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A phenomenological dynamic model of a magnetorheological damper using a neuro-fuzzy
system
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2013 Smart Mater. Struct. 22 125013
(http://iopscience.iop.org/0964-1726/22/12/125013)
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IOP PUBLISHING SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 22 (2013) 125013 (14pp) doi:10.1088/0964-1726/22/12/125013
A phenomenological dynamic model ofa magnetorheological damper using aneuro-fuzzy system
Mohammadjavad Zeinali1, Saiful Amri Mazlan1,Abdul Yasser Abd Fatah1 and Hairi Zamzuri2
1 VSE Research Laboratory, Malaysia–Japan International Institute of Technology (MJIIT), UniversitiTeknologi Malaysia (UTM), 54100 Kuala Lumpur, Malaysia2 Razak School of Engineering and Advanced Technology, Universiti Teknologi Malaysia (UTM), JalanSemarak, 54100 Kuala Lumpur, Malaysia
E-mail: javad [email protected], [email protected], [email protected] and [email protected]
Received 22 August 2013, in final form 10 October 2013Published 5 November 2013Online at stacks.iop.org/SMS/22/125013
AbstractA magnetorheological (MR) damper is a promising appliance for semi-active suspensionsystems, due to its capability of damping undesired movement using an adequate controlstrategy. This research has been carried out a phenomenological dynamic model of two MRdampers using an adaptive-network-based fuzzy inference system (ANFIS) approach. Twokinds of Lord Corporation MR damper (a long stroke damper and a short stroke damper) wereused in experiments, and then modeled using the experimental results. In addition, aninvestigation of the influence of the membership function selection on predicting the behaviorof the MR damper and obtaining a mathematical model was conducted to realize therelationship between input current, displacement, and velocity as the inputs and force asoutput. The results demonstrate that the proposed models for both short stroke and long strokeMR dampers have successfully predicted the behavior of the MR damper with adequateaccuracy, and an equation is presented to precisely describe the behavior of each MR damper.
(Some figures may appear in colour only in the online journal)
1. Introduction
The capability of a magnetorheological (MR) damper as asemi-active system to produce high force capacity and widedynamic range has attracted researchers to focus more onMR dampers. Some comprehensive reviews have considereda wide variety of studies involving MR dampers: design andmodeling for a rotary MR damper (Imaduddin et al 2013) andstructure design and analysis (Zhu et al 2012).
Figure 1 shows a schematic of an MR damper with anaccumulator. The applied input current produces a magneticflux in which the flux lines are perpendicular to thesymmetric axis (the horizontal line in figure 1). The producedmagnetic field influences MR fluid particles to increase theMR fluid viscosity in terms of magnetic flux density (themagnified ellipse in figure 1). This phenomenon generates
a complex relation between the effective input parameterssuch as displacement, which represents the behavior of theaccumulator as a spring, velocity, which corresponds to thedamping behavior of the MR damper, and input current.
To control an MR damper, an accurate mathematicalmodel of the MR damper must be implemented in thecontrolling strategy to describe this complex relationship. Themulti-physics and nonlinearity behaviors of an MR dampercreate a challenging objective in order to achieve the mostprecise model. All designed models can be divided into twodifferent types: predictive quasi-static and phenomenologicaldynamic models. In predictive quasi-static modeling, theconventional model is the Bingham model, which is utilized inmuch research (Bhatnagar 2013, Yang et al 2002, Chew et al2006). The Hershel–Bulkley model has been presented witha higher accuracy (Wang and Gordaninejad 2007). Between
10964-1726/13/125013+14$33.00 c© 2013 IOP Publishing Ltd Printed in the UK & the USA
Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
Figure 1. Schematic of an MR damper model.
Figure 2. (a) Bouc–Wen model. (b) Modified Bouc–Wen model (Spencer et al 1997).
these models, the phenomenological dynamic model producesa thorough characterization of an MR damper. This type ofmodel outperforms the predictive quasi-static model in allaspects of the damper, such as force–velocity behavior.
The Bouc–Wen model has been used widely as aphenomenological dynamic model in a large number ofstudies (Weber 2013, Ismail et al 2009) (see figure 2(a)).Spencer et al (1997) presented a modified Bouc–Wen modelwhich is based on a mixture of elastic springs and a viscousdamper (see figure 2(b)). Spaggiari and Dragoni (2012)suggested an efficient dynamic model in which the proposedmodel by Spencer et al (1997) is modified to investigate asimpler and efficient model.
There are number of studies which modeled RD-8040-1(short stroke) and RD-8041-1 (long stroke) MR dampersusing a feed-forward neural network (Ekkachai et al 2011), atime-delayed adaptive-network-based fuzzy inference system(ANFIS) model under high impact forces (Sarp Arsava et al2013), and a polynomial model (Bin Pokaad and Nasir 2011).In recent years, the outstanding learning capability of fuzzysystems and neural networks has led researchers to apply theseapproaches in predicting the complex relationship between
MR damper parameters (Patil et al 2012, Azwadi et al2013, Guneri et al 2011). In addition, the model of the MRdamper can be used in control strategies and simulation ofsemi-active suspension systems. Therefore, the response time(fast solution) and accuracy of the model are priority targetsto utilize in the suspension system. Both mentioned targetscan be achieved by the ANFIS approach due to its capabilityto quickly predict the phenomena with reasonable accuracy(Jang 1993, Nazari et al 2012).
