Optimization design of magnetorheological damper based on multi-objective whale algorithm

Abstract

An efficient optimization design method for magnetorheological (MR) dampers, aimed at enhancing the damping force output and the adjustable coefficient, is explored in this study. The structural parameters of the double-rod MR damper, which significantly influence dynamic performance, were systematically analyzed and determined through Sobol Sensitivity Analysis. On this basis, the critical parameters were automatically optimized using Non-Dominated Sorting Whale Optimization Algorithm. By analyzing the unified Pareto front, the optimal structural parameters of the MR damper are determined and verified through numerical simulations and experimental comparisons. The results show that the key parameters affecting the mechanical performance of MR dampers can be reduced to five. The MR damper designed with these optimal parameters demonstrated a 17.1% increase in the adjustable coefficient and a 1.6-fold increase in damping force. Additionally, the optimization design method exhibited notable computational efficiency with superior global convergence characteristics, effectively solving the challenges in the optimization design of MR dampers. This study further deepens the optimization design theory of MR dampers and broadens the potential for diverse engineering applications.

Article Highlights

Sobol sensitivity analysis pinpoints critical parameters to boost optimization efficiency;

Integrated Sobol-NSWOA methodology advances MR damper optimization;

High-precision rapid-response method enables scalable MR damper production and applications.

Introduction

In recent years, smart damping devices have gained prominence as a critical research focus in vibration mitigation studies, with magnetorheological (MR) dampers receiving particular attention [1, 2, 3, 4–5]. These devices employ MR fluids that exhibit field-dependent rheological characteristics when subjected to external magnetic excitation, thus enabling real-time damping tunability. Systematic optimization of MR damper configurations proves essential for achieving optimal performance metrics while ensuring operational reliability and energy efficiency. Furthermore, properly implemented optimization frameworks not only improve experimental protocol precision but also enable reliable prediction of system responses under varying operational conditions.

Numerous researchers have focused on parameter optimization of MR dampers. Olivier et al. [6] utilized response surface methodology and Box-Behnken design with yield stress as the objective function to optimize the parameters of MR dampers. The simulation results indicated that the optimized MR damper exhibited higher damping force and an adjustable range. Dong et al. [7] optimized the geometric dimensions of bridge vibration MR dampers using genetic algorithms, considering output damping force, adjustable range, response time, and magnetic flux density as optimization objectives. Gao et al. [8] employed particle swarm optimization to refine damper structure, reducing energy consumption by 1.34 J and weight by 0.2 kg for the optimized MR fluid damper. Hu et al. [9] transformed the multi-objective problem into a single-objective one with different weighting coefficients for output damping force and adjustable range, combining particle swarm optimization with BP neural networks for optimization. Naserimojarad et al. [10] used a combination of mathematical optimization and finite element analysis to achieve optimal power consumption and weight for MR dampers, considering manufacturing constraints to ensure practical results. Jiang et al. [11] identified a constraint relationship between the adjustable range of MR dampers and the number of coils turns, employing the NSGA-II algorithm for multi-objective optimization, and achieving a larger adjustable range while reducing the number of coils turns.

Sensitivity analysis has been established as an essential methodology for quantitatively assessing the parametric influences on the dynamic performance of MR dampers. Several studies have employed different models and methods to conduct sensitivity analysis, thereby generating critical insights regarding parameter sensitivity indices and their complex interrelationships. Zhao et al. [12] conducted a sensitivity analysis of seven key structural parameters of MR dampers using the Bingham model. Zhang et al. [13] explored the sensitivity and contribution of key parameters to the output damping force of an MR damper with energy harvesting capabilities. Although sensitivity analysis was effectively combined with optimization techniques, the complexity of the models may limit practical implementation. For parameter identification of the Bouc–Wen model for MR dampers, sensitivity analysis and a modified Particle Swarm Optimization (PSO) algorithm were used [14], addressing issues of multi-identification parameters and low accuracy. Xue et al. [15] emphasized the importance of accurate parameter estimation and sensitivity analysis in the modeling of MR dampers, using a modified genetic algorithm and a simplified Bouc–Wen model to address non-linear hysteresis characteristics efficiently. To address the Bouc–Wen model’s shortcomings, Jiang et al. [16] established an optimized Bouc–Wen model with sensitive parameters to ensure calculation accuracy. The sensitivity of parameters was analyzed using the one-at-a-time method based on mechanical property tests. Boreiry et al. [17] used sensitivity analysis to identify key parameters affecting chaotic vibrations of a nonlinear 7-degree-of-freedom full vehicle model equipped with an MR damper. The sensitivity analysis results highlighted the critical parameters that need to be optimized to enhance the damper’s performance, providing valuable insights for MR damper design and optimization. Nanthakumar et al. [18] conducted a response surface analysis of parameters and investigated the influence of parameter correlations on the performance of MR dampers. Their study indicated that the distance between the piston rod and coil and the length of the magnetic pole had a more significant impact on the response of MR dampers. Hou et al. [19] employed a proposed single-rod dual-cylinder dual-coil MR damper damping force model. They calculated the sensitivity of the parameters by the damping force variation caused by a single parameter change and determined the key parameters affecting damper performance.

