CN103926947B - Semi-active vibration control method of nonlinear system of long-span cable bridge structure - Google Patents

Abstract

The invention relates to a semi-active vibration control method of a nonlinear system of a long-span cable bridge structure. The method includes the following steps that first, a buffeting force load model acting on the long-span cable bridge structure is established; second, a long-span cable bridge structure vibration control system model is established; third, composite performance indexes described by the formula 3 are selected; fourth, the maximum principle is carried out on the formula 2 and the formula 3; fifth, the optimum vibration control law is solved. According to the method, the nonlinear dynamic characteristics of the long-span cable bridge structure are taken into consideration, and thus a control system closer to the reality is established; a reasonable solution of a controller is used, the optimum vibration control law has good vibration reducing effect on the nonlinear system of the long-span cable bridge structure, and thus the method can adapt to control over long-space cable bridges with large-amplitude vibration better.

Description

Semi-active vibration control method for nonlinear system of large-span cable bridge structure

Technical Field

The invention relates to a vibration control method, in particular to a semi-active vibration control method for a nonlinear system of a large-span cable bridge structure.

Background

The bridge plays an irreplaceable role in a traffic system and has important values in political, economic, cultural and military aspects. Although the traffic system in China has been greatly developed, it is still difficult to meet the increasing traffic demand, and the traffic infrastructure is under great pressure. In addition, extreme weather and geological disasters frequently occur in China in recent years, and accidents and dangerous cases frequently occur in bridge structures in traffic systems in China. The bridge structure bears various load effects to cause the structure to vibrate violently, the vibration of the bridge structure not only reduces the quality of the driving environment, but also accelerates the mechanical damage and fatigue failure of the structure, and reduces the safety of the bridge. Long-term vibration of the structure can lead to bridge overturning, causing significant economic losses and adverse social impact. Therefore, constructing a bridge vibration control system to reduce the vibration of the bridge and improve the reliability and safety of the bridge is a problem to be solved urgently.

In recent years, a great number of theoretical analyses and experimental researches are carried out on the dynamic action response between different loads and bridges and the vibration control of the bridges by many scholars at home and abroad. With the continuous perfection of the structure dynamics theory and the finite element theory and the wide application of high-speed large-capacity computers, the axle vibration research is greatly developed. At present, people can establish a vehicle and bridge calculation model closer to actual engineering, can consider the influences of various factors such as the mass of a vehicle body and a bogie, the action of a damper and a spring, the driving speed, the mass, the rigidity and the damping of an upper structure and a lower structure of a bridge, the dynamic interaction between wheels and tracks, and between tracks and beams, and the irregularity of the tracks, and adopt a numerical simulation method to research the spatial vibration control of an axle time-varying system.

In the construction process of a vibration control system of a bridge structure, a linear model is suitable for the vibration condition with small amplitude, and for strong vibration loads (such as earthquakes and the like) acting on the structure, in addition, the large-span cable bridge structure has the characteristics of material nonlinearity, geometric nonlinearity and the like, and the linear model is not suitable any more. Therefore, it is a problem to be solved to find a vibration control method suitable for the vibration control with nonlinear dynamic characteristics.

In view of the above, the present invention is particularly proposed.

Disclosure of Invention

The invention aims to provide a semi-active vibration control method suitable for a large-span cable bridge structure with nonlinear dynamic characteristics.

In order to solve the technical problems, the invention adopts the technical scheme that:

a semi-active vibration control method for a nonlinear system of a large-span cable bridge structure comprises the following steps:

(1) establishing a shaking force load model acting on a large-span cable bridge structure:

the harmonic waves formed by a plurality of trigonometric functions are mutually superposed to simulate the shaking force load acting on the large-span cable bridge structure, and the shaking force acting on the large-span cable bridge structure of the jth formed wave can be given by the following formula:

j=1,2,…,r

wherein A is_j，ω_jRespectively representing the amplitude, frequency and initial phase angle of the jth component wave,

order toTo obtain

Wherein,order toCan obtain the product

Wherein,I_ris a unitMatrix, 0 ∈ R^r×rIs a zero matrix;

the shaking force load model acting on the large-span cable bridge structure is an external system described by formula 1:

in the formula 1, the compound is shown in the specification,

wherein: p (t) is the shaking force acting on the large-span cable bridge structure;

