CN111783200A - Rapid analysis method for damping characteristics of large-span suspension bridge - Google Patents

Rapid analysis method for damping characteristics of large-span suspension bridge Download PDF

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CN111783200A
CN111783200A CN202010570251.9A CN202010570251A CN111783200A CN 111783200 A CN111783200 A CN 111783200A CN 202010570251 A CN202010570251 A CN 202010570251A CN 111783200 A CN111783200 A CN 111783200A
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韩飞
邓子辰
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Abstract

The invention discloses an analysis method for damping characteristics of a large-span suspension bridge, which comprises the following steps of firstly providing a simplified dynamic model and a motion differential equation of the large-span suspension bridge; solving a dynamic stiffness matrix and a frequency equation of the system by a dynamic stiffness method; solving the frequency equation based on a numerical iteration algorithm to further obtain the undamped modal frequency of the suspension bridge; and finally, giving a closed-type solution of the system modal damping ratio and the damping frequency, and establishing a functional relation among the system modal damping ratio, the viscous damping coefficient and the undamped modal frequency. The method has simple process and definite physical significance, all intermediate variables are given in a closed form, and the method has higher calculation precision and efficiency. The method has high solving precision and high efficiency, can be reliably applied to the damping characteristic analysis of the large-span suspension bridge, and provides a theoretical basis for the research work of the dynamic topics of the structure, such as vibration control, health monitoring and the like.

