research article impact of cross-tie properties on the

Research ArticleImpact of Cross-Tie Properties on the Modal Behavior ofCable Networks on Cable-Stayed Bridges

Javaid Ahmad Shaohong Cheng and Faouzi Ghrib

Department of Civil and Environmental Engineering University of Windsor 401 Sunset Avenue Windsor ON Canada N9B 3P4

Correspondence should be addressed to Shaohong Cheng shaohonguwindsorca

Received 22 January 2015 Accepted 18 March 2015

Academic Editor Vincenzo Gattulli

Copyright copy 2015 Javaid Ahmad et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Dynamic behaviour of cable networks is highly dependent on the installation location stiffness and damping of cross-ties Thusthese are the important design parameters for a cable network While the effects of the former two on the network responsehave been investigated to some extent in the past the impact of cross-tie damping has rarely been addressed To comprehendour knowledge of mechanics associated with cable networks in the current study an analytical model of a cable network will beproposed by taking into account both cross-tie stiffness and damping In addition the damping property of main cables in thenetwork will also be considered in the formulation This would allow exploring not only the effectiveness of a cross-tie design onenhancing the in-plane stiffness of a constituted cable network but also its energy dissipation capacity The proposed analyticalmodel will be applied to networks with different configurations The influence of cross-tie stiffness and damping on the modalresponse of various types of networks will be investigated by using the corresponding undamped rigid cross-tie network as areference base Results will provide valuable information on the selection of cross-tie properties to achieve more effective cablevibration control

1 Introduction

Stay cables are important load carrying structural elementson cable-stayed bridges Due to high tension frictionbetween the composing wires or strands of stay cables ischanged considerably which significantly reduces their struc-tural damping [1ndash3] This along with their long and flexiblefeature renders cables on cable-stayed bridges very vulnera-ble to various dynamic excitations Over the past few decadesthe length of stay cables has been increasing noticeably owingto the rapid growth of bridge span length Therefore usingexternal dampers to suppress cable vibrations turns to be lesseffective because of the constraint on installation locationBut rather the cross-tie solution which connects a vulnerablecable with its neighbouring ones using transverse cross-tiesto form a cable network becomes more popular in designingnew cable-stayed bridges and rehabilitating the existing ones[4 5]

Though the geometric form of a cable network appearsto be relatively simple its behaviour is highly sophisticated

and is yet to be fully understoodThe effectiveness of a cross-tie solution in improving stiffness and damping propertiesof a vulnerable cable greatly depends on the propertiesof cross-ties and the connected neighbouring cables Uptill now much of the research effort was dedicated toinvestigate the enhancement on network in-plane stiffnessresulting from the cross-tie solution [6ndash9] only very fewresearchers explored how it would help to redistribute energyamong different consisting cables [10] and affect the dampingproperty of the network [11 12] While the undamped rigidcross-tie assumption was made in most of the cable networkanalytical models [6 13] a few experimental and analyticalstudies were conducted to investigate the effect of cross-tiestiffness on the network behaviour Scaled cable networkmodels were tested by Yamaguchi and Nagahawatta [3] aswell as Sun et al [14] It was found in both tests that the stifftype of cross-tie was more effective in improving networkin-plane stiffness whereas the flexible one was more helpfulin dissipating energy and increasing network damping Astudy by Caracoglia and Jones [7] on a cable network on

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 989536 14 pageshttpdxdoiorg1011552015989536

2 The Scientific World Journal

the Fred Hartman Bridge showed that if flexible cross-tieswere selected instead of the rigid ones the network funda-mental frequency would decrease by 3 On the contraryBosch and Park [15] pointed out in a numerical study thatusing oversized (too stiff) cross-tie would excite more localmodes This was confirmed by Ahmad and Cheng [8] whoproposed an analytical model to investigate the impact ofcross-tie stiffness on the dynamic response of a cable networkIn the model the main cables were assumed as taut cablesand the flexible cross-tie was assumed to behave like areversible tensioncompression linear spring connector Itwas observed that when amore flexible cross-tie was used thelocalmodes dominated by either the left or the right segmentsof a corresponding cable network using stiffer cross-tiewould evolve into global modes Giaccu and Caracoglia [16]addressed the nonlinear interaction between main cablesand cross-ties with behaviour of the latter described by ageneralized power-law stiffness model They also reportedthat the equivalent nonlinear effects in network highermodescould be responsible for the transition from a local mode to aglobal one

It is clear from the above literature review that connectinga vulnerable cable with its neighbouring ones through trans-verse cross-ties would alter its in-plane stiffness and dampingpropertyThe level of such alteration highly depends on cross-tie properties When designing a cable network the selectionof cross-tie properties should be based on their combinedimpact on the network in-plane stiffness and damping Toproperly assess the latter the damping property of maincables as well as the stiffness and damping of cross-ties shouldbe included in the analysis The few available experimentalstudies (eg [3 14]) only discussed how the first modaldamping ratio of a vulnerable cable would be affected in acable network but such an impact on higher order modeswas not addressed Although a recent analytical study by theauthors [12] examined this issue the proposed model onlyconsidered damping in main cables whereas the cross-tiewas assumed to be undamped and rigid Thus the responsebetween a vulnerable cable and its neighbouring ones wasldquocommunicatedrdquo directly without the influence of cross-tieproperties However in reality when transmitting responsesbetween these two parties cross-tie would behave like aldquofilterrdquo which alters the transmitted response by its stiffnessand damping properties Therefore to properly assess theeffect of cross-tie solution on the dynamic response of a vul-nerable cable and the formed cable network it is imperativeto describe the cross-tie behaviour more accurately

In view of the above needs the current paper aimsat extending the analytical model proposed earlier by theauthors [12] to include both the stiffness and damping prop-erties of cross-tie in the formulation The network systemcharacteristic equation will be derived analytically and themodal characteristics of the cable networkwill be determinedby solving the associated complex eigenvalue problem Theproposed model would allow evaluating the impact of cross-tie solution on the in-plane frequency and also the dampingproperty of a vulnerable cable not only for the fundamentalmode but for the higher order modes as well The respectiverole of cross-tie stiffness and damping in affecting network

dynamic behaviour will be examined These would furtherenhance our understanding of cable network mechanics andprovide valuable information to improve the current practiceof cross-tie design

2 Formulation of Analytical Model

The proposed analytical model consists of two horizontallylaidmain cables connected by a transverse cross-tie as shownschematically in Figure 1The twomain cables are assumed tobe damped taut cables and are fixed at both ends The lengthof the two cables is 119871

1and 119871

2(1198711ge 1198712) respectively It is

assumed that the longer cable (main cable 1) is vulnerableto dynamic excitations In the following discussion it willbe referred to as the ldquotarget cablerdquo whereas the shorter cable(main cable 2) will be referred to as the ldquoneighbouring cablerdquo119874119871and119874

119877denote respectively the left and the right horizon-

tal offset of the neighbouring cable In general they are notequal to each other The unit mass tension and structuraldamping ratio of the two main cables are denoted by 119898

119895

119867119895 and 120585

119895(119895 = 1 2) respectively The transverse cross-tie

is assumed to be located at 1198971(1198971lt 1198972) from the left end

of the target cable Its axial stiffness property is representedby a linear spring connector with an associated stiffnessconstant of 119870

119888 where the subscript ldquo119888rdquo refers to ldquocross-tierdquo

The structural damping which comes from various types ofenergy dissipation mechanisms is usually represented by ahighly idealized form in structural analysis The equivalentlinear viscous damping model proposed by Rayleigh [17]assumes the damping force to be linearly proportional tothe motion velocity This model is commonly adopted dueto the simplicity of its form and the amenability in derivinganalytical solutionTherefore in the current model dampingproperty of the two main cables and that of the cross-tieare all assumed to be the linear viscous type and uniformlydistributed along the member length The correspondingdamping coefficients are denoted 119862

119895(119895 = 1 2) for the 119895th

main cable and 119862119888for the cross-tie The additional tension

in the main cables caused by vibration is neglected in theproposed model

When the cable network in Figure 1 is excited to vibratewithin its plane all four main cable segments oscillate inthe transverse direction whereas the cross-tie moves alongits longitudinal direction The motion of each main cablesegment can be described by the equation of motion of a tautcable subjected to in-plane damped free vibration that is

1198671205972V1205971199092

= 1198981205972V1205971199052+ 119862120597V120597119905

(1)

or

1198671205972V1205971199092

= 1198981205972V1205971199052+ 2119898120585120596

119900

120597V120597119905 (2)

where V 119867 and 119898 are respectively the transverse dis-placement the tension and the unit mass of the taut cable119862 = 2119898120585120596

