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vibration

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Nader Jalili
Mem. ASMEAssistant Professor,Robotics and Mechatronics Laboratory,Department of Mechanical Engineering,Clemson University,Clemson, SC 29634-0921e-mail: jalili@clemson.edu
A Comparative Study and Analysisof Semi-Active Vibration-ControlSystems
Semi-active (SA) vibration-control systems are those which otherwise passively generated damping or spring forces are modulated according to a parameter tuning policy with onlya small amount of control effort. SA units, as their name implies, ﬁll the gap between purely passive and fully active vibration-control systems and offer the reliability of pas-sive systems, yet maintain the versatility and adaptability of fully active devices. Duringrecent years there has been considerable interest towards practical implementation of these systems for their low energy requirement and cost. This paper brieﬂy reviews thebasic theoretical concepts for SA vibration-control design and implementation, and sur-veys recent developments and control techniques for these systems. Some related practicalapplications in vehicle suspensions are also presented.
DOI: 10.1115/1.1500336
1 Introduction
In most of today’s mechatronic systems a number of possibledevices, such as reaction or momentum wheels, rotating devicesand electric motors are essential to the system’s operation andperformance. These devices, however, can also be sources of det-rimental vibrations that may signiﬁcantly inﬂuence the missionperformance, effectiveness and accuracy of operation. Severaltechniques are utilized to either limit or alter the vibration re-sponse of such systems. In vibration isolation either the source of vibration is isolated from the system of concern
also called‘‘force transmissibility,’’ see Fig. 1
a
, or the device is protectedfrom vibration of its point of attachment
also called ‘‘displace-ment transmissibility,’’ see Fig. 1
b
. Unlike the isolator, a vibra-tion absorber consists of a secondary system
usually mass-spring-damper trio
added to the primary device to protect it fromvibrating
see Fig. 1
c
. By properly selecting absorber mass,stiffness, and damping, the vibration of the primary system can besuppressed
1
.A vibration-control system, either as an isolator or an absorber,is said to be active, passive, or semi-active depending on theamount of external power required for the system to perform itsfunction, see Fig. 2,
2
. A passive vibration-control unit consistsof a resilient member
stiffness
and an energy dissipator
damper
to either absorb vibratory energy or load the transmission path of the disturbing vibration
3
, Fig. 2
a
. This conﬁguration has sig-niﬁcant limitations in structural applications where broadband dis-turbances of highly uncertain nature are encountered. In order tocompensate for these limitations, active vibration-control systemsare utilized. With an additional active force introduced as a part of a suspension unit,
u
(
t
) in Fig. 2
b
, the vibration-control systemis then controlled using different algorithms to make it more re-sponsive to sources of disturbance
2,4–6
. A combination of active/passive treatment is intended to reduce the amount of external power necessary to achieve the desired performancecharacteristics
7
.In view of these systems, it often occurs that the system isrequired to operate over a wide band load and frequency rangewhich is impossible to meet with a single choice of stiffness anddamping. If the desired response characteristics cannot be ob-tained, an active vibration-control system may provide an attrac-tive alternative for such broadband disturbances. However, suchactive conﬁgurations suffer from control-induced instability in ad-dition to the large control effort requirement. This is a seriousconcern that prevents common usage in most industrial applica-tions. On the other hand, passive vibration-control systems
espe-cially vibration absorbers
are often hampered by a phenomenonknown as ‘‘de-tuning.’’ This occurs due to deterioration of thestructural parameters and/or variations in the excitation frequency,and the passive system will no longer be effective.Semi-active
also known as adaptive-passive
conﬁguration ad-dresses these limitations by effectively integrating a tuning controlscheme with tunable passive devices. For this, active force gen-erators are replaced by modulated variable compartments such asvariable rate damper and stiffness, see Fig. 2
c
8–10
. Thesevariable components are referred to as ‘‘tunable parameters’’ of the vibration-control system, which are re-tailored via a tuningcontrol thus resulting in semi-actively inducing optimal operation.Much attention is being paid to these arrangements for their lowenergy requirement and cost. Recent advances in smart materialsand adjustable dampers and absorbers have signiﬁcantly contrib-uted to applicability of these systems
11–13
.The remainder of the paper is organized as follows. Section 2brieﬂy reviews semi-active vibration-control system design as avibration absorber as well as an isolator. Adjustable suspensionelements including variable rate dampers and springs are pre-sented in Section 3. Section 4 overviews the fundamental prin-ciples of automotive semi-active suspensions, followed by recentadvances in this area. Application of control techniques to semi-active vibration-control systems is presented in Section 5 and ﬁ-nally Section 6 concludes the study.
