Design and Parameter Optimization of the Soil-Structure Interaction on Structures with Electromagnetic Damper

Abstract

Currently, the application of electromagnetic dampers in structural vibration control and energy harvesting has become increasingly widespread. The optimization research of electromagnetic dampers in building design has also received more attention. Previous studies on vibration control of building structures with electromagnetic dampers have been conducted under fixed foundations, neglecting the effect of soil-structure interaction on building structures with electromagnetic dampers. The main contribution of this paper is to fill the research gap in the study of building structural vibration control with electromagnetic dampers considering soil-structure interaction. An effective design and parameter optimization method for building structures with both soil-structure interaction and electromagnetic energy harvesting is explored. The soil-structure interaction is taken into account, and the building model with electromagnetic dampers is improved to form a coupled vibration reduction system with both structural vibration control and energy harvesting functions. The dynamic equations of the system with both structural vibration control and energy harvesting are derived and then optimized using the $H_2$ norm criterion and Monte Carlo-mode search method. A single-layer building structure is used as an example to study the influence of soil-structure interaction on building structures equipped with electromagnetic dampers under strong earthquake action. The dynamic response and energy harvesting of building structures under earthquake action considering soil-structure interaction are analyzed and evaluated. The results show that the influence of soil-structure interaction on building structures equipped with electromagnetic dampers needs to be considered. As the soil density decreases, the dynamic response of the building structure under earthquake action becomes larger using the electromagnetic damper system. Compared to the use of fixed foundations, the energy harvesting effect of building structures with electromagnetic dampers is weakened when considering soil-structure interactions.Definition:

1. Introduction

With the acceleration of urbanization, research on earthquake-resistant building structures has become a hot topic. Structural vibration control is a research direction that focuses on reducing the dynamic response of building structures under dynamic loads such as wind and earthquakes. From the development of structural vibration control techniques from the early days of passive control techniques, the role of passive control systems in the mitigation of seismic damage was explored [1]. Researchers have considered the damping effect of three different passive control devices (TMD, TLC, and TLCD) for structural vibration control and the performance at different total masses [2]. These research results provide a practical reference for the application of passive control systems in seismic hazards. To the later active control techniques [3,4], the biggest difference with passive control is that it requires a damping control technique with the help of external loads. As for the current semi-active control technology [5], it is generally adjusted automatically by means of control mechanisms to achieve the seismic loads that the building structure can withstand. Symans et al. provide a comprehensive evaluation and analysis of the use of semi-active control systems for the protection of structures from earthquakes [6]. However, passive controllers have been widely used due to their advantages, such as simple structure, low cost, and easy maintenance.

Traditional passive controllers usually use linear or nonlinear dampers, such as viscous dampers, friction dampers, and viscoelastic dampers [7,8]. These dampers can effectively dissipate the vibration energy of the structure and thus reduce the displacement and acceleration response of the building structure. However, these dampers also have some drawbacks [9,10], such as the damping force is proportional to the relative velocity or displacement, which cannot adjust the damping force magnitude adaptively; the damping force is in phase with the structural response, which cannot improve the stiffness and ductility of the building structure; the dampers themselves cannot generate energy, but can only convert energy into forms such as thermal or acoustic energy. To overcome these drawbacks, new dampers such as MR damper [11,12] and electromagnetic dampers [13,14,15,16,17,18,19,20,21,22] based on electromagnetic effects have been widely used in structural vibration control. Among them, electromagnetic dampers have attracted the attention of researchers. An electromagnetic damper is a device that uses the principle of electromagnetic induction to generate viscous damping forces. Electromagnetic dampers have the following advantages: (1) electromagnetic dampers can adjust the size and direction of damping force according to different load types and sizes; (2) electromagnetic dampers can generate damping force in anti-phase with the structural response, thus improving the stiffness and ductility of building structures; (3) electromagnetic dampers can convert the vibration energy of the structure into electrical energy and store or utilize it through the load. In recent years, electromagnetic dampers have been widely studied and applied as a device with dual functions of structural vibration damping and energy harvesting [22,23,24]. Hongping Zhu proposed the first solution to simultaneously mitigate vibration and energy harvesting through the use of EIMD [19]. Wenai Shen proposed the Tuned Inertial Mass Electromagnetic Dampers (TIMED) to effectively mitigate the seismic response of building structures [20].

However, most of the current studies have been conducted based on fixed base structure models [19,20,21,22,25,26,27,28], neglecting the effects on building structures using electromagnetic dampers when soil-structure interactions are considered. Soil-structure interaction refers to the additional inertia forces and additional stiffnesses resulting from the interaction between the soil and the building structure. Researchers presented the seismic response of foundation-isolated structures considering soil-structure interaction, which was verified using shake table tests [29] and numerical simulations to investigate the effect of dynamic soil-structure interaction in shake table tests [30]. In addition, the displacement and acceleration response of the structure is also affected by the soil elasticity [31,32]. The soil-structure interaction changes the properties of the structure, such as the intrinsic frequency, vibration type, damping ratio, displacement, and acceleration [33], and affects the design and optimization of the controller parameters [34,35,36,37,38]. Therefore, it is important to consider soil-structure interactions for structural vibration control under practical engineering conditions. During the 1970s, researchers were interested in the effect of soil-structure interaction on the dynamic response of single- and multi-degree-of-freedom structures [39,40,41,42]. Researchers have analyzed the effect of soil-structure interaction on the seismic performance of structures using different damping techniques [43,44,45,46,47,48,49,50]. In recent years, there has been much interest in the control of structural vibrations in high-rise buildings considering soil-structure interaction under earthquake or wind loads. Different control strategies have been investigated, including seismic pile-raft foundations, tuned mass dampers (TMDs), and foundation isolation, all aiming to improve the seismic capacity and structural stability of high-rise buildings [51,52,53]. In all the above studies, the influence of soil-structure interaction effects in building structures with electromagnetic dampers was not considered.