Description

Rapid analysis method for damping characteristics of large-span suspension bridge
Technical Field
The invention belongs to the field of bridge engineering, relates to a method for quickly analyzing damping characteristics of a large-span suspension bridge, and is particularly suitable for quickly analyzing damping ratio and modal frequency of the suspension bridge.
Background
The large-span suspension bridge is very sensitive to the action of wind due to low damping, small rigidity and light self weight, so that the large-amplitude wind effect can be easily generated under the action of external wind load, and even the structure can generate wind-induced disasters. The main beam and other components of the suspension bridge are easy to generate vortex-induced vibration at low wind speed, so that the fatigue damage of the components is easy to cause, and the service life of the structure is further influenced. In addition, vortex-induced vibration can also reduce driving comfort and cause unnecessary panic to the issuer. In order to reveal the vortex vibration mechanism of the suspension bridge and reduce the vortex amplitude value, researchers have conducted extensive research on the mechanism, and it is a generally accepted view that the mass and inherent dynamic characteristics of the structure, including frequency and damping, are decisive factors affecting the vortex vibration generation condition and maximum response.
In order to further discuss the induction reason and vibration suppression measures of vortex vibration of the Tiger-Men suspension bridge, the expert group carries out serious review on the submitted suspension bridge vibration reason analysis and vibration suppression measure research report, and considers that the vibration reason is as follows: the tiger door bridge is a large-span steel box girder suspension bridge, belongs to a typical flexible structure, and the water horses continuously arranged along the side guardrails of the bridge change the pneumatic appearance of the steel box girder, so that the suspension bridge is induced to generate vertical vortex-induced resonance under the condition of specific wind conditions. However, even if the water horse is dismantled, the occurrence of vortex vibration of the unloaded bridge is still observed under certain wind conditions. By further analysis it is believed that: the eddy vibration lasting for a long time and with large amplitude causes the damping of the bridge structure to be remarkably reduced, so that the vibration control of the tiger-door suspension bridge needs to be realized by improving the structural damping ratio. However, the existing research work cannot provide an expression of the damping ratio of the suspension bridge in an analytic or semi-analytic mode, and the expression can only be obtained through actual measurement, so that it is difficult to provide an evolution rule of the damping characteristic of the suspension bridge in a mechanism manner.
In view of the current demand for a suspension bridge damping characteristic analysis means, a set of high-precision and high-efficiency theoretical rapid analysis method is urgently needed to be researched and developed, so that the technical bottleneck of the existing research work is broken through, and a theoretical basis is provided for the analysis and research of the damping characteristics of the bridge.
Disclosure of Invention
The technical problem solved by the invention is as follows: the invention aims to provide a rapid analysis method which is more suitable for practical conditions and can be more reliably applied to the damping characteristics of a large-span suspension bridge aiming at the defects of the prior art.
The technical scheme of the invention is as follows: a method for rapidly analyzing damping characteristics of a large-span suspension bridge comprises the following steps:
the method comprises the following steps: based on the simplified dynamic modeling of the suspension bridge, a free vibration differential equation set of the system is established, and the method comprises the following sub-steps:
the first substep: defining a suspension bridge dynamic model, wherein a curved beam above the model represents a suspension bridge main cable, the main cable is subjected to the action of horizontal tension H, has initial sag d under the action of self-weight, and the initial static configuration of the main cable can be described by a quadratic parabolic function y (x), and the (x, y) is a whole coordinate system of the suspension bridge; the straight beam below the main cable is a main beam, and the span of the straight beam is l; the main cable and the main beam are connected through a plurality of hanging rods;
and a second substep: neglecting the axial elongation of the boom, a differential equation of motion of the system is established:
Figure BDA0002549336940000021
wherein EI represents the bending stiffness of the main beam, m represents the sum of the mass of the main cable and the main beam in unit length, H is the horizontal tension borne by the main cable, c represents the viscous damping coefficient of the system, u represents the vertical displacement function of the system, H is the additional cable force caused by elastic extension in the vibration process of the main cable, and is equal to the product of the cable section dynamic strain (t) and the axial stiffness EA, wherein A is the cross-sectional area of the main cable, namely:
Figure BDA0002549336940000022
in the formula
Figure BDA0002549336940000023
Is the vertical span ratio, g is the acceleration of gravity; leIs the curved length of the main cable.
Step two: calculating a dynamic stiffness matrix of a suspension bridge undamped system;
the first substep: function of mode of vibration
Figure BDA0002549336940000024
Setting the damping coefficient c to 0, and separating the damping coefficient c into two parts by a separation variable method
Figure BDA0002549336940000025
And
Figure BDA0002549336940000026
can be obtained by substituting into the formula (1)
Figure BDA0002549336940000031
The general solution of the above formula is:
Figure BDA0002549336940000032
wherein ξ is x/l,
Figure BDA0002549336940000033
B＝b0·[b1b2b3b4]，
Figure BDA0002549336940000034
Figure BDA0002549336940000035
Φ(ξ)＝[e-pξe-p(1-ξ)cos(qξ) sin(qξ)](5)
Figure BDA0002549336940000036
wherein gamma is2＝Hl2/EI，
Figure BDA0002549336940000037
And a second substep: according to the node displacement continuous condition:
Figure BDA0002549336940000038
and node force balance conditions:
Figure BDA0002549336940000039
the relationship between nodal displacement and nodal force can be found as follows:
K·[αaθal αbθal]T＝[VaMa/l VbMb/l]T(9)
αaand thetaaVertical displacement and corner displacement, V, of the left end point of the suspension bridgeaAnd MaRespectively the shearing force and bending moment of the left end of suspension bridge αbAnd thetabVertical displacement and corner displacement, V, of the right end point of the suspension bridge, respectivelybAnd MbRespectively is the node shear force and the bending moment of the left end of the suspension bridge. K is the dynamic stiffness matrix of the suspension bridge undamped system.
Step three: and calculating the undamped modal frequency omega of the suspension bridge. After the integral stiffness matrix K is obtained, a numerical iteration algorithm is applied to solve a characteristic equation det (K (omega)) of the system to be 0, and then the undamped modal frequency of each stage of the system is obtained.
Step four: and calculating the damping ratio zeta of the suspension bridge. After obtaining the undamped modal frequency omega of each step of the system, the following formula is used
