Improved real bridge inhaul cable vortex-induced vibration prediction method

Abstract

The invention discloses an improved real bridge inhaul cable vortex-induced vibration prediction method, which effectively solves the problem that the dynamic response of an inhaul cable under multi-dominant-frequency load excitation cannot be accurately calculated at present, and considers the damping force of the inhaul cable for resisting translational motion and the damping torque of the inhaul cable for resisting bending motion. The dynamic model capable of reflecting the damping characteristics and the frequency characteristics of the inhaul cable structure at the same time is obtained, and the dynamic characteristics of the inhaul cable can be reflected more accurately. An improved dynamic model is combined with a wake flow vibrator model, an improved inhaul cable vortex-induced vibration-wake flow vibrator coupling model is obtained, main vortex-induced vibration characteristics such as the locking characteristic and the vibration amplitude of structure vortex-induced vibration are predicted, the dynamic characteristics of an inhaul cable can be reflected more accurately, and the performance of the inhaul cable is improved. And the vortex-induced vibration response of the inhaul cable under the full wind speed, which is matched with a field monitoring result, is obtained.

Description

Technical Field

[0001] The invention relates to the field of wind-induced vibration of bridges, and in particular to an improved method for predicting vortex-induced vibration of cables of real bridges. Background Art

[0002] With the development of science and technology, the span capacity of bridges is constantly breaking through. The main span of cable-stayed bridges has now exceeded 1100m, and the main span of suspension bridges has exceeded 2000m. As the main load-bearing components of long-span bridges, the cables and suspenders are hundreds of meters long and have a very large slenderness ratio. The large slenderness ratio makes their stiffness and damping extremely small, and the dynamic effect is more prominent, which makes them prone to vortex-induced vibration under the action of wind loads. The starting wind speed of the vortex-induced vibration of the cable is low and the vibration frequency is high, which may cause fatigue damage to the cable, affecting the safety of the bridge and the comfort of driving. Studying the dynamic effect of the cable and accurately predicting its vortex-induced vibration is the basis for ensuring the safe service of the cable, which has important scientific and engineering significance.

[0003] Based on Newtonian mechanics, D'Alembert's principle and Hooke's law, the dynamic model of the cable can be derived by analyzing the force of the cable microelement. At present, the research on the cable dynamic model mainly focuses on stiffness, and the damping is not fully considered. Damping is the main factor affecting the dynamic response of the structure. Since the current cable dynamic model cannot accurately reflect the damping ratio characteristics of the cable, there are major defects in calculating the vibration characteristics of the multi-modal vortex-induced vibration of the cable and the multi-modal coupling vibration characteristics. Proposing a dynamic model that can accurately reflect the damping characteristics of the cable structure is the key to calculating the vortex-induced vibration response of the cable. In all the dynamic modeling of ultra-long flexible cable structures such as cables and marine risers at home and abroad, the damping term is derived by adding a damping force opposite to the motion in the microelement force analysis. Under the assumption that the damping coefficient is proportional to the mass, the distribution characteristics of the modal damping ratio are that the higher the modal order, the lower the modal damping ratio. This damping distribution characteristic cannot reflect the actual damping characteristics of the cable, resulting in the failure to obtain ideal results when calculating the vortex-induced vibration response of the flexible cable structure. For example, the on-site monitoring results of the inclined cable show that the vortex-induced vibration of the cable structure only occurs within a certain wind speed range, and vortex-induced vibration does not occur at high wind speeds. The vortex-induced vibration response of the inclined cable calculated using the theoretical model shows that the vortex-induced vibration acceleration amplitude of the inclined cable will continue to increase with the increase of wind speed, which is seriously inconsistent with the on-site monitoring results, indicating that the current cable dynamic model cannot reflect the true state of the cable, and therefore cannot effectively calculate its vortex-induced vibration response. The method of using mode separation cannot solve the problem of multi-mode coupling. Improving the motion equation of the cable dynamic model is the best and only way to solve the current problem of being unable to accurately calculate the vortex-induced vibration response of the cable. Summary of the invention

[0004] Based on the above shortcomings, the purpose of the present invention is to propose an improved method for predicting vortex-induced vibration of real bridge cables, which effectively solves the current problem that it is impossible to accurately calculate the dynamic response of cables under multi-main frequency load excitation, and can predict the vortex-induced vibration response of real bridge cables under full wind speed.

