Jtam.dvi

JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 4, pp. 893-917, Warsaw 2007

EXPERIMENTAL STUDY OF VIBRATION CONTROL

OF A CABLE WITH AN ATTACHED MR DAMPER

Marcin Maślanka

Bogdan Sapiński

AGH University of Science and Technology, Department of Process Control, Cracow, Poland

e-mail: masmar@agh.edu.pl; deep@agh.edu.pl

Jacek Snamina

Cracow University of Technology, Institute of Applied Mechanics, Cracow, Poland

e-mail: js@mech.pk.edu.pl

The paper presents experimental investigation of a horizontally suspen-
ded cable with anMR damper attached transversally near the support.
The algorithm proposed to MR damper control employs the concept of
emulation of a viscous damper with an optimal viscous damping coeffi-
cient. The algorithm is realized using a velocity feedback with damping
force tracking control is applied. Free vibration of the cable with anMR
damper operating in passive and controlledmodes are investigated. The
obtained results indicate that an appropriately controlled MR damper
ensures a nearly constant damping level in a wide range of cable vibra-
tion amplitudes.

Key words: cable vibration, magnetorheological (MR) damper, control

1. Introduction

Since the 1960s, numerous cable-stayed bridges have been built all over the
world, their spans and cables made longer and longer. In the early 1990s,
the Skarnsund Bridge was opened – the first cable-stayed bridge with the
main span length exceeding 500m (Norway, 1991, main span 530m). Other
outstanding achievements of the 1990s include: Yang PuBridge (China, 1993,
main span 602m),NormandieBridge (France, 1994,main span 856m),Tatara
Bridge (Japan, 1999,main span 890m). They are light structures, constructed

894 M. Maślanka et al.

with the use of state-of-the-art technologies (Virlogeux, 1999). These structu-
res, particularly cables, are susceptible to vibration mainly due to weather
conditions because of their low internal damping, relatively small mass, large
flexibility andconsiderable length.Dangerous, high-amplitudecable vibrations
caused serious damages.

Cable vibrations in cable-stayed bridgeshave received agreat deal of atten-
tion recently. Many research teams study the complex nature of these vibra-
tions and explore potential mitigatingmeasures.Major research achievements
were reported e.g. in (Kumarasena et al., 2005; Virlogeux, 2005) and in inter-
national conference proceedings (AIM, 2005; MoDOT, 2006).

Most cable vibrations are induced by specific rain and wind interactions.
Such vibrations were first recognised on the Meiko-Nishi Bridge in Japan
(Hikami and Shiraishi, 1988). They occur in conditions of light to modera-
te winds (5,15)m/s and light rain. These rain-wind induced cable vibrations
usually take formofoneof several firstnatural cablemodes,with the frequency
range (1,3)Hz, andmaximumamplitude approaching 1m (Kumarasena et al.,
2005).

Reported are cable vibrationswith larger amplitudes, too.One of themost
intensive cable vibrationswere registered on theOresundBridge, linkingDen-
mark and Sweden, during severe snow storms. These vibrations might have
been caused by wet snow accumulating on the cable surface. The registered
amplitude of cable vibrations approached 3m, the cable kept on vibrating for
over an hour (Larsen andLafreniére, 2005). In similarweather conditions, vio-
lent cable vibrationswere observed twice on theDubrovnikBridge inCroatia,
which damaged the cable protective sheathing and some anchoring elements
(Savor et al., 2006).

Tomitigate cable vibrations, external viscous dampers are often attached
to cables near lower anchorages (Persoon and Noorlander, 1999; Main and
Jones, 2001) – the main purpose is to improve cable damping. In order to
ensure the required damping level, viscous dampers should be carefully tuned,
individually for each cable and each vibration mode to be damped. Too low
external damping will not guarantee a sufficient cable damping level, whilst
too high value shall cause the cable to be supported at the location of the
damper which results in cable vibration with slightly increased frequency and
no appreciable damping effect (Krenk, 2000). In practice, it is difficult to
determine a priori the dominant vibrationmode to which the viscous damper
should be tuned. In specific weather conditions, the dominant mode might
be different for individual cables. The occurrence of a specific cable vibration
mode strongly depends also on the type of excitations.

