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Research ArticleVibration Mitigation without Dissipative DevicesFirst Large-Scale Testing of a State Switched Inducer

Daniel Tirelli

European Laboratory for Structural Assessment (ELSA) Joint Research Centre of the European Commission Italy

Correspondence should be addressed to Daniel Tirelli danieltirellijrceceuropaeu

Received 15 July 2013 Accepted 10 March 2014 Published 12 August 2014

Academic Editor Miguel M Neves

Copyright copy 2014 Daniel TirelliThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new passive device for mitigating cable vibrations is proposed and its efficiency is assessed on 45-meter long taut cables througha series of free and forced vibration tests It consists of a unilateral spring attached perpendicularly to the cable near the anchorageBecause of its ability to change the cable dynamic behaviour through intermittent activation the device has been called stateswitched inducer (SSI)The cable behaviour is shown to be deeplymodified by the SSI the forced vibration response is anharmoniccand substantially reduced in amplitude whereas the free vibration decay is largely sped up through a beating phenomenon Thevibration mitigation effect is mainly due to the activation and coupling of various vibration modes as evidenced in the responsespectra of the equipped cableThis first large-scale experimental campaign shows that the SSI outperforms classical passive devicesthus paving the way to a new kind of low-cost vibration mitigation systems which do not rely on dissipation

1 Introduction

Resonances of slender structures such as bridges towersor cables are usually mitigated by dampers of various typesorand dynamic absorbers [1ndash7] In the case of cables alsocrossties are sometimes used but they totally change theoriginal characteristics of the system [8] Occasionally thesepassive devices may not be sufficient and are then substitutedby active or semiactive devices All existing passive devicesincluding nonlinear energy sinks (NES) [9ndash13] are based onthe same principle part of the kinetic energy is transferredfrom the structure to the device where it is dissipated

An original strategy of vibration mitigation is proposedwhich consists in attaching a unilateral spring perpendicu-larly to the cable near the anchorage Because of its abilityto change the cable dynamic behaviour through intermittentactivation this new passive device has been called stateswitched inducer (SSI) The scope of the present work is toassess the efficiency of the proposed device through large-scale testing and to understand how it operates In fact unlikeexisting passive devices the SSI cannot mitigate vibrationsthrough energy dissipation since it is purely elastic A goodunderstanding of the phenomena involved is essential foroptimizing the device and identifying its limitations

This paper is organised as follows In Section 2 the SSIconcept is explained and its effect on the cable behaviouris anticipated by using some established properties of non-linear dynamic systems Then the experimental campaignis described with special emphasis on the difficulties arisingfrom the very nature of the system In Sections 4 and 5 a fewrepresentative free and forced vibration tests are presentedand analysed The observed behaviour of the equipped cableis tentatively interpreted in Section 6 by means of an ldquoequiv-alentrdquo SDoF bilinear oscillator Finally the SSI optimisationis addressed in Section 7 where an empirical formula forthe most efficient switching position is proposed and fairlyverified experimentally

2 The State Switched Inducer (SSI)

21 Basic Concept Structural resonances are due to thesuperposition of travelling waves In the case of linear struc-tures this superposition results in a stationary wave Sincetravelling waves are symmetric each reflected wave at theanchor of a cable for example superposes perfectly withthe incident one giving a linear increment of the resonantamplitude (Figure 1(a)) This is why any resonance needs

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 135767 14 pageshttpdxdoiorg1011552014135767

2 Shock and VibrationD

ispla

cem

ent (

mm

)

Time (s)10 12 14 16 18

0

10

20

30

40

minus10

minus20

minus30

minus40

s07 cable 1 Fexi = 100N

(a)

SSI active

SSI inactive

Ap = Anchorage positionElastic wire

Load cell

Switch positiong =

Screw for gapregulation

(b)

Figure 1 (a) Cable resonance with low damping (blue curve fitting) (b) SSI principle

a minimum number of excitation cycles to develop possiblyup to a dangerous level

Now if travelling waves are not perfectly symmetricincident and reflected waves will not perfectly superposeThe amplitude increment will be smaller and the resonancephenomenon is reduced Asymmetric travelling waves can beobtained if the structure is in one dynamic state 119878 (stiffness119870andmass119872) for half a period and in another dynamic state 1198781015840(stiffness1198701015840 and possiblymass1198721015840) for the successive (nearly)half of the period In a cable this state switching can beeasily achieved by attaching a unilateral spring of stiffness119870

119904

between the cable and the ground or the deck (Figure 1(b))This explains the name given to the proposed device stateswitched inducer (SSI)

A linear structure equipped with a SSI becomes a bilinearoscillator a system extensively studied in the literature Abilinear oscillator admits more basins of energy (peaks) inthe high frequency range than the corresponding linear oscil-lator This property naturally yields a positive effect of theSSI according to the Parseval theorem conservative systemsreceiving the same input energy have the same integral in thefrequency domain thus if peaks are more numerous for thenonlinear oscillator their amplitude and thus the resonancesare smaller than those for the corresponding linear oscillator

Another interesting property of the SSI is the suddenchange of dynamic state at the switch time Any such non-linear behaviour (in electricity acoustics mechanics etc) isknown to generate harmonics The amplitude of these har-monics depends on the degree of nonlinearity introduced inthe system but it is generally very small with respect to theexcited frequency Thus the proportion of energy subtractedto the main resonant mode is usually insignificant Howeverif these harmonics are closed to natural modes of the struc-ture a substantial proportion of the energy can be transferredto them this phenomenon is called internal resonance Tautcables are good candidates to internal resonance since theirnatural modal distribution is harmonic (119899th frequency =119899 lowast 1st frequency) at least for the first ten modes Internalresonances have been observed in other types of (geo-metrically) nonlinear system with symmetric section when

the symmetry is broken [14 15] In particular the energy isthen shown to be distributed among several modes even ifonly one of them is excited

22 Expected Effects A cable equipped with a SSI is a con-tinuous (MDoF) structure combining two nonlinearitiesnamely a second-order effect in the cable (smooth nonline-arity) and a unilateral contact in the SSI (piecewise nonline-arity)The dynamic behaviour of nonlinear systems is knownto be complex even in the simplest case of the SDoF bilin-ear oscillator In fact depending on the time variation andamplitude of the loading different kinds ofmotions periodicquasiperiodic chaotic stable or unstable may coexist withbifurcations leading to them Through approximated meth-ods closed form solutions have occasionally been derived butonly for one or two DoF systems submitted to harmonic orimpulsive loading [16 17] In the present case even numericalmodels could easily fail to produce reliable solutions sincethere is a huge uncertainty on the loading

This explains why the design and assessment of the SSIhave initially been addressed in a purely experimental waytaking into account the practical aspects and limitations ofthis type of device Essentially the stiffness increment dueto the SSI should be acceptable for the cable anchorage Inparticular this excludes the case of an impact oscillator andleads to a weakly bilinear system (Δ119870 ≪ 119870

119888 with 119870

119888=

Cable stiffness and Δ119870 = (119870119904+ 119870119888)119870119888) also extensively

studied in the literatureCable resonances are mainly due to parametric excita-

tions through the anchorage motion or to direct excitationsby a combination of wind and rain In both cases the loadingusually assumedharmonic can last from some seconds to oneminute and is followed by a free decay period

