Calculation method for hydrodynamic coefficient of internal damping component of tuned liquid damper

Abstract

A calculation method for the hydrodynamic coefficients of the internal damping component of a tuned liquid damper, comprising the following steps: Step S11, under the action of a random load, respectively collect the wave height response signals of the free surface of the TLD through a plurality of wave gauges; Step S12, use the data-driven stochastic subspace method to identify the parameters of the wave height response signals to obtain the natural frequency and damping ratio of the TLD system; Step S13, establish the natural frequency ω T formula and damping ratio ζ T formula, Step S14, substitute the natural frequency of the TLD system into the natural frequency ω T formula for calculation to obtain the added mass coefficient C of the internal damping component of the TLD m , substitute the damping ratio of the TLD system into the damping ratio ζ T formula for calculation to obtain the drag coefficient C of the internal damping component of the TLD d . The hydrodynamic coefficients of the damping component obtained by the method of the present invention can be used to calculate the added mass and damping ratio generated by the damping component under the action of a random load, and for the equivalent mechanical model to analyze the dynamic response of the TLD under the action of a random load; and the method is simple and convenient, easy to implement, and applicable to the determination of the optimization design scheme of the damping component for the wind-induced vibration control of super high-rise buildings by the TLD.

Description

Technical Field

[0001] The present invention relates to the technical field of wind-induced vibration control of high-rise buildings, and specifically relates to a calculation method for the hydrodynamic coefficients of the internal damping members of a tuned liquid damper. Background Technique

[0002] The tuned liquid damper (TLD) has the advantages of simple structure, low installation and maintenance costs, etc., and has been successfully applied to the wind-induced vibration control of super high-rise buildings. The TLD generally consists of a water tank and internal damping members, among which the damping members can improve the energy dissipation capacity of the TLD and optimize the vibration reduction performance of the TLD. When the wind-induced load causes the vibration of a super high-rise building, the TLD uses the inertial force generated by the sloshing liquid to suppress the vibration response of the super high-rise building. In order to study the basic principle of liquid sloshing and to facilitate the design of TLDs in actual engineering, the TLD is usually represented as an equivalent mechanical model. However, in the equivalent mechanical model, the calculation of the natural frequency and damping ratio requires prior determination of the hydrodynamic coefficients of the damping members (including the drag coefficient C d and the added mass coefficient C m ).

[0003] Currently, there are mainly two methods for obtaining the hydrodynamic coefficients of the internal damping members of the TLD: (1) calculated according to the measured hydrodynamic pressure received by the damping members in one sloshing period using theoretical formulas; (2) fitting the frequency response curve of the equivalent mechanical model with the measured data of the TLD, and determining by minimizing the error between the two curves. The above methods are currently only applicable to the case where the external excitation is a periodic load, and cannot be directly applied to determine the hydrodynamic coefficients of the internal damping members of the TLD of a super high-rise building under random wind loads. Summary of the Invention

[0004] Based on this, the purpose of the present invention is to provide a calculation method for the hydrodynamic coefficients of the internal damping members of a tuned liquid damper to solve the problem of solving the hydrodynamic coefficients of the internal damping members of the TLD of a super high-rise building under random loads.

[0005] A calculation method for the hydrodynamic coefficients of the internal damping members of a tuned liquid damper includes the following steps:

[0006] Step S11, under random loads, respectively collect the wave height response signals η(t) = [η1(t), η2(t), …, η k (t)] Τ of the free surface of the TLD through a plurality of wave gauges, where k is the number of wave gauges, the minimum value is 1, and the maximum value is equal to the modal order of the TLD;

[0007] Step S12: Use the data-driven stochastic subspace method to identify the parameters of the wave height response signal to obtain the natural frequency and damping ratio of the TLD system;

[0008] Step S13: Establish the natural frequency ω T formula and damping ratio ζ T formula, where

[0009]

[0010] ζ T = ζ w + ζ p ;

[0011] In the formula, ω w and m eq are the natural frequency and equivalent mass of the pure water tank respectively, m p,eq is the additional mass of the damping member, ζ w is the damping ratio caused by the viscosity of water, ζ p is the damping ratio caused by the damping member;

[0012] Step S14: Substitute the natural frequency of the TLD system into the natural frequency ω T formula for calculation to obtain the additional mass coefficient C m of the internal damping member of the TLD. Substitute the damping ratio of the TLD system into the damping ratio ζ T formula for calculation to obtain the resistance coefficient C d of the internal damping member of the TLD.