In this study, an ANFIS approach is implemented toachieve an accurate prediction of the MR damper force byusing the input current, unwanted displacement, and unwantedvelocity as the prediction system’s inputs. In the vehiclesemi-active suspension system, the unwanted displacementand velocity are equivalent to displacement and velocity. TheANFIS model has been trained by the experimental results oftesting the magnetorheological damper under various speedsand input currents.
This paper is structured as follows. The experimentalsetup is explained in section 2. In section 3, the ANFISmethod and parameters are described. Section 4 gives the
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Table 1. Specification of variables in the experiment.
Input Short stroke MR damper Long stroke MR damper
Displacement (mm) ±5 ±10Frequency (Hz) 3.2, 6.3, 9.5, 12.6 1.6, 3.2, 4.8, 6.3, 7.9Current (A) 0, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1 0, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1
Figure 3. Side view of RD-8040-1 and RD-8041-1 MR dampers.
prediction results, and the accuracy of the ANFIS approachis discussed. Conclusions are presented in section 5.
2. Experimental setup
Two types of MR damper are investigated in this study,RD-8040-1 (short stroke) and RD-8041-1 (long stroke),shown in figure 3, supplied by Lord Corporation. Thedifference between these two products is the length ofthe stroke, which is related to the application of thedamper. However, the function of the dampers is similarand helps validate the proposed ANFIS model. The lengthof the dampers at full extension is 207.8 ± 1.9 mm and248.15 ± 2.25 mm for the short stroke MR damper and thelong stroke MR damper, respectively. The stroke length is55.45 ± 0.65 mm and 73.7 ± 0.7 mm in the short and longstrokes, respectively. The maximum value for the input currentexerted to the coils is 2 A.
The experiments were done using a damper test systemMTS 850, which is shown in figure 4. The experimentalresults were evaluated and developed in the vehicle systemengineering laboratory at Universiti Teknologi Malaysia(UTM). The test rig is able to measure three parameters ofthe damper (relative displacement, velocity, and damper force)by using a transducer and a load cell. Figure 4 shows theexperimental setup, the MR damper, and a device controllerprovided by Lord Corporation to control the input currentof the coils. Table 1 presents the input variables in theexperiment, which are displacement, frequency, and inputcurrent. There are three significant parameters, which are thevariation of displacement, velocity, and peak velocity, duringeach test. As can be seen in table 1, the displacement is variedfrom −5 to +5 mm for the short stroke damper and from−10 to +10 mm for the long stroke damper. The frequencies,in table 1, represent the variation of both velocity and peakvelocity. As as example, the 1.6 value of frequency for thelong stroke MR damper shows that the velocity is varied from−0.1 to 0.1 m s−1 and that the magnitude of the peak velocityis 0.1 m s at a displacement of 0 mm. Thus, the initial positionof the piston is in displacement of −5 mm for short stroke
Figure 4. Experimental setup for testing short and long stroke MRdampers.
MR damper and displacement of −10 mm for long strokeMR damper. The actual and predicted values of dampingforce in terms of displacement, velocity and peak velocity areshown in section 4. The input variables have been specifiedfor each MR damper on the basis of the specifications of eachindividual damper.
3. ANFIS model
The ANFIS model is a combination of a fuzzy systemand a neural network to accurately predict the behavior ofnonlinear and complex relationships in a phenomenon (Jang1993). An if–then rule is employed in this model, proposedby Takagi and Sugeno (1985). This structure comprisesfive layers, which are the input membership function, rulelayers (layers 2 and 3), output membership function, andsummation layer. In this study, three inputs, namely the inputcurrent, displacement, and velocity, were assumed to realizethe amount of force as output (see figure 5). The inputswere divided into various spaces by different membershipfunctions (MFs), as shown in tables 2 and 3. In the firstlayer, the parameters of the input MFs which describe theconfiguration of the MFs are defined; these parameters arecalled premise parameters. An AND rule was utilized tomultiply the incoming signals from the first layer to the secondlayer. The general equation of this layer can be written as
wi = µAi(I)× µBi(d)× µCi(v), (1)
where wi is the output of each second layer’s node and µAi ,µBi , and µCi are the incoming signals from the applied MFs
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Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
Figure 5. Schematic of the ANFIS method.
on the inputs (input current (I), displacement (d), and velocity(v)) to the individual second layer’s node.
In the third layer, the relative value of each rule weight isacquired by dividing wi by the total amount of all rule weights:
wi =wi∑n
i=1(wi), (2)
where wi is the relative rule weight of ith node.The proposed Takagi and Sugeno if–then rule is applied
in this layer (Takagi and Sugeno 1985), whereby the nodefunction of the fourth layer is
wifi = wi(piI + qid + riv+ si), (3)
where pi, qi, ri, and si are the corresponding coefficients ofeach input in the if–then rule and are known as consequentparameters. The premise and consequent parameters are theadjustable parameters of the ANFIS structure which areplaced in different layers.