Figure BDA00025493369400000310
The damping ratio of the suspension bridge is calculated. Thereafter, can be based on
Figure BDA0002549336940000041
To calculate the damping frequency omega of the systemD。
Effects of the invention
The invention has the technical effects that:
1. at present, a rapid and effective rapid analysis method is lacked in solving the damping characteristic of the large-span suspension bridge, so that the analysis mostly adopts an actual measurement mode, and therefore, the degradation evolution rule of the damping characteristic of the suspension bridge is difficult to be provided from a mechanism. The method provided by the invention is a frequency domain solution method, and the solving process is in a closed form, so that the method has high calculation efficiency and precision, and is expected to provide a theoretical basis for the inhibition and monitoring of the vortex-induced vibration of the large-span suspension bridge.
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 can obtain the undamped modal frequency of the system by solving the frequency equation. Then, the invention provides an analytic expression of the modal damping ratio and the damping frequency of the large-span suspension bridge, and the analytic expression can be used for the dynamic analysis of the damping characteristic analysis, the vibration control and the like of the large-span suspension bridge.
Drawings
FIG. 1 simplified dynamics model diagram of suspension bridge
FIG. 2 analysis flow chart
Detailed Description
Referring to fig. 1-2, a method for rapidly analyzing damping characteristics of a large-span suspension bridge is characterized by comprising the following steps:
the first step is as follows: according to the dynamic model shown in fig. 1, the damping coefficient c is first set to 0, and the additional cable force h and the dimensionless mode-shape function of the main cable are calculated
Figure BDA0002549336940000042
The second step is that: according to the node displacement continuous condition and the node force balance condition, by
Figure BDA0002549336940000043
Calculating a dynamic stiffness matrix K of the suspension bridge undamped system;
the third step: solving a system frequency equation det (K (omega)) to be 0 by using a numerical iterative algorithm, wherein the root of the overrunning equation corresponds to the undamped modal frequency omega of the suspension bridge;
the fourth step: calculating damping ratio zeta and modal damping frequency omega of suspension bridgeD。
It can be further described as:
1. calculating the additional cable force h of the main cable of the suspension bridge, establishing a free vibration differential equation of the suspension bridge in an integral coordinate system, transforming the equation to a frequency domain by adopting a separation variable method, and solving a dimensionless vibration mode function
Figure BDA0002549336940000051
2. The solution matrix B is calculated and,
3. and calculating a dynamic stiffness matrix K by combining the node displacement continuous condition and the force balance condition:
wherein
Figure BDA0002549336940000052
4. Solving a system frequency equation det (K (omega)) to be 0 by using numerical iterative algorithms such as a Newton method, a Muller method or a dichotomy, wherein the root of the transcendental equation is the undamped modal frequency omega of each order of the system;
5. calculating the damping ratio zeta of the suspension bridge according to the zeta as c/2 omega and then passing through
Figure BDA0002549336940000053
Calculating modal damping frequency omega of suspension bridgeD。
The technical solution of the present invention will be described in detail by one, but the scope of the present invention is not limited to the embodiments.
As shown in fig. 1, the method for rapidly analyzing the damping characteristic of a large-span suspension bridge according to the present invention includes the following steps:
1. the motion differential equation of each cable section and each beam section of the suspension bridge under a local coordinate system is established as follows:
Figure BDA0002549336940000054
wherein EI represents the bending stiffness of the main beam, m represents the sum of the mass of the main cable and the main beam in unit length, H is the horizontal tension borne by the main cable, c represents the viscous damping coefficient of the system, u represents the vertical displacement function of the system, H is the additional cable force caused by elastic extension in the vibration process of the main cable, and is equal to the product of the cable section dynamic strain (t) and the axial stiffness EA, wherein A is the cross-sectional area of the main cable, namely:
Figure BDA0002549336940000055
in the formula
Figure BDA0002549336940000061
Is the vertical span ratio, g is the acceleration of gravity; leIs the curved length of the main cable.
2. Dimensionless mode function
Figure BDA0002549336940000062
Is solved for
The invention solves the problem of the free vibration of the stay cable expressed by the formula (11) by applying a dynamic stiffness theory. . Setting the damping coefficient c to 0, and separating the damping coefficient c into two parts by a separation variable method
Figure BDA0002549336940000063
And
Figure BDA0002549336940000064
substituting the formula (11) to obtain:
Figure BDA0002549336940000065
introducing a dimensionless parameter of ξ ═ x/l,
Figure BDA0002549336940000066
a dimensionless form of the above formula can be obtained:
Figure BDA0002549336940000067
wherein gamma is2＝Hl2/EI，
Figure BDA0002549336940000068
The solution (14) can be obtained
Figure BDA0002549336940000069
Wherein
Φ(ξ)＝[e-pξe-p(1-ξ)cos(qξ) sin(qξ)](16)
Figure BDA00025493369400000610
B＝b0·[b1b2b3b4]，
Figure BDA00025493369400000611
A1A2A3A4Is a pending coefficient related to the boundary condition, which can be eliminated by substitution in the subsequent analysis process.
3. Solving of dynamic stiffness matrix K
According to the node displacement continuous condition:
Figure BDA00025493369400000612
and node force balance conditions:
Figure BDA00025493369400000613
substituting (15) into (18) and (19), respectively, and eliminating A1A2A3A4A relationship between nodal displacement and nodal force can be obtained:
K·[αaθal αbθal]T＝[VaMa/l VbMb/l]T(20)
α thereinaAnd thetaaVertical displacement and corner displacement, V, of the left end point of the suspension bridgeaAnd Maα being respectively the node shearing force and the bending moment of the left end of the suspension bridgebAnd thetabRespectively the vertical displacement and the corner displacement of the right end point of the suspension bridge,Vband MbRespectively is the joint shearing force and the bending moment of the left end of the suspension bridge. K is a dynamic stiffness matrix of the suspension bridge undamped system:
Figure BDA0002549336940000071
4. and solving the undamped modal frequency omega.
After the overall stiffness matrix K is obtained, a numerical iterative algorithm, such as a Newton method, a Muller method, a dichotomy, etc., is applied, so that the characteristic equation det (K (ω)) ═ 0 of the system can be solved. The root of this equation corresponds to the undamped modal frequency ω of the suspension bridge.
5. And calculating a system damping ratio zeta.
After obtaining the undamped modal frequency omega of each stage of the system, the following formula can be used
Figure BDA0002549336940000072
The damping ratio of the suspension bridge is calculated. Thereafter, can be based on
Figure BDA0002549336940000073
To calculate the damping frequency omega of the systemD。

