Efficiency evaluation of gearboxes for parallel hybrid vehicles Theory and applications

Ef ficiency evaluation of gearboxes for parallel hybrid vehicles:Theory and applicationsEttore Pennestrìa,⁎,Lorenzo Mariti a ,Pier Paolo Valentini a ,Victor H.Mucino ba Dipartimento di Ingegneria Meccanica,Universitàdi Roma Tor Vergata,via del Politecnico,100133Roma -Italy bMechanical and Aerospace Engineering Department,West Virginia University,Morgantown,WV26506-6106USAa r t i c l e i n f o ab s t r ac tArticle history:Received 21February 2011Received in revised form 16October 2011Accepted 25October 2011Available online 26November 2011In this investigation is presented a systematic approach for the modelling and analysis of power split transmissions which include an epicyclic gear train,in various configurations,as they are used in hybrid vehicles.Emphasis is placed on the efficiency of the epicyclic gear trains and the associated power-flow in the transmission.The approach is based on the graph based representation of the kinematic chains and numerical examples are provided to further illustrate the applicability to hybrid vehicle transmissions with epicyclic gear trains and CVT elements.The graph-theory approach is shown to be a practical way to discern all possible configurations and their associated efficiencies.©2011Elsevier Ltd.All rights reserved.Keywords:DrivetrainHybrid vehiclesMechanical efficiency Planetary gears CVT1.IntroductionThe efficiency of gear trains has acquired added significance in industry in the light of current trends for efficient vehicles,ligh-ter engines and the proliferation of hybrid vehicles design concepts.The efficiency of gear trains depends on many factors such as the type and number of gears,the number and type of bearings and friction elements used to transfer power from a power source to specific outputs.In the case of hybrid vehicles two sources of power may be employed requiring systems with two or more kinematic degrees of freedom (such as differentials or planetary gear trains).Studies by Martin [1],by Anderson and Loewenthal [2],by Kahraman et al.[3,4],among others,provided models for mechan-ical efficiency analysis of ordinary gear trains.While the efficiency of gear elements has been established for various types of gears (spur,helical,worm,hypoid etc.),the combined efficiency of these gears in the arrangement of a planetary of differential gear trains has not yet been fully deployed in a systematic approach.Some studies by Hedman [5],del Castillo [6],Nelson and Cipra [7],Chen and Angeles [8]and,more recently,by Mathis and Remond [9],Chen and Liang [10]have effectively addressed the efficiency of epicyclic gear trains.However,a systematic approach,expanded from previous work by the authors [11],that can be specifically applied to parallel transmissions used in hybrid vehicles,is a much needed capability for the efficient design of gear train systems in industry today,and is the aim of this paper.In planetary or differential gear trains power recirculation or a power split configuration occurs.In the former,parasite power recirculation may further and significantly reduce the efficiency [12,13].In the case of continuously variable transmission (CVT)applications,with planetary or differential gear sets,the issue of power recirculation plays a very important role in the efficiency as parasite power recirculation does not produce useful output work [14].Mechanism and Machine Theory 49(2012)157–176⁎Corresponding author.E-mail addresses:pennestri@mec.uniroma2.it (E.Pennestrì),mariti@ing.uniroma2.it (L.Mariti),valentini@ing.uniroma2.it (P.P.Valentini),victor.mucino@ (V.H.Mucino).0094-114X/$–see front matter ©2011Elsevier Ltd.All rights reserved.doi:10.1016/j.mechmachtheory.2011.10.012Contents lists available at SciVerse ScienceDirectMechanism and Machine Theoryj ou r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /m e c h mt158 E.Pennestrìet al./Mechanism and Machine