CN111783198A - Suspension bridge refined power analysis method based on double-beam model - Google Patents

Suspension bridge refined power analysis method based on double-beam model Download PDF

Info

Publication number
CN111783198A
CN111783198A CN202010570239.8A CN202010570239A CN111783198A CN 111783198 A CN111783198 A CN 111783198A CN 202010570239 A CN202010570239 A CN 202010570239A CN 111783198 A CN111783198 A CN 111783198A
Authority
CN
China
Prior art keywords
cable
ith
main
suspension bridge
dynamic
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN202010570239.8A
Other languages
Chinese (zh)
Other versions
CN111783198B (en
Inventor
韩飞
邓子辰
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Northwestern Polytechnical University
Original Assignee
Northwestern Polytechnical University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Northwestern Polytechnical University filed Critical Northwestern Polytechnical University
Priority to CN202010570239.8A priority Critical patent/CN111783198B/en
Publication of CN111783198A publication Critical patent/CN111783198A/en
Application granted granted Critical
Publication of CN111783198B publication Critical patent/CN111783198B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/13—Architectural design, e.g. computer-aided architectural design [CAAD] related to design of buildings, bridges, landscapes, production plants or roads
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/13—Differential equations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

Abstract

The invention discloses a dynamic modeling and dynamic characteristic solving method of a single-span suspension bridge, which adopts a double-beam model connected by a plurality of discrete springs to dynamically model the suspension bridge, considers the bending rigidity of a main cable, the rigidity of a suspender and the axial force action possibly borne by a main beam, and is used for dynamic analysis of self-anchored and ground-anchored suspension bridges; establishing a coupling vibration differential equation of the dynamic model, solving the method by applying a dynamic stiffness method and obtaining a closed form suspension bridge frequency equation; the solution of the frequency equation is realized through a numerical iteration algorithm, and further the modal frequency and the mode shape of the system are obtained. The method has the advantages of simple process and clear physical significance, and can be more reliably applied to the dynamic analysis of the engineering structure due to the fact that the method is a frequency domain solution in nature, is good in stability and high in calculation efficiency, and has higher calculation precision compared with the traditional numerical solution.

