A method for calculating broadband consistent damping force in time-domain dynamic analysis

Abstract

This invention discloses a method for calculating broadband uniform damping force in time-domain dynamic analysis. The method includes the following steps: constructing a finite element model of a hydropower plant; assigning material properties to the model; obtaining the initial stiffness matrix of the hydropower plant using the finite element method; idealizing the finite element model into an idealized model consisting of a main spring and multiple combined elements composed of a slug and a spring connected in parallel; calculating the stiffness coefficient of each combined element using the least squares method based on the initial stiffness matrix, the dynamic stiffness model of the combined elements, and the damping ratio in the time-domain dynamic analysis; calculating the viscosity coefficient of the combined elements based on their main frequency and the target damping ratio when the main frequency of the combined elements is uniformly distributed over a broadband range; and calculating the uniform damping force of the hydropower plant over a broadband range using a time-domain damping force model based on the stiffness coefficient and viscosity coefficient.

Description

Technical Field

[0001] This invention relates to structural safety assessment techniques, specifically to a method for calculating broadband consistent damping force in time-domain dynamic analysis. Background Technology

[0002] In the dynamic response analysis of structures, damping has a significant impact, directly determining the energy dissipation and attenuation behavior of the structure. Hydraulic powerhouses, especially those on land, are complex structures, comprising not only slabs, beams, and columns, but also large volumes of concrete such as gate piers, machine piers, and spiral casings. The natural frequencies of these components differ considerably. In time-history analysis, Rayleigh damping can only guarantee accurate damping ratios at two frequencies; other frequencies deviate significantly from the target damping ratio, leading to substantial biases in the dynamic response assessment of other structural components. Employing a damping model with a broadband, consistent damping ratio can significantly increase the accuracy of dynamic time-history analysis results for multi-component structures like hydropower plants, which is crucial for improving the reliability of structural safety assessments.

[0003] Currently, Rayleigh damping is commonly used in the dynamic time history calculation of hydropower plant structures. Rayleigh damping is a viscous damping model, and its damping parameters are simple and easy to integrate, thus leading to its widespread application. Rayleigh damping is the most widely used damping model in structural dynamic analysis, but its biggest drawback is that the damping ratio it describes is a frequency-dependent function, which does not reflect the actual situation where the damping ratio is independent of the load excitation frequency. The damping ratio directly determines the damping force, and the damping force directly determines the energy dissipation and attenuation behavior of the structure in the dynamic response, thus affecting the reliability of the dynamic response calculation results for hydropower plant structures.

[0004] Specific combination Figure 1 To explain, from Figure 1 It can be seen that when the frequency is ω i and ω j The target damping ratio ξ is determined by the time. aim When less than ω i and greater than ω j When the damping ratio is greater than ξ, the damping ratio is greater than ξ. aim When greater than ω i Less than ω j When the damping ratio is less than ξ aim This dependency is inconsistent with numerous experimental results, which show that damping force and test frequency are almost independent. Therefore, the Rayleigh damping model is unsuitable for dynamic time-history analysis of multi-component structures such as hydropower plants. Summary of the Invention

[0005] To address the aforementioned shortcomings in the existing technology, this invention provides a method for calculating broadband consistent damping force in time-domain dynamic analysis. This method solves the problem that existing hydropower plant dynamic time-history analysis is based on Rayleigh damping, which makes it difficult to maintain consistent damping force across a wide frequency range, resulting in low reliability of the calculated hydropower plant structure.

[0006] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows:

[0007] A method for calculating broadband uniform damping force in time-domain dynamic analysis is provided, comprising the following steps:

[0008] S1. Construct a finite element model of the hydropower plant, assign material properties to the model, and then use the finite element method to obtain the initial stiffness matrix of the hydropower plant.

[0009] S2. The finite element model is idealized into an idealized model consisting of a main spring and multiple combined elements composed of sticky pots and springs connected in parallel. The stiffness coefficient of the main spring is the initial stiffness matrix.

[0010] S3. Based on the initial stiffness matrix, the dynamic stiffness model of the combined elements, and the damping ratio in the time-domain dynamic analysis, the stiffness coefficient of each combined element is calculated using the least squares method.

[0011] S4. When the main frequency of the combined components is uniformly distributed over a wide frequency range, calculate the viscosity coefficient of the combined components based on the main frequency of the combined components and the target damping ratio.

[0012] S5. Based on the stiffness coefficient and viscosity coefficient, the time-domain damping force model is used to calculate the consistent damping force of the hydropower plant over a wide frequency range.

