Calculation method for transverse natural vibration frequency of water body in aqueduct with U-shaped section

Abstract

The invention provides a method for calculating the transverse natural vibration frequency of a water body in an aqueduct with a U-shaped section, and relates to the technical field of aqueduct anti-seismic analysis. The method comprises the steps that the section size information of the U-shaped aqueduct is obtained, and the target order of the calculation frequency is obtained; obtaining a frequency order coefficient of the transverse natural vibration frequency of the water body in the aqueduct with the U-shaped section through calculation on the basis of the calculation frequency target order; calculating by utilizing the section size information of the U-shaped aqueduct to obtain a section shape coefficient of a U-shaped section under a fixed order; and determining the transverse natural vibration frequency of the water body in the aqueduct with the U-shaped cross section under the corresponding order through calculation by utilizing the cross section shape coefficient of the U-shaped cross section under the fixed order and the calculation frequency target order, thereby completing the calculation of the transverse natural vibration frequency of the water body in the aqueduct with the U-shaped cross section in the aqueduct with the U-shaped cross section. According to the method, the problems that an existing calculation method for the transverse natural vibration frequency of the water body in the U-shaped aqueduct is complex in calculation formula, large in calculation amount and small in frequency order range capable of being calculated are solved.

Description

Technical Field

[0001] The present invention relates to the technical field of seismic analysis of aqueducts, and in particular to a method for calculating the lateral natural frequency of water in a U-shaped cross-section aqueduct. Background Art

[0002] Aqueducts are widely used water conveyance structures in water diversion projects. The biggest difference between aqueducts and ordinary bridges is that aqueducts need to carry a large amount of water, resulting in the mass of the aqueduct superstructure and supporting structures (frames, piers, trusses, towers and cables, etc.) being equivalent to the water in the aqueduct. Under the action of an earthquake, the ground movement will cause the movement of the aqueduct through the supporting structure, and the involved movement of the aqueduct will drive the sloshing of the water in the aqueduct. The sloshing of the water in the aqueduct will in turn affect the movement of the aqueduct and the supporting structure. Therefore, the seismic calculation of the aqueduct is a rather complex dynamic problem. In this case, obtaining the various lateral natural frequencies of the water in the aqueduct is crucial for the design and sloshing control of the aqueduct.

[0003] The existing calculation method of the lateral natural frequency of water in a U-shaped aqueduct has the disadvantages of complex calculation formula, large amount of calculation and small range of frequency orders that can be calculated. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a method for calculating the lateral natural frequency of water in a U-shaped cross-section aqueduct, which solves the problems of the existing method for calculating the lateral natural frequency of water in a U-shaped aqueduct having complex calculation formulas, large amount of calculations, and a small range of frequency orders that can be calculated.

[0005] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: a method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct, comprising:

[0006] S1: Obtain the cross-sectional dimension information of the U-shaped aqueduct and calculate the target frequency order;

[0007] S2: Based on the target order of the calculated frequency, the frequency order coefficient of the lateral natural frequency of the water body in the U-shaped cross-section aqueduct is obtained by calculation;

[0008] S3: using the cross-sectional dimension information of the U-shaped aqueduct, the frequency order coefficient of the transverse natural frequency of the water body in the aqueduct of the U-shaped cross-sectional area, and the target order of the calculated frequency, to obtain the cross-sectional shape coefficient of the U-shaped cross-sectional area at a fixed order through calculation;

[0009] S4: Using the cross-sectional shape coefficient of the U-shaped cross-section at the fixed order and the target order of the calculation frequency, the transverse natural frequency of the water body in the U-shaped cross-section aqueduct at the corresponding order is determined through calculation, and the calculation of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct is completed.

[0010] Furthermore, the expression of the frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct is:

[0011]

[0012] Among them, k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, j represents the order of the natural frequency, J1 represents the first-order Bessel function, and n represents the summation subscript.

[0013] Furthermore, the expression of the cross-sectional shape coefficient of the U-shaped cross-section under the fixed order is:

[0014]

[0015]

[0016] Among them, a Aj represents the first cross-sectional shape coefficient of the U-shaped section under odd order, b Aj represents the second cross-sectional shape coefficient of the U-shaped section under odd order, c Aj represents the third cross-sectional shape coefficient of the U-shaped cross section under odd order, a Sj represents the first cross-sectional shape coefficient of the U-shaped section under even order, b Sj represents the second cross-sectional shape coefficient of the U-shaped section under even order, c Sj represents the third cross-sectional shape coefficient of the U-shaped section under even order, a represents the radius of the semicircular part of the U-shaped section, k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water body in the U-shaped section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water body in the U-shaped section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water body in the U-shaped section aqueduct, H represents the liquid level height of the rectangular part of the U-shaped section, and j represents the order of the natural frequency.

