Stochastic Response-Based Optimization of a Bilinear ECD Model for Seismic Analysis

Abstract

The eddy current damper (ECD), as a high-performance seismic protection device, has been increasingly applied in the field of civil engineering. However, most existing finite element software lacks a nonlinear constitutive model for this type of damper, and an equivalent simplified model capable of accurately capturing its real damping characteristics under complex seismic excitations has not yet been established. To address this gap, this paper proposes a stochastic optimization framework for calibrating a bilinear ECD model, using the Wouterse model as a reference. Specifically, the core of this framework employs a Monte Carlo Simulation (MCS) approach, in which a suite of spectrum-compatible seismic excitations is generated and analyzed to evaluate the corresponding stochastic structural responses. The Grey Wolf Optimizer (GWO) is then utilized to minimize the discrepancy between the dynamic responses predicted by the proposed bilinear model and those of the reference model. A key outcome of this study is a practical bilinear model in which these parameters can be treated as constants, enabling designers to focus exclusively on the physical parameters $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $v_{\mathrm{c}\mathrm{r}}$. This study bridges the gap between high-fidelity simulations and practical engineering design, providing a feasible pathway for integrating ECDs into existing finite element software for dynamic analysis.

1. Introduction

The safety and integrity of civil infrastructure are persistently threatened by seismic excitations, underscoring the critical need for effective vibration control strategies [1,2,3]. Eddy current dampers (ECDs) have emerged as a promising non-contact passive solution [4,5,6]. Their operation relies on the relative motion of a conductor through a magnetic field, which induces eddy currents that generate a velocity-dependent damping force [7,8,9]. This friction-free mechanism, combined with exceptional durability and fatigue resistance, makes ECDs a robust, low-maintenance option for civil structures, thereby expanding the range of available engineering solutions.

The promising characteristics of ECDs have been validated through extensive full-scale applications across diverse structural types and under various vibration scenarios. For long-span bridges, ECDs have been effectively deployed to mitigate seismic response [10], train-induced displacements [11], multi-mode vortex-induced vibrations [12], and complex stay-cable oscillations [13]. In super-tall buildings, eddy-current tuned mass dampers (TMDs) have demonstrated significant efficacy in mitigating both wind-induced and seismic responses [14]. Beyond conventional structures, ECDs have shown great adaptability in specialized applications including offshore wind turbines [15,16], floating offshore wind turbines [17], high-voltage transmission towers [18], and advanced hybrid damping systems for seismic protection [19,20]. These diverse applications highlight the importance of developing accurate and readily implementable models to simulate the complex damping behavior of ECDs for design and analysis purposes.

Initially developed for electromechanical eddy current brakes, the analytical model proposed by Wouterse [21] offers a robust physical basis for describing velocity-dependent damping. Owing to its high accuracy, the model has been widely adopted as a benchmark for simulating ECDs in civil engineering applications [22,23]. The model’s effectiveness has been extensively validated across various ECD configurations and implementation scenarios. It consequently serves as the foundation for both high-fidelity research simulations [24,25] and practical performance evaluations [26,27]. This widespread reliance underscores its status as the de facto standard for modeling ECD behavior in the technical literature. While the Wouterse model holds an established role in academic and research settings, its practical application in routine engineering design is hindered by a fundamental limitation. Its specialized velocity–damping force relationship, characterized by a unique single-peak curve, cannot be represented using the standard input options available in commercial finite element (FE) software. Consequently, implementing the model necessitates complex customization in advanced packages, or it is entirely infeasible in design-oriented FE software that lacks such functionality. This incompatibility with standard FE workflows presents a significant practical barrier to the model’s adoption in engineering design.

In response to these implementation challenges, researchers have developed simplified models for ECDs. Although multi-term power function combinations can achieve high mathematical precision [28], their practical adoption is often limited by poor compatibility with commercial finite-element software, which may not support complex function definitions. The quasi-bilinear behavior observed in certain ECD configurations [29] offers a more universally compatible alternative, making it advantageous for broad engineering applications. However, a critical challenge remains: the current literature lacks a systematic, generalizable method for determining the key yield-point parameters of the bilinear model to accurately approximate the behavior of a high-fidelity reference model under stochastic dynamic loads. Furthermore, the robustness and applicability limits of such bilinear approximations under diverse loading conditions remain unquantified.