The overall output is obtained in the last layer byaggregating the incoming signals from layer 4. In thisstructure, a hybrid algorithm of gradient descent andleast square estimates is employed to update the premiseparameters in the first layer and consequent parameters in thefourth layer, respectively.
3.1. Membership function selection
Various configurations of MFs are used to investigate the besttype and number of MFs for all inputs by comparing the leastroot mean square error (RMSE). The RMSE can be calculatedas
RMSE =
√√√√ 1N
N∑i=1
(Actual Force− Predicted Force)2, (4)
where N is the number of testing sets.In this aim, two diverse types of membership function,
(bell shaped and Gaussian shaped), and 64 combinations of
number of MFs are employed in order to realize the mostaccurate model (see tables 2 and 3). It can be noted thatthe RMSE value for 6-6-6 input MFs (bell shaped) in theshort stroke damper is extremely close to the RMSE valuefor 4-6-6 input MFs (Gaussian shaped), while its number ofrules is many more than the second configuration, and it willbe shown that the rules of the second configuration can besimplified. Therefore, the configuration of 4-6-6 input MFswith Gaussian-shaped MF has been selected for the shortstroke damper. In the long stroke MR damper, the best ANFISstructure is similar to the structure of the selected ANFISmodel in the short stroke MR damper, which is 4–6–6 inputMFs with Gaussian-shaped function as the input membershipfunction.
All proposed ANFIS approaches were trained for 1000epochs to generate the structure. Figure 6 shows theconvergence trend of the RMSE for the training data in termsof number of epochs for both MR dampers. The experimentaldata are split into two different data sets, training and testingdata sets. Seventy per cent of the experimental data arerandomly selected to train the ANFIS structures, and theremaining data are used to evaluate the constructed ANFISstructures. The selected ANFIS models produce 144 if–thenrules for both short stroke and long stroke MR dampers.
4. Results and discussion
A study of the prediction of two magnetorheologicaldampers has been carried out by examining diverse ANFISmodel configurations. The selected ANFIS models forboth short stroke and long stroke MR dampers havethe same configurations of 4-6-6 MFs for inputs with aGaussian-shaped membership function. Figures 7 and 8 depictthe plots of force in terms of displacement at 1.6 and 3.2 Hzfrequencies for the long stroke MR damper and the shortstroke MR damper, respectively. The input current is variedfrom 0 to 1 A. As can be seen, the values of force have been
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Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
Table 2. Configuration of MFs applied in the ANFIS model. (Note: a, b, c, and σ are the MF parameter sets.)
Membershipfunction Equation Figure
Gaussian shapede−(x−c)2
2σ2
Bell shaped 1
1+∣∣∣ x−c
a
∣∣∣2b
Table 3. Comparison of output RMSE for diverse numbers and types of MF.
Number ofMFs for Short stroke Long stroke Number of MFs for Short stroke Long stroke
(I) (d) (v)Bellshaped
Gaussianshaped
Bellshaped
Gaussianshaped (I) (d) (v)
Bellshaped
Gaussianshaped
Bellshaped
Gaussianshaped
3 3 3 111.08 124.60 147.59 151.31 5 3 3 108.89 118.97 160.87 171.393 3 4 95.07 78.54 94.37 64.33 5 3 4 89.21 104.08 66.77 124.293 3 5 99.54 106.22 102.08 127.48 5 3 5 106.25 106.07 112.89 80.693 3 6 85.92 72.36 64.33 105.33 5 3 6 93.90 97.14 73.17 97.883 4 3 105.67 123.10 146.39 142.14 5 4 3 110.49 120.37 157.38 147.663 4 4 84.70 92.95 66.60 110.04 5 4 4 88.62 77.97 97.80 77.523 4 5 96.49 79.49 55.94 102.08 5 4 5 96.03 71.12 106.00 91.603 4 6 71.17 84.08 68.13 59.55 5 4 6 85.48 90.09 72.51 55.543 5 3 105.49 111.02 156.92 142.45 5 5 3 106.80 109.00 155.63 138.633 5 4 75.24 65.41 80.52 63.11 5 5 4 74.91 70.11 77.66 101.123 5 5 78.64 74.31 75.83 81.25 5 5 5 78.40 82.65 100.86 83.123 5 6 71.26 66.02 68.24 57.46 5 5 6 65.44 80.21 63.31 74.163 6 3 115.53 99.90 151.11 139.31 5 6 3 115.74 100.32 142.01 141.163 6 4 66.10 74.86 75.89 66.60 5 6 4 66.41 75.74 61.75 71.133 6 5 72.52 75.81 76.33 82.08 5 6 5 77.56 76.99 73.19 79.873 6 6 64.05 69.52 64.88 71.77 5 6 6 65.52 71.02 51.32 54.344 3 3 110.88 130.59 155.79 170.33 6 3 3 104.35 117.80 152.81 169.554 3 4 95.91 103.85 89.89 124.38 6 3 4 94.73 106.75 69.56 79.624 3 5 85.74 81.38 85.45 95.87 6 3 5 106.02 104.20 83.83 117.624 3 6 94.26 92.04 66.28 102.27 6 3 6 97.66 99.52 75.58 63.714 4 3 129.40 124.32 154.27 127.98 6 4 3 109.97 129.43 164.33 149.574 4 4 86.60 69.74 87.97 72.05 6 4 4 90.43 78.50 90.35 58.004 4 5 71.17 93.71 70.39 86.85 6 4 5 97.44 92.33 101.52 92.514 4 6 80.93 90.33 52.80 59.00 6 4 6 86.26 94.33 73.07 61.104 5 3 104.55 111.66 150.65 139.28 6 5 3 107.46 109.63 155.11 137.504 5 4 67.90 83.69 83.01 61.55 6 5 4 68.31 85.24 74.65 62.674 5 5 87.78 86.65 83.02 83.24 6 5 5 65.08 88.62 98.89 83.934 5 6 74.39 75.78 54.93 53.44 6 5 6 77.55 64.48 67.17 62.374 6 3 103.01 100.31 160.58 124.89 6 6 3 116.94 100.50 148.92 141.514 6 4 63.20 75.20 70.76 94.74 6 6 4 66.05 76.28 63.18 51.684 6 5 77.87 76.56 79.71 83.86 6 6 5 77.72 77.53 98.68 85.034 6 6 64.21 60.47 60.66 48.26 6 6 6 59.34 71.86 67.52 73.17
