Reliability models of belt drive systems under slipping failure mode

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1 Specil Issue rticle Relibility models of belt drive systems under slipping filure mode dvnces in Mechnicl Engineering 1, Vol. 9(1) 1 1 Ó The uthor(s) 1 DOI: 1.11/11119 journls.sgepub.com/home/de Peng Go 1 nd Liyng Xie bstrct Conventionl relibility ssessment nd relibility-bsed optiml design of belt drive re bsed on the stress strength interference model. However, the stress strength interference model is essentilly sttic model, nd the sensitivity nlysis of belt drive relibility with respect to design prmeters needs further investigtions. In this rticle, timedependent fctors tht contribute the dynmic chrcteristics of relibility re pointed out. Moreover, dynmic relibility models nd filure rte models of belt drive systems under the filure mode of slipping re developed. Furthermore, dynmic sensitivity models of belt drive relibility bsed on the proposed dynmic relibility models re proposed. In ddition, numericl exmples re given to illustrte the proposed models nd nlyze the influences of design prmeters on dynmic chrcteristics of relibility, filure rte, nd sensitivity functions. The results show tht the sttisticl properties of design prmeters hve different influences on relibility nd filure rte of belt drive in cses of different vlues of design prmeters nd different opertionl durtions. Keywords Dynmic relibility, belt drive, filure rte, dynmic sensitivity, filure rte Dte received: 3 June 1; ccepted: December 1 cdemic Editor: Yongming Liu Introduction Belt drives re comprehensively used to trnsmit power nd motion in vriety of mchinery, such s utomobile, household pplinces, officchinery, nd so on. s common form of mechnicl trnsmission, the schemtic structure of the belt drive system is shown in Figure 1. typicl belt drive system is minly composed of drive pulley, driven pulley, nd belts. Motion nd power re trnsferred from the drive pulley to the driven pulley vi the friction between the belt nd pulleys. 1 The relibility of the belt drive system plys n importnt role in the risk ssessment nd optiml design of mechnicl products. Therefore, it is necessry to develop ccurte relibility models of belt drive systems. In this rticle, we will be concentrted on dynmic relibility nlysis of belt drive systems with input power trnsmitted by friction between belt nd pulleys, such s the flt-belt drive systems nd the V-belt drive systems. In the lst few decdes, mny investigtions hve been mde into proposing relibility models of belt drive systems. n et l. nlyzed the influences of belt length devition on the power trnsmitted by the multiple belt drive systems nd defined the filure criterion of the multiple belt drive systems. Then, relibility models were developed vi modeling the system s the weighted k-out-of-n system bsed on the universl generting function technique. Sun et l. presented new method for life clcultion of the V-belt with different relibility. Sun nd Zho 9 proposed n pproch to 1 School of Mechnicl Engineering, Lioning Shihu University, Fushun, P.R. Chin Key Lbortory of ero Power Equipment Vibrtion nd Control of Ministry of Eduction, Northestern University, Shenyng, P.R. Chin Corresponding uthor: Peng Go, School of Mechnicl Engineering, Lioning Shihu University, Fushun 1131, Lioning, P.R. Chin. Emil: gogopeng13.com Cretive Commons CC-BY: This rticle is distributed under the terms of the Cretive Commons ttribution 3. License ( which permits ny use, reproduction nd distribution of the work without further permission provided the originl work is ttributed s specified on the SGE nd Open ccess pges ( open-ccess-t-sge).

