### 3.1 MR Brake Structure

A MR brake is a type of controlled brake that creates a torque by changing the viscosity of the MR fluid inside the brake. The apparent viscosity of the MR fluid can be changed very quickly (in milliseconds) by applying a magnetic field [22]. The braking torque of the rotary MR brake is generated by the shear stress of the MR fluid placed in the gap between the two rotating surfaces [23]. The MR fluid can be modeled using nonlinear models such as the Bingham plastic model [24]:

$$\left\{ {\begin{array}{*{20}c} {\tau = \tau_{y} (H){\text{sgn}} (\dot{\gamma }) + \mu \dot{\gamma },} & {\tau > \tau_{y} ,} \\ {\dot{\gamma } = 0,} & {\tau \le \tau_{y} ,} \\ \end{array} } \right.$$

(7)

where *τ* is the shear stress, *H* is the magnetic flux intensity, \(\dot{\gamma }\) is the shear rate, *τ*_{y} is the yield stress, which changes with the magnetic flux intensity, and *μ* is the plastic viscosity of the MR fluid.

Different types of MR fluids exhibit particular characteristics according to their rheological properties. The MR fluid used in this study is MRF-140CG, which was acquired from the Lord Corporation.

The MR brake structure determines the direction of the magnetic field lines. To achieve torques with smaller magnetomotive forces in the brake, the MR brake should have a large effective area and a short magnetic circuit in the rotor and shell. There are several types of MR brakes with different rotor shapes such as the disc-type [25], multiple disc-type [26], and drum-type [27] MR brakes.

To achieve a larger torque in a rotary MR brake without increasing the size of the brake, a larger surface in the MR brake rotor should be activated by the magnetic flux. This objective is achieved in this study by designing the rotor with a T-shape and strategically placing excitation coils to provide a larger effective area on the surface of the rotor. Figure 4 shows a diagram of the MR brake components. The designed rotary MR brake consists of four main part, namely, the MR fluid, the MR fluid sealing, the rotor, and the stator. The magnetic field lines pass through the stator, MR fluid, and rotor and form a closed loop. The rotor of the brake is T-shaped so that the brake has multiple effective areas as shown in Figure 5, where areas (a) to (d) are respectively called effective areas 1 to 4. The rotor serves as the steel core for the electromagnet and the transmission element for the torque. Excitation coils are placed in the end caps that are installed on both ends of the shell, and the end caps and shell are connected to form the stator.

The two excitation coils generate magnetic fields in opposite directions to form a magnetic circuit with two parallel magnetic paths. The magnetic path is kept short because a short magnetic path can effectively reduce the volume and weight of the brake.

A sealing approach using MR fluid sealing is applied on both sides of the brake shaft. The MR fluid itself functions as the sealing element. Compared to a conventional dynamic seal, MR fluid sealing has a long lifetime and a good sealing effect. Two O-rings are employed between the end caps and the shell, and the gap between the stator and rotor is filled by MR fluid.

### 3.2 Torque Calculation and Design Optimization of MR Brake

DT4E-type magnetic pure iron was chosen as the magnetic core material to obtain reasonable values of the MR brake structural parameters. The yield stress of the MR fluid and the magnetic induction intensity of the magnetic core material should be determined based on the maximum brake torque. The required magnetic field intensity and the magnetic induction intensity of the MR fluid can then be obtained from the manufacturer technical data sheet, while the magnetic field strength of the magnetic core material can be obtained from a material handbook [28]. The brake parameters for each part are shown in Figure 5.

The magnetic flux around the magnetic circuit is constant and has the value of \(\Phi = B \cdot S\). Based on Ohm’s law for magnetic circuits, two principles should be followed in the magnetic circuit calculations: (1) parts that have the same magnetic conductivity should have equal magnetic flux areas, and (2) the ratio of the magnetic flux areas of parts that have different magnetic conductivities should be equal to the reciprocal of the ratio of their magnetic induction intensities in the initial design. According to these two magnetic circuit calculation principles, the relationships between the structural parameters of the MR brake can be expressed as

$$\left\{ \begin{gathered} \frac{{S_{{1}} }}{{S_{{5}} }} = \frac{{B_{t} }}{{\sigma B_{y} }}, \hfill \\ S_{1} = S_{2} + S_{3} + S_{4} , \hfill \\ S_{5} = S_{6} , \hfill \\ \end{gathered} \right.$$

