applied sciences Article Article EfficiencyEfficiency EstimationEstimation ofof RollerRoller ChainChain PowerPower TransmissionTransmission SystemSystem

Sheng-Peng Zhang and Tae-Oh Tak * Sheng-Peng Zhang and Tae-Oh Tak * DepartmentDepartment of of Mechanical Mechanical and and Biomedical Biomedical Engineer Engineering,ing, Kangwon Kangwon National Un University,iversity, Chun Cheon 24341, Korea;Chun [email protected] 24341, Korea; [email protected] ** Correspondence: [email protected] Received: 30 September 2020; Accepted: 29 October 2020; Published: date


Received: 30 September 2020; Accepted: 29 October 2020; Published: 31 October 2020


Abstract:Abstract: InIn the the present present study, study, a a novel novel approach approach to to estimating estimating the the efficiency efficiency of of roller roller chain chain power power transmissiontransmission systems systems is is proposed proposed based based on on sliding sliding friction friction losses losses and and damping damping force. force. The The dynamics dynamics modelmodel is is taken taken into into account account between between chain chain links links wi withth lateral lateral offset offset owing owing to to the the derailleur derailleur system. system. FrictionalFrictional losses losses were were calculated calculated according according to to Coulomb’s Coulomb’s law law of of friction, friction, and and the the damping damping force force was was dependentdependent on on the the damping damping coefficient. coefficient. The The effects effects of of rotational rotational speed, speed, load, load, derailleur derailleur system, system, and and dampingdamping coefficient coefficient on on transmission transmission efficiency efficiency were were analyzed. analyzed. The The test test stan standd of of the the roller roller chain chain powerpower transmission transmission system system was was set set up up to to verify verify the the estimated estimated efficiency, efficiency, and and the the results results showed showed a a goodgood correlation, correlation, demonstrating demonstrating the the validity validity of of th thee chain chain power power transmission efficiency efficiency estimation.

Keywords:Keywords: rollerroller chain; chain; chain chain efficiency; efficiency; sliding sliding friction; friction; lateral lateral offset; offset; damping force

1.1. Introduction Introduction ChainsChains areare widelywidely used used in bicycles,in bicycles, motorcycles, motorcycles, personal personal mobility mobility devices, devices, and various and industrial various industrialequipment equipment since they providesince they reliable, provide effi reliable,cient, and efficient, durable and power durable transmission power withtransmission relatively with low relativelyeconomic low cost. economic Figure1 presentscost. Figure a schematic 1 presents of a a schematic roller chain, of a the roller most chain, common the most and popularcommon type and popularof chain, type in which of chain, the innerin which link the is connectedinner link byis connected a pin on the by outera pin link,on the and outer the link, pin is and covered the pin by is a coveredbush and by roller a bush that and contacts roller withthat thecontacts mating with sprocket. the mating The transmissionsprocket. The e ffitransmissionciency of the efficiency chain drive of thesystem, chain which drive is system, the ratio which of the is input the ratio power of ofthe the in drivingput power sprocket of the to driving the output sprocket power to of the the output driven powersprocket, of the depends driven on sprocket, the configuration depends on of the the config chainuration and the of operation the chain conditions and the operation such as torqueconditions and suchspeed. as Intorque the process and speed. of power In the transmission, process of energypower losstransmission, occurs due energy to sliding loss friction occurs between due to sliding the pin, frictionthe roller, between and the the bush pin, during the roller, the rotationand the ofbush articulation, during the vibration rotation inof the articulation, chain spans, vibration impact in when the chainthe roller spans, engages impact and when disengages the roller with engages the sprocket, and disengages and contacting with the force sprocket, between and thecontacting inner and force the betweenouter side the plates inner of and the the derailleur outer side system plates for of gear the shifting.derailleur system for gear shifting.

Figure 1. Schematic of roller chain and its components. Figure 1. Schematic of roller chain and its components. 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,

Appl. Sci. 2020, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7729; doi:10.3390/app10217729 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7729 2 of 13 load, lateral offset between driving and driven sprockets, deflection of the tight side of the chain, etc. In 1999, Kidd 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. James C. Conwell and Johnson G.E. [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] James B. Spicer 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. Hollingworth, N.E. [5] and Hills, D.A. [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. Burgess, S.C. [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, Lodge [10] and Burgess [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. The results of efficiency were in the range of 9–99% in high torque transmission. Troedsson, I. [12] and Vedmar, L. [13] in 2001 modeled a complete chain transmission with two sprockets and a chain with tight and slack spans. Damping force is considered as an important factor in the tight span that transmits the most power, and it is assumed to be proportional to the relative velocity between the two end points at the tight side. In 2004, Stuart Burgess and Chris Lodge [14] demonstrated that the efficiency of the chain is significantly influenced by the center distance between two sprocket axes, and showed that chain center distance is a key design parameter for motorcycles, although experimental verification was not achieved. Wragge-Morley, R. et al. [15], in 2018, estimated the drive transmission efficiency of the roller chain according to the sliding friction force among the pin, bush, and roller of a chain; however, the experimental test was only validated in the condition of low rotational speed. In 2018, Aleksey Egorov et al. developed a method that determined the transmission energy consumed to accelerate a rotating chain system [16–19]. In addition, chain efficiency was estimated by measuring the time of angular acceleration in the rotation axis of the drive shaft with and without load during the speeding-up of a chain drive. This method, however, could not discern the characteristics of a steady-state for a chain system, such as various affected parameters. In computing the roller chain system efficiency, energy loss should be always considered. While some energy stored as elastic deformation of components is recovered, other energy losses due to friction and damping cannot be recovered. James B. Spicer et al. [3] estimated the bicycle chain drive efficiency based on the energy losses due to sliding friction. Theoretical efficiency of the chain system was found to be between 97 and 99%, however, the results from the experiment and theoretical analysis showed some discrepancy. In order to estimate the energy losses based on damping force, Troedsson, I. and Vedmar, L. [12] proposed a method to determine the load distribution during power transmission considering the damping factor. The damping force was modeled according to the damping coefficient and rotational speed of the chain span. Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 13

