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First submission on 7th Nov. 2013, and minor revised submission on 8th Mar. 2014, to Mechanism and Machine Theory.Numerical and Experimental Investigation of Drag Torque in a Two-speed Dual Clutch Transmissionby*Xingxing Zhou1, Paul Walker1, Nong Zhang1,2, Bo Zhu1,3, Jiageng Ruan11 School of Electrical, Mechanical and Mechatronic Systems, University of Technology, Sydney (UTS), Sydney, NSW, Australia 20072 State Key Laboratory of Advanced Design and Manufacturing for Vehicle Design, Hunan University, Changsha, China 4100823 BAIC Motor Electric Vehicle Co Ltd, BAIC Motor Electric Vehicle Co Ltd, Beijing, China102606Xingxing ZhouTel: +61-2-9514 2517; Fax: +61-2-9514 2655.E-mail address: Xingxing.Zhou@student.uts.edu.au Paul Walker Tel: +62-2-9514 2412; Fax: +61-2-9514 2655.E-mail address: Paul.Walker@uts.edu.au (Dr. Paul Walker)Nong ZhangTel: +61-2-9514 2662; Fax: +61-2-9514 2655.E-mail address: Nong.Zhang@uts.edu.au (Prof. Nong Zhang)Bo ZhuTel: +62-2-9514 2517; Fax: +61-2-9514 2655.E-mail address: zhubo@.cn Jiageng RuanTel: +62-2-9514 2412; Fax: +61-2-9514 2655.E-mail address: Jiageng.Ruan@student.uts.edu.au Corresponding Author in SubmissionTel: +61-2-9514 2517; Fax: +61-2-9514 2655.E-mail address: Xingxing.Zhou@student.uts.edu.au (Xingxing Zhou)Corresponding Author in PublicationTel: +62-2-9514 2412; Fax: +61-2-9514 2655.E-mail address: Paul.Walker@uts.edu.au (Dr. Paul Walker)*Author to whom all further correspondence should be addressedThis paper is submitted for possible publication in Mechanism and Machine Theory. It has not been previously published, is not currently submitted for review to any other journal, and will not be submitted elsewhere during the peer review.Numerical and Experimental Investigation of Drag Torquesin a Two-speed Dual Clutch TransmissionAbstract:The theoretical analysis of drag torques within a two-speed dual clutch transmission is presented in this article. The numerical models are developed to study the different sources of drag torques in dual clutch transmission. Simulations are performed in Matlab/Simulink platform to investigate the variation of drag torques under different operating conditions. Then an experimental investigation is conducted to evaluate the proposed model using an electric vehicle powertrain test rig. Outcomes of experimentation confirm that simulation results agree well with test data. Therefore the proposed model performs well in the prediction of drag torque for the transmission, and can be applied to assess the efficiency of the transmission. Results demonstrate that the entire drag torque is dominated by the viscous shear in the wet clutch pack and gear churning losses. This lays a theoretical foundation to future research on reducing drag torque and applications of drag torque in powertrain system efficiency optimization. Keywords: Drag torque; power loss; mathematical model; dual clutch transmission; experimental investigation.IntroductionIn recent years there has been significant attention drawn towards reducing fossil fuel consumption and emissions in automotive industry. Improvement of the overall energy efficiency of existing technologies is one of the most important subjects for developing new vehicle technologies. As a consequence of this, the development of commercially viable hybrid electric vehicles (HEVs), fuel cell vehicles (FCVs) for using in the short to mid term, and pure electric vehicles (EVs) in the long term are one of the major contributions in the automotive industry to solve related issues ADDIN EN.CITE <EndNote><Cite><Author>Chan</Author><Year>2007</Year><RecNum>7</RecNum><DisplayText>[1]</DisplayText><record><rec-number>7</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">7</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Chan, C.C.</author></authors></contributors><titles><title>Electric, Hybrid & Fuel Cell Vehicles: Overview, State-of-the-art, Key Technologies & Issues</title><secondary-title>Proceedings of the IEEE</secondary-title></titles><periodical><full-title>Proceedings of the IEEE</full-title></periodical><pages>704-718</pages><volume>95</volume><number>4</number><dates><year>2007</year></dates><urls><related-urls><url>;[1]. Pure EVs currently being used in the market are mainly equipped with single speed transmissions, with tradeoffs between dynamic (such as climbing ability, top speed, and acceleration) and economic performance (drive range). Nowadays, more and more EV researchers and designers are paying attention to application of multiple speed transmissions instead of traditional single speed transmissions, expecting to improve the EV performance. The usage of multispeed transmissions for electric vehicles is likely to improve average motor efficiency and range capacity, or even can reduce the required motor size. The detail advantages of two-speed transmission over single speed are demonstrated in previously reported work ADDIN EN.CITE <EndNote><Cite><Author>Zhou</Author><Year>2013</Year><RecNum>8</RecNum><DisplayText>[2]</DisplayText><record><rec-number>8</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">8</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Xingxing Zhou</author><author>Paul Walker</author><author>Nong Zhang</author><author>Bo Zhu </author></authors></contributors><titles><title>Performance Improvement of a Two Speed EV through Combined Gear Ratio and Shift Schedule Optimization</title><secondary-title>SAE Technical Paper 2013-01-1477</secondary-title></titles><periodical><full-title>SAE Technical Paper 2013-01-1477</full-title></periodical><dates><year>2013</year></dates><urls></urls><electronic-resource-num>doi: 10.4271/2013-01-1477</electronic-resource-num></record></Cite></EndNote>[2].As an important part of electric vehicle powertrain system design and optimization, it is of great importance to predict the transmission efficiency early in the design process, leading to improved powertrain efficiency as the system is refined. 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ADDIN EN.CITE.DATA [7-9] and oil churning ADDIN EN.CITE <EndNote><Cite><Author>Changenet</Author><Year>2006</Year><RecNum>15</RecNum><DisplayText>[10, 11]</DisplayText><record><rec-number>15</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">15</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>C. Changenet</author><author>P. Velex</author></authors></contributors><titles><title>A model for the prediction of churning losses in geared transmissions—preliminary results</title><secondary-title>J. Mech. Des. </secondary-title></titles><periodical><full-title>J. Mech. Des.</full-title></periodical><pages>128-133</pages><volume>129</volume><number>1</number><dates><year>2006</year></dates><urls></urls></record></Cite><Cite><Author>Changenet</Author><Year>2006</Year><RecNum>16</RecNum><record><rec-number>16</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">16</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Changenet, C.</author><author>Oviedo-Marlot, X.</author><author>Velex, P.</author></authors></contributors><titles><title>Power loss predictions in geared transmissions using thermal networks-application to a six speed manual gearbox</title><secondary-title>Trans. Am. Soc. Mech. Eng.</secondary-title></titles><periodical><full-title>Trans. Am. Soc. Mech. Eng.</full-title></periodical><pages>618–625 </pages><volume>128</volume><dates><year>2006</year></dates><urls></urls></record></Cite></EndNote>[10, 11]. Other important drag torque sources have to be considered as well, comprising bearings and sealPEVuZE5vdGU+PENpdGU+PEF1dGhvcj5DaGFuZ2VuZXQ8L0F1dGhvcj48WWVhcj4yMDA2PC9ZZWFy