Experimental study of vibration control... 895

It has to be emphasised that the longer the cables, themore difficult their
vibration damping. The current expertise allows for construction of cable-
stayed bridges with the main span length exceeding 1000m and the cable
length of 550m. There are so large bridges under construction: Stonecutters
Bridge (Hong Kong, to be commissioned in 2008, main span length 1018m,
maximal cable length 536m) and Sutong Bridge (China, to be commissioned
in 2009,main span1088m,maximal cable length577m).For practical reasons,
external dampers are attached to cables at the distance of 1-2% of their length
from the lower anchorage, which limits the capability of effective damping.
Thus, precise tuning of dampers is of primary importance. New solutions are
sought as e.g. those utilising MR dampers attached to cables in a similar
manner as viscous dampers.The benefit ofMRdampers is that their damping
characteristic can be easily adapted to particular conditions.

The concept of using MR dampers in cable vibration reduction systems
has received a great deal of attention. Applications of controlled semi-active
elements to cable vibration control were first explored by Johnson et al. (1999,
2000). Their research focusedon the taut stringmodel andan ideal semi-active
element (withno limitationonthemaximumforce). Simulation tests confirmed
good performance of a sub-optimal (with passive constraints) LQ controller.
Further testswere runonamodelof a inclined cable,where cable sagwas taken
into account (Johnson et al., 2003). Experimental verification was conducted
in (Christenson et al., 2001) on a laboratory cable model, 12.6m in length.

A pioneering implementation of MR dampers to cable vibration control
systems tookplace in 2002, on theDongtingLakeBridge inChina (Chen et al.,
2003; Duan et al., 2006). Those dampers operated in the passivemode (under
constant current). In this case, the MR damper force depends on the piston
velocity in a minor degree only, which seems to be a major drawback. MR
dampers capabilities can be fully utilised in the controlled mode of operation,
provided an appropriate control algorithm is selected. The study of a control
algorithm with experimental verification on the Dongting Lake Bridge was
reported by Duan et al. (2005).

Research on semi-active vibration control systems completewithMRdam-
pers was undertaken by other research teams, too. Results of experiments run
on a horizontally suspended cable 215m in length (a prototype cable for the
Sutong Bridge) were compiled in (Sun et al., 2004; Zhou and Sun, 2005). In
those experiments, the performance of MR dampers operating in the passive
mode were compared with the performance of other types of dampers.

Results of experiments on a cable 15.5m in length were summarised by
Weber et al. (2005c,d). This programwas further pursued in the implementa-

896 M. Maślanka et al.

tion of a single MR damper on the Eiland Bridge in the Netherlands (Weber
et al., 2005b). The tests were run on a specially designed and fabricated MR
damper, generating a force up to 40kN. This type of damper was utilised in
2006 on the Dubrovnik Bridge to counteract high-amplitude cable vibrations
previously observed on this bridge (EMPA, 2006). That was the first imple-
mentation of a semi-active vibration control system on a cable-stayed bridge
world-wide, utilising specially designed and controlled MR dampers.
This study summarises the experimental data gathered during laborato-

ry tests of a cable 30m in length. Both passive and controlled modes of MR
damper operation are considered. The algorithm chosen for control purposes
is that proposed byWeber et al. (2005a), which was first tested on the Eiland
Bridge. The purpose of the tests was to explore potential applications of MR
dampers operating in the passive mode to the cable vibration control and
to find advantages of the applied control algorithm. Damping of cable vibra-
tions usingMR dampers both in passive and controlled modes is investigated
thoroughly.
Thearticle is organized as follows. Section1provides an introduction to the

cable vibration problem and mitigation countermeasures, particularly using
MRdampers. The considered cable-MR system is described in Section 2. Sec-
tion 3 presents simple and inverse models of theMR damper. Section 4 deals
with implementation of the control algorithm. Selected results are compiled
in Section 5. Results obtained for cable vibration reduction systems complete
with the controlled MR damper are preceded by the analysis of dynamic be-
haviour of the cable with the MR damper in the passive mode and the cable
with no damper.Discussion of results and conclusions are provided in Sections
6 and 7.

2. System description

Of particular interest is a horizontally suspended and stretched cable of the
length L, linear density m and static tension force T , with an MR damper
attached transversely to the cable at the distance xd from the support (Fig.1).
Themain research area covers vibration control algorithms for this system.
Thework byKrenk (2000) gives an approximate formula to determine the

optimal viscous damping coefficient copt of a viscous damper attached to the
cable at the point xd, at a small distance from the support

cnopt =
1

nπ

√
Tm (2.1)

where: n – mode number.