[0013] Preferably, the step S12 includes:

[0014] Step S121: Establish a Hankel matrix according to the wave height response signal;

[0015]

[0016] In the formula, Y 0|2i-1 is a matrix composed of 2i block rows and j columns. "0|2i - 1" are the subscripts of y0 and y 2i-1 respectively. y i is the response data measured by all wave height gauges at time i. j satisfies 2i + j - 1 < N, where N is the total data length. Y p is the upper half of the Hankel matrix, called the past matrix, and Y f is the lower half of the Hankel matrix, called the future matrix;

[0017] Among them, another expression of Y 0|2i-1 is;

[0018]

[0019] In the formula, compared with Y p one row is added, compared with Y f one row is reduced;

[0020] Step S122: Calculate the projection matrix P according to the Hankel matrix i ;

[0021]

[0022] Step S123: Introduce the Kalman filter sequence and the observability matrix O i to transform the projection matrix;

[0023]

[0024] Step S124: Perform singular value decomposition on P i to obtain the observability matrix O i and the Kalman filter sequence

[0025]

[0026]

[0027]

[0028]

[0029] In the formula, S is a diagonal matrix with singular values arranged from large to small, S1 is the non-zero part of S, U and V are orthogonal matrices, U1, U2 and V1, V2 are matrices composed of the internal elements of U and V respectively, which are specifically related to the number of rows and columns of S2, and the superscripts "T" and "+" represent the transpose and pseudo-inverse of the matrix;

[0030] Step S125: Calculate the TLD system matrix according to the Kalman filter sequence ;

[0031]

[0032] Y i|j = [y i y i+1 … y i+j-1 ;

[0033] In the formula, A is the discrete state matrix and C is the output matrix;

[0034] Step S126, calculate the natural frequency and damping ratio of the TLD system according to the TLD system matrix.

[0035] Preferably, the step S126 includes:

[0036] Step S1261, perform eigenvalue decomposition on the discrete state matrix A;

[0037] A = ΨΛΨ -1 ;

[0038] where Ψ is the discrete system eigenvector matrix, Λ is the diagonal matrix composed of discrete system eigenvalues, Λ = diag(μ i ), μ i is the discrete system eigenvalue;

[0039] Step S1262, calculate according to the discrete system eigenvalue and the wave height meter sampling time interval to obtain the continuous system eigenvalue, and at the same time calculate the continuous system eigenvalue to obtain the natural frequency and damping ratio of the TLD system:

[0040]

[0041]

[0042]

[0043] where Δt is the wave height meter sampling time interval, a i is the real part of the continuous system eigenvalue, b i is the imaginary part of the continuous system eigenvalue.

[0044] Preferably, the step S13 includes:

[0045] Step S131, calculate the sloshing force f acting on a unit column length at any position of the internal damping member of the TLD;

[0046]

[0047]

[0048] where C d is the resistance coefficient of the damping member, ρ is the density of water, A0 is the projected area of the unit height of the damping member, U p is the water particle velocity orthogonal to the damping member, C m is the added mass coefficient of the damping member, V0 is the drainage volume of the unit height of the damping member, q is the generalized coordinate, x and z are the position coordinates of the damping member, L is the length of the water tank, and h is the static liquid height of the water tank;

[0049] Step S132: Based on the principle of virtual displacement, calculate the virtual work generated by all damping members according to the sloshing force.