The multi-objective optimization for MR dampers provides an effective solution to simultaneously mitigate the critical issues of excessive energy dissipation and insufficient damping capacity. Jameel and Abouhawwash [20] integrated the optimal Latin hypercube design, ellipsoidal basis function neural network (EBFNN), and multi-objective particle swarm optimization (MOPSO) to improve the damping performance of MR dampers. The combination of experimental data and computational models provides a robust framework for optimization. Khodadadi et al. [11] employed the Non-Dominated Sorting Genetic Algorithm version II (NSGA-II) to optimize MR dampers for maximum dynamic range and minimum coil turns. The genetic algorithm effectively explores a wide range of design configurations, demonstrating high research quality. By establishing a relationship model between structural parameters and optimization objectives, Mirjalili et al. [21] used a multi-objective genetic algorithm to optimize the structural parameters of MR dampers, aiming to improve their performance and efficiency. Liang et al. [22] employed a multi-objective optimization algorithm and finite element modeling to investigate the non-parallel plate magnetic circuit characteristics of the CFC-MRD. Experimental results demonstrated a maximum deviation of 6.77% from theoretical values, validating the design and optimization process. Wei et al. [23] utilized multi-objective optimization techniques to balance the damping performance of MRD and vehicle handling stability. By integrating the Non-Dominated Sorting Genetic Algorithm II (NSGA-II), the structural parameters of MR dampers were optimized to achieve improved handling stability and ride comfort. Olivier et al. [24] optimized the design parameters to enhance the damping force of a novel hybrid-type MR (MR) damper, which incorporates both an electromagnet and permanent magnets. The comprehensive optimization design and numerical analysis underscore the potential of hybrid designs in enhancing vibration control performance. During optimization, multi-objective optimization algorithms based on Pareto solutions [25, 26–27] can achieve more precise results. In recent years, the improved whale optimization algorithm, known for its global search capability and fast convergence, has been widely applied in various fields [28, 29].

Systematic optimization of key parameters is essential for enhancing the performance of MR dampers. The selection of optimization parameters should be dynamically adapted to specific structural configurations and operational scenarios. Current research lacks in-depth exploration of parameter sensitivity analysis for MR dampers. Although their optimization involves multi-dimensional coordination of structural parameters, mechanical model parameters, and control algorithms, existing approaches predominantly rely on empirical assumptions, local sensitivity analyses, or limited numerical simulations. The critical tool of global sensitivity analysis remains underutilized. This methodology enables quantitative evaluation of the integrated effects of parameters on system outputs, thereby establishing a robust decision-making foundation for design optimization.

The application of NSWOA in the optimization of MR dampers is relatively scarce. Compared with conventional algorithms, NSWOA eliminates the need for weight assignment to reduce multi-objective problems to single-objective formulations. It directly generates high-precision Pareto front solutions with demonstrated advantages: (1) fast convergence speed, (2) escape from local optima constraints, and (3) improvement in solution diversity metrics, effectively balancing exploration–exploitation trade-offs.

Building on this foundation, this study innovatively integrates Sobol global sensitivity analysis with NSWOA, supported by simulation analysis and numerical modeling, to develop a targeted theoretical and methodological framework for MR damper optimization design. By analyzing the unified Pareto front, the optimal structural parameters of the MR damper are determined and verified through numerical simulations and experimental comparisons. This approach significantly reduces resource expenditures while providing actionable guidance for the engineering realization of high-performance MR dampers.

Sensitivity analysis

Analysis model

In MR damper research, various nonlinear constitutive models have been developed, with the Bingham model occupying a unique position by unifying electromagnetic principles with fluid dynamics through systematic integration of Navier–Stokes equations. This formulation establishes it as the cornerstone theoretical framework in MR damper studies through its distinctive physical interpretability—each parameter embodies measurable material properties or operational boundary conditions, rather than abstract curve-fitting coefficients. The model exhibits three cardinal virtues: conceptual parsimony (minimalist topology), mathematical tractability (closed-form solutions), and parametric determinism (one-to-one mapping between physical mechanisms and model parameters). These characteristics enable first-principle parameter identification through constitutive law decomposition, while maintaining a complete yet non-redundant parameter space adequate for fully characterizing MR dampers' nonlinear hysteretic behavior.

Compared with higher-order phenomenological models prone to parameter covariance, the Bingham model's axiomatically orthogonal parameterization fundamentally eliminates collinearity artifacts in Sobol global sensitivity analyses. Furthermore, this inherent parametric efficiency enables strategic complexity management in multi-objective optimization frameworks: By constraining the design space dimension to physically essential variables, the approach achieves rigorous dimensionality reduction without compromising model fidelity, thereby critically enhancing computational efficiency for population-based metaheuristic optimizers like Whale Optimization Algorithm (WOA). These mutually reinforcing advantages in analytical tractability, sensitivity resolution, and numerical feasibility motivated our selection of the Bingham model as the foundation for subsequent parameter sensitivity profiling and performance optimization in this investigation. Figure 1 illustrates a schematic diagram of the MR damper.

Fig. 1 [Images not available. See PDF.]

Double-rod MR damper. 1. Chamber 2. End cap 3. Cylinder 4. Piston 5. Excitation coil 6. MR fluid 7. Piston rod

Figure 2 presents a detailed schematic representation of the parametric configuration of the MR damper system. Figure 3 demonstrates the optimized magnetic circuit design, highlighting the key structural components and flux distribution characteristics.

Fig. 2 [Images not available. See PDF.]

Schematic diagram of MR damper

Fig. 3 [Images not available. See PDF.]

Schematic diagram of the magnetic circuit of MR damper

The magnetic flux density is provided below,

1

$$B=\frac{\phi }{S}$$where φ represents the magnetic flux in webers (Wb), and S denotes the area vector in square meters (m²) that is perpendicular to the magnetic field. According to the magnetic circuit Ohm's law, the magnetic flux can be calculated using Eq. (2).

2

$$\varphi =\frac{\mathit{NI}}{R_{\text{m}}}$$where N represents the number of turns in the excitation coil, I denotes the current flowing through the input excitation coil, and R_m signifies the total magnetic reluctance of the magnetic circuit. Typically, the number of turns in the excitation coil can be calculated based on the relationship between the coil slot and the wire diameter:

3

$$N=\frac{\mathit{hb}}{a^2}$$where h represents the coil slot height, b represents the coil slot length, and a represents the diameter of the excitation coil wire. The magnetic reluctance of each part of the magnetic circuit in the MR damper can be calculated using the following equation:

4

$$R_i=\frac{l}{\mu S}$$where l represents the length of the magnetic circuit, and μ represents the magnetic permeability. The magnetic reluctance of each part is as follows: R₁ is the magnetic reluctance of the central shaft segment, R₂ is the side wing segment, R₃ is the cylinder segment and R_MR is the damping gap segment.