To address this research gap, this paper proposes a stochastic optimization framework for calibrating a simplified bilinear model of ECDs, aiming to achieve both high fidelity and practical implementability. The Wouterse model is adopted as the high-fidelity benchmark. The core methodology employs a Monte Carlo Simulation (MCS) stochastic optimization framework. A suite of spectrum-compatible seismic excitations is first generated to serve as the input ground motions for the subsequent analysis. The Grey Wolf Optimizer (GWO) is then employed to identify the optimal bilinear model parameters by minimizing the discrepancy in the stochastic dynamic responses of a structure equipped with an ECD. Furthermore, a comprehensive sensitivity analysis is conducted to rigorously evaluate the robustness of the optimized parameters against variations in seismic intensity and spectral characteristics.

2. Bilinear Approximation of the Velocity-Force Behavior of ECDs

ECDs exhibit unique mechanical properties, and their velocity-damping force relationship is characterized by a nonlinear curve with a distinct peak. To facilitate the integration of ECDs into existing finite element software, this section proposes the use of a simplified bilinear model to approximate the velocity-force behavior of ECDs.

2.1. Velocity-Force Behavior of ECDs

Figure 1 shows a cross-sectional schematic of a typical ECD. As illustrated, the damper consists primarily of two cylindrical components made of carbon steel: the stator and the rotor. A magnet array is fixed to the outer wall of the stator cylinder, while a conductor cylinder is embedded within the rotor. The magnets are typically fabricated from high-strength permanent magnetic materials such as NdFeB, and the conductor is often made of highly conductive materials such as copper or aluminum. According to the principle of electromagnetic induction, this changing flux induces eddy currents in the conductor, leading to Joule heating. From an energy perspective, kinetic energy is converted into thermal energy and dissipated. Mechanically, the relative motion produces an opposing damping force due to eddy current generation. This describes the fundamental operating principle of an eddy current damper.

Eddy current dampers are velocity-dependent devices. The relationship between the relative velocity across their ends and the resulting damping force exhibits a characteristic nonlinear, single-peak profile, as illustrated in Figure 2. It can be observed that the damping force initially increases with relative velocity until it reaches a maximum value at a critical velocity. Beyond this point, the damping force gradually decreases with further increases in velocity. Notably, the velocity-force relationship is nearly linear at low relative velocities. Furthermore, the rate of decrease in damping force beyond the peak is relatively slow, resulting in a gentle decline. These two characteristics of the velocity-force curve align well with a bilinear model, justifying its use for approximation.

Currently, the most widely used model is the semi-empirical, semi-analytical model proposed by Wouterse. The curve in Figure 2 is plotted based on this model, and its expression is given as follows [23]:

$$f_{ECD}=f_{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \frac{2}{\frac{v}{v_{\mathrm{c}\mathrm{r}}}+\frac{v_{\mathrm{c}\mathrm{r}}}{v}}}$$

(1)

where $f_{ECD}$ is the nonlinear damping force of the eddy current damper; $v$ is the relative velocity between the two ends of the damper; $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the designed maximum damping force; and $v_{\mathrm{c}\mathrm{r}}$ is the critical relative velocity at which the damping force reaches $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$. Evidently, this expression is not intuitive, which complicates interpretation for engineering practitioners. Furthermore, the compound fractional form is relatively complex to implement in most finite element software, often requiring customizing development.

Therefore, employing a more intuitive bilinear model to approximate the mechanical behavior of the eddy current damper holds significant practical value in engineering applications. Accordingly, the Wouterse model can serve as the target for approximation, and parameter optimization studies based on the bilinear model can be conducted.

2.2. Bilinear Model Approximation

The bilinear model is a widely adopted approach for characterizing the mechanical behavior of dampers. Its characteristics align well with the velocity-force relationship of eddy current dampers, making it suitable for approximating their damping properties. To enhance universality and comparability while eliminating dimensional effects, the target model in Equation (1) is normalized. The dimensionless damping force is defined as $f_{ECD}/f_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and the dimensionless relative velocity as $v/v_{\mathrm{c}\mathrm{r}}$. The dimensionless form of the Wouterse constitutive model is expressed as follows:

$${\displaystyle \frac{f_{ECD}}{f_{\mathrm{m}\mathrm{a}\mathrm{x}}}}={\displaystyle \frac{2}{\frac{v}{v_{\mathrm{c}\mathrm{r}}}+\frac{1}{\frac{v}{v_{\mathrm{c}\mathrm{r}}}}}}$$

(2)