Description

Suspension bridge refined power analysis method based on double-beam model
Technical Field
The invention belongs to the field of structural dynamics, and particularly relates to a suspension bridge refined dynamic analysis method based on a double-beam model.
Background
The suspension bridge is known as the main structural form of the bridge with extra-large span due to simple structure, definite stress, strong spanning capability and the like. The dynamic characteristic analysis of the large-span suspension bridge is usually carried out by adopting a finite element method, the accuracy of an analysis result is directly influenced if the structural rigidity, the quality and the boundary condition are correctly simulated, and the dynamic characteristic is the basis of the wind-resistant and earthquake-resistant analysis of the structure. Since the main cable of the suspension bridge is not replaceable and the safety of the main cable is crucial to the bearing capacity and safety of the full bridge, it is necessary to perform a fine modeling analysis on the dynamic characteristics of the main cable. However, in the existing research, when a suspension bridge is subjected to modeling analysis, a main cable is generally simplified, namely, a rod unit is used for modeling, and geometric nonlinearity caused by the sag effect is generally processed by an Ernst equivalent elastic modulus formula. The following problems can be caused when the main cable is subjected to dynamic characteristic analysis by the treatment mode: (1) due to the fact that bending rigidity of the main cable is ignored, when the main cable or the main cable with small span is analyzed, the calculation result is likely to have large errors; (2) it is difficult to account for the additional cable force effect due to elastic elongation when the main cable vibrates.
Because the main cable has very big initial tension under the dead load effect, provide powerful "gravity rigidity" to follow-up structural shape, and the amount of deflection of stiffening beam is subordinate to the main cable, along with the increase of span, the influence of stiffening beam's bending rigidity to structural rigidity also reduces gradually. Therefore, it is necessary to perform refined modeling on the main cable of the suspension bridge and develop a set of refined dynamic analysis theory of the suspension bridge, which needs to improve the existing mechanical model and analysis theory of the suspension bridge.
Disclosure of Invention
The technical problem solved by the invention is as follows: the invention aims to provide a suspension bridge refined dynamic analysis model and a solving method aiming at the defects of the prior art, wherein the model can better accord with the actual situation and can be more reliably used for the occasions of structural design, dynamic characteristic analysis, vibration control and the like of a bridge structure.
The technical scheme of the invention is as follows: a suspension bridge refined power analysis method based on a double-beam model comprises the following steps:
the method comprises the following steps: the suspension bridge dynamics modeling based on the double-beam model establishes a system motion differential equation set, and comprises the following substeps:
the first substep: the double-beam model is composed of two beams and a plurality of discrete spring setsThe two beams are connected through a spring and are respectively used for simulating a main cable and a main beam of the suspension bridge. The springs being adapted to simulate the suspension rod, k, of a suspension bridgeiRepresenting the stiffness coefficients of the i springs, wherein the stiffness coefficients are equal to the axial stiffness of the ith suspension rod; lsiRepresenting the distance of the ith boom from the left end point; the upper beam in the model with sag represents the main cable of the suspension bridge, the lower beam is used for simulating the main beam of the suspension bridge, and the main span of the suspension bridge is l0(ii) a In the vibration process, each unit (respectively called cable section and beam section) of the main cable and the main beam divided by the suspension rod follows different motion configurations, so that a local coordinate system needs to be respectively established on each cable section and each beam section; defining the ith cable segment and the beam segment as S respectivelyciAnd SgiThe local coordinate systems of the two are respectively (X)ci,Yci) And (X)giY), the overall coordinate system of the system is (x, y);