This paper focuses on the effect of soil-structure interaction on the damping performance and energy harvesting effect of building structures with electromagnetic dampers. First, we need to build a model of a single-degree-of-freedom system considering soil-structure interaction, which is connected by a transducer electromagnet. The structure and the soil are jointly considered as a composite system, and the dynamics formulation is established. In terms of parameter optimization, parameter optimization methods based on the $H_2$ parametric criterion [37,38] and the Monte Carlo-mode search method (MCSM) [54,55,56] are applied. The parameters are evaluated and adjusted in combination with the seismic performance index of the structure to improve the vibration resistance of the building structure. For the numerical analysis in the frequency domain, the effects on the vibration-damping performance of the structure were compared under different control systems and different mass ratios. The stability of the system is studied by adjusting the soil type. In the time domain numerical analysis, the dynamics formulation is established. After setting the simulation parameters for the building structure, we analyze the effect on displacement and acceleration response in different cases according to the response characteristics under different seismic excitations. The effect of soil-structure interaction effect on the energy harvesting effect is also verified.

2. Theoretical Analysis

2.1. Analytical Models

Figure 1a shows the single degree of freedom structural mechanics model on a fixed foundation. Figure 1b,c show the mechanical models considering electromagnetic dampers. In Figure 1b,c, according to the linear generator principle, it can be equivalent to an inductance L and an internal resistance $R$ (where $R_e$ and $R_i$ are connected in series).

In Figure 1a,b, it is assumed that the soil behaves as a rigid body, and the seismic excitation is directly transmitted to the structure.

In Figure 1c, it is necessary to incorporate soil factors into the system and consider the damping and stiffness effects of the soil. The commonly used method is to treat the structure and soil as a composite system and add coupling dampers to mitigate the impact of seismic excitation on the soil and structure. This model can reduce the response of vibration and improve the seismic performance of actual building structures.

The structural coupling model under seismic excitation with ${\ddot{x}}_g$ is shown in Figure 1. The controlled mode has a mass of $m_1$, stiffness of $k_1$, and damping of $c_1$. For the soil-structure system, the foundation has a mass of $m_0$, a horizontal vibration stiffness of $k_0$, a damping coefficient of $c_0$ for the horizontal vibration of the foundation, a rotational vibration stiffness of $k_{\phi }$, a damping coefficient of $c_{\phi }$ for the rotational vibration of the foundation, and a height of $h$. $e$ is the induced electromotive force generated when the coil moves. The magnitude of the induced electromotive force is proportional to the magnetic field strength and the relative moving speed and $e_{EMF}=k_v\dot{x}$, where $k_v$ is the velocity-dependent counter-electromotive force coefficient $\dot{x}$ is the relative velocity; then, the current in the loop generates a counter-electromotive force, which is proportional to the current in the loop, and the direction of the force is always opposite to the direction of the current in the loop The direction of the force is always opposite to the direction of the current in the loop. The counter-electromotive force is $f_{EMF}=k_f\dot{q}$,where $k_f$ is the force constant. In the ideal model $k_f=k_v$

2.2. Governing Equation of Motion

Research has shown that compared to fixed foundation systems, soil-structure interaction systems have longer natural periods. Therefore, the effect of soil-structure interaction should be considered when considering the structural earthquake response and the damping effect of energy-dissipating devices. Usually, two analytical methods, the substructure method and the direct method, are used when considering soil-structure interactions. The substructure method separates the soil and structure for analysis and then gradually analyzes them. Finally, considering the compatibility of the contact surface between the soil and structure, they are combined into a soil-structure model. The direct method analyzes the soil and structure as a whole without considering the contact surface between the soil and structure. The direct method has a smaller calculation amount compared to the substructure method, but the accuracy is relatively low. In practical engineering, the choice of which method to use depends on specific circumstances, taking into account factors such as calculation accuracy and efficiency.

In this paper, it is assumed that the structure undergoes small displacement vibration, and the impact of the soil on the upper building structure is simulated based on Richard’s classical theory and the swing damping stiffness system. The model can be considered a dual-degree-of-freedom system structure. Based on the d’Alembert principle, the dynamic equation of the coupled structure is established as follows:

$$\left\{\begin{array}{c}{m}_1({\ddot{x}}_0+{\ddot{x}}_1+\ddot{\theta }h)+c_1{\dot{x}}_1+k_1{x}_1+k_{f}I=-m_1{\ddot{x}}_g\\ k_v({\dot{x}}_1-{\dot{x}}_0-\dot{\theta }h)+R_{c}I+L\dot{I}=0\\ m_1({\ddot{x}}_0+{\ddot{x}}_1+\ddot{\theta }h)+m_0{\ddot{x}}_0+c_0{\dot{x}}_0+k_0{x}_0=-(m_1+m_0){\ddot{x}}_g\\ m_{1}h({\ddot{x}}_0+{\ddot{x}}_1+\ddot{\theta }h)+(I_0+I_1)\ddot{\theta }+c_{\theta }\dot{\theta }+k_{\theta }\theta =-m_{1}h{\ddot{x}}_g\end{array}\right.$$

(1)

where and $I_1$ and $I_0$ are the moment of inertia of the structure and soil.

