Abstract
The consistency of magnetic flux density of damping gap (CMDG) represents the balancing magnetic flux density in each damping gap of magnetorheological (MR) dampers. It can make influences on the performances of MR dampers and the accuracy of relevant objective functions. In order to improve the mechanical performances of the MR damper with a two-stage coil, the function for calculating CMDG needs to be found. By establishing an equivalent magnetic circuit model of the MR damper, the CMDG function is derived. Then, the multiobjective optimization function and the working flow of optimal design are presented by combining the parallel-plate model of the MR damper with the function posed before. Taking the damping force, the dynamic range, the response time, and the CMDG as the optimization objective, and the external geometric dimensions of the SG-MRD60 damper as the bound variable, this paper optimizes the internal geometric dimensions of MR damper by using a NSGA-III algorithm on the PlatEMO platform. The results show that the obtained scheme in Pareto-optimal solutions has existed with better performance than that of SG-MRD60 scheme. According to the results of the finite element analysis, the multiobjective optimization design including the CMDG function can improve the uniformity of magnetic flux density of the MR damper in damping gap, which meets the requirements of manufacture and application.
1. Introduction
As a semiactive controller for vibration attenuation, the magnetorheological (MR) dampers with excellent electromagnetic controllability have been widely used in various engineering applications [1–4]. A MR damper with high performance not only needs excellent MR fluid [5] but also should have a great damping force and a large dynamic range under certain constraints of geometric dimensions [6]. In order to improve the performance of MR dampers, many methods have been proposed to improve the internal structure of MR dampers, such as multistage piston [7], annular radial channel design [8], variable resistance gap [9], parallel double coil arrangement [10], and meandering magnetic circuit design [11]. However, the new structures resulted from these methods are usually complicated MR damper structures with expensive manufacturing costs and many maintenance difficulties, which significantly limit their engineering applications.
Considering the problems arised from the change of the internal structure, many scholars have turned to the multiobjective optimization design method based on the original structure of MR dampers [12]. As we can see, they basically used the objective function optimization method and the finite element optimization method. Guan et al. [13] and Parlak et al. [14, 15] used the damping force and the dynamic range as the optimization objectives to optimize the performances of the MR damper, while Parlak et al. [16] used the damping force and the magnetic flux density as the optimization objectives based on the multiobjective genetic algorithm, Taguchi experimental method, finite element analysis, and CFD analysis technique. Compared to these studies that involve only two-objective optimization for MR dampers, Nguyen and Choi [17] used the specific volume of a vehicle MR damper as a geometric dimension limiting factor and transformed the three-objective optimization problem which involves the damping force, the dynamic range, and the response time into a single-objective optimization problem by means of a comprehensive evaluation function. By combining the results of the Bingham model with finite element analysis, they have identified the geometric dimension of the damper. As for the finite element optimization method, Ferdaus et al. [18] and Hu et al. [19] have explored the impact of different piston shapes on the performance of MR dampers by means of the finite element analysis method. It suggests optimization of the performances of MR dampers without changing its original structure on the basis of changing the internal geometric dimensions of that. Taken the temperature effects into consideration, besides, Dong et al. [20] used the Six Sigma robust method to optimize the main geometrical parameters of MR dampers. All the aforementioned optimization design methods for MR dampers only take two even less performance indexes into account at the same time. In addition, they usually take the middle piston thickness as twice of the flank piston thickness and do not specifically deduce and use any function to optimize the thickness of flank piston, which can influence the performance of MR dampers to some extent. Therefore, the optimization results of precise performance of MR damper are difficult to obtain.
This paper takes the damping force, the dynamic range, the response time, and the function of the consistency of magnetic flux density of damping gap (CMDG) as the optimization objectives into account simultaneously and proposes a multiobjective optimization design method for MR dampers based on NSGA-III algorithm [21]. The SG-MRD60 damper applied in sea-crossing bridges is used as a subject, and the piston diameter, rod diameter, coil slot length, coil groove depth, cylinder body thickness, and flank piston thickness are used as variables of the geometric dimension. The formula for calculating CMDG performance of MR dampers is derived by establishing an equivalent magnetic circuit model of MR damper, which has reverse exciting current in each coil. After integrating this formula with the MR damper parallel-plate model, the multiobjective optimization function and the working flow of optimal design are proposed. Thereafter, the comprehensive performance of MR dampers is further optimized by using PlatEMO [22], which is a MATLAB Platform for evolutionary multiobjective optimization. The results show that the Pareto solution set formed by the optimization method provides better solutions for those with performance indexes (damping force, dynamic range, and response time) higher than those of SG-MRD60 scheme. And the performance index of CMDG decreases by 11.27%, compared to the corresponding result of SG-MRD60 scheme. According to the finite element analysis, these results demonstrate that the introduction of CMDG into multiobjective optimization design for MR damper is of significant importance.