and a second substep: according to the Hamilton principle, the motion differential equation of each cable section and beam section of the suspension bridge under a local coordinate system is established as follows:
Figure BDA0002549336750000026
Figure BDA0002549336750000027
wherein E1I1(E2I2)、m1(m2)、H1(H2) Bending rigidity, mass per linear meter and horizontal tension (compression) force borne by the main cable (main beam) are respectively provided; u. of1iAnd u2iRespectively the displacement function of the ith unit of the main cable and the ith unit of the main beam; () ' representing the spatial coordinate X in the local coordinate systemciOr XgiDerivation, (. cndot.) denotes derivation over time t; (. is a dirac function; y isiThe initial static configuration of the ith cable section of the main cable; thetaiAn acute angle is formed between the ith suspension rod and the normal direction of the main cable;
in the formula hiFor the additional cable force caused by the elastic elongation when the ith cable section of the main cable vibrates, the calculation formula is as follows：
Figure BDA0002549336750000021
Wherein A is1Andi(t) represents the cross-sectional area of the main cable and the dynamic strain of the cable section,/iIs the axial length of the ith cable segment, and can be defined byi＝lsi-lsi-1Determining;
Figure BDA0002549336750000022
represents the curve length of i cable segments;
step two: applying a separation variable method to the formulas (1) and (2) and solving the separation variable method to obtain the vibration mode function of the main cable and the main beam after the dimensionless
Figure BDA0002549336750000023
And
Figure BDA0002549336750000024
the following were used:
Figure BDA0002549336750000025
Figure BDA0002549336750000031
wherein mui＝li/l0，ξ1i＝Xci/x，ξ2i＝Xgi/x，
Figure BDA0002549336750000032
Figure BDA0002549336750000033
Wherein n is 1,2, 1 represents a main cable and 2 represents a main beam;
Figure BDA0002549336750000034
Figure BDA0002549336750000035
wherein
Figure BDA0002549336750000036
μsi＝lsi/l0；
(4) And the coefficients in the formulae (5)
Figure BDA0002549336750000037
Unknown constants which depend on the boundary conditions of the structure can be eliminated in the subsequent analysis process by substitution, and the system modal frequency omega is determined after the system modal frequency omega is obtained.
Step three: calculating a unit dynamic stiffness matrix K(j)The method comprises the following substeps:
the first substep: for convenience of expression, expressions (4) and (5) are further written in the form of a matrix as follows:
Figure BDA0002549336750000038
Figure BDA0002549336750000039
wherein
Figure BDA00025493367500000310
Figure BDA00025493367500000311
Wherein
Figure BDA00025493367500000312
Figure BDA00025493367500000313
Figure BDA00025493367500000314
Obtaining B from the formula (9)(i)Then, according to the node displacement U(i)The relation with the displacement function can be used for displacing the node of the ith cable segment and the beam segment by a vector U(i)Uniformly expressed as:
U(i)＝C(i)·A(i)(10)
wherein
Figure BDA00025493367500000315
Figure BDA00025493367500000316
And
Figure BDA00025493367500000317
respectively showing the displacement and the rotation angle of the left end node of the ith cable segment,
Figure BDA00025493367500000318
and
Figure BDA00025493367500000319
respectively representing the displacement and the rotation angle of a node at the right end of the ith cable segment;
Figure BDA00025493367500000320
and
Figure BDA0002549336750000041
respectively showing the displacement and the rotation angle of the left end node of the ith beam section,
Figure BDA0002549336750000042
and
Figure BDA0002549336750000043
respectively representing the displacement and the rotation angle of the right end node of the ith beam section;
Figure BDA0002549336750000044
Figure BDA0002549336750000045
Figure BDA0002549336750000046
Cni＝cos(qnμi),Sni＝sin(qnμi),n＝1,2。
combining with the node force balance condition
Figure BDA0002549336750000047
Wherein
Figure BDA0002549336750000048
Figure BDA0002549336750000049
Wherein ν ═ E2I2/E1I1. Formula (11) can be further written as
F(i)＝K(i)·U(i)(12)
Wherein the cell stiffness matrix K(i)Can be determined by
Figure BDA00025493367500000410
Step four: for each unit rigidity matrix K(i)And performing grouping to obtain a total dynamic stiffness matrix K under the integral coordinate system.
The matrix K is a square matrix of system modal frequencies ω, which is the root of the frequency equation | K (ω) | 0. Where | is a determinant symbol. The solution of the frequency equation | K (ω) | 0 can be realized by means of a conventional numerical iterative algorithm, such as a Newton method or a Muller method, so as to obtain the modal frequency ω of each order of the system;
