# Modal Interaction-Induced Parametric Resonance of Stayed Cable: A Combined Theoretical and Experimental Investigation

Authors: Qi-Chang Zhang; Su-Yu Cui; Zhi Fu; Jian-Xin Han
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The cable-stayed bridge is widely used due to its strong spanning capacity and navigability. However, flexible cables parametrically resonated by external excitation may result in instability or even damage to the bridge. To prevent such undesirable resonance, this paper discusses an in-plane modal interaction-induced parametric resonance of the stayed cable excited by the bridge deck vibration via nonlinear dynamic analysis. Based on the nonlinear distributed model, two modal governing equations of the cable are established via the Galerkin method. A certain working condition, when the external excitation frequency is close to the second-order natural frequency of the stay cable while nearly twice the first-order natural frequency, is theoretically and experimentally investigated. Specifically, the frequency response equations are obtained by the multiscale method, and the stability of solutions is examined through the Routh Hurwitz criterion. Theoretical and experimental results show that bridge deck vibration can induce not only the primary and superharmonic resonance of the cable but also the principal parametric resonance. Parametric resonance-induced bifurcations are also observed in the system. Particularly, the energy exchange from second-order primary resonance to first-order principal parametric resonance is found, which can induce the parametric resonance with the response amplitude one to three times higher than that of the primary resonance. This paper also validates the superiority of the present modal interaction model over the traditional single-mode model in practical engineering applications.


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## Body

## 1. Introduction

The cable-stayed bridge has a strong spanning ability, and the cable is a very efficient structural member due to its high strength and lightweight. However, because of its low mass, stiffness, and damping, the cable is prone to appear strong lateral vibration under the excitation of wind or the supporting end. Besides, secondary axial flow vibration, vortex-induced vibration behavior, and violent vibration excited by breeze can also be observed in practice [1]. To prevent such undesirable phenomena, plenty of scholars have made indispensable contributions to cable dynamics. Results show that the flexible cable parametrically resonated by excitations may result in instability or even damage of the bridge [2–5], which show greater destructiveness than that caused by the primary resonance [6, 7]. Dynamic analysis and design of parametrically excited cables are significant in the construction and application of the bridge [8–10].The vibration of the cable can be classified into in-plane and out-plane vibration. Each vibration involves one-modal or multimodal vibration. Many researchers study the in-plane and out-plane vibration of the cable through just a single-mode model. For example, Lilien et al. [11] simplified the cable to a tension string, assuming that the displacement excitation at the end of the cable was a cosine function. Costa et al. [12] studied the amplitude of the parametric vibration and the internal force of the cable with different inclination angles under vertical excitation. Gattulli et al. [13] confirmed that the sag affects only in-plane eigenvalues. Xia et al. [14] found that when subjected to a small dynamic vertical random displacement, only in-plane linear motion of the cable-stayed beam system occurs. Zhang et al. [15] developed a 1 : 1 principal parametric resonances model considering cable-beam coupling. Kang et al. [16] used a numerical method to study the first-order primary resonance and first-order parametric resonance among in-plane, out-plane, and bridge deck. The research results have a good reference value for the future research on how to suppress parametric resonance-induced cable vibration. Liu et al. [17] discussed a single mode in-plane vibration of the stay cable under axial excitation based on sag curve. Besides, Li et al. [18] numerically investigated the in-plane and out-plane coupled responses of a stay cable under the combined rain-wind and support excitations, compared with the results induced by single-source excitation. The result shows that the cable response amplitude under the combined excitations is smaller than that induced only by support excitation, while it is larger than that induced only by rain-wind excitation. Moreover, theoretical and experimental investigations on multicable are also carried out. Meng [19] studied two cables stayed beam structure. The numerical results show that the energy transfer between cables caused by the movement of deck exists and may lead to larger vibration of cables when the lower natural frequencies of cables are close to each other. Therefore, it has a good guiding significance on how to avoid safety hazards and pay attention to the design of cable-stayed bridge. Chen et al. [20] and Su et al. [21] designed the Xiangshan Harbor Model on a scale of 1 : 250, established the one-element cable system (OECS) and multielement cable system (MECS) finite element models, and compared it with the experimental data under vertical harmonic excitation. The analysis shows that the mathematical model of the multicable structure of the cable-stayed bridge is closer to the experimental results than that of the single-cable structure. However, ignoring modal interaction, the single-mode vibration response of the cable can only reflect the vibration of the corresponding mode, which may be not consistent with the actual phenomenon under a certain condition.Recently, the multimodal interaction induced parametric vibration of the cable of cable-stayed bridge has attracted much attention. Some meaningful works have been done in order to explore the complex parametric resonance among multimodal of member of the cable-stayed bridge. To study the stability regions in parameter space, Gonzalez-Buelga et al. [22] developed an analytical model, which included nonlinear coupling effects with other modes (first out-plane mode, second out-plane mode, and second in-plane mode). And their research reflects the importance of including the nonlinear coupling terms when studying the stability boundaries close to the 2 : 1 resonance region. Macdonald et al. [23] used the scaling and averaging method to calculate the steady-state amplitude of the cable under nonlinear modal interaction when the excitation frequency is close to any order natural frequency of the in-plane and out-plane models. Douthe et al. [24] proposed the simplest possible analytical model to study the coupling between parametric resonance and external resonances in stay cables affected by an anchorage motion with components both along and perpendicular to their axis. The gap between the complex mathematical interaction models and the actual cables design in structural engineering is bridged. Ying et al. [25] illustrated that the multimode coupling needs to be considered for the stochastic stability of the parameter-excited cable vibration under the random and periodic support disturbances. Gattulli et al. [26] have extensively explored the rich diversity and challenging complexity of cable excitation mechanisms and employed the exact linear eigensolution to formulate a nonlinear coupled two-degree-of-freedom beam-cable-beam model, which was defined in the reduced space of the modal amplitudes corresponding to a global and a local mode interaction. Demsic et al. [27] defined the expressions for the interaction resonance primary region, where the solution is unique, as well as the hysteresis regions, where the cable response depends on the initial conditions in the excitation parameter space. The expressions and conclusions derived in this paper are of universal significance. Kang et al. [28] investigated the multimodal interaction among the lowest in-plane and out-plane modes of cables and the lowest mode of the shallow arch by analyzing the frequency response and force-response curves of the system.To the best of our knowledge, researchers have carried out a lot of studies on the dynamic characteristics of the stay cable. However, the study on the mechanism of energy exchange between different modes and the bifurcation characteristics under parametric vibration is still insufficient. In this paper, the mechanism of energy exchange between modes of a stay cable under 2 : 1 internal resonance is revealed, and the bifurcation parameter characteristics of the system are theoretically calculated and experimentally studied. The purpose is to provide a better theoretical basis for dynamic design of the cable-stayed bridge, reducing the parametric amplitude [29–31] and even suppressing the parametric vibration [32] of stay cables. Firstly, the cable model is simplified moderately. Considering the cable sag [33], damping, and the nonlinear influence of tension variation during the cable vibration, the nonlinear coupled motion equation of the cable is obtained [34]. Then, Galerkin discretization [35] and the multiple scale method [36–38] are used to obtain the averaged equations. Stabilities of solutions are determined by the Routh Hurwitz criterion. The nonlinear coupling equations of the stay cable are numerically simulated by using Wolfram Mathematica [39]. The natural frequencies of the stay cable are simulated by COMSOL [40]. Finally, the parametric vibration of the stay cable is observed and analyzed through the designed experiment. Discussions and conclusions are drawn in the last section.