[0013] Furthermore, step S3 further includes:

[0014] S31. Determine the dynamic stiffness model related to the spring's stiffness coefficient:

[0015]

[0016] Where S(ω) is the dynamic stiffness of the hydropower plant unit; N is the number of combined components; α j τ is the stiffness coefficient of the j-th composite element; j Let ω be the relaxation time of the j-th combined element; ω be the dominant frequency; K be the initial stiffness matrix; and i be the imaginary unit.

[0017] S32. Within a wide frequency range, determine the relationship between each element in the matrix x and vector y based on the selected frequency:

[0018]

[0019] Where, x mn Let be the element in the m-th row and n-th column of matrix x, 1 ≦ m ≦ N, 1 ≦ n ≦ N; ω1 and ω2 are the lower and upper limits of the wideband range; ω m and ω n The frequency is randomly selected within a wide frequency range; ξ(·) is the damping ratio function; y n ξ is the nth element in vector y; aim Let be the target damping ratio; Im(·) be the imaginary part of the function; Re(·) be the real part of the function;

[0020] S33. Based on equations (1) to (4), the least squares method is used to solve the relationship between the stiffness coefficient and the matrix x and vector y, and the stiffness coefficient of each composite element is obtained. The relationship is as follows:

[0021] α opt =x -1 y

[0022] Where, α opt This is a matrix consisting of the stiffness coefficients of all the combined elements.

[0023] The beneficial effects of the above technical solution are as follows: This solution establishes the objective function of the damping ratio by defining the damping ratio and expressing it in terms of the stiffness and viscosity terms of the constructed equivalent element. This transforms the solution of the complex transcendental equation into a fitting problem, which significantly reduces the difficulty of the solution and improves the practicality of the method.

[0024] Furthermore, the expression for calculating the relaxation time is:

[0025] τ j =2ξ aim ω j ω j =ω1+(ω2-ω1) / (N-1)*(j-1)

[0026] Where, τ j α j and η j These represent the relaxation time, stiffness coefficient, and viscosity coefficient of the j-th composite element, respectively; ω j Let be the main frequency of the j-th combined element.

[0027] Furthermore, the expression for calculating the viscosity coefficient of the combined components is as follows:

[0028] η j =2ξ aim ω j α j K, ω j =ω1+(ω2-ω1) / (N-1)*(j-1)

[0029] Where, ω j ωj represents the main frequency of the j-th combined element; ω1 and ω2 are the lower and upper limits of the wideband range; N is the number of combined elements; ξ aim α is the target damping ratio; j and η j ...

[0030] The beneficial effects of the above technical solution are as follows: This solution provides a principle for selecting the fitting frequency, and selects points within the set cutoff frequency range, avoiding the blindness and arbitrariness of frequency selection. Since the selected frequency is closely related to the structural response of interest, it ensures that the obtained damping ratio can meet the structural dynamic response analysis.

[0031] Furthermore, the expression for the time-domain damping force model is:

[0032]

[0033] Where t is time; f(t) is the uniform damping force at time t; K is the initial stiffness matrix; N is the number of combined elements; τ j Let α be the relaxation time of the j-th combined element; j ω is the stiffness coefficient of the j-th composite element; ω is the dominant frequency; ω j The frequency of the j-th combined element; Let u(t) be the velocity of the hydropower plant at time t; u(t) be the displacement of the hydropower plant at time t.

[0034] The beneficial effects of the above technical solution are as follows: This solution provides an explicit expression of the time-domain damping force. Compared with the traditional form based on modal superposition or differential equations, each parameter has a clear physical meaning, the parameter determination is simpler and clearer, and the tedious solution is avoided, so that the damping force with a consistent damping ratio can be easily obtained when the structure is solved in the time-domain dynamic solution.

[0035] Furthermore, the combined elements are formed by connecting a glue pot and a spring in series, and the number of them is 5 to 7.

[0036] The beneficial effects of this invention are as follows: By introducing springs and sticky pot elements and combining multiple elements, this solution achieves an idealized representation of the elastic restoring force and damping force of the unit. By utilizing the dynamic stiffness model of the combined elements and the damping ratio in time-domain dynamic analysis, the optimal stiffness coefficient can be constructed through the least squares method. This achieves the goal of keeping the damping force basically consistent within the frequency range sensitive to the structural response in the time-domain analysis of the structure. In addition, since the damping force directly determines the energy dissipation and attenuation behavior of the structure in the dynamic response, the reliability of the dynamic response calculation results of the hydropower plant structure is ensured. Attached Figure Description

[0037] Figure 1 This is a schematic diagram illustrating the relationship between Rayleigh damping and frequency.

