A parameter optimization design method of negative stiffness-inerter tuned liquid column damper considering SSI effect under random earthquake

Abstract

This invention discloses a parameter optimization design method for a negative stiffness-inertial volume tuned liquid column damper considering the soil-structure interaction (SSI) effect under random earthquakes. The method specifically includes: S1: a coupled dynamic model of the supporting foundation, the upper working platform, and the negative stiffness-inertial volume tuned liquid column damper considering soil-structure interaction effects; S2: deriving the transfer function and root mean square response of the NS-TLCDI system considering the SSI effect based on the NS-TLCDI system dynamic model; S3: optimizing the displacement and acceleration indices of the working platform based on the root mean square response using the NSGA-II multi-objective algorithm, and determining the optimal design parameters of the NS-TLCDI system considering the SSI effect. This invention comprehensively considers the influence of site conditions and seismic motion spectrum characteristics on the dynamic characteristics of the upper working platform considering soil-structure interaction and the optimal parameters of the vibration reduction device, improving the accuracy, reliability, and engineering applicability of seismic and vibration reduction design of high-altitude working platforms under complex site conditions.

Description

Technical Field

[0001] This invention relates to the field of structural vibration control and seismic damping technology, specifically to a parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes. Background Technology

[0002] With the increasing prevalence of flexible structures and engineering structures in complex site conditions, such as wind turbines, photovoltaic systems, and aerial work platforms, the vibration control of structures under seismic loading has received increasing attention. Tuned liquid column dampers (TLCDs) can suppress structural vibration through a combination of liquid oscillation and nonlinear damping energy dissipation, offering advantages in reducing structural displacement and acceleration response. However, traditional TLCDs typically rely on precise parameter tuning, and detuning can easily occur when the structural dynamic characteristics or the input seismic motion spectrum changes, leading to a decline in vibration reduction performance. Furthermore, TLCDs still have limitations in controlling low-frequency vibrations, broadening the effective control bandwidth, and balancing both structural displacement and absolute acceleration response indicators. To further improve vibration reduction performance, negative stiffness and inertial capacitance elements are introduced into the nonlinear liquid column damping system, forming a negative stiffness-inertial capacitance tuned liquid column damper (NS-TLCDI), providing a new approach to structural vibration control. Negative stiffness elements can adjust the equivalent stiffness of the system, while inertial capacitance elements can enhance the equivalent inertial effect without significantly increasing the added mass. The synergistic effect of the two is expected to improve the dynamic response of the structure under seismic loading.

[0003] Existing research and design methods are mostly based on the assumption of a fixed base (FB) to establish models. However, in actual engineering, the superstructure is built on a foundation with finite stiffness and damping, resulting in a significant soil-structure interaction (SSI) effect between the foundation and the superstructure. This effect alters the overall dynamic characteristics of the structure, including equivalent stiffness, natural frequency, damping level, and resonance range, thus affecting the parameter configuration and control effect of additional vibration damping devices—a phenomenon often overlooked in current research. Furthermore, existing parameter design methods often use a single response index as the optimization objective, such as minimizing only the structural displacement response or only the structural acceleration response. However, for engineering structures, displacement control and acceleration control often need to be considered simultaneously. When considering soil-structure interaction, there is usually a conflict between the two types of response objectives, making it difficult to obtain a well-balanced parameter result through single-objective optimization. Therefore, it is necessary to introduce multi-objective optimization methods to comprehensively design the parameters of vibration damping devices to improve the engineering applicability of the parameter scheme.

[0004] Existing technologies include schemes that combine inertial capacitive elements with tuned liquid dampers and apply them to the parameter design of vibration control systems. However, these existing technologies conduct structural vibration control research under rigid foundation conditions. Related patents mainly focus on random wind loads, while related papers mainly focus on random seismic excitation simplified to white noise processes, which have some reference value. However, the aforementioned existing research has not yet addressed the vibration control problem of the superstructure considering soil-structure interaction under seismic acceleration excitation with different spectral characteristics, nor has it established a dynamic model that includes the coupling effect of foundation translation, foundation rotation, platform response, and nonlinear damping of the liquid column; nor has it introduced negative stiffness elements to enhance the low-frequency control effect of the vibration damper. Furthermore, there is a lack of corresponding research on multi-objective collaborative optimization targeting the root mean square response of relative displacement and root mean square response of absolute acceleration in the frequency domain of the aerial work platform. Summary of the Invention