2 Semi-Active Vibration-Control System Design
SA vibration-control systems can achieve the majority of theperformance characteristics of fully active systems, thus allowingfor a wide class of applications. The idea of SA conﬁguration isvery simple: to replace active force generators with continuallyadjustable elements which can vary and/or shift the rate of theenergy dissipation in response to instantaneous condition of mo-tion. This section presents the basic understanding, fundamentalprinciples and design issues for SA vibration-control systems.
2.1 Semi-Active Vibration Absorption Design.
A vibra-tion absorber is elastically attached to the vibrating body to alle-viate detrimental oscillations from its point of attachment
see Fig.2
. The underlying proposition for SA absorber is to properly ‘‘ad- just’’ the absorber parameters such that it becomes absorbent of the vibratory energy within the frequency interval of interest.
Contributed by the Technical Committee on Vibration and Sound for publicationin the J
OURNAL OF
V
IBRATION AND
A
COUSTICS
. Manuscript received May 2001;Revised April 2002. Associate Editor: R. L. Clark.
Copyright
©
2002 by ASMEJournal of Vibration and Acoustics
OCTOBER 2002, Vol. 124
Õ
593
In order to explain the SA absorber concept, a single-degree-of-freedom
SDOF
primary system with an SDOF absorber attach-ment is considered as shown in Fig. 3. It is easy to show that thetransfer function between the excitation force and primary systemdisplacement can be expressed as
TF
s
X
p
s
F
s
m
a
s
2
c
a
s
k
a
H
s
(1)where
H
s
m
p
s
2
c
p
c
a
s
k
p
k
a
m
a
s
2
c
a
s
k
a
c
a
s
k
a
2
, (2)
x
p
(
t
) is the primary system displacement,
f
(
t
) is the externalforce,
k
a
and
c
a
are the adjustable absorber stiffness and dampingcoefﬁcients and
X
p
(
s
) and
F
(
s
) are the Laplace transformationsof
x
p
(
t
) and
f
(
t
), respectively.The steady state displacement of the primary system due to aharmonic excitation is then
X
p
j
F
j
k
a
m
a
2
jc
a
H
j
(3)where
is the disturbance frequency and
j
1. Utilizing ad- justable properties of SA unit
i.e., variable rate damper
c
a
andspring
k
a
, an appropriate parameter tuning scheme is selected tominimize the vibration of the primary system subject to externaldisturbance
f
(
t
).
2.1.1 Harmonic Excitation.
When excitation is tonal, the ab-sorber is generally tuned at the disturbance frequency. For com-plete attenuation, the steady state
X
p
(
j
)
must equal zero. Con-sequently from Eq.
3
, the ideal stiffness and damping of the SAabsorber are adjusted as
k
a
m
a
2
,
c
a
0 (4)Notice this tuned condition is only a function of the absorberelements
m
a
,
k
a
, and
c
a
. That is, the absorber tuning does notneed information from the primary system except the vibratingfrequency and hence its design is theoretically stand-alone. Fortonal application, ideally zero damping in the absorber subsectionresults in improved performance, for the damped or undampedprimary structure. In practice, however, damping is incorporatedin order to maintain a reasonable trade-off between the absorbermass and its displacement. Hence, the design effort for this classof application is focused on having precise tuning of absorber tothe disturbance frequency and controlling damping to an appro-priate level. Referring to Snowdon
14
, it can be proven that theabsorber, in the presence of damping, can be most ‘‘favorably’’tuned if adjustable stiffness and damping are selected as
Fig. 1 Schematic of
„
a
…
force transmissibility for foundation isolation,
„
b
…
displacement trans-missibility for protecting device from vibration of the base, and
„
c
…
application of vibrationabsorber for suppressing primary system vibrationFig. 2 A typical primary structure equipped with three versions of vibration-control systems:
„
a
…
passive,
„
b
…
active, and
„
c
…
semi-active conﬁguration
594
Õ
Vol. 124, OCTOBER 2002
Transactions of the ASME
k
opt
m
a
m
p
2
2
m
a
m
p
2
,
c
opt
m
a
3
k
opt
2
m
a
m
p
(5)
2.1.2 Broadband Excitation.
In broadband vibration control,the absorber subsection is generally designed to add damping toand change the resonant characteristics of the primary structure inorder to maximally dissipate vibrational energy over a range of frequencies. The objective of SA absorption system design istherefore to adjust the
absorber parameters
to minimize the peak magnitude of the frequency transfer function (
FTF
(
)
TF
(
s
)
s
j
) over the absorber variable suspension parameters
p
c
a
k
a
T
. That is, we seek
p
tomin
p
sup
min
max
FFT
(6)Alternatively, one may select the mean square displacement re-sponse
MSDR
of the primary system for vibration suppressionperformance. That is, the absorber variable parameters vector
p
isselected such that the MSDR
E
x
¯
p
2
0
FTF
2
S
d
(7)is minimized over a desired wide band frequency range.