Experimental study of vibration control... 897

Fig. 1. Cable with an attachedMR damper

The optimal value of the viscous damping coefficient for a cable with spe-
cified parameters and for the given damper location is different for eachmode.
As stated in Section 1, a viscous damper optimally tuned to damp one mode
of vibrations may not optimally damp other modes. For a given cable and
specific weather conditions, it is difficult to determine the dominant vibration
mode to accurately tune the damper.

Capability of tuning of an MR damper to the occurring vibration mode
proves to be its major advantage over conventional viscous dampers. When
the appropriate control algorithm is chosen, one MR damper generating a
sufficient force might be well utilised to damp several modes of vibrations of
cables differing in lengths and with variable damper locations.

3. MR damper

MRdampers are controllable, semi-active actuators utilisingMRfluids in their
operation (Jolly et al., 1999;Goncalves et al., 2006).MRfluidshave aproperty
to reversibly change fromthe liquid to semi-solid state, inacontinuousmanner,
under the action of a magnetic field.

Thedamperused for thepurposeof the researchprogram is anMRdamper
series RD-1097-01 (Lord Co., 2002). Its construction is different from typical
linear stroke dampers filled with MR fluids and operating in the valve mode.
Themain difference lies in the presence of an absorbentmatrix saturatedwith
an MR fluid in the RD-1097-01 damper (Carlson, 1999; Chrzan and Carlson,
2001). A schematic diagram of the damper is shown in Fig.2.

The key functional parameters of the damper RD-1097-01 listed by the
manufacturer are: maximum force 100N (for current 1A and piston velocity
51mm/s), stroke ±25mm, response time< 25ms (time required to reach 90%
of the steady-state value of force under a step change of the current from 0
to 1A, for 51mm/s). The force in the passive-off mode (0A) is less than 9N
(at piston velocity 200mm/s).

898 M. Maślanka et al.

Fig. 2. Structure of the RD-1097-01 damper

The damper RD-1097-01 was first utilised in a cable vibration control
system in laboratory tests run on a cable 7.2m in length (Wu andCai, 2006).

3.1. Visco-plastic model with hysteresis

AnMRdampermaybeapproximately consideredasa frictiondamperwith
an additional force component proportional to velocity (viscous component).
Accordingly, in steady statemotion, the restoring force Fd of theMRdamper
is governed by the Binghammodel (Spencer et al., 1997)

Fd =Fc sgn(ẇd)+ c0ẇd (3.1)

where: Fc is the friction force, ẇd is the piston velocity and c0 is the viscous
damping coefficient.

TheBinghammodel accurately approximates the steady-state relationship
between the restoring force and relative velocity of the piston in theMRfluid
post-yield region (|Fd| Fc). The model fails to capture the real nature of
the damper restoring force in the pre-yield region (|Fd| < Fc), in which the
MRfluid exhibits features of a quasi-solid body (Gandhi andBullough, 2005).

Inorder tobetter fit themodel to experimental data in thepre-yield region,
the signum function in Eq. (3.1) is replaced by a hyperbolic tangent function
(Guo et al., 2006; Kwok et al., 2006), and a hysteresis is added to the model

Fd =Fc tanh[µ(ẇd+pwd)]+ c0(ẇd+pwd) (3.2)

where: µ is the scaling parameter enabling representation of smooth transition
in the pre-yield region from negative to positive velocities and the other way
round, p denotes the scaling parameter of the hysteresis. In the general case,
themodel parameters are functions of the current applied to the damper. For
parameters p=0, µ→∞, the model governed by Eq. (3.2) is reduced to the
Bingham model.

Experimental study of vibration control... 899

Identification of the model given by Eq. (3.2) is performed on the basis of
experimental data. The design of the laboratory setup is described in more
detail in the Subsection 5.1.
Figure 3 shows the results of model identification obtained bymethods of

parameter optimisation. Relationships between the parameters Fc and c0 and
the current I are approximated by linear functions

Fc =C1I+C2 c0 =C3I+C4

where: C1 = 62N/A, C2 = 1.5N, C3 = 48Ns/Am, C4 = 14Ns/m. The
remainingmodel parameters are: µ=130 s/m, p=1 s−1.