2 dvnces in Mechnicl Engineering Figure 1. Schemtic structure of belt drive. optiml design of belt drive systems by combining the conventionl crrying cpcity nlysis of belts nd relibility-bsed optimiztion design theory. These relibility models provide the theoreticl frmework for relibility evlution of belt drive systems. Conventionl relibility models of belt drive systems under the filure mode of slipping rinly sttic models with the ultimte tension nd the working stress tken s two rndom vribles. However, the fluctuting working lod nd the degrdtion of friction coefficient contribute to the dynmic behvior of relibility of belt drive systems. In this rticle, the concept of dynmic relibility is dopted to indicte tht the relibility is the function of time. Mny reserches hve been performed to investigte dynmic relibility models of vrious technologicl systems. Wng nd Xie 1 developed dynmic component relibility model using order sttistics nd probbility differentil equtions. Zhu et l. 11 presented ftigue relibility models bsed on the estimtion of the evolution nd probbilistic distribution of ftigue dmge over time. Wng et l. 1 introduced two types of dynmic relibility models considering the ftigue crck growth. Hu et l. 13 proposed method for time-dependent relibility nlysis of composite hydrokinetic turbine bldes. Furthermore, bsed on dynmic relibility nlysis, it is necessry to develop dynmic sensitivity models, which re impertive to the optiml design of belt drive systems. In this rticle, we re concentrted on the relibility estimtion under the slipping filure mode. In the opertionl process, the friction between the belts nd the pulleys, which is usully termed s effective tension, provides the power to complete intended mission. The friction cnnot exceed its ultimte vlue tht is determined by the initil tension on the belt drive. When the working lod exceeds the ultimte effective tension, slipping could occur, which significntly ffects the norml opertion of the belt drive systems. In prctice, slipping tkes plce frequently nd lwys ppers before the occurrence of ftigue frcture, which could result in lrge economic loss. Therefore, it is impertive to perform the relibility nlysis of belt drive systems under the filure mode of slipping. In current relibility models of belt drive, relibility is clculted using the stress strength interference (SSI) model. The ultimte effective tension is tken s the generlized strength, nd the working lod is treted s the generlized stress. Filure could occur in the cse where there exists n interference re between the probbility density function (PDF) of the ultimte effective tension nd the PDF of trnsmitted lod. 1 Then, relibility cn be obtined by clculting the probbility tht the system normlly opertes. However, in some circumstnces, it is difficult to directly utilize the existing relibility models for relibility ssessment of belt drive systems, which is listed in the following. Oil contmintion nd brsion could cuse the qulity decline of the contct surfces, which leds to the decrese in ultimte effective tension. Moreover, the decrese in ultimte effective tension tkes plce grdully with time. Therefore, relibility presents obvious dynmic chrcteristics. It is necessry to develop dynmic relibility models of the belt drive systems under the slipping filure mode. In ddition, the sensitivity nlysis of the relibility to design prmeters of belt drive systems, which hs not been reported, is criticl to the optiml design of belt drive. The purpose of the sensitivity nlysis is to explore the response of system relibility to the chnge of the design vribles. Most relibility-bsed sensitivity models re sttic models. Considering the dynmic chrcteristics of relibility, the sensitivity function should lso be time dependent. Dynmic sensitivity nlysis is useful for dynmic optiml design nd decision mking of mintennce nd replcement strtegies. To ddress these problems, time-dependent fctors tht contribute the dynmic chrcteristics of relibility re pointed out. Moreover, dynmic relibility models of belt drive systems re developed in this rticle. Furthermore, both sttic sensitivity models nd dynmic sensitivity models of belt drive systems re estblished. The proposed models re helpful for stble design of belt drive systems. The reminder of this rticle is orgnized s follows: In section Sttic relibility nd sensitivity models of belt drive systems, stress nlysis nd sensitivity nlysis of sttic relibility of belt drive systems re crried out. In section Dynmic relibility models of belt drive systems, dynmic relibility models nd dynmic sensitivity models of belt drive re developed. In section Numericl exmples, numericl exmples re given to illustrte the proposed models nd nlyze the influences of design prmeters on dynmic chrcteristics of relibility, filure rte, nd sensitivity functions. Finlly, conclusions re summrized in section Conclusion.

3 Go nd Xie 3 Figure. Force digrm of belt. Sttic relibility nd sensitivity models of belt drive systems In the instlltion process of the belt drive device, the belts hve to be preloded to ssure sufficient friction between the belts nd pulleys for the trnsmission of motion nd force. The initil tension is denoted by F. s shown in Figure, t the beginning of the opertion of the belt drive, belt spn 1 nd belt spn re subjected to identicl force of F. When the belt drive device strts to work, the force on belt spn 1 increses from F to F 1, while the force on belt spn decreses from F to F. Provided tht the totl length of belt keeps constnt nd the deformtion occurs within the elstic rnge, F, F 1, nd F stisfy the following condition 9 F 1 + F = F 1 When the belt drive device opertes normlly, the trnsferred lod is less thn the ultimte effective tension, nd slipping does not occur. In this cse, the vrition of force in belt spn 1 nd belt spn cn be expressed by DF = Ku 1 r 1 u r where u 1 nd u re the rottion ngles of the drive pulley nd the driven pulley, respectively; r 1 nd r represent the rdius of the drive pulley nd the rdius of the driven pulley, respectively; nd K stnds for the longitudinl stiffness of the belt. Hence, the F 1 nd F cn be further expressed s follows F 1 = F + DF = F + Ku 1 r 1 u r 3 F = F DF = F Ku 1 r 1 u r Hence, the effective tension cn be derived s follows F e = F 1 F = F + DF F DF = DF = Ku 1 r 1 u r Denote the cross section of the belt by, the stress on belt spn 1 nd belt spn cused by F 1, F, nd F e cn be clculted by s 1 = F 1 = F + K u i r i u j r j s = F = F K u i r i u j r j s e = F e = K u ir i u j r j In ddition, when the effective tension reches its ultimte vlue, slipping tends to tke plce. In this circumstnce, ccording to the Euler formul, the reltionship between F 1 nd F cn be given by 9 F 1 = 9 F where m is the friction coefficient between the belt nd the pulleys nd is the wrp ngle tht is determined by the center distnce nd the difference between the rdius of the drive pulley nd the rdius of the driven pulley. Thus, ultimte tension in belt spn 1 nd belt spn cn be expressed s F 1m = F F m = F The ultimte effective tension cn be computed s F x = F The ultimte stress cn be clculted by s x = F x = F When considering the rndomness of the working stress s e nd the ultimte stress s x, ccording to the SSI model, the performnce function cn be given by M 1 = s x s e = F em K u 1r 1 u r = F 1 + Ku 1 r 1 u r 1 From the theory of mteril mechnics, the longitudinl stiffness of belt cn be given by 1 K = E 1 where E is the elstic modulus of belt nd is the center distnce between the drive pulley nd the driven pulley. Thus, eqution (1) cn be further expressed by

4 dvnces in Mechnicl Engineering M 1 = F 1 + E u 1r 1 u r 1 For descriptive convenience, denote (u 1 r 1 u r )by s nd define rndom vector s follows! X = ½ x1, x Š= ½F, sš 1 Denote the men vlue nd stndrd devition of x i (, ) bym i nd s i, respectively. Then, the men vlue nd stndrd devition of M 1 cn be expressed s follows m M1 = m E m 1 s M1 = 1 + s 1 + E s 19 The relibility cn be further derived s follows m e R 1 = 1 P f 1 = 1 F m E 1 m s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 1 + s 1 + E C s The corresponding sensitivity with respect to the men vlue nd the devition of x 1 nd x cn be given by f 11 = R e = m m 1 pffiffiffiffiffiffi exp m E 1 3 m B C p s s s s E Dynmic relibility models of belt drive systems In prctice, oil contmintion nd brsion could result in the decline in the qulity of the contct surfces, which is reflected by the grdul degrdtion of the friction coefficient between the belt nd the pulleys. Hence, from eqution (1), it cn be lernt tht the ultimte effective tension will lso grdully decrese. The decrese in the cpcity of trnsferring working lod significntly influences the relibility of the belt drive systems. Moreover, the stress cused by externl lod could increse due to the existence of the grdul growth of the micro-crcks. To mesure the relibility vrition with time, dynmic relibility models of belt drive systems re developed in this section. Moreover, the influences of design prmeters on dynmic relibility re lso nlyzed in this section. The initil tension F is n importnt design prmeter of the belt drive, which could vry in prcticl instlltion process nd should be treted s rndom vrible. Denote the PDF of F by f F (F ). The initil distribution of s x, chrcterized by the PDF of f sx (s x ), cn be obtined vi eqution (1). Besides, the friction coefficient in the time instnt t is represented by m(t). Then, the stochstic degrdtion process of s x cn be determined by f sx (s x ) nd E f 1 = R s e = m m E m s 1 1 pffiffiffiffiffiffi e p m 1 + s 1 + E 3 exp m E 1 3 m B s 1 + s 1 + E C s 1 f 1 = R 1 = m p E ffiffiffiffiffiffi p 1 + s 1 + E exp m E 1 3 m B s 1 + s 1 + E C s E f = R s m e = m E m s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s pffiffiffiffiffiffi p 1 + s 1 + E!3 exp m E 1 3 m B s 1 + s 1 + E C s 3