Equation (1) is obtained by the Laplace transform, which is expressed as:

$$\left\{\begin{array}{c}{m}_1(x_0+x_1+\theta h)s^2+c_1{x}_{1}s+k_1{x}_1+k_{f}I=-m_1{\ddot{x}}_g\\ k_v(x_1-x_0-\theta h)s+R_{c}I+LIs=0\\ m_1(x_0+x_1+\theta h)s^2+m_0{x}_0{s}^2+c_0{x}_{0}s+k_0{x}_0=-(m_1+m_0){\ddot{x}}_g\\ m_{1}h(x_0+x_1+\theta h)s^2+(I_0+I_1)\theta s^2+c_{\theta }\theta s+k_{\theta }\theta =-m_{1}h{\ddot{x}}_g\end{array}\right.$$

(2)

The parameters are defined as:

$$\begin{gathered} s=j\omega \mu =\frac{m_0}{m_1} {\omega }_1^2=\frac{k_1}{m_1} \\ {\omega }_0^2=\frac{k_0}{m_0} {\zeta }_e=\frac{R_c}{2L{\omega }_1} {\zeta }_0=\frac{c_0}{2m_0{\omega }_0} \\ {\zeta }_1=\frac{c_1}{2m_1{\omega }_1} {\omega }_{\theta }^2=\frac{k_{\theta }}{m_1{h}^2} {\zeta }_{\theta }=\frac{c_{\theta }}{2m_0{\omega }_0} \end{gathered}$$

(3)

A preliminary dimensionless analysis of Equation (2) by substituting Equation (3) can be simplified as:

$$\left\{\begin{array}{c}(x_0+x_1+\theta h){\left(j\omega \right)}^2+{2\omega }_1{{\zeta }_{1}x}_1\left(j\omega \right)+{\omega }_1^2{x}_1+\frac{k_{f}I}{m_1}=-{\ddot{x}}_g\\ k_v(x_1-x_0-\theta h)\left(j\omega \right)+{2\omega }_{1}LI{\zeta }_e+LI\left(j\omega \right)=0\\ \left(x_0+x_1+\theta h\right){\left(j\omega \right)}^2+\mu x_0{\left(j\omega \right)}^2+2\mu {\omega }_0{\zeta }_0{x}_0\left(j\omega \right)+\mu {{\omega }_0^{2}x}_0=\\ -(1+\mu ){\ddot{x}}_g\\ h\left(x_0+x_1+\theta h\right){\left(j\omega \right)}^2+\left(\mu r_0^2+r_1^2+h^2\right)\theta {\left(j\omega \right)}^2+2\mu {\zeta }_{\theta }{\omega }_0\theta \left(j\omega \right)\\ +{\omega }_{\theta }^2\theta h^2=-h{\ddot{x}}_g\\ .\end{array}\right.$$

(4)

The parameters are defined as:

$$\alpha =\frac{\omega }{{\omega }_1};f_1=\frac{{\omega }_1}{{\omega }_0};f_{\theta }=\frac{{\omega }_{\theta }}{{\omega }_0}$$

(5)

Substituting Equation (5) into Equation (4), obtain:

$$\begin{array}{c}(x_0+x_1+\theta h){\left(j\alpha \right)}^2+2{{\zeta }_{1}x}_1\left(j\alpha \right)+x_1+\frac{k_{f}I}{k_1}=-\frac{{\ddot{x}}_g}{{\omega }_1^2}\\ k_v(x_1-x_0-\theta h)\left(j\alpha \right)+2LI{\zeta }_e+LI\left(j\alpha \right)=0\\ (x_0+x_1+\theta h){\left(j\alpha \right)}^2+\mu x_0{\left(j\alpha \right)}^2+\frac{2\mu {\zeta }_0{x}_0\left(j\alpha \right)}{f_1}+\frac{\mu x_0}{f_1^2}=-\frac{(1+\mu ){\ddot{x}}_g}{{\omega }_1^2}\\ h\left(x_0+x_1+\theta h\right){\left(j\alpha \right)}^2+\left(\mu r_0^2+r_1^2+h^2\right)\theta {\left(j\alpha \right)}^2+\frac{2\mu {\zeta }_{\theta }\theta \left(j\alpha \right)}{f_1}\\ +\frac{f_{\theta }^2{h}^2\theta }{f_1^2}=-\frac{h{\ddot{x}}_g}{{\omega }_1^2}\\ .\end{array}$$

(6)

In Equation (6), where μ is the mass ratio between the foundation and the main structure; ${\omega }_1$ is the frequency of the main structure; ${\zeta }_1$ is the damping ratio of the main structure; ${\omega }_0$ is the frequency of the foundation; ${\zeta }_0$ is the damping ratio of the foundation; ${\zeta }_e$ is the electronic damping ratio; $\alpha$ is the external excitation frequency ratio; $f_1$ is the structural frequency ratio; $f_{\theta }$ is the frequency ratio of the foundation.

The relative motion between the structure and the soil is the focus of this study. The small relative motion between the upper structure and the soil ensures that the upper structure withstands lower levels of horizontal vibration, making it safer. The normalized transfer function from ground motion ${\ddot{x}}_g/{\omega }_1^2$ to the relative displacement between the upper structure and the soil can be solved from Equation (6) and derived as Equation (7).

$$\begin{gathered} H_i\left(j\alpha \right)=\frac{X_1-X_0}{{\ddot{x}}_g/{\omega }_1^2} \\ =\frac{B_0{\left(j\alpha \right)}^5+B_1{\left(j\alpha \right)}^4+B_2{\left(j\alpha \right)}^3+B_3{\left(j\alpha \right)}^2+B_4\left(j\alpha \right)+B_5}{{A_0\left(j\alpha \right)}^7+A_1{\left(j\alpha \right)}^6+A_2{\left(j\alpha \right)}^5+A_3{\left(j\alpha \right)}^4+A_4{\left(j\alpha \right)}^3+A_5{\left(j\alpha \right)}^2+A_6\left(j\alpha \right)+A_7} \end{gathered}$$

(7)

where the electromechanical coupling coefficient is ${\mu }_k$, and j is the imaginary unit.

$${\mu }_k=\frac{k_v{k}_f}{k_{1}L};j=\sqrt{-1}$$

(8)

The numerator coefficients in Equation (7) are:

$$B_0=f_1^4{h}^2\mu +f_1^4\mu \left(\mu r_0^2+r_1^2\right)$$

(9)

$$B_1=2f_1^4{h}^2\mu {\zeta }_e+2f_1^4\mu {\zeta }_e\left(\mu r_0^2+r_1^2\right)$$