[0005] The technology adopted by the present invention is as follows: An improved method for predicting vortex-induced vibration of cables in real bridges is as follows:

[0006] First, a Cartesian coordinate system is established. The horizontal coordinate coincides with the line segment with the two fixed ends of the cable as endpoints. The origin of the coordinate is located at one of the fixed ends. The horizontal coordinate is represented by Z, the vertical coordinate is represented by Y, and the displacement of the cable along the vertical axis is represented by r. y It means that any microelement of the cable is taken, and the force analysis of the microelement is performed based on Newtonian mechanics. The damping force of the microelement against translational motion and the damping moment against bending deformation are considered. The constitutive equation is established according to the D'Alembert principle. By balancing the forces and ignoring high-order small quantities and gravity, the following improved nonlinear partial differential equation of the cable is obtained:

[0007]

[0008] Where EI is the bending stiffness, a1 is the stiffness proportional factor related to the material strain velocity impedance, I is the moment of inertia, T is the time, H is the cable force, h is the dynamic cable force, m is the cable line density, a0 is the mass proportional factor related to the structural damping coefficient, g′ is the component of gravity along the longitudinal axis, and f Y For external load.

[0009] Formula (1) contains six terms, from left to right, namely, the stiffness term related to bending stiffness, the damping term related to bending stiffness, the nonlinear stiffness term related to cable force, the inertial force term, the damping term generated by the cable resisting translational motion, and the load term. Compared with the traditional dynamic model, the present invention adds the damping term related to bending stiffness, which represents the damping moment of the cable resisting bending deformation, and can accurately represent the damping ratio characteristics of the cable when high-modal vibration occurs;

[0010] The improved cable nonlinear partial differential equation is combined with the three-dimensional wake oscillator model to obtain the improved cable vortex-induced vibration-wake oscillator coupling model, which is expressed as follows:

[0011]

[0012] Among them, ρ is the air density, U is the incoming wind speed, D is the cable diameter, C L is the lift coefficient, q is the vortex lift coefficient, q = 2C L / C L0 ,ω f is the vortex shedding frequency, C L0is the initial value of the lift coefficient, which is determined by measuring the lift coefficient of the structure at rest, ε r is the van der Pol coefficient, which is 0.3 for a circular structure, A r is the coupling coefficient, which is 12 for the circular structure;

[0013] By using the wake oscillator model to describe the lift acting on the structure, that is, using the van der Pol equation to represent the unstable lift of the structure, and establishing the coupling model equation between the structural vibration and the lift, the main vortex-induced vibration characteristics such as the locking characteristics and vibration amplitude of the structural vortex-induced vibration can be predicted.

[0014] Furthermore, after determining the damping proportional factors a0 and a1, the coupled model can be used to calculate the damping ratio distribution of cables under different cable forces and cable lengths.

[0015] Another object of the present invention is to provide a real bridge cable vortex-induced vibration prediction system, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the computer program is executed by the processor, the real bridge cable vortex-induced vibration prediction method as described above is implemented.

[0016] Another object of the present invention is to provide a computer-readable storage medium, on which is stored an implementation program for information transmission, and when the program is executed by a processor, a method for predicting vortex-induced vibration of a real bridge cable as described above is implemented.

[0017] Advantages and beneficial effects of the present invention: The cable dynamic model of the present invention takes into account the damping force of the cable against translational motion and the damping moment against bending motion, and can simultaneously reflect the damping ratio characteristics and frequency characteristics of the cable structure, and can more accurately reflect the dynamic characteristics of the cable. The cable dynamic model of the present invention is combined with the three-dimensional wake oscillator model to calculate the vortex-induced vibration response of the real bridge cable at full wind speed, and the calculation results are in good agreement with the on-site monitoring results. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 is a schematic diagram of the cable.

[0019] Figure 2 This is the schematic diagram of the deformation of the cable microelement.

[0020] Figure 3 It is the force diagram of the cable microelement.

[0021] Figure 4 The following is a comparison chart of the vortex-induced vibration response of the actual bridge cable calculated by the present invention and the on-site monitoring result.

