CN111651907B - Modeling analysis method of complex cable net system - Google Patents
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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
Technical Field
The invention belongs to the field of structural vibration control, and relates to a modeling analysis method of a complex cable network vibration reduction system consisting of a inhaul cable, a damper and a connecting piece.
Background
Vibration control of flexible structures such as guy ropes and suspenders is an important guarantee for safety and durability of long-span bridge structures (cable bearing bridges, underbearing arch bridges and middle bearing arch bridges) and high-rise building structures (such as mast structures). The guy cable and the suspender are used as main bearing members of the structure, and because the transverse rigidity and the self damping are small, the guy cable and the suspender are easy to vibrate under the actions of wind, wind and rain, and the service life of the guy cable and the suspender is influenced. As bridge spans increase, the length of stay cables also increases, and the length of stay cables used on cable-stayed bridges nowadays is already close to 600 meters; pneumatic measures and the manner in which the damper is mounted at the cable ends make it difficult to completely dampen the multi-modal/multi-mechanical vibrations of the cable. The vibration reduction and suppression of a cable network system formed by connecting adjacent inhaul cables and combining cable end dampers are an effective method; meanwhile, the mechanical behaviors of the damper/connecting piece have certain nonlinearity; cable multi-order vibration control based on semi-active dampers such as magnetorheological dampers is also used in practical engineering. Parameters and installation positions of the damper/connecting piece in the vibration reduction system need to be finely designed and optimized to meet the multi-modal vibration control requirement of the cable, and a set of efficient and accurate modeling analysis method is needed.
The traditional method for analyzing the cable net mainly obtains the self-vibration frequency and the damping ratio of the cable net through multi-modal analysis and takes the self-vibration frequency and the damping ratio as the basis for evaluating the vibration control effect of the cable net. In the analysis, complex overrun equations with infinite solutions are needed to be deduced and solved according to different arrangements and parameters of the cable network, the solving process of the complex cable network is extremely complex, and the characteristic value analysis is more convenient by adopting a numerical model; the complex modal method is not suitable for a nonlinear cable network system, and a numerical model is needed to be adopted when the nonlinear behavior of the system is considered; in addition, even if nonlinearity in the cable net vibration reduction system is ignored, the deformation amplitude of the damper/connecting piece in cable vibration needs to be determined, parameters such as damper stroke and the like are designed, and a numerical model is needed to perform forced vibration and random vibration analysis of the system.
The inhaul cable numerical model commonly used at present mainly comprises a finite difference model and a modal expansion model. In order to achieve the required calculation accuracy, the finite difference model needs to divide the cable into a plurality of sections, so that 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 mainly aimed at a single cable system at present, and the number of degrees of freedom of a numerical model can be reduced by combining a static correction function, but the method is not popularized to a complex cable network system. Therefore, a modeling analysis method suitable for various complex cable network systems and capable of meeting the requirements on calculation efficiency and accuracy is urgently needed.
Aiming at the defects of the prior art, a new technical scheme is necessary to be provided.
Disclosure of Invention
The invention aims to provide a modeling analysis method of a complex cable network system, so that cable network structures with different arrangements can adopt a unified method to perform vibration characteristics and forced vibration analysis, and a general tool is provided for the design of the complex cable network system. Meanwhile, a dimensionless model is established, and the method is suitable for optimizing and researching the position and parameters of the damper and the cable-to-cable connecting piece.
In order to solve the technical problems, the invention adopts the following technical scheme:
providing a cable net knot construction model frame based on a substructure modal synthesis method, considering single cables in a cable net as a substructure, and further performing model coupling according to the mechanical characteristics of the damper/cable connecting piece, the deformation continuity of the cable at the position of the additional damper/cable connecting piece and the stress balance condition; in the modeling of the substructure, a single-cable numerical model is obtained by using a modal cut-off method and a generalized static correction method, so that model order reduction is realized.
A modeling analysis method of a complex cable network system adopts a substructure modal synthesis method and a model order reduction method to form the functions of dynamic characteristics and dynamic response analysis of the cable network system. Firstly, each inhaul cable in a cable net structure is considered as a substructure, and the substructure adopts a modeling scheme of combining modal truncation with generalized static modal correction to realize model order reduction. The modal shape and the static correction function of each cable are obtained by adopting an analytical theory, the mass, rigidity and damping matrix of the single cable reduced model are calculated by adopting a Galerkin method to calculate an expression, and numerical integration is applied to calculate the expression. Further, vibration of each cable is coupled to form a comprehensive reduced-order model of the whole cable network structure according to the mechanical properties of the damper and the cable-to-cable connecting piece attached to each cable and the displacement continuity and stress balance conditions of the cable at the position where the damper/connecting piece is installed. The method is particularly applicable to cable nets formed when stay cables are connected with towers/beams by using end dampers and are connected with adjacent stay cables by using connecting pieces for vibration suppression and vibration reduction. 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 to optimize the vibration control effect of the cable network system.