Fig. 3. Parameters of theMR damper model vs. current: (a) friction force (Fc),
(b) viscous damping coefficient (c0)

3.2. Inverse model

Realization of MR damper force tracking control systems often utilise an
inversemodel ofMRdampers (Xia, 2003; Du et al., 2005; Tsang et al., 2006).
In practical implementations of digital control systems, an inversemodelmay
be representedby look-up-table basedon the steady-state relationshipbetween
the current, damper force and piston velocity. An inverse model, formulated
analytically, has a number of advantages, though it is difficult to obtain. The
approach used here involves a linear approximation of relationships between
model parameters and current (Fig.3).
Assuming the approximating functions of model parameters, an inverse

formof the presentedmodel (Eq. (3.2)) is developed by direct transformations
yielding the following equation (for ẇd 6=0)

900 M. Maślanka et al.

I =
Fd−C2 tanh[µ(ẇd+pwd)]−C4(ẇd+pwd)
C1 tanh[µ(ẇd+pwd)]+C3(ẇd+pwd)

(3.3)

When the hysteresis is neglected (p=0) Eq. (3.3) is reduced to the form

I =
Fd−C2 tanh(µẇd)−C4ẇd
C1 tanh(µẇd)+C3ẇd

(3.4)

4. Control system design

Themainpurposeof the considered control algorithm is to ensure that theMR
damper should emulate operation of a viscous damper with the desired value
of the viscousdamping coefficient (cdes). It is assumed that the force generated
by the controlledMRdamper shouldbeproportional to piston velocity (unlike
in the passive mode of damper operation when the force depends on velocity
in a minor degree only). For the given cable parameters, vibration mode and
damper location, the desired viscous damping coefficient should be derived
from Eq. (2.1) to ensure the maximal damping. A simplified diagram of the
control algorithm is shown in Fig.4.

Fig. 4. A simplified diagram of the control algorithm

The algorithm utilises the feedback loop from the piston velocity. The
velocity, in real conditions, might be reconstructed from the acceleration or
piston displacement signals. For the purpose of the research program, the ve-
locity signal was estimated on-line (Signal Processing) on the basis of piston
displacement measurements. Knowing the velocity, the desired force is obta-
ined (Primary controller), which is processed with the force tracking control
system (Secondary controller) to obtain the current applied to the damper.
Themajor task in the implementation of the control algorithm is to ensure

a satisfactory accuracy of the MR damper force tracking control. Damping
force control might be handled either by applying an internal feedback from

Experimental study of vibration control... 901

the damper force or with no force feedback, by applying an inverse model
of the damper. The second approach is adopted throughout this study. The
inverse model of the damper, proposed in Subsection 3.2, is applied to the
force tracking control system. The piston velocity and desired force are fed
to the model input. The model output is a current which should ensure the
desired damping force.
When the inverse model of the MR damper is employed, its accuracy is

of key importance in the whole considered range of the piston velocity and
current. However, degradation of force tracking capabilities might be caused
by a phase shift in the feedback loop.

5. Experiments

5.1. Experimental setup

Experimental tests were run in a laboratory setup specially designed for
thepurposeof testingvibration reduction systemsof a cablewithMRdampers
(Sapiński et al., 2006). A diagram of the experimental setup and a photo of
theMR damper installation are shown in Fig.5 and Fig.6.

Fig. 5. A diagram of the experimental setup

There is a horizontally suspended steel cable, clamped at the ends. The
length of the control section is L = 30m, cable mass per unit length is
m = 1.8kg/m. The cable is tensioned using a lever mechanism. The maxi-
mum tension force approaches 50kN. AnMRdamper of the RD-1097-01 type

902 M. Maślanka et al.

Fig. 6. A photograph of theMR damper installation

is attached near one of the cable supports.Thedamper is slide-mountedwhich
allows one to investigate its performance at various positions determined by
its distance from the support (1,2.5)m.

Themeasurement and control system comprises a PCwith multi I/O bo-
ard andMATLAB/Simulink. Transverse cable accelerations can bemeasured
atmaximally 12 locations, transverse cable displacements aremeasured at two
points and thedamper force ismeasured along the damper axis.TheMRdam-
per is controlled using a power controller operating in the analogue, voltage
input – current output mode (Rosół and Sapiński, 2006).

On account of the adopted MR damper control algorithm, optimal values
of viscous damping coefficients have to be estimated for given parameters
of the facility. For the given cable parameters (L, m), the optimal viscous
damping coefficient depends on the static tension force T , damper attachment
location xd and the cable vibration mode. The tension force is taken to be
27kN and xd = 1m, yielding copt of about 2100Ns/m for the first mode,
1050Ns/m for the second mode and 700Ns/m for the third one. The force
range of theRD-1097-01MRdamper allows the optimal viscous damper force
to be tracked only in a certain amplitude range for the given mode (Fig.7).