[0017] Furthermore, the expression of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct at the corresponding order is:

[0018]

[0019] in, represents the lateral natural frequency of the water in the U-shaped aqueduct at the corresponding order, a Aj represents the first cross-sectional shape coefficient of the U-shaped section under odd order, b Aj represents the second cross-sectional shape coefficient of the U-shaped section under odd order, c Ajrepresents the third cross-sectional shape coefficient of the U-shaped cross section under odd order, a Sj represents the first cross-sectional shape coefficient of the U-shaped section under even order, b Sj represents the second cross-sectional shape coefficient of the U-shaped section under even order, c Sj represents the third cross-sectional shape coefficient of the U-shaped cross-section under even orders, j represents the order of the natural frequency, and g represents the gravitational acceleration.

[0020] The beneficial effects of the present invention are as follows: (1) It provides a method for calculating the transverse natural frequency of water in a U-shaped aqueduct, which solves the problems of the existing method for calculating the transverse natural frequency of water in a U-shaped aqueduct, which is complex in calculation formula, large in calculation amount, and small in the range of frequency orders that can be calculated; (2) This method realizes the separation of the complex terms represented by the frequency order coefficient that are only related to the required frequency order when calculating the cross-sectional shape coefficient. When calculating the transverse natural frequency of water in a U-shaped aqueduct of different sizes, the frequency order coefficient only needs to be calculated once and can be applied to the calculation of the transverse natural frequency of water in a U-shaped aqueduct of any size. (3) This method is different from numerical methods such as the finite element method and the finite volume method. It is an analytical calculation method for the transverse natural frequency of water in a U-shaped aqueduct. Compared with numerical methods such as the finite element method and the finite volume method, it significantly improves the calculation speed and ensures the calculation accuracy, reduces the calculation cost of the transverse natural frequency of water in a U-shaped aqueduct, and meets the actual needs of the project. The transverse natural vibration frequency of the water body in the U-shaped aqueduct obtained by this method is in good agreement with the experimental and numerical simulation results, and can be used as a reference for analyzing the transverse natural vibration characteristics of the water body in the U-shaped aqueduct, which has practical engineering value. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] This specification will be further described in the form of exemplary embodiments, which will be described in detail by the accompanying drawings. These embodiments are not restrictive, and in these embodiments, the same number represents the same structure, wherein:

[0022] Figure 1 is an exemplary flow chart of a method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct according to some embodiments of this specification;

[0023] Figure 2 is an exemplary schematic diagram of a U-shaped aqueduct cross section according to some embodiments of the present specification. DETAILED DESCRIPTION

[0024] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0025] Example

[0026] In order to facilitate the understanding of the present application, the present application will be described more comprehensively with reference to the relevant drawings below.

[0027] The core idea of ​​the present invention: Based on the existing method for calculating the natural frequency of water bodies in two-dimensional containers of arbitrary cross-sectional shapes derived based on Rayleigh quotients, a method for calculating the transverse natural frequency of water bodies in U-shaped aqueducts is derived through Taylor series expansion and the introduction of mathematical means such as β functions and Γ functions. This method avoids the complex transcendental integral problem in the original method and converts it into an algebraic calculation formula, so that when calculating the cross-sectional shape coefficient, the complex terms represented by the frequency order coefficient that are only related to the required frequency order are separated. When calculating the transverse natural frequency of water bodies in U-shaped aqueducts of different sizes, the frequency order coefficient only needs to be calculated once and can be applied to the calculation of the transverse natural frequency of water bodies in U-shaped aqueducts of any size. This method significantly reduces the amount of calculation and can meet engineering needs.

[0028] Figure 1 This is an exemplary flow chart of a method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct according to some embodiments of this specification. Figure 1 As shown, the process includes the following steps.

[0029] S1: Obtain the U-shaped aqueduct cross-sectional size information and calculate the target frequency order.

[0030] The U-shaped aqueduct cross-sectional dimension information is information reflecting the cross-sectional size and capacity of the U-shaped aqueduct. Figure 2 As shown, the U-shaped aqueduct cross-sectional dimension information may include the width of the U-shaped cross-section, the liquid level height of the rectangular part of the U-shaped cross-section, and the radius of the semicircular part of the U-shaped cross-section.

[0031] In some embodiments, as shown in Table 1, the cross-sectional dimension information of the U-shaped aqueduct can be obtained from the design drawings.

[0032] Table 1 Cross-sectional dimensions of 35 U-shaped aqueducts with different cross-sectional dimensions.

[0033]

[0034] The target order of the calculated frequency is the target order of the natural frequency of the U-shaped aqueduct water body that needs to be calculated.

[0035] S2: Based on the target order of the calculated frequency, the frequency order coefficient of the lateral natural frequency of the water body in the U-shaped cross-section aqueduct is obtained by calculation.

[0036] The frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct is a correlation coefficient of the current frequency order. The frequency order coefficient includes the first frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, the second frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, and the third frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct.

[0037] In some embodiments, the frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct is expressed as:

[0038]

[0039] Among them, k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, j represents the order of the natural frequency, J1 represents the first-order Bessel function, and n represents the summation subscript.

[0040] The first four frequency order coefficients of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct are shown in Table 2.

[0041] Table 2 The first four frequency order coefficients of the lateral natural frequency of the water body in the U-shaped cross-section aqueduct.

[0042]

[0043] S3: Using the cross-sectional dimension information of the U-shaped aqueduct, the frequency order coefficient of the lateral natural frequency of the water body in the U-shaped aqueduct and the target order of the calculated frequency, the cross-sectional shape coefficient of the U-shaped cross-section at a fixed order is obtained through calculation.

[0044] The cross-sectional shape coefficient of the U-shaped cross-section at a fixed order may include the cross-sectional shape coefficient of the first U-shaped cross-section at an odd order, the cross-sectional shape coefficient of the second U-shaped cross-section at an odd order, the cross-sectional shape coefficient of the third U-shaped cross-section at an odd order, the cross-sectional shape coefficient of the first U-shaped cross-section at an even order, the cross-sectional shape coefficient of the second U-shaped cross-section at an even order, and the cross-sectional shape coefficient of the third U-shaped cross-section at an even order, etc.

[0045] The expression of the cross-sectional shape coefficient of the U-shaped cross section at a fixed order is:

[0046]

[0047] Among them, a Aj represents the first cross-sectional shape coefficient of the U-shaped section under odd order, b Aj represents the second cross-sectional shape coefficient of the U-shaped section under odd order, c Aj represents the third cross-sectional shape coefficient of the U-shaped cross section under odd order, a Sj represents the first cross-sectional shape coefficient of the U-shaped section under even order, b Sj represents the second cross-sectional shape coefficient of the U-shaped section under even order, c Sj represents the third cross-sectional shape coefficient of the U-shaped section under even order, a represents the radius of the semicircular part of the U-shaped section, k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water body in the U-shaped section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water body in the U-shaped section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water body in the U-shaped section aqueduct, H represents the liquid level height of the rectangular part of the U-shaped section, and j represents the order of the natural frequency.

[0048] Substituting the cross-sectional dimension information of 35 U-shaped aqueducts with different cross-sectional dimensions shown in Table 1 and the first four frequency order coefficients of the transverse natural frequency of the water body in the U-shaped aqueduct shown in Table 2 into the expression of the U-shaped cross-sectional shape coefficient, the cross-sectional shape coefficients of the U-shaped cross-sections required for the calculation of the first four transverse natural frequencies can be obtained.

[0049] S4: Using the cross-sectional shape coefficient of the U-shaped cross-section at the fixed order and the target order of the calculation frequency, the transverse natural frequency of the water body in the U-shaped cross-section aqueduct at the corresponding order is determined through calculation, and the calculation of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct is completed.

[0050] In some embodiments, the processor can use the cross-sectional shape coefficient of the U-shaped cross-section obtained at the first four orders and the target order of the calculated frequency to determine the transverse natural frequency of the water body in the U-shaped cross-section aqueduct at the corresponding order through calculation, and complete the calculation of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct.

[0051] The expression of the transverse natural frequency of the water body in the U-shaped aqueduct under the corresponding order is:

[0052]

[0053] in, represents the lateral natural frequency of the water in the U-shaped aqueduct at the corresponding order, a Aj represents the first cross-sectional shape coefficient of the U-shaped section under odd order, b Aj represents the second cross-sectional shape coefficient of the U-shaped section under odd order, cAj represents the third cross-sectional shape coefficient of the U-shaped cross section under odd order, a Sj represents the first cross-sectional shape coefficient of the U-shaped section under even order, b Sj represents the second cross-sectional shape coefficient of the U-shaped section under even order, c Sj represents the third cross-sectional shape coefficient of the U-shaped cross-section under even orders, j represents the order of the natural frequency, and g represents the gravitational acceleration.

[0054] For the 35 U-shaped aqueducts with different cross-sectional sizes in Table 1, the calculation results of the first four-order transverse natural frequencies of the water body and the calculation results of the CFD model as well as the difference between the two are shown in Table 3.

[0055] Table 3 Calculation results of the first four lateral natural frequencies of water obtained by this method And CFD model calculation results And the difference between the two

[0056]

[0057]

[0058]

[0059] It can be seen that among the 35 models, the maximum difference is 2.89% and the average difference is 0.87%. The calculation accuracy of this method is relatively high. In addition, the U-shaped aqueduct model with size information a = 0.1m, H = 0.015m is used to compare the difference between the calculation method and the model test results. Table 4 lists the calculation results of this calculation method and the model test results and the difference between the two. It can also be seen that the calculation accuracy of this method is relatively high.

[0060] Table 4 Calculation results of this method under the U-shaped aqueduct model with a = 0.1m, H = 0.015m And the model test results and the difference rate between the two.

[0061]

[0062] (1) A method for calculating the transverse natural frequency of water in a U-shaped aqueduct is provided. This method solves the problems of the existing method for calculating the transverse natural frequency of water in a U-shaped aqueduct, which is complex in calculation, large in calculation amount, and small in the range of frequency orders that can be calculated. (2) This method separates the complex terms represented by the frequency order coefficient that are only related to the required frequency order when calculating the cross-sectional shape coefficient. When calculating the transverse natural frequency of water in a U-shaped aqueduct of different sizes, the frequency order coefficient only needs to be calculated once and can be applied to the calculation of the transverse natural frequency of water in a U-shaped aqueduct of any size. (3) This method is different from numerical methods such as the finite element method and the finite volume method. It is an analytical calculation method for the transverse natural frequency of water in a U-shaped aqueduct. Compared with numerical methods such as the finite element method and the finite volume method, it significantly improves the calculation speed and ensures the calculation accuracy, reduces the calculation cost of the transverse natural frequency of water in a U-shaped aqueduct, and meets the actual needs of the project. The transverse natural vibration frequency of the water body in the U-shaped aqueduct obtained by this method is in good agreement with the experimental and numerical simulation results, and can be used as a reference for analyzing the transverse natural vibration characteristics of the water body in the U-shaped aqueduct, which has practical engineering value.

Claims

1. A method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct, characterized in that: include: S1: Obtain the cross-sectional dimension information of the U-shaped aqueduct and calculate the target frequency order; S2: Based on the target order of the calculated frequency, the frequency order coefficient of the lateral natural frequency of the water body in the U-shaped cross-section aqueduct is obtained by calculation; S3: using the cross-sectional dimension information of the U-shaped aqueduct, the frequency order coefficient of the transverse natural frequency of the water body in the aqueduct of the U-shaped cross-sectional area, and the target order of the calculated frequency, to obtain the cross-sectional shape coefficient of the U-shaped cross-sectional area at a fixed order through calculation; S4: Using the cross-sectional shape coefficient of the U-shaped cross-section at the fixed order and the target order of the calculation frequency, the transverse natural frequency of the water body in the U-shaped cross-section aqueduct at the corresponding order is determined through calculation, and the calculation of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct is completed.

2. The method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct according to claim 1, characterized in that: The expression of the frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct is: Among them, k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct, j represents the order of the natural frequency, J1 represents the first-order Bessel function, and n represents the summation subscript.

3. The method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct according to claim 1, characterized in that: The expression of the cross-sectional shape coefficient of the U-shaped cross section under the fixed order is: Among them, a Aj represents the first cross-sectional shape coefficient of the U-shaped section under odd order, b Aj represents the second cross-sectional shape coefficient of the U-shaped section under odd order, c Aj represents the third cross-sectional shape coefficient of the U-shaped cross section under odd order, a Sj represents the first cross-sectional shape coefficient of the U-shaped section under even order, b Sj represents the second cross-sectional shape coefficient of the U-shaped section under even order, c S j represents the third cross-sectional shape coefficient of the U-shaped section under even order, a represents the radius of the semicircular part of the U-shaped section, k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water body in the aqueduct of the U-shaped section, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water body in the aqueduct of the U-shaped section, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water body in the aqueduct of the U-shaped section, H represents the liquid level height of the rectangular part of the U-shaped section, and j represents the order of the natural frequency.

4. The method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct according to claim 1, characterized in that: The expression of the transverse natural frequency of the water body in the U-shaped cross-section aqueduct under the corresponding order is: in, represents the lateral natural frequency of the water in the U-shaped aqueduct at the corresponding order, a Aj represents the first cross-sectional shape coefficient of the U-shaped section under odd order, b Aj represents the second cross-sectional shape coefficient of the U-shaped section under odd order, c A j represents the third cross-sectional shape coefficient of the U-shaped cross section under odd order, a S j represents the first cross-sectional shape coefficient of the U-shaped cross section under even order, b Sj represents the second cross-sectional shape coefficient of the U-shaped section under even order, c Sj represents the third cross-sectional shape coefficient of the U-shaped cross-section under even orders, j represents the order of the natural frequency, and g represents the gravitational acceleration.

Images