5

$$R_1=\frac{L+2b}{2{\mu }_1\mathrm{\pi }\left[{\left(R-h\right)}^2-r^2\right]}$$

6

$$R_2=\frac{2\left(R-r\right)}{{\mu }_1\mathrm{\pi }RL}$$

7

$$R_3=\frac{L+2b}{{\mu }_2\mathrm{\pi }\left(4R+2g+t\right)t}$$

8

$$R_{\text{MR}}=\frac{g}{{\mu }_{\text{MR}}\mathrm{\pi }\left(2R+g\right)L}$$where L is the piston's effective length(L = 2L₁); R is the piston radius, r is the piston rod radius, t is the cylinder wall thickness, μ₁ is the magnetic permeability of the piston material, μ₂ is the magnetic permeability of the cylinder material, and μ_MR is the magnetic permeability of the MR material. The total magnetic reluctance can be expressed as:

9

$$R_{\text{m}}=R_1+2R_2+R_3+2R_{\text{MR}}$$

Combining Eqs. (5)–(9):

10

$$R_m=\frac{L_1+b}{{\mu }_1\mathrm{\pi }\left[{\left(R-h\right)}^2-r^2\right]}+\frac{2(R-r)}{{\mu }_1\mathrm{\pi }R{L}_1}+\frac{L_1+b}{{\mu }_2\mathrm{\pi }\left(2R+g+\frac{t}{2}\right)t}+\frac{2g^2}{{\mu }_{\mathit{MR}}\mathrm{\pi }\left[{\left(R+g\right)}^2-R^2\right]L_1}$$

Substituting Eqs. (2) and (3) into Eq. (1), we obtain the relationship between magnetic flux density and coil dimensions:

11

$$B=\frac{\mathit{hbIg}}{a^2{R}_{\text{m}}\mathrm{\pi }[\left({R+g)}^2-R^2\right]L_1}$$

For enhanced magnetic flux concentration and mechanical strength, DT4 electrical pure iron was selected as the piston material, while 45# steel was chosen for the cylinder housing due to its superior mechanical properties and cost-effectiveness. The MR fluid formulation, incorporating multi-walled carbon nanotubes encapsulating magnetic particles (as developed by Xu [30]), was employed as the rheological medium to achieve optimal field-responsive characteristics. The rheological properties of this advanced MR fluid formulation, demonstrating the shear stress versus shear rate relationship under varying magnetic field intensities, are quantitatively presented in Fig. 4.

Fig. 4 [Images not available. See PDF.]

The rheological characteristic curve of MR fluid

Based on the relationship curve shown in Fig. 4 between shear yield stress (τ_y) and magnetic flux density (B), it can be nonlinearly fitted to the following polynomial:

12

$${\tau }_y=1366.48B^4-1275.05B^3+347.31B^2-8.86B+0.19$$

Substituting Eq. (11) into Eq. (12) yields:

13

$$\begin{array}{cc}\hfill {\tau }_{\text{y}}& =1366.48{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^4-1275.05{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^3\hfill \\ \hfill & +347.31{(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1})}^2-8.86(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1})+0.19\hfill \\ \hfill \end{array}$$

Considering the stringent spatial constraints inherent in practical applications, the selection of coil diameter requires careful optimization to achieve a balance between magnetic performance and dimensional compatibility, while maintaining structural integrity within the allocated volume. To systematically address these design constraints, Eq. (14) was developed as a comprehensive mathematical model for determining the optimal coil diameter, incorporating magnetic field requirements, thermal dissipation characteristics, and spatial limitations to ensure both design rationality and operational accuracy.

14

$$a=2\sqrt{\frac{I_{\text{max}}}{\mathrm{\pi }j}}$$

The current during operation is set to 2A, and the current density j is set to 8A/mm². After calculating, a is found to be 0.56mm. To prevent damage to the coil due to excessive current, copper core enameled wire with a diameter of 0.8mm is used as the excitation coil. Table 1 presents the values of the material parameters mentioned in the above formulas.

Table 1. Material parameter

Parameters	Value
Current I (A)	2
Piston rod velocity v0 (mm/s)	20
zero-field viscosity η(Pas)	0.5
Magnetic permeability of DT4 electrical pure ironμ1(H/m)	2 × 10–3
Magnetic permeability of 45# steelμ2(H/m)	1 × 10–3
Magnetic permeability of MR fluidμMR(H/m)	2.5 × 10–6

The damping gap of the MR damper is much smaller than the thickness of the piston and cylinder wall, allowing it to be considered a plate model. According to the Bingham plate model, the damping force F of the MR damper is represented by the following equation:

15

$$F=\frac{12\eta L{A}_p^2}{\pi D{g}^3}{v}_o+\frac{3L{\tau }_y}{g}{A}_p$$

The adjustable coefficient K is calculated using Eq. (16):

16

$$K=1+\frac{\mathrm{\pi }D{g}^2{\tau }_y}{4\eta A_p{v}_0}$$where η is the zero-field viscosity of the MR fluid; A_p is the effective piston area, A_P = π(D²-d²)/4, D is the piston diameter and d is the diameter of the piston rod; g is the damping gap; v₀ is the velocity of the piston rod movement; and τ_y is the shear yield stress of the MR fluid.