step five: substituting the obtained modal frequency omega into the formulas (4) and (5), and determining the undetermined coefficient by combining boundary conditions
Figure BDA0002549336750000051
Further, the modal shape of each order of the system can be obtained
Figure BDA0002549336750000052
And
Figure BDA0002549336750000053
the further technical scheme of the invention is as follows: the matrix K in the fourth step is a square matrix related to the system modal frequency ω, where ω is each root of the frequency equation | K (ω) | 0; | · | is a determinant symbol; the accurate solution of the equation can be solved by adopting conventional numerical iterative algorithms such as a Netwon method, a Muller algorithm and the like, and further, the modal frequency omega of each order of the system can be obtained.
Effects of the invention
The invention has the technical effects that:
1. at present, a quick and effective analysis method is lacked for solving the dynamic characteristics of the suspension bridge, so that the dynamic analysis mostly adopts a numerical solution represented by a finite element method, the calculation efficiency is low, and the batch parameter analysis is inconvenient. The method provided by the invention is a frequency domain solution, and the solving process is in a closed form, so that the method has higher calculation efficiency and precision compared with the traditional time domain solution.
2. The method has simple process, provides the closed solution of the frequency equation of the suspension bridge according to the dynamic stiffness method, and realizes the solution of the modal frequency and the mode shape of the system by solving the frequency equation. The invention establishes a set of complete and fine dynamic models which fully consider the influence of the rigidity of the main cable, the main beam and the suspender of the suspension bridge, and provides a general solving process of the dynamic characteristic of the system, so that engineering personnel can be conveniently applied to the optimized design of the structure of the suspension bridge, the health monitoring, the vibration control and the like.
3. Description of the drawings
FIG. 1 is a flow chart of the calculation
FIG. 2 is a simplified dynamics model diagram of a single span suspension bridge
Detailed Description
Referring to fig. 1-2, the refined dynamic analysis method for the large-span suspension bridge according to the present invention includes the following steps:
1. the suspension bridge is dynamically modeled according to the double-beam model provided by the invention (figure 2), and a motion differential equation set of the suspension bridge is established according to the Hamilton principle;
2. calculating the additional cable force h of each cable section of the main cable divided by the suspenderiEstablishing motion differential equation of each cable segment under local coordinate system, transforming the motion differential equation to frequency domain by adopting separation variable method and solving vibration mode function of the motion differential equation
Figure BDA0002549336750000061
3. From hjCalculating a sag matrix B of each cable section of the main cable(i)Calculating an excessive matrix C by combining the node displacement continuous condition and the force balance condition(i)And D(i)Finally calculating the dynamic stiffness matrix K of the unit(i)：
Figure BDA0002549336750000062
Wherein
Figure BDA0002549336750000063
n-1 stands for the main cable, and n-2 stands for the main beam.
4. Grouping the dynamic stiffness matrixes of the units to obtain an integral dynamic stiffness matrix K of the suspension bridge;
5. calculating a frequency equation | K (omega) | 0 according to a Wittrick-Williams algorithm, wherein the root of the transcendental equation is the modal frequency of each order of the system;
6. substituting the obtained modal frequency omega into the general solution
Figure BDA0002549336750000064
Method for solving undetermined coefficient by combining boundary conditions
Figure BDA0002549336750000065
And then the corresponding mode shape is obtained.
The technical solution of the present invention is described in detail below, but the scope of the present invention is not limited to the embodiments. As shown in FIG. 1, the fast fine analysis method for the single span suspension bridge of the present invention comprises the following steps:
1. establishing a suspension bridge dynamic model as shown in the attached figure 2, and establishing a suspension bridge motion differential equation according to the Hamilton principle as follows:
Figure BDA0002549336750000066
Figure BDA0002549336750000067
wherein E1I1(E2I2)、m1(m2)、H1(H2) Bending rigidity, mass per linear meter and horizontal tension (compression) force borne by the main cable (main beam) are respectively provided; u. of1iAnd u2iRespectively the displacement function of the ith unit of the main cable and the ith unit of the main beam; k is a radical ofiThe stiffness of the ith boom; () ' represents the derivation of the spatial coordinate x, (-) represents the derivation of time t; lsiThe abscissa position of the ith suspension rod is shown; (. is a dirac function; y isiThe initial configuration of the ith cable segment of the main cable; thetaiiThe included angle between the normal direction of the main cable and the suspension rod is an acute angle.
Figure BDA0002549336750000071
The additional cable force caused by elastic elongation when the ith cable segment of the main cable vibrates.
2. Main cable and main beam vibration mode function
Figure BDA0002549336750000072
And
Figure BDA0002549336750000073
is solved for
Will be provided with
Figure BDA0002549336750000074
And
Figure BDA0002549336750000075
substituting equations (15) and (16) and performing dimensionless processing
Figure BDA0002549336750000076
Solving the general solution to obtain the vibration mode functions of the units of the main cable and the main beam as follows:
Figure BDA0002549336750000077
Figure BDA0002549336750000078
wherein
Figure BDA0002549336750000079
liIs the length of the ith cell (see FIG. 2), l0In order to realize the main span of the suspension bridge,
Figure BDA00025493367500000710
Figure BDA00025493367500000711
Figure BDA00025493367500000712
wherein
Figure BDA00025493367500000713
8 undetermined coefficients
Figure BDA00025493367500000714
The method can be determined by boundary conditions of the cable unit and the beam unit, namely the undetermined coefficient should satisfy the following relation:
Figure BDA00025493367500000715
Figure BDA00025493367500000716
Cni＝cos(qnμi),Sni＝sin(qnμi),n＝1,2。
3. unit dynamic stiffness matrix K(i)Is solved for
(4) And (5) the equations may be further expressed in matrix form as follows:
Figure BDA00025493367500000717
Figure BDA0002549336750000081
wherein
Figure BDA0002549336750000082
Figure BDA0002549336750000083
Wherein
Figure BDA0002549336750000084
Figure BDA0002549336750000085
Figure BDA0002549336750000086
Find B(i)Then, according to the node displacement U(i)The relationship to the displacement function may represent the node displacement as:
U(i)＝C(i)·A(i)(24)
wherein
Figure BDA0002549336750000087
Figure BDA0002549336750000088
Figure BDA0002549336750000089
Figure BDA00025493367500000810
Cni＝cos(qnμi),Sni＝sin(qnμi),n＝1,2。
Combining with the node force balance condition
Figure BDA00025493367500000811
Wherein
Figure BDA0002549336750000091
Figure BDA0002549336750000092
Wherein ν ═ E2I2/E1I1. Formula (25) can be further written as
F(i)＝K(i)·U(i)(26)
Wherein
Figure BDA0002549336750000093
4. Solving of integral dynamic stiffness matrix K
According to the vertical rigidity k of the suspenderiAnd a unit dynamic stiffness matrix K(j)Namely, the integral rigidity matrix K can be obtained by superposing the rigidity contribution of each unit to the integral structure according to the method which is the same as the finite element method unit rigidity matrix grouping process,and then obtaining an integral dynamic stiffness matrix K under an integral coordinate system.
Figure BDA0002549336750000094
The above formula is an overall dynamic stiffness matrix of a suspension bridge with dual booms, wherein,
Figure BDA0002549336750000095
matrix K of unit rigidity and significance of upper and lower marks of each element in matrix(j)Are identical.
5. Solving of frequency equations
After the overall stiffness matrix K is obtained, solving the characteristic equation det (K (ω)) -0 to obtain the modal frequency ω of each order of the system. The equations are typically transcendental equations that can be solved by numerical iterative algorithms, such as the Newton and Muller methods, among others.
6. Solving for vibration pattern
Calculating the ith order modal frequency omega of the systemiThen, it can be substituted back to the formulas (4) and (5)
Figure BDA0002549336750000101
And
Figure BDA0002549336750000102
then substituting the expressions of the two into the expression (20) to obtain the coefficient
Figure BDA0002549336750000103
Finally determining the modal vibration modes of the main cable and the main beam
Figure BDA0002549336750000104
And
Figure BDA0002549336750000105

