# Theoretical and Numerical Analysis of 1 : 1 Main Parametric Resonance of Stayed Cable Considering Cable-Beam Coupling

Authors: Li-Na Zhang; Feng-Chen Li; Xiao-Yong Wang; Peng-Fei Cui
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

For the 1 : 1 main parametric resonances problems of cable-bridge coupling vibration, a main parametric resonances model considering cable-beam coupling is developed and dimensionless parametric resonances differential equations are derived. The main parametric resonances characteristics are discussed by means of multiscale approximation solution methods. Using an actual cable of cable-stayed bridge project for research object, numerical simulation analysis under a variety of conditions is illustrated. The results show that when the coupling system causes 1 : 1 parametric resonance, nonlinear main parametric resonances in response are unrelated to initial displacement of the cable, but with the increase of deck beam end vertical initial displacement increases, accompanied with a considerable “beat” vibration. When the vertical initial displacement of deck beam end is 10−6 m order of magnitude or even smaller, “beat” vibration phenomenon of cable and beam appears. Displacement amplitude of the cable is small and considerable amplitude vibration may not occur at this time, only making a slight stable “beat” vibration in the vicinity of the equilibrium position, which is different from 2 : 1 parametric resonance condition of cable-bridge coupling system. Therefore, it is necessary to limit the initial displacement excitation amplitude of beam end and prevent the occurrence of amplitude main parametric excitation resonances.


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## Body

## 1. Introduction

The excitation of parametric resonance is dependent on time and is used as a parameter in the vibration equation of the system [1–3]. According to whether the amplitude and frequency of the excitation change with time or not, the parametric resonance of the stay cable is divided into two cases: When the quality of deck or tower is far greater than the cable, without considering the effect of cables on the deck or tower, as an ideal incentive, the incentive is not affected by the response and the system vibration is simplified as spring-mass [4] and cable-mass [5–9] model, in which the input energy of the stay cable is from the support point; when the vibration of cables and bridge deck or tower are coupled with each other, the system will be coupled with resonance in initial condition of cable-deck or cable-tower system. Finally, the large vibration of these two can be aroused, and the system vibration is simplified as the [10–12] model of the cable flexible beam.In this paper, the displacement time history response and amplitude-frequency characteristics of the cable are mainly discussed in the case of the 1 : 1 main parametric resonance in the cable-beam coupling system, the approximate solutions of motion equations are solved by the method of multiple scales, and the results are verified by numerical simulation. However, the 2 : 1 parametric resonance and 1 : 2 and 1 : 3 superharmonic resonance response problems are not the focus of discussion here.