5 Go nd Xie m(t). In ddition, in order to tke into ccount the vrition of the working stress, denote the PDF of the working stress in the ith opertionl cycle by (s, ). Becuse the chnge process of m(t) is slow, s x in ech opertionl cycle is considered to be constnt, nd m( ) is used to represent the vlue of m(t) in the ith opertionl cycle. The relibility of the belt drive systems fter operting for n cycles under the filure mode of slipping cn be clculted by l 11 Ð f F F S 11 F e m( ) e m( ) Ð Ð f F F R 11 f F F Y n F e m( ) e m( ) s, ds df The procedures for dynmic relibility modeling cn be summrized in Figure 3. The corresponding filure rte cn be further derived s follows s, ds df Ð F e m( ) e Ð m( ) f F F Q n + 1 F s, ds df e m( ) e m( ) Ð s, ds df Therefore, the dynmic sensitivity of the belt drive system with respect to the men vlue nd the stndrd devition of F cn be expressed s follows 9 F e m( ) 1 Ð Q f F F n 1 + e Ð m( ) s, ds df S 1 Ð f F F F mf e m( ) e m( ) Ð sf 9 s, ds df Provided tht the degrdtion of the friction coefficient is not tken into considertion, eqution () reduces to R 1 f F F Y n F s, ds df 9 The corresponding filure rte cn be derived s follows l 11 Ð f F F Ð F Ð s, ds df Ð f F F Ð F F n Q + 1 Ð s, ds df f F F s, ds df 3

6 dvnces in Mechnicl Engineering Figure 3. Procedure flowchrt for dynmic relibility modeling. Figure. Flowchrt of Monte Crlo simultion. The dynmic sensitivity of the belt drive system with respect to the men vlue nd the stndrd devition of F cn be expressed s follows S 13 S 1 Ð Ð f F F f F F Ð F mf Ð F sf 9 s, ds df 31 9 s, ds df 3 From equtions (9) to (3), it cn be lernt tht even if the friction coefficient stys constnt, the relibility could chnge with time owing to the rndomness of F nd the dynmic sttisticl chrcteristics of working stress. It should be noted tht the proposed relibility models re dependent on the degrdtion lw of the friction coefficient. For the fitting of the degrdtion lw of the friction coefficient, it is ssumed tht the friction coefficient degrdtion occurs grdully. The opertionl durtion is divided into series of subintervls with the friction coefficient in ech subintervl remining constnt. Then, curve fitting techniques re dopted to obtin the degrdtion lw of the friction coefficient by testing the friction coefficient in ech subintervl. Numericl exmples In this section, numericl exmples re given to illustrte the proposed models nd nlyze the influences of design prmeters on relibility, filure rte, nd sensitivity functions. Thteril nd geometric prmeters of the belt drive system nd the working lod prmeters re listed in Tble 1. s nd F follow the norml distribution. The degrdtion rule of the friction coefficient is expressed s follows, where i represents the number of opertionl cycles m = : 1 i 1 33

7 Go nd Xie Tble 1. Mteril nd geometric prmeters of the belt drive nd lod prmeters. m(f ) (N) s(f ) (N) (mm) E (MP) (mm ) m(s) (mm) s(s) (mm) Figure. Relibility from Monte Crlo simultion nd relibility from the proposed models. Figure. Filure rte of belt drive. Figure. Relibility of belt drive. To vlidte the proposed dynmic relibility models, Monte Crlo simultion is crried out in this section with the flowchrt shown in Figure. The results from the Monte Crlo simultion nd the results from the proposed models re shown in Figure. From Figures nd, it is lernt tht the simultion results show good greement with the proposed model. In ddition, the sttic relibility, the dynmic relibility without degrdtion of friction coefficient, nd the relibility considering friction coefficient degrdtion re shown in Figure. The filure rte without degrdtion of friction coefficient nd the filure rte Figure. Relibility of belt drive in cses of