Theory49(2012)157–176Hybrid vehicles(HVE)use different power sources with various powertrain configurations,including internal combustion en-gines,electric motors,electric generators,batteries and belt transmissions.When a CVT element is used,the two power sources of hybrid configuration can have variable speed,yet,a variable output speed is also possible through the CVT variator while maintaining the engine speed constant.Gear trains installed on car models such as Toyota Prius and Opel Astra can be identified as planetary gear hybrid powertrains(PGHP).Because of the outstanding performances in fuel savings and exhaust emissions,PGHP are considered among the most promising alternatives of powertrain solutions on hybrid vehicles.Technical literature provides several`focused on the mechanical efficiency of powertrains for hybrid vehicles.However,some of these thoughtful contributions consider the mechanical efficiency of the planetary gears to be constant[15]or equal to unity [16].An effective evaluation of a PGHP configuration requires both a kinematic power-flow and a mechanical efficiency analysis. This paper provides all the theoretical tools necessary for such analyses.The graph based computation approach herein discussed requires the minimum amount of data and can be applied to PGHPs with complex topological structures.In the proposed approach power losses due to churning of lubricant,friction in the clutches and bearings are neglected.The only power power loss considered is due to gear meshing under stationary working conditions.In the presence of a belt CVT,the variable efficiency of this component can be included in the computations.Hence,the purpose of this paper is to provide a systematic method of gear train analysis that can be applied to a variety of PGHP.The methods herein proposed are an adaptation to PGHP of those presented in Pennestrìand Freudenstein[17,11].All basic equations are written in an algebraic simpler format.Moreover,a detailed discussion on the mechanical efficiency of ordi-nary gear train kinematic inversions is offered.Mechanical efficiency analysis of some innovative PGHPs is herein presented for the first time.The sequence of topics in the paper is organized as follows:-description of the main topological properties of the gear trains analysed;-discussion of graph based methods for kinematic and power flow analysis of epicyclic gear trains;-mechanical efficiencies of inversions of a simple ordinary gear train unit;-mechanical efficiency of a basic two degrees of freedom basic gear train;-discussion of a graph based method for mechanical efficiency analysis of epicyclic gear trains;-applications of such method to PGHPs.2.Representation and topological properties of gear trainsThe topological structure of a mechanism can be rationally represented by means of a graph.According to this representation, links are denoted by labelled vertices and kinematic pairs by means of edges.These are also labelled to represent their nature(e.g. R:revolute joints,G:gear joints,etc.).In order to identify the level of the axis of the revolute pairs,in the graph representation of a gear train,an extra label is added to the symbol R.Fig.1offers an example of graph representation for a basic epicyclic gear train(BEGT),where the labels R(a)end R(b)denote revolute pairs whose axes are at level a and b,respectively as shown.One of the most useful consequences of the correspondence between mechanisms and graphs is the extension of relationships from graph theory to mechanisms.This investigation is limited to gear trains:-obeying the Gruebler-Kutzbach degree-of-freedom equation;-the number of gear pairs is equal to the number L ind of independent circuits;-represented by planar graphs whose topological features are outlined by Freudenstein[18].Fig.1.Basic epicyclic gear train and its graph representation.For this class of PGHPs,each gear pair corresponds to a fundamental circuit[19,20]where there exists only one vertex(called