Method and device for determining equivalent damping ratio of longitudinal damping device to vertical bending vibration

Abstract

This invention discloses a method and apparatus for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration, relating to the field of structural dynamic analysis technology. The method includes: determining the generalized mass based on the mass and mode shape vector of each mass point in the structural system; determining the generalized vertical bending stiffness based on the generalized mass and vibration frequency; and determining the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and the resistance of the longitudinal damping device. By equating the multi-degree-of-freedom system to a generalized one-dimensional vibration system, the calculation efficiency of the equivalent damping ratio is high, the numerical value is stable, and the results are accurate. This solves the problems of long solution time, large fluctuations in the equivalent damping ratio, and poor performance in existing multi-degree-of-freedom nonlinear time-history finite element analysis methods considering large deformations.

Description

Technical Field

[0001] This invention relates to the field of structural dynamic analysis technology, specifically to a method and apparatus for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration. Background Technology

[0002] While vortex-induced vibration (VID) in long-span bridges is insufficient to affect the structural safety of the bridge, excessive amplitude can cause public panic and undermine public confidence in the construction of long-span bridges, which is a crucial issue in the construction of such bridges. VID primarily causes vertical vibration in the main girder. Most bridge structures can only accommodate longitudinal damping devices at the bridge ends; therefore, it is necessary to determine the equivalent damping ratio of the longitudinal damping devices for the vertical bending vibration of the structural system.

[0003] Vortex-induced vibration causes dozens of vertical vibration modes in the main beam, with longitudinal-to-vertical coupling resulting in a large deformation effect. In existing technologies, the conventional method for determining the equivalent damping ratio, employing a multi-degree-of-freedom nonlinear time-history finite element analysis method considering large deformation, suffers from problems such as long solution time, large fluctuations in the equivalent damping ratio, and poor performance. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a method and apparatus for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration. This method solves the problems of long solution time, large fluctuations in the equivalent damping ratio, and poor performance associated with existing multi-degree-of-freedom nonlinear time-history finite element analysis methods that consider large deformations.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] On the one hand, this solution provides a method for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration, including:

[0007] The generalized mass is determined based on the mass and mode shape vector of each mass in the structural system.

[0008] The generalized vertical bending stiffness is determined based on the generalized mass and vibration frequency.

[0009] The equivalent damping ratio is determined based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance.

[0010] In some alternative solutions, determining the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance includes:

[0011] The amplitude displacement is obtained based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance.

[0012] Based on the amplitude displacement, obtain the amplitude displacement attenuation curve and determine the equivalent damping ratio.

[0013] In some alternative solutions, according to the formula: Obtain amplitude displacement;

[0014] Where M is the generalized mass, K is the generalized stiffness, f is the resistance of the longitudinal damping device, α is the influence factor of vertical bending deformation on longitudinal deformation, and z is the amplitude displacement. For amplitude velocity, This represents the amplitude acceleration.

[0015] In some alternative solutions, according to the formula: Obtain the resistance of the longitudinal damping device;

[0016] Among them, f q This represents the greatest resistance.

[0017] In some alternative schemes, the range of the influence factor of vertical bending deformation on longitudinal deformation is: 2.4≤α≤2.5.

[0018] In some alternative solutions, according to the formula: Determine generalized quality;

[0019] Where, m i Let y be the mass of the i-th particle. i Let be the mode shape vector of the i-th mass point, where i = 1 to n, and n is the number of mass points.

[0020] In some alternative solutions, according to the formula: K = ω 2 M determines the generalized vertical bending stiffness;

[0021] Where ω is the vibration frequency.

[0022] In some alternative schemes, the mass of each mass point is obtained through a finite element model of the structural system.

[0023] In some alternative approaches, the vibration frequencies and mode shape vectors of each mass point are obtained through modal analysis of the structural system.

[0024] On the other hand, this solution provides a device for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration, comprising:

[0025] The generalized mass determination module is used to determine the generalized mass based on the mass and mode shape vector of each mass in the structural system.