[0005] The purpose of this invention is to address the limitations of existing TLCD (Thickness-Inertial Capacity-Tunneled Liquid Column) dampers, the neglect of SSI (Seismic Inertial Siphon) effect in parameter design, and the difficulty in simultaneously considering both displacement and acceleration control objectives. This invention proposes a parameter optimization design method for negative stiffness-inertial capacitive tuning liquid column dampers under random earthquakes, taking into account the SSI effect. This method introduces negative stiffness to propose a negative stiffness-inertial capacitive tuning liquid column damper and establishes a system coupled dynamic model considering the translational and rotational degrees of freedom of the foundation, the supporting foundation, and the NS-TLCDI (Non-Stiffness-Tunneled Liquid Column Damper). By combining equivalent linearization processing, frequency domain multi-objective optimization, and actual ground motion time history analysis, the key design parameters of the NS-TLCDI are rationally determined, thereby improving the seismic and vibration reduction performance and parameter design reliability of aerial work platforms under different site conditions.

[0006] Technical Solution: To achieve the above objectives, this invention proposes a parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes. This method includes the following steps:

[0007] Step 1) Establish a system coupling dynamics model considering the SSI effect for the operating platform, supporting foundation, and NS-TLCDI;

[0008] Step 2) Derive the transfer function and root mean square response of the NS-TLCDI system considering the SSI effect based on the NS-TLCDI system dynamic model;

[0009] Step 3) Use the NSGA-II multi-objective algorithm to optimize the displacement and acceleration indices of the work platform based on the root mean square of the response, and determine the optimal design parameters of the NS-TLCDI system considering the SSI effect.

[0010] Furthermore, the specific method for step 1) is as follows:

[0011] Step 1.1) The mass of the working platform is m1, which is connected to the support foundation of the working platform through a support arm with horizontal stiffness k1 and horizontal damping c1. The mass of the support foundation is m0, and its interaction with the ground is through the horizontal stiffness k. h and horizontal damping c h and rotational stiffness k r and rotational damping c r Equivalent representation:

[0012] (1)

[0013] Where R is the equivalent circular base radius; G, ν, ρ s and V s These are the soil's shear modulus, Poisson's ratio, density, and shear wave velocity, respectively.

[0014] The designed negative stiffness-inertia-capacity tuned liquid column damper NS-TLCDI is placed on the working platform. It consists of a U-shaped liquid column storage tank and its connecting elements. The cross-sectional area of ​​the U-shaped storage tank is A, the length of the horizontal section is B, and the total length of the internal U-shaped liquid column is denoted as L. A grid is provided at the center of the horizontal section of the storage tank, and its local fluid resistance characteristics are determined by the head loss coefficient. Characterization; let the mass of the container itself be M. C The internal fluid density is The total mass of NS-TLCDI is expressed as AL+M CThe NS-TLCDI's storage tank is connected to the work platform via a connector with horizontal stiffness k2 and horizontal damping c2, and is connected in parallel to an inertial element with inertial capacity b and horizontal negative stiffness k. n The negative stiffness element is connected to the support foundation of the work platform; h represents the vertical height of the work platform from its support foundation, a g This represents the random ground acceleration excitation experienced by the system; the work platform, support foundation, and NS-TLCDI as a whole are denoted as SSI+NS-TLCDI;

[0015] Step 1.2) Considering the interaction between the support foundation of the work platform and the ground, the entire system is represented as a multi-degree-of-freedom coupled dynamic system, including the horizontal displacement x of the work platform relative to its support foundation, the horizontal displacement y of the liquid storage tank relative to the support foundation of the work platform, the vertical vibration displacement z of the liquid column, the horizontal translational displacement u of the support foundation, and the rotational displacement θ of the support foundation. Its motion control equations are as follows:

[0016] (2)

[0017] Where g is the gravitational acceleration, J0 is the moment of inertia based on the equation, and the dots above the variables represent the time derivative of the variables. A single dot represents the first derivative, and two dots represent the second derivative.

[0018] Furthermore, the specific method for step 2) is as follows:

[0019] Step 2.1) To calculate the transfer function of the NS-TLCDI system, an equivalent linearization coefficient λ is introduced. eq Let represent the coefficient of the original nonlinear fluid damping term. Then, the error ε between the nonlinear term and its equivalent linearized form is expressed as:

[0020] (3)

[0021] (4)

[0022] in, and Let λ represent the natural circular frequency and equivalent liquid damping ratio of the liquid column in the storage tank, respectively. To minimize the mean square error ε, according to the extreme value theory, λ eq satisfy:

[0023] (5)

[0024] in, λ represents the expectation operation. eq By solving equation (5), we obtain:

[0025] (6)

[0026] Assumption If the signal is a stationary Gaussian signal with zero mean, then:

[0027] (7)