A bilinear model is established to approximate the target model described by Equation (2), as illustrated in Figure 3. The model introduces a yield point $(\alpha , \beta )$, where the slope is $\beta /\alpha$ for dimensionless relative velocities less than or equal to $\alpha$, and zero when the velocity exceeds $\alpha$. The expression of the bilinear constitutive model is given by:

$$f_{ECD}^B=\left\{\begin{array}{c}{\displaystyle \frac{\beta f_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\alpha v_{\mathrm{c}\mathrm{r}}}}v v\le \alpha v_{\mathrm{c}\mathrm{r}}\\ \beta f_{\mathrm{m}\mathrm{a}\mathrm{x}} v>\alpha v_{\mathrm{c}\mathrm{r}}\end{array}\right.$$

(3)

This expression retains the parameters $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $v_{\mathrm{c}\mathrm{r}}$, preserving the intuitive physical meaning of the target model. Thus, the only undetermined parameters are $\alpha$ and $\beta$. Consequently, the accurate simulation of the ECD’s velocity-force characteristics can be simplified to the problem of determining the optimal values of these two parameters.

This section has outlined the nonlinear velocity-force behavior of ECDs and introduced a bilinear model that offers improved usability and interpretability. By incorporating the yield point parameters $\alpha$ and $\beta$, the model simplifies implementation while preserving the key physical meanings of $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $v_{\mathrm{c}\mathrm{r}}$. Optimizing these parameters is essential for accurately representing ECD performance.

3. Stochastic Optimization of Bilinear Model Parameters for Seismic Control

This section presents a stochastic framework for parameter optimization of the bilinear model for ECDs applied to a SDOF system under seismic excitations. The optimization is conducted through a stochastic seismic response analysis via MCS, with the objective of identifying the parameter set that minimizes the discrepancy in the stochastic dynamic responses between the bilinear model and the reference Wouterse model. The resulting optimized model can thus serve as a high-fidelity substitute for complex models in seismic analysis and control design frameworks.

3.1. Seismic Excitation Generation

Before performing stochastic parameter optimization, it is necessary to generate a set of representative seismic excitation samples that accurately capture the statistical characteristics of real earthquake events. This subsection describes the methodology employed for synthesizing seismic ground motions, ensuring their suitability for subsequent probabilistic analysis.

Given the non-stationary stochastic nature of seismic motion, the seismic acceleration random process can be represented as the product of a stationary stochastic process and a uniform modulation function. The stationary stochastic process can be generated using the spectral representation method (SRM) based on a specified power spectrum [30]. After generating the artificial seismic excitations, multiplying them by the uniform modulation function yields seismic excitation samples.

Furthermore, in accordance with Chinese design code [31], artificially generated seismic waves must satisfy the requirements of the corresponding response spectrum. Since the response spectrum cannot be directly applied in the SRM, it must first be converted into a power spectrum. The approximate relationship between the power spectrum and the response spectrum is given as follows [32]:

$$S\left(\omega \right)\approx {\displaystyle \frac{\xi }{\pi \omega }}{S}_a^2\left(\omega \right){\left[-\ln \left(-{\displaystyle \frac{\pi }{\omega T_d}}\ln \left(p\right)\right)\right]}^{-1}$$

(4)

where $S_a\left(\omega \right)$ represents the design acceleration response spectrum; $\omega$ denotes the circular frequency; $\xi$ denotes the structural damping ratio; $T_d$ represents the duration of the seismic excitation; and $p$ denotes the exceedance probability. Since the adopted response spectrum corresponds to a median spectrum, the parameter p is set to 0.5.

Due to the approximate nature of the relationship, the response spectrum of seismic waves generated using this method may still deviate from the target spectrum, necessitating iterative optimization. The iterative optimization process is illustrated in Figure 4. As shown, the procedure involves the following steps: first, generating seismic wave samples and calculating the response spectrum for each sample; second, computing the average response spectrum and evaluating its deviation from the target spectrum; if the error exceeds the tolerance, the power spectrum is adjusted according to $S_{\mathrm{new}}\left(\omega \right)={S\left(\omega \right)(S_a\left(\omega \right)/S_a^{\prime }\left(\omega \right))}^2$); finally, repeating the process until a power spectrum satisfying the response spectrum requirements is obtained, thereby yielding compliant seismic excitation samples.

The seismic excitation parameters were determined in accordance with the design code, featuring a characteristic period of 0.55 s and a peak ground acceleration (PGA) of 0.05 g. Based on these parameters, 800 seismic excitation samples were generated, and their power spectra were iteratively adjusted using the algorithm illustrated in Figure 4. The iteration termination criterion was defined as a mean squared error of less than 0.001 between the average response spectrum and the target response spectrum. Figure 5 presents a comparison between the target response spectrum and the final average response spectrum obtained after iteration. The results demonstrate close agreement between the two spectra. Additionally, Figure 5 displays the time history of a representative seismic wave sample, which has a duration of 30 s.