[0050]

[0051]

[0052]

[0053]

[0054]

[0055] In the formula, ω w and m eq are the natural frequency and equivalent mass of the pure water tank respectively, ζ p and m p,eq are the damping ratio and added mass of the damping member respectively, Г is the modal participation factor, n p is the number of damping members, B is the width of the water tank, g is the acceleration due to gravity, δυ is the virtual horizontal displacement of the damping member, and x j is the horizontal distance between the damping member and the side wall of the water tank.

[0056] Step S133: Establish the formula for the natural frequency ω T and the formula for the damping ratio ζ T of the equivalent mechanical model of the TLD according to the virtual work.

[0057] Preferably, when the damping member inside the TLD specifically refers to the paddle column, calculate according to the sloshing force and virtual work to obtain the damping ratio ζ p and the added mass m p,eq ;

[0058]

[0059]

[0060] In the formula, a p is half of the paddle column length, and σ q is the root mean square of the sloshing response of the TLD.

[0061] Preferably, the damping ratio ζ w caused by the viscosity of water is:

[0062]

[0063] In the formula, ν is the kinematic viscosity of the water inside the water tank.

[0064] Preferably, the model of the TLD system is a scaled model or a CFD model.

[0065] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0066] (1) This method solves the problem of solving the hydrodynamic coefficient of the internal damping component of the TLD under the action of random loads;

[0067] (2) The hydrodynamic coefficient of the damping component obtained by this method can be used to calculate the added mass and damping ratio generated by the damping component under the action of random loads;

[0068] (3) The hydrodynamic coefficient of the damping component obtained by this method can be used for the dynamic response analysis of the TLD under the action of random loads by using an equivalent mechanical model;

[0069] (4) This method is simple and convenient, easy to implement, and is applicable to determining the optimization design scheme of the damping component for controlling the wind-induced vibration of super high-rise buildings by the TLD. Description of the Drawings

[0070] Figure 1 It is a schematic flow chart of the calculation method of the hydrodynamic coefficient of the internal damping component of the tuned liquid damper proposed by the present invention;

[0071] Figure 2 It is a schematic diagram of the built-in paddle column TLD in the present invention;

[0072] Figure 3 For Figure 2 the CFD model diagram of the TLD in;

[0073] Figure 4 For Figure 2 the time history curve diagram of the random excitation at the base of the TLD in.

[0074] The following specific embodiments will further illustrate the present invention in conjunction with the above drawings. Specific Embodiments

[0075] The following description is used to disclose the present invention so that those skilled in the art can implement the present invention. The preferred embodiments described below are only examples, and other obvious variations can be thought of by those skilled in the art.

[0076] Please refer to Figures 1 to 3 , the embodiment of the present invention provides a calculation method for the hydrodynamic coefficient of the internal damping component of a tuned liquid damper, including the following steps:

[0077] Step S11, under the action of random loads, respectively collect the wave height response signals η(t) = [η1(t), η2(t), …, η k (t)] Τ of the free surface of the TLD through a plurality of wave gauges, where k is the number of wave gauges, the minimum value is 1, and the maximum value is equal to the modal order of the TLD;

[0078] Step S12: Use the data-driven stochastic subspace method to identify the parameters of the wave height response signal, so as to obtain the natural frequency and damping ratio of the TLD system;

[0079] Step S13: Establish the natural frequency ω T formula and damping ratio ζ T formula of the equivalent mechanical model of the TLD, where

[0080]

[0081] ζ T =ζ w +ζ p ;

[0082] In the formula, ω w and m eq are the natural frequency and equivalent mass of the pure water tank respectively, m p,eq is the additional mass of the damping member, ζ w is the damping ratio caused by the viscosity of water, ζ p is the damping ratio caused by the damping member;

[0083] Step S14: Substitute the natural frequency of the TLD system into the natural frequency ω T formula for calculation to obtain the additional mass coefficient C m of the internal damping member of the TLD. Substitute the damping ratio of the TLD system into the damping ratio ζ T formula for calculation to obtain the drag coefficient C d of the internal damping member of the TLD.