Appl.power Sci. 2020transmission, 10, 7729 considering the damping factor. The damping force was modeled according3 of 13 to the damping coefficient and rotational speed of the chain span. The aim of the present study is to estimate the efficiency of the roller chain transmission system The aim of the present study is to estimate the efficiency of the roller chain transmission system considering energy losses due to sliding friction and damping force in the tight span. The sliding considering energy losses due to sliding friction and damping force in the tight span. The sliding frictional force between the pin, the bush, and the roller was established. The dynamics between links frictional force between the pin, the bush, and the roller was established. The dynamics between links with lateral offset owing to the derailleur system are also modeled. In addition, the damping force with lateral offset owing to the derailleur system are also modeled. In addition, the damping force between the driving sprocket and the driven sprocket in the tight span is considered theoretically between the driving sprocket and the driven sprocket in the tight span is considered theoretically according to the choice of damping coefficient. The effects of rotational speed, torque, offset angle, according to the choice of damping coefficient. The effects of rotational speed, torque, offset angle, and and damping coefficient are analyzed. The test stand of the roller chain power transmission system damping coefficient are analyzed. The test stand of the roller chain power transmission system is set up to is set up to verify the estimated efficiency. The results demonstrate the experimental measurements verify the estimated efficiency. The results demonstrate the experimental measurements of the efficiency of the efficiency of the roller chain drive under various operation conditions to validate the theoretical of the roller chain drive under various operation conditions to validate the theoretical estimation. estimation. 2. Chain Drive Efficiency 2. Chain Drive Efficiency Chain drive efficiency, which is the ratio of the input power of the driving sprocket to the output Chain drive efficiency, which is the ratio of the input power of the driving sprocket to the output power of the driven sprocket, depends on torque and speed. In other words, power is transferred from power of the driven sprocket, depends on torque and speed. In other words, power is transferred the driving shift to the driven shift. Therefore, part of the total energy extracted from the system cannot from the driving shift to the driven shift. Therefore, part of the total energy extracted from the system be recovered due to such factors as frictional loss and damping. The sliding friction force includes the cannot be recovered due to such factors as frictional loss and damping. The sliding friction force coplanar drive transmission, which is calculated as the surface contact pressure, and the lateral offset includes the coplanar drive transmission, which is calculated as the surface contact pressure, and the drive, which is the two-points contact owing to the derailleur system. lateral offset drive, which is the two-points contact owing to the derailleur system. 2.1. Sliding Friction Force 2.1. Sliding Friction Force Figure2 shows the interaction between the chain and the sprocket, where the roller is engaged Figure 2 shows the interaction between the chain and the sprocket, where the roller is engaged with the sprocket tooth, resulting in normal contact force N1 and friction force F1 between the pin and thewith bush. the sprocket Articulation tooth, from resulting the chain in normal span to contact the seated force sprocket 𝑁 and turnsfriction a maximumforce F1 between angle θthe. Normal pin and the bush. Articulation from the chain span to the seated sprocket turns a maximum angle 𝜃. Normal contact force N1 between the pin and the bush is generated, where Tc represents the link tension with contact force 𝑁 between the pin and the bush is generated, where 𝑇 represents the link tension the chain span, θN is the angle of normal contact force; Rp and Rbi are the radius of the pin and the bushwith inner the chain surface, span, respectively; 𝜃 is the angleθ isthe of normal articulation contact angle force; of the𝑅 pinand between𝑅 are the axes radius of the of chainthe pin span and andthe thebush seated inner link surface, in the respectively; sprocket; and 𝜃 t is the articulation width of the angle bush. of the pin between axes of the chain span and the seated link in the sprocket; and t is the width of the bush.

FigureFigure 2. 2.Schematic Schematic of of contact contact for for pin pin articulation. articulation.

Hollingworth,Hollingworth, N.E. N.E. [ 5[5]] and and Hills, Hills, D.A. D.A. [ 6[6]] determined determined that that sliding sliding frictional frictional losses losses during during pin pin articulationarticulation are are dependent dependent on on the the pin pin sliding sliding against against the the bush. bush. Static Static or or slip slip motion motion between between the the pin pin andand the the bush bush depends depends on on the the magnitude magnitude of of the the friction friction angle angle of ofθ N𝜃compared compared with with the the articulation articulation angleangleθ .𝜃 The. The frictional frictional force force is is su sufficientlyfficiently large large enough enough to to ensure ensure pin pin stiction stiction under under the the condition condition of of tantanθ 𝜃< tan< tanθ N𝜃.. In In the the case case of of the the pin pin slipping, slipping, it it should should be be satisfied satisfied with with the the condition condition of of tan tan( θ𝜃) ≥ 𝜇 and 𝜇=𝐹/𝑁 tan𝜃), where 𝜇 is the friction coefficient between the pin and the bush. The relationship of contact force between the pin and the bush can be written as follows: Appl. Sci. 2020, 10, 7729 4 of 13