119900is the cable damping coefficient per unit

length 120585 is the cable structural damping ratio and 120596119900is

The Scientific World Journal 3

Cable 1

C

A B

D Cable 2

x1 x2

y1

x3

y3

x4

y4

y2

L2

L1l2l1

l3 l4

N2

N1

CcKc

OLOR

Figure 1 Schematic layout of a two-cable network with dampedflexible cross-tie

the undamped circular frequency of the taut cable Separatingthe temporal and spatial variables contained in the cabletransverse displacement V(119909 119905) using the Bernoulli-Fouriermethod it can be expressed as V(119909 119905) = V(119909)119890119894120596119905 where V(119909)is the shape function and 120596 is the complex circular frequencyof cable vibration Substituting this expression into (2) itbecomes

V10158401015840 (119909) + 1205722V (119909) = 0 (3)

and its general solution would be

V (119909) = 119860 cos (120572119909) + 119861 sin (120572119909) (4)

where 119860 and 119861 are constants determined from the boundaryconditions and 120572 is a complex wave number of the form

120572 = radic1198981205962minus 119894 sdot 2119898120585120596120596

119900

119867 (5)

Since all four main cable segments in Figure 1 have oneend fixed that is V(0) = 0 constant 119860 in (4) would be zeroTherefore their transverse motion shape functions can bereduced to

V2119895minus1(1199092119895minus1) = 1198612119895minus1

sin (1205721198951199092119895minus1) 119895 = 1 2 (6a)

V2119895(1199092119895) = 1198612119895sin (120572

21198951199092119895) 119895 = 1 2 (6b)

where V2119895minus1

and V2119895represent respectively the shape function

of the left and the right segments of the 119895th main cable (119895 =1 2) and 119861

2119895minus1and 119861

2119895are the corresponding shape function

constantsThemass of the cross-tie is not considered in the proposed

model since it is usually very small compared to that of themain cables The behaviour of the damped flexible cross-tieis described by a linear tensioncompression reversal springconnector in parallel with a linear viscous damper Whenthe cross-tie oscillates along its axial direction the forcedeveloped in it can be expressed as

119865119888(119905) = 119870

119888119906 (119905) + 119862

119888

120597119906

120597119905 (7)

where 119906(119905) is the change in cross-tie length that is

119906 (119905) = V1(1198971 119905) minus V

3(1198973 119905) = [V

1(1198971) minus V3(1198973)] 119890119894120596119905 (8)

At point1198731where the cross-tie connects with the target cable

(Figure 1) the equilibrium requires the force exerted by thecross-tie on the target cable to be equal to the transverse forcein the left and the right segments of the target cable inducedby its tension that is

[(120597V1

1205971199091

100381610038161003816100381610038161003816100381610038161199091=1198971

+120597V2

1205971199092

100381610038161003816100381610038161003816100381610038161199092=1198972

)1198671] 119890119894120596119905= minus119865119888(119905) (9)

Plug (7) and (8) into (9) it gives

12057211198671[1198611cos (120572

11198971) + 1198612cos (120572

11198972)]

= [1198613sin (120572

21198973) minus 1198611sin (120572

11198971)] [119870119888+ 119894120596119862

119888]

(10)

Moreover longitudinal equilibrium of the isolated cross-tieshould satisfy

(120597V1

1205971199091

100381610038161003816100381610038161003816100381610038161199091=1198971

+120597V2

1205971199092

100381610038161003816100381610038161003816100381610038161199092=1198972

)1198671

+ (120597V3

1205971199093

100381610038161003816100381610038161003816100381610038161199093=1198973

+120597V4

1205971199094

100381610038161003816100381610038161003816100381610038161199094=1198974

)1198672= 0

(11)

Substitute (6a) and (6b) into (11) the following equation isobtained

12057211198671[1198611cos (120572

11198971) + 1198612cos (120572

11198972)]

+ 12057221198672[1198613cos (120572

21198973) + 1198614cos (120572

21198974)] = 0

(12)

The transverse displacement compatibility between the leftand the right main cable segments at nodes119873

1and119873

2gives

V2119895minus1(1198972119895minus1) = V2119895(1198972119895) 119895 = 1 2 (13)

which by considering (6a) and (6b) yields

1198611sin (120572

11198971) minus 1198612sin (120572

11198972) = 0 (14a)

1198613sin (120572

21198973) minus 1198614sin (120572

21198974) = 0 (14b)

Combining (10) (12) (14a) and (14b) and expressing in amatrix form we get

[119878] 119883 = 0 (15)


where

[119878] =

[[[[[

[

sin (01) minus sin (0

2) 0 0

0 0 sin (03) minus sin (0

4)

1205951198771cos (0

1) + sin (0

1) 1205951198771cos (0

2) minus sin (0

3) 0

1205741cos (0

1) 120574

1cos (0

2) 1205742cos (0

3) 1205742cos (0

4)

]]]]]

]

(16)

is the coefficient matrix 119883 = [1198611 1198612 1198613 1198614 ]119879 is a vectorcontaining all four unknown shape function constants and0 is the null vector In the coefficient matrix [119878] 0

2119895minus1=

1198771198951205762119895minus1

and 02119895= 1198771198951205762119895apply respectively to the left and the

right segment of the 119895th main cable (119895 = 1 2) 119877119895= 120572119895119871119895is a

complex parameter 120572119895is the complex wave number defined

in (5) 1205762119895minus1

= 1198972119895minus1119871119895and 1205762119895= 1198972119895119871119895are the segment ratio

parameters for the left and the right cable segments of the119895th main cable (119895 = 1 2) and 120574

119895is the mass-tension ratio

parameter of the 119895th cable which is defined by

120574119895=

119867119895119877119895119871119895

119867111987711198711

(17)

120595 is the nondimensional complex cross-tie parameter havingthe form of

120595 =1198671

1198711[119870119888+ 119894120596119862

119888] (18)

DefineΩ = 1205871198911198911as the nondimensional complex frequency

of the cable network and 120578119895= 1198911119891119895as the frequency

ratio of the 119895th (119895 = 1 2) main cable where 119891 and 119891119894are

respectively the complex frequency of the cable network andthe undamped fundamental frequency of the 119895th (119895 = 1 2)main cable the complex parameter 119877

119895can be rewritten as

119877119895= radic[(Ω120578

119895)2

minus 119894 sdot 2120587120585119895Ω120578119895] 119895 = 1 2 (19)

where 120585119895(119895 = 1 2) is the structural damping ratio of the 119895th

cableTo find the nontrivial solution to (15) the determinant of

the coefficient matrix [119878] should be set to zero This leads tothe characteristic equation of the two-cable network shown inFigure 1 which consists of two horizontally laid damped tautmain cables interconnected by a transverse damped flexiblecross-tie that is

1205741sin (119877

1) sin (0

3) sin (0

4)

+ 1205742sin (119877

2) sin (0

1) sin (0

2)

+ 120595119877112057411205742sin (119877

1) sin (119877

2) = 0

(20)

If we neglect the damping in the two main cables andthe cross-tie the three complex parameters 119877

119895 120574119895(119895 =

1 2) and 120595 in (17) to (19) would reduce to 119877119895= Ω120578

119895

120574119895= radic11986711989511989811989511986711198981 and 120595119900 = 11986711198711119870119888 Therefore (15)

would be the same as the system characteristic equationof an undamped two-cable network connected through anundamped flexible cross-tie derived earlier by the authors [8]and 120595

119900is the cross-tie stiffness parameter defined in [8]

3 Case Studies

In this section the proposed analytical model will be val-idated by finite element simulations and applied to studydynamic behaviour of cable networks having different lay-outs By using the rigid cross-tie case as a reference base theimpact of cross-tie stiffness and damping on network modalresponses will be evaluated

A finite element model of the cable network shownin Figure 1 will be developed using the commercial finiteelement analysis software Abaqus 610 The behaviour ofthe main cables will be simulated by the two-node linearB21 beam element whereas the flexible damped cross-tiewill be modeled by the CONN2D2 connector element TheRayleigh viscous damping model will be applied to simulatethe structural damping in the main cables and cross-tie

31 Case 1 Twin-Cable Network In a twin-cable networktwo main cables of the same geometric and physical prop-erties are arranged in parallel with each other and inter-connected by a transverse cross-tie This type of cablenetwork is not common in practice However due to itsunique characteristics seeking analytical solution of networkmodal response would be possible Thus studying dynamicbehaviour of this special type of cable network has the meritof deepening our understanding of phenomena associatedwith cable networks

Since the two main cables in a twin-cable network havethe same length unit mass tension and damping it gives01= 03 02= 04 1198771= 1198772 and 120574

1= 1205742= 1 By inserting

these conditions into (20) the system characteristic equationcan be reduced to

2 sin (1198771) sin (0

1) sin (0

2) + 120595119877

1sin2 (119877

1) = 0 (21)

or

sin (1198771) [2 sin (0

1) sin (0

2) + 120595119877

1sin (119877

1)] = 0 (22)

Three sets of roots can be determined from (22)The first setyielded from sin(119877

1) = 0 describes network global modes

and is independent of cross-tie properties Since the damping


ofmain cables and cross-tie is considered in the current studythe network complex frequency can be expressed as

