dynamic characteristics and vibration control of a cable system with sub-structural interactions

8/14/2019 Dynamic Characteristics and Vibration Control of A Cable System with Sub-Structural Interactions

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Engineering Structures, 23(19), pp.1348-1358, Mar. 2001

Dynamic Characteristics and Vibration Control of A Cable Systemwith Sub-Structural Interactions

H. Yamaguchi*, Md. Alauddin**and N. Poovarodom***

* Department of Civil and Environmental Engineering, Saitama University

Urawa, Saitama 338, Japan

** Formerly ditto

*** Department of Civil Engineering, Thammasat University

Pathumthani 12121, Thailand

Corresponding author: Professor Hiroki Yamaguchi

Department of Civil and Environmental Engineering

Saitama University255 Shimo-Ohkubo, Saitama 338-8570, Japan

Phone: 81-48-858-3552Fax: 81-48-858-3552

Email: [email protected]

Abstract

Free and forced vibration analyses of a cable system, which consists of two identical

sagged cables in parallel connected by another sagged cross cable, are conducted

using modal synthesis method with sub-structural formulation in order to investigate

the system behavior paying much attention to the sub-structural interaction. It is

shown that the secondary cable contribution in the system modal damping can bedominant and can cause greater damping in the case of coupled motion of main and

secondary cables. The importance of secondary cable in transferring the energy from

one substructure to another through the sub-structural interaction is also discussed for

the passive control of harmonically forced vibration in the main cable.

Keywords

cable system, sub-structural interaction, secondary cable, modal damping, passivecontrol, modal synthesis

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INTRODUCTION

Civil engineering structures are, normally, composed of a number of components or

substructures. Each component has its own dynamic characteristics. But in the

complete system, the dynamic characteristics of a component may be different fromits dynamic behavior when the component vibrates individually. Actually, for the

prediction of dynamic characteristics of a structural system, only the knowledge about

the dynamic characteristics of individual member or component is not sufficient. An

adequate knowledge about the interaction among various members of the system is

also necessary because the sub-structural behavior can be changed in the system dueto interaction among different substructures.

Because of the rationality in using steel for tension members, cable has become

one of the most important structural members in the field of civil engineering. A

number of cable elements are sometimes used to form a component of large civil

engineering structures such as suspension bridges, cable-stayed bridges, guyed towers,transmission lines and so on. However, various types of vibration problems can occur

easily in cable systems due to their flexibility, relatively small mass and less energy-

dissipative characteristics. In the past, many researches [1-6] have been done to

investigate the dynamic behavior of single cable. Single cable behavior is important

in understanding the system behavior to some extent. But complete understanding of

the system dynamic characteristics requires additional knowledge regarding the

modifications encountered in the individual behavior in system in order to explain the

system behavior in relation to that of individual characteristics. In the existing

literature, the research on dynamic behavior of cable system on the basis of sub-

structural characteristics is not sufficient, while some works have appeared

concerning dynamic analysis of cable system. For example, Yamaguchi andJayawardena [7] studied the damping behavior of cable-tie system in cable-stayed

bridges; Morris [8] applied the direct modal superposition method in the analysis of

cable networks and suspension bridges. As far as interaction among various

substructures in a cable system and sub-structural contributions in different system

characteristics concerned, however, more detailed investigations still remain to be

done.

On the other hand, many interesting and useful studies have been made to control

the cable vibrations [9]. Among various types of countermeasures such as

aerodynamic means, oil dampers, visco-elastic dampers, and high damping rubber,

use of secondary cables interconnecting the main functional cables can serve as the

controlling measure for cable vibration in some case in a better way [10]. Oneexample is so-called cross-tie (secondary taut cable) which connects several stay

cables in cable-stayed bridges in order to suppress the rain-wind-induced vibrations of

some of the stay cables [11]. Another example is an attempt to suppress the man-

induced lateral vibration problem of the catwalk, which is a temporal cable structure

for the erection of main cables in suspension bridges, by using cross-rope (secondary

sagged cable) [12-14]. In both cases, the addition of secondary cable in a cable system

can change the dynamical behavior of system in a favorable way whereby the

vibration of the primary cables in the system can be controlled. Interactions among

the sub-structural cables play an important role on control performance of the adopted

control measures. Better understanding of this interaction may help greatly in the

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complete description of the control mechanism, which is necessary to design a cable

system more confidently and appropriately.