2. Performance Calculation Model
The performance calculation function mainly reflects the functional relation between the design variables and performance indexes of MR dampers. In order to optimize and compare the design variables, we select SG-MRD60 two-stage coil-type MR dampers manufactured by NingBo ShanGong Intelligent Safety Technology Co., Ltd. as the subject (see in Figure 1). The SG-MRD60 has a very high degree of representativeness in engineering, and its entity structure model is shown in Figure 2. As for materials, according to the SG-MRD60, the piston rod, piston, and cylinder block are chosen by 45# steel, electrician pure iron, and Q235, respectively. The diameter of the cylinder cavity is 60 mm. The maximum stroke of the damper is ±40 mm. The rated input current is 0–2A, and the maximum damping force is 5 kN. On the premise of maintaining the basic structure, material, and cavity dimension of the damper, the optimization of the internal geometric dimension parameters is studied. The design variables to be optimized include diameter of the piston , diameter of the rod , length of the coil slot , depth of the coil groove , thickness of the cylinder , and thickness of the flank piston , which are shown in Figure 3.
2.1. Design Theory of Magnetic Circuit Performance
2.1.1. Response Time Model
As shown in Figure 3, the magnetic circuit area is divided into piston axial area (Area11), rod area (Area12), piston flank radial area (Area21), piston middle radial area (Area22), flank damping gap area (Areaf1), middle damping gap area (Areaf2), and cylinder area (Area3). The distribution of the corresponding magnetic circuits and areas is shown in Figure 4. In this figure, the red arrows represent the magnetic induction line direction of the magnetic circuit when the MR damper coil works with reverse exciting current.
According to Gauss’s law [23], the sum of magnetic flux of the circuit can be written aswhere is the magnetic flux density and varies with the change of area and is the cross-sectional area formed by the passage of magnetic induction lines through various parts of the area.
On the basis of Figure 4, an equivalent magnetic circuit model (shown in Figure 5) is established. In this model, the magnetic reluctance of most regions is connected in series. However, since the piston rod and piston are made from different materials, the magnetic reluctance in the corresponding areas “Area11” and “Area12” is connected in parallel.
According to the model in Figure 5, neglecting the influence of magnetic flux leakage, the magnetic reluctances of each area can be written aswhere , , , , , , and represent the magnetic reluctance of the flank damping gap, middle damping gap, piston axial, rod axial, flank piston radial, middle piston radial, and damper cylinder, respectively. The meaning of other parameters appearing in the formula is given in Table 1.
According to Ohm’s law and Ampere’s law of magnetic circuit [23], the number of coil turns can be expressed aswhere is the number of single stage coil and is the current of coil. Due to the fact that the piston is symmetrically arranged with double coils, represents the total magnetic reluctance of one coil and is the parallel magnetic reluctance. The magnetic flux of flank damping gap in (3) is given bywhere is the magnetic flux density of flank damping gap.
According to the rated current range of MR dampers, let A; the number of single-stage coil can be rewritten as
According to the ratio of coil turns to coil slot length, the value of coil layer is expressed as
The coil resistance of the MR damper can be expressed aswhere is the resistivity, is the wire radius, and is the wire length of coil and it is given by
The response time of MR damper can be expressed as
2.1.2. CMDG Model
According to the structure of MR dampers with a two-stage coil, the damping gap can be divided into three effective parts in magnetic field. And its magnetic flux equals the sum of each coil. When the current directions of the two exciting coils are opposite to each other, the middle part has the highest magnetic flux [24]. Therefore, the consistency of magnetic flux density between damping gaps in different areas (such as Areaf1 and Areaf2, as shown in Figure 4) should be taken into account during the design process.
The key to control CMDG is to determine the geometric dimensions of W1 and W2 (as shown in Figure 3). The traditional model [13] simply defines the size of W2 as twice as W1, which affects the magnetic circuit performance of MRD. Therefore, this study does not adopt the traditional way and deduces the optimization objective function of CMDG. The derivation process is as follows:
According to Gauss’s law and Ohm’s law of magnetic circuit [23], taking the left piston exciting coil as the research subject, the formulas for calculating the magnetic flux generated by the coil in the middle of the piston and the right side of the piston are expressed, respectively, aswhere and are the magnetic flux of middle and right damping gap, respectively.
