Calculation method of broadband consistent damping force in time domain dynamic analysis

Abstract

The invention discloses a method for calculating broadband consistent damping force in time-domain dynamic analysis, which comprises the following steps of: constructing a finite element model of a hydroelectric plant, endowing the model with material attributes, and then obtaining an initial stiffness matrix of the hydroelectric plant by adopting a finite element method; idealizing the finite element model as a main spring, and connecting a plurality of combined elements consisting of sticky kettles and springs in parallel to form an idealized model; according to the initial stiffness matrix, the dynamic stiffness model of the combined element and a damping ratio in time domain dynamic analysis, calculating a stiffness coefficient of each combined element by adopting a least square method; when the dominant frequency of the combined element is uniformly distributed in the broadband range, calculating the viscosity coefficient of the combined element according to the dominant frequency of the combined element and the target damping ratio; and according to the rigidity coefficient and the viscosity coefficient, a time domain damping force model is adopted to calculate the consistent damping force of the hydroelectric plant in the broadband range.

Description

Technical Field

[0001] The present invention relates to a structural safety assessment technology, and in particular to a calculation method of broadband consistent damping force in time-domain dynamic analysis. Background Art

[0002] In the dynamic response analysis of structures, damping has an important influence and directly determines the energy consumption and attenuation behavior of the structure. The structure of a hydraulic ground-type powerhouse is complex, including components such as plates, beams, and columns, as well as large-volume concrete such as gate piers, machine piers, and volutes. The natural frequencies of each component vary greatly. In time-history analysis, the use of Rayleigh damping can only ensure that the damping ratio at two frequency points is accurate. The other frequencies differ greatly from the target damping ratio, which will lead to large deviations in the dynamic response assessment of other structural components. The use of a damping model with a wide-band consistent damping ratio can significantly increase the accuracy of the dynamic time-history analysis results of multi-component structures such as hydropower plants, which is of great significance to improving the reliability of structural safety evaluation.

[0003] Currently, Rayleigh damping is commonly used in dynamic time-history calculations of hydropower plant structures. Rayleigh damping is a viscous damping model with simple damping parameters and easy integration, leading to its widespread application. Rayleigh damping is the most widely used damping model in structural dynamic analysis, but its major drawback is that the damping ratio it describes is a frequency-dependent function, which does not conform to the actual situation where the damping ratio is independent of the load excitation frequency. The damping ratio directly determines the damping force, which in turn directly determines the energy dissipation and attenuation behavior of the structure during dynamic response, thus affecting the reliability of the dynamic response calculation results of 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 aim , when less than ω i and greater than ω j When 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 dependence is inconsistent with numerous test results, which show that the damping force and test frequency are almost unrelated. Therefore, the Rayleigh damping model is not suitable for dynamic time history analysis of multi-component structures such as hydropower plants. Summary of the Invention

[0005] In response to the above-mentioned deficiencies in the prior art, the present invention provides a method for calculating broadband consistent damping force in time-domain dynamic analysis, which solves the problem that the existing hydropower plant dynamic time-history analysis is based on Rayleigh damping, and the calculated damping force is difficult to remain consistent within a wide frequency range, resulting in low reliability of the calculated hydropower plant structure.

[0006] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0007] A method for calculating broadband consistent damping force in time-domain dynamic analysis is provided, which comprises 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. Idealizing the finite element model into an idealized model consisting of a main spring and a plurality of combination elements consisting of a viscometer and a spring in parallel, wherein the stiffness coefficient of the main spring is the initial stiffness matrix;

[0010] S3. Calculate the stiffness coefficient of each composite element 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;

[0011] S4. When the main frequency of the combined element is evenly distributed in a wide frequency range, the viscosity coefficient of the combined element is calculated based on the main frequency of the combined element 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 in a wide frequency range.

[0013] Furthermore, step S3 further includes:

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

[0015]

[0016] Where S(ω) is the dynamic stiffness of the hydropower plant unit; N is the number of combined elements; α j is the stiffness coefficient of the jth combined unit; τ j is the relaxation time of the jth composite element; ω is the main frequency; K is the initial stiffness matrix; i is the imaginary unit

[0017] S32. In a wide frequency range, according to the selected frequency, determine and calculate the relationship between each element in the matrix x and the vector y based on the damping ratio:

[0018]

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

[0020] S33. According to equations (1) to (4), the least square method is used to solve the relationship between the stiffness coefficient and the matrix x and the vector y to obtain the stiffness coefficient of each combined element. The relationship is:

[0021] α opt =x -1 y

[0022] Among them, α opt is the matrix of stiffness coefficients of all combined elements.

[0023] The beneficial effects of the above technical solution are as follows: this solution defines the damping ratio and expresses it with the stiffness term and viscosity term of the constructed equivalent unit, thereby establishing the objective function of the damping ratio and converting the solution of the complex transcendental equation into a fitting problem, which significantly reduces the difficulty of solution and improves the practicality of the method.

