# Development of Power-Assist Device for a Manual Wheelchair Using Cycloidal Reducer
## 2 Materials and Methods
### Design of Cycloidal Reducer  
In the conventional cycloidal reducer, all the teeth are in contact with the pin rollers, however, for the proposed reducer there is only a single-contact condition between the housing gear and cycloidal disks, hence the different dynamic characteristics when compared with the existing cycloidal reducer. Therefore, it is important to analyze the dynamic characteristics of the proposed reducer. In addition, a dynamic simulation was performed using an MSC ADAMS 12.0 (MSC Software, Newport Beach, CA, USA) multi-body dynamics analysis program for even and odd reduction ratios. The even and odd reduction ratios used in this study were 50:1 and 49:1, respectively. In general, when performing a dynamic analysis, a 3D CAD program such as SolidWorks converts the model to parasolid format, and the converted model is loaded into the ADAMS. However, in the case of a very diversified curve, such as a cycloidal gear, the gear surface of the parasolid-converted model cannot be accurately represented for simulation, making accurate contact force analysis difficult. For a more accurate dynamic simulation, using the cycloidal gear design program developed in the laboratory, the teeth of the cycloidal disk and housing gear were fabricated at 120 points in a tooth and the gear curve was expressed as a polyline. Therefore, the greater the number of discrete points on the tooth profiles of the cycloidal gears, the more accurate are the contact analysis results. The contact stiffness between the cycloid disk and housing gear can be calculated using [21]:  
\[K_{hc}=\frac{4H}{3\pi(h_{c}+h_{p})}\left(\frac{r_{c}r_{h}}{r_{c}+r_{h}}\right) ^{1/2} \tag{14}\]  
\[h_{c}=\frac{1-v_{c}^{2}}{\pi E_{c}}\,\ h_{h}=\frac{1-v_{h}^{2}}{\pi E_{h}} \tag{15}\]  
Figure 4: Cycloidal tooth profiles according to tooth thickness ratio. where \(K_{hc}\) represents the contact stiffness between the cycloidal gear and housing gear. \(r_{c}\) is the radius of curvature of the cycloidal tooth profile, \(E_{c}\) and \(v_{c}\) denote Young's modulus and Poisson's ratio of the material used in the cycloidal disk, respectively, and \(E_{h}\) and \(v_{h}\) are the corresponding properties for the material used to make the housing gear, respectively. Here, \(H\) is an adjusting coefficient used to control the magnitude of the contact stiffness. The description of parameters with values is shown in Table 1.  
When the input shaft rotates once at a low speed (10 deg/s), a vibration is generated and transmitted to the ground owing to the contact force between the cycloidal disk and housing gear, and the results differ depending on the reduction ratio (Figure 5). At an even reduction ratio (50:1), the two disks are aligned exactly 180\({}^{\circ}\) to each other; however, the contact points of each disks have a half-tooth phase difference based on the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & & **Value** \\ \hline Contact between the housing gear and cycloidal disk & Stiffness (N/mm) & 7.0 \(\times\) 10\({}^{6}\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} Contact between the cycloidal disk and output roller \\ \end{tabular} } & \multirow{3}{*}{\
\begin{tabular}{c} Splitting (N\(\cdot\)/mm) \\ Force Exponent \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} \(7.0\times 10^{6}\) \\ 2.2 \\ \end{tabular} } \\  & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used in the dynamic simulations. Figure 5: Simulation results of cycloidal reducer with 49:1 (**top**), 50:1 (**below**). housing gear owing to the odd teeth of the housing gear. Therefore, the resultant force in the x- and y-axis directions of the two contact forces cannot be canceled, leaving a residual force and causing vibration. Such vibrations may cause discomfort and fatigue when a user travels (Figure 6).