# Efficiency Estimation of Roller Chain Power Transmission System
## 3 Results  
The chain system consists of the driving sprocket, the driven sprocket, and the roller chain. The driving sprocket and the driven sprocket are selected with teeth number 32T-32T, which is the maximum ratio in a sprocket, and avoids the chain from falling off of the sprocket due to the lateral offset angle. The type of roller chain is specified in ISO 606 [21]. The friction coefficient for lubricated steel-steel interfaces is assumed to be 0.11 [22]. Driving torque is applied to the driving sprocket, and braking torque is exerted on the driven sprocket. Efficiency was calculated with different parameter values of rotational speed, torque, lateral offset angle, and damping coefficient. The range of parameters is as follows: A rotational speed 40-80 RPM with an interval of 10 RPM, torque 5-9 nm with an interval of 1 nm, a lateral offset angle of 0/0.0174/0.0349/0.0523 rad (0/1/2/3 deg), and a damping coefficient of 40/60/80 ns/m. The list of drive train components and their parameters is presented in Table 1.  
Figure 6 shows the following: Transmission efficiency as a function of rotational speed and offset angle; the damping coefficient, in which the range of efficiency is 86.3-93.1%; and demonstrates that efficiency decreases with increasing rotational speed and damping coefficient for a given lateral offset angle. In addition, for a given rotational speed, chain efficiency decreases with increasing damping coefficient and lateral offset angle.  
\begin{table}
\begin{tabular}{l l c c} \hline \hline
**Parts** & **Parameters** & **Symbol [Unit]** & **Value** \\ \hline \hline \multirow{8}{*}{Roller chain} & Friction coefficient between the pin and the & \multirow{2}{*}{\(\mu_{1}/\mu_{2}\) [-]} & 0.11/0.11 \\  & bush/among the pin, the bush, and the roller & & \\  & Radius of inner bush & \multirow{2}{*}{\(\tau_{\text{bi}}\) [m]} & \multirow{2}{*}{\(1.74\times 10^{-3}\)} \\  & Maximum rotation angle of chain link & & \\  & Average mass of one chain link & & \\  & Chain pitch & & \\  & Radius of inner roller & \multirow{2}{*}{\(\tau_{\text{bo}}\) [m]} & \multirow{2}{*}{\(2.83\times 10^{-3}\)} \\  & Damping coefficient & & \\  & Chain angle related to x-axle of link in & & \\  & driving/driven sprocket & \multirow{2}{*}{\(\theta_{\text{in,1/2}}\) [rad]} & \multirow{2}{*}{\(0.00012\)/0.00012} \\  & Tension in the chain span & & \\ \hline \hline \multirow{8}{*}{Sprocket} & Radius of driving/driven sprockets & \multirow{2}{*}{\(\text{R}_{s,1/2}\) [m]} & 62.43/62.43 \(\times 10^{-3}\) \\  & Tenth number of driving-driven sprocket & \multirow{2}{*}{\(\text{N}_{s,1-2}\) [-]} & 32–32 \\ \cline{1-1}  & Rotational speed of driving/driven sprocket & \multirow{2}{*}{\(\text{w}_{s,1/2}\) [rad/s]} & 4.18/5.23/6.28/ \\ \cline{1-1}  & Coordinate for position of driving/driven & & \\ \cline{1-1}  & sprocket & \multirow{2}{*}{\(\sigma_{1/2}\) [rad]} & \multirow{2}{*}{\(3.09\)/1.52} \\ \cline{1-1}  & Lateral offset angle between sprockets & \multirow{2}{*}{\(\gamma\) [rad]} & \multirow{2}{*}{\(0\)/0.0174/0.0349/0.0523} \\ \cline{1-1}  & Power of driven sprocket & \multirow{2}{*}{\(\text{P}_{s,2}\) [W]} & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Drive train components and parameters for roller chain system.  
Figure 7 presents the effect of torque, lateral offset angle, and damping coefficient to transmission efficiency, demonstrating that efficiency increases with torque, and the range of efficiency is 90.8-93.0%. Similar to the results from Figure 6, chain efficiency decreases with increasing damping coefficient and offset angle for a given torque.  
Figure 6: Transmission efficiency as a function of rotational speed, offset angle, and damping coefficient.  
Figure 7: Transmission efficiency as function of torque, offset angle, and damping coefficient.