Substituting Eq. (13) into Eqs. (15) and (16) and rearranging, we can derive the expressions for the output damping force and the adjustable coefficient:

17

$$\begin{array}{cc}\hfill F& =\frac{6L_1{A}_p}{g}[\frac{2\eta A_p{v}_0}{\mathrm{\pi }R{g}^2}+1366.48{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^4-1275.05{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^3\hfill \\ \hfill & +347.31{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^2-8.86\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)+0.19]\hfill \\ \hfill \end{array}$$

18

$$\begin{array}{c}K=1+\frac{\mathrm{\pi }R{g}^2}{2\eta A_p{v}_0}[1366.48{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^4-1275.05{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^3\\ +347.31{\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)}^2-8.86\left(\frac{\mathit{hbI}}{a^2{R}_{\text{m}}\mathrm{\pi }(2R+g)L_1}\right)+0.19]\\ \end{array}$$

Sensitivity analysis

To thoroughly examine the mechanisms by which parameters influence the performance of MR dampers, this study employs the Sobol [31] sensitivity analysis method. The Sobol method, a variance-based sensitivity analysis technique, uses the variance decomposition method to decompose the total output variance into contributions attributable to each input parameter and their interactions.

19

$$D={\sum }_{i=1}^m{D}_i+{\sum }_{i\le j\le m}^m{D}_{\mathit{ij}}+\cdots +{\sum }_{i\le m}^m{D}_{1\cdots m}$$where, D represents the total variance of the output, D_i is the variance caused by the ith parameter X_i, D_ij is the variance resulting from the interaction between X_i and X_j, and m is the number of parameters.

Within the Sobol sensitivity analysis framework, the quantitative analysis of the relationship between parameters and the objective function typically relies on the calculation of first-order sensitivity indices (S_i) and total sensitivity indices (S_T). The first-order sensitivity index (S_i) aims to measure the direct impact of a single parameter X_i variation on the objective function, without involving potential interactions with other parameters. The total sensitivity index (S_T) provides a more comprehensive evaluation of the overall influence of parameter X_i on the objective function, considering the potential interactions between X_i and all other parameters. This evaluation approach enables us to insightfully discern synergistic or antagonistic effects among parameters, thereby more accurately grasping the global impact of parameters on the objective function. The first-order sensitivity index and total sensitivity index for parameter X_i are respectively given by Eqs. (20) and (21).

20

$$S_i=\frac{D_i}{D}=\frac{Va{r}_{X_i}\left(E_{X_{\sim i}}\left(Y\left|X_i\right)\right)\right)}{D}$$

21

$$S_{\text{T}}=1-\frac{D_{\sim i}}{D}=1-\frac{Va{r}_{X\sim i}(E_{X_i}(Y\left|X_{\sim i}\right)))}{D}=\frac{E_{X\sim i}\left(V_{\mathit{Xi}}\left(Y\left|X_{\sim i}\right)\right)\right)}{D}$$where D_∼i represents the variance caused by the interaction of all parameters except the ith parameter. According to Eq. (19), the total variance of the MR damper's seven structural parameters is:

22

$$D={\sum }_{i=1}^7{D}_i+{\sum }_{i\le j\le 7}^7{D}_{\mathit{ij}}+\cdots +{\sum }_{i\le 7}^7{D}_{1\cdots 7}$$

The total sensitivity index serves to evaluate the comprehensive impact of parameters on model output. This index not only encompasses the first—order effects of individual parameters but also accounts for their interactions with other parameters. Within this framework, parameters with higher total sensitivity indices are considered more sensitive to the model. In this study, this analysis process is implemented using the Python programming language. The specific workflow is illustrated in Fig. 5.

Fig. 5 [Images not available. See PDF.]

Flowchart of global sensitivity analysis

Sensitivity analysis results

To guarantee the stability and reliability of the analysis results, the parameter N was set to 10,000. The total sensitivity index is of particular significance in the sensitivity analysis of MR dampers. Thus, we specifically concentrated on this key indicator. Figure 6 presents the detailed results of the sensitivity analysis, with the total sensitivity indices of parameters F and K elaborated.

Fig. 6 [Images not available. See PDF.]

Sobol method total sensitivity index results

As shown in Fig. 6, when exploring the factors influencing the output damping force, the damping gap (g) is identified as the most critical parameter, showing significant sensitivity. Following closely are the piston radius (R), the ratio of the piston rod radius to the piston radius (W), and the ratio of the coil slot height to the piston radius (Z). In comparison, the sensitivity of the piston's effective length (L) and the coil slot length (b) is relatively lower, while the cylinder wall thickness (t) exhibits the lowest sensitivity.

Regarding the adjustable coefficient, the damping gap (g) still maintains the highest sensitivity, followed by the ratio of the coil slot height to the piston radius (Z) and the piston radius (R). In terms of the total sensitivity index, the influence of the ratio of the piston rod radius to the piston radius (W) surpasses the coil slot length (b) and the piston's effective length (L). Once again, the cylinder wall thickness (t) is confirmed as the least sensitive parameter.

By applying the Sobol method for sensitivity analysis, the damping gap is clearly identified as the parameter with a decisive impact on the viscous damping of shear valve-type MR dampers. As the damping gap decreases, the viscous damping tends to increase. Given the definition of the adjustable coefficient, under the condition of constant Coulomb damping force, an increase in viscous damping force will lead to a decrease in the adjustable coefficient, thereby revealing a negative correlation between the damping gap and the adjustable coefficient of the MR damper.

Although sensitivity analysis reveals that the damper gap has a significant impact on the maximum damping force and the adjustable coefficient of the MR damper, even a slight increase in clearance causes a sharp decline in the maximum damping force. Simultaneously, excessively small clearance significantly reduces the adjustable coefficient and poses considerable challenges to machining and industrial assembly. Therefore, taking into account the precision of machining, the ease of assembly, and the balance between the maximum damping force and the adjustable coefficient, its range is often constrained by manufacturing processes and commonly set at 1mm or 2mm. Therefore, further adjustments to the damping gap are typically not considered during the optimization design process. On the other hand, the cylinder wall thickness does not have a direct impact on the flow of MR fluid in the channel but primarily affects the local characteristics of the magnetic field distribution. During the design process, the cylinder wall thickness serves more as a parameter to control the size of the MR damper, with minimal direct impact on the output damping force and adjustable coefficient. Taking into account the considerations mentioned above, to achieve optimal performance in terms of output damping force and the adjustable coefficient of the MR damper, the key parameters affecting the mechanical properties of MR dampers are as follows: piston radius (R), piston's effective length (L), coil slot length (b), the ratio of coil slot height to the piston radius (Z), and the ratio of piston rod radius to the piston radius (W). We will conduct appropriate optimization designs targeting these parameters.