Ω = Ωre + 119894 sdot Ωim (23)

where Ωre = Ω119900radic1 minus 1205852

eq and Ωim = Ω119900120585eq are respectivelythe real and the imaginary parts of Ω Ω

119900is the nondimen-

sional undampednetwork frequency and 120585eq is the equivalentdamping ratio of the system The real part of the complexfrequency describes network vibration frequency whereasthe imaginary part gives the system energy dissipationcapacity Substitute (23) into sin(119877

1) = 0 we obtain Ωre =

119899120587radic1 minus (120585119899)2 andΩim = 120587120585 where 119899 and 120585 are respectively

the mode number and damping ratio of an isolated singlemain cable The nondimensional modal frequency Ω

0and

modal damping ratio 120585eq of the corresponding network globalmode can thus be computed from

Ω0= radicΩ2re + Ω

2

im = 119899120587 119899 = 1 2 3 (24a)

120585eq =Ωim

radicΩ2re + Ω2

im

=120585

119899119899 = 1 2 3 (24b)

This set of network modal property is exactly the same asthose of an isolated single cable which suggests that in thiskind of global modes the two main cables would oscillate in-phase with the same shape and the modal properties are notaffected by the presence of cross-tie

The other two sets of roots can be obtained by setting thesummation of the two terms in the square bracket of (22) tozero that is

2 sin (01) sin (0

2) + 120595119877

1sin (119877

1) = 0 (25)

It is important to note that should a rigid cross-tie be used ina twin-cable network that is 120595 = 0 the second term in (25)would vanish Therefore the two remaining sets of roots of(22) can be directly obtained from sin(0

1) = 0 and sin(0

2) =

0 which represent respectively the network local modesdominated by its left segments (LS) or right segments (RS)However if the cross-tie has certain flexibility and dampingthe second term in (25) would reflect the effect of cross-tieproperties (stiffness damping and position) on the networklocal modes Not only would their modal frequencies anddamping be ldquomodifiedrdquo but also their mode shapes wouldevolve from that dominated by the oscillations of either thenetwork left or right segments to the out-of-phase globalmodesThis agrees with the earlier findings by the authors [8]when studying two identical taut main cables interconnectedby an undamped flexible cross-tie

When installing a cross-tie it can be placed either at thenodal point of a specific main cable mode or off the nodalpoint Noticing119877

1= 011205761= 021205762 (25) can also be expressed

as

2 sin (01) sin (0

2) + 120595119877

1sin [ 1

119898

01

1205761

+ (1 minus1

119898)02

1205762

] = 0

(26)

where 119898 is a positive integer or fraction which would satisfy1198981205761= 1 and thus (1 minus 1119898) = 120576

2when cross-tie happens to

be placed at the nodal point of a particular main cable modeTherefore (26) can be further simplified as

sin (01) 2 sin (0

2) + 120595119877

1[cos (0

2) + cos (0

1)sin (02)

sin (01)]

= 0

(27)

In (27) the condition of sin(01) = 0 describes the counterpart

of LSmodes in a rigid cross-tie network themodal frequencyand damping ratio of which are

Ω0=119899120587

120576119899 = 1 2 3 (28a)

120585eq =120576120585

119899119899 = 1 2 3 (28b)

This implies that if a damped flexible cross-tie is placed atthe nodal point of a particular main cable mode althoughthe change in cross-tie properties would lead to evolutionof mode shapes into out-of-phase globe modes the modalfrequency and damping of these modes will not be affected

The roots obtained by setting the term enclosed by thecurly bracket in (27) to zero reflect how modal properties oflocal RS modes in a rigid cross-tie case would be influencedby the flexibility and damping of a cross-tie They wouldnot only contribute to reduce modal frequency and increasemodal damping but also excite more sizable oscillationsof the left segments Therefore a network local RS modewould evolve into a global one should a rigid cross-tiebe replaced by a damped flexible one The same modeevolution phenomenon was reported earlier [8] for a twin-cable network with an undamped flexible cross-tie located atthe quarter span of the main cables which is the nodal pointof the second antisymmetric main cable mode

From the above discussion it is clear that in a twin-cablenetwork the in-phase global modes are independent of thecross-tie position stiffness and damping However modalproperties of local RS modes in the rigid cross-tie case wouldbe ldquomodifiedrdquo by the cross-tie stiffness and damping andevolve into globalmodesThe impact of cross-tie stiffness anddamping on the modal properties of local LS modes dependson the cross-tie installation location If the cross-tie is locatedat the main cable nodal point the frequency and damping ofthe LS modes would be independent of cross-tie stiffness anddamping However if the cross-tie is not placed at the nodalpoint the presence of cross-tie stiffness and damping wouldalter modal frequency and damping of the LS modes In bothcases such a change in cross-tie properties would render alocal LS mode evolving to an out-of-phase global mode

311 Numerical Example To validate the proposed cablenetwork analytical model and further discuss the modalcharacteristics associatedwith twin-cable networks a numer-ical example is presented Both main cables are assumed tohave a length of 72m a unit mass of 50 kgm a tension of


Table 1 Comparison of modal properties of a twin-cable network with damped flexible or rigid cross-tie at 120576 = 13

Mode

Damped flexible cross-tie Rigid cross-tie(119870119888= 3054 kNm 119862

119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)

119891 (Hz) 120585eq () Mode shape 119891 (Hz) 120585eq () Mode shapeAnalytical FEA Analytical FEA

1 146 146 050 050 GM in-phase 146 050 GM in-phase2 163 163 334 335 GM out-of-phase 218 033 LM RS3 291 291 025 025 GM in-phase 291 025 GM in-phase4 302 302 249 249 GM out-of-phase 437 017 LM RS5 437 437 017 017 GM in-phase 437 017 GM in-phase6 437 437 017 017 GM out-of-phase 437 017 LM LS7 583 582 013 013 GM in-phase 583 013 GM in-phase8 588 587 119 119 GM out-of-phase 656 011 LM RS9 728 727 010 010 GM in-phase 728 010 GM in-phase10 733 731 103 103 GM out-of-phase 874 008 LM RSGM global mode LM local mode LS left segment mode and RS right segment mode

2200 kN and a structural damping ratio of 05The stiffnesscoefficient of cross-tie is assumed to be 119870

119888= 3054 kNm

and its damping coefficient is 119862119888= 10 kNsdotsm Two cross-tie

installation locations of 120576 = 13 and 120576 = 25 are consideredin the example

(a) Cross-Tie Installed at 120576 = 13 In this case a flexibledamped cross-tie is placed at the one-third span of the twomain cables which happens to be the nodal point of the 3rdmode of an isolated single main cable The modal propertiesof the first ten network modes obtained from the proposedanalytical model and finite element simulation are listed inTable 1 and the mode shapes are depicted in Figure 2 Thetwo sets of results are found to agree well Given also in thesame table are the modal properties of the correspondingrigid cross-tie twin-cable network It can be seen from thetable that for all five in-phase global modes that is Modes1 3 5 7 and 9 their modal frequencies modal dampingratios and mode shapes not only are independent of thecross-tie flexibility and damping but also are not affectedby the presence of cross-tie The properties of these modesremain the same as those of an isolated single cable Sincethe cross-tie is placed at the main cable one-third span themodal frequency and damping of the first local LS modeMode 6 remain the same when the rigid cross-tie is replacedby a damped flexible one but the mode shape evolves to anout-of-phase global mode Moreover this particular cross-tie position also renders the modal frequency and modaldamping ratio of Mode 6 to be the same as those of Mode5 which is the 3rd in-phase network global mode In the caseof Modes 2 4 8 and 10 which are pure local RS modes inthe rigid cross-tie case the adoption of a flexible dampedcross-tie is found to not only considerably affect their modalfrequencies and damping ratios but also alter their modeshapes For example in the case of Mode 2 such a changein the cross-tie properties would excite the left segments ofthe network so that the mode shape becomes an out-of-phase global mode The modal frequency is reduced by 25from 218Hz to 163Hz whereas the modal damping ratio

increases substantially from 033 to 334 by roughly tentimes One possible reason for such a drastic increment inthe network modal damping ratio could be the relativelyhigh damping coefficient (119862

119888= 10 kNsdotsm) and relatively

low stiffness coefficient (119870119888= 3054 kNm) assumed in the

example Besides it is also important to note that since linearviscous type of damping model is used for the cross-tiethe energy dissipation due to damped cross-tie would onlyoccur when the two ends of the cross-tie oscillate at differentvelocities Therefore in the case of network in-phase globalmodes (Modes 1 3 5 7 and 9) of which the twin cables vibratewith the same shape the oscillating velocities at the cross-tietwo ends are the same so that a flexible damped cross-tie isnot capable of dissipating energy Thus all the network in-phase global modes have the same modal damping ratio asthat of an isolated single cable vibrating in the samemodeOnthe other hand when a network out-of-phase global mode isexcited velocities at the two ends of the cross-tie are equal butopposite in direction so the flexible damped cross-tie wouldmanifest the maximum possible energy dissipation capacitySimilar pattern that is decrease in modal frequency andsignificant increase in modal damping can also be found inhigher order out-of-phase global modes (Modes 4 8 and 10)