the transfer vertex)separating revolute pairs on two different axis levels.The mechanical counterpart of a transfer vertex is the gear or planet carrier.3.Kinematic and power-flow analysisThe underlying concept of our approach is that the fundamental circuit corresponds to a basic epicyclic gear train(see Fig.1). Thus the partition of the PGHP graph into fundamental circuits provides evidence for the elementary components of the PGHP. The solution of the system formed by all equations valid for each elementary component(i.e.the basic epicyclic gear train) and by other equations which follow from phisycally consistent conditions,allows a detailed and systematic analysis of the PGHP.This reasoning is along the same line as the pioneering studies about the applications of network concepts to the analysis of gear trains due to Polder[21,22],Freudenstein[18]and Hedman[23,5].3.1.Kinematic analysisThe representation of a gear train by means of a graph allows the prompt deduction of the equations for kinematic analysis [18,24-27].For each basic epicyclic gear train in the PGHP one can write the Willis’equation as followsωi−R mωjþR m−1ðÞωk¼0;ð1Þwhereωi,ωj andωk denote the angular velocities of the wheels i,j and gear carrier,respectively,and the gear ratioR m¼Æz jmz im¼ÆNumber o f t eeth o f w heel jNumber o f t eeth o f wheel i;ð2Þwith m=1,2,…,L ind.The method of kinematic analysis of the PGHP involves the following steps:I.Draw and label the graph of the PGHP.II.Remove all the kinematically redundant gear pairs and identify all the fundamental circuits.III.For each fundamental circuit obtain the meshing wheels i and j,the gear carrier k and the gear ratio.IV.For each fundamental circuit particularize Willis’equation(1).V.Set equal to zero the angular velocity of the frame link and prescribe the angular velocities of driving links.VI.Solve the system of equations just obtained.The belt CVT should be considered as a BEGT where the gear carrier is fixed and the gear ratio R m is set to the current velocity ratio between the two pulleys.3.2.Power-flow analysisIn compound epicyclic gear trains either power split or recirculating power configuration occurs.Since the recirculating power may exceed the transmitted power,this condition should be avoided in order to prevent low mechanical efficiency.Buckingham reports a case where the overlooked excessive amount of recirculating power caused the catastrophic destruction of a planetary gear train[28].This type of analysis is typically carried out assuming the absence of power losses.In fact,usually these losses are not large enough to change the pattern of power-flow distribution significantly in planetary gear transmissions.This problem has been addressed by several authors(e.g.[13,29-34]).Because of its generality and the similarities with the kinematic analysis method discussed in the previous section,the method proposed by Pennestrìand Freudenstein[35,17]is summarized in this section.The main equations required for this analysis are:•the power balance condition of the m th BEGTP imþP jmþP km¼0ð3aÞwhere P im,P jm and P km are,respectively,the powers through the links i,j and k;•the power ratios[35,17]P jm im ¼−R mωjωið3bÞP km P jm ¼1−R mR mωkωið3cÞ159E.Pennestrìet al./Mechanism and Machine Theory49(2012)157–176The steps of the power flow analysis are as follows:plete the kinematic analysis of the PGHP by means of the method discussed in the previous subsection.II.Represent the PGHP in term of blocks.Each block corresponding to a fundamental circuit or to a BEGT.The block has three nodes and each node corresponds to a link.The blocks are connected through nodes representing the same link.III.For each fundamental circuit write two equations arbitrarily chosen from (6).If one of the links is the frame then write onlyEq.