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Figure 6. Training data RMSE versus number of epochs. (a) Long stroke MR damper. (b) Short stroke MR damper.
Figure 7. Force versus displacement in different currents for thelong stroke MR damper.
successfully predicted in the whole range of displacementfrom −10 to 10 mm for the long stroke damper and −5 to5 mm for the short stroke damper.
It is obvious that the numerical result of force versusdisplacement in the long stroke MR damper (figure 7) hasbetter agreement with the experimental result than is thecase for the numerical result of the short stroke MR damper(figure 8). In particular, the proposed ANFIS model ofthe long stroke MR damper outperforms the short strokeMR damper’s model around displacement boundaries. Theseportions of displacement can be defined as the last 20%of both positive and negative displacements in which thedirection of motion is going to be changed.
Displacement is one of the effective input parametersof the force produced by the damper that is considered inthe proposed ANFIS model. All effective input parametersmust be assessed to entirely evaluate the performance of theproposed prediction models.
Velocity is another effective parameter which has beenapplied as an input of the ANFIS model. Figures 9 and 10show the predicted force in terms of velocity in different input
Figure 8. Force versus displacement in different currents for theshort stroke MR damper.
Figure 9. Force versus velocity in different currents for the longstroke MR damper.
currents. As can be seen, the prediction results which are closeto peak velocities in each input current outperform the othervelocities.
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Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
Figure 10. Force versus velocity in different currents for the shortstroke MR damper.
Figure 11. Force versus peak velocity in different currents for thelong stroke MR damper.
Figure 12. Force versus peak velocity in different currents for theshort stroke MR damper.
Figures 11 and 12 demonstrate the accuracy of theprediction model in the long stroke MR damper and the shortstroke MR damper, respectively. These figures show that both
Figure 13. Regression plot of the prediction model for the longstroke MR damper.
Figure 14. Regression plot of the prediction model for the shortstroke MR damper.
Table 4. The applicable intervals of the MR damper inputs.
Inputcurrent, I (A)
Displacement,d (m)
Velocity,v (m s−1)
Short stroke MRdamper
[0 1] [−0.005 0.005] [−0.4 0.4]
Long stroke MRdamper
[0 1] [−0.01 0.01] [−0.5 0.5]
prediction models have precisely predicted the force in termsof peak velocity for every input current.
In addition, a regression analysis is provided betweenthe actual force and predicted force for further evaluation.Figure 13 shows the comparison of the actual force andpredicted force values for the long stroke MR damperwith a regression value of 0.9989. Figure 14 shows thecomparison of the actual force and predicted force values forthe short stroke MR damper with a regression value of 0.9961.Therefore, the regression plots demonstrate the trustableability of the proposed model in predicting the force withrespect to the actual value of the force which is obtained byexperiment.
Among models presented in previous studies, theaccuracy of the proposed model is compared to that ofthe Bingham, Bouc–Wen, polynomial, and neural networkmodels. It is obvious that the Bingham model is unable topredict the nonlinear hysteresis behavior of an MR damper.Spencer et al presented a modified Bouc–Wen model with
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Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
Figure 15. Plots of long stroke MR damper MFs ((a), (c) and (e)) and short stroke MR damper MFs ((b), (d) and (f)).
Table 5. The values of the premise parameters for the short strokeMR damper.