(b) Cross-Tie Installed at 120576 = 25 Modal analysis of the sametwin-cable network is conducted by relocating the cross-tie position to 120576 = 25 Table 2 summarizes the modalproperties of the first ten network modes and the modeshapes are portrayed in Figure 3 A good agreement betweenthe modal results determined by the proposed analyticalmodel and finite element simulations can be clearly observedfrom Table 2 For the convenience of comparison the modalproperties of the corresponding rigid cross-tie network arealso listed in Table 2 Results show that similar to theprevious case of 120576 = 13 the modal characteristics (modalfrequency modal damping ratio and mode shape) of thein-phase global modes that is Modes 1 3 5 7 and 9 arenot affected by the presence of cross-tie They remain thesame as the flexibility and damping of the cross-tie change



Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)


1 146 146 050 050 GM in-phase 146 050 GM in-phase2 168 168 411 412 GM out-of-phase 243 030 LM RS3 291 291 025 025 GM in-phase 291 025 GM in-phase4 296 296 130 130 GM out-of-phase 364 020 LM LS5 437 437 017 017 GM in-phase 437 017 GM in-phase6 440 437 082 082 GM out-of-phase 486 015 LM RS7 583 582 013 013 GM in-phase 583 013 GM in-phase8 589 589 149 149 GM out-of-phase 728 010 LM RS9 728 727 010 010 GM in-phase 728 010 GM in-phase10 728 727 010 010 GM out-of-phase 728 010 LM LS

Mode 1 (GM in-phase) Ω = 10120587 120585eq = 050





Mode 2 (GM out-of-phase) Ω = 112120587 120585eq = 334





Figure 2 First ten modes of a twin-cable network with a damped flexible cross-tie (119870119888= 3054 kNm 119862

119888= 10 kNsdotsm) at 120576 = 13

When using a rigid cross-tie network as a reference base byincreasing cross-tie flexibility and dampingmodal frequencyof local RS modes decreases whereas modal damping ratioincreases In addition their mode shapes evolve to out-of-phase global modes Take Mode 2 as an example whenreplacing the rigid cross-tie by a flexible damped one with119870119888= 3054 kNm and 119862

119888= 10 kNsdotsm its frequency

drops from 243Hz to 168Hz by 31 but the associatedmodal damping ratio increases approximately 125 times from

033 to 411 and the oscillation extends from the rightsegments to the entire network The same phenomenon canbe observed in Mode 6 and Mode 8 In the case of Mode 4and Mode 10 both of which are local LS modes in the rigidcross-tie network a cross-tie position of 120576 = 25 is off thenodal point of the 1st antisymmetricmode of an isolated cablein Mode 4 but it happens to be the nodal point of the 3rdsymmetric mode of an isolated cable in Mode 10 Thus forMode 4 the change in cross-tie properties would not only













119888= 10 kNsdotsm) at 120576 = 25

cause evolution of its mode shape into an out-of-phase globalmode but also alter its modal frequency and damping ratiowhereas forMode 10 its frequency and damping ratio remainthe same as those of the rigid cross-tie case although themode evolution phenomenon occurs

32 Case 2 Symmetric DMT Cable Network In majority ofcable networks on real cable-stayed bridges the consistingmain cables have different length unit mass and tensionwhich results in different mass-tension (DMT) ratio Besidessince the spacing between cables is typically closer on thepylon side than on the deck side the geometric layout ofa real cable network is generally asymmetric However it isunderstood from earlier studies [8 12] that if rigid or flexibleundamped cross-ties are used in a symmetric cable networkpure local modes dominated by oscillations of individualmain cables could form Therefore before analyzing a morerealistic asymmetric DMT cable network the impact of usingflexible damped cross-ties on the modal response of a DMTcable network having symmetric layout is studied first

When the cable network in Figure 1 has a symmetriclayout the left and the right offsets of main cable 2 are thesame that is 119874

119871= 119874119877 and the flexible damped cross-tie

locates at the mid-span of the two main cables Thereforethe segment parameters would satisfy 0

1= 02= 11987712

and 03= 04= 11987722 Substitute these relations into (20)

the system characteristic equation of a symmetric DMT two-cable network can be expressed as

sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)

By setting each of the three terms on the left side of (29) tozero that is the two sine terms and the one enclosed by thesquare bracket three sets of roots can be determined Thefirst two sets yielded respectively from sin(119877

12) = 0 and

sin(11987722) = 0 describe the local modes dominated by the

target or the neighbouring cable They are as follows

Local modes of the target cable

Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)


Local modes of the neighboring cable

Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)

First of all it is interesting to note that the form of (30a)(30b) and (31a) (31b) implies that these two types of networklocal modes have the samemodal properties as the respectiveisolated single cable antisymmetric modes Secondly thesame two types of network localmodes with exactly the samemodal frequencies andmodal damping ratios were identifiedearlier [12] when analyzing modal behaviour of a symmetricDMT two-cable network using rigid cross-tie Clearly sincethe cross-tie is placed at the nodal point of the main cablesthe network local modes dominated by an individual maincable would not be affected by the stiffness and damping ofthe cross-tie

The third set of roots describing the modal properties ofnetwork global modes can be found by setting the term inthe square bracket of (29) to zero While the first two termsinside the square bracket show the interaction between thetwo main cables and the coupling in their motions the thirdterm reflects the role of cross-tie properties in ldquomodifyingrdquothe frequency and damping of network global modes Shoulda rigid cross-tie be used this term would vanish suggestingthat the properties of network global modes would only beaffected by the main cable properties This set of solution canbe determined by separating the real and imaginary termsand using the same procedure explained in Section 31 or [12]

321 Numerical Example Consider a symmetric DMT cablenetwork with the following properties

Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)

Modal properties of the first ten modes obtained from theproposed analytical model and finite element simulationare given in Table 3 The corresponding mode shapes areillustrated in Figure 4 Again the two sets of results arefound to agree well with each other To assess the impactof cross-tie stiffness and damping on the modal behaviourof the studied symmetric DMT network modal response ofthe same network but using rigid cross-tie is also listed inthe same table Noticing that the fundamental frequency of

the target cable is 146Hz and the associated modal dampingratio is 05 results in Table 3 show that when the target cableis connected to the neighbouring one using a rigid cross-tieits fundamental frequency increases from 146Hz to 168Hzby 15 and the modal damping ratio from 05 to 061 by22 However if a flexible damped cross-tie with a stiffnesscoefficient of 119870

119888= 3054 kNm and damping coefficient of

119862119888= 10 kNsdotsm is used instead the fundamental frequency

of the target cable would be increased by 62 to 155Hzwhereas the modal damping ratio would be increased to171 by 34 timesThese suggest that using more rigid cross-tie would further enhance the in-plane stiffness of a cablenetwork and thus the target cable which agrees with theexperimental observations by Yamaguchi and Nagahawatta[3] and Sun et al [14] Although the network fundamentalmode is an in-phase global mode the velocities at the cross-tie two ends are different due to the difference in the dynamicproperties of the twomain cablesThus unlike the twin-cablenetwork case damping existing in the cross-tie would offernonzero damping force and help to dissipate more energyduring oscillation

In the case of the first out-of-phase global mode theflexibility in the cross-tie reduces its modal frequency so itis advanced from the third network mode (119891 = 3375Hz)in the rigid cross-tie case to the second network mode(119891 = 2153Hz) should a damped flexible cross-tie be usedBesides since the relative velocity between the two cross-tieends reaches its maxima in this oscillation mode the largedamping offered by the cross-tie leads to a drastic incrementof its modal damping ratio from 034 in the rigid cross-tiecase to 401 for a flexible damped cross-tie case

The modal properties of the network local modes dom-inated by either the target cable or its neighbouring oneare found to be independent of the cross-tie stiffness anddamping (Modes 3 4 6 9 and 10) In these cases one ofthe main cables vibrates in an antisymmetric shapeThus thecross-tie happens to locate at the nodal point of the modeshape and would not have a role in altering modal propertiesHowever it is interesting to note that although the frequencyof the 5th network mode 442Hz is very close to the thirdmodal frequency of the isolated target cable which is 438Hz(Ω119900= 30120587) the modal damping ratio jumps by roughly

7 times from 016 (120585eq = 12058513) for a single cable to 111when it is networked As can be seen from Figure 4 whenthe network oscillates in this mode the target cable vibratesin its 3rd mode and is dominant The cross-tie is locatedat the maximum deformation location of the target cablewhereas the neighbouring cable is almost at rest Thereforein terms of energy dissipation the damped cross-tie acts likea dashpot damper installed at themid-span of the target cableand ldquorigidlyrdquo supported by the neighbouring cable Similarphenomenon is also observed in the 7th and 8th modes ofthe cable network of which the motion is mainly dominatedby one cable with the other cable almost at rest