## 2. Problem Formulation

### 2.1. Reduced-Order Modeling

The cable is a flexible structure and has a certain sag due to its own weight. The geometric nonlinearity can obviously increase with the increase of cable length. To establish the distributed parameter model of cable system, the following assumptions are introduced [17].(1)
The influence of bending, torsion, and shear stiffness of cables is not considered.(2)
The gravity sag curve of the cable is a parabola.(3)
The deformation constitutive relation of the cable obeys Hooke's law, and the stress on each point is uniform.(4)
The influence of cable gravity on chordal tension is not considered.In Figure1, the x-axis coordinate coincides with the axial direction of the cable. The vibration of the cable in the x-y plane is called in-plane vibration. The length of the cable is l. The vertical and horizontal distances between the two ends of the cable are W and L, respectively. The inclination angle between the cable and the horizontal direction is α. Next, we intercept a microsegment for analysis.Figure 1
Spatial diagram of cable considering sag.The deformation and stress in thez-direction are ignored [12]. In the static case, the deformation of the cable is shown in the microelement section at the top of Figure 2. T is the tension along the tangent direction of the cable. M is the mass of the unit length of the cable. s is the arc length coordinate of the cable. According to Newton's second law, the static equilibrium equation can be obtained as(1)ddsTdyds=−Mgcosα,Figure 2
Stress analysis of the cable element.For the stay cable in a cable-stayed bridge, the ratio of sag to span is usually very small. Therefore,ds≈dx is taken [41], By using the boundary condition and integral constant, the sag curve of the cable in static state can be derived as(2)y=Mgl2cosα2Txl−x2l2,where g is the acceleration of gravity, and y is the sag value.The differential equations of motion are derived using the local coordinate system shown in Figure1. When the cable vibrates, the infinitesimal segment at the bottom of Figure 2 shows the displacement and stress of the cable element. t′ is the change of tangential tension relative to the initial tension during vibration. ξ1 and ξ2 are the viscous damping coefficients of the cable along the x and y axes, respectively. u′ and v′ are the displacements of the cable element at the x-axis and y-axis deviating from the static equilibrium position. Substituting equation (1) into equation (3), the in-plane vibration differential equation of the cable is obtained by Newton's second law [42].(3)∂∂sT+t′dyds+dv′ds=M∂2v′∂2t+ξ2∂v′∂t−Mgcosα.Simplifying the differential equation of motion yields(4)∂∂xT+t′∂v′∂x+t′dydx=M∂2v′∂2t+ξ2∂v′∂t.The connection between the cable and the pylon is assumed to be fixed. The lower end of the cable is excited by the vertical sine excitationDB=YBsinΩt, where YB, Ω are the external excitation amplitude and frequency, respectively. The displacements are labeled uB=DBsinα and vB=DBcosα for the longitudinal and transverse components, respectively. Then, we obtain(5)u′=u+sluB,v′=v+slvB,where u and v are the shape variables of the cable element relative to the initial state in the dynamic coordinates of the cable.An ideally flexible cable is considered, and therefore, the deformation due to bending, torsion, and shear is neglected, and the strain is a result of stretching only. Further, it is considered that longitudinal displacements and static strain are small (compared to the transverse displacements). Therefore, its higher-order terms can be ignored [13, 27]. Thus, the partial derivative of s is obtained:(6)∂u′∂s=uBl,∂v′∂s=∂v∂s+vBl.The length of the cable element before and after deformation can be expressed as(7)ds2=dx2+dy2,ds′2=dx+du′2+dy+dv′2.Assuming thatΔ is infinitesimal and ds′=ds+Δ, then the dynamic strain ε can be given by(8)ε=ds′−dsds≈ds′2−ds22ds2.The dynamic analysis of the intercepted infinitesimal segment is carried out, and the expression oft′ is obtained from t′=εEA by considering ds≈dx,(9)t′=EA∂v∂xdydx+12∂v∂x2+vBl∂v∂x+uBl+12uBl2+vBldydx+12vBl2,where EA is the axial stiffness.In order to establish the first two order modal interaction equations, letvx,t=∑i=12ϕixqit. According to the cable boundary conditions, we assume ϕ1x=sinπx/l, ϕ2x=sin2πx/l [43]. Substituting equations (2), (5), and (9) into equation (4), the equation of motion of the dynamic system is obtained:(10)c1q1″t+c2q1′t+c3q1t+c4q12t+c5q13t+d1q2″t+d2q2′t+d3q2t+d4q22t+d5q23t+e1q1tq2t+e2q12tq2t+e3q1tq22t+fx,t=0,where ci, di, ei are the coefficients containing ϕ1x and ϕ2x. fx,t is the external excitation (the full details are given in Appendix A). Then, applying Galerkin method on equation (10) and then normalizing can yield the following ordinary differential equations:(11)A1q1″t+A2q1′t+A3q1t+A4q12t+A5q13t+A6q2t+A7q22t+A8q1tq2t+A9q1tq22t+A10=0,B1q2″t+B2q2′t+B3q2t+B4q22t+B5q23t+B6q1t+B7q12t+B8q1tq2t+B9q2tq12t+B10=0,where A1,A2,A4,A5,A7,A9,B1,B2,B4,B5,B8,B9 are constant coefficients (the full details are given in Appendix B), A3,A6,A8,A10,B3,B6,B7,B10 contain parametric excitation terms, and A10,B10 are the external excitation terms.After separating the constant coefficients and the parameter excitation coefficients, the nonlinear modal interaction equations of continuous cable are obtained:(12)q1″t+ω12q1t+F1t+J11q1t+J12q2t+J13q1tq2t+A4q12t+A5q13t+A7q22t+A9q1tq22t+A2q1′t=0,(13)q2″t+ω22q2t+F2t+J21q2t+J22q1t+J23q12t++B5q23t+B8q1tq2t+B9q12tq2t+B2q2′t=0.In this model,Fkt=pk1sinΩt+pk2 (k=1,2 denotes the kth mode) (the full details are given in Appendix C) is the forced term of element deformation caused by the movement of cable end. Jkn=jknsinΩtn=1,2,3… (the full details are given in Appendix C) is the parametric excitation coefficient.Traditional single-mode investigation without consideration of internal resonance can be realized by introducingvkx,t=ϕkxqkt. Similarly, the Galerkin method is used to obtain the single-mode equation, and then the primary resonance and parametric resonance of the first-order mode and second-order mode can be analyzed, respectively.