Claims (2)

1. a suspension bridge fine power analysis method based on a double-beam model is characterized by comprising the following steps: the suspension bridge dynamics modeling based on the double-beam model establishes a system motion differential equation set, and comprises the following substeps:
the first substep: the double-beam model is connected by a plurality of discrete springs, and each spring is used for simulating a suspender, k of the suspension bridgeiRepresenting the stiffness coefficients of the i springs, wherein the stiffness coefficients are equal to the axial stiffness of the ith suspension rod; lsiRepresenting the distance of the ith boom from the left end point; the upper beam in the model with sag represents the main cable of the suspension bridge, the lower beam is used for simulating the main beam of the suspension bridge, and the main span of the suspension bridge is l0(ii) a In the vibration process, each unit (respectively called cable section and beam section) of the main cable and the main beam divided by the suspension rod follows different motion configurations, so that a local coordinate system needs to be respectively established on each cable section and each beam section; defining the ith cable segment and the beam segment as S respectivelyciAnd SgiThe local coordinate systems of the two are respectively (X)ci,Yci) And (X)giY), the overall coordinate system of the system is (x, y);
and a second substep: according to the Hamilton principle, the motion differential equation of each cable section and beam section of the suspension bridge under a local coordinate system is established as follows:
Figure FDA0002549336740000011
Figure FDA0002549336740000012
wherein E1I1(E2I2)、m1(m2)、H1(H2) Bending rigidity, mass per linear meter and horizontal tension (compression) force borne by the main cable (main beam) are respectively provided; u. of1iAnd u2iRespectively the displacement function of the ith unit of the main cable and the ith unit of the main beam; () ' representing the spatial coordinate X in the local coordinate systemciOr XgiDerivation, (. cndot.) denotes derivation over time t; (. is a dirac function; y isiThe initial static configuration of the ith cable section of the main cable; thetaiAn acute angle is formed between the ith suspension rod and the normal direction of the main cable;
in the formula hiFor the additional cable force due to elastic elongation when the ith cable section of the main cable vibrates, the calculation formula is as follows:
Figure FDA0002549336740000013
wherein A is1Andi(t) represents the cross-sectional area of the main cable and the dynamic strain of the cable section,/iIs the axial length of the ith cable segment, and can be defined byi＝lsi-lsi-1Determining;
Figure FDA0002549336740000014
represents the curve length of i cable segments;
step two: applying a separation variable method to the formulas (1) and (2) and solving the separation variable method to obtain the vibration mode function of the main cable and the main beam after the dimensionless
Figure FDA0002549336740000015
And
Figure FDA0002549336740000016
the following were used:
Figure FDA0002549336740000021
Figure FDA0002549336740000022
wherein mui＝li/l0，ξ1i＝Xci/x，ξ2i＝Xgi/x，
Figure FDA0002549336740000023
Figure FDA0002549336740000024
Wherein n is 1,2, 1 represents a main cable and 2 represents a main beam;
Figure FDA0002549336740000025
Figure FDA0002549336740000026
wherein
Figure FDA0002549336740000027
μsi＝lsi/l0；
(4) And the coefficients in the formulae (5)
Figure FDA0002549336740000028
Is an unknown constant;
step three: calculating a unit dynamic stiffness matrix K(j)Comprising the following substeps:
the first substep: for convenience of expression, expressions (4) and (5) are further written in the form of a matrix as follows:
Figure FDA0002549336740000029
Figure FDA00025493367400000210
wherein
Figure FDA00025493367400000211
Figure FDA00025493367400000212
Wherein
Figure FDA00025493367400000213
Figure FDA00025493367400000214
Figure FDA00025493367400000215
Obtaining B from the formula (9)(i)Then, according to the node displacement U(i)The relation with the displacement function can be used for displacing the node of the ith cable segment and the beam segment by a vector U(i)Uniformly expressed as:
U(i)＝C(i)·A(i)(10)
wherein
Figure FDA00025493367400000216
Figure FDA00025493367400000217
And
Figure FDA00025493367400000218
respectively showing the displacement and the rotation angle of the left end node of the ith cable segment,
Figure FDA00025493367400000219
and
Figure FDA00025493367400000220
respectively representing the displacement and the rotation angle of a node at the right end of the ith cable segment;
Figure FDA00025493367400000221
and
Figure FDA0002549336740000031
respectively showing the displacement and the rotation angle of the left end node of the ith beam section,
Figure FDA0002549336740000032
and
Figure FDA0002549336740000033
respectively representing the displacement and the rotation angle of the right end node of the ith beam section;
Figure FDA0002549336740000034
Figure FDA0002549336740000035
Figure FDA0002549336740000036
Cni＝cos(qnμi),Sni＝sin(qnμi),n＝1,2。
combining with the node force balance condition
Figure FDA0002549336740000037
Wherein
Figure FDA0002549336740000038
Figure FDA0002549336740000039
Wherein ν ═ E2I2/E1I1. Formula (11) can be further written as
F(i)＝K(i)·U(i)(12)
Wherein the cell stiffness matrix K(i)Can be determined by
Figure FDA00025493367400000310
Step four: for each unit rigidity matrix K(i)And performing grouping to obtain a total dynamic stiffness matrix K under the integral coordinate system.
Step five: substituting the obtained modal frequency omega into the formulas (4) and (5),determining undetermined coefficients by combining boundary conditions
Figure FDA00025493367400000311
Further, the modal shape of each order of the system can be obtained
Figure FDA00025493367400000312
And
Figure FDA00025493367400000313
2. the method for fine power analysis of a suspension bridge based on a dual-beam model as claimed in claim 1, wherein the matrix K in the fourth step is a square matrix related to the system modal frequency ω, where ω is each root of the frequency equation | K (ω) | 0; | · | is a determinant symbol; the accurate solution of the equation can be solved by adopting conventional numerical iterative algorithms such as a Netwon method, a Muller algorithm and the like, and further, the modal frequency omega of each order of the system can be obtained.
CN202010570239.8A 2020-06-21 2020-06-21 Suspension bridge refined power analysis method based on double-beam model Active CN111783198B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202010570239.8A CN111783198B (en) 2020-06-21 2020-06-21 Suspension bridge refined power analysis method based on double-beam model

