POWER FLOW AND EFFICIENCY ANALYSIS OF A MULTI-PATH TRANSMISSION WITH PLANETARY GEAR TRAIN

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1 International Journal of Mechanical Engineering an Technology (IJMET) Volume 9, Issue 13, December 2018, pp , Article ID: IJMET_09_13_043 Available online at ISSN Print: an ISSN Online: IAEME Publication Scopus Inexe POWER FLOW AND EFFICIENCY ANALYSIS OF A MULTI-PATH TRANSMISSION WITH PLANETARY GEAR TRAIN Essam L. Esmail Assistant Professor, University of Al-Qaisiyah Diwaniyah, Iraq Anahe H. Juber Assistant Professor, University of Al-Qaisiyah, Diwaniyah, Iraq Hame A. Hussen Assistant Professor, Mile Technical University, Bagha, Iraq ABSTRACT Power flow through ifferent paths of planetary gear trains may result in power amplification when two powers meet together or it may lea to power re-circulation when the power flows in a close path. A new easy way is suggeste to know the irection of the flow of power an to ensure the presence or absence of power circulation. A typical compoun planetary gear arrangement is stuie to illustrate the application of the current methos. The power flow analysis showe that there is power re-circulation through the train, leaing to power amplification. For this reason the total efficiency is very low although local efficiencies are high. Detaile analysis of power re-circulation is provie. Conitions for no power circulation are etermine an new configuration are propose. The analytical expressions for the powers, power losses an efficiency are erive. In general, by carefully controlling the gear ratios, the flow of power can change its irection, thus improving the overall efficiency. Keywors: The power flow analysis, the gear ratios Cite this Article: Essam L. Esmail, Anahe H. Juber an Hame A. Hussen, Power Flow an Efficiency Analysis of a Multi-Path Transmission with Planetary Gear Train, International Journal of Mechanical Engineering an Technology, 9(13), 2018, pp INTRODUCTION Planetary gears are wiely use to transmit power in machinery an vehicles. The efficiency of planetary gear trains is sometimes low compare to conventional gear trains. This is ue to the fact that the power can flow through ifferent paths of the train. Power flow through ifferent paths may result in power amplification when two powers meet together or it may lea to power re-circulation when the power flows in a close path. Power amplification an power re eitor@iaeme.com

2 Essam L. Esmail, Anahe H. Juber an Hame A. Hussen circulation are very unesirable phenomena in PGTs because they increase friction losses, reuce total efficiency an weaken the strength of gears. Power recirculation in specific power split hybri transmissions is analyze by [Schulz, Villeneuve]. Schulz [1] analyze the operating ranges of a Dual-E transmission to ensure that the power is not circulate. Villeneuve [2] inspecte the power split characteristics of the transmission evelope by Renault. Namoo Kim et.al. [3] Investigate the power flow in ual moe power split transmission, an foun that there is power circulation in ual moe transmission. They also foun that when power circulation happens the efficiency ecreases. White [4] analyze power recirculation in one egree of freeom PGTs to ientify the conitions for least power losses. Nevertheless, it i not aress the stuy of power losses or transmission efficiency. Gupta et al. [5] iscusse a metho for eucing the power flow irections in two egree of freeom PGTs. Unfortunately, their analysis inclues some basic errors [6]. In our previous work [6] which inclues some comments on the work of Gupta et al. [5], we specifie the conitions that will cause power re-circulation in two inputs PGTs. It is true that power recirculation is unesirable in PGTs, but the power loss, an important concern in PGTs, must be carefully calculate to know to what extent its value will affect the system. Most of the previous stuies, incluing our stuies [1-6] isregare the calculation of power losses an efficiency analysis an focuse on power-recirculation. However, etermining power losses an estimating the efficiency are necessary to etermine the conitions uner which the system shoul operate efficiently. Esmail evelope a methoology base on nomography for the power-circulation analyses of planetary gear trains in the absence of friction [7]. The main avantage of the metho is that it allows quickly moifying the system into an alternative which is free of power-recirculation. A new easy way is suggeste to know the irection of the flow of power an to ensure the presence or absence of power circulation. A typical planetary gear arrangement, shown in Figure 1, is synthesize to illustrate the application of the current metho. It consists of two interconnecte gear train entities (GTEs) with one input, one output, an a fixe link. The planetary GTE is forme from gear pairs one (GP1) an two (GP2) along with the planet carrier 2, while the orinary gear train is forme from GP3 an GP4 along with the fixe planet carrier f. Figure 1 A typical planetary gear arrangement consisting of two interconnecte gear train entities eitor@iaeme.com