### 2.2. Perturbation Analysis

After getting the modal interaction nonlinear equations of continuous cable, the multiple scale method is used to obtain the averaged equations. Firstly, we introduce the time scalesTm=ηmtm=1,2,3,…, where η is a small nondimensional parameter of the order of the amplitude of the solution. Let the solution be the following form:(14)qkt=xk0T0,T1+ηxk1T0,T1.Because of the modal interaction, even if the solving process is consistent, the results of each step have different expressions. Therefore, the first-order principal parametric resonance is taken as an example to restore the solving process.LetΩ=2ω1+ηρ,ω2=2ω1+ησ (ρ, σ are detuning parameters of order O (1)). There is no internal resonance near the primary resonance. Thus, F1t in equation (12) is put into η0, and F2t in equation (13) is put into η1, yielding(15)q1″t+ω12q1t+F1t+ηJ11q1t+J12q2t+J13q1tq2t+A4q12t+A5q13t+A7q22t+A9q1tq22t+A2q1′t=0,q2″t+ω22q2t+ηF2t+J21q2t+J22q1t+J23q12t+B5q23t+B8q1tq2t+B9q12tq2t+B2q2′t=0.By substituting equation (14) into equation (15), a polynomial about η is obtained. Finding out the term η0 and solving it yield(16)x10T0,T1=A10T1eiω1T0+A¯10T1e−iω1T0+1Ω2−ω12−ip11eiΩT0+p12+ip11e−iΩT0+p12,x20T0,T1=A20T1eiω2T0+A¯20T1e−iω2T0,where A¯ and A are conjugate relations. By substituting equation (16) into the polynomial of the order of η1, we can find out the secular term. Making it zero yields(17)6A5p11Ω2−ω122A10T1+iA2ω1A10T1+3A¯10T1A5A102T1+2iω1A10′T1+−i2A¯10T1j11−A¯10T1A42ieip12p11Ω2−ω12⋅eiρT1+2A¯20T1A9A10T1A20T1+i2j13A10T1A20T1eiσ−ρT1−12iA¯20T1j13A10T1ei−σ+ρT1=0,2B9p112A20T1Ω2−ω122+2A¯10T1B9A10T1A20T1+3A¯20T1B5A202T1−ij23p112Ω2−ω122+ij23p1122Ω2−ω122e2ip12+ieip22p21+ij23A¯10T1A10T1ei−σ+ρT1+iB2ω2A20T1+2iω2A20′T1−A¯20T1B9p112Ω2−ω122e2i−σ+ρT1+2ip12=0.LetA10T1=a10T1eiθ10T1/2,A20T1=a20T1eiθ20T1/2, where a and θ are real. And substitute them into the equation (17). The real and imaginary parts are separated and are equal to zero, respectively. For the steady solution, let a10′T1=a20′T1=0, γ1′=γ2′=0, in which(18)γ1=ρT1−2θ10T1,γ2=ρT1−σT1−θ20T1,Then(19)θ10′T1=ρ2,θ20′T1=ρ−σ.The averaged equations of the steady solution obtained by the multiple scale method are shown in the following equation:(20)−a10T1j11cosγ14ω1−cosp12+γ1a10T1A4p11ω1Ω2−ω12+12A2a10T1=0,−sinγ1a10T1j114ω1−sinp12+γ1a10T1A4p11ω1Ω2−ω12−sinγ2a20T1j13a10T18ω1−3A5p112a10T1ω1Ω2−ω122−3A5a103T18ω1−A9a10T1a202T14ω1−sinγ2j13a10T1a20T18ω1+a10T1ρ2=0,−cosγ2j23p112ω2Ω2−ω122−cos2p12+γ2j23p1122ω2Ω2−ω122−cosp22+γ2p21ω2−cosγ2a102T1j234ω2+12B2a20T1−a20T1B9p1122Ω2−ω122ω2sin2p12+2γ2=0,−sinγ2j23p112ω2Ω2−ω122−sin2p12+γ2j23p1122ω2Ω2−ω122−sinp22+γ2p21ω2−sinγ2a102T1j234ω2−B9p112a20T1ω2Ω2−ω122−a102T1B9a20T14ω2−3B5a203T18ω2+a20T1ρ−σ+a20T1B9p1122Ω2−ω122ω2cos2p12+2γ2=0.In the same way, the averaged equations of theΩ≈ω1, Ω≈2ω2 in modal interaction and the averaged equations under four excitation conditions in the single mode can also be obtained.Observing secular terms in the single mode, when the primary resonance of the cable occurs, the secular terms are mainly related to the forcing termspk1 and the cubic nonlinearities ak03, and the resonance effect is mainly caused by the forcing term. When the principal parametric resonance of the cable occurs, the parametric coefficient jkn in the secular terms affects the averaged equation, and the resonance effect is mainly caused by the parametric terms. Under the modal interaction, when Ω≈ω2, it can be seen from equations (17) and (20) that the secular term is related to both the parametric terms jkn and the forced terms pk1.


## 2.1. Reduced-Order Modeling

The cable is a flexible structure and has a certain sag due to its own weight. The geometric nonlinearity can obviously increase with the increase of cable length. To establish the distributed parameter model of cable system, the following assumptions are introduced [17].(1)
The influence of bending, torsion, and shear stiffness of cables is not considered.(2)
The gravity sag curve of the cable is a parabola.(3)
The deformation constitutive relation of the cable obeys Hooke's law, and the stress on each point is uniform.(4)
The influence of cable gravity on chordal tension is not considered.In Figure1, the x-axis coordinate coincides with the axial direction of the cable. The vibration of the cable in the x-y plane is called in-plane vibration. The length of the cable is l. The vertical and horizontal distances between the two ends of the cable are W and L, respectively. The inclination angle between the cable and the horizontal direction is α. Next, we intercept a microsegment for analysis.Figure 1
Spatial diagram of cable considering sag.The deformation and stress in thez-direction are ignored [12]. In the static case, the deformation of the cable is shown in the microelement section at the top of Figure 2. T is the tension along the tangent direction of the cable. M is the mass of the unit length of the cable. s is the arc length coordinate of the cable. According to Newton's second law, the static equilibrium equation can be obtained as(1)ddsTdyds=−Mgcosα,Figure 2
Stress analysis of the cable element.For the stay cable in a cable-stayed bridge, the ratio of sag to span is usually very small. Therefore,ds≈dx is taken [41], By using the boundary condition and integral constant, the sag curve of the cable in static state can be derived as(2)y=Mgl2cosα2Txl−x2l2,where g is the acceleration of gravity, and y is the sag value.The differential equations of motion are derived using the local coordinate system shown in Figure1. When the cable vibrates, the infinitesimal segment at the bottom of Figure 2 shows the displacement and stress of the cable element. t′ is the change of tangential tension relative to the initial tension during vibration. ξ1 and ξ2 are the viscous damping coefficients of the cable along the x and y axes, respectively. u′ and v′ are the displacements of the cable element at the x-axis and y-axis deviating from the static equilibrium position. Substituting equation (1) into equation (3), the in-plane vibration differential equation of the cable is obtained by Newton's second law [42].(3)∂∂sT+t′dyds+dv′ds=M∂2v′∂2t+ξ2∂v′∂t−Mgcosα.Simplifying the differential equation of motion yields(4)∂∂xT+t′∂v′∂x+t′dydx=M∂2v′∂2t+ξ2∂v′∂t.The connection between the cable and the pylon is assumed to be fixed. The lower end of the cable is excited by the vertical sine excitationDB=YBsinΩt, where YB, Ω are the external excitation amplitude and frequency, respectively. The displacements are labeled uB=DBsinα and vB=DBcosα for the longitudinal and transverse components, respectively. Then, we obtain(5)u′=u+sluB,v′=v+slvB,where u and v are the shape variables of the cable element relative to the initial state in the dynamic coordinates of the cable.An ideally flexible cable is considered, and therefore, the deformation due to bending, torsion, and shear is neglected, and the strain is a result of stretching only. Further, it is considered that longitudinal displacements and static strain are small (compared to the transverse displacements). Therefore, its higher-order terms can be ignored [13, 27]. Thus, the partial derivative of s is obtained:(6)∂u′∂s=uBl,∂v′∂s=∂v∂s+vBl.The length of the cable element before and after deformation can be expressed as(7)ds2=dx2+dy2,ds′2=dx+du′2+dy+dv′2.Assuming thatΔ is infinitesimal and ds′=ds+Δ, then the dynamic strain ε can be given by(8)ε=ds′−dsds≈ds′2−ds22ds2.The dynamic analysis of the intercepted infinitesimal segment is carried out, and the expression oft′ is obtained from t′=εEA by considering ds≈dx,(9)t′=EA∂v∂xdydx+12∂v∂x2+vBl∂v∂x+uBl+12uBl2+vBldydx+12vBl2,where EA is the axial stiffness.In order to establish the first two order modal interaction equations, letvx,t=∑i=12ϕixqit. According to the cable boundary conditions, we assume ϕ1x=sinπx/l, ϕ2x=sin2πx/l [43]. Substituting equations (2), (5), and (9) into equation (4), the equation of motion of the dynamic system is obtained:(10)c1q1″t+c2q1′t+c3q1t+c4q12t+c5q13t+d1q2″t+d2q2′t+d3q2t+d4q22t+d5q23t+e1q1tq2t+e2q12tq2t+e3q1tq22t+fx,t=0,where ci, di, ei are the coefficients containing ϕ1x and ϕ2x. fx,t is the external excitation (the full details are given in Appendix A). Then, applying Galerkin method on equation (10) and then normalizing can yield the following ordinary differential equations:(11)A1q1″t+A2q1′t+A3q1t+A4q12t+A5q13t+A6q2t+A7q22t+A8q1tq2t+A9q1tq22t+A10=0,B1q2″t+B2q2′t+B3q2t+B4q22t+B5q23t+B6q1t+B7q12t+B8q1tq2t+B9q2tq12t+B10=0,where A1,A2,A4,A5,A7,A9,B1,B2,B4,B5,B8,B9 are constant coefficients (the full details are given in Appendix B), A3,A6,A8,A10,B3,B6,B7,B10 contain parametric excitation terms, and A10,B10 are the external excitation terms.After separating the constant coefficients and the parameter excitation coefficients, the nonlinear modal interaction equations of continuous cable are obtained:(12)q1″t+ω12q1t+F1t+J11q1t+J12q2t+J13q1tq2t+A4q12t+A5q13t+A7q22t+A9q1tq22t+A2q1′t=0,(13)q2″t+ω22q2t+F2t+J21q2t+J22q1t+J23q12t++B5q23t+B8q1tq2t+B9q12tq2t+B2q2′t=0.In this model,Fkt=pk1sinΩt+pk2 (k=1,2 denotes the kth mode) (the full details are given in Appendix C) is the forced term of element deformation caused by the movement of cable end. Jkn=jknsinΩtn=1,2,3… (the full details are given in Appendix C) is the parametric excitation coefficient.Traditional single-mode investigation without consideration of internal resonance can be realized by introducingvkx,t=ϕkxqkt. Similarly, the Galerkin method is used to obtain the single-mode equation, and then the primary resonance and parametric resonance of the first-order mode and second-order mode can be analyzed, respectively.