The effect of the SSI on the cable response can be inferredfrom published experimental andor numerical analyses ofother nonlinear systems submitted to comparable inputsNonlinear responses are remarkably well processed in [18ndash20] mainly through wavelet analysis For the same inputwhile energy concentrates on one modefrequency in linear

Shock and Vibration 3

SMA wire and spring

Length = 45m

(a)

G5

T4

T3

T2G2

G1

A5

A4

A3

A1

5

4

3

12

V4

V3

V2

ELSAreaction wall A2

Taut cable number 1Length = 447m

The proportions of the figureare not in scale

(SMA length = 42m)

Load cell

Spring or SMA wire

(b)

Figure 2 (a) ELSA cable facility with zoomed views of the SSI attachments (b) Transducers type and positions on the cable no 1

systems the same energy is spread over different frequenciesin bilinear oscillators and most other nonlinear systemsThisdistribution may occur contemporaneously (different fre-quencies at the same time) or in cascade (different frequenciesat successive times) In both cases resonances are mitigatedsince the highest peak in the response spectrum is cappedThe energy transfer to higher modes has been evidenced instructures with nonlinear attachment [11] or presenting geo-metrical symmetry and light nonlinearity (eg plates shellsand cymbals) and when this symmetry is slightly broken[14 15] A very good agreement has been found between com-puted and measured energy flow even if the significance ofthis transfer is somehowhidden in the log graphsThis energytransfer is however essential froma vibrationmitigation pointof view since at equivalent energy vibration amplitudes arelower at higher frequency

Whether and to which extent the SSI triggers the above-mentioned effects in the cable can be assessed by comparingthe cable responses in different configurations with SSI with-out SSI (free cable) and possibly with the SSI spring attachedpermanently In the sequel these three cable configurationsare respectively referred to as ldquoSSI cablerdquo ldquofree cablerdquo andldquorestraint cablerdquo

3 The Test Campaign

31 Description of the Specimens In the ELSA facility fourreal cables of 45 meters and mass sim450 kg each are installed(Figure 2(a)) The performance of the SSI has been assessedon two of them

(i) Cable no 1 grouted with wax and under a tensionof 250KN was instrumented with in-plane (verticalplane containing the cable) displacement transduc-ers and accelerometers located at midspan at theattachment of the SSI and in three other locations(Figure 2(b)) Since the SSI was expected to modifythe modal content of the cable the output loca-tions were chosen on or nearby the antinodes ofthe first three modes An out-of-plane accelerometerwas located on the cable at the same point of theSSI attachment (72m) to measure the variation ofthe ratio in-planeout-of-plane acceleration (and dis-placement deduced)

(ii) Cable no 2 grouted with cement and under a tensionof 500KN was instrumented at 11m from the bottomanchoragewith one in-plane displacement transducerand four accelerometers two in-plane and two out-of-plane In fact the tests performed on cable no 1showed that the contribution of the first three modescould be adequately measured at this particular posi-tion and that the out-of-plane cable motions weresubstantial An in-plane displacement transducer wasmaintained at the attachment of the SSI

In both cases the SSI was attached to the cable at 72mfrom the bottom anchorage and at the other end to a fixedfoundation or to a movable steel mass of 1300 kg so thatthe attachment position could be changed easily along thecable The tension in the SSI was recorded by a load cell soas to detect the switching times The input force was appliedat 10m from the bottom anchorage and measured with adynamometer It is worth underlining that such an inputlocation allows the effective excitation of any mode until the3rd one at least

As mentioned earlier the SSI is a unilateral spring Inpractice it is made of a linear spring and a unilateral contactsystem connected in series The unilateral contact system isshown in Figure 1(b)The load cell and the screw are requiredto regulate the gap (switching position) which can be set to apositive or negative value For positive gaps (clearance) thespring is unloaded at equilibrium whereas for negative gaps(interference) the spring is in tension at equilibrium

Two different SSI have been tested On cable no 1 the SSIspring was a nitinol (nickel-titanium alloy) wire of diameter25mm and length 42m (Figure 2(a) left) Initially the wirewas intended to work as a shape memory alloy (SMA) andhad therefore been characterized and stabilized (Figures 3(a)and 3(b)) However during the cable tests the wire turnedout to work simply as a super elastic spring (Figure 3(c))Nevertheless it is not excluded that the hysteretic behaviourof the SMA could be activated in the SSI in case of exceptionalexcitation (eg tornado or heavy storm) so that the SMAdamping property at high strain (2 to 6) could then alsocontribute to mitigate large amplitude vibrations In our casethe SMA wire used was not adapted at the ldquohighrdquo frequenciesof cables oscillations In order to fulfil the condition of

4 Shock and Vibration

42 m SMA wire

(a)

Range of usein the SSI

Load

(N)

Displacement (mm)

NITI 246mm characterisation curvesstrain rate =14e minus 4s

00

1

50

50

100 150 200 250 300

500

1000

1500

2000

2500

2

3

(b)

0 10 200

100

200

300

400

500

600

700

800

900

Displacement (mm)

Forc

e (N

)

minus20 minus10

s07 Fexi = 100 N

(c)

Figure 3 (a) SMA characterisation setup (b) SMA training curve (c) SMA behaviour during a cable dynamic test

significant hysteretic dissipation the wire type and setupmust be optimized as described in [21] but it was not the aimof this work

On cable no 2 a classical steel spring was used in the SSI(Figure 2(a) right) Initially the spring was connected to theunilateral system through a steel bar which had however asubstantial mass likely to perturb the behaviour of the SSI Ithas been subsequently replaced by a much lighter steel cableof diameter 4mm

Whether based on a nitinol wire or a steel spring the SSIhas a negligible mass thus unlike TMD and NES it doesnot represent an additional DoF Each SSI is defined by twoparameters spring stiffness and unilateral gap Once installedon a cable the attachment position constitutes a third param-eter

The stiffness increase induced in the cable by the SSImay be characterised by the transversal forcedisplacementrelationship at the attachment point It can be computed

andormeasured on the equipped cable In this latter case theattachment point can be lifted up with a crane (Figure 4(a))or pulled down with the gap regulation screw The forcedisplacement curve is directly given by the load cell anddisplacement transducer installed on the SSI In Figure 4(b)the difference between the lift-up and pull-down slopesreveals a substantial relative stiffness increase for cable no 1(Δ119870119870

119888asymp 60) which seems inconsistent with the weak

bilinearity hypothesis but this is a merely static (and local)value In fact static and dynamic stiffness generally differfor systems with more than one DoF In dynamics a moreappropriate measure of the (global) stiffness increase is givenby the increase of the squared fundamental frequency whichcan be computed andor measured between the free andthe restraint configurations For cable no 1 the fundamentalfrequency is found to increase by 11 between the free andrestraint configurations which corresponds to a dynamicstiffness increase of 23 For cable no 2 the increase is


(a)

Load

(N)

Lift up with a crane

Pull down with the screw

Displacement (mm)

Stiffness test for a switch position in zero

Free cableCable with spring

0

0

200

400

600

800

minus200

minus30 minus20 minus10 10

minus1000

minus800

minus400

minus600

(b)

Figure 4 (a) Measurement of the static stiffness increase in cable no 1 (b) Measured forcedisplacement curve

approximately the same (+10 in frequency and thus +21 indynamic stiffness) The weak nonlinearity hypothesis is thusreasonably verified