Parameter optimization

Multi-objective whale optimization algorithm

The Non-Dominated Sorting Whale Optimization Algorithm (NSWOA) [32] is a multi-objective optimization version of the whale algorithm, which is based on non-dominated sorting and crowding distance to evaluate the optimal solution. The proposed algorithm demonstrates superior performance compared to established multi-objective optimization algorithms in terms of solution quality and computational efficiency, particularly in handling complex engineering optimization problems characterized by nonlinear constraints and multiple conflicting objectives. Non-dominated sorting is the most popular and effective technique in multi-objective optimization. A non-dominated solution in multi-objective optimization is defined as a solution where all objective function values are non-inferior to those of another solution, and at least one objective function value is strictly superior, establishing a Pareto dominance relationship that enables the classification of solution quality without requiring explicit weighting of multiple objectives. In this context, the former solution dominates the latter, while non-dominated solutions are those that cannot be dominated by any other solution. Non-dominated sorting is used to determine the order of solutions in the Pareto front. The main drawback of non-dominated sorting is its high computational cost, which is addressed in the NSGA-II algorithm. Therefore, the NSWOA algorithm used in this study is implemented based on the non-dominated sorting in NSGA-II during programming.

1. Initialization: Initialize the whale population by generating a random set of whales and their position vectors, representing combinations of independent variables. These position vectors are represented in matrix form, and the fitness of each combination of independent variables is computed based on the objective functions of the problem, selecting the best search whale.

2. Position Update: Update whale positions utilizing equations for either shrinking encircling or spiral motion, determining the numerical values of the next positions for each whale based on the current optimal solution. It is assumed that there is a 50% probability for whales to choose either the shrinking encircling or spiral path toward the position of the optimal solution for updating.

3. Iterative Process: During iterations, calculate the crowding distance at each step to determine the reference position for the next position update based on the calculation results. This approach helps avoid local optima during optimization, continuously updating the positions of whales towards the optimal solution, ensuring the convergence of the search to the best solution.

4. Guiding Individual Selection: The NSWOA algorithm stores the Pareto optimal solutions in a collection, similar to the "archive" used in NSGA-II and the "shoreline" in the water wave algorithm. This collection serves as a repository for storing the best non-dominated solutions obtained so far. During the optimization process, dominated solutions are gradually eliminated, and non-dominated solutions fill the collection. When the collection is full, i.e., it exceeds a predefined size, some solutions with higher crowding distances will be removed from the collection based on the crowding distance mechanism.

Figure 7 illustrates this process. The crowding distance mechanism measures the crowding degree of each solution in the Pareto front, evaluating the local density of solutions on the front. Typically, the crowding degree of a solution is related to the distance between it and its neighboring solutions in the objective space. The greater the distance between solutions, the higher the crowding degree, indicating that the solution is in a sparser region. This concept is crucial for maintaining diversity and balance on the Pareto front. In multi-objective optimization algorithms, the combination of non-dominated solution sorting and crowding distance is often used to guide individual selection strategies during the evolution process, aiming to retain both better solutions and sparser solutions in the solution set, thus maintaining better diversity in the Pareto front. The workflow of this algorithm is depicted in Fig. 8.

Fig. 7 [Images not available. See PDF.]

Non-dominated solution selection process

Fig. 8 [Images not available. See PDF.]

Flowchart of NSWOA

Optimize condition settings

To improve the efficiency of parameter optimization based on the sensitivity analysis results f, five optimization parameters were selected, piston radius R, piston's effective length L, coil slot length b, the ratio of the coil slot height to the piston radius Z, and the ratio of the piston rod radius to the piston radius W. These parameters significantly influence the damping force and adjustable coefficient. The optimization objective is to maximize the output damping force and the adjustable coefficient of the MR damper. It has been observed that an increase in damping force results in a decrease in the adjustable coefficient; however, a high adjustable coefficient is essential for achieving diversified vibration control and maintaining structural stability. Therefore, during the optimization process, while similar levels of damping force are maintained, the focus is placed on maximizing the adjustable coefficient. This enhancement improves the adaptability of the MR damper to different engineering requirements and varying operating conditions, providing more flexible damping adjustment (Table 2). The ranges of each parameter are as follows:

Table 2. Parameter value range

Parameters	R(mm)	L(mm)	b(mm)	Z	W
Value range	5–50	5–50	5–50	0.2–0.99	0.2–0.99

The damping clearance is the channel through which the MR fluid flows, significantly affecting the performance of the MR damper. As the damping clearance increases, the adjustable coefficient will increase significantly, but the damping force will decrease significantly. For ease of processing, it is set to 1 mm. The cylinder wall thickness, as the least sensitive parameter, has almost no effect on the performance of the MR damper, so it is set to 4 mm. Optimization is carried out using the NSWOA optimization algorithm in MATLAB software. In the optimization process, the relationship among parameters, damping force, and the adjustable coefficient is established using the Bingham model. The number of iterations is set to 1000, with 100 search whales, and the capacity of the collection storing non-dominant solutions is set to 100. The coordinates of the search whales are controlled by five optimization parameters, where X = [R, L, b, Z, W]. The optimal search whale is determined based on the values of the output damping force and adjustable coefficient, guiding other whales to find the optimal solution. According to practical requirements, in the study of shock absorption performance, the MR damper requires a damping force of approximately 400N, with the additional requirement of maximizing the adjustable coefficient. Meanwhile, during optimization, constraints are applied to the piston rod diameter values based on Eq. 23.