33 Case Study 3 Asymmetric DMT Cable Network Thesame two main cables in the symmetric DMT cable networkof Section 32 are rearranged in this section such that the leftand the right offsets of the neighbouring cable with respect


Table 3 Comparison of modal properties of a symmetric DMT two-cable network with damped flexible or rigid cross-tie at 120576 = 12

Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)

119891 (Hz) 120585eq () Mode shape 119891 (Hz) 120585eq () Mode shapeAna FEA Ana FEA

1 155 155 171 172 GM in-phase 168 061 GM in-phase2 215 215 401 401 GM out-of-phase 291 025 LM cable 13 291 291 025 025 LM cable 1 337 034 GM out-of-phase4 398 398 040 040 LM cable 2 398 040 LM cable 25 442 441 111 112 LM cable 1 504 021 GM in-phase6 583 582 012 012 LM cable 1 583 013 LM cable 17 603 602 129 129 LM cable 2 674 017 GM out-of-phase8 731 730 072 072 LM cable 1 797 020 LM cable 29 797 796 020 020 LM cable 2 841 013 GM in-phase10 874 872 008 008 LM cable 1 874 008 LM cable 1


Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)

Figure 4 First ten modes of a symmetric DMT two-cable network with a damped flexible cross-tie (119870119888= 3054 kNm 119862

119888= 10 kNsdotsm) at

120576 = 12

to the target cable (Figure 1) are respectively 3m and 9mIn addition the cross-tie is relocated to one-third span ofthe target cable from its left support that is 120576 = 13 Thesechanges in the layout lead to an asymmetric DMT cablenetwork Table 4 lists the modal properties of the first tennetworkmodes obtained from the proposed analytical modeland numerical simulation A good agreement between thetwo sets can be clearly seen In addition the modal analysisresults of a corresponding rigid cross-tie network are also

given in the same table for the convenience of comparisonThemode shapes of these ten modes are depicted in Figure 5

Results in Table 4 indicate that by replacing the rigidcross-tie with a damped flexible one the frequency of thenetwork fundamental mode which is an in-phase globalmode decreases whereas its modal damping ratio increasesdrastically The target cable has a fundamental frequency of146Hz and a modal damping ratio of 050 When it isconnectedwith the neighbouring cable using a rigid cross-tie


Table 4 Comparison of modal properties of an asymmetric DMT two-cable network with damped flexible or rigid cross-tie at 120576 = 13

Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)


1 153 153 149 149 GM in-phase 165 058 GM in-phase2 211 211 288 289 GM out-of-phase 255 044 GM out-of-phase3 297 297 142 142 LM cable 1 349 035 GM in-phase4 404 403 148 148 LM cable 2 437 017 LM cable 15 437 437 017 017 LM cable 1 500 023 GM out-of-phase6 585 585 066 066 LM cable 1 597 026 GM out-of-phase7 598 597 030 030 LM cable 2 643 014 GM out-of-phase8 730 729 051 051 LM cable 1 754 014 GM in-phase9 800 799 094 094 LM cable 2 874 008 LM cable 110 874 872 008 008 LM cable 1 905 014 GM out-of-phase


Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)

Mode 10 (LM cable 1)

Figure 5 First ten modes of an asymmetric DMT two-cable network with a damped flexible cross-tie (119870119888= 3054 kNm 119862

119888= 10 kNsdotsm)

at 120576 = 13

the modal frequency is increased to 165Hz by 13 and thedamping ratio to 058 by 16 However if the cross-tie hasproperties of 119870

119888= 3054 kNm and 119862

119888= 10 kNsdotsm the

increment of its fundamental frequency and damping ratiobecomes 48 to 153Hz and approximately three times to149 respectively The same phenomenon can be observedin Mode 2 which is an out-of-phase global mode that isalthough using a damped flexible cross-tie would reduce thegain in network stiffness to some extent it could greatly

improve the energy dissipation capacity of the formed cablenetwork This is consistent with the existing experience (eg[3 14]) In addition it should be noted that similar to thesymmetric layout case in Section 32 such a change in cross-tie properties would lead to excitation of more local modesdominated by one of the main cables This could be mainlyattributed to the increased flexibility in cross-tie which offersmore freedom to one cable from the constraint of the otherso it can oscillate more independently Among the first ten


modes listed in Table 4 the number of local modes increasesfrom 2 to 8 Mode 4 and Mode 9 in the rigid cross-tie caseare dominated respectively by the 3rd and the 6th mode ofan isolated target cable The position of cross-tie at 120576 = 13happens to coincide with the nodal point of these single cablemodes Thus the modal properties of these two local modesare not affected by the cross-tie stiffness and damping exceptthat they become the 5th and the 10th modes when a dampedflexible cross-tie is used instead

Besides a parametric study is conducted for this asym-metric DMT cable network to better understand the effectof cross-tie stiffness and damping on the modal frequencyand damping ratio of the network global modes Figure 6depicts the modal property variation of the lowest networkin-phase global mode and out-of-phase global mode withrespect to the undamped cross-tie stiffness parameter 120595

119900

In the analysis the cross-tie damping coefficient is assumedto be 119862

119888= 10 kNsdotsm whereas 120595

119900varies from 0 (rigid)

to 10 which is a typical range of cross-tie stiffness on realcable-stayed bridges [7] It can be seen from Figure 6 thatoverall the modal properties of the out-of-phase global modeare more sensitive to the cross-tie stiffness As expected thefrequencies of both global modes decrease monotonicallywith the increase of cross-tie flexibility Within the studiedrange of 120595

119900 the frequency of the in-phase global mode

decreases by 7 while that of the out-of-phase global modedrops roughly by 18 In terms of modal damping ratiosince the linear viscous dampingmodel is used for describingcross-tie damping property a more flexible cross-tie wouldresult in higher relative motion velocity between the cross-tietwo ends and thus more contribution to energy dissipationof the oscillating main cables in the network It is alsointeresting to note from the figure that while the dampingratio increment rate of the in-phase global mode is moresteady when 120595

119900increases from 0 to 1 that of the out-of-

phase global mode appears to be gradually decreasing asthe cross-tie becomes more and more flexible In generalthe patterns of 120595

119900-Ω and 120595

119900-120585eq curves in Figure 6 imply

that although using a more flexible cross-tie would causesome loss in network in-plane stiffness the energy dissipationcapacity could be greatly improved which is beneficial forcable vibration control

The influence of cross-tie damping level on the modalproperties of the two lowest network global modes is shownin Figure 7 The modal frequencies of the two global modesare independent of the cross-tie damping level and remainas constants whereas their modal damping ratios increasealmost linearly with the increase of cross-tie damping Againthe out-of-phase global mode is found to be more sensitiveto change in cross-tie damping By increasing 119862

119888from 0 to

10 kNsdotsm the modal damping ratio of the lowest out-of-phase global mode increases from 044 to 288 by roughly65 times whereas that of the in-phase global mode increasesalmost three times from 058 to 149

4 Conclusions

Cross-tie solution has been successfully applied on site to sup-press large amplitude bridge stay cable vibrations Although it

Cross-tie stiffness parameter 120595o

00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency

Modal frequency (in-phase) Modal damping (in-phase)Modal damping (out-of-phase)Modal frequency

(out-of-phase)

Figure 6 Effect of undamped cross-tie stiffness parameter 120595119900

on modal frequency and modal damping ratio of the lowest in-phase and out-of-phase global modes of an asymmetric DMT cablenetwork

00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency

Modal frequency (in-phase)Modal frequency

Modal damping (in-phase)Modal damping (out-of-phase)

00

05

10

15

20

25

0 100 200 300 400 500 600 700 800 900 1000Damping coefficient C

(out-of-phase)

Figure 7 Effect of cross-tie damping coefficient 119862 on modalfrequency and modal damping ratio of the lowest in-phase and out-of-phase global modes of an asymmetric DMT cable network

is understood that the dynamic behaviour of a cable networkis highly dependent on the installation location stiffness anddamping of cross-ties only the effects of the former two onthe network response have been investigated whereas theimpact of cross-tie damping has rarely been addressed inthe past To have a more comprehensive description of thenetwork properties and better predict its dynamic responsein the current study an analytical model of a cable networkhas been proposed by considering the cross-tie stiffness anddamping as well as the damping of the constituting maincables in the formulationThe impact of cross-tie stiffness anddamping on cable networks having different configurationshas been investigated by using the corresponding undamped


rigid cross-tie networks as the reference base The findingsobtained from the current study are summarized as follows

(1) In the case of a twin-cable network the modal prop-erties of its in-phase global modes are independent ofcross-tie installation location stiffness and dampingHowever replacing rigid cross-tie with a dampedflexible one could have a significant impact on thelocal LS and RSmodes and render them to evolve intoout-of-phase global modes While the frequenciesand damping ratios of local RS modes would all bealtered by the cross-tie stiffness and damping thoseassociated with local LS modes depend on the cross-tie position They would remain the same as thosein the rigid cross-tie case if the cross-tie positioncoincides with the nodal point of a specific isolatedmain cable mode but they would be altered if thecross-tie is installed at the off-nodal point location