(10)

$$\begin{gathered} B_2=f_1^2{h}^2\mu f_{\theta }^2-f_1^4{h}^2\mu {\mu }_k+2f_1^4{h}^2\mu -f_1^2{h}^2\mu +f_1^4{h}^2+f_1^4\mu \left(\mu r_0^2+r_1^2\right) \\ +f_1^4\left(\mu r_0^2+r_1^2\right)-f_1^2\mu \left(\mu r_0^2+r_1^2\right) \end{gathered}$$

(11)

$$\begin{array}{ll}{B}_3=2f_1^2{h}^2\mu {\zeta }_e{f}_{\theta }^2& + 4f_1^4{h}^2\mu {\zeta }_e-2f_1^2{h}^2\mu {\zeta }_e+2f_1^4{h}^2{\zeta }_e\\ & +2f_1^4\mu {\zeta }_e\left(\mu r_0^2+r_1^2\right)+2f_1^4{\zeta }_e\left(\mu r_0^2+r_1^2\right)\\ & -2f_1^2\mu {\zeta }_e\left(\mu r_0^2+r_1^2\right)\end{array}$$

(12)

$$B_4=f_1^2{h}^2\mu f_{\theta }^2-h^2\mu f_{\theta }^2+f_1^2{h}^2{f}_{\theta }^2+f_1^2{h}^2\mu {\mu }_k$$

(13)

$$B_5=2f_1^2{h}^2\mu {\zeta }_e{f}_{\theta }^2-2h^2\mu {\zeta }_e{f}_{\theta }^2+2f_1^2{h}^2{\zeta }_e{f}_{\theta }^2$$

(14)

The numerator coefficients in Equation (8) are:

$$A_0=f_1^4{h}^2\mu -f_1^4\mu \left(-\mu r_0^2-r_1^2\right)$$

(15)

$$A_1=2f_1^4{h}^2\mu {\zeta }_e-2f_1^4\mu {\zeta }_e\left(-\mu r_0^2-r_1^2\right)$$

(16)

$$\begin{array}{ll}{A}_2=f_1^2{h}^2\mu f_{\theta }^2 - & f_1^4{h}^2\left(3\mu +2\right)\mu k+2f_1^4{h}^2\mu +f_1^2{h}^2\mu +f_1^4{h}^2\\ & +f_1^4\left(\mu +2\right)\mu k\left(-\mu r_0^2-r_1^2\right)-f_1^4\mu \left(-\mu r_0^2-r_1^2\right)\\ & -f_1^4\left(-\mu r_0^2-r_1^2\right)+f_1^2\mu \left(\mu r_0^2+r_1^2\right)\end{array}$$

(17)

$$\begin{array}{ll}{A}_3=2f_1^2{h}^2\mu {\zeta }_e{f}_{\theta }^2& + 4f_1^4{h}^2\mu {\zeta }_e+2f_1^2{h}^2\mu {\zeta }_e+2f_1^4{h}^2{\zeta }_e\\ & -2f_1^4\mu {\zeta }_e\left(-\mu r_0^2-r_1^2\right)-2f_1^4{\zeta }_e\left(-\mu r_0^2-r_1^2\right)\\ & +2f_1^2\mu {\zeta }_e\left(\mu r_0^2+r_1^2\right)\end{array}$$

(18)

$$\begin{array}{ll}{A}_4=-f_1^2{h}^2(\mu + & 2)\mu k{f}_{\theta }^2+f_1^2{h}^2\mu f_{\theta }^2+h^2\mu f_{\theta }^2+f_1^2{h}^2{f}_{\theta }^2-3f_1^2{h}^2\mu \mu k\\ & +2f_1^2{h}^2\mu -f_1^2\mu \mu k\left(\mu r_0^2+r_1^2\right)+f_1^2\mu \left(\mu r_0^2+r_1^2\right)\end{array}$$

(19)

$$\begin{array}{ll}{A}_5=2f_1^2{h}^2\mu & {\zeta }_e{f}_{\theta }^2+2h^2\mu {\zeta }_e{f}_{\theta }^2+2f_1^2{h}^2{\zeta }_e{f}_{\theta }^2+4f_1^2{h}^2\mu {\zeta }_e\\ & +2f_1^2\mu {\zeta }_e\left(\mu r_0^2+r_1^2\right)\end{array}$$

(20)

$$A_6=h^2\mu f_{\theta }^2-h^2\mu \mu k{f}_{\theta }^2$$

(21)

$$A_7=2h^2\mu {\zeta }_e{f}_{\theta }^2$$

(22)

3. Parameter Optimization

To facilitate the study of the damping effect of the system, the controlled structure is set to an undamped state.

3.1. Parameter Optimization Based on $H_2$

In this subsection, the $H_2$ parametric criterion is used for the solution of Equation (7) to obtain the frequency domain characteristics of the building structure system. Thus, we can obtain the parametric evaluation index $PI$ of the relative displacement $H_i$ of the main structure.

$$PI=\frac{E\left[x_s^2\right]}{2\pi {\omega }_1{S}_0}=\frac{〈x_s^2〉}{2\pi {\omega }_1{S}_0}$$

(23)

where $E\left[·\right]$ represents the root mean square value, $〈·〉$ represents the instantaneous mean value, $S_0$ represents the energy intensity of the external excitation, and the unit is ${(m}^2·s)/rad$.

The root mean square value of the relative displacement $x_s$ of the structure can be defined as:

$$〈x_s^2〉={\omega }_1{S}_0{\int }_{-\infty }^{\infty }{\left|H_i\left(j\alpha \right)\right|}^{2}d\alpha$$

(24)

By substituting Equation (24) into Equation (23), the performance index $PI$ can be expressed as:

$$PI=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|H_i\left(j\alpha \right)\right|}^{2}d\alpha$$

(25)

The main principle of the $H_2$ criterion is referred to in Appendix A.