Because of the symmetry of the exciting coils in both sides of the piston, the magnetic flux density of the damping gap in each side is the same. Therefore, the left excitation coil of the piston is taken as the subject for research. According to the superposition principle of magnetic field, ignoring the direction of the magnetic field, the formula for calculating the magnetic flux density of the flank and the middle damping gap can be written aswhere and are the magnetic flux density of flank and middle of damping gap, respectively.
Combining (12) and (13), the formula of CMDG is given bywhere is the magnetic leakage coefficient, which is given bywhere and are the simulation results corresponding to and in physical meaning, respectively.
2.2. Design Theory of Mechanical Performance
On the basis of the MR damper parallel-plate model [25, 26], damping force can be expressed aswhere is the viscous force, is the controllable force, and is the friction. Considering that friction force is hard to model and calculate, we assume that there is no friction force:where is the dimension of damping gap, is the mean circumference of the damper’s annular flow path, is the effective axial pole length, is the effective extrusion surface on the side of the piston, and is the volumetric flow rate; these variables are given by
Combining (16)–(19), the damping force of MR damper can be rewritten as
The dynamic range is defined as the ratio of the damping force to the viscous force, and it is given by
3. Optimization Design Method
3.1. Background of Multiobject Optimization
A multiobjective optimization problem, as the name suggests, has many objective functions. Its general form [27] can be expressed aswhere is the objective function and and are the constraint function. M, J, and K are the numbers of each kind of function. is a vector of decision variables, whose range of value is limited within a lower bound and an upper bound .
In this procedure of optimization design, represents four objective functions, the damping force, the dynamic range, the response time, and the CMDG. The vector represents all the parameters of geometric dimension. If we convert all the into a single-objective optimization problem, the objective function will be written aswhere is the objective function for the problem of single-objective optimization and is the weighting factor for each objective function.
As shown in (23), the result of a single-objective optimization depends on the value of and there is only one optimal solution which can be obtained after optimization. It is impossible for the decision-maker to compare the solutions in the optimal solution set and select the required optimal solution from them. Therefore, the appropriate evolutionary algorithm needs to be selected and used to obtain the optimal solutions from the feasible solutions, which is the other focus in this paper.
3.2. Premise and Hypothesis
(1)The cavity diameter () is determined as 60 mm.(2)In order to keep the piston stroke constant when the length of the cavity is constant, let piston length mm.(3)According to the measured data of SG-MRF2035MR fluid, the relationship between yield stress and magnetic flux density is shown in Figure 6, and the curve of the yield stress is fitted by the least-squares method and can be written as According to the relevant parameters of SG-MRD60 under the maximum operating current I = 2 A, we assume that the maximum magnetic flux density of damping gap T. Then, the yield stress of MR fluid in damping gap can be calculated by (20) with a value of 26 kPa.(4)Substituting the finite element simulation results of SG-MRD60 into (15), we get the magnetic leakage coefficient .(5)Set the piston linear velocity mm/s, the optimization parameters are listed in Table 1.
3.3. Relative Permeability of Materials
In fact, the permeability of the material tends to decrease with the increase of magnetic field strength, and different materials have different relative permeabilities at different temperatures [14]. Generally, the magnetization curve of DT4 pure iron and 45# steel can be seen in Figure 7, which shows the nonlinearity relationship between B and H. Since the relative permeability is the derivative of H to B, it can be approximated to a constant when the magnetic flux density B of the materials is less than 1.5 T.
To simplify the calculation, it is assumed that the relative permeability of each material is a constant which does not vary with temperature. And the values of relative permeability [26] of MR fluid (), DT4 pure iron (), 45# steel (), and Coil () are listed in Table 1.
3.4. Constraint Condition
(1)Diameter of the piston must be smaller than diameter of the inner cavity, that is, .(2)According to the GB/T699-1999 standard, 45# steel with a diameter of 15 mm can withstand the static axial force of 106 kN, which is much larger than that of the SG-MRD60 design with a damping force 5 kN; therefore, let mm.(3)According to MR damper geometric relations, the constraint conditions of the geometric dimension and can be expressed as(4)The remaining constraints are selected with reference to the SG-MRD60 geometric dimensions.
Based on the geometric relationship between the constraint conditions and the geometric dimension parameters, six geometric dimension parameters are determined as design variables to be optimized. The corresponding meaning and range of values are shown in Table 2.