[0024] Furthermore, the calculation expression of relaxation time is:

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

[0026] Among them, τ j , α j and η j are the relaxation time, stiffness coefficient and viscosity coefficient of the j-th combined unit respectively; ω j is the main frequency of the jth combination element.

[0027] Furthermore, the expression for calculating the viscosity coefficient of the composite element is:

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

[0029] Among them, ω j is the main frequency of the jth combination element; ω1 and ω2 are the lower and upper limits of the wide frequency range; N is the number of combination elements; ξ aim is the target damping ratio; α j and η j are the stiffness coefficient and viscosity coefficient of the j-th combined unit respectively; K is the initial stiffness matrix.

[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, thereby 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 of 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 composite elements; τ j is the relaxation time of the jth composite element; α j is the stiffness coefficient of the j-th combined unit; ω is the main frequency; ω j is the main frequency of the j-th combination element; is the velocity of the hydropower plant at time t; u(t) is 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 damping force in the time domain. 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 tedious solution is avoided, so that the damping force with a consistent damping ratio can be easily obtained when solving the structure in the time domain dynamics.

[0035] Furthermore, the combination element is formed by connecting a sticky pot and a spring in series, and the number of the combination element is 5 to 7.

[0036] The beneficial effects of the present invention are as follows: this scheme realizes an idealized representation of the unit elastic restoring force and damping force by introducing spring and viscoelastic pot elements and combining multiple elements. By utilizing the dynamic stiffness model of the combined elements and the damping ratio in the time-domain dynamic analysis, the optimal stiffness coefficient can be constructed through the least squares method, thereby achieving 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 consumption and attenuation behavior of the structure in the dynamic response, the reliability of the calculation results of the dynamic response of the hydropower plant structure is ensured. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 Schematic diagram of the relationship between Rayleigh damping and frequency.

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

[0039] Figure 3 Schematic diagram of the idealized model of this scheme.

[0040] Figure 4 Schematic diagram of obtaining a broadband consistent damping ratio using this scheme. DETAILED DESCRIPTION

[0041] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

[0042] refer to Figure 2 , Figure 2 A flow chart showing a method for calculating broadband consistent damping force in time domain dynamic analysis is shown in FIG. 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, and material properties are assigned to the model. Then, the finite element method is used to obtain the initial stiffness matrix of the hydropower plant; the material properties include elastic modulus, Poisson's ratio, density and other information.

[0044] In step S2, the finite element model is idealized as an idealized model consisting of a main spring and multiple combination elements consisting of a sticky pot and a spring in parallel; the combination elements are formed by connecting the sticky pot and the spring in series, and the number of the combination elements 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 hydropower plant, and the stiffness of the combined element is the product of its stiffness coefficient and the initial stiffness matrix, where the stiffness coefficient of the spring is α i The viscosity coefficient of the sticky pot is an unknown coefficient that needs to be solved.

[0046] In step S3, the stiffness coefficient of each composite element is calculated using the least squares method according to 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 stiffness coefficient:

[0049]

[0050] Where S(ω) is the dynamic stiffness of the hydropower plant unit; N is the number of combined elements; α j is the stiffness coefficient of the jth combined unit; τ j is the relaxation time of the jth composite element; ω is the main frequency; K is the initial stiffness matrix; i is the imaginary unit

[0051] S32. In a wide frequency range, according to the selected frequency, determine and calculate the relationship between each element in the matrix x and the vector y based on the damping ratio:

[0052]

[0053]

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

[0055] S33. According to equations (1) to (4), the least square method is used to solve the relationship between the stiffness coefficient and the matrix x and the vector y to obtain the stiffness coefficient of each combined element. The relationship is:

[0056] α opt =x -1 y

[0057] Among them, α opt is the matrix of stiffness coefficients of all combined elements.

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

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

[0060] Among them, τ j , α j and η j are the relaxation time, stiffness coefficient and viscosity coefficient of the j-th combined unit respectively; ω j is the main frequency of the jth combination element.

[0061] In step S4, when the main frequency of the combined element is evenly distributed in a wide frequency range, the viscosity coefficient of the combined element is calculated according to the main frequency of the combined element 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] Among them, ω j is the main frequency of the jth combination element; ω1 and ω2 are the lower and upper limits of the wide frequency range; N is the number of combination elements; ξ aim is the target damping ratio; α j and η j are the stiffness coefficient and viscosity coefficient of the j-th combined unit respectively; K is the initial stiffness matrix.

[0064] In step S5, the time domain damping force model is used to calculate the uniform damping force of the hydropower plant in a wide frequency range based on the stiffness coefficient and the viscosity coefficient. The expression of the time domain damping force model is:

[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 composite elements; τ j is the relaxation time of the jth composite element; α j is the stiffness coefficient of the j-th combined unit; ω is the main frequency; ω j is the main frequency of the j-th combination element; is the velocity of the hydropower plant at time t; u(t) is the displacement of the hydropower plant at time t.