The results presented in this studyapply to thefirstmodeof freevibration.
The chief advantage of the free vibration test is that the basic analysis of

Experimental study of vibration control... 903

Fig. 7. Force-velocity characteristics for various linear viscous dampers
in comparison to theMR damper (RD-1097-01) force range

registered data reveals relationships of damping and frequency as functions of
the amplitude. Such observations are of primary importance in the analysis of
dynamic properties of cables withMR dampers.

5.2. Cable with no damper

Free vibrationmeasurement data are compiled in Fig.8. There is no dam-
per attached to the cable. The onlymeasured quantity is displacement at the
cable mid-point. The time when free vibration began is denoted as ts. In the
time range (0, ts) the cable was excited manually at the cable mid-point.

Cable vibration frequency and themodal damping ratio are identified ba-
sing on the analysis of the displacement envelope, yielding relationships be-
tween the amplitude, frequency anddamping ratio (Figs. 8b and 8c).Dynamic
component of the cable tension force increases with the amplitude, which is
revealed as a slight increase in the free vibration frequency (Fig.8b).Thedam-
ping ratio for the firstmode varies from 0.2·10−3 for the amplitude 5·10−3m,
to 1.05·10−3 for the amplitude 80·10−3m.The obtained relationship between
the damping ratio and amplitude is approximately linear in the investigated
range.

5.3. Cable with MR damper in passive mode

Selected results of measurements of the vibration decay are shown in Fig.9
and Fig.10. Of particular interest is the first mode of free vibration of the

904 M. Maślanka et al.

Fig. 8. Free vibration of a cable with no damper: (a) displacement decay at the
cable mid-point, (b) relationship between the amplitude and frequency,

Fig. 9. Free vibration of the cable with theMR damper for I =0.25A: (a) cable
displacement at the damper location, (b) cable displacement at the mid-point,

cable with the MR damper operating in the passive mode. The damper lo-
cation is defined by the coordinate xd = 1m and two values of the applied
current are considered: 0.25A and 0.5A. The measured quantities are: cable
displacement and acceleration at the point the damper is attached, mid-point
cable displacement and the force acting along the damper axis.

Free vibration of the cable with the attached damper tends to decaymuch
faster than in the case of a cable with no damper as shown in Subsection 5.2.
One has to bear in mind, however, that the decay is accompanied by a mi-

Experimental study of vibration control... 905

Fig. 10. Free vibration of the cable with theMR damper for I =0.5A: (a) cable
displacement at the damper location, (b) cable displacement at the mid-point,

nor variation of the damper force amplitude (Fig.9c and Fig.10c). This very
feature of the MR damper force leads to an adverse blocking effect of piston
movement in theMR damper.

A rapid decay of cable vibration (the higher the current, the faster the
vibration decay) is observed from the time when free vibration begins right
till the moment when complete blocking of the MR damper occurs (denoted
as tc in Figs.9 and 10). Thedamper blocking is observedwhena force of cable
action upon the damper is less than the force required to change the pistonpo-
sition in thewhole period of vibration. As the vibration amplitude decays, the
force of cable action upon the damper is reducedwhilst the current-dependent
friction component of the damper restoring force (Fc) remains unchanged.
Comparison of these two forces allows for formulating a theoretical condition
of theMR damper blocking (Weber et al., 2005a; Maślanka, 2006). Graphs of
cable displacements at the damper location and at the mid-point (Figs.9a,b
and 10a,b) indicate that when the damper stops working, the remaining part
of the cable still vibrates with no appreciable damping. The point where the
damper is attached becomes a node of a new, undamped mode with a sligh-
tly higher frequency. The observed frequency shift is accompanied by minor
beating phenomenon revealed in the measured force (Fig.10c).

MR damper characteristics describing the damper force as a function of
piston velocity for the considered current are shown in Fig.11. They are ob-
tained in the time range from the 5th to 10th period of free vibration. The
damper force is determined from force measurements along the damper axis
(Figs. 9c and 10c). Presented force-velocity characteristics are well captured

906 M. Maślanka et al.

by the damper model with hysteresis (Eq. (2.3)). The inertial force compo-
nent, associated with themass of damper attachment elements, has been sub-
tracted.

