A method for calculating the transverse self-vibration frequency of water in a U-shaped section aqueduct

Abstract

This invention provides a method for calculating the transverse natural frequency of water within a U-shaped cross-section aqueduct, relating to the field of aqueduct seismic resistance analysis technology. The method comprises: obtaining the cross-sectional dimensions of the U-shaped aqueduct and the target frequency order; calculating the frequency order coefficient of the transverse natural frequency of the water within the U-shaped cross-section aqueduct based on the target frequency order; calculating the cross-sectional shape factor of the U-shaped cross-section at a fixed order using the cross-sectional dimensions of the U-shaped aqueduct; and determining the transverse natural frequency of the water within the U-shaped cross-section aqueduct at the corresponding order using the cross-sectional shape factor of the U-shaped cross-section at the fixed order and the target frequency order, thus completing the calculation of the transverse natural frequency of the water within the U-shaped cross-section aqueduct. This invention solves the problems of existing methods for calculating the transverse natural frequency of water within U-shaped aqueducts, which involve complex formulas, large computational loads, and a limited range of calculable frequency orders.

Description

Technical Field

[0001] This specification relates to the field of seismic resistance analysis technology for aqueducts, and in particular to a method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct. Background Technology

[0002] Aqueducts, widely used water conveyance structures in water diversion projects, differ from ordinary bridges primarily in that they carry a large volume of water. This results in the superstructure and supporting structures (frames, piers, trusses, towers, and cables, etc.) having a mass comparable to the water within the aqueduct. Under seismic loads, ground motion causes movement in the aqueduct via the supporting structures. This movement in turn causes sloshing of the water within the aqueduct, which in turn affects the movement of the aqueduct and its supporting structures. Consequently, seismic calculations for aqueducts constitute a highly complex dynamic problem. In this context, obtaining the various transverse natural frequencies of the water within the aqueduct is crucial for aqueduct design and sloshing control.

[0003] Existing methods for calculating the transverse natural frequency of water in U-shaped aqueducts have drawbacks such as complex formulas, large computational load, and a small range of frequency orders that can be calculated. Summary of the Invention

[0004] To address the aforementioned shortcomings in existing technologies, this invention provides a method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct. This method solves the problems of existing methods for calculating the transverse natural frequency of water in a U-shaped aqueduct, which suffer from complex formulas, large computational load, and a small range of possible frequency orders.

[0005] To achieve the aforementioned objectives, the technical solution adopted by this invention is as follows: a method for calculating the transverse natural frequency of water in a U-shaped cross-section aqueduct, comprising:

[0006] S1: Obtain the cross-sectional dimensions 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 transverse natural frequency of the water in the U-shaped cross-section aqueduct is obtained through calculation.

[0008] S3: Using the cross-sectional dimensions of the U-shaped aqueduct, the frequency order coefficient of the transverse natural frequency of the water in the U-shaped aqueduct, and the target order of the calculated frequency, the cross-sectional shape coefficient of the U-shaped cross section under a fixed order is obtained through calculation.

[0009] S4: Using the cross-sectional shape factor of the U-shaped cross section under the fixed order and the target order of the calculated frequency, the transverse natural frequency of the water in the U-shaped cross section aqueduct under the corresponding order is determined by calculation, thus completing the calculation of the transverse natural frequency of the water in the U-shaped cross section aqueduct.

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

[0011]

[0012] Where k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water 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 for the cross-sectional shape factor of the U-shaped cross-section at the fixed order is:

[0014]

[0015]

[0016] Among them, a Aj b represents the first section shape factor of the U-shaped section in odd-order cases. Aj c represents the second section shape factor of the U-shaped section in odd-order cases. Aj a represents the third section shape factor of the U-shaped cross-section for odd orders. Sj b represents the first section shape factor of the U-shaped section in even-order cases. Sj c represents the second section shape factor of the U-shaped section in even-order cases. Sj Let represent the third cross-sectional shape factor of the U-shaped cross-section at even-numbered orders, 'a' represent the radius of the semicircular part of the U-shaped cross-section, 'k1(j)' represent the first frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, 'k2(j)' represent the second frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, 'k3(j)' represent the third frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, 'H' represent the height of the liquid surface in the rectangular part of the U-shaped cross-section, and 'j' represent the order of the natural frequency.

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

[0018]

[0019] in, This represents the transverse natural frequency of the water within the U-shaped cross-section aqueduct at the corresponding order, a. Aj b represents the first section shape factor of the U-shaped section in odd-order cases. Aj c represents the second section shape factor of the U-shaped section in odd-order cases. Aja represents the third section shape factor of the U-shaped cross-section for odd orders. Sj b represents the first section shape factor of the U-shaped section in even-order cases. Sj c represents the second section shape factor of the U-shaped section in even-order cases. Sj denoted by , j represents the order of the natural frequency of the U-shaped cross section, and g represents the gravitational acceleration.