3 Power Flow an Efficiency Analysis of a Multi-Path Transmission with Planetary Gear Train Many transmission systems use in reality are similar to the one shown in Figure 1, an can be broaly represente as shown in Figure 2. The egree of freeom (DOF) of the system is controlle actively by connecting the three basic links of the gear pair entities in various manners. Since one of the three links of the compoun PGT (Figure 2) is fixe, the DOF of the train is reuce to one. Figure 2 Two gear train entities connecte in parallel with a fixe link. 2. NOMOGRAPH The PGT shown in Figure 1 consists of two GTEs, one of which is a PGT, the other is a conventional train. The nomograph of the planetary part of the gear train is shown to the left of Figure 3, while the nomograph of the orinary gear train is shown to the right. For more etails on how to raw a nomograph for a gear train, one can refer to references [8-11]. In general, the nomograph is rawn epening on the planet gear ratios. For any gear pair entity, the planet gear ratio is expresse mathematically as follows: R, = = (1) Z An Z are the number of teeth on gears q an p, respectively anω, ω an ω are the velocities of a p, q, an c, respectively. The value of R, an R, can be obtaine from equation (1) an are equal to 0 an 1, respectively. Figure 3 Nomographs of the planetary an conventional parts of the mechanism in Figure 1. Links 1, 2, 3, an 4 are compoun links consisting of two elements on the same shaft. The rotational velocity of compoun links is the same as its constituent elements. This means thatω =ω,ω =ω,ω =ω, an ω =ω. The otte lines are rawn between the two nomographs in the common links 2 an 4. Depening on the rotational velocities of compoun links, a unifie nomograph can be obtain by shifting the axes of the links other than the compoun links from the right nomograph to the left one or vice versa eitor@iaeme.com

4 Essam L. Esmail, Anahe H. Juber an Hame A. Hussen Figure 4 The unifie nomograph of the planetary mechanism shown in Figure 1. The otte lines are rawn where the axes to be shifte. Figure 4 shows the unifie nomograph for the planetary gear train uner consieration. When an are two share links, the R, can be foun from the following Equation [12]: R, =,,,,! ( 2 ) Let" =# #," =# #," =Z % Z, an " =Z % Z, then from Eq. (2), we can write An =' R,& ( ) * )+, (3) =- R,& ) ) ) ) ( ) * )+. (4) The overall reuction ratio is: / 0 =/ 1, = ( ) * )+ ) ) ( ) * )+ (5) / 0, R,&, an R,& epens on the gear ratios ","," an " which epen only on the number of teeth on the gears. 3. POWER FLOW THROUGH THE MECHANISM Accoring to Esmail's metho [13], the torque vector pointing upwar is positive while the ownwar torque vector is negative. Also the velocity is positive above the zero line an negative below it. Thus, an input power is a result of multiplying positive torque at positive velocity, or negative torque at negative velocity. When gear 2 > gear 3 an gear 4 5 > gear2 5, the torque vectors are rawn in Figure eitor@iaeme.com

5 Power Flow an Efficiency Analysis of a Multi-Path Transmission with Planetary Gear Train Figure 4 The power balance of the planetary mechanism shown in Figure 1. The torque vectors at ω an ω both are negative while ω an ω are positive, this means that the power is flowing out of links 1 an 2 of the first gear train entity (GTE1). This can be best shown on the GTEs iagram shown in Figure 2. Since link 2 (2') is a common link between GTE1 an GTE2, then the output power from GTE1 (P ) is the input power to the GTE2 (P ). This implies that Link 4 5 must be an output link anp +P =0. Figure 5 Power flow through the mechanism. For the present configuration, there is power circulation through links4,3 5,2,2 5,5,5 5,4 5, an 4. Also, there is power amplification through links4, an 3 5 where the power coming from planet gear 5 converges with the power of the motor, but it soon splits into two parts, one goes to the output link (link 1) an the other returns back to recirculate again. Figure 6 Power circulation through the mechanism. 4. EFFICIENCY FORMULA DERIVATION The efficiency of a conventional GTE can be written as [14] as: η (6)? CD eitor@iaeme.com