(2) For more general two-cable networks having eithersymmetric or asymmetric layout although comparedto rigid cross-tie case the adoption of a dampedflexible cross-tie would decrease the frequencies ofnetwork global modes considerable increase of theirmodal damping ratio has been observed Therefore acareful balance between the loss in network in-planestiffness and the gain in energy dissipation capacityshould be achieved when selecting cross-tie stiffnessand damping in the network design In addition sucha change in the cross-tie properties is found to excitemore local modes dominated by one of the maincables

(3) Results obtained from a parametric study indicatethat the flexibility of a cross-tie would affect bothfrequency and damping of different cable networkmodes whereas cross-tie damping would only affectthe damping ratio of these modes An approximatelinear relation between the cross-tie damping anddamping ratio of network global modes is observed

(4) Compared to network in-phase global modes modalproperties associated with out-of-phase global modesare more sensitive to the change of cross-tie stiffnessand damping

Overall the findings of the current study imply thatalthough using a more flexible cross-tie would cause someloss in the network in-plane stiffness its effectiveness inimproving the energy dissipation capacity couldmake itmorebeneficial in cable vibration control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theauthors are grateful to theNatural Sciences andEngineer-ing Research Council of Canada (NSERC) for supporting thisproject

References

[1] A R Hard and R O Holben ldquoApplication of the vibrationdecay test to transmission line conductorsrdquo IEEE Transactionson Power Apparatus and Systems vol PAS-86 no 2 pp 189ndash1991967

[2] H Yamaguchi and Y Fujino ldquoModal damping of flexural oscil-lation in suspended cablesrdquo Structural EngineeringEarthquakeEngineering vol 4 no 2 pp 413ndash421 1987

[3] H Yamaguchi and H D Nagahawatta ldquoDamping effects ofcable cross ties in cable-stayed bridgesrdquo Journal of WindEngineering and Industrial Aerodynamics vol 54-55 pp 35ndash431995

[4] S Kangas A Helmicki V Hunt R Sexton and J SwansonldquoCable-stayed bridges case study for ambient vibration-basedcable tension estimationrdquo Journal of Bridge Engineering vol 17no 6 pp 839ndash846 2012

[5] A B Mehrabi C A Ligozio A T Ciolko and S T WyattldquoEvaluation rehabilitation planning and stay-cable replace-ment design for the hale boggs bridge in Luling LouisianardquoJournal of Bridge Engineering vol 15 no 4 pp 364ndash372 2010

[6] L Caracoglia and N P Jones ldquoIn-plane dynamic behavior ofcable networks Part 1 formulation and basic solutionsrdquo Journalof Sound and Vibration vol 279 no 3ndash5 pp 969ndash991 2005

[7] L Caracoglia and N P Jones ldquoIn-plane dynamic behavior ofcable networks Part 2 prototype prediction and validationrdquoJournal of Sound and Vibration vol 279 no 3-5 pp 993ndash10142005

[8] J Ahmad and S Cheng ldquoEffect of cross-link stiffness on thein-plane free vibration behaviour of a two-cable networkrdquoEngineering Structures vol 52 pp 570ndash580 2013

[9] J Ahmad and S Cheng ldquoImpact of key system parameters onthe in-plane dynamic response of a cable networkrdquo Journal ofStructural Engineering vol 140 no 3 Article ID 04013079 2014

[10] F Ehsan and R H Scanlan ldquoDamping stay cables with tiesrdquo inProceedings of the 5th US-Japan Bridge Workshop pp 203ndash217Tsukuba Japan 1989

[11] H Yamaguchi and L Jayawardena ldquoAnalytical estimation ofstructural damping in cable structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 43 no 1ndash3 pp 1961ndash1972 1992

[12] J Ahmad S Cheng and F Ghrib ldquoAn analytical approachto evaluate damping property of orthogonal cable networksrdquoEngineering Structures vol 75 pp 225ndash236 2014

[13] J Ahmad and S Cheng ldquoAnalytical study on in-plane freevibration of a cable network with straight alignment rigid cross-tiesrdquo Journal of Vibration and Control vol 21 no 7 pp 1299ndash1320 2015

[14] L Sun Y Zhou and H Huang ldquoExperiment and dampingevaluation on stay cables connected by cross tiesrdquo inProceedingsof the 7th International Symposium on Cable Dynamics PaperID 50 AIM (Electrical Engineers Association of theMontefioreInstitute) Liege Belgium 2007

[15] H R Bosch and S W Park ldquoEffectiveness of external dampersand crossties in mitigation of stay cable vibrationsrdquo in Proceed-ings of the 6th International Symposium on Cable Dynamicspp 115ndash122 AIM (Electrical Engineers Association of theMontefiore Institute) Liege Belgium 2005


[16] G FGiaccu andLCaracoglia ldquoGeneralized power-law stiffnessmodel for nonlinear dynamics of in-plane cable networksrdquoJournal of Sound and Vibration vol 332 no 8 pp 1961ndash19812013

[17] L RayleighTheory of Sound vol 1-2 Dover Publications NewYork NY USA 2nd edition 1945

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the Fred Hartman Bridge showed that if flexible cross-tieswere selected instead of the rigid ones the network funda-mental frequency would decrease by 3 On the contraryBosch and Park [15] pointed out in a numerical study thatusing oversized (too stiff) cross-tie would excite more localmodes This was confirmed by Ahmad and Cheng [8] whoproposed an analytical model to investigate the impact ofcross-tie stiffness on the dynamic response of a cable networkIn the model the main cables were assumed as taut cablesand the flexible cross-tie was assumed to behave like areversible tensioncompression linear spring connector Itwas observed that when amore flexible cross-tie was used thelocalmodes dominated by either the left or the right segmentsof a corresponding cable network using stiffer cross-tiewould evolve into global modes Giaccu and Caracoglia [16]addressed the nonlinear interaction between main cablesand cross-ties with behaviour of the latter described by ageneralized power-law stiffness model They also reportedthat the equivalent nonlinear effects in network highermodescould be responsible for the transition from a local mode to aglobal one

It is clear from the above literature review that connectinga vulnerable cable with its neighbouring ones through trans-verse cross-ties would alter its in-plane stiffness and dampingpropertyThe level of such alteration highly depends on cross-tie properties When designing a cable network the selectionof cross-tie properties should be based on their combinedimpact on the network in-plane stiffness and damping Toproperly assess the latter the damping property of maincables as well as the stiffness and damping of cross-ties shouldbe included in the analysis The few available experimentalstudies (eg [3 14]) only discussed how the first modaldamping ratio of a vulnerable cable would be affected in acable network but such an impact on higher order modeswas not addressed Although a recent analytical study by theauthors [12] examined this issue the proposed model onlyconsidered damping in main cables whereas the cross-tiewas assumed to be undamped and rigid Thus the responsebetween a vulnerable cable and its neighbouring ones wasldquocommunicatedrdquo directly without the influence of cross-tieproperties However in reality when transmitting responsesbetween these two parties cross-tie would behave like aldquofilterrdquo which alters the transmitted response by its stiffnessand damping properties Therefore to properly assess theeffect of cross-tie solution on the dynamic response of a vul-nerable cable and the formed cable network it is imperativeto describe the cross-tie behaviour more accurately

In view of the above needs the current paper aimsat extending the analytical model proposed earlier by theauthors [12] to include both the stiffness and damping prop-erties of cross-tie in the formulation The network systemcharacteristic equation will be derived analytically and themodal characteristics of the cable networkwill be determinedby solving the associated complex eigenvalue problem Theproposed model would allow evaluating the impact of cross-tie solution on the in-plane frequency and also the dampingproperty of a vulnerable cable not only for the fundamentalmode but for the higher order modes as well The respectiverole of cross-tie stiffness and damping in affecting network

dynamic behaviour will be examined These would furtherenhance our understanding of cable network mechanics andprovide valuable information to improve the current practiceof cross-tie design

2 Formulation of Analytical Model

The proposed analytical model consists of two horizontallylaidmain cables connected by a transverse cross-tie as shownschematically in Figure 1The twomain cables are assumed tobe damped taut cables and are fixed at both ends The lengthof the two cables is 119871

1and 119871

2(1198711ge 1198712) respectively It is

assumed that the longer cable (main cable 1) is vulnerableto dynamic excitations In the following discussion it willbe referred to as the ldquotarget cablerdquo whereas the shorter cable(main cable 2) will be referred to as the ldquoneighbouring cablerdquo119874119871and119874

119877denote respectively the left and the right horizon-

tal offset of the neighbouring cable In general they are notequal to each other The unit mass tension and structuraldamping ratio of the two main cables are denoted by 119898