[0028] in, The root mean square of the liquid column velocity inside the storage tank is represented by equation (7). Substituting equation (7) into equation (6), we get λ. eq Simplified to:

[0029] (8)

[0030] Substituting equation (8) into equation (4) yields the equivalent liquid damping ratio. :

[0031] (9)

[0032] Step 2.2) The equivalent liquid damping ratio shown in equation (9) Substituting into equation (2), the linearized motion control equation of the system is rewritten as:

[0033] (10)

[0034] Let the state variable vector x be [xyzu θ] T The governing equation (10) is rewritten in the following dimensionless form:

[0035] (11)

[0036] Where M, C, K, and F are the mass matrix, damping matrix, stiffness matrix, and seismic excitation vector of the NS-TLCDI system considering the SSI effect, respectively:

[0037] Step 2.3) Solve the dimensionless governing equations using the Laplace transform to obtain several transfer functions required for the calculations. Their expressions are as follows:

[0038] Horizontal displacement transfer function of the working platform relative to its supporting foundation:

[0039] (12)

[0040] Liquid velocity transfer function within the storage tank:

[0041] (13)

[0042] Translational displacement transfer function of the supporting foundation:

[0043] (14)

[0044] Rotational displacement transfer function of the supporting foundation:

[0045] (15)

[0046] in, A represents the angular frequency in the frequency domain. i B i C i D i E i The polynomial expression is as follows:

[0047] ;

[0048] ;

[0049] ;

[0050] ;

[0051] ;

[0052] ;

[0053] Step 2.4) Considering the SSI effect, the horizontal absolute acceleration transfer function of the working platform relative to its supporting foundation. :

[0054] (16)

[0055] Let the power spectral density function of the input seismic acceleration be... To simulate random ground motion input, the Clough-Penzien (CP) power spectrum model is used, and its expression is as follows:

[0056] (17)

[0057] in, and These are the superior circular frequency and damping ratio of the soil on which the supporting foundation is placed; and These are the CP spectrum filtering angular frequency and damping ratio, respectively; where the CP spectrum intensity parameter S0 is determined based on the given peak ground acceleration.

[0058] (18)

[0059] in, Representing the peak ground acceleration due to random seismic motion, the root mean square of the relative displacement response of the work platform relative to its supporting foundation is... Root mean square of absolute acceleration response They are represented as follows:

[0060] (19)

[0061] (20).

[0062] Furthermore, the calculation methods for M, C, K, and F are as follows:

[0063] ,

[0064] ,

[0065] , ;

[0066] in, The mass ratio of the liquid in the storage tank. ;

[0067] The mass ratio of the storage tank. ;

[0068] The NS-TLCDI quality ratio ;

[0069] The ratio of the supporting foundation quality of the operating platform. ;

[0070] The SSI effect scaling ratio. ;

[0071] The ratio of liquid column lengths, ;

[0072] The inherent circular frequency of the working platform, ;

[0073] This is the inherent angular frequency of NS-TLCDI.

[0074] The natural circumfrequency of the liquid. ;

[0075] The horizontal angular frequency of SSI. ;

[0076] The angular frequency in the direction of SSI rotation. ;

[0077] The damping ratio of the work platform, ;

[0078] For NS-TLCDI damping ratio, ;

[0079] The horizontal damping ratio of SSI. ;

[0080] The SSI rotational direction damping ratio, ;

[0081] Inertia-mass ratio, ;

[0082] It is a negative stiffness ratio. ;

[0083] The frequency ratio of NS-TLCDI ;

[0084] The ratio of liquid frequencies. .

[0085] Furthermore, the specific method for step 3) is as follows:

[0086] Step 3.1) To unify the dimensions and orders of magnitude of the root mean square of relative displacement and root mean square of absolute acceleration of the working platform in multi-objective optimization, dimensionless root mean square of relative displacement J1 and root mean square of absolute acceleration J2 are introduced based on equations (19) and (20):

[0087] (twenty one)

[0088] in, and Let represent the root mean square of the relative displacement response and the root mean square of the absolute acceleration response of the work platform when the system is not controlled by NS-TLCDI, respectively. Before optimizing the NS-TLCDI system considering the SSI effect, the range of values ​​of the variables needs to be determined. First, the positive definite condition of the stiffness matrix is ​​used to ensure the stability of the system, that is, the stiffness matrix K in equation (11) should satisfy the positive definiteness requirement det(K)>0, and the NS-TLCDI frequency ratio is obtained. Compared with negative stiffness The constraints that should be satisfied between them are as follows:

[0089] (twenty two)

[0090] Secondly, the total length L of the liquid column is limited by the structural dimensions of the storage tank and must meet the following requirements: min ≤L≤L max , where L min and L max Given the minimum and maximum dimensions of the storage tank, respectively, the liquid frequency ratio is... The constraints are as follows:

[0091] (twenty three)

[0092] The multi-objective optimization problem under consideration is formulated as follows:

[0093] (twenty four)

[0094] Where vars represents the design variable vector ( , , , ), lb and ub are the upper and lower bounds of the design variables, determined by the constraints described in equations (22) and (23), that is:

[0095] (25)

[0096] That is, the four design variables mentioned above , , , Each of the following conditions must be met:

[0097] < ≤1, 10 −3 ≤ ≤1、 ≤ ≤ 10 −3 ≤ ≤1;

[0098] Step 3.2) Solve the multi-objective optimization problem described in equation (24) using the NSGA-II algorithm to obtain the Pareto solution set that reflects the trade-off between the optimization objectives J1 and J2. Define the point on the Pareto solution set with the largest vertical distance between it and the chord connecting the two extreme points as the knee point, and denote the point K on the Pareto solution set as K( , The perpendicular distance from the line L: ax + by + c = 0 connecting the two extreme Pareto solutions is:

[0099] (26)

[0100] In the formula, K ( , () represents any point on the Pareto solution set, the line L is the chord connecting the two extreme Pareto solutions, a, b, and c are the coefficients of this line, and the perpendicular distance d(K, L) is used to measure the degree of deviation of point K from the line connecting the extreme points. The Pareto solution with the largest deviation is defined as the knee point; the design parameter vars corresponding to the knee point. opt As a representative optimal solution of the NS-TLCDI system considering the SSI effect;

[0101] Step 3.3) Calculate the optimal NS-TLCDI parameters vars at the knee of the Pareto solution set obtained by NSGA-II optimization. opt Substituting the dynamic equations shown in equation (11) and solving them using the step-by-step integration method, we obtain the time history responses of the relative displacement and absolute acceleration of the work platform in the NS-TLCDI system under external excitation, considering the SSI effect. By comparing and analyzing the response results with those of the uncontrolled SSI system, we can verify the effectiveness of the designed vibration damper and its parameters in suppressing the response of the work platform.

[0102] Beneficial effects: Compared with existing technical solutions, the technical solution of the present invention has the following beneficial technical effects:

[0103] (1) This invention proposes a novel vibration reduction system configuration of a negative stiffness-inertial volume tuned liquid column damper and establishes a coupled dynamic model of the support foundation, the working platform and the NS-TLCDI that simultaneously considers the translational motion of the foundation, the rotation of the foundation, the vibration of the working platform and the nonlinear damping effect of the liquid column. Compared with the existing design method based on the assumption of rigid foundation, it can more realistically reflect the dynamic coupling characteristics of the working platform, the support foundation and the additional control device under complex site conditions, and improve the accuracy of parameter design and the applicability to actual engineering.

[0104] (2) This invention introduces a frequency domain multi-objective optimization method with the relative displacement response and root mean square response of absolute acceleration of the working platform as the objectives, and combines the Pareto solution set and knee point criterion to determine the representative parameter solution. This can overcome the shortcomings of traditional single-objective design methods that are difficult to take into account both displacement control and acceleration control, and improve the comprehensive vibration reduction performance and engineering applicability of the parameter design scheme.

[0105] (3) This invention is applicable to the design and performance evaluation of NS-TLCDI parameters under different soil conditions. It can analyze the influence of SSI effect on the optimal design parameters and control effect of NS-TLCDI, and further verify the effectiveness of the designed parameters through actual ground motion time history response. It provides theoretical basis and design reference for the seismic and vibration reduction design of work platforms under complex site conditions. Attached Figure Description

[0106] Figure 1 A dynamic model of a negative stiffness-inertial-capacitance tuned liquid column damper considering the SSI effect (SSI+NS-TLCDI);

[0107] Figure 2 The dynamic model of the negative stiffness-inertial volume tuned liquid column damper under the assumption of rigid foundation (FB+NS-TLCDI) is presented.

[0108] Figure 3 When the inertia-to-mass ratio γ = 0.10 and the negative stiffness ratio Pareto solution set for multi-objective optimization of NS-TLCDI vibration damper considering SSI effect under different soil types when =−0.10;

[0109] Figure 4 When the inertia-to-mass ratio γ = 0.10 and the negative stiffness ratio Comparison of the relative displacement transfer functions of the working platform for each model under different soil types when =−0.10;

[0110] Figure 5 When the inertia-to-mass ratio γ = 0.10 and the negative stiffness ratio Comparison of the absolute acceleration transfer functions of the working platform for each model under different soil types when =−0.10;

[0111] Figure 6 When the inertia-to-mass ratio γ = 0.10 and the negative stiffness ratio When =−0.10, the root mean square of the response of the operating platform under earthquake in soft soil site considering SSI effect is compared between uncontrolled and NS-TLCDI controlled platforms.