6 Essam L. Esmail, Anahe H. Juber an Hame A. Hussen Esmail [14] efine the carrier-potential power is the power that will evelop in a link when the carrier is relatively fixe. When the carrier is stationary, the carrier-potential power is the same as the actual power that passes through the train. For GTE2, since carrier E is fixe then the efficiency can be written as: η η =? *? (7) The power loss L can be estimate simply as follows [15]: L= Hη IJKL 1NP J (8) For GP3 an GP4, the losses may be written as [16]: L =Iη 1L P an L = Iη 1L η P, therefore: L +L = Iη η 1L P (9) For GTE1,? O J =R?,& O an Eq. (6) can be re-written as: L= Hη IJKL 1NR J,& P J (10) Where link x in the input link in the moving reference frame (MRF). Figure 7 shows the planetary part of the gear train in the fixe reference frame FRF an MRF. In both frames link 4 is an input an link 1 is an output link. Figure 7 The planetary part of the train in the FRF an MRF. Accoring to Eq. (6), the efficiency of the PGT can be written as: P P = Q Q (11) * But Q * = / Q,1 an Q = / * Q,1, therefore Eq. (11) can be written as: P P = Q Q * * (12) Using Eq. (10), the power losses can be written as: S = I1 P L T /,1 IP P P L T /,1. Combining these two equations, the power losses is: ans = eitor@iaeme.com

7 Power Flow an Efficiency Analysis of a Multi-Path Transmission with Planetary Gear Train S +S =IP P 1 L T /,1 (13) The total power losses L is S=S +S +S +S =IP P 1LT /,1 +IP P 1 L P (14) For GTE1 an uner steay-state operation conitions, the input power is equal to the output power(s) plus the power losses: T = IS +S +P +P L (15) By substituting T from Eq. (12) an IS +S L from Eq. (13) into Eq. (15) an simplifying, T can be written in terms of T as: T = IU U L U U /,1 + ButT = T, then T = IU U L U U /,1 U U + * 1!T (16) U U * 1!T (17) By substituting T from Eq. (12) an T from Eq. (17) into Eq. (14), we get: S=V IP P 1L IP U U P 1 L IU U L / U U,1 + U U * 1!WT (18) The efficiency is written as follows: η X = Y Z (19) By substituting T \]^ =T an S from Eq. (18) into Eq. (19) an simplifying, we get: η X = U + U * Y _ `` aiu U LU + U * Y I(`+`* L _ * b (20) The overall efficiency epens on the local efficiencies of the gear pairs (P, P, P, P ) an on their gear ratios (",","," ). 5. RESULTS AND DISCUSSION: 5.1. Total Efficiency The overall reuction ratior c, which ranges from 0 to 1, epens only on the gear ratios an * +. Both gear ratios an * + are larger than one. Figure 8 shows the 3D-plot among the reuction ratior c, an * eitor@iaeme.com

8 Essam L. Esmail, Anahe H. Juber an Hame A. Hussen Figure 8 The 3D-plot among the reuction ratior c, an * +. From Eq. (5), the overall reuction ratio will increase when e an e * ecreases but the effect e e + of e is ominant as seen from Figure 8. e Since both of the overall reuction ratio an the total efficiency epen on the gear ratios an *, we can then explain how the overall reuction ratio affects the total efficiency. + Figure 9 The 3D-plot among the total efficiencyη X, reuction ratior c, an. Figure 9 shows that when the total reuction ratio is about 0.5, the overall efficiency ecreases regarless of the value of the gear ratios,, while the efficiency remains high for all values e * e +, as seen from Figure 10. From these figures, it is clear that the efficiency ecreases rapily with eitor@iaeme.com