In the present study, therefore, attempts have been made to analyze a cable system

with secondary sagged cable in order to explain the system characteristics and the

control mechanism of secondary cable on the basis of sub-structural characteristics.Equations of motion for the cable system consisting of several sagged cables are first

derived using modal synthesis method [15-17] with sub-structural formulation.

Comprehensive study is then done to investigate the dynamic characteristics and

behavior of the cable system under both the free and forced vibration conditions. For

free vibration, experimental investigations for both the single cables and the systemare also done besides the method of energy based damping analysis [11], in order to

discuss the system damping in terms of sub-structural contributions. In the case of

forced vibration analysis, the dependency of controlling effect of one sub-structural

cable on the other is discussed carefully by paying much attention to the energy

interaction among different sub-structural cables.

MODEL FOR INVESTIGATION

A cable system consisting of two identical, sagged cables in parallel connected by

another sagged cross cable, which is originally a scaled model of the catwalk system

studied for controlling its man-induced lateral vibrations [12-14], is investigated in

this paper. The schematic diagram of this model is shown in Figure 1 and the

specification of each cable is given in Table 1. The cross cable for the vibration

control is designated as secondary cable and its weight and rigidity are very small in

comparison with those of the two parallel cables, while the parallel cables are termed

as main cables. It should be noted that the sag of secondary cable is set to be same asthat of main cable in order to make the natural frequencies of the secondary cable

close to those of the main cable. We can expect the effects of Tuned Mass Damper

(TMD) for several modes of the main cable by utilizing this multi-mode-tuning

characteristics of the secondary cable.

DERIVATION OF SYSTEM EQUATION OF MOTION

As mentioned earlier, sub-structural formulation with modal synthesis approach [15-

17] is applied to solve the dynamic interaction problem of the cable system. This is

because it is convenient to explain the effect of one component or substructure on the

other on the basis of sub-structural response solution. Since the behavior of sub-structural cables is geometrically nonlinear in nature, some modifications have been

incorporated in the sub-structural formulation in this study.

Static configuration of the cable system is first determined by nonlinear finite

element analysis [18] using 3-node elements with the shape function matrix, [N(s)],

where the curvilinear coordinate, s, is used to designate the undeformed arc length of

cable element. The position vector of the r-thsubstructure during dynamic response,

{X}r, is then decomposed into the static position vector, {X0}r, and the dynamic

displacement vector, {u}r, as follows:

X{ }r = X

0

r

+ u{ }r (1)

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According to the principle of modal synthesis approach, the dynamic displacement

vector, {u}r, is represented in the modal space as

u{ }r = !cnstrnt !normal[ ]r p{ }r (2)

where sub-matrices,!

cnstrnt

[ ]r and!

normal[ ] r , are sub-structural constraint-modematrix and truncated normal-mode matrix [16], respectively. {p}r in Equation (2) is

the sub-structural generalized coordinate vector. In the case of linear structures, thesystem matrices corresponding to the sub-structural generalized coordinate vector are

derived from the direct assembly of the sub-structural matrices and transferred into

those corresponding to the system generalized coordinates by applying the kinematic

compatibility conditions at the junctions of substructures. In the case of nonlinear

system, however, this type of linear transformation on the system matrices is not

possible and the sub-structural displacement vector inEquation (2)is first represented

in terms of the system generalized coordinate vector, {q}, by using kinematiccompatibility conditions:

u{ }r= ![ ]

r q{ } (3)

Applying the principle of virtual work for initial stress problem [19] to an element

with lengthL, we get

!u{ }eT

0

L

fD

{ }eds + !unodal{ }e

Tr{ }e # A$s!%s( )e

0

L

" ds# A$0!%snl( )

e0

L

" ds = 0 (4)

where {fD}, {r} and A are distributed load vector, point load vector and cross-

sectional area, respectively [20].!0 and!sare the static stress with self-weight and

Kirchhoffs dynamic stress, respectively. "s in Equation (4) is corresponding to

Lagrangian strain given by

!s( ) e = Xnodal0{ }

e

T

"N[ ]T "N[ ] #[ ]e q{ } + 1

2q{ }T #[ ]e

T"N[ ]T "N[ ] #[ ]e q{ } (5)

where the sign ( !) indicates the differentiation with respect to the arc length, s, [9].

!s

nlin Equation (4) represents the nonlinear part of the Lagrangian strain in the

second term of Equation (5). Considering only linear stiffness terms and taking intoaccount of an appropriate damping matrix, the final expression for linearized equation

of motion can be derived as

M[ ] q{ } + C[ ] q{ } + K[ ] q{ }! R{ } = 0 (6)where

M[ ] = me B[ ] e

0

L

! dse = 1" (7.a)

C[ ]= Appropriate damping matrix (7.b)

K[ ] = A!0

"#

$%e

&B[ ]e

0

L

' ds + AE( )e C1()*+,-e

C1()*+,-e

T

0

L

' ds.

/

00

1

2

33

e = 1

4 (7.c)

R{ }= ![ ]T r{ } (7.d)

B[ ]e

= ![ ]e

T

N[ ] T N[ ] ![ ]e (7.e)

!B[ ]e= "[ ]

e

T !N[ ] T !N[ ] "[ ]e (7.f)

C1!"#$%&e

T

= Xnodal

0!"#$%&e

T

'N[ ]T 'N[ ] ([ ]e (7.g)

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LINEAR DYNAMIC CHARACTERISTICS OF THE CABLE SYSTEM

Natural Frequencies and System Modes

Undamped free vibration analysis of the cable system shown in Figure 1 has beendone using the linear stiffness and mass matrices in Equation (6). Six sub-structural

normal modes and necessary constraint modes for main and secondary cables, as

shown in Figure 2, are considered for the sub-structural formulation in the previous

section. The first and second sub-structural modes for the main cable, C1 andC2, in

the figure are constraint modes which are derived from the deflected shape of thecable by introducing unit displacement at the junction point of main and secondary

cables in Y and Z directions, respectively. On the contrary, the first to fourth

constraint modes for the secondary cable, C1-C4, in Figure 2 are obtained by

introducing unit displacement at one of the ends of secondary cable in Y and Z

directions with the other end remaining fixed. Since the axial displacement of themain cable in X direction is very small, the constraint mode corresponding to this

degree-of-freedom is not considered. As for the normal modes, the first six sub-

structural modes,N1-N6, in Figure 2 are used in the analysis after confirming that the

first fifteen natural frequencies and the corresponding mode shapes of the cable

system are in very good agreement with those obtained by the conventional finite

element analysis [18].

The numerically obtained natural frequencies of the cable system along with the

sub-structural natural frequencies and corresponding mode shapes are shown in

Figure 3. It is seen from Figure 3 that there exist several groups of closely spaced

frequencies in the system due to connecting two individual main cables by the

secondary cable. Each group consists basically of three system-modes in which thesubstructures are internally resonated in specific sub-structural modes with nearly

equal sub-structural frequencies. The first group of system frequency, for example,

comes from the first symmetric out-of-plane modes of main and secondary cables

whose natural frequencies coincide with each other, as shown in Figure 3. The groups

of the first asymmetric in-plane and out-of-plane sub-structural modes, however, are

grouped into one in Figure 3because the natural frequencies of the first in-plane and

out-of-plane modes are nearly equal for the asymmetric modes of single cable. It is

noted that the natural frequency of the first in-plane symmetric mode in the secondary

cable differs relatively largely from that in the main cable because of the modal

transition due to the sag effect [1, 2].

The first fifteen mode shapes of the cable system and their composition ofgeneralized coordinates, or modal contribution in the groups, are shown in Figure 4.

Each figure of system-mode shape consists of plan view, elevation and side view. In

the bar plots of Figure 4 which depicts the contribution of sub-structural modes,

darker bar indicates the in-plane sub-structural mode and lighter bar corresponds to

the out-of-plane sub-structural mode. The abscissas of these bar plots represent the

sub-structural mode number in which C is used to designate the constraint modes

while N indicates the normal modes. The upper diagram of each sub-plot designated

by i is for two main cables and the bottom one indicated by ii is for the secondary

cable.

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From these figures, the characteristics of groups of system modes as observed

from Figure 3can be directly discussed. The first group in Figure 4(a)corresponds to

the system modes in which the first symmetric out-of-plane mode of substructure is

dominant. It is clearly indicated in the figure that there are two types of main-cable-

dominant modes; in-phase (Mode-1) and out-of-phase (Mode-3) modes, and one typeof secondary-cable-dominant mode (Mode-2). Insignificant contributions from other

sub-structural modes can be observed in this group while the main-cable-dominant,

out-of-phase mode (Mode-3) is affected largely by the constraint mode of secondary

cable. The fourth group of system modes in Figure 4(d), which is dominated by the

second symmetric out-of-plane mode of substructure, has almost same characteristicsas the first group.

The second group of system modes in Figure 4(b) is dominated by the first

asymmetric in-plane and out-of-plane modes of substructures (sub-structural modes

N2andN3in the bar plot 'i'). Due to very closely spaced natural frequencies of these

sub-structural in-plane and out-of-plane modes, all of main-cable-dominant modes(Mode-5, 7, 8 and 9) are more or less composed of coupled motions between the in-

plane and out-of-plane modes of main cables. The secondary-cable-dominant modes,

Mode-4 and 6, on the contrary, consist of purely in-plane and out-of-plane motions of

secondary cable (N2andN3in the bar plot 'ii'), respectively. It is worthy to mention

here that the main-cable-dominant modes (Mode-5 and 7), in which the out-of-plane

motions of two main cables are out-of-phase, contain the symmetric in-plane mode of

secondary cable (indicated by N4) which are coupled with its constraint modes

(indicated by C).

For the third group of system modes in Figure 4(c),the dominant contribution is

from the first symmetric in-plane modes of substructures (indicated by N4). Small

contributions from the N3 and the N5 sub-structural normal modes, which indicatelittle coupling between in-plane and out-of-plane motions of main cables, are

observed.

The secondary cable motion is important for controlling the main cable vibration

and is now discussed in detail. As can be seen from Figure 4, out-of-plane motion of

the secondary cable in system modes (Mode-2, 6 and 13) is uncoupled with any type

of main cable motion. In-plane motion of the secondary cable is, on the contrary,

coupled largely with the main cable motion and mostly dependent on the phase

relationship of the two main cable motions in the system. Out-of-phase motions of

two main cables in system modes cause the N2 and N4 sub-structural modal

contribution of secondary cable to its in-plane motion to be additive, and secondary

cable response becomes greater. Due to this large response of the secondary cable, themain cable motions become much smaller. This is true for the system modes (Mode-3,

10 and 14) in which the main cable motions are due to symmetric sub-structural

modes. On the other hand, this type of controlled motion of main cables is not

observed for system modes (Mode-5 and 7) with dominating contribution from the

asymmetric sub-structural modes in main cable motion.

Modal DampingThe modal damping ratios of the single main cable and the cable system were first

measured by conducting free vibration experiment in order to check the damping

effect of secondary cable. The modes whose damping ratios are measured are the first

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out-of-plane mode of single cable and the system modes in the first group of natural

frequency, which are dominated by the first out-of-plane sub-structural motion of

main cable. The damping measurement for each mode was repeated several times to

take into account of possible experimental errors. Figure 5shows the experimentally

measured modal damping ratios versus response amplitude evaluated from a fewrecords of free vibration in each mode. It is clearly indicated in the figure that the

third system modal damping becomes much larger than the modal damping of single

main cable, while the first system modal damping is not changed from the original

cables modal damping. This means that the system damping effect can be expected

by using the secondary cable in the case of the third system mode that is the out-of-phase mode with large motion of the secondary cable.

Energy-based method of damping evaluation[11] is now applied to analyze the

characteristics of system-modal damping. The damping ratio corresponding to the n-

thnormal mode, !n, of a cable system consisting of msubsystems is given by

!n= 1

2" VnU

tn+V

n

#$%

&'(i

m

)i

(8)

where Utn

and Vnare the modal strain energy of substructure due to the initial tension

and the strain energy of substructure caused by the system modal vibration per cycle,

respectively, and given by

Utn=

1

2A!0

0

l

"#uj

#s

#uj

#sds

j=1

3

$ dt0

2% &

" , Vn =1

2EA!

2ds

0

l

" dt0

2# $

" (9.a, b)

where ujis thej-thcomponent of the normal mode vector of an arbitrary point in the

cable. # inEquation (8)is known as loss factor [21], a measure of damping, which is

generally defined for materials as the ratio of energy dissipated to energy stored percycle. In this study, however, the loss factor is defined for each sub-structural cable as

its damping parameter and is evaluated experimentally by using the free vibration

record of each sub-structural cable.

Modal damping ratio of each sub-structural mode is first computed by applying the

above mentioned energy-based method to each substructure with assumed loss factor

and with analyzed modal strain energies, and then compared with the experimental

data. The results are depicted in Figure 6where both the analytically (solid line) andexperimentally obtained modal damping ratios for the first out-of-plane mode of main

cable, the first asymmetric and symmetric in-plane modes of secondary cable are

presented. The loss factors of cables are identified from Figure 6as 0.09 for the first

symmetric out-of-plane mode of the main cable, and 0.03 and 0.055 for the firstasymmetric and first symmetric in-plane modes, respectively, of the secondary cable.

Although the different values of loss factor are identified for different modes of the

secondary cable, this result is acceptable in the sense that the difference is not so large

and might be caused by some other experimental damping sources. Therefore the

lower value 0.03 is selected as the loss factor of the secondary cable.

By using thus evaluated loss factors of main and secondary cables (!m

= 0.09 ,

!s = 0.03), the damping ratios of the first and third system-modes are calculated by

Equations (8)and (9). The analytical results are presented with experimental data in

Figure 7.It can be seen from the figure that the analytical curve of damping ratio for

the system mode-1 is in very good agreement with the experimental damping values.

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In the case of the system mode-3, however, the modal damping ratio is evaluated very

largely in the analysis, while the analytical result explains qualitatively very well the

damping characteristics of the mode-3 such as the amplitude-dependency. A possible

reason of having this quantitative discrepancy might be the over estimation of the

secondary cable's response amplitude which is evaluated from the analyticallyevaluated mode of the undamped system.

As it is understood from the previous discussions, the energy distribution among

substructures in a system mode, which is different for different modes, is the main

factor in determining the system modal damping ratio. Figure 8(a)shows analytically

calculated sub-structural energy distributions in several important system-modespreviously given in Figure 4. Figure 8(b) depicts corresponding modal damping

ratios as the summation of main and secondary cable contributions evaluated

analytically by the energy-based theory. Each bar plot in both figures, corresponding

to the specified system mode, consists of main-cable contribution and secondary-

cable contribution. The horizontal dotted line is also given for comparison in each barof Figure 8(b), indicating analytically evaluated modal damping ratio of single main

cable for the sub-structural mode whose contribution is dominant in the respective

system mode. It should be noted that the modal damping ratio of single main cable

corresponding to the system mode-11 is relatively very large because the dominant

mode in the substructure of main cable is in-plane symmetric mode [7] as shown in

Figure 4. From careful observation of Figures 8(a) and(b), it becomes obvious that

the system modal damping is more or less larger than the single main cable for all the

modes and that the modal damping ratio can be increased significantly for the modes

(Mode-3 and 8) in which the secondary cable energy contributions are very large.

Even when the secondary cable energy contribution in the system mode is smaller

than that of main cable in the case of the system mode-7, the secondary cablecontribution in the system damping is greater than that of main cable. From the above

discussions it can be concluded that the increase in the modal damping ratio of the

system due to additional energy dissipation from the secondary cable is one of the

factors in control performance of the secondary cable.

SYSTEM BEHAVIOR UNDER HARMONICALLY VARYING LOAD

One of the main cables is excited by applying harmonically varying concentrated

load in the out-of-plane direction of main cable at a distance of 0.75 % of span from

the support (Figure 1). Steady state responses are computed by solving the equation

of motion in Equation (6). Assumed force amplitude is about 6% of the horizontaltension in the main cable and its frequency is set around the natural frequency

corresponding to the third out-of-plane normal mode of the single main cable; N5-

mode in Figure 2(a). Figures 9(a)and (b)show the frequency response curves of themid-span, out-of-plane response of two main cables and of the mid-span, in-plane

response of the secondary cable, respectively. The abscissa of each graph is the ratio

of excitation frequency to the natural frequency of the N5-mode of main cable; 7.43

Hz, and the ordinate is the response amplitude normalized by the span length of themain cable. Due to the addition of secondary cable, the response pattern of main cable

is changed into two-peak type as shown in Figure 9(a)where the frequency response

curve of single main cable is also depicted for comparison. One additional smaller

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peak in the frequency response curves is observed, which is also found in the

experimental investigation [13]. These peak responses, corresponding to the third and

fourth groups of system natural frequencies, are smaller than the original peak

response of single main cable, and the secondary cable response plays an important

role by becoming larger as shown in Figure 9(b). This can be considered as TMD-likephenomenon in which the secondary cable with large single mode response acts as a

TMD, as was expected.

The mechanism of controlling the main cable response, however, is somewhat

different from ordinary SDOF system with a TMD. Some modes other than the

controlled mode of single main cable, N5, are also excited having significantcontribution in the frequency region near the smaller peak, as shown in Figure 10(a)

where the response amplitudes in the generalized coordinate of sub-structural mode

are depicted for the excited main cable. The frequency response curve of secondary

cable is also different from that of TMD, while the secondary cable has almost single

mode response as shown in Figure 10(b). These differences are, of course, caused bythe existence of non-excited main cable and able to lead to better control performance.

One possibility is to use stiffer secondary cable with smaller sag in order to transfer

the energy from excited cable to non-excited cable and hence to reduce the response

of excited cable by inducing several modal responses of substructure [20]. The

differently sagged secondary cable, however, loses its multi-tuning characteristics and

cannot be effective for multi-modes of main cable. It should be noted that better

control performance can be obtained without losing the multi-tuning characteristics if

the secondary cable can have larger damping.

CONCLUDING REMARKS

A cable system consisting of two main cables with one secondary cable was

studied under the condition of identical sags by applying the modal synthesis method.

The findings from this study are summarized as follows:

(1) Corresponding to each mode of sub-structural cable, there exists a group of

system modes whose natural frequencies are closely spaced near the natural

frequency of sub-structural cable. The number of system modes in one group is

basically equal to the number of sub-structural cables, and main cable motion can be

small in one of the system modes in which two main cable motions are out-of-phase

and coupled with in-plane symmetric motion of the secondary cable.

(2) Modal damping of system modes depends on the composition of system

modes as energy contribution from various substructures. Greater energy contributionfrom secondary cable in system modes indicates greater contributions in the system

modal damping from secondary cable and greater is the system modal damping.

(3) TMD-like effect of secondary cable in reducing the main cable response is

observed and the secondary cable performs as a media of energy transfer from the

oscillating cable. The appropriate design of the damping property and the multi-

mode-tuning characteristics in the secondary cable will lead to better control

performance.

(4) The characteristics of secondary cable has significant effects on the main

cable oscillation within the linear response range, while the secondary cable is

susceptible to nonlinear response which can affect on its control performance.

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16 Craig, R. R. Jr., Bampton, M. C. C. Coupling of substructures for dynamicanalysis. AIAA Journal 1968; 6: 1313-1319.

17 Benfield, W. A., Hruda R. F. Vibration analysis of structures by component modesubstitution. AIAA Journal 1971; 7: 1255-1261.

18 Henghold, W. M., Russel, J. J. Equilibrium and natural frequencies of cablestructures (A nonlinear finite element approach). Journal of Computers and

Structures 1976; 6: 267-271.

19 Washizu, K. Variational methods in elasticity and plasticity. 2nd Edition,Pergamon Press, 1974.

20 Alauddin, Md. Vibration control and its mechanism in cable system withsecondary cable. Doctoral dissertation, 1998, Saitama University, Japan.

21 Nashif, A.D., Jones, D.I.G., Henderson, J.P. Vibration damping. Wiley, NewYork, 1985. p. 45-51.

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Table 1. Specifications of main and secondary cables

Cables Total Mass

(g)

Span

(m)

Sag

(m)

Sag ratio

(%)

Horizontal

tension (N)

EA

(kN)

Main cable 547 3.10 0.050 1.6 41.5 353

Secondary cable 11.5 0.60 0.050 8.3 0.169 103

Engineering Structures

Yamaguchi, H., Alauddin, Md. and Poovarodom, N.

Table 1

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Captions of figures

Figure 1 Schematic diagram of the model of cable systemFigure 2 Sub-structural modes for main and secondary cablesFigure 3 Natural frequencies of cable system with sub-structural natural

frequencies

Figure 4 System normal modes and their composition of generalized coordinates.i: modal contribution in main cables, ii: that in secondary cable

Figure 5 Experimentally obtained modal damping ratios for some of the normalmodes of single main cable and cable system

Figure 6 Experimental and analytical (solid line) modal damping ratios for singlemain and secondary cables for different modes

Figure 7 Experimental and analytical (solid line) modal damping ratios for cablesystem in the first and third system modes

Figure 8 Analytically evaluated strain energy and damping ratio of importantsystem mode as summation of sub-structural contributions

Figure 9 Frequency response curves for main cables and secondary cable underharmonic and lateral excitation to one of the main cables

Figure 10 Response amplitude in generalized coordinates for excited main cableand secondary cable under harmonic excitation

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Figure 1

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Figure 2

(a) Main cables (C1-C2 constraint modes, N1-N6 normal modes)

(b) Secondary cable (C1-C4 constraint modes, N1-N6 normal modes)

N3 (out-of-plane) N4 (in-plane) N5 (out-of-plane) N6 (in-plane)

f=9.52Hz

f=7.43Hz

f=6.71Hz

f=4.95 Hz

N5 (out-of-plane) N6 (in-plane)

f=7.40Hz

f=9.82Hz

N2 (in-plane) N3 (out-of-plane) N4 (in-plane)

f=7.04Hz

f=4.94 Hzf=4.81 Hz

C1 (in-plane) C2 (in-plane) C3 (in-plane) C4 (in-plane) N1 (out-of-plane)

f=2.48 Hz

C1 (in-plane) C2 (out-of-plane) N1 (out-of-plane) N2 (in-plane)

f=2.48

f=4.94 Hz

X-displacement Y-displacement Z-displacement

X-displacement Y-displacement Z-displacement

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Figure 3

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(a) The first group of system-modes with the first symmetric, out-of-plane

substructural mode



Figure 4

-1

0

1

i

-1

0

1

C N1 N2 N3 N4 N5 N6

Modalcontribution

Substructural modes

i i

Mode-1, f=2.485 Hz

i

C N1 N2 N3 N4 N5 N6

Substructural modes

i i

Mode-2, f=2.490 Hz

i

C N1 N2 N3 N4 N5 N6

Substructural modes

i i

Mode-3, f=2.520 Hz

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(b) The second group of system-modes with the first asymmetric, in-

plane and out-of-plane substructural modes



Figure 4

-1

0

1i

C N1 N2 N3 N4 N5 N6

-1

0

1

Substructural modes

i i

Modalcontribution

Mode-4, f=4.775 Hz

i

C N1 N2 N3 N4 N5N6Substructural modes

i i

Mode-5, f=4.934 Hz

i

C N1 N2N3N4 N5 N6Substructural modes

i i

Mode-6, f=4.949 Hz

-1

0

1i

C N1 N2 N3 N4 N5 N6-1

0

1

Substructural modes

i i

Modalcontribution

Mode-7, f=5.002 Hz

i

C N1 N2 N3 N4 N5 N6

Substructural modes

i i

Mode-8, f=5.007 Hz

i

C N1 N2 N3 N4 N5 N6

Substructural modes

i i

Mode-9, f=5.064 Hz

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(c) The third group of system-modes with the first symmetric, in-

plane substructural-mode



Figure 4

-1

0

1i

C N1 N2 N3 N4 N5 N6-1

0

1

Substructural modes

i i

Modalcontribution

Mode-10, f=6.702 Hz

i

C N1 N2 N3 N4 N5 N6

Substructural modes

i i

Mode-11, f=6.805 Hz

i

C N1 N2 N3 N4 N5 N6

Substructural modes

i i

Mode-12, f=6.861 Hz

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(d) The fourth group of system-modes with the second symmetric,out-of-plane substructural mode



Figure 4

-1

0

1

i

-1

0

1

C N1 N2 N3 N4 N5 N6

Mod

alcontribution

Substructural modes

i i

Mode-13, f=7.418 Hz

i

C N1 N2 N3 N4 N5 N6

Substructural modes

i i

Mode-14, f=7.551 Hz

i

C N1 N2 N3 N4 N5N6

Substructural modes

i i

Mode-15, f=7.566 Hz

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Figure 5

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25

Dampingratio(%)

Amplitude to span ratio (%)

single main cable, 1st out-of-plane mode

cable system, 3rd mode

cable system, 1st mode

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Figure 6

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Figure 6

23


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Figure 7

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Figure 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 3 7 8 11 15

Secondary cable Main cable

Modalstrainenergy(N-m

m)

System mode number

(a) Sub-structural contribution to modal strain energy

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 3 7 8 11 15

Secondary cableMain cable

Modaldampingr

atio(%)

System mode number

Single main cable

(b) Sub-structural contribution to modal damping ratio

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Figure 9

0

0.001

0.002

0.003

0.8 0.9 1 1.1 1.2

Excited cable

Non-excited cableSingle main cable

Normalizedresponseamplitude

Excitation frequency ratio

(a) Out-of-plane response of main cable at mid-span

0

0.002

0.004

0.006

0.008

0.010

0.012

0.8 0.9 1 1.1 1.2

Normalizedresponseamplitude


(b) In-plane response of secondary cable at mid-span

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Figure 10

0

1

2

3

4

5

6

7

0.8 0.9 1 1.1 1.2

C1N2

N3N4N5N6

Responseamplitudeinmodalc

oordinate


(a) Response of main cable

0

5

10

15

20

25

30

35

0.8 0.9 1 1.1 1.2

C1C2C3C4

N1N2N3N4N5N6

Responseamplitudeinmodalcoordinate


(b) Response of secondary cable

dynamic characteristics and vibration control of a cable system with sub-structural interactions

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