[0084] Preferably, the step S12 includes:

[0085] Step S121: Establish a Hankel matrix according to the wave height response signal;

[0086]

[0087] In the formula, Y 0|2i-1 is a matrix composed of 2i block rows and j columns. "0|2i - 1" are the subscripts of y0 and y 2i-1 respectively. y i is the response data measured by all wave gauges at the i-th moment. j satisfies 2i + j - 1 < N, where N is the total data length. Y p is the upper half of the Hankel matrix, which is called the past matrix, and Y f is the lower half of the Hankel matrix, which is called the future matrix;

[0088] Among them, another expression of Y 0|2i-1 is;

[0089]

[0090] wherein, compared with Y p one row is added; compared with Y f one row is reduced;

[0091] Step S122: Calculate the projection matrix P according to the Hankel matrix i ;

[0092]

[0093] Step S123: Introduce the Kalman filter sequence and the observable matrix O i to transform the projection matrix;

[0094]

[0095] Step S124: Perform singular value decomposition on P i to obtain the observable matrix O i and the Kalman filter sequence

[0096]

[0097]

[0098]

[0099]

[0100] wherein, S is a diagonal matrix with singular values arranged from large to small, S1 is the non-zero part of S, U and V are orthogonal matrices, U1, U2 and V1, V2 are matrices composed of the internal elements of U and V respectively, specifically related to the number of rows and columns of S2, and the superscripts "T" and "+" represent the transpose and pseudo-inverse of the matrix;

[0101] Step S125: Calculate the TLD system matrix according to the Kalman filter sequence ;

[0102]

[0103] Y i|j =[y i y i+1 ... y i+j-1 ;

[0104] wherein, A is the discrete state matrix and C is the output matrix;

[0105] Step S126, calculate the natural frequency and damping ratio of the TLD system according to the TLD system matrix.

[0106] Preferably, the step S126 includes:

[0107] Step S1261, perform eigenvalue decomposition on the discrete state matrix A;

[0108] A = ΨΛΨ -1 ;

[0109] where Ψ is the discrete system eigenvector matrix, Λ is the diagonal matrix composed of discrete system eigenvalues, Λ = diag(μ i ), μ i is the discrete system eigenvalue;

[0110] Step S1262, calculate according to the discrete system eigenvalue and the wave height meter sampling time interval to obtain the continuous system eigenvalue, and at the same time calculate the continuous system eigenvalue to obtain the natural frequency and damping ratio of the TLD system:

[0111]

[0112]

[0113]

[0114] where Δt is the wave height meter sampling time interval, a i is the real part of the continuous system eigenvalue, b i is the imaginary part of the continuous system eigenvalue.

[0115] Preferably, the step S13 includes:

[0116] Step S131, calculate the sloshing force f acting on a unit column length at any position of the internal damping member of the TLD;

[0117]

[0118]

[0119] where C d is the resistance coefficient of the damping member, ρ is the density of water, A0 is the projected area of the unit height of the damping member, U p is the water particle velocity orthogonal to the damping member, C m is the added mass coefficient of the damping member, V0 is the drainage volume of the unit height of the damping member, q is the generalized coordinate, x and z are the position coordinates of the damping member, L is the length of the water tank, and h is the static liquid height of the water tank;

[0120] Step S132: Based on the principle of virtual displacement, calculate the virtual work generated by all damping members according to the sloshing force.

[0121]

[0122]

[0123]

[0124]

[0125]

[0126] In the formula, ω w and m eq are respectively the natural frequency and the equivalent mass of the pure water tank, ζ p and m p,eq are respectively the damping ratio and the added mass of the damping member, Г is the modal participation factor, n p is the number of damping members, B is the width of the water tank, g is the acceleration due to gravity, δυ is the virtual horizontal displacement of the damping member, and x j is the horizontal distance between the damping member and the side wall of the water tank.

[0127] Step S133: Establish the formula for the natural frequency ω T and the formula for the damping ratio ζ T of the equivalent mechanical model of the TLD according to the virtual work.