To obtain the optimal design parameters, the following two conditions need to be satisfied in the derivation of the results containing the target parameters in Equation (25):

$$\frac{\partial PI}{\partial f_1}=0;\frac{\partial PI}{\partial f_{\theta }}=0$$

(26)

In general, by solving Equation (26), the optimal solutions (${f_1}_{opt}$ and ${f_{\theta }}_{opt}$) for parameters $f_1$ and $f_{\theta }$ can be obtained. However, it was found during the solving process that Equation (26) is a higher-order equation group, and explicit expressions for ${f_1}_{opt}$ and ${f_{\theta }}_{opt}$ cannot be obtained (see Appendix B). Therefore, this paper will combine the Monte Carlo-pattern search method for parameter optimization, which can obtain the theoretically optimal solutions (${f_1}_{opt}$ and ${f_{\theta }}_{opt}$) for parameters $f_1$ and $f_{\theta }$.

3.2. Parameter Optimization Based on Monte Carlo-Mode Search Method

Monte Carlo-Mode Seeking (MCMS) is an approach commonly used to solve optimization problems. Its main logic is to generate a random set of parameters in the parameter space and then decide whether to accept new parameters according to the value of the objective function and thus iterate continuously to find the optimal solution. The core of the MCMS method is to divide the search process into two parts: Monte Carlo search and mode search.

In the Monte Carlo search stage, a set of parameters is randomly generated in the parameter space, and the value of the objective function is calculated. The new parameters are accepted or rejected based on a certain probability, which is usually calculated based on the difference between the current objective function value and the new objective function value.

In the mode search phase, the mode (i.e., gradient direction) and the mode center (i.e., extreme value point) of the current point are determined by calculating the first- and second-order derivatives of the objective function. Then, based on the mode center and mode direction, a new set of parameters is searched in the parameter space. The advantage of the mode search method is that it has no restriction on the dimensionality of the parameter space and is suitable for high-dimensional parameter optimization. However, the pattern search method also has its own drawbacks, which are that it can only find the local optimum and it is difficult to reach the global optimum of the function, the selection of the initial base point is closely related to the search result, a large number of stochastic simulations are required, the computation is large, and the convergence speed is slow. In addition, the results of the method are also affected by random factors and are not easily reproducible. Generally combined with the optimization process of Monte Carlo, the resulting optimization result is not optimal, but it is an acceptable quasi-optimal result.

For Equation (7), under the undamped condition of the controlled structure, since the partial derivatives of each parameter in it are difficult to calculate, it is more convenient to use the mode search method than other methods for this type of function. In the mode search method, we don’t need to calculate the derivatives or gradients but determine the search direction and step size directly based on the function value at the current point, so it can be applied to some functions that are difficult to find derivatives and have better robustness and global convergence. Here, we use the Monte Carlo-pattern search method mentioned by Xiaofang Kang [56] and apply it to the numerical optimization in this paper. The number of random initial points is set to m = 100,000.

The optimization process of the Hooke–Jeeves pattern search method for an objective function $f\left(x_1,x_2,x_3,\cdots ,x_n\right)$ with $n$ variables is as follows:

(1)

First, a starting point $x^{\left(1\right)}$, unit search vectors $m_1,m_2,m_3,\cdots ,m_n$, step length $\delta$, acceleration factor $\alpha \ge 1$, step length reduction rate $\beta \in \left(\mathrm{0,1}\right)$, and allowable error $\epsilon >0$ are given. Let $y^{\left(1\right)}=x^{\left(1\right)}$, $k=1$,$j=1$. The axial search vectors $m_1,m_2,m_3,\cdots ,m_n$ can be expressed by the Equation (27)

$$\left\{\begin{array}{c}\begin{array}{c}{\mathrm{m}}_1={\left(\begin{array}{ccc}1,& 0,& \begin{array}{cc}\cdots ,& 0\end{array}\end{array}\right)}_{1\ast n}\\ {\mathrm{m}}_2={\left(\begin{array}{ccc}0,& 1,& \begin{array}{cc}\cdots ,& 0\end{array}\end{array}\right)}_{1\ast n}\\ ⋮\end{array}\\ ⋮\\ {\mathrm{m}}_n={\left(\begin{array}{ccc}0,& 1,& \begin{array}{cc}\cdots ,& 0\end{array}\end{array}\right)}_{1\ast n}\end{array}\right.$$

(27)

(2)

If $f(y^{\left(j\right)}+\delta m_j)<f\left(y^{\left(j\right)}\right)$, then set $y^{\left(j+1\right)}=y^{\left(j\right)}+\delta m_j$ and proceed to (4); otherwise, proceed to (3);

(3)

If $f(y^{\left(j\right)}-\delta m_j)<f\left(y^{\left(j\right)}\right)$, then set $y^{\left(j+1\right)}=y^{\left(j\right)}+\delta m_j$; otherwise, set $y^{\left(j+1\right)}=y^{\left(j\right)}$. If j < n, then set j = j + 1 and proceed to (2);

(4)

If $f\left(y^{\left(n+1\right)}\right)<f\left(x^{\left(k\right)}\right)$, then proceed to (5); otherwise, proceed to (6);

(5)

Set $x^{(k+1)}=y^{(n+1)}$,and $y^{\left(1\right)}=x^{(k+1)}+\alpha (x^{\left(k+1\right)}-x^{\left(k\right)})$. Proceed to (7);

(6)

If $\delta \le \epsilon$, end the iteration and obtain $x^{\left(k\right)}$; otherwise, set $\delta =\beta \delta$, $y^{\left(1\right)}=x^{\left(k\right)}$, and $x^{(k+1)}=x^{\left(k\right)}$. Proceed to (7);

(7)

Set $k=k+1$, and $j=1$. Proceed to (2).

Based on the contour map of $PI$ in Appendix B, the Monte Carlo Pattern Search method can be used to obtain:

Figure 2 and Figure 3 indicate that the range of values for $PI$ is basically consistent between the two. After applying the Monte Carlo Pattern Search method, a set of data was obtained. Mean normalization of the data yields: $f_1=0.3866$, $f_{\theta }=0.2732$.