Further, the invention comprises the following steps:
and firstly, carrying out dimensionless treatment on the cable net system, and determining key parameters. Considering a total of N cables in the cable net, the cable numbers are denoted s=1, 2,3, …, N. Ginseng of rope sThe number includes the string length L s Mass per unit length ρ s Axial stiffness (EA) s Inclination angle theta s The sag can be measured by adopting the following dimensionless parameters
The stay cables in an actual cable-stayed bridge,and a value of between 0 and 2.5. The length of the above-mentioned middle rope after elongation under the action of gravity is denoted as (L) e ) s Calculated as follows
Constant g=9.8 m/s 2 Gravitational acceleration. When the single cable does not consider the sag effect, the first order circle frequency has the following analytical expression
Subscript 0 considers it to the first order circular frequency of sagDifferentiated from each other.
In order to obtain a unified single cable model building method and develop optimization and characteristic analysis of a cable network structure system, the following dimensionless scheme of cable network system parameters is provided. First, timeLateral dynamic displacement of the cable>And damping force/connecting force acting on cable s +.>The dimensionless treatment is carried out according to the parameters of the cable 1,
wherein the method comprises the steps ofCoordinates along the chord line; the same variable symbol is provided with an upper transverse line in a dimensional form, and the same variable symbol is provided with a non-dimensional form without a transverse line; where j is the number of the forces on the cable s, < >> Representing the total amount of force on the cable s generated by the damper/connector. Other variables include the coordinates along the chord line>Modal frequency of rope->Self-damping coefficient of cable->External force->Dimensionless based on the parameters and dynamic characteristics of each cable
Where i is the modality order number. In addition, the ratio of each cable parameter to the corresponding cable 1 parameter is defined,
and secondly, determining the modal frequency and the vibration mode of each cable structure. The power characteristics of each cable after dimensionless can be expressed by the same formula, and the vibration frequency of the cable s without any additional damper/connecting piece is obtained by the following two formulas
ω s,i ＝i,i＝2,4,6,…
For odd modes, a nonlinear equation needs to be solved to obtain dimensionless cable frequency, and sag parameters of the cable are consideredThe sag has a more obvious effect on the first-order mode only, and the frequency of the odd-order mode above the third order is approximately equal to
When the influence of sagging is not considered, for all modes,
ω s,i ＝i
mode shape function of each order of corresponding cable sIs that
C in the above s,i The specific value of the vibration mode function is a constant, the modeling method and the analysis precision are not affected, and the amplitude value of the vibration mode function is 1.0 along the length of a cable by setting the parameter. The mode shape function subscript i is the mode order, and the superscript mod represents the mode (mode) mode shape function, as distinguished from the generalized static correction function below.
When only the mode shape function is used for analyzing the response of the single cable-single damper system, the action force of the damper can cause the discontinuity of cable response, so that at least 200-order modes are needed to meet the precision requirement.
And thirdly, deducing a generalized static correction (correction) function to be used as a basis function, so that the degree of freedom of the inhaul cable reduced-order model is reduced. First analyzing and analyzing the installation position x of a pendulous cable at a damper/connecting piece s,j Dimensionless deformation function y of the cable when acting on a unit force s,j (x s ) The following expression is obtained
The deformation function is not orthogonal to the modal function of the cable, and orthogonalization processing is performed when forming a single-cable expansion basis function. Consider reservation for cable sThe mode shape function is generally preserved>And (5) step. Assume that there is a common +.>A damper/connector is connected to each connection point, and more than one damper/connector can be connected to the same connection point. Accordingly, the number of forces acting on the damper/connector on the cable s is related to the number of connection points, +.>Orthogonalization is performed as follows
Wherein a is s,i Andis y s,j (x) And the correlation coefficient of the corresponding function, n is a number. />Is a generalized static correction function after orthogonalization. Reserved->Individual mode shape functions and total +.>The generalized static correction functions form a basis function vector phi of corresponding expansion of a single cable (s) (x s ) The following are listed below
The number of basis functions, i.e. the number of degrees of freedom for single-cable modeling, isAfter the generalized static correction function is added, the modal shape function of the cable can reach the analysis precision generally required by the cable with 20 orders.
And step four, obtaining a single-cable reduced order model. The displacement of the cable s vibration is approximately developed into a form using the above-described basis function,
wherein the method comprises the steps ofAnd->Is the generalized displacement corresponding to the basis function; similarly, the displacement vectors are arranged as follows
Where the superscript T is the matrix/vector transpose symbol.