(3a).IV.Locate the nodes of the g driving and h driven links.Denote the input powers by P in x (x =1,2,…,g )and the output powersby P out y (y =1,2,…,h ).V.Apply the condition of power flow balance balance to each node.VI.Assign the values of prescribed powers.The number of prescribed input or output powers must be equal to the degree-of-freedom F .VII.Solve the system of equations obtained.Powers with the negative algebraic sign are leaving the block.4.Mechanical ef ficiency of one d.o.f.BEGT kinematic inversionsFor preliminary calculations,under average operating conditions,the percentage of power loss can be approximated to 1%or 2%of input power,for internal or external ordinary spur-gear trains,respectively.The simplest formula for estimation of mechanical efficiency ηij oof an ordinary spur-gear train is as follows [36]ηo¼1−151z a Æ1z b;ð4Þwhere z a and z b are the number of teeth and the ±sign refers to external and internal gear pairs,respectively.For a more accuratecomputation one may refer to the formulas offered in [37,3,4].In this section it will be shown how the efficiency of a conventional spur-gear train is related to the efficiency of its epicyclic inversions.The classical reference for this type of analysis is the textbook of Merritt [38],where a table with twelve entries is reported.A similar table has been deduced by Macmillan [39],but does not include all possible cases.The table produced by Müller [30]is identical with the one herein deduced.Shown that only ten entries are necessary,some particular cases omitted by Merritt will also be discussed.The following terms are first defined-ηij o the efficiency of spur-gear trains with a fixed gear carrier and the power flowing from wheel i to wheel j ;-T i ,T j and T k the torques acting on wheels i and j and gear carrier k ,respectively;-P i ,P j and P k the powers flowing through wheels i and j and gear carrier k ,respectively;If the power has a positive algebraic sign,then the link is driving,otherwise is driven link.When the link is fixed it cannot transmit power.This reasoning is based on the following observations:•The efficiency of a gear train is not influenced by the motion of an observer on the gear carrier as the reference link.•The torques T i ,T j and T k ,acting on links i ,j and k ,respectively,are in equilibrium,regardless of the observer's motion,thusT i þT j þT k ¼0:ð5ÞThe first observation above allows to reduce the computation of an epicyclic inversion efficiency to the one of an ordinary spur-gear train.However,under a kinematic inversion,the direction of power flow may change.Thus a driving link may become a driven link or viceversa.The analyst must choose the appropriate relationship between torques according to the direction of power flow observed in the kinematic inversion.For the purpose of this paper it appears useful to report the relative angular velocities among the links assuming ωk =0(gear carrier of the BEGT fixed):ωij ¼ωi −ωj ¼ωi1−1R;ð6a Þωkj ¼ωk −ωj ¼−ωj ¼−ωi;ð6b Þωik ¼ωi −ωk ¼ωi :ð6c Þ160 E.Pennestrìet al./Mechanism and Machine Theory 49(2012)157–176Case 1.Driving:i ,Driven:j ,Fixed:kThe power balance equation,the Willis ’equation and the torque equation can be,respectively,written as follows:T j ωj þηoij T i ωi ¼0;ð7a Þωi ¼R ωj ;ð7b ÞT i þT j þT k ¼0:ð7c ÞAfter some algebraic substitutions the following equations resultT j ¼−T i R ηo ij ð8a ÞT k ¼T i R ηoij −1ð8b ÞCase 2.Driving:j ,Driven:i ,Fixed:kThe power balance equation,Willis ’equation and torque equation can be,respectively,particularized as follows:ηoji T j ωj þT i ωi ¼0;ð9a Þωi ¼R ωj ;ð9b ÞT i þT j þT k ¼0:ð9c ÞAfter some algebraic substitutions,the following equations result:T i ¼−T jηoji R;ð10a ÞT k ¼T jηo ji R−1 !:ð10b ÞCase 3.Driving:i ,Driven:k ,Fixed:jIn this case the efficiency isηj i −k ðÞ¼−P k P i ¼−T k ωk T i ωi:ð11ÞBy imposing ωj =0in (1)the following results:ωi ¼ωk 1−R ðÞ;whereas,in the kinematic inversion with the gear carrier fixed,the following relationship holds (see Eq.