[0112] Figure 7 When the inertia-to-mass ratio γ = 0.10 and the negative stiffness ratio Comparison of time history response of uncontrolled and NS-TLCDI controlled operating platforms under earthquake conditions in soft soil sites when considering SSI effect at 0.10. Specific implementation methods

[0113] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. The present invention will be further elaborated below with reference to specific embodiments.

[0114] like Figure 1 As shown, this invention proposes a parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes. The implementation of this invention will be described in detail below with reference to the accompanying drawings and specific examples, so that the advantages and features of this invention can be more easily understood by those skilled in the art. In the example, the relevant parameters of the working platform are shown in Table 1, and the system parameters are selected as: μ d =0.05, μ C =0.01, α=0.9, γ=0.10 and κ=−0.10.

[0115] Table 1. Parameters of the operating platform and its supporting foundation in the NS-TLCDI system considering the SSI effect.

[0116]

[0117] Step 1):

[0118] The characteristic parameters of the site soil used in step 1.1) include shear wave velocity, Poisson's ratio and density, as well as the translational and rotational stiffness and damping of the foundation calculated by equation (1), which are listed in Table 2.

[0119] Table 2 Soil parameter values ​​for the system considering SSI effect

[0120]

[0121] Step 1.2) Consider the interaction between the support foundation of the work platform and the ground, and the motion control equation of the whole system is shown in equation (2).

[0122] Step 2):

[0123] Step 2.1) Linearize the nonlinear damping term of the liquid by using the equivalent transformations described in equations (3) to (9).

[0124] Step 2.2) The equivalent liquid damping ratio shown in equation (9) Substituting the above motion control equations, the linearized motion control equations of the system can be rewritten as shown in equation (10). Let the state variable vector x be [xyzu θ]. T Then the governing equations can be rearranged into a dimensionless form and substituted with specific numerical values ​​(taking parameter combinations under soft soil as an example):

[0125]

[0126] in,

[0127] ,

[0128] ,

[0129] ,

[0130]

[0131] Step 2.3) Solve the dimensionless governing equations by Laplace transform to obtain several transfer functions required for subsequent calculations, and their expressions are shown in equations (12) to (15).

[0132] Step 2.4) Considering the SSI effect, the horizontal absolute acceleration transfer function of the working platform relative to its supporting foundation is shown in equation (16). Let the one-sided power spectral density function of the input seismic acceleration be... To simulate random ground motion input, the Clough-Penzien (CP) power spectral density model is used, with the following power spectral density function: It is given by equation (17).

[0133] Among them, the peak ground acceleration of random ground motion is taken as 0.1g, and the CP spectrum parameters and the spectrum intensity S0 calculated by formula (18) under different soil types are listed in Table 3.

[0134] Table 3. Values ​​of CP spectral parameters under different soil types

[0135]

[0136] The root mean square of the relative displacement response of the working platform relative to its supporting foundation is then determined. Root mean square of absolute acceleration response They can be represented as equations (19) to (20) respectively.

[0137] Step 3:

[0138] Step 3.1) Introduce the dimensionless root mean square of relative displacement J1 and root mean square of absolute acceleration J2 of the working platform shown in equation (21), and calculate the optimization parameter constraints based on the stability condition in equation (22) and the size constraint of the storage tank in equation (23) (considering that the minimum and maximum size of the storage tank are 5 m and 50 m respectively). and The multi-objective optimization problem under consideration can then be expressed as shown in formula (24). The corresponding upper and lower limits lb and ub, substituted with specific values, are:

[0139] ;

[0140] Step 3.2) Solve the multi-objective optimization problem described in equation (24) using the NSGA-II algorithm to obtain the Pareto solution set that reflects the trade-off between optimization objectives J1 and J2, and select the knee point on the Pareto solution set according to equation (26). The design parameters corresponding to the knee point are taken as the optimal solution of the NS-TLCDI system design parameters when considering the SSI effect. Figure 3 For the inertial mass ratio =0.10, negative stiffness ratio When =−0.10, the multi-objective optimization results of the NS-TLCDI system considering the SSI effect are compared under different soil types. It can be seen from the figure that there are significant differences in the location of the Pareto solution set and the distribution of the knee point under different soil types. Among them, the Pareto solution set under hard soil conditions is closer to the origin of the J1-J2 target plane coordinates, followed by medium-hard soil, and the farthest is soft soil.