## 2.2. Perturbation Analysis

After getting the modal interaction nonlinear equations of continuous cable, the multiple scale method is used to obtain the averaged equations. Firstly, we introduce the time scalesTm=ηmtm=1,2,3,…, where η is a small nondimensional parameter of the order of the amplitude of the solution. Let the solution be the following form:(14)qkt=xk0T0,T1+ηxk1T0,T1.Because of the modal interaction, even if the solving process is consistent, the results of each step have different expressions. Therefore, the first-order principal parametric resonance is taken as an example to restore the solving process.LetΩ=2ω1+ηρ,ω2=2ω1+ησ (ρ, σ are detuning parameters of order O (1)). There is no internal resonance near the primary resonance. Thus, F1t in equation (12) is put into η0, and F2t in equation (13) is put into η1, yielding(15)q1″t+ω12q1t+F1t+ηJ11q1t+J12q2t+J13q1tq2t+A4q12t+A5q13t+A7q22t+A9q1tq22t+A2q1′t=0,q2″t+ω22q2t+ηF2t+J21q2t+J22q1t+J23q12t+B5q23t+B8q1tq2t+B9q12tq2t+B2q2′t=0.By substituting equation (14) into equation (15), a polynomial about η is obtained. Finding out the term η0 and solving it yield(16)x10T0,T1=A10T1eiω1T0+A¯10T1e−iω1T0+1Ω2−ω12−ip11eiΩT0+p12+ip11e−iΩT0+p12,x20T0,T1=A20T1eiω2T0+A¯20T1e−iω2T0,where A¯ and A are conjugate relations. By substituting equation (16) into the polynomial of the order of η1, we can find out the secular term. Making it zero yields(17)6A5p11Ω2−ω122A10T1+iA2ω1A10T1+3A¯10T1A5A102T1+2iω1A10′T1+−i2A¯10T1j11−A¯10T1A42ieip12p11Ω2−ω12⋅eiρT1+2A¯20T1A9A10T1A20T1+i2j13A10T1A20T1eiσ−ρT1−12iA¯20T1j13A10T1ei−σ+ρT1=0,2B9p112A20T1Ω2−ω122+2A¯10T1B9A10T1A20T1+3A¯20T1B5A202T1−ij23p112Ω2−ω122+ij23p1122Ω2−ω122e2ip12+ieip22p21+ij23A¯10T1A10T1ei−σ+ρT1+iB2ω2A20T1+2iω2A20′T1−A¯20T1B9p112Ω2−ω122e2i−σ+ρT1+2ip12=0.LetA10T1=a10T1eiθ10T1/2,A20T1=a20T1eiθ20T1/2, where a and θ are real. And substitute them into the equation (17). The real and imaginary parts are separated and are equal to zero, respectively. For the steady solution, let a10′T1=a20′T1=0, γ1′=γ2′=0, in which(18)γ1=ρT1−2θ10T1,γ2=ρT1−σT1−θ20T1,Then(19)θ10′T1=ρ2,θ20′T1=ρ−σ.The averaged equations of the steady solution obtained by the multiple scale method are shown in the following equation:(20)−a10T1j11cosγ14ω1−cosp12+γ1a10T1A4p11ω1Ω2−ω12+12A2a10T1=0,−sinγ1a10T1j114ω1−sinp12+γ1a10T1A4p11ω1Ω2−ω12−sinγ2a20T1j13a10T18ω1−3A5p112a10T1ω1Ω2−ω122−3A5a103T18ω1−A9a10T1a202T14ω1−sinγ2j13a10T1a20T18ω1+a10T1ρ2=0,−cosγ2j23p112ω2Ω2−ω122−cos2p12+γ2j23p1122ω2Ω2−ω122−cosp22+γ2p21ω2−cosγ2a102T1j234ω2+12B2a20T1−a20T1B9p1122Ω2−ω122ω2sin2p12+2γ2=0,−sinγ2j23p112ω2Ω2−ω122−sin2p12+γ2j23p1122ω2Ω2−ω122−sinp22+γ2p21ω2−sinγ2a102T1j234ω2−B9p112a20T1ω2Ω2−ω122−a102T1B9a20T14ω2−3B5a203T18ω2+a20T1ρ−σ+a20T1B9p1122Ω2−ω122ω2cos2p12+2γ2=0.In the same way, the averaged equations of theΩ≈ω1, Ω≈2ω2 in modal interaction and the averaged equations under four excitation conditions in the single mode can also be obtained.Observing secular terms in the single mode, when the primary resonance of the cable occurs, the secular terms are mainly related to the forcing termspk1 and the cubic nonlinearities ak03, and the resonance effect is mainly caused by the forcing term. When the principal parametric resonance of the cable occurs, the parametric coefficient jkn in the secular terms affects the averaged equation, and the resonance effect is mainly caused by the parametric terms. Under the modal interaction, when Ω≈ω2, it can be seen from equations (17) and (20) that the secular term is related to both the parametric terms jkn and the forced terms pk1.


## 3. Modal Analysis at Ω ≈ω2

After obtaining the solutions of the averaged equations and Jacobian matrix, the stability of the solutions is judged by the Routh Hurwitz criterion. After that, we use Wolfram Mathematica to solve the differential equation (11) by the Runge–Kutta method and get the phase diagrams and power spectral density diagrams. Firstly, the parameters of the cable are set as shown in Table 1 (the parameters are consistent with the experimental model parameters in Chapter 4).Table 1
Model parameters for the cable.
Cable parametersValueUnitLength,l2.4(m)Diameter,d1.2 × 10–3(m)Acceleration of gravity,g9.8(m/s2)Compressive rigidity,EA1.2 × 105(N)Mass per unit length,M0.015(kg/m)Tension of static cable atx-axis, T160(N)Linear damping per unit length aty-axis, ξ20.001(-)Angle between cable and horizontal direction,aπ/6(-)The natural frequencies of each mode can be calculated asω1=135.217rad/s, ω2=270.421rad/s. Obviously, ω2≈2ω1.