Substituting the above into the dimensionless partial differential equation of the vibration of the cable s,
wherein ε is s Delta () is a dirac function, i.e. the function value is 1 when the argument is 0 and zero when it is other times;representing the second and first derivatives of the time variable versus t; () "represents the second derivative of the variable pair. The method for obtaining the reduced order second ordinary differential equation of the vibration of the single inhaul cable by utilizing Galerkin (Galerkin) has the following form
Wherein M is (s) 、C (s) 、K (s) Is the mass and resistance of the rope sDamping and stiffness matrix, q (s) (t) generalized displacement vector of the cable s, f (s) (t) is the load vector g acting on the cable s (s) (t) is the acting force of the damper/connector on the cable s, superscript (s) Representing the variables and parameters associated with the cords s. In the above, M (s) 、C (s) 、K (s) The matrix is integrated over the entire length of the cable according to the basis function, f (s) (t) is obtained by integrating the external load acting on the cable and the value of the basis function. The calculation expressions are as follows, respectively. Quality matrix:
stiffness matrix:
damping matrix:
distributing external load vectors:
transverse concentrated connection force vector applied by damper and connection:
where i and j are the element numbers,for a specific system, a numerical integration method is adopted to calculate the values of each matrix and vector, and x is calculated s The integration is carried out by 10000 sections between 0 and 1, thus ensuringPrecision.
Fifthly, synthesizing the vibration equation of each substructure (inhaul cable) to obtain the following cable net integral vibration differential equation,
the M, C, K is obtained by arranging mass, damping and rigidity matrixes of each cable along diagonal lines according to cable numbers, and q (t) is a generalized displacement vector of a cable network structure, namely, the generalized displacement vectors of the cables are connected in sequence. Similarly, f (t) contains the external load vector acting on all cables, and g (t) is the combination of the forces of all dampers/connectors on the cables. Respectively 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
And sixthly, establishing an integral coupling model. G (t) in the equation is determined by the mechanical property of the damper/connector and is a function of the relative displacement, speed and acceleration of the two ends, namely the generalized displacement of the cable net, and the linear part of the generalized displacement of the cable net can be moved into the left side of the equation to obtain
Wherein g nl (t) represents the portion of the damper/connector where the force on the cable to which it is attached is non-linearly related to the generalized displacement. In the aboveTo take into account the system mass, damping and stiffness matrix of the coupling effect between the cables. To build the coupling model, define displacement vectors for all cable locations connected to the damper/connector
U in the formula s,j For the displacement of the j-th connection point of the s-number inhaul cable, the matrix W is converted pos The expression of (2) is as follows
Its dimension isWherein the connection vector corresponding to the j-th connection position of the s-number inhaul cable is
Likewise, the speed of the connection pointAnd acceleration->Respectively according to the following calculation
According to the mechanical model of the damper/auxiliary cable, the relation between the damper/auxiliary cable force and the cable generalized displacement is established, and the following general relation expression is adopted for expression
Wherein the method comprises the steps ofp s,j Is the pairing vector of the damping/connecting force and the cable connecting point, and has the dimension ofP when j damper/connector connects a cable to a fixed structure s,j Only one element being 1, i.e. p s,j ＝[0 … 1 … 0] T The non-zero element corresponds to the connection point of the damper and the cable; when the j-damper/connector connects two cables, p s,j With one element being 1 and the other element being-1 and the remaining elements being zero, i.e. p s,j ＝[0 … 1 … -1 … 0] T The non-zero element corresponds to the point of connection of the damper/connector to the two cables.
Further, consider a generic model modeling the damper/connector, i.e., a model in which the spring, damping element and inertial container are connected in parallel, as shown in FIG. 2.
The mass, stiffness, damping matrix of the resulting monolithic coupling model has the following general form (taking into account the inertial, stiffness and damping parameters of the damper/connector):
wherein the method comprises the steps of
Wherein b s,j 、c s,j And k s,j The inertia coefficient, the damping coefficient and the rigidity coefficient of the j connection of the s-inhaul cable in the figure 2 are respectively; p (P) b 、P c 、P k A coefficient matrix for the damper/connector; w (W) for A matrix of forces acting as a damper/connector with dimensions ofSince one cable may connect a plurality of dampers or auxiliary cables at one connection point, the vector +.>In matrix W for May be repeated.
And seventhly, the second-order ordinary differential equation can directly adopt numerical integration to analyze the response of the cable network system under any known external load. Frequently, the above equations are written as state space equations, then numerical integration solutions are performed,
wherein the method comprises the steps of
Wherein A is a system matrix, I is an unit diagonal matrix, z is a state variable, 0 is a zero matrix, and G is an action matrix. In particular, when the force of the damper/linkage is linear with respect to the relative displacement, velocity, acceleration of its ends, i.e. g nl =0, and the frequency, damping and vibration mode of the cable-network structure can be obtained by performing eigenvalue analysis on the system matrix a.
The external load acting on the cable is typically a wind load, which is distributed along the length of the cable, and 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 the vibration characteristics of the cable, periodic loads are also applied to the whole cable or to one or more points on the cable. The model can theoretically take any form of load into account.