Input MF σ c
Current 11 0.141 75 0.000 1912 0.141 53 0.333 3913 0.141 71 0.666 1014 0.141 87 0.999 76
Displacement 21 0.00270 −0.0184022 0.000 87 0.000 6123 −0.000 23 −0.013 4924 0.003 45 0.018 0125 0.001 46 0.016 8926 −0.003 49 0.021 49
Velocity 31 0.069 07 −0.395 2032 0.068 54 −0.235 9233 0.056 77 −0.081 3234 0.058 60 0.081 6135 0.068 94 0.236 0436 0.068 72 0.394 97
14 parameters which need to be obtained by curve fittingof experimental results (Spencer et al 1997). In this model,the number of parameters is much less than in the ANFISmodel. In addition, the methods of obtaining the parametersof the ANFIS model are more precise and faster than in theBouc–Wen model. Choi et al proposed a polynomial model
Table 6. The values of the premise parameters for the long strokeMR damper.
Input MF σ c
Current 11 0.141 71 0.000 1912 0.141 21 0.333 2013 0.142 09 0.665 7614 0.142 20 0.999 50
Displacement 21 −0.000 75 −0.014 8722 −0.000 80 −0.019 0323 −0.001 47 0.002 2624 −0.000 45 0.013 6925 −0.005 68 0.035 6626 −0.001 41 0.015 69
Velocity 31 0.091 33 −0.525 1132 0.093 82 −0.310 4133 0.064 50 −0.109 5434 0.066 68 0.120 8335 0.092 48 0.323 4236 0.090 99 0.536 64
to describe the behavior of an MR damper (Choi et al 2001).This model has successfully predicted the nonlinear behaviorof an MR damper. The capability of the ANFIS model isto predict all damping, stiffness, and hysteresis behavior ofa damper in the whole input range, while the polynomialmodel cannot accurately predict the damper as well as the
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103
−6.
885×
101
4.23
6×
102
−8.
054×
103
104
−4.
894×
103
3.09
1×
101
−7.
112×
102
5.58
2×
103
332.
468×
104
5.63
4×
102
−1.
206×
103
6.77
7×
104
105
−1.
736×
104
−2.
525×
102
8.28
7×
103
−2.
049×
104
34−
6.85
3×
103
−7.
654×
101
−4.
303×
102
−8.
966×
103
106
−2.
167×
103
6.50
4×
101
−1.
264×
104
−1.
086×
103
35−
9.53
3×
101
1.27
0×
101
2.25
5×
101
1.43
4×
103
107
−9.
387×
103
−1.
102×
102
−2.
904×
103
−1.
441×
104
362.
605×
101
7.70
1×
10−
1−
1.95
0×
101
8.76
6×
101
108
1.34
4×
102
3.14
0×
100
−1.
818×
101
5.09
7×
102
37−
1.17
4×
104
9.66
7×
101
−6.
381×
103
−9.
594×
103
109
−2.
013×
104
8.69
1×
101
2.75
1×
102
−1.
785×
104
38−
1.78
1×
103
−8.
714×
103
−1.
445×
104
−2.
834×
103
110
−1.
101×
103
−5.
839×
103
−3.
733×
104
−6.
987×
103
39−
9.84
7×
102
−4.
604×
104
−4.
320×
103
−7.
024×
102
111
3.25
1×
103
−1.
645×
104
−8.
021×
103
−5.
080×
103
40−
7.38
5×
101
3.99
6×
105
−1.
112×
103
2.12
5×
103
112
−2.
215×
103
−9.
915×
104
3.68
9×
103
1.73
6×
103
41−
3.20
5×
103
2.81
2×
104
3.07
1×
103
1.16
4×
103
113
1.25
0×
102
−3.
666×
104
3.54
9×
104
−5.
295×
103
42−
4.89
4×
103
1.43
7×
102
7.78
0×
103
−9.
458×
102
114
−1.
171×
105
2.02
8×
102
−2.
734×
104
−1.
072×
105
43−
9.31
1×
102
−1.
209×
104
4.75
7×
102
−4.
686×
102
115
4.71
5×
103
2.48
9×
103
1.10
0×
102
−6.
683×
103
44−
1.06
4×
103
−6.
355×
103
5.54
7×
102
−3.
204×
102
116
3.58
3×
103
−6.
343×
103
2.38
1×
102
−5.
303×
103
45−
9.79
4×
102
−2.
825×
103
2.26
1×
102
−3.
169×
102
117
3.62
8×
103
−1.
372×
104
−1.
928×
103
−5.
344×
103
469.
359×
102
−6.
197×
103
1.30
7×
102
3.21
2×
102
118
−4.
194×
103
5.15
2×
103
−3.
261×
102
5.79
7×
103
471.
308×
103
−3.
459×
103
5.67
0×
102
2.46
6×
102
119
−2.
963×
103
3.07
7×
103
1.12
0×
103
4.49
4×
103
487.
775×
102
−8.
023×
103
6.87
5×
102
4.51
4×
102
120
−5.
280×
103
−1.
463×
104
1.21
5×
103
6.86
6×
103
55−
3.50
9×
104
−5.
915×
10−
13.
926×
103
−1.
444×
104
127
3.34
7×
104
3.81
9×
102
−1.
618×
104
9.21
4×
104
562.
041×
103
1.65
9×
104
−2.
507×
104
−6.
298×
103
128
6.14
7×
103
−1.
353×
104
1.18
8×
104
−1.