23

$$d\ge \sqrt{\frac{4F}{\mathrm{\pi }{\sigma }_1}}$$where F represents the damping force generated by the MR damper, σ₁ is the allowable stress of the steel material, calculated as σ₁ = σ/n, where σ is the tensile stress of the steel material, taken as 235 MPa, and n is the safety factor, taken as 3.

Optimization results

Figure 9 illustrates the obtained Pareto front, representing the set of optimal solutions in multi-objective optimization space where improvement in one objective function inevitably leads to degradation in at least one other objective, thereby establishing the trade-off relationship between competing design goals. The optimal solutions for multi-objective optimization form a Pareto-optimal set distributed across the Pareto front, capturing the full spectrum of trade-off possibilities between conflicting objectives. Through comprehensive evaluation considering practical application constraints and design priorities, an optimal solution can be systematically selected from the Pareto-optimal set using decision-making criteria tailored to specific engineering requirements. The optimized solution set demonstrates uniform distribution characteristics (diversity index > 0.85), ensuring comprehensive coverage of the Pareto front and facilitating effective decision-making across the entire trade-off spectrum.

Fig. 9 [Images not available. See PDF.]

Pareto front

From the perspective of mathematical optimization theory, all parameters on the Pareto front are theoretically optimal solutions. However, in practical engineering applications, a balance must be struck between mechanical machining precision and assembly feasibility. Under the premise of ensuring robustness against parameter variations to meet technical requirements, the prioritized solution set should incorporate integer-valued parameters to maximize the operability of industrial manufacturing processes, the optimized results selected from the Pareto front are presented in the Table 3.

Table 3. Values taken on the Pareto front surface

Parameters	R/mm	L/mm	t/mm	g/mm	b/mm	h/mm	r/mm	F/N	K
1	10	25	4	1	27	4	3	153.85	4.1521
2	11	25	4	1	28	4	3	196.14	3.8439
3	12	25	4	1	26	4	3	242.61	3.5701
4	13	25	4	1	27	4	4	278.74	3.4595
5	14	25	4	1	23	4	4	330.60	3.1925
6	15	24	4	1	22	5	5	359.2	3.1358
7	16	25	4	1	15	6	5	440.60	2.9525
8	16	25	4	1	23	4	4	465.58	2.8903
9	17	25	4	1	13	7	5	518.58	2.8268
10	17	25	4	1	15	6	3	561.78	2.7224
11	18	25	4	1	23	4	5	602.52	2.7112
12	19	25	4	1	30	3	6	655.01	2.6332
13	19	25	4	1	12	8	5	700.29	2.5967
14	20	25	4	1	30	3	6	751.78	2.5361
15	20	25	4	1	30	3	4	810.07	2.4555

For the 15 sets of structural parameter values mentioned above, ANSYS APDL finite element software will be employed to model and conduct magnetic field simulation analysis on these configurations. This analysis aims to observe whether the structural settings are reasonable and whether the magnetic field distribution is appropriate.

Magnetic field analysis

To address the magnetic field distribution and magnetic flux density of the MR-piston valve, a finite element model was built using ANSYS MAXWELL software. A 2D axisymmetric model was developed for the 15 sets of structural parameter values mentioned above. This analysis aims to observe whether the structural settings are reasonable and whether the magnetic field distribution is appropriate. The material for the piston part of the damper is DT4 electric pure iron, while the cylinder part utilizes 45# steel. Material parameters are defined by importing the B-H curve of the magnetic material (refer to Fig. 10). The piston rod and coil slot parts are made of non-magnetic materials, with a magnetic permeability set to 1.

Fig. 10 [Images not available. See PDF.]

B-H curves of materials

Finite element analysis (FEA) has been conducted on the 15 optimized parameter sets, resulting in the magnetic field intensity distribution plots presented in Fig. 11. Figure 11 shows the magnetic field distribution of 15 groups of MR dampers after parameter optimization under an applied current of 1.0 A. The figure reveals a symmetric magnetic field distribution in the MR damper. The magnetic flux is generated by the core, exits the magnetic-conductive component through the upper piston edge, passes through the damping gap, flows across the cylinder barrel, traverses the damping gap again, enters the lower piston, and eventually returns to the core. As shown in Fig. 11a–o, the core exhibits the highest magnetic flux density. The flux density at the edges of the upper and lower pistons is nearly symmetric, with higher values observed near the coil (inner piston side) compared to the outer piston side. The optimized MR damper demonstrates reasonable magnetic field distribution, no magnetic leakage, and absence of magnetic saturation, indicating an improved magnetic design. However, despite all field distributions being based on the optimized parameters in Table 3, certain discrepancies in field strength and uniformity are observed at the damping gaps of the pistons. In Fig. 11e–l, the magnetic field in the damping gaps exhibits higher uniformity and strength compared to other cases. These observations provide critical insights for selecting structural parameters in subsequent MR damper designs.

Fig. 11 [Images not available. See PDF.]

Magnetic flux density contour plot: a–o correspond to the magnetic field distribution contour maps of the MR damper for the parameters 1—15 in Table 3

To obtain the average magnetic flux density within the damping gap, the magnetic flux density at each node within the gap is first extracted. Subsequently, for each quadrilateral element within the gap, the average magnetic flux density is calculated based on the values at its four nodes. This process yields the magnetic flux density for each element within the gap. Next, the magnetic flux density values for all elements within the damping gap are summed, and the result is divided by the total number of elements within the gap to obtain the average magnetic flux density. The average magnetic flux density obtained from the ANSYS simulation analysis is then compared with the calculated magnetic flux density from the Bingham model. The numerical analysis results serve as reference values, while the simulation results act as observed values. The error between the two is listed in Table 4.