And eighth step, when analyzing the concrete cable net structure, according to the definition of the dimensionless parameters of the system, obtaining the dimensionless vibration characteristics and response of the system for design.
Due to the adoption of the technical scheme, the invention has the following beneficial effects:
(1) The method is simple and convenient, has universality and can be applied to modeling of the existing complex cable network systems in most forms;
(2) The model established by the method is a dimensionless model, has generality, and can be used for researching and optimizing parameters and positions of the damper/connecting piece in the cable network system;
(3) The method can efficiently analyze the frequency and the vibration mode of the complex cable network system when the damper/connecting piece are linear components;
(4) According to the method, the nonlinear mechanical characteristics of the damper/connecting piece can be considered, and the influence of the nonlinear mechanical behavior on the vibration characteristics and response of the cable network system can be analyzed;
(5) The method can efficiently and accurately analyze the forced vibration and random vibration response of the complex cable network system;
(6) The method allows for taking into account the influence of the sagging of the cable in the cable web system, and the bending stiffness and the boundary properties of the cable can also be taken into account within the framework of the method.
Drawings
FIG. 1 is a flow chart of modeling analysis of the complex cable network system of the present invention.
Fig. 2 is a schematic diagram of a general model of a linear damper/connector commonly found in complex cable network systems.
Fig. 3 is a schematic diagram of a cable system for the analysis of example 1.
Fig. 4 is an example of a displacement function and corresponding generalized static correction function of the cable 1 after a unit force is applied to the connector position.
Fig. 5 is a first order mode shape of the cable network system of example 1.
Fig. 6 is a displacement response of the cable web system of example 1 in each cable span under cyclic loading.
FIG. 7 is a displacement response of the cable network system of example 1 at the damper mounting location for each cable under cyclic loading.
Fig. 8 is a cable-dual damper system analyzed in example 2.
Fig. 9 is a plot of the pre-cable fourth-order modal damping ratio versus damper coefficient for the analysis of example 2.
Detailed Description
The invention provides a cable network knot construction model frame based on a substructure mode synthesis method, wherein single cables in a cable network are considered as a substructure, and model coupling is carried out according to the mechanical characteristics of a damper/cable connecting piece, the deformation continuity of the cable at the position of an additional damper/cable connecting piece and the stress balance condition; in the modeling of the substructure, a single-cable numerical model is obtained by using a modal cut-off method and a generalized static correction method, so that model order reduction is realized.
The invention is further described below with reference to the drawings and specific examples.
The technology of the present invention will be described in detail first by way of a preferred example (example 1). Now, a complex cable network system is assumed, and a viscoelastic damper is respectively added to the end parts of three cables, wherein the rigidity and the damping coefficient of the damper are constants, and the table 1 shows. Simultaneously, three inhaul cables are connected by two elastic auxiliary cables, and the specific arrangement mode is shown in figure 3.
The modeling analysis using the present invention is performed as follows (omitting part of the derivation step):
first, key parameters of dimensionless and modeling of the cable network system are determined. As shown in fig. 2, the total three ropes in the rope net are n=3, and the installation positions of the damper and the connecting piece are that
x s,1 ＝l s,1 /L s
Wherein l s,1 ,l s,2 Is the distance parameter in fig. 1. The relative mounting positions of the damper/connector are given directly in this example, see table 1.
Table 1 example 1 parameters of a cable network system
Sag press calculation of cable in cable net system
In addition to the parameters in table 1, the sag parameters are also related to the axial stiffness and tilt angle of the cable. For 500 meters of guy cable, the sag parameter is generally around 2, in this case the sag parameter of each cable is set1.86, 1.97 and 2.14, respectively, are listed in table 2. Dimensionless forces of time, displacement of individual cables, connector/damper
The parameters for dimensionless representation are
L 1 ＝562.6m
π 2 T 1 ＝64152kN
The other parameters of each cable are each dimensionless. The relative parameters of the three cables are calculated as follows
The calculation results are shown in Table 2. As shown in fig. 2, each cable has two connection points, i.e. there areAccording to literature and analytical experience, each cable retains the first 20 order single cable vibration mode shape, i.e +.>The other two ropes except the rope 2 are provided with the acting force of 2 dampers/connecting pieces, and the rope 2 bears the acting force of 3 dampers/connecting pieces, namely The number of degrees of freedom per cable is +.>The number of degrees of freedom of the entire system is 66. The modeling parameters described above are listed in table 2.
Table 2 non-dimensional Cable network System modeling parameters
The damping of the cable itself is small and can be ignored, in this case considered
And secondly, determining the frequency and the vibration mode function of each cable. After determining the sag parameters of each cable, the dimensionless frequency of each cable can be calculated according to the following formula
ω s,i ＝i,i＝2,4,6,…
The first-order and third-order dimensionless frequencies of the three cables are calculated to be respectively
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
Other order modal frequencies of the three cables have omega s,i =i. Sag only has an effect on the fourth bit after the frequency decimal point of the odd-order mode. After the frequency is obtained, expressions of the mode shape functions can be written
In the above, C is firstly taken s,i =1.0, and then each function is normalized.