[0022] Figure 5 The vortex-induced vibration response of the cable-stayed bridge of a real bridge calculated using the traditional dynamic model. DETAILED DESCRIPTION

[0023] The present invention is further explained below with reference to the accompanying drawings:

[0024] Example 1

[0025] (I) The cable dynamic model is constructed as follows:

[0026] Figure 1 The figure shows a schematic diagram of a cable. The cable inclination angle is θ, L represents the length of the cable, the two fixed ends are represented by A and B respectively, the origin of the coordinate is located at A, the horizontal axis coincides with the line segment AB, and the cable will have a certain sag under the action of gravity. Y represents the static configuration of the cable. Take any infinitesimal body of the cable, Figure 2 The figure shows the deformation diagram of the microelement. The displacement of the microelement along the horizontal and vertical axes is represented by r and z and r y Indicates that the original length and the length after deformation of the microelement are ds and ds′ respectively, and have the following relationship:

[0027] ds 2 =dY 2 +dZ 2

[0028] ds′ 2 =(dY+r y ) 2 +(dZ+r z ) 2 (3)

[0029] Considering the axial force and viscous damping force of the cable, the force diagram of the cable micro-segment is as follows: Figure 3 As shown, where f I is the inertia force of the micro segment, which can be expressed as m is the cable linear density, f D is the damping force, expressed as c is the cable damping coefficient, f Y is the load density, T e and τ are the initial cable force and dynamic cable force respectively, the shear force and bending moment are represented by V and M respectively, and g′ is the component of gravity acceleration along the longitudinal axis.

[0030] From the force balance in the longitudinal direction:

[0031]

[0032] H and h are used to represent the axial components of the cable force and dynamic cable force respectively, and their relationship with the cable force is:

[0033]

[0034] Taking the moment of the center of the microelement and ignoring high-order small quantities, we can obtain:

[0035]

[0036] The bending moment consists of three parts: the bending moment caused by the bending of the cable under the action of its own weight, the bending moment caused by the bending of the cable under the action of dynamic load, and the damping moment caused by the viscous impedance of the material strain:

[0037]

[0038] Where E is the elastic modulus, c s is the impedance of material strain velocity, I is the moment of inertia, and EI represents the bending stiffness. Introducing the damping proportional factor a1, let:

[0039]

[0040] Substituting formula (6) into formula (4) to eliminate the shear force V, introducing formulas (5), (7) and (8) at the same time, and expressing the damping coefficient c as the product of mass m and stiffness proportional factor a0, the improved nonlinear dynamic partial differential equation of the cable can be obtained.

[0041]

[0042] Ignoring the effect of gravity, we can get:

[0043]

[0044] The axial component of the dynamic cable force is calculated using the following formula:

[0045]

[0046] Where A is the cross-sectional area of ​​the cable.

[0047] (II) Modal decomposition and damping ratio calculation

[0048] First, the vibration mode and frequency are determined by solving the undamped free vibration equation of the cable. Ignoring the sag and dynamic cable force caused by the deadweight, the following vibration equation can be obtained:

[0049]

[0050] The analytical solution of the cable vibration mode, amplitude and frequency under ideal conditions can be obtained by using the separation of variables method.

[0051]

[0052] Where n is the vibration order, P n and ω n is the vibration mode and frequency of the nth order vibration, A n and is the vibration amplitude and vibration phase, which depends on the initial excitation conditions. According to Betti's law, the work done by the inertial force of the nth vibration mode on the displacement of the kth vibration mode is equal to the work done by the inertial force of the kth vibration mode on the displacement of the nth vibration mode. Two orthogonality conditions can be derived, namely:

[0053]

[0054] Substituting the orthogonality condition into the cable dynamic equation and ignoring the dynamic cable force, we can obtain the generalized mass M n , generalized stiffness K n and generalized damping C n The expression is:

[0055]

[0056] It can be seen that generalized damping does not have complete orthogonality. In structural dynamic analysis, it is usually assumed that the damping coefficient c is related to the mass, that is, let c = a0m (a0 is a constant, called the mass proportional factor), and the generalized damping can be written as:

[0057]

[0058] The damping of the structure can be calculated by simply determining the damping proportional factors a0 and a1. For a specific structure, the damping proportional factor can be identified by measuring the first few damping ratios. Since the damping proportional factor is only related to the material, changing the cable tension or length will not change the size of the damping proportional factor.

[0059] (III) Improved prediction method of vortex-induced vibration of real bridge cables

[0060] Wind flowing through the cable will produce alternating vortex shedding behind it. The periodic vortex shedding will cause periodic vortex excitation force on the cable. When the frequency of the vortex excitation force is close to a certain order vibration frequency of the cable, vortex-induced vibration may occur. In the theoretical model of structural vortex-induced vibration calculation, the wake oscillator model is currently used to describe the vortex excitation force on the structure, that is, the van der Pol equation is used to represent the unstable lift of the structure, and the coupling equation between the structural vibration and the lift is established to calculate the locking characteristics and vibration amplitude of the vortex-induced vibration of the cable. The expression of the improved real bridge cable vortex-induced vibration-wake oscillator coupling model proposed in this embodiment is:

[0061]

[0062] Among them, ρ is the air density, U is the incoming wind speed, D is the cable diameter, C L is the lift coefficient, q is the vortex lift coefficient, q = 2C L / C L0 ,ω f is the vortex shedding frequency, CL0 is the initial value of the lift coefficient, which is determined by measuring the lift coefficient of the structure at rest, ε r is the van der Pol coefficient, which is 0.3 for a circular structure, A r is the coupling coefficient, which is 12 for a circular structure.

[0063] Example 2

[0064] (I) Calculation of damping ratio of flexible cable model under experimental conditions

[0065] A flexible cable model was made in the laboratory, and then a dynamic test was performed on it. The cable diameter and length were 0.08m and 5.6m respectively, the cable line density was 1.802kg / m, the cable force under the test condition was 1950N, and the measured first two vibration frequencies were 2.87Hz and 5.83Hz respectively, corresponding to the first two damping ratios of 0.18% and 0.85% respectively. The two damping proportional factors a0 and a1 were calculated by the cable dynamic equation proposed in the present invention, which were equal to 0.03 and 0.28 respectively. Since the damping proportional factor is only related to the material, the identified parameters can be used to calculate the damping ratio of the cable model under different cable forces and lengths. Four groups of test conditions were set up by changing the cable force and frequency of the cable model to measure the damping ratio. The accuracy of the theoretical model was verified by comparing the calculated results and the measured results of the damping ratio. The results are shown in Table 1. The error between the damping ratio obtained by theoretical calculation and the measured damping ratio is within 10%, which proves that the dynamic model proposed in the present invention can reflect the damping characteristics of the cable structure.

[0066] Table 1 Comparison of calculated and experimental damping ratios of flexible cable models

[0067]

[0068] At the same time, the damping ratio of the cable model was calculated using the traditional dynamic model. In the traditional dynamic model, only the damping proportional factor a0 needs to be determined. The a0 calculated based on the first-order damping ratio is equal to 6.49. The second-order damping ratio calculated based on this coefficient is 0.09%. The measured value of the second-order damping ratio under the test condition is 0.85%, and the error reaches 89%, indicating that the traditional dynamic model cannot reflect the damping characteristics of the cable model. Table 2 shows the comparison results of the calculated and measured damping ratios under four groups of test conditions. The maximum error reaches 40%, and the error will be greater when the cable length or cable force changes more.

[0069] Table 2 Comparison of the calculated damping ratio values ​​obtained by traditional methods and the experimental values

[0070]

[0071] (II) Calculation of vortex-induced vibration response of cables in real bridges

[0072] The improved real bridge cable vortex-induced vibration prediction method proposed in the present invention is used to calculate the vortex-induced vibration response of the Sutong Bridge cable, and compared with the on-site monitoring results. The improved real bridge cable vortex-induced vibration-wake oscillator coupling model is solved by the central difference method to calculate the vortex-induced vibration response of the cable at full wind speed. The monitored cable is not equipped with a damper, and the structural damping is provided by the structure itself, which meets the prerequisites of the improved dynamic model proposed in the present invention. The cable length is 214.88m, the diameter is 0.116m, the inclination angle is 51.2°, the linear density is 50.7kg / m, the cable force is 3300kN, and the tensile stiffness and bending stiffness are 1.19×10 9 N and 5.51×10 5 N·m 2 The initial value of the lift coefficient in the equation is taken as 0.13, and the damping proportional factors a0 and a1 are 0.014 and 0.0012 respectively. According to the "Highway Bridge Design Code" (JTG / T 3360-01-2018), the surface characteristics of the Sutong Yangtze River Bridge are a wide river surface, the wind speed profile conforms to the wind speed distribution of Class A surface classification, the wind speed is exponentially distributed along the height, and the surface roughness coefficient is 0.12. The vibration acceleration of the cable was measured on-site, and the distance from the installation position of the acceleration sensor to the lower end of the cable was about 10m.

[0073] The calculation results of the cable vortex-induced vibration wake oscillator coupling model and the field monitoring results are shown in Figure 2. Figure 4 As shown in the figure, on-site monitoring shows that the reference wind speed range for vortex-induced vibration of the cable is 3-10m / s, and the maximum vibration acceleration is about 20m / s 2 , and there are two main vibration ranges, 4-6m / s and 7-10m / s. The results calculated by the theoretical model are basically consistent with the on-site monitoring results, proving the superiority of the present invention. The results calculated by the traditional dynamic model are as follows Figure 5 As shown in Figure 2, the vibration acceleration of the structure increases with the increase of wind speed. When the reference wind speed is 50 m / s, the acceleration amplitude exceeds 4000 m / s. 2 , which is completely inconsistent with the actual results, further proving the accuracy and superiority of the present invention.

Claims

1. A method for predicting vortex-induced vibration of a real bridge cable, characterized in that: Method steps: First, a Cartesian coordinate system is established. The horizontal coordinate coincides with the line segment with the two fixed ends of the cable as endpoints. The origin of the coordinate is located at one of the fixed ends. The horizontal coordinate is represented by Z, the vertical coordinate is represented by Y, and the displacement of the cable along the vertical axis is represented by r. y It means that taking any microelement of the cable, considering the damping force of the microelement to resist translational motion and the damping moment to resist bending deformation, the improved nonlinear partial differential equation of the cable is obtained by balancing the forces and ignoring high-order small quantities and gravity: Where EI is the bending stiffness, a1 is the stiffness proportional factor related to the material strain velocity impedance, I is the moment of inertia, T is the time, H is the cable force, h is the dynamic cable force, m is the cable line density, a0 is the mass proportional factor related to the structural damping coefficient, g′ is the component of gravity along the longitudinal axis, and f Y is the external load; The nonlinear partial differential equation of the cable is combined with the three-dimensional wake oscillator model to obtain the improved cable vortex-induced vibration-wake oscillator coupling model equation, which is expressed as follows: Among them, ρ is the air density, U is the incoming wind speed, D is the cable diameter, C L is the lift coefficient, q is the vortex lift coefficient, q = 2C L / C L0 ,ω f is the vortex shedding frequency, C L0 is the initial value of the lift coefficient, which is determined by measuring the lift coefficient of the structure at rest, ε r is the van der Pol coefficient, which is 0.3 for a circular structure, A r is the coupling coefficient, which is 12 for the circular structure; The wake oscillator model is used to describe the vortex-induced force on the structure, that is, the van der Pol equation is used to represent the unstable lift of the structure. The locking characteristics and vibration amplitude of the vortex-induced vibration of the structure are predicted through the cable vortex-induced vibration-wake oscillator coupling model.

2. The improved method for predicting vortex-induced vibration of cables in real bridges according to claim 1 is characterized in that: After determining the damping proportional factors a0 and a1, the coupled model can be used to calculate the damping ratio distribution of cables under different cable forces and cable lengths.

3. A real bridge cable vortex-induced vibration prediction system, comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the computer program, when executed by the processor, implements a method for predicting vortex-induced vibration of a real bridge cable as claimed in claim 1.

4. A computer-readable storage medium having stored thereon a program for implementing information transmission, characterized in that: When the program is executed by a processor, a method for predicting vortex-induced vibration of a real bridge cable as claimed in claim 1 is implemented.

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