of control effort. For such cases, alternatives methods are utilized to change the effective tuning

ratio ðt ¼

ffiffiffiffiffiffiffi

ka=ma

=vprimary Þ; thus resulting in a tunable resonant frequency.

In Liu et al. (2000), a SA flutter-suppression scheme was proposed using differential changes of

external store stiffness. As shown in Figure 23.11, the motor drives the guide screw to rotate with slide

block, G; moving along it, thus changing the restoring moment and resulting in a change of store

FIGURE 23.9 A schematic configuration of an MR damper. (Source: From Spencer, B.F. et al., Proc. 2nd World Conf.

on Structural Control, 1998. With permission.)

23-10 Vibration and Shock Handbook

pitching stiffness. Using a double-ended cantilever

beam carrying intermediate lumped masses, a SA

vibration absorber was recently introduced (Jalili

and Esmailzadeh, 2002), where positions of

moving masses are adjustable (see Figure 23.12).

Figure 23.13 shows an SA absorber with an

adjustable effective inertia mechanism (Jalili et al.,

2001; Jalili and Fallahi, 2002). The SA absorber

consists of a rod carrying a moving block and a

spring and damper, which are mounted on a

casing. The position of the moving block, rv ; on

the rod is adjustable which provides a tunable

resonant frequency.

23.2.5.3 Other Variable-Rate Elements

Recent advances in smart materials have led to the

development of new SA vibration-control systems using indirect influence on the suspension elements.

Wang et al. used a SA piezoelectric network (1996) for structural-vibration control. The variable

resistance and inductance in an external RL circuit are used as real-time adaptable control parameters.

Another class of adjustable suspensions is the so-called hybrid treatment (Fujita et al., 1991).

The hybrid design has two modes, an active mode and a passive mode. With its aim of lowering the

Spring

driving gear Spring

collar

Absorber

spring

Absorber

mass

Guide

rod

Absorber base

Motor and geartrain (Potentiometer not shown)

FIGURE 23.10 The application of a variable-stiffness vibration absorber to a four-DoF building. (Source: From

Franchek, M.A. et al., J. Sound Vib., 189, 565 – 585, 1995. With permission.)

G Left wing tip

FIGURE 23.11 A SA flutter control using adjustable pitching stiffness. (Source: From Liu, H.J. et al., J. Sound Vib.,

229, 199 – 205, 2000. With permission.)

C K

q(t) M

f(t) = F0 sin(wet)

m m

FIGURE 23.12 A typical primary system equipped

with the double-ended cantilever absorber with adjustable

tuning ration through moving masses, m. (Source:

From Jalili, N. and Esmailzadeh, E., J. Multi-Body Dyn.,

216, 223 – 235, 2002. With permission.)

Vibration Control 23-11

control effort, relatively small vibrations are reduced in active mode, while passive mode is used for large

oscillations. Analogous to hybrid treatment, the semiautomated approach combines SA and active

suspensions to benefit from the advantages of individual schemes while eliminating their shortfalls

(Jalili, 2000). By altering the adjustable structural properties (in the SA unit) and control parameters

(in the active unit), a search is conducted to minimize an objective function subject to certain

constraints, which may reflect performance characteristics.

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