[0141] Figure 4 and Figure 5 The optimal transfer function curves obtained under three soil types were compared under two design conditions: one based on the rigid foundation assumption and the other considering the SSI effect. The corresponding peak values ​​of the transfer functions are listed in Table 4. Figure 4 (a), (b) and Figure 5 As shown in (a) and (b), under hard and medium-hard soil conditions, regardless of whether the SSI effect is considered in the optimal parameter design, the peak values ​​of the relative displacement transfer function and the absolute acceleration transfer function of the working platform are very close. This indicates that under relatively hard site conditions, the SSI effect has a limited impact on the dominant resonance characteristics of the system, and the NS-TLCDI design based on the rigid foundation assumption can still maintain good control. However, under soft soil conditions, significant differences appear between the two types of design results, such as... Figure 4 (c) and Figure 5 As shown in (c) of the table. When considering the SSI effect, the equivalent stiffness of the coupling system of the working platform, supporting foundation, and ground is significantly reduced, and its dominant resonance range will shift to the low-frequency range. Affected by this, the NS-TLCDI parameters optimized based on the rigid foundation assumption will be detuned with the resonance characteristics of the actual system considering the SSI effect. This detuning will cause the NS-TLCDI designed for rigid foundations to be directly applied to soft soil foundations, and the low-frequency response of the system will be significantly amplified. Thus, the designed NS-TLCDI fails to achieve the best vibration reduction effect. Taking the results in Table 4 as an example, the peak value of the relative displacement transfer function of the working platform under soft soil increases from 0.9280 to 1.0477, and the peak value of the absolute acceleration transfer function increases from 0.9398 to 1.2171, indicating that the vibration control performance has deteriorated significantly. In contrast, when the SSI effect is explicitly considered in the optimization design, the obtained NS-TLCDI can better match the actual dynamic characteristics of the coupling system, and the resonance peak value of the working platform is effectively reduced. The corresponding results are shown in Table 4. Figure 4 (c) and Figure 5 As shown by the green dashed line in (c).

[0142] The above results indicate that the rigid foundation assumption remains applicable to hard and medium-hard soil conditions, but it no longer holds true for soft soil conditions. Ignoring the SSI effect in this case will lead to significant detuning in the NS-TLCDI optimization design, thereby greatly weakening its vibration reduction performance. Therefore, when conducting NS-TLCDI parameter studies on soft soil sites, the SSI effect must be incorporated into the analysis framework.

[0143] Table 4. Peak values ​​of the optimal transfer function considering the rigid foundation assumption and SSI effect.

[0144]

[0145] Step 3.3) Figure 6 The root mean square (RMS) responses of the uncontrolled system (SSI in the legend) and the work platform system controlled by NS-TLCDI (SSI+NS-TLCDI in the legend) under selected seismic events are presented under soft soil conditions, considering the SSI effect. For the uncontrolled case, the response dispersion between different seismic records is quite significant; however, after installing NS-TLCDI, the RMS responses of the work platform corresponding to most seismic records under soft soil are significantly reduced, indicating that NS-TLCDI can effectively and stably reduce the relative displacement and absolute acceleration responses of the work platform. To illustrate the time history response differences between the two systems under seismic events where the RMS responses are still relatively large in some cases, the 14th record under soft soil (the Iwate earthquake) is selected as a representative example; the corresponding time history responses are shown below. Figure 7 .Depend on Figure 7 As shown in (a) of the diagram, under the Iwate earthquake, the peak relative displacement response of the uncontrolled work platform was 0.0308 m, which was reduced to 0.0179 m by NS-TLCDI; Figure 7 As can be seen in (b), the peak absolute acceleration of the uncontrolled work platform is 0.0828 m·s⁻¹. −2 After being controlled by NS-TLCDI, the concentration can be further reduced to 0.0609 m·s. −2 Furthermore, starting from approximately 40 seconds, the NS-TLCDI significantly promoted the vibration attenuation of the work platform, with its long-term response attenuation rate being significantly faster than that of the uncontrolled platform. This indicates that although the NS-TLCDI-controlled work platform exhibited a relatively large root mean square response during the duration of the earthquake in the Iwate earthquake, its response still showed a faster attenuation trend.

[0146] In summary, under the action of the selected actual seismic records, the optimal NS-TLCDI based on the multi-objective optimization framework of this invention can achieve excellent vibration control performance at both the root mean square response and peak response levels.