µ1 and µ1 = F1/N1 = tan(θN), where µ1 is the friction coefficient between the pin and the bush. The relationship of contact force between the pin and the bush can be written as follows: Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 13 1 1 N1 = Tc cos(θN) = Tc q = Tc p (1) · · 112 · 1 + µ 2 NT=⋅cos(θ ) =⋅ T1 + tan (θ =⋅N T) 1 1 cNc++22θμ c (1) 1tan()N 1 1 Normally, a normal force can be found by integrating the contact pressure over the area. The link tensionNormally, is then related a normal to the force surface can contactbe found pressure by integr andating area. the James contact B. Spicer pressure [3] calculatedover the area. pressure The relatedlink tension with contactis then surfacerelated betweento the surface the pin contact and the pressure bush. It and was area. assumed James that B. the Spicer contact [3] pressurecalculated is apressure constant, related and the with contact contact angle surface between between the pinthe andpin and the bushthe bush. is also It was a constant. assumed Here, that thethe normalcontact contactpressure force is a Nconstant,1 between and the the pin contact and the angle bush between is considered the pin to beand a the normal bush force. is also The a constant. relationship Here, of contactthe normal pressure contact and force normal N forcebetween is shown the pin as Equationand the bush (2), where is considered it is assumed to be that a normal contact force. pressure The Prelationshipcos(θ) acts of radially contact and pressure is a constant: and normal force is shown as Equation (2), where it is assumed that contact· pressure P ∙ cos𝜃 acts radially and is a constant: π Zπ2 = 2 ( ) N1 NRtP=⋅⋅⋅Rp t P cos(cosθθθ ) ddθ (2) 1 ·p ·  · (2) ππ −2 − 2 Then,Then, frictionfriction forceforce betweenbetween thethe pinpin andand thethe bushbush cancan bebe givengiven byby EquationEquation (3): (3): F =⋅⋅μ ()PdA F =11µ (P dA) (3)(3) 1 1 · · The tension in the chain links owing to centripetal acceleration is considered to have little effect The tension in the chain links owing to centripetal acceleration is considered to have little effect on friction force. The energy losses W of pin articulation over the contact area between the pin and on friction force. The energy losses W of pin articulation over the contact area between the pin and the bush are given by: pin the bush are given by:   π π 1  = =⋅ 1 ⋅ ⋅⋅μθ ⋅ WpinWTRpin pT cc µ1 1 biRbi θ (4)(4) 2  22  ·2 11++µμ1 · · · · 1 Figure3 shows that the chain is driven under lateral o ffset owing to the derailleur system. The Figure 3 shows that the chain is driven under lateral offset owing to the derailleur system. The contact force between the pin and the bush with lateral offset angle γ is generated. This analysis differs contact force between the pin and the bush with lateral offset angle γ is generated. This analysis from that in Kidd [1] and James B. Spicer [3], in that their analysis did not consider the contact of chain differs from that in Kidd [1] and James B. Spicer [3], in that their analysis did not consider the contact links. In fact, contact between the pin and the bush is not generated along the vertical direction of of chain links. In fact, contact between the pin and the bush is not generated along the vertical the inner bush surface, but rather occurs at the boundary of the bush plate, shown as normal force in direction of the inner bush surface, but rather occurs at the boundary of the bush plate, shown as one side N1 and other side N2 due to lateral offset. When the link engages the sprocket, the sliding normal force0 in one side 𝑁0 and other side 𝑁 due to lateral offset. When the link engages the friction force of pin articulation F1 and F2 are generated. The pressure angles of both normal forces are sprocket, the sliding friction force0 of pin0 articulation 𝐹 and 𝐹 are generated. The pressure angles assumed to be equal, and the normal forces can be expressed as follows: of both normal forces are assumed to be equal, and the normal forces can be expressed as follows: 1 1 1 212 ' 11 N0 = NNNT0 ===⋅Tc cos(cos(θN0 )θ = )T =⋅c Tq =⋅= TTc p (5) ooc· Nc· 2' c· 2 2 (5) 11tan()+++tan2(θμθ ) 1 1 + µ1 NN0 1

Figure 3. Schematic of contact for chain links with lateral offset (above view) on the left side and contact Figure 3. Schematic of contact for chain links with lateral offset (above view) on the left side and of pin articulation (side view) on the right side. contact of pin articulation (side view) on the right side.

The energy losses W of pin articulation with a lateral offset between the pin and the bush are given by: Appl. Sci. 2020, 10, 7729 5 of 13

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 13 The energy losses Wpin of pin articulation with a lateral offset between the pin and the bush are given by:  1  WTr' =⋅2 1 ⋅ ⋅μθ ⋅ ⋅ W =pin  2 Tcbi 1 r (6) pin0 2  p+ μ c µ1 bi θ (6) ·  1 12 ·  · · · 1 + µ1 FigureFigure4 4presents presents the the schematic schematic of of forces forces and and motion motion between between the the bush bush and and the the roller. roller. The The tooth tooth contactcontact forceforce transferstransfers toto thethe bushbush whenwhen thethe rollerroller isisengaged engagedwith withthe the sprocket sprocket tooth. tooth. ContactContact forfor bushbush articulation,articulation, whichwhich isis locatedlocated atat thethe pin-bushpin-bush contactcontact andand thethe bush-rollerbush-roller contact,contact, isis generated.generated. DiDifferentfferent fromfrom thethe relationshiprelationship betweenbetween thethe pinpin andand thethe bush,bush, thethe surfacesurface contactcontact forceforce betweenbetween thethe bushbush andand the the roller roller occurred occurred with with lateral lateral off offsetset or coplanaror coplanar off setoffset when when the linkthe link engages engages the teeth, the teeth, due todue the to independent the independent roller. Therefore,roller. Therefore, the friction theF10 frictionbetween F the between pin and thethe bush,pin and and Fthe2 between bush, and the bush𝐹 between and the the roller, bush are and generated. the roller,W arebush generated.is the energy 𝑊 loss is of the bush energy articulation loss of bush owing articulation to friction; owingRri is theto friction; inner radius 𝑅 ofis thethe roller;inner Tradius1 is the of link the tension roller; seated 𝑇 is inthe the link sprocket; tensionNs seatedrepresents in the the sprocketsprocket; tooth𝑁 represents contact force; the sprocketβ is the nominaltooth contact pressure force; angle; β is and the ϕnominalis the friction pressure angle angle; for normaland φ is contact the friction force betweenangle for the normal bush contact and the force roller. between the bush and the roller.