Table 4. Comparison of magnetic flux density between numerical analysis and simulation analysis

Parameters	Bm/T	Ba/T	Absolute value of error (%)
1	0.3015	0.3171	5.17
2	0.3244	0.3395	4.65
3	0.3126	0.3265	4.45
4	0.3257	0.3395	4.24
5	0.2872	0.2991	4.14
6	0.3334	0.3474	4.20
7	0.2872	0.3011	4.84
8	0.2914	0.3039	4.29
9	0.2918	0.3066	5.07
10	0.2918	0.3065	5.04
11	0.2933	0.3063	4.43
12	0.2812	0.2922	3.91
13	0.3113	0.3221	3.47
14	0.2817	0.2931	4.05
15	0.2814	0.2935	4.30

The comparison results, as shown in Table 4, indicate that the error in magnetic flux density does not exceed 10%. The alignment between the optimization results and the finite element simulation results is relatively high. From the results, it is observed that the optimized magnetic flux density is around 0.3 Tesla, which correlates with the characteristics of the MR fluid. As previously described, the segment where the shear yield strength of the MR fluid increases rapidly has a significant impact on damping force. Within this range, optimizing parameters to maximize the shear yield strength of the MR fluid is considered the optimal strategy. However, when the magnetic flux density exceeds 0.3 Tesla, the growth rate of the shear yield strength of the MR fluid slows down, resulting in minimal change in the Coulomb damping force and minimal influence on the change in the viscous damping force with adjustment of the structural parameters. This leads to a decrease in the controllable coefficient. Therefore, in optimization algorithms seeking the optimal solution, controlling the magnetic flux density around the end stage of the rapid increase in shear yield strength, approximately 0.3 Tesla, is preferable as a reference point for adjustment to ensure that the output damping force and controllable coefficient are maximized as much as possible. According to the rationality of MR damper design and the expected damping force, the 6th set of values are selected from the solution set of Table 3, as shown in Table 5.

Table 5. Parameter combinations that meet the expected damping force

Parameters	R/mm	L/mm	t/mm	g/mm	b/mm	h/mm	r/mm	F/N	K
6	15	24	4	1	22	5	5	359.2	3.1358

Figure 12 presents the magnetic field analysis results of the optimized MR damper. It can be observed that the magnetic field exhibits a symmetrical distribution, with the magnetic field intensity increasing closer to the coil. The magnetic field is uniformly distributed across the damping gap and is effectively utilized, with no magnetic field passing through the gap between the coil and the cylinder. The magnetic flux density at the damping gap decreases as the distance from the coil increases, which is consistent with Ampère's circuital law.

Fig. 12 [Images not available. See PDF.]

Magnetic flux density

Figure 13 illustrates the magnetic flux density at the damping gap of the MR damper under different currents. It can be observed from the figure that the magnetic field strength at the damping gap increases significantly with the increase in current, but the rate of increase gradually decreases. When the current increases from 0 to 1.0A, the magnetic flux density at the damping gap increases from 0 to 0.27 T. However, when the current increases from 1.0 to 2.0A, the growth rate of the magnetic flux density is lower, with the average magnetic flux density at the damping gap increasing from 0.27 to 0.34 T, representing a 26% increase. This indicates that as the current increases, the magnetic field at the damping gap gradually approaches magnetic saturation.

Fig. 13 [Images not available. See PDF.]

Magnetic flux density at different positions of the damping gap

Performance analysis

Dynamic performances

Incorporation of the optimized parameters into the Bingham model has been conducted, and the model is utilized to establish a structural dynamics model. By varying the amplitude and frequency of the sinusoidal excitation, as well as the magnitude of the input current, the variation in damping force is observed.

When the influence of current on the damping force of the MR damper is observed, the amplitude is fixed at 10 mm and the frequency is maintained at 0.1 Hz.

As shown in Fig. 14, the displacement-damping curves exhibit symmetric distribution about the zero-damping-force axis. The damping force follows a consistent increasing trend across all applied currents, and the enclosed area of the displacement-damping curves expands with higher input currents, indicating enhanced energy dissipation capacity of the MR damper as the current increases. At zero current, the displacement-damping curves adopt an elliptical shape. This occurs because the MR fluid remains in a liquid state, and the damping force primarily originates from viscous damping effects, resulting in linear viscous damper behavior. When a current is applied, the MR fluid transitions from a fluid to a semi-solid state. The total damping force is composed of viscous damping and Coulomb friction contributions, leading to nonlinear hysteresis characteristics in the MR damper. The Coulomb damping force rapidly increases with higher currents, ensuring an elevated MR adjustable coefficient. As depicted in Fig. 15, within the current range of 0–1 A, the Coulomb damping force rises proportionally, causing a sharp increase in the total damping force of the MR damper. However, beyond 1.0 A, the growth rate of the maximum damping force significantly diminishes. This phenomenon arises because the MR fluid approaches magnetic saturation as the applied magnetic field intensifies. At currents exceeding 1.6 A, the MR damper nearly reaches full magnetic saturation, thereby halting further rapid growth in damping force. This observation confirms that the optimization process thoroughly accounts for the magnetic saturation behavior of the MR fluid.

Fig. 14 [Images not available. See PDF.]

Displacement-Damping force curves under different currents

Fig. 15 [Images not available. See PDF.]

Relationship between damping force and current

The displacement-damping force curve of the MR damper under sinusoidal excitation with different amplitudes is shown in Fig. 16, with a current of 1A and a frequency of 1Hz. The analysis reveals that under constant current and excitation frequency, as the displacement amplitude increases, the maximum output damping force also increases significantly. This is primarily due to the increase in piston displacement, which leads to a rise in the pressure difference of the MR fluid at both ends of the damping gap. In the force–displacement curve, the increase in amplitude expands the enclosed area of the curve, which intuitively reflects the enhanced energy dissipation capability of the damper with increasing amplitude. Further observation shows that as the amplitude increases, not only does the displacement increase, but the output damping force also grows significantly. The primary reason is that at a fixed frequency, higher amplitude induces faster excitation speed, thereby increasing the displacement speed of the piston rod, leading to a consistent increment in viscous damping force.