9 Power Flow an Efficiency Analysis of a Multi-Path Transmission with Planetary Gear Train Figure 10 the 3D-plot among the total efficiencyη X, reuction ratior c, an e * e Power circulation Let us assume that the gear ratios are:" = 36 28," = 28 36, " = 30 34, an " = Then from Eqs. (3) An (4),& = an R,& = The local efficiencies are assume to be the same an have a value of 0.9. From Equation (20), the total efficiency can be calculate asη X =25.14%. ForT j\^\0 =1, thent klml0n^\0 =T = ; the negative sign refers to an output power. From Eq. (17), (7), (12) an S the powers are calculate ast =3.152,P = , T = an T = The gear pair losses are:s = ,S = , L = an L = Figure 11 shows the power flowing through the PGT an the power losses. Figure 11 power flow an power losses through the PGT eitor@iaeme.com

10 Essam L. Esmail, Anahe H. Juber an Hame A. Hussen As expecte, there is power circulation through the train, leaing to power amplification. The amplification power is approximately 3 times the input power so that the losses are proportional to the amplifie power in the loop in which they circulate. The output power from link 4 5 is the input power to link 4 with opposite sign. The power entering link 4 from 4 5 meets with the input power from the motor an thus amplifies the power at link 4 (T =T + T ). More than half of the input power is lost in gear pairs 3 an 4, where the amplifie power goes through. For this reason the total efficiency is very low although local efficiencies are high Conitions for no power circulation When gear 3 > gear 2 an gear 4 5 > gear2 5, an since both of R, an /, are negative, then R, >/, an the power balance of the planetary mechanism can be visualize as shown in Figure 12. Figure 12 The power balance of the planetary mechanism when Gear 1 > gear 4 an gear 2 5 > gear4 5. Now, the torque an angular velocity of link 2 both are positive, this means that the power is flowing into the first gear train entity (GTE1), which implies that link 4 5 must be an input link. Let us assume that" = 28 36, an" = 36 28, while", an " are as before. Uner this conition, power will not circulate an will not be amplifie. The total efficiency is over 93% which excees the local efficiencies that are assume to be 0.9. Figure 13 shows a GTE iagram on which the powers flowing through the elements of the system an the power losses are inicate. Figure 13 Power flow through the mechanism. Figure 14 shows the planetary mechanism shown in Figure 1 when gear 2 > gear 3 an gear 2 5 is an internal gear eitor@iaeme.com

11 Power Flow an Efficiency Analysis of a Multi-Path Transmission with Planetary Gear Train Figure 14 The planetary mechanism shown in Figure 1 when Gear 4 > gear 1 an gear 4 5 is an internal gear. Figure 15 shows the power balance of the planetary mechanism when gear 4 > gear 1 an gear 4 5 is an internal gear. Figure 15 The power balance of the planetary mechanism when Gear 4 > gear 1 an gear 4 5 is an internal gear. Clearly, the power flow irections is as shown in figure 13. In general, by carefully controlling the gear ratios, the power flow in the system can be change, thus improving the overall efficiency. 6. CONCLUSIONS A simple algorithm has been propose to know the irection of the power flow an to ensure the presence or absence of power circulation. A typical planetary gear arrangement has been analyze to illustrate the application of the propose methos. The analytical expressions for the powers, power losses an efficiency are erive. As expecte, there is power circulation through the train, leaing to power amplification. For this reason the total efficiency is very low although local efficiencies are high. Conitions for no power circulation are etermine an new configuration are propose. In general, by carefully controlling the gear ratios, the power flow in the system can be change, thus improving the overall efficiency eitor@iaeme.com

