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A new magnetorheological damper for seismic control

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2013 Smart Mater. Struct. 22 115003

(http://iopscience.iop.org/0964-1726/22/11/115003)

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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 22 (2013) 115003 (12pp) doi:10.1088/0964-1726/22/11/115003

A new magnetorheological damper forseismic controlYang Ding1,2, Lu Zhang1, Hai-Tao Zhu1,2 and Zhong-Xian Li1,2

1 School of Civil Engineering, Tianjin University, Tianjin 300072, Peoples Republic of China2 Key Laboratory of Coast Civil Structure Safety (Tianjin University), Ministry of Education,Tianjin 300072, Peoples Republic of China

E-mail: [email protected]

Received 5 April 2013, in final form 16 July 2013Published 19 September 2013Online at stacks.iop.org/SMS/22/115003

AbstractThis paper proposes a new MR damper with bidirectional adjusting damping forces toenhance the fail-safe property of the MR damper. The structure of the composite magneticcircuits is improved for the new damper. Four prototype dampers are fabricated and tested bymagnetic field tests and dynamic tests. The magnetic field distribution in the damping path andthe dynamic properties of the dampers with different input currents are obtained. TheGompertz model is proposed to portray the dynamic behavior of the prototype dampers. Thestudy shows that, due to the improved structure of composite magnetic circuits, the prototypedampers can maintain a medium damping force with zero current input. This behavior mayensure a better fail-safe property and avoid settlement of MR fluid compared withconventional MR dampers. Furthermore, the minimum and maximum output powers of theproposed dampers can be obtained at the states of the negative peak and positive peak ofcurrents inputs, respectively. In addition, the dynamic range of controllable force is wider thanthat of conventional MR dampers. The analysis further shows that the proposed Gompertzmodel can precisely portray the nonlinear hysteretic behavior of the proposed dampers withoutcomplicated function forms.

(Some figures may appear in colour only in the online journal)

1. Introduction

Over the past few decades, magnetorheological (MR) dampershave attracted much attention because they can significantlyreduce the response of structures excited by seismic andwind loadings. MR dampers possess the excellent inherentcharacteristics of stability and adaptability. Much researchhas been done on the control strategy and models for MRdampers. Dyke et al proposed a clipped-optimal controlstrategy for controlling MR dampers to reduce structuralresponses due to seismic loadings. The strategy was basedon the acceleration feedback. Both numerical simulation andexperiments have been conducted to verify the effectivenessof MR dampers on the reduction of seismic response [1, 2].After studying several idealized mechanical models ofcontrollable fluid dampers, Spencer et al proposed aphenomenological model which can portray the behaviorof a typical MR damper effectively and precisely [35].

Yang et al investigated a full-scale MR damper with amaximum damping force of 200 kN and a dynamic rangeof 10 [68]. They also proposed both a quasi-static modeland a dynamic model for this damper. Li and Xu designedand manufactured a double-ended shear mode combined witha valve mode MRF-04K damper with a maximum dampingforce at a full magnetic field strength of 20 kN. The maximumpower required was less than 50 W [9]. Tse and Changdesigned, manufactured, and tested a small-scale rotary typeof MR damper [10]. They also developed a simplified andrelatively accurate inverse dynamic model. This model candirectly relate the damper force to the input voltage. Chooiand Oyadiji derived a method for designing, modeling andtesting MR dampers using analytical flow solutions [11]. Theeffectiveness of this method was validated by simulation andtest results. The test results came from a double-tube MRdamper fabricated at the University of Manchester. Gavin et alstudied the optimal design of MR dampers [12]. They also

10964-1726/13/115003+12$33.00 c 2013 IOP Publishing Ltd Printed in the UK & the USA

Smart Mater. Struct. 22 (2013) 115003 Y Ding et al

developed an algebraic model utilizing the hyperbolic tangentfunction.

MR dampers possess the fail-safe property. This propertyis one of the most important advantages of MR dampers,making them superior to fully active control devices. Thisis because the power supply of fully active control devicesmay fail during the excitation of extreme seismic and windloadings. On the other hand, analytical and experimentalstudies have already shown that MR dampers can work witha power supply as passive control viscous dampers do. Evenunder such situation, MR dampers can still offer considerablecontrol effects. When working without a power supply, MRdampers cannot provide the best performance compared withother passive states with power supply at fixed magnitudemost times. This limitation shows that there is still an optimalstate for MR dampers to provide the best passive controleffect. Carlson first theoretically discussed this issue andwhether it was possible to use a permanent magnet to biasa MR fluid valve or device at a mid-range condition [13].By this means, MR dampers can be designed as the optimalpassive state for achieving better performance without a powersupply. Du et al and Yan et al utilized a similar structure ofcomposite magnetic circuit to manufacture a MR damper. It iscalled an inverse MR damper, and is capable of maintainingthe damping force at its maximum value due to a magneticfield excited by the incorporated permanent magnet withoutcurrent input. After that, the current can be applied to theaccompanying electromagnetic coil to cancel the magneticfield and decrease the damping force [14, 15]. However,the test results on the prototype damper showed that thedynamic range of the damping force provided by the inverseMR damper is poor, even below 1.5. This dynamic range isobviously of no practical significance. The results also showedthat the magnetic field excited by the permanent magnet isdifficult to be reduced. This is due to the magnetic saturationaccruing at the magnetic core, which may be the mainlimitation of the composite magnetic structures proposed bythe researchers above.

In this study, a new prototype MR damper is proposed.The proposed MR damper is developed with bidirectionaladjusting damping forces. The structure of the compositemagnetic circuits is improved for the proposed MR damperto avoid magnetic saturation accruing at the magnetic core.Therefore, the dynamic range will not be affected comparedto conventional MR dampers. Four prototype dampers arefabricated and tested by magnetic field tests and dynamictests. The magnetic field distribution in the damping pathand the dynamic properties of the dampers with differentinput currents are obtained in the tests. In terms of theobtained test results, the Gompertz model is proposed toportray the dynamic behavior of the prototype dampers. Thestudy shows that, due to the improved structure of compositemagnetic circuits, the prototype dampers can maintain amedium damping force with zero current input. This behaviormay ensure a better fail-safe property and avoid the settlementof MR fluid compared with conventional MR dampers. Thecomparison further shows that the proposed Gompertz modelcan precisely portray the nonlinear hysteretic behavior of theproposed dampers without complicated function forms.

Figure 1. Composite magnetic circuit.

2. The improved structure of composite magneticcircuit

Figure 1 shows a schematic diagram of the proposedcomposite magnetic circuit and figure 2 shows the equivalentmagnetic circuit with different current inputs. In figure 2,NI and Fm are the magnetomotive forces induced by theelectromagnetic coil and the permanent magnet, respectively.Rm and Rw represent the reluctance of the permanent magnetand the damping path, respectively, while R0 represents thevariable reluctance associated with the magnetic saturation atthe magnetic core. 81,82,83 and 84 are the magnetic fluxaccording to states with different currents inputs, respectively.

As shown in figure 2, the composite magnetic circuit isgoverned by

Fm = 82R0 +81rw + (81 +82)Rm (1)Fm NI1 = 83(R0 + Rm) (2)Fm + NI2 = 84(Rm + Rw), (3)

where I1 represents the negative peak value of current inputsand I2 represents the positive peak value of current inputs.(1)(3) correspond to the working states with current inputsat zero, negative peak and positive peak, respectively.

Improvements to the structure of the composite magneticcircuit is proposed herein. One improvement concernsavoiding the difficulties of demagnetization. The secondarygap proposed by Carlson [13] incorporated in the compositemagnetic circuit is removed. Instead, by means of controllingthe thickness of the magnetic core and utilizing the magneticsaturation, the magnetic flux can be restricted when passingthrough the magnetic core. When there is no current input,the magnetic flux of the permanent magnet is supposed to beproportionally distributed between the circuit passing throughthe magnetic core and the other circuit passing throughthe damping path. In this way it is possible to maintain amedium damping force, as shown in figure 2(a). A secondimprovement addresses how to obtain the minimum andmaximum output powers of the dampers. When the damperswork in semi-active mode, the minimum and maximum outputpower of the dampers can be obtained at the stage of currentinput being at negative peak and positive peak, respectively.When negative current is applied to the electromagnetic coil,the operating point of the permanent magnet falls along itsoperating line as shown in figure 3. The magnetic flux passing

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Figure 2. Equivalent magnetic circuit. (a) Zero. (b) Negative peak. (c) Positive peak.

Figure 3. Operating point motion of permanent magnet.

through the damping path decreases until the left magneticflux completely passes through the magnetic core as shownin figure 2(b). In contrast, when positive current is applied tothe electromagnetic coil, the operating point of the permanentmagnet rises along its operating line as shown in figure 4.The magnetic flux passing through the damping path increasesuntil the maximum value is achieved as shown in figure 2(c).That is, the minimum, medium and maximum values ofthe magnetic flux passing through the damping path can beobtained at the stage of currents input being at negative peak,zero and positive peak values, respectively. The correspondingminimum, medium and maximum output power of the dampercan be also obtained.

As shown in figure 3, in order to avoid magnetic propertyvariation under repeated magnetizing and demagnetizingoperations, the permanent magnet must possess ideal lineardemagnetization property to eliminate the possibility of

permanent demagnetization. For this reason, NdFeBpermanent magnets are chosen. Because of high intrinsiccoactivity of NdFeB permanent magnets, the thicknessof the permanent should be restricted to 12 mmfor implementing fully magnetizing and demagnetizingoperations. This performance can be fulfilled within limitedturns of electromagnetic coils and limited magnitude ofcurrents.

In accordance with the above assumption, multiple designschemes are proposed. As shown in figures 4(a) and (b),magnets with axial distribution magnetic field are chosenfor schemes 1 and 2. The permanent magnets are installedoutside the magnetic core in scheme 1, which ensures abigger cross section area of the permanent magnets. Abigger magnetomotive force of the permanent magnets can beobtained. In scheme 2, the permanent magnets are installedinside the magnetic core, which means a smaller crosssection area. A relatively smaller magnetomotive force of thepermanent magnets can be produced. Because of the differentstructural arrangement, schemes 1 and 2 can satisfy differentrequirements of MR dampers depending on the magnitude ofdamping force. Besides the former two schemes, permanentmagnets with radial distribution magnetic field are chosen forscheme 3, as shown in figure 4(c). This scheme is consideredto be a particularly simple scheme. However, the fabricationof permanent magnets with radial distribution magnetic fieldis still complicated and relatively costly.

In this paper scheme 1 is used for the damper, consideringthe design purpose and process cost. In order to evaluatethe proposed improved structure of the composite magneticcircuit, finite element analysis is conducted. Figure 5 showsthe magnetic flux distributions of a single section of theproposed dampers at the stage of current input being at zero,negative peak and positive peak values, respectively. Figure 6

Figure 4. Design schemes. (a) Scheme 1. (b) Scheme 2. (c) Scheme 3.

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Figure 5. Distributions of magnetic flux. (a) Zero. (b) Negative peak. (c) Positive peak.

Figure 6. Distributions of magnetic induction. (a) Zero. (b) Negative peak. (c) Positive peak.

Figure 7. Schematic diagram of a MR damper with bidirectional adjusting damping forces. (a) Overall structure. (b) Local structure.

shows the induction of the magnetic distributions. As shownin figures 5 and 6, the simulated results agree well with theequivalent magnetic circuit diagram shown in figure 2. It isnoted that when current input is at negative peak value, themagnetic flux passing through the damping path completelyvanishes. This means that the demagnetization is fullyimplemented, which ensures a comparable minimum dampingforce and a dynamic range compared with conventional MRdampers.

3. Design and fabrication of MR dampers

Following the above conceptual design and theoreticalanalysis, four prototype dampers with bidirectional adjustingdamping forces are designed and fabricated. Two conventionalprototype dampers are also fabricated to provide a compar-ison. Figure 7 shows a schematic diagram of the proposedMR damper with bidirectional adjusting damping forces. The

Table 1. Geometry parameters of the prototype dampers.

h(mm)

d(mm)

D(mm)

L(mm)

t(mm)

Turns of theelectromagnetic coils

1 20 66 20 5 500

basic geometry of the proposed damper is consistent withthe large-scale seismic MR fluid damper fabricated by Yanget al [6] and Lord Company. The major difference is that apiece of permanent magnet is inserted in the middle of thepiston and the thickness of the left magnetic core is reducedto a small value, as shown in figure 7(b).

By means of finite element simulation on the electro-magnetic field, most of the geometry parameters can bedetermined. These geometry parameters are listed in table 1.An illustration of the geometry parameters is shown infigure 7(b). Two geometry parameters are very important for

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Figure 8. The curves of B0 versus h1.

Figure 9. The curves of B0 versus h1.

the proposed improved structure of the composite magneticcircuit. One is the thickness of the magnetic core h1 which isrelated to the residual magnetic induction B0 in the dampingpath without current input. The other is the thickness of thepermanent magnet t1 which is related to the peak value ofdemagnetization current I1. Both relationships are obtainedby means of finite element simulation on electromagneticfield. The numerical results are shown in figures 8 and 9.As shown in figure 8, permanent magnets with both 1.26 Tand 1.37 T remanence Br are selected and B0 is controlledby h1. As 500 mT is considered to be the saturation valueof the selected MR fluid (supplied by Chongqing InstrumentMaterials Research Institute of China, denoted MR-J), therange of B0 from 200 to 300 mT is considered to beappropriate for the medium output powers of the damper.Considering the difficulty of simulation process and theproper value of B0, 1.5 and 3 mm are selected for h1. As shownin figure 9, I1 mainly depends on t1 and two processes withdifferent h1 and Br are simulated. As a dc power supply withmaximum output power of 30 V/3 A is used for the tests, 2 mis selected for t1 ensuring that I1 for both processes will notexceed the maximum output power of the dc power supply.

According to the simulated results, MR dampers withdifferent Br and h1 are fabricated. Meanwhile, the proposedimproved structure of composite magnetic circuit is appliedto dampers with single piston section and multiple pistonsections. Dampers with different numbers of piston sectionsn are fabricated. Serial numbers and other kernel geometryparameters are listed in table 2. Figure 10 is a photo

Figure 10. Photo of prototype dampers.

Figure 11. The equipment setup for magnetic field tests.

Table 2. Serial numbers and kernel geometry parameters.

Serialnumber n h1 (mm) Br (T) t1 (mm)

MR-B1 1 3 1.26 2MR-B2 1 1.5 1.37 2MR-B3 3 3 1.26 2MR-B4 3 1.5 1.37 2

of fabricated prototype dampers with single piston sectionand multiple piston sections, respectively. Two conventionalprototype dampers of the same geometry parameters aredesigned to provide a comparison. The two dampers aredenoted as MR-C1 and MR-C2 with single piston section andmultiple piston sections, respectively.

4. Magnetic field tests

In order to examine the effectiveness of the proposedimproved structure of the composite magnetic circuit, a seriesof magnetic field tests are conducted utilizing the speciallycustomized testing device designed by the authors as shown infigure 11. A LZ-610H teslameter is employed in conjunctionwith a dc power supply with a maximum output power of30 V/3 A. Because the designed thickness of the dampingpath for the MR dampers is only 1 mm, a specially customizedHall probe is adopted with a width of 1 mm and a thickness

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Figure 12. Comparison between measured results and predictedresults using a MR-B1 damper.

Figure 13. Comparison between measured results and predictedresults using a MR-B2 damper.

of 0.32 mm. Considering that the length of the Hall probeis limited, only the MR dampers fabricated with singlepiston section are tested, i.e., MR-B1, MR-B2 and MR-C1,respectively. The positive peak value of the input currents is2.5 A, which is associated with the maximum output powerof the dc power supply. The negative peak value of thecurrents input is related to the full demagnetization of thedamping path. That is, the negative peak value leads to thedisappearance of magnetic flux passing through the dampingpath. In order to eliminate the influence of the fabrication erroron the thickness of the damping path around the piston, fourtesting points are uniformly distributing around the piston.The average thickness value of these four testing points iscalculated and chosen as the final result.

Figures 12 and 13 show the magnetic field test results ofMR-B1 and MR-B2 dampers and the comparison betweenthe measured values and the predicted values, respectively.As shown in figures 12 and 13, the measured values ofthe four testing points around the piston are consistent witheach other. The uniform distribution of the magnetic fieldin the damping path shows that the fabrication error can benegligible and the thickness of the damping path around thepiston is nearly constant. It is also noted that the predictions

Figure 14. Comparison of measured results among a MR-B1damper, a MR-B2 damper and a MR-C1 damper.

of the magnetic induction in the damping path are slightlylower than the measured values. This difference may be dueto the fact that there is flux leakage in the piston shaft. Thepiston shaft is made of nonmagnetic stainless iron. The shaftbecomes slightly magnetizable after machining. Figure 14shows a comparison of measured values among MR-B1damper, MR-B2 damper and MR-C1 damper. As shown infigure 14, the measured values of magnetic induction ofMR-B1 and MR-B2 with zero current input achieve 117 mTand 215 mT, respectively. However, no magnetic field existsin the damping path of MR-C1 damper. Due to the permanentmagnet of higher remanence and lower-thickness magneticcore, the residual magnetic induction in the damping pathof MR-B2 damper is obviously higher than that of MR-B1damper. This phenomenon indicates that the residual magneticinduction in the damping path can be changed to be in thedesired state by controlling the remanence of the permanentmagnet and the thickness of the magnetic core. Moreover, themagnetic induction in the damping path of MR-B1 damperand MR-B2 damper reach zero with 1.2 A and 2.5 A negativecurrents input, respectively. Therefore, full demagnetizationcan be achieved with limited negative current input and higherresidual magnetic induction at the state of zero current input.In addition, the maximum magnetic induction in the dampingpaths of MR-B1 damper and MR-B2 damper is almost thesame. The maximum induction is still slightly lower than thatof MR-C1 damper. Regarding magnetic induction increaserate, the increase rates of MR-B1 damper and MR-B2 damperare lower than that of MR-C1 damper. There is biggerreluctance of the permanent magnet installed in MR-B1damper and MR-B2 damper than that of the pure iron pistonof MR-C1 damper. Also, all measurements are obtained inair circumstance. This condition is different from the actualoperation states of the damping path being full of MR fluid.The actual magnetic induction will be much higher due to thehigher permeability of MR fluid. Thus, the improvement issufficient for the maximum output power of the dampers.

5. Dynamic tests

Laboratory tests are further conducted to study the dynamicperformance of the proposed MR dampers with bidirectional

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Table 3. Maximum output capacities and dynamic ranges of a MR-B1 damper, a MR-B2 damper and a MR-C1 damper.

Seriesnumber

Zero current inputF0 (kN)

Positive peak currentsinput Fmax (kN)

Negative peak currentinput Fmin (kN) F0/Fmax (%)

Dynamicrange

MR-B1 1.729 5.789 0.591 29.9 9.8MR-B2 2.465 6.382 0.484 38.6 13.1MR-C1 0.661 6.285 10.5 9.5

Table 4. Maximum output capacities and dynamic ranges of a MR-B3 damper, a MR-B4 damper and a MR-C2 damper.

Seriesnumber

Zero current inputF0 (kN)

Positive peak current inputFmax (kN)

Negative peak currentinput Fmin (kN) F0/Fmax (%)

Dynamicrange

MR-B3 3.893 20.295 1.171 19.2 17.3MR-B4 7.593 18.721 0.882 40.6 21.2MR-C2 2.35 21.332 11.0 9.1

Figure 15. The setup of a dynamical test equipment.

adjusting damping forces. A MTS-810 electro-hydraulic servotester is adopted. A MTS Teststar data acquisition systemand dc power supply are used as shown in figure 15. Thedampers are tested under sinusoidal displacement excitation atthe frequencies of 0.25 Hz, 0.5 Hz and 1 Hz with magnitudesbeing 5 mm, 10 mm and 15 mm, respectively. Several constantinput currents are adopted. The positive peak value of theinput current is 2.5 A when the maximum output power ofthe dc power supply is reached. The negative peak value ofthe input current depends on the occurrence of the minimumoutput power of the dampers. Figure 16 shows the peak valuecurves of damping force versus input current for MR-B1damper, MR-B2 damper and MR-C1 damper. Figure 17 showsthe peak value curves of damping force versus input currentfor MR-B3 damper, MR-B4 damper and MR-C2 damper.Table 3 gives the maximum output power versus differentcurrent input and the dynamic range for MR-B1 damper,MR-B2 damper and MR-C1 damper. Table 4 presents themaximum output power versus different current input andthe dynamic range for MR-B3 damper, MR-B4 damper andMR-C2 damper.

Comparison is made in these figures and tables. MR-B1damper and MR-B3 damper with zero current input maintain20%30% of the maximum output power when currents inputis at positive peak. MR-B2 damper and MR-B4 damper withzero current input maintain about 40% of the maximum

Figure 16. Maximum output capacities of a MR-B1 damper, aMR-B2 damper and a MR-C1 damper.

Figure 17. Maximum output capacities of a MR-B3 damper, aMR-B4 damper and a MR-C2 damper.

output power when current input is at positive peak. Thisincrease is due to the permanent magnet of higher remanenceand lower-thickness magnetic core. However, MR-C1 damperand MR-C2 damper, having conventional magnetic structure,only maintain about 10% of the maximum output powerwhen current input is at positive peak. The comparisonshows that the proposed improved structure of compositemagnetic circuit is effective. The output power of MR

7


Figure 18. Comparison between logistic function and Gompertzfunction.

dampers without current input is significantly increased. Thisadvantage can ensure a much better fail-safe property andavoid the settlement of MR fluid. Moreover, because fulldemagnetization can be achieved, the proposed MR damperswith bidirectional adjusting damping forces can reach loweroutput power with current input being at negative peakthan that of the conventional MR dampers. The maximumoutput powers of all the tested dampers are almost the same.Therefore, the proposed MR dampers have a bigger dynamicrange. In addition, the negative peak of current input forMR-B2 damper and MR-B4 damper are higher than thoseof MR-B1 damper and MR-B3 damper. This result is inaccordance with that of the above magnetic field tests. Finally,the maximum output powers of MR dampers with threepiston sections are almost three times as large as those ofthe MR dampers with single piston section. This indicatesthat the proposed improved structure of composite magneticcircuit can be applicable to the dampers with either singlepiston section or multiple piston sections. The output poweris proportional to piston section numbers.

6. Dynamic modeling of MR dampers

MR dampers have inherent hysteretic characteristics, whichhave to be considered in the dynamic model of MR dampers.For the stressstrain behavior, the Bingham viscoplasticmodel was proposed by Shames and Cozzarelli [16].Similarly, Stanway et al proposed a simplified mechanicalmodel, denoted the Bingham model, describing the behaviorof electro-rheological (ER) and MR dampers [17, 18]. TheBingham model has a simplified function form, which isapplicable for the initial design of MR dampers. However,this model is not adequate for the control analysis dueto its deficiency in describing the nonlinear hystereticcharacteristics of MR dampers. Spencer et al proposed aphenomenological model of MR dampers which is ableto precisely portray the nonlinear hysteretic behavior ofMR dampers [3]. This phenomenological model considersforcedisplacement relationship and forcevelocity relation-ship using a complicated function form. This limitation pre-vents the practical application of the phenomenological modelto semi-active control systems. Several different models for

Figure 19. Comparison between dynamic models based on theGompertz function and the logistic function.

MR dampers have been proposed by researchers [1923]. Liand Li proposed a double-sigmoid model with a symmetricalsigmoid function [24]. The corresponding experimentalverification was also conducted.

This study presents an improved dynamic model denotedthe Gompertz model. This model is adopted to simulate thedynamic behavior of MR dampers with bidirectional adjustingdamping forces. A Gompertz function, named after BenjaminGompertz, is a sigmoid function. It is a mathematical modelof a time series with the growth being slowest at the start andthe end of a period. The left-hand or lower-value asymptote ofthe function can be approached much more gradually by thecurve than the upper right-hand or future-value asymptote asshown in figure 18. This differs from the logistic function. Thelogistic function has symmetrical asymptotes. The formula ofGompertz function is given by

y(t) = aebect , (4)where a is the upper asymptote, c is the growth rate, b and care negative numbers and e is Eulers number. Consideringthe asymmetry of the Gompertz function, it seems tobe more suitable for dynamic modeling of MR dampers.Figure 19 shows a comparison between dynamic modelsbased on the Gompertz function and the logistic function. Theexperimental result comes from the MR-B4 damper responseunder a 15 mm-amplitude sinusoidal displacement excitationof the frequency of 0.5 Hz. The magnitude of input current is2.5 A. As shown in figure 19, the dynamic model based on theGompertz function is superior to the dynamic model based onthe logistic function, especially in the regions of low velocityand high velocity.

The proposed Gompertz model is governed by

F = 2fd(0.5 e ln 2axsgn(x)x0 )sgn(x)+ xcd, (5)where fd represents the controllable Coulomb damping force;cd and control the viscous damping at large velocitiesand small velocities, respectively; x and x are pistondisplacements and velocities, respectively; and xd is the pistonvelocity when the damping force is zero. A total of fourparameters (fd, cd, , and xd) are needed to characterize theMR damper. Equation (5) can only simulate the damper

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Figure 20. The curves of model parameters versus input current I. (a) fd0. (b) cd0. (c) xd0.

Table 5. Relations between model parameters and input current I.

I fd0 cd0 xd0

2.5 0.6 0.003 51.5 1.45 0.006 8

0 6 0.018 141.5 14.5 0.027 19.52.5 15.5 0.035 21

response when the applied current, frequency and amplitudeare held at a constant level. As MR dampers are always usedas semi-active control devices, the adopted dynamic modelhas fluctuating current, frequency and amplitude. Thereforethe parameters are defined as the functions of input current Iand maximum velocity vmax.

First, the relationships between model parameters andapplied current I are studied. The relationships between modelparameters and applied current I are formulated according tothe MR-B4 damper response. A 15 mm-amplitude sinusoidaldisplacement excitation of the frequency of 0.5 Hz wasapplied. The magnitude of an input current is 2.5 A. Table 5presents the results. A logistic function is used to fit the dataas shown in figure 20. From the test results and the fittingfunction, is kept constant and the other three parameters aregiven by

fd0 = Af + Af Bf1+ e(ICf )/Df (6)

cd0 = Ac + Ac Bc1+ e(ICc)/Dc (7)

xd0 = Ax + Ax Bx1+ e(ICx)/Dx , (8)

where Af ,Bf ,Cf ,Df ,Ac,Bc,Cc,Dc,Ax,BxCx and Dx areundefined parameters depending on the test results and thefitting results. These parameters are listed in table 6.

Second, the relationships between model parameters andmaximum velocity vmax are introduced. The model parametersdepending on vmax are given by

fd = (kf vmax + tf )fd0 (9)cd = (kcvmax + tc)cd0 (10)xd = (kxvmax + tx)xd0 (11) = kvmax + t, (12)

where kf , tf , kc, tc, kx, tx, k and t are undetermined parame-ters related to vmax, and their values are obtained though thelinear regression analysis of test results. The regression resultsof the MR-B4 damper are listed in table 7.

Finally, a total of 20 parameters (Af ,Bf ,Cf ,Df , kf ,tf ,Ac,Bc,Cc,Dc, kc, tc,Ax,BxCx,Dx, kx, tx, k and t) areneeded to characterize the MR dampers with bidirectionaladjusting damping forces. The Gompertz model with inputcurrent I and maximum velocity vmax is given by

fd = Af + Af Bf1+ e(ICf )/Df (kf vmax + tf ) (13)

cd = Ac + Ac Bc1+ e(ICc)/Dc (kcvmax + tc) (14)

xd = Ax + Ax Bx1+ e(ICx)/Dx (kxvmax + tx) (15)

= kvmax + t. (16)

Table 6. Fitting results of fd0, cd0 and xd0.

Af Bf Cf Df Ac Bc Cc Dc Ax Bx Cx Dx

0.76 15.78 0.31 0.52 0.007 0.052 0.771 1.962 2.23 22.64 0.39 1.15 0.9

Table 7. Fitting results of undetermined parameters related to vmax.

kf tf kc tc kx tx k t

0.0013 0.8784 0.0141 2.3376 0.0093 0.1123 0.0026 0.6568

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Figure 21. The behavior of a MR-B4 damper under a sinusoidal excitation of 15 mm-amplitude and 1 Hz-frequency.(a) Forcedisplacement curve. (b) Forcevelocity curve.

Figure 22. The behavior of a MR-B4 damper under a sinusoidal excitation of 10 mm-amplitude and 0.5 Hz-frequency.(a) Forcedisplacement curve and (b) forcevelocity curve.

Table 8. Identification results of a MR-B1 damper.

Af Bf Cf Df Ac Bc Cc Dc Ax Bx Cx Dx kf tf kc tc kx tx k t

0.49 4.90 0.45 0.38 0.0008 0.0104 0.315 0.66 4.79 12.07 0.35 0.40 0.0023 0.78 0.026 3.43 0.0095 0.12 0.0034 0.49

A comparison between the measured and predictedresults of the MR-B4 damper is plotted in figures 21 and22. The red solid line and the black dashed line representthe measured results and the predicted results, respectively.The results in the cases of 15 mm-amplitude sinusoidaldisplacement excitation at 1 Hz and 10 mm-amplitudesinusoidal displacement excitation at 0.5 Hz are plotted.

The modeling of the MR-B1 damper is conducted usingthe same method. The identification results of the MR-B1damper are listed in table 8. The results in the cases of15 mm-amplitude sinusoidal displacement excitation at 1 Hzand 10 mm-amplitude sinusoidal displacement excitation at1 Hz are presented in figures 23 and 24.

As shown in figures 2124, the Gompertz model can tracethe measured results well. The Gompertz model can portraythe nonlinear dynamic characteristics of the proposed MRdampers precisely using a relatively simplified function formcompared with the phenomenological model [3].

7. Conclusions

In this study, four prototype MR dampers with bidirectionaladjusting damping forces were designed, fabricated and

tested. An improved structure of the composite magneticcircuit is proposed. Performance tests, including magneticfield tests and dynamic tests, were conducted in order toverify the effectiveness of the proposed improved structure.For practical application, a Gompertz model is proposed andadopted to describe the nonlinear hysteretic behavior of theproposed MR damper.

The test results show that, due to the improvedstructure, the proposed dampers are capable of maintaininga medium damping force with zero currents input. Thisadvantage may ensure a better fail-safe property and avoidsettlement of the MR fluid compared with conventionalMR dampers. Furthermore, the minimum and maximumoutput powers of the dampers can be obtained when inputcurrents reach negative peak and positive peak values,respectively. Moreover, the dynamic range of controllableforce is even larger than that of conventional MR dampers.Meanwhile, by controlling the remanence of the permanentmagnet and the thickness of the magnetic core, the residualmagnetic induction in the damping path can be changedto accommodate the actual application requirements. Thestudy further shows that the proposed improved structure of

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composite magnetic current is applicable to dampers witheither single piston section or multiple piston sections. Theproposed Gompertz model can precisely portray the nonlinearhysteretic behavior of the proposed dampers with a simplefunction form.

Acknowledgments

The present work is jointly supported by the SpecializedResearch Fund for the Doctoral Program of Higher Educationof China under Grant No. 20110032110042 and the NationalBasic Research Program of China (973 Program) under GrantNo. 2011CB013606.

References

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A new magnetorheological damper for seismic controlIntroductionThe improved structure of composite magnetic circuitDesign and fabrication of MR dampersMagnetic field testsDynamic testsDynamic modeling of MR dampersConclusionsAcknowledgmentsReferences

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