32 Description of the Approach The dynamic testing of theSSI cable presents specific difficulties which are absent in theother two configurations (free and restraint)

The first difficulty is to apply the most critical excitationthat is to say the input inducing the largest response of thenonlinear structure and thus revealing the efficiency limit ofthe SSI Since such a critical input is asymmetric (frequencyalternatively tuned to each dynamic state) it is difficult to usean electrodynamic actuator also because the cable responsehas not the same frequency content as the input Two typesof excitation have therefore been chosen a snap-back testwhich corresponds to an ldquoautotunedrdquo input and a manualshaking inducing resonance a technique commonly usedeven for very long cables (up to 350m) as mentioned in [22]The loading and all cable displacements and accelerations arerecorded by a dynamic data recorder (TEAC) so that criticalexcitations and corresponding anharmonic cable responsescan be identified During manual shaking the input force isalso processed online by a dynamic signal spectrum analyser(HP) different parameters (load frequency maximum loadper cycle and load integration on cycles) can be checked toensure immediately that the manual shaking is unbiased

The second difficulty is to process and compare nonlin-ear outputs (SSI cable) and linear ones (free and restraintcables) The signals are processed mainly by Fourier trans-form with an automatic modal extraction toolbox describedin [23] implemented under MATLAB Even if some peaks in

the Fourier transform of a nonlinear response do not nec-essarily represent actual modes they nevertheless quantifythe resonances of a fictitious linear system having the sameresponse Comparing the Fourier coefficients (frequenciesdamping ratios) and the amplitude of the cable response indifferent configurations allows a better understanding of howand how much the cable vibrations are mitigated by the SSIThe SSI cable outputs are also processed in the time domainFrom the recorded switching times any SSI cable signal canbe split up into two intermittent subsignals corresponding toeach dynamic state The frequency and damping evolutionof each subsignal are then computed by a particular imple-mentation of the logarithmic decrement method allowingprocessing asymmetric signals as described in [24]

The third difficulty is to deal with tricky phenomenainherent to nonlinear dynamics such as instabilities andbifurcations To avoid experimental errors spurious effectsandmisinterpretations the tests have been repeated formanydifferent configurations (cable tension and grouting SSIdevice inputoutput locations loading intensity etc) and asmentioned before the nonlinear outputs have also been pro-cessedwith differentmethods both in the time and frequencydomains Since all results were checked to be consistentonly a few selected tests are presented to support the drawnconclusions

4 Free Vibration Tests

41 Effect of the SSI The same snap-back test (sudden releaseof 700N at 10m from the lower anchorage) has been repeatedin the three configurations of cable no 1 The displacement


10 15 20 25 30

0

2

4

6

8

Time (s)

Gef

Disp

(m

m)

9 95 10 105 11 115 12

SSI

minus2

minus4

FreeRestrained

s09 release of 700N F in xL = 022 (x = 10m)

(a)

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

1800

Frequency (Hz)

SSI

FreeRestrained

(b)

Figure 5 (a) Displacement and (b) associated spectrum at 378m on the free constraint and SSI cable no 1

signals recorded at 378mof the lower anchorage (sim30m fromthe SSI attachment) and the corresponding displacementspectra are displayed in Figure 5

The initial displacement is the same for the free (blue)and SSI (green) cables because the spring is then inactivewhile it is lower for the constraint (red) cable because thespring is always active In the free and restraint cables thevibrations fade out very slowly but at a similar rate whereasin the SSI cable they fade out much faster especially at thebeginning In fact after only 7 to 8 seconds the vibrationsof the SSI cable are already damped (more than 25 s for freeand restraint cables) and remain the lowest in amplitudeThe response spectra for the free and restraint cables aremore or less proportional In both cases the first modeis largely dominant but the higher modes are also visibleHowever because of its higher dynamic stiffness (+23) andfundamental frequency (+11) the restraint cable exhibits areduced spectrum with a shift to the right Conversely theresponse spectrum for the SSI cable is markedly different inamplitude the first and secondmodes are substantially lowerin amplitude and are approximately at the same level Thisconfirms the capping of the fundamental frequency throughenergy transfer towards the higher modes mainly from thefirst to the second In the time domain this effect looks like astrong damping enhancement but a closer look at the signals(zooming view in Figure 5(a)) confirms that the vibrationreduction is actually due to the strong activation of highermodes mainly the second one

Tests repeated on cable no 2 (Figure 6) are similar butshow a slightly weaker effect of the SSI which might be due tothe lower relative stiffness of the SSI or to less suitable value ofthe switch position Nevertheless the principle is confirmedSMA hysteretic behaviour is not required at all in oppositionto what is reported in [25]

42 Importance of the Output Location The test of Figure 5is now represented at midspan (225m) in Figure 7 The SSIeffect (apparent damping enhancement) is also visible andmore generally is visible along the entire cable as confirmedby the other three output locations installed on cable no 1However the amplitude and frequency content of the cablemotion are known to vary along the cable in relation to thenodes and antinodes of the activated modes At 378m fromthe lower anchorage the four first modes are well detectablewhereas at midspan only odd modes can be detected Thisexplains the difference observed between the two sets ofsignals of Figures 5 and 7

43 Equivalent Stiffening and Damping of the SSI Since theSSI appears to both reduce the vibration amplitude andincrease the damping its effect can be quantified by comput-ing and comparing the equivalent dynamic stiffness anddamping ratio of the cable in the different configurationsThe equivalent dynamic stiffness is computed as a mean ofthe modal stiffness weighted by the modal participationLikewise the equivalent damping ratio is computed as ameanof the modal damping ratios weighted by the modal partic-ipation

In practice the modal parameters are extracted fromacceleration signals because of their quality at high frequencyFree vibration tests are particularly suitable for this taskbecause the results are not perturbed by input irregularities

In Figure 8 the results are shown for the snap-back testcarried out on cable no 2 in the restrained and SSI con-figurations The equivalent stiffness and damping have beenderived from the in-plane acceleration signal recorded at111m from the lower anchorage that is between nodes ofmodes 4 and 5Themodal parameters have been extracted for18 peaks in a frequency range of 0ndash45Hz with a software of


Time (s)5 10 15 20 25

024

Disp

lace

men

t (m

m)

SSI

minus2

minus4

minus6

minus8

minus10

minus12

s14 snap back (1130N) in x = 10m

FreeRestrained

(a)

Frequency (Hz)2 4 6 8 10

0

1000

2000

3000

4000

5000

6000

7000

SSI


FreeRestrained

(b)


10 15 20 25

0

1

2

3

Tro

Disp

(cm

)

Time (s)

minus2

minus1

minus3

minus4

s09 release of 700N F in x = 10m

Cable freeCable restrainedSSI cable

(a)

2 4 6 8 100

200

400

600

800

1000


Frequency (Hz)

(b)

Figure 7 (a) Displacement and (b) associated spectrum at 225 (midspan) on the free constraint and SSI cable no 1

automatic modal extraction implemented for the fast impacthammer testing method (FIHT) described in [23] underMATLAB The positive effect of the SSI is substantial evenif it refers to the less favorable case since the differencebetween the restrained and SSI responses of cable no 2 isthe least one and the switch position is not optimized for thiscase

