Modal analysis of cabletower system of twinspan suspension bridge
Libin Wang^{1} , Xiaoyi Guo^{2} , Mohammad Noori^{3} , Jie Hua^{4}
^{1, 2, 4}School of Civil Engineering, Nanjing Forestry University, Nanjing, Jiangsu, 210037, China
^{3}California Polytechnic State University, Mechanical Engineering Department, San Luis Obispo, CA, 93407, USA
^{1}Corresponding author
Journal of Vibroengineering, Vol. 16, Issue 4, 2014, p. 19771991.
Received 8 March 2014; received in revised form 1 May 2014; accepted 7 May 2014; published 30 June 2014
JVE Conferences
A threedimensional finite element model is developed in order to analyze the free vibration characteristics of the towercable system of a tripletower twinspan suspension bridge during the construction phase and right after the erection of the main cable. The dynamic characteristics of each component in the towercable system, the isolated side span, main span cables and freestanding towers, are first analyzed separately. The natural frequencies and the vibration modes of the isolated side span and the main span cables obtained from the finite element analysis closely matched the analytical solution from the linear free vibration analysis and verified the validity of the finite element model. The local natural frequency and global natural frequency were defined to categorize the characteristics of the free vibration. The calculation results show that not only does the towercable system maintains the information on the modal characteristics of each component in the system, but it also contains its own unique modal characteristics and other important information about the dynamics characteristics of the system. At lower natural frequency range, the swinging motion and inplane motion are uncoupled. The coupled modal information of the towers and the cables are separated into two groups based on the natural frequencies of the vibration of the main component of the towercable system. Some additional natural frequencies and vibration modes are obtained from the finite element analysis depicting the dynamic interaction between the towers and the cables. Furthermore, it is observed that the lower order modes of side span cables couple with the higher order modes of the main span cables. Either in phase or out of phase, local or global modes, the towercable system exhibits many new coupled mode combinations that reveal useful information.
Keywords: towercable system, modal analysis, local natural frequency, global natural frequency, combined modes.
1. Introduction
In recent years, in order to meet the increasing demand for crossing long distances over water, a new era in building super long span bridges, namely multitower suspension bridges, has begun. In China, the first two triple tower suspension bridges, the Taizhou Yangtze River bridge and Ma’anshan Yangtze River bridge, with twin span of 1008 meter each, have just been completed and the first triple tower four span suspension bridge, the Yingwuzhou bridge, is currently under construction. Moreover, there are plans under way for building a multitower suspension bridge connecting Taiwan and the mainland China. The New Millennium Bridge in Korea, and the Chacao Channel Bridge in Chile, which will be finished in 2018 and 2019 on schedule respectively, are also triple tower suspension bridges.
In 1983, a comprehensive study was conducted proposing why a two adjacent suspension bridge shared with one common anchor should be adopted instead of a fourspan suspension bridge, as the final alternative for the San FranciscoOakland Bay Bridge. Niels J. Gimsing studied the stiffness characteristic of the multispan suspension bridge, furthermore, he presented four additional types of nontraditional multispan suspension bridges. His work was a comprehensive pioneering study on multisuspension bridges [1]. In 1991, T. Y. Lin presented an innovative concept for bridging Gibraltar Strait, which is also a combined multispan suspension bridge [2]. In 2001, Torben Forsberg discussed some of the specific technical aspects of multispan suspension bridges including his own experience in the predesign of Gibraltar Strait Crossing and Gibraltar Strait link [3]. In 2004, Osamu Yoshida investigated the deformation characteristics of a fourspan suspension bridge with two main 2000 m spans and pointed out that live load deflection of the girder can be reduced to less than 1/200 of the main span length by stiffening the bending coefficient of the center tower. Based on his comprehensive research, he concluded that the fourspan suspension bridge is the best, most economical solution when a deep and extra longspan water crossing is required [4].
Since only a limited number of suspension bridges with a double span longer than 1000 meter have so far been constructed and utilized, most investigations on suspension bridges reported in the literature have been limited in scope. To the best of the authors’ knowledge a comprehensive analysis considering all aspects, including the dynamic characteristics, of multispan bridges had been lacking until Taizhou Yangtze River bridge, Ma’anshan Yangtze River bridge and New Millennium Bridge began to be designed and erected [510]. Moreover, no research has been reported in literature investigating the wind resistance of twin span suspension bridges during the erection period [1115]. Investigating the dynamic performance of multispan suspension bridges during the erection process is critically important due to the fact that many of these bridges will be built in straits frequently subjected to violent typhoon or hurricane and wind or earthquake during the construction period. Therefore, assessment of the bridge resistance to wind load during construction is more favorable than post construction state.
In this paper, the modal characteristics of towercable system of the twin span Ma’anshan Yangtze River bridge during the erection is studied. Vibration frequencies of the side span and the main span are compared with the results from finite element analysis to verify the accuracy of the FEM model. Subsequently, local and global frequencies are defined for convenience and to clearly describe the uncoupling of the free vibration, both in plane and out of plane. Based on these definitions the modal analysis of the single main span cable and single tower are carried out respectively in order to identify the vibration characteristics of each component in the tower cable system separately. Finally, the vibration characteristics of the tower cable system and its relationship with each single component are studied, and the unique desired modal characteristics for multitower cable system are clarified. This work lays down the foundation for further study on the wind resistance of multispan bridges.
2. The finite element model
2.1. Construction details of the twinspan suspension bridge
Ma’anshan Bridge is a bridge over the Yangtze River in Ma’anshan, Anhui, China. It ties with Taizhou Bridge as the longest double span suspension bridge with a span arrangement of 360+2×1080+360 m as shown in Fig. 1.
Fig. 1. General layout of Ma’anshan Yangtze River bridge (unit: m)
Two main cables of 35 m apart are made of galvanized hightensile steel wires with a diameter of 5.2 mm and an ultimate tensile strength of 1670 MPa by prefabricated parallelwire strands (PPWS). Each main cable consists of 154 strands and each strand weight 49T with 91 wires built into it. The outer diameter is approximately 0.688 m, and, the volumeweight and elasticity modulus of the main cable are 78.5 kN/m^{3} and 200 kN/mm^{2} respectively.
The double hanger rope is also made of hightensile galvanized parallel wire bundles. Each hanger is composed of 109 parallel steel wires with a diameter of 5.0 mm and an ultimate tensile strength of 1670 MPa. The spacing between two adjacent hangers is 16.0 m while the separation distance from tower center to the nearest hanger is 20.0 m.
The midtower and two side towers are portaltype frames with two crossbeams installed between two tower legs. As Fig. 2 shows, the height of the tower is 178.8 m measured from the top surface of the pile cap to the pylon top. Besides 10.5 m decoration segment part at the top, the 127.8 m steel upper part from the bottom of the saddle downwards of the midtower is connected to the 40.5 m lower concrete part by the steelconcrete connection joint segments just beneath the bottom steel crossbeam. The centrelines of the tower legs are 35 m apart at the top tower increasing to 43.5 m at the pile cap. The width of the leg is constant at 6 m in the tower plane, but in the plane of bridge tapers from 11 m at top steelconcrete joint section to 7 m at the top tower. For the prestressed reinforced concrete bottom tower, the widths reduces from 12 m in tower plane and 25 m in the bridge plane at the pile cap to 9.2 m and 17 m at the joint section , and, the corresponding thicknesses are 1.6 m and 2.0 m of the box section respectively, while the joint is a solid block. As for the side towers, the width of the legs are both constant at 6 m in tower plane and at 8 m in the plane of the bridge. Other dimensions are nearly the same as those of the midtower.
Fig. 2. The principal dimension of midtower
Fig. 3. FEM of cabletower system of twin span suspension bridge
2.2. Finite element modelling
A threedimensional finite element model is established for Ma’anshan suspension bridge using MIDAS Civil (2012) software. Subsequently, the supports, girders, and the hangers are dismantled following the reverse process of the construction in order to obtain the FE model of the towercable system as shown in Fig. 3. In this model, three dimensional Timoshenko beam elements are used to model the thick crossbeams and legs fixed at the base of the tower. The stiffening girders are modeled with beam elements passing through the centroids of the girder sections. The anchor points of the hangers on the stiffening girders are connected to the central beam by imaginary rigid beams. All hangers and main cables are modeled with truss elements with Ernst’s equivalent elastic modulus. The model has 913 nodes and 1177 elements. The modulus of elasticity of the upper midtower and side towers are 210 kN/mm^{2} and 34.5 kN/mm^{2} respectively. The Poisson ratio of midtower is 0.3 and that of the side towers is 0.2. The volumeweight of the upper midtower and the side towers are 76.98 kN/m^{3} and 25 kN/m^{3} respectively [13, 16].
The towercable system described above is slightly modified with the consideration of the actual construction conditions associated with the completion of the erection process. For instance, the masses of the working cranes, cable saddles and platforms connected to the top of the towers are added to the corresponding nodes as additional lumped masses. The results of the static analysis of the towercable system show that the midspan sag of the main span is 111.663 m which is 8.337 m smaller than that of the post construction state.
Based on this model, an extensive numerical analysis of the modal characteristics of the isolated cable, free standing tower and towercable coupling system are carried out and presented in the following sections.
3. Isolated cables and freestanding towers
3.1. Isolated main span cable
Due to the symmetry of the structure, only the isolated left main span cable is studied in this section. The first fifteen natural frequencies for inplane and outplane vibration of the isolated suspended cable based on IrvineCaughey method [17] and FEA software respectively are tabulated in Table 1,and the mode shapes are shown in Fig. 4. As the results show, the inplane and outofplane vibration modes are uncoupled. The relative difference of natural frequency between the two methods is less than 3.48 %. As can be observed from Fig. 4, the vertical component of the first inplane vibration mode of the isolated left main span cable is antisymmetric single wave while the first mode of the outofplane vibration is symmetric halfwave, and the first natural frequency of inplane vibration of the cable is almost twice the first natural frequency of outofplane vibration. These results are consistent with IrvineCaughey free vibration theory [17]. The consistency between the two results above indicates the accuracy of the FE model.
Fig. 4. First mode shapes of isolated main span cable
a) The first inplane mode
b) The first outofplane mode
Table 1. Natural frequencies of left isolated main cable
Mode no.

Natural frequency (Hz)


Inplane vibration

Outofplane vibration


FEM

Irvine

Difference (%)

FEM

Irvine

Difference (%)


1

0.1075

0.1054

–1.95

0.0524

0.0527

0.52

2

0.1520

0.1502

–1.17

0.1042

0.1054

1.17

3

0.2138

0.2109

–1.38

0.1561

0.1582

1.33

4

0.2610

0.2582

–1.08

0.2079

0.2109

1.43

5

0.3199

0.3163

–1.11

0.2595

0.2636

1.54

6

0.3670

0.3639

–0.86

0.3111

0.3163

1.64

7

0.4252

0.4217

–0.82

0.3624

0.3690

1.79

8

0.4700

0.4673

–0.58

0.4137

0.4217

1.91

9

0.5297

0.5272

–0.48

0.4644

0.4745

2.11

10

0.5574

0.5539

–0.64

0.5154

0.5272

2.24

11

0.6016

0.5980

–0.60

0.5654

0.5799

2.50

12

0.6330

0.6326

–0.06

0.6159

0.6326

2.64

13

0.6874

0.6889

0.22

0.6651

0.6853

2.96

14

0.7350

0.7380

0.42

0.7151

0.7380

3.11

15

0.7860

0.7923

0.80

0.7632

0.7908

3.48

3.2. Isolated side span cable
Similarly, the first ten natural frequencies for inplane and outplane vibration of isolated side span cable are tabulated in Table 2, and, the mode shapes are shown in Fig. 5. As can be observed from Fig. 5, the vertical component of the first inplane vibration mode of isolated left side span cable is antisymmetric single wave, while the first mode of the outofplane vibration is symmetric halfwave.
The $i$th natural frequencies of the isolated side span cable are always larger than either inplane or outofplane natural frequencies of the isolated main span cable. The lower natural frequencies of the isolated side span cable are closed to the higher natural frequencies of the isolated main span cable. Therefore, it can be predicted that, in the towercable coupling system, some internal resonances exist between the higher vibration modes of the main span cables and the lower vibration modes of the side span cables.
For instance, because the first inplane natural frequency, 0.3270 Hz, of the isolated side span cable is between the fifth and the sixth inplane natural frequency of the isolated main span cable, it will excite the vibration of the side span cable during the fifth vibration mode of main span cable and turns that to the sixth vibration mode. However, the first five vibration modes of the main span cable cannot cause free vibration of the side span cable.
Fig. 5. First mode shapes of isolated side span cable
a) The first inplane mode
b) The first outofplane mode
Table 2. Natural frequencies of left isolated side cable
Mode no.

Natural frequency (Hz)


Inplane vibration

Outofplane vibration


1

0.3270

0.1513

2

0.4321

0.3014

3

0.5856

0.4497

4

0.6459

0.5953

5

0.8098

0.7372

6

0.9568

0.9140

7

1.0945

1.0067

8

1.3650

1.2561

9

1.6584

1.5133

10

2.0039

1.7904

3.3. Freestanding towers
Due to the symmetry of the left and the right side tower, only the left side and the middle tower are studied herein. The first nine natural frequencies of the left side tower and the first six natural frequencies of the midtower, including the bending mode, torsional mode and sway mode, are listed in Table 3, and, the first bending, torsional and sway modes of the left side tower are shown respectively in Fig. 6.
From Tables 13, we can observe that the first sway frequency of the left side tower, 0.2589 Hz, is between the first and the second natural frequency, 0.1513 Hz and 0.3014 Hz, of the outofplane vibration of the side span cable, is close to the fifth natural frequency 0.2595 Hz of the outofplane vibration of the main span cable. The first sway frequency of midtower, 0.6052 Hz, is close to the twelfth natural frequency of the outofplane vibration of the main span cable and the fourth natural frequency, 0.5953 Hz, of the outofplane vibration of the side span cable. Hence, it can be predicted that the first sway mode of towers appears along with the higher outofplane vibration mode of the main span cables in the towercable system. Due to the dynamic interaction between the towers and the cables in the coupled system, the inplane modal properties of the cables and the towers will be changed.
Fig. 6. First mode shapes of freestanding left side tower
a) The first bending mode
b) The first sway mode
c) The first torsion mode
Table 3. Natural frequencies of freestanding towers
Mode no.

Left side tower

Midtower


Frequencies (Hz)

Mode type

Frequencies (Hz)

Mode type


1

0.1787

Bending

0.5406

Bending

2

0.2589

Sway

0.6052

Sway

3

0.6219

Torsion

1.0855

Torsion

4

1.2822

Bending

2.8354

Bending

5

1.4837

Sway

3.1002

Sway

6

1.6706

Torsion

3.7919

Torsion

7

3.3091

Sway


8

3.4790

Bending


9

3.9517

Torsion

4. Towercable system
A towercable system is the system when the cable erection is just finished but no girder segments are yet erected. The modal analysis shows that the swinging motion and the inplane motion are uncoupled at a lower natural frequency range. In order to describe the modal characteristics and the dynamic interaction of the system, the inplane vibration modes of the system are divided into the local and the global inplane vibration modes. The local modes are further subdivided into three types that consist of the vibration of the cables only, the vibration of the cables and the side towers, and the vibration of the cables together with the midtower only. If all three towers vibrate together with two cables, that is defined as the global inplane modes of vibration.
4.1. Local inplane vibration
4.1.1. Local inplane vibration of cables
The natural frequencies of local inplane vibration modes for main span cables and side span cables in towercable system are listed in Table 4 and Table 5 respectively. Table 4 shows the first thirtythree natural frequencies of vibration modes. The fourth and eighth columns are the abbreviated description for the vibration modes of cables. For the purpose of better clarification two capital letters are defined to describe the modal combination of cables. The first letter stands for the Symmetric mode, with S, and Antisymmetric mode, with A, for any single span. The second explains the relationship between any two spans with A and C to denote Axialsymmetry and Centralsymmetry modes about the midtower respectively.
Table 4. Natural frequencies of local inplane modes of vibration of main span cables
Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

1

0.1075

in phase

AA

18

0.4252

out of phase

AC

2

0.1075

out of phase

AA

19

0.5297

in phase

AA

3

0.1075

in phase

AC

20

0.5297

out of phase

AA

4

0.1075

out of phase

AC

21

0.5297

in phase

AC

5

0.1517

out of phase

SA

22

0.5620

in phase

SA

6

0.2138

in phase

AA

23

0.5625

out of phase

SA

7

0.2138

out of phase

AA

24

0.5962

in phase

SA

8

0.2138

in phase

AC

25

0.5966

out of phase

SA

9

0.2138

out of phase

AC

26

0.6330

in phase

AA

10

0.3198

out of phase

AA

27

0.6330

out of phase

AA

11

0.3198

in phase

AA

28

0.6856

in phase

SA

12

0.3199

out of phase

AC

29

0.6861

out of phase

SA

13

0.3199

in phase

AC

30

0.7340

in phase

AA

14

0.3682

in phase

SA

31

0.7354

out of phase

AA

15

0.4252

in phase

AA

32

0.7386

in phase

AA

16

0.4252

out of phase

AA

33

0.7854

in phase

SA

17

0.4252

in phase

AC

Comparing the data of Table 4 with the natural frequencies of isolated main span cables in Table 1, it is evident that Table 4 almost includes all the natural frequencies in Table 1. It indicates that the local inplane vibration modes of towercable system still holds the modal characteristics of each component of towercable coupled system, but it also has new unique modal characteristics different from the isolated single cable. For example, all the first four natural frequencies in Table 4 are 0.1075 Hz, and the corresponding mode shapes of single main span cables are still single wave as the Fig. 7 shows. There are four types of modal combination, two of the modes are single span Antisymmetry with double span Axialsymmetry (AA) inphase mode and outofphase mode of vibrations, the other two are single span Antisymmetry with double span Centralsymmetry (AC) of inphase and outofphase modes. Thus, the frequencies of towercable system in Table 4 can be divided into some groups according to frequencies of isolated cables in Table 1. Such as the first four natural frequencies in Table 4 according to the first inplane natural frequency 0.1075 Hz in Table 1, the fifth natural frequency in Table 4 according to the second natural frequency 0.1520 Hz in Table 1 and the sixth to ninth natural frequency in Table 4 according to the third natural frequency 0.2138 Hz in Table 1 and so on. Similarly, from Table 5 and Table 1, the natural frequencies of local inplane vibration of side span cables are divided into groups.
There are several coupling resonances between higher order modes of main span and the modes of side span according to Table 4 and Table 5. For instance, the first local inplane vibration in which the corresponding natural frequency is 0.3198 Hz of side span cables excites the tenth local inplane vibration of main span cables. However, as Fig. 8 shows, the fifteenth vibration mode of main span cable is single span Symmetric mode in which the additional tension of cable is zero which means it doesn’t excite the vibration of side span cable, thus, there is no natural frequency in Table 5 that matches that. Similarly, the natural frequency for the third vibration mode of side span cables in Table 1 does not have a corresponding frequency in Table 4.
Table 5. Natural frequencies of local inplane modes of vibration of side span cables
Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

1

0.3198

out of phase

SA

10

0.5966

out of phase

SA

2

0.3270

out of phase

AC

11

0.6458

in phase

AC

3

0.3271

in phase

AA

12

0.6458

in phase

AA

4

0.3271

in phase

AC

13

0.6856

in phase

SA

5

0.3276

out of phase

AA

14

0.6861

out of phase

SA

6

0.3682

in phase

SA

15

0.7340

in phase

SA

7

0.5620

in phase

SA

16

0.7354

out of phase

SA

8

0.5625

out of phase

SA

17

0.7386

in phase

SA

9

0.5962

in phase

SA

18

0.7854

in phase

SA

It is interesting that, along with the local inplane vibration, all modal combinations of side span cables are single span Symmetry with double spans Axialsymmetry (SA) in Table 5. It can also be observed from Fig. 7 and Fig. 8 that the towers generate some relatively small motion when the local natural frequencies of cables are close to the local frequencies of towers or the global frequencies of the system.
Fig. 7. The first four local inplane mode shapes of main span cables
a) In phase (AA)
b) Out of phase (AA)
c) In phase (AC)
d) Out of phase (AC)
Fig. 8. The local inplane mode shapes of main span cables and side span cables
a) The fifteenth modal motion of main span cables
b) The third modal motion of side span cables
4.1.2. Local inplane vibration of two side towers
The natural frequencies of local inplane vibration of two side towers of towercable system are listed in Table 6. Affected by the main cables in the local inplane vibration of towercable system, some new natural frequencies of bending mode and torsional mode of side tower are generated as compared with Table 3, and, the lower order frequencies are closely spaced. The result also shows that all the modal properties of bending mode and torsional mode of two side towers are outofphase in the local inplane vibration mode of side towers, and, when bending is the dominant mode of two side towers, all the modal properties of two cables are inphase, while when torsion is the dominant mode of two side towers all the modal properties of two cables are outofphase.
Table 6. Natural frequencies of local inplane modes of vibration of two side towers
Mode no.

Bending mode (out of phase)

Torsion mode (out of phase)


Natural frequency (Hz)

Two cables

Natural frequency (Hz)

Two cables


Modal property

Modal combination

Modal property

Modal combination


1

0.1465

in phase

SA

0.2595

out of phase

SA

2

0.1767

in phase

SA

0.3207

out of phase

SA

3

0.2631

in phase

SA

0.3693

out of phase

SA

4

0.4626

in phase

SA

0.4654

out of phase

SA

5

0.4814

in phase

SA

0.4900

out of phase

SA

It is interesting that, for either the bending mode or the torsional mode, all the modal combinations of two cables are single span Symmetry with double span Axialsymmetry (SA). The second bending and torsional mode of side towers in local inplane vibration are shown in Fig. 9. It can be seen that the second bending frequency 0.1767 Hz of side towers excites the second vibration mode of main span cables and the halfwave mode of side span cable. The second torsional frequency 0.3207 Hz of side towers excites the fifth vibration mode of main span cables and the first mode of side span cable. Therefore, it can be concluded that, in the local inplane vibration of tripletower twinspan towercable system, the larger out of phase bending vibration mode of two side towers will excite the main cable as an inphase modal combination which is single span Symmetry with double span Axialsymmetry (SA). While the larger out of phase torsional vibration of the side towers will excite a similar modal combination but outofphase.
Fig. 9. The second local inplane mode shapes of two side towers
a) The second bending mode (in phase) (SA)
b) The second torsional mode (out of phase) (SA)
4.1.3. Local inplane vibration of midtower
Table 7 lists the natural frequencies of local inplane vibration modes for the midtower. Similarly, compared with table 3, some new natural frequencies of bending mode and torsional mode of side tower are generated and are closely spaced. In the local inplane vibration modes of midtower, for the bending mode of midtower all the modal properties of two cables are inphase, while for the torsional mode of midtower the modal properties of two cables are outofphase.
It is obvious that either for the bending modes or the torsional modes, all the modal combinations of two cables are single span Symmetry with double span Centralsymmetry (SC) and the side span cables have no motion or only slight motion. Fig. 10 shows the first bending and torsional mode of the local inplane vibration mode of midtower. The first bending frequency 0.3735 Hz excites the sixth vibration mode of main span cables, and, the first torsional frequency 0.1499 Hz excites the second vibration mode of main span cables while the side span cables experience almost no motion at all. Hence, it can be concluded that, in the local inplane vibration of threetower and twospan towercable system, the larger bending vibration mode of midtower will excite an inphase modal combination which is single span Symmetry with double span Centralsymmetry (SC) of main cables. While the larger torsional vibration will excite a similar modal combination but outofphase mode of the main cable.
Table 7. Natural frequencies of local inplane modes of vibration of midtower
Mode no.

Bending mode

Torsion mode


Natural frequency (Hz)

Two cables

Natural frequency (Hz)

Two cables


Modal property

Modal combination

Modal property

Modal combination


1

0.3735

in phase

SC

0.1499

out of phase

SC

2

0.4777

in phase

SC

0.5820

out of phase

SC

3

0.5813

in phase

SC

0.6838

out of phase

SC

Fig. 10. The first local inplane mode shapes of midtower
a) The first bending mode (in phase) (SC)
b) The first torsional mode (out of phase) (SC)
4.2. Global inplane vibration
The natural frequencies of global inplane vibrations are listed in Table 8. It is still clear that the lower order natural frequencies present are closely spaced. Both the bending and torsional modes of two side towers are in phase. For the bending mode the modes of cable are in phase while for torsional mode of the side tower the modes of cable are out of phase. All the modal combinations of the twin cables are single span Symmetry with double span Centralsymmetry (SC). In contrast to the local inplane vibration mode of midtower, the vibration of side span cables are excited in the global inplane vibration of system due to the vibration of side towers.
The first bending and torsional mode of global inplane vibration are shown in Fig. 11. The first bending frequency of the towercable system 0.1297 Hz is a new outcome just between the first and the second natural frequency of isolated main span cable, and, smaller than the first natural frequency of isolated side span cable. The fourth vibration mode of main span cables and the halfwave mode of side span cables are excited by the first torsional frequency, 0.2497 Hz, close to the fourth natural frequency of isolated main span cable. Fig. 12 shows the second bending and torsional mode of global inplane vibration respectively. Similarly, the second vibration mode shape of main span cables and the halfwave mode shape of side span cables are excited by the second bending frequency of 0.1696 Hz. While the second torsional vibration mode excites the fifth vibration mode of the main span cables and the similar mode of the side span cables. The new frequency of the towercable system is the result of coupling effect of the components with each other. From Table 8, it can be observed that in global inplane vibration of twinspan towercable system, large bending or torsional mode with two side towers in phase, will excite the vibrations in phase mode of two cables with the single span Symmetry and with double spans Centralsymmetry (SC). As the torsional mode of the side tower is concerned, the same SC modes are excited but out of phase only. That is much more relevant to the local inplane vibration of towers.
Table 8. Natural frequencies of global inplane modes of vibration of cabletower system
Mode no.

Bending mode (in phase)

Torsion mode (in phase)


Natural frequency (Hz)

Two cables

Natural frequency (Hz)

Two cables


Modal property

Modal combination

Modal property

Modal combination


1

0.1297

in phase

SC

0.2497

out of phase

SC

2

0.1696

in phase

SC

0.2940

out of phase

SC

3

0.2294

in phase

SC

0.3625

out of phase

SC

4

0.2714

in phase

SC

0.3863

out of phase

SC

5

0.4901

in phase

SC

0.4791

out of phase

SC

6

0.5111

out of phase

SC

Fig. 11. The first bending mode and torsion mode of global inplane vibration of cabletower system
a) In phase (SC)
b) Out of phase (SC)
Fig. 12. The second bending mode and torsion mode of global inplane vibration of cabletower system
a) In phase (SC)
b) Out of phase (SC)
4.3. Local outofplane vibration
Similar to the inplane vibration, the outofplane vibration of threetower and twospan towercable system is divided into the local and global outofplane vibration. The local outofplane vibration is subdivided into the local outofplane vibration of cables, side towers and the midtower.
4.3.1. Local outofplane vibration of cables
The local outofplane vibration frequencies of the system are listed in Table 9 and Table 10. The natural frequencies of the local outofplane vibration of cables can be separated into groups according to the frequency of the isolated cables. For example, the first four natural frequencies of local outofplane vibrations of cables of towercable system can be classified into one group corresponding to the first natural frequency of outofplane vibration of isolated main span cables in Table 1. The first four vibration modes are the combinations of the modes of the two main span cables. It also indicates that, because the interaction of outofplane vibrations between cables and towers is not as strong as in the inplane vibrations, thus, the outofplane vibrations of main cables still completely maintain the characteristics of isolated cables and it is combined with the unique characteristics of twinspan towercable system.
Table 9. Natural frequencies of local outofplane modes of vibration of main span cables
Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

1

0.0522

in phase

SA

17

0.3043

in phase

AA

2

0.0524

in phase

SC

18

0.3044

in phase

AC

3

0.0524

out of phase

SA

19

0.3111

out of phase

AA

4

0.0524

out of phase

SC

20

0.3111

out of phase

AC

5

0.1038

in phase

AA

21

0.3618

in phase

SA

6

0.1040

in phase

AC

22

0.3624

out of phase

SA

7

0.1042

out of phase

AA

23

0.3624

out of phase

SC

8

0.1042

out of phase

AC

24

0.3629

in phase

SC

9

0.1554

in phase

SA

25

0.4136

out of phase

AA

10

0.1558

in phase

SC

26

0.4137

out of phase

AC

11

0.1560

out of phase

SA

27

0.4140

in phase

AC

12

0.1560

out of phase

SC

28

0.4644

out of phase

SA

13

0.2078

out of phase

AA

29

0.4647

in phase

SC

14

0.2079

out of phase

AC

30

0.5153

out of phase

AA

15

0.2595

out of phase

SA

31

0.7298

in phase

AA

16

0.2595

out of phase

SC

32

0.7632

in phase

SA

Nevertheless, due to the interactions of system some new natural frequencies of local outofplane vibrations are generated such as the first six natural frequencies of outofplane of side span cables in Table 10. It must noted that, due to the growing interactions with the increasing frequencies, the separation of frequencies in groups will be not apparent, as can be seen, for instance, from the 31st, 32nd frequency etc. in Table 9.
Table 10. Natural frequencies of local outofplane modes of vibration of side span cables
Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

Mode no.

Natural frequency
(Hz)

Modal property

Modal combination

1

0.1501

in phase

SA

8

0.3044

in phase

AC

2

0.1501

in phase

SC

9

0.4497

out of phase

SA

3

0.1512

out of phase

SA

10

0.4506

in phase

SC

4

0.1512

out of phase

SC

11

0.4506

in phase

SC

5

0.1554

in phase

SA

12

0.7298

in phase

AA

6

0.1558

in phase

SC

13

0.7374

in phase

SA

7

0.3043

in phase

AA

4.3.2. Local outofplane vibration of two side towers
The natural frequencies of local outofplane vibration of side towers in Table 11 also appear in pairs. The sway modes of two side towers in one pair are in phase and out of phase respectively, while the modes of two cables are both in phase. For the Axialsymmetry mode of the two main cables, the mode of the side tower is in phase, while for the Centralsymmetry mode of the main cables, the mode of side tower is out of phase.
From Table 3 and Table 11, we know the third and fourth natural frequencies are almost equal to the first sway frequency of the isolated cable in Table 3, while the first two and the 5th frequencies are the newly generated frequencies. This indicates that the two side towers have the modal properties of freestanding towers and generate adjacent natural frequencies due to the influence of vibration of two cables in the local outofplane vibration.
Table 11. Natural frequencies of local outofplane modes of vibration of two side towers
Mode no.

Natural frequency (Hz)

Two cables

Sway mode property of side towers


Mode property

Mode combination


1

0.2065

in phase

AA

in phase

2

0.2070

in phase

AC

out of phase

3

0.2512

in phase

SA

in phase

4

0.2515

in phase

SC

out of phase

5

0.3125

in phase

AC

out of phase

4.3.3. Local outofplane vibration of midtower
The natural frequencies of local outofplane of midtower present are not so closely spaced as the side tower are, as shown in Table 12. Affected by the vibration of cables in towercable system, the midtower has new natural frequencies near the first sway natural frequency 0.6052 Hz of freestanding midtower in Table 3. The mode of main cables in the local outofplane vibrations are all in phase and Axialsymmetry about the midtower. This indicates that, for the midtower with larger stiffness, a resonance is excited in the sway Axialsymmetry mode of the main span cables and the midtower near the first sway frequency of freestanding midtower in Table 3.
Table 12. Natural frequencies of local sway modes of vibration of midtower
Mode no.

Natural frequency (Hz)

Two cables


Modal property

Modal combination


1

0.5122

in phase

AA

2

0.5591

in phase

SA

3

0.5957

in phase

SA

4

0.5974

in phase

SA

5

0.6171

in phase

AA

6

0.6275

in phase

AA

7

0.6695

in phase

SA

8

0.7171

in phase

AA

9

0.7864

in phase

AA

10

0.8160

in phase

SA

4.4. Global outofplane vibration
Table 13 lists the natural frequencies of global outofplane vibration of the towercable system. It is interesting that the sway natural frequencies of the system are just between the first sway natural frequency of the side tower and the midtower. The modes of two cables are in phase Axialsymmetry modes. Thus, the sway modal properties of two side towers must be in phase and the corresponding natural frequencies always excite the outofplane vibration of the midtower. Subsequently, with the interactions between three towers and two cables, the sway natural frequencies of two side towers and the midtower tend to couple. This results in increasing the sway natural frequencies of the side towers and decreases the sway natural frequencies of the midtower.
From the above modal analysis, we conclude that the inplane and outofplane dynamic response of the towercable system to wind excitation should be computed using the coupled system model rather than the separate freestanding tower model and isolated model. As the interaction relationship between the towers and the main cables indicate, the natural frequencies of the towercable system are more closely spaced and are lower than those of the freestanding towers. The crosscorrelation terms and the higher modes may have to be taken into account as the wind response is considered.
Table 13. Natural frequencies of global outofplane modes of vibration of cabletower system
Mode no.

Natural frequency (Hz)

Two cables

Sway modal property of side towers


Modal property

Modal combination


1

0.3116

in phase

AA

in phase

2

0.4125

in phase

AA

in phase

3

0.4626

in phase

SA

in phase

The modes of the cabletower system are categorized and described in details, and important issues such as modal combination problems have been elaborated. However, for the initial curvature and stretching parameters of the cables being neglected, the nonlinearity effects due to quadratic and cubic items were not introduced into the cabletower dynamic system above. It is speculated and possibly expected that more nonlinear characteristics such as internal resonance or mode bifurcation may develop when certain conditions such as yielding the inner resonance is satisfied, for the twinspan suspension bridge with large span is always a flexible structure especially in the construction period. The complex nonlinear modal effects will be subject of further research by the authors.
5. Conclusions
By considering and taking into account the conditions and the parameters encountered in the construction process, a threedimensional dynamic finite element model is established to analyze the modal characteristics of the free vibration of the towercable system of a triple tower twin span suspension bridge during the construction process and after the main cable erection is completed. The results indicate that not only the towercable system still maintains the modal characteristics of the components of the towercable coupling system, but it also has its own unique modal characteristics. The swinging motion and the inplane motion of the coupled system are uncoupled in lower natural frequency range.
Based on the frequency of the isolated tower or the cables, the combination mode of the towercable system can be categorized into several distinct groups. The system generates several new natural frequencies and vibration modes because of the dynamic interaction between the towers and the cables, and, the lower order modes of side span cables couple with the higher order modes of the main span cables, either in phase or out of phase, local or global modes. The towercable system exhibits many new mode combinations.
The modal analysis of the towercable system of tripletower twinspan suspension bridge reveals some new characteristics for free vibration of multispan towercable coupled systems, and it is useful for further study on the wind excited dynamic response of multispan bridges during the erection period.
Acknowledgements
The work reported is supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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