3.5. Algorithm Principle and Optimization Method
In multiobjective optimization problem, it is difficult to achieve the best optimal results simultaneously for all optimal objectives. The NSGA-III algorithm [21] is used in this study. This algorithm seeks to find all solutions with at least one target having better solution while making no target worse, namely, Pareto-optimal solutions. Each performance index is overall optimal in the front surface formed by the Pareto-optimal solution set, which is convenient for decision-makers to filter out the optimal solution to meet their own needs.
3.5.1. Principle of NSGA-III Algorithm
In order to handle the multiobjective optimization problems, NSGA-III algorithm [21] is proposed as an evolutionary multiobjective optimization, which, compared with the NSGA-II algorithm [28], achieves further improvement of population selection mechanism. NSGA-III algorithm can ensure diversity of obtained solutions through a predefined set of reference points. As for the operating steps of the NSGA-III algorithm, it can be written as follows. Firstly, we use the genetic algorithm to initialize the population and layer the population at different levels according to the nondomination sorting algorithm, so a new population can be sorted from each level of the population. Secondly, on the basis of the minimum value of each objective function and the reference point supplied by a systematic approach or user, we construct the reference line and associate the population with it. According to the niching algorithm, niche counts that determine the next generation of populations are obtained by calculating the distance between population and reference line. Finally, the iterative process is repeated and the algorithm does not stop until it reaches the maximum iterations. When the algorithm stops, it outputs the Pareto-optimal solutions.
3.5.2. Procedure of Optimization Design
According to the needs of engineering, the objectives of the damping force and the dynamic range are necessary to be maximized. However, the NSGA-III algorithm is used to minimize the objective function in the PlatEMO [22]. Therefore, we transfer the optimization problem of max into the problem of min . Taking the damping force, the dynamic range, the response time, and the CMDG as the optimization objectives simultaneously, as is mentioned in Section 3.1, the multiobjective optimization problem can be rewritten aswhere , , , and are the optimization objective function about the damping force, the dynamic range, the response time, and the CMDG, respectively.
Equation (28) is written to a MRD file, and the above optimization algorithm is implemented by setting parameters (shown as Table 3) in the PlatEMO [22], a MATLAB-based open source platform with more than 50 popular evolutionary multiobjective optimization algorithms including the NSGA-III algorithm, is easily used by the MATLAB GUI.
At the end of each iteration in the running NSGA-III algorithm, it is necessary to judge all the design variables whether the depth of the coil groove is equal to or greater than the product of the number of coil layers and the wire radius . If a set of design variables satisfies , the set of design variables will be defined as an invalid variable and eliminated at the next iteration of NSGA-III algorithm.
After the calculation of NSGA-III algorithm, the variable Population including both the Pareto-optimal solutions and its geometric dimensions is outputted by the PlatEMO. And then, the scheme index which is required by engineering will be filtered from Population by using Excel. Besides, the corresponding geometric dimension vectors are found from Population. The working flow of optimal design is shown in Figure 8.
4. Results and Discussion
4.1. Results of Optimization
4.1.1. Results of Algorithm Calculation
The NSGA-III algorithm generates a total of 5986 Pareto-optimal solutions after 30000 iterations. The variables of geometric dimension corresponding to the Pareto-optimal solutions are taken into (10), (14), (20), and (21) respectively, and then, the abnormal solutions in the course of preserving the operation precision are removed. With damping force less than 35 kN, 4844 sets of data points are retained. The distribution of 3D Pareto-optimal front is drawn and shown as Figure 9. Then, we divide the Pareto-optimal solutions into 3 layers according to the range of response time and CMDG from the 4844 sets of data points. And the distribution of points is shown in Figures 10 and 11.
As shown in Figure 9, the damping force is negatively correlated with the dynamic range, which is in accordance with results of similar research [13]. According to Figure 10, the Pareto-optimal solutions are concentrated in a region with low damping force and high dynamic range when response time is less than 160 ms. With the increase of response time, the Pareto-optimal solutions tend to move to the region with high damping force and low dynamic range. And the comprehensive mechanical performance which refers to damping force and dynamic range will be improved. As shown in Figure 11, the design scheme with CMDG less than 0.2% has a large distribution area of Pareto-optimal solutions and the frontier surface formed by the Pareto solution set is uniformly distributed, which meets the needs of design. Besides, with the significant increase of CMDG, the improvement of mechanical performance of MR damper is not obvious.
4.1.2. Analysis of Parametric Correlation
In order to analyze the uncertainty in MR damper parameters and to find which parameter will influence the performances of MR dampers most, we have done some research studies to obtain the correlation coefficient between each value of the optimal objective and the geometric dimension parameter in the Pareto-optimal solutions.