[0147] The above description is merely an embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural or procedural transformations made based on the content of the present invention's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. A parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes, characterized in that, The method includes the following steps: Step 1) Establish a system coupling dynamics model considering the SSI effect for the operating platform, supporting foundation, and NS-TLCDI; Step 2) Derive the transfer function and root mean square response of the NS-TLCDI system considering the SSI effect based on the NS-TLCDI system dynamic model; Step 3) Use the NSGA-II multi-objective algorithm to optimize the displacement and acceleration indices of the work platform based on the root mean square of the response, and determine the optimal design parameters of the NS-TLCDI system considering the SSI effect.

2. The parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes, as described in claim 1, is characterized in that... The specific method for step 1) is as follows: Step 1.1) The mass of the working platform is m1, which is connected to the support foundation of the working platform through a support arm with horizontal stiffness k1 and horizontal damping c1. The mass of the support foundation is m0, and its interaction with the ground is through the horizontal stiffness k. h and horizontal damping c h and rotational stiffness k r and rotational damping c r Equivalent representation: (1) Where R is the equivalent circular base radius; G, ν, ρ s and V s These are the soil's shear modulus, Poisson's ratio, density, and shear wave velocity, respectively. The designed negative stiffness-inertia-capacity tuned liquid column damper NS-TLCDI is placed on the working platform. It consists of a U-shaped liquid column storage tank and its connecting elements. The cross-sectional area of ​​the U-shaped storage tank is A, the length of the horizontal section is B, and the total length of the internal U-shaped liquid column is denoted as L. A grid is provided at the center of the horizontal section of the storage tank, and its local fluid resistance characteristics are determined by the head loss coefficient. Characterization; let the mass of the container itself be M. C The internal fluid density is The total mass of NS-TLCDI is expressed as AL+M C The NS-TLCDI's storage tank is connected to the work platform via a connector with horizontal stiffness k2 and horizontal damping c2, and is connected in parallel to an inertial element with inertial capacity b and horizontal negative stiffness k. n The negative stiffness element is connected to the support foundation of the work platform; h represents the vertical height of the work platform from its support foundation, a g This represents the random ground acceleration excitation experienced by the system; the work platform, support foundation, and NS-TLCDI as a whole are denoted as SSI+NS-TLCDI; Step 1.2) Considering the interaction between the support foundation of the work platform and the ground, the entire system is represented as a multi-degree-of-freedom coupled dynamic system, including the horizontal displacement x of the work platform relative to its support foundation, the horizontal displacement y of the liquid storage tank relative to the support foundation of the work platform, the vertical vibration displacement z of the liquid column, the horizontal translational displacement u of the support foundation, and the rotational displacement θ of the support foundation. Its motion control equations are as follows: (2) Where g is the gravitational acceleration, J0 is the moment of inertia based on the equation, and the dots above the variables represent the time derivative of the variables. A single dot represents the first derivative, and two dots represent the second derivative.

3. The parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes, as described in claim 2, is characterized in that... The specific method for step 2) is as follows: Step 2.1) To calculate the transfer function of the NS-TLCDI system, an equivalent linearization coefficient λ is introduced. eq Let represent the coefficient of the original nonlinear fluid damping term. Then, the error ε between the nonlinear term and its equivalent linearized form is expressed as: (3) (4) in, and Let λ represent the natural circular frequency and equivalent liquid damping ratio of the liquid column in the storage tank, respectively. To minimize the mean square error ε, according to the extreme value theory, λ eq satisfy: (5) in, λ represents the expectation operation. eq By solving equation (5), we obtain: (6) Assumption If the signal is a stationary Gaussian signal with zero mean, then: (7) in, The root mean square of the liquid column velocity inside the storage tank is represented by equation (7). Substituting equation (7) into equation (6), we get λ. eq Simplified to: (8) Substituting equation (8) into equation (4) yields the equivalent liquid damping ratio. : (9) Step 2.2) The equivalent liquid damping ratio shown in equation (9) Substituting into equation (2), the linearized motion control equation of the system is rewritten as: (10) Let the state variable vector x be [xyzu θ] T The governing equation (10) is rewritten in the following dimensionless form: (11) Where M, C, K, and F are the mass matrix, damping matrix, stiffness matrix, and seismic excitation vector of the NS-TLCDI system considering the SSI effect, respectively: Step 2.3) Solve the dimensionless governing equations using the Laplace transform to obtain several transfer functions required for the calculations. Their expressions are as follows: Horizontal displacement transfer function of the working platform relative to its supporting foundation: (12) Liquid velocity transfer function within the storage tank: (13) Translational displacement transfer function of the supporting foundation: (14) Rotational displacement transfer function of the supporting foundation: (15) in, A represents the angular frequency in the frequency domain. i B i C i D i E i The polynomial expression is as follows: ； ； ； ； ； ； Step 2.4) Considering the SSI effect, the horizontal absolute acceleration transfer function of the working platform relative to its supporting foundation. : (16) Let the power spectral density function of the input seismic acceleration be... To simulate random ground motion input, the Clough-Penzien (CP) power spectrum model is used, and its expression is as follows: (17) in, and These are the superior circular frequency and damping ratio of the soil on which the supporting foundation is placed; and These are the CP spectrum filtering angular frequency and damping ratio, respectively; where the CP spectrum intensity parameter S0 is determined based on the given peak ground acceleration. (18) in, Representing the peak ground acceleration due to random seismic motion, the root mean square of the relative displacement response of the work platform relative to its supporting foundation is... Root mean square of absolute acceleration response They are represented as follows: (19) (20)。 4. The parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes, as described in claim 3, is characterized in that... The calculation methods for M, C, K, and F are as follows: ， ， ， ； in, The mass ratio of the liquid in the storage tank. ; The mass ratio of the storage tank. ; The NS-TLCDI quality ratio ; The ratio of the supporting foundation quality of the operating platform. ; The SSI effect scaling ratio. ; The ratio of liquid column lengths, ; The inherent circular frequency of the working platform, ; This is the inherent angular frequency of NS-TLCDI. The natural circumfrequency of the liquid. ; The horizontal angular frequency of SSI. ; The angular frequency in the direction of SSI rotation. ; The damping ratio of the work platform, ; For NS-TLCDI damping ratio, ; The horizontal damping ratio of SSI. ; The SSI rotational direction damping ratio, ; Inertia-mass ratio, ; It is a negative stiffness ratio. ; The frequency ratio of NS-TLCDI ; The ratio of liquid frequencies. .