[0128] Preferably, when the damping member inside the TLD specifically refers to a paddle column, calculate according to the sloshing force and the virtual work to obtain the damping ratio ζ p and the added mass m p,eq ;

[0129]

[0130]

[0131] In the formula, a p is half of the paddle column length, and σ q is the root mean square of the sloshing response of the TLD.

[0132] Preferably, the damping ratio ζ w caused by the viscosity of water is:

[0133]

[0134] In the formula, ν is the kinematic viscosity of the water inside the water tank.

[0135] In a preferred embodiment of the present invention, the model of the TLD system is a scaled model or a CFD model.

[0136] The method of the present invention will be described in detail below with specific embodiments:

[0137] Please refer to Figure 2 and Figure 3 , take an internal paddle-column TLD. The length, width, and height of the water tank are 2.1 m, 0.64 m, and 0.9 m respectively, and the static liquid height is 0.44 m. Nine paddle columns are arranged at equal intervals inside the water tank. The connecting lines of the central points of the paddle columns bisect the length and width of the water tank respectively, and the length of each paddle column is 0.077 m.

[0138] Please refer to Figure 3 and Figure 4 , and use the computational fluid dynamics (CFD) method to simulate the sloshing response of the above TLD under Gaussian white noise random excitation. Four wave gauges are installed at 0.1 m, 0.78 m, 1.3 m, and 2 m away from the side wall of the water tank respectively, for real-time recording of the wave height response data η(t) = [η1(t), η2(t), η3(t), η4(t)] T . After the wave height data is collected, the data-driven stochastic subspace method is used for modal parameter identification. Since the equivalent mechanical model of the TLD is a first-order sloshing model, when i = 1, the first-order modal parameter results of the TLD system are the required parameters for solving the hydrodynamic coefficient of the damping component. Finally, the first natural frequency of the TLD system is ω1 = 2.834 rad / s, and the first damping ratio of the TLD system is ζ1 = 3.09%. Further, substitute the first-order modal parameter identification results (the first natural frequency and the first damping ratio) into the natural frequency ω T formula and the damping ratio ζ T formula of the TLD equivalent mechanical model, and obtain the added mass coefficient C m = 1.52, and the drag coefficient C d = 8.54.

[0139] The above shows and describes the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. What is described in the above embodiments and the specification is only the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of protection claimed by the present invention is defined by the appended claims and their equivalents.

Claims

1. A calculation method for the hydrodynamic coefficient of the internal damping member of a tuned liquid damper, characterized in that, Including the following steps: Step S11, under the action of random loads, respectively collect the wave height response signals of the free surface of the TLD through a plurality of wave gauges , where k is the number of wave gauges, the minimum value is 1, and the maximum value is equal to the modal order of the TLD; Step S12, using the data-driven stochastic subspace method to identify the parameters of the wave height response signal to obtain the natural frequency and damping ratio of the TLD system; Step S13, establish the natural frequency and damping ratio of the equivalent mechanical model of the TLD ω T formulas ζ T formulas, where ； ； In the formula, ω w and m eq are the natural frequency and equivalent mass of the pure water tank respectively, m p,eq is the additional mass of the damping member, ζ w is the damping ratio caused by the viscosity of water, ζ p is the damping ratio caused by the damping member; Among them, the damping ratio caused by the viscosity of water ζ w is ； In the formula, ν is the kinematic viscosity of the water inside the water tank; When the TLD internal damping member is a paddle column, calculations are performed according to the sloshing force and virtual work to obtain the damping ratio of the paddle column ζ p and the added mass m p,eq ; ； ； In the formula, a p is half of the length of the paddle column, and σ q is the root mean square of the sloshing response of the TLD; Step S14, substitute the natural frequency of the TLD system into the natural frequency ω T formula for calculation to obtain the additional mass coefficient of the internal damping member of the TLD C m , substitute the damping ratio of the TLD system into the damping ratio ζ T formula for calculation to obtain the drag coefficient of the internal damping member of the TLD C d .