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ADDIN EN.CITE.DATA [11-13], synchronizer and free-pinion losses ADDIN EN.CITE <EndNote><Cite><Author>Walker</Author><Year>2011</Year><RecNum>18</RecNum><DisplayText>[14, 15]</DisplayText><record><rec-number>18</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">18</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Paul D. Walker</author><author>Nong Zhang</author><author>Ric Tamba</author><author>Simon Fitzgerald</author></authors></contributors><titles><title>Simulations of drag torque affecting synchronisers in a dual clutch transmissions</title><secondary-title>Japan J. Indust. Appl. Math.</secondary-title></titles><periodical><full-title>Japan J. Indust. Appl. Math.</full-title></periodical><pages>119-140</pages><volume>28</volume><dates><year>2011</year></dates><urls></urls></record></Cite><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[14, 15]. If the clutch is immersed in oil, torsional resistance and its influences caused by the viscous shear between wet clutch plates should be considered as well ADDIN EN.CITE <EndNote><Cite><Author>Kato</Author><Year>1993</Year><RecNum>20</RecNum><DisplayText>[16, 17]</DisplayText><record><rec-number>20</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">20</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kato, Y.</author><author>Murasugi, T.</author><author>Hirano, H.</author></authors></contributors><titles><title>Fuel Economy Improvement through Tribological Analysis of the Wet Clutches and Brakes of an Automatic Transmission</title><secondary-title>Society of automotive Engineers of Japan</secondary-title></titles><periodical><full-title>Society of automotive Engineers of Japan</full-title></periodical><pages>57-60</pages><volume>16</volume><number>12</number><dates><year>1993</year></dates><urls></urls></record></Cite><Cite><Author>Iqbal</Author><Year>2012</Year><RecNum>21</RecNum><record><rec-number>21</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">21</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>S. Iqbal</author><author>T. Janssens</author><author>W. Desmet</author><author>F. Al-Bender</author></authors></contributors><titles><title>Transmitted power and energy flow behaviour of degrading wet friction clutches</title><secondary-title>International Journal of Applied Research in Mechanical Engineering</secondary-title></titles><periodical><full-title>International Journal of Applied Research in Mechanical Engineering</full-title></periodical><pages>94-100</pages><volume>2</volume><number>1</number><dates><year>2012</year></dates><urls></urls></record></Cite></EndNote>[16, 17]. Several researchers have done some specific studies regarding the drag torque within disengaged wet clutches PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5LaXRhYnlhc2hpPC9BdXRob3I+PFllYXI+MjAwMzwvWWVh
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ADDIN EN.CITE.DATA [18-21], which will be analysed in the 2nd section. However, in order to comprehensively improve the whole automotive powertrain system efficiency, it is necessary to consider all aspect of the transmission power losses ADDIN EN.CITE <EndNote><Cite><Author>Lechner</Author><Year>1999</Year><RecNum>26</RecNum><DisplayText>[22]</DisplayText><record><rec-number>26</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">26</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Lechner, G.</author><author>Naunheimer, H.</author></authors></contributors><titles><title>Automotive Transmissions—Fundamentals, Selection, Design and Application</title></titles><edition>1st</edition><dates><year>1999</year></dates><pub-location>Berlin</pub-location><publisher>Springer</publisher><urls></urls></record></Cite></EndNote>[22]. There are only a few reported works on the entire transmission power lossesPEVuZE5vdGU+PENpdGU+PEF1dGhvcj5QYXRlbDwvQXV0aG9yPjxZZWFyPjIwMTI8L1llYXI+PFJl
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ADDIN EN.CITE.DATA [23-27], which are mostly focus on manual transmission (MT) gear-train. There is limited, if any, published work to combine the gearbox components and wet clutch losses together to study. Especially, there is no published report on study of entire drag torque within wet dual clutch transmissions (DCT). In this paper, drag torque within a two-speed dual clutch transmission is discussed. A general two-speed wet DCT is suggested to be equipped into a pure electric vehicle (EV), as shown in Figure 1. The system under consideration is modified from a 6-speed DCT (DQ250) into a two-speed DCT. The two-speed DCT housing is made from an aluminium alloy. And the DCT is made up of two clutches, the inner clutch (C1) and the outer clutch (C2). The two clutches have a common drum attached to the same input shaft from the electric motor, and the friction plates are independently connected to 1st or second gear. C1, shown in green, hereby connects the outer input shaft engaged with 1st gear, and C2, shown in red, connects the inner input shaft engaged with 2nd gear. In order to make transmission control system simpler and save manufacture fees, there are no synchronisers in this new type of two-speed DCT. Thus, the transmission can be looked at as two clutched gear pairs, and, in this sense, shifting is realised through the simultaneous shifting between these two halves of the transmission. For this special layout, vehicle equipped with a DCT can not only change speed smoothly with nearly no power hole, identified by Goetz ADDIN EN.CITE <EndNote><Cite><Author>Goetz</Author><Year>2005</Year><RecNum>31</RecNum><DisplayText>[28]</DisplayText><record><rec-number>31</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">31</key></foreign-keys><ref-type name="Thesis">32</ref-type><contributors><authors><author>Manuel Goetz</author></authors></contributors><titles><title>Integrated Powertrain Control for Twin Clutch Transmissions</title></titles><volume>PHD</volume><dates><year>2005</year></dates><publisher>University of Leeds</publisher><urls></urls></record></Cite></EndNote>[28], but improve the EV efficiency as well.DiffC11st GearC2S1 & S22nd Gear S3S1 & S2S3DiffElectric MachineWheel(1)(5)(4)(7)(6)(9)(8)(2)(3)Figure 1. Schematic of a two-speed DCT powertrain system.This paper is organised as follows. In the 2nd section, the different source of drag torque in the DCT is theoretically analysed and modelled including torsional resistance caused by viscous shear caused between wet clutch plates and concentrically aligned shafts, gear mesh friction and windage, oil churning, and bearing losses. Then experimental aspects comprising of the description of the UTS test rig, and the drag torque test procedures are presented and discussed in the 3rd section. In the 4th section, simulation and experimental results are compared and analysed to validate the effectiveness of the numerical model. Finally, some significant conclusions drawn from the study are summarized in the 5th section.Theoretical analysis of drag torque in wet dual clutch transmissionsThe sources of power loss in a two-speed DCT, or drag torque, can be divided into five major origins: gears related meshing, windage and churning losses, bearings and oil seal related losses, concentric shaft related viscous shear loss, and disengaged wet clutch caused drag torque losses, as shown in Equation (1). For each individual power losses sources, the following formulae have been implemented in the simulation code.PL=PCon+PB+PG+PCh+PCl (1)Where PLis total power losses in DCT, PCon is power losses caused by concentric shaft drag torque, PB is power losses caused by bearings drag torque, PG is power losses caused by gear meshing drag torque, PCh is power losses caused by gear churning, PCl is power losses caused by wet clutch plates drag torque. If engaged with 1st gear, clutch 1 is closed, while clutch 2 will be open and subsequently develop a drag torque. (Tm-Tcon-TB1,2)*r1st=T1st_output_outer (2)T1st_output_outer-TGM1st_pair-TB6,7-TGM2nd_pair+TClC2+TB3,4,5*r2nd=T2nd_outputr3rd (3)T2nd_output-TGM3rd_pair-TCh-TB8,9=Tfinal_output (4)Where Tm is motor output torque; Tcon is drag torque caused by concentric shafts viscous shear resistance; TB is drag torque caused by bearings; and r is gear ratio, T1st_output_outerrepresents the output torque of the outer concentric shaft. TGM is drag torque caused by gear pairs meshing, TCl is the drag torque caused by wet clutch packs. TCh is the drag torque caused by churning. And the individual power losses can be calculated in rotational speed multiplied by drag torque:P=T*n9550 (5)For the power losses caused by concentric shaft raised shear resistance is:PCon=Tcon*nmotor9550 (6)The total power loss caused by bearings and gear friction are:PB=TB1,2*nmotor+TB3,4,5*nmotorr2ndr1st+TB6,7*nmotorr1st+TB8,9*nmotorr1st*r3rd/9550 (7)PG=TGM1st_pair*nmotorr1st+TGM2nd_pair*nmotorr2ndr1st+TGM3rd_pair*nmotorr1st*r3rd/9550 (8)The power losses resulting from gear windage and churning is:PCh=TCh*nmotorr1st*r3rd*9550 (9)The overall power loss caused by clutch package is:PCl=TClC2*nmotor*r2ndr1st*9550 (10)Similarly, if engaged with 2nd gear, clutch 2 is closed, while clutch 1 open.(Tm-Tcon-TB3,4,5)*r1st=T1st_output_inner (11)T1st_output_inner-TGM2nd_pair-TB6,7-TGM1st_pair+TClC1+TB1,2*r1st=T2nd_outputr3rd (12)T2nd_output-TGM3rd_pair-TCh-TB8,9=Tfinal_output (13)PCon=Tcon*nmotor9550 (14)PB=TB1,2*nmotor*r1str2nd+TB3,4,5*nmotor+TB6,7*nmotorr2nd+TB8,9*nmotorr2nd*r3rd/9550 (15)PG=TGM1st_pair*nmotorr1str2nd+TGM2nd_pair*nmotorr2nd+TGM3rd_pair*nmotorr2nd*r3rd/9550 (16)PCh=TCh*nmotorr2nd*r3rd*9550 (17)PCl=TClC2*nmotor*r1str2nd*9550 (18)The overall efficiency of a DCT or any transmission in general can be obtained by monitoring the input and output speed and torque respectively.nfinal*Tfinal_outputnmotor*Tmotor*100%=EDCT (19)2.1 Drag Torques Caused By GearsDrag torques caused by gears can be generally divided into gear meshing loss, and gear windage and churning loss. Meshing losses are a result of rolling and sliding friction in mating gears and is, as such, a load dependent loss, whilst windage and churning losses are generated from the gears rotating in fluids, and are therefore speed dependent.2.1.1 Gear meshThe gear meshing losses in gear pairs are dependent on both rolling and sliding friction. Methods for studying these losses reported in works PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5EaWFiPC9BdXRob3I+PFllYXI+MjAwNjwvWWVhcj48UmVj
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ADDIN EN.CITE.DATA [3, 4, 10] on theoretically analysis and modelling the friction coefficient are divided into two methods. One majority of works study the friction coefficient variation over the gears geometry and the contact forces vector ADDIN EN.CITE <EndNote><Cite><Author>Li</Author><Year>2009</Year><RecNum>4</RecNum><DisplayText>[5, 11]</DisplayText><record><rec-number>4</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">4</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Li, Sheng</author><author>Vaidyanathan, Aarthy</author><author>Harianto, Jonny</author><author>Kahraman, Ahmet</author></authors></contributors><titles><title>Influence of Design Parameters on Mechanical Power Losses of Helical Gear Pairs</title><secondary-title>Journal of Advanced Mechanical Design, Systems, and Manufacturing</secondary-title></titles><periodical><full-title>Journal of Advanced Mechanical Design, Systems, and Manufacturing</full-title></periodical><pages>13</pages><volume>3</volume><number>2</number><section>146</section><keywords><keyword>Helical Gear Power Losses, Gear Mechanical Efficiency, Gear Design</keyword></keywords><dates><year>2009</year></dates><urls></urls><electronic-resource-num>10.1299/jamdsm.3.146 </electronic-resource-num></record></Cite><Cite><Author>Changenet</Author><Year>2006</Year><RecNum>16</RecNum><record><rec-number>16</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">16</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Changenet, C.</author><author>Oviedo-Marlot, X.</author><author>Velex, P.</author></authors></contributors><titles><title>Power loss predictions in geared transmissions using thermal networks-application to a six speed manual gearbox</title><secondary-title>Trans. Am. Soc. Mech. Eng.</secondary-title></titles><periodical><full-title>Trans. Am. Soc. Mech. Eng.</full-title></periodical><pages>618–625 </pages><volume>128</volume><dates><year>2006</year></dates><urls></urls></record></Cite></EndNote>[5, 11], which seem impractical for wide application. Other works ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15] considers the gear pairs only loss which is more compact and applicable to general simulation and practical experiments. Consider the rigorous development of British Standards ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15], model presented in the standards is chosen to calculation the gear meshing caused drag torques, as shown below: pm=fmT1n1cos2βw9549M (20)Where pm is meshing power loss, kw; fm is mesh coefficient of friction; T1 is pinion torque, N.m; n1 is pinion rotational speed, rpm; βw is operating helix angle, degree; M is mesh mechanical advantage. Meshing mechanical advantage can be calculated using Equation (21). This equation is a function of the sliding ratios. For external gears, the sliding ratio at the start of approach action, Hs, is calculated using Equation (22), and the sliding ratio at the end of recess action, Ht, is obtained with Equation (23). M=2cosaw(Hs+Ht)Hs2+Ht2 (21)where αw is transverse operating pressure angle, degree; Hs is sliding ratio at start of approach; Ht is sliding ratio at end of recess.Hs=(r+1)(ro22rw22-cos2αw)0.5-sinαw (22)Ht=(r+1r)(ro12rw12-cos2αw)0.5-sinαw (23)where r is gear ratio, which can be calculated by Equation (24); Ro2 is gear outside radius, mm; Rw2 is gear operating pitch radius, mm; Ro1 is pinion outside radius, mm; Rw1 is pinion operating pitch radius, mm.r=z2z1 (24)where z2 is number of gear teeth; z1 is number of pinion teeth;if the pitch line velocity, V, is 2m/s ≤ V ≤ 25 m/s and the load intensity (K-factor) is 1.4 N/mm2 <K≤ 14 N/mm2, then the gear meshing coefficient of friction, fm, can be expressed by Equation (25). Outside these limits, values for fm must be determined by experience. K-factor can be calculated using Equation (26). Exponents, j, g and h modify viscosity, ν, K-factor and tangential line velocity, V, respectively.fm=νjKgC1Vh (25)where ν is kinematic oil viscosity at operating sump temperature, cSt (mm2/s); K is K-factor, N/mm2; C1 is a constant; V is tangential pitch line velocity, m/s.K=1000T1(z1+z2)2bw(rw1)2z2 (26)where bw is face width in contact, mm. Values to be used for exponents j, g and h and constant C1 are as follows: J=-0.223, g=-0.40, h=0.70, C1=3.239. The coefficient varies from 0.07 to 0.03 during the simulation.2.1.2 Gear windage and churningAs alluded before, research on gear windage had been done by Dawson ADDIN EN.CITE <EndNote><Cite><Author>Dawson</Author><Year>1984</Year><RecNum>12</RecNum><DisplayText>[7]</DisplayText><record><rec-number>12</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">12</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Dawson, P.H.</author></authors></contributors><titles><title>Windage loss in large high speed gears</title><secondary-title>Proc. Inst. Mech. Eng. Part J: J. Engineering Tribology</secondary-title></titles><periodical><full-title>Proc. Inst. Mech. Eng. Part J: J. Engineering Tribology</full-title></periodical><pages>51-59</pages><volume>198A</volume><number>1</number><dates><year>1984</year></dates><urls></urls></record></Cite></EndNote>[7] and Diab ADDIN EN.CITE <EndNote><Cite><Author>Diab</Author><Year>2004</Year><RecNum>5</RecNum><DisplayText>[9]</DisplayText><record><rec-number>5</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">5</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author> Y. Diab</author><author> F. Ville</author><author> H. Houjoh</author><author> P. Sainsot</author><author> P. Velex</author></authors></contributors><titles><title>Experimental and numerical investigations on the air pumping phenomenon in high speed spur and helical gears</title><secondary-title>Proc. Inst. Mech. Eng., Part C: J. Mechanical Engineering Sciences</secondary-title></titles><periodical><full-title>Proc. Inst. Mech. Eng., Part C: J. Mechanical Engineering Sciences</full-title></periodical><pages>785-790
</pages><volume>219</volume><number>G8</number><dates><year>2004</year></dates><urls></urls></record></Cite></EndNote>[9] on larger high speed gears, and Eastwick ADDIN EN.CITE <EndNote><Cite><Author>Eastwick</Author><Year>2008</Year><RecNum>13</RecNum><DisplayText>[8]</DisplayText><record><rec-number>13</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">13</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Eastwick, C.N.</author><author>Johnson, G.</author></authors></contributors><titles><title>Gear windage: a review</title><secondary-title>J.Mech. Des. </secondary-title></titles><periodical><full-title>J.Mech. Des.</full-title></periodical><volume>130</volume><dates><year>2008</year></dates><urls></urls></record></Cite></EndNote>[8] concludes that the effects of gear windage loss in low and mid speed is not evident, while it will becomes gradually prominent in high speed. For the churning loss, both Changenet ADDIN EN.CITE <EndNote><Cite><Author>Changenet</Author><Year>2006</Year><RecNum>15</RecNum><DisplayText>[10]</DisplayText><record><rec-number>15</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">15</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>C. Changenet</author><author>P. Velex</author></authors></contributors><titles><title>A model for the prediction of churning losses in geared transmissions—preliminary results</title><secondary-title>J. Mech. Des. </secondary-title></titles><periodical><full-title>J. Mech. Des.</full-title></periodical><pages>128-133</pages><volume>129</volume><number>1</number><dates><year>2006</year></dates><urls></urls></record></Cite></EndNote>[10] and British Standard ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15] present a study for a general gear box. However, the formulas presents in Changenet ADDIN EN.CITE <EndNote><Cite><Author>Changenet</Author><Year>2006</Year><RecNum>15</RecNum><DisplayText>[10]</DisplayText><record><rec-number>15</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">15</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>C. Changenet</author><author>P. Velex</author></authors></contributors><titles><title>A model for the prediction of churning losses in geared transmissions—preliminary results</title><secondary-title>J. Mech. Des. </secondary-title></titles><periodical><full-title>J. Mech. Des.</full-title></periodical><pages>128-133</pages><volume>129</volume><number>1</number><dates><year>2006</year></dates><urls></urls></record></Cite></EndNote>[10] assumed that all the gears are submerged in the fluids, and too many parameters are required to validate to get a reasonable practical dimensionless drag torque coefficient Cm. Considers the rigours and universe of standards models, British Standard formulas are adopted here.As shown in British Standards ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15], a gear dip factor, fg, must be considered before obtain gear windage and friction loss. This factor is based on the amount of dip that the element has in the oil. When the gear or pinion does not dip in the oil, fg=0. When the gear dips fully into the oil, fg=1. When the element is partly submerged in the oil, linearly interpolate between 1 and 0. The primary dip fact of the final gear is 0.21. The higher the speed, the smaller the dip factor becomes. When the speed increases, it will range from 0.21 to 0.1 according to the speed. The primary depth value with 0.21 is calculated via opening the transmission, and measuring the height of shaft, gear size and the oil tube. The final 0.1 is an estimated value after debugging the model parameters and referring from a Conference paper ADDIN EN.CITE <EndNote><Cite><Author>Yang</Author><Year>2013</Year><RecNum>310</RecNum><DisplayText>[29]</DisplayText><record><rec-number>310</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">310</key></foreign-keys><ref-type name="Conference Paper">47</ref-type><contributors><authors><author> Likun Yang </author><author> Heyan Li </author><author>Biao Ma </author></authors></contributors><titles><title>Prediction Model of No-load Power loss for a DSG Transmission</title><secondary-title>The 5th TM Symposium China, ICE, HEV and EV transmission</secondary-title></titles><pages>44-52</pages><dates><year>2013</year></dates><pub-location>Suzhou, China</pub-location><urls></urls></record></Cite></EndNote>[29]. The power loss equation for the tooth surface is named as a roughness factor, Rf. Table 12.5 of reference ADDIN EN.CITE <EndNote><Cite><Author>Dudley</Author><Year>1991</Year><RecNum>32</RecNum><DisplayText>[30]</DisplayText><record><rec-number>32</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">32</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Dudley, D.W.</author><author>Townsend, D.P.</author></authors></contributors><titles><title>Dudley's gear handbook</title></titles><dates><year>1991</year></dates><publisher>McGraw-Hill</publisher><isbn>9780070179035</isbn><urls><related-urls><url>;[30] presents several values based on gear tooth size. Equation (27) is a reasonable approximation of the values from Dudley’s model.Rf=7.93-4.648Mt (27)Where Rf is roughness factor, Mt is transverse tooth module, mm.Gear windage and churning loss includes three kinds of loss. For those loss associated with a smooth outside diameter, such as the outside diameter of a shaft, use Equation (28). For those loss associated with the smooth sides of a disc, such as the faces of a gear, use Equation (29). It should be pointed out that Equation (29) includes both sides of the gear, so do not double the value. For those loss associated with the tooth surfaces, such as the outside diameter of a gear pinion, use Equation (30).For smooth outside diameters,PGW=7.37fgγn3D4.7LAg1026 (28)For smooth sides of discusPGW=1.474fgγn3D5.7Ag1026 (29)For tooth surfacesPGWi=7.37fgγn3D4.7FRftanβAg1026 (30)Where PGWi is power loss for each individual gear, kw; fg is gear dip factor; D is outside diameter of the gear, mm; A is arrangement constant, here set as 0.2; F is total face width, mm; L is length of the gear, mm; β is generated helix angle, degrees. For helix angles less than 10o, use 10o in Equation (30).For a common output shaft assembly, Equation (28) would be used for the OD of the shaft outside of the gear between the bearings, Equation (29) for the smooth sides of the gear, and Equation (30) for the tooth surfaces. After calculating the individual gears for each shaft assembly in a reducer, they must be added together for the total loss. 2.2 Bearing and Oil Seal ModelAs an important part of the power losses in transmission, bearing loss have been analysed by Harris ADDIN EN.CITE <EndNote><Cite><Author>Harris</Author><Year>1966</Year><RecNum>1</RecNum><DisplayText>[12]</DisplayText><record><rec-number>1</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">1</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author> T. A. Harris</author></authors></contributors><titles><title>Rolling Bearing Analysis</title></titles><edition>1st</edition><dates><year>1966</year></dates><orig-pub>John Wiley & Sons, Inc.,</orig-pub><urls></urls></record></Cite></EndNote>[12] for a variety of bearing designs, considering both viscous friction caused torque and applied load generated torque. This work is widely accepted as the forefront on the topic, with similar results being applied in work ADDIN EN.CITE <EndNote><Cite><Author>Heingartner</Author><Year>2005</Year><RecNum>10</RecNum><DisplayText>[4, 11]</DisplayText><record><rec-number>10</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">10</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Heingartner, P.</author><author>Mba, D.</author></authors></contributors><titles><title>Determining Power Losses in the Helical Gear Mesh</title><secondary-title>Gear Technol.</secondary-title></titles><periodical><full-title>Gear Technol.</full-title></periodical><pages>32-37</pages><dates><year>2005</year></dates><urls></urls></record></Cite><Cite><Author>Changenet</Author><Year>2006</Year><RecNum>16</RecNum><record><rec-number>16</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">16</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Changenet, C.</author><author>Oviedo-Marlot, X.</author><author>Velex, P.</author></authors></contributors><titles><title>Power loss predictions in geared transmissions using thermal networks-application to a six speed manual gearbox</title><secondary-title>Trans. Am. Soc. Mech. Eng.</secondary-title></titles><periodical><full-title>Trans. Am. Soc. Mech. Eng.</full-title></periodical><pages>618–625 </pages><volume>128</volume><dates><year>2006</year></dates><urls></urls></record></Cite></EndNote>[4, 11], And a similar bearing model is presented in reference ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15], which is widely used in German gear industry. Hence, the equations from the work ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15] are chosen and presented below. Load dependent bearing loss is:Pbl=fLFbdm (31)Speed dependent loss is:Pbv=1.6*10-8f0dm3 &νn<2000010-10f0(νn)2/3dm3, &νn≥2000 (32)Where Pbv is no-load torque of the bearing, only speed independent, Nm; f0 is bearing dip factor. Factor f0 adjusts the torque based on the amount that the bearing dips in the oil and varies from f0(min) to f0(max). Use f0(min) if the rolling elements do not dip into the oil and f0(max) if the rolling elements are completely submerged in the oil, linearly interpolate between f0(min) and f0(max). Values for the dip factor range can be found in reference ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15].The oil seal power losses Poil is calculated from reference ADDIN EN.CITE <EndNote><Cite><Author>British Standards Institute BS ISO/TR 14179-1:2001</Author><Year>2001</Year><RecNum>19</RecNum><DisplayText>[15]</DisplayText><record><rec-number>19</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">19</key></foreign-keys><ref-type name="Standard">58</ref-type><contributors><authors><author>British Standards Institute BS ISO/TR 14179-1:2001,</author></authors></contributors><titles><title>Gears-thermal capacity-Part 1: rating gear drives with thermal equilibrium at 95?C sump temperature. Part 2:Thermal load-carrying capacity</title></titles><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>[15].Poil=7.69*10-6*Voil*dm2 (33)WhereVoil is the velocity, and dm is the diameter. The total oil seal losses and bearing losses are expressed inPb.Pb=Pbl+Pbv+Poil (34)2.3 Concentric Shaft Drag TorqueThe DCT has two concentric arranged shafts which connect the gears and clutches respectively. When one of the concentric shaft running, the other concentric one will run in a different speed due to the different engaged gear ratios. Therefore, there exists relative speed between two concentric shafts, which cause viscous shear resistance. This concentric shafts shear torque phenomenon has been studied by Schlichting presented in reference ADDIN EN.CITE <EndNote><Cite><Author>Schlichting</Author><Year>2000</Year><RecNum>33</RecNum><DisplayText>[31]</DisplayText><record><rec-number>33</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">33</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Schlichting, H.</author><author>Gersten, K.</author></authors></contributors><titles><title>Boundary-Layer Theory</title></titles><dates><year>2000</year></dates><publisher>MacGraw-Hill</publisher><isbn>9783540662709</isbn><urls><related-urls><url> app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">33</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Schlichting, H.</author><author>Gersten, K.</author></authors></contributors><titles><title>Boundary-Layer Theory</title></titles><dates><year>2000</year></dates><publisher>MacGraw-Hill</publisher><isbn>9783540662709</isbn><urls><related-urls><url>;[31] acting as an example of Couette flow. When the DCT input speed is at 4000 rpm (test speed maximum range), the Taylor’s number is 779 which is smaller than critical one 1708. Hence the assumption for Couette flow is reasonable. Therefore, the drag torque caused by concentric shafts can be expressed as:Tcon=4πμhconRcon_o2Rcon_i2Rcon_o2-Rcon_i2?ω (35)whereRo , Ri are the inner radius of the outer shaft and out radius of the inner shaft. ?ω is the relative speed between two concentric shafts. The concentric length is expressed with h. It is assumed here that the annual area is lubricated with continuous flow.2.4 Drag Torque within Multi-plates Wet Clutches The theories for drag torque within wet multi-plate clutches has been discussed by several authors in the past decades. In 1984, the governing equations presented by Hashimoto ADDIN EN.CITE <EndNote><Cite><Author>Hashimoto</Author><Year>1984</Year><RecNum>17</RecNum><DisplayText>[13]</DisplayText><record><rec-number>17</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">17</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Hashimoto, H.</author><author>Wada, S.</author><author>Murayama, Y.</author></authors></contributors><titles><title>The performance of a Turbulent Lubricated Sliding Bearing Subject to Centrifugal Effect</title><secondary-title>Trans. Jpn, Soc. Mech. Eng.,Ser, C </secondary-title></titles><periodical><full-title>Trans. Jpn, Soc. Mech. Eng.,Ser, C</full-title></periodical><pages>1753-1761</pages><volume>49</volume><number>446</number><dates><year>1984</year></dates><urls></urls></record></Cite></EndNote>[13] describe the flow between adjacent flat rotating plates which laid down a frame work for subsequent clutch studies. Then Kato et al. ADDIN EN.CITE <EndNote><Cite><Author>Kato</Author><Year>1993</Year><RecNum>20</RecNum><DisplayText>[16]</DisplayText><record><rec-number>20</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">20</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kato, Y.</author><author>Murasugi, T.</author><author>Hirano, H.</author></authors></contributors><titles><title>Fuel Economy Improvement through Tribological Analysis of the Wet Clutches and Brakes of an Automatic Transmission</title><secondary-title>Society of automotive Engineers of Japan</secondary-title></titles><periodical><full-title>Society of automotive Engineers of Japan</full-title></periodical><pages>57-60</pages><volume>16</volume><number>12</number><dates><year>1993</year></dates><urls></urls></record></Cite></EndNote>[16] explained oil film shrinking between two adjacent clutch plates due to centrifugal force which is widely accepted and referred. Kitabayashi et al. ADDIN EN.CITE <EndNote><Cite><Author>Kitabyashi</Author><Year>2003</Year><RecNum>22</RecNum><DisplayText>[18]</DisplayText><record><rec-number>22</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">22</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kitabyashi, H.</author><author>Li, C.</author><author>Hiraki, H. </author></authors></contributors><titles><title>Analysis of the Various Factors Affecting Drag Torque in Multiplate Plate Wet Clutches</title><secondary-title>JSAE Paper No.2003-01-1973.</secondary-title></titles><periodical><full-title>JSAE Paper No.2003-01-1973.</full-title></periodical><dates><year>2003</year></dates><urls></urls></record></Cite></EndNote>[18]provides demonstration of the drag torque which is accurate in low speed ranges, but it is poor with predicting drag torque at high speed region as it can only show the rising portion of a typical drag torque curve at the low speed region when the clearance if full of oil film. It was demonstrated by Yuan et al. ADDIN EN.CITE <EndNote><Cite><Author>Yuan</Author><Year>2003</Year><RecNum>23</RecNum><DisplayText>[19]</DisplayText><record><rec-number>23</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">23</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Yuan, Y.</author><author>Attibele, P.</author><author>Dong, Y.</author></authors></contributors><titles><title>CFD Simulation of the Flows Within Disengaged Wet Clutches of an Automatic Transmission</title><secondary-title>SAE Technical Paper 2003-01-0320</secondary-title></titles><periodical><full-title>SAE Technical Paper 2003-01-0320</full-title></periodical><dates><year>2003</year></dates><urls><related-urls><url>;[19] using the commercial computational fluid dynamics (CFD) code FLUENT. As Yuan et al. ADDIN EN.CITE <EndNote><Cite><Author>Yuan</Author><Year>2010</Year><RecNum>25</RecNum><DisplayText>[21]</DisplayText><record><rec-number>25</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">25</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Yuan, Shihua</author><author>Peng, Zengxiong</author><author>Jing, Chongbo</author></authors></contributors><titles><title>Experimental Research and Mathematical Model of Drag torque in Single-plate Wet clutch</title><secondary-title>Chinese Journal of Mechanical Engineering</secondary-title></titles><periodical><full-title>Chinese Journal of Mechanical Engineering</full-title></periodical><pages>1-8</pages><volume>23</volume><dates><year>2010</year></dates><urls></urls></record></Cite></EndNote>[21] then provides an improved model which has a reasonable accurate at both low and high speeds, this drag torque model will be compared here with a new shrinking model, which is conducted by Li et al. ADDIN EN.CITE <EndNote><Cite><Author>Li</Author><Year>2013</Year><RecNum>34</RecNum><DisplayText>[32]</DisplayText><record><rec-number>34</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">34</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author> Li, Heyan</author><author>Jing,Qi </author><author>Ma,Biao </author></authors></contributors><titles><title>Modeling and Parametric Study on Drag Torque of Wet Clutch,</title><secondary-title><style face="normal" font="default" size="100%">Proceedings of the FISITA 2012 World Automotive Congress</style><style face="normal" font="default" charset="134" size="100%">, Lecture Notes in Electrical Engineering</style></secondary-title></titles><periodical><full-title>Proceedings of the FISITA 2012 World Automotive Congress, Lecture Notes in Electrical Engineering</full-title></periodical><pages>21-30</pages><volume>193</volume><dates><year>2013</year></dates><urls></urls></record></Cite></EndNote>[32]based on analytical and experimental investigation.2.4.1 Surface Tension ModelYuan et al. ADDIN EN.CITE <EndNote><Cite><Author>Yuan</Author><Year>2003</Year><RecNum>23</RecNum><DisplayText>[19]</DisplayText><record><rec-number>23</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">23</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Yuan, Y.</author><author>Attibele, P.</author><author>Dong, Y.</author></authors></contributors><titles><title>CFD Simulation of the Flows Within Disengaged Wet Clutches of an Automatic Transmission</title><secondary-title>SAE Technical Paper 2003-01-0320</secondary-title></titles><periodical><full-title>SAE Technical Paper 2003-01-0320</full-title></periodical><dates><year>2003</year></dates><urls><related-urls><url>;[19] has presented a reasonable logic to describe the phenomenon in wet clutch plates. When the wet DCT clutch packages is full of dual clutch transmission fluid (DCTF) with viscous, a schematic of a DCT clutch pack is shown in Figure 2. Within the clutch pack, there feeding pressure (inner diameter) is approximately the same as the exiting pressure (outer diameter). The fluid is continually pumped through the clutch pack and flow between two clutches plates is essentially driven by the centrifugal force, while viscous shear force and surface tension forces tend to resist this motion. hSeparate PlateFriction PlateQr1r2Figure 2. Schematic of an open wet clutch.The effective radius within the clutches plates is determined by the centrifugal force, viscous force and surface tension forces. When rotational speed is low, the viscous and surface tension forces are larger than that of centrifugal force and full immersion is maintained. Thus, clutches plates are full of fluid with r2 as effective radius. With the rotational speed increasing, the centrifugal force will increase. Once the centrifugal force is larger than viscous and surfaces tensions forces, the effective radius ro will become smaller. 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ADDIN EN.CITE.DATA [21] continues solving the Reynolds equation, which can be expressed in polynomial form. The effective clutch plate outer radius ro can be achieved as the root of the Equation (35) if less than the existing outer radius.ρoil?ω22f+14ro2-μQ2πrmhc3Grro+μQ2πrmhc3Grri-2σcosθhc-ρoil?ω22f+14ri2=0 (36)The Reynolds number is used to determine the turbulent flow efficiency f shown in Eq. (36), as shown in reference YuanPEVuZE5vdGU+PENpdGU+PEF1dGhvcj5ZdWFuPC9BdXRob3I+PFllYXI+MjAxMDwvWWVhcj48UmVj
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ADDIN EN.CITE.DATA [21]. According to reference ADDIN EN.CITE <EndNote><Cite><Author>Kitabyashi</Author><Year>2003</Year><RecNum>22</RecNum><DisplayText>[18]</DisplayText><record><rec-number>22</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">22</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kitabyashi, H.</author><author>Li, C.</author><author>Hiraki, H. </author></authors></contributors><titles><title>Analysis of the Various Factors Affecting Drag Torque in Multiplate Plate Wet Clutches</title><secondary-title>JSAE Paper No.2003-01-1973.</secondary-title></titles><periodical><full-title>JSAE Paper No.2003-01-1973.</full-title></periodical><dates><year>2003</year></dates><urls></urls></record></Cite></EndNote>[18], the effective drag torque within the clutch packs then can be expressed as follows:TCl=2πNμ?ωhcriror3dr+0.0024πNμ?ω1.94hc(ρhcμ)0.94riror3.94dr (37)Where ?ω represents the relative speed between two adjacent clutch plates. From Figure 2, ?ω can also means the relative speed between the input clutch plates and the output clutch plates. 2.4.2 Alternative Shrinking ModelAfter studying the numerical and experimental investigated wet clutch drag torque model conducted by Yuan et al. ADDIN EN.CITE <EndNote><Cite><Author>Yuan</Author><Year>2010</Year><RecNum>25</RecNum><DisplayText>[21]</DisplayText><record><rec-number>25</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">25</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Yuan, Shihua</author><author>Peng, Zengxiong</author><author>Jing, Chongbo</author></authors></contributors><titles><title>Experimental Research and Mathematical Model of Drag torque in Single-plate Wet clutch</title><secondary-title>Chinese Journal of Mechanical Engineering</secondary-title></titles><periodical><full-title>Chinese Journal of Mechanical Engineering</full-title></periodical><pages>1-8</pages><volume>23</volume><dates><year>2010</year></dates><urls></urls></record></Cite></EndNote>[21], Li et al. ADDIN EN.CITE <EndNote><Cite><Author>Li</Author><Year>2013</Year><RecNum>34</RecNum><DisplayText>[32]</DisplayText><record><rec-number>34</rec-number><foreign-keys><key app="EN" db-id="2dra5zszs9tzthea2scvstd1dzxre0fpzszr">34</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author> Li, Heyan</author><author>Jing,Qi </author><author>Ma,Biao </author></authors></contributors><titles><title>Modeling and Parametric Study on Drag Torque of Wet Clutch,</title><secondary-title><style face="normal" font="default" size="100%">Proceedings of the FISITA 2012 World Automotive Congress</style><style face="normal" font="default" charset="134" size="100%">, Lecture Notes in Electrical Engineering</style></secondary-title></titles><periodical><full-title>Proceedings of the FISITA 2012 World Automotive Congress, Lecture Notes in Electrical Engineering</full-title></periodical><pages>21-30</pages><volume>193</volume><dates><year>2013</year></dates><urls></urls></record></Cite></EndNote>[32] improved the model with a new way to obtain the equivalent radius considering oil film shrinking phenomenon, which will be discussed as below.When the clearance is full of oil film, i.e., r=R1, then the required input flow rate, can be obtainedQ=6μπhc3lnRiRo+(6μπhc3lnRiRo)2-81ρ2?ω2Ro-2-Ri-2Ri2-Ro2-540ρRo-2-Ri-2?p700π2hc227ρ70π2hc2Ro-2-Ri-2 (38)Therefore Equation (38) can be used to calculate the required input flow rate enable the clearance being full of oil film. It shows that the demanded input flow is influenced by the angular speed of the clutches. A trending figure to show the needed flow rate can be plotted by simulation shown in Figure 3.As shown in Figure 3, the required input flow for full oil film between clutch plates grows with the increasing of the clutch plate’s speed. However, in the real practice, particular in the two-speed DCT, the actual flow rate is constant. Therefore, the oil film will shrink when the flow rate can not meet the required flow rate with the increasing of the speed. Then define the required flow rate as Q, the actual flow rate is Qi, and the effective outer oil film radius as R0.When Qi≥Q, R0=R2 (39)When Qi<Q, the relationship of oil flow rate and volume between import and export, the equivalent radius of oil can be presented as:QiQ=Ro2-R12R22-R12 (40)Equation (40) can be rearranged asRo=QiQR22+R12(1-QiQ) (41)Figure 3. Relationship of ideal and actual required flow rate.With the obtained equivalent effective radius from Equation (41), the drag torque can be achieved as follows:τθz=μ?Vθ?Z|z=hc=μ?ωrhc (42)TCl=2πNRiRorτθzrdr=πμ?ωN2hc(Ro4-Ri4) (43)Comparisons are made between the two kinds of wet clutch drag torque model, and the simulation results are shown in Figure 4.Figure 4. Relationship of drag torque and angular velocity (2nd model) with clearance 0.20 mm.Figure 4 shows the simulation results for the first surface tension and the second alternative shrinking model for predicting wet clutch drag torque. It shows the relationship between drag torque and angular velocity. The sold line shows the drag torque for the 1st surface tension model, and the dashed line for the 2nd new shrinking model. Both of the line show that the drag torque is linearly increasing with the angular velocity in the first part from 0 to the peak value. However, after the speed with the peak drag torque, the drag torque reaches almost zero for the 1st surface tension model. And the drag torque for the second alternative shrinking model doesn’t decrease so dramatically. In real practice, in order to continue cool the clutch packages, the follow rates should not be diminished, therefore, the drag torque should not be drop down to zero in high speed, i.e., higher than 3000 rpm, consequently. The second model is more likely to present actual conditions. Therefore, the second shrinking model will be chosen for further modelling and simulation within this work.2.4.3 Drag Torque within two-speed DCTC1C1Motor output speedDifferentialω0ω1ω2r2ndΔω1C2r1st(a)DifferentialMotor output speedω0ω1ω2r2ndC2r1stΔω2(b)S3S1S2S1S2S3Figure 5. Schematics of powertrain equipped with two-speed DCT,(a) clutch 1 closed, clutch 2 open; (b) clutch 2 closed, clutch 1 open.For steady state conditions, if the Clutch 1 (C1) is engaged, as shown in Figure 5 (a), and the C2 is in open state, the vehicle is running with 1st gear. The clutch 1 input shaft (S1) is rotating via mating of the input clutch, and its speed is the same as the output speed of EM (ωEM). Then the 1st gear is engaged with the pinions in the layshaft (S3). The layshaft speed is ωEM/r1st. Then the layshaft (S3) will rotate with the 1st gear, causing the inner shaft (S2) rotating as well. The inner shaft (S2) speed becomes ωEM*r2nd/r1st. Therefore, the C2 output clutch plates will rotate freely caused by the engaged 2nd gear, with speed ωEM*r2nd/r1st. The relative speed between the input clutch plates and the 2nd output clutches plates is:?ωc2=ωEM*(1-r2ndr1st) (44)Similarly, if the C2 is engaged, as shown in Figure 5(b), and the vehicle runs with the 2nd gear, the relative speed between the input clutch plates and the 1st output clutch plates is ωEM*(1-r1st/r2nd).?ωc1=ωEM*(1-r1str2nd) (45)Equation (43) shows that the drag torque is directly affected by the relative rotational speed between the input and output clutch plates. Substituting of Equation (44) and Equation (45) into Equation (43) respectively, the wet clutch drag torqueTClr1st=πμ?ωc22hcRo4-Ri4=ωEM*(1-r2ndr1st)*πμN2hcRo4-Ri4 (46)TClr2nd=πμ?ωc12hcRo4-Ri4=ωEM*(1-r1str2nd)*πμN2hcRo4-Ri4 (47)In transmissions, if the input speed from the motor is the same, as r1st is larger than that of G2, it comes out?ωc2>?ωc1. Thus the relative rotational speed for clutch 2 (run with 2nd gear) is always larger than that of C1 (1st gear). If the effective radius of the clutch 1 and clutch 2 are the same, the drag torque engaged 2nd gear TClG2 should be larger than that of engaged with 1st gear. However, the effective radius of plates in clutch 2 (C2) of this particular DCT are larger than that of C1. Therefore, the finally drag torque will be determined by both the differences in effective radius and gear ratios.Experimental ApparatusIn this section, test facility and hardware, test instrumentation, data acquisition, and test operation will be briefly presented. 3.1 Test Facility and HardwareThe test facility used for this study is shown in Figure 6 with several changes to the original UTS powertrain system test rig. The new test rig is based on an electric vehicle power train system. The original Engine is replaced with a 330V DC drive permanent magnet Motor which is converted from AC high voltage electric power; the original AT transmission is replaced with a modified two-speed DCT, and the detail transmission parameters are shown from Table 1 to Table 4. A group of flywheels and tires are used to simulate the vehicle inertia. The dynamometer, HPA engine stand 203, a load drag on the flywheels. This part of test rig consists of two wheels contacted with flywheels, another final drive connected with shaft before dynamometer. The DCT has a separate supply pump. The lubricate fluid will be instead with DCTF rather ATF. And the vehicle control unit (VCU) and transmission control unit (TCU) will be built based on dSPACE Micro-Auto Box. Figure 6. Modified test rig of two-speed DCT powertrain, A) DCT; B) Electric motor; C) Motor controller; D) Groups of flywheels; E) Dynamometer; F) High voltage power supply.Table 1. DCT clutch plates geometric parametersClutch 1 platesClutch 2 platesClutch ClearancePlates NumberInner RadiusOuter RadiusInner RadiusOuter Radius57.5 mm69.5 mm81.5 mm96.5 mm0.5 mm8Table 2. DCT gears geometrical dataGears 1st gear?2nd gear?DifferentialGearPinionGearPinionGearPinion Ratio2.04551.4524.058Pitch circle diameter108.2553.2595.566212.552.5Tooth depth2.8752.8752.752.753.753.75Centre distance80.7580.7580.7580.75132.5132.5Outside Diameter1145910171.522060Face width181615153537Number of teeth452245316917Helix angle303032323232Transverse operating pressure angle0.3980.3980.4050.4050.4050.40522.79622.79623.22823.22823.22823.228Module2.4052.4202.1222.129Table 3. DCT Bearings InformationBearings12345678TypeNeedleNeedleNeedleDeep groove ballRoller, angular contactRoller, angular contactRoller, angular contactRoller, angular contactSealsnononoNon-contactnonononoCasingexternalnonoallallallallall ID34.0024.0035.0028.0030.0038.0028.0040.00OD56.0028.0042.0068.0050.0083.0062.0072.00Thickness14.0024.0026.0018.0018.0018.0013.0019.00 dm45.9026.0538.6150.7840.8363.2947.1457.52Table 4. Dual Clutch Transmission Fluid (DCTF) Properties and Test conditionsFlow rate (m^3/s)Oil density (kg/m3)Oil viscosity (Ns/m2)Temperature (oC)0.00005853.40.029040During the test, the temperature is between 35oC and 45oC, normally approximate 40 oC. Because every test point is running about 2 - 3 minutes then taking a breaking. Hence it is assumed that the oil temperature does not change too much during this period.3.2 Instrumentation and Data AcquisitionThe test instrumentation used in this study only required measurement of the input and output speed and torque of the DCT respectively, and the DCT inside oil case temperature. Two wireless torque sensors are installed in the front half drive shafts, shown in Figure 7, which are used to acquire the output torque data of the DCT. The wireless torque sensor was bought from ATI Technologies. And the oil temperature is tested by the temperature sensor embedded in the transmission itself. The torque sensors are calibrated before testing. And the DCT input torque is equal to the output torque of electric motor, which can be collected by the motor feedback torque. This data was collected at 1000 per second throughout the test via dSPACE Control Desk Recorder, and then processed in computer.Electric MotorWireless Torque sensorDCTFlywheelsMotorControllerTorque Sensors Data RecorderElectrical ConnectionMechanical ConnectionFigure 7. Schematic test rig system3.3 Test OperationThe test procedure used for collecting the data was the following. For a given group of operating conditions (gear, speed, motor torque, and lubricating oil temperature) the test rig was operated and stabilized last for at least 2 minutes. Then average one is chosen. And the error of efficiency is ±1%. Via comparing the input and output torque of transmission under different driving conditions, the difference of total drag torque power loss can be measured by torque sensor. Results and analysisIn order to validate the proposed model under the conditions of Table 1 and Table 2, simulation and experimental tests are made. The following contents are results and analysis regarding experimental and simulation results.Figure 8 and Figure 9 show the DCT efficiency for 1st gear (a) and 2nd gear (b) from simulation prediction and during testing using constant input torque and constant input speed respectively. In Figure 8, constant input torque (60 Nm) and various input speed from electric motor is set in this group of study. The DCT average efficiency is higher than 95% in both gears. And the efficiency in the 2nd gear shown in Figure 8 (b) is slightly higher than that of in the 1st gear shown in Figure 8 (a). Both of the efficiency decreases until reach their bottom values respectively. After a certain speed, i.e. critical speed, the drag torque will increase gradually with the speed continually increasing in high speed. The predicted critical speeds for 1st and 2nd gear are 2950 rpm and 2530 rpm respectively, which are very close to the speeds obtained from test results. The differences between simulation and test results are smaller, less than 0.5%. It appears that the simulation results are always a little higher than test results by 0.2% in efficiency. It is likely there are some other minor sources affecting the test final results, such as the windage loss from the shaft and 1st and 2nd gear pairs by oil-air mixture resistance. (a) 1st gear 2nd gearFigure 8. Comparison of DCT efficiency between simulation and test results with input torque 60 NmIn Figure 9, results for constant input speed (3000 rpm) and variation of input torque from electric motor is presented. The input torque for first gear (a) changes from 20 Nm to 60 Nm, whilst for the second ranges from 30 Nm to 60 Nm, because second gear ratio (G2=5.36) is smaller than first one (G1=8.45), i.e., smaller gear ratio requiring larger torque to start up or maintaining stable speed. Both Figure 9 (a) and (b) show that the DCT efficiency increases with the input torque continually increasing. And the efficiency in the second gear shown in Figure 9 (b) is slightly higher than that of in the first gear shown in Figure 9 (a). The differences between simulation and test results are smaller, especially in higher input torque with less than 0.5%. From Figure 8 and Figure 9, it can be concluded that the results from the predicted drag torque model agrees well with that from experimental test. 1st gear2nd gearFigure 9. Comparison of DCT efficiency between simulation and test results with input speed 3000 rpm(a) 1st gear (b) 2nd gearFigure 10. Individual drag torque power loss from simulation result with input torque 60 Nm.Figure 10 shows the individual drag torque losses from simulation results at stable statue. The clutch drag torque is the major loss in both gears, followed by gear churning and windage losses and bearing losses. And the clutch drag torque loss in 1st gear is larger than that of 2nd gear if under the same input speed. The reason for that is when 1st gear engaged, the larger clutch (C2) will open, which generates higher torque than that of the smaller clutch (C1) when the 2nd gear is engaged. Figure 11 shows the percentage of losses by individual drag torque sources. It shows that the percent of loss generated by clutch drag increases gradually before reach the critical speed. After the critical speed, the percent of clutch drag loss drops down slightly. It is closely corresponding with the shape in Figure 8 and Figure 10. The percent of power loss in 1st gear by clutch drag is higher than that of in 2nd gear. And gear churning and windage loss holds the reverse condition, that is, the percent of power losses in 1st gear by gear churning and windage losses is lower than that of in 2nd gear.1st gear2nd gearFigure 11. Percent of Individual drag torque power loss from simulation results with input torque 60 Nm.1st gear2nd gearFigure 12. Individual drag torque power loss from simulation result with input speed 3000 rpm.Figure 12 shows the individual drag torque losses from simulation results with constant input speed 3000 rpm. Input torque from electric motor changes from 20 Nm to 70 Nm. Both Figure 12 (a) and (b) demonstrate that the clutch drag torque is the major loss in both gears, followed by gear churning and windage losses and bearing losses. And the gear meshing loss shares only a small part of the global resisting torque. Also the clutch drag torque loss in 1st gear is larger than that of 2nd gear, which can explain why the efficiency of DCT first gear is smaller than that of second gear. Moreover, only bearing and oil seal, and gear meshing loss increase slightly with the increasing of input torque, as both represent only a small part of the total drag torque losses, which can explain why the DCT efficiency will increase with increasing of input torque at constant input speed.5 ConclusionsThe purpose of this paper is to develop a model of drag torque in a dual clutch transmission to predict its operating efficiency. After theoretical analysis of five major sources of drag torque within a two-speed DCT, this article then implemented the developed mathematical model of the transmission and applied it to steady state simulation. To support the theoretical study an experimental investigation is conducted to compare the physical results with that of obtained from simulation under various operating conditions. The test results for DCT efficiency agrees reasonable accurately with the simulation results, especially the critical speed prediction. The average error in DCT efficiency is less than 0.5%. It can be concluded that the proposed model performs well in the prediction of drag torques for the transmission, and is can be applied to assess the efficiency of the transmission. Results demonstrate that (1) the mean two-speed DCT efficiency can be reach 95%. (2)And the efficiency of DCT grows with the input torque increasing, and decreases with the rise of input speed. Additionally, (3) the entire drag torque is dominated by the viscous shear in the wet clutch pack, followed by the differential gear churning and windage losses. And the gear meshing loss holds only a small contribution of the global resisting torque. (4) Moreover, when the vehicle run in 2nd gear, the drag torque generated by clutch shear stress is smaller than that of run in 1st gear in the same input speed. (5) The gear churning and windage losses for 2nd gear, however, are larger than that of in 1st gear. This work can be looked as a reference to future research on reducing drag torque, applications of drag torque in two-speed or multi-speed electric vehicle powertrain system efficiency optimization and torque estimated shift control. With some slight adjustments according to the detail transmissions layout, this numerical method for predicting the two-speed DCT drag torque can also be applied to other transmissions equipped with wet clutch packages, let alone ordinary manual transmission. But the limitation of this work does not consider the influences of temperature on drag torque, which will be analysed on our future work regarding transmission thermal behaviours analysis.AcknowledgementSupport from Beijing Electric Vehicle Co. Ltd., NTC Powertrain, and the Ministry of Science and Technology of China is gratefully acknowledged. The first author would also like to express his gratitude to China Scholarship Council and University of Technology, Sydney (UTS) for their scholarship support.NomenclatureAg =arrangement constant for gearingbw =face width in contactC1 =clutch 1C2 =clutch 2D =outside diameter of the geardm =diameterEDCT =efficiency of DCTF =total face widthFb =applied loadf =turbulent flow coefficientsf0 =bearing dip factorfg =gear dip factorfm =mesh coefficient of frictionfL =bearing load empirical factorHs =sliding ratio at the start of approach actionHt =sliding ratio at the end of recess actionh =tangential line velocity modifying exponenthc =clutch plates clearancehcon =length of concentric shaftj =Viscosity modifying exponentK = load intensity L =length of the gearM =mesh mechanical advantageMt =transverse tooth moduleN =number of frictional surface n1 =pinion rotational speednmotor =motor speedPB =power losses caused by bearings drag torquePCon =power losses caused by concentric shaft drag torque,Pbl =load independent power lossPbv =speed independent power lossPCh =power losses caused by gear churningPCl =power losses caused by wet clutch plates drag torquePG =power losses caused by gear meshing drag torquePGW =individual gear windage and churning lossPL =total power losses in DCTPoil =oil seal power lossesQ =flow rateT1 =pinion torqueTB =drag torque caused by bearingsTB1,2 =drag torque caused by bearing (1) and (2)TCl =drag torque caused by wet clutch packsTCh =drag torque caused by churningTcon =drag torque caused by concentric shafts viscous shear resistanceT1st_output_outer =output torque of the outer concentric shaftT1st_output_inner =output torque of the inner concentric shaftTfinal_output =final output torque from DCTTGM =drag torque caused by gear pairs meshingTGM1st_pair =drag torque caused by 1st gear pair meshingTm =motor output torqueV = pitch line velocityVoil=velocity for oil sealRcon_i =outer radius of the inner shaftRcon_o =inner radius of the outer shaftRf =roughness factorRo2 =gear outside radiusRw2 =gear operating pitch radiusRo1 =pinion outside radiusRw1 =pinion operating pitch radiusr =gear ratiorm =mean radiusro =outer radius of the clutchr1st =1st gear ratior2nd =2nd gear ratioz1 =number of pinion teethz2 =number of gear teethαw =transverse operating pressure angleβ =generated helix angleβw =operating helix angleμ =viscosity of the oilν =kinematic oil viscosity ρoil =density of oilωEM =output speed of the electric motor ?ω =relative speed between two concentric shafts?ωc1 =relative speed within clutch 1?ωc2 =relative speed within clutch 2?p =pressure difference between the input and output of clutch pairReferences ADDIN EN.REFLIST 1.Chan CC. 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