Electro-rheological Cylinders used as Impact Energy Absorbers

X. W. ZHANG,1 T. X. YU1,* AND W. J. WEN

2

1Department of Mechanical Engineering, Hong Kong University of Science and TechnologyClear Water Bay, Kowloon, Hong Kong, China

2Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

ABSTRACT: Smart fluids, that is, electro-rheological (ER) and magneto-rheological (MR)fluids have been studied widely in vibration control, seismic isolations, and rotation transmis-sion, where the velocity is low and the motions are periodic. However, very few investigationsconcern about the dynamic response and energy absorption of ER dampers under impactloadings. To explore the feasibility of using ER cylinders as impact energy absorbers, twodifferent ER fluids were first characterized by using a capillary rheometer with rectangularduct. Then, a double-ended ER cylinder with two parallel annular ducts was designed, and itsperformance in response to a mass impact was tested. The experiments show that a typicaldynamic response of the ER cylinder consists of three distinct stages, namely, an initial shockstage, a transition stage, and a stable flow stage. Afterwards, the dynamic response is analyzedtheoretically, in which the contact between the impinger and the piston rod, the viscous andER effects as well as the inertia and response time of the ER fluid are considered. It is revealedthat, the controllability of the ER impact energy absorber greatly depends on the impactvelocity and ER fluid’s yield stress, and that when the impact velocity increases, its controll-ability deteriorates due to viscosity and response time.

Key Words: electro-rheological, characterization, response time, impact, energy absorbers.

INTRODUCTION

SMART fluids, including electro-rheological (ER) andmagneto-rheological (MR) fluids, are special sus-

pensions which consist of large mounts of solid particlesdispensed in certain carrier liquids. When subjected to ahigh-intensity electric or magnetic field, the flow proper-ties of these fluids can be changed promptly and thisprocess is reversible. Therefore, the flow resistance ofthe smart fluids can be controlled by adjusting the inten-sity of applied electric/magnetic field. Due to their fastresponse and controllability, the smart fluids haveattracted great attentions of engineers and researcherssince their inventions (Wang and Meng, 2001;Stanway, 2004). In the past decades, the applicationsof these fluids have been investigated extensively invibration control (Stanway et al., 1996), seismic isola-tion (Makris et al., 1996), torque transmission (Choi andLee, 2005), and so on. However, most of the previousstudies merely considered the low-velocity and periodicmovements, while very few investigations concernedabout the dynamic response of the ER/MR devicesunder impact loadings.

Different from those in response to mechanical vibra-tions, the behaviors of ER dampers under impact load-ings exhibit characteristics of high shear-rate,singe-stroke, large resistant force, and short duration.El-Wahed et al. (1999) studied the dynamic behaviorsof MR and ER fluids in squeeze mode, but the effectiveenergy absorbing stroke in this mode was very short andnot suitable for impact energy absorbers. Lee et al.(2002) employed the Herschel-Bulkley model to analyzethe dynamic response of the ER impact dampers, whileYeo et al. (2002) consider the response time and gaveseveral numerical results for a semi-active ER damper.Song et al. (2002); Nam and Park (2007) studied ER andMR shock dampers under impulsive loadings, respec-tively. Batterbee et al. (2007a, 2007b) designed andinvestigated the response of MR landing gears.Although, the devices were designed for realisticimpact velocities, experimental tests were performed atvelocities below 0.5m/s which were much lower than thereal impact cases. On the other hand, Ahmadian andNorris (2004, 2008) conducted a series of impact testsby using two types of MR dampers with the maximumvelocity as high as 7m/s. From their experimentalresults, they found that the average crushing force ofthe MR dampers could be adjusted by the applied cur-rent. They also pointed out that the MR damper is con-trollable only when the impact velocity is lower than a

*Author to whom correspondence should be addressed. E-mail: metxyu@ust.hkFigures 2�12, 15�17, A2 and A4 appear in color online: http://jim.sagepub.com

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 0— 2010 1

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Journal of Intelligent Material Systems and Structures OnlineFirst, published on March 19, 2010 as doi:10.1177/1045389X10366316

certain value because of high Reynolds Number.Although their conclusion seems partly reasonable,the definition of critical Reynolds number ofnon-Newtonian fluids is doubtful. According to Nouarand Frigaard (2001), the critical Reynolds Numberdepends on the Bingham Number, while in some previ-ous papers (Ahmadian and Norris, 2008; Batterbeeet al., 2007a), they assumed the critical ReynoldsNumber as 2000 or 1000. However, this problem isstill unclear.In this article, to study the performance of ER fluids

in impact scenarios, the characterization of ER fluidsflowing in parallel duct under high shear rate is firstdiscussed. Both a nominal and exact characterizationmethods are evaluated and compared. Then, two typesof ER fluids were experimentally characterized by meansof a capillary rheometer with rectangular ducts, andtheir nominal yield stresses and viscosities under dif-ferent electric fields determined. Afterwards, adouble-ended ER cylinder with two parallel annularducts was designed, and its dynamic responses to amass impact under different electric field intensity andimpact velocity were tested. Based on the experimentalobservations, the dynamic response of the ER cylinderwas analyzed and a theoretical model proposed, inwhich the initial yielding, the inertia effect, and theresponse time of the ER fluid are mainly considered.Finally, the theoretical predictions of the dynamicresponse are compared with the experimental results,and the controllability of the ER cylinder is discussed.

CHARACTERIZATION OF ER FLUIDS UNDER

HIGH SHEAR RATE

According to the previous study (Stanway et al.,1996), when subjected to high intensity electric field,the flowing behaviors of ER fluid can be approximatedto be Bingham-plastic, whose constitutive equation is asfollows:

� ¼ �0 þ � _� ð1Þ

where � is the shear stress, _� is the shear rate, � is theviscosity, and �0 denotes the yield stress of the ER fluid.

Usually, it is assumed that �0 and � are functions of theelectric field intensity which is denoted by E (kv/mm).For theoretical modeling, most of the researchersemploy the ideal yield stress and viscosity to evaluatethe performance of ER fluid devices. However, in theexperimental characterizations (Park et al., 1999; Chenand Wei, 2006; Goncalves et al., 2006) only the nominalyield stress and viscosity are measured by rheometersthrough the similar method as that for Newtonianfluids. Although, Choi et al. (2005) analytically evalu-ated the errors of rheological properties of ER fluidcalculated from three different property models forboth rotational co-axial cylinder and parallel-diskrheometers, there is still a lack of detailed analysis ofcapillary rheometer, which is suitable for high shearrate flowing.

Consider a capillary rheometer with a rectangular ERduct as illustrated in Figure 1(a) which has length L andwidth b, and the distance between the electrodes is h. If bis quite larger than h (e.g., b� 10 h), the boundary effectat the narrow side can be ignored and the flow in theduct can be assumed as to be 1D flow. For thequasi-steady flow of ER fluids through the parallelduct, when it is free of electric field, the flowing behaviorof the ER fluid is close to Newtonian and the contour ofthe stream lines has a parabola shape S1 as shown inFigure 1(b). After an electric field is applied, the streamlines change from S1 to S2. For the case S2, the ER fluidin the region between A and B does not yield and itsshear rate is zero, but in the region between A, B, andthe boundaries, the ER fluid yields. For ideal Binghamplastic flow, the analytical relation between the flow rateQ and the pressure drop �P through this duct can beobtained from (Stanway et al., 1996):

�P3 � 3�0L

hþ12�LQ

bh3

� ��P2 þ 4�30

L

h

� �3

¼ 0 ð2Þ

If the viscosity of the Bingham plastic fluid is known,for specified flow rate Q and pressure drop �P, the yieldstress �0 can be obtained by the above equation. In mostof the previously published papers about the character-ization of ER/MR fluids, either Equation (2) was usedto calculate the yield stress by assuming the viscosity to

(b)(a)

hb

L

ERF

S2

A

+

–

B

S1up

–to

t

to

Q

Figure 1. The flowing behavior of ER fluid in parallel duct: (a) configuration of the ER duct; (b) cross-section of the duct.

2 X. W. ZHANG ET AL.

be constant (Lee and Choi, 2002; Goncalves et al., 2006)or similar method with Newtonian fluids was applied tocalculate the shear rate and shear stress (Chen and Wei,2006). However, for a new ER fluid whose yield stressand viscosity both need to be determined, Equation (2)alone is not sufficient. In the characterization tests, theflow of the fluid within the duct is quasi-steady, theshear stress along the boundary can be calculated by:

�w ¼ �P�h

2L: ð3Þ

If the stream-lines are assumed to be similar to thosein a Newtonian flow, the nominal shear rate at theboundary of the ERF duct can be found as:

_�n ¼6Q

bh2: ð4Þ

Hence, by employing Equations (3) and (4), the nom-inal flow curves of the ER fluid under different electricfields can be obtained. However, it should be noted thatthis method can only obtain exact shear stress and shearrate for Newtonian fluids. To obtain the true yield stressand viscosity of Bingham-plastic fluids, a similar methodwith Rabinowitsch’s correction (Ferguson andKemblowski, 1991) should be used.Recalling to Equations (1) and (3), the true shear rate

at the boundary should be:

_�w ¼�w � �0�¼ �

p

��

h

2þ�0�, ð5Þ

where p ¼ �P=L is the pressure drop within a unitlength of an ER duct. By taking the derivative of theboth sides of Equation (2) and combining it withEquations (3) and (5), the true shear rate is found to be:

_�w ¼4Q

bh2þ

2p

bh2�

dQ

dp: ð6Þ

In the characterization tests, if the relationshipbetween Q and p under a certain electric field is mea-sured, by using Equations (3) and (6), the flow curve ofthis ER fluid can be calculated. As a result, the true yieldstress and viscosity of the ER fluid can be obtained byfitting the data. Compared with the previous nominalcharacterization method, this method is denoted asexact characterization method in the subsequent text.In order to verify the above exact characterization

method and evaluate the errors of the nominalmethod, a numerical validation procedure is proposedas follows. First, for this rheometer, take the same set ofthe geometric parameters as those adopted by Zhanget al. (2008); that is, the width and the gap of the ERduct are b¼ 10mm and h¼ 1.2mm, respectively; and the

piston area is Ap¼ 1963mm2. Then, assume the ERfluids are ideal Bingham plastic fluids, and differentrheological properties are evaluated, which are �¼ 0.1,0.5, 1.5 Pa s and �0¼ 1.5, 3.0, and 8.0 kPa; choose agroup of piston velocities as Vp¼ 10, 50, 100, 200, 300,400, and 500mm/min. For a certain combination of �and �0, the relationship between flow rate Q and pres-sure drop p can be obtained by means of Equation (2);according to the true Q-p relations, both the nominaland exact characterization methods are employed to cal-culate the shear rate-stress curves. Finally, by the linearfitting of these flow curves, the nominal and true rheo-logical properties are obtained and compared with thetheoretical values.

Figure 2(a) illustrates the results of the relationshipsbetween Q and p for �¼ 0.1 Pa s and �0¼ 1.5, 3.0,8.0KPa. It is seen that when the flow rate Q is small,the Q�p curve is non-linear. However, when Q becomesvery large, the Q�p curves approach to linear relations.Figure 2(b) depicts the comparison of the two character-izations for �¼ 0.1 Pas and �0¼ 3KPa, showing thatwith the same flowing conditions, the shear rate mea-sured by the exact method is much larger than that by

(a)Q-p relationship for m = 0.1Pas

Pressure drop p (kPa/mm)

Flo

w r

ate

Q (

mm

2*m

/s)

ty=3 kPa

ty=8 kPa

ty=1.5 kPa

(b)

Comparison between nominal and true flow curves,m = 0.1 Pa s, ty = 3 kPa

y = 0.1044x + 2.9059

y = 0.2048x + 3.2141

Shear rate (1000 1/s)

She

ar s

tres

s (k

Pa)

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 16 18 20 22 24

00.5

11.5

22.5

33.5

44.5

5

0 2 4 6 8 10 12 14

Nominal TRUELinear (TRUE)Linear (Nominal)

Figure 2. Numerical comparison between two characterizationmethods: (a) typical relations between Q and p; (b) comparisonbetween the nominal and exact characterization methods for ERFwith �¼ 0. Pa s and �0¼3 kPa.

Electro-rheological Cylinders used as Impact Energy Absorbers 3

the nominal one. Compared with the theoretical values,while the yield stress obtained by the nominal method isacceptable, the nominal viscosity has an error more than100%. On the other hand, the exact characterizationmethod can achieve very good agreement with thetheoretical values.More numerical characterization examples are listed

in Table 1, which demonstrates that for the given rheo-logical properties, by using the exact characterizationmethod, the errors in the viscosity is always smallerthan 5%, while the errors in the yield stress is smallerthan 7%. However, the errors of nominal properties arevery large. The results also show that with the sameviscosity, the errors in the nominal viscosity increaseswith the increase of yield stress, whilst the error in thenominal yield stress decreases. Similar observations aremade when the yield stress remains unchanged.For ideal Bingham-plastic fluids, if Equation (2) is

non-dimensionalized by �0=h, the non-dimensional flowrate and pressure drop are as follows:

�q ¼12�Q

bh3�

1

�0=hð Þ, �p ¼

p

�0=h, ð7Þ

Then, Equation (2) can be rewritten as:

�p3 � ð3þ �qÞ � �p2 þ 4 ¼ 0 ð8Þ

The relationship between �p and �q for �q from 0 to 10are plotted in Figure 3(a), which indicates that when theflow rate Q approaches to 0 and infinite, the relationshipbetween Q and p can be approximated by a liner rela-tionships as follows:

Q! 0, p!12�Q

bh3þ2�0h

, ð9Þ

Q!1, p!12�Q

bh3þ3�0h: ð10Þ

Therefore, for any flow conditions, we have:

p ¼12�Q

bh3þ K �

�0h, where 25K5 3: ð11Þ

Recalling Equation (5), it is found that when K¼ 2,Equation (11) becomes Equation (4), which means thatthe nominal method is corresponding to Equation (11)with K¼ 2, i.e., Equation (9). To make use of the nom-inal rheological properties, for given flow rate Q,Equation (9) is used to calculate the pressure drop andthe results are compared with the values obtained fromtrue yield stress and viscosity and the non-linearEquation (2). It can be seen from Figure 3 that byusing the nominal rheological properties, the pressuredrop obtained by the linear Equation (9) can achieve

(a)

Non-dimensional relationship between flow rate andpressure drop

0

2

4

6

8

10

12

14

16

0

Flow rate q

Pre

ssur

e dr

op p

Equation (8)

Equation (9)

Equation (10)

(b)

Comparison between the nominalmethod and the accurate values

0

1

2

3

4

5

6

7

8

9

Flow rate Q (mm2*m/s)

Pre

ssur

e dr

op (

kPa/

mm

)

True pressureBy nominal values

12108642

0 1814128 1062 4 16

Figure 3. Analysis of the nominal characterization method: (a) rela-tionship between non-dimensional flow rate and pressure drop;(b) error evaluation of the pressure drop from nominal rheologicalproperties.

Table 1. Comparison between the nominal and exact characterization methods.

Nominal l

(Pa s)True l

(Pa s)Nominal q0

(kPa)True q0

(kPa)

�¼ 0.1 Pa s �0¼ 1.5 KPa 0.163 (63%) 0.104 (4%) 1.65 (10%) 1.4 (6.7%)�0¼ 3.0 KPa 0.205 (105%) 0.104 (4%) 3.21 (7%) 2.91 (3%)�0¼ 8.0 KPa 0.299 (200%) 0.104 (4%) 8.35 (4.4%) 7.84 (2%)

�0¼ 3 KPa �¼ 0.1 Pa s 0.205 (105%) 0.104 (4%) 3.21 (7%) 2.91 (3%)�¼ 0.5 Pa s 0.64 (28%) 0.52 (4%) 3.45 (15%) 2.83 (5.7%)�¼ 1.5 Pa s 1.64 (9.3%) 1.53 (2%) 3.70 (23%) 2.80 (6.7%)

Note: The percentages in the brackets are the errors compared with the theoretical values.

4 X. W. ZHANG ET AL.

very good agreement with the real values. Therefore,although the nominal characterization method can notobtain the real rheological properties, in the theoreticalmodeling of ER fluid flows, the nominal properties arestill useful and can greatly simplify the calculation of thepressure drop.For the experimental study of the ER impact energy

absorber, two kinds of ER fluids are employed. The firstone is labeled by Giant-ER fluid, which was produced inthe HKUST by dispersing nano-particle coatedBa0.8(Rb)0.4TiO(C2O4)2 particles in silicon oil (Wenet al., 2004). The mass concentration of the particles isC¼ 44.5%. The other ER fluid named as SMT-ER fluidwas purchased from Smart Technology Limited andlabeled by LID3354S, which is density-matched andhas a mass concentration C¼ 37.5%. After characteriza-tion by means of the capillary rheometer (Zhang et al.,2008), the nominal yield stress and viscosity of these twoER fluids with respect to different electric field intensityare plotted in Figure 4(a) and (b), respectively. In the

subsequent experiments and modeling, these rheologicalproperties will be used.

DESIGN OF THE ER FLUID IMPACT

ENERGY ABSORBER

Up to now, two typical designs for ER cylinders havebeen reported in literature (Ahmadian and Norris,2008), which are single-ended and double-ended, respec-tively. The former is compact in structure, but requires achamber filled with high pressure gas, which makes itcomplicated, while the double-ended design is relativelysimple in design and implementation. Hence, in this arti-cle, for simplicity, the double-ended design is chosen.

Considering that the density and plastic viscosityof our ER fluids are about �¼ 0.15 Pas andq¼ 1.0� 103 kg/m3, and their maximum mean flowspeeds in the duct are lower than 10m/s, the Reynoldsnumber, Re ¼ �vQ=�b, (Batterbee et al., 2007), will belower than 200. Therefore, the flow can be assumed tobe laminar. If ignoring the influences of the compressi-bility and inertia effect of the ER fluid, the resistantforce of a typical ER fluid cylinder can beapproximated by:

Fd ¼12�nQ

bh3þ 2 �

�0�nh

� �� L � Ap þ FR, ð12Þ

where the �n and �0�n are the nominal viscosity andyield stress of the ER fluid, and FR is the frictionforce. In the subsequent sections, �n and �0�n will bereplaced by � and �0 for convenience. The first andsecond terms in the bracket on the right-hand side ofEquation (12) are the contributions of viscosity andyield stress, which will be termed as the viscous andER effects, respectively. For impact scenarios, sincethe shear rate is high, to reduce the viscous effect andincrease the controllability of the ER fluid cylinder, thewidth b of ER duct should be as large as possible.Accordingly, a double-ended ER cylinder with two par-allel annular ducts was designed, whose cross-sectionalong its axis is as shown in Figure 5(a). This ER cylin-der is mainly composed of three coaxial cylindrical tubessealed by insulated plates at their two ends. The gapsbetween the neighboring tubes form the ER ducts.During the tests, inner-most and outer tubes are con-nected to the ground (negative electrode), while themiddle tube is connected to the positive electrode.When the piston rod moves downwards, the ER fluidin the lower chamber will be forced to move down andgo back to the upper chamber through the ER ducts. Byadjusting the electric field applied to the ER ducts, theflow behaviors of the ER fluid as well as the resistantforce of the rod can be controlled. The important dimen-sions of the ER cylinder are depicted in Figure 5, inwhich the gap and the effective length of the duct are

(a)

0.00.0

0.5

1.0

1.5

2.0

2.5

Y=0.47E

Giant-ER fluid C=44.5%SMT-ER fluid C=37.5%

Electric-field intensity E (kv/mm)

Yie

ld s

tres

s (k

Pa)

Y=0.20E1.83

(b)

00.0

0.1

0.2

0.3

0.4

0.5

Giant-ER fluid C=44.5%SMT-ER fluid C=37.5%

Vis

cosi

ty (

Pa

s)

Electric field intensity (kV/mm)

4.03.53.02.52.01.51.00.5

1 2 3 4

Figure 4. (a) The nominal yield stresses of the ER fluids;(b) nominal viscosity of the ER fluids.

Electro-rheological Cylinders used as Impact Energy Absorbers 5

h¼ 1.5mm and L¼ 65mm, respectively. Therefore, theeffective cross-sectional areas of the piston and theducts, the width of ER duct are as follows:

Ap ¼�

4� d 2

1 � d 20

� �¼ 1206:4mm2, ð13Þ

Ad ¼�

4d22 � ðd2 � 2hÞ2 þ d 2

3 � ðd3 � 2hÞ2� �

¼ 461:8mm2 and b ¼Ad

h¼ 308mm: ð14Þ

If assuming the ER fluids in the two channels have thesame velocity, then the velocity ratio between the pistonand the ER fluid inside the ducts is:

Rs ¼Ap

Ad¼ 2:61 ð15Þ

In our experimental study, the two types of ER fluidsmentioned in the previous section were filled into the ERcylinder and tested, respectively. A Dynatup 8250Drop-weight Tester was used for the impact tests anda diagram of the testing system is shown in Figure 5(b).The ER cylinder and its supporting platform wereplaced at the bottom of the Drop-weight Tester, whilea weight with mass M was employed to impinge thepiston rod of the ER cylinder. Before the impact test,the piston rod was first pulled to its top limited locationand an electric field was applied, then the weightdropped down from height H and impinged the pistonrod connected to the ER cylinder. The loading historywas recorded by the load cell. In order to avoid excessiveshock and protect the devices, a piece of rubber wasplaced between the crosshead and the piston rod, anda honeycomb block was also employed as shown in

Figure 5(b). The total drop mass including the weight,the load cell, and the crosshead in this test wasM¼ 5.67 kg.

EXPERIMENTAL RESULTS

Quasi-steady Tests

The ER cylinder was first compressed quasi-steadilywith a loading speed 40mm/min on the UTMSINTECH 10/D installed at HKUST. During theexperiments, the UTM was first turned on to let thecrosshead compress the piston rod of the ER cylinder,and then the electric field was applied to the ER ductwhen the displacement was about 5mm. This procedurecan avoid the damage of the device, because the staticyield stress of the ER fluid may be very large so that theER duct may get completely jammed. Considering theamplifying ratio of the ER fluid velocity was Rs¼ 2.61and the loading speed of the machine was 40mm/min,the real flow speed of the ER fluids inside the channelwould be Vd¼ 1.93mm/s. By using Equation (12), it isfound that the viscous force is lower than 0.5N. Hence,in the quasi-steady tests the viscous force can be ignoredcompared with the ER force.

Figure 6(a) shows the quasi-steady loading curves ofthe ER cylinder filled with Giant-ER fluid under differ-ent electric field intensities. It can be seen that even whenthe electric field intensity is zero (i.e., E¼ 0 kv/mm),there is still a resistant force FR� 100N. Since at thisloading speed, the viscous and ER forces are both neg-ligible, this resistant force is attributed to the frictionbetween the piston rod and the inner wall of the tube.With the increase of E, the resistant force increases, too.However, the fluctuation of the force is very obvious,because the real flow inside the ER duct is not

h = 1.5 F

d1 = 40

d2 = 47

d3 = 54

d0 = 8(a) (b)

L+ H

ER cylinder

Weight

M

Honeycomb block

Platform

xs

Figure 5. (a) Cross-section of the ER cylinder; (b) schematic diagram of the experimental setup.

6 X. W. ZHANG ET AL.

continuous, which was observed during the characteriza-tion tests (Zhang et al., 2008). When E¼ 4.0 kv/mm, theaverage resistant force of the ER cylinder is about350N. On the other hand, the friction force ofSMT-ER fluid is 80N and the maximum electric fieldintensity is 3.5 kv/mm. It should be noted that in theexperiments, the particles of ER fluids were verycoarse and the lubricant of the ER cylinder was notsatisfactory.After subtracting the friction force, the experimental

resistant forces with respect to the electric field for theER cylinders filled with Giant-ER and SMT-ER fluidsare depicted in Figure 6(b). For comparison, the resis-tant forces predicted by Equation (12) are also plotted.It is revealed that although the dispersion of the exper-imental results is relatively large, the accuracy of theresults is acceptable within this range of electric fieldintensity.

Impact Tests of the ER Cylinders

The impact tests were conducted on a Dynatup-8250testing machine as shown in Figure 5(b). For each type

of ER fluids, except the case with E¼ 0, four other elec-tric field intensities from 1.33 to 3.33 kv/mm wereapplied to the ER duct, and for every intensity, threeimpact velocities, about 1.2m/s, 2.0m/s, and 3.0m/s,were tested. Two typical loading and velocity historiesfor the impact tests of the ER cylinder filled withGiant-ER fluid under E¼ 0 are plotted in Figure 7(a)and (b), in which the impact velocities are about 1.2 and2.0m/s, respectively. For each impact condition, two orthree repeat tests were conducted, which were denotedby A, B, and C. The results confirmed that the repeat-ability of the tests was good. A typical loading curve ofthis dynamic response possesses an initial peak forcefollowed by some fluctuations, and then the force grad-ually decreases. After a certain displacement, thedecrease of the resistant force becomes very slow. Theinitial peak force for the two impact conditions is aboutFpeak¼ 800N and 1600N, respectively. It should benoted that for the impact velocity of about V0¼ 2m/s,the effective energy absorbing stroke was finished at thedisplacement xs¼ 50mm. However, at this moment, theweight still had a velocity of about 0.5m/s and the cross-head was finally stopped by the honeycomb as shown inFigure 5(b). Consequently, when the displacement

00

50

100

150

200

250

300

350

400

450

E=4.0

E=3.33

E=0.0

E=1.33

E=0.67

E=2.67

Res

ista

nt fo

rce

(N)

Displacement (mm)

Unit: kv/mm

E=2.0

(b)

(a)

00

50

100

150

200

250

300

Dam

ping

forc

e (N

)

Electric field intensity (kv/mm)

Giant ERFSMT ERF

605040302010

1 2 3 4

Figure 6. (a) Loading curves for Giant-ER fluid under differentelectric fields; (b) comparison between experimental resistantforces and those predicted by the material properties.

Vel

ocity

(m

/s)

Vel

ocity

(m

/s)

(a)

Impact tests for E=0, V=1.2 m/s

0

100

200

300

400

500

600

700

800

900

0

Displacement (mm)

For

ce (

N)

0

0.2

0.4

0.6

0.8

1

1.2

1.4F-A

F-B

F-C

V-A

V-B

V-C

(b)Impact tests for E=0, V=2.0 m/s

0

200

400

600

800

1000

1200

1400

1600

1800

0

Displacement (mm)

For

ce (

N)

0

0.5

1

1.5

2

2.5F-V3AF-V3BVAVB

45403530252015105

605040302010

Figure 7. Typical loading and velocity histories for the impact testswith V0¼ 1 and 2 m/s.

Electro-rheological Cylinders used as Impact Energy Absorbers 7

exceeded 50mm, the load increased greatly whilst thevelocity was soon reduced to zero.Figure 8 exhibits the experimental results for the ER

cylinder filled with Giant-ER fluid under impact velocityV0¼ 1.2m/s and different electric fields. It is seen fromFigure 8(a) that the configurations of the load-displacement curves are similar to those with E¼ 0,and higher the electric field intensity, the larger the resis-tant force. Also, it is revealed by Figure 8(b) that in theinitial impact stage, the decrease of the velocity is fasterthan that in the subsequent stage and the final displace-ment decreases with the increase of the electric fieldintensity. The curves of the load history are plotted inFigure 8(c), which indicate that most of the impactslasted no more than 50ms.However, it is difficult to fully explore the effect of the

electric field simply by comparing the loading and veloc-ity histories, because the resultant force always containsboth the viscous and ER effects. To decouple thesetwo effects, the force-velocity curves are re-plotted inFigure 8(d). With the initial velocity of this testV0¼ 1.2m/s, the curves in Figure 8(d) should be readfrom right to left and the difference between the resistantforces under the same velocity represents the ER effect.

It also reveals that the entire impact process has threedistinct stages. In the first stage, there is an initial peak,after which the load fluctuates, and then decreases grad-ually. In the second stage, the effect of the electric field isquite prominent. After a transition between the secondand the third impact stages, the loading curve becomessmooth.

The test results for this ER cylinder under the impactvelocity V0¼ 3m/s are also illustrated in Figures9(a)�(d). Compared with the results for V0¼ 1.2m/s,the initial peaks under V0¼ 3m/s are much higher. Atthe end of the energy absorbing stroke of 50mm, theinitial kinetic energy in some cases has not been fullyabsorbed by the ER cylinder. As a result, the residualenergy has to be dissipated by the honeycomb block.From the load history curves, the difference betweenthe cases with different E is very small. However, inFigure 9(b), at the displacement 50mm, the residualvelocity under more intense electric field is larger thanthat under weaker electric field. Also, the duration of theimpact response is about 30�40ms, which is shorterthan that in the case of V0¼ 1.2m/s. Figure 9(d)shows that under high impact velocity, the ER effectin the second stage seems to be very small, while that

(a) Comparison of the impact test with V0 = 1.2 m/s

0

200

400

600

800

1000

1200

1400

1600

0

Displacement (mm)

For

ce (

N)

E=0 kv/mm

E=1.33 kv/mm

E=2 kv/mm

E=2.67 kv/mm

E=3.335 kv/mm

E=4 kv/mm

(b) Velocity–displacement curves for V0 = 1.2 m/s

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Displacement (mm)

Vel

ocity

(m

/s)

E=0 kv/mm

E=1.33 kv/mm

E=2 kv/mm

E= 2.67 kv/mm

E=3.33 kv/mm

(c) Force history of V0 = 1.2 m/s

0

200

400

600

800

1000

1200

1400

0

Time (ms)

For

ce (

N)

E=0 kv/mmE=1.33 kv/mmE=2 kv/mmE=2.67 kv/mmE=3.33 kv/mm

(d) Force–velocity curves for V0 = 1.2 m/s

0

200

400

600

800

1000

1200

0

Velocity (m/s)

For

ce (

N)

E=0 kv/mm

E=1.33 kv/mm

E=2 kv/mmE=2.67 kv/mm

E=3.33 kv/mm

30252015105 0 252015105 403530

8070605040302010 1.41.210.80.60.40.2

Figure 8. The test results for the ER cylinder filled with Giant-ER fluid under V0¼ 1.2 m/s: (a) F�x; (b) V�x; (c) F�s; (d) F�V.

8 X. W. ZHANG ET AL.

in the third stage it almost disappears. Therefore, com-pared with the test under V0¼ 1.2m/s, the ER effectunder 3m/s becomes weaker.In addition, the impact tests for the ER cylinder filled

with SMT-ER fluid were also conducted, and the initialpeak forces for the cylinders filled with the two ERfluids (Giant-ER and SMT ER fluids) under differentimpact velocities are plotted in Figure 10(a) and (b). Itcan be seen that, although the electric field intensity hascertain influence on the peak force, the dominant factoris the impact velocity; and with the increase of theimpact velocity, the initial peak force increases greatly.The mean crushing forces during the impact tests are

also calculated based on the experimental results. Thecalculation method is as follows:

Fm ¼

12M� V2

0

xf, xf 5 xs

12M V2

0 � V2s

� �xs

, xf 4 xs

8>>><>>>:

ð16Þ

where xf is the final displacement of the piston, xs is thelimit of the energy absorbing stroke (i.e., displacement

of the piston when the impinger contacted the honey-comb block, which was set to be 50mm) and Vs is theresidual velocity of the piston at x¼xs. The results ofmean forces for the ER cylinder filled with the two ERfluids are depicted in Figure 11(a) and (b), respectively.In these figures, the results for V0¼ 0m/s are from thequasi-steady compression tests. It is shown that with theincrease of electric field intensity, the mean crushingforce increases. However, for larger impact velocitythis ER effect becomes weaker. As shown inFigure 11(a), under V0¼ 3m/s the mean crushing forceof cylinder with Giant-ER fluid nearly has no increase,even when E increases from 0 to 4 kv/mm. On the otherhand, although the Fm for SMT-ER fluid has a slightincrease, the enhancement due to electric field underhigher velocity is weaker than that under smallervelocity.

THEORETICAL MODELING FOR THE IMPACT

RESPONSE OF ER CYLINDER

Based on the experimental observations as shown inFigures 8 and 9, it is found that an entire impactresponse of the ER cylinder consists of three distinct

(a) Loading–displacement curves for V0=3 m/s

0

500

1000

1500

2000

2500

3000

3500

4000

Displacement (mm)

For

ce (

N)

E=0 kv/mm

E=1.33 kv/mm

E=2 kv/mm

E=2.67 kv/mm

E=3.33 kv/mm

(b)3.5

3

2.5

2

1.5

1

0.5

0

Velocity–displacement curves for V0=3.0 m/s

Displacement (mm)

Vel

coci

ty (

m/s

)

E=0 kv/mm

E=1.33 kv/mm

E=2.0 kv/mm

E=2.67 kv/mm

E=3.33 kv/mm

(c) Force history of the tests with V0=3 m/s

Time (ms)

E=0 kv/mm

E=1.33 kv/mm

E=2 kv/mm

E=2.67 kv/mm

E=3.33 kv/mm

(d) Force–velocity curves with V0=3 m/s

0

500

1000

1500

2000

2500

3000

3500

4000

Velocity (m/s)

For

ce (

N)

E=0 kv/mm

E=1.33 kv/mm

E=2.0 kv/mm

E=2.67 kv/mm

E=3.33 kv/mm

6050403020100 6050403020100

0

500

1000

1500

2000

2500

3000

3500

4000

For

ce (

N)

6050403020100 32.521.510.50

Figure 9. The test result for the ER cylinder filled with Giant-ER fluid under V0¼3 m/s: (a) F�x; (b) V�x; (c) F�s; (d) F�V.

Electro-rheological Cylinders used as Impact Energy Absorbers 9

stages, namely, the initial shock stage, a transition stage,and a smooth flow stage. In this section, the initial shockand the subsequent two stages will be analyzed stepby step.

Modeling of the Initial Shock Stage

The initial shock stage is characterized by a peak forcefollowed by some fluctuations. Before the impact, theimpinger has a velocity V0, while the piston rod is sta-tionary; after they contact each other, the impingerdecelerates and the piston rod accelerates until theygain the same velocity. To avoid a rigid-to-rigidimpact, a rubber plate of thickness 4mm was placedbetween the impinger and the piston rod. As the fluctua-tions are caused by the wave propagation and vibrationof the set-up, rather than the dynamic behavior of theER cylinder, these effects are ignored in the modelinganalysis. Besides, according to the preliminary analysisas shown in the Appendix, the compressibility of the ERfluid only affects the initial impact stage, while the iner-tial effect functions through the entire process, especiallywhen the density of the fluid is large (e.g., MR fluids).Therefore, the influences of the four factors, that is, the

compression of the rubber, the initial yielding, compres-sibility, and the inertial effect of the ER fluid, will beinvestigated in this section.

The proposed theoretical model is sketched inFigure 12(a), in which X1 and X2 are the displacementsof the crosshead and the piston rod, respectively, whilethe governing equations of the system are given by:

M €X1 ¼Mg� Fk, ð17aÞ

m €X2 ¼ mgþ Fk � Fd, ð17bÞ

where M denotes the mass of the weight together withthe load cell and crosshead, m is the mass of the pistonand ER fluid, Fk is the compressive force of the rubberplate and Fd is the resistant force of the ER cylinder.

To determine Fk, a quasi-static compression test for therubber plate was conducted, in which the contact condi-tions were kept the same as those in the impact tests. Theresult is plotted in Figure 12(b), showing that itsload�displacement relationship can be approximated by:

Fk ¼ 1068�� 1311�2 þ 656�3, where � ¼ X1 � X2:

ð18Þ

(a) The initial peak forces for Gaint-ERF

00 1 2 3 4 5

500

1000

1500

2000

2500

3000

3500

4000

Electric field intensity (kv/mm)

0 1 2 3 4 5 6

Electric field intensity (kv/mm)

For

ce (

N)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

For

ce (

N)

(b) Peak forces of the commercial ERF cylinder

V0=1.2 m/s

V0=1.4 m/s

V0=2.0 m/s

V0=3 m/s

V0=1.2 m/s

V0=1.4 m/s

V0=2.0 m/s

V0=3 m/s

Figure 10. Peak forces of the ER cylinders under different impactconditions: (a) filled with Giant-ER fluid; (b) filled with SMT-ER fluid.

(a) Mean force of the impact tests with different velocity

0

100

200

300

400

500

600

700

Electric field (kV/mm)

Electric field (kV/mm)

For

ce (

N)

(b) Mean damping force for different velocity

0

100

200

300

400

500

600

700

800

Mea

n fo

rce

(N)

V=1 m/s

V=1 m/s

V=0 m/s

V=2 m/s

V=3 m/s

V=2 m/s

V=3 m/s

543210

43210

V0

Figure 11. Mean crushing force of the ER cylinders: (a) filled withGiant-ER fluid; (b) filled with SMT-ER fluid.

10 X. W. ZHANG ET AL.

In the unloading process, the compressive force dropssharply and the deformation of the rubber is irrecover-able. Although the dynamic properties of the rubberplate may have some differences between the quasi-staticresults, it will not influence the dynamic response verymuch.Relevant to the resistant force Fd, it is mainly contrib-

uted by the viscous and ER effects of the ER fluid. Twodifferent types of yield stresses for the ER fluid, i.e., thequasi-static yield stress �YS and the dynamic yield stress�YD (¼ �0), should be distinguished. Before starting toflow, the ER fluid has to overcome the quasi-static yieldstress and its deformation tends to be viscous-elastic.The constitutive relation of the ER fluid can beexpressed by:

� ¼G� þ � _�, �5 �S�0 þ � _�, �4 �S

�, ð19Þ

where G is the shear modulus of the ER fluid beforeyielding and �S¼ �YS/G is the yield shear strain. Onthe other hand, recalling the Equations (A1) and (A4)for the fluid compressibility, the flow rate Q is:

Q ¼ Ap_X2 �ðL� X2Þ

�_P

, ð20Þ

where � is the bulk modulus of the ER fluid.

Besides, considering the inertia effect of the ER fluidin Equation (A9), and combining Equations (12) and(19), we have:

_X2 ¼ 0, ðmgþ FkÞ5FR

Fd ¼12�Q

bh3þ 2 �

G�

hþ �

Ap

bh� X2

���� �

� L � Ap þ FR,

�5 �S

Fd ¼12�Q

bh3þ 2 �

�0hþ �

Ap

bh� X2

���� �

� L � Ap þ FR,

�4 �S

8>>>>>>>>>><>>>>>>>>>>:

ð21Þ

At the end of the initial shock stage, the compressionof the rubber plate has exceeded its elastic range.However, to simplify the model, the spring-back of therubber plate is ignored, so that the weight and the pistonrod will move together after they gain the same velocityat the end of the initial shock stage.

Analysis of the Second and Third Stages

In the second impact stage, the weight and the pistonrod move together, so that the motion of the system isrepresented by Figure 13 and the governing equationbecomes:

ðMþmÞ €X2 ¼ ðMþmÞ g� Fd: ð22Þ

Because the compressibility of the ER fluid onlyaffects the initial stage, in the second and third impactstages, it will be ignored, which means that Equation(20) becomes Q ¼ Ap

_X2. Also, the inertia effect of ERfluid is still considered by Equation (21). However, sincethe duration of the impact process is of the same scale asthe response time of the ER fluid to the applied electricfield, the effect of this response time should be takeninto account.

Different from the quasi-steady tests under low veloc-ity, as shown in Figure 14(a), there are two types of ER

(a)

(b)

0.00

200

400

600

800

1000 LoadingFitting for loading Unloading

Displacement (mm)

Com

pres

sive

forc

e (N

)

F=–7.1+881µΔ–1076µΔ2+529µΔ3

M

m

V0

X1

X2

Fk

Fd =FY+FN

1.61.20.80.4

Figure 12. (a) Analytical model for the initial shock stage; (b) com-pression curve for the rubber plate.

Fc

X2

M+m

Fd =FY+FN

Figure 13. Analytical model for the second and third impact stages.

Electro-rheological Cylinders used as Impact Energy Absorbers 11

fluids inside the ducts, namely, the old ER fluid whichhas been in the duct before the impact, and the new ERfluid which just enters the ducts pushed by the piston.The yield stress of the old ER fluid under the electricfield has been developed, but the yield stress of the newER fluid has not been fully developed, because theirdwell time under the electric field is too short. In theanalysis of the previous stage, because the duration ofthe initial shock is small, the new ER fluid and the effectof the response time are negligible. However, when thisnew ER fluid section becomes longer, it should be con-sidered. Assume the response time of the ER fluid to theapplied electric field is tR (which is usually 30�50ms)and the yield stress increases linearly with the dwelltime, then the instantaneous yield stress is:

�Y ¼ �0 �mintdtR

, 1

� �ð23Þ

where td is the dwell time of the ER fluid within the duct.As shown in Figure 14(a), it is assumed that at a certainmoment t, the interface between the old and new ERfluids locates at x*. A diagram relationship of x*� t isillustrated in Figure 14(b), which indicates that the dwelltime td for any ER fluid at the position x0 can beobtained from another coordinate system x0 � td by:

x0 ¼

Z t

t�td

_xdt or td ðx0Þ ¼ tðx�Þ � tðx� � x0Þ, ð24Þ

where x� ¼ RsX2. Therefore, in Equation (21) theresistant force Fd in the second impact stage can beexpressed as:

Fd ¼ Ap

"�p0ðL� x�Þ þ

Z x�

0

�pdxþþ�Ap

bh� L � X2

���

#

þ FR, ð25Þ

where:

�p0 ¼12�Q

bh3þ2�0h

and �p ¼12�Q

bh3þ2�0h

mintdtR

,1

� �:

ð26Þ

When x* is equal to L, all the old ER fluid has beenpushed out of the duct, so the dynamic response entersthe third impact stage, namely, the smooth flow stage, inwhich the resistant force becomes:

Fd ¼ Ap

Z L

0

�p dxþ FR, ð27Þ

where �p is still defined by Equations (24)�(26).In the second and third impact stages, the compressive

force applied to the load-cell can be obtained from:

Fc ¼Mð g� €X2Þ ð28Þ

For given initial conditions, the theoretical predic-tions of the dynamic response of the ER cylinder canbe obtained by means of Runge�Kutta method.

Results for the Theoretical Modeling

To verify this model and investigate the influences ofdifferent parameters, the geometric conditions adoptedin the experiments are used in theoretical calculation.On the other hand, the density and bulk modulus ofthe ER fluids are assumed to be q¼ 1.0� 103 kg/m3

and �¼ 1.0GPa. However, three unknown parametersG, �S, and tR, have to be estimated. To evaluate theeffects of these parameters, the yield stress of the ERFare assumed to be �YS¼Kt�YD and �S¼ 2, 5, 10, so Gcan be obtained by: G¼ �YS/�S accordingly. The shearrate and shear stain in the visco-plastic deformationstage of the ERF are approximated by Newtonianflow as governed by Equation (4).

A typical impact event with V0¼ 2m/s, M¼ 5.67 kg,m¼ 0.25 kg, �¼ 0.15 Pa s, �YD¼ 1.5 kPa, and FR¼

100N is first analyzed by this model. Figure 15(a)plots the predicted responses of the initial shock stagewith Kt¼ 1.5, 3, 6, and �S¼ 2, 5, 10. It shows that thestatic yield stress influences the value of the initial peakforces, and the yield shear strain affects the displacementof the piston. It should be pointed out that since theenergy dissipation during this stage is relatively small,these factors do not affect the subsequent two stages.

(a) (b)

x*L

New ER fluid

Old ER fluid

x„

x*

td

td

tO

O

t*

Figure 14. (a) Status of the ERF inside the duct; (b) diagram for the calculation of dwell time.

12 X. W. ZHANG ET AL.

Figure 15(b) shows the dynamic responses of the initialshock stage with three fluid densities applied. It can beseen that the fluid inertia effect greatly influences theinitial peak force, and larger the density of the fluid,the higher the initial peak force will be. In addition,three values of the bulk modulus of the fluid were exam-ined, and the results in Figure 15(c) show that the influ-ence of the fluid compressibility is smaller than 0.5%. Itshould be noted that the use of the rubber plate

significantly reduced the influence of the fluid compres-sibility. Therefore, the compressibility of the ER fluid inthis model could be ignored.

The entire predicted response is obtained by usingKt¼ 1.5, �S¼ 2 and tR¼ 30ms, and compared with theexperimental results in Figure 16. The comparison con-firms that the initial peak force can be predicted. Sincethe vibration and wave propagation within the impingersystem are ignored, and also the impinger and piston rodare assumed to move together after they obtained thesame velocity, the fluctuations and possible secondimpact cannot be predicted. After the peak force, themagnitude of the resistant force and final displacementalso agree with the experiment very well. It is noted thatin the rear part of the loading curve, the theoretical loadis over-estimated, which may be because of the fact thatin the third impact stage, friction force becomes smallerthan that obtained by the quasi-steady tests. However,dynamic measurement of the friction force is stilldifficult.

The theoretical model is also employed to study theinfluences of the impact velocity and response time (ofthe ER fluid to the applied electric field) on the meancrushing force. For given response time tR¼ 30ms, themean crushing forces under different impact velocitiesand different electric field intensities are plotted inFigure 17(a). It reveals that with the increase of theyield stress, the mean crushing force of the dynamicresponse increases, but at higher impact velocity, thistendency becomes weaker. On the other hand, asshown in Figure 17(b), with the same impact velocity,longer the response time, the smaller the mean crushingforce. It should be noted that when the response timeapproaches to infinite, the mean crushing forceapproaches under �Y¼ 0 kPa. Therefore, the theoreticalanalysis reveals that with the increase of the impactvelocity, the controllability of the ERF cylinder willdeteriorate, because of not only the viscous effect, butalso the response time.

(a)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

Displacement (mm)

For

ce (

N)

Simplest

K=1.5, rs=5

K=3.0, rs=5

K=6.0, rs=5

K=3.0, rs=2

K=3.0, rs=10

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Displacement (mm)

For

ce (

N)

Rou=0

Rou=1.0

Rou=4.0

(c)

0

200

400

600

800

1000

1200

1400

1600

1800

Displacement (mm)

For

ce (

N)

Beta=0.3 GPa

Beta=1.0 GPa

Beta=4.0 GPa

32.521.510.5

0 32.521.510.5

0 2.521.510.5

Figure 15. Investigation of the initial shock stage: (a) the effect ofinitial yielding the ER fluid; (b) the inertia effect; (c) influence of thefluid compressibility.

0

500

1000

1500

2000

2500

0

Displacement (mm)

For

ce (

N)

0

0.5

1

1.5

2

2.5

Vel

ocity

(m

/s)

5040302010

F-Experimental

V-ExperimentV-Theoretical

F-Theoretical

Figure 16. Comparison between the theoretical analysis andexperimental results for the typical dynamic response.

Electro-rheological Cylinders used as Impact Energy Absorbers 13

CONCLUDING REMARKS

In this article, the application of ER fluids underimpact scenarios is investigated. First, the characteriza-tion of ER fluids in parallel duct under high shear rate isdiscussed, and the errors caused by the nominal and exactmethods are analyzed. It is found that although the nom-inal method can not obtain the accurate viscosity andyield stress, the nominal values can be used to greatlysimplify the modeling of the ER devices. Secondly, adouble-ended ER cylinder with two coaxial annularducts was designed and manufactured. Its quasi-steadybehavior and dynamic responses under a mass impact aretested. The results show that the dynamic response hasthree distinct stages, namely, the initial shock stage, atransition stage, and finally the smooth flow stage. Theinitial shock stage is characterized by a peak force fol-lowed by force fluctuations. In the transition stage, twotypes of ER fluid in the duct are distinguished, namely,the old ER fluid whose yield stress has been fully devel-oped before impact, and new ER fluid with its yield stresspartly developed. In this stage, the resistant force of theER cylinder decreases, but the ER effect is very apparent.After a transition point, the second stage ends and thedecrease of the resistant force slows down.

Based on the experimental results, an analytical modelis proposed, in which the initial contact, yielding, inertiaeffect as well as the response time of the ER fluid are allconsidered. By comparison with the experimentalresults, it is validated that this model can predict thedynamic response of the ER cylinder well. It is foundthat the performance of the ER cylinder is mainly deter-mined by the viscous effect, ER effect, and the impactvelocity, while the inertia effect and response time of theER fluid also have some influences. The controllabilityof the ER cylinder becomes weaker under higher impactvelocity, because of the viscous effect and the responsetime of the ER fluid.

APPENDIX: INFLUENCE OF THE FLUID

COMPRESSIBILITY

As shown in Figure A1, when the compressibility ofthe fluid is concerned, the governing equation of the ERfluid in the lower chamber of this ER cylinder can beexpressed by:

dV

dtþV

�

dP

dt¼ �Q ðA1Þ

where � is the bulk modulus of the ER fluid, V is thevolume of the ER fluid in the lower chamber of thecylinder, and Q is the flow rate through the duct.When the piston moves downwards, the ER fluid willflow through the duct into the upper chamber. Hence,the pressure drop through the ER duct can be approxi-mated by:

�P ¼12�LQ

bh3þ2L

h� �0, where �P ¼ P1 � P2: ðA2Þ

When the compressibility of the ER fluid in the lowerchamber is considered, there will be some vacancy inthe upper chamber, which means P2¼ 0, and �P¼P1

(i.e., P). Also, we have the damping force:

Fd ¼ P � Ap þ FR ðA3Þ

Then, the volume of the lower chamber is:

V ¼ ðL� xÞAp: ðA4Þ

Therefore, the motion equation of the piston is:

M €x ¼Mg� P � Ap � FR, ðA5-aÞ

P ¼12�L

bh3� _x�

ðL� xÞ

�� _P

� Ap þ

2L

h� �0, ðA5-bÞ

(b) The effect of response time

0

100

200

300

400

500

600

Response time (ms)

Mea

n fo

rce

(N)

V=2 m/s

(a) The mean crushing force under diferent electirc fieldand impact velocity

0

100

200

300

400

500

600

0

Yield stress (kPa)

Mea

n cr

ushi

ng fo

rce

(N)

V=1 m/s

V=2 m/s

V=3 m/s

2.521.510.5

6050403020100

Figure 17. Effects of the yield stress and the response time.

14 X. W. ZHANG ET AL.

It can be seen that the first term on the right-hand sideof Equation (A5-b) is from the viscous effect, the thirdterm is due to the yield stress or ER effect, while thesecond term is related to the compressibility of the ERfluid. By means of Equations (A5-a) and (A5-b), forgiven initial impact conditions, the dynamic responseof the ER cylinder can be obtained.To investigate the influences of the compressibility of

the ER fluid, three values of bulk modulus are exam-ined, that is, �¼ 0.3, 1.0, 5.0GPa. The mass of theweight is assumed as M¼ 5 kg, and the impact velocityis V0¼ 2m/s. The geometry of the cylinder is the same asthose in the Section 3, while the properties of the ERfluid are �¼ 0.15 Pa s, and �y¼ 8 kPa.Figure A2 plotted the comparison of the dynamic

responses with different bulk modulus �. The resultsshowed that the compressibility of the ER fluid onlyinfluences the very initial impact stage and nearly doesnot affect the mean crushing force when the displace-ment is relatively large. In some degree, the compressi-bility of ER fluid can explain the gradual increase of theimpact force in the initial stage. However, it can notexplain the initial peak force which was observed inthe experiments.

THE INERTIAL EFFECT OF ER FLUIDS

If the acceleration (or the inertia effect) of the ERfluid is considered, Equation (3) will be incorrect. Asshown in Figure A3, the motion equation of the fluidwithin the duct of length L can be expressed as:

ðP1 � P2Þ � bh� 2� � Lb ¼ � €XdLbh, ðA6Þ

where � is the shear stress at the two boundaries, xd isthe displacement of the ER fluid in the duct, and b is the

width of the duct in the direction normal to this page.Hence, xd can be obtained by:

Xd ¼Ap

bh� X: ðA7Þ

After simplifying Equation (A7), we have:

� ¼ �P�h

2L��h

2�

Ap

bh� €X: ðA8Þ

If the inertial effect of the fluid (i.e., the second termon the right-hand side of the equation) is ignored,Equation (A8) can be simplified to Equation (3) in thisarticle.

Recalling Equation (11) in the article, the pressuredrop can be approximated as:

P ¼2L

h�

6�

bh2� _X� Ap þ �0 þ

1

2� �h �

Ap

bh� €X

: ðA9Þ

By combining Equation (A5-a), we have:

Mþ � � LAp �Ap

bh

� �€X

¼Mg� Ap �12�L

bh3� _Xþ

2L

h�0

� �� Ap � FR: ðA10Þ

x

O

L

P2

P1

V0

Figure A1. Cross-section of a typical ER cylinder.

0

0.2

0.4

0.6

0.8

1

1.2

0

Displacement (mm)

Pre

ssur

e dr

op (

MP

a)

Beta=5.0GPa

Beta=1.0GPa

Beta=0.3GPa

21.81.61.41.210.80.60.40.2

Figure A2. Dynamic responses of the ER cylinder with different fluidbulk modulus.

h

L

VP2P1

Xd

Figure A3. Flowing of the ER fluid within the duct.

Electro-rheological Cylinders used as Impact Energy Absorbers 15

According to the above equation, the inertia effect ofER fluid can be evaluated by the mass ratio�LA2

p=ðM� bhÞ. If this ratio is small enough, the inertiaeffect can be ignored. By using q¼ 1.0� 103 kg/m3,L¼ 65mm, Ap¼ 1206mm2, M¼ 5 kg, this mass ratio isabout 0.04.To further investigate the influence of the ER fluid

inertial effect, an impact response is analyzed bymeans of Equation (A10). The geometric and impactconditions are the same with those for the investigationof compressibility, and the dynamic responses of the ERcylinder under q¼ 1.0� 103 and 4� 103 kg/m3, withV0¼ 2m/s, were analyzed. The force history andvelocity-displacement relations for different q areshown in Figure A4(a) and (b), in which q¼ 0 meansthe inertia effect is ignored. It is seen that larger thevelocity, the more serious the influence of the inertiaeffect will be. For q¼ 1.0� 103 kg/m3, the influence ofthe resistant force at the very beginning is about 4%,which agrees well with the estimation in the previousparagraph. However, when the density of the fluid isq¼ 4.0� 103 kg/m3, the influence of the inertia effectwill be significant.

ACKNOWLEDGEMENTS

This study is supported by the Hong Kong ResearchGrant Council (RGC) under CERG grant No.621505and the National Natural Science Foundation ofChina under Key Project No.10532020. Their financialsupports are gratefully acknowledged. The authors alsowould like to thank C.B. Zhang, Alex Fok, J.B. Wu andX.Q. Gong for their help.

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(a)

(b)

0

100

200

300

400

500

600

Time (ms)

For

ce (

N)

F-Rou=0

F-Rou=1000

F-Rou=4000

0

0.5

1

1.5

2

2.5

Displacement (mm)

Vel

ocity

(m

/s)

Rou=0

Rou=1000

Rou=4000

35302520151050

35302520151050

Figure A4. Investigation of the inertia effect of the ER fluid.

16 X. W. ZHANG ET AL.

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Electro-rheological Cylinders used as Impact Energy Absorbers 17