728×
103
57−
2.95
6×
102
6.78
7×
104
−1.
008×
104
−4.
260×
103
129
−2.
668×
102
−3.
791×
104
4.91
2×
103
5.17
2×
103
581.
117×
103
−1.
360×
104
−7.
008×
103
1.24
4×
103
130
−3.
457×
103
1.32
0×
103
−8.
200×
103
4.91
2×
103
591.
119×
103
−2.
230×
103
−1.
905×
104
4.04
5×
103
131
−8.
694×
102
−6.
787×
102
−3.
552×
104
9.26
6×
103
9
Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
Tabl
e7.
(Con
tinue
d.)
Rul
ep
qr
sR
ule
pq
rs
60−
4.57
9×
103
2.46
5×
102
−1.
386×
104
1.55
7×
104
132
1.37
8×
104
1.53
0×
102
−1.
237×
104
2.08
9×
103
67−
3.33
0×
102
1.31
5×
100
1.50
7×
101
6.45
4×
101
139
4.80
1×
102
4.89
6×
100
−2.
339×
102
1.21
3×
103
682.
121×
104
4.52
5×
102
−7.
641×
103
5.75
4×
104
140
−4.
732×
104
−4.
121×
102
6.75
8×
103
−5.
085×
104
694.
463×
104
1.61
5×
103
−2.
861×
104
2.15
9×
105
141
−1.
430×
105
−1.
173×
103
2.72
9×
104
−1.
441×
105
70−
4.74
2×
103
−2.
408×
102
−5.
467×
103
−3.
366×
104
142
1.16
4×
104
1.19
3×
102
−1.
059×
104
1.07
6×
104
71−
1.60
1×
103
−3.
581×
101
−1.
652×
103
−5.
275×
103
143
−4.
477×
103
−4.
069×
101
−1.
718×
103
−4.
695×
103
721.
240×
102
5.09
4×
100
8.45
2×
100
7.80
0×
102
144
5.35
3×
102
2.78
8×
100
−2.
971×
101
4.10
5×
102
10
Smart Mater. Struct. 22 (2013) 125013 M Zeinali et alTa
ble
8.T
heva
lues
ofth
eco
nseq
uent
para
met
ers
for
the
long
stro
keM
Rda
mpe
r.
Rul
ep
qr
sR
ule
pq
rs
1−
2.78
9×
103
8.90
1×
101
2.31
7×
103
−9.
761×
103
73−
2.75
5×
104
3.49
6×
102
1.06
9×
104
−4.
016×
104
26.
573×
102
−4.
846×
103
−5.
042×
103
−1.
482×
103
745.
740×
103
−9.
481×
103
−4.
006×
104
−1.
329×
104
38.
331×
101
7.28
3×
104
3.20
9×
102
5.70
1×
102
754.
559×
103
8.53
8×
104
−4.
945×
103
−3.
044×
103
43.
385×
102
1.17
0×
105
7.35
7×
102
1.23
8×
103
76−
7.97
0×
102
4.00
6×
105
−2.
616×
103
4.58
0×
103
5−
3.61
7×
103
−2.
352×
103
1.13
5×
104
−2.
392×
103
77−
5.92
6×
103
7.98
8×
103
−1.
263×
104
6.77
0×
103
6−
1.39
5×
103
1.06
7×
102
−3.
691×
103
−1.
263×
104
788.
976×
103
−1.
388×
102
5.24
2×
103
1.77
7×
104
135.
355×
102
6.32
1×
102
1.08
2×
103
−1.
400×
102
855.
507×
103
−4.
250×
103
1.06
6×
103
−4.
302×
103
143.
631×
102
1.44
0×
103
1.11
4×
103
−6.
098×
101
864.
977×
103
−2.
668×
103
1.01
0×
103
−3.
983×
103
15−
7.73
5×
100
1.92
8×
103
1.04
2×
103
−5.
522×
101
874.
263×
103
−3.
516×
103
9.95
4×
102
−3.
554×
103
163.
576×
101
1.86
4×
103
1.02
1×
103
3.79
2×
101
88−
4.59
7×
103
−6.
321×
101
7.97
9×
102
3.77
0×
103
17−
3.27
5×
102
−1.
471×
103
1.14
6×
103
3.86
9×
101
89−
5.18
2×
103
−4.
515×
103
7.65
6×
102
4.19
6×
103
18−
5.54
7×
102
−1.
680×
103
1.14
8×
103
1.26
1×
102
90−
5.08
4×
103
−3.
370×
103
7.30
3×
102
4.24
8×
103
25−
7.13
9×
102
−1.
977×
103
1.34
0×
104
−3.
813×
104
97−
1.63
3×
104
−5.
938×
102
1.42
0×
104
−3.
820×
104
265.
041×
103
−4.
022×
104
5.56
7×
103
8.76
8×
102
98−
1.19
1×
104
−5.
786×
104
−7.
834×
104
−1.
512×
104
27−
8.94
1×
102
−1.
474×
104
1.20
8×
103
−5.
864×
102
994.
886×
103
1.11
2×
105
−3.
590×
104
−5.
052×
103
285.
409×
102
9.92
2×
103
1.59
6×
102
2.38
3×
101
100
−1.
563×
102
1.95
9×
105
−1.
327×
104
4.14
0×
102
29−
3.64
3×
103
−3.
415×
104
−1.
360×
103
1.60
8×
103
101
4.50
2×
103
5.29
5×
104
−2.
742×
104
6.42
2×
103
302.
700×
102
−8.
033×
102
8.85
6×
103
2.44
3×
104
102
7.38
7×
103
1.20
5×
103
5.29
4×
103
1.55
4×
104
312.
219×
103
−4.
333×
101
2.25
8×
103
−5.
898×
103
103
1.10
4×
104
8.99
5×
101
−2.
114×
103
9.39
7×
103
32−
7.93
5×
103
−2.
938×
103
−4.
779×
104
−1.
028×
104
104
−9.
009×
103
−4.
698×
103
−6.
149×
104
−3.
724×
103
337.
808×
102
−4.
205×
104
−4.
935×
103
7.18
2×
102
105
5.71
4×
102
−3.
566×
104
2.09
7×
103
3.36
1×
102
34−
5.26
0×
102
1.67
8×
104
−6.
068×
102
−2.
734×
100
106
−7.
317×
103
2.59
5×
104
−8.
240×
103
5.15
0×
103
351.
892×
103
1.21
8×
103
−1.
646×
104
3.49
0×
103
107
3.98
0×
103
−1.
557×
102
−1.
101×
105
2.24
4×
104
361.
199×
103
−2.
240×
101
−1.
156×
103
−3.
213×
103
108
1.50
0×
104
1.91
9×
102
6.82
2×
103
2.29
3×
104
37−
8.39
7×
103
2.49
3×
102
7.02
8×
103
−2.
717×
104
109
−3.
734×
104
3.44
9×
102
1.04
3×
104
−3.
956×
104
386.
172×
102
−5.
256×
103
−1.
562×
104
−4.
250×
103
110
8.88
9×
103
−1.
295×
104
−4.
941×
104
−2.
133×
104
394.
086×
102
4.51
2×
104
−1.
651×
103
−2.
906×
102
111
8.87
0×
103
1.01
1×
105
−6.
023×
103
−9.
622×
103
40−
1.70
3×
102
3.59
1×
105
−1.
270×
103
3.65
4×
103
112
−1.
944×
103
6.39
5×
105
−2.
079×
103
8.31
4×
103
41−
4.63
4×
103
1.43
9×
104
−4.
025×
103
3.27
3×
103
113
−1.
412×
104
5.18
1×
103
−1.
523×
104
1.79
5×
104
423.
035×
103
2.93
1×
101
−5.
334×
102
−1.
412×
103
114
−6.
693×
103
6.15
4×
101
−1.
907×
103
−7.
134×
103
497.
015×
102
−4.
126×
103
1.02
3×
103
−7.
413×
102
121
1.04
0×
104
2.70
6×
103
1.30
8×
103
−1.
192×
104
506.
200×
102
−6.
602×
102
9.71
8×
102
−7.
119×
102
122
9.54
2×
103
−1.
665×
102
1.11
0×
103
−1.
114×
104
512.
659×
102
−9.
878×
102
8.57
7×
102
−5.
897×
102
123
8.02
7×
103
−2.
969×
103
6.61
8×
102
−9.
665×
103
52−
2.90
8×
102
1.65
9×
103
7.04
8×
102
5.99
8×
102
124
−9.
083×
103
−2.
449×
103
7.58
6×
102
1.07
4×
104
53−
5.67
5×
102
−3.
176×
103
9.13
3×
102
6.96
6×
102
125
−1.
039×
104
−5.
181×
103
8.85
5×
102
1.21
5×
104
54−
5.98
9×
102
−4.
203×
103
8.77
1×
102
8.20
7×
102
126
−1.
031×
104
−3.
757×
103
1.03
1×
103
1.20
9×
104
61−
2.56
6×
104
−1.
071×
103
2.48
5×
104
−6.
906×
104
133
−1.
790×
104
−2.
889×
102
6.39
2×
103
−1.
492×
104
626.
381×
103
−4.
961×
104
−6.
590×
104
−1.
952×
104
134
−1.
241×
104
−1.
951×
104
−1.
197×
105
−1.
929×
104
63−
7.96
3×
102
2.88
8×
104
−1.
520×
104
−1.
128×
103
135
2.12
4×
104
9.34
1×
104
−4.
499×
104
−2.
458×
104
64−
1.01
4×
102
1.63
1×
104
−7.
119×
103
1.04
9×
103
136
−5.
321×
103
2.42
6×
105
−1.
792×
104
6.10
3×
103
659.
879×
102
1.93
6×
104
−2.
700×
104
8.16
0×
103
137
2.35
8×
103
1.58
0×
105
−3.
357×
104
7.23
9×
103
11
Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
Tabl
e8.
(Con
tinue
d.)
Rul
ep
qr
sR
ule
pq
rs
661.
591×
104
6.68
1×
102
1.55
3×
104
4.29
7×
104
138
1.63
7×
104
1.65
5×
103
4.53
2×
103
1.57
2×
104
67−
9.43
6×
102
7.65
1×
100
−6.
851×
101
9.76
6×
102
139
5.38
8×
103
7.37
4×
101
−2.
222×
103
8.57
5×
103
68−
5.60
0×
103
−9.
012×
103
2.10
1×
104
7.46
5×
103
140
−1.
228×
103
−1.
421×
103
−1.
063×
104
2.39
4×
103
697.
331×
102
−1.
365×
105
−1.
152×
101
1.27
1×
103
141
−2.
918×
103
4.31
8×
104
9.35
4×
103
3.26
3×
103
70−
6.74
7×
102
2.00
4×
104
−4.
221×
103
5.35
8×
102
142
−1.
277×
104
1.91
6×
104
−6.
622×
103
1.40
8×
104
718.
742×
102
6.61
6×
102
−5.
965×
104
1.28
0×
104
143
8.37
1×
103
−2.
397×
103
−1.
137×
105
1.79
0×
104
727.
099×
103
1.69
8×
102
5.61
2×
103
1.97
3×
104
144
2.40
3×
104
2.02
6×
102
7.55
4×
103
2.48
5×
104
12
Smart Mater. Struct. 22 (2013) 125013 M Zeinali et al
ANFIS model can. The graphs of damping force in termsof displacement, velocity, and peak velocity, figures 7–12,illustrate the accuracy of the proposed ANFIS model incomparison with other artificial intelligent models such as aneural network (Wang and Liao 2005).
In damping force criteria, the accuracy of the controlsystem is directly related to the damper model (Choi et al2001). To illustrate this, Choi et al presented an open-loopcontrol system to examine the performance of the controlstrategy by using a polynomial model (Choi et al 2001). Thefuture work of this study is to implement the proposed modelin a control strategy and show the improvement of controllerperformance by using the ANFIS model.
The accuracy of the proposed prediction models in termsof input current supplied to coils has been examined in parallelwith assessing the prediction performance of force versusdisplacement (figures 7 and 8) and velocity (figures 9 and 10).
The last section presents the examination of variousANFIS model configurations. Therefore, the most accurateANFIS model is obtained by comparing the root mean squareerror of all experimental data. The graphs of the input MFsfor the selected ANFIS model of short stroke and long strokedampers are shown in figure 15.
As can be seen in figures 15(c) and (d), the output valuesof MF21, MF22, MF24, MF25, and MF26 for both shortstroke and long stroke MR dampers are close to zero inspecified displacements. In addition, the premise parametersfor both MR dampers are similar, and show that the generalbehavior of the short stroke MR damper is extremely close tothat of the long stroke MR damper.
Another significant characteristic is the consequentparameters. A study of the effect of each layer’s consequentparameters showed that a number of each layer’s consequentparameters have no influence on the predicted output, dueto their small value. Therefore, the relationship betweenthe force (as the output) and the inputs was simplifiedby eliminating ineffectual rules because of the insignificantvalues of the consequent parameters. All rules related toMF23 and MF25 in the short stroke MR damper modeland MF22 and MF24 in the long stroke MR dampermodel were eliminated; the result of the prediction modelremained constant. The consequent parameters for the rulesin accordance with MF23 and MF25 are extremely small, andeliminating them helps simplify the equation of the predictionmodel.
Equation (5) can be used to obtain the predicted dampingforce in terms of input current (I), displacement (d), andvelocity (v) in specified intervals (see table 4).
F =
((4∑
i=1
6∑j=1
6∑k=1
(µ1i × µ2j × µ3k)
× (p36(i−1)+6(j−1)+kI + q36(i−1)+6(j−1)+kd
+ r36(i−1)+6(j−1)+kv+ s36(i−1)+6(j−1)+k)
))
×
(4∑
i=1
6∑j=1
6∑k=1
(µ1i × µ2j × µ3k)
)−1
, (5)
where µ is the Gaussian-shaped membership function (seetable 2) of the input MFs and the parameters of the input MFsare given in tables 5 and 6 and drawn in figure 5. p, q, r and sare the consequent parameters. Both premise and consequentparameters are shown in tables 5–8.
5. Conclusion
In this research, a phenomenological dynamic model ispresented for short stroke and long stroke MR dampers usingan adaptive-network-based fuzzy inference system (ANFIS)approach. The structure of the proposed models is identicalwith diverse values of parameters. Short stroke and longstroke MR dampers supplied by Lord Corporation were usedin the experiments, and the results obtained were used toconstruct the ANFIS structure. In addition, a study of theeffect of membership function selection, premise parameters,and consequent parameters on realizing the most accurate andsimplified relationship between the inputs and the outputs ofthe MR dampers was done. The results show that the proposedprediction models for both short stroke and long stroke MRdampers can precisely predict the MR damper’s behavior. Anequation was presented to numerically describe the proposedmodels for each MR damper.
Acknowledgments
The authors would like to thank the Malaysia–JapanInternational Institute of Technology through the Ministryof Higher Education (MOHE), Exploratory Research GrantScheme (ERGS) Grant No. 9L094, for supporting thisresearch.
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