# Nonlinear Dynamical Analysis for the Cable Excited with Parametric and Forced Excitation

Authors: C. Z. Qian; C. P. Chen; G. W. Zhou
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Considering the deck vibration effect on the cable in cable-stayed bridge, using nonlinear structure dynamics theory, the nonlinear dynamical equation for the stayed cable excited with deck vibration is proposed. Research shows that the vertical vibration of the deck has a combined parametric and forced excitation effect on the cable when the angle of the cable is taken into consideration. Using multiscale method, the 1/2 principle parametric resonance is studied and the bifurcation equation is obtained. Despite the parameters analysis, the bifurcation characters of the dynamical system are studied. At last, by means of numerical method and software MATHMATIC, the effect rules of system parameters to the dynamical behavior of the system are studied, and some useful conclusions are obtained.


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## Body

## 1. Introduction

Along with the new materials, new technology continues to improve, large span cable is applied widespread, and the cable-stayed bridge becomes one of the main forms of long-span bridges in the word. Because of the flexibility, quality, and small damping characteristic, the dynamical problem of the cable structure is relatively prominent. In recent decades, the dynamics of stayed cable has gained widespread concern in the bridge engineering. Researches have shown that the breeze and light rain conditions may induce the cable having dramatic large amplitude vibrations [1–4]. Such sharp vibration may cause the fatigue fracture of cable or junction of fatigue cracks in the cable anchor, undermine the claim of anticorrosion system, and cause harm to the safety and durability of the bridge.In the earlier 1970s, some researchers began to study the cable vibration in cable-stayed bridge [5]. Variety induced vibration theory is proposed, such as vibration induced with rain and wind, secondary axial flow vibration, three-dimensional vortex shedding, and wake galloping. But so far stayed cable vibration mechanism was not studied clearly. It is shown that the cable vibration is a complex phenomenon affected by multiple factors. The aerodynamic characteristics caused by the wind and rain and the geometric nonlinearity of the bridge structure. The parametric vibration about cable is studied in the literature [6], in which a single mass is used to simulate continue cable. The parametric vibration about cable is also studied in literature [7], in which the cable is considered a vertical chord. Considering the initial deflection, the tension, and external excitation, the coupling dynamics of the cable about parametric vibration and forced vibration is studied [8].The vertical direction vibration of the deck will be induced with the wind loads or other live loads role. When an order of the natural frequency of the deck is similar to the twice the cable’s natural frequency, slight vibration of the bridge deck will inspire cable vibration sharply [9, 10]. This phenomenon is significant feature of the parametric vibration. Therefore, studying this vibration problem is meaningful to design and protection of the cable structure. In the literature [11], the equations governing the linearized dynamics of the cable-stayed beam are presented and the main features of the parametric vibration phenomenon are investigated. The literature [12] investigates the coupled nonlinear vibration behaviors of the cable/deck system by numerical analysis of a proposed three-degree-of-freedom model and using the Runge-Kutta integration method reveal some fundamental characteristics of the nonlinear cable vibration.Parametric vibration system is the incentive depending on the time and appears as a parameter in the control equation. A feature of such a vibration system is the system parameter with the time change. When the natural frequency of the system has a multiple relationship with the excitation frequency, even a small excitation can cause complex responses [13, 14] of the system which is different from the external excitation.Based on the literature [6], using bifurcation theory and nonlinear vibration theory, the coupling effect on parametric excitation and external force to the cable vibration is studied. Using numerical method, the bifurcation response is obtained, in which some nonlinear phenomenon is shown with figures.


## 2. Nonlinear Dynamical Equation of the Stay Cable

In order to study vibration mechanism of the stay cable bridge, the cable vibration system is modeled as Figure1. In this mechanics model, the initial deflection and off-plane motion are not considered. Using nonlinear vibration theory [13], the dynamical equation of the cable in plan xy is expressed as (1). Consider
(1)ρA∂2w∂t2-T∂2w∂x2+2μ∂w∂t=EAL[D(t)cos⁡γ+12∫0L(∂w∂x)2dx]∂2w∂x2.
With the boundary condition
(2)w(0,t)=0,w(L,t)=D(t)sinγ,
in which w denotes the displacement of cable in y direction, L denotes the length, A is the section area of the cable, E denotes elastic modulus, ρ denotes the density of the material, and γ is the angle between cable and vertical direction. D(t)=Dcos⁡ωt is the vertical displacement of the connection point of the cable and the bridge deck. If only the first order modal is considered, the solution of (1) can be denoted by
(3)w(x,t)=xLD(t)sinγ+y(t)sinπxL.
For purpose of analysis, some nondimensional parameters are introduced as follows:
(4)w^=wL,d=DL,η=xL,τ=ωt,ω0=πLTρA,λ=ω0ω,c=πωLEρ.
Then (1)–(3) can be rewritten as
(5)∂2w^∂τ2-TρAL2ω2∂2w^∂η2+2μρALω∂w^∂τ=EρL2ω2[dcos⁡τcos⁡γ+12∫01(∂w^∂η)2dη]∂2w^∂η2,(6)w^(0,τ)=0,w^(1,τ)=dcos⁡τsinγ,(7)w^(η,τ)=ηdcos⁡τsinγ+y(τ)sinπη.
Substituting (7) into (5) and adding the viscous damping item, using Galerkin integration method, the differential equation about y(t) can be simplified as
(8)d2ydτ2+2ξdydτ+[c12+c22(dcos⁡τcos⁡γ+12d2cos⁡2τsin2γ)]×y(τ)+c222y3(τ)-2dπsinγcos⁡τ=0
when γ=0 and (8) has not forced vibration item, which means the system (1) has only parametric vibration and this case has been introduced in literature [6]. In general case, γ≠0, forced vibration and parametric vibration items exist in (8) at the same time and system (1) should be considered a dynamical system with combined forced and parametric excitation. Considering d2 is higher order small quantities than d, (1/2)d2cos⁡2τsin2γ can be ignored and (8) can be simplified as
(9)d2ydτ2+2ξdydτ+[λ2+c22dcos⁡τcos⁡γ]y(τ)+c222y3(τ)-2πdsinγcos⁡τ=0.Figure 1
Dynamical modules of cable.


## 3. Bifurcation Response Equations

In order to take advantage of multiscale method analysis, ignoring the effect of deflect and using the substitution parameters,(10)ξ⟶εμ,c222⟶ε2β,c22dcos⁡γ⟶εf,-2dπsinγ⟶p.
Then, (9) can be rewritten as
(11)d2ydτ2+2εμdydτ+λ2y+ε2βy3+εfycos⁡τ+pcos⁡τ=0.
Based on the multiple scale method, the perturbation solution of (11) can be expensed as the power series about small parameter ε:
(12)y(ε,τ)=y0(T0,T1,T2)+εy1(T0,T1,T2)+ε2y2(T0,T1,T2)+⋯,
in which Ti=εiτ. Considering the differential connections,
(13)ddτ=∂∂T0+ε∂∂T1+ε2∂∂T2=D0+εD1+ε2D2.
Substituting (12) into (11), a series perturbation equations can be obtained as follows:
(14)D02y0+λ2y0=-pcos⁡T0,(15)D02y1+λ2y1=-2D0D1y0-2μD0y0-fy0cos⁡T0,(16)D02y2+λ2y2=-2D0D2y0-D12y0-2D0D1y1-2μD0y1-2μD1y0-βy03-fy1cos⁡T0.
Solving (14), y0 can be expressed as
(17)y0=A(T1,T2)eiλT0+A-(T1,T2)e-iλT0+G(eiT0+e-iT0),
in which G=p/2(1-λ2). Substituting (17) into (15), then (15) can be rewritten as
(18)D02y1+λ2y1=-2iλD1AeiλT0+2iλD1A-e-iλT0-2iμλAeiλT0+2iμλA-e-iλT0-2iμGeiT0+2iμGe-iT0-f2Aei(λ+1)T0-f2A-e-i(λ+1)T0-f2Ge2iT0-fG-f2A-e-i(λ-1)T0-f2Aei(λ-1)T0-f2Ge-2iT0.
Considering the 1/2 main parametric resonance condition, assuming
(19)λ=ω0ω=12-εσ,
in which σ is tuning parameter, substituting (19) into (18), and eliminating the long-term items, then (20) is obtained as follows:
(20)2iλD1A+2iμλA+f2A-e2iσT1=0.
Solving (18), y1 can be obtained as
(21)y1=2iμG1-λ2eiT0-2iμG1-λ2e-iT0+fA2(2λ+1)ei(λ+1)T0+fG2(4-λ2)e2iT0-fGλ2+fA-2(2λ+1)e-i(λ+1)T0+fG2(4-λ2)e-2iT0.
Substituting (17), (21) into (16) and eliminating the long-term items, then (22) is obtained as follows:
(22)-2iλD2A-D12A-2μD1A-β(3A2A-+6G2A)-f2A4(2λ+1)=0.
Associating (20), eliminating the D12A and D1A items from (22), and considering 2λ≈1, then (23) can be obtained as follows:
(23)iD2A-μ2A-fσA-e2iσT1+β(3A2A-+6G2A)+3f2A8=0.
Considering the differential connection,
(24)dAdτ=εD1A+ε2D2A.
Equation (24) can be rewritten as
(25)i(dAdτ+εμA)+12εfA-e2iσT-ε2×[μ2A-fσA-e2iσT1+β(3A2A-+6G2A)+3f2A8]=0.
Assuming that A=(1/2)a(τ)eiφ(τ) and is substituted into (25), separating the real part and the imaginary part, (26) can be obtained as
(26)dadτ=-εμa-(12εf+fσε2)asinθ,12adθdτ=2εσ+2ε2μ2a-12ε2βG2a-6ε2βa3-(εf-2fσε2)acos⁡θ,
where
(27)θ=εστ-2φ.
For da/dτ=dθ/dτ=0 corresponding to the steady-state solution of the system, (26) can be rewritten as
(28)εμa+(12εf+fσε2)asinθ=0,-εσa2-ε2μ2a+6ε2βG2a+3ε2βa3+(12εf-fσε2)acos⁡θ=0.
Eliminating θ from (28), then
(29)μ2a2+(12σ-εμ2+6εβG2+3εβa2)2a2=(12f-fσε)2a2.
In addition to ordinary solution a=0, the nonordinary solution of (29) meets the following equation:
(30)μ2+(12σ-εμ2+6εβG2+3εβa2)2=(12f-fσε)2.
Solving (30), the nonordinary solution can be obtained as
(31)a2=-((1/2)σ-εμ2+6εβG2)±[((1/2)f-fσε)2-μ2]1/23εβ.
Considering different parameters, there may exist some possible solutions as follows:(I)
((1/2)f-fσε)2-μ2=0, (1/2)σ-εμ2+6εβG2<0, and system has one nonzero solution;(II)
((1/2)f-fσε)2-μ2>0,  (1/2)σ-εμ2+6εβG2<0,  ((1/2)σ-εμ2+6εβG2)2>((1/2)f-fσε)2-μ2, and system has only two nonzero real solutions;(III)
((1/2)f-fσε)2-μ2>0,(1/2)σ-εμ2+6εβG2<0,((1/2)σ-εμ2+6εβG2)2≤((1/2)f-fσε)2-μ2,   and system has only one nonzero real solution;(IV)
((1/2)f-fσε)2-μ2>0,(1/2)σ-εμ2+6εβG2>0,((1/2)σ-εμ2+6εβG2)2>((1/2)f-fσε)2-μ2,   and system has only one nonzero real solution;(V)
other conditions, system exist only one zero solution.From the above-mentioned discussion, it is sure that there are critical excitation amplitudefcrit and fold bifurcation point fT. In order to avoid saddle-node bifurcation, the critical bifurcation point fcrit is important. Consider
(32)fcrit=±μεω.
At this time, the system parameters meet with
(33)(12σ-εμ2+6εβG2)2=(12fT-fTσε)2-μ2.
The supercritical bifurcation is decided with bifurcation point number fT in (33).


## 4. Numerical Analysis for the Cable Vibration

In this section, the numerical method and software MATHMATIC are used to solve a cable system. It aims at analyzing the vibration frequency of deck and inclination angle of the cable effect on the cable vibration. The parameters of the structure are defined as Table1, in which the damping is not considered and the vibration amplitude of the anchorage point is defined as D=5 mm. the first natural vibration frequency is obtained f=2.6 Hz and ω0=16.352.Table 1
The essential parameters of the cable.
Mass per unit length (ρA)
0.391 kg/m
Length of cable (L)
11.90 m
Axial tensile force (T)
1.5 kN
Section area (A)
7.28 mm2
Elastic modulus (E)
8.242× 104 MPa
Extensional rigidity (EA)
6.0× 105 NUsing the numerical method, the bifurcation equation (29) and the different equation (9) are solved, in which the deck vibration amplitude is assumed as constant. The parametric resonance bifurcation curves corresponding to various inclination angles of the cables can be obtained as Figure 2.Figure 2
Frequency-amplitude curves with various inclination angles of the cables.Results of bifurcation points of parametric resonance corresponding to various cables’ inclination angles are given in Figure2. It shows that inclination angles change the dislocation of bifurcation points.When the inclination angles of the cable are made small, two bifurcation points become closer and the amplitude of resonance becomes larger. The parametric resonance becomes a dominant factor compared to the forced oscillation. In order to observe the effect of the inclination angle of amplitude response, the amplitude-inclination angle curves are given in Figure3; in those curves, the vibration frequencies of the anchor point are fixed.Figure 3
Amplitude-inclination angle curves with various deck vibration frequencies.From Figure3, it can be found that tune frequency parameter σ affects not only the bifurcation point values but also the numbers of bifurcation points. Based on the numerical analyzing, the rules are found as follows. (1) When σ≤0, the system has only one critical bifurcation point and the system has only one steady state response solution. (2) The nonsteady response solution occurred as σ increase, and a fold bifurcation occurred. (3) When σ reach a fixed value, the systems’ response changed to be more complicated.In order to observe the effect of parametric excitation, the curves time domain and phase plan result of some systems are given in Figures4 and 5.Time domain curve and phase plane about the systemsσ=0, γ=π/3.
(a)
(b)Time domain curve and phase plane about the systemsσ=0, γ=π/6.
(a)
(b)From Figures4 and 5, some interesting phenomenon is observed. As the case of 0<γ<π/2, the vibration responses are caused by parametric excitation and forced excitation or the corresponding joint action. As γ decreases, the parametric excitation plays a major role; this conclusion meets well with [8]. When the value of γ is larger, parametric excitation and forced excitation combined effects are obvious; its dynamic behavior is more complex and will need further research. So, this paper will not repeat them.


## 5. Conclusion

In this paper, nonlinear dynamical characters of the cable in cable-stayed bridge are studied. First, using nonlinear structure dynamics, a nonlinear model of a cable in cable-stayed bridge excited with deck vibration is proposed. Secondly, using multiscale method, the 1/2 principle parametric resonance is studied and the bifurcation equation is obtained. And then the bifurcation characters of the dynamical system are studied. At last, the effect rules of system parameters to the dynamic behavior of the system are studied. Through study of this dynamical system, some useful conclusions are obtained.(1)
Parametric vibration of the cables in the cable-stayed bridge is induced with the vertical vibration of the deck. At the same time, the vertical vibration of the deck has the forced excitation effect on the cables.(2)
In general, the vibration response of the cable is joint effect of the parametric and forced excitation.(3)
Asγ decreases, the parametric excitation plays a major role. When the value of γ is larger, parametric excitation and forced excitation combined effects are obvious.



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*Source: 183257-2014-10-19.xml*