4. Verification in the Frequency Domain

The Frequency domain analysis is a classical method for studying control systems.

Frequency domain analysis is based on the frequency characteristics of signals. By analyzing the frequency response of a system, the stability, performance, and reliability of a system can be evaluated. Commonly used tools include frequency response functions, amplitude-frequency characteristic curves, and phase-frequency characteristic curves. Through these tools, the frequency characteristics, phase characteristics, gain margin, and other indicators of the system can be analyzed so as to evaluate and optimize the performance of the control system. Frequency domain analysis has the advantages of theoretical simplicity, convenient calculation, and ease of understanding, so it has been widely used in parameter optimization verification.

4.1. Displacement Frequency Response under Different Systems

The relative displacement frequency response of the building structure is compared under uncontrolled conditions, without considering soil-structure interaction, and with consideration of soil-structure interaction, when the quality ratio is at 0.1, as shown in Figure 4. It can be observed from Figure 4 that, under the same quality ratio, the displacement response of the structure considering soil conditions is much larger than that without considering soil conditions. Without considering soil conditions, soil-structure interaction can be regarded as a fixed boundary condition imposed by the external environment, assuming that the interaction between the structure and soil is rigid and unchangeable and is not affected by soil deformation, settlement, and type, which cannot truly reflect the dynamic response of the structure under actual conditions. This assumption may be feasible in specific cases, such as when the soil layer is relatively hard, the structural stiffness is large, and the seismic action is small. However, in most cases, soil deformation and settlement will exert a reactive force on the structure, thereby affecting its stability and safety. Therefore, the effect of soil state on soil-structure interactions needs to be considered when SSI effects are considered.

4.2. Displacement Frequency Response under Different Mass Ratios Considering Soil-Structure Interaction

According to Figure 5, by comparing the vibration reduction performance under different mass ratios considering soil-structure interaction, it can be observed that the mass ratio has a certain effect on the vibration reduction effect. Specifically, when the mass ratio is larger, the vibration reduction effect is better. This is because a structure with a larger mass has a greater inertia force against vibration, which can absorb more energy and reduce the amplitude of the structure’s vibration. Conversely, when the mass ratio is smaller, the vibration reduction effect becomes worse, as the structure’s inertia force is smaller and can absorb less energy, making it difficult to effectively reduce the vibration amplitude. Therefore, when designing vibration reduction measures, appropriate measures and plans should be selected based on the size of the structure’s mass to achieve the best vibration reduction effect.

4.3. Robustness of Soil Parameters

In this section, the main focus is on the uncertain factors of soil types. By analyzing the seismic performance indicators of the system, such as maximum displacement, the vibration reduction effect of the building structure can be evaluated.

Figure 6 illustrates the impact of changes in soil hardness on the structure. The values of $f_1$ and $f_{\theta }$ are influenced by the stiffness of the soil. When the soil becomes softer, that is, when $f_1$ increases or $f_{\theta }$ decreases, the displacement of the structure increases compared to medium soil, indicating a greater swinging amplitude. At this point, the soil has characteristics such as high water content, high porosity, high compressibility, low shear strength, high disturbance, and complex layer distribution. These factors lead to a lower ultimate bearing capacity of soft soil, and the influence of soil-structure interaction is also small. Therefore, in areas with soft soil, special attention should be paid to the stability and bearing capacity of the soil to ensure the safety of the structure.

As the soil becomes harder, the stiffness of the soil increases, and the rigid constraint effect on the main structure is enhanced, resulting in a decrease in $f_1$ or an increase in $f_{\theta }$, thereby reducing the free vibration amplitude of the structure. Compared to medium soil, the relative displacement of the structure decreases when the soil becomes harder, indicating that the soil-structure interaction effect is more pronounced. In this case, the structure is less affected by vibrations, but at the same time, it is necessary to consider whether the bearing capacity of the soil can meet the needs of the structure.

In conclusion, considering the stability analysis of the soil-structure interaction system’s parameters is a crucial aspect of structural design and analysis. It can help evaluate the system’s ability to withstand vibrations and provide guidance and reference for engineering practices. In the design and analysis process, appropriate parameter selection and analysis methods should be employed to improve the system’s stability. For example, the seismic performance of the system can be enhanced by selecting suitable dampers and optimizing the mass ratio of the structure.

4.4. Energy Harvesting Theory Analysis

In order to maximize the energy harvesting efficiency of the system, it is hoped to amplify the energy of the external resistance $R$, and the instantaneous power applied to the external resistance $R$ is:

$$P\left(t\right)=R{I}^2$$

(28)

Similarly, when the system is under seismic wave acceleration excitation, the energy harvesting evaluation index can be defined as:

$${PI}_e=\frac{R}{2\pi }{\int }_{-\infty }^{\infty }{\left|I_n\right|}^{2}d\alpha$$

(29)

Bringing Equation (29) into Equation (28) can be obtained, as shown in Appendix C.

5. Verification in the Time Domain

This section investigates the influence of soil-structure interaction on structural response using numerical examples of a single-story building. The response characteristics of the structure and the soil-structure interaction effects are considered under seismic waves in three types of soil: soft soil, medium-hard soil, and hard soil. Numerical simulations and analyses are conducted to compare the seismic response of the structure under different soil types and the extent of the influence of soil-structure interaction on the structure. The detailed parameters of the numerical model structure and foundation are presented in

Table 1. Parameters of the first mode for a single-story building structure.

$\mathbf{Mass} {\mathit{m}}_1\left(\mathbf{k}\mathbf{g}\right)$	$\mathbf{Damping} {\mathit{c}}_1$$(\mathbf{N}\mathbf{s}·{\mathbf{m}}^{-1})$	$\mathbf{Stiffness} {\mathit{k}}_1$$(\mathbf{N}·{\mathbf{m}}^{-1})$
$1.76\times {10}^7$	$1.15\times {10}^6$	$4.73\times {10}^7$

and

Table 2. Parameters of example soils.