(8)

where *B*_{t} is the magnetic induction intensity of the magnetic pure iron, *B*_{y} is the magnetic induction intensity of the MR fluid in the effective area, and

$$S_{1} = 2\uppi R_{1} l_{1} ,$$

$$S_{2} = {\uppi }R_{1}^{2} - {\uppi }(R_{1} - l_{2} )^{2} ,$$

$$S_{3} = 2{\uppi }(R_{1} - l_{2} - {\text{g}})(l_{1} - l_{3} - g),$$

$$S_{4} = {\uppi }(R_{1} - l_{2} - g)^{2} - {\uppi }(R_{1} - l_{2} - g - l_{4} )^{2} ,$$

$$S_{5} = {\uppi }(R_{1} + l_{9} + g + l_{5} )^{2} - {\uppi }(R_{1} + g + l_{9} )^{2} ,$$

$$S_{6} = 2{\uppi }(R_{2} { + }l_{7} )l_{6} ,$$

where *σ* is the magnetic flux leakage coefficient. The inequality

where \(S_{7} = {\uppi }(R_{2} + l_{7} )^{2} - {\uppi }R_{2}^{2}\) should also be simultaneously satisfied.

The design configuration of the MR brake is optimized to maximize the braking torque and minimize the weight. Because the operational mode of the MR fluid in the brake corresponds a shear model, the braking torque of the MR brake can be calculated using the Bingham plastic model of a MR fluid and the configuration parameters of the MR brake. The plastic viscosity of the MR fluid is ignored because the speed of revolution of the MR brake is low. The braking torques occur in effective areas 1, 2, 3, and 4, and can be calculated using the following formulae:

$$\left\{ \begin{gathered} T_{1} = \tau_{1} S_{{1}} (R_{1} + 0.5g), \hfill \\ T_{{2}} { = }\int_{{r_{1} }}^{{r_{2} }} {\tau_{2} \cdot 2{\uppi }r^{2} \cdot {\text{d}}r,} \hfill \\ T_{3} = \tau_{3} S_{3} (R_{1} - l_{2} - 0.5g), \hfill \\ T_{{4}} { = }\int_{{r_{3} }}^{{r_{4} }} {\tau_{4} \cdot 2{\uppi }r^{2} \cdot {\text{d}}r,} \hfill \\ \end{gathered} \right.$$

(10)

where \(\tau_{1}\), \(\tau_{2}\), \(\tau_{3}\), and \(\tau_{4}\) are the yield stresses of the MR fluid, and

$$r_{1} = R_{1} - l_{2} ,$$

$$r_{3} = R_{1} - l_{2} - g - l_{4} ,$$

$$r_{4} = R_{1} - l_{2} - g.$$

The braking torque of the brake can be obtained by combining the torques from the four effective areas:

$$T = 2 \cdot (T_{1} + T_{2} + T_{3} + T_{4} ).$$

(11)

The volume of the brake can then be calculated using the radius of the rotor (*R*_{1}) and Eqs. (8), (10), and (11). A given constraint torque can be satisfied by multiple MR structures with different volumes (*V*) and different values of *R*_{1}; therefore, there may exist a value of *R*_{1} at which the volume is minimized. The relationship between *V* and *R*_{1} can be represented as

$$V = {\uppi }(R_{1} + g + l_{9} + l_{5} )^{2} \cdot (l_{1} + l_{6} + l_{10} ) \cdot 2,$$

(12)

where the parameters are shown in Figure 5. \(l_{1}\), \(l_{5}\), and \(l_{6}\) can be calculated using Eqs. (8), (10), and (11) after the other parameter values are predefined to the values shown in Table 1.

Under the constraint \(R_{{1}} \in [30,60]\), the value of *V* was calculated with MATLAB. The relationship between *V* and *R*_{1} is shown in Figure 6.

Figure 6 shows that as *R*_{1} increases, the volume first decreases and then increases. There is therefore a value of *R*_{1} at which the volume is minimized. This particular value was calculated using MATLAB. It was confirmed that inequality (9) is satisfied at this value of *R*_{1}. The parameters for the MR brake are hence obtained.

Using the structural parameters of the brake, the magnetomotive force (*F*) can be calculated as

$$F = IN = \sum\limits_{i = 1}^{6} {H_{i} L_{i} } ,$$

(13)

where *N* is the number of coil turns, *I* is the current in the coil, and *L* is the length of the magnetic path.

The calculated value of *F* is *F* = 240 AN. Considering the installation space of the coil in the MR brake, *N* was set to 60 N, and *I* to 4.0 A.