Thus, a suite of spectrum-compatible seismic excitations has been successfully generated and will be employed in the subsequent stochastic response analysis of the structure.

3.2. Stochastic Seismic Response Analysis

This section presents the stochastic seismic response analysis of an SDOF system equipped with an ECD, subjected to the artificially generated seismic excitations described previously. The schematic of the SDOF-ECD system is shown in Figure 6, and its equation of motion is given by:

$$m\ddot{u}+c\dot{u}+ku+f_{\mathrm{ECD}}\left(\dot{u}\right)=-m{\ddot{u}}_g$$

(5)

where $m$, $c$, $k$ represent the mass, damping coefficient, and stiffness of the SDOF system, respectively; $\ddot{u}$, $\dot{u}$, $u$ denote the acceleration, velocity, and displacement of the system, respectively; ${\ddot{u}}_g$ is the ground acceleration; and $f_{\mathrm{ECD}}\left(\dot{u}\right)$ is the damping force provided by the ECD, which is a function of the system velocity.

The parameters of the SDOF system, summarized below for clarity, are derived from a prototype long-span suspension bridge to ensure practical relevance. The parameters of the SDOF system are summarized as follows (

Table 1. Model parameters and assumptions.

Parameter	Symbol	Value	Description
Girder Mass	$m$	1.4 × 107 kg	Equivalent mass of the bridge girder
Fundamental Frequency	$f$	0.12 Hz	Longitudinal floating mode frequency
Damping Ratio	$\xi$	2%	Structural damping ratio

).

This modeling approach ensures practical relevance, as it reflects a realistic engineering scenario for controlling the longitudinal seismic response of a suspension bridge girder. The SDOF model is employed as it is a validated simplification for seismic analysis of long-span suspension bridges [33,34,35], thereby significantly reducing the computational cost for the numerous iterations required in our parameter optimization. Equation (5) is solved numerically for each seismic excitation sample using the Runge-Kutta method within the MATLAB (R2019a) environment. Based on the MCS, the stochastic seismic responses of the SDOF-ECD system are obtained for subsequent statistical analysis.

The influence of sample size on response statistics is a critical consideration in MCS. Figure 7 compares the time histories of displacement standard deviation (STD) under four different sample sizes. These responses were computed using the Wouterse model, with parameters set to $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ = 1000 kN and $v_{\mathrm{c}\mathrm{r}}$ = 0.2 m/s. As shown in the figure, when the sample size is small, the time history curve of displacement STD exhibits significant instability with substantial fluctuations in the calculated results. As the sample size increases, the displacement standard deviation gradually stabilizes, showing markedly improved convergence characteristics. When the sample size reaches 500, the results demonstrate essentially converged behavior. Therefore, a sample size of 500 is adopted for subsequent MCS analysis.

The subsequent MCS analysis will also compute the mean peak (MP) values alongside the STD. This is particularly relevant for the ECD due to its critical velocity parameter, $v_{\mathrm{c}\mathrm{r}}$. The MP velocity response serves as a direct indicator of the maximum velocity experienced by the damper. Consequently, it provides the primary metric for validating the adequacy of the selected $v_{\mathrm{c}\mathrm{r}}$ value and for ensuring the damper operates within its intended design range.

This stochastic response analysis establishes the essential statistical foundation for evaluating the control effectiveness of the ECD and for validating the optimal design parameters identified in this study.

3.3. Stochastic Optimization of Bilinear Model Parameters

Following the stochastic response analysis presented in Section 3.2, the objective is to develop a simplified bilinear model suitable for practical design applications. The stochastic optimization process is formulated to minimize the discrepancy between the response statistics of the bilinear model and those of the Wouterse model. The optimization problem is defined as follows:

$$\left\{\begin{array}{cc}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}:& \alpha , \beta \\ \mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}:& g_0\left(\alpha , \beta \right)\\ \mathrm{s}.\mathrm{t}.:& \begin{array}{cc} & f_{\mathrm{m}\mathrm{a}\mathrm{x}}^l\le f_{\mathrm{m}\mathrm{a}\mathrm{x}}\le f_{\mathrm{m}\mathrm{a}\mathrm{x}}^u\end{array}; v_{\mathrm{c}\mathrm{r}}^l\le v_{\mathrm{c}\mathrm{r}}\le v_{\mathrm{c}\mathrm{r}}^u\end{array}\right.$$

(6)

where $g_0\left(\alpha , \beta \right)$ is the objective function, defined as $\left|1-{\mathrm{R}}^2\right|$, where ${\mathrm{R}}^2$ is the coefficient of determination between the STD time histories of displacement calculated by the bilinear model and the Wouterse model. A value of ${\mathrm{R}}^2$ close to 1 indicates a high degree of agreement between the two displacement STD time histories.