(2) establishing a large-span cable bridge structure vibration control system model:

the cable-bridge coupling vibration power system under the action of the shaking force load comprises:

wherein W (t) is the displacement from the equilibrium position; y (t) is the displacement of the cable end, namely the displacement of the bridge deck along the cable direction; omega₁And ω₂Natural frequency of cable and bridge deck, ξ damping ratio of bridge deck, a_i(i =1,2, …,6) is the system parameter, u (t) is the semi-active control force, p (t) is the buffeting force loading, generated by equation 1;

selecting a state variable:

then the model of the large-span cable bridge structure vibration control system is as follows:

formula 2; wherein,

(3) selecting a composite performance index as described in a formula 3;

formula 3;

(4) the maximum principle is implemented for equations 2 and 3:

first, a Hamiltonian is constructed as formula 4:

H(·)=x^TQx₁+Ru²+λ^T(Ax+Bu+f(x)+Dp_ω) Formula 4;

and further converting the problem of solving the controller u (t) under the constraint of the performance index formula 3 in the formula 1 into a problem of solving the following nonlinear two-point boundary values in the formula 5 according to the extreme value condition:

formula 5;

(5) solving an optimal vibration control law:

solving the iterative solution of the nonlinear two-point boundary value problem described in equation 5 by using a successive approximation method, and enabling

λ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)

Where g (t) is a common mode vector, for λ (t) = P₁x(t)+P₂p(t)+P₃p_ω(t) + g (t) derivation, followed by taking the first form of formula 2Substituted therein to obtain

From formula 2 and formula 5 to obtain

In view of

By comparisonThe matrix equation 7 and the sequence equations 8 and 9 are obtained, and the optimal vibration control law described by equation 6 can be obtained

Formula 7;

formula 8;

formula 9;

the approximate optimal vibration control law is obtained from the above equations 7, 8 and 9:

u^(k)(t)=-R^-1B^Tλ^(k)(t)=-R^-1B^T[P₁x^(k)(t)+P₂p(t)+P₃p_ω(t)+g^(k)(t)]formula 6;

wherein,and P is₁，P₂And P₃The following equation 7: x is the number of^(k)And g^(k)(t) is determined from the following formulae 8 and 9.

Further, after the step (2), the method comprises the following steps: simplifying the large-span cable bridge structure vibration control system described by the large-span cable bridge structure vibration control system model of the formula (2) into a single-degree-of-freedom system.

The invention has the beneficial effects that: the invention considers the nonlinear power characteristics in the large-span cable bridge structure and establishes a control system which is closer to reality; the optimal vibration control law has a good vibration damping effect on a nonlinear system of a large-span cable bridge structure and is more suitable for control of a large-amplitude vibrating large-span cable bridge, the analytic solution of directly solving the nonlinear two-point boundary value problem is avoided by using a successive approximation algorithm, only the solution of a linear vector difference equation sequence needs to be solved, the whole control law of linear iteration is subjected to transposition iteration common-state vector, and the calculated amount is greatly reduced.

Drawings

FIG. 1 is a flow chart of the present invention;

FIG. 2 is a mechanical model diagram of a guy cable and a bridge deck;

FIG. 3 is a graph of the displacement of the large span cable bridge structure of the present invention;

fig. 4 is a velocity profile of the long span cable bridge structure of the present invention.

Detailed Description

In order that those skilled in the art will better understand the disclosure, the invention will be described in further detail with reference to the accompanying drawings and specific embodiments.