### 3.1. Modal Interaction Analyses

LetΩ=287rad/s, YB=0.20mm. Three groups of stable solutions can be obtained by solving equation (20). The corresponding phase diagrams and the power spectral density diagrams can be obtained by changing the initial values q10 and q20 with all other parameters unchanged. Equation (11) is simulated, as shown in Figure 3.Figure 3
Vibration responses of the first two modes of the cable (the dotted line and the solid line represent the numerical solution and the analytical solution, respectively, and the different colors represent different initial values): (a) the phase diagram of the first-order modal vibration, (b) the power spectral density diagram of the first-order modal vibration, (c) the phase diagram of the second-order modal vibration, and (d) the power spectral density diagram of the second-order modal vibration.
(a)(b)(c)(d)It can be seen from Figure3 that the numerical solution is in good agreement with the analytical solution. If Ω≈ω2, the second-order primary resonance and the first-order principal parametric resonance and superharmonic resonance (it can be seen from Figure 3(b) that the response amplitude is very small) will occur by changing q10 and q20.(1)
When the initial valueq10=0,q20=0, the analytical solution a20T1=0.50812mm is almost consistent with the linear solution 2P21/Ω2−ω22=0.49145mm. Therefore, it can be seen from Figures 3(a) and 3(c) that it is a nonresonant case, and the cable hardly vibrates.(2)
The initial values ofq10=0 and q20=15mm are set for analysis. Figure 3(a) shows that the vibration displacement is biased towards the negative direction of the x-axis. The factor leading to this phenomenon is that high-order terms are small in magnitude and are deleted when simplifying the theoretical equation. If it is a linear problem [34], the primary resonance will only occur when the external excitation frequency is close to the natural frequency. However, in the nonlinear equation (11), in addition to the external excitation term, the quadratic nonlinearities and the cubic nonlinearities also contribute to the secular terms. Therefore, the resonance period in the primary resonance is increased. At this time, the first-order mode will not resonate, while the second-order modal amplitude is more than 9 mm.(3)
If the initial valueq10=25mm, q20=4mm, the first-order principal parametric resonance of the cable occurs. And the amplitude of the first-order principal parametric resonance is more than twice that of the second-order primary resonance. This shows that when the frequency of parametric terms approximately equals twice the first-order natural frequency, a strong principal parametric resonance occurs, and the vibration frequency is near the first-order natural frequency. This phenomenon also reflects the characteristics of parametric resonance.


### 3.2. Modal Interaction vs. Single Mode

After verifying the correctness of the analytical method, we compare vibration responses of the first two order modes in modal interaction of the cable with that of the single mode under the excitation condition:YB=0.20mm, Ω=287rad/s, given the initial values q10=25mm, q20=4mm, as shown in Figure 4.Figure 4
The vibration responses of the first two modes of the cable under modal interaction and single mode are obtained by the analytical method and the numerical method.Figure4 shows that the vibration responses of the cable under two different modal processing conditions obtained by numerical method are almost the same. And the vibration response of the cable under modal interaction obtained by analytical method is almost consistent with that obtained by numerical method. However, the second-order modal response in a single mode obtained by the analytical method is quite different from the numerical solution. Because the modes of the cable are coupled with each other, the vibration state of the cable will change with the change of external excitation frequency and given initial value, as shown in Figure 5. Compared with Liu et al. [17] intercepting the first mode of the cable and numerically analyzing the primary resonance and principal parametric resonance of the cable under axial excitation, considering the modal interaction, we can more accurately solve the parametric vibration of the cable and further understand the dynamic characteristics of the cable.Figure 5
The power spectrum density of the first two modes of the cable is obtained under different excitation and initial values.WhenΩ≈ω2, Figure 5 shows that not only the second-order primary resonance occurs, but the first-order principal parametric resonance occurs as well. At this time, the energy exchange between the two modes can be clearly seen by changing the initial values q10,q20, and the response amplitude increases with the increase of external excitation frequency.The energy exchanges from the second-order mode to the first-order mode, and finally, the amplitude of the two modes is stable by determining the initial valuesq10,q20 and external excitation frequency. When the external excitation amplitude of the cable is constant, it can also be seen from the analytical solutions that the amplitude of the principal parametric resonance is far greater than that of the primary resonance.


### 3.3. Parametric Resonance-Induced Bifurcation

Previous studies mainly focused on the changes of boundary curves in the main region of parametric resonance, which are affected by the forced vibration caused by the primary resonance. Demsic et al. [27] deduced the analytical expression of the boundary curve of the interaction resonance region under the first two modes coupling. On this basis, for a certain excitation frequency, we have a more in-depth discussion on the multimodal interaction caused by simultaneous parametric resonance and primary resonance and find that there are energy exchange and parameter bifurcation, as shown in Figures 6 and 7.Figure 6
Frequency-amplitude curves of the first two modes: (a) the first-order mode and (b) the second-order mode (the dotted line shows the unstable solution).
(a)(b)Figure 7
The relationship curves between the first two-modal amplitude and the external excitation amplitude (the dotted line shows the unstable solution): (a) the first-order mode and (b) the second-order mode.
(a)(b)The vibration frequency and amplitude of the connection point of the beam cable are taken as the variable parameters. The vibration law of the first-order principal parametric resonance is consistent with that of the second-order principal parametric resonance. Thus, the averaged equation is used to make the parameter bifurcation diagrams when the first-order principal parametric resonance occurs.GivenYB=0.20mm, the initial excitation frequency Ω=265rad/s. Figure 6 shows that the amplitude of the cable increases with the increase of Ω. For the first-order mode, when the Ω decreases to 272.56 rad/s, the amplitude a10 suddenly changes from 0 to 9.33 mm. That is, the bifurcation point of the theoretical model is 272.56 rad/s, and the effective value of the amplitude after jumping at the bifurcation point is 6.60 mm and then decreases to 0 along the first curve. Figure 6(b) shows the frequency-amplitude curves of the second-order mode.Figure7 shows the relationship between the external excitation amplitude and the response amplitude of the cable, where Ω=287rad/s. There are two jumping points in Figure 7(a). When the external excitation amplitude YB gradually increases from 0 to 1.55 mm, a10 will suddenly change from 0 to 25.96 mm and then increase with the increase of YB. When YB decreases to 0 mm, a10 is 18.46 mm. Then, a10 mutated to 0 with the further decrease of YB. For Figure 7(b), YB=0,a10=9.22mm and YB=1.40mm,a10=5.68mm are two jumping points.These figures show that when the frequency and amplitude of the external excitation are continuously changed, the vibration of the cable will have different bifurcation points in different directions of change. Moreover, when the frequency and amplitude are constant, the cable vibration will have different theoretical solutions for different initial values.


## 3.1. Modal Interaction Analyses

LetΩ=287rad/s, YB=0.20mm. Three groups of stable solutions can be obtained by solving equation (20). The corresponding phase diagrams and the power spectral density diagrams can be obtained by changing the initial values q10 and q20 with all other parameters unchanged. Equation (11) is simulated, as shown in Figure 3.Figure 3
Vibration responses of the first two modes of the cable (the dotted line and the solid line represent the numerical solution and the analytical solution, respectively, and the different colors represent different initial values): (a) the phase diagram of the first-order modal vibration, (b) the power spectral density diagram of the first-order modal vibration, (c) the phase diagram of the second-order modal vibration, and (d) the power spectral density diagram of the second-order modal vibration.
(a)(b)(c)(d)It can be seen from Figure3 that the numerical solution is in good agreement with the analytical solution. If Ω≈ω2, the second-order primary resonance and the first-order principal parametric resonance and superharmonic resonance (it can be seen from Figure 3(b) that the response amplitude is very small) will occur by changing q10 and q20.(1)
When the initial valueq10=0,q20=0, the analytical solution a20T1=0.50812mm is almost consistent with the linear solution 2P21/Ω2−ω22=0.49145mm. Therefore, it can be seen from Figures 3(a) and 3(c) that it is a nonresonant case, and the cable hardly vibrates.(2)
The initial values ofq10=0 and q20=15mm are set for analysis. Figure 3(a) shows that the vibration displacement is biased towards the negative direction of the x-axis. The factor leading to this phenomenon is that high-order terms are small in magnitude and are deleted when simplifying the theoretical equation. If it is a linear problem [34], the primary resonance will only occur when the external excitation frequency is close to the natural frequency. However, in the nonlinear equation (11), in addition to the external excitation term, the quadratic nonlinearities and the cubic nonlinearities also contribute to the secular terms. Therefore, the resonance period in the primary resonance is increased. At this time, the first-order mode will not resonate, while the second-order modal amplitude is more than 9 mm.(3)
If the initial valueq10=25mm, q20=4mm, the first-order principal parametric resonance of the cable occurs. And the amplitude of the first-order principal parametric resonance is more than twice that of the second-order primary resonance. This shows that when the frequency of parametric terms approximately equals twice the first-order natural frequency, a strong principal parametric resonance occurs, and the vibration frequency is near the first-order natural frequency. This phenomenon also reflects the characteristics of parametric resonance.


