吉林大學(xué)汽車(chē)?yán)碚撈?chē)動(dòng)力性

1、第一章汽車(chē)動(dòng)力性Vehicle Performance主要研究?jī)?nèi)容: 分析受力; 建立行駛方程式; 以圖、表(或編程)形式，按照評(píng)價(jià)指標(biāo)，確定汽車(chē)的動(dòng)力性。定義在平直良好路面上行駛時(shí)所能達(dá)到的平均行駛速度。Definition : At the good and straight and even track, the auto driving can reach at the average speed.Main contents:Analysis is subjected to the reactions;Establishing the motion equation of vehicl

2、e;With the form of the diagram, the table or the plait distance), press the vehicle tractive evaluation index, make sure the motion of the vehicle.汽車(chē)動(dòng)力性Veh. Perf.第一章 汽車(chē)動(dòng)力性1.1 汽車(chē)動(dòng)力性評(píng)價(jià)指標(biāo)1.2 汽車(chē)驅(qū)動(dòng)力和行駛阻力1.3 汽車(chē)驅(qū)動(dòng)力附著條件以及附著力1.4 汽車(chē)驅(qū)動(dòng)力行駛阻力平衡圖1.5 汽車(chē)功率平衡1.1 Traction performance evaluation index1.2 Traction forc

3、es and the resistance as vehicle driving1.3 Driving traction force , adhesive condition and its forces1.4 Balance diagram between the traction force & driving resistance1.5 Power balanceVehicle traction performance1.1 汽車(chē)動(dòng)力性評(píng)價(jià)指標(biāo)最大車(chē)速 umax加速時(shí)間 t最大爬坡度 imaxIndex Evaluating Vehicle Traction Performance Ma

4、ximum SpeedAcceleration TimeMaximum grade加速時(shí)間評(píng)價(jià)方法超車(chē)加速時(shí)間 Overtaking acceleration time 以最高檔或次高檔，以amax加速至某一高速所用的時(shí)間。原地起步加速時(shí)間 The start acceleration time 由I或II檔起步，以amax，并考慮換檔時(shí)間，一般用0400m或0100km/h的時(shí)間表示原地起步的加速時(shí)間。Method evaluating acceleration time 最高車(chē)速 Max Speed最高車(chē)速，是指汽車(chē)在平直的良好道路(混凝土或柏油)上所能達(dá)到的平均最高行駛車(chē)速。Max. s

5、peed, means that the vehicle drives on the straight and even and good (concrete or asphalt) roadway can reach at the average and largest speed.汽車(chē)加速過(guò)程曲線VelocityTimeDiagram of velocity vs timevehicle acceleration history 車(chē)速Gear shifting爬坡能力的評(píng)價(jià) 以滿載、良好路面上的imax來(lái)表示: 商用車(chē)30或16.5； 越野汽車(chē)60或31。 轎車(chē)最高車(chē)速較大，且通常在良好的

6、市區(qū)道路行駛，其爬坡度很大。對(duì)其無(wú)這項(xiàng)要求。 有的國(guó)家要求汽車(chē)在常遇坡道上汽車(chē)必須保持的速度來(lái)表明其加速能力。The sloping ability evaluationSome nations request the MV on often meeting the road of slope the speed that the MV must keep to expresses that it accelerates the ability.發(fā)動(dòng)機(jī)傳動(dòng)系離合器、變速器、分動(dòng)器、傳動(dòng)軸、主減速器、半軸驅(qū)動(dòng)輪驅(qū)動(dòng)力行駛阻力滾動(dòng)阻力坡度阻力加速阻力空氣阻力汽車(chē)動(dòng)力傳遞路線：發(fā)動(dòng)機(jī)離合器變速器副變

7、速器傳動(dòng)軸主減速器差速器半軸輪邊減速器車(chē)輪1.2 汽車(chē)驅(qū)動(dòng)力和行駛阻力1.2 Traction forces and resistances as vehicle driving平衡1.2 汽車(chē)驅(qū)動(dòng)力和行駛阻力Traction force and resistances as vehicle driving驅(qū)動(dòng)力滾動(dòng)阻力空氣阻力加速阻力爬坡阻力驅(qū)動(dòng)轉(zhuǎn)矩行駛阻力發(fā)動(dòng)機(jī)轉(zhuǎn)矩Traction forcetraction effortdriving force圖1-2 汽車(chē)驅(qū)動(dòng)力MV Traction ForceNormal reactionGravitySpeedRadiusTorque部分負(fù)荷特性

8、：節(jié)氣門(mén)部分開(kāi)啟時(shí)，轉(zhuǎn)矩或功率等與轉(zhuǎn)速的關(guān)系使用負(fù)荷特性曲線：即帶有附件時(shí)的負(fù)荷特性，通常汽油機(jī)下降15，而柴油機(jī)下降10外特性曲線：節(jié)氣門(mén)(油門(mén))全開(kāi)時(shí)，轉(zhuǎn)矩或功率等與轉(zhuǎn)速的關(guān)系發(fā)動(dòng)機(jī)速度特性full throttle/acceleratorrpm:revolution per minute 圖1-3 汽油發(fā)動(dòng)機(jī)外特性Braking Power ,kWRevolution, rpmCharacteristic of engine speed at full throttleBraking Torque, .m發(fā)動(dòng)機(jī)過(guò)渡工況的速度特性在過(guò)渡工況，功率和轉(zhuǎn)矩均下降約5%6%。Characteri

