Experimental and theoretical study on cable-supporting system

Mechanical Systems and Signal Processing 140 (2020) 106638

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Experimental and theoretical study on cable-supporting system Han Fei a, Dan Danhui b,c,⇑, Zou Yiqing d, Lei Huan d a

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710129, China School of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China Institute of Architectural Engineering, Xinjiang University, Urumchi 830046, China d Liuzhou OVM Machinery Co., LTD, Guangxi 545005, China b c

a r t i c l e

i n f o

Article history: Received 31 July 2019 Received in revised form 4 January 2020 Accepted 10 January 2020

Keywords: Cable dynamics Full-scale cable test Suspension cable system Closed-form solution Dynamic characteristic analysis

a b s t r a c t Suspension type cable system is an important type of cable bearing system, its dynamic problems have always been the key to the structural design, health monitoring, and vibration control. In view of this, an exact dynamic analysis method for suspension type cables is proposed by the author. The dynamic characteristic of the suspension cable system is studied by a full-scale cable test. By comparing with experimental results and finite element solutions, the accuracy and universality of the proposed method are verified. The results show that the calculation result agrees well with the measured results, and the maximum relative error of each condition is basically no more than 2%. Through parameter identification, the bending stiffness, cable force, and the installation position of the weights are identified and corrected, and the calculation errors after parameter identification are controlled within 1%, which significantly improves the calculation accuracy and further verifies the accuracy of the proposed method. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction Cable structure is an important type of bearing system in modern structures like voltage transmission, stay cables, suspended cables, and mooring cables. With the increase of the span and height of modern engineering structures, the structural form of the cable system becomes more complicated. To make the mechanical behavior of the cable system more reasonable and maximize its bearing and spanning capacity, modern flexible cables are often attached with several lateral components, such as elastic supports, dampers, and lumped mass. For convenience, this type of cable-supporting systems is collectively referred to as the suspension type cable system (STCS). Due to the presence of lateral components, the dynamic analysis of the STCS become more difficult and require a unified dynamic analysis method with high precision and efficiency. Suspension cables can meet the demands of large spans, lightweight, high strength, ease of construction and aesthetic appearance at the same time [1]. For cable structures, the dynamic analysis model has been developed from the initial string model based on the inextensible assumption [2,3] to extensible elastic beam model considering the effect of bending stiffness, sag, inclination, and lateral components [4]. However, for STCS considering the effects of multiple factors, their dynamic characteristics analysis is still based on numerical analysis methods such as finite element method, and there is no universal analytical or semi-analytical analysis method to satisfy the analysis accuracy and efficiency simultaneously. For example, recently, a continuum model is proposed by Gwon for static and dynamic analyses of a three-span suspension bridge based

⇑ Corresponding author at: School of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China. E-mail address: [email protected] (D. Danhui). https://doi.org/10.1016/j.ymssp.2020.106638 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

on the Galerkin method [5]; Guo et al. propose a reduced cable-support coupled model to investigate the modal resonant dynamics of cables with a flexible support [6]; Xiang Shi and Songye Zhu investigates the dynamic characteristics of a stay cable with an inerter damper by numerical approaches [7]; Javanbakht. M et al. proposes a control-oriented numerical model for evaluating the dynamic response of a stay cable based on the mode superposition method [8]; Achref Mansour et al. investigates the free undamped vibrations of cables with arbitrary sag and inclination based on catenary string model [9]; Sun and Chen present a reduced-order model for a shallow cable attached with dampers and investigate the sag effects on optimal performance of nonlinear cable dampers [10,11]; Han et al. investigate the free vibration of a cable with several lateral components attached at intermediate locations [12]. Zipeng Kan et al. proposed a linear complementarity approach for both static and dynamic analysis of sliding cable with friction [13]; Zipeng Kan and Haijun Peng analyzes the nonlinear dynamic behavior of clustered tensegrity structures using a positional formulation finite element method [14]; To analyze the nonlinear dynamic deploy of Clustered Tensegrity, a sliding cable element of multibody dynamics by Zipeng Kan et al, results shows that the multibody dynamic methodology does provide a powerful tool for analyzing the mechanical properties of such systems [15]. Although the above works have drawn some meaningful conclusions, the analysis methods used are difficult to extend to more general situations, such as considering the effect of cable bending stiffness and multiple attached components. This is mainly due to that when the cable is attached with several lateral components, different cable segments divided by lateral components will follow different dynamic configurations in motion [16,17]. As the number of cable segments increases, the number of dynamic degrees of freedom will increase rapidly, thus making the dynamic analysis of the STCS more difficult. Another difficulty is the calculation of the additional cable force of the STCS [18,19]. Because the dynamic configuration of the STCS show significant segmented characteristic, it is necessary to calculate the additional cable force for each cable segment separately. However, existing studies can only give the explicit expression of additional cable force for one [20] or two segments cable systems, and cannot to be extended to multi-segment cable systems. To solve these tricky problems and proposed a unified dynamic analysis method for STCS, this paper first derive the general expression of the additional cable force of the suspension cable with multiple lateral components. Then, a unified modal analysis method for STCS is proposed based on the dynamic stiffness method (DSM) [21–23], which considers the effect of the cable bending stiffness, sag, and additional cable force. Finally, the accuracy of the proposed method is verified by comparing with experiment and FEM results. This paper is organized as follows: Section 2 states the theoretical model of suspension type cable systems and then presents a closed form solution for the frequency equation considering multiple factors; Section 3 is the introduction of the full-scale cable test used for experimental verification of the theoretical part; Section 4 investigates and discusses the experimental results by comparing with the theoretical solutions, the parameter identification is used to analyse the possible cause of errors; Finally, main results obtained in this effort are summarized in Section 5. 2. Basic theory 2.1. Statement of the problem In existing dynamic analysis theories, the DSM has been widely used in recent years and has achieved certain results, which transforms the dynamic characteristics analysis problem of the structure into the solution of a transcendental frequency equation [24–26]. However, due to the geometric nonlinearity of the shallow sagged cable system and the complexity of the problem obtaining the additional cable force of the suspension type cable system, there are two key problems in the implementation of DSM: (1) the establishment of the accurate dynamic stiffness matrix and frequency equation; (2) how to solve the transcendental frequency equation without missing any root [27]. To obtain the dynamic stiffness matrix of the STCS, it is first necessary to obtain the governing differential equation (GDE) of the system. In the modeling process, the suspension cable is simulated by Euler beam element, which can consider the effects of bending stiffness, sag, and elastic elongation. The additional weights are attached on the cable, which is used to consider the effect of hanging loads. Considering that most suspension cables in actual engineering are shallow sagged cables (sagh
 ih i 2 2 to-span ratio less than 1/8), the quadratic parabola yðxÞ ¼ mgl =ð2HÞ ðx=lÞ  ðx=lÞ is usually used to describe the cable static profile which considers the effect of the bending stiffness and initial sag [20,28]. Accordingly, the initial static configuration of the jth cable segment divided by weights can be expressed in local coordinate as i X 4e yi ðxi Þ ¼  xi þ lk  li l0 k¼1

!

xi þ

i X

!

lk  l0  li

ð1Þ

k¼1

0 where l0 is the chordwise length of the cable, e ¼ mgl is the sag-to-span ratio, xi is the local coordinates. The meaning of the 8H remaining symbols can be seen in Fig. 1, where the cable is divided into n segment by n  1 attached weights. The chordwise length, static, and vibration function of the ith cable segment are denoted as li , yi and ui ðxi ; t Þ respectively; H is axial force of the cable (Fig. 1). To consider the geometric nonlinear of the cable, the general expression of the additional (dynamic) strain eðtÞ of a suspension cable has been derived in [20] as

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

3

Fig. 1. Suspension type cable system divided by transverse mass.

Fig. 2. The calculation flowchart of the improved W-W algorithm.

eðtÞ ¼

1 e l0

Z

l0

0

"  2  2 # @u dy 1 @u 1 @w þ þ @x dx 2 @x 2 @x

ð2Þ e

where u and w are planar and out-of-plane vibration function of the cable, and l0 is the cable length in the static deformed configuration. However, based on small sag-to-span ratio assumption, a linear dynamic model, which takes into account additional cable tension during motion, can be obtained neglecting in Eq. (2) the two quadratic terms [28]:

eðtÞ ¼

1 e l0

Z

l0

0

  @u dy @x dx

ð3Þ

According to Eq. (3), the additional cable tension hðt Þ ¼ EAeðtÞ of a suspension cable without any attached component can be expressed as [20]

hðt Þ ¼

mgEA e l0 H

Z

l0

udx 0

ð4Þ

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

Now consider the effect of attached weights during the motion, as shown in Fig. 1, when a transverse displacement of size

aBi occurs at point Bi , the weight mi will produce a lateral force of size V Bi ¼ mi x2 aBi , where x is the modal frequency of the  system. More generally, if there is an elastic support with a stiffness of ki at point Bi , the lateral force V Bi ¼ mi x2  ki aBi . For the sake of simplicity, this paper only discusses the case of additional weights, but we should keep in mind that the proposed analytical model can be extended to more generalized cases in actual engineering. Then, the governing differential equation (GDE) of the jth cable segment divided by lateral force of a suspension type cable system can be expressed uniformly as [4]. 2

EI

d yj @ 4 uj @ 2 uj @ 2 uj  H  h þ m ¼0 j dxj 2 @x4j @x2j @t 2

ð5Þ

where EI and m are the flexural stiffness and mass per unit length of the cable; hj is the additional cable force of the jth cable segment produced by the elastic extension during the vibration. Existing researches can only give the explicit expression of the additional cable force for the integral cable (no lateral force applied on the cable), see Eq. (4) or two-segment cable (only one lateral force) [29], failed to be extended to multisegment cable system. For this purpose, the author has derived the explicit expression of hj for multi-segment cable system as follows

8EAe hj ¼ e l0 lj where

(Z

lj

    uj xj ; t dxj þ 0:5l0  lsj uj lj ; t þ lsj1  0:5l0 uj ð0; tÞ

)

ð6Þ

0

lj ¼ lj =l0 , lsj ¼

j P

lk is the sum length of the first j cable segments. Particularly, it is not difficult to find that lsn ¼ l0 ,

k¼1

lsn ¼ 1, ls0 ¼ ls0 ¼ 0. More details of the derivation of hj for multi-segment cable system can be found in Appendix I. 2.2. Development and solution of the frequency equation According to the calculation procedure of the DSM, the dynamic stiffness matrix of the jth cable segment KðjÞ can be obtained from the Eq. (5).   With separation of variables, by substituting uj xj ; t ¼ u xj eixt into Eq. (5) the GDE of the jth cable segment with respect  to the vibration mode function u xj can be expressed as





2



^ ^ 0000 nj  c2 u ^ 00 nj  x u ^ nj ¼ h u j

ð7Þ


   2  l20 ^j ¼ h ^j cosh=H. ^ nj ¼ / xj  EI= mgl4 ; h where the dimensionless parameters c2 ¼ HlEI0 ; x ¼ x pffiffiffiffiffiffiffi ; nj ¼ xj =l0 , / 0 EI=m

The general solution of Eq. (7) is











2

^ =x ^ nj ¼ Aj1 epnj þ Aj2 epðlj nj Þ þ Aj3 cos qnj þ Aj4 sin qnj þ h u j

ð8Þ

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o
2 2 Hl0 ml40 Hl20 p 2 where q ¼ þ x EI  2EI , the dimensionless additional cable force is 2EI

^ ¼g h j j

"Z 0

lj














 



^ nj j¼ 0 ^ nj dnj þ 0:5  lsj u ^ nj ¼ lsj þ lsj1  0:5 u u

Fig. 3. Cable cross section.

# ð9Þ

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

where gj ¼ 8EAe , which is totally determined by structural parameters of the system. l le 0 j

Eq. (8) can also be written as following matrix form






 n

^ nj ¼ UðjÞ þ Bð jÞ  AðjÞ AðjÞ AðjÞ AðjÞ u 1 2 3 4

oT

ð10Þ

h    i where UðjÞ ¼ epnj epðlj nj Þ cos qnj sin qnj sin qnj is the mode shape matrix, the sag matrix B(j) is given in Appendix I The displacement vector and the nodal forces vector of the jth cable segment are represented by the dimensionless mode shape function, i.e.

h

aðajÞ hðajÞ l0 aðbjÞ hðajÞ l0

iT

¼

4

 iT  mgl0 h  ^ 0 nj j ¼ 0 u ^ n j j ¼ lj ^ 0 nj j ¼ lj ^ nj j ¼ 0 u u u EI

8     000   3 > > ^ nj  c2 u ^ 0 nj  eixt < V xj ; t ¼ EI @@xu3  H @@xuj  eixt ¼ mgl0 u j

  2 > > ^ 00 nj eixt : M xj ; t ¼ EI @@xu3 eixt ¼ mgl20 u

ð11Þ

ð12Þ

j

Substituting Eq. (10) into Eq. (11) and Eq. (12) yields

h KðjÞ  aðajÞ

hðajÞ l0

aðbjÞ hðajÞ l0

iT

h ¼ V ðajÞ

M ðajÞ =l0

ðjÞ

Vb

ðjÞ

M b =l0

iT

ð13Þ

where K(j) is namely the element dynamic stiffness matrix of suspension type cable systems. The explicit expression of the K(j) are shown in Appendix II. After obtaining the dynamic stiffness matrix of each cable segment, the global dynamic stiffness matrix of the suspension type cable system can be easily obtained by assembling the contribution of each cable element similar to the finite element method. Take a n-segment cable as an example, when the endpoints A and B of the cable are clamped, the global dynamic stiffness matrix K(0) considering the effect of the clamped boundary and intermediate attached weight can be obtained, which is shown in Appendix II. The frequency equation corresponds to the value of the determinant of K(0) is zero, i.e.


 det Kð0Þ ¼ 0

ð14Þ

Because the above equation is a complicated transcendental equation, conventional iterative methods such as Muller and Newton method are likely to missing some roots [30]. To avoid this problem, it is necessary to introduce the WittrickWilliams (W-W) algorithm which can not only obtain the roots of the frequency equation with the required precision, but also perfectly solve the root-missing problem [31]. The W-W algorithm is essentially a root counting method based on the sturm sequence properties, which does not directly solve the frequency equation, but first estimates the upper and lower bounds of modal frequencies by calculating the number of modal frequencies below a trail frequency x , denoted by J ðx Þ [27]. By repeating this step and continuously adjusting the trail frequency x , a proper upper and lower bounds ðxu ; xl Þ for the kth order modal frequency can be selected by making it satisfies J ðxu Þ P k and J ðxl Þ < k. Then, the modal frequency xk can be determined using the dichotomy or Newton method by narrowing the range of ðxu ; xl Þ until it meets xl  xu 6 Tol, where Tol is the allowable error defined by users. In general, J is difficult to solve directly for most structures, but it can be expressed by the sum of other two counts

n o J ¼ J 0 þ s KD ðx Þ ¼ J 0 þ J K

ð15Þ

where J 0 is called the clamped-clamped frequency count, which is the number of modal frequencies of the structure below the trail frequency x with all boundaries restrained; J K is the number of negative elements in the leading diagonal of KD ðx Þ, and KD ðx Þ is the upper triangular matrix obtained by applying the usual form of Gaussian elimination to Kðx Þ; Kðx Þ is the overall DS matrix of the structure at x ¼ x . The premise of this transformation is that the dynamic stiffness matrix Kðx Þ is symmetrical, that is, the structure is linear elastic. It can be seen from the solution process of the W-W algorithm that the method is only suitable for linear system, however when considering the sag effect of the cable, the geometric nonlinearity will make the dynamic stiffness no longer strictly symmetrical, and the W-W algorithm cannot be directly applied for frequency calculation. Because this paper mainly focuses on flat-sag cables, whose sag-to-length ratio e is less than 1/8 and the deformation is small, such as the stay cables on cablestayed bridges and the main cable on a suspension bridge. For these cables, due to the small sag-span-ratio e, the static  e length of the cable l0 ¼ l0 1 þ 8e2 approximately equals to the chordwise length l0 , thus has little effect on Irvine parameter 0 . Due to this weak nonlinear characteristic, the influence of sag on modal frequencies is limited. Therefore, some k2 ¼ 64e2 EAl Hle 0

improvements of the original W-W algorithm have been done by the author in our previous studies, which makes it still effective to multi-segment cable systems. The calculation flow chart of the improved W-W algorithm is given in Fig. 2

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

3. Introduction of the cable test 3.1. Cable and sensor characteristics To study the dynamic characteristic of the STCS, a full-scale cable test is carried out in a cable factory. The horizontal tension tests of 20 m cable is extruded by PEJ15B-12 strand, the cross section and design parameters of the cable are shown in Fig. 3 and Table 1 respectively. The exciter is a simply artificial excitation device as shown in Fig. 4. Acceleration sensors shown in Fig. 5 use the three-axis wireless acceleration sensor AX-3D-XRANGE manufactured by BEANAIR, France. The sensor parameters are as follow (Table 2) The pressure ring shown in Fig. 6 consists of a CL-YB-M4 MN type sensor with 400 t measurement range and a 2050 type acquisition instrument with 50 Hz sampling frequency. The overall layout of this cable test is shown in Fig. 7. The boundary condition of the cable was clamped, the photo of the anchorage, sensors, and collection device can be seen in Fig. 8.

Fig. 4. Artificial excitation device.

Table 1 Parameters of the test cable. Specifications PEJ15B-12

l (m) 19.715

D (mm) 85

m (kg m1) 16.6

I (m4) 6.1757  10

A(m2) -7

l: the free length between anchorage ends. D: the diameter of the cable. m: mass per unit length I: the sum of inertia mo.ments of all strands relative to the central axis of cable section. A: effective section, the sum of areas of all strands. H*: the breaking force of cable.

Fig. 5. Acceleration sensor and weight

2:178  10

3

H*(kN)

E (MPa)

3120

1.9  105

7

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638 Table 2 Parameters of acceleration sensors. Type

Range

Sampling frequency

Size

Operating temperature

AX-3D-RANGE

2g

1 Hz ~ 4.5 kHz

80*55*21

20 ~ +65

Acceleration sensor and weight. Pressure ring and lifting jack.

Fig. 6. Pressure ring and lifting jack.

Fig. 7. Layout of test cable.

Fig. 8. Full-scale cable test site layout.

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

3.2. Tension schemes 1) By controlling the oil pressure gauge (as shown in 3.1 Fig. 6), the test cable is tensioned under five different levels. From the 3rd level, the pressure should be maintained five minutes after tensioning to next level. The tension speed needs to be uniform, and the duration from one level to the next should be no less than 2 min.

Fig. 9. Sket alh map for Case 2.

Fig. 10. Sket alh map for Case 3.

Fig. 11. Sket alh map for Case 4.

Fig. 12. Sket alh map for Case 4.

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

2) During the tension process, the difference between the actual elongation value and theoretical elongation value should be controlled within 6%. Otherwise, the tension should be suspended until the cause is identified and measures are taken

Fig. 13. Acceleration spectrum for Case 1.

Table 3 Case 1 Level 4, H ¼ 1:2487  106 N (Hz). Mode

Measured

TS

Error (%)

BT

Error (%)

This paper

Error (%)

1 2 3 4 5 6

7.24 14.42 21.77 29.38 36.75 44.40

6.96 13.91 20.87 27.82 34.78 41.74

3.88 3.59 4.14 5.30 5.34 5.96

6.96 13.96 21.05 28.25 35.60 43.15

3.79 3.22 3.32 3.86 3.10 2.78

7.16 14.37 21.65 29.06 36.62 44.39

1.03 0.45 0.55 1.10 0.32 0.02

Table 4 Case 1 Level 5: H ¼ 1:4068  106 N (Hz). Mode

Measured

TS

Error (%)

BT

Error (%)

This paper

Error (%)

1 2 3 4 5 6

7.67 15.26 23.20 31.12 38.90 46.95

7.38 14.77 22.15 29.53 36.92 44.30

3.74 3.24 3.99 5.10 5.10 5.61

7.39 14.82 22.32 29.93 37.69 45.63

3.66 2.91 3.26 3.83 3.11 2.77

7.59 15.22 22.92 30.74 38.71 46.86

1.06 0.29 0.65 1.23 0.49 0.14

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

The test contains five working conditions, each condition is divided into five levels for tensioning, the specific arrangements are as follows:  Case 1: Pure cable, i.e., no weight attached on the cable. The value of cable force H for each tension level is0:7645  106 N, 0:9209  106 N, 1:0842  106 N, 1:2487  106 N, 1:4068  106 N respectively;  Case 2: Single weigh with a mass of m1 ¼ 8:3 kg attached near the midspan, the installation position of weight and sensors are shown in Fig. 9. The cable force for each tension level is0:7806  106 N, 0:9381  106 N, 1:0909  106 N, 1:2467  106 N, 1:4046  106 N respectively;  Case 3: Single weigh with a mass of m2 ¼ 2m1 ¼ 16:6 kg attached near the midspan, the installation diagram and picture are shown in Fig. 10 and Fig. 5 respectively. The cable force for each tension level is 0:7838  106 N, 0:9435  106 N, 1:0898  106 N, 1:2489  106 N, 1:4003  106 N respectively;

Fig. 14. Comparison of different calculation results of the first six modes under each tension level for Case 1.

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

 Case 4: Two weights m2 attached near the midspan, the installation position of the two weights is shown in Figs. 5 and 11. The cable force for each tension level is 0:7616  106 N, 0:9327  106 N, 1:0930  106 N, 1:2462  106 N, 1:4007  106 N;  Case 5: Three weights m1 attached near the midspan and quarter span of the cable, where the installation position of the weights

is

shown

in

Fig.

6

12.

The

cable

6

force

for

each

tension

level

is0:7789  106 N,

0:9350  106 N,

6

1:0935  10 N,1:2514  10 N, and 1:4061  10 N; 4. Experimental study In this section, without loss of generality, the measured acceleration spectrum and error analysis table of the 4th and 5th level tension condition are given at the beginning of each working condition. The results of the remaining levels are given in the form of a histogram. Finally, according to the measured results, the influence of the mass and installation position of the weight on the modal frequency of the system is discussed. 4.1. Case 1: pure cable For pure cable system, it is a special case of the STCS, such as stay cables in engineering structures. In actual engineering, the taut string (TS) theory or the beam theory (BT) considering the effect of the cable is usually used to estimate the cable force. The relationship between the cable force and the cable frequency based on the two method is given as follow

xn ¼

np l

np xn ¼ l

rffiffiffiffiffi H m

ð16Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   H n2 p2 EI 1þ 2 m H l

ð17Þ

where n is the order of the mode, H is the measured cable force. The meaning of the rest of the symbols is the same as above. Since there is a one-to-one correspondence between the cable force and the frequency for a given system, the cable force is usually solved by means of an average value of the frequency differences. In the following analysis process of this case, the measured cable force will be substituted into Eqs. (14)–(17) respectively to calculate the cable frequency. The results calculated from the TS, BT, and this paper are compared with the measured results. To compare the accuracy of the three methods, v aluecalculated v alue. the relative error is defined asd ¼ MeasuredMeasured v alue

Fig. 15. Acceleration spectrum for Case 2.

Table 5 Comparison of results for Case 2 (Hz). Mode

Level 4H ¼ 1:245  106 N Measured

This paper

Error (%)

Measured

This paper

Error (%)

1 2 3 4 5 6

7.05 14.39 21.25 29.34 35.90 44.30

7.04 14.35 21.29 29.03 36.03 44.35

0.10 0.25 0.20 1.05 0.36 0.12

7.47 15.24 22.52 31.08 37.91 46.86

7.46 15.20 22.54 30.71 38.08 46.83

0.10 0.24 0.09 1.18 0.45 0.07

Level 5H ¼ 1:405  106 N

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

As shown in Fig. 13, the acceleration spectrum at different positions of the test cable can be obtained by applying Fast Fourier Transformation on measured acceleration data. Then, the modal frequency can be obtained from acceleration spectrums by the peak picking method (PPE). It is worth noting that the peak of some modes may become difficult to identified when employing the PPE if some sensors were installed near the mode nodes of the cable. To this end, this paper determines the modal frequencies by comparing the acceleration spectrum of multiple sensors in the following analysis. For example, from Fig. 13(a) and (c) we can clearly determine the 2nd order modal frequency of the cable is 14.42. For the sake of simplicity, acceleration spectrum of one sensor is shown here as an example. The corresponding results calculated by different methods are shown in Tables 3 and 4. It can be seen from Tables 3 and 4 that the error of the result calculated from the TS and BT is non-negligible. In particular, for the TS solution, the error of the high-order mode is significant; the relative error of the results obtained by this method is less than 1.5%, and it still has high precision for high-order modes. To further illustrate the accuracy of the proposed method, Fig. 14 gives the histogram of the results and error of the three methods for different tension levels (Fig. 15).

Fig. 16. Comparison of calculation results of the first six modes under each tension level for Case 2.

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

It can be seen from Fig. 14 that (1) for different tension levels, the calculation results of each order mode are very close to the measured values, and the maximum relative error is only 1.22%; (2) Both the TS and BT have large errors, the calculation error of the former increases with the increment of the modal order, and reduces almost linearly with the decrease of the cable force; The calculation error of the latter has a decreasing trend with the increase of the modal order. When the cable force decreases, the error of the first three modes is nearly linear, while the error of the 4th to 6th mode is basically unchanged. The main reason for the error is that for a short cable, the effect of the flexural stiffness and boundary conditions cannot be ignored. This working condition once again verifies the shortcomings of TS and BT in the identification of the cable force of a short cable. 4.2. Case 2: cable attached with one weight m1 It can be seen from Fig. 15 and Table 5 that when the weight m1 is added near the midspan, the calculation results of each mode are very close to the measured results, and the maximum error is less than 1.2%. Fig. 16 further shows the comparison between the measured and calculation results of five tension levels. It can be seen that for the first three modes, the relative error of this paper is less than 0.4%, and the error is increased for the higher-order modes but do not exceed 1.2% (Fig. 17). 4.3. Case 3: cable attached with one weight m2 It can be seen from Fig. 17 and Table 6 that when the mass of the weight varies, the results of this paper still have high precision, and the maximum error is 1.12%. Fig. 18 shows the results comparison under other tension levels and the relative error are given as well, where the maximum error is 1.3%, which indicates the correctness of the method (Fig. 19). 4.4. Case 4: cable attached with two weights m2 In this working condition, two weights m2 are added adjacently in the midspan of the cable, Fig. 19 and Table 7 gives the calculation and measured results of the level 4 and level 5. It can be seen that the method in this paper is still in good agreement with the measured results. From Fig. 20 we can see that the calculation error of the odd-order mode is significantly higher than that of the even-order mode. This is because the position of the weight is close to the midspan, i.e., at the modal node of the even-order mode, the effect of the weights on the modal frequency is significant smaller than the odd-order. In addition, the maximum calculation error in this working condition is 3.6%, which is slightly higher than the above two single-weight cases. This is mainly due to

Fig. 17. Acceleration spectrum for Case 3.

Table 6 Comparison of results for Case 3 (Hz). Mode

Level 4H ¼ 1:249  106 N Measured

This paper

Error (%)

Measured

This paper

Error (%)

1 2 3 4 5 6

6.87 14.41 20.83 29.33 35.12 44.22

6.94 14.37 20.99 29.05 35.36 44.37

1.00 0.31 0.77 0.95 0.68 0.34

7.27 15.35 22.00 31.01 37.10 46.69

7.34 15.18 22.17 30.66 37.30 46.74

0.90 1.10 0.78 1.12 0.54 0.11

Level 5H ¼ 1:4  106 N

14

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

Fig. 18. Comparison of calculation results of the first six modes under each tension level for Case 3.

Fig. 19. Acceleration spectrum for Case 4.

15

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

the inevitable error in measuring the installation position of the weight. As the number of weights increases, the measurement error will accumulate and thus have an impact on the accuracy of the calculation, but the error is still within the acceptable range (Fig. 21).

Table 7 Comparison of results for Case 4 (Hz). Mode

1 2 3 4 5 6

Level 4H ¼ 1:246  106 N

Level 5H ¼ 1:401  106 N

Measured

This paper

Error (%)

Measured

This paper

Error (%)

6.55 14.40 20.05 29.21 34.15 43.91

6.72 14.34 20.41 28.95 34.79 44.10

2.65 0.41 1.82 0.90 1.86 0.44

6.94 15.23 21.19 30.97 36.15 46.45

7.12 15.17 21.59 30.59 36.72 46.51

2.54 0.38 1.88 1.22 1.59 0.14

Fig. 20. Comparison of calculation results of the first six modes under each tension level for Case 4.

16

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

4.5. Case 5: cable attached with three weights m1 It can be seen from Fig. 21, Table 8 and Fig. 22 that when the cable is attached three weights, the calculation error of the first five order modal frequency of the system under each tension level is basically below 2%, and the error of the 6th order mode is slightly larger but no more than 4%. These prove the accuracy of the proposed method again. In this real cable test, five cases and five tension levels in each case were used to verify the accuracy of the proposed method. The effectiveness and universality of the proposed method are repeatedly verified by changing the mass, number and installation position of the weight. The experimental results show that the results of this paper are consistent with the measured results for the pure cable condition; for single-weigh conditions (Case 2 and 3), the calculation error does not exceed 1.3%; For multi-weights condition (Case 4 and 5), the calculation errors of each condition are basically no more than 2%, while the maximum error is less than 4%, thus verifying the accuracy of the method of this paper in analysis of the dynamic characteristics of the STCS. 4.6. Discussion of experimental results and error analysis In general, most design parameters of a prefabricated cable may be measured accurately except the bending stiffness. The bending stiffness depends on the cross-section characteristic, processing technology, and construction technology. Thus, it is difficult to calculate quantitatively. Besides, with the increase in cable force, the radial force between wires increases, which will then increase the friction between wires and thus increasing the bending stiffness of cables. Additionally, due to the instability of the pressure ring, the reading of tension force will drift after the tension process is completed. In view of this, firstly, this section will take Case 1 as an example to discuss the influence of bending stiffness and cable force on modal frequencies. 

 

During the investigation, the design parameters under consideration are the ratios H =H and E I =EI, where the superscript ‘‘~” indicates the trial value. In this case the modal frequencies have been set equal to the measured values, and the variation intervals of the two parameters are set in their common ranges. When these material parameters are considered as continuous variables, the above analysis method allows us to plot a surface for each modal frequency. Fig. 23(a) and (b) shows the surface related to the first two modal frequencies when the two design parameters vary in intervals. An inverse procedure 

 

based on work [32] is used to identify the parameters H =H and E I =EI. For each ith frequency, intersecting the surfaces of 



 

Fig. 23 with a horizontal plane of equation xi ¼ xi enables us to determine infinite couples of values H =H and E I =EI

Fig. 21. Acceleration spectrum for Case 5.

Table 8 Comparison of results for Case 5 (Hz). Mode

Level 4H ¼ 1:251  106 N Measured

This paper

Error (%)

Measured

This paper

Error (%)

1 2 3 4 5 6

6.96 13.94 20.88 28.96 35.81 43.41

6.88 13.71 20.69 29.50 35.24 42.30

1.00 1.68 0.92 1.87 1.56 2.56

7.29 14.49 21.87 31.15 37.29 44.66

7.36 14.75 22.08 30.60 37.80 45.78

1.02 1.78 0.96 1.80 1.34 2.41

Level 5H ¼ 1:406  106 N

17

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

Fig. 22. Comparison of calculation results of the first six modes under each tension level for Case 5.



providing the given value of measured frequency xi . The intersection gives the solution to the inverse problem of determining the material parameters when two modal frequencies are given. 

 



As seen in Fig. 24, it shows the possible couples of H =H and E I =EI, corresponding to given values of xi for i ¼ 1; :::; 6. In the absence of test and measurement errors, the six frequency curves cross in a small region (a single point theoretically). The central point of the intersection region allows the correct identification of the design parameters. Fig. 24(a) shows that  

the corrected bending stiffness E I is about 1.01 times the estimated value EI, which can also be seen in Fig. 24(b), and the actual cable force is a little less than measured value because of the loss of tension. Fig. 24(b) shows that the corrected bending stiffness is 1.02 times EI and a little larger than Level 4, this is because the increase of tension will increase the friction between wires and thus increasing the bending stiffness. Through this identification process, a more accurate bending stiff 



ness E I and cable force H can be obtained.

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

As seen in Table 9, the relative error d before and after parameter identification are given. It can be seen that by correcting the bending stiffness and cable force, the error of each order frequency of the system is controlled within 1%, and the calculation accuracy is significantly improved compared with before the identification. Another factor affecting the test results of this experiment is the measurement accuracy of the weight installation position, which is actually difficult to be measured accurately. In view of this, Case 4 is taken as an example to discuss the measurement error of the weigh position on results. As can be seen in Fig. 25, the curves representing modal frequencies of the system basically intersect at one point, additionally, the coordinates of the two intersections in Fig. 25 are also very close, which shows that the identification results are credible. From Table 10 we can find that, by identifying the installation location of the weight, the calculation error is significantly reduced. Especially, the error of the first-order mode is reduced from 2.65% (2.54) to 0.45% (0.58%), and the errors of other modes are also within 1%.



 

Fig. 23. First two modal frequencies x rad/s as a function of H =H and E I =EI for Case 1 level 4.



 

Fig. 24. Project onto the H =H and E I =EI plane of the curves providing given values of first 6 modal frequency.

Table 9 Error comparison of calculated natural frequencies before and after parameter identification (Hz). Mode

1 2 3 4 5 6

Level 4

Level 5

Measured

Before

jdj

Corrected

jd j

Measured

Before

jdj

Corrected

jd j

7.24 14.42 21.77 29.38 36.75 44.40

7.16 14.37 21.65 29.06 36.62 44.39

1.03 0.45 0.55 1.10 0.32 0.02

7.19 14.43 21.78 29.27 36.97 44.62

0.63 0.004 0.02 0.36 0.64 0.54

7.67 15.26 23.07 31.12 38.9 46.93

7.59 15.22 22.92 30.74 38.71 46.86

1.06 0.29 0.65 1.23 0.49 0.14

7.63 15.28 23.04 30.95 39.04 47.18

0.52 0.13 0.12 0.55 0.37 0.53

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

4.7. Comparison with FEM Because the tensile force in the experiment is large and the vertical deformation of the cable is very small, to further illustrate the effectiveness of the method for shallow sagged cable, the finite element model of the cable is established through the software ANSYS. During the modeling process, the cable profile (or sag-to-span ration e) is adjusted by changing the tensile force. The other parameters of the cable are the same as those in Table 1. As shown in Fig. 26, the entire cable is modeled by Beam 3 element and divided into 20 elements. To verify the reliability of the numerical model above, firstly, the solutions of the FE model are compared with the test results of Case 5. Table 11 shows the results and relative error of the numerical model. It can be seen that the FE solutions agree well with the test results and the maximum relative error is only 1.44%, thus verifying the reliability of the numerical mode.





Fig. 25. Project onto the l 1 =l1 and l 2 =l2 plane of the curves providing given values of the first 6 order frequencies.

Table 10 Error comparison of calculated natural frequencies before and after parameter identification (Hz). Mode

1 2 3 4 5 6

Level 4

Level 5 

Measured

Before

jdj

Corrected

jd j

Measured

Before

jdj

Corrected

jd j

6.55 14.40 20.05 29.21 34.15 43.91

6.72 14.34 20.41 28.95 34.79 44.10

2.65 0.41 1.82 0.90 1.86 0.44

6.52 14.38 20.03 28.97 34.38 44.08

0.46 0.14 0.10 0.82 0.67 0.38

6.94 15.23 21.19 30.97 36.15 46.45

7.12 15.17 21.59 30.59 36.72 46.51

2.54 0.38 1.88 1.22 1.59 0.14

6.9 15.21 21.18 30.79 36.38 46.44

0.58 0.13 0.05 0.58 0.64 0.02

Fig. 26. Finite element model of a shallow cable with parabolic profile.

Table 11 Verification of the numerical model (Hz). Mode

Level 4 H ¼ 1:251  106 N Measured

FE model

Error (%)

Measured

This paper

Error (%)

1 2 3 4 5 6

6.96 13.94 20.88 28.96 35.81 43.41

6.86 13.75 20.62 29.31 35.62 42.93

1.44 1.36 1.24 1.21 0.53 1.11

7.29 14.49 21.87 31.15 37.29 44.66

7.26 14.54 21.80 30.95 37.57 45.23

0.41 0.35 0.32 0.64 0.75 1.28

Level 5 H ¼ 1:406  106 N

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

To further illustrate the application of the proposed method, the cable-support system with different sag-span ratio are investigated by the established FE model. During the analysis, three lumped weights are modeled by mass21 element and arranged equidistantly on the cable. It can be seen from Table 12 that the results of this paper agree well with the FE solutions. When the sag-to-span ratio e takes different values (e > 1=8), the maximum relative error d is only 0.13%, which further verifies the accuracy and applicability of the proposed method. 4.8. Discussion of test results To further discuss the influence of mass, number, and installation position of the weight on modal frequency, Fig. 27 shows the first six measured modal frequencies corresponding to different tension levels under the five cases. From Fig. 27 we can see that: (1) the cable force and the cable frequency are basically the linear under various conditions. For the odd-order mode, because the mass of the additional weight in Case 4 is the largest (4m1 ) the corresponding modal frequency is the smallest among the five cases. Besides, the frequency value of Case 1, Case 3, and Case 5 are very close and less than Case 2; (2) For even-order modes, due to midspan point is exactly the modal node, the weight added at this position will not affect the dynamic characteristics of the system, and thus the modal frequencies of the first four cases are basically the same. In case 5, three weights are installed near the quarter span of the cable, thus the modal frequencies of the 2nd and 6th order of the STCS can be reduced but the influence on the 4th mode is little. Suspension type cable system is an important type of cable bearing system, its dynamic problems have always been the key to the structural design, health monitoring, and vibration control. In view of this, an exact dynamic analysis method for Table 12 Comparison with FE solutions for different cable profiles. Mode

1 2 3 4 5 6

e ¼ 0:0128

e ¼ 0:0114

e ¼ 0:0107

FEM

Present

d(%)

FEM

Present

d(%)

FEM

Present

d(%)

1.3587 3.0099 5.0706 8.2156 11.4033 15.2659

1.3605 3.0136 5.0743 8.2226 11.4110 15.2749

0.13 0.12 0.07 0.08 0.07 0.06

1.4140 3.1072 5.1952 8.3684 11.5626 15.4332

1.4154 3.1102 5.1983 8.3739 11.5685 15.4393

0.10 0.09 0.06 0.07 0.05 0.04

1.4494 3.1701 5.2766 8.4685 11.6673 15.5432

1.4508 3.1728 5.2792 8.4733 11.6722 15.5479

0.10 0.09 0.05 0.06 0.04 0.03

Fig. 27. Comparison of measured cable frequencies for different conditions.

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

21

suspension type cables is proposed by the author. The dynamic characteristic of the suspension cable system is studied by a full-scale cable test. By comparing with experimental results and finite element solutions, the accuracy and universality of the proposed method are verified. The results show that the calculation result agrees well with the measured results, and the maximum relative error of each condition is basically no more than 2%. Through parameter identification, the bending stiffness, cable force, and the installation position of the weights are identified and corrected, and the calculation errors after parameter identification are controlled within 1%, which significantly improves the calculation accuracy and further verifies the accuracy of the proposed method. 5. Discussion The main purpose of this paper is to propose a method for dynamic analysis of cable-supporting system, and a real cable test is used to verify the accuracy of the proposed method. The experimental cable in this paper is a real cable which can be used as the hanger of an arch bridge or suspension bridge, or a short stay cable of a cable-stayed bridge. Although the experimental cable is different from the main cables of suspension bridge, as long as the cable-supporting system being analyzed is shallow sagged (the sag-span ratio is less than 1/8), the method in this article is applicable. Actually, most tensioned stayed cables and main cables of suspension bridges in actual engineer are shallow sagged cables and within the scope of this paper, therefore, the analysis method proposed in this paper has a wide range of applications. 6. Conclusion To meet the demand for exact dynamic analysis of complicated cable systems in engineering, an exact dynamic analysis method for suspension type cables is proposed in this paper. Besides, the explicit expression of the additional cable force of the multi-segment cable system is derived in this paper considering both vertical and inclined supports. Finally, an accurate method for cable parameter identification is proposed. The accuracy of the proposed method is investigated by experiment. To further illustrate the effectiveness of the method for shallow sagged cable, the results of this paper are compared with FE solutions. Results show that when the sag-to-span ratio e takes different values, the maximum relative error d is only 0.13%, which further verifies the accuracy and applicability of the proposed method Some main conclusions can be summarized as follow: 1. For a short cable, the effect of the flexural stiffness and boundary conditions cannot be ignored. The frequency method for cable force measurement based on TS and BT will bring significant error; 2. When the position of the weight is close to the modal nodes of a certain mode, the mass of the weight will have little effect on the mode; 3. A parameter identification method is proposed to identify the design parameter of the cable, and the installation position of the weights. It is worth noting that the method proposed by the author is not only highly accurate but also have general application. In this test, the cable is designed divided by several weights, while for the actual engineering structures, the number of lateral components and even the degree of dynamic freedom of the structure will be very large. Since the intermediate derivation of the method is all given in analytical form, the computational efficiency is guaranteed. In this paper, the additional weights are taken as an example for experimental study of suspension type cable systems. In fact, the proposed method is still applicable to the multi-segment cable system with multiple attached elastic supports or dampers, thus it has certain promotion and application value. CRediT authorship contribution statement Dan Danhui: Project administration. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by the National Nature Science Foundation of China (Grant No. 51878490); the National key R&D Program of China (2017YFF0205605); Shanghai Urban Construction Design Research Institute Project ‘Bridge Safe Operation Big Data Acquisition Technology and Structure Monitoring System Research’; and the Ministry of Transport Construction Science and Technology Project ‘Medium-Small Span Bridge Structure Network Level Safety Monitoring and Evaluation’.

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H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

Appendix I The explicit expression of the additional cable force hj According to existing studies, the additional cable force of the jth segment can be expressed as

hj ðtÞ ¼ EAej ðtÞ ¼

EA e lj

Z

lej



0

 @uj dyj dxj @xj dxj

ð18Þ

For convenience, the following derivation still ignore the effect of lateral support on static configurations of the cable, at this time, the static profile of the cable can be described by quadratic parabola [28] (valid for sag-to-span ratio e < 1=8). Thus, combined with Eq. (1) the additional cable force hj can be obtained by

hj ¼ EAej R lj ¼ EA 0 le

@uj ðxj ;t Þ dyðxj Þ @xj

j

dxj

dxj

 2xj þ 2lsj1  l0 dxj n  l l o   Rl  ¼  4EAe 2 uj xj ; t xj 0j  2 0j uj xj ; t dxj þ 2lsj1  l0 uj xj ; t 0j l0 lej nR l  o    j ¼ 8EAe 0 uj xj ; t dxj þ 0:5l0  lsj uj lj ; t þ lsj1  0:5l0 uj ð0; t Þ l le ¼  4EAe l le 0 j

@uj ðxj ;t Þ

R lj

ð19Þ

@xj

0

0 j

The additional cable force hj of a multi-segment cable system considering the effect of the flexural stiffness, sag, inclination, et al. can be expressed as

hj ¼

8EAe e l0 lj

(Z

lj

    uj xj ; t dxj þ 0:5l0  lsj uj lj ; t þ lsj1  0:5l0 uj ð0; tÞ

) ð20Þ

0 e

where e is the sag-to-span ratio defined by e ¼ mgl0 cosh=8H, lj ¼ lj =l0 is the relative length of jth cable segment, lj is the arch length of the jth cable segment

2 12 3 12 3 0

 6 6 B 1 þ 4e 1  lsj tanh C 7 B 1 þ 4e 1  lsj1 tanh C 7 6 e C 7  l0 6 C 7 lj ¼ l0 6lsj þ 8e2 l3sj B 6lsj1 þ 8e2 l3sj1 B @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 A 7
2ffiA 7 5 5 4 4 1 þ 16e2 1  lsj 1 þ 16e2 1  lsj1 2

0

ð21Þ

Eq. (20) is the generalized expression of the additional cable force suitable for arbitrary boundary conditions. In particular, when the transverse displacement of the cable at the endpoint is constrained, then u1 ðx1 j¼ 0; t Þ ¼ un ðxn j¼ ln ; t Þ ¼ 0 and we have

8EAe h1 ¼ e l0 lj 8EAe hn ¼ e l0 lj

(Z

l1

) u1 ðx1 ; t Þdx1 þ ð0:5l0  l1 Þu1 ðx1 j¼ l1 ; tÞ

ð22Þ

0

(Z

ln

) un ðxn ; tÞdxn þ ðlsn1  0:5l0 Þun ðxn j¼ 0; t Þ

ð23Þ

0

For a two-segment cable system, it can be seen that the Eqs. (22) and (23) will degenerate into the expression given in [29]. Considering a suspension cable with several radially aligned hangers as shown in Fig. 28(a), take the ith cable segment as an example, the formula of additional cable force Eq. (20) can be modified as follows according to the horizontal force balance condition 

hj ¼ EAej þ Hj

ð24Þ

where the superscript * denotes the corrected additional cable force of a suspension cable considering inclined supports. Hj is the horizontal component of the spring force V Bi , which can be expressed by

Hi ¼ V Bi  coshi ¼ ki  ui ð0; t Þ

sin2hi 2

ð25Þ

where V Bi ¼ ki  ui ð0; t Þ  sinhi . hi is the angle between the hanger and cable segment at point Bi , ki is the axial stiffness of the ith hanger.

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

23

Fig. 28. Diagram of suspension cables with inclined hangers.

For Warren truss hangers shown in Fig. 28(b), since the horizontal component of the spring force of the two hangers connected at the same node (see Bj and Bm) are balanced with each other, the calculation of the additional cable force of the cable is not affected. The explicit expression of BðjÞ 2 ^ =x Assume the particular solution h in Eq. (8) has the following form j

^ h j



x

n

¼ BðjÞ 

2

ðjÞ

ðjÞ

h then Bð jÞ ¼ bð1jÞ

ðjÞ

A2

A1 ðjÞ

ðjÞ

b2

ðjÞ

b3

ðjÞ

A3 b4

i

oT

ð26Þ

A4

, (i ¼ 1; 2; 3; 4) can be determined as

 When j ¼ 1 ð1Þ


pl 
pl   ð1Þ ð1Þ ð1Þ ð1Þ g1 ; b ¼ b0 1ep 1 þ 0:5  l1 el1 p ; b2 ¼ b0 1ep 1 þ 0:5  l1 ;  ðx2 0:5g1 Þ 1         ð1Þ sinðql1 Þ ð1Þ ð1Þ 1cosðql1 Þ cos q ; b sin q ¼ b0 þ 0:5  l l ¼ b þ 0:5  l l 1 1 1 1 4 0 q q

b0 ¼ ð1Þ

b3

 When j ¼ n ðnÞ


  pl ðnÞ ðnÞ þ 0:5  ln eln p ; b2 ¼ b0 1ep n þ 0:5  ln ; ð Þ         ðnÞ sinðqln Þ ðnÞ ðnÞ 1cosðqln Þ ¼ b0 þ 0:5  l l ¼ b þ 0:5  l l cos q ; b sin q n n n n 4 0 q q

ðnÞ

b3

ðnÞ

ðnÞ

gn x2 0:5gn

; b1 ¼ b0

gj  x2

¼ b0

b0 ¼



 Others b0 b3

ðjÞ

¼

ð jÞ

¼ b0

; b1 

ðjÞ

ð jÞ

sinðqlj Þ q

ð jÞ

þ










1epln p

plj

1e p

þ















  þ 0:5  lsj þ lsj1  0:5 eplj 

 ðjÞ 1cosðqlj Þ sin q ¼ b0 þ 0:5  l l sj j q

lsj1  0:5 þ 0:5  lsj eplj ; b2 ð jÞ ¼ bð0jÞ









lsj1  0:5 þ 0:5  lsj cos qlj

 ð jÞ ; b4




plj

1e p

3

where gi ¼ 64 AlIl0e e2 i

Appendix II Element dynamic stiffness matrix KðjÞ

2 KðjÞ ¼

ðjÞ

k11

6 ðjÞ EI 6 6 k21 3 6 ðjÞ l 6 4 k31 ðjÞ k41

ðjÞ

k12 ðjÞ

k22 ðjÞ

k32 ðjÞ

k42

ðjÞ

k13 ðjÞ

k23 ðjÞ

k33 ðjÞ

k43

ðjÞ

k14

3

7 ðjÞ k24 7 7 7 ðjÞ k34 7 5 ðjÞ k44

Then the coefficients in matrix KðjÞ can be determined by

ð27Þ

24

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

 
ðjÞ  ðjÞ ðjÞ ðjÞ k11 ¼ p p2  c2 c11  ei c21  q q2 þ c2 c41  
ðjÞ  ðjÞ ðjÞ ðjÞ k12 ¼ p p2  c2 c12  ei c22  q q2 þ c2 c42  
ðjÞ  ðjÞ ðjÞ ðjÞ k13 ¼ p p2  c2 c13  ei c23  q q2 þ c2 c43  
ðjÞ  ðjÞ ðjÞ ðjÞ k14 ¼ p p2  c2 c14  ei c24  q q2 þ c2 c44

 ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ k21 ¼ p2 c11 þ ej c21 þ q2 cj31 ; k22 ¼ p2 c12 þ ei c22 þ q2 cj32

 ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ k23 ¼ p2 c13 þ ei c23 þ q2 cj33 ; k24 ¼ p2 c14 þ ei c24 þ q2 cj34    ðjÞ 
ðjÞ ðjÞ ðjÞ ðjÞ k31 ¼ p2  c2 pei c11 þ p c2  p2 c21 þ q3 þ qc2 Si c31 þ C i c41    ðjÞ 
ðjÞ ðjÞ ðjÞ ðjÞ k32 ¼ p2  c2 pej c12 þ p c2  p2 c22 þ q3 þ qc2 Sj c32 þ C i c42    ðjÞ 
ðjÞ ðjÞ ðjÞ ðjÞ k33 ¼ p2  c2 pej c13 þ p c2  p2 c23 þ q3 þ qc2 Si c33 þ C i c43    ðjÞ 
ðjÞ ðjÞ ðjÞ ðjÞ k34 ¼ p2  c2 pej c14 þ p c2  p2 c24 þ q3 þ qc2 Sj c34 þ C i c44 ðjÞ

k41 ¼ p2 ðjÞ

k43 ¼ p2

ðjÞ






































ej cð11jÞ þ cð21jÞ  q2 C j cð31jÞ þ Si cð41jÞ ; kð42jÞ ¼ p2 ej cð12jÞ þ cð22jÞ  q2 C j cð32jÞ þ Sj cð42jÞ ej cð13jÞ þ cð23jÞ  q2 C j cð33jÞ þ Sj cð43jÞ ; kð44jÞ ¼ p2 ej cð14jÞ þ cð24jÞ  q2 C j cð34jÞ þ Sj cð44jÞ

c ¼ 2

where c11 ¼




p q

Hl0 2 , EI



sr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 Hl20 ml40 Hl20 2 ¼ þ x EI  2EI , 2EI

qðpðb3 þC j ÞðC j ej 1Þðqpej b4 þb2 qÞ

Sj þej pS2j

ej ¼ eplj ,


 C j ¼ cos qlj ,


 Sj ¼ sin qlj

Þ

yj

ðjÞ c12

ðjÞ c13

¼

¼

ðjÞ

c14 ¼

ðjÞ

c21 ¼

ðjÞ c22

ðjÞ c23



 
 ðjÞ  ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ  ðjÞ ðjÞ pb4 C j  1 þ C j 1 þ b2 þ b3  C j b2  b3 þ C j ej pSj  b4 ej  1 qSj  b2 þ ej qS2 yj





  ðjÞ ðjÞ  ðjÞ q b2 þ ej qSj  p 1 þ b3 1 þ C j ej þ b4 ej Sj yj


 ðjÞ  ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ b4 C j  1 ej p  1 þ b2 þ b3 q þ b2 C j q þ b3 þ C j ej q þ 1 þ b3 ej pSj yj



   ðjÞ ðjÞ ðjÞ q b3 þ C j C j  ej p þ b4 p þ b1 þ ej q Sj þ pS2j yj

¼

¼




 ðjÞ  ðjÞ ðjÞ ðjÞ  ðjÞ ðjÞ  ðjÞ b4 C j  1 ej p þ C j b1  C j  b1 C j þ b3 ej  1 þ ej q þ 1 þ b3 ej pSj þ b4 ej  1 qSj  1 þ b1 qS2j yj

 ðjÞ  ðjÞ ðjÞ 1 þ b3 C j  ej pq þ q b4 p þ q þ b4 q Sj


ðjÞ

c24 ¼

yj

;


 ðjÞ  ðjÞ ðjÞ ðjÞ  ðjÞ b4 p  C j p þ b1  C j  b1 C j þ b3 ej  1 þ ej q þ 1 þ b3 pSj yj

and

25

H. Fei et al. / Mechanical Systems and Signal Processing 140 (2020) 106638

ðjÞ c31

¼


 




 ðjÞ ðjÞ ðjÞ  ðjÞ p b4 e2j  1 p þ  2 þ b2 ej þ b1 C j ej  1 þ C j 1 þ b2 þ e2j q þ e2j  1 pSj yj

ðjÞ 

ðjÞ c32

ðjÞ c33

¼

¼

b4

ðjÞ c41

ðjÞ c42

ðjÞ c43

ðjÞ c44

¼

¼

¼

¼

¼
















pSj

;

yj 




    ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ p b4 e2j  1 p  1 þ b1  C i b2 þ b2  2 þ b1 C j ej þ e2j q yj ðjÞ 

ðjÞ c34



ej  1 2 p þ C j ej  1 1 þ bð1jÞ þ bð2jÞ þ ej q þ 1 þ bð1jÞ þ ej bð2jÞ þ ej

b4
















;

 

ej  1 2 p  ej  1 1 þ bð1jÞ þ bð2jÞ þ ej q  bð2jÞ þ 2 þ bð1jÞ ej pSj yj





  ðjÞ ðjÞ ðjÞ p  b3 þ C j e2j  1 p þ 1 þ b2 þ b1 ej þ e2j qSj yj

;

;


 
  2

ðjÞ ðjÞ  ðjÞ ðjÞ ðjÞ ðjÞ  b1 C j  1 þ C j þ b3 ej  1  2 þ b2 ej þ C j ej b2 þ ej p þ ej  1 1 þ b1 þ b2 þ ej qSj yj




   ðjÞ ðjÞ ðjÞ 1 þ e2j p  b2 þ 2 þ b1 ej qSj p 1 þ b3 yj

;


  2
ðjÞ  ðjÞ  ðjÞ  1 þ b2 1 þ C j  b3 1 þ ej þ b1 1 þ C j þ 2C j  ej ej p yj


  2  
ðjÞ  ðjÞ  ðjÞ  yj ¼ b3 1 þ C j 1 þ ej þ b1 1 þ C j 1 þ C j ej þ C j  ej 2 þ b2 1 þ C j þ 2C j ej pq 
 

  

ðjÞ  ðjÞ ðjÞ ðjÞ ðjÞ 1 þ b3 1 þ ej p2  1 þ b1 þ b2 þ ej q2 Sj þ b2 þ 2 þ b1 ej pqS2j  1 þ ej
 
 2  ðjÞ þb4 p 1 þ C j 1 þ e2j p  1 þ ej qSj The global dynamic stiffness matrix Kð0Þ for an n-segment cable system with the clamped boundary condition

2 6 6 6 6 6 6 6 6 6 6 6 6 6 ð0Þ K ¼6 6 6 6 6 6 6 6 6 6 6 6 6 4

ð1Þ

ð2Þ

ð1Þ

ð2Þ

ð1Þ k44

ð2Þ þ k22

ð2Þ

k33 þ k11  x2 m1 k34 þ k12 ð1Þ k43

ð2Þ þ k21 ð2Þ

ð2Þ

ð2Þ

ð2Þ

k13

k14

0

0

 

0



ð2Þ k23

ð2Þ k24

0

0

  0

 .. .

 .. .

0 .. .

0 .. .

 .. .

ð3Þ

ð2Þ

ð3Þ

ð3Þ

ð3Þ

ð2Þ k44

ð3Þ þ k22

ð3Þ k23

k24

k33 þ k11  x2 m2 k34 þ k12 k13 k14

k31

k32

ð2Þ k41

ð2Þ k42

0

0

k31

0 .. .

0 .. .

k41 .. .

k42 .. .

0 .. .

0 .. .

 .. .

0 .. .

0

0



0

ð2Þ k43

ð3Þ þ k21 ð3Þ ð3Þ

ð3Þ

k32

..

ð3Þ

. ..

ð3Þ

.

0 ðn2Þ

ðn1Þ

 

0

 

0

 

0

k31

 

0

k41

k33

þ k11 ðn2Þ

k43

0 þ k12

ðn1Þ

þ k21

ðn2Þ

þ k22

ðn1Þ ðn1Þ

k44

ðn1Þ

ðn1Þ

k32

ðn1Þ

k42

ðn1Þ

k13

ðn1Þ

ðn1Þ

k23 ðn1Þ

k33

ðnÞ

þ k11  x2 mn1 ðn1Þ

k43

7 7 7 7 7 7 7 7 7 0 7 7 .. 7 . 7 7 7 7 0 7 7 7 ðn1Þ 7 k14 7 7 ðn1Þ k24 7 7 7 ðn1Þ ðnÞ 7 k34 þ k12 5 0 .. .

0

ðn2Þ

 x2 mn2 k34

3

0

ðnÞ

þ k21

ðn1Þ

k44

ðnÞ

þ k22

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