## 3.2. Modal Interaction vs. Single Mode

After verifying the correctness of the analytical method, we compare vibration responses of the first two order modes in modal interaction of the cable with that of the single mode under the excitation condition:YB=0.20mm, Ω=287rad/s, given the initial values q10=25mm, q20=4mm, as shown in Figure 4.Figure 4
The vibration responses of the first two modes of the cable under modal interaction and single mode are obtained by the analytical method and the numerical method.Figure4 shows that the vibration responses of the cable under two different modal processing conditions obtained by numerical method are almost the same. And the vibration response of the cable under modal interaction obtained by analytical method is almost consistent with that obtained by numerical method. However, the second-order modal response in a single mode obtained by the analytical method is quite different from the numerical solution. Because the modes of the cable are coupled with each other, the vibration state of the cable will change with the change of external excitation frequency and given initial value, as shown in Figure 5. Compared with Liu et al. [17] intercepting the first mode of the cable and numerically analyzing the primary resonance and principal parametric resonance of the cable under axial excitation, considering the modal interaction, we can more accurately solve the parametric vibration of the cable and further understand the dynamic characteristics of the cable.Figure 5
The power spectrum density of the first two modes of the cable is obtained under different excitation and initial values.WhenΩ≈ω2, Figure 5 shows that not only the second-order primary resonance occurs, but the first-order principal parametric resonance occurs as well. At this time, the energy exchange between the two modes can be clearly seen by changing the initial values q10,q20, and the response amplitude increases with the increase of external excitation frequency.The energy exchanges from the second-order mode to the first-order mode, and finally, the amplitude of the two modes is stable by determining the initial valuesq10,q20 and external excitation frequency. When the external excitation amplitude of the cable is constant, it can also be seen from the analytical solutions that the amplitude of the principal parametric resonance is far greater than that of the primary resonance.


## 3.3. Parametric Resonance-Induced Bifurcation

Previous studies mainly focused on the changes of boundary curves in the main region of parametric resonance, which are affected by the forced vibration caused by the primary resonance. Demsic et al. [27] deduced the analytical expression of the boundary curve of the interaction resonance region under the first two modes coupling. On this basis, for a certain excitation frequency, we have a more in-depth discussion on the multimodal interaction caused by simultaneous parametric resonance and primary resonance and find that there are energy exchange and parameter bifurcation, as shown in Figures 6 and 7.Figure 6
Frequency-amplitude curves of the first two modes: (a) the first-order mode and (b) the second-order mode (the dotted line shows the unstable solution).
(a)(b)Figure 7
The relationship curves between the first two-modal amplitude and the external excitation amplitude (the dotted line shows the unstable solution): (a) the first-order mode and (b) the second-order mode.
(a)(b)The vibration frequency and amplitude of the connection point of the beam cable are taken as the variable parameters. The vibration law of the first-order principal parametric resonance is consistent with that of the second-order principal parametric resonance. Thus, the averaged equation is used to make the parameter bifurcation diagrams when the first-order principal parametric resonance occurs.GivenYB=0.20mm, the initial excitation frequency Ω=265rad/s. Figure 6 shows that the amplitude of the cable increases with the increase of Ω. For the first-order mode, when the Ω decreases to 272.56 rad/s, the amplitude a10 suddenly changes from 0 to 9.33 mm. That is, the bifurcation point of the theoretical model is 272.56 rad/s, and the effective value of the amplitude after jumping at the bifurcation point is 6.60 mm and then decreases to 0 along the first curve. Figure 6(b) shows the frequency-amplitude curves of the second-order mode.Figure7 shows the relationship between the external excitation amplitude and the response amplitude of the cable, where Ω=287rad/s. There are two jumping points in Figure 7(a). When the external excitation amplitude YB gradually increases from 0 to 1.55 mm, a10 will suddenly change from 0 to 25.96 mm and then increase with the increase of YB. When YB decreases to 0 mm, a10 is 18.46 mm. Then, a10 mutated to 0 with the further decrease of YB. For Figure 7(b), YB=0,a10=9.22mm and YB=1.40mm,a10=5.68mm are two jumping points.These figures show that when the frequency and amplitude of the external excitation are continuously changed, the vibration of the cable will have different bifurcation points in different directions of change. Moreover, when the frequency and amplitude are constant, the cable vibration will have different theoretical solutions for different initial values.


## 4. Experimental Validation

By establishing a single-cable model similar to the actual cable, as shown in Figure8, the parametric vibration of the cable under the displacement excitation in the vertical direction is reproduced from the perspective of model test. The vibration spectrum diagrams of the cable in the first two-order primary resonance region and the first two-order principal parameter resonance region are tested by using DASP software. Taking the external excitation frequency and amplitude as the variable parameters, the response of the cable is measured by the accelerometers, and parameter bifurcation diagrams are drawn by using the tracing point method and compared with the theoretical results.Figure 8
Schematic diagram of the experimental device.


### 4.1. Experiment Setup

The test assembly diagrams are shown in Figures8 and 9. An accelerometer is arranged at the beam-cable connection to read the amplitude and frequency of external excitation. Another accelerometer, in order to avoid the fixed point of the second-order mode, is arranged at the quarter node of the cable.Figure 9
Test equipment.All the experimental model parameters are shown in Table1 (in which the damping is given), and the testing process is given in Figure 10. The basic parameters of the experimental device are consistent with those used in the analytical method. The natural frequencies of the cable measured by DASP software are ω1=135.09rad/s, ω2=267.04rad/s.Figure 10
The test process diagram.In order to verify the correctness of the experiment results, the natural frequency of a single prestressed cable is calculated by using the truss structure in COMSOL software. The material parameters, constraints, and initial conditions are consistent with the experimental parameters. The specific settings are as follows: Young's modulusE=106GPa, Poisson's ratio μ=0.3, density ρc=13270kg/m3, damping ξ = 0.001, and the initial axial stress σ0=141542817N/m2. The first two order natural frequencies of the cable are obtained, as shown in Figure 11, ω1=135.19rad/s and ω2=270.39rad/s.Figure 11
The first two order modes of vibration of the cable: (a) the first-order mode of vibration of cable and (b) the second-order mode of vibration of cable.
(a)(b)


### 4.2. Vibration Response at Ω ≈ω2

The vibration frequency of the beam-cable connection is 263.70 rad/s, and the amplitude is 0.20 mm. The second-order primary resonance occurs in the cable system, and the double peaks are at the 1/4 and 3/4 nodes of the cable, respectively. The response frequency and amplitude of the measuring point of the cable are shown in Figure12(a).Figure 12
Spectrum diagrams of the measuring point vibration of the cable obtained from the experiment: (a)Ω=263.70rad/s,YB=0.20mm, (b) Ω=269.79rad/s,YB=0.20mm, and (c) Ω=288.25rad/s,YB=0.15mm.
(a)(b)(c)When the vibration frequency of the beam-cable connection increases slowly to about 269.79 rad/s, two kinds of vibration appear at the same time. As shown in Figure12(b), the vibration frequency and amplitude of corresponding measuring points are given.The amplitude of beam-cable connection is set as 0.15 mm. As shown in Figure12(c), when the vibration frequency increases slowly to 288.25 rad/s, the first-order principal parametric resonance will obviously occur in the cable system, forming a single wave peak at the midpoint of the cable.It can be seen from Figure12 and Table 2 (the f, A represent response frequency, response amplitude, respectively) that when the vibration frequency of the beam-cable connection increases slowly, the vibration behavior of the system changes. When the vibration frequency increases to about 263.76 rad/s, the second-order primary resonance occurs. With the increase of frequency, the energy will gradually transform to the first-order mode, and the vibration form will change from the second-order primary resonance to the first-order principal parametric resonance. Finally, the amplitude near the second-order mode will be reduced to zero, and only the first-order principal parametric resonance will occur in the system. The vibration mode of the cable will gradually change from double peak to single peak, and the final amplitude follows the law of the first-order principal parametric resonance.Table 2
Experimental results.
Vibration behaviors of the cableExternal excitationMeasuring pointNearω1Nearω2Ω (rad/s)YB (mm)f1 (rad/s)A1 (mm)f2 (rad/s)A2 (mm)Ω ≈ω1Primary resonance134.890.20134.894.89Superharmonic resonance128.740.55128.743.52263.760.53Ω ≈ω2Primary resonance263.700.20263.704.86Primary resonance + principal parametric resonance269.790.20136.284.48269.796.34Principal parametric resonance288.250.15147.2013.77


### 4.3. Bifurcation Properties at Ω ≈ω2

As shown in Figure13(a), YB=0.20mm. When Ω increases gradually, the response amplitude of the measuring point of the cable increases continuously with the conic curve. When Ω increases to 314.16 rad/s, the amplitude of the measuring point reaches the maximum value of 18.39 mm and then suddenly decreases to 0. On the contrary, when Ω decreases steadily to 273.32 rad/s, the amplitude of the measuring point suddenly rises to about 9.30 mm and then decreases with the conic curve.Figure 13
Parameter bifurcation diagrams: (a) the amplitude-frequency response diagram of the measuring point of the cable (○ representsΩ=270rad/s and YB=0.20mm) and (b) the relationship between amplitude of the external excitation and amplitude of the measuring point of the cable.
(a)(b)Ω=270rad/s, and the parameter bifurcation of the cable is tested by adjusting the YB in 0.1 mm step. As shown in Figure 13(b), when YB increases steadily to 0.80 mm, the response amplitude of the measuring point suddenly rises to 9.19 mm and then increases continuously with the increase of YB. On the contrary, when YB decreases to 0.20 mm, the vibration amplitude of the cable measuring point suddenly drops to about 0.Figure13 shows that when the principal parametric resonance of the cable occurs, its vibration amplitude does not change continuously with the continuous change of vibration frequency or vibration amplitude of the bridge deck. And when the parameter value increases or decreases to one special value, the vibration amplitude of the cable will change suddenly, and bifurcation points are different. Thus, different initial values will cause different vibration responses, which also reflects the mechanism of parametric vibration.In addition, we also study the second-order principal parametric vibration of the cable when Ω ≈ 2ω2. It is found that the case is similar to that when Ω ≈ ω2, so we will not repeat it here.


### 4.4. Theoretical and Experimental Comparison

The natural frequency of the cable is calculated by analytical method, experiment, and finite element simulation, as shown in Table3. AS, ER, and FES represent analytical solution, experimental result, and finite element simulation, respectively.Table 3
The natural frequency of the cable.
ASERFESErrorAS/FES (%)ER/FES (%)ω1 (rad/s)135.22135.08135.190.020.08ω2 (rad/s)270.42269.79270.390.010.20Ahmad [33] found that the cable considering sag will have higher fundamental frequency. It can be seen from Table 3 that the natural frequency of the system calculated by approximate analytical method, experiment, and finite element simulation is consistent, which shows that the model simplification is reasonable, and the approximate analytical solutions are accurate. Therefore, the cable sag needs to be considered in the subsequent dynamic analysis of stay cables.The analytical solutions and the experimental results are shown in Table4. It should be noted that the amplitude value at l/4 of the cable can be obtained by multiplying the amplitude obtained by the above theoretical calculation by the mode shape.Table 4
The comparison between analytical solutions and experimental results.
External excitationResponse atl/4Ω (rad/s)YB (mm)f (rad/s)A (mm)Ω ≈ω1AS135.000.20134.574.71ER134.890.20134.894.89Error (%)0.0800.243.68Ω ≈ω2AS277.000.20279.116.32ER269.790.20269.796.34Error (%)2.6703.450.32Ω ≈ 2ω1AS287.000.15143.5013.55ER288.250.15146.9513.77Error (%)0.4302.351.60Ω ≈ 2ω2AS560.000.18279.117.67ER533.800.18263.768.10Error (%)4.9005.825.30It can be seen from Table4 that the error between the analytical solution and the experimental result is about 5%, so the experiment is reasonable and effective. At the same time, it is observed that the amplitude of the principal parametric resonance is 1–3 times that of the primary resonance of the same order under the same external excitation amplitude.When the primary resonance occurs, the secular term of the cable is mainly related to the forced term and the cubic term contained in the equation itself. The resonance effect is mainly caused by the forced term. When parametric resonance occurs, the secular term is mainly related to the parametric excitation coefficient of the equation, and the resonance effect is mainly caused by the parametric excitation coefficient. Douthe et al. [24] studied the influence of external excitation frequency on the parametric vibration region in the numerical model ignoring the higher-order term of the equation of motion. On this basis, we retain all high-order terms and obtain good agreement with the experimental results. By observing the parameter bifurcation diagrams from experiments, we found that even if the external excitation frequency is away from the cable's natural frequency, the parametric resonance amplitude of the cable will continue to increase with the increase of the external excitation frequency, and the resonance phenomenon still exists. Therefore, the damage to the bridge caused by the parametric resonance of the cable is often worse than that of the primary resonance. On the basis of Gonzalez-Buelga et al. [22] proving that nonlinear out-plane multimodal coupling needs to be considered in the study of stay cables, we study the dynamic characteristics of stay cables under the coupling of first-order in-plane mode and second-order in-plane mode, which provides a solid theoretical basis for further preventing parametric resonance of stay cables and reducing vibration amplitude.


## 4.1. Experiment Setup

The test assembly diagrams are shown in Figures8 and 9. An accelerometer is arranged at the beam-cable connection to read the amplitude and frequency of external excitation. Another accelerometer, in order to avoid the fixed point of the second-order mode, is arranged at the quarter node of the cable.Figure 9
Test equipment.All the experimental model parameters are shown in Table1 (in which the damping is given), and the testing process is given in Figure 10. The basic parameters of the experimental device are consistent with those used in the analytical method. The natural frequencies of the cable measured by DASP software are ω1=135.09rad/s, ω2=267.04rad/s.Figure 10
The test process diagram.In order to verify the correctness of the experiment results, the natural frequency of a single prestressed cable is calculated by using the truss structure in COMSOL software. The material parameters, constraints, and initial conditions are consistent with the experimental parameters. The specific settings are as follows: Young's modulusE=106GPa, Poisson's ratio μ=0.3, density ρc=13270kg/m3, damping ξ = 0.001, and the initial axial stress σ0=141542817N/m2. The first two order natural frequencies of the cable are obtained, as shown in Figure 11, ω1=135.19rad/s and ω2=270.39rad/s.Figure 11
The first two order modes of vibration of the cable: (a) the first-order mode of vibration of cable and (b) the second-order mode of vibration of cable.
(a)(b)


## 4.2. Vibration Response at Ω ≈ω2

The vibration frequency of the beam-cable connection is 263.70 rad/s, and the amplitude is 0.20 mm. The second-order primary resonance occurs in the cable system, and the double peaks are at the 1/4 and 3/4 nodes of the cable, respectively. The response frequency and amplitude of the measuring point of the cable are shown in Figure12(a).Figure 12
Spectrum diagrams of the measuring point vibration of the cable obtained from the experiment: (a)Ω=263.70rad/s,YB=0.20mm, (b) Ω=269.79rad/s,YB=0.20mm, and (c) Ω=288.25rad/s,YB=0.15mm.
(a)(b)(c)When the vibration frequency of the beam-cable connection increases slowly to about 269.79 rad/s, two kinds of vibration appear at the same time. As shown in Figure12(b), the vibration frequency and amplitude of corresponding measuring points are given.The amplitude of beam-cable connection is set as 0.15 mm. As shown in Figure12(c), when the vibration frequency increases slowly to 288.25 rad/s, the first-order principal parametric resonance will obviously occur in the cable system, forming a single wave peak at the midpoint of the cable.It can be seen from Figure12 and Table 2 (the f, A represent response frequency, response amplitude, respectively) that when the vibration frequency of the beam-cable connection increases slowly, the vibration behavior of the system changes. When the vibration frequency increases to about 263.76 rad/s, the second-order primary resonance occurs. With the increase of frequency, the energy will gradually transform to the first-order mode, and the vibration form will change from the second-order primary resonance to the first-order principal parametric resonance. Finally, the amplitude near the second-order mode will be reduced to zero, and only the first-order principal parametric resonance will occur in the system. The vibration mode of the cable will gradually change from double peak to single peak, and the final amplitude follows the law of the first-order principal parametric resonance.Table 2
Experimental results.
Vibration behaviors of the cableExternal excitationMeasuring pointNearω1Nearω2Ω (rad/s)YB (mm)f1 (rad/s)A1 (mm)f2 (rad/s)A2 (mm)Ω ≈ω1Primary resonance134.890.20134.894.89Superharmonic resonance128.740.55128.743.52263.760.53Ω ≈ω2Primary resonance263.700.20263.704.86Primary resonance + principal parametric resonance269.790.20136.284.48269.796.34Principal parametric resonance288.250.15147.2013.77


## 4.3. Bifurcation Properties at Ω ≈ω2

As shown in Figure13(a), YB=0.20mm. When Ω increases gradually, the response amplitude of the measuring point of the cable increases continuously with the conic curve. When Ω increases to 314.16 rad/s, the amplitude of the measuring point reaches the maximum value of 18.39 mm and then suddenly decreases to 0. On the contrary, when Ω decreases steadily to 273.32 rad/s, the amplitude of the measuring point suddenly rises to about 9.30 mm and then decreases with the conic curve.Figure 13
Parameter bifurcation diagrams: (a) the amplitude-frequency response diagram of the measuring point of the cable (○ representsΩ=270rad/s and YB=0.20mm) and (b) the relationship between amplitude of the external excitation and amplitude of the measuring point of the cable.
(a)(b)Ω=270rad/s, and the parameter bifurcation of the cable is tested by adjusting the YB in 0.1 mm step. As shown in Figure 13(b), when YB increases steadily to 0.80 mm, the response amplitude of the measuring point suddenly rises to 9.19 mm and then increases continuously with the increase of YB. On the contrary, when YB decreases to 0.20 mm, the vibration amplitude of the cable measuring point suddenly drops to about 0.Figure13 shows that when the principal parametric resonance of the cable occurs, its vibration amplitude does not change continuously with the continuous change of vibration frequency or vibration amplitude of the bridge deck. And when the parameter value increases or decreases to one special value, the vibration amplitude of the cable will change suddenly, and bifurcation points are different. Thus, different initial values will cause different vibration responses, which also reflects the mechanism of parametric vibration.In addition, we also study the second-order principal parametric vibration of the cable when Ω ≈ 2ω2. It is found that the case is similar to that when Ω ≈ ω2, so we will not repeat it here.


## 4.4. Theoretical and Experimental Comparison

The natural frequency of the cable is calculated by analytical method, experiment, and finite element simulation, as shown in Table3. AS, ER, and FES represent analytical solution, experimental result, and finite element simulation, respectively.Table 3
The natural frequency of the cable.
ASERFESErrorAS/FES (%)ER/FES (%)ω1 (rad/s)135.22135.08135.190.020.08ω2 (rad/s)270.42269.79270.390.010.20Ahmad [33] found that the cable considering sag will have higher fundamental frequency. It can be seen from Table 3 that the natural frequency of the system calculated by approximate analytical method, experiment, and finite element simulation is consistent, which shows that the model simplification is reasonable, and the approximate analytical solutions are accurate. Therefore, the cable sag needs to be considered in the subsequent dynamic analysis of stay cables.The analytical solutions and the experimental results are shown in Table4. It should be noted that the amplitude value at l/4 of the cable can be obtained by multiplying the amplitude obtained by the above theoretical calculation by the mode shape.Table 4
The comparison between analytical solutions and experimental results.
External excitationResponse atl/4Ω (rad/s)YB (mm)f (rad/s)A (mm)Ω ≈ω1AS135.000.20134.574.71ER134.890.20134.894.89Error (%)0.0800.243.68Ω ≈ω2AS277.000.20279.116.32ER269.790.20269.796.34Error (%)2.6703.450.32Ω ≈ 2ω1AS287.000.15143.5013.55ER288.250.15146.9513.77Error (%)0.4302.351.60Ω ≈ 2ω2AS560.000.18279.117.67ER533.800.18263.768.10Error (%)4.9005.825.30It can be seen from Table4 that the error between the analytical solution and the experimental result is about 5%, so the experiment is reasonable and effective. At the same time, it is observed that the amplitude of the principal parametric resonance is 1–3 times that of the primary resonance of the same order under the same external excitation amplitude.When the primary resonance occurs, the secular term of the cable is mainly related to the forced term and the cubic term contained in the equation itself. The resonance effect is mainly caused by the forced term. When parametric resonance occurs, the secular term is mainly related to the parametric excitation coefficient of the equation, and the resonance effect is mainly caused by the parametric excitation coefficient. Douthe et al. [24] studied the influence of external excitation frequency on the parametric vibration region in the numerical model ignoring the higher-order term of the equation of motion. On this basis, we retain all high-order terms and obtain good agreement with the experimental results. By observing the parameter bifurcation diagrams from experiments, we found that even if the external excitation frequency is away from the cable's natural frequency, the parametric resonance amplitude of the cable will continue to increase with the increase of the external excitation frequency, and the resonance phenomenon still exists. Therefore, the damage to the bridge caused by the parametric resonance of the cable is often worse than that of the primary resonance. On the basis of Gonzalez-Buelga et al. [22] proving that nonlinear out-plane multimodal coupling needs to be considered in the study of stay cables, we study the dynamic characteristics of stay cables under the coupling of first-order in-plane mode and second-order in-plane mode, which provides a solid theoretical basis for further preventing parametric resonance of stay cables and reducing vibration amplitude.


## 5. Conclusions

The vibration of the cable-stayed bridge deck causes the movement of the beam-cable connection, which will lead to the large-scale resonance of the cable. This paper focuses on this practical problem, establishes an experimental model, and studies it through analytical method, numerical simulation, experiment, and finite element simulation. Some interesting conclusions are drawn as follows:(1)
Through further theoretical research, it is found that, under the same external excitation and given initial value, the calculation result of the parametric vibration of the cable considering in-plane modal interaction is better than that of the cable considering single mode.(2)
In the nonlinear vibration system, when the external excitation frequency is near the first-order natural frequency, the system will have the primary resonance at the first-order mode, and the superharmonic resonance will appear in the region of twice the first-order natural frequency. When the external excitation frequency is near the second-order natural frequency, not only the primary resonance occurs at the second-order mode, but also the superharmonic resonance occurs in the region of twice the second-order natural frequency, and the first-order principal parametric resonance occurs at the first-order mode. When the external excitation frequency is equal to the fourth natural frequency of the cable, the vibration of the cable is similar to that of the above situation.(3)
There is an exchange of energy between the second-order primary resonance and the first-order principal parametric resonance by changing the initial value and external excitation frequency. Besides, from parametric bifurcation diagrams, we can see that when the frequency and amplitude of the external excitation are continuously changed in different directions, the vibration of the cable will have different bifurcation points. And even if the frequency of the external excitation is increased, far away from the frequency region of the primary resonance, parametric vibration of the system still exists, and the amplitude will not decline but will increase. Moreover, when the frequency and amplitude are constant, the vibration of the cable will have different solutions for different initial values.(4)
In addition, it is found that the system amplitude caused by principal parametric resonance will be 1 to 3 times that of the primary resonance of the same order under the same external excitation amplitude. Therefore, it can be explained that the harm of parametric resonance caused by the bridge deck vibration is more serious than that of the primary resonance.



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*Source: 5797761-2021-10-05.xml*