Fig. 11. Damper force vs. velocity for: (a) I =0.25A, (b) I =0.5A

The study of cable vibrationwith theMRdamper in a passivemode invo-
lves analysis of the equivalent viscous damping coefficient. For a given damper
restoring force (Fd) and piston velocity (ẇd), the energy dissipated during
one period of vibration (T0) is equal to

∆E =

t+T0∫

Fdẇd dt (5.1)

Assuming that the displacement wd is described by a sine function with
an amplitude A, the same amount of energy per cycle shall be dissipated by
a viscous damper with the equivalent viscous damping coefficient

ceq =∆E
T

2π2A2
(5.2)

Figure 12a compiles the computed equivalent viscous damping coefficient
ceq obtained on the basis of measurements of the free vibration decay of the
cablewith theMRdamper under the current 0.5A.The calculation procedure
is definedbyEqs. (5.1) and (5.2).Theamplitude A is estimatedbyanenvelope
of the cable displacement signal at the point xd. The velocity ẇd is estimated
on the basis of the cable acceleration signal measured at the point xd. The
obtained results are encumberedwith aminor error only, because the velocity
signal is close to the purely sine one. However, the lower the amplitude, the
greater the error.

Experimental study of vibration control... 907

Fig. 12. Amplitude-dependent damping of the cable with theMR damper in the
passivemode, I =0.5A: (a) equivalent viscous damping coefficient,

(b) damping ratio

The coefficient ceq is stronglydependenton the amplitude.That is because
various amplitudes of piston displacements are associated with similar values
of damper force amplitudes (compare Figs. 10a and 10c).

Figure 12a shows the optimal viscous damping coefficient copt obtained for
the 1st mode and given parameters of the laboratory setup. For the assumed
current, the equivalent viscous damping coefficient ceq is equal to copt for
the amplitude denoted as Aopt(xd). When the amplitude exceeds Aopt(xd),
the value of ceq decreases, without providing the optimal damping.When the
vibration amplitude is less than Aopt(xd), the value of ceq sharply increases
and, finally, the cable gets clamped at the point the damper is attached.

A thorough analysis of Fig.12b allows for interpretation of properties of
cable vibration dampingusing anMRdamper in the passivemode.This graph
is plotted on the basis of the displacement signal measured at the cable mid-
point. Damping ratio values are obtained in a narrow time window shifted
along the time axis.

Thedampingcurve inFig.12bhas themaximumfor theamplitudedenoted
as Aopt(L/2). This amplitude is registeredwhen ceq equals approximately copt
(Fig.12a). When the amplitude of cable vibration is larger than Aopt(L/2),
thedampingperformance is lower.When the amplitude is less than Aopt(L/2),
the damping ratio of the cable assumes lower values. This is a consequence of
getting closer to the blocking of the damper piston.

The amplitude denoted as Ac(L/2) (see Fig.12b) corresponds to the fully
blocked damper. TheMRdamperwhen blocked does not dissipate the energy
of cable vibration.That iswhy, for vibrationamplitudes less than Ac(L/2), the
damping isminimal, approximately on the level of the cable internal damping.

908 M. Maślanka et al.

It has to be emphasised that, for practical purposes, the characteristics in
Fig.12 seemvery unfavourable. There is an obvious need to provideMRdam-
per control in the vibration reduction system. The control should ensure that
variations of vibration amplitudes should not produce anymajor deterioration
of vibration damping performance.

5.4. Cable with MR damper in controlled mode

Figure 13 compiles measurement data collected for the first mode of free vi-
bration of the cable with the MR damper controlled in accordance with the
algorithm presented in Section 4. The damper location is given by the coordi-
nate xd =1m. The results are obtained for cdes =500Ns/m.

Fig. 13. Free vibration of the cable with the feedback controlledMR damper,
cdes =500Ns/m: (a) cable displacement at the damper location, (b) cable

displacement at the cable mid-point, (c) force along the damper axis, (d) current

Experimental study of vibration control... 909

Vibration decay plots shown in Fig.13 vastly differ from cable vibration
decay patterns collected when the damperwas operating in the passivemode.
The vibration decay envelope has a shape similar to an exponential curve.
Furthermore, the displacement decay at the point the damper is attached
(Fig.13a) and at the cable mid-point (Fig.13b) proceeds in the samemanner
throughout the whole considered amplitude range. Application of the control
algorithm allowed the damper blocking effect to be almost entirely elimina-
ted. The system comprising a cable with a controlled MR damper exhibits
properties of a viscous-damped system, as planned.

Figure 13c compiles the force measured along the damper axis and the
desired force proportional to piston velocity. TheMR damper force was gene-
rated by the current, plotted in Fig.13d. At maximum cable displacements,
the desired force determined by the control algorithm is slightly higher than
the maximum force to be generated by the MR damper RD-1097-01. Hence
the observed difference between the two forces in this interval. This also ap-
plies to the difference between the current levels: that required to produce the
desired force and themeasured one.