5. The parameter optimization design method for a negative stiffness-inertia-capacity tuned liquid column damper considering the SSI effect under random earthquakes, as described in claim 4, is characterized in that... The specific method for step 3) is as follows: Step 3.1) To unify the dimensions and orders of magnitude of the root mean square of relative displacement and root mean square of absolute acceleration of the working platform in multi-objective optimization, dimensionless root mean square of relative displacement J1 and root mean square of absolute acceleration J2 are introduced based on equations (19) and (20): (21) in, and Let represent the root mean square of the relative displacement response and the root mean square of the absolute acceleration response of the work platform when the system is not controlled by NS-TLCDI, respectively. Before optimizing the NS-TLCDI system considering the SSI effect, the range of values ​​of the variables needs to be determined. First, the positive definite condition of the stiffness matrix is ​​used to ensure the stability of the system, that is, the stiffness matrix K in equation (11) should satisfy the positive definiteness requirement det(K)>0, and the NS-TLCDI frequency ratio is obtained. Compared with negative stiffness The constraints that should be satisfied between them are as follows: (22) Secondly, the total length L of the liquid column is limited by the structural dimensions of the storage tank and must meet the following requirements: min ≤L≤L max , where L min and L max Given the minimum and maximum dimensions of the storage tank, respectively, the liquid frequency ratio is... The constraints are as follows: (23) The multi-objective optimization problem under consideration is formulated as follows: (24) Where vars represents the design variable vector ( , , , ), lb and ub are the upper and lower bounds of the design variables, determined by the constraints described in equations (22) and (23), that is: (25) That is, the four design variables mentioned above , , , Each of the following conditions must be met: < ≤1、10 −3 ≤ ≤1、 ≤ ≤ 、10 −3 ≤ ≤1； Step 3.2) Solve the multi-objective optimization problem described in equation (24) using the NSGA-II algorithm to obtain the Pareto solution set that reflects the trade-off between the optimization objectives J1 and J2. Define the point on the Pareto solution set with the largest vertical distance between it and the chord connecting the two extreme points as the knee point, and denote the point K on the Pareto solution set as K( , The perpendicular distance from the line L: ax + by + c = 0 connecting the two extreme Pareto solutions is: (26) In the formula, K ( , () represents any point on the Pareto solution set, the line L is the chord connecting the two extreme Pareto solutions, a, b, and c are the coefficients of this line, and the perpendicular distance d(K, L) is used to measure the degree of deviation of point K from the line connecting the extreme points. The Pareto solution with the largest deviation is defined as the knee point; the design parameter vars corresponding to the knee point. opt As a representative optimal solution of the NS-TLCDI system considering the SSI effect; Step 3.3) Calculate the optimal NS-TLCDI parameters vars at the knee of the Pareto solution set obtained by NSGA-II optimization. opt Substitute the dynamic equation shown in equation (11) and solve it using the step-by-step integration method to obtain the time history response of the relative displacement and absolute acceleration of the working platform in the NS-TLCDI system under external excitation considering the SSI effect.

Images