This study employs the Grey Wolf Optimizer (GWO) to address the optimization problem [36,37,38]. The selection of GWO is based on its well-documented effectiveness for nonlinear inverse problems and the following specific considerations. First, the optimization problem—minimizing $\left|1-{\mathrm{R}}^2\right|$—is inherently nonlinear and potentially multimodal. The GWO’s search mechanism, guided by the social hierarchy of $\alpha$, $\beta$, and $\delta$ wolves, promotes a balance between global exploration and local exploitation. This characteristic helps mitigate the risk of premature convergence, enhancing the reliability of locating a near-optimal solution. Second, compared to algorithms such as Particle Swarm Optimization (PSO) or Genetic Algorithms (GA), GWO features a simpler structure with fewer critical parameters to adjust. The computational cost per iteration is also typically lower than that of GA. These attributes collectively reduce the complexity and computational overhead of the optimization process. Therefore, GWO is well-suited for this problem and is expected to provide reliable solutions for identifying the optimal parameters of the bilinear model.

Figure 8 presents a flowchart that succinctly illustrates the optimization procedure, providing a clear overview of the iterative process from parameter initialization to final validation. The process begins with the definition of the objective function and parameter constraints. At its core is an iterative loop in which each candidate solution is evaluated via Monte Carlo simulation; the simulated response is compared against the Wouterse model baseline to compute the objective function value. The GWO algorithm subsequently updates the parameter population based on this feedback. This loop continues until the convergence criterion is met, ultimately yielding an optimal parameter set that ensures the high-fidelity approximation capability of the bilinear model.

The optimal parameter set for the bilinear model was successfully identified by solving the proposed stochastic optimization framework. The optimization process converged successfully, yielding the optimal values of $\left(\alpha , \beta \right)$ = (0.4839, 0.8934).

The optimization results are comprehensively validated in Figure 9. Figure 9 compares the displacement STD time histories between the bilinear model and the target Wouterse model, demonstrating strong agreement. This agreement is quantified by a coefficient of determination R² of 0.9999, indicating that the optimized bilinear model achieves a high degree of consistency with the reference model in capturing the stochastic damping behavior of the ECD. Furthermore, The MP displacements are 0.1762 m for the Wouterse model and 0.1761 m for the bilinear model. The MP velocities are 0.1914 m/s for the Wouterse model and 0.1918 m/s for the bilinear model, resulting in negligible relative errors. These results further confirm, from an engineering response perspective, that the optimized bilinear model accurately simulates the damping performance of the ECD.

The velocity-damping force relationship for the bilinear model, obtained by substituting the optimal parameters $\left(\alpha , \beta \right)$ = (0.4839, 0.8934), is presented in Figure 10. Figure 10 shows that the yield point setting ensures a close match between the bilinear model and the Wouterse model in the low-velocity range. When the dimensionless velocity approaches the value of $\alpha$, the dimensionless damping force of the bilinear model slightly exceeds that of the Wouterse model. As the dimensionless velocity increases further, the dimensionless damping force of the bilinear model becomes lower than that of the Wouterse model.

To further investigate the influence of parameters $\left(\alpha , \beta \right)$ on the bilinear model’s capability to simulate the damping characteristics of the ECD, a contour map of the coefficient of determination R² was computed for the $\left(\alpha , \beta \right)$ values within the black box region indicated in Figure 10. This coefficient R² quantifies the agreement between the time-history of displacement standard deviation for the bilinear model and the target Wouterse model. The resulting contour map is presented in Figure 11. Analysis of Figure 11 reveals that displacing the optimal yield point along paths corresponding to minimal changes in slope results in only a minor decrease in R². Conversely, displacement along paths associated with substantial slope changes leads to a marked reduction in R². Notably, all R² values within the boxed region are close to 1. This indicates that even non-optimal parameter sets enable the bilinear model to effectively simulate the ECD’s damping behavior.