Soil Type	${\mathit{m}}_0\left(\mathbf{k}\mathbf{g}\right)\times {10}^6$	${\mathit{I}}_0$$(\mathbf{k}\mathbf{g}·{\mathbf{m}}^2$$) {\times 10}^8$	${\mathit{k}}_0$$(\mathbf{N}/\mathbf{m})$$\times {10}^{10}$	${\mathit{c}}_0(\mathbf{N}\mathbf{s}/\mathbf{m}$$)\times {10}^8$	${\mathit{k}}_{\mathit{\theta }}$$(\mathbf{N}/\mathbf{m}$$)\times {10}^{12}$	${\mathit{c}}_{\mathit{\theta }}$$(\mathbf{N}\mathbf{s}/\mathbf{m})$$\times {10}^{10}$
Soft soil	$1.96$	$1.96$	$0.19$	$2.19$	$0.75$	$2.26$
Medium soil	1.96	1.96	$1.80$	$6.90$	$7.02$	$7.02$
Dense soil	1.96	1.96	$5.75$	$13.2$	$19.1$	$11.5$

, respectively.

This section will conduct a time-history analysis of the building structure considering soil-structure interaction using four types of earthquake waves. Then, the damping effect and energy-collecting effect of the building structure will be analyzed.

5.1. Theoretical Model

To better analyze the time-domain response, the current I is expressed in terms of electric charge $\dot{q}$, and Equation (1) can be expressed as:

$$\left\{\begin{array}{c}{m}_1({\ddot{x}}_0+{\ddot{x}}_1+\ddot{\theta }h)+c_1{\dot{x}}_1+k_1{x}_1+k_f\dot{q}=-m_1{\ddot{x}}_g\\ k_v({\dot{x}}_1-{\dot{x}}_0-\dot{\theta }h)+R_c\dot{q}+L\ddot{q}=0\\ m_1({\ddot{x}}_0+{\ddot{x}}_1+\ddot{\theta }h)+m_0{\ddot{x}}_0+c_0{\dot{x}}_0+k_0{x}_0=-(m_1+m_0){\ddot{x}}_g\\ m_{1}h({\ddot{x}}_0+{\ddot{x}}_1+\ddot{\theta }h)+(I_0+I_1)\ddot{\theta }+c_{\theta }\dot{\theta }+k_{\theta }\theta =-m_{1}h{\ddot{x}}_g\end{array}\right.$$

(30)

The matrix form of Equation (31) is:

$$M\left[\begin{array}{c}{\ddot{x}}_0\\ \begin{array}{c}{\ddot{x}}_1\\ \begin{array}{c}\ddot{\theta }\\ \ddot{q}\end{array}\end{array}\end{array}\right]+C\left[\begin{array}{c}{\dot{x}}_0\\ \begin{array}{c}{\dot{x}}_1\\ \begin{array}{c}\dot{\theta }\\ \dot{q}\end{array}\end{array}\end{array}\right]+K\left[\begin{array}{c}{x}_0\\ \begin{array}{c}{x}_1\\ \begin{array}{c}\theta \\ q\end{array}\end{array}\end{array}\right]=-N{\ddot{x}}_g$$

(31)

where:

$$M=\left[\begin{array}{cccc}{m}_1& m_1& m_{1}h& 0\\ 0& 0& 0& L\\ m_1+m_0& m_1& m_{1}h& 0\\ m_{1}h& m_{1}h& 2m_1{h}^2+I_0+I_1& 0\end{array}\right]$$

(32)

$$C=\left[\begin{array}{cccc}0& c_1& 0& k_f\\ -k_v& k_v& -k_{v}h& R_c\\ c_0& 0& 0& 0\\ 0& 0& c_{\theta }& 0\end{array}\right]$$

(33)

$$K=\left[\begin{array}{cccc}0& k_1& 0& 0\\ 0& 0& 0& 0\\ k_0& 0& 0& 0\\ 0& 0& k_{\theta }& 0\end{array}\right]$$

(34)

$$N=\left[\begin{array}{c}{m}_1\\ 0\\ \begin{array}{c}{m}_1+m_0\\ m_{1}h\end{array}\end{array}\right]$$

(35)

Definition:

$$x={\left(\begin{array}{cccccccc}{x}_0& x_1& \theta & q& {\dot{x}}_0& {\dot{x}}_1& \dot{\theta }& \dot{q}\end{array}\right)}^T$$

(36)

The state equation is then:

$$\dot{x}=Ax+B{\ddot{x}}_g$$

(37)

where:

$$A=\left[\begin{array}{cc}{0}_{4\times 4}& I_{4\times 4}\\ -M^{-1}K& -M^{-1}C\end{array}\right],B=\left[\begin{array}{c}{0}_{4\times 1}\\ -M^{-1}N\end{array}\right]$$

(38)

5.2. Simulation Parameters

This article adopts the relevant parameters of the building structure in [47]. The detailed parameters of the building structure and three types of soil are shown in Table 1 and Table 2. Electromagnetic sensor constant $k_f=k_v=50$, The internal resistance $R_i$ and external resistance $R_e$ are 0.14 $\mathsf{\Omega }$. Inductance $L$ is 9.6$\mathrm{m}\mathrm{H}$.

5.3. Simulation Result Analysis

To make the structural response more apparent, certain adjustments are made to the amplitude of the earthquake wave. The four types of earthquake waves (Figure 7) are respectively: (a) Chi-Chi, (b) EL Centro, (c) Loma−Prieta, and (d) Mexico City. The displacement and acceleration are compared for each wave, considering the soil-structure interaction and different soil types.

Figure 8 represents a comparison of the displacement response results of the building structure with and without soil considerations and with and without electromagnetic dampers. Figure 9 represents a comparison of the displacement response results for the building structure different soil conditions. Figure 10 represents a comparison of the acceleration response results of the building structure with and without the soil factor and with and without the electromagnetic damper. Figure 11 represents a comparison of the acceleration response of the building structure under different soil conditions. The effects of different soil conditions on the forces on the electromagnetic dampers in building structures with electromagnetic damping are considered in Figure 12 and Figure 13, respectively.