(6a)):ωik ¼ωi −ωk ¼ωk 1−R ðÞ−ωk ¼−ωk R :Comparing ωi and ωij ,two cases can be identified as follows:a)If R b 0or R >1then ωi and ωik have the same algebraic sign.Therefore,in the kinematic inversion,wheel i maintains its role ofdriving link and the torques are related by means of (10).Hence (14)can be rewritten as follows:ηj i −k ðÞ¼−−T i R ηo ij −1 ωiR T i 1−1Rωi¼R ηo ij −1R −1:ð12Þ161E.Pennestrìet al./Mechanism and Machine Theory 49(2012)157–176b)If0b R b1thenωi andωik have different algebraic sign.In the kinematic inversion wheel i is driving and the torque relation-ships(8)apply.Eq.(14)can be rewritten as follows:ηj i−k ðÞ¼−−T i Rηoji−1!ωiRT i1−1Rωi¼R−ηojiηojiR−1ðÞð13ÞCase4.Driving:k,Driven:i,Fixed:j In this case the efficiency isηj k−i ðÞ¼−P ik¼−T iωikωk:ð14ÞAssumingωj=0(1)givesωk¼ωi 1−R:In the kinematic inversion that makes the gear carrier fixed,the relative velocity of k with respect to j is(see Eq.(6b))ωkj¼−ωi R :Comparing the expressions ofωk andωkj two cases can be identified as follows:a)If R b0or R>1thenωk andωkj have the same algebraic sign.Also in the kinematic inversion,wheel i keeps its role of drivenlink.Hence torque Eq.(10)must be applied.From(24)the following results:ηj k−i ðÞ¼−T i1−1Rωi−T i Rηoji−1!ωiR¼R−1ðÞηojiR−ηoji:ð15Þb)If0b R b1thenωk andωkj have opposite algebraic sign.In the kinematic inversion wheel i is a driving link,hence torque re-lationships(8)apply.From(14)the following results:ηj k−i ðÞ¼−T i1−1R−T i Rηoij−1ωiR¼R−1Rηoij−1:ð16ÞCase5.Driving:j,Driven:k,Fixed:i In this case the efficiency isηj k−i ðÞ¼−P kj¼−T kωkjωj:ð17ÞAssumingωi=0the following results from Eq.(1)ωj¼ωk R−1;In the kinematic inversion that makes the gear carrier the fixed link,the relative velocity of j with respect to k is(see Eq.(6b))ωjk¼ωj−ωk¼ωk R−1R −ωk¼−ωkR:Comparing the expressions ofωj andωjk,two cases can be identified as follows:a)If R b1thenωj andωjk have the same algebraic sign.In the kinematic inversion,wheel j is a driving link hence torque relationships(21)apply.From(17)the following results:ηi j−k ðÞ¼−T jηo ji−1!ωkT jR−1ωk¼R−ηojiR−1:ð18Þ162 E.Pennestrìet al./Mechanism and Machine Theory49(2012)157–176b)If R >1then ωj and ωjk have opposite algebraic sign.In the kinematic inversion,wheel i is a driving link hence torque relation-ships (8)apply.From (17)the following results:ηi j −k ðÞ¼−T j1−ηo ij RR ηo ijωk T jRωk ¼1−ηoij R ηoij :ð19ÞCase 6.Driving:k ,Driven:j ,Fixed:iIn this case the efficiency is given byηi k −j ðÞ¼−P j k ¼−T j ωjk ωk:ð20ÞAssuming ωi =0,the following results from Eq.(1):ωj ¼ωkR −1R:Under a kinematic inversion that makes the gear carrier k the fixed link the following results:ωjk ¼ωj −ωk ¼−ωkR:Comparing the expressions of ωj and ωjk two cases can be identified as follows:a)If R b 1then ωj and ωjk have the same algebraic sign.In the kinematic inversion wheel j is a driven link.Hence torque relation-ships (8)apply.From (20)the following resultsηi k −j ðÞ¼−−T i R ηo ij ωjT i R ηoij −1 ωjR R −1¼R −1R −1ηo ij :ð21Þb)If R >1then ωj and ωjk have different algebraic sign.In the kinematic inversion,wheel i is a driven link hence torque relation-ships (10)apply.From (20)the following results:ηi k −j ðÞ¼−T kRo ji !ωj −T kR −1ωj ¼R −1R −ηo ji :ð22ÞThe results obtained in this secion are summarized in Table 1.The cases not considered by Merritt [40]are those with the gear ratio R within the range 0b R b 1.5.Mechanical ef ficiency of basic two d.o.f.gear trainsWith the term basic two degrees-of-freedom gear train ,it is herein denoted the simplest gear train having two degrees of freedom (d.o.f.)and composed of a single pair of meshing wheels.In a basic two d.o.f.gear train,the absolute angular velocity of a link p can be expressed as a linear combination of the absolute angular velocities of other two links,namely m and n .The links p ,m and n can be the gear wheels or the gear carrier.Therefore,with the aid of Willis ’equation,the following equality can be establishedωp ¼a ωm þb ωn ¼ω′p þω′′pð23Þwhere a and b are coefficients depending on gear ratio R m ,ωp′=a ωm and ωp ′′=b ωn .An example of a basic two d.o.f.gear train is shown in Fig.2.By letting p =j ,m =i and n =k ,and taking into account Willis ’formula,Eq.(23)can be expressed in the following form:ωj ¼ωi þR −1ωk ;ð24Þwhere R is the gear ratio.Hence,in this case,a =1/R ,b =(R −1)/R ,ω′p ¼ωi Rand ω′′p ¼R −1R ωk .Let ηE be denoted as the overall efficiency of the basic two d.o.f.gear train and ηm (n −p )as the efficiency of this gear train,when link m is fixed,and the power flow going from n to p .163E.Pennestrìet al./Mechanism and Machine Theory 49(2012)157–176With reference to the gear train represented in Fig.3,two different working conditions can be identified as follows:­dual input power single output power (DISO);­single input power dual output power (SIDO).Assuming relative motion between the parts,the efficiency can be expressed by means of the following expressions [41,42]:-for DISO systemsηE ¼ω′p þω′′pω′p ηm n −p ðÞþω′′pηn m −p ðÞ;ð25a Þ-for SIDO systemsηE ¼ω′p ηm p −n ðÞþω′′p ηn p −m ðÞω′p þω′′p;ð25b Þwhere ωp ′and ωp ′′are de fined from Eq.(23).Table 1Efficiencies of single epicyclic spur-gear trains.Case Driving Driven Fixed η1i j k ηij o 2j i k ηjio 3a (R b 0)(R >1)i k j R ηo ij −1R −13b (0b R b 1)i k j R −ηo ji ηo ji R −1ðÞ4a (R b 0)(R >1)k i j R −1ðÞηo ji R −ηo ji 4b (0b R b 1)k i j R −1R ηo ij −15a (R b 1)j k i R −ηo ij R −15b (R >1)j k i 1−R ηo ij ηo ij 1−R ðÞ6a (R b 1)k j i R −1ðÞηo ij R ηo ij −16b(R >1)kjiR −1R −ηo jiFig.2.A basic 2d.o.f.gear train with externally meshing gears.164 E.Pennestrìet al./Mechanism and Machine Theory 49(2012)157–176A demonstration of Eq.(25),based on general principles,and numerical applications of relationships are discussed in [42].For a two d.o.f.BEGT,after combining (25)with the Willis ’equation and the appropriate power balance condition (see Table 2),the results summarized in Table 3are obtained.Observing the expressions derived for ηE ,it can be concluded that,under the simplifying hypotheses adopted,ηE depends only on gear ratios,on angular velocities and on the efficiencies of the corresponding gear train kinematic inversions.When ωi =ωj =ωk ,there is not relative motion and meshing losses are absent.In this case ηE =1must be assumed.6.Mechanical ef ficiency analysis of PGHPsIn this section a graph based method of mechanical efficiency analysis of PGHPs will be proposed.Mechanical efficiency analysis of EGTs is often adressed in specialized textbooks [40,43,44,30,29,45,31,46].Recent contributions on systematic methods of analysis are due to Hedman [23,5],Pennestrìand Freudenstein [11],del Castillo [6],Pennestrìand Mantriota [47],Pennestrìand Valentini [42],Chen and Angeles [8],Chen and Liang [10].The method presented here is similar to the one discussed in [11].However,the following differences are observed:•the method is based on algebraically simpler equations;•a more extended table of mechanical efficiencies of one d.o.f.gear train kinematic inversions is embodied;•the method has been extended to planetary gear hybrid powertrains with CVT elements.The validity of results requires that,in the presence of meshing losses,the power flow directions are the same of those com-puted without any energy loss.The fulfillment of this condition is a posteriori ascertained.The PGHP will be represented as explained in the section on power flow analysis.Given a BEGT,by combining the appropriate balance equation from Table 2with the Willis ’Eq.(1)and the torque equilibrium Eq.(5)the following power balance relationship is obtained:αP i þβP j ¼0;ð26Þwhere the coefficients αand βare summarized in Table 4.Alternative relationships between P j and P k have been also derived.The steps for the estimation of mechanical efficiency analysis are as follows:plete the kinematic analysis of the PGHP by means of the method discussed in the previous plete the power flow analysis assuming no power losses.III.Taking into account the power flow directions,for each BEGT calculate the efficiency ηE from Table 3.IV.Taking into account the power flow directions,for each fundamental circuit,write Eq.