To explain this phenomenon, an analysis of the ECD’s actual velocity distribution is essential. The critical velocity $v_{\mathrm{c}\mathrm{r}}$ of the ECD is 0.2 m/s, while the MP velocity obtained from the bilinear model is 0.1918 m/s. Consequently, the damper operates in the region where the dimensionless velocity $v/v_{\mathrm{c}\mathrm{r}}$ is less than 1, as shown in Figure 10. The ECD’s characteristic velocity range was determined by histogram analysis of seismic response time histories. As presented in Figure 12, the damper’s velocity remains within ±0.1 m/s ($v/v_{\mathrm{c}\mathrm{r}}\in \left[-0.5, 0.5\right]$) for 91.0% of the operating time and within ±0.2 m/s ($v/v_{\mathrm{c}\mathrm{r}}\in \left[-1, 1\right]$) for 98.5% of the time. Thus, the ECD operates primarily within the velocity region where the velocity-damping force curves of the bilinear and Wouterse models are in close agreement. This finding effectively explains why the coefficient of determination R² for the displacement STD time history between the bilinear and target Wouterse models is consistently close to 1. Therefore, even when parameters $\alpha$ and $\beta$ vary within a certain range, the bilinear model maintains high accuracy, underscoring its practical utility for simulating ECD.

This section has detailed the stochastic optimization framework developed for calibrating the bilinear model parameters of the ECD and successfully identified a parameter set that ensures close agreement with the target Wouterse model. A key finding is the model’s significant robustness; it exhibits only a minimal reduction in accuracy when the yield point deviates along paths associated with minor slope variations. This robustness stems from the damper’s predominant operation within a low-velocity range of ±0.2 m/s, where the velocity-damping force relationships of both models are similar. However, it is crucial to note that this finding is based on an optimization under a specific condition where the ratio $v/v_{\mathrm{c}\mathrm{r}}$ is close to 1. Therefore, the stability of this parameter set under varying seismic inputs requires further investigation, which motivates the sensitivity analysis presented in the following section.

4. Sensitivity Analysis of Bilinear Model Parameters

The stochastic optimization analysis presented in the previous section successfully identified $\left(\alpha , \beta \right)$ = (0.4839, 0.8934) as the optimal parameter set for the ECD’s bilinear model. Furthermore, it was demonstrated that values within a neighborhood of this optimum also yield a highly accurate simulation of the ECD’s damping characteristics. However, this conclusion was drawn under the specific condition that the ratio of the MP velocity to the critical velocity $v_{\mathrm{c}\mathrm{r}}$ is approximately 1. Due to the inherent uncertainties in seismic excitations, variations in seismic intensity and spectral characteristics can alter the effective operational velocity range of the ECD. To address this issue, a sensitivity analysis is conducted in this section to evaluate the robustness of the bilinear model’s accuracy in capturing the ECD’s damping behavior across a wide range of seismic conditions.

4.1. Bilinear Parameter Optimization Sensitivity to Seismic Intensity

To investigate the influence of seismic intensity on the optimal parameters of the bilinear model, a suite of seismic input motions was generated. This was achieved by scaling the seismic excitation samples previously developed in Section 3. The scaling factor ranged from 0.8 to 1.5 in increments of 0.1, resulting in eight distinct cases. For each case, the optimal parameters $\left(\alpha , \beta \right)$ for the bilinear model were determined using the stochastic optimization methodology described previously. Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 present the following results for the optimal parameters in each case: displacement STD time histories, velocity-damping force relationships, and the distribution of the ECD’s operational velocity range. The corresponding statistical results are summarized in

Table 2. Statistics of the stochastic optimization under varying seismic intensity.

Case	Scaling Factor	$\mathbf{MP} \mathbf{Vel}./{\mathit{v}}_{\mathbf{c}\mathbf{r}}$	R2	$\left(\mathit{\alpha }, \mathit{\beta }\right)$	Vel. Range (−0.5, 0.5)
1	0.8	0.7491	0.9999	(0.4338, 0.8132)	97.1%
2	0.9	0.8519	0.9999	(0.4691, 0.8691)	95.4%
3	1.0	0.9589	0.9999	(0.4839, 0.8934)	93.5%
4	1.1	1.0692	0.9999	(0.4987, 0.9095)	91.4%
5	1.2	1.1842	0.9999	(0.5093, 0.9224)	89.2%
6	1.3	1.2992	0.9999	(0.5248, 0.9404)	87.1%
7	1.4	1.4207	0.9999	(0.5179, 0.9356)	84.9%
8	1.5	1.5406	0.9999	(0.5235, 0.9405)	82.8%

.

The analysis reveals that as the scaling factor increases, the MP velocity rises proportionally. Consequently, the ratio of the MP velocity to the critical velocity $v_{\mathrm{c}\mathrm{r}}$ shifts from 0.7491 to 1.5406. This result demonstrates that this investigation comprehensively captures the ECD’s operational states across both the ascending and descending branches of its velocity-damping force relationship curve. Thus, it effectively encompasses the typical working range considered in damper design.