Building structures considering soil-structure interaction will experience greater displacements of the main structure under the same external excitation. The transfer of seismic energy will be affected by the stiffness and damping of the foundation, which will affect the response of the structure, and the acceleration will often decrease. This is because soil-structure interaction can increase the energy dissipation capacity of the structure, thereby increasing the displacement of the structure, reducing the acceleration response, and making the vibration characteristics of the structure more gentle. Specifically, when the building structure vibrates due to external excitation, the deformation of the soil will absorb a part of the vibration energy of the structure, thereby increasing the displacement response of the structure and reducing the acceleration response. The speed at which seismic waves propagate between the structure and the foundation will also be affected by the foundation, which will cause the peak time of the structure response to be advanced or delayed. Among the four time-history response results mentioned above, the effect of the EL Centro earthquake wave loading seismic excitation is the most significant throughout the process. Although the focus of this paper is on displacement, the acceleration response also decreases slightly. In addition, since the characteristics of the foundation can affect the propagation of seismic waves between the structure and the foundation, the waveform of the structural response damping time-history chart may change. For example, the damping characteristics of the foundation can weaken the vibration of the structure, thereby reducing the amplitude of the damping time-history chart or making the waveform smoother. Through theoretical and model analysis, it is fully proved that considering soil-structure interaction has an impact on building structures.

The research findings can provide reference and guidance for engineering design, helping designers to better understand the nature and characteristics of soil-structure interaction, as well as how to take reasonable measures to reduce the impact of soil-structure interaction and improve the seismic performance of civil engineering structures.

5.4. Energy Harvesting Simulation

In Figure 14 and Figure 15 we consider the energy harvesting diagrams for building structures with electromagnetic dampers with and without consideration of SSI, respectively.

Energy harvesting refers to the conversion of the vibration energy of a structure into other forms of energy and storage through some devices during an earthquake in order to reduce the amplitude of vibration of the structure and reduce the degree of damage to the structure, improve the self-damping and energy dissipation capacity of the structure, reduce the resonance phenomenon of the structure, and thus guarantee the safety performance of the structure.

From Figure 16 and

Table 3. Effects of soil factors on average energy harvesting under different seismic waves.

Earthquake	SSI:Power (W)	Fixed:Power (W)	Percentage
Chi-Chi	56,506.3410	88,117.2393	35.8737
EL Centro	135,505.7216	213,990.6605	36.6768
Loma-Prieta	133,938.2377	203,140.6428	34.0663
Mexico City	344,925.5130	509,157.8335	32.2557

, it can be seen that the energy harvesting of the soil-structure interaction system is also ideal. Under the simulation of four types of seismic loads, such as in Mexico City, the average power collected is $5.95\times {10}^5$ W. At the same time, the energy collection on a fixed foundation has improved, indicating that some energy is consumed in the soil-structure interaction.

To consider the energy harvesting of a building structure under soil-structure interaction, it is necessary to analyze the interaction between soil and structure in the building. Therefore, when considering this situation, factors such as soil temperature, humidity, and density will affect the transmission and distribution of energy in the soil, thereby affecting energy harvesting. Therefore, after considering soil factors, the amount of energy collected will change.

To sum up, the energy harvesting of soil-structure interaction systems is influenced by multiple factors and requires a comprehensive consideration of these factors to accurately evaluate the energy harvesting situation.

6. Conclusions

In order to investigate the effects on the damping and energy harvesting of building structures with electromagnetic dampers under consideration of soil-structure interaction, the dynamic response of building structures with and without consideration of soil-structure interaction is compared separately. The classical structural model with electromagnetic dampers is improved under consideration of soil-structure interaction. The parameters of the coupled model are optimized based on the $H_2$ parametric criterion and MCSM to obtain the theoretically optimal solution. Numerical simulations are performed using the obtained parameters to analyze the damping and energy-harvesting effects of the coupled model. The following specific conclusions can be obtained:

(1)

Under single ground motion conditions, the H₂ parametric criterion and the MCSM optimization algorithm design are able to provide theoretically optimal solutions, but the optimal parameters are influenced by soil type and structural form. Although the frequency domain analysis obtained from the design based on the optimization algorithm deviates from the time domain analysis under different seismic conditions, they are still very close.

(2)

The displacement frequency response considering the soil-structure interaction increases by 48% compared to the displacement frequency response without considering the soil-structure interaction. With each 0.1 increase in the mass ratio, the displacement frequency response decreases by 8.7%.

(3)

Neglecting the soil-structure interaction effect may result in an unrealistic seismic response of the structure in the soil. Compared to structures built on fixed foundations, structures considering soil-structure interaction effects have higher displacement and acceleration responses. As the soil density decreases, the displacement and acceleration responses also decrease, indicating that the effectiveness of soil-structure interaction effects depends on the soil density.

(4)

When considering soil-structure interactions in building structures, the average energy harvesting power with fixed foundations increases by about 30%, improving the energy harvesting effect.

(5)

As the soil density increases, there is a smaller effect of soil-structure interaction on the electromagnetic damper system. An increase in soil density leads to a smaller support area of the damper, which reduces its support force on the structure and decreases the damping effect of the electromagnetic damper.

The limitation of this thesis is that the study only addresses structural vibration control and energy harvesting in single-story buildings. Additionally, the conclusions of the paper are only applicable to the electromagnetic dampers selected for this study, and other types of electromagnetic dampers may have different results. In addition, the paper only considered the effect of soil-structure interaction on the building structure and did not consider the effect of other factors on structural vibration and energy harvesting, such as temperature, humidity, and wind. Therefore, further research on the effects of other factors on structural vibration and energy harvesting and exploration of more types of electromagnetic dampers are needed to more fully and accurately evaluate the application of electromagnetic dampers for structural vibration control and energy harvesting.