5 Forced Vibration Tests

51 Effect of the SSI Cable no 1 has been submitted to forcedvibration tests in the free restrained and SSI configurationsA rope was attached to the cable at 10m from the anchorageand was manually pulled down initially at the estimatedfrequency of the fundamental mode or of a higher mode


10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)

Mean values for 18 freq peaks

Equivalent stiffness

Equivalent damping

Restrained cable

163 037

SSI cable 269 098

Ratio = SSIrestrained 165 264

Frequency (Hz)

s14 snap back in x = 10m

Restrained cableSSI cable

Figure 8 Equivalent stiffness and damping for the SSI cable no 2 as compared to the restrained configuration

Then the shaking was adapted so as to fit the responsefrequency that is the pulling force was applied only duringthe downward motion of the rope The pulling force mea-sured with a dynamometer could be 100N or 200N and wasapplied for either a short (5 s) or a long period (40 s) In thelatter case the steady state response could be reached

The effect of the SSI is again assessed by comparingthe cable responses under forced vibrations in the freerestrained and SSI configurations Of particular interest arethe amplitude and frequency of the steady state response InFigure 9 the responses of the free and SSI cable no 1 are com-pared for a 40 s loading period at 100N on the first modeTheamplitude of the steady state response is drastically reducedby the SSI but only in displacement whereas the response inacceleration is increased by the SSI especially for negativevalues that is when the SSI is activated Again this is due to atransfer of energy from the fundamental mode to the highermodes as evidenced on the acceleration spectrum in Figure 9the first mode is tremendously reduced while the 2nd and 3rdmodes are enhanced Similar results obtained with an inputon the 2ndmode prove the robustness of the device energy isalways transferred from the excitedmode to the highermodeswith however different sharing among the modes

52 Equivalent Damping Estimate After the shaking periodthe response of the cable was still recorded This free decaytest differs from the snap-back test by the initial conditionsthe snap-back test starts from a static configuration under agiven load while the free decay test starts from the dynamic(modal) configuration However it can be processed as inSection 43 to derive another equivalent stiffness and damp-ing

The SSI operates as long as the displacement amplitudeis higher than the absolute value of the gap For smaller dis-placement amplitudes the SSI cable oscillates as the free cable

(resp restrained cable) if the gap is positive (resp negative)In Figure 10 the first graph shows the output of a short periodof shaking on cable no 1 in the free restrained and SSIconfigurations The input force is 100N the mode excited isthe first one and the output position is at midspan Since theSSI gap was set to zero the device works during all the decayperiod but is not very efficientThe second graph refers to thefree decay following a long period of shaking and an outputposition at 72m The SSI gap was set to 16mm For clarityonly envelopes of displacements are reported in the amplituderange 10ndash35mm The SSI is particularly efficient in dampingthe vibrations as long as the amplitude is higher than 15mmBelow this amplitude value the damping is less important andwhen the amplitude falls below the gap position the dampingis the same as that for the restrained cable for this switchposition chosen

6 Tentative Explanation of the SSI Effect

The SSI concept relies on the mismatch (detuning) betweenthe excitation and the response In fact a sinusoidal inputforce at a given frequency induces an unharmonic responseof a cable equipped with a SSI device owing to the suddenstiffness change at the switching times Hereafter this detun-ing effect is tentatively explained and quantified on the basisof a SDoF bilinear model of the equipped cable

A cable equipped with a SSI device is a MDoF bilinearsystem According to [16 26] the most critical sinusoidalinputs for such systems are obtained for the so-called bilinearfrequencies The 119894th bilinear frequency 119891119887

119894is approximately

the harmonic average of the 119894th free and restrained frequen-cies 119891minus

119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60

Long input period Fexi = 100N

Acc V1 [A]Free cable

Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)

s07 Fexi = 100Ntimes104


Acc V1 [A]SSI cable

(c)

Figure 9 Displacement (a) acceleration (b) and acceleration spectrum (c) for the free and SSI cable no 1 (119865 = 100N on 1st mode)

This formulation is available for a zero clearance the exactformulation for a nonzero clearance is more complex and canbe found in [26] However depending on the switch position119891119887

119894may vary in the [119891minus

119894 119891+

119894] range Similarly the 119894th mode

shape is a combination of the 119894th free and restrained modeshapes and varies with the switch position Basically bilinearfrequenciesmodes are to bilinear systems what eigenfre-quencies are to linear systems However bilinear frequenciesmodes depend not only on the system characteristics (mas-ses stiffness and switching position) but also on the input

amplitude Moreover for increasing amplitude completelydifferentmode shapesmay appear by bifurcation for the samevalue of the bilinear frequency These mode shapes combinefree and restrained mode shapes of different order (eg 119891minus

119894

and 119891+119894+1

) This phenomenon called internal resonance istypical of nonlinear systems

During any type of test with varying amplitude (eg freedecay after a snap-back or forced vibrations) the response ofa MDoF bilinear system is therefore extremely complex sinceit is a varying combination of varying frequenciesmodes


60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100

S10 10 shakes + decay s07 comparison of decays after 40 s excitation (Fexi = 100N)

Acc position

(a) (b)

Am

ppk

-pk

mm

55

Figure 10 SSI damping effect during a free decay after a short (a) or long (b) loading period on mode number 1

372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect

s07 Fsma = 100 Fexi = 100N s07 Fsma = 100 Fexi = 100N

minus5

minus10

minus5

minus10

minus15minus15

Figure 11 Nearly steady state response of SSI cable no 1 at 72m during a long shaking period

In particular the influence of the switching position evolvesconstantly which makes any optimisation attempt difficultThe task is much easier in the case of a test (or test period)at nearly constant amplitude as the one shown in Figure 11where despite a slight beating the cable displacement at theSSI attachment can be considered periodic of period close to119891119887

1(sim2Hz in our case) During a period the displacement

appears distinctly composed of two parts separated by theswitching position the bottom part (SSI spring inactivated)recalls a sinusoidal curve of frequency 119891minus

1(=183Hz in our

case) whereas the upper displacement (SSI spring activated)is rather close to a sinusoidal curve of frequency119891+

2(=407Hz

in our case)Therefore the steady state response is actually aninternal resonance of the mode no 1 (free) and of the modeno 2 (restrained) Its shape is not the composition of two halfsines but the superposition of several harmonics giving a nonsinusoidal and asymetric wave

To quantify the detuning effect due to this internal res-onance a bilinear SDoF system is considered which is sup-posed to reproduce the observed steady state response of

the SSI cable under forced vibrations To this end the SDoFsystem includes the main ldquoingredientsrdquo of the observedresponse that is to say the free mode 1 and restrained mode 2The characteristics (mass main stiffness spring stiffness andswitch position) are thus chosen so that

(i) the SDoF free frequency coincides with the first freecable frequency 119891minus

1

(ii) the SDoF restrained frequency coincides with thesecond restrained cable frequency 119891+

2

(iii) a small damping ratio (measured on the first freemode) is added through a linear dashpot

(iv) the switching position measured from the equilib-rium position can be varied

The steady state response of this SDoF under a sinusoidalinput force of 100N has been computed for switching posi-tions ranging from minus4 cm to +4 cm and for input frequenciesranging from 15Hz to 45Hz thus including the free andrestrained mode frequencies The numerical detuning effect


FrequencySwitch

position

Mag

nitu

dera

tio

Optimal position of the SSI device for the smaller magnitude ratio

0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2

Magnitude ratio (model 1 DoF) between a system equipped with a SSI device with respectsystem with bi-lateral spring