FigureFigure 4.4. SchematicSchematic ofof contactcontact forfor bushbush articulation.articulation.

AccordingAccording toto thethe frictionfriction forces,forces, thethe frictionalfrictional losseslosses betweenbetween thethe pin-bushpin-bush andand thethe bush-rollerbush-roller cancan bebe calculated.calculated. TheThe slidingsliding frictionfriction forceforce willwill occuroccur atat thethe bushbush andand rollerroller interfaceinterface firstfirst becausebecause itit isis locatedlocated atat aa smallsmall radius.radius. Therefore,Therefore, itit isis assumedassumed thatthat therethere isis nono slidingsliding forceforce betweenbetween thethe toothtooth andand rollers,rollers, andand thethe frictionfriction lossloss occursoccurs betweenbetween thethe rollerrollerand andthe thebush. bush. ForFor bushbush articulation,articulation, thisthis isis becausebecause thethe tensiontension ofof thethe chainchain inin thethe tighttight spanspan isis muchmuch greater greater than than that that in in the the slack slack span. span. ForFor thethe slipslip conditioncondition ofof pinpin articulation,articulation, itit shouldshould bebe satisfiedsatisfied withwith thethe conditioncondition ofof tantan(𝜃θ) ≥ µ𝜇1,, µ𝜇2 andandµ 𝜇2 ==𝐹F2/𝑁/N2=tantanφ(ϕ, )where, where 𝜇µ 2representsrepresents the the friction friction coefficient coefficient between the bushbush andand thethe roller.roller. WeWe assume assume the the following: following: N N = N2 (7) sN = 2 s cos(ϕϕ) (7) cos( ) and and N sin(θ ) s = (8) Tc N sinsin(（βθ )θ) s = − T sin(βθ− ) (8) Substituting Equation (7) into Equation (8),c the normal contact force N2 between the bush and the roller can be written as follows: Substituting Equation (7) into Equation (8),cos the(ϕ normal) sin(θ contact) force 𝑁 between the bush and N2 = Tc · (9) the roller can be written as follows: · sin(β θ) − cos(ϕθ )⋅ sin( ) NT=⋅ 2 c βθ− (9) sin( ) Due to the surface contact of bush articulation, the energy loss between the bush and the roller in the tight span using the relationship between pressure and normal force can be given by the following: Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 13

πθ =⋅sin ⋅⋅γμ ⋅⋅ ⋅ θ WTRbush ccos( ) 2 bo (10) 2 1sin()+⋅μβθ2 − 2

2.2. Damping Force Appl. Sci. 2020, 10, 7729 6 of 13 Viscous damping is an important characteristic in the friction force. The damping coefficient is dependent on lubrication, contacting surface, damping force, and contacting speed. In this study, the magnitudeDue to of the the surface damping contact coefficient of bush refers articulation, to the work the energy of Troedsson, loss between I. and the Vedmar, bush and L [12]. the roller Power in duethe tightto damping span using force the in relationship the chain drive between system pressure is mainly and normaltransmitted force through can be given the tight by the span. following: In the work of Yu, S. [20] and Lodge, C,π J. and Burgess,sin θ S, C. [10], magnitude of tension at the slack and tight span were investigated,W whichbush = showedp that tension in theT slackc cos span(γ) µ is2 muchRbo θ smaller than tension(10) in 2 · 1 + µ 2 sin(β θ) · · · · · the tight span. Therefore, the damping force2 · is assumed− to occur at the tight span, and the damping force2.2. Damping in the slack Force span is neglected, since tension at the tight span is larger than that at the slack span. The damping force is assumed to be proportional to the relative velocity between the end points from the drivingViscous sprocket damping to isthe an driven important sprocket characteristic at the tight in span the frictionin Figure force. 5. Therefore, The damping the damping coefficient force is isdependent expressed on as: lubrication, contacting surface, damping force, and contacting speed. In this study, the magnitude of the damping coefficient refers to the work of Troedsson, I. and Vedmar, L [12]. Power due to damping force𝐹 in 𝐷 the chain𝑅, drive𝜎 ∙ cos𝜎system 𝜃 is mainly,𝑅 transmitted,𝜎 ∙ cos𝜎 through𝜃, the tight span. In(11) the work of Yu, S. [20] and Lodge, C, J. and Burgess, S, C. [10], magnitude of tension at the slack and tight where Dchain is the damping coefficient in the tight span of the chain; subscripts 1 and 2 describe the span were investigated, which showed that tension in the slack span is much smaller than tension in driving and the driven sprocket, respectively; Rs,1 and Rs,2 are the radius of the driving and the the tight span. Therefore, the damping force is assumed to occur at the tight span, and the damping driven sprocket, respectively; 1 is the angle between the circle line of the first seated tooth into the force in the slack span is neglected, since tension at the tight span is larger than that at the slack span. links and coordinate line of the driving sprocket; 2 is the angle between the circle line of the last The damping force is assumed to be proportional to the relative velocity between the end points from seated tooth into the links and coordinate line of the driven sprocket; 1 is the rotational speed with the driving sprocket to the driven sprocket at the tight span in Figure5. Therefore, the damping force respect to 1; 2 is the rotational speed with respect to 2; and θin,1 and θin,2 are the chain angles is expressed as: related to the x-axle of the first hlinks. in contact with the sprocket. in the drivingi and the driven F = D R σ cos(σ θ ) Rs σ cos(σ θ ) (11) sprocket, respectively. d chain s,1 1· 1 − in,1 − ,2 2· 2 − in,2 where Dchain is the damping coefficient in the tight span of the chain; subscripts 1 and 2 describe the 2.3. Efficiency of Chain Drive System driving and the driven sprocket, respectively; Rs,1 and Rs,2 are the radius of the driving and the driven sprocket,The magnitude respectively; ofσ the1 is efficiency the angle of between the chain the driv circlee system line of theis given first seatedby the toothoutput into energy the links divided and bycoordinate the overall line energy, of the drivingincluding sprocket; output σenergy2 is the and angle transmission between the loss circle energy. line ofThe the total last transmission seated tooth . energyinto the losses links in and the coordinate chain drive line can of be the calculated driven sprocket; by summingσ1 is the sliding rotational frictional speed withloss and respect damping to σ1; . forceσ2 is thein the rotational tight span. speed The with sliding respect friction to σ2; andforceθ in,1includesand θ in,2theare coplanar the chain drive angles transmission, related tothe which x-axle is calculatedof the first as links the insurface contact contact with thepre sprocketssure and in the the lateral driving offset and drive, the driven which sprocket, is the two-points respectively. contact.

