# Efficiency Estimation of Roller Chain Power Transmission System
## 1 Introduction  
Chains are widely used in bicycles, motorcycles, personal mobility devices, and various industrial equipment since they provide reliable, efficient, and durable power transmission with relatively low economic cost. Figure 1 presents a schematic of a roller chain, the most common and popular type of chain, in which the inner link is connected by a pin on the outer link, and the pin is covered by a bush and roller that contacts with the mating sprocket. The transmission efficiency of the chain drive system, which is the ratio of the input power of the driving sprocket to the output power of the driven sprocket, depends on the configuration of the chain and the operation conditions such as torque and speed. In the process of power transmission, energy loss occurs due to sliding friction between the pin, the roller, and the bush during the rotation of articulation, vibration in the chain spans, impact when the roller engages and disengages with the sprocket, and contacting force between the inner and the outer side plates of the derailleur system for gear shifting. Even though the roller chain drive system has been utilized for many years, little is known concerning how to estimate efficiency, which is affected by various factors, such as transmission speed,  
Figure 1: Schematic of roller chain and its components. load, lateral offset between driving and driven sprockets, deflection of the tight side of the chain, etc. In 1999, [PERSON] et al. [1], performed an experimental examination in which tension was statically loaded for a bicycle chain with a sprocket under conditions of in-plane loading and out-plane loading. However, since only steady-state analysis for tension was carried out, the mechanical interaction of the roller chain and its components was not addressed. [PERSON] and [PERSON]. [2] in 1995 investigated the tension of link and impact force on roller sprockets in roller chains under various operation conditions and summarized the results by regression analysis. The impact force increased as the rotational speed and tension of link increased. However, the results did not provide thorough theoretical basis. In 2001, [3] [PERSON] found that chain transmission is primarily affected by frictional energy loss between 97 and 99%, and compared and validated this result under various rotational speed, power, and lubrication conditions. However, regarding the aspect of chain drive to transmission efficiency with lateral offset, that study only theoretically modeled frictional loss, and experimental verification was not performed. Furthermore, [1,3,4] designed and analyzed the effects of lateral offset for a roller chain system, but inner contact of the chain link with lateral offset owing to the derailleur type system was not reported. [PERSON] [5] and [PERSON] [6] in 1986 modeled and analyzed the link tension and theoretical efficiency of a chain and formulated the contact force of the pin and bush articulation. However, it is only used in a cranked chain, and does not apply to a roller chain. Although the roller chain system is lubricated, friction occurs at the contacts between the sprockets and the rollers. One advantage of presence of friction force is that it damps out high peak forces during the contact [7,8]. Additionally, Coulomb friction modeling is simple because it only requires the kinetic coefficient of friction. [PERSON] [9] in 1998 calculated chain power transmission efficiency based on the energy loss between the chain links and sprockets and investigated the effects of size of sprockets. It was found that average chain transmission efficiency was 98.8%. However, this model is limited to a race cycling bike chain links that only include pin and sleeve without roller. In 2001, [PERSON] [10] and [PERSON] [11] assessed the efficiency of roller chains by modeling frictional losses due to pin sliding against the bush and bush sliding against the roller, in which sliding friction was based on Coulomb's law of friction.