The linear correlation coefficient is calculated by using the linear correlation analysis formula (29). The calculating results are listed in Table 4 and shown in Figure 12:where is the linear correlation coefficient and and are the ith parameter of the optimal objective and the structural parameter, respectively.
(a)
(b)
(c)
(d)
As shown in Figure 12, the piston diameter makes a great influence on the performances of the damping force and the dynamic range in all the geometric dimension parameters. And the flank piston thickness has higher influence on the performances of the response time, and the CMDG also performs apparently than other parameters. Besides, the and are useless to improve the optimization objective of the MR dampers, as shown in Table 4. It may be partly because of the small values of geometric dimension in the rod diameter and the cylinder body thickness compared to other dimensions.
4.1.3. Scheme Comparison
To compare the influence of CMDG difference, we filter out solutions where the performance indexes of the damping force F, the dynamic range K, and the response time T are better than those of SG-MRD60 scheme from the Pareto-optimal solutions set by using Excel. According to the optimization results of CMDG, the solution whose CMDG is approximately 0%, 15%, and 30% are selected and named as scheme A, B, and C, respectively. All schemes including SG-MRD60 scheme are shown in Table 5.
According to Table 5, compared with the SG-MRD60 scheme, the damping force, the dynamic range, and the response time in each scheme are increased by up to 2.2%, 4.5%, and 11.6%, respectively.
4.2. Results of Simulation
In order to compare the optimization results with the simulation results of CMDG, we import the parameters of geometric dimension (see in Table 5) into ANSYS finite element analysis software. The Nephogram magnetic flux density distribution for each scheme is obtained and shown in Figure 13. And the simulation results of magnetic flux density of the damping gap are listed in Table 6. Besides, the average magnetic flux density of the damping gap is given by
(a)
(b)
(c)
(d)
As can be seen in Figure 13(a)–13(c), the original scheme from SG-MRD60, whose magnetic flux density of middle damping gap is significantly greater than that of in the flank damping gap, has poor performance in CMDG. The CMDG of schemes A and B are better than that of SG-MRD60. It is in accordance with the result in Table 6. Besides, the trend of CMDG in Table 6 is the same as that in Table 5. The average deviation between the optimization result and the simulation result is 0.4%, which reflects that the CMDG function can express the distribution effect of the magnetic flux density in damping gap well.
As shown in Table 6, the CMDG in scheme C is 28.93%, which is the largest of all schemes. It leads to the consequence that the simulation result of CMDG in scheme C is the worst as shown in Figure 13(d), which means CMDG with maximum value has the worst performance. The results show that the worse the CMDG performance is, the more unbalanced the magnetic flux density of flank damping gap and middle damping gap will be. It will affect the calculation result of the optimization objective function of damping force and make the optimization scheme invalid. Therefore, the introduction of CMDG into multiobjective optimization design for MR dampers is very important.
5. Conclusions
This study optimizes the geometric dimensions of the selected MR damper using NSGA-III algorithm in the PlatEMO platform. The objectives of the optimization are maximizing the damping force and the dynamic range and minimizing the response time and the CMDG of the damper. This study also proposes a CMDG formula and introduces the CMDG into the process of optimization design. The main conclusions can be listed as follows.
Optimized by NSGA-III algorithm, the Pareto-optimal front shows that the damping force is negatively correlated with the dynamic range and the response time in the Pareto-optimal solutions. And the CMDG index is greatly influenced by the geometric dimensions of the and . The Pareto-optimal front also shows that the damping force and the dynamic range have no correlation with CMDG, which indicates that during the process of final design scheme selection from the Pareto-optimal solutions, the performance of CMDG does not need to be ensured by reducing other performances of the MR dampers.
Compared with the SG-MRD60 scheme, the optimized design scheme achieves higher performance of the MR damper. The finite element analysis results of CMDG are close to the optimization results with an average deviation of 0.4%. Besides, the simulation results of CMDG decrease by 11.27%, which is superior to the simulation results using the SG-MRD60 scheme. It indicates that the introduction of the CMDG index can make the procedure of multiobjective optimization design more valuable for engineering application.
In addition, the optimization results of actual damping force, dynamic range, and response time need to be further tested and verified after the MR damper is manufactured.
Data Availability
All the data used to support the findings of this study are included within the article. And these data are available to all researchers.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to express their special appreciation to all the participants of the expert survey. The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (nos. 51675286 and 51505237), Natural Science Foundation of Ningbo (no. 2017A610081), and the K. C. Wong Magna Fund in Ningbo University.