Thirdly, obtaining a cable static deformation function according to the position of the connecting point and the sag parameter
Orthogonalizing the static deformation function and the modal function to obtain a generalized static correction function
Wherein the correlation coefficient is obtained by numerical integration
Calculated as aboveNormalization is also required. In this example, the generalized correction function pair after orthogonalization of the displacement function of the cable per unit force is shown in fig. 4. Finally form the basis function->
And fourthly, determining the rigidity, the mass and the damping matrix of each cable. After the parameters and the basis functions are determined, the mass, the rigidity and the damping matrix of each cable are obtained by numerical integration according to the following formula
In this example calculation, x will be s Divided into 10000 sections between 0 and 1, integrated by using a trapezoidal formula, and calculated by using a computer program.
Meanwhile, the calculation expressions of the external force vector and the damper/connector force vector are as follows
Fifthly, constructing a cable network system model. Firstly, constructing a mass, rigidity and damping matrix of the cable network system. The calculated mass stiffness matrices of the cables are combined to form a mass, stiffness and damping matrix of the cable network system (note c=0 in this example). Press type combination
Meanwhile, the generalized displacement vector, the external force vector and the damper/connecting piece acting force of the cable net system are constructed by programming a computer program according to the following formula,
q＝[q (1) q (2) q (3) ] T
f＝[f (1) f (2) f (3) ] T
g＝[g (1) g (2) g (3) ] T
and sixthly, establishing an integral coupling model. In this example, the displacement vectors of all connection points in the cable-net system are
Wherein matrix W pos Is that
Wherein the connection vector w corresponding to the j-th connection position of the s-number inhaul cable s,j Is that
Likewise, the speed of the connection pointAnd acceleration->Respectively according to the following calculation
And establishing the relation between the damper/auxiliary cable force and the cable generalized displacement according to the mechanical model of the damper/auxiliary cable. The damper/connector in this example can be considered as a parallel model of a spring and a damping unit, i.e. there are
Wherein the method comprises the steps ofk s,j And c s,j Is a dimensionless parameter, is the rigidity coefficient and the damping coefficient of the damper/connecting piece corresponding to the acting force, and can be based on g s,j The definition of (c) is obtained,
in this example of the present invention, pairing vector p of damping/connecting force and cable connecting point s,j 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 obtaining equation and pair of cable network system coupling modelThe mass, stiffness, damping matrix (note g in this example nl ＝0)
Wherein the method comprises the steps of
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 ]
And seventhly, the second-order ordinary differential equation can directly adopt numerical integration to analyze the response of the cable network system under any known external load. Frequently, the above equations are written as a first order state space equation, then numerical integration solutions are performed,
wherein the method comprises the steps of
The state matrix A can be analyzed by adopting a characteristic value decomposition method, and the dimensionless frequency omega of the ith-order mode of the cable network system can be obtained according to the characteristic value i The first six orders of natural vibration frequencies and damping ratios of the cable network system are shown in table 3. Wherein the frequency is as followsAnd (5) calculating.
TABLE 3 first six order self-vibration frequencies and damping ratios of cable network systems
| Order of | 1 | 2 | 3 | 4 | 5 | 6 |
| Dimensionless frequency omega 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 |
Complex frequency omega of cable net system i Substituting the second-order ordinary differential equation (f (t) =0) of the system in the sixth step to obtain the corresponding feature vector, and combining the basis functions of the system to obtain the vibration mode of the system. The first order mode of vibration of the cable network system in this example is shown in fig. 5.
And eighth, forced vibration analysis of the cable net system. In this example, the periodic load is defined as follows
Wherein f a For the load amplitude, consider the distribution form of the load along each cable and the first order vibration mode function of the single cableSimilarly, Ω is the dimensionless frequency of the external load. Taking f in this example a ＝10 -5 ,Ω＝ω 1 =1.139. Numerical integration using a calculation program as follows
The numerical value of the external force vector can be obtained, then the fourth-order Longguge tower method is adopted to carry out numerical integration on the first-order state space model of the seventh step, and the solution is obtained
In this example, the time step is taken to be 0.001 and the loading time is 300 pi. Each cable being in its position x s The dimensionless displacement of the position is obtained by
v s (x s ,t)＝[φ (s) ] T q (s)
In the 287 pi to 293 pi period, each cable is at mid-span position (x s Non-dimensional displacement of =0.5) as shown in fig. 6, each cable is at the damper position (x s = 0.0221) is shown in fig. 7. The cable net system has the largest amplitude of the No. 1 cable with the longest length when vibrating at the first step, and is most likely to be damaged and bumped; while the amplitude of the No. 3 cable damper position is greatest, care should be taken in the damper setting to ensure that the No. 3 cable additional damper has sufficient travel.