Fig. 16 [Images not available. See PDF.]

Displacement-Damping force curves under different amplitudes

Additionally, Fig. 17 illustrates the impact of different current frequencies on the displacement-damping force curve under conditions of 1A current and 10mm amplitude. The results indicate that under the same current and displacement amplitude, higher frequency results in a larger area enclosed by the curve, providing more substantial damping force. This phenomenon occurs because increasing the current excitation frequency under fixed displacement amplitude accelerates the motion speed of the piston rod, thereby increasing the viscous damping force. It is noteworthy that the relationship between speed and damping force in the MR damper at different frequencies exhibits a pattern similar to that observed under different amplitudes. As the frequency increases, the maximum speed gradually increases, leading to a linear increase in the damping force. In conclusion, the in-depth analysis of the above results allows us to conclude that the optimized MR damper exhibits reasonable dynamic mechanical characteristics in terms of both viscous damping force and Coulomb damping force, providing solid theoretical support for its practical application.

Fig. 17 [Images not available. See PDF.]

Velocity-Damping force curves under different amplitudes

Comparison

To validate the effectiveness of the optimized design, the parameters of the optimized MR damper were applied to a finite element simulation model to calculate the magnetic field intensity of the optimized magnetorheological damper. Subsequently, numerical calculations were performed using the MR damping force mechanical model to determine the damping force and current-dependent adjustable coefficient of the optimized MR damper. The results were then compared with experimental data from a non-globally-optimized MR damper to evaluate the efficacy of the proposed optimization method. The non-globally-optimized MR damper was tested on a servo-hydraulic testing machine. The current applied to the MR damper was incrementally increased from 0 to 2 A, while the hydraulic actuator was driven by a sinusoidal signal with predefined amplitude and frequency. The amplitude and excitation frequency were set to 15 mm and 0.2 Hz, respectively—values intentionally kept low to ensure testing stability. To achieve stable measurements, each test was repeated for 10 sinusoidal cycles. The servo-hydraulic testing machine’s data acquisition system directly recorded the damper’s hysteresis force and corresponding displacement. Numerical simulations of the optimized MR damper’s mechanical characteristics were conducted under identical testing conditions and loading protocols, with consistent material parameters maintained. The mechanical properties of the optimized MR damper are compared with the initial MR damper (as shown in Fig. 18). The results are shown in Fig. 19.

Fig. 18 [Images not available. See PDF.]

MR damper: a MR damper components and b assembled MR damper

Fig. 19 [Images not available. See PDF.]

Comparison of the optimized damping force

Figure 19 clearly illustrates the variation trend of damping force with increasing current for both the original and optimized MR dampers. Notably, the optimized MR damper exhibits a more significant enhancement in the growth of damping force compared to the initial damper. Specifically, the damping force of the initial MR damper gradually stabilizes and show evident magnetic saturation when the current reaches 1A. In contrast, the optimized MR damper, despite the deceleration in the growth rate of damping force, does not exhibit a clear saturation trend, indicating more efficient magnetic field utilization and a more fully leveraged performance of the MR damper. Under a current of 1A, the output damping force of the initial MR damper is 125.3N, with an adjustable coefficient of 2.078. The optimized MR damper shows significant improvements in both maximum damping force and adjustable coefficient, as shown in Table 6. Notably, the adjustable coefficient of optimized MR damper reaches 2.435 under 1A current, indicating an increase of 17.1% compared to the initial MR damper, while the damping force of optimized MR damper increases by 1.6 times, suggesting a broader application control range for the optimized MR damper. These results also reflect the high efficiency and stability advantages of the NSWOA optimization design method for coupling parameter sensitivity analysis proposed in this paper. It can achieve a larger adjustable coefficient, thereby better meeting the needs of vibration control.

Table 6. Comparison of the mechanical properties between the optimized MR damper and the Initial damper

Current /A	0	0.2	0.4	0.6	0.8	1	1.5	2
Initial/N	60.3	67.3	88.5	96.7	115.4	125.3	127.5	128.1
F6/N	108.4	120.6	141.0	178.2	221.9	264.0	318.4	333.1
K6	1	1.113	1.301	1.644	2.047	2.435	2.937	3.073

Conclusions

This study employs a global sensitivity analysis method to meticulously select critical optimization parameters for a MR damper. Subsequently, the Non-dominated Sorting Whale Optimization Algorithm (NSWOA) is applied to optimize these key parameters, and the optimization results are evaluated. The main conclusions are as follows:

1. The Sobol global sensitivity analysis reveals that the key parameters influencing the mechanical performance of the MR damper are ranked as follows: piston radius, effective piston length, coil slot length, ratio of coil slot height to piston radius, and ratio of piston rod radius to piston radius. Consequently, the number of parameters requiring optimization is reduced from 7 to 5, decreasing the optimization workload by 28.6%.

2. The multi-objective NSWOA algorithm achieves efficient optimization of the structural parameters of the MR damper. Compared to the initial MR damper, the optimized damper exhibits a 17.1% improvement in the adjustable coefficient and a 1.6-fold increase in damping force.

3. The NSWOA optimization algorithm integrated with Sobol parameter sensitivity analysis demonstrates higher search efficiency and superior solution quality in the design optimization of MR dampers, offering technical references for their mass production and engineering applications.

Author contributions

Conceptualization, W.L. and Y.Z.; Methodology, C.L.; Validation, X.H.; Resources, Y.Z.; Data curation, Y.X. and X.C.; Writing—original draft, X.C.; Writing—review and editing, Y.Z. All authors reviewed the manuscript and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data availability

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Declarations

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent to publication

Not applicable.

Competing interests

The authors declare no competing interests.

Abbreviations

MR damper

Magnetorheological (MR) damper

NSWOA

Non-Dominated Sorting Whale Optimization Algorithm

Wb

Weber

T

Tesla

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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