Figure 5. SchematicSchematic of of damping damping force force between two sprockets.

2.3. EUsingfficiency the of power Chain of Drive the Systemdriven sprocket, the transmission efficiency η of the chain in the coplanar driveThe can magnitudebe given by of Equation the efficiency (12): of the chain drive system is given by the output energy divided by the overall energy, including output energy and⋅ transmission loss energy. The total transmission Twos,2 η = (12) energy losses in the chain driveTw can⋅ be + calculated N ⋅⋅ w() Wby summing + W +⋅⋅ the Fw sliding R frictional loss and damping force in the tight span. The slidingos,2 friction s ,1 force s ,1 includes pinbushds the coplanar ,1 drive s ,1 transmission, which is calculatedTransmission as the surface efficiency contact 𝜂, considering pressure and lateral the lateral offset, off canset be drive, written which as isEquation the two-points (13): contact. Using the power of the driven sprocket, the transmission efficiency η of the chain in the coplanar drive can be given by Equation (12):

To ws,2 η = · (12) To ws + N w (W + W ) + F w R · ,2 s,1 · s,1 · pin bush d · s,1 · s,1 Appl. Sci. 2020, 10, 7729 7 of 13

Transmission efficiency η0, considering lateral offset, can be written as Equation (13):

To ws,2 η0 = · (13) To ws + N w (W + W ) + F w R · ,2 s,1 · s,1 · pin0 bush d · s,1 · s,1 where η is the chain drive efficiency with the coplanar drive; To is the torque of the driven sprocket; ws,1 and ws,2 are the rotational speed of the driving and the driven sprocket, respectively; and Ns,1 is the number of teeth on the driving sprocket.

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 Table1.

Table 1. Drive train components and parameters for roller chain system.

Parts Parameters Symbol [Unit] Value Friction coefficient between the pin and the µ /µ [-] 0.11/0.11 bush/among the pin, the bush, and the roller 1 2 Radius of inner bush r [m] 1.74 10 3 bi × − Maximum rotation angle of chain link αm [rad] 0.19 Average mass of one chain link m [kg] 0.0024 Roller chain Chain pitch p [m] 12.7 10 3 × − Radius of inner roller r [m] 2.83 10 3 bo × − Damping coefficient Dchain [Ns/m] 40/60/80 Chain angle related to x-axle of link in θ [rad] 0.00012/0.00012 driving/driven sprocket in,1/2 Tension in the chain span Tc [N] - Radius of driving/driven sprockets R [m] 62.43/62.43 10 3 s,1/2 × − Teeth number of driving-driven sprocket Ns,1 2 [-] 32–32 − 4.18/5.23/6.28/ Sprocket Rotational speed of driving/driven sprocket w [rad/s] s,1/2 7.32/9.37 Coordinate for position of driving/driven σ [rad] 3.09/1.52 sprocket 1/2 Lateral offset angle between sprockets γ [rad] 0/0.0174/0.0349/0.0523 Power of driven sprocket Ps,2 [W] -

Figure6 shows the following: Transmission e fficiency 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. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13

Appl. FigureSci. 2020 ,6 10 shows, x FOR the PEER following: REVIEW Transmission efficiency as a function of rotational speed and offset8 of 13 angle; the damping coefficient, in which the range of efficiency is 86.3–93.1%; and demonstrates that efficiencyFigure decreases 6 shows withthe following: increasing Transmission rotational spe efficiened andcy damping as a function coefficient of rotational for a given speed lateral and offsetoffset angle.angle; Inthe addition, damping for coefficient, a given rotational in which thespeed, range chain of efficiency efficiency is decreases 86.3–93.1 with%; and increasing demonstrates damping that coefficientefficiency decreasesand lateral with offset increasing angle. rotational speed and damping coefficient for a given lateral offset angle. In addition, for a given rotational speed, chain efficiency decreases with increasing damping Appl.coefficient Sci. 2020, 10 and, 7729 lateral offset angle. 8 of 13

Figure 6. Transmission efficiency as a function of rotational speed, offset angle, and damping coefficient. FigureFigure 6. Transmission6. Transmission efficiency efficiency as a function as a offunction rotational of speed, rotational offset angle,speed, and offset damping angle, coe andfficient. damping Figurecoefficient. 7 presents the effect of torque, lateral offset angle, and damping coefficient to Figure7 presents the e ffect of torque, lateral offset angle, and damping coefficient to transmission transmission efficiency, demonstrating that efficiency increases with torque, and the range of efficiency, demonstrating that efficiency increases with torque, and the range of efficiency is 90.8–93.0%. efficiencyFigure is 790.8–93.0%. presents theSimilar effect to of the torque, results late fromral Figureoffset angle,6, chain and efficiency damping decreases coefficient with to Similar to the results from Figure6, chain e fficiency decreases with increasing damping coefficient and increasing damping coefficient and offset angle for a given torque. offtransmissionset angle for aefficiency, given torque. 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.