S
(
) isthe power spectral density of the excitation force
f
(
t
), and FTFwas deﬁned earlier.This optimization is subjected to some constraints in
p
space,where only positive elements are acceptable. Once the optimalabsorber suspension properties,
c
a
and
k
a
, are determined, theycan be implemented using adjustment mechanisms on the springand the damper elements. The conceptual devices for such adjust-able suspension elements will be discussed later in Section 3.
2.1.3 Simulations.
To better demonstrate the effectiveness of the SA absorber over the passive and optimum passive absorbersettings, the simple system shown in Fig. 3 with the followingnominal structural parameters
marked by an overscore
is taken.
m
¯
p
5.77 kg,
k
¯
p
251.132
10
6
N/m,
c
¯
p
197.92 kg/s(8)
m
¯
a
0.227 kg,
k
¯
a
9.81
10
6
N/m,
c
¯
a
355.6 kg/sThese values are from an actual test setting for vibration controlof high-frequency disturbances in submarine hulls. As shown inFig. 4, the absorber is a piezoelectric actuator with reaction masswhere its operating frequencies are in the range of 600–10,000 Hz
15,16
. That is, the peak of FTF is minimized
see thin lines inFig. 5
. When the primary stiffness and damping increase 5%
forinstance during the operation
, the FTF of the primary systemdeteriorates considerably
dashed line in Fig. 5
, and the absorberis no longer an optimum one for the present primary. When theabsorber is optimized based on optimization problem
6
, the re-tuned setting is reached as
k
a
10.29
10
6
N/m,
c
a
364.2 kg/s (9)which yields a much better frequency response
see thick line inFig. 5
.
2.2 Semi-Active Vibration Isolation Design.
The param-eter tuning control scheme for the SA isolator is similar to that of the SA vibration absorber, with the only difference being in thederivation of the transfer function. The classical isolator systemshown in Figs. 1
a
and 1
b
consists of a rigid body of mass
m
,linear spring
k
and viscous damping
c
. Conversely to the vibrationabsorber, the function of the isolator is to reduce the amplitude of motion transmitted from a moving support to the body
Fig. 1
b
,or to reduce the magnitude of the force transmitted from the bodyto the foundation to an acceptable level
Fig. 1
a
.The transfer functions between isolated mass displacement andbase displacement—or transmitted force to foundation and exci-tation force—are expressed as
F
T
F
0
X
s
Y
s
2
n
s
n
2
s
2
2
n
s
n
2
(10)
X
s
F
s
1/
ms
2
2
n
s
n
2
(11)where
c
/2
km
is the damping ratio,
n
k
/
m
is the naturalfrequency, and
F
T
is the amplitude of the transmitted force to thefoundation
see Fig. 1
a
.Figure 6 shows the transmissibility
T
A
(
T
A
F
T
/
F
0
X
/
Y
)as a function of the frequency ratio
/
n
and the damping ratio
,where the low frequency range in which the mass displacementessentially follows the base excitation,
X
Y
, is separated fromthe high frequency range of isolation,
X
Y
. Near resonance,
T
A
is controlled completely by the value of the damping ratio. Afundamental problem is that while a high value of damping ratiosuppresses the resonance, it also compromises the isolation forhigh frequency region (
n
). An optimal setting for isolatorcan be obtained similar to optimum vibration absorber problem
6
except with the transfer function
10
17
. The frequency
Fig. 3 Application of a semi-active absorber to SDOF primarysystem with adjustable stiffness
k
a
and damping
c
a
Fig. 4 PCB series 712 PZT inertial actuator
„
left
…
, schematic of operation
„
middle
…
, and a simpleSDOF mathematical model
„
right
… „
from
†
16
‡…
Journal of Vibration and Acoustics
OCTOBER 2002, Vol. 124
Õ
595
response plot of this transfer function shown in Fig. 7 indicatesthat the damping values sufﬁcient to control the resonance haveno adverse effect on high frequency isolation as opposed to Fig. 6.An SA isolator can also be utilized for disturbances with time-varying frequency. The variation of natural frequency
which is afunction of suspension stiffness
with the transmissibility
T
A
, inthe absence of damping, is given as
n
T
A
/
1
T
A
, 0
T
A
1 (12)With variable disturbance frequency,
, and desired transmissibil-ity
T
A
, the natural frequency
n
or the suspension stiffness
k
can be changed in accordance with Eq.
12
to arrive at optimalperformance operation
18
. The conceptual devices for adjustablesuspension elements are brieﬂy reviewed next.
Fig. 5 Frequency transfer functions
„
FTF
…
for nominal absorber
„
thin-solid
…
; de-tuned absorber
„
thin-dashed
…
; and re-tuned absorber
„
thick-solid
…
settings
„
from
†
15
‡…
Fig. 6 Frequency response plot of transmissibility
T
A
for the semi-active isolator as a function ofvariable damping ratio
596
Õ
Vol. 124, OCTOBER 2002
Transactions of the ASME

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