9、stic of the engine speed in transient conditionIn transient condition, power and torque all descend about 5%-6%.外負(fù)荷特性的使用和制作方法表格法（輔助插值）曲線族法數(shù)學(xué)模型法n1n2n3nnPe1Pe2Pe3PenTt1Tt2Tt3Ttnbe1be2be3benFull-load characteristic diagram methodInterpolation with tableCurvesModeling外特性及負(fù)荷特性數(shù)學(xué)描述Modeling for Full- & par

10、t-load characteristicm7 mopt5Braking Power ,kW圖1-4 汽油發(fā)動(dòng)機(jī)外特性及部分負(fù)荷特性Characteristic of engine speed at full and part loadengine speed, r/minBraking Torque, N.m發(fā)動(dòng)機(jī)負(fù)荷特性經(jīng)驗(yàn)公式Approximation for Full- & part-load characteristic傳動(dòng)系功率損失機(jī)械損失液力損失摩擦潤(rùn)滑油齒輪傳動(dòng)副間摩擦軸承摩擦。潤(rùn)滑油的攪動(dòng)潤(rùn)滑油與零件表面間摩擦。傳動(dòng)系機(jī)械效率T Transmissions efficienc

11、yPower lossLiquid L.Mecha. L.RubLubr.傳動(dòng)系機(jī)械效率T Transmissions efficiency損失主要部件: Parts causing to lose 變速器和主減速器(含差速器)損失主要形式: Forms causing to lose 液力損失和機(jī)械摩擦損失 Mechanical loss & liquid loss液力損失,如攪動(dòng)和磨擦。與潤(rùn)滑油的性能、溫度、轉(zhuǎn)速、油面高度等有關(guān)。46檔變速器 4-6speed gearbox 0.9668檔變速器 6-8speed gearbox 0.95傳動(dòng)軸 drive shaft 0.98離合器 C

12、lutch 0.98主減速器 differential 0.96-0.92半軸 semi-shaft 0.99汽車(chē)機(jī)械效率 MV mechanical efficiency轎車(chē) Passenger car T=0.900.92商用車(chē) Commercial Veh.T=0.820.85越野車(chē)(SUV) 4W drive Veh.T=0.800.85典型的傳動(dòng)系效率值Typical drive system efficiency某汽車(chē)變速器的機(jī)械效率Mechanical efficiency of MV gearbox輪胎半徑r Tire radius自由半徑 r Unloaded radius靜

13、力半徑 rs Static radius with load滾動(dòng)半徑 rr Dynamic radius with load rr= rs = r=s/(2nw)S行駛距離 trip，n 轉(zhuǎn)動(dòng)圈數(shù) revolutions歐洲輪胎與輪輞委員會(huì)(ETRTO)推薦 rr= Fd/(2) F=3.05 radial F=2.99 bias德國(guó)橡膠工業(yè)協(xié)會(huì)推薦 CR=CR(1+ua/1000),mmCircumference length輪胎半徑r（mm)車(chē)型規(guī)格自由靜力滾動(dòng)桑塔納2000195/60R1429532685286紅旗CA7228l85/70HRl431232855301奧迪100195/

14、65Rl531732915308躍進(jìn)NJl0617.00-2045244306448解放CA10919.00-2050954856492東風(fēng)EQl0929.R2050954766493黃河JNll8111.00-2054265177525大 小 中定義：用Ft-ua曲線圖來(lái)全面地描述汽車(chē)的驅(qū)動(dòng)力。若已知外特性曲線、傳動(dòng)系速比、傳動(dòng)系機(jī)械效率、車(chē)輪半徑就可求計(jì)算驅(qū)動(dòng)力。汽車(chē)驅(qū)動(dòng)力圖 MV traction force diagramDefinition:Use the Ft-ua curves to describe completely the MV traction force. If on

15、e has already known the full load characteristic curve, transmission , system efficiency and rolling radius, one can calculates the traction forces.轉(zhuǎn)速速度線性變換轉(zhuǎn)矩驅(qū)動(dòng)力線性變換驅(qū)動(dòng)力圖制作 Drawing diagram of tractionSp ISp IISp IIITractionMV VelocityMV traction diagram驅(qū)動(dòng)力汽車(chē)行駛阻力滾動(dòng)阻力Ff :輪胎內(nèi)部摩擦產(chǎn)生的遲滯損失。這種遲滯損失表現(xiàn)為阻礙車(chē)輪運(yùn)動(dòng)的阻

