# Investigation of Segmented Halbach Array Design for Passive Magnetorheological Brake System
## 3 Segmented Halbach Array Design
### Simulation of a Segmented Halbach Configuration Validated in a Magnetorheological Brake
\[{\rm F}_{\rm shear}=\sigma\times{\rm A} \tag{2}\]  
Shear stress (\(\sigma\)) is exerted by the magnetorheological fluid due to the applied magnetic field strength:  
\[\sigma=k\;{\rm B}^{2} \tag{3}\]  
where \(k\) is the magnetorheological fluid's material constant and B is the magnetic flux density in MRF. \[{\rm F}_{\rm shear,\;conv}=(k\times{\rm B}^{2})\times{\rm A} \tag{4}\]  
\[{\rm F}_{\rm shear,\;proposed}\;=\;\left(k\;\times\;{\rm B}^{2}_{\rm avg}\right) \;\times\;2\pi{\rm r}{\rm h} \tag{5}\]  
In the conventional cylindrical PM Halbach array, the shear force is given by F\({}_{\rm shear,\;conv}\), where \(k\) is a constant related to the magnetorhological fluid's properties, B represents the magnetic flux density, and A is the cross-sectional area of the magnetic field interacting with the fluid. For the proposed design of NdFeB permanent magnets in a segmented Halbach array, the shear force is described by F\({}_{\rm shear,\;proposed}\), the magnetic flux density is indicated as B\({}_{\rm avg}\), and the area of contact where the magnetic field affects the MRF is denoted as A = 2\(\pi\)rh. Here, r is the radius of the cylinder, and h is the height of the cylindrical section. The equation for torque is generated by shear force, where \(R\) is the distance from the axis of rotation. \[{\rm T}={\rm F}_{\rm shear}\times R \tag{6}\]  
Braking torque (T) is generated by the shear force in a magnetorheological brake utilizing permanent magnets. \[{\rm T}_{\rm conv}=k\times{\rm B}^{2}\times{\rm A}\times R \tag{7}\]  
\({\rm T}_{\rm conv}\) is the torque generated in a conventional cylindrical Halbach array, where \(k\) is a constant dependent on the magnetorheological fluid properties, B is the magnetic flux density, A is the area through which the magnetic field acts, and \(R\) is the effective radius at which the force is applied. In the proposed segmented cylindrical Halbach array, \(N\) segments of NdFeB permanent magnets are arranged around the circumference. The magnetization vector in each segment follows a specific Halbach sequence to amplify the magnetic field on one side. The expression for the magnetization 'M\({}_{\rm n}\)' of the n-th segment:  
\[{\rm M}_{\rm n}\;={\rm M}_{\rm s}\cos(k\theta_{\rm n})\hat{\rm r}+{\rm M}_{\rm s }\sin(k\theta_{\rm n})\hat{\rm\phi} \tag{8}\]  
where  
\({\rm M}_{\rm s}\): saturation magnetization of the material. \(k\): wave number of the Halbach array  
\(\theta_{\rm n}\): angular position of the n-th segment. The governing equation for magnetic field B is  
\[\
abla\text{B}=0,\,\
abla\,\times\text{B}=\mu_{0}\,\,\
abla\,\times\text{M} \tag{9}\]  
Here, M is the magnetization vector of the Halbach array. Solving maxwell's equations in cylindrical coordinates (r,\(\phi\),z), the magnetic field generated by the Halbach array can be derived. For a Halbach array, the field component solution in the working region inside the Halbach cylinder yields that the magnetic flux density B at the working surface is primarily radial (Br) and azimuthal (B\(\phi\)). \[\text{B}_{\text{r}}\text{(r,}\phi)=\text{B}_{0}\text{cos}(\text{k}\phi) \tag{10}\]  
\[\text{B}_{\phi}\text{(r,}\phi)=\text{B}_{0}\text{sin}(\text{k}\phi) \tag{11}\]  
where B\({}_{0}\) is proportional to remanence B\({}_{\text{r}}\) of the NdFeB magnets and the geometry of the Halbach array. The MR fluid's apparent yield stress \(\tau_{y}\) is proportional to the square of the magnetic flux density:  
\[\sigma=k\;{\rm B}^{2} \tag{3}\]0  
where \(\alpha\) is a material-dependent constant determined by the fluid properties. Inserting the expression for B into this relation gives  
\[\sigma=k\;{\rm B}^{2} \tag{3}\]1  
The total shear stress \(\tau\) in the MR fluid is related to the yield stress and the brake geometry:  
\[\sigma=k\;{\rm B}^{2} \tag{3}\]2  
where h is the thickness of the MR fluid gap and r is the radial position within the gap. A position adjustment of the segmented magnets enables dynamic control of the flux density distribution, offering flexibility.