Figure 12 Detuning effect computed on a SDoF model in function of the switch position and input frequency

is then computed by dividing the amplitude of this responseby the amplitude of the restrained SDoF response under thesame input A 3D plot of the results obtained is shown inFigure 12

The computed detuning effect is substantial for thechosen free and restrained frequencies Other computationsperformed for closer free and restrained frequencies (eg firstfree and first restrained) lead to much lower detuning effectsIt can be noticed that very small increment in frequency andchiefly in time (gt20000 iterationspoints of the curve) mustbe used to obtain accurate results

The largest detuning effect is obtained for two particularswitching positions of opposite sign the negative one isslightly more effective than the positive one and for the mostcritical frequency gives a reduction of 18with respect to therestrained system and a much higher reduction with respectto the free systemHowever both computed values underesti-mate the experimentalmeasurements on the cable A possibleexplanation could be that the SDoF system does combine theappropriate modes in a nonlinear way but inappropriatelythe mode shape associated with the bilinear frequency doesnot result from an internal resonance phenomenon Thisis why the resonant frequency varies continuously with theswitching position whereas in the experiment it remainsalmost constant for a wide range of switching positions

7 Optimisation of the Switch Position

Independent of the input frequency the proposed SSI devicehas always been found to mitigate the cable vibrations ina more or less effective way though In particular the SSIefficiency appears to depend on the form and amplitude ofthe input signal The problem thus remains to design theSSI device that is to say to determine its most appropriatecharacteristics to mitigate potentially dangerous vibrationsinduced in a given cable by a set of possible excitations Inthe following an empirical design formula is proposed whichgives the best switching position once all other parameters

(cable characteristics SSI stiffness and maximum vibrationamplitude) are fixed

From the experimental results obtained so far the follow-ing conclusions can be drawn

(i) A stiffening of the cable is noticeable mainly dur-ing the excitation phase and occurs through energytransfer from the excited mode to higher modes Thecable motions are reduced in displacement but not inacceleration

(ii) An increase of the cable damping ratio is noticeableduring the free decay phase and occurs through thesimilar energy transfer phenomenon

(iii) In weakly bilinear MDoF systems the 119894th bilinearfrequency resulting from the combination of the 119894thfree and restrainedmodes is accompanied by sub- andsuperharmonics which may activate higher bilinearfrequencies and trigger internal resonances betweenfree and restrained modes of different orders

(iv) Under sinusoidal input the detuning effect is substan-tial if internal resonances are activated

The aforementioned effects increase with the increasingnonlinearity and also with the increasing velocity at theswitch time For a given spring stiffness 119870

119904and a given SSI

attachment position 119889 (far from the nodes of the first cablemodes) the increase Δ119870

119894of the dynamic stiffness 119870

119894for the

119894th mode can be derived from a Galerkin approximation [27]as follows

Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)

where 119871 is the cable length and 119909119889is the distance of the

device attachment from the lower anchorage However thisstiffness increase should remain small to avoid excessiveenergy transfer to the cable anchoragesThe design of the SSIdevice thus reduces to the optimisation of the switch positionSP for the maximum amplitude 119860

119894allowable on the 119894th cable


50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)

State 2 stiff cableState 1 free cable

Free cable env maxFree cable env min

70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90

Free cableLower and upper cycles (black)

Cable with SSIMotion in stiff state (upper cycles) Motion in soft state (lower cycles)

minus10

0minus10

Input = 200N

Max of reduction factor = 33 Pos = 166mmt = 33

SSI optimized disp equality in both states soft and stiff

Reduction factor = 23



s07 Fexi = 200N switch pos = 791mmReduction factor = 23


Pos = 368mmt = 30

Figure 13 Variation of the reduction factor with respect to the switch position (4 different tensions of the spring)

mode (provided by the bridge designer) and a given stiffnessincrease Δ119870

119894 For a given spring (Δ119870

119894) and a given amplitude

total 119860119894 the amplitude is distributed between the two states

(1199041 soft 1199042 stiff) of the cable and it reads as follows

119860119894= 1198601199041+ 1198601199042 (3)

Two effects contribute to obtain a smaller displacement

(1) themaximum transfer of energy to higher harmonicsto transfer the maximum of energy on the harmonicsof the mode governing the motion the velocity ofthe cable has to be the maximum at the switchevent (impulse dFdt maximum) The velocity at theswitching time is the highest when the switch positionis at equilibrium of the cable (SP = 0) It meansthat the displacement in the stiff state (1199042) is lightly(because Δ119870

119894is small) smaller than in the soft state

(1199041)(2) the critical sinusoidal input produces an increment of

displacement by resonance effect in each of the statesof the cables To obtain the smaller increment of thedisplacement in both states the amplitudes 119860

1199041and

1198601199042of the signal in each state must be equal If the

stiffness is lightly different in each state the switch

position must be an interference [26] which meansthat the spring will have a small tension (SP = minus119909) atthe equilibrium position To calculate it we start fromthe switch position in zero if we applied the sameforce on the cable in each direction (or each state) wehave

1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)

It appears clearly that the displacement amplitude 1198601199041is

greater than 1198601199042of the quantity 2SP = (Δ119870

119894119870119894) sdot 1198601199042 To

obtain the same amplitude in each state it is sufficient tostretch the spring of a value equal to the switch position

SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)

Then using (3) and (4) to substitute 1198601199042 we obtain the

switch position which gives the equal displacement in eachstate

SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)

It is clear from (6) that the switch position SP is amplitudedependent for this second effect Now to reach the best


efficiency of the device we should respect the optima of thetwo effects Equation (6) shows that SP is a small proportionof the amplitude maximum 119860

119894

In the example of Figure 11 the amplitude 119860119894is about

30mm for an increment of stiffness Δ119870119894equal to 23 The

switch position is therefore equal to 10 of the amplitudewhich is about 3mm It is a small value and in that caseit could be adjusted with the value of SP = 0 of the first effect(energy transfer) to benefit of both optima at the half sum (3and 0) =15mm

This value gives the best SSI effect only on the given ampli-tude of a given mode For other modes andor other ampli-tudes of the same mode the SSI effect is lower but remainspositive

The validity of these two assumptions has been checkedby repeating the same forced vibration test with four differentvalues of the switching position In Figure 13 the resultsconfirm that the greater displacement reductions with inputon mode 1 are obtained for the switch position SP given by(6) and SP = 0

The optimum is for SP = 16mm The same tests wereconducted with a different level of excitation not included inthis paper showing again that when the two displacements ineach state are equal the reduction is optimal

Even these results are in agreement with the previousassumptions much more cases should be studied to betterunderstand the behaviour of the cable equipped with the SSIdevice

8 Conclusion

The state switched inducer (SSI) is a unilateral spring whichslightly stiffens in an intermittent way the structure it isconnected toThe equipped structure thus becomes a bilinearoscillator An experimental campaign conducted on two full-scale cables has shown that SSI devices reduce significantlythe steady state vibration amplitude under forced vibrationsand shorten drastically the free decay period To the authorrsquosbest knowledge it is the first time that cable resonances couldbe mitigated by a passive device without involving any dis-sipation process but relying exclusively on some establishedproperties of nonlinear dynamic systems Thanks to thespecific harmonic modal distribution of cables SSI devicestrigger a substantial transfer of energy from the excitedmodeto the higher modes through the so-called internal resonan-ces