2. The calculation method of the hydrodynamic coefficient of the internal damping member of the tuned liquid damper according to claim 1, characterized in that, The said step S12 includes: Step S121, establishing a Hankel matrix according to the wave height response signal; ； In the formula, Y 0|2i-1 is a matrix composed of 2 i block rows and j columns. "0|2 i -1” are respectively y 0 and y 2i-1 subscripts, y i is i the response data measured by all wave gauges at time j satisfies 2 i + j -1 < N, N is the total data length, Y p is the upper half of the Hankel matrix and is called the past matrix, Y f is the lower half of the Hankel matrix and is called the future matrix; Among them, Y 0|2i-1 Another expression of; ； In the formula, compared with Y p one row is added, compared with Y f one row is reduced; Step S122, calculate the projection matrix according to the Hankel matrix P i ; ； Step S123, introduce the Kalman filter sequence and the observable matrix O i , transform the projection matrix; ； Step S124, perform P i singular value decomposition to obtain the observable matrix O i and the Kalman filter sequence ; ； ； ； ； Wherein, S is a diagonal matrix with singular values arranged from large to small, S 1 is S the non-zero part, U and V are orthogonal matrices, U 1, U 2 and V 1, V 2 are matrices respectively composed of the internal elements of U and V , specifically related to the number of rows and columns of S 2. The superscripts " T " and " + " represent the transpose and pseudo-inverse of the matrix; Step S125, according to the Kalman filter sequence , calculate the TLD system matrix; ； ； In the formula, A is the discrete state matrix, C is the output matrix; Step S126, calculating the natural frequency and damping ratio of the TLD system according to the TLD system matrix.

3. The calculation method of the hydrodynamic coefficient of the internal damping member of the tuned liquid damper according to claim 2, wherein The said step S126 includes: Step S1261, perform eigenvalue decomposition on the discrete state matrix A ; ； wherein, Ψ is the discrete system eigenvector matrix, Λ is the diagonal matrix composed of the discrete system eigenvalues, Λ = diag( μ i ), μ i are the discrete system eigenvalues; Step S1262, calculating according to the discrete system eigenvalues and the wave height meter sampling time interval to obtain the continuous system eigenvalues, and at the same time calculating the continuous system eigenvalues to obtain the natural frequency and damping ratio of the TLD system: ； ； ； In the formula, is the sampling time interval of the wave height meter, is the real part of the eigenvalue of the continuous system, is the imaginary part of the eigenvalue of the continuous system.

4. The calculation method of the hydrodynamic coefficient of the internal damping member of the tuned liquid damper according to claim 3, characterized in that, The said step S13 includes: Step S131, calculate the sloshing force acting on a unit column length at any position of the internal damping member of the TLD f ; ； ； In the formula, C d is the resistance coefficient of the damping member, is the density of water, A 0 is the projected area per unit height of the damping member, U p is the velocity of the water particle orthogonal to the damping member, C m is the added mass coefficient of the damping member, V 0 is the drainage volume per unit height of the damping member, q is the generalized coordinate, x and z are the position coordinates of the damping member, L is the length of the water tank, h is the static liquid height of the water tank; Step S132, based on the principle of virtual displacement, calculating the virtual work generated by all damping members according to the sloshing force; ； ； ； ； ； In the formula, ω w and m eq are the natural frequency and equivalent mass of the pure water tank respectively, ζ p and m p,eq are the damping ratio and additional mass of the damping member respectively, Г is the modal participation factor, n p is the number of damping members, B is the width of the water tank, g is the acceleration due to gravity, δυ is the imaginary horizontal displacement of the damping member, x j is the horizontal distance between the damping member and the side wall of the water tank; Step S133, establish the natural frequency and damping ratio formulas of the TLD equivalent mechanical model according to the virtual work ω T formulas ζ T formulas.

5. The calculation method of the hydrodynamic coefficient of the internal damping member of the tuned liquid damper according to any one of claims 1 to 4, characterized in that, The model of the TLD system is a scaled model or a CFD model.

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