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202010570239.8A CN111783198B (en) 2020-06-21 2020-06-21 Suspension bridge refined power analysis method based on double-beam model

Publications (2)

Publication Number Publication Date
CN111783198A true CN111783198A (en) 2020-10-16
CN111783198B CN111783198B (en) 2022-08-12

Family

ID=72756105

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202010570239.8A Active CN111783198B (en) 2020-06-21 2020-06-21 Suspension bridge refined power analysis method based on double-beam model

Country Status (1)

Country Link
CN (1) CN111783198B (en)

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112575672A (en) * 2020-11-17 2021-03-30 南京理工大学 Method for designing and restraining bridge vortex vibration force vibration absorber based on network synthesis method
CN112651072A (en) * 2021-01-06 2021-04-13 华南理工大学 Suspension bridge double-sling parameter identification method based on cable net model
CN114065429A (en) * 2021-11-18 2022-02-18 哈尔滨工业大学 Method for solving inherent characteristics of single-symmetrical-section wind turbine blade
CN115017681A (en) * 2022-05-10 2022-09-06 西北工业大学 Fine analysis method for dynamic characteristics of small-sag stay cable
CN115758505A (en) * 2022-09-27 2023-03-07 中国海洋大学 Mooring system solving method based on global shape function

Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US7386428B1 (en) * 2004-12-29 2008-06-10 Livermore Software Technology Corp. Method and system for guided cable contact in finite element analysis
CN102735386A (en) * 2012-07-14 2012-10-17 福州大学 Bending stiffness-considered numerical computation method for stay cable forces
CN106096257A (en) * 2016-06-06 2016-11-09 武汉理工大学 A kind of non-linear cable elements analyzes method and system
CN108268729A (en) * 2018-01-22 2018-07-10 武汉理工大学 The elasticity modulus frequency sensitivity analysis method and system of transmission pressure
CN109033656A (en) * 2018-08-03 2018-12-18 华北水利水电大学 The existing prestressed method of pre-stressed bridge is identified based on two-point step size gradient algorithm
US20190243935A1 (en) * 2017-06-26 2019-08-08 Dalian University Of Technology A sensor placement method using strain gauges and accelerometers for structural modal estimation
CN111310269A (en) * 2020-02-25 2020-06-19 东南大学 Method for analyzing nonlinear dynamic characteristics of clamped beam structure by considering boundary displacement

Patent Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US7386428B1 (en) * 2004-12-29 2008-06-10 Livermore Software Technology Corp. Method and system for guided cable contact in finite element analysis
CN102735386A (en) * 2012-07-14 2012-10-17 福州大学 Bending stiffness-considered numerical computation method for stay cable forces
CN106096257A (en) * 2016-06-06 2016-11-09 武汉理工大学 A kind of non-linear cable elements analyzes method and system
US20190243935A1 (en) * 2017-06-26 2019-08-08 Dalian University Of Technology A sensor placement method using strain gauges and accelerometers for structural modal estimation
CN108268729A (en) * 2018-01-22 2018-07-10 武汉理工大学 The elasticity modulus frequency sensitivity analysis method and system of transmission pressure
CN109033656A (en) * 2018-08-03 2018-12-18 华北水利水电大学 The existing prestressed method of pre-stressed bridge is identified based on two-point step size gradient algorithm
CN111310269A (en) * 2020-02-25 2020-06-19 东南大学 Method for analyzing nonlinear dynamic characteristics of clamped beam structure by considering boundary displacement

Non-Patent Citations (5)

* Cited by examiner, † Cited by third party
Title
HAN FEI 等: "A novel method for dynamic analysis of complex multi-segment cable systems", 《SCIENCE DIRECT》 *
HAN FEI 等: "Free vibration of the complex cable system An exact method using symbolic computation", 《MECHANICAL SYSTEMS AND SIGNAL PROCESSING》 *
HAN YUAN 等: "Elastodynamic analysis of cable-driven parallel manipulators considering dynamic stiffness of sagging cables", 《2014 IEEE INTERNATIONAL CONFERENCE ON ROBOTICS AND AUTOMATION (ICRA)》 *
康俊涛 等: "大跨径自锚式悬索桥成桥状态动力特性参数分析", 《桥梁建设》 *
曹懿 等: "一种适用于小垂度拉索的精确动力分析方法", 《西北工业大学学报》 *