[0026] The generalized vertical bending stiffness determination module is used to determine the generalized vertical bending stiffness based on the generalized mass and vibration frequency.

[0027] The equivalent damping ratio determination module is used to determine the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance.

[0028] Compared with existing technologies, the advantages of this invention are as follows: This scheme determines the generalized mass based on the mass and mode shape vector of each mass point in the structural system; it determines the generalized vertical bending stiffness based on the generalized mass and vibration frequency; and it determines the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and the resistance of the longitudinal damping device. By equating the multi-degree-of-freedom system to a generalized one-dimensional vibration system, the calculation efficiency of the equivalent damping ratio is high, the value is stable, and the results are accurate. This solves the problems of long solution time, large fluctuations in the equivalent damping ratio, and poor performance associated with existing multi-degree-of-freedom nonlinear time-history finite element analysis methods that consider large deformations. Attached Figure Description

[0029] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0030] Figure 1 This is a flowchart illustrating the method for determining the equivalent damping ratio of the longitudinal damping device for vertical bending vibration in an embodiment of the present invention.

[0031] Figure 2 This is a schematic diagram of a simply supported beam with a uniform cross-section in an embodiment of the present invention;

[0032] Figure 3 This is a schematic diagram of the time history decay curve of the amplitude displacement of the first-order positive symmetric vertical bending mode in an embodiment of the present invention;

[0033] In the diagram: 1. Fixed support; 2. Movable support; 3. Mass point; 4. Main beam. Detailed Implementation

[0034] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0035] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0036] like Figure 1 As shown, in one aspect, the present invention provides a method for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration, comprising:

[0037] S1: Determine the generalized mass based on the mass and mode shape vector of each mass point in the structural system.

[0038] In some alternative embodiments, according to the formula: Determine generalized quality;

[0039] Where, m i Let y be the mass of the i-th particle. i Let be the mode shape vector of the i-th mass point, where i = 1 to n, and n is the number of mass points.

[0040] In this embodiment, the generalized mass is obtained by multiplying the mass of each mass point by the square of the corresponding mass point mode vector and summing the products of all mass points.

[0041] S2: Determine the generalized vertical bending stiffness based on the generalized mass and vibration frequency.

[0042] In some alternative embodiments, according to the formula: K = ω 2 M determines the generalized vertical bending stiffness;

[0043] Where ω is the vibration frequency.

[0044] In this embodiment, the product of the generalized mass and the square of the vibration frequency of the vertical bending vibration mode is the generalized vertical bending stiffness.

[0045] S3: Determine the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance.

[0046] Step S3 specifically includes:

[0047] S31: Obtain the amplitude displacement based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance.

[0048] In some alternative embodiments, according to the formula: Obtain amplitude displacement;

[0049] Where M is the generalized mass, K is the generalized stiffness, f is the resistance of the longitudinal damping device, α is the influence factor of vertical bending deformation on longitudinal deformation, and z is the amplitude displacement. For amplitude velocity, This represents the amplitude acceleration.

[0050] In this embodiment, the amplitude displacement is solved by step-by-step integration, and the attenuation value of the amplitude displacement is recorded to obtain the amplitude displacement attenuation curve.

[0051] In some alternative embodiments, according to the formula: Obtain the resistance of the longitudinal damping device;

[0052] Among them, f q This represents the greatest resistance.

[0053] In some optional embodiments, the influence factor of vertical bending deformation on longitudinal deformation ranges from 2.4 to α, which is 2.5.

[0054] S32: Based on the amplitude displacement, obtain the amplitude displacement attenuation curve and determine the equivalent damping ratio.

[0055] In this embodiment, the coordinates of the positive extreme point are extracted from the amplitude displacement decay curve and substituted into the exponential function: y = ae bx By fitting the data, the equivalent damping ratio is determined according to the formula: ξ=-b / ω;

[0056] Where y is the ordinate of the exponential function, i.e., amplitude displacement, x is the abscissa of the exponential function, i.e., time, a is the first fitting parameter, b is the second fitting parameter, and ξ is the equivalent damping ratio.

[0057] In some alternative embodiments, the mass of each mass point is obtained through a finite element model of the structural system.

[0058] In some alternative embodiments, the vibration frequencies and mode shape vectors of each mass point are obtained by modal analysis of the structural system.

[0059] In this embodiment, the structural system modal analysis includes multiple vertical bending vibration modes, each corresponding to a vibration frequency. At different vibration frequencies, the equivalent damping ratio under different vertical bending vibration modes is determined.

[0060] On the other hand, the present invention provides a device for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration, comprising:

[0061] The generalized mass determination module is used to determine the generalized mass based on the mass and mode shape vector of each mass in the structural system.

[0062] The generalized vertical bending stiffness determination module is used to determine the generalized vertical bending stiffness based on the generalized mass and vibration frequency.

[0063] The equivalent damping ratio determination module is used to determine the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance.

[0064] In summary, this invention determines the generalized mass based on the mass and mode shape vector of each mass point in the structural system; it determines the generalized vertical bending stiffness based on the generalized mass and vibration frequency; and it determines the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and the resistance of the longitudinal damping device. By equating the multi-degree-of-freedom system to a generalized one-dimensional vibration system, the calculation efficiency of the equivalent damping ratio is high, the numerical value is stable, and the results are accurate. This solves the problems of long solution time, large fluctuations in the equivalent damping ratio, and poor performance in existing multi-degree-of-freedom nonlinear time-history finite element analysis methods that consider large deformations.

[0065] This invention, based on the relationship between vertical bending deformation and longitudinal deformation and the energy principle, establishes a generalized one-dimensional dynamic equation for the vertical bending vibration of a structural system considering longitudinal resistance. This equation conforms to fundamental dynamic theory, transforming the nonlinear dynamic equation of a multidimensional system into a one-dimensional nonlinear dynamic equation. The equivalent damping ratio determined using this method is calculated in seconds, demonstrating high efficiency. The obtained amplitude-displacement attenuation curve is stable and without fluctuations, facilitating accurate calculation of the equivalent damping ratio.

[0066] The following specific examples will facilitate understanding of the present invention.

[0067] like Figure 2 The diagram shows a simple supported beam structure with uniform cross-section. The simple supported beam with uniform cross-section corresponds to the structural system in this invention, comprising: a main beam 4, with fixed supports 1 and movable supports 2 arranged below both ends of the main beam 4, wherein the movable supports 2 correspond to the longitudinal damping device in this invention. There are nine mass points 3 between the fixed supports 1 and the movable supports 2. The main beam 4 has a length of 100m, a dead load of 300kN / m, and an in-plane bending stiffness EI of 3.91×10⁻⁶. 8 kN.m 2 The extension direction is divided into units of 10m each, with a total of nine mass points 3 between the fixed support 1 and the movable support 2. The maximum frictional force of the movable support 2 is 1500kN.

[0068] A finite element model of a simply supported beam with uniform cross-section is established. The mass of each mass point is obtained from the finite element model, forming a mass matrix of [3.06,3.06,3.06,3.06,3.06,3.06,3.06,3.06,3.06]×10 5 kg.

[0069] Modal analysis of vertical bending vibration was performed on a simply supported beam with uniform cross-section. The mode shapes of the first four vertical bending vibrations were first-order symmetric vertical bending, first-order antisymmetric vertical bending, second-order symmetric vertical bending, and second-order antisymmetric vertical bending, respectively. Taking the first-order symmetric vertical bending as an example, its natural frequency is 3.529 rad / s, and the normalized mode shape vectors formed by the mode shape vectors of each mass point are [0.309, 0.588, 0.809, 0.951, 1.000, 0.951, 0.809, 0.588, 0.309].

[0070] To calculate the generalized mass and generalized vertical bending stiffness of a first-order positive symmetric vertical bend, multiply the mass of each mass point by the square of the corresponding mode shape vector, and sum the products of all mass points. The generalized mass is 1,529,676 kg. Multiplying the generalized mass by the square of the vibration frequency of the corresponding mode, the generalized vertical bending stiffness is 19,050 kN / m.

[0071] Calculate the first-order positive symmetric vertical bending equivalent damping ratio. Substitute the generalized mass, generalized vertical bending stiffness, and support resistance into the formula for determining the amplitude displacement in this invention. Perform stepwise integration to solve for the amplitude displacement, record the attenuation amplitude of the amplitude displacement, and obtain the following... Figure 3 The amplitude-displacement attenuation curve shown can be used to determine the equivalent damping ratio of this first-order positively symmetric vertical bending mode as 12.2 × 10⁻⁶. -5 .

[0072] Based on the above steps, calculations under different vertical bending vibration modes yielded equivalent damping ratios of 6.0 × 10⁻⁶ for the first-order antisymmetric vertical bend, the second-order orthosymmetric vertical bend, and the second-order antisymmetric vertical bend. -5 4.2×10 -5 3.6×10 -5 .

[0073] When the movable support is replaced with other longitudinal damping devices, the method for determining the equivalent damping ratio is the same.

[0074] In the description of this application, it should be noted that the terms "upper," "lower," etc., indicating the orientation or positional relationship are based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this application. Unless otherwise expressly specified and limited, the terms "installed," "connected," and "linked" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication between two elements. For those skilled in the art, the specific meaning of the above terms in this application can be understood according to the specific circumstances.

[0075] It should be noted that in this application, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0076] The above description is merely a specific embodiment of this application, enabling those skilled in the art to understand or implement this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features claimed herein.

Claims

1. A method for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration, characterized in that, include: The generalized mass is determined based on the mass and mode shape vector of each mass in the structural system. The generalized vertical bending stiffness is determined based on the generalized mass and vibration frequency. The equivalent damping ratio is determined based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance. According to the formula: Determine generalized quality; in, For the first The mass of a point mass For the first The mode shape vector of a mass point , The number of particles; According to the formula: Determine the generalized vertical bending stiffness; in, The vibration frequency; the determination of the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance includes: The amplitude displacement is obtained based on the generalized mass, the generalized vertical bending stiffness, and the resistance of the longitudinal damping device. Based on the amplitude displacement, obtain the amplitude displacement attenuation curve and determine the equivalent damping ratio.

2. The method for determining the equivalent damping ratio of the longitudinal damping device for vertical bending vibration as described in claim 1, characterized in that, According to the formula: To obtain the amplitude displacement; in, For quality in a broad sense, For generalized stiffness, For the longitudinal damping device resistance, The influence factor of vertical bending deformation on longitudinal deformation. For amplitude displacement, For amplitude velocity, This represents the amplitude acceleration.

3. The method for determining the equivalent damping ratio of the longitudinal damping device for vertical bending vibration as described in claim 2, characterized in that, According to the formula: To obtain the resistance of the longitudinal damping device; in, This represents the greatest resistance.

4. The method for determining the equivalent damping ratio of the longitudinal damping device for vertical bending vibration as described in claim 2, characterized in that, The range of values ​​for the influence factor of vertical bending deformation on longitudinal deformation is: .

5. The method for determining the equivalent damping ratio of the longitudinal damping device for vertical bending vibration as described in claim 1, characterized in that, The mass of each particle is obtained through the finite element model of the structural system.

6. The method for determining the equivalent damping ratio of the longitudinal damping device for vertical bending vibration as described in claim 5, characterized in that, By performing modal analysis on the structural system, the vibration frequency and the mode shape vectors of each mass point are obtained.

7. A device for determining the equivalent damping ratio of a longitudinal damping device for vertical bending vibration, which implements the method of claim 1, characterized in that, include: The generalized mass determination module is used to determine the generalized mass based on the mass and mode shape vector of each mass point in the structural system. The generalized vertical bending stiffness determination module is used to determine the generalized vertical bending stiffness based on the generalized mass and vibration frequency. The equivalent damping ratio determination module is used to determine the equivalent damping ratio based on the generalized mass, generalized vertical bending stiffness, and longitudinal damping device resistance.

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