different dispersions of initil tension. considering degrdtion of friction coefficient re shown in Figure. From Figures nd, it cn be seen tht in the relibility estimtion under the filure mode of slipping, it could cuse lrge error to utilize trditionl relibility models due to neglecting the time-dependent fctors tht hve gret influences on relibility. The ultimte effective tension grdully decreses owing to friction coefficient degrdtion, which obviously ccelertes the drop speed of relibility. Besides, even without the friction coefficient degrdtion, relibility could decline due to the time-dependent sttisticl chrcteristics of working stress in different time periods. These

8 dvnces in Mechnicl Engineering Figure 9. Filure rte of belt drive in cses of different dispersions of initil tension. time-dependent fctors contribute the dynmic chrcteristics of relibility nd should be tken into considertion in the optiml design of belt drive device. In ddition, friction coefficient degrdtion results in n obvious increse in the filure rte of belt drive system nd mkes the filure rte curve present the feture of the wer-out period of the bthtub curve, while only the feture in the infnt mortlity period nd the rndom filure period of the bthtub curve is found in the filure rte curve of the belt drive system without friction coefficient degrdtion tken into considertion. In ddition, considering the friction coefficient degrdtion, the relibility nd the filure rte in the cses of different dispersions of F re shown in Figures nd 9, respectively. The dispersion of F comes from the instlltion process nd mnufcture process of the belt drive. From Figures nd 9, it cn be seen tht the dispersion of F hs considerble effects on relibility nd filure rte. The increse in the dispersion of F mkes the relibility lower nd the filure rte higher, which significntly ffects the effectiveness of the originl design of the belt drive system. Thus, in prcticl instlltion process nd mnufcture process, ttention should be pid to lower this dispersion to ensure the belt drive systems normlly complete their intended mission. dditionlly, the sensitivity of sttic relibility with respect to the men vlue nd stndrd devition of F is shown in Figures 1 nd 11, respectively. The sensitivity of the dynmic relibility with respect to the men vlue nd stndrd devition of F is shown in Figures 1 1, respectively. In generl, n intervl of design vrible with low sensitivity is desirble for robust system design. In this section, sttic sensitivity curves presented in the numericl exmples cn provide guidnce for reders to find the robust re for system design. From Figures 1 1, it cn be lernt tht there exists lrge difference between Figure 1. Sensitivity of the sttic relibility with respect to m(f ). Figure 11. Sensitivity of the sttic relibility with respect to s(f ). Figure 1. Sensitivity of the dynmic relibility with respect to m(f ) without friction coefficient degrdtion.

9 Go nd Xie 9 Figure 13. Sensitivity of the dynmic relibility with respect to s(f ) without friction coefficient degrdtion. Figure 1. Sensitivity of the dynmic relibility with respect to s(f ) with friction coefficient degrdtion. Figure 1. Sensitivity of the dynmic relibility with respect to m(f ) with friction coefficient degrdtion. the sensitivity of sttic relibility nd the sensitivity of dynmic relibility with respect to the sttisticl prmeters of F. The influences of m(f ) on relibility vry with the chnge in m(f ). Generlly, in the cse of smll m(f ), the sensitivity of relibility with respect to m(f ) decreses with time, while the sensitivity of relibility with respect to m(f ) increses with time in the cse of lrge m(f ). Menwhile, t fixed time instnt, for smll m(f ), the sensitivity of relibility with respect to m(f ) increses with the increse in m(f ), while the sensitivity of relibility with respect to m(f ) decreses with the increse in m(f ). In ddition, t fixed time instnt, the sensitivity of relibility with respect to s(f ) increses with the increse in s(f ). Furthermore, the sensitivity of relibility with respect to s(f ) increses with time. Compred with the sitution where friction coefficient remins constnt, the sttisticl prmeters of F hve more significnt influences on dynmic relibility in the cse where friction coefficient degrdes. Conclusion In this rticle, the sensitivity models of the belt drive bsed on trditionl relibility models re proposed in this rticle, which cn be used to nlyze the influences of the sttisticl properties of design prmeters on sttic relibility nd optiml design of belt drive. Furthermore, time-dependent fctors tht contribute the dynmic chrcteristics of relibility re pointed out, nd dynmic relibility models s well s filure rte models of belt drive systems re developed. When considering the rndom lod fluctution nd the degrdtion of friction coefficient, dynmic vrition of system sensitivity with time hs to be tken into ccount. Thus, dynmic sensitivity models of the belt drive re estblished, which cn be used to clculte the system sensitivity in different time periods of opertion nd demonstrte the time-dependent vrition of the robust re for life-cyclngement nd design of belt drive systems. Besides, numericl exmples re given to illustrte the proposed models nd nlyze the influences of design prmeters on dynmic chrcteristics of relibility, filure rte, nd sensitivity functions. The results show tht the sttisticl properties of design prmeters hve different influences on relibility nd filure rte of belt drive in cses of different vlues of design prmeters nd different opertionl durtions. Therefore, dynmic influences of the design prmeters on system relibility nd filure rte in the full life cycle of belt drive should be tken into considertion, which re importnt in the decision mking of design, mnufcturing, instlltion, nd mintennce of belt drive. The equivlent coefficient of friction of the belt drive system is ffected by mny fctors. In this rticle, the trnsmission rtio nd the speed of belt trnsmission, which hve significnt influences on equivlent coefficient of friction, re ssumed to be deterministic when performing relibility nlysis. further investigtion

10 1 dvnces in Mechnicl Engineering of the influences of these key fctors on relibility of belt drive systems will be crried out in our next work. Declrtion of conflicting interests The uthor(s) declred no potentil conflicts of interest with respect to the reserch, uthorship, nd/or publiction of this rticle. Funding The uthor(s) disclosed receipt of the following finncil support for the reserch, uthorship, nd/or publiction of this rticle: This work is supported by Ntionl Nturl Science Foundtion of Chin (grnt no. 1), Scientific Reserch Fund of Lioning Provincil Eduction Deprtment of Chin (grnt no. L19), Lioning Provincil Nturl Science Foundtion of Chin (grnt no. 11), nd Ntionl Nturl Science Foundtion of Chin (grnt no. 1333). References 1. Rvindr V, Pdmnbhn C nd Sujth C. Sttic nd free vibrtion studies on pulley-belt system with ground stiffness. J Brz Soc Mech Sci 1; 3: 1.. Png X, Li J nd Kou Z. Longitudinl vibrtions of the viscoelstic moving belt. Shock Vib 1; 1: Ding H. Periodic responses of pulley belt system with one-wy clutch under inerti excittion. J Sound Vib 1; 33: Liu CS nd Chng WT. Frictionl behviour of beltdriven nd periodiclly excited oscilltor. J Sound Vib ; :.. Singru PM nd Modk JP. Computer simultion of the dynmic nd vibrtion response of belt drive pulley. J Sound Vib 1; : 93.. Iwtsubo T, Hsegw K, rii S, et l. The formultion nd dynmic nlysis of multiple belt system. J Sound Vib 199; : n ZW, Hung HZ nd Lin D. n pproch to relibility evlution of multiple V-belt drives considering the devition of belt length. Proc IMechE, Prt O: J Risk nd Relibility 9; 3: Sun C, Ren, Sun G, et l. The clcultion of the Clssicl V-Belt life with different relibility. Proced Eng 11; 1: Sun Z nd Zho N. ppliction of relibility design in the belt drives. Mech Des Mnuf 1991; : (in Chinese). 1. Wng Z nd Xie L. Dynmic relibility model of components under rndom lod. IEEE T Relib ; : Zhu SP, Hung HZ, Li Y, et l. Probbilistic modeling of dmge ccumultion for time-dependent ftigue relibility nlysis of rilwy xle steels. Proc IMechE, Prt F: J Ril nd Rpid Trnsit 1; 9: Wng KS, Chng ST nd Shen YC. Dynmic relibility models for ftigue crck growth problem. Eng Frct Mech 199; : Hu Z, Li H, Du X, et l. Simultion-bsed timedependent relibility nlysis for composite hydrokinetic turbine bldes. Struct Multidiscip O 13; : Xie L, Zhou J nd Ho C. System-level lod strength interference bsed relibility modeling of k-out-of-n system. Relib Eng Syst Sfe ; : Gere JM nd Goodno BJ. Mechnics of mterils. Cengge Lerning: Independence, KY, 9.