12 Essam L. Esmail, Anahe H. Juber an Hame A. Hussen REFERENCES [1] M. Schulz, CIRCULATING MECHANICAL POWER IN A POWER-SPLIT HYBRID ELECTRIC VEHICLE TRANSMISSION. Proceeings of the institution of mechanical engineers, Part D: J. Automobile Eng. 218 (2004) [2] A. VILLENEUVE, DUAL MODE ELECTRIC INFINITELY VARIABLE TRANSMISSION, SAE 04CVT-19. [3] NAMDOO KIM, JEONGMIN KIM, AND HYUNSOO KIM, CONTROL OF DUAL MODE POWER SPLIT TRANSMISSION FOR A HYBRID ELECTRIC VEHICLE. The Worl Electric Vehicle Journal, Volume: 2, Issue: 4, Pages: DOI: /wevj [4] G. WHITE, DERIVATION OF HIGH EFFICIENCY TWO-STAGE EPICYCLIC GEAR. Mech. Mach. Theory 38 (2003) [5] A.K. GUPTA, C.P. RAMANARAYANAN, ANALYSIS OF CIRCULATING POWER WITHIN HYBRID ELECTRIC VEHICLE TRANSMISSIONS. Mech. Mach. Theory 64 (June) (2013), Pages: [6] ISAM ELAIBI ISMAEL, Comments on Analysis of circulating power within hybri electric vehicle transmissions. Mechanism an Machine Theory 64 (2013) , by A.K. Gupta, C.P. Ramanarayanan. Mechanism an Machine Theory, Volume: 121, Pages: [7] ESMAIL, E. L., AND HASSAN, S. S., AN APPROACH TO POWER-FLOW AND STATIC FORCE ANALYSIS IN MULTI-INPUT MULTI-OUTPUT EPICYCLIC-TYPE TRANSMISSION TRAINS. ASME J. Mech. Des., 132(1), p [8] ESSAM L. ESMAIL, TAMATHER N. ALI, HAMED A. HUSSEN an ANAHED H. JUBER, KINEMATIC ANALYSIS OF AUTOMATIC TRANSMISSIONS WITH PLANETARY GEAR. International Journal of Mechanical Engineering & Technology (IJMET) - Scopus Inexe. Volume: 9, Issue: 11, Pages: [9] ESMAIL EL, NOMOGRAPHS AND FEASIBILITY GRAPHS FOR ENUMERATION OF RAVIGNEAUX-TYPE AUTOMATIC TRANSMISSIONS. Avances Mechanical Engineering, Volume 2013, 2013, Article number oi: /2013/ [10] E. L. ESMAIL, NOMOGRAPHS FOR SYNTHESIS OF EPICYCLIC-TYPE AUTOMATIC TRANSMISSIONS, Meccanica, 48 (8) (2013) DOI /s z. [11] ESMAIL, E. L., NOMOGRAPHS FOR ENUMERATION OF CLUTCHING SEQUENCES ASSOCIATED WITH EPICYCLIC-TYPE AUTOMATIC TRANSMISSION MECHANISMS. ASME International Mechanical Engineering Congress an Exposition Proceeings, Volume 17, 2009, Pages [12] ESMAIL, E. L., A UNIVERSAL KINEMATIC ANALYSIS OF GEARED MECHANISMS. J. Braz. Soc. Mech. Sci. Eng., 39(6), pp [13] E. L. ESMAIL, H. A. HUSSEN, NOMOGRAPHS FOR KINEMATIC, STATICS AND POWER FLOW ANALYSIS OF EPICYCLIC GEAR TRAINS. ASME International Mechanical Engineering Congress an Exposition (IMECE09), Lake Buena Vista, Floria, USA, Volume 13, paper No , pp , November [14] ESMAIL, E. L., MESHING EFFICIENCY ANALYSIS OF TWO DEGREE-OF- FREEDOM EPICYCLIC GEAR TRAINS. ASME J. Mech. Des., 138(8), p [15] ESMAIL E.L., PENNESTRÌ E., AND JUBER A.H., POWER LOSSES IN TWO- DEGREES-OF-FREEDOM PLANETARY GEAR TRAINS: A CRITICAL ANALYSIS OF RADZIMOVSKY S FORMULAS. Mech. Mach. Theory, 128(2), 2018, pp [16] PENNESTR_I, E., AND VALENTINI, P. P., A REVIEW OF FORMULAS FOR THE MECHANICAL EFFICIENCY ANALYSIS OF TWO DEGREES-OF-FREEDOM EPICYCLIC GEAR TRAINS. ASME J. Mech. Des., 125(3), pp eitor@iaeme.com