FigureFigure 7. 7.Transmission Transmission effi ciencyefficiency as function as function of torque, of torq offue,set offset angle, angle, and damping and damping coefficient. coefficient.

Figure 7. Transmission efficiency as function of torque, offset angle, and damping coefficient. Appl. Sci. 2020, 10, x 7729 FOR PEER REVIEW 9 of 13

4. Experimental Measurement 4. Experimental Measurement An experiment is performed in order to measure and verify the estimated efficiency. The layout An experiment is performed in order to measure and verify the estimated efficiency. The layout of of the experiment is shown schematically in Figure 8. Driving sprocket axes and driven sprocket axes the experiment is shown schematically in Figure8. Driving sprocket axes and driven sprocket axes are are connected via the test chain. In the driving sprocket axes, an electric motor with a maximum connected via the test chain. In the driving sprocket axes, an electric motor with a maximum rotational rotational speed of 1700 RPM and a maximum torque capability of 13 nm provides propulsion. speed of 1700 RPM and a maximum torque capability of 13 Nm provides propulsion. Driving torque Driving torque from the electric motor is measured through a torque transducer (accuracy ± 0.3%), from the electric motor is measured through a torque transducer (accuracy 0.3%), which has a rated which has a rated capacity of 0–100 kgf-m. In the driven sprocket axes, a friction± disk brake that can capacity of 0–100 kgf-m. In the driven sprocket axes, a friction disk brake that can regulate speed and regulate speed and torque is attached. The torque of the driven sprocket axes is measured with the torque is attached. The torque of the driven sprocket axes is measured with the torque transducer of torque transducer of the same type as that with the driving axes. Support bearings are used in the the same type as that with the driving axes. Support bearings are used in the rotational axes between rotational axes between the electric motor, torque transducers, brake disk, and driving/driven sprocketthe electric axes. motor, In order torque to take transducers, into consideration brake disk, friction and driving loss due/driven to support sprocket bearings, axes. In friction order toloss take of individualinto consideration support friction bearings loss is duemeasured to support with bearings,varying rotational friction loss speeds of individual in the range support of 50 bearings RPM–280 is RPM.measured The lateral with varying offset angle rotational between speeds sprockets in the can range be ofadjusted 50 RPM–280 by moving RPM. the The driven lateral sprocket offset angle axes laterally.between sprockets can be adjusted by moving the driven sprocket axes laterally.

Figure 8. Schematic of setup and test stand equipment.

EfficiencyEfficiency is measuredmeasured withwith didifferentfferent operatingoperating conditions.conditions. The The lateral lateral offset offset angle is 0/0.0174/0.0349/0.05230/0.0174/0.0349/0.0523 rad with an increment of 1 degree,degree, the torquetorque rangerange isis 5–95–9 Nmnm with an increment ofof 11 nm, nm, and and the the rotational rotational speed speed is inis thein the range range of 40–80 of 40–80 RPM RPM with with an increment an increment of 10 RPM.of 10 RPM.Each testEach takes test approximatelytakes approximately 20 s and 20 iss and repeated is repeated seven times.seven times.

5. Experiment Results Figures9 9 and and 10 10 show show the the chain chain drive drive e ffi efficiencyciency under under diff differenterent conditions conditions (rotational (rotational speeds speeds and andlateral lateral offset offset angle) angle) and compares and co themmpares with them theoretical with theoretical results. The results. experimental The experimental results demonstrate results demonstratethat chain drive that e ffichainciency drive decreased efficiency with decreased increasing with rotational increasing speed rotational for a given speed offset for angle; a given whereas, offset angle;for a given whereas, rotational for a speed,given rotational chain drive speed, efficiency chain decreased drive efficiency with increasing decreased off withset angles. increasing It is foundoffset angles.that both It is results found exhibit that both the results same tendency exhibit the in same terms tendency of the eff ectin terms of rotational of the effect speed of androtational lateral speed offset andangle. lateral These offset results angle. agree These with theresults theoretical agree with results. the The theoretical efficiency re issults. dependent The efficiency on rotation is dependent speed and onthe rotation offset angle speed in theand Equationsthe offset (11)angle and in (12).the Equati Figureons 10 (11)shows and the (12). diff Figureerence 10 between shows thethe theoreticaldifference betweenand experimental the theoretical results and with experimental an error range results of 0.35–1.81%. with an error The range results of reveal 0.35–1.81%. that, when The results the rotational reveal that,speed when is high, the the rotational error value speed is small.is high, However, the error error value increases is small. with However, decreasing error rotational increases speed. with Thisdecreasing is attributable rotational to that,speed. when This the is testattributable was carried to that, out withwhen low the rotational test was speedcarried and out off withset angle, low vibrationrotational andspeed impact and offset occur angle, between vibration the chain and and impact the sprocket. occur between the chain and the sprocket. Appl.Appl. Sci. Sci. 20202020,, 1010,, x 7729 FOR PEER REVIEW 1010 of of 13 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13

Figure 9. Transmission efficiency of theoretical and experimental results. FigureFigure 9. 9. TransmissionTransmission efficiency efficiency of of theoretical theoretical and and experimental experimental results. results.

Figure 10. The difference between theoretical and experimental results. Figure 10. The difference between theoretical and experimental results. Figure 10. The difference between theoretical and experimental results. Figures 11 and 12 show the chain drive efficiency under different output torques and offset angles, Figures 11 and 12 show the chain drive efficiency under different output torques and offset and compares them with the theoretical results. The results demonstrate that the chain drive efficiency angles,Figures and compares11 and 12 them show with the thechain theoretical drive efficien results.cy The under results different demonstrate output thattorques the chainand offset drive decreased with increasing output torque for a given offset angle; whereas, for a given output torque, the angles,efficiency and decreased compares withthem increasingwith the theoretical output torque results. for The a givenresults offset demonstrate angle; whereas, that the chainfor a drivegiven chain drive efficiency decreased with the increasing offset angle. It is found that both results possess efficiencyoutput torque, decreased the chain with drive increasing efficiency output decreased torque withfor a the given increasing offset angle; offset whereas,angle. It isfor found a given that the same trend in terms of the effect of torque and lateral offset angle. Figure 12 presents the difference outputboth results torque, possess the chain the samedrive trendefficiency in terms decreased of the efwithfect theof torque increasing and lateraloffset angle.offset angle.It is found Figure that 12 between theoretical and experimental results, with an error range between 0.23–3.77%. The reason for bothpresents results the possess difference the samebetween trend theoretical in terms ofand the ex efperimentalfect of torque results, and lateralwith an offset error angle. range Figure between 12 this is that since the test is carried out with small torque and large offset angle, vibration and impact presents0.23–3.77%. the Thedifference reason between for this istheoretical that since andthe testexperimental is carried outresults, with with small an torque error andrange large between offset become more apparent. Thus, the contact surface becomes large between the chain links and sprockets. 0.23–3.77%.angle, vibration The reasonand impact for this become is that more since apparent the test . isThus, carried the outcontact with surface small torquebecomes and large large between offset angle,the chain vibration links andand sprockets.impact become more apparent. Thus, the contact surface becomes large between the chain links and sprockets. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13 Appl. Sci. 2020, 10, 7729 11 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13

Figure 11. The transmission efficiency of theoretical and experimental results. FigureFigure 11. 11. TheThe transmission transmission efficiency efficiency of of theo theoreticalretical and and experimental experimental results. results.

Figure 12. The difference between theoretical and experimental results. Figure 12. The difference between theoretical and experimental results. Figure 13 showsFigure the 12. effi Theciency difference under between the diff erenttheoretica centerl and distance experimental between results. two sprockets axes and a diFigurefferent 13 off setshows angle the and efficiency compares under them the with differen the theoreticalt center distance results. The between experiment two sprockets was carried axes out andwith aFigure different an interval 13 showsoffset of 0.05 anglethe m efficiency inand the compares test under stand. thethem The differen with results tthe center demonstrate theoreti distancecal results. that between the The effi twociency experiment sprockets of the was chainaxes carriedanddrive a systemdifferentout with increased anoffset interval angle with of and decreasing0.05 compares m in the center test them distancestand. with The betweenthe results theoreti sprocket demonstracal results. axeste forthatThe a the givenexperiment efficiency offset angle.was of thecarriedThe chain range out drive ofwith measurement system an interval increased resultsof 0.05 with ism 89.34–93.24%.indecreasing the test stand. center The distance results between demonstra sprockette that axes the efficiencyfor a given of offsetthe chain angle. drive The systemrange of increased measurement with decreasingresults is 89.34–93.24%. center distance between sprocket axes for a given offset angle. The range of measurement results is 89.34–93.24%. Appl.Appl. Sci. Sci. 20202020, 10, 10, x, 7729FOR PEER REVIEW 1212 of of 13 13

Figure 13. Chain drive efficiencies for different center distances between sprockets with an offset angle.

6. Conclusions AFigure novel 13. approach Chain drive for estimatingefficiencies for chain different transmission center distances efficiency, between which sprockets takes into with account an offset combined sliding friction losses due to pin articulation, bushangle. articulation, and damping force in the tight span, was proposed. The frictional losses due to pin articulation motion and bush articulation motion were 6. Conclusions calculated according to Coulomb’s law of friction. The dynamics model was applied to investigate chainA linksnovel with approach lateral offforset estimating owing to the chain derailleur transm system.ission Theefficiency, damping which force, whichtakes existsinto account between combinedthe driving sliding sprocket friction and drivenlosses due sprocket to pin in articula the tighttion, span, bush was articulation, also considered and accordingdamping force to the in choice the tightof damping span, was coe proposed.fficient. The frictional losses due to pin articulation motion and bush articulation motionThe were theoretical calculated value according of chain to transmission Coulomb’s law effi ciencyof friction. was calculatedThe dynamics with model a range was of applied 86.3–93.1%, to investigatedepending chain on the links driving with lateral conditions. offset Transmissionowing to the derailleur loss in each system. parameter The damping was analyzed force, which by the existsvarious between operation the conditions driving sprocket of the chain and drive driven system, sprocket including in the rotational tight span, speed, was torque, also considered lateral offset accordingangle, and to damping the choice coe offfi dampingcient. Tests coefficient. were performed to measure efficiency and compared with the theoreticalThe theoretical results, withvalue a of di chainfference tran rangesmission of 0.23–3.77%, efficiency was which calculated demonstrates with a goodrange agreement of 86.3–93.1%, with dependingtheoretical on values. the driving conditions. Transmission loss in each parameter was analyzed by the various operation conditions of the chain drive system, including rotational speed, torque, lateral Author Contributions: offset angle, and dampinT.-O.T.g coefficient. conceived Tests and designed were performed the experiments; to measure S.-P.Z. efficiency performed and the experiments,compared analyzed the data and wrote the paper. All authors have read and agreed to the published version of the manuscript. with the theoretical results, with a difference range of 0.23–3.77%, which demonstrates good Funding: This work was supported by Ministry of Trade, Industry and Energy (20001447). agreement with theoretical values. Conflicts of Interest: The authors declare no conflict of interest. Author Contributions: T.-O.T. conceived and designed the experiments; S.-P.Z. performed the experiments, analyzedReferences the data and wrote the paper.

Funding:1. Kidd, This M.D.; work Loch, was N.E.;supported Reuben, by Ministry R.L. Experimental of Trade, Industry Examination and ofEnergy Bicycle (20001447). Chain Forces. Exp. Mech. 1999, 39 Conflicts, of 278–283. Interest: [CrossRef The authors] declare no conflict of interest. 2. Conwell, J.C.; Johnson, G.E. Experimental investigation of link tension and roller-sprocket impact in roller Referenceschain drives. Mech. Mach. Theory 1995, 31, 533–544. [CrossRef] 3. James, B.S.; Christopher, J.K.R.; Michael, J.E.; Johanna, R.B.; Masahiko, F.; Masao, T. Effects of Frictional Loss 1. Kidd,on Bicycle M.D.; ChainLoch, DriveN.E.; Reuben, Efficiency. R.L.J. Mech.Experimental Des. 2001 Examination, 123, 598–605. of Bicycle Chain Forces. Exp. Mech. 1999, 4. 39(4)Wang,, 278–283. C.C.; Tseng, C.H.; Fong, Z.H. A method for improving bicycle shifting performance. Int. J. Vehicle Des. 2. Conwell,1997, 18 ,J.C.; 100–117. Johnson, G.E. Experimental investigation of link tension and roller-sprocket impact in roller 5. chainHollingworth, drives. Mech. N.E.; Mach. Hills, Theory D.A. Force 1995, in31 a, 533–544. heavy-duty drive chain during articulation. J. Mech. Eng. Sci. 1986, 3. James,200, 367–374. B.S.; Christopher [CrossRef ]J.K.R.; Michael, J.E.; Johanna, R.B.; Masahiko, F.; Masao, T. Effects of Frictional 6. LossHollingworth, on Bicycle Chain N.E.;Hills, Drive D.A. Efficiency. Theoretical J. Mech. effi Des.ciency 2001 of, a 123(4) cranked, 598–605. link chain drive. J. Mech. Eng. Sci. 1986, 4. Wang,200, 375–377. C.C.; Tseng, [CrossRef C.H.;] Fong, Z.H. A method for improving bicycle shifting performance. Int. J. Vehicle Des. 1997, 18(1), 100-117. 5. Hollingworth, N.E.; Hills, D.A. Force in a heavy-duty drive chain during articulation. J. Mech. Eng. Sci. 1986, 200(5), 367–374. Appl. Sci. 2020, 10, 7729 13 of 13

7. Jalon, J.G.D.; Bayo, E. Kinematic and Dynamic Simulation of Multibody Systems; Springer: Berlin/Heidelberg, Germany, 1994. 8. Marques, F.; Flores, P.; Pimenta Claro, J.C.; Hamid, M.L. Modeling and analysis of friction including rolling effects in multibody dynamics: A review. Multibody Syst. Dyn. 2019, 45, 223–244. [CrossRef] 9. Burgess, S.C. Improving cycling performance with large sprockets. Sports Eng. 1998, 1, 107–113. [CrossRef] 10. Lodge, C.J.; Burgess, S.C. A model of the tension and transmission efficiency of a bush roller chain. Proc. Inst. Mech. Eng. 2001, 216, 385–394. [CrossRef] 11. Lodge, C.J.; Burgess, S.C. An investigation into the selection of optimum chain and sprocket size. J. Eng. Des. 2004, 15, 536–580. [CrossRef] 12. Troedsson, I.; Vedmar, L. A method to determine the dynamic load distribution in a chain drive. Proc. Inst. Mech. Eng. 2001, 215, 569–579. [CrossRef] 13. Troedsson, I.; Vedmar, L. A method to determine the static load distribution in a chain drive. J. Mech. Des. 1999, 121, 402–408. [CrossRef] 14. Burgess, S.; Lodge, C. Optimisation of the chain drive system on sports motorcycles. Sports Eng. 2004, 7, 65–73. [CrossRef] 15. Wragge-Morley, R.; Yon, J.; Lock, R.; Alexander, B.; Burgess, S. A novel pendulum test for measuring roller chain efficiency. Meas. Sci. Technol. 2018, 29.[CrossRef] 16. Egorov, A.; Kozlov, K.; Belogusev, V. A method for evaluation of the chain drive efficiency. J. Appl. Eng. Sci. 2015, 34, 13–14. [CrossRef] 17. Kozlov, K.; Belogusev, V.; Egorov, A.; Syutov, N. Development of method of evaluate friction losses of chain drives. In Proceedings of the 17th International Scientific Conference Engineering for Rural Development, Jelgava, Latvia, 23–25 May 2018; pp. 930–936. 18. Egorov, A.; Kozlov, K.; Belogusev, V. Experimental identification of the electric motor moment of inertia and its efficiency using the additional inertia. ARPN J. Eng. Appl. Sci. 2016, 11, 10582–10588. 19. Kozlov, K.E.; Egorov, A.V.; Belogusev, V.N. Experimental evaluation of chain transmissions lubricants quality using a new method based on additional inertia moment use. Procedia Eng. 2017, 206, 617–623. [CrossRef] 20. Huo, J.; Yu, S.; Yang, J.; Li, T. Static and dynamic characteristic of the chain drive system of a heavy duty apron feeder. Open Mech. Eng. J. 2013, 7, 121–128. [CrossRef] 21. Short-Pitch Transmission Precision Roller and Bush Chains: Attachments and Associated Chain Sprockets; ISO606; ISO: Vernier, Switzerland, 2015. 22. Lee, C.H.; Polycarpou, A.A. Static Friction Experiments and Verification of an Improved Elastic-Plastic Model Including Roughness Effects. J. Tribol. 2007, 129, 754–760. [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).