[0020] The beneficial effects of this invention are as follows: (1) It provides a method for calculating the transverse natural frequency of water in a U-shaped aqueduct. This method solves the problems of complex calculation formulas, large calculation volume, and small range of frequency orders that can be calculated in existing methods for calculating the transverse natural frequency of water in a U-shaped aqueduct. (2) This method separates the complex terms represented by the frequency order coefficient, which are only related to the frequency order to be calculated, 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 engineering. The transverse natural frequencies of the water in the U-shaped aqueduct obtained by this method are in good agreement with the experimental and numerical simulation results. They can be used as a reference for analyzing the transverse natural vibration characteristics of the water in the U-shaped aqueduct and have practical engineering value. Attached Figure Description

[0021] This specification will be further described by way of exemplary embodiments, which will be described in detail with reference to the accompanying drawings. These embodiments are not limiting; in these embodiments, the same reference numerals denote the same structures, wherein:

[0022] Figure 1 This is an exemplary flowchart illustrating 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 This is an exemplary schematic diagram of the cross-section of a U-shaped aqueduct according to some embodiments of this specification. Detailed Implementation

[0024] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0025] Example

[0026] To facilitate understanding of this application, a more comprehensive description of this application will be provided below with reference to the accompanying drawings.

[0027] The core idea of ​​this invention is as follows: Building upon existing methods for calculating the natural frequencies of water in two-dimensional containers with arbitrary cross-sectional shapes, derived based on Rayleigh's quotient, this invention derives a method for calculating the transverse natural frequencies of water in U-shaped aqueducts through Taylor series expansion and the introduction of mathematical techniques such as the β and Γ functions. This method avoids the complex transcendental integral problem in the original method, transforming it into an algebraic formula. This allows for the separation of complex terms, represented by the frequency order coefficients, that are only related to the desired frequency order when calculating the cross-sectional shape coefficients. When calculating the transverse natural frequencies of water in U-shaped aqueducts of different sizes, the frequency order coefficients only need to be calculated once, making this method applicable to the calculation of transverse natural frequencies of water in U-shaped aqueducts of any size. This method significantly reduces the computational load and meets engineering requirements.

[0028] Figure 1 This is an exemplary flowchart illustrating 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 cross-sectional dimensions of the U-shaped aqueduct and calculate the target frequency order.

[0030] The cross-sectional dimensions of a U-shaped aqueduct reflect its size and capacity. For example, such as... Figure 2 As shown, the cross-sectional dimensions of the U-shaped aqueduct may include the width of the U-shaped cross-section, the liquid level height of the rectangular portion of the U-shaped cross-section, and the radius of the semi-circular portion of the U-shaped cross-section.

[0031] In some embodiments, as shown in Table 1, the cross-sectional dimensions 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 sizes.

[0033]

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

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

[0036] The frequency order coefficients of the transverse natural frequency of the water in the U-shaped cross-section aqueduct are correlation coefficients for the current frequency order. The frequency order coefficients include the first frequency order coefficient, the second frequency order coefficient, and the third frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct.

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

[0038]

[0039] Where k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water 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 order coefficients of the transverse natural frequency of the water in the U-shaped cross-section aqueduct are shown in Table 2.

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

[0042]

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

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

[0045] The expression for the section shape factor of a U-shaped section at a fixed order is:

[0046]

[0047] Among them, a Aj b represents the first section shape factor of the U-shaped section in odd-order cases. Aj c represents the second section shape factor of the U-shaped section in odd-order cases. Aj a represents the third section shape factor of the U-shaped cross-section for odd orders. Sj b represents the first section shape factor of the U-shaped section in even-order cases. Sj c represents the second section shape factor of the U-shaped section in even-order cases. Sj Let represent the third cross-sectional shape factor of the U-shaped cross-section at even-numbered orders, 'a' represent the radius of the semicircular part of the U-shaped cross-section, 'k1(j)' represent the first frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, 'k2(j)' represent the second frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, 'k3(j)' represent the third frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, 'H' represent the height of the liquid surface in the rectangular part of the U-shaped cross-section, and 'j' represent the order of the natural frequency.

[0048] By substituting the cross-sectional dimensions of the 35 different U-shaped aqueducts shown in Table 1 and the first four order frequency coefficients of the transverse natural frequencies of the water in the U-shaped aqueducts shown in Table 2 into the expression for the U-shaped cross-section shape coefficient, the cross-sectional shape coefficients of each section of the U-shaped cross-section required for calculating the first four transverse natural frequencies can be obtained.

[0049] S4: Using the cross-sectional shape factor of the U-shaped cross section under the fixed order and the target order of the calculated frequency, the transverse natural frequency of the water in the U-shaped cross section aqueduct under the corresponding order is determined by calculation, thus completing the calculation of the transverse natural frequency of the water in the U-shaped cross section aqueduct.

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

[0051] The expression for the transverse natural frequency of the water in the U-shaped cross-section aqueduct at the corresponding order is:

[0052]

[0053] in, This represents the transverse natural frequency of the water within the U-shaped cross-section aqueduct at the corresponding order, a. Aj b represents the first section shape factor of the U-shaped section in odd-order cases. Aj c represents the second section shape factor of the U-shaped section in odd-order cases.Aj a represents the third section shape factor of the U-shaped cross-section for odd orders. Sj b represents the first section shape factor of the U-shaped section in even-order cases. Sj c represents the second section shape factor of the U-shaped section in even-order cases. Sj denoted by , j represents the order of the natural frequency of the U-shaped cross section, and g represents the gravitational acceleration.

[0054] Table 3 shows the calculated results of the first four transverse natural frequencies of the water body and the CFD model calculation results for the 35 U-shaped aqueducts with different cross-sectional dimensions in Table 1, as well as the differences between the two.

[0055] Table 3 shows the calculation results of the first four transverse natural frequencies of water obtained by this method. and CFD model calculation results and the difference rate between the two

[0056]

[0057]

[0058] Therefore, among the 35 models, the maximum difference was 2.89%, and the average difference was 0.87%, indicating that the calculation accuracy of this method is relatively high. Furthermore, a U-shaped aqueduct model with dimensions a = 0.1m and H = 0.015m was used to compare the differences between the calculation method and the model test results. Table 4 lists the calculation results of this method and the model test results, as well as the differences between the two, further demonstrating the high calculation accuracy of this method.

[0059] Table 4 shows the calculation results of this method under the U-shaped aqueduct model with a = 0.1m and H = 0.015m. and model test results And the difference rate between the two.

[0060]

[0061] (1) A method for calculating the transverse natural frequency of water in a U-shaped aqueduct is provided. This method solves the problems of complex calculation formulas, large calculation volume, and small range of frequency orders that can be calculated in existing methods for calculating the transverse natural frequency of water in a U-shaped aqueduct. (2) This method separates the complex terms represented by the frequency order coefficient, which are only related to the frequency order to be calculated, when calculating the cross-sectional shape factor. When calculating the transverse natural frequency of water 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 in U-shaped aqueducts 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 U-shaped aqueducts. Compared with numerical methods such as the finite element method and the finite volume method, it significantly improves the calculation speed, ensures the calculation accuracy, and reduces the calculation cost of the transverse natural frequency of water in U-shaped aqueducts, which meets the actual needs of engineering. The transverse natural frequencies of the water in the U-shaped aqueduct obtained by this method are in good agreement with the experimental and numerical simulation results. They can be used as a reference for analyzing the transverse natural vibration characteristics of the water in the U-shaped aqueduct and have 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 dimensions 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 transverse natural frequency of the water in the U-shaped cross-section aqueduct is obtained through calculation. S3: Using the cross-sectional dimensions of the U-shaped aqueduct, the frequency order coefficient of the transverse natural frequency of the water in the U-shaped aqueduct, and the target order of the calculated frequency, the cross-sectional shape coefficient of the U-shaped cross section under a fixed order is obtained through calculation. S4: Using the cross-sectional shape factor of the U-shaped cross section under the fixed order and the target order of the calculated frequency, the transverse natural frequency of the water in the U-shaped cross section aqueduct under the corresponding order is determined by calculation, thus completing the calculation of the transverse natural frequency of the water in the U-shaped cross section aqueduct.

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 for the frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct is as follows: Where k1(j) represents the first frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k2(j) represents the second frequency order coefficient of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k3(j) represents the third frequency order coefficient of the transverse natural frequency of the water 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 for the cross-sectional shape factor of the U-shaped cross-section at the fixed order is: Among them, a Aj b represents the first section shape factor of the U-shaped section in odd-order cases. Aj c represents the second section shape factor of the U-shaped section in odd-order cases. Aj a represents the third section shape factor of the U-shaped cross-section for odd orders. Sj b represents the first section shape factor of the U-shaped section in even-order cases. Sj c represents the second section shape factor of the U-shaped section in even-order cases. S j represents the shape factor of the third section of the U-shaped cross-section at even-numbered orders, a represents the radius of the semicircular part of the U-shaped cross-section, k1(j) represents the first frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k2(j) represents the second frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, k3(j) represents the third frequency order factor of the transverse natural frequency of the water in the U-shaped cross-section aqueduct, H represents the liquid level height of the rectangular part of the U-shaped cross-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 for the transverse natural frequency of the water in the U-shaped cross-section aqueduct at the corresponding order is: in, This represents the transverse natural frequency of the water within the U-shaped cross-section aqueduct at the corresponding order, a. Aj b represents the first section shape factor of the U-shaped section in odd-order cases. Aj c represents the second section shape factor of the U-shaped section in odd-order cases. A j represents the shape factor of the third section of the U-shaped cross-section in odd-order cases, a S j represents the first section shape factor of the U-shaped section at even-order values, b Sj c represents the second section shape factor of the U-shaped section in even-order cases. Sj denoted by , j represents the order of the natural frequency of the U-shaped cross section, and g represents the gravitational acceleration.

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