Referring to fig. 1, the invention relates to a semi-active vibration control method of a nonlinear system of a large-span cable bridge structure, which comprises the following steps:

s100, establishing a shaking force load model acting on a large-span cable bridge structure:

the shaking force loads acting on the large span cable bridge structure can be constructed using harmonic synthesis. According to the harmonic synthesis method, the shaking force load acting on the large-span cable bridge structure can be simulated by mutually superposing the harmonics formed by a plurality of trigonometric functions, and the shaking force acting on the large-span cable bridge structure of the jth component wave can be given by the following formula:

j=1,2,…,r

wherein A is_j，ω_jRespectively representing the amplitude, frequency and initial phase angle of the jth component wave,

order toTo obtain

Wherein,order toCan obtain the product

Wherein,I_ris an identity matrix, 0 ∈ R^r×rIs a zero matrix;

the shaking force load model acting on the large-span cable bridge structure is an external system described by formula 1:

formula 1, wherein: p (t) is the shaking force acting on the large-span cable bridge structure;

s101, establishing a large-span cable bridge structure vibration control system model:

for the convenience of research, the bridge deck is simplified into concentrated masses M acting on the cable ends, the stiffness of the deck is simulated by springs K, and the damping of the deck is simulated by C. The cable-bridge coupling vibration under the action of the shaking force load can be decomposed into two motions along the axial direction and perpendicular to the cable axis, the axial vibration is a research object in the text, and the mechanical model of the cable and the bridge deck is shown in figure 2, wherein: l is the length of the cord.

As the basic mode occupies the main position in the vibration system formed by the guy cable and the bridge deck, and the first-order vibration mode is adopted, the cable-bridge coupling vibration power system under the action of buffeting force load is as follows:

wherein W (t) is the displacement from the equilibrium position; y (t) is the displacement of the cable end, namely the displacement of the bridge deck along the cable direction; omega₁And ω₂Natural frequency of cable and bridge deck, ξ damping ratio of bridge deck, a_i(i =1,2, …,6) is the system parameter, u (t) is the semi-active control force, p (t) is the buffeting force loading, generated by equation 1;

selecting a state variable:

then the model of the large-span cable bridge structure vibration control system is as follows:

formula 2;

wherein,

s102, selecting a composite performance index described as a formula 3 to ensure that the input control force and the input control time are minimum;

formula 3;

s103, implementing a maximum value principle for the formulas 2 and 3: converting the nonlinear optimal control problem into a nonlinear non-homogeneous two-point edge value

First, a Hamiltonian is constructed as formula 4:

H(·)=x^TQx₁+Ru²+λ^T(Ax+Bu+f(x)+Dp_ω) Formula 4;

and further converting the problem of solving the controller u (t) under the constraint of the performance index formula 3 in the formula 1 into a problem of solving the following nonlinear two-point boundary values in the formula 5 according to the extreme value condition:

formula 5;

s104, solving an optimal vibration control law:

because the analytic solution of the nonlinear two-point boundary value problem is difficult to obtain, the iterative solution of the nonlinear two-point boundary value problem described by the formula 5 is solved by using a successive approximation method, so that

λ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)

Where g (t) is a common mode vector, for λ (t) = P₁x(t)+P₂p(t)+P₃p_ω(t) + g (t) derivation, followed by taking the first form of formula 2Substituted therein to obtain

From formula 2 and formula 5 to obtain

In view of

By comparisonThe matrix equation 7 and the sequence equations 8 and 9 are obtained, and the optimal vibration control law described by equation 6 can be obtained

Formula 7;

formula 8;

formula 9;

the approximate optimal vibration control law is obtained from the above equations 7, 8 and 9:

u^(k)(t)=-R^-1B^Tλ^(k)(t)=-R^-1B^T[P₁x^(k)(t)+P₂p(t)+P₃p_ω(t)+g^(k)(t)]formula 6;

wherein,and P is₁，P₂And P₃The following equation 7: x is the number of^(k)And g^(k)(t) is determined from the following formulae 8 and 9.

Preferably, the method comprises the following steps after the step (2): simplifying the large-span cable bridge structure vibration control system described by the large-span cable bridge structure vibration control system model of the formula (2) into a single-degree-of-freedom system.

The invention considers the nonlinear power characteristics in the large-span cable bridge structure and establishes a control system which is closer to reality; the optimal vibration control law has a good vibration damping effect on a nonlinear system of a large-span cable bridge structure and is more suitable for control of a large-amplitude vibrating large-span cable bridge, the analytic solution of directly solving the nonlinear two-point boundary value problem is avoided by using a successive approximation algorithm, only the solution of a linear vector difference equation sequence needs to be solved, the whole control law of linear iteration is subjected to transposition iteration common-state vector, and the calculated amount is greatly reduced.

In order to verify the effectiveness of the control law, fig. 3 and 4 show the system open loop and the displacement and speed curves of the system under the control of the control law, and it can be seen from the graphs that the control law has a good damping effect on the nonlinear system of the large-span cable bridge structure. The displacement and the speed acceleration of the structure are respectively reduced to 19 percent and 17 percent of the open loop state of the system under the control of the control law.

The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.

Claims (2)

1. A semi-active vibration control method for a nonlinear system of a large-span cable bridge structure is characterized by comprising the following steps:

(1) establishing a shaking force load model acting on a large-span cable bridge structure:

the harmonic waves formed by a plurality of trigonometric functions are mutually superposed to simulate the shaking force load acting on the large-span cable bridge structure, and the shaking force acting on the large-span cable bridge structure of the jth formed wave can be given by the following formula:

j=1,2,…,r

wherein A is_j，ω_jRespectively representing the amplitude, frequency and initial phase angle of the jth component wave,

order to $\stackrel{\‾}{p}\left(t\right)=[\stackrel{\‾}{p_1},\stackrel{\‾}{p_2},\·\·\·,\stackrel{\‾}{p_r}{]}^T,$ To obtain

$\stackrel{\·\·}{p_j}=-{\mathrm{\ω}}^2\stackrel{\‾}{p_j},j=\mathrm{1,2},\·\·\·,r,$

$\begin{array}{c}\stackrel{\stackrel{\·\·}{~}}{p}\left(t\right)=-\mathrm{diag}\{{{\mathrm{\ω}}_1}^2,{{\mathrm{\ω}}_2}^2,\·\·\·,{{\mathrm{\ω}}_r}^2\}\tilde{p}\left(t\right)\\ =-{\stackrel{\‾}{\mathrm{\Ω}}}^2\tilde{p}\left(t\right)\end{array}$

Wherein, $\stackrel{\‾}{\mathrm{\Ω}}=\mathrm{diag}\{{\mathrm{\ω}}_{1,}{\mathrm{\ω}}_2,\·\·\·,{\mathrm{\ω}}_r\}.$ order to $w\left(t\right)={\left[\begin{array}{cc}\stackrel{\‾}{p}\left(t\right)& \stackrel{\stackrel{\·}{\‾}}{p}\left(t\right)\end{array}\right]}^T$ Can obtain the product

$\stackrel{\·}{w}\left(t\right)=\left[\begin{array}{cc}0& I_r\\ -{\stackrel{\‾}{\mathrm{\Ω}}}^2& 0\end{array}\right]w\left(t\right)=\stackrel{\‾}{G}w\left(t\right),$

$\tilde{p}\left(t\right)=\left[\begin{array}{cc}{I}_r& 0\end{array}\right]w\left(t\right),$

Wherein,I_ris an identity matrix, 0 ∈ R^r×rIs a zero matrix;

the shaking force load model acting on the large-span cable bridge structure is an external system described by formula 1:

$\begin{array}{c}\stackrel{\·}{w}\left(t\right)=\stackrel{\‾}{G}w\left(t\right),\\ p\left(t\right)=\stackrel{\‾}{F}w\left(t\right)\end{array}$ in the formula 1, the compound is shown in the specification,

wherein: p (t) is the shaking force acting on the large-span cable bridge structure;

(2) establishing a large-span cable bridge structure vibration control system model:

the cable-bridge coupling vibration power system under the action of the shaking force load comprises:

$\stackrel{\·\·}{W}\left(t\right)+({\mathrm{\ω}}_1^2+a_{3}Y\left(t\right))W+a_1{W}^3\left(t\right)+a_2{W}^2\left(t\right)+a_{4}Y\left(t\right)=0,$

$\stackrel{\·\·}{Y}\left(t\right)+2{\mathrm{\ω}}_2\mathrm{\ξ}\stackrel{\·}{Y}\left(t\right)+{\mathrm{\ω}}_2^{2}Y\left(t\right)+a_{5}W\left(t\right)+a_6{W}^2\left(t\right)=u\left(t\right)+p\left(t\right)$

wherein W (t) is the displacement from the equilibrium position; y (t) is the displacement of the cable end, namely the displacement of the bridge deck along the cable direction; omega₁And ω₂Natural frequency of cable and bridge deck, ξ damping ratio of bridge deck, a_i(i =1,2, …,6) is a system parameterA number, u (t) is the semi-active control force, p (t) is the buffeting force load, generated by equation 1;

selecting a state variable:

$x_1\left(t\right)=W\left(t\right),x_2\left(t\right)=\stackrel{\·}{W}\left(t\right),x_3\left(t\right)=Y\left(t\right),x_4\left(t\right)=\stackrel{\·}{Y}\left(t\right),$

then the model of the large-span cable bridge structure vibration control system is as follows:

$\begin{array}{c}\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\end{array}$ formula 2;

wherein,

$A=\left[\begin{array}{cccc}0& 1& 0& 0\\ -{\mathrm{\ω}}_1^2& 0& {-a}_4& 0\\ 0& 0& 0& 1\\ -a_5& 0& -{\mathrm{\ω}}_2^2& -2{\mathrm{\ω}}_2\mathrm{\ξ}\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right],D=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right],$

$f\left(x\right)=[\begin{array}{c}\begin{array}{cccc}0& -a_1{x}_1^2\left(t\right)-a_3{x}_3\left(t\right)x_1\left(t\right)& 0& -a_6{x}_1^2\left(t\right).\end{array}\end{array}{]}^T$

(3) selecting a composite performance index as described in a formula 3;

$J=\underset{T\→\∞}{\lim }\frac{1}{T}{\∫}_0^T[x^T\left(t\right)\mathrm{Qx}\left(t\right)+{\mathrm{Ru}}^2\left(t\right)]\mathrm{dt},$ formula 3;

(4) the maximum principle is implemented for equations 2 and 3:

first, a Hamiltonian is constructed as formula 4:

H(·)=x^TQx₁+Ru²+λ^T(Ax+Bu+f(x)+Dp_ω) Formula 4;

and further converting the problem of solving the controller u (t) under the constraint of the performance index formula 3 in the formula 1 into a problem of solving the following nonlinear two-point boundary values in the formula 5 according to the extreme value condition:

$\begin{array}{c}-\stackrel{\·}{\mathrm{\λ}}\left(t\right)=\mathrm{Qx}\left(t\right)+A^T\mathrm{\λ}\left(t\right)+f_x^T\left(x\right)\mathrm{\λ}\left(t\right),\\ \stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)-\mathrm{S\λ}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right),\\ x\left(0\right)=x_0,\\ \mathrm{\λ}(\∞)=0\end{array}$ formula 5;

(5) solving an optimal vibration control law:

solving the iterative solution of the nonlinear two-point boundary value problem described in equation 5 by using a successive approximation method, and enabling

λ(t)=P₁x(t)+P₂p(t)+P₃p_ω(t)+g(t)

Where g (t) is a common mode vector, for λ (t) = P₁x(t)+P₂p(t)+P₃Derivation of p ω (t) + g (t), followed by taking the first formula of formula 2 $\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+f\left(x\right)+\mathrm{Dp}\left(t\right):$ Substituted therein to obtain

$\begin{array}{c}\stackrel{\·}{\mathrm{\λ}}\left(t\right)=P_1\stackrel{\·}{x}\left(t\right)+P_2\stackrel{\·}{p}\left(t\right)+P_3\stackrel{\·\·}{p}\left(t\right)+\stackrel{\·}{g}\left(t\right)\\ =(P_{1}A-P_1{\mathrm{SP}}_1)x\left(t\right)+(P_{1}D-P_1{\mathrm{SP}}_2)p\left(t\right)\\ +(P_2-P_1{\mathrm{SP}}_3)\stackrel{\·}{p}\left(t\right)+P_3\stackrel{\·\·}{p}\left(t\right)-P_1\mathrm{Sg}\left(t\right)+P_{1}f\left(x\right)+\stackrel{\·}{g}\left(t\right).\end{array}$