Cited By (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112575672A (en) * 2020-11-17 2021-03-30 南京理工大学 Method for designing and restraining bridge vortex vibration force vibration absorber based on network synthesis method
CN112651072A (en) * 2021-01-06 2021-04-13 华南理工大学 Suspension bridge double-sling parameter identification method based on cable net model
CN112651072B (en) * 2021-01-06 2023-08-22 华南理工大学 Suspension bridge double-sling parameter identification method based on cable network model
CN114065429A (en) * 2021-11-18 2022-02-18 哈尔滨工业大学 Method for solving inherent characteristics of single-symmetrical-section wind turbine blade
CN114065429B (en) * 2021-11-18 2023-04-18 哈尔滨工业大学 Method for solving inherent characteristics of single-symmetrical-section wind turbine blade
CN115017681A (en) * 2022-05-10 2022-09-06 西北工业大学 Fine analysis method for dynamic characteristics of small-sag stay cable
CN115017681B (en) * 2022-05-10 2023-06-23 西北工业大学 Fine analysis method for dynamic characteristics of small-sag inhaul cable
CN115758505A (en) * 2022-09-27 2023-03-07 中国海洋大学 Mooring system solving method based on global shape function
CN115758505B (en) * 2022-09-27 2023-05-16 中国海洋大学 Global shape function-based mooring system solving method

Also Published As

Publication number Publication date
CN111783198B (en) 2022-08-12

Similar Documents

Publication Publication Date Title
CN111783198B (en) Suspension bridge refined power analysis method based on double-beam model
Banerjee et al. Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects
CN111783201B (en) Rapid analysis method for dynamic characteristics of three-span self-anchored suspension bridge
CN111695188B (en) Rapid and fine analysis method for dynamic characteristics of ground anchor type suspension bridge
CN111783199A (en) Refined rapid solving method for dynamic characteristics of multi-section cable structure
CN110162826B (en) Analysis method for thermoelasticity dynamic response of thin-wall structure
CN114757070A (en) New WENO format construction method under trigonometric function framework for numerical simulation
CN113076677B (en) Flexible body structure high-order nonlinear finite element numerical simulation method based on quintic hermite function
CN114638046B (en) Railway pier digital twin variable cross-section simulation calculation method
CN108268729B (en) Elastic modulus frequency sensitivity analysis method and system for power transmission conductor
Burlon et al. Coupled bending and torsional free vibrations of beams with in-span supports and attached masses
CN102880769B (en) The construction control method of Long span concrete deck cable stayed bridge and camber computing method
Jun et al. Coupled bending and torsional vibration of axially loaded Bernoulli–Euler beams including warping effects
CN111985018A (en) Calculation method for designing wind load of ultrahigh large-span tower and line based on inertia force method and tower line separation method and considering tower line coupling influence
CN111783200A (en) Rapid analysis method for damping characteristics of large-span suspension bridge
CN106773782B (en) Pneumatic servo elastic hybrid modeling method
CN107220678B (en) Multi-degree-of-freedom beam type structure nonlinear type determination method
CN115310188B (en) Method for determining equivalent parameters of stiffened plate based on minimum potential energy theory
CN117271949B (en) Suspension cable self-vibration analysis method and system considering elastic boundary and large sag influence
CN104978464A (en) Cable force measuring method for suspender cable of suspension bridge
CN115935465A (en) Long-span bridge full-life cycle state evaluation method combined with digital twinning technology
CN108493882B (en) transmission conductor galloping mechanism research method
Vihar et al. Design and analysis for the flutter behaviour of different selected wing plan forms computationally
Rao et al. Torsional vibrations of pre-twisted blades using artificial neural network technology
Fallah et al. A displacement formulation based on finite volume method for analysis of Timoshenko beam

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant