Analysis and design of a cylindrical magneto-rheological fluid brake

J. Huang*, J.Q. Zhang, Y. Yang, Y.Q. WeiDepartment of Engineering Mechanics, Chongqing University, 400044 Chongqing, China

Abstract

A magneto-rheological (MR) fluid brake is a device to transmit torque by the shear force of an MR fluid. An MR rotary brake has the

property that its braking torque changes quickly in response to an external magnetic field strength. In this paper, the design method of the

cylindrical MR fluid brake is investigated theoretically. The equation of the torque transmitted by the MR fluid within the brake is derived to

provide the theoretical foundation in the cylindrical design of the brake. Based on this equation, after mathematical manipulation, the

calculations of the volume, thickness and width of the annular MR fluid within the cylindrical MR fluids brake are yielded.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: MR fluids; Brake; Design method

1. Introduction

Magneto-rheological (MR) fluids consist of stable suspen-

sions of micro-sized, magnetizable particles dispersed in a

carrier medium such as silicon oil or water. When an external

magnetic field is applied, the polarization induced in sus-

pended particles results in the MR effect of the MR fluids. The

MR effect directly influences the mechanical properties of the

MR fluids. The suspended particles in the MR fluids become

magnetized and align themselves, like chains, with the direc-

tion of the magnetic field. The formulation of these particle

chains restricts the movement of the MR fluids, thereby

increasing the yield stress of the fluids. The change is rapid,

reversibleandcontrollablewith themagneticfieldstrength [1].

The mechanical properties of the MR fluids can be used in the

construction of magnetically controlled devices such as the

MR fluid rotary brake, or clutch [2–7]. To design the MR fluid

brake for a given specification, one must establish the relation-

ship between the torque developed by MR fluids and the

parameters of the structure and the magnetic field strength.

In this paper the fundamental design method of the

cylindrical MR brake is investigated theoretically. A Bing-

ham model is used to characterize the constitutive behavior of

the MR fluids subject to an external magnetic field strength.

The theoretical method is developed to analyze the torque

transmitted by the MR fluid within the brake. An engineering

expression for the torque is derived to provide the theoretical

foundation in the design of the brake. Based on this equation,

after algebraic manipulation, the volume and thickness of the

annular MR fluid within the brake is yielded.

2. Operational principle

An MR fluid brake is a device to transmit torque by the

shear stress of MR fluid. A MR rotary brake has the property

that its braking torque changes quickly in response to an

external magnetic field strength. The operational principle of

the cylindrical MR brake is shown in Fig. 1. The MR fluid

fills the working gap between the fixed outer cylinder and the

rotor. The rotor rotates at a rotational speed of o.

In the absence of an applied magnetic field, the suspended

particles of the MR fluid cannot restrict the relative motion

between the fixed outer cylinder and the rotor. However, in

the course of operation, a magnetic flux path is formed when

electric current is put through the solenoidal coil. As a result,

the particles are gathered to form chain-like structures, in the

direction of the magnetic flux path. These chain-like struc-

tures restrict the motion of the MR fluid, thereby increasing

the shear stress of the fluid. The brake can be achieved by

utilizing the shear force of the MR fluid. The braking torque

values can be adjusted continuously by changing the exter-

nal magnetic field strength [8].

3. Properties of MR fluids

MR fluids are suspensions of micron-sized, magnetizi-

able particles in a carrier fluid. They mainly consist of the

Journal of Materials Processing Technology 129 (2002) 559–562

* Corresponding author.

E-mail address: jhuangcq@yahoo.com.cn (J. Huang).

0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 6 3 4 - 9

following three components: magnetizable particles, a

carrier fluid, and some additives. The magnetizable parti-

cles in MR fluids induce polarization upon the application

of an external magnetic field, which results in the MR

effect of the MR fluids. The carrier fluid serves as a

dispersed medium and ensures the homogeneity particles

in the fluid. The additives include stabilizers and surfac-

tants. The stabilizers serve to keep the particles suspended

in the fluid, whilst the surfactants are adsorbed on the

surface of the magnetic particles to enhance the polariza-

tion induced in the suspended particles upon the applica-

tion of a magnetic field.

In the absence of an applied magnetic field, the particles in

MR fluid disperse randomly in the carrier fluid. MR fluid

flows freely through the working gap between the fixed outer

cylinder and the rotor. MR fluid exhibits a Newtonian-like

behavior, where the shear stress of MR fluids can be

described as

t ¼ Z _g (1)

in which t is the shear stress, Z the viscosity of the MR fluid

with no applied magnetic field, and _g the shear rate.

When the magnetic field is applied, the behavior of the

controllable fluid is often represented as a Bingham fluid

having a variable yield strength. In this model, the consti-

tutive equation is derived by the least-squares method [1]:

t ¼ tB þ Z _g (2)

where tB is the yield stress developed in response to the

applied magnetic field. Its value is dependent upon the

magnetic induction field B.

Fig. 2 shows the relationship obtained from experiment

between shear rate and shear stress, depending upon the

applied magnetic field strength. As can be seen, the MR

fluids have a variable yield strength, the shear stress increas-

ing with the applied magnetic field strength. The shear rate

has little influence on the shear stress. This result indicates

that the MR fluid exhibits Bingham behavior.

4. Analysis of torque

The key question in the design of MR fluid brake is to

establish the relationship between the torque and the para-

meters of the structure and the magnetic field strength.

Fig. 3 shows the flow of the MR fluid in the MR brake.

When the magnetic field is applied, the braking torque T

developed by the MR fluid can be calculated by

T ¼ 2pr2wt (3)

where w is the effective width of the MR effect developed by

the MR fluid, and r the radius of the annular MR fluid. The

shear stress t is proportional to the shear rate _g as described

by Eq. (2). The shear rate _g can be calculated by [9]

_g ¼ rdor

dr(4)

where or is the rotational speed in the MR fluid at radius r.

The differential of the rotational speed or can be obtained by

Eqs. (2)–(4) as follows:

dor ¼1

ZT

2pwr3� tB

r

� �dr (5)

Integrating Eq. (5) and applying the boundary conditions of

the MR fluid brake: r ¼ r1 at or ¼ o, and r ¼ r2 at or ¼ 0;

Fig. 1. The operational principle of the MR brake.

Fig. 2. Shear stress versus shear rate.

Fig. 3. The analysis of the torque.

560 J. Huang et al. / Journal of Materials Processing Technology 129 (2002) 559–562

the braking torque T developed by the MR fluid can be

calculated, to yield

T ¼ 4pwtBr21r2

2 lnðr2=r1Þr2

2 � r21

þ 4pZwr21r2

2or2

2 � r21

(6)

where r1 and r2 are the radius of the rotor and the outer

cylinder, respectively, and o is the rotational speed of the

rotor.

As shown in Fig. 3, the thickness h of the annular MR fluid

between the rotor and outer cylinder can be given by

h ¼ r2 � r1 (7)

If it is assumed that the thickness is much smaller than the

radius of the rotor (h=r1 ! 1), Eqs. (6) and (7) can be

manipulated mathematically to yield:

T ¼ 2pwtBr21 þ

2pZwr31o

h(8)

Eq. (8) shows that the braking torque developed in the

cylindrical MR fluid brake can be divided into a mag-

netic-field-dependent induced yield stress component TB

and a viscous component TZ:

TB ¼ 2pwtBr21 (9)

TZ ¼2pZwr3

1oh

(10)

The total torque T is the sum of TB and TZ, i.e.

T ¼ TB þ TZ (11)

5. Thickness and width of the MR fluid

The active volume of annular MR fluid in the cylindrical

MR brake can be obtained through the integration of the

radius of the annular MR fluid as follows:

V ¼ 2pw

Z r2

r1

r dr (12)

Eq. (12) can be manipulated mathematically to yield

V ¼ 2pr1wh (13)

Eqs. (9)–(13) can be further manipulated to yield

V ¼ Zt2

B

� �TB

TZ

� �ðTBoÞ (14)

Eq. (14) gives the minimum active MR fluid volume that is

necessary within the brake in order to achieve the desired

control torque ratio (TB/TZ) at a given rotational speed o, and

the specified controllable torque TB.

Eqs. (9) and (10) can be manipulated algebraically to

derive the thickness of the annular MR fluid as follows:

h ¼ ZtB

� �TB

TZ

� �r1o (15)

Eq. (15) provides geometric constraints for the MR fluid

brake based on the MR fluid material properties (Z/tB), the

desired control torque ratio (TB/TZ) at a given rotational

speed o, and the radius r1 of the rotor.

The effective width of the MR fluid w can be obtained

from Eqs. (13) and (14).

6. Design example of cylindrical MR brake

6.1. Original data

In most brake applications, the maximum mechanical

power level Pm and the maximum rotational speed of the

rotor om will be specified. For this example, the following

parameters are given: Pm ¼ 500 W andom ¼ 100 s�1.

6.2. Desired control torque ratio

Based on the anticipated performance of the brake, and

the properties of the MR fluid material when the magnetic

circuit is capable of magnetically saturating the MR fluid,

the desired control torque ratio (TB/TZ) may be chosen. For

this example, the desired control torque ratio is chosen to be

TB

TZ¼ 15

6.3. MR fluid material

In Eqs. (9) and (10), the parameters (Z=t2B) and (Z=tB) have

a bearing on the MR fluid material properties. In order to

reduce the dimensions of the brake, a designer should select

the highest yield stress developed in response to an applied

magnetic field tB under the MR fluid saturation magnetization

and the lowest viscosity of the MR fluid Z possible. With this

knowledge, one can turn to an MR fluid specification sheet

and choose appropriate fluid parameters. For this example,

assume that the MR fluid can be magnetically saturated, the

values of tB and Z are tB ¼ 56 kPa and Z ¼ 0:33 Pa s.

6.4. MR fluid’s volume

The maximum mechanical power level of the brake can be

calculated by

Pm ¼ ðTB þ TZÞom ¼ TB 1 þ TZ

TB

� �om (16)

Now, the calculation of the MR fluid volume V in the MR

fluid brake can be calculated from Eqs. (13) and (16) as

V ¼ 0:74 � 10�6 m3.

6.5. Rotor’s radius

The minimum dimension of the radius of the rotor r1 is

constrained to the inner structure dimensions of the brake.

For this example, r1 is chosen as r1 ¼ 50 mm.

J. Huang et al. / Journal of Materials Processing Technology 129 (2002) 559–562 561

6.6. MR fluid’s thickness

The thickness h of annular MR fluid can be calculated

from Eq. (14) as h ¼ 0:442 mm.

6.7. MR fluid’s effective width

The effective width w of the MR effect developed by the

MR fluid can be calculated from Eq. (12) as w ¼ 5:33 mm.

7. Conclusions

The geometric design method of a cylindrical MR fluid

brake is investigated theoretically in this paper. The braking

torque developed by the MR fluid within the brake under

differentmagneticfield strengthconditionshas been analyzed.

The engineering design calculations of the volume, thickness

and width of the annular MR fluid within the brake are derived.

The parameters of the thickness and width of the fluid in the

brake can be calculated from the equations obtained, when the

required mechanical power level, the rotational speed of the

rotor, and the desired control torque ratio are specified.

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