The bilinear model parameters identified for all cases yielded excellent agreement with the displacement STD history curves of the target Wouterse model, as evidenced by coefficients of determination R² consistently exceeded 0.9999. For Case 3 to 8, the optimal parameter $\alpha$ ranged from 0.4839 to 0.5248, while $\beta$ ranged from 0.8934 to 0.9405. This narrow distribution suggests strong clustering of the parameter values. Cases 1 and 2 could be considered exceptions, corresponding to amplitude scaling factors of 0.8 and 0.9, respectively. In these cases, the damper operated predominantly within the dimensionless velocity range below 0.5, accounting for 97.1% and 95.4% of the operational time, respectively. Under these low-velocity conditions, moving the yield point along the ascending branch of the bilinear model had a minimal impact on the simulation accuracy. The contour distribution characteristics observed in Figure 11 further corroborate this phenomenon. Notably, after proportional adjustment, the parameters for Case 1 shifted from (0.4338, 0.8132) to (0.4900, 0.9186), and those for Case 2 from (0.4691, 0.8691) to (0.4900, 0.9078). These adjusted values ultimately aligned with the parameter ranges observed in the other cases. The alignment demonstrates the high stability of the bilinear model’s optimal parameters across a wide range of seismic intensities.

4.2. Bilinear Parameter Optimization Sensitivity to Seismic Site Characteristic Period

The site characteristic period of an earthquake is a key parameter that reflects the spectral properties of ground motions. In Section 3, the analysis was conducted using seismic excitations with a site characteristic period of 0.55 s. This section extends thies investigation by conducting a sensitivity analysis of the optimal bilinear model parameters to the seismic site characteristic period for simulating the ECD’s mechanical behavior.

To investigate the influence of spectral characteristics on the parameter optimization process, five characteristic period values—ranging from 0.35 s to 0.75 s—were selected to generate the corresponding seismic excitations. The resulting acceleration response spectra are presented in Figure 21. As shown in the figure, an increase in the characteristic period leads to a gradual intensification of the long-period components in the acceleration response spectra, demonstrating significant variations in spectral characteristics.

A stochastic parameter optimization study for the bilinear model was conducted using five sets of seismic excitations, each containing 500 samples, which exhibited distinct acceleration response spectrum characteristics. The statistical results of the optimization are summarized in

Table 3. Statistics of the stochastic optimization under varying site characteristic periods.

Case	Characteristic Period	$\mathbf{MP} \mathbf{Vel}./{\mathit{v}}_{\mathbf{c}\mathbf{r}}$	R2	$\left(\mathit{\alpha }, \mathit{\beta }\right)$	Vel. Range (−0.5, 0.5)
I	0.35	0.6027	0.9999	(0.3859, 0.7433)	99.1%
II	0.45	0.7767	0.9999	(0.4618, 0.8534)	96.8%
III	0.55	0.9589	0.9999	(0.4832, 0.8948)	93.5%
IV	0.65	1.1298	0.9999	(0.4951, 0.9102)	89.8%
V	0.75	1.3073	0.9999	(0.5150, 0.9344)	86.1%

. Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26 present, for each case, the time histories of displacement STD, the velocity-damping force relationships under the optimal parameters, and the distribution of the damper’s operational velocity range.

Analysis of the displacement STD time histories in Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26 reveals significant differences in both the shapes and peak values of the curves resulting from seismic excitations with different characteristic periods. This result confirms the significant influence of ground motion spectral characteristics on structural response. Nevertheless, under their respective optimal parameters, the displacement STD curves of the bilinear model showed strong agreement with those of the target Wouterse model across all cases, achieving a coefficient of determination R² of 0.9999. These results demonstrate that, despite variations in the spectral characteristics of the input ground motions, the proposed bilinear model can accurately replicate the damping properties of the ECD through parameter optimization.

An observation of the optimal parameters across the different cases indicates that their values exhibit low sensitivity to the spectral characteristics of the ground motions. For cases with longer characteristic periods (Cases III, IV, and V: 0.55, 0.65, and 0.75 s), the ECD operates at higher velocities, with the MP velocity approaching or exceeding the critical velocity $v_{\mathrm{c}\mathrm{r}}$. Correspondingly, the distributions of the optimal parameters are highly concentrated, with $\alpha$ ranging from 0.4832 to 0.5150 and $\beta$ from 0.8948 to 0.9344.