[0038] Figure 2 This is a flowchart of a method for calculating broadband consistent damping force in time-domain dynamic analysis.

[0039] Figure 3 This is a schematic diagram of the idealized model of this scheme.

[0040] Figure 4 This diagram illustrates how to obtain a broadband consistent damping ratio using this scheme. Detailed Implementation

[0041] 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.

[0042] refer to Figure 2 , Figure 2 A flowchart illustrating a method for calculating broadband consistent damping force in time-domain dynamic analysis is shown, as follows: Figure 2 As shown, the method S includes steps S1 to S5.

[0043] In step S1, a finite element model of the hydropower plant is constructed based on the geometric dimensions of the hydropower plant structure. Material properties are assigned to the model, and then the initial stiffness matrix of the hydropower plant is obtained using the finite element method. The material properties include information such as elastic modulus, Poisson's ratio, and density.

[0044] In step S2, the finite element model is idealized into an idealized model consisting of a main spring and multiple combined elements composed of glue pots and springs connected in parallel; the combined elements are formed by glue pots and springs connected in series, and the number of them is preferably 5 to 7.

[0045] Idealized models such as Figure 3 As shown, the stiffness of the main spring is the initial stiffness matrix of the hydroelectric powerhouse, and the stiffness of the combined element is the product of its stiffness coefficient and the initial stiffness matrix, where the stiffness coefficient α of the spring is... i Both the viscosity coefficients of the sticky pot and the viscosity coefficients of the sticky pot are undetermined coefficients that need to be solved.

[0046] In step S3, the stiffness coefficient of each composite element is calculated using the least squares method based on the initial stiffness matrix, the dynamic stiffness model of the composite element, and the damping ratio in the time-domain dynamic analysis.

[0047] In one embodiment of the present invention, step S3 further includes:

[0048] S31. Determine the dynamic stiffness model related to the spring's stiffness coefficient:

[0049]

[0050] Where S(ω) is the dynamic stiffness of the hydropower plant unit; N is the number of combined components; α j τ is the stiffness coefficient of the j-th composite element; j Let ω be the relaxation time of the j-th combined element; ω be the dominant frequency; K be the initial stiffness matrix; and i be the imaginary unit.

[0051] S32. Within a wide frequency range, determine the relationship between each element in the matrix x and vector y based on the selected frequency:

[0052]

[0053]

[0054] Where, x mn Let be the element in the m-th row and n-th column of matrix x, 1 ≦ m ≦ N, 1 ≦ n ≦ N; ω1 and ω2 are the lower and upper limits of the wideband range; ω m and ω n The frequency is randomly selected within a wide frequency range; ξ(·) is the damping ratio function; y n ξ is the nth element in vector y; aim Let be the target damping ratio; Im(·) be the imaginary part of the function; Re(·) be the real part of the function;

[0055] S33. Based on equations (1) to (4), the least squares method is used to solve the relationship between the stiffness coefficient and the matrix x and vector y, and the stiffness coefficient of each composite element is obtained. The relationship is as follows:

[0056] α opt =x -1 y

[0057] Where, α opt This is a matrix consisting of the stiffness coefficients of all the combined elements.

[0058] In implementation, the preferred expression for calculating the relaxation time in this scheme is:

[0059] τ j =2ξ aim ω j ω j =ω1+(ω2-ω1) / (N-1)*(j-1)

[0060] Where, τ j α j and η j These represent the relaxation time, stiffness coefficient, and viscosity coefficient of the j-th composite element, respectively; ω j Let be the main frequency of the j-th combined element.

[0061] In step S4, when the main frequency of the combined element is uniformly distributed over a wide frequency range, the viscosity coefficient of the combined element is calculated based on the main frequency and the target damping ratio.

[0062] η j =2ξ aim ω j α j K, ω j =ω1+(ω2-ω1) / (N-1)*(j-1)

[0063] Where, ω j ωj represents the main frequency of the j-th combined element; ω1 and ω2 are the lower and upper limits of the wideband range; N is the number of combined elements; ξ aim α is the target damping ratio; j and η j ...

[0064] In step S5, based on the stiffness coefficient and viscosity coefficient, the uniform damping force of the hydropower plant over a wide frequency range is calculated using a time-domain damping force model. The expression for the time-domain damping force model is as follows:

[0065]

[0066] Where t is time; f(t) is the uniform damping force at time t; K is the initial stiffness matrix; N is the number of combined elements; τ j Let α be the relaxation time of the j-th combined element; j ω is the stiffness coefficient of the j-th composite element; ω is the dominant frequency; ω j The frequency of the j-th combined element; Let u(t) be the velocity of the hydropower plant at time t; u(t) be the displacement of the hydropower plant at time t.