[0067] The following calculation method of this scheme is combined with the calculation of the broadband consistent damping ratio of the hydropower plant:

[0068] This example sets the target damping ratio for the seismic analysis of a hydropower plant to 10%, selects the calculation frequency to be 1-100Hz, and the frequency intervals are 1Hz, 10Hz, 20Hz, 30Hz, 40Hz, 50Hz, 60Hz, 70Hz, 80Hz, 90Hz, and 100Hz. The plant is simulated using three-dimensional solid elements, and the concrete material has an elastic modulus of 28GPa, a Poisson's ratio of 0.2, and a bulk density of 24kN / m 3 The specific implementation plan is to establish a finite element model according to the calculation method of broadband uniform damping force in this plan, select the above frequency interval, calculate the dynamic stiffness of each frequency point, use the least squares method to fit each parameter, and put it into the expression of damping ratio to obtain the damping ratio of each point. After plotting along the frequency, it is as follows Figure 4 shown.

[0069] The uniform damping force calculated by the method of this solution 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. This shows that the damping force calculation method of this solution is beneficial for ensuring the reliability of the dynamic time history analysis of complex structures with multiple components.

Claims

1. A method for calculating broadband consistent damping force in time domain dynamic analysis, characterized in that: Including 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. Idealize the finite element model into an idealized model consisting of a main spring and multiple combination elements consisting of sticky pots and springs connected in parallel; S3. Calculate the stiffness coefficient of each composite element 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; S4. When the main frequency of the combined element is evenly distributed in a wide frequency range, the viscosity coefficient of the combined element is calculated based on the main frequency of the combined element 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 in a wide frequency range.

2. The method for calculating broadband consistent damping force in time domain dynamic analysis according to claim 1, characterized in that: Step S3 further comprises: S31. Determine the dynamic stiffness model related to the spring stiffness coefficient: Where S(ω) is the dynamic stiffness of the hydropower plant unit; N is the number of combined elements; α j is the stiffness coefficient of the jth combined unit; τ j is the relaxation time of the jth composite element; ω is the main frequency; K is the initial stiffness matrix; i is the imaginary unit S32. In a wide frequency range, according to the selected frequency, determine and calculate the relationship between each element in the matrix x and the vector y based on the damping ratio: Among them, x mn is the element in the mth row and nth column of the matrix x, 1≦m≦N, 1≦n≦N; ω1 and ω2 are the lower and upper limits of the wide frequency range; ω m and ω n is a randomly selected frequency within a wide frequency range; ξ(·) is the damping ratio function; y n is the nth element in vector y; ξ aim is the target damping ratio; Im(·) is the imaginary part of the function; Re(·) is the real part of the function; S33. According to equations (1) to (4), the least square method is used to solve the relationship between the stiffness coefficient and the matrix x and the vector y to obtain the stiffness coefficient of each combined element. The relationship is: a opt =x -1 y Among them, α opt is the matrix of stiffness coefficients of all combined elements.

3. The method for calculating broadband consistent damping force in time domain dynamic analysis according to claim 2, characterized in that: The calculation expression of relaxation time is: t j =2ξ aim oh j Oh, oh j =ω1+(ω2-ω1) / (N-1)*(j-1) Among them, τ j , α j and η j are the relaxation time, stiffness coefficient and viscosity coefficient of the j-th combined unit respectively; ω j is the main frequency of the jth combination element.

4. The method for calculating broadband consistent damping force in time domain dynamic analysis according to claim 1, characterized in that: The expression for calculating the viscosity coefficient of the composite element is: or j =2ξ aim oh j a j K, oh j =ω1+(ω2-ω1) / (N-1)*(j-1) Among them, ω j is the main frequency of the jth combination element; ω1 and ω2 are the lower and upper limits of the wide frequency range; N is the number of combination elements; ξ aim is the target damping ratio; α j and η j are the stiffness coefficient and viscosity coefficient of the j-th combined unit respectively; K is the initial stiffness matrix.

5. The method for calculating broadband consistent damping force in time domain dynamic analysis according to claim 1, characterized in that: The expression of the time domain damping force model is: 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 composite elements; τ j is the relaxation time of the jth composite element; α j is the stiffness coefficient of the j-th combined unit; ω is the main frequency; ω j is the main frequency of the j-th combination element; is the velocity of the hydropower plant at time t; u(t) is the displacement of the hydropower plant at time t.

6. The method for calculating broadband consistent damping force in time domain dynamic analysis according to any one of claims 1 to 5, characterized in that: The combination elements are formed by connecting a sticky pot and a spring in series, and the number of the combination elements is 5 to 7.

Images