(26)and the appropriate power bal-ance equation from Table 4.If one of the links is the frame then write only this last equation.The coefficients αand βare chosen from Table 4.V.Locate the nodes of the g driving and h driven links.Let us denote by P in x (x =1,2,…,g ),the input powers and P out y (y =1,2,…,h )the output powers.VI.Apply the condition of power flow balance to each node.VII.Assign the values of prescribed powers.The number of prescribed input or output powers must be equal to the kinematic degree of freedom F .VIII.Solve the system of equations obtained.IX.Verify that the direction of powers in each block is not altered by the presence of friction.X.The mechanical efficiency is obtained as the ratio between the sum of output and the sum of input powers.Fig.3.Block representation of DISO (left)and SIDO (right)systems.Table 2Enumeration of power balance equations for a BEGT.Case Driving links Driven links Power balance equation1i ,k j T j ωj +ηE (T i ωi +T k ωk )=02j ,k i T i ωi +ηE (T j ωj +T k ωk )=03i ,j k T k ωk +ηE (T i ωi +T j ωj )=04i k ,j T k ωk +T j ωj +ηE T i ωi =05j i ,k T k ωk +T i ωi +ηE T j ωj =06ki ,jT i ωi +T j ωj +ηE T k ωk =0165E.Pennestrìet al./Mechanism and Machine Theory 49(2012)157–1767.ApplicationsFor simplicity,in all the analyses it is assumed that:•the efficiencies of a single gear mesh ordinary gear train are η0=0.98and η0=0.99,for external and internal gears,respectively;•the mechanical efficiency of the ordinary gear train is independent of the driving gear (pinion or wheel).More accurate values could be obtained using one of the the models discussed by Anderson and Lowenthal [37]and Kahraman et al.[3,4].In particular,one could adjust the friction coefficient taking into account the meshing speeds.This choice would com-plicate the model and require specific data about the meshing gears not always available during preliminary calculations.8.Tsai-Schultz parallel hybrid transmissionThe topology and control strategy of this powertrain,depicted in Fig.4,have been proposed by Tsai and Schultz [48].It is an improved design of a previous solution also introduced by Tsai et al.[49,50].The motor-integrated transmission mechanism can be installed in parallel hybrid electric front-wheel drive or rear-wheel drive vehicles.Only one internal combustion engine (ICE)and one electric motor/generator (EM)are required.There are five basic modes of operation:Motor,Power,Engine,Engine/Charge and Regenerative.Only the mechanical efficiency analysis will be considered herein,by means of the methodology previously described.For a comparison of results obtained in this paper,the numbering of links is identical with the one adopted in reference [48].The gears and the gear carriers of the BEGTs in the PGHP under analysis are reported in Table 5.8.1.Mechanical ef ficiency in power modeThe control system switch to power mode when the maximum acceleration is needed.There are three submodes:•Submode P1:Clutch C1and brake B1are both engaged and all other clutches disengaged.•Submode P2:Clutch C2and brake B1are both engaged and all other clutches disengaged.•Submode P3:Clutches C1and C2are both engaged and all other clutches disengaged.In this case ICE and EM supply power with one-to-one gear ratio.The analysis here will initially focus on submode P1.Since in this submode the EM and the ICE shafts are connected to a com-mon link,the PGHP has only one d.o.f.the output being on link 3.Table 4Table of coefficients αand βof Eq.(26).Driving links Driven links αβi ,k j R ηE ωj (ωj −ωi )ωi ωj (1−R )+ηE ωi (R ωj −ωi )j ,k i ωi ωj (1−R )+ηE ωj (R ωj −ωi )ηE ωi (ωj −ωi )i ,j k ωj (ωi −R ωj )+ηE ωi ωj (R −1)R ωi ωj (ηE −1)+ωi (ωi −ηE ωj )i k ,j ωj (ωi −R ωj )+ηE ωi ωj (R −1)ωi (ωi −ωj )j i ,k R ωj (ωj −ωi )ωi (R ωj −ωi )+ηE ωi ωj (1−R )ki ,jωi ωj (1−R )+ηE ωj (R ωj −ωi )R ωi ωj (ηE −1)+ωi (ωj −ηE ωi )166 E.Pennestrìet al./Mechanism and Machine Theory 49(2012)157–176。