The SSI efficiency depends on the expected vibrationamplitude An empirical formula for optimising the SSIswitching position for a given vibration amplitude has beenproposed and fairly verified experimentally However furtherwork based on numerical models with two or more DoFs isneeded to characterizemore accurately the SSI behaviour andto improve its efficiency

This first large-scale experimental campaign shows thatthe SSI outperforms classical passive devices and opens theway to a new kind of vibration mitigation systems Last butnot least SSI devices are cheap very simple to install and easyto maintain However before any commercial used furthertesting is recommended so as to exclude undesirable effects

such as unexpected response to irregular excitation (windrain) or large out-of-plane vibrations or whirling amplitudes

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research leading to these results has received fundingfrom the European Communityrsquos Seventh Framework Pro-gram (FP72007ndash2013) The author expresses his gratitude tothe Scientific Committee of the Institute for the Protectionand the Security of the Citizen (IPSC) for supporting thisexploratory research project and to the European ScienceFoundation (ESF) for supporting the Project Shape MemoryAlloys to Regulate Transient Responses in Civil Engineering(SMARTER) during which the first experiments of this paperwere carried out The author thanks several colleagues of theELSA Laboratory for the discussions and the different pointof views of this complex mechanical oscillator The author isgrateful to Pr V Torra (Universitat Politecnica de CatalunyaBarcelona Spain) for the collaboration during the ProjectSMARTER fromwhich the first idea of a SSI device was born

References

[1] F Weber J Hogsberg and S Krenk ldquoOptimal tuning of ampli-tude proportional Coulomb friction damper for maximumcable dampingrdquo Journal of Structural Engineering vol 136 no2 pp 123ndash134 2010

[2] F Casciati L Faravelli and C Fuggini ldquoCable vibration miti-gation by added SMA wiresrdquo Acta Mechanica vol 195 no 1ndash4pp 141ndash155 2008

[3] X Y Wang Y Q Ni J M Ko and Z Q Chen ldquoOptimal designof viscous dampers for multi-mode vibration control of bridgecablesrdquo Engineering Structures vol 27 no 5 pp 792ndash800 2005

[4] G Cazzulani F Resta and F Ripamonti ldquoActive modal tunedmass damper for smart structuresrdquo Engineering Letters vol 19no 4 p 297 2011

[5] J POu andH Li ldquoThe state-of-the-art andpractice of structuralcontrol of civil structures for hazardmitigation inmainlandrdquo inProceedings of the 14th World Conference on Earthquake Engi-neering Beijing China 2008

[6] H Yamaguchi and Md Alauddin ldquoControl of cable vibrationsusing secondary cable with special reference to nonlinearity andinteractionrdquo Engineering Structures vol 25 no 6 pp 801ndash8162003

[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003

[8] L Caracoglia and N P Jones ldquoPassive hybrid technique for thevibrationmitigation of systems of interconnected staysrdquo Journalof Sound and Vibration vol 307 no 3ndash5 pp 849ndash864 2007

[9] A Y Kozrsquomin Y V Mikhlin and C Pierre ldquoLocalization ofenergy in nonlinear systems with two degrees of freedomrdquo In-ternational Applied Mechanics vol 43 no 5 pp 568ndash576 2007

[10] G Kerschen D M McFarland J J Kowtko Y S Lee L ABergman and A F Vakakis ldquoExperimental demonstration of


transient resonance capture in a system of two coupled oscilla-tors with essential stiffness nonlinearityrdquo Journal of Sound andVibration vol 299 no 4-5 pp 822ndash838 2007

[11] T P Sapsis D D Quinn A F Vakakis and L A BergmanldquoEffective stiffening and damping enhancement of structureswith strongly nonlinear local attachmentsrdquo Journal of Vibrationand Acoustics vol 134 no 1 Article ID 011016 12 pages 2012

[12] B Vaurigaud A T Savadkoohi and C-H Lamarque ldquoTargetedenergy transfer with parallel nonlinear energy sinks Part Idesign theory and numerical resultsrdquo Nonlinear Dynamics vol66 no 4 pp 763ndash780 2011

[13] B Vaurigaud L I Manevitch and C-H Lamarque ldquoPassivecontrol of aeroelastic instability in a long span bridge modelprone to coupled flutter using targeted energy transferrdquo Journalof Sound and Vibration vol 330 no 11 pp 2580ndash2595 2011

[14] C Touze O Thomas and M Amabili ldquoTransition to chaoticvibrations for harmonically forced perfect and imperfect circu-lar platesrdquo International Journal of Non-Linear Mechanics vol46 no 1 pp 234ndash246 2011

[15] E C Carvalho P B Goncalves J G N Del Prado Zenon andG Rega ldquoThe influence of symmetry breaking on the non-planar vibrations of slender beamsrdquo in Proceedings of the 15thInternational Symposium on Dynamic Problems of Mechanics(DINAME 13) M A Savi Ed ABCM Rio de Janeiro BrazilFebruary 2013

[16] A VDyskin E Pasternak and E Pelinovsky ldquoPeriodicmotionsand resonances of impact oscillatorsrdquo Journal of Sound andVibration vol 331 no 12 pp 2856ndash2873 2012

[17] Z K Peng Z Q Lang S A Billings and Y Lu ldquoAnalysisof bilinear oscillators under harmonic loading using nonlinearoutput frequency response functionsrdquo International Journal ofMechanical Sciences vol 49 no 11 pp 1213ndash1225 2007

[18] S Tsakirtzis G Kerschen P N Panagopoulos and A FVakakis ldquoMulti-frequency nonlinear energy transfer from lin-ear oscillators to mdof essentially nonlinear attachmentsrdquo Jour-nal of Sound and Vibration vol 285 no 1-2 pp 483ndash490 2005

[19] T M Nguyen Non-linear dynamics of coupled mechanicalsystems model reduction and identification [PhD thesis] EcoleNationale des Ponts et Chaussees 2007 (French)

[20] F Nucera D M McFarland L A Bergman and A F VakakisldquoApplication of broadband nonlinear targeted energy transfersfor seismic mitigation of a shear frame computational resultsrdquoJournal of Sound and Vibration vol 329 no 15 pp 2973ndash29942010

[21] D Tirelli and SMascelloni ldquoCharacterisation and optimizationof shape memory alloys for seismic applicationsrdquo Journal dePhysique IV France vol 10 2000

[22] Federal Highway Administration (FHWA) ldquoChapter 3 Anal-ysis evaluation and testing wind-induced vibration of staycablesrdquo Tech Rep FHWA-HRT-05-083 United States Depart-ment of Transportation 2007

[23] D Tirelli ldquoModal analysis of small amp medium structures byfast impact hammer testing methodrdquo Tech Rep EUR 24964EN Joint Research Centre Publications Office of the EuropeanUnion Luxembourg 2010

[24] D Tirelli ldquoA fast automated impact hammer test method formodal parameter extraction (FIHT) implementation on a com-posite bridge beamrdquo in Proceedings of the International Sym-posium on Nondestructive Testing of Materials and Structures(NDTMS 11) Istanbul Turkey May 2011

[25] V Torra C Auguet A Isalgue G Carreras P Terriault and FC Lovey ldquoBuilt in dampers for stayed cables in bridges via SMA

The SMARTeR-ESF project a mesoscopic and macroscopicexperimental analysis with numerical simulationsrdquo EngineeringStructures vol 49 pp 43ndash57 2013

[26] E A Butcher ldquoClearance effects on bilinear normal modefrequenciesrdquo Journal of Sound and Vibration vol 224 no 2 pp305ndash328 1999

[27] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


2 Shock and VibrationD

ispla

cem

ent (

mm

)

Time (s)10 12 14 16 18

0

10

20

30

40

minus10

minus20

minus30

minus40

s07 cable 1 Fexi = 100N

(a)

SSI active

SSI inactive

Ap = Anchorage positionElastic wire

Load cell

Switch positiong =

Screw for gapregulation

(b)

Figure 1 (a) Cable resonance with low damping (blue curve fitting) (b) SSI principle

a minimum number of excitation cycles to develop possiblyup to a dangerous level

Now if travelling waves are not perfectly symmetricincident and reflected waves will not perfectly superposeThe amplitude increment will be smaller and the resonancephenomenon is reduced Asymmetric travelling waves can beobtained if the structure is in one dynamic state 119878 (stiffness119870andmass119872) for half a period and in another dynamic state 1198781015840(stiffness1198701015840 and possiblymass1198721015840) for the successive (nearly)half of the period In a cable this state switching can beeasily achieved by attaching a unilateral spring of stiffness119870

119904

between the cable and the ground or the deck (Figure 1(b))This explains the name given to the proposed device stateswitched inducer (SSI)

A linear structure equipped with a SSI becomes a bilinearoscillator a system extensively studied in the literature Abilinear oscillator admits more basins of energy (peaks) inthe high frequency range than the corresponding linear oscil-lator This property naturally yields a positive effect of theSSI according to the Parseval theorem conservative systemsreceiving the same input energy have the same integral in thefrequency domain thus if peaks are more numerous for thenonlinear oscillator their amplitude and thus the resonancesare smaller than those for the corresponding linear oscillator

Another interesting property of the SSI is the suddenchange of dynamic state at the switch time Any such non-linear behaviour (in electricity acoustics mechanics etc) isknown to generate harmonics The amplitude of these har-monics depends on the degree of nonlinearity introduced inthe system but it is generally very small with respect to theexcited frequency Thus the proportion of energy subtractedto the main resonant mode is usually insignificant Howeverif these harmonics are closed to natural modes of the struc-ture a substantial proportion of the energy can be transferredto them this phenomenon is called internal resonance Tautcables are good candidates to internal resonance since theirnatural modal distribution is harmonic (119899th frequency =119899 lowast 1st frequency) at least for the first ten modes Internalresonances have been observed in other types of (geo-metrically) nonlinear system with symmetric section when

the symmetry is broken [14 15] In particular the energy isthen shown to be distributed among several modes even ifonly one of them is excited

22 Expected Effects A cable equipped with a SSI is a con-tinuous (MDoF) structure combining two nonlinearitiesnamely a second-order effect in the cable (smooth nonline-arity) and a unilateral contact in the SSI (piecewise nonline-arity)The dynamic behaviour of nonlinear systems is knownto be complex even in the simplest case of the SDoF bilin-ear oscillator In fact depending on the time variation andamplitude of the loading different kinds ofmotions periodicquasiperiodic chaotic stable or unstable may coexist withbifurcations leading to them Through approximated meth-ods closed form solutions have occasionally been derived butonly for one or two DoF systems submitted to harmonic orimpulsive loading [16 17] In the present case even numericalmodels could easily fail to produce reliable solutions sincethere is a huge uncertainty on the loading

This explains why the design and assessment of the SSIhave initially been addressed in a purely experimental waytaking into account the practical aspects and limitations ofthis type of device Essentially the stiffness increment dueto the SSI should be acceptable for the cable anchorage Inparticular this excludes the case of an impact oscillator andleads to a weakly bilinear system (Δ119870 ≪ 119870

119888 with 119870

119888=

Cable stiffness and Δ119870 = (119870119904+ 119870119888)119870119888) also extensively

studied in the literatureCable resonances are mainly due to parametric excita-

tions through the anchorage motion or to direct excitationsby a combination of wind and rain In both cases the loadingusually assumedharmonic can last from some seconds to oneminute and is followed by a free decay period

The effect of the SSI on the cable response can be inferredfrom published experimental andor numerical analyses ofother nonlinear systems submitted to comparable inputsNonlinear responses are remarkably well processed in [18ndash20] mainly through wavelet analysis For the same inputwhile energy concentrates on one modefrequency in linear


SMA wire and spring

Length = 45m

(a)

G5

T4

T3

T2G2

G1

A5

A4

A3

A1

5

4

3

12

V4

V3

V2




(SMA length = 42m)

Load cell

Spring or SMA wire

(b)




3 The Test Campaign








42 m SMA wire

(a)


Load

(N)

Displacement (mm)


00

1

50

50

100 150 200 250 300

500

1000

1500

2000

2500

2

3

(b)

0 10 200

100

200

300

400

500

600

700

800

900

Displacement (mm)

Forc

e (N

)

minus20 minus10

s07 Fexi = 100 N

(c)










(a)

Load

(N)



Displacement (mm)



0

0

200

400

600

800

minus200


minus1000

minus800

minus400

minus600

(b)











10 15 20 25 30

0

2

4

6

8

Time (s)

Gef

Disp

(m

m)

9 95 10 105 11 115 12

SSI

minus2

minus4

FreeRestrained


(a)

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

1800

Frequency (Hz)

SSI

FreeRestrained

(b)










Time (s)5 10 15 20 25

024

Disp

lace

men

t (m

m)

SSI

minus2

minus4

minus6

minus8

minus10

minus12


FreeRestrained

(a)


0

1000

2000

3000

4000

5000

6000

7000

SSI


FreeRestrained

(b)


10 15 20 25

0

1

2

3

Tro

Disp

(cm

)

Time (s)

minus2

minus1

minus3

minus4



(a)

2 4 6 8 100

200

400

600

800

1000


Frequency (Hz)

(b)






10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)



Equivalent damping

Restrained cable

163 037

SSI cable 269 098


Frequency (Hz)














119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










SMA wire and spring

Length = 45m

(a)

G5

T4

T3

T2G2

G1

A5

A4

A3

A1

5

4

3

12

V4

V3

V2




(SMA length = 42m)

Load cell

Spring or SMA wire

(b)




3 The Test Campaign








42 m SMA wire

(a)


Load

(N)

Displacement (mm)


00

1

50

50

100 150 200 250 300

500

1000

1500

2000

2500

2

3

(b)

0 10 200

100

200

300

400

500

600

700

800

900

Displacement (mm)

Forc

e (N

)

minus20 minus10

s07 Fexi = 100 N

(c)










(a)

Load

(N)



Displacement (mm)



0

0

200

400

600

800

minus200


minus1000

minus800

minus400

minus600

(b)











10 15 20 25 30

0

2

4

6

8

Time (s)

Gef

Disp

(m

m)

9 95 10 105 11 115 12

SSI

minus2

minus4

FreeRestrained


(a)

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

1800

Frequency (Hz)

SSI

FreeRestrained

(b)










Time (s)5 10 15 20 25

024

Disp

lace

men

t (m

m)