Fig. 14. Illustration of force tracking control accuracy, cdes =500Ns/m: (a) force vs.
velocity, (b) force acting along the damper axis (zoomed window 1 of Fig.13c),

Tracking accuracy of the MR damper force is presented in more detail
for two selected time intervals. The first one (Fig.14) covers subsequent free
vibrationperiods (from8 to 12) designated aswindow1 inFig.13.The error of
the desired force representation is minimal in this interval. The other interval
covers free vibration periods 32-36 (window2). The force tracking error in this
range takes the maximum value.

910 M. Maślanka et al.

Fig. 15. Illustration of force tracking control accuracy, cdes =500Ns/m: (a) force vs.
velocity, (b) force acting along the damper axis (zoomed window 2 in Fig.13c),

Force-velocity relationships are given for these two considered time inte-
rvals. There are also plots of the force and applied current in function of time.
The measured data are compared with the desired values generated by the
control system. Plots in Fig.14 and Fig.15 confirm the adequacy of the ap-
plied control algorithm. When the appropriate control algorithm is adopted,
the MR damper well emulates a linear viscous damper with the given ope-
rational characteristics. The resultant force-velocity relationship is similar to
the desired characteristic and does not exhibit any hysteresis typical for MR
dampers operating in the passive mode (Fig.11). The force tracking error ob-
served inFigs. 15a and15b results fromthe fact that the relationships between
the dampermodel parameters and the applied current are approximated by a
linear function (Fig.3). In the current interval where the approximation error
was the largest (0.1,0.2)A, the force tracking error would bemaximal, too.

It is worthwhile to mention that underlying the experiments was the in-
verse model of an MR damper with no hysteresis (Eq. (3.4)). As the force
tracking control accuracy is satisfactory, the application of the inverse model
with hysteresis does not seem justified.

Figure 16a shows the equivalent viscous damping coefficient (ceq) derived
from Eqs. (5.1) and (5.2). In an ideal case, the value ceq should be equal to
the desired value cdes, nomatter what amplitude. In our case, for amplitudes
larger than 7·10−3m, the observed difference between ceq and cdes is a conse-
quence of the maximum force constraints in theMR damper. For amplitudes
lower than 10−3m, current stabilises on 0A, a further amplitude decay will
not produce the desired reduction of the damper force (see Fig.13), which
corresponds to increase of ceq in this interval, in accordance with Eq. (5.2).

Experimental study of vibration control... 911

Fig. 16. Amplitude-dependent damping for the cable with the feedback controlled
MR damper, cdes =500Ns/m: (a) equivalent viscous damping coefficient,

(b) damping ratio

Figure 16b gives a plot of the cable damping ratio obtained on the basis
of the mid-point cable displacement signal. This characteristic is associated
with that in Fig.16a and takes a fixed value (6 · 10−3, 8 · 10−3) in a wide
range of amplitudes. The obtained damping level corresponds to the desired
value cdes. The considered value of the desired viscous damping coefficient
cdes =500Ns/mmakes up for about 25% of the optimal damping level.

6. Discussion of obtained results

The research data compiled in this studyare achieved for the firstmodeof free
vibration and damper position defined by the coordinate xd =1m. The me-
asurement data obtained for the cablewith an attachedMRdamper operating
in thepassivemodeare summarised inSubsection5.3, for twovalues of current.
The assumedvalue of the desired viscous damping coefficient cdes =500Ns/m
is about 25% of the optimal value for the first mode.
The laboratory tests were conducted for various current levels from the

range (0,0.5)A and for several values of desired viscous damping coefficients,
ranging from 100 to 2000Ns/m. The obtained data became the starting point
for comparative analysis of cable vibration damping usingMRdampers in the
two operating modes. The applied criterion is the rate of vibration decay, as
used by Weber et al. (2005a). The mid-point displacement signal is utilised
to find the time necessary to decay of preset percentage fraction of the initial

912 M. Maślanka et al.

amplitude. The assumed factor t90% determines the time required for 90%
decay of the initial amplitude.

Figure 17a shows a plot of the decay time versus the current applied to
theMR damper. This relationship has a minimum at 0.4A. At current levels
exceeding 0.4A, the decay time tends to increase due to the damper blocking
effect. In terms of the assumed criterion, the current 0.4A appears to be
optimal. The value of the decay time t90% is 16s for this current.