A further general example (example 2) is provided, where two dampers are mounted near the anchored end of a cable, a high damping rubber damper is mounted at the sleeve opening nearer to the anchored end, and an oil damper is mounted farther away, as shown in fig. 8. The damping coefficient of the oil damper is now optimally designed given the rubber damper parameters.
And step one, determining all parameters of the inhaul cable, and carrying out dimensionless treatment. In this example, the number 1 cable of example 1 was used as the cable, and parameters of the sleeve damper and mounting positions of the respective dampers are shown in table 4.
Table 4 oil damper and casing damper parameters
Dimensionless parameters other than the sleeve damper are carried out according to the principle in the embodiment 1, and the fundamental frequency of the inhaul cable
In this exampleLikewise take->In this case only one damper is attached to each attachment point on the cable, i.e. +.>Neglecting the cable damping itself. The rigidity of the rubber damper of the sleeve opening is +.>Representing the corresponding dimensionless stiffness coefficient as
To obtain the damping of the inhaul cable along with the coefficient c of the damper d (dimensionless) change curve, assume a c d Then the following steps are performed.
The second step is the same as the second step of 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 of the present invention,
W pos ＝[w 1,1 w 1,2 ]
the force expression of the oil damper is the same as that in example 1, and the force of the high damping rubber damper is as follows
Lower expression calculation
In this exampleW for ＝W pos 。
Seventh, in the same way as in the seventh step of embodiment 1, a state space equation of the system is obtained, the eigenvalue decomposition is performed on the system matrix a, the complex frequency of the system is obtained, and damping is obtained according to the real part and the imaginary part of the complex frequency.
Eighth step, change c d And (3) repeating the second to seventh steps to finally obtain the pre-cable fourth-order modal damping random c d The change curve of (2) is shown in fig. 9. Considering the design for the second to fourth order vibrations of the cable, the optimal dimensionless damper coefficient is about c d =1.5, corresponding damper coefficient is
The embodiments are described above in order to facilitate the understanding and application of the present invention by those of ordinary skill in the art. It will be apparent to those skilled in the art that various modifications can be readily made to these embodiments and the generic principles described herein may be applied to other embodiments without the use of the inventive faculty. Therefore, the present invention is not limited to the embodiments described herein, and those skilled in the art, based on the present disclosure, should make improvements and modifications within the scope of the present invention.
Claims (15)
1. A modeling analysis method of a complex cable network system is characterized by comprising the following steps of: adopting a substructure modal synthesis method and a model order reduction method to form the functions of dynamic characteristics and dynamic response analysis of the cable network system;
firstly, carrying out dimensionless on a cable net system, and determining modeling parameters according to the geometric constitution of a cable net and the characteristics of each cable;
secondly, determining the modal frequency and the mode shape function of each cable substructure by adopting an analysis method, and reserving a certain number of modal mode shape functions to construct a basis function;
thirdly, deriving a generalized static correction function for each position on the cable where a damper/connector is connected: applying a unit transverse force on each connecting point respectively, analyzing to obtain the static displacement of the inhaul cable, and orthogonalizing each generalized static correction function with a cable vibration type function and other correction functions reserved during modal cutting; the reserved modal shape functions and the generalized static correction functions jointly form a basis function for approximate expansion of single-cable vibration response, and the number of the basis functions is the number of degrees of freedom of a single-cable model;
fourthly, adopting the basis function to make the dynamic displacement response v of the cable s in the cable network s (x s T) is approximately expanded as follows
Wherein s is the number of the rope in the rope net, t is the time, and x s For taking one end point of the cable s as the origin of coordinatesAlong the axis of the chord line;is the i-th order mode function of the cable s, < ->Generalized static correction function corresponding to the j-th junction of cable s,>and->Is the corresponding generalized displacement; />For the number of mode shapes reserved when the cable s mode is truncated, +.>The total number of connection points of the s-shaped inhaul cable and the damper/connecting piece, namely the number of generalized static correction functions; it can be seen that after the above-mentioned expansion, the number of degrees of freedom of the single rope is +.>The basis functions and the displacements are arranged in the form of vectors according to the following formula
Middle and upper mark T Transpose the symbols for the matrix/vector;
fifthly, substituting the above formula into a partial differential equation of vibration of a single cable, and deducing a reduced differential equation of vibration of the single cable by adopting a Galerkin method, wherein the reduced differential equation has the following form
Wherein M is (s) 、C (s) 、K (s) Is a mass, damping and stiffness matrix of the cable s, q (s) (t) generalized displacement vector of the cable s, f (s) (t) is the load vector g acting on the cable s (s) (t) is the force of the damper/connector on the cable s,representing the second and first derivatives of the time variable pair t, superscript (s) Representing variables and parameters associated with the cords; in the above, M (s) 、C (s) 、K (s) The matrix is obtained by numerical integration at the total length of the cable according to the basis function, f (s) (t) obtained by numerical integration according to the external load and the basis function acting on the cable;