119895

119867119895 and 120585

119895(119895 = 1 2) respectively The transverse cross-tie

is assumed to be located at 1198971(1198971lt 1198972) from the left end

of the target cable Its axial stiffness property is representedby a linear spring connector with an associated stiffnessconstant of 119870

119888 where the subscript ldquo119888rdquo refers to ldquocross-tierdquo

The structural damping which comes from various types ofenergy dissipation mechanisms is usually represented by ahighly idealized form in structural analysis The equivalentlinear viscous damping model proposed by Rayleigh [17]assumes the damping force to be linearly proportional tothe motion velocity This model is commonly adopted dueto the simplicity of its form and the amenability in derivinganalytical solutionTherefore in the current model dampingproperty of the two main cables and that of the cross-tieare all assumed to be the linear viscous type and uniformlydistributed along the member length The correspondingdamping coefficients are denoted 119862

119895(119895 = 1 2) for the 119895th

main cable and 119862119888for the cross-tie The additional tension

in the main cables caused by vibration is neglected in theproposed model

When the cable network in Figure 1 is excited to vibratewithin its plane all four main cable segments oscillate inthe transverse direction whereas the cross-tie moves alongits longitudinal direction The motion of each main cablesegment can be described by the equation of motion of a tautcable subjected to in-plane damped free vibration that is

1198671205972V1205971199092

= 1198981205972V1205971199052+ 119862120597V120597119905

(1)

or

1198671205972V1205971199092

= 1198981205972V1205971199052+ 2119898120585120596

119900

120597V120597119905 (2)

where V 119867 and 119898 are respectively the transverse dis-placement the tension and the unit mass of the taut cable119862 = 2119898120585120596

119900is the cable damping coefficient per unit

length 120585 is the cable structural damping ratio and 120596119900is


Cable 1

C

A B

D Cable 2

x1 x2

y1

x3

y3

x4

y4

y2

L2

L1l2l1

l3 l4

N2

N1

CcKc

OLOR



V10158401015840 (119909) + 1205722V (119909) = 0 (3)


V (119909) = 119860 cos (120572119909) + 119861 sin (120572119909) (4)


120572 = radic1198981205962minus 119894 sdot 2119898120585120596120596

119900

119867 (5)



sin (1205721198951199092119895minus1) 119895 = 1 2 (6a)

V2119895(1199092119895) = 1198612119895sin (120572

21198951199092119895) 119895 = 1 2 (6b)




2119895minus1and 119861




119865119888(119905) = 119870

119888119906 (119905) + 119862

119888

120597119906

120597119905 (7)


119906 (119905) = V1(1198971 119905) minus V

3(1198973 119905) = [V

1(1198971) minus V3(1198973)] 119890119894120596119905 (8)



[(120597V1

1205971199091

100381610038161003816100381610038161003816100381610038161199091=1198971

+120597V2

1205971199092

100381610038161003816100381610038161003816100381610038161199092=1198972

)1198671] 119890119894120596119905= minus119865119888(119905) (9)


12057211198671[1198611cos (120572

11198971) + 1198612cos (120572

11198972)]

= [1198613sin (120572

21198973) minus 1198611sin (120572

11198971)] [119870119888+ 119894120596119862

119888]

(10)


(120597V1

1205971199091

100381610038161003816100381610038161003816100381610038161199091=1198971

+120597V2

1205971199092

100381610038161003816100381610038161003816100381610038161199092=1198972

)1198671

+ (120597V3

1205971199093

100381610038161003816100381610038161003816100381610038161199093=1198973

+120597V4

1205971199094

100381610038161003816100381610038161003816100381610038161199094=1198974

)1198672= 0

(11)


12057211198671[1198611cos (120572

11198971) + 1198612cos (120572

11198972)]

+ 12057221198672[1198613cos (120572

21198973) + 1198614cos (120572

21198974)] = 0

(12)


1and119873

2gives

V2119895minus1(1198972119895minus1) = V2119895(1198972119895) 119895 = 1 2 (13)


1198611sin (120572

11198971) minus 1198612sin (120572

11198972) = 0 (14a)

1198613sin (120572

21198973) minus 1198614sin (120572

21198974) = 0 (14b)


[119878] 119883 = 0 (15)


where

[119878] =

[[[[[

[


2) 0 0


4)

1205951198771cos (0

1) + sin (0

1) 1205951198771cos (0

2) minus sin (0

3) 0

1205741cos (0

1) 120574

1cos (0

2) 1205742cos (0

3) 1205742cos (0

4)

]]]]]

]

(16)


2119895minus1=

1198771198951205762119895minus1




in (5) 1205762119895minus1





120574119895=

119867119895119877119895119871119895

119867111987711198711

(17)


120595 =1198671

1198711[119870119888+ 119894120596119862

119888] (18)






119877119895= radic[(Ω120578

119895)2

minus 119894 sdot 2120587120585119895Ω120578119895] 119895 = 1 2 (19)




1205741sin (119877

1) sin (0

3) sin (0

4)

+ 1205742sin (119877

2) sin (0

1) sin (0

2)

+ 120595119877112057411205742sin (119877

1) sin (119877

2) = 0

(20)


119895 120574119895(119895 =


119895




3 Case Studies







2 sin (1198771) sin (0

1) sin (0

2) + 120595119877

1sin2 (119877

1) = 0 (21)

or

sin (1198771) [2 sin (0

1) sin (0

2) + 120595119877

1sin (119877

1)] = 0 (22)






Ω = Ωre + 119894 sdot Ωim (23)








0and



2

im = 119899120587 119899 = 1 2 3 (24a)

120585eq =Ωim

radicΩ2re + Ω2

im

=120585

119899119899 = 1 2 3 (24b)



2 sin (01) sin (0

2) + 120595119877

1sin (119877

1) = 0 (25)


1) = 0 and sin(0

2) =




as

2 sin (01) sin (0

2) + 120595119877

1sin [ 1

119898

01

1205761

+ (1 minus1

119898)02

1205762

] = 0

(26)




sin (01) 2 sin (0

2) + 120595119877

1[cos (0

2) + cos (0

1)sin (02)

sin (01)]

= 0

(27)



Ω0=119899120587

120576119899 = 1 2 3 (28a)

120585eq =120576120585

119899119899 = 1 2 3 (28b)







Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




119888= 3054 kNm











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)














119888= 10 kNsdotsm) at 120576 = 13

















119888= 10 kNsdotsm) at 120576 = 25






1= 02= 11987712



sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)


12) = 0 and




Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)



Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Cable 1

C

A B

D Cable 2

x1 x2

y1

x3

y3

x4

y4

y2

L2

L1l2l1

l3 l4

N2

N1

CcKc

OLOR



V10158401015840 (119909) + 1205722V (119909) = 0 (3)


V (119909) = 119860 cos (120572119909) + 119861 sin (120572119909) (4)


120572 = radic1198981205962minus 119894 sdot 2119898120585120596120596

119900

119867 (5)



sin (1205721198951199092119895minus1) 119895 = 1 2 (6a)

V2119895(1199092119895) = 1198612119895sin (120572

21198951199092119895) 119895 = 1 2 (6b)




2119895minus1and 119861




119865119888(119905) = 119870

119888119906 (119905) + 119862

119888

120597119906

120597119905 (7)


119906 (119905) = V1(1198971 119905) minus V

3(1198973 119905) = [V

1(1198971) minus V3(1198973)] 119890119894120596119905 (8)



[(120597V1

1205971199091

100381610038161003816100381610038161003816100381610038161199091=1198971

+120597V2

1205971199092

100381610038161003816100381610038161003816100381610038161199092=1198972

)1198671] 119890119894120596119905= minus119865119888(119905) (9)


12057211198671[1198611cos (120572

11198971) + 1198612cos (120572

11198972)]

= [1198613sin (120572

21198973) minus 1198611sin (120572

11198971)] [119870119888+ 119894120596119862

119888]

(10)


(120597V1

1205971199091

100381610038161003816100381610038161003816100381610038161199091=1198971

+120597V2

1205971199092

100381610038161003816100381610038161003816100381610038161199092=1198972

)1198671

+ (120597V3

1205971199093

100381610038161003816100381610038161003816100381610038161199093=1198973

+120597V4

1205971199094

100381610038161003816100381610038161003816100381610038161199094=1198974

)1198672= 0

(11)


12057211198671[1198611cos (120572

11198971) + 1198612cos (120572

11198972)]

+ 12057221198672[1198613cos (120572

21198973) + 1198614cos (120572

21198974)] = 0

(12)


1and119873

2gives

V2119895minus1(1198972119895minus1) = V2119895(1198972119895) 119895 = 1 2 (13)


1198611sin (120572

11198971) minus 1198612sin (120572

11198972) = 0 (14a)

1198613sin (120572

21198973) minus 1198614sin (120572

21198974) = 0 (14b)


[119878] 119883 = 0 (15)


where

[119878] =

[[[[[

[


2) 0 0


4)