Claims (1)

1. A method for rapidly analyzing damping characteristics of a large-span suspension bridge is characterized by comprising the following steps:
the method comprises the following steps: based on the simplified dynamic modeling of the suspension bridge, a system free vibration differential equation set is established, and the method comprises the following substeps:
the first substep: defining a suspension bridge dynamic model, wherein a curved beam above the model represents a suspension bridge main cable, the main cable is subjected to the action of horizontal tension H, has initial sag d under the action of self-weight, and the initial static configuration of the main cable can be described by a quadratic parabolic function y (x), and the (x, y) is a whole coordinate system of the suspension bridge; the straight beam below the main cable is a main beam, and the span of the straight beam is l; the main cable and the main beam are connected through a plurality of hanging rods;
and a second substep: neglecting the axial elongation of the boom, a differential equation of motion of the system is established:
Figure FDA0002549336930000011
wherein EI represents the bending stiffness of the main beam, m represents the sum of the linear masses of the main cable and the main beam in unit length, H is the horizontal tension borne by the main cable, c represents the viscous damping coefficient of the system, u represents the vertical displacement function of the system, H is the additional cable force caused by elastic extension in the vibration process of the main cable, and is equal to the product of the dynamic strain (t) of the cable section and the axial stiffness EA, wherein A is the section area of the main cable, namely:
Figure FDA0002549336930000012
in the formula
Figure FDA0002549336930000013
Is the vertical span ratio, g is the acceleration of gravity; leIs the curved length of the main cable.
Step two: calculating a dynamic stiffness matrix of a suspension bridge undamped system;
the first substep: function of mode of vibration
Figure FDA0002549336930000014
Setting the damping coefficient c to 0, and separating the damping coefficient c into two parts by a separation variable method
Figure FDA0002549336930000015
And
Figure FDA0002549336930000016
can be obtained by substituting into the formula (1)
Figure FDA0002549336930000017
The general solution of the above formula is:
Figure FDA0002549336930000018
wherein ξ is x/l,
Figure FDA0002549336930000019
B＝b0·[b1b2b3b4]，
Figure FDA00025493369300000110
Figure FDA0002549336930000021
Φ(ξ)＝[e-pξe-p(1-ξ)cos(qξ) sin(qξ)](5)
Figure FDA0002549336930000022
wherein gamma is2＝Hl2/EI，
Figure FDA0002549336930000023
And a second substep: according to the node displacement continuous condition:
Figure FDA0002549336930000024
and node force balance conditions:
Figure FDA0002549336930000025
the relationship between nodal displacement and nodal force can be found as follows:
K·[αaθal αbθal]T＝[VaMa/l VbMb/l]T(9)
αaand thetaaVertical displacement and corner displacement of the left end point of the suspension bridge respectively，VaAnd Maα being respectively the node shearing force and the bending moment of the left end of the suspension bridgebAnd thetabVertical displacement and corner displacement, V, of the right end point of the suspension bridge, respectivelybAnd MbRespectively is the node shear force and the bending moment of the left end of the suspension bridge. K is the dynamic stiffness matrix of the suspension bridge undamped system.
Step three: and calculating the undamped modal frequency omega of the suspension bridge. After the integral stiffness matrix K is obtained, a numerical iteration algorithm is applied to solve a characteristic equation det (K (omega)) of the system to be 0, and then the undamped modal frequency of each stage of the system is obtained.
Step four: and calculating the damping ratio zeta of the suspension bridge. After obtaining the undamped modal frequency omega of each step of the system, the following formula is used
Figure FDA0002549336930000026
The damping ratio of the suspension bridge is calculated. Thereafter, can be based on
Figure FDA0002549336930000027
To calculate the damping frequency omega of the systemD。
CN202010570251.9A 2020-06-21 2020-06-21 Rapid analysis method for damping characteristics of large-span suspension bridge Pending CN111783200A (en)

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CN117271949A (en) * 2023-11-23 2023-12-22 华东交通大学 Suspension cable self-vibration analysis method and system considering elastic boundary and large sag influence

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CN113848009A (en) * 2021-09-22 2021-12-28 浙江浙交检测技术有限公司 Detection method and detection equipment for constant-load cable force of arch bridge suspender
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