CN111651907B - Modeling analysis method of complex cable net system - Google Patents

Abstract

The invention discloses a modeling analysis method of a complex cable network system, which adopts a substructure modal synthesis method and a model order reduction method to form the functions of dynamic characteristic and dynamic response analysis of the cable network system. And taking each main rope in the rope net structure into a substructure, and establishing a model of the overall structure by coupling deformation continuity and stress balance of each substructure at the connecting position of the damper/connecting piece and mechanical properties of the damper/connecting piece. The established model can be used for analyzing the response of the actual cable network under the action of external load, and can be used for complex eigenvalue analysis to obtain the frequency, damping and vibration mode of the system; the built model is a dimensionless model, and can be used for researching general characteristics of a cable network structure and designing parameters of a damper and a cable-to-cable connecting piece so as to optimize the vibration control effect of the cable network system.

Description

A modeling and analysis method for complex cable net system

Technical Field

The invention belongs to the field of structural vibration control and relates to a modeling and analysis method of a complex cable net vibration reduction system composed of cables, dampers and connectors.

Background Art

Vibration control of flexible structures such as cables and hangers is an important guarantee for the safety and durability of long-span bridge structures (cable-supported bridges, bottom-supported and mid-supported arch bridges) and tall building structures (such as mast structures). As the main load-bearing components of the structure, cables and hangers are very prone to vibration under the action of wind, rain, etc. due to their low lateral stiffness and self-damping, which affects their service life. As the span of the bridge increases, the length of the cable also increases. Today, the length of the cable used on the cable-stayed bridge is close to 600 meters; pneumatic measures and the method of installing dampers at the cable ends are difficult to completely suppress the multi-modal/multi-mechanism vibration of the cable. Using connectors to connect adjacent cables and combining them with cable end dampers to form a cable net system for vibration reduction and vibration suppression is an effective method; at the same time, the mechanical behavior of the damper/connector has certain nonlinearity; multi-order vibration control of cables based on semi-active dampers such as magnetorheological dampers has also been applied in actual engineering. The parameters and installation positions of the dampers/connectors in the above-mentioned vibration reduction system need to be carefully designed and optimized to meet the multi-modal vibration control requirements of the cable, which requires a set of efficient and accurate modeling and analysis methods.

The traditional method of analyzing cable nets is mainly to obtain the natural frequency and damping ratio of the cable net through complex modal analysis as the basis for evaluating the vibration control effect of the cable net. In the analysis, it is necessary to derive and solve complex transcendental equations with infinite solutions according to the different arrangements and parameters of the cable net. The solution process is extremely complicated for complex cable nets, and it is more convenient to use numerical models for eigenvalue analysis; the complex modal method is not suitable for nonlinear cable net systems. When considering the nonlinear behavior of the system, a numerical model must be used; in addition, even if the nonlinearity in the cable net vibration reduction system is ignored, it is necessary to determine the deformation amplitude of the damper/connector in the cable vibration and design parameters such as the damper stroke, which also requires a numerical model for forced vibration and random vibration analysis of the system.

The commonly used numerical models of cables currently include finite difference models and modal expansion models. In order to achieve the required calculation accuracy, the finite difference model needs to divide the cable into multiple segments, so the number of degrees of freedom of the established model is large and the calculation efficiency is low. The method based on modal expansion is currently mainly aimed at single cable systems. It can reduce the number of degrees of freedom of the numerical model by combining it with the static correction function, but this method has not been extended to complex cable network systems. Therefore, there is an urgent need for a modeling and analysis method that is applicable to various complex cable network systems and can meet the requirements of calculation efficiency and accuracy.

In view of the shortcomings of existing technologies, it is necessary to propose new technical solutions.

Summary of the invention

The purpose of the present invention is to provide a modeling and analysis method for a complex cable net system, so that cable net structures with different arrangements can use a unified method to analyze vibration characteristics and forced vibration, and provide a universal tool for the design of complex cable net systems. At the same time, a dimensionless model is established, which is suitable for the location and parameter optimization research of dampers and cable connectors.

In order to solve the above technical problems, the technical solution adopted by the present invention is:

A cable net structure modeling framework based on the substructure modal synthesis method is proposed. A single cable in the cable net is considered as a substructure, and then the model is coupled according to the mechanical properties of the damper/cable connector, the deformation continuity of the cable at the position of the additional damper/cable connector, and the force balance conditions. In the modeling of the substructure, the modal truncation method and the generalized static correction method are used to obtain the numerical model of the single cable to achieve model reduction.

A modeling and analysis method for a complex cable net system adopts a substructure modal synthesis method and a model reduction method to form the function of dynamic characteristics and dynamic response analysis of the cable net system. First, each cable in the cable net structure is considered as a substructure, and the substructure adopts a modeling scheme of modal truncation combined with generalized static modal correction to achieve model reduction. Among them, the modal vibration shape and static correction function of each cable are obtained by analytical theory, and the mass, stiffness and damping matrix of the single cable reduced-order model are derived by the Galerkin method and obtained by numerical integration. Further, according to the mechanical properties of the dampers attached to each cable and the connecting parts between cables, the displacement continuity and force balance conditions of the cables at the locations where the dampers/connectors are installed, the vibrations of each cable are coupled to form a comprehensive reduced-order model of the entire cable net structure. This method is particularly applicable to the cable net formed when the inclined cable is connected to the tower/beam by the end damper and connected to the adjacent cables by the connector for vibration suppression and reduction. The established model can be used to analyze the response of the actual cable net under external loads, and can be used for complex eigenvalue analysis to obtain the frequency, damping and vibration mode of the system; the established model is a dimensionless model, which can be used to study the general characteristics of the cable net structure, design parameters of dampers and cable connectors, and optimize the vibration control effect of the cable net system.

Furthermore, the present invention comprises the following steps:

The first step is to non-dimensionalize the cable net system and determine the key parameters. Consider that there are N cables in the cable net, and the cable numbers are s=1,2,3,…,N. The parameters of the cable s include its chord length L _s , unit length mass ρ _s , axial stiffness (EA) _s , inclination angle θ _s , and the cable sag can be measured using the following non-dimensional parameters

The cables in the actual cable-stayed bridge are The value of is between 0 and 2.5. In the above formula, the length of the cable after extension under the action of gravity is recorded as (L _e ) _s , and is calculated as follows:

The constant g = 9.8m/ ^s2 is the acceleration due to gravity. When the sag effect is not considered for a single cable, its first-order circular frequency has the following analytical expression:

The subscript 0 is used to compare it with the first-order circular frequency when the cable is considered sag Distinguish them.

In order to obtain a unified single cable model establishment method and to carry out optimization and characteristic analysis of the cable net structure system, the following cable net system parameter dimensionless scheme is proposed. Transverse dynamic displacement of the cable and the damping force/connection force acting on the cables The parameters of cable 1 are uniformly dimensionless.

in is the coordinate along the cable chord; in this paper, the same variable symbol with a horizontal line above is a dimensional form, and the variable without a horizontal line is a dimensionless form; here j is the number of the force on the cable s, represents the total amount of force on the cable s caused by the damper/connector. Other variables include the coordinates along each cable chord Modal frequency of the cable The cable's own damping coefficient external force According to the parameters and dynamic characteristics of each cable, it is dimensionless.

Where i is the modal order number. In addition, the ratio of each cable parameter to the corresponding parameter of cable 1 is defined,

The second step is to determine the modal frequency and vibration mode of each cable structure. After non-dimensionalization, the dynamic characteristics of each cable can be expressed by the same formula. The vibration frequency of the cable s without considering any additional dampers/connectors is obtained by the following two formulas:

ω _s,i ＝i,i＝2,4,6,…

For odd modes, it is necessary to solve nonlinear equations to obtain dimensionless cable frequencies, taking into account the cable sag parameter The value of is between 0 and 2.5. The sag has a significant effect only on the first-order mode. The odd-order modal frequencies above the third order are approximately

When the sag effect is not considered, for all modes,

ω _s,i ＝i

The modal vibration functions of each order of cable s for

In the above formula, C _s,i is a constant, and its specific value does not affect the modeling method and analysis accuracy. In this method, by setting this parameter, the amplitude of the vibration function along the cable length will be set to 1.0. The subscript i of the vibration function is the modal order, and the superscript mod represents the modal vibration function, which is distinguished from the generalized static correction function described below.

When only the modal vibration function is used to analyze the response of a single cable-single damper system, at least 200 modes are required to meet the accuracy requirements because the force of the damper will cause discontinuity in the cable response.

The third step is to derive the generalized static correction function as the basis function to reduce the number of degrees of freedom of the cable reduction model. First, the dimensionless deformation function y _s,j (x _s ) of the cable with a sag when a unit force is applied to the damper/connector installation position x _s,j is analyzed, and the following expression is obtained:

The above deformation function is not orthogonal to the modal function of the cable. When forming the single cable expansion basis function, an orthogonalization process is performed. modal vibration shape functions, generally retain the previous Assume that there are a total of Each connection point is connected to a damper/connector, and the same connection point can be connected to more than one damper/connector. Accordingly, the number of damper/connector forces acting on the cable s is related to the number of connection points as follows: Orthogonalization is performed as follows

where a _s,i and is the correlation coefficient between _ys,j (x) and the corresponding function, and n is the number. is the orthogonalized generalized static correction function. mode shape functions and a total of The generalized static correction functions form the basis function vector φ ^(s) (x _s ) of the corresponding expansion of a single cable as follows:

The number of basis functions, i.e. the number of degrees of freedom for single cable modeling, is After adding the generalized static correction function, the cable's modal vibration function can achieve the generally required analysis accuracy by retaining 20th order.

The fourth step is to obtain the single cable reduced-order model. The displacement of the cable s vibration is approximately expanded into the following form using the above basis functions:

in and is the generalized displacement corresponding to the basis function; similarly, the displacement vectors are arranged as follows

Where superscript T is the matrix/vector transpose symbol.

Substituting the above equation into the dimensionless partial differential equation of the vibration of cable s,

Where ε _s is the integral variable, δ() is the Dirac function, that is, when the independent variable is 0, the function value is 1 and otherwise it is zero; Denotes the second and first order derivatives of the time variable with respect to t; ()″ denotes the second order derivative of the variable with respect to t. The reduced order quadratic ordinary differential equation for the vibration of a single cable is obtained by applying the Galerkin method, which has the following form:

Where M ^(s) , C ^(s) , K ^(s) are the mass, damping and stiffness matrices of cable s, q ^(s) (t) is the generalized displacement vector of cable s, f ^(s) (t) is the load vector acting on cable s, g ^(s) (t) is the force of the damper/connector on cable s, and the superscript ^(s) represents the variables and parameters related to cable s. In the above formula, the M ^(s) , C ^(s) , K ^(s) matrices are obtained by integrating the basis functions over the entire length of the cable, and f ^(s) (t) is obtained by numerically integrating the external loads acting on the cable and the basis functions. The calculation expressions are as follows. Mass matrix:

Stiffness matrix:

Damping matrix:

Distributed external load vector:

Transverse concentrated connection force vector applied by the damper and the connection:

Where i and j are element numbers, For a specific system, the numerical integration method is used to obtain the values of each matrix and vector. The accuracy can be guaranteed by dividing _xs into 10,000 segments between 0 and 1 for integration.

The fifth step is to combine the vibration equations of each substructure (cable) to obtain the following overall vibration differential equation of the cable net:

Where M, C, and K are obtained by arranging the mass, damping, and stiffness matrices of each cable along the diagonal according to the cable number. q(t) is the generalized displacement vector of the cable net structure, that is, the generalized displacement vectors of each cable are connected in sequence. Similarly, f(t) contains the external load vector acting on all cables, and g(t) is the combined force of all dampers/connectors on the cables. They are expressed as follows:

q＝[q ⁽¹⁾ … q ^(s) … q ^(N) ] ^T

f＝[f ⁽¹⁾ … f ^(s) … f ^(N) ] ^T

g＝[g ⁽¹⁾ … g ^(s) … g ^(N) ] ^T

Step 6: Establish the overall coupling model. The g(t) in the above equation is determined by the mechanical properties of the damper/connector and is a function of the relative displacement, velocity, and acceleration at both ends, that is, a function of the generalized displacement of the cable net. The linear part can be moved to the left side of the above equation to obtain

Where g _nl (t) represents the part of the force exerted by the damper/connector on the cable to which it is connected that is nonlinear with respect to the generalized displacement. The mass, damping and stiffness matrices of the system considering the coupling effects between the cables are given. To establish the coupling model, the displacement vectors of all cables connected to the dampers/connectors are defined.

Where u _s,j is the displacement of the jth connection point of cable s, and the expression of the transformation matrix W _pos is as follows:

Its dimension is The connection vector corresponding to the jth connection position of cable s is

Likewise, the speed of the connection point and acceleration Calculate according to the following formula

According to the mechanical model of the damper/auxiliary cable, the relationship between the damper/auxiliary cable force and the generalized displacement of the cable is established, and the following general relationship is used to express it:

in _ps,j is the pairing vector of the damping/connection force and the cable connection point, and its dimension is Pairing vector of: when damper/connector No. j connects a cable to a fixed structure, only one element of _ps,j is 1, that is, _ps,j = [0 … 1 … 0] ^T , and the non-zero elements correspond to the connection points between the damper and the cable; when damper/connector No. j connects two cables, _ps,j has one element that is 1, another element that is -1, and the rest of the elements are zero, that is _{, ps,j} = [0 … 1 … -1 … 0] ^T , and the non-zero elements correspond to the connection points between the damper/connector and the two cables.

Furthermore, we consider a general model to simulate the damper/connection, that is, a model in which a spring, a damping element and an inertia vessel are connected in parallel, as shown in Figure 2.

The mass, stiffness, and damping matrices of the overall coupled model are formed in the following general form (taking into account the inertia, stiffness, and damping parameters of the damper/connector):

in

Where bs _,j , cs _,j and ks _,j are the inertia coefficient, damping coefficient and stiffness coefficient of cable s and connection j in Figure 2 respectively; _Pb , _Pc , _Pk are the coefficient matrices of the damper/connector; _Wfor is the action matrix of the damper/connector force, and its dimension is Because a cable may be connected to multiple dampers or auxiliary cables at one connection point, the vector There may be repetitions in the matrix W _for .

Step 7: The above second-order ordinary differential equation can be directly integrated numerically to analyze the response of the cable net system under any known external load. Often, the above equation is written as a state space equation and then numerically integrated to solve it.

in

Where A is the system matrix, I is the unit diagonal matrix, z is the state variable, 0 is the zero matrix, and G is the force matrix. In particular, when the force of the damper/connector is linearly related to the relative displacement, velocity, and acceleration at both ends, that is, g _nl = 0, the frequency, damping, and vibration mode of the cable net structure can be obtained by performing eigenvalue analysis on the system matrix A.

The external load acting on the cable is generally a wind load, which is distributed along the length of the cable. In order to ensure the accuracy of the simulation, it is necessary to sample the wind load more densely along the length of the cable. In the analysis of cable vibration characteristics, periodic loads are also used to act on the entire cable or on one or more points on the cable. In theory, this model can consider any form of load.

The eighth step is to analyze the specific cable net structure and obtain the dimensional vibration characteristics and responses of the system according to the definition of the dimensionless parameters of the system for design.

Due to the adoption of the above technical solution, the present invention has the following beneficial effects:

(1) This method is simple and universal and can be applied to modeling most existing complex cable network systems;

(2) The model established by this method is a dimensionless model, which is general and can be used to study and optimize the parameters and positions of dampers/connectors in cable net systems;

(3) This method can efficiently analyze the frequency and vibration mode of complex cable net systems when the dampers/connectors are linear components;

(4) This method can consider the nonlinear mechanical characteristics of the damper/connector and analyze the impact of nonlinear mechanical behavior on the vibration characteristics and response of the cable net system;

(5) This method can efficiently and accurately analyze the forced vibration and random vibration responses of complex cable-net systems;

(6) This method can take into account the influence of the sag of the cables in the cable net system. The bending stiffness and boundary characteristics of the cables can also be considered within the framework of this method.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a flow chart of modeling and analysis of a complex cable network system of the present invention.

Figure 2 is a schematic diagram of the general model of linear dampers/connectors commonly found in complex cable net systems.

FIG. 3 is a schematic diagram of the cable net system analyzed in Example 1.

FIG. 4 is an example of the displacement function and the corresponding generalized static correction function of cable 1 after being subjected to a unit force at the connection position.

FIG. 5 is a first-order modal vibration shape of the cable net system of Example 1.

FIG6 is a diagram showing the displacement response of the cable-net system of Example 1 in each cable span under cyclic load.

FIG. 7 is a diagram showing the displacement response of each cable at the damper installation position of the cable net system of Example 1 under cyclic load.

FIG. 8 is a cable-double damper system analyzed in Example 2.

FIG. 9 is a relationship curve between the fourth-order modal damping ratio before the cable and the damper coefficient obtained by analyzing Example 2.

DETAILED DESCRIPTION

The present invention proposes a cable net structure modeling framework based on the substructure modal synthesis method, in which a single cable in the cable net is considered as a substructure, and then the model is coupled according to the mechanical properties of the damper/cable connector, the deformation continuity of the cable at the position of the additional damper/cable connector, and the force balance conditions; in the modeling of the substructure, the modal truncation method and the generalized static correction method are used to obtain the numerical model of the single cable, thereby realizing model reduction.

The present invention is further described below in conjunction with the accompanying drawings and specific embodiments.

First, the technology of the present invention is described in detail through a best example (Example 1). Now assume that there is a complex cable network system, and three cables are respectively attached with a viscoelastic damper at the end, and the stiffness and damping coefficient of the damper are constants, see Table 1. At the same time, two elastic auxiliary cables are used to connect the three cables, and the specific arrangement is shown in Figure 3.

The steps of modeling analysis using the present invention are as follows (some derivation steps are omitted):

The first step is to determine the key parameters for dimensionless modeling and modeling of the cable net system. As shown in Figure 2, there are three cables in the cable net, that is, N = 3, and the installation positions of the dampers and connectors are

x _s,1 = l _s,1 /L _s

Where l _s,1 and l _s,2 are the distance parameters in Figure 1. In this example, the relative installation positions of the damper/connector are directly given, see Table 1.

Table 1 Parameters of the cable net system of Example 1

The sag of the cable in the cable net system is calculated as follows:

In addition to the parameters in Table 1, the sag parameter is also related to the axial stiffness and inclination of the cable. For cables with a length of more than 500 meters, the sag parameter is generally around 2. In this example, the sag parameter of each cable is set to They are 1.86, 1.97 and 2.14 respectively, listed in Table 2. The time, the displacement of each cable, and the force of the connector/damper are dimensionless.

The parameters used for dimensionless transformation are

L ₁ =562.6m

π ² T ₁ ＝64152 kN

The other parameters of each cable are dimensionless. The relative parameters of the three cables are calculated as follows:

The calculation results are listed in Table 2. As shown in Figure 2, each cable has two connection points, namely According to the literature and analytical experience, each cable retains the first 20 single-cable vibration mode shapes, that is, Except for cable 2, the other two cables are subjected to the forces of two dampers/connectors, and cable 2 is subjected to the forces of three dampers/connectors, that is, Then the number of degrees of freedom of each cable is The number of degrees of freedom of the entire system is 66. The above modeling parameters are listed in Table 2.

Table 2 Dimensionless cable net system modeling parameters

The damping of the cable itself is very small and can be ignored.

The second step is to determine the frequency and vibration mode function of each cable. After determining the sag parameters of each cable, the dimensionless frequency of each cable can be calculated as follows:

ω _s,i ＝i,i＝2,4,6,…

The first-order and third-order dimensionless frequencies of the three cables are calculated as follows:

ω _1,1 =1.074, ω _1,1 = 1.078, ω _1,1 = 1.084

ω _1,3 ＝3.003, ω _2,3 ＝3.003, ω _3,3 ＝3.003

The other modal frequencies of the three cables are ω _s,i = i. The sag only affects the fourth decimal place of the odd-order mode frequency. After obtaining the frequency, the expression of each mode vibration function can be written

In the above formula, C _s,i = 1.0 is first taken, and then each function is normalized.

The third step is to obtain the static deformation function of the cable according to the position of the connection point and the sag parameter.

Orthogonalize the static deformation function and the modal function to obtain the generalized static correction function

The correlation coefficient is obtained by numerical integration.

Calculated according to the above formula Normalization is also required. In this example, the displacement function of the cable under unit force and the orthogonalized generalized correction function are compared as shown in Figure 4. The basis function is finally formed

The fourth step is to determine the stiffness, mass and damping matrix of each cable. After the parameters and basis functions are determined, the mass, stiffness and damping matrix of each cable are obtained by numerical integration according to the following formula:

In this calculation, the _xs is divided into 10,000 intervals between 0 and 1, and the trapezoidal formula is used for integration and the calculation is performed using a computer program.

At the same time, the calculation expressions of the external force vector and the damper/connector force vector are as follows

Step 5: Build the cable net system model. First, build the mass, stiffness and damping matrix of the cable net system. Combine the mass and stiffness matrices of the cables calculated to form the mass, stiffness and damping matrices of the cable net system (note that C = 0 in this example). Combine according to the following formula:

At the same time, the generalized displacement vector, external force vector and damper/connector force of the cable net system are compiled into a computer program according to the following formulas:

q＝[q ⁽¹⁾ q ⁽²⁾ q ⁽³⁾ ] ^T

f＝[f ⁽¹⁾ f ⁽²⁾ f ⁽³⁾ ] ^T

g＝[g ⁽¹⁾ g ⁽²⁾ g ⁽³⁾ ] ^T

The sixth step is to establish the overall coupling model. In this example, the displacement vectors of all connection points in the cable net system are

The matrix W _pos is

The connection vector w _s,j corresponding to the jth connection position of cable s is

Likewise, the speed of the connection point and acceleration Calculate according to the following formula

According to the mechanical model of the damper/auxiliary cable, the relationship between the damper/auxiliary cable force and the generalized displacement of the cable is established. The damper/connector in this example can be considered as a parallel model of spring and damping unit, that is,

in k _s,j and c _s,j are dimensionless parameters, which are the stiffness coefficient and damping coefficient of the damper/connector corresponding to the force. They can be obtained according to the definition of g _s,j.

In this example, The pairing vector _ps,j of the damping/connection force and the cable connection point can be obtained according to the definition of u:

p _1,1 = [1 0 0 0 0 0] ^T

p _1,2 = [0 1 0 -1 0 0] ^T

p _2,1 = [0 1 0 0 0 0] ^T

p _2,2 = [0 -1 0 1 0 0] ^T

p _2,3 = [0 0 0 1 0 -1] ^T

p _3,1 = [0 0 0 0 1 0] ^T

p _3,2 = [0 0 0 -1 0 1] ^T

Further, we can obtain the equations of the cable-net system coupling model and the corresponding mass, stiffness, and damping matrices (note that g _nl = 0 in this example).

in

P _c = [c _1,1 p _1,1 c _1,2 p _1,2 c _2,1 p _2,1 c _2,2 p _2,2 c _2,3 p _2,3 c _3,1 p _{3 ,1} c _3,2 p _3,2 ]

P _k =[k _1,1 p _1,1 k _1,2 p _1,2 k _2,1 p _2,1 k _2,2 p _2,2 k _2,3 p _2,3 k _3,1 p _{3 ,1} k _3,2 p _3,2 ]

Step 7: The above second-order ordinary differential equation can be directly numerically integrated to analyze the response of the cable net system under any known external load. Often, the above equation is written as a first-order state space equation and then numerically integrated to solve it.

in

The state matrix A can be analyzed by the eigenvalue decomposition method. According to the eigenvalue, the dimensionless frequency ω _i of the i-th order mode of the cable net system can be obtained. The first six natural frequencies and damping ratios of the cable net system are shown in Table 3. calculate.

Table 3 The first six natural frequencies and damping ratios of the cable net system

Degree 1 2 3 4 5 6 Dimensionless frequency ω _i 1.139 1.175 1.196 2.140 2.197 2.244 Frequency (Hz) 0.25 0.26 0.27 0.48 0.50 0.51 Modal damping ratio (%) 0.48 0.07 0.04 0.91 0.00 0.07

Substitute the complex frequency ω _i of the cable net system into the second-order ordinary differential equation of the system in step 6 (set f(t) = 0 for free vibration) to obtain the corresponding eigenvector, and combine the basis function of the system to obtain the vibration mode of the system. The first-order vibration mode of the cable net system in this example is shown in Figure 5.

Step 8: Forced vibration analysis of the cable net system. In this example, the following periodic loads are defined:

Where f _a is the load amplitude, considering the distribution of load along each cable and the first-order vibration function of a single cable The same, Ω is the dimensionless frequency of the external load. In this example, f _a = 10 ^-5 , Ω = ω ₁ = 1.139. The numerical integration is performed using the calculation program according to the following formula

The value of the external force vector can be obtained, and then the fourth-order Runge-Kutta method is used to numerically integrate the first-order state space model in the seventh step to obtain

In this example, the time step is 0.001 and the loading time is 300π. The dimensionless displacement of each cable at its position _xs is obtained as follows:

v _s (x _s ,t)＝[φ ^(s) ] ^T q ^(s)

In the period from 287π to 293π, the dimensionless displacement of each cable at the mid-span position ( _xs = 0.5) is shown in Figure 6, and the dimensionless displacement of each cable at the damper position ( _xs = 0.0221) is shown in Figure 7. It can be seen that in the first-order vibration of the cable net system, the longest cable No. 1 has the largest amplitude and is most likely to be damaged and hit in the cable net system; while the amplitude at the damper position of the No. 3 cable is the largest. In the damper setting, attention should be paid to ensuring that the damper attached to the No. 3 cable has sufficient travel.

Another common example (Example 2) is provided, where two dampers are installed near the anchor end of a cable, the one installed near the anchor end at the casing mouth is a high damping rubber damper, and the one farther away is an oil damper, as shown in Figure 8. Now, given the parameters of the rubber damper, the damping coefficient of the oil damper is optimized.

The first step is to determine the parameters of the cable and make them dimensionless. In this example, the cable is the No. 1 cable in Example 1, and the parameters of the casing damper and the installation positions of each damper are shown in Table 4.

Table 4 Parameters of oil damper and casing damper

According to the principle of Example 1, all parameters except the casing port damper are dimensionless, and the fundamental frequency of the cable is

In this example Also take In this example, each connection point on the cable is connected to only one damper, that is, Ignore the cable's own damping. The stiffness of the rubber damper at the casing mouth is The corresponding dimensionless stiffness coefficient is

In order to obtain the variation curve of cable damping with the damper coefficient c _d (dimensionless), first assume a value of c _d and then perform the following steps.

The second step is the same as the second step in Example 1.

The third step is the same as the third step of Example 1.

The fourth step is the same as the fourth step of Example 1.

The fifth step is the same as the fifth step of Example 1.

The sixth step is the same as the sixth step of Example 1. In this example,

W _pos = [w _1,1 w _1,2 ]

The expression of the force of the oil damper is the same as that in Example 1, and the force of the high damping rubber damper is as follows:

The following expression is evaluated

In this example W _for =W _pos .

The seventh step is the same as the seventh step of Example 1, obtaining the state space equation of the system, performing eigenvalue decomposition on the system matrix A, obtaining the complex frequency of the system, and obtaining the damping according to the real part and imaginary part of the complex frequency.

In the eighth step, change the c _d value and repeat the second to seventh steps. Finally, the variation curve of the first four-order modal damping of the cable with c _d is shown in Figure 9. Considering the design for the second to fourth order vibration of the cable, the optimal dimensionless damper coefficient is approximately c _d = 1.5, and the corresponding damper coefficient is

The above description of the embodiments is to facilitate the understanding and application of the present invention by those skilled in the art. It is obvious that those skilled in the art can easily make various modifications to these embodiments and apply the general principles described herein to other embodiments without creative work. Therefore, the present invention is not limited to the embodiments herein, and improvements and modifications made to the present invention by those skilled in the art based on the disclosure of the present invention should be within the scope of protection of the present invention.

Claims (15)

1. A modeling and analysis method for a complex cable net system, characterized by: using a substructure modal synthesis method and a model reduction method to form the function of dynamic characteristics and dynamic response analysis of the cable net system; The first step is to make the cable net system dimensionless and determine the modeling parameters according to the geometric structure of the cable net and the characteristics of each cable; In the second step, the modal frequencies and vibration shape functions of each cable structure are determined by analytical methods, and a certain number of modal vibration shape functions are retained to construct basis functions; The third step is to derive the generalized static correction function for each position on the cable connected with a damper/connector: apply a unit lateral force to each connection point, analyze and obtain the static displacement of the cable, and orthogonalize each generalized static correction function with the cable vibration shape function retained during modal truncation and other correction functions; the retained modal vibration shape function and the generalized static correction function together constitute the basis function of the approximate expansion of the single cable vibration response, and the number of basis functions is the number of degrees of freedom of the single cable model; In the fourth step, the above basis functions are used to approximately expand the dynamic displacement response v _s (x _s , t) of the cable s in the cable net as follows: Where s is the number of the cable in the cable network, t represents time, and _xs is the coordinate axis along the cable chord with one end point of the cable s as the origin; is the i-th order vibration mode function of cable s, The generalized static correction function corresponding to the j-th connection point of cable s, and is the corresponding generalized displacement; is the number of mode shapes retained when the cable s is modally truncated, is the total number of connection points between the s-cable and the damper/connector, that is, the number of generalized static correction functions; it can be seen that after the above formula is expanded, the number of degrees of freedom of a single cable is The basis functions and displacements are arranged in the form of vectors as follows Where superscript ^T is the matrix/vector transpose symbol; The fifth step is to substitute the above equation into the partial differential equation of the vibration of a single cable and use the Galerkin method to derive the reduced-order differential equation of the vibration of a single cable, which has the following form: Where M ^(s) , C ^(s) , K ^(s) are the mass, damping and stiffness matrices of cable s, q ^(s) (t) is the generalized displacement vector of cable s, f ^(s) (t) is the load vector acting on cable s, g ^(s) (t) is the force acting on cable s by the damper/connector, represents the second and first order derivatives of the time variable with respect to t, and the superscript ^(s) represents the variables and parameters related to the cable s; in the above formula, the M ^(s) , C ^(s) , and K ^(s) matrices are obtained by numerical integration of the basis functions over the entire length of the cable, and f ^(s) (t) is obtained by numerical integration based on the external load acting on the cable and the basis functions; The sixth step is to combine the vibration equations of each substructure to obtain the following ordinary differential equation for the overall vibration of the cable net: The mass, damping and stiffness matrices (M, C, K) of the overall system model are obtained by arranging the mass, damping and stiffness matrices of each cable along the diagonal according to the cable number. q(t) is the generalized displacement vector of the cable net structure, that is, the generalized displacement vectors of each cable are connected in sequence. Similarly, f(t) contains the external load vector acting on all cables, and g(t) is the combined force of all dampers/connectors on the cables. The total number of cables in the cable net system is recorded as N, that is, s = 1, 2, 3, ..., N. Step 7. g(t) in the above equation is determined by the mechanical properties of the damper/connector and is a function of the relative displacement, velocity, and acceleration at both ends, that is, a function of the generalized displacement of the cable net. The linear part can be moved to the left side of the above equation to obtain where g _nl (t) represents the part of the force exerted by the damper/connector on the cable to which it is connected that is nonlinear with respect to the generalized displacement; The mass, damping and stiffness matrices of the system considering the coupling effect between the cables; the above second-order ordinary differential equation can be directly solved by numerical integration method to analyze the response of the cable net system under any known external load; often, the above equation is written as a first-order state space equation, and then numerically integrated and solved. The state space equation is as follows in Where A is the system matrix, I is the unit diagonal matrix, z is the state variable, 0 is the zero matrix, and G is the force matrix; The eighth step is to obtain the dimensional vibration characteristics and responses of the system for the specific cable net structure according to the definition of the system's dimensionless parameters for use in actual engineering design. 2. The modeling and analysis method of a complex cable net system according to claim 1 is characterized in that each main cable in the cable net structure is considered as a substructure, and the model of the overall structure is established by coupling the deformation continuity and force balance of each substructure at the damper/connector connection position and the mechanical properties of the damper/connector. 3. The modeling and analysis method for a complex cable net system according to claim 1 is characterized in that: when each cable in the cable net structure is considered as a substructure, the substructure adopts a modeling scheme of modal truncation combined with generalized static modal correction to achieve model reduction; according to the mechanical properties of the dampers attached to each cable and the connecting parts between cables, the displacement continuity and force balance conditions of the cables at the locations where the dampers/connectors are installed, the vibrations of each cable are coupled to form a comprehensive reduced-order model of the entire cable net structure. 4. The modeling and analysis method for a complex cable net system according to claim 3 is characterized in that the modal vibration shape and static correction function of each cable are obtained by analytical theory, and the mass, stiffness and damping matrix of the single cable reduced-order model are expressed by the Galerkin method and obtained by numerical integration method. 5. The modeling and analysis method for a complex cable net system according to claim 1 is characterized in that: in the seventh step, when the force of the damper/connector is linearly related to the relative displacement, velocity and acceleration of its two ends, that is, g _nl = 0, the frequency, damping and vibration mode of the cable net can be obtained by performing eigenvalue analysis on the system matrix A. 6. The modeling and analysis method of a complex cable network system according to claim 1 is characterized by: considering the influence of cable sag and inclination; the influence of these two aspects on the cable dynamic characteristics can be expressed by dimensionless parameters To measure, it is defined as Where _θs is the cable inclination angle, g is the gravitational acceleration, (EA) _s is the cable axial stiffness, (L _e ) _s is the length of the cable after it sags and extends under its own weight, _Ls is the length of cable s, _Ts is the tension of cable s, and _ρs is the mass per unit length of cable s; the sag parameter of the cable is It is mainly related to the cable length and the axial force of the cable. According to the parameters of the cables currently used in cable-stayed bridges, The value of is between 0 and 2.5. When determining the cable basis function including the vibration mode function and the generalized static correction function, the sag of the cable is considered to perform analytical modal analysis and static linear analysis. 7. The modeling and analysis method of a complex cable network system according to claim 1 is characterized in that: a dimensionless scheme is proposed: time, displacement of each cable, and force of the connector/damper are uniformly dimensionless according to the dynamic characteristics of cable 1 (s=1), as follows The variables with an upper horizontal line in the above formula are the dimensional forms of the corresponding parameters. g _s,j is the force of the damper/connector on the jth connection point of cable s. Other variables include the coordinates along the chord of each cable, the cable's own damping, and the external force, which are dimensionless according to the vibration characteristics of each cable: Where _cs is the cable's own damping coefficient, is the first-order circular frequency of cable s, that is In addition, the ratios of the following index parameters to the corresponding parameters of index 1 are defined: Using the dimensionless scheme above, the dynamic characteristics of the cable-net system are determined by the sag parameter of each cable. The damper/connector installation position xs _,j is determined by the three parameters _γL,s , γT _,s and γρ _,s . 8. The modeling and analysis method for a complex cable network system according to claim 1 is characterized in that the basis functions of each cable mainly include the modal vibration function of a single cable without a damper/connector, and these vibration functions have analytical expressions, wherein the frequency of a single cable is obtained by the following two equations: Where ω _s,i is the dimensionless frequency of the cable i is the modal order; for odd modes, it is necessary to solve the above equation to obtain the dimensionless cable frequency, considering the cable sag parameter The value of is between 0 and 2.5. Only the first-order modal frequency of the cable is greatly affected by the sag. The odd-order modal frequencies above the third order are approximately When the sag effect is not considered, all modes have ω _s,i ＝i The modal vibration functions of each order of cable s for Where C _s,i is a constant. 9. The modeling and analysis method for a complex cable net system according to claim 8, characterized in that: _Cs,i is set to 1 and then the modal vibration shape is normalized. 10. The modeling and analysis method of a complex cable network system according to claim 1 is characterized in that: when the response of the cable is approximately expanded using a basis function, the basis function also includes a generalized static correction function; the generalized static correction function is derived based on the static linear shape of the cable after a unit concentrated force is applied at the damper/connector connection point; when the sag of the cable is considered, the static deformation of the cable after a unit force is applied at any connection position has an analytical expression: Where xs _,j is the relative position of the jth connection point on cable s, and _ys,j ( _xs ) is the dimensionless static displacement of cable s at position _xs,j after being subjected to unit concentrated force. 11. A modeling and analysis method for a complex cable network system according to claim 1, characterized in that: in the process of reducing the order of a single cable model by using modal truncation and generalized static correction function, the cable modal vibration mode and the correction vibration mode function are orthogonalized as follows: where a _s,i and is the correlation coefficient between y _s,j (x) and the corresponding function, n is the number; Normalization is also required. 12. The modeling and analysis method for a complex cable network system according to claim 1 is characterized in that: the expressions of the mass, stiffness and damping matrices of each cable structure are derived by the Galerkin method; the mass, stiffness, damping and distributed external load vector in the fifth step have the following expressions: The specific cable net structure is directly obtained by numerical integration according to the above expressions and basis functions, and the method is universal; the external load vector needs to be determined and then numerically integrated to obtain it. 13. The modeling and analysis method for a complex cable network system according to claim 1 is characterized in that: the forces of each cable at these positions are calculated according to the responses of each cable at its position relative to the damper/auxiliary cable and the mechanical characteristics of the damper/auxiliary cable, so as to achieve coupling of the responses of each cable and form an overall model; The following general steps are included: In the first step, the displacement response of each substructure at the damper/auxiliary cable connection position is extracted to form a displacement vector u, as shown in the following formula: Where u _s,j is the displacement of the jth connection point of cable s, and the transformation matrix W _pos is defined as Its dimension is The connection vector corresponding to the jth connection position of cable s is Similarly, the speed of each connection point and acceleration Calculate according to the following formula The second step is to establish the relationship between the damper/auxiliary cable force and the generalized displacement of the cable based on the mechanical model of the damper/auxiliary cable. The following general relationship is used: Here j = 1, 2, ..., is the number of forces acting on the damper/connector on cable s, Because there may be multiple dampers/connectors connected to the same position; _ps,j is the pairing vector of the damping/connection force and the cable connection point, and its dimension is When the j-th damper/connector connects a cable to a fixed structure, only one element of _ps,j is 1, that is, _ps,j = [0…1…0] ^T , and the non-zero elements correspond to the connection points between the damper/connector and the cable; when the j-th damper/connector connects two cables, _ps,j has one element 1, another element -1, and the rest are zero, that is, _ps,j = [0…1…-1…0] ^T , and the non-zero elements correspond to the connection points between the damper/connector and the two cables; the force of the damper/connector acting on cable s is expressed as 14. The modeling and analysis method for a complex cable net system according to claim 1 is characterized in that the mass, stiffness and damping matrices of the formed overall coupling model have the following general form: in Where bs _,j , cs _,j and ks _,j are the inertia coefficient, damping coefficient and stiffness coefficient of cable s and connection j in Figure 2 respectively; _Pb , _Pc , _Pk are the coefficient matrices of the damper/connector; _Wfor is the action matrix of the damper/connector force, and its dimension is Because a cable may be connected to multiple dampers or auxiliary cables at one connection point, the vector There may be repetitions in the matrix W _for . 15. The modeling and analysis method for a complex cable net system according to claim 1 is characterized in that: the cable net is a cable net formed when the inclined cables are connected to the towers/beams by end dampers and connected to adjacent cables by connectors for vibration suppression and reduction.