From formula 2 and formula 5 to obtain

$\stackrel{\·}{\mathrm{\λ}}\left(t\right)=-(Q+A^T{P}_1)x\left(t\right)-A^T{P}_{2}p\left(t\right)-A^T{P}_3{p}_{\mathrm{\ω}}\left(t\right)-A^{T}g\left(t\right)-f_x^T\left(x\right)\mathrm{\λ}\left(t\right)$

In view of

$\stackrel{\·}{p}\left(t\right)=\stackrel{\‾}{F}\stackrel{\·}{w}\left(t\right)=\stackrel{\‾}{F}\stackrel{\‾}{G}w\left(t\right)\stackrel{A}{=}{p}_{\mathrm{\ω}}\left(t\right)$

By comparisonThe matrix equation 7 and the sequence equations 8 and 9 are obtained, and the optimal vibration control law described by equation 6 can be obtained

$\begin{array}{c}{A}^T{P}_1+P_{1}A-P_{1}S{P}_1+Q=0,\\ {(A^T-P_{1}S)}^2{P}_2\stackrel{\‾}{F}+P_2\stackrel{\‾}{F}{\mathrm{\Ω}}^2=-(A^T-P_{1}S)P_{1}D\stackrel{\‾}{F}\\ {(A^T-P_{1}S)}^2{P}_3\stackrel{\‾}{F}+P_3\stackrel{\‾}{F}{\mathrm{\Ω}}^2=P_{1}D\stackrel{\‾}{F}\end{array},,$ Formula 7;

$\begin{array}{c}{x}^{\left(0\right)}\left(t\right)=\mathrm{\Φ}\left(t\right)x_0+{\∫}_0^t\mathrm{\Φ}(r-t)[(D-{\mathrm{SP}}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\ω}}\left(r\right)-{\mathrm{Sg}}^{\left(0\right)}\left(r\right)]\mathrm{dr},\\ x^{\left(k\right)}\left(t\right)=\mathrm{\Φ}\left(t\right)x_0+{\∫}_0^t\mathrm{\Φ}(r-t)[(D-S{P}_2)p\left(r\right)-{\mathrm{SP}}_3{p}_{\mathrm{\ω}}\left(r\right)-{\mathrm{Sg}}^{\left(k\right)}\left(r\right)\\ +f\left(x^{(k-1)}\left(r\right)\right)]\mathrm{dr},k=\mathrm{1,2},\·\·\·,\end{array}$ formula 8;

$\begin{array}{c}{g}^{\left(0\right)}\left(t\right)={\∫}_t^{\∞}{\mathrm{\Φ}}^T(r-t)f_x^T\left(0\right)[P_{2}p\left(r\right)+P_3{p}_{\mathrm{\ω}}\left(r\right)]\mathrm{dr},\\ g^{\left(k\right)}\left(t\right)={\∫}_t^{\∞}{\mathrm{\Φ}}^T(r-t)\{P_{1}f\left(x^{(k-1)}\left(r\right)\right)+f_x^T\left(x^{k-1)}\left(r\right)\right)[P_1{x}^{(k-1)}\left(r\right)\\ +P_{2}p\left(r\right)+P_3{p}_{\mathrm{\ω}}\left(r\right)+g^{(k-1)}\left(r\right)\left]\right\}\mathrm{dr},k=\mathrm{1,2}\·\·\·,\end{array}$ formula 9;

the approximate optimal vibration control law is obtained from the above equations 7, 8 and 9:

u^(k)(t)=-R^-1B^Tλ^(k)(t)=-R^-1B^T[P₁x^(k)(t)+P₂p(t)+P₃p_ω(t)+g^(k)(t)]formula 6;

wherein,and P is₁，P₂And P₃The following equation 7: x is the number of^(k)And g^(k)(t) is determined from the following formulae 8 and 9.

2. The method for controlling semi-active vibration of nonlinear system of long-span cable bridge structure according to claim 1, characterized by comprising after step (2): simplifying the large-span cable bridge structure vibration control system described by the large-span cable bridge structure vibration control system model of the formula (2) into a single-degree-of-freedom system.