[0067] The following calculations, based on the methods used in this scheme, calculate the broadband uniform damping ratio of the hydropower plant:

[0068] In this embodiment, the target damping ratio for the seismic analysis of a hydropower plant is set to 10%, and the calculation frequency is selected from 1 to 100 Hz, with frequency intervals of 1 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, and 100 Hz. The plant is simulated using three-dimensional solid elements, with the concrete material having an elastic modulus of 28 GPa, a Poisson's ratio of 0.2, and a unit weight of 24 kN / m³. 3 The specific implementation plan involves establishing a finite element model according to the broadband consistent damping force calculation method of this scheme, selecting the above-mentioned frequency intervals, calculating the dynamic stiffness at each frequency point, obtaining each parameter through least squares fitting, and substituting it into the damping ratio expression to obtain the damping ratio at each point. The result is plotted along the frequency as shown below. Figure 4 As shown.

[0069] The uniform damping force calculated using the method in this scheme is as follows: Figure 4 As shown, from Figure 4 It can be seen that the damping ratio remains basically consistent within the range of 1Hz to 100Hz. Therefore, using this method for damping force calculation is beneficial for ensuring the reliability of dynamic time-history analysis of complex multi-component structures.

Claims

1. A method for calculating broadband uniform damping force in time-domain dynamic analysis, characterized in that, Including the following steps: S1. Construct a finite element model of the hydropower plant, assign material properties to the model, and then use the finite element method to obtain the initial stiffness matrix of the hydropower plant. S2. The finite element model is idealized into an idealized model consisting of a main spring and multiple combined elements composed of glue pots and springs connected in parallel; S3. Based on the initial stiffness matrix, the dynamic stiffness model of the combined elements, and the damping ratio in the time-domain dynamic analysis, the stiffness coefficient of each combined element is calculated using the least squares method. S4. When the main frequency of the combined components is uniformly distributed over a wide frequency range, calculate the viscosity coefficient of the combined components based on the main frequency of the combined components and the target damping ratio. S5. Based on the stiffness coefficient and viscosity coefficient, the time-domain damping force model is used to calculate the consistent damping force of the hydropower plant over a wide frequency range. Step S3 further includes: S31. Determine the dynamic stiffness model related to the spring's stiffness coefficient: in, N represents the dynamic stiffness of the hydropower plant unit; N represents the number of combined components. Let be the stiffness coefficient of the j-th composite element; Let be the relaxation time of the j-th combined element; The dominant frequency; K is the initial stiffness matrix; i is the imaginary unit. ; S32. Within a wide frequency range, determine the matrix for calculating the damping ratio based on the selected frequency. and vector The relational expression for each element in the formula: in, For matrix The element in the m-th row and n-th column of the array, 1≦m≦N, 1≦n≦N; and These are the lower and upper limits of the wideband range; and The frequency is randomly selected within a wide frequency range; The damping ratio function; For vectors The nth element in; The target damping ratio; The imaginary part of the function; The real part of the function; S33. According to equations (1) to (4), the least squares method is used to compare the stiffness coefficient with the matrix. and vector Solving the relationship yields the stiffness coefficient of each composite element, as shown in the following formula: in, This is a matrix consisting of the stiffness coefficients of all the combined elements.

2. The method for calculating broadband uniform damping force in time-domain dynamic analysis according to claim 1, characterized in that, The expression for calculating relaxation time is: ， in, and These are the relaxation time and stiffness coefficient of the j-th composite element, respectively. Let be the main frequency of the j-th combined element.

3. The method for calculating broadband uniform damping force in time-domain dynamic analysis according to claim 1, characterized in that, The expression for calculating the viscosity coefficient of the combined components is: ， in, The frequency of the j-th combined element; and Here, N represents the lower and upper limits of the wideband range; N is the number of combined components. The target damping ratio; and ...

4. The method for calculating broadband uniform damping force in time-domain dynamic analysis according to claim 1, characterized in that, The expression for the time-domain damping force model is: Where t is time; Let be the uniform damping force at time t; K be the initial stiffness matrix; and N be the number of combined elements. Let be the relaxation time of the j-th combined element; Let be the stiffness coefficient of the j-th composite element; Main frequency; The frequency of the j-th combined element; Let t be the velocity of the hydroelectric power plant at time t; Let t be the displacement of the hydropower plant at time t.

5. The method for calculating broadband consistent damping force in time-domain dynamic analysis according to any one of claims 1-4, characterized in that, The combined components are formed by connecting a glue pot and a spring in series, and the number of them is 5 to 7.

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