16、力偶。MV driving resistanceRolling resistance Ff: the tire inner part rub causes to loss, this kind of lose performance shows the resistance moment of the wheel motion.輪胎徑向變形曲線 Tire radius deforming curveDamping loss輪胎/路面接觸輪胎變形載荷遲滯損失阻力偶loadunload從動(dòng)輪Driven Wheel滾動(dòng)阻力系數(shù) f rolling resistance coefficient輪胎內(nèi)

17、部因摩擦產(chǎn)生的遲滯損失表現(xiàn)為阻礙車(chē)輪運(yùn)動(dòng)的阻力偶.The damping loses caused by tire inner inter-rubs are shown as the resistance moment against wheel motion滾動(dòng)阻力系數(shù):車(chē)輪在一定條件下，滾動(dòng)所需要推力Fp1與負(fù)荷W1之比，即單位重力的推力。Rolling resistance coefficient: wheel under the certain condition rolls over with the ratio of push force Fp1 to weight W1, nam

18、ely push force for the unit gravity .阻力偶一般用滾動(dòng)阻力描述roll resistance moment is depicted by rolling resistance.滾動(dòng)阻力無(wú)法在受力圖上畫(huà)出。它是一個(gè)數(shù)值,在受力圖上它是切向反力。Roll resistance cant be drawn in being subjected to the reaction diagram, It is only one value. There is a longitudinal reaction in the force diagram.在實(shí)際計(jì)算時(shí)可不必考慮

19、阻力偶，而用滾動(dòng)阻力描述。While computing physically need not consider the resistance pair, but use to roll resistance.試驗(yàn)確定法牽引法、滑行法和轉(zhuǎn)鼓法經(jīng)驗(yàn)法 通過(guò)以往試驗(yàn)結(jié)果歸納的經(jīng)驗(yàn)公式滾動(dòng)阻力系數(shù)的確定方法室內(nèi)試驗(yàn)道路試驗(yàn)滾動(dòng)阻力系數(shù)的影響因素1. 速度 ua 對(duì) f 的影響creating a phenomenon of standing wave, heat, shed off, burst.Effect of velocity ua on resistance coefficient f.駐

20、波彈性輪胎在滾動(dòng)時(shí)，胎面各部份都反復(fù)處于受力變形和恢復(fù)原形的狀態(tài)。這個(gè)運(yùn)動(dòng)過(guò)程使輪胎的變形恢復(fù)到原狀的力有彈力、胎壓和離心力等。輪胎從接地受力變形，到離開(kāi)地面后恢復(fù)原狀，存在滯后時(shí)間。其長(zhǎng)短與輪胎負(fù)荷、轉(zhuǎn)速、工作溫度和環(huán)境溫度等有關(guān)。負(fù)荷大，胎面變形大; 轉(zhuǎn)速高，胎面離心力大。離心力使胎面反向變形，這樣胎面在正、反方向力的交替作用下造成不同胎面部位半徑的波動(dòng)。當(dāng)輪胎行駛速度超過(guò)輪胎的撓屈變形極限速度后，此胎面波動(dòng)看上去變形移動(dòng)的狀態(tài)恰似停止，稱(chēng)作駐波。常發(fā)生在胎肩與胎側(cè)交界區(qū)。 2.輪胎結(jié)構(gòu)、材料、簾線對(duì) f 的影響也很大。子午線輪胎 f 小，天然橡膠 f 低。The tire structu

21、re, material, the influence of the lines upon the f is also very big. The meridian tire f is small, the crude rubber f is low.f 經(jīng)驗(yàn)公式 fit for fCommercial veh.Passenger carPassenger carCommercial veh.While turning normally驅(qū)動(dòng)輪受力圖Forces reacting on drive wheel切向力Circumferential Force真正驅(qū)動(dòng)車(chē)輪前進(jìn)的力是地面切向反力FX2

22、。其在數(shù)值上等于汽車(chē)驅(qū)動(dòng)力Ft與滾動(dòng)阻力Ff之差驅(qū)動(dòng)力與地面縱向（切向）力定義：汽車(chē)直線行駛時(shí)受到的空氣阻力在汽車(chē)行駛方向上的分力。分類(lèi)：壓力阻力和摩擦阻力壓力阻力由形狀阻力、擾動(dòng)阻力和誘導(dǎo)阻力組成。形狀阻力主要與汽車(chē)的形狀有關(guān)，約占58?？諝庾枇rag or air resistanceCD對(duì)高速汽車(chē)動(dòng)力性和燃料經(jīng)濟(jì)性影響極大! 干擾阻力：汽車(chē)上的突出部件，如后視鏡、門(mén)把手、導(dǎo)水槽、驅(qū)動(dòng)軸、懸架導(dǎo)向桿等，約占14。 內(nèi)循環(huán)阻力：發(fā)動(dòng)機(jī)冷卻系、車(chē)身通風(fēng)氣流等流過(guò)汽車(chē)內(nèi)部形成的，占12。 誘導(dǎo)阻力：空氣升力在水平方向的分力，占7。 摩擦阻力：9?？諝庾枇w正比于氣流相對(duì)運(yùn)動(dòng