sixth, synthesizing the vibration equation of each substructure to obtain the following ordinary differential equation of the integral vibration of the cable network,
the mass, damping and rigidity matrix (M, C, K) of the system integral model is obtained by arranging the mass, damping and rigidity matrix of each cable along a diagonal line according to cable numbers, and q (t) is a generalized displacement vector of a cable network structure, namely, the generalized displacement vectors of each cable are connected in sequence; likewise, f (t) contains the external load vector acting on all cables, g (t) is the combination of the forces of all dampers/connectors on the cables; the total number of cables in the cable network system is denoted as N, i.e. s=1, 2,3, …, N;
seventh, g (t) in the above equation is determined by the mechanical property of the damper/connector, and is a function of the relative displacement, speed and acceleration of the two ends, namely the generalized displacement of the cable net, and the linear part of the generalized displacement of the cable net can be moved into the left side of the equation to obtain
Wherein g nl (t) represents the portion of the damper/connector where the force of the cable to which it is attached is non-linearly related to the generalized displacement; in the aboveA system mass, damping and stiffness matrix to account for coupling effects between the cables; the second-order ordinary differential equation can be directly solved by adopting a numerical integration method, and the response of the cable network system under any known external load effect is analyzed; frequently, the above equation is written as a state space equation, and then numerical integration solution is performed, the state space equation is as follows
Wherein the method comprises the steps of
Wherein A is a system matrix, I is a unit diagonal matrix, z is a state variable, 0 is a zero matrix, and G is an action matrix;
eighth, aiming at a specific cable net structure, according to definition of dimensionless parameters of the system, the dimensionless vibration characteristics and response of the system are obtained and are used for actual engineering design.
2. The modeling analysis method of the complex cable network system according to claim 1, wherein: and taking each main rope in the rope net structure into a substructure, and establishing a model of the overall structure by coupling the deformation continuity and the stress balance of each substructure at the connecting position of the damper/connecting piece and the mechanical characteristics of the damper/connecting piece.
3. The modeling analysis method of the complex cable network system according to claim 1, wherein: when each inhaul cable in the cable net structure is considered as a substructure, the substructure adopts a modeling scheme of combining modal interception with generalized static modal correction, so that model order reduction is realized; and according to the mechanical properties of the damper and the inter-cable connecting piece which are additionally arranged on each cable and the displacement continuity and stress balance conditions of the cable at the position where the damper/connecting piece is arranged, the vibration of each cable is coupled to form a comprehensive reduced-order model of the whole cable net structure.
4. A method of modeling analysis of a complex cable network system in accordance with claim 3, wherein: the modal shape and static force correction function of each cable are obtained by adopting an analytical theory, the mass, rigidity and damping matrix of the single cable reduced model are calculated by adopting a Galerkin method to calculate an expression, and a numerical integration method is used to calculate the model.
5. The modeling analysis method of the complex cable network system according to claim 1, wherein: in the seventh step, when the force of the damper/connector is in linear relation to the relative displacement, velocity, acceleration of the two ends, i.e., g nl =0, the frequency, damping and mode of the cable network can be obtained by eigenvalue analysis of the system matrix a.
6. The modeling analysis method of the complex cable network system according to claim 1, wherein: considering the influence of cable sag and inclination;the influence of the two aspects on the dynamic characteristic of the cable can be realized by using dimensionless parametersMeasured by definition as
Wherein θ is s The inclined angle of the inhaul cable, g is gravity acceleration, (EA) s Is the axial rigidity of the inhaul cable, (L) e ) s L is the length of the guy cable after falling and extending under the dead weight s For the length of the cable s, T s Is the tension of the s-shaped rope, ρ s The mass of the s-shaped inhaul cable in unit length; sag parameter of the ropeMainly related to cable length and cable axial force, according to the parameters of the cable currently used in cable-stayed bridge,/v>The value of (2) is between 0 and 2.5; when the cable base function is determined to comprise the vibration mode function and the generalized static correction function, analyzing mode analysis and static linear analysis are carried out by considering the sag of the cable.
7. The modeling analysis method of the complex cable network system according to claim 1, wherein: a dimensionless scheme is proposed: the time, displacement of each cable, and force of the connector/damper are unified into dimensionless ones according to the dynamic characteristics of the cable 1 (s=1), as follows
The variables with upper horizontal lines are in the dimensional form of corresponding parameters, g s,j Acting force for the j-th connecting point of the damper/connecting piece to the cable s; other variables include coordinates along the chord line of each cable, self damping of the cable, dimensionless external forces based on the vibration characteristics of each cable:
in c s Is the damping coefficient of the cable itself,for first order circular frequency of ropes s, i.e.