## 2. Theoretical Model of Parametric Resonance Considering Cable-Beam Coupling

### 2.1. Differential Equation of Motion with Parametric Resonance Considering Cable-Beam Coupling

Several basic assumptions are made before establishing the equation of motion for the cable-beam composite structure [13]:(1)
The material nonlinearity of the cable and beam is not considered.(2)
The gravity sag curve of cable is considered as a parabola.(3)
The bending stiffness, torsional stiffness, and shear stiffness of the cable are not considered.(4)
The change of cable force along the length direction is not considered.(5)
Cable is always in elastic state during vibration.(6)
The axial deformation of beam is not considered.On the premise of the above assumptions, the model is established in Figure1, considering the refined model of the cable-beam coupling.Figure 1
Parametric resonance refined model of cable-beam coupling.The motion trajectory of the cable and the deck beam is described by local coordinate systemo-xbyb ando-xcyc, respectively, as shown in Figure 1. According to the Hamilton principle, the differential equation of motion for the cable-beam coupling structure is [14] (1)mc∂2ucxc,t∂t2+cc∂ucxc,t∂t=∂∂sT∂ucxc,t+Xcxc∂s+mcgcos⁡θmc∂2vcxc,t∂t2+cc∂vcxc,t∂t=∂∂sT∂vcxc,t+Ycxc∂s+mcgsin⁡θmb∂2vbxb,t∂t2+cb∂vbxb,t∂t+EbIb∂4vbxb,t∂xb4=∂∂xbN∂vbxb,t+Ybxb∂xb-Tdcos⁡θδx+mbg.For the convenience of research, the variables with subscriptsc andb are defined as the variables of the cable and the deck beam, respectively. Variablesuandvindicate the longitudinal and transverse vibration displacement, respectively, and they are the function of the position coordinatex and timet.X andY represent the longitudinal and transverse sag curves function, respectively; moreover, Y=4z0[x/L-x/L2]. The meaning of other variables in formula (1) is as follows: m, ω, c, ξ, E, L, I, A, z0, z, and g are the quality of unit length, natural frequency, damping coefficient, damping ratio, elastic modulus, length, moment of inertia, the cross section area, the transverse initial deflection at the midspan, the sag at the midspan, and gravitational acceleration, respectively. Units of each variable are taken from the international unit system. In addition, T is the tangential tension of the stay cable, Td is the dynamic tension of cable in vibration process,Nis axial force of deck beam,s is thr coordinate by the arc length, and δ(x) is Dirac function: (2)δx=∞x=00x≠0.By using the static equilibrium relation of stay cable, formula (1) can be simplified as follows: (3)mc∂2uc∂t2+cc∂uc∂t-∂∂sT0∂uc∂s+EcAcεcdXcdxc+∂vc∂s=0mc∂2vc∂t2+cc∂vc∂t-∂∂sT0∂vc∂s+EcAcεcdYcdxc+∂vc∂s=0mb∂2vb∂t2+cb∂vb∂t+EbIb∂4vb∂xb4-∂∂xbN∂vb∂xb+EbAbεbdYbdxc+∂vb∂xb+EcAcεccos⁡θδx=0,where T0 is the static tangential tension of stay cable in formula (3); moreover, H0 is static axial tension and εc and εb are the axial strains of stay cable and beam, respectively. The axial strains can be expressed as follows: (4)εc=∂uc∂s+dYcdxc∂vc∂s+12∂uc∂s2+12∂vc∂s2εb=∂ub∂s+dYbdxb∂vb∂s+12∂ub∂s2+12∂vb∂s2,where ub represents the longitudinal displacement of beam in formula (4); the two-order small quantity of longitudinal elastic strain for the cable and beam can be neglected, because the longitudinal deformation of the cable and beam is much less than that of transverse deformation. It can be approximate to take ds≈dx for the stay cable with small sag, and formula (3) can be simplified as(5)mc∂2uc∂t2+cc∂uc∂t-∂∂xcEcAc∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2∂vc∂xc=0mc∂2vc∂t2+cc∂vc∂t-∂∂xcH0∂vc∂xc+EcAcdYcdxc+∂vc∂xc∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2=0mb∂2vb∂t2+cb∂vb∂t+EbIb∂4vb∂xb4-N∂2vb∂xb2+EcAc∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2cos⁡θδx=0.In the case of low modal vibration for the cable, the coupling effect of transverse and longitudinal vibration for the cable is not considered. Meanwhile, the first equation of formula (5) can be simplified as (6)et=∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2.The second equation of formula (5) can be simplified as(7)mc∂2vc∂t2+cc∂vc∂t-∂∂xcH0∂vc∂xc+EcAcdYcdxc+∂vc∂xcet=0.From the simplified assumptions, the geometric boundary condition of the stay cable is as follows:(8)ucLc=0;uc0-ub0cos⁡θ=0.On both sides of formula (6) is integral on [0,Lc] range for xc at the same time, and it is brought into formula (8), which can be obtained as follows: (9)et=vb0,tLccos⁡θ+1Lc∫0LcdYcdxc∂vc∂xc+12∂vc∂xc2dxc.The axial dynamic force increment of the cable in the process of transverse vibration can be expressed as follows:(10)Hd=EcAcLcevb0,tcos⁡θ+∫0LcdYcdxc∂vc∂xc+12∂vc∂xc2dxc10aHω=EcAcLcevb0,tcos⁡θ10bHp=EcAcLce∫0LcdYcdxc∂vc∂xc+12∂vc∂xc2dxc.The axial force of bridge deck beam in vibration is as follows:(11)N=H+Hdsin⁡θEquations (9) and (10) are simultaneous. The third equation of formula (5) is simplified; differential equations of motion for the cable-beam coupled vibration are as follows: (12)mcv¨c+ccv˙c-H0+Hd∂2vc∂xc2+Hd8z0Lc2=0mbv¨b+cbv˙b+EbIb∂4vb∂xb4-H0+Hdsin⁡θ∂2vb∂xb2+Hdcos⁡θδx=0.


### 2.2. Dimensionless Method of Parametric Resonance Equations Considering Cable-Beam Coupling

The differential equation (12) of motion for cable-beam coupled vibrations is derived by using dimensionless method, and the following dimensionless quantities are defined [15–17]:(13)t-=ωct;v-c=vcLc;v-b=vbLb;ω-c=ωceωc;ω-b=ωbωc,where ωc and ωce are the first natural frequencies of the cable without considering or considering the sag; ωb is the first natural frequency of bridge deck beam; γ is the sag-span ratio of cable; k2 is the stiffness coefficient of cable; Lc and Lb are the span of cable and beam, respectively; λ2 is Irvine parameter; βb indicates the ratio coefficient of first-order natural frequency for bridge deck beam; t- denotes the dimensionless time; v-c and v-b are the transverse relative displacements of the cable and beam, respectively. ω-c and ω-b are the relative frequencies of the cable and beam, respectively.The expressions of each parameter are(14)ωc=πLcH0mc;ωce=ωc1+2/π4λ221/2;ωb=βb2EbIbmbLb4;ξc=cc2mcωc;ξb=cb2mbωb;γ=z0Lc;k2=EcAcH0;λ2=64k2γ21+8γ2;βb4=ωb2mbLb4EbIb.Equation (12) is derived by using the dimensionless quantity of the above definition.(15)∂2v-c∂t-2+2ξcω-c∂v-c∂t--1π21+H-ω+H-p∂2v-c∂x-c2+H-ω+H-p8γπ2=0∂2v-b∂t-2+2ξbω-b∂v-c∂t-+1βb4ω-b2∂4v-b∂x-b4-1π21+H-ω+H-pmcmbLcLb2sin⁡θ∂2v-b∂x-b2+1π2H-ω+H-pmcmbLc2Lbcos⁡θδx=015aH-ω=k21+8γ2v-b0,t-cos⁡θ15bH-p=λ2648∫011γv-cdx-c+12∫011γ∂v-c∂x-c2dx-c.When considering that the first-order mode is only the main vibration, and constraint on the boundary condition of formula (8) is satisfied, the dimensionless displacement function of the cable is as follows: (16)v-cx-c,t-=v-b0,t-sin⁡θ1-x-c+sin⁡πx-cqct-.Moreover, the geometric boundary condition of the fixed end for the bridge deck beam is as follows:(17)vbLb=0;v˙bLb=0;v¨b0=0.The physical boundary condition of the connection between cable and bridge deck beam is as follows:(18)EbIbvb′′′0=-kcvb0,while the dimensionless modal function of in-plane vibration for the deck beam can be expressed as follows:(19)ψ-bx-b=A1sin⁡βbx-b+A2cos⁡βbx-b+A3sinh⁡βbx-b+A4cosh⁡βbx-b,where A1, A2, A3, and A4 are constant coefficients; they are determined by formulas (17) and (18). Moreover, βb is determined by the following characteristic equation:(20)kcEbIb=βb31+cos⁡βbLbcosh⁡βbLbcos⁡βbLbsinh⁡βbLb-sin⁡βbLbcosh⁡βbLb.And the dimensionless displacement function of the deck beam can be expressed as follows:(21)v-bx-b,t-=ψ-bx-bq-bt-.Formulas (16) and (21) are brought into formula (12) by the Galerkin method, on both sides multiplied by sin⁡(πx-c) and ψ-b at the same time, and integral on [0,1]; it can be obtained by using the mode orthogonality principle: (22)q¨c+2ξcω-cq˙c+ω-c2qc+α11qcqb+α12qcqb2+α2qc2+α3qc3+α4qb+α5qb2+γ1q˙b+γ2q¨b=0q¨b+2ξbω-bq˙b+ω-b2qb+β1qb+β11qcqb+β12qc2qb+β2qb2+β3qb3+β4qc+β5qc2=0,where α11, α12, α2, α3, α4, α5, γ1, γ2, ζ1, ζ2, μ1, μ2, β1, β11, β12, β2, β3, β4, and β5 are parameters.


## 2.1. Differential Equation of Motion with Parametric Resonance Considering Cable-Beam Coupling

Several basic assumptions are made before establishing the equation of motion for the cable-beam composite structure [13]:(1)
The material nonlinearity of the cable and beam is not considered.(2)
The gravity sag curve of cable is considered as a parabola.(3)
The bending stiffness, torsional stiffness, and shear stiffness of the cable are not considered.(4)
The change of cable force along the length direction is not considered.(5)
Cable is always in elastic state during vibration.(6)
The axial deformation of beam is not considered.On the premise of the above assumptions, the model is established in Figure1, considering the refined model of the cable-beam coupling.Figure 1
Parametric resonance refined model of cable-beam coupling.The motion trajectory of the cable and the deck beam is described by local coordinate systemo-xbyb ando-xcyc, respectively, as shown in Figure 1. According to the Hamilton principle, the differential equation of motion for the cable-beam coupling structure is [14] (1)mc∂2ucxc,t∂t2+cc∂ucxc,t∂t=∂∂sT∂ucxc,t+Xcxc∂s+mcgcos⁡θmc∂2vcxc,t∂t2+cc∂vcxc,t∂t=∂∂sT∂vcxc,t+Ycxc∂s+mcgsin⁡θmb∂2vbxb,t∂t2+cb∂vbxb,t∂t+EbIb∂4vbxb,t∂xb4=∂∂xbN∂vbxb,t+Ybxb∂xb-Tdcos⁡θδx+mbg.For the convenience of research, the variables with subscriptsc andb are defined as the variables of the cable and the deck beam, respectively. Variablesuandvindicate the longitudinal and transverse vibration displacement, respectively, and they are the function of the position coordinatex and timet.X andY represent the longitudinal and transverse sag curves function, respectively; moreover, Y=4z0[x/L-x/L2]. The meaning of other variables in formula (1) is as follows: m, ω, c, ξ, E, L, I, A, z0, z, and g are the quality of unit length, natural frequency, damping coefficient, damping ratio, elastic modulus, length, moment of inertia, the cross section area, the transverse initial deflection at the midspan, the sag at the midspan, and gravitational acceleration, respectively. Units of each variable are taken from the international unit system. In addition, T is the tangential tension of the stay cable, Td is the dynamic tension of cable in vibration process,Nis axial force of deck beam,s is thr coordinate by the arc length, and δ(x) is Dirac function: (2)δx=∞x=00x≠0.By using the static equilibrium relation of stay cable, formula (1) can be simplified as follows: (3)mc∂2uc∂t2+cc∂uc∂t-∂∂sT0∂uc∂s+EcAcεcdXcdxc+∂vc∂s=0mc∂2vc∂t2+cc∂vc∂t-∂∂sT0∂vc∂s+EcAcεcdYcdxc+∂vc∂s=0mb∂2vb∂t2+cb∂vb∂t+EbIb∂4vb∂xb4-∂∂xbN∂vb∂xb+EbAbεbdYbdxc+∂vb∂xb+EcAcεccos⁡θδx=0,where T0 is the static tangential tension of stay cable in formula (3); moreover, H0 is static axial tension and εc and εb are the axial strains of stay cable and beam, respectively. The axial strains can be expressed as follows: (4)εc=∂uc∂s+dYcdxc∂vc∂s+12∂uc∂s2+12∂vc∂s2εb=∂ub∂s+dYbdxb∂vb∂s+12∂ub∂s2+12∂vb∂s2,where ub represents the longitudinal displacement of beam in formula (4); the two-order small quantity of longitudinal elastic strain for the cable and beam can be neglected, because the longitudinal deformation of the cable and beam is much less than that of transverse deformation. It can be approximate to take ds≈dx for the stay cable with small sag, and formula (3) can be simplified as(5)mc∂2uc∂t2+cc∂uc∂t-∂∂xcEcAc∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2∂vc∂xc=0mc∂2vc∂t2+cc∂vc∂t-∂∂xcH0∂vc∂xc+EcAcdYcdxc+∂vc∂xc∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2=0mb∂2vb∂t2+cb∂vb∂t+EbIb∂4vb∂xb4-N∂2vb∂xb2+EcAc∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2cos⁡θδx=0.In the case of low modal vibration for the cable, the coupling effect of transverse and longitudinal vibration for the cable is not considered. Meanwhile, the first equation of formula (5) can be simplified as (6)et=∂uc∂xc+dYcdxc∂vc∂xc+12∂vc∂xc2.The second equation of formula (5) can be simplified as(7)mc∂2vc∂t2+cc∂vc∂t-∂∂xcH0∂vc∂xc+EcAcdYcdxc+∂vc∂xcet=0.From the simplified assumptions, the geometric boundary condition of the stay cable is as follows:(8)ucLc=0;uc0-ub0cos⁡θ=0.On both sides of formula (6) is integral on [0,Lc] range for xc at the same time, and it is brought into formula (8), which can be obtained as follows: (9)et=vb0,tLccos⁡θ+1Lc∫0LcdYcdxc∂vc∂xc+12∂vc∂xc2dxc.The axial dynamic force increment of the cable in the process of transverse vibration can be expressed as follows:(10)Hd=EcAcLcevb0,tcos⁡θ+∫0LcdYcdxc∂vc∂xc+12∂vc∂xc2dxc10aHω=EcAcLcevb0,tcos⁡θ10bHp=EcAcLce∫0LcdYcdxc∂vc∂xc+12∂vc∂xc2dxc.The axial force of bridge deck beam in vibration is as follows:(11)N=H+Hdsin⁡θEquations (9) and (10) are simultaneous. The third equation of formula (5) is simplified; differential equations of motion for the cable-beam coupled vibration are as follows: (12)mcv¨c+ccv˙c-H0+Hd∂2vc∂xc2+Hd8z0Lc2=0mbv¨b+cbv˙b+EbIb∂4vb∂xb4-H0+Hdsin⁡θ∂2vb∂xb2+Hdcos⁡θδx=0.


## 2.2. Dimensionless Method of Parametric Resonance Equations Considering Cable-Beam Coupling

The differential equation (12) of motion for cable-beam coupled vibrations is derived by using dimensionless method, and the following dimensionless quantities are defined [15–17]:(13)t-=ωct;v-c=vcLc;v-b=vbLb;ω-c=ωceωc;ω-b=ωbωc,where ωc and ωce are the first natural frequencies of the cable without considering or considering the sag; ωb is the first natural frequency of bridge deck beam; γ is the sag-span ratio of cable; k2 is the stiffness coefficient of cable; Lc and Lb are the span of cable and beam, respectively; λ2 is Irvine parameter; βb indicates the ratio coefficient of first-order natural frequency for bridge deck beam; t- denotes the dimensionless time; v-c and v-b are the transverse relative displacements of the cable and beam, respectively. ω-c and ω-b are the relative frequencies of the cable and beam, respectively.The expressions of each parameter are(14)ωc=πLcH0mc;ωce=ωc1+2/π4λ221/2;ωb=βb2EbIbmbLb4;ξc=cc2mcωc;ξb=cb2mbωb;γ=z0Lc;k2=EcAcH0;λ2=64k2γ21+8γ2;βb4=ωb2mbLb4EbIb.Equation (12) is derived by using the dimensionless quantity of the above definition.(15)∂2v-c∂t-2+2ξcω-c∂v-c∂t--1π21+H-ω+H-p∂2v-c∂x-c2+H-ω+H-p8γπ2=0∂2v-b∂t-2+2ξbω-b∂v-c∂t-+1βb4ω-b2∂4v-b∂x-b4-1π21+H-ω+H-pmcmbLcLb2sin⁡θ∂2v-b∂x-b2+1π2H-ω+H-pmcmbLc2Lbcos⁡θδx=015aH-ω=k21+8γ2v-b0,t-cos⁡θ15bH-p=λ2648∫011γv-cdx-c+12∫011γ∂v-c∂x-c2dx-c.When considering that the first-order mode is only the main vibration, and constraint on the boundary condition of formula (8) is satisfied, the dimensionless displacement function of the cable is as follows: (16)v-cx-c,t-=v-b0,t-sin⁡θ1-x-c+sin⁡πx-cqct-.Moreover, the geometric boundary condition of the fixed end for the bridge deck beam is as follows:(17)vbLb=0;v˙bLb=0;v¨b0=0.The physical boundary condition of the connection between cable and bridge deck beam is as follows:(18)EbIbvb′′′0=-kcvb0,while the dimensionless modal function of in-plane vibration for the deck beam can be expressed as follows:(19)ψ-bx-b=A1sin⁡βbx-b+A2cos⁡βbx-b+A3sinh⁡βbx-b+A4cosh⁡βbx-b,where A1, A2, A3, and A4 are constant coefficients; they are determined by formulas (17) and (18). Moreover, βb is determined by the following characteristic equation:(20)kcEbIb=βb31+cos⁡βbLbcosh⁡βbLbcos⁡βbLbsinh⁡βbLb-sin⁡βbLbcosh⁡βbLb.And the dimensionless displacement function of the deck beam can be expressed as follows:(21)v-bx-b,t-=ψ-bx-bq-bt-.Formulas (16) and (21) are brought into formula (12) by the Galerkin method, on both sides multiplied by sin⁡(πx-c) and ψ-b at the same time, and integral on [0,1]; it can be obtained by using the mode orthogonality principle: (22)q¨c+2ξcω-cq˙c+ω-c2qc+α11qcqb+α12qcqb2+α2qc2+α3qc3+α4qb+α5qb2+γ1q˙b+γ2q¨b=0q¨b+2ξbω-bq˙b+ω-b2qb+β1qb+β11qcqb+β12qc2qb+β2qb2+β3qb3+β4qc+β5qc2=0,where α11, α12, α2, α3, α4, α5, γ1, γ2, ζ1, ζ2, μ1, μ2, β1, β11, β12, β2, β3, β4, and β5 are parameters.


## 3. Discussion on the Equations with Dimensionless Parametric Resonance

### 3.1. Approximate Solution of the Method of Multiple Scales

The approximate solution of formula (22) is solved by the method of multiple scales, setting the form of solution as follows: (23)qc=εqc1T0,T1,T2+ε2qc2T0,T1,T2+ε3qc3T0,T1,T2qb=εqb1T0,T1,T2+ε2qb2T0,T1,T2+ε3qb3T0,T1,T2.Before the multiscale method is adopted to solve the equation, the parameters are as follows:(24)ξc=ε2μc;ξb=ε2μb;α4=εη1;γ1=ε2η2;γ2=ε2η3;β1=εη4;β4=εη5.Formula (23) is brought into formula (22), and the same power coefficient of ε is sorted out, which can be obtained as follows:ε order is(25)D02qc1+ω-c2qc1=0D02qb1+ω-b2qb1=0.ε 2 order is(26)D02qc2+ω-c2qc2=-2D0D1qc1-α11qc1qb1-α2qc12-η1qb1-α5qb12D02qb2+ω-b2qb2=-2D0D1qb1-η4qb1-β11qc1qb1-β2qb12-η5qc1-β5qc12.ε 3 order is(27)D02qc3+ω-c2qc3=-2D0D1qc2-D12+2D0D2qc1-2μcω-cD0qc1-α11qc1qb2+qc2qb1-α12qc1qb12-2α2qc1qc2-α3qc13-η1qb2-2α5qb1qb2-η2D0qb1-η3D02qb1D02qb3+ω-b2qb3=-2D0D1qb2-D12+2D0D2qb1-2μbω-bD0qb1-β11qc1qb2+qc2qb1-η4qb2-β12qc12qb1-2β2qb1qb2-β3qb13-η5qc2-2β5qc1qc2.From formula (27), the first solution of the equation can be expressed as(28)qc1=Q1T1,T2eiω-cT0+ccqb1=B1T1,T2eiω-bT0+cc,wherecc represents the front of the conjugate.Formula (28) is brought into the second equations of formula (27) and can be obtained as follows:(29)D02qc2+ω-c2qc2=-i2ω-c∂Q1∂T1eiω-cT0-α2Q12ei2ω-cT0-η1B1eiω-bT0-α5B12ei2ω-bT0-α11Q1B1eiω-c+ω-bT0-α11Q-1B1eiω-b-ω-cT0-α2Q1Q-1+α5B1B-1+ccD02qb2+ω-b2qb2=-i2ω-b∂B1∂T1eiω-bT0-η4B1eiω-bT0-β2B12ei2ω-bT0-η5Q1eiω-cT0-β5Q12ei2ω-cT0-β11Q1B1eiω-c+ω-bT0-β11Q-1B1eiω-b-ω-cT0-β5Q1Q-1+β2B1B-1+cc.It can be obtained by eliminating the periodic term of formula (29):(30)∂Q1∂T1=0i2ω-b∂B1∂T1+η4B1=0.By using the method that the equation contains a nonperiodic term of multiple scales, when the internal resonance is not considered, the solution of formula (29) is as follows:(31)qc2=α2Q123ω-c2ei2ω-cT0+η1B1ω-b2-ω-c2eiω-bT0+α5B124ω-b2-ω-c2ei2ω-bT0+α11Q1B1ω-b2+2ω-cω-beiω-c+ω-bT0+α11Q-1B1ω-b2-2ω-cω-beiω-b-ω-cT0-α2Q1Q-1+α5B1B-1ω-c2+ccqb2=β2B123ω-b2ei2ω-bT0-η5Q1ω-b2-ω-c2eiω-cT0-β5Q12ω-b2-4ω-c2ei2ω-cT0+β11Q1B1ω-c2+2ω-cω-beiω-c+ω-bT0+β11Q-1B1ω-c2-2ω-cω-beiω-b-ω-cT0-β5Q1Q-1+β2B1B-1ω-b2+cc.


### 3.2. Discussion on Resonance Characteristics of Parametric Excitation

From formula (31), η1B1/ω-b2-ω-c2eiω-bT0 items contained in the expression of the displacement modal component qc2 of the cable will make the first equation show the nonsingular solution when the frequency ratio between the bridge and the cable is satisfied with ω-b:ω-c= 1 : 1. It can be seen from the second equations that -η5Q1/ω-b2-ω-c2eiω-cT0 items contained in the expression of the displacement modal components qb2 of the bridge deck beam will make the second equation show the nonsingular solution, and this moment the cable-beam coupling system will present the 1 : 1 principal parametric resonance phenomenon with displacement modal component qc2 and qb2 as the main form.Similarly, when the frequency ratio between the bridge deck beam and the cable is satisfied with theω-b : ω-c= 2 : 1 and ω-b : ω-c= 1 : 2, α11Q-1B1/(ω-b2-2ω-cω-b)eiω-b-ω-cT0 or α5B12/(4ω-b2-ω-c2)ei2ω-bT0 items of the first equation and -β5Q12/(ω-b2-4ω-c2)ei2ω-cT0 or β11Q-1B1/(ω-c2-2ω-cω-b)ei(ω-b-ω-c)T0 items of the second equation of formula (31) led to the equations showing nonsingular solution.At this moment, the cable-beam coupling system will present the large vibration phenomenon of the 2 : 1 principal parametrical resonance and 1 : 2 superharmonic resonance with displacement modal componentqc2 and qb2 as the main form.


## 3.1. Approximate Solution of the Method of Multiple Scales

The approximate solution of formula (22) is solved by the method of multiple scales, setting the form of solution as follows: (23)qc=εqc1T0,T1,T2+ε2qc2T0,T1,T2+ε3qc3T0,T1,T2qb=εqb1T0,T1,T2+ε2qb2T0,T1,T2+ε3qb3T0,T1,T2.Before the multiscale method is adopted to solve the equation, the parameters are as follows:(24)ξc=ε2μc;ξb=ε2μb;α4=εη1;γ1=ε2η2;γ2=ε2η3;β1=εη4;β4=εη5.Formula (23) is brought into formula (22), and the same power coefficient of ε is sorted out, which can be obtained as follows:ε order is(25)D02qc1+ω-c2qc1=0D02qb1+ω-b2qb1=0.ε 2 order is(26)D02qc2+ω-c2qc2=-2D0D1qc1-α11qc1qb1-α2qc12-η1qb1-α5qb12D02qb2+ω-b2qb2=-2D0D1qb1-η4qb1-β11qc1qb1-β2qb12-η5qc1-β5qc12.ε 3 order is(27)D02qc3+ω-c2qc3=-2D0D1qc2-D12+2D0D2qc1-2μcω-cD0qc1-α11qc1qb2+qc2qb1-α12qc1qb12-2α2qc1qc2-α3qc13-η1qb2-2α5qb1qb2-η2D0qb1-η3D02qb1D02qb3+ω-b2qb3=-2D0D1qb2-D12+2D0D2qb1-2μbω-bD0qb1-β11qc1qb2+qc2qb1-η4qb2-β12qc12qb1-2β2qb1qb2-β3qb13-η5qc2-2β5qc1qc2.From formula (27), the first solution of the equation can be expressed as(28)qc1=Q1T1,T2eiω-cT0+ccqb1=B1T1,T2eiω-bT0+cc,wherecc represents the front of the conjugate.Formula (28) is brought into the second equations of formula (27) and can be obtained as follows:(29)D02qc2+ω-c2qc2=-i2ω-c∂Q1∂T1eiω-cT0-α2Q12ei2ω-cT0-η1B1eiω-bT0-α5B12ei2ω-bT0-α11Q1B1eiω-c+ω-bT0-α11Q-1B1eiω-b-ω-cT0-α2Q1Q-1+α5B1B-1+ccD02qb2+ω-b2qb2=-i2ω-b∂B1∂T1eiω-bT0-η4B1eiω-bT0-β2B12ei2ω-bT0-η5Q1eiω-cT0-β5Q12ei2ω-cT0-β11Q1B1eiω-c+ω-bT0-β11Q-1B1eiω-b-ω-cT0-β5Q1Q-1+β2B1B-1+cc.It can be obtained by eliminating the periodic term of formula (29):(30)∂Q1∂T1=0i2ω-b∂B1∂T1+η4B1=0.By using the method that the equation contains a nonperiodic term of multiple scales, when the internal resonance is not considered, the solution of formula (29) is as follows:(31)qc2=α2Q123ω-c2ei2ω-cT0+η1B1ω-b2-ω-c2eiω-bT0+α5B124ω-b2-ω-c2ei2ω-bT0+α11Q1B1ω-b2+2ω-cω-beiω-c+ω-bT0+α11Q-1B1ω-b2-2ω-cω-beiω-b-ω-cT0-α2Q1Q-1+α5B1B-1ω-c2+ccqb2=β2B123ω-b2ei2ω-bT0-η5Q1ω-b2-ω-c2eiω-cT0-β5Q12ω-b2-4ω-c2ei2ω-cT0+β11Q1B1ω-c2+2ω-cω-beiω-c+ω-bT0+β11Q-1B1ω-c2-2ω-cω-beiω-b-ω-cT0-β5Q1Q-1+β2B1B-1ω-b2+cc.


## 3.2. Discussion on Resonance Characteristics of Parametric Excitation

From formula (31), η1B1/ω-b2-ω-c2eiω-bT0 items contained in the expression of the displacement modal component qc2 of the cable will make the first equation show the nonsingular solution when the frequency ratio between the bridge and the cable is satisfied with ω-b:ω-c= 1 : 1. It can be seen from the second equations that -η5Q1/ω-b2-ω-c2eiω-cT0 items contained in the expression of the displacement modal components qb2 of the bridge deck beam will make the second equation show the nonsingular solution, and this moment the cable-beam coupling system will present the 1 : 1 principal parametric resonance phenomenon with displacement modal component qc2 and qb2 as the main form.Similarly, when the frequency ratio between the bridge deck beam and the cable is satisfied with theω-b : ω-c= 2 : 1 and ω-b : ω-c= 1 : 2, α11Q-1B1/(ω-b2-2ω-cω-b)eiω-b-ω-cT0 or α5B12/(4ω-b2-ω-c2)ei2ω-bT0 items of the first equation and -β5Q12/(ω-b2-4ω-c2)ei2ω-cT0 or β11Q-1B1/(ω-c2-2ω-cω-b)ei(ω-b-ω-c)T0 items of the second equation of formula (31) led to the equations showing nonsingular solution.At this moment, the cable-beam coupling system will present the large vibration phenomenon of the 2 : 1 principal parametrical resonance and 1 : 2 superharmonic resonance with displacement modal componentqc2 and qb2 as the main form.


## 4. Analysis on Numerical Example

### 4.1. Basic Parameters

The cable in practical engineering of a cable-stayed bridge can be taken as the study object, in order to further verify the characteristics of 1 : 1 main parametrical resonance for cable-beam coupled system. The geometric parameters and material properties of the stay cable are shown in Table1.Table 1
Parameter of cables and bridge deck beam.
Cable length (m) Deck beam length (m) Cable quality (kg·m−1) Bridge deck beam quality (kg·m−1) 178 30 120.4 1.6 × 10 6 Elastic modulus of cable (GPa) Elastic modulus of bridge deck (GPa) Cable section area (cm2) Angle of cable (°) 190 35 141.24 62.1 Natural frequency of cableωc (Hz) Cable damping ratioξc Damping ratio of bridge deckξb Time stepΔt- 0.768 0.001 0.01 0.01


### 4.2. The 1 : 1 Principal Parametrical Resonance Response Characteristics of Cable-Beam Coupled System

#### 4.2.1. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=3.33×10-4

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 0.01 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupled system are shown in Figures2 and 3.Figure 2
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.Figure 3
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From cable’s dimensionless displacement response time history and spectrum curve in Figure2, stay cable presents large beat vibration near the equilibrium position at its natural frequency ωc. Due to the small damping of the cable and the slow decay of parametrical resonance displacement response, the sharp vibration has been continued. From the dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end in Figure 3, beam presents the obvious beat vibration near the equilibrium position at initial displacement, the coupling effect is obvious, and the amplitude-frequency characteristic curve of the deck beam end is in single peak value, which says that it does harmonic vibration at its natural frequency.We could find that, from the dimensionless displacement response amplitude of cable-beam coupled system, due to deck beam’s large quality and stiffness, its energy during the vibration is larger than the energy during cable’s parametrical resonance process when its excitation amplitude of vertical initial displacement is small; at this moment, the constraints of the cable on the deck beam movement are very small, while the effect of the deck beam on the cable movement is large, so cable presents large beat vibration, and deck beam presents small beat vibration at given initial conditions.


#### 4.2.2. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=1.66×10-3

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 0.05 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures4 and 5.Figure 4
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.Figure 5
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From Figures5 and 6, compared with working condition qb(0)=3.3×10-4, parametrical resonance’s coupling effect of cable-beam composite structure is much more obvious under this working condition. Both cable and deck beam present obvious beat vibration phenomenon, but cable’s beat vibration has greater displacement response; the maximal displacement amplitude is increased from 1.18% of the cable length to 2.42%, which says that cable midspan’s parametrical resonance displacement amplitude is also increased as the excitation amplitude of vertical initial displacement of deck beam end increased; this is fully consistent with the principle of conservation of energy.Figure 6
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.


#### 4.2.3. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=3.3×10-8

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 10−6 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures 6 and 7.Figure 7
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From Figures6 and 7, under the condition of beam’s small initial displacement, both cable and beam present obvious beat vibration phenomenon; the coupling effect is obvious, but at this point the cable’s displacement amplitude is small and there is no significant vibration, which is only a slight stable “beat” vibration in the vicinity of the equilibrium position. Analysis of the reason is as follows: from the multiscale differential equations (29) of stay cable and beam’s displacement components qc2 and qb2, resonance terms of both parametrical resonance and primary resonance -α11Q-1B1ei(ω-b-ω-c)T0 and -η1B1eiω-bT0 are related to the coefficient α11 affected by deck beam’s excitation amplitude. By derivation, we conclude that the coefficient 32γ/π3 which is the difference between coefficients η1 and α11 is much affected by sag. So the main parametrical resonance will be not significant vibration under beam’s small initial condition.


## 4.1. Basic Parameters

The cable in practical engineering of a cable-stayed bridge can be taken as the study object, in order to further verify the characteristics of 1 : 1 main parametrical resonance for cable-beam coupled system. The geometric parameters and material properties of the stay cable are shown in Table1.Table 1
Parameter of cables and bridge deck beam.
Cable length (m) Deck beam length (m) Cable quality (kg·m−1) Bridge deck beam quality (kg·m−1) 178 30 120.4 1.6 × 10 6 Elastic modulus of cable (GPa) Elastic modulus of bridge deck (GPa) Cable section area (cm2) Angle of cable (°) 190 35 141.24 62.1 Natural frequency of cableωc (Hz) Cable damping ratioξc Damping ratio of bridge deckξb Time stepΔt- 0.768 0.001 0.01 0.01


## 4.2. The 1 : 1 Principal Parametrical Resonance Response Characteristics of Cable-Beam Coupled System

### 4.2.1. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=3.33×10-4

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 0.01 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupled system are shown in Figures2 and 3.Figure 2
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.Figure 3
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From cable’s dimensionless displacement response time history and spectrum curve in Figure2, stay cable presents large beat vibration near the equilibrium position at its natural frequency ωc. Due to the small damping of the cable and the slow decay of parametrical resonance displacement response, the sharp vibration has been continued. From the dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end in Figure 3, beam presents the obvious beat vibration near the equilibrium position at initial displacement, the coupling effect is obvious, and the amplitude-frequency characteristic curve of the deck beam end is in single peak value, which says that it does harmonic vibration at its natural frequency.We could find that, from the dimensionless displacement response amplitude of cable-beam coupled system, due to deck beam’s large quality and stiffness, its energy during the vibration is larger than the energy during cable’s parametrical resonance process when its excitation amplitude of vertical initial displacement is small; at this moment, the constraints of the cable on the deck beam movement are very small, while the effect of the deck beam on the cable movement is large, so cable presents large beat vibration, and deck beam presents small beat vibration at given initial conditions.