1205951198771cos (0

1) + sin (0

1) 1205951198771cos (0

2) minus sin (0

3) 0

1205741cos (0

1) 120574

1cos (0

2) 1205742cos (0

3) 1205742cos (0

4)

]]]]]

]

(16)


2119895minus1=

1198771198951205762119895minus1




in (5) 1205762119895minus1





120574119895=

119867119895119877119895119871119895

119867111987711198711

(17)


120595 =1198671

1198711[119870119888+ 119894120596119862

119888] (18)






119877119895= radic[(Ω120578

119895)2

minus 119894 sdot 2120587120585119895Ω120578119895] 119895 = 1 2 (19)




1205741sin (119877

1) sin (0

3) sin (0

4)

+ 1205742sin (119877

2) sin (0

1) sin (0

2)

+ 120595119877112057411205742sin (119877

1) sin (119877

2) = 0

(20)


119895 120574119895(119895 =


119895




3 Case Studies







2 sin (1198771) sin (0

1) sin (0

2) + 120595119877

1sin2 (119877

1) = 0 (21)

or

sin (1198771) [2 sin (0

1) sin (0

2) + 120595119877

1sin (119877

1)] = 0 (22)






Ω = Ωre + 119894 sdot Ωim (23)








0and



2

im = 119899120587 119899 = 1 2 3 (24a)

120585eq =Ωim

radicΩ2re + Ω2

im

=120585

119899119899 = 1 2 3 (24b)



2 sin (01) sin (0

2) + 120595119877

1sin (119877

1) = 0 (25)


1) = 0 and sin(0

2) =




as

2 sin (01) sin (0

2) + 120595119877

1sin [ 1

119898

01

1205761

+ (1 minus1

119898)02

1205762

] = 0

(26)




sin (01) 2 sin (0

2) + 120595119877

1[cos (0

2) + cos (0

1)sin (02)

sin (01)]

= 0

(27)



Ω0=119899120587

120576119899 = 1 2 3 (28a)

120585eq =120576120585

119899119899 = 1 2 3 (28b)







Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




119888= 3054 kNm











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)














119888= 10 kNsdotsm) at 120576 = 13

















119888= 10 kNsdotsm) at 120576 = 25






1= 02= 11987712



sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)


12) = 0 and




Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)



Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

[119878] =

[[[[[

[


2) 0 0


4)

1205951198771cos (0

1) + sin (0

1) 1205951198771cos (0

2) minus sin (0

3) 0

1205741cos (0

1) 120574

1cos (0

2) 1205742cos (0

3) 1205742cos (0

4)

]]]]]

]

(16)


2119895minus1=

1198771198951205762119895minus1




in (5) 1205762119895minus1





120574119895=

119867119895119877119895119871119895

119867111987711198711

(17)


120595 =1198671

1198711[119870119888+ 119894120596119862

119888] (18)






119877119895= radic[(Ω120578

119895)2

minus 119894 sdot 2120587120585119895Ω120578119895] 119895 = 1 2 (19)




1205741sin (119877

1) sin (0

3) sin (0

4)

+ 1205742sin (119877

2) sin (0

1) sin (0

2)

+ 120595119877112057411205742sin (119877

1) sin (119877

2) = 0

(20)


119895 120574119895(119895 =


119895




3 Case Studies







2 sin (1198771) sin (0

1) sin (0

2) + 120595119877

1sin2 (119877

1) = 0 (21)

or

sin (1198771) [2 sin (0

1) sin (0

2) + 120595119877

1sin (119877

1)] = 0 (22)






Ω = Ωre + 119894 sdot Ωim (23)








0and



2

im = 119899120587 119899 = 1 2 3 (24a)

120585eq =Ωim

radicΩ2re + Ω2

im

=120585

119899119899 = 1 2 3 (24b)



2 sin (01) sin (0

2) + 120595119877

1sin (119877

1) = 0 (25)


1) = 0 and sin(0

2) =




as

2 sin (01) sin (0

2) + 120595119877

1sin [ 1

119898

01

1205761

+ (1 minus1

119898)02

1205762

] = 0

(26)




sin (01) 2 sin (0

2) + 120595119877

1[cos (0

2) + cos (0

1)sin (02)

sin (01)]

= 0

(27)



Ω0=119899120587

120576119899 = 1 2 3 (28a)

120585eq =120576120585

119899119899 = 1 2 3 (28b)







Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




119888= 3054 kNm











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)














119888= 10 kNsdotsm) at 120576 = 13

















119888= 10 kNsdotsm) at 120576 = 25






1= 02= 11987712



sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)


12) = 0 and




Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)



Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Ω = Ωre + 119894 sdot Ωim (23)








0and



2

im = 119899120587 119899 = 1 2 3 (24a)

120585eq =Ωim

radicΩ2re + Ω2

im

=120585

119899119899 = 1 2 3 (24b)



2 sin (01) sin (0

2) + 120595119877

1sin (119877

1) = 0 (25)


1) = 0 and sin(0

2) =




as

2 sin (01) sin (0

2) + 120595119877

1sin [ 1

119898

01

1205761

+ (1 minus1

119898)02

1205762

] = 0

(26)




sin (01) 2 sin (0

2) + 120595119877

1[cos (0

2) + cos (0

1)sin (02)

sin (01)]

= 0

(27)



Ω0=119899120587

120576119899 = 1 2 3 (28a)

120585eq =120576120585

119899119899 = 1 2 3 (28b)







Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




119888= 3054 kNm











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)














119888= 10 kNsdotsm) at 120576 = 13

















119888= 10 kNsdotsm) at 120576 = 25






1= 02= 11987712



sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)


12) = 0 and




Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)



Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




119888= 3054 kNm











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)














119888= 10 kNsdotsm) at 120576 = 13

















119888= 10 kNsdotsm) at 120576 = 25






1= 02= 11987712



sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)


12) = 0 and




Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)



Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)














119888= 10 kNsdotsm) at 120576 = 13

















119888= 10 kNsdotsm) at 120576 = 25






1= 02= 11987712



sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)


12) = 0 and




Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)



Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















119888= 10 kNsdotsm) at 120576 = 25






1= 02= 11987712



sin(11987712) sin(1198772

2) [cos(1198771

2) sin(1198772

2)

+ 1205742sin(1198771

2) cos(1198772

2)

+ 212059511987711205742sin (119877

1)] = 0

(29)


12) = 0 and




Ω119900= 2119899120587 119899 = 1 2 3 (30a)

120585eq =1205851

(2119899)119899 = 1 2 3 (30b)



Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Ω119900=2119899120587

1205782

119899 = 1 2 3 (31a)

120585eq =1205852

(2119899)119899 = 1 2 3 (31b)




Main cable 1

1198711= 72m 119867

1= 2200 kN 119898

1= 50 kgm

1205851= 05

(32)

Main cable 2

1198712= 60m 119867

2= 2400 kN 119898

2= 42 kgm

1205852= 08

(33)

Cross-tie

120576 =1

2 119870

119888= 3054 kNm 119862

119888= 10 kN sdot sm (34)












Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Mode 3 (LM cable 1) Ω = 20120587 120585eq = 025

Mode 5 (LM cable 1) Ω = 303120587 120585eq = 111

Mode 7 (LM cable 2) Ω = 414120587 120585eq = 129

Mode 9 (LM cable 2) Ω = 547120587 120585eq = 020


Mode 4 (LM cable 2) Ω = 274120587 120585eq = 040

Ω = 40120587 120585eq = 012Mode 6 (LM cable 1)

Mode 8 (LM cable 1) Ω = 502120587 120585eq = 072

Ω = 60120587 120585eq = 008Mode 10 (LM cable 1)



120576 = 12






Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Mode


119888= 10 kNsdotsm) (119870

119888= infin 119862

119888= 0)




Ω = 204120587 120585eq = 142

Ω = 30120587 120585eq = 017

Ω = 410120587 120585eq = 030

Ω = 549120587 120585eq = 094


Ω = 277120587 120585eq = 148

Ω = 402120587 120585eq = 066

Ω = 502120587 120585eq = 051

Ω = 60120587 120585eq = 008

Mode 3 (LM cable 1)

Mode 5 (LM cable 1)

Mode 7 (LM cable 2)

Mode 9 (LM cable 2)

Mode 4 (LM cable 2)

Mode 6 (LM cable 1)

Mode 8 (LM cable 1)




at 120576 = 13


119888= 3054 kNm and 119862







119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119900









119900-Ω and 120595




119888from 0 to


4 Conclusions



00

05

10

15

20

25

30

35

00

05

10

15

20

25

00 01 02 03 04 05 06 07 08 09 10

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency


(out-of-phase)



00

05

10

15

20

25

30

35

Equi

vale

nt d

ampi

ng ra

tio (

)

Non

dim

ensio

nal f

requ

ency



00

05

10

15

20

25


(out-of-phase)












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















Acknowledgment


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









research article impact of cross-tie properties on the

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