Fig. 17. Free vibration decay time for the cable with the attachedMR damper,
xd =1m: (a) passivemode, (b) controlledmode

Figure 17b shows a plot of the decay time obtained frommeasurements of
cable vibration with a controlled damper. Values of t90% are given in function
of the desired viscous damping coefficient cdes. The minimum value of the
decay time t90% (achieved for cdes =2000Ns/m) is equal to 14s.

A comparison of plots in Figs. 17a and 17b does not reveal major impro-
vements of the vibration reduction performancewhen the control algorithm is
applied. It has to be emphasised, however, that for cdes exceeding 500Ns/m,
the damping force tracking involves a certain error due to the fact that no
larger force can be realized than the maximum force of theMR damper used
in the study. For example, for cdes = 2000Ns/m, the maximum value of the
desired force is 220N, whilst the force generated by the MR damper appro-
aches 45N. For the given cdes, the error is greater for higher amplitudes. The
plot inFig.17b does not fully portray the cable vibrationdamping capacity for
the adopted control algorithm.When theMRdamper is capable of generating
larger forces, the decay time should take lower values.

Experimental study of vibration control... 913

The decay time of the cable with no damper equals t90% =400s (Fig.8a).
Connecting thedamper at thepoint xd =1mallows for approximately 25-fold
(passivemode) and 28-fold (controlled mode) reduction of the time needed to
decay 90% of the initial amplitude for the first mode of free vibration.

7. Summary

Thepaper compiles the results of laboratory testing of a cable vibration reduc-
tion systemcompletewith anMRdamper.Damper operationwas investigated
both in the passive and controlled mode.

Measurements taken on the cablewith the damper operating in the passive
mode reveal strongly nonlinear properties of the system. The main source of
nonlinearity is theMR damper.
The analysis of damping of the cable-damper system utilises an equivalent

viscous damping and a non-dimensional damping ratio. Results of analysis
justify the need to implement an appropriate algorithm to control the MR
damper to ensure the desired damping level (independent of amplitude).

The paper outlines the implementation of the algorithm based on a feed-
back loop from the piston velocity. The control system structure comprises
a primary controller with the desired damping force as the output and a se-
condary controller to control the MR damper force. Application of an MR
damper inverse model in the force control system ensures sufficient precision
of control without a need of another feedback loop from the damper force.
The inverse model was created on the basis of the MR damper model with
linear approximations of relationships between themodel parameters and the
applied current. It has to be pointed out that in the case of MR dampers,
where the relationship between themodel parameters and the current is non-
linear, this approach might lead to major errors. Delays in the measurement
and control system are the source of an additional error in the force tracking
control.

Results ofmeasurements of free vibration of a cable with an attached con-
trolled MR damper confirm the adequacy of algorithm implementation. The
analysis of measurement data indicates that an appropriately controlled dam-
per ensures a nearly constant damping level in a wide range of amplitudes,
which is a major benefit of viscous dampers. However, themaximal, theoreti-
cally predicted damping level is impossible to achieve during the experiments.
These constraints are associated with the limited force range of theMR dam-
per used in the tests.

914 M. Maślanka et al.

It is worthwhile tomention that the investigated control conceptmight be
implemented in analternativemanner, byperiodic changing of the current, de-
pending on the vibration amplitude (Maślanka, 2008). The current is changed
once per one period of vibration, in response to the instantaneous amplitude
of the piston displacement and vibration frequency. The thus achieved cable
damping performance is similar to that reported in this study.

Acknowledgements

The work is supported by the State Committee for Scientific Research (Poland)

as a part of the research programNo. 4T07B00830.

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Badania eksperymentalne sterowania drganiami liny z dołączonym

tłumikiem MR

Streszczenie

W artykule przedstawiono wyniki badań eksperymentalnych układu poziomo za-
wieszonej liny z tłumikiemMRdołączonympoprzeczniewpobliżu podpory.Do stero-
wania tłumikiemMR przyjęto koncepcję emulacji tłumika wiskotycznego o optymal-
nymwspółczynniku tłumienia.Algorytmsterowaniazrealizowanoprzywykorzystaniu
sprzężenia od prędkości oraz nadążnego układu sterowania siłą tłumienia. Analizie
poddano drgania swobodne liny z tłumikiem MR pracującym w trybie pasywnym
oraz sterowanym. Wyniki badań wskazują, że opracowany algorytm sterowania po-
zwala na uzyskanie w przybliżeniu stałego poziomu tłumienia drgań liny w szerokim
zakresie wartości amplitudy drgań.

Manuscript received March 26, 2007; accepted for print July 4, 2007