### 4.2.2. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=1.66×10-3

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 0.05 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures4 and 5.Figure 4
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.Figure 5
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From Figures5 and 6, compared with working condition qb(0)=3.3×10-4, parametrical resonance’s coupling effect of cable-beam composite structure is much more obvious under this working condition. Both cable and deck beam present obvious beat vibration phenomenon, but cable’s beat vibration has greater displacement response; the maximal displacement amplitude is increased from 1.18% of the cable length to 2.42%, which says that cable midspan’s parametrical resonance displacement amplitude is also increased as the excitation amplitude of vertical initial displacement of deck beam end increased; this is fully consistent with the principle of conservation of energy.Figure 6
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.


### 4.2.3. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=3.3×10-8

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 10−6 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures 6 and 7.Figure 7
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From Figures6 and 7, under the condition of beam’s small initial displacement, both cable and beam present obvious beat vibration phenomenon; the coupling effect is obvious, but at this point the cable’s displacement amplitude is small and there is no significant vibration, which is only a slight stable “beat” vibration in the vicinity of the equilibrium position. Analysis of the reason is as follows: from the multiscale differential equations (29) of stay cable and beam’s displacement components qc2 and qb2, resonance terms of both parametrical resonance and primary resonance -α11Q-1B1ei(ω-b-ω-c)T0 and -η1B1eiω-bT0 are related to the coefficient α11 affected by deck beam’s excitation amplitude. By derivation, we conclude that the coefficient 32γ/π3 which is the difference between coefficients η1 and α11 is much affected by sag. So the main parametrical resonance will be not significant vibration under beam’s small initial condition.


## 4.2.1. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=3.33×10-4

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 0.01 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupled system are shown in Figures2 and 3.Figure 2
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.Figure 3
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From cable’s dimensionless displacement response time history and spectrum curve in Figure2, stay cable presents large beat vibration near the equilibrium position at its natural frequency ωc. Due to the small damping of the cable and the slow decay of parametrical resonance displacement response, the sharp vibration has been continued. From the dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end in Figure 3, beam presents the obvious beat vibration near the equilibrium position at initial displacement, the coupling effect is obvious, and the amplitude-frequency characteristic curve of the deck beam end is in single peak value, which says that it does harmonic vibration at its natural frequency.We could find that, from the dimensionless displacement response amplitude of cable-beam coupled system, due to deck beam’s large quality and stiffness, its energy during the vibration is larger than the energy during cable’s parametrical resonance process when its excitation amplitude of vertical initial displacement is small; at this moment, the constraints of the cable on the deck beam movement are very small, while the effect of the deck beam on the cable movement is large, so cable presents large beat vibration, and deck beam presents small beat vibration at given initial conditions.


## 4.2.2. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=1.66×10-3

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 0.05 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures4 and 5.Figure 4
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.Figure 5
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From Figures5 and 6, compared with working condition qb(0)=3.3×10-4, parametrical resonance’s coupling effect of cable-beam composite structure is much more obvious under this working condition. Both cable and deck beam present obvious beat vibration phenomenon, but cable’s beat vibration has greater displacement response; the maximal displacement amplitude is increased from 1.18% of the cable length to 2.42%, which says that cable midspan’s parametrical resonance displacement amplitude is also increased as the excitation amplitude of vertical initial displacement of deck beam end increased; this is fully consistent with the principle of conservation of energy.Figure 6
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the cable midspan.


## 4.2.3. When the Working Conditionsqc(0)=5.618×10-6 and qb(0)=3.3×10-8

Under this working condition, the actual initial displacements of cable midspan and deck beam end are about 0.001 m and 10−6 m. The dimensionless displacement response time history and amplitude-frequency characteristic curve of coupling system are shown in Figures 6 and 7.Figure 7
Dimensionless displacement response time history and amplitude-frequency characteristic curve of the deck beam end.From Figures6 and 7, under the condition of beam’s small initial displacement, both cable and beam present obvious beat vibration phenomenon; the coupling effect is obvious, but at this point the cable’s displacement amplitude is small and there is no significant vibration, which is only a slight stable “beat” vibration in the vicinity of the equilibrium position. Analysis of the reason is as follows: from the multiscale differential equations (29) of stay cable and beam’s displacement components qc2 and qb2, resonance terms of both parametrical resonance and primary resonance -α11Q-1B1ei(ω-b-ω-c)T0 and -η1B1eiω-bT0 are related to the coefficient α11 affected by deck beam’s excitation amplitude. By derivation, we conclude that the coefficient 32γ/π3 which is the difference between coefficients η1 and α11 is much affected by sag. So the main parametrical resonance will be not significant vibration under beam’s small initial condition.


## 5. Conclusions

(1) Based on the establishment of the model of the cable-beam coupling dimensionless parametrical resonance, the main parametrical resonance characteristics of cable-beam coupled system are discussed emphatically in terms of the frequency ratio relation of the deck beam and cable meetsω-b : ω-c= 1 : 1. Then, the possibility of the occurrence of large main parametrical resonance of cable is proved theoretically.(2) The nonlinear main parametrical resonance response of cable is irrelevant to its initial displacement, which is increased with the bigger initial displacement of the deck beam end; also, the cable midspan displacement is increased obviously and presents obvious beat vibration phenomenon.(3) When the vertical initial displacement of the deck beam end is 10−6 m or even smaller, both the cable and beam present obvious beat vibration phenomenon, but, at this time, the cable displacement amplitude is small and there is no significant vibration, only making a slight stable “beat” vibration in the vicinity of the equilibrium position, which is different from the situation of 2 : 1 parametrical resonance.



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*Source: 6948081-2017-01-26.xml*