Conversely, for cases with shorter characteristic periods (Cases I and II), the damper operates primarily within the low-velocity range. The ratios of MP velocity to critical velocity $v_{\mathrm{c}\mathrm{r}}$ are 0.6027 and 0.7767, respectively. Furthermore, the damper operates within the dimensionless velocity range < 0.5 for 99.1% and 96.8% of the time in these two cases. Since the damper’s response in this low-velocity region on the ascending branch is governed primarily by the slope, the precise location of the yield point has a minimal impact on simulation accuracy. This explains the relatively scattered distribution of the initial optimal parameters $\left(\alpha , \beta \right)$ identified for these two cases: Case I: (0.3859, 0.7433) and Case II: (0.4618, 0.8534). After proportional adjustment, these parameters converge to (0.4900, 0.9438) and (0.4900, 0.9055), respectively, aligning them closely with the optimal parameter sets identified for Cases III, IV, and V.

This sensitivity analysis robustly demonstrates that the optimal parameters $\left(\alpha , \beta \right)$ converge to a stable, narrowly distributed region (approximately $\alpha \in [0.48,0.53]$, $\beta \in [0.89,0.94]$) across a wide range of seismic conditions, particularly when the damper is driven into its nonlinear regime. The observed stability under these conditions, coupled with the explainable and reconcilable scatter in the low-velocity range, provides a solid foundation for treating $\left(\alpha , \beta \right)$ as fixed constants. This key finding liberates the designer from additional parameter calibration burdens, shifting the focus entirely to the determination of the primary physical design variables, $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $v_{\mathrm{c}\mathrm{r}}$, thereby significantly enhancing the model’s practicality for engineering design

5. Conclusions

This study aimed to develop and validate a simplified bilinear model for representing the nonlinear velocity-damping force of ECDs, facilitating their integration into finite element software for vibration control design. A comprehensive stochastic framework was employed for parameter identification and robustness assessment. The main conclusions derived from this work are summarized as follows:

• A bilinear model was proposed to approximate the complex velocity–damping force relationship of ECDs as described by the Wouterse model. By introducing two dimensionless parameters $\left(\alpha , \beta \right)$, the model preserves the key physical meanings of the maximum damping force $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and the critical velocity $v_{\mathrm{c}\mathrm{r}}$ while providing a bilinear formulation that is significantly more amenable to implementation in finite element software.
• An optimization framework integrating MCS and the GWO algorithm was developed to identify the optimal parameters of the bilinear model. The objective was to minimize the discrepancy in the stochastic dynamic response—specifically, the displacement STD time histories—between the bilinear model and the reference Wouterse model for a SDOF system subjected to spectrum-compatible seismic excitations. The algorithm successfully converged to an optimal parameter set of $\left(\alpha , \beta \right)$ = (0.4839, 0.8934), achieving an exceptional coefficient of determination (R² > 0.9999) for the time history of displacement STD.
• The sensitivity analysis demonstrated the convergence of the optimal parameters $\left(\alpha , \beta \right)$ to a stable and narrowly distributed region under conditions that drive the damper into its nonlinear regime. This occurs specifically when the ratio of the MP velocity to the critical velocity $v_{\mathrm{c}\mathrm{r}}$ approaches or exceeds unity (approximately 0.9 to 1.55 in this study). Within this range, the identified parameters exhibited low variance, with $\alpha$ clustered between 0.48 and 0.53 and $\beta$ between 0.89 and 0.94. Although apparent scatter was observed under low velocity range (where MP velocity/$v_{\mathrm{c}\mathrm{r}}$ < 0.8), this behavior is both explainable and reconcilable. It arises because the damping force in this regime is governed by the overall slope $\alpha /\beta$, making the individual parameters less critical. A proportional adjustment of these parameters confirms their alignment with the stable region, underscoring the model’s robustness.
• The primary practical outcome is that the bilinear model can be implemented by treating $\left(\alpha , \beta \right)$ as fixed constants, leaving only the physical parameters $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $v_{\mathrm{c}\mathrm{r}}$ as the primary design variables. This approach simplifies the workflow by removing the complex numerical implementation step associated with conventional models. It enables the direct use of the bilinear model in standard commercial software, increasing its practicality. Consequently, the designer’s focus shifts entirely to determining $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $v_{\mathrm{c}\mathrm{r}}$ based on the device and project requirements while ensuring that the anticipated maximum operational velocity is appropriately related to the chosen $v_{\mathrm{c}\mathrm{r}}$ to engage the nonlinear behavior as intended. This simplification drastically enhances the model’s practicality for engineering design.