SSI

minus2

minus4

minus6

minus8

minus10

minus12


FreeRestrained

(a)


0

1000

2000

3000

4000

5000

6000

7000

SSI


FreeRestrained

(b)


10 15 20 25

0

1

2

3

Tro

Disp

(cm

)

Time (s)

minus2

minus1

minus3

minus4



(a)

2 4 6 8 100

200

400

600

800

1000


Frequency (Hz)

(b)






10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)



Equivalent damping

Restrained cable

163 037

SSI cable 269 098


Frequency (Hz)














119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










42 m SMA wire

(a)


Load

(N)

Displacement (mm)


00

1

50

50

100 150 200 250 300

500

1000

1500

2000

2500

2

3

(b)

0 10 200

100

200

300

400

500

600

700

800

900

Displacement (mm)

Forc

e (N

)

minus20 minus10

s07 Fexi = 100 N

(c)










(a)

Load

(N)



Displacement (mm)



0

0

200

400

600

800

minus200


minus1000

minus800

minus400

minus600

(b)











10 15 20 25 30

0

2

4

6

8

Time (s)

Gef

Disp

(m

m)

9 95 10 105 11 115 12

SSI

minus2

minus4

FreeRestrained


(a)

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

1800

Frequency (Hz)

SSI

FreeRestrained

(b)










Time (s)5 10 15 20 25

024

Disp

lace

men

t (m

m)

SSI

minus2

minus4

minus6

minus8

minus10

minus12


FreeRestrained

(a)


0

1000

2000

3000

4000

5000

6000

7000

SSI


FreeRestrained

(b)


10 15 20 25

0

1

2

3

Tro

Disp

(cm

)

Time (s)

minus2

minus1

minus3

minus4



(a)

2 4 6 8 100

200

400

600

800

1000


Frequency (Hz)

(b)






10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)



Equivalent damping

Restrained cable

163 037

SSI cable 269 098


Frequency (Hz)














119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

Load

(N)



Displacement (mm)



0

0

200

400

600

800

minus200


minus1000

minus800

minus400

minus600

(b)











10 15 20 25 30

0

2

4

6

8

Time (s)

Gef

Disp

(m

m)

9 95 10 105 11 115 12

SSI

minus2

minus4

FreeRestrained


(a)

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

1800

Frequency (Hz)

SSI

FreeRestrained

(b)










Time (s)5 10 15 20 25

024

Disp

lace

men

t (m

m)

SSI

minus2

minus4

minus6

minus8

minus10

minus12


FreeRestrained

(a)


0

1000

2000

3000

4000

5000

6000

7000

SSI


FreeRestrained

(b)


10 15 20 25

0

1

2

3

Tro

Disp

(cm

)

Time (s)

minus2

minus1

minus3

minus4



(a)

2 4 6 8 100

200

400

600

800

1000


Frequency (Hz)

(b)






10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)



Equivalent damping

Restrained cable

163 037

SSI cable 269 098


Frequency (Hz)














119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 15 20 25 30

0

2

4

6

8

Time (s)

Gef

Disp

(m

m)

9 95 10 105 11 115 12

SSI

minus2

minus4

FreeRestrained


(a)

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

1800

Frequency (Hz)

SSI

FreeRestrained

(b)










Time (s)5 10 15 20 25

024

Disp

lace

men

t (m

m)

SSI

minus2

minus4

minus6

minus8

minus10

minus12


FreeRestrained

(a)


0

1000

2000

3000

4000

5000

6000

7000

SSI


FreeRestrained

(b)


10 15 20 25

0

1

2

3

Tro

Disp

(cm

)

Time (s)

minus2

minus1

minus3

minus4



(a)

2 4 6 8 100

200

400

600

800

1000


Frequency (Hz)

(b)






10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)



Equivalent damping

Restrained cable

163 037

SSI cable 269 098


Frequency (Hz)














119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Time (s)5 10 15 20 25

024

Disp

lace

men

t (m

m)

SSI

minus2

minus4

minus6

minus8

minus10

minus12


FreeRestrained

(a)


0

1000

2000

3000

4000

5000

6000

7000

SSI


FreeRestrained

(b)


10 15 20 25

0

1

2

3

Tro

Disp

(cm

)

Time (s)

minus2

minus1

minus3

minus4



(a)

2 4 6 8 100

200

400

600

800

1000


Frequency (Hz)

(b)






10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)



Equivalent damping

Restrained cable

163 037

SSI cable 269 098


Frequency (Hz)














119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 20 30 400

500

1000

1500

2000

2500Ac

cele

ratio

n (m

ss

)



Equivalent damping

Restrained cable

163 037

SSI cable 269 098


Frequency (Hz)














119894and 119891+

119894 that is

119891119887

119894=2119891minus

119894119891+

119894

119891minus

119894+ 119891+

119894

(1)


20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20 40 60 80 100

0

20

40

60

Time (s)

Disp

lace

men

t (m

m)

minus20

minus40

minus60



Acc V1 [A]SSI cable

(a)

0 20 40 60 80 100

0

5

10

15

Acce

lera

tion

(ms

s)

Time (s)

minus5

minus10

minus15

minus20

minus25

minus30

s07 Fexi = 100N


Acc V1 [A]SSI cable

(b)

2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Frequency (Hz)

Acce

lera

tion

(ms

s)



Acc V1 [A]SSI cable

(c)




119894 119891+




119894

and 119891+119894+1




60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










60 65 70 75 8010

15

20

25

30

35

Time (s)

Time (s)0 10 20 30

0

50

100

Disp

lace

men

t (m

m)


L2 ~L6

Free cable

minus50

minus100


Acc position

(a) (b)

Am

ppk

-pk

mm

55


372 374 376 378 38 382

0

5

10

15

Time (s)

Switch position

34 36 38 40 42 44

0

5

10

15

Time (s)

Disp

lace

men

t (m

m)

Beating effect


minus5

minus10

minus5

minus10

minus15minus15





1(=183Hz in our


2(=407Hz





1


2





FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










FrequencySwitch

position

Mag

nitu

dera

tio


0051

15

15

2

minus004

minus0018

minus003minus002

minus0010

001002

003004

0014

21073

28736

45

4

35

3

25

2


















119894for the


Δ119870119894= 119870119904sdot sin2 (119894120587119909119889

119871) (2)





50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 60 70 80 90

100

203040

708090

Time (s) Time (s)

Time (s) Time (s)

Disp

lace

men

t (m

m)

Disp

lace

men

t (m

m)



70 80 90

0 10 20 30 40 50 60 70 80 90

0102030405060708090

0 10 20 30 40 50 60 70 80 90



minus10

0minus10

Input = 200N








Pos = 368mmt = 30







119860119894= 1198601199041+ 1198601199042 (3)






1199041and




1198701198941198601199041= (119870119894+ Δ119870119894) sdot 1198601199042 (4)



119894119870119894) sdot 1198601199042 To


SP =Δ119870119894

2119870119894

sdot 1198601199042 (5)



SP =Δ119870119894

2119870119894+ Δ119870119894

sdot 119860119894 (6)




119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894








8 Conclusion







Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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Research Article Vibration Mitigation without Dissipative ...downloads.hindawi.com/journals/sv/2014/135767.pdf · Research Article Vibration Mitigation without Dissipative Devices: