# Influences of Two Calculation Methods about Dynamic Tension on Vibration Characteristics of Cable-Bridge Coupling Model

Authors: Guangyun Min; Xiaohui Liu; Chuan Wu; Shuguang Yang; Mengqi Cai
Journal: Discrete Dynamics in Nature and Society
 (2021)

---

## Abstract

For the cable-bridge coupling model, the dynamic tension of cables is an important parameter to study the vibration characteristics of the model. Based on this concept, two calculation methods about dynamic tension of cables were introduced in great detail, and the influences of these two calculation methods on the vibration characteristics of cable-bridge coupling model were systematically investigated. Firstly, the vibration equation of the cable was derived based on the variational principle for Hamiltonian, and the vibration equation of the bridge deck was obtained by Newton’s law. Then, the vibration equation of the cable and bridge deck was transformed into ordinary differential vibration equation by the Galerkin method. In addition, the differences of the coefficients in the ordinary differential vibration equation obtained by these two calculation methods about dynamic tension were compared, and a parameter analysis was listed. Finally, the resonance mode of the cable-bridge coupling model was analyzed by a multiple scales method, and an example analysis was listed. The results of parameter analysis show that there are obvious differences in the linear coefficient and nonlinear coefficient of the ordinary differential vibration equation obtained by these two calculation methods. The results of example analysis show that, for the cable-bridge coupling model with 1 : 1 resonance, the amplitude of the model would not be different because of the two calculation methods about dynamic tension, but the amplitude of the cable would be affected by the calculation method significantly. It can be found that the research conclusions here can be helpful to the perfection of theoretical modeling and has certain guiding value for practical engineering.


---

## Body

## 1. Introduction

Because cables have the characteristics of light-weight, large-flexibility, and small-damping, those are used in civil engineering, navigation engineering, aerospace engineering, and other fields usually [1–4]. In civil engineering, from the investigation of long-span stay-cable bridges, it can find that large-scale vibration of stay-cables may occur under the loads of winds, rains, pedestrians, or vehicles [5–7]. In addition, the vibration characteristics of stay-cables are also affected by the tower column and bridge deck, which would make the dynamic behavior of stay-cables be more complex. For example, when the vibration frequency of stay-cables is proportional to the vibration frequency of the bridge deck, the resonance would occur in the cable-bridge coupling system. And in this condition, even small disturbances would also let stay-cables generate large displacement.Scholars have made lots of contributions to the study of the vibration characteristics of the cable-bridge coupling model. In 1998, the authors in [8] established the cable-bridge coupling model and the vibration equation of it is derived. The research results showed that, under certain conditions, the stay-cables may resonate with the bridge deck, which would bring adverse effects to the safety of the bridge. In 2001, the parametric vibration model of stay-cables was established in [9], and the vibration characteristics of stay-cables were analyzed. Then, it proposed to apply dampers on the bridge deck to control the large-amplitude vibration of stay-cables. In 2003, the cable-bridge coupling model was established in [10], and the displacement response of stay-cables was obtained by the fourth-order Runge–Kutta method. The research results showed that the vibration characteristics of stay-cables and bridge deck are related to the sag and inclination of the stay-cables. In [11], the coupling model of bridge deck and double-cables is established, and the interaction between one cable and the other cable is further considered. The research results showed that, by changing the parameters of one cable, the vibration characteristics of another cable would be greatly affected; that is, the coupling interaction between cable and cable is significant. In 2012, the continuous nonlinear refined vibration model of tower-cable-bridge deck was established in [12], and the influences of frequency ratio of the bridge deck to cables, amplitude of excitation, force of cables, and damping on the vibration characteristics of the model were studied. In 2016, considering the synergistic effects of cable-tower-beam, the authors in [13] analyzed the parameter vibration mechanism of cables and calculated the dynamic characteristics of the bridge deck and cables. The research results can provide a theoretical value for the installation of vibration suppression devices for cables. In 2017, Zhao et al. [14] were concerned with the temperature effect on the vibration of a cable-stayed beam, the research results showed that the effect of temperature plays a dominant role on the vibration characteristics of the cable-stayed beam, and the effect is closely related with the initial tension force and the stiffness ratio.For investigating the vibration characteristics of arbitrary vibration models, such as cables [15, 16], bridges [17, 18], iced conductors [19, 20], and plates [21–24], the partial differential vibration equation of those vibration models should be established firstly. As known for us, for the cable-bridge coupling model, the dynamic tension of cables is an important parameter to study the vibration characteristics of the model. Based on this concept, two calculation methods about dynamic tension of cables were presented in this paper. Under the two calculation methods about dynamic tension of cables, the partial differential vibration equation of the model can be converted into ordinary differential vibration equation by two methods. One is to average the dynamic tension of cables over the span length of cables, and then, the Galerkin method is used to convert the partial differential vibration equation into ordinary differential vibration equation directly. The other is not to average the dynamic tension of cables over the span length, and then, the Galerkin method is used. There are different calculation methods about dynamic tension of cables, the partial differential vibration equation of the cable-bridge coupling model would be different, and then, the ordinary differential vibration equation of the model would also be different. Since the two calculation methods about dynamic tension both are reasonable and have their own advantages, whether these two calculation methods about dynamic tension would affect the coefficients of vibration equation of the cable-bridge coupling model or lead to the vibration characteristics of the model to be different has not been investigated by scholars.Based on these conclusions above, the influences of the two calculation methods about dynamic tension on the vibration characteristics of a cable-bridge coupling model has been systematically studied in this paper. Firstly, the cable-bridge coupling model is established, and the vibration equation of the model is derived in this paper. Second, the differences of the coefficients in the ordinary differential equation obtained by the two calculation methods of dynamic tension were compared. Furthermore, the resonance mode of the cable-bridge coupling model is analyzed by the multiple scales method. Finally, the influence of the two calculation methods of dynamic tension on the vibration characteristics of cable-bridge coupling model is further analyzed. The research results of this paper can be good for improving the theoretical modeling and offering some reference values to the actual engineering.


## 2. Vibration Equation

In order to facilitate the study and reflect the essence of the research problem in this paper, the following assumptions are made:(1)
The bending stiffness, torsional stiffness, and shear stiffness of the cable are not considered.(2)
The static equilibrium configuration of the cable is represented by parabola.(3)
The constitutive relation of the cable satisfies Hooke’s law.In this paper, the bridge deck is simplified as an equivalent mass block with certain stiffness. And, based on this concept, the cable-bridge coupling model can be established, which is shown in Figure1.Figure 1
Cable-bridge coupling model.In Figure1, y is the static equilibrium configuration of the cable, ξ is the dynamic displacement of the cable deviating from the static equilibrium configuration, θ is the angle between the cable and the horizontal direction, and X is the displacement of the bridge deck.According to [20], it can be obtained that the vibration of the cable would mainly occur in the y-axis direction (in-plane), so this paper would only consider the vibration of the cable in the y-axis direction (in-plane). Based on the variational principle for Hamiltonian, the following results can be obtained:(1)∫t1t2δKV−δ∏+δWdt=0,where KV represents the kinetic energy of the cable, ∏ represents the potential energy of the cable, W represents the work done by the non-conservative forces, t represents the time, t1 is the initial time, and t2 is the final time.The items in equation (1) are as follows:(2)KV=12∫0Lm∂ξ∂t2dx;∏=∫0LHh+12EAh2dx;W=∫0Lmgcosθ+ςC∂ξ∂tdx,where H represents the initial static tension, EA represents the tensile stiffness, h is the dynamic tension, m is the unit mass of the cable, ςC is the damping coefficient of the cable, L is the span length of the cable, and g is the acceleration of gravity.By substituting equations (2) into (1), the in-plane vibration equation of the cable can be obtained as follows:(3)H+h∂2ξ∂x2+hd2ydx2=m∂2ξ∂t2+ςC∂ξ∂t.According to Newton’s law, the vibration equation of bridge deck is(4)M∂2X∂t+C∂X∂t+KX+h=0,where K is the equivalent stiffness of the bridge deck, C is the viscous damping of the bridge deck, and M is the equivalent mass of the bridge deck.The vibration equation of the cable-bridge coupling model can be obtained by combining equations (3) and (4), which is(5)H+h∂2ξ∂x2+hd2ydx2=m∂2ξ∂t2+ςC∂ξ∂t,M∂2X∂t+C∂X∂t+KX+h=0.


## 3. Two Calculation Methods

### 3.1. The First Calculation Method

In order to facilitate the analysis about vibration characteristics of the cable-bridge coupling model, the calculation equation of dynamic tension about the cable should be obtained firstly. The microelement length ds on the static equilibrium configuration of the cable and the microelement length ds’ on the dynamic equilibrium configuration of the cable are taken for the research object, respectively (see Figure 2 for details).Figure 2
Microelement diagram of cable.The cable can be regarded as a continuum, and then, the calculation equation about dynamic tension of the cable can be obtained according to the continuum theory, which is(6)h=EAex=ds′+dsds′−dsdsds′+ds≈ds′2−ds22ds2≈EAdydx∂ξ∂x+12∂ξ∂x2,where e(x) is the strain.Equation (6) is the dynamic tension caused by the change in the configuration of the cable, and the change in dynamic tension caused by the vibration of bridge deck is not considered. If the influence of the vibration of bridge deck on dynamic tension is considered, equation (2) can be rewritten as follows:(7)h1=EAε≈EAdydx∂ξ∂x+12∂ξ∂x2+XL.Equation (7) is the first calculation method of dynamic tension.By substituting equation (7) into (5), the vibration equation of cable-bridge coupling model under the first calculation method about dynamic tension can be obtained, which is(8)H+EAdydx∂ξ∂x+12∂ξ∂x2+XL∂2ξ∂x2+EAdydx∂ξ∂x+12∂ξ∂x2+XLd2ydx2=m∂2ξ∂t2+ςC∂ξ∂t,M∂2X∂t+C∂X∂t+KX+EAdydx∂ξ∂x+12∂ξ∂x2+XL=0.


### 3.2. The Second Calculation Method

In practical engineering, the axial vibration of the cable is far less than its transverse vibration; that is to say, the effect of axial inertia force of the cable can be ignored. Based on this concept, equation (7) can be averaged over the range of 0 to L; that is, equation (7) can be rewritten as follows:(9)h2=EAε≈EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+X.Equation (9) is the second calculation method of dynamic tension.By substituting equation (9) into (5), the vibration equation of the cable-bridge coupling model under the second calculation method about dynamic tension can be obtained, which is(10)H+EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+X∂2ξ∂x2+EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+Xd2ydx2=m∂2ξ∂t2+ςC∂ξ∂t,M∂2X∂t+C∂X∂t+KX+EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+X=0.


### 3.3. Differences in Coefficients

According to [20], the vibration characteristics of the cable are mainly affected by the first-order mode shape. Therefore, the first-order mode truncation method is used here, and the influences of higher-order mode shape on the vibration behaviors of the cable would be considered in subsequent research.According to the first-order mode truncation method, the dynamic displacement (ξ) of the cable can be written as the product of time function and mode shape function, which is(11)ξ=qtϕx.By substituting equation (11) into (8), based on the Galerkin method, the ordinary differential vibration equation of the cable-bridge coupling model under the first calculation method about dynamic tension can be obtained:(12)ν1q¨t+ν2q˙t+ν3qt+ν4q2t+ν5q3t+ν6X+ν7Xqt=0,ν8X¨t+ν9X˙t+ν10Xt+ν11qt+ν12q2t=0,where the dot represents the derivative of t and the calculation equations of coefficients involved in equation (12) are as follows:(13)ν1=−∫0Lmϕ2xdx;ν2=−∫0LςCϕ2xdx;ν3=∫0LHϕ″xϕxdx+∫0LEAy′y″ϕ′xϕxdx;ν4=∫0LEAy′ϕ′xϕxϕxdx+∫0LEA2y″ϕ′2xϕxdx;ν5=∫0LEA2ϕ′2xϕ″xϕxdx;ν6=EAL∫0Ly″ϕxdx;ν7=EAL∫0Lϕ″xϕxdx;ν8=∫0LMϕxdx;ν9=∫0LCϕxdx;ν10=∫0LEALϕxdx+∫0LKϕxdx;ν11=∫0LEAy′ϕ′xϕxdx;ν12=∫0LEA2ϕ′2xϕxdx.In the same way, by substituting equation (11) into (10), based on the Galerkin method, the ordinary differential vibration equation of the cable-bridge coupling model under the second calculation method about dynamic tension can be obtained:(14)ϑ1q¨t+ϑ2q˙t+θ3qt+ϑ4q2t+ϑ5q3t+ϑ6X+ϑ7Xqt=0,ϑ8X¨t+ϑ9X˙t+ϑ10Xt+ϑ11qt+ϑ12q2t=0.The calculation equations of coefficients involved in equation (14) are as follows:(15)ϑ1=−∫0Lmϕ2xdx;ϑ2=−∫0LςCϕ2xdx;ϑ3=∫0LHϕ″xϕxdx+∫0LEAL∫0Ly′ϕ′xdxy″ϕxdx;ϑ4=∫0LEAL∫0Ly′ϕ′xdxϕx″ϕxdx+∫0LEA2L∫0Lϕ′2xdxy″ϕxdx;ϑ5=∫0LEA2L∫0Lϕ′2xdxϕ″xϕxdx;ϑ6=EAL∫0Ly″ϕxdx;ϑ7=EAL∫0Lϕ″xϕxdx;ϑ8=∫0LMϕxdx;ϑ9=∫0LCϕxdx;ϑ10=∫0LKϕxdx+∫0LEALϕxdx;ϑ11=∫0L∫0LEALy′ϕ′xdxϕxdx;ϑ12=∫0L∫0LEA2Lϕ′2xdxϕxdx.By comparing equations (13) and (15), it can be found that the coefficients (ν3 and ϑ3; ν4 and ϑ4; ν5 and ϑ5; ν11 and ϑ11; ν12 and ϑ12) in the ordinary differential vibration equation of cable-bridge coupling model obtained by the first calculation method about dynamic tension and the second calculation method about dynamic tension have obvious differences. The differences of these coefficients may lead to the differences in vibration characteristics of the cable-bridge coupling model, and the differences between them would also lead to different natural frequencies of the model. Therefore, from the analysis here, it can also be found that it is necessary to consider the two calculation methods about dynamic tension when studying the vibration characteristics of cable-bridge coupling model.


### 3.4. Parameter Analysis

From the analysis form in the section above, it can be seen that there are some differences in some coefficients of the ordinary differential vibration equation obtained by the two calculation methods about dynamic tension. The differences of some coefficients in vibration equation of cable-bridge coupling model under the two calculation methods about dynamic tension are studied systematically based on the physical parameters in [11]. Also, based on Matlab, the curves about influences of span length on coefficients are obtained, which is shown in Figure 3.Figure 3
Influences of span length on coefficients: (a) coefficients ofν3 and ϑ3; (b) coefficients of ν4 and ϑ4; (c) coefficients of ν5 and ϑ5; (d) coefficients of ν11 and ϑ11; (e) coefficients of ν12 and ϑ12.
(a)(b)(c)(d)(e)It can be seen from Figure3 that, with the increasing L, the coefficients (ν4 and ϑ4, ν5 and ϑ5, and ν11 and ϑ11) would increase; with the increasing L, the coefficients (ν3 and ϑ3 and ν12 and ϑ12) would decrease; therefore, the coefficients obtained by the two calculation methods have the same variation law; that is, these two calculation methods would not affect the variation law of the coefficients. It can also be obtained from Figure 3 that when L is greater than 300 m, the ratio of coefficients obtained by these two calculation methods is a certain value; that is, different calculation methods of dynamic tension would affect the value of coefficients; when L is less than 300 m, the ratio of coefficients obtained by the two methods is a variable. Moreover, with the increasing L, the ratio of coefficients would increase, and finally, the ratio would tend to be constant; these two calculation methods would also affect the vibration frequency of the cable, which may lead to the vibration characteristics of the cable-bridge coupling model to be different. Therefore, it has certain theoretical significances and engineering values to study the influence of these two different calculation methods about dynamic tension on the vibration characteristics of the cable-bridge coupling model. In the following section, the influence of these two calculation methods about dynamic tension on the internal resonance of the cable-bridge coupling model would also be studied based on a numerical example.


## 3.1. The First Calculation Method

In order to facilitate the analysis about vibration characteristics of the cable-bridge coupling model, the calculation equation of dynamic tension about the cable should be obtained firstly. The microelement length ds on the static equilibrium configuration of the cable and the microelement length ds’ on the dynamic equilibrium configuration of the cable are taken for the research object, respectively (see Figure 2 for details).Figure 2
Microelement diagram of cable.The cable can be regarded as a continuum, and then, the calculation equation about dynamic tension of the cable can be obtained according to the continuum theory, which is(6)h=EAex=ds′+dsds′−dsdsds′+ds≈ds′2−ds22ds2≈EAdydx∂ξ∂x+12∂ξ∂x2,where e(x) is the strain.Equation (6) is the dynamic tension caused by the change in the configuration of the cable, and the change in dynamic tension caused by the vibration of bridge deck is not considered. If the influence of the vibration of bridge deck on dynamic tension is considered, equation (2) can be rewritten as follows:(7)h1=EAε≈EAdydx∂ξ∂x+12∂ξ∂x2+XL.Equation (7) is the first calculation method of dynamic tension.By substituting equation (7) into (5), the vibration equation of cable-bridge coupling model under the first calculation method about dynamic tension can be obtained, which is(8)H+EAdydx∂ξ∂x+12∂ξ∂x2+XL∂2ξ∂x2+EAdydx∂ξ∂x+12∂ξ∂x2+XLd2ydx2=m∂2ξ∂t2+ςC∂ξ∂t,M∂2X∂t+C∂X∂t+KX+EAdydx∂ξ∂x+12∂ξ∂x2+XL=0.


## 3.2. The Second Calculation Method

In practical engineering, the axial vibration of the cable is far less than its transverse vibration; that is to say, the effect of axial inertia force of the cable can be ignored. Based on this concept, equation (7) can be averaged over the range of 0 to L; that is, equation (7) can be rewritten as follows:(9)h2=EAε≈EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+X.Equation (9) is the second calculation method of dynamic tension.By substituting equation (9) into (5), the vibration equation of the cable-bridge coupling model under the second calculation method about dynamic tension can be obtained, which is(10)H+EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+X∂2ξ∂x2+EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+Xd2ydx2=m∂2ξ∂t2+ςC∂ξ∂t,M∂2X∂t+C∂X∂t+KX+EAL∫0Ldydx∂ξ∂x+12∂ξ∂x2dx+X=0.


## 3.3. Differences in Coefficients

According to [20], the vibration characteristics of the cable are mainly affected by the first-order mode shape. Therefore, the first-order mode truncation method is used here, and the influences of higher-order mode shape on the vibration behaviors of the cable would be considered in subsequent research.According to the first-order mode truncation method, the dynamic displacement (ξ) of the cable can be written as the product of time function and mode shape function, which is(11)ξ=qtϕx.By substituting equation (11) into (8), based on the Galerkin method, the ordinary differential vibration equation of the cable-bridge coupling model under the first calculation method about dynamic tension can be obtained:(12)ν1q¨t+ν2q˙t+ν3qt+ν4q2t+ν5q3t+ν6X+ν7Xqt=0,ν8X¨t+ν9X˙t+ν10Xt+ν11qt+ν12q2t=0,where the dot represents the derivative of t and the calculation equations of coefficients involved in equation (12) are as follows:(13)ν1=−∫0Lmϕ2xdx;ν2=−∫0LςCϕ2xdx;ν3=∫0LHϕ″xϕxdx+∫0LEAy′y″ϕ′xϕxdx;ν4=∫0LEAy′ϕ′xϕxϕxdx+∫0LEA2y″ϕ′2xϕxdx;ν5=∫0LEA2ϕ′2xϕ″xϕxdx;ν6=EAL∫0Ly″ϕxdx;ν7=EAL∫0Lϕ″xϕxdx;ν8=∫0LMϕxdx;ν9=∫0LCϕxdx;ν10=∫0LEALϕxdx+∫0LKϕxdx;ν11=∫0LEAy′ϕ′xϕxdx;ν12=∫0LEA2ϕ′2xϕxdx.In the same way, by substituting equation (11) into (10), based on the Galerkin method, the ordinary differential vibration equation of the cable-bridge coupling model under the second calculation method about dynamic tension can be obtained:(14)ϑ1q¨t+ϑ2q˙t+θ3qt+ϑ4q2t+ϑ5q3t+ϑ6X+ϑ7Xqt=0,ϑ8X¨t+ϑ9X˙t+ϑ10Xt+ϑ11qt+ϑ12q2t=0.The calculation equations of coefficients involved in equation (14) are as follows:(15)ϑ1=−∫0Lmϕ2xdx;ϑ2=−∫0LςCϕ2xdx;ϑ3=∫0LHϕ″xϕxdx+∫0LEAL∫0Ly′ϕ′xdxy″ϕxdx;ϑ4=∫0LEAL∫0Ly′ϕ′xdxϕx″ϕxdx+∫0LEA2L∫0Lϕ′2xdxy″ϕxdx;ϑ5=∫0LEA2L∫0Lϕ′2xdxϕ″xϕxdx;ϑ6=EAL∫0Ly″ϕxdx;ϑ7=EAL∫0Lϕ″xϕxdx;ϑ8=∫0LMϕxdx;ϑ9=∫0LCϕxdx;ϑ10=∫0LKϕxdx+∫0LEALϕxdx;ϑ11=∫0L∫0LEALy′ϕ′xdxϕxdx;ϑ12=∫0L∫0LEA2Lϕ′2xdxϕxdx.By comparing equations (13) and (15), it can be found that the coefficients (ν3 and ϑ3; ν4 and ϑ4; ν5 and ϑ5; ν11 and ϑ11; ν12 and ϑ12) in the ordinary differential vibration equation of cable-bridge coupling model obtained by the first calculation method about dynamic tension and the second calculation method about dynamic tension have obvious differences. The differences of these coefficients may lead to the differences in vibration characteristics of the cable-bridge coupling model, and the differences between them would also lead to different natural frequencies of the model. Therefore, from the analysis here, it can also be found that it is necessary to consider the two calculation methods about dynamic tension when studying the vibration characteristics of cable-bridge coupling model.


## 3.4. Parameter Analysis

From the analysis form in the section above, it can be seen that there are some differences in some coefficients of the ordinary differential vibration equation obtained by the two calculation methods about dynamic tension. The differences of some coefficients in vibration equation of cable-bridge coupling model under the two calculation methods about dynamic tension are studied systematically based on the physical parameters in [11]. Also, based on Matlab, the curves about influences of span length on coefficients are obtained, which is shown in Figure 3.Figure 3
Influences of span length on coefficients: (a) coefficients ofν3 and ϑ3; (b) coefficients of ν4 and ϑ4; (c) coefficients of ν5 and ϑ5; (d) coefficients of ν11 and ϑ11; (e) coefficients of ν12 and ϑ12.
(a)(b)(c)(d)(e)It can be seen from Figure3 that, with the increasing L, the coefficients (ν4 and ϑ4, ν5 and ϑ5, and ν11 and ϑ11) would increase; with the increasing L, the coefficients (ν3 and ϑ3 and ν12 and ϑ12) would decrease; therefore, the coefficients obtained by the two calculation methods have the same variation law; that is, these two calculation methods would not affect the variation law of the coefficients. It can also be obtained from Figure 3 that when L is greater than 300 m, the ratio of coefficients obtained by these two calculation methods is a certain value; that is, different calculation methods of dynamic tension would affect the value of coefficients; when L is less than 300 m, the ratio of coefficients obtained by the two methods is a variable. Moreover, with the increasing L, the ratio of coefficients would increase, and finally, the ratio would tend to be constant; these two calculation methods would also affect the vibration frequency of the cable, which may lead to the vibration characteristics of the cable-bridge coupling model to be different. Therefore, it has certain theoretical significances and engineering values to study the influence of these two different calculation methods about dynamic tension on the vibration characteristics of the cable-bridge coupling model. In the following section, the influence of these two calculation methods about dynamic tension on the internal resonance of the cable-bridge coupling model would also be studied based on a numerical example.


## 4. Multiple Scales Method

In practical engineering, when the frequency of the cable is proportional to the frequency of the bridge deck, it would let the cable-bridge coupling system generate resonance. The resonance would greatly let the amplitude of the cable and bridge deck be much bigger, which would damage the stability of structure and threaten the safety of personal. Therefore, it is necessary to analyze the resonance mode of the cable-bridge coupling system. In recent decades, many effective methods have been proposed to solve the nonlinear vibration equation. The multiple scales method is a common method to solve the nonlinear vibration equation and is one of the effective methods to analyze the nonlinear resonance mode. Based on this concept, the multiple scales method is used to solve the resonance mode of the cable-bridge coupling system here.From equations (14) and (12), it can be obtained that the form of vibration equation under the two calculation methods of dynamic tension is the same even though the coefficients of some items in the vibration equation are different. For the form of the vibration equation is exactly the same, only the solution process of resonance mode of the cable-bridge coupling system under the first calculation method of dynamic tension is listed as shown in the following section. And the resonance mode of the cable-bridge coupling system under the second calculation method of dynamic tension can be deduced by analogy.In order to satisfy the solution form of multiple scales method, equation (12) should be rewritten as follows:(16)Q¨1+ωQ12Q1=εξQ1Q˙1+ξQ2Q12+ξQ3Q13+ξQ4Q2+ξQ5Q1Q2,Q¨2+ωQ22Q2=εξQ6Q˙2+ξQ7Q1+ξQ8Q12,where Q1 is q(t), Q2 is X(t), and ε is a dimensionless small parameter. The calculation formulas of other coefficients are as follows:(17)ξQ1=−ν2ν1;ξQ2=−ν4ν1;ξQ3=−ν5ν1;ξQ4=−ν6ν1;ξQ5=−ν7ν1;ξQ6=−ν9ν8;ξQ7=−ν11ν8;ξQ8=−ν12ν8;ωQ12=ν3ν1;ωQ22=ν10ν8.According to [20, 25–28], the solution of equation (16) can be set as(18)Q1=Q11T0,T1+εQ12T0,T1+Οε2,Q2=Q21T0,T1+εQ22T0,T1+Οε2,where Q11 and Q21 are the periodic solutions of the system, Q12 and Q22 are the modified solutions of the system, O(ε2) is a higher-order small quantity, and T0 and T1 are two time scales (T0=t and T1=εt).The operator formula is introduced, which is(19)D0=∂∂T0,D1=∂∂T1.By substituting equations (18) and (19) into equation (16) and equating the coefficients of ε0 and ε1 to zero independently, it can be obtained that(20)ε0:D02Q11+ωQ12Q11=0,D02Q21+ωQ22Q21=0,(21)ε1:D02Q12+ωQ12Q12=−2D0D1Q11+ξQ1D0Q11+ξQ2Q112+ξQ3Q113+ξQ4Q21+ξQ5Q11Q21,D02Q22+ωQ22Q22=−2D0D1Q21+ξQ6D0Q21+ξQ7Q11+ξQ8Q112.According to the solvability condition of differential equation, the solution of equation (20) can be set as follows:(22)Q11=AQ1T1expiωQ1T0+CC,Q21=AQ2T1expiωQ2T0+CC,where AQ1 is the amplitude of the cable, AQ2 is the amplitude of the bridge deck, i is the imaginary unit, and CC is conjugate terms.(23)D02Q12+ωQ12Q12=−2iωQ1AQ1′expiωQ1T0+ξQ1AQ1iωQ1expiωQ1T0+ξQ2AQ12exp2iωQ1T0+AQ1A¯Q1+ξQ3AQ13exp3iωQ1T0+3AQ12A¯Q1expiωQ1T0+ξQ4AQ2expiωQ2T0+ξQ5AQ1AQ2expiωQ1+ωQ2T0+AQ1A¯Q2expiωQ1−ωQ2T0+CC,D02Q22+ωQ22Q22=−2iωQ2AQ2′expiωQ2T0+ξQ6AQ2iωQ2expiωQ2T0+ξQ7AQ1expiωQ1T0+ξQ8AQ12exp2iωQ1T0+AQ1A¯Q1+CC,where “'” in equation (23) represents the derivation of the time scale T1.According to equation (23), there would be a 1 : 1 resonance mode in the cable-bridge coupling system. In the following section, the influence of the two calculation methods about dynamic tension on the resonance of the cable-bridge coupling system would be analyzed.


## 5. Example Analysis

When equivalent stiffness of bridge deck isK = 5 × 106 N/m, equivalent mass of bridge deck is M = 6.13 × 105 kg, tension is H = 105.88 N, and span length L is 300 m. At this time, there is a 1 : 1 resonance mode in the cable-bridge coupling system under the first calculation methods of dynamic tension.When equivalent stiffness of bridge deck isK = 5 × 106 N/m, equivalent mass of bridge deck is M = 4 × 105 kg, tension is H = 105.88 N, and span length L is 300 m. At this time, there is a 1 : 1 resonance mode in the cable-bridge coupling system under the second calculation methods of dynamic tension.When equivalent stiffness of bridge deck isK = 5 × 106 N/m, equivalent mass of bridge deck is M = 5 × 105 kg, tension is H = 1.25 × 106 N, and span length L is 300 m. At this time, there is no resonance mode in the cable-bridge coupling system under the two calculation methods of dynamic tension.Supposing that the initial displacement of the cable is 0.1 m and that of the bridge deck is 0.05 m. According to Matlab, the displacement response curves of the cable-bridge coupling system under the two calculation methods of dynamic tension can be obtained, which is shown in Figures4 and 5.Figure 4
Displacement response (with 1 : 1 response mode): (a) the first calculation method of dynamic tension; (b) the second calculation method of dynamic tension.
(a)(b)Figure 5
Displacement response (without 1 : 1 response mode): (a) the first calculation method of dynamic tension; (b) the second calculation method of dynamic tension.
(a)(b)From Figure4, it can be obtained that the beat frequency of displacement response would be different due to the differences of the calculation methods of dynamic tension. The amplitudes of the bridge deck under the two calculation methods of dynamic tension are 0.05 m; the amplitude of the cable under the first calculation method of dynamic tension is 0.366 m, and the amplitude of the cable under the second calculation method of dynamic tension is 0.224 m, which is a noteworthy phenomenon. The reason for this phenomenon can be explained as follows: amplitude is a form of energy; that is, when the resonance occurs in the cable-bridge coupling system, the energy of the cable-bridge coupling system is certain, and the energy would not be different due to the different calculation methods about dynamic tension. However, the second calculation method about dynamic tension has ignored the inertial force on the cable; that is to say, it has ignored a part of energy, which would make the amplitude of the cable decrease.From Figure5, it can be obtained that when the cable-bridge coupling system does not meet the 1 : 1 resonance condition, the displacement response curves obtained by the first calculation method about dynamic tension and the second calculation method about dynamic tension are almost the same; that is, any calculation method of dynamic tension is feasible for the nonresonance condition.Therefore, it is reasonable to choose the first calculation method of dynamic tension to solve the displacement response of the cable due to the large energy generated by the resonance (the amplitude would also be large) when analyzing the resonance of the cable-bridge coupling system. For the cable-bridge coupling system without resonance, the above two calculation methods are reasonable to solve the displacement response of the cable.


## 6. Conclusion

Based on the theoretical analysis and numerical simulation, the influences of the two calculation methods about dynamic tension of cables on the linear coefficient and nonlinear coefficient in the ordinary differential vibration equation are analyzed. Also, the displacement response of the cable-bridge coupling model under the two calculation methods about dynamic tension is compared. The following conclusions can be listed:(1)
There are obvious differences in some coefficients in the ordinary differential vibration equation of the cable-bridge coupling model obtained by the two calculation methods about dynamic tension. Because the vibration frequency of the cable is dominated by coefficientsν3 and ϑ3, the differences of coefficients ν3 and ϑ3 would make the vibration characteristics of cable-bridge coupling model be different.(2)
When theL is greater than 300 m, the ratios of coefficients obtained by these two calculation methods all are certain values which range from 1.5 to 2.0; while when the L is less than 300 m, the ratio of coefficients ν3 and ϑ3 is a change value, which increases with the increasing L and finally tends to be 1.62.(3)
The energy of the cable-bridge coupling model is certain, and the energy would not be different due to the different calculation methods about dynamic tension. However, the second calculation method about dynamic tension has ignored the inertial force on the cable; that is to say, a part of the energy is ignored, which would reduce the amplitude of the cable.(4)
For the cable-bridge coupling model with 1 : 1 resonance mode, because the energy generated by 1 : 1 resonance is large (the amplitude of cables would be large), it is reasonable to choose the first calculation method of dynamic tension to solve the displacement response of the cable.(5)
For the cable-bridge coupling model without a resonance mode, because the energy generated by nonresonance mode is small (the amplitude of cables would be small), these two calculation methods about dynamic tension both are reasonable to solve the displacement response of the cable.The influence of two calculation methods about dynamic tension on the vibration characteristics of the cable-bridge coupling model has been considered in this paper, and then, it can be found that the conclusions here can be helpful to the perfection of the cable-bridge coupling model and have some guiding value for practical engineering.



---

*Source: 6681954-2021-02-08.xml*