# A Tiny Haptic Knob Based on Magnetorheological Fluids
## 2 Design of Proposed Haptic Knob
### Operating Principle of Proposed Haptic Knob and Its Magnetic Flux Path Simulation  
Figure 5a shows the design of the proposed haptic knob. We filled the gap between the housing and rotary shaft with MR fluid. Before a magnetic field was applied to the MR fluid in the gap, the MR fluid was in a fluid state (initial state). Because the MR fluid serves as a lubricant in the initial state, the rotary shaft rotates smoothly (Figure 5b). If we apply a magnetic field to the MR fluid through the solenoid coil, magnetic particles in the MR fluid form chain structures in the direction of the applied magnetic field (Figure 5c). The most important aspect in designing haptic knobs based on MR fluids is that multiple operating modes of MR fluids contribute to the actuation. We designed a haptic knob whose structure supports the two operating modes of MR fluids simultaneously. The magnetic chain disturbs the rotation of the rotary shaft, creating a resistive torque when the user rotates the dial plate (shear mode) (Figure 5d). As previously mentioned, there are bumps on the surface of the rotary shaft. The rotation of the bumps accelerates the flow of MR fluids. This accelerated MR fluid flow produces yield stress (flow mode) (Figure 5e) in the gap. Because the total resistive torque is determined by summing up all the torque feedback obtained from the two modes, we can maximize the resistive torque in a small-sized haptic knob. A finite element method simulation using commercial software (Ansys Maxwell) was conducted to investigate the magnetic flux path (Figure 5f). The result shows  
Figure 4: Magnetomotive force according to the wire diameter of the solenoid coil and limited power consumption. that there is little magnetic flux leakage in the haptic knob; hence, almost all the magnetic flux can contribute toward the formation of a magnetic chain. As we mentioned before, MR fluid is filled in the gap between the rotary shaft and the housing in the proposed haptic knob. Therefore, the resistive torque occurs in four regions (I, II, III, and IV), as shown in Figure 6a. In regions I and II (Figure 6b,c), only shear mode contributes to the actuation, and in regions III and IV (Figure 6d,e), two modes (shear and flow modes) contribute to the actuation at the same time. The resistive torque (\(\tau_{shear}\)) generated by the shear mode is calculated by Equations (2) and (3) [17; 18; 19; 20]. \[\tau_{shear}=\frac{\mu bL}{d}v+bL\sigma \tag{2}\]  
where \(\mu\) is the viscosity (0.28) of MR fluid, \(b\) is the circumference of the shaft, \(L\) is the length of the cross-sectional area of the shaft through which magnetic flux passes (see Figure 6b-e), \(v\) is the angular velocity of the rotary shaft, \(d\) is the gap thickness between the shaft and the housing (or between the shaft and the cover), and \(\sigma\) is the yield stress generated by the MR fluid. This yield stress (\(\sigma\)) is calculated by Equation (3)  
\[\sigma=C271,700\cdot\Phi^{1.5239}\cdot\tanh\left(6.33\times 10^{-6}\cdot H\right) \tag{3}\]  
where \(C\) is the carrier fluid constant (\(C=1\) in the case of hydrocarbon oil), \(\Phi\) is the particle volume percent (\(\Phi=40\)) in MR fluid, and \(H\) is the magnetic field intensity (\(H=\) At/m, ampere-turns per meter). Moreover, the resistive torque (\(\tau_{flow}\)) generated by the flow mode is expressed by Equation (4) [17; 18; 19; 20]. \[\tau_{flow}=\frac{12A^{2}\mu L}{bd^{3}}v+\frac{2 AL\sigma}{d} \tag{4}\]  
where \(A\) is the average cross-sectional area of the cone-shaped shaft. The total resistive torque (\(\tau_{total}\)) becomes the sum of the torques created in the four regions (I, II, III, and IV) (Equation (5)). Figure 5: Operating principle of the proposed haptic knob and its finite element method simulation: (**a**) Design of the proposed haptic knob; (**b**) magnetorheological (MR) fluids before magnetic field; (**c**) MR fluids after magnetic field; (**d**) shear mode of MR fluids; (**e**) flow mode of MR fluids; (**f**) finite element method simulation of the proposed haptic knob.