Furthermore, the ratio of each cable parameter to the corresponding cable 1 parameter is defined,
by adopting the dimensionless scheme, the dynamic characteristic of the cable net system is determined by the sag parameter of each cableDamper/connector mounting location x s,j And three parameters gamma L,s 、γ T,s 、γ ρ,s And (5) determining.
8. A method of modeling analysis of a complex cable network system according to claim 1, wherein the basis functions of each cable mainly comprise modal shape functions of a single cable undamped/connection, the modal shape functions having analytical expressions, wherein the single cable frequency is obtained as follows
Omega in s,i For cable dimensionless frequencyi is the modal order; for odd modes, the equation is needed to be solved to obtain the dimensionless cable frequency, and the sag parameter of the cable is considered +.>The number of the mode frequency is between 0 and 2.5, and only the mode frequency of the first order of the cable is greatly influenced by sag, and the mode frequency of the odd order above the third order is approximately as follows
Irrespective of sagging effect, all modes are presented
ω s,i ＝i
Mode shape function of each order of corresponding cable sIs that
Wherein C is s,i Is constant.
9. The modeling analysis method of the complex cable network system according to claim 8, wherein: wherein C is s,i Taking 1 and then normalizing the mode vibration mode.
10. The modeling analysis method of the complex cable network system according to claim 1, wherein: when the response of the cable is approximately unfolded by adopting a basis function, the basis function also comprises a generalized static correction function; the generalized static correction function is derived based on the static line shape of the cable after applying unit concentrated force at the damper/connecting piece connecting point; when the sag of the rope is considered, the static deformation of the rope after the rope receives unit force at any connecting position has an analytical expression
Wherein x is s,j Is the relative position of the j-th connecting point on the s-shaped cable, y s,j (x s ) Is S-shaped cable in x s,j The position is subjected to dimensionless static displacement after unit concentrated force.
11. The modeling analysis method of a complex cable network system according to claim 1, wherein the modeling analysis method comprises the following steps: in the process of reducing the order of a single cable model by adopting a modal cut-off and generalized static correction function, the cable modal shape and the correction shape function are orthogonalized, and the method is as follows
Wherein a is s,i Andis y s,j (x) Correlation coefficients with corresponding functions, n being a number; />Normalization is also required.
12. The modeling analysis method of the complex cable network system according to claim 1, wherein: the expressions of the mass, rigidity and damping matrix of each cable substructure are deduced by a Galerkin method; the mass, stiffness, damping and distributed external load vector in the fifth step have the following expression
The specific cable net structure is directly obtained through numerical integration according to the expression and the basis function, and the method has universality; the external load vector is obtained by numerical integration after the external load function is determined.
13. The modeling analysis method of the complex cable network system according to claim 1, wherein: calculating the stress of each cable at the positions according to the response of each cable at the positions of the cable and the damper/auxiliary cable and the mechanical characteristics of the damper/auxiliary cable, so as to realize the coupling of the responses of each cable and form an integral model;
comprises the following general steps:
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
U in the formula s,j For the displacement of the j-th connection point of the s-number inhaul cable, the matrix W is converted pos Is defined as
Its dimension isWherein the connection vector corresponding to the j-th connection position of the s-number inhaul cable is
Likewise, the speed of each connection pointAnd acceleration->Respectively according to the following calculation
Secondly, establishing a relation between the damper/auxiliary cable force and cable generalized displacement according to a mechanical model of the damper/auxiliary cable; using the following general relation
Where j=1, 2, …,for the number of damper/connector forces acting on the s-wire,because multiple dampers/connectors may be attached at the same location; p is p s,j Is the pairing vector of the damping/connecting force and the cable connecting point, and the dimension is +.>When j damper/connector connects a cable to a fixed structure p s,j Only one element being 1, i.e. p s,j ＝[0…1…0] T The non-zero element corresponds to the point of connection of the damper/connector to the cable; when the j-damper/connector connects two cables, p s,j With one element being 1 and the other element being-1 and the remaining elements being zero, i.e. p s,j ＝[0…1…-1…0] T The non-zero element corresponds to the connection point of the damper/connecting piece and the two inhaul cables; the force of the damper/connector acting on the cable s is expressed as
14. The modeling analysis method of a complex cable network system according to claim 1, wherein the mass, rigidity and damping matrix of the formed integral coupling model has the following general form:
wherein the method comprises the steps of
Wherein b s,j 、c s,j And k s,j The inertia coefficient, the damping coefficient and the rigidity coefficient of the j connection of the s-inhaul cable in the figure 2 are respectively; p (P) b 、P c 、P k A coefficient matrix for the damper/connector; w (W) for A matrix of forces acting as a damper/connector with dimensions ofSince one cable may connect a plurality of dampers or auxiliary cables at one connection point, the vector +.>In matrix W for May be repeated.
15. The modeling analysis method of the complex cable network system according to claim 1, wherein: the cable net is formed when the stay cables are connected with the tower/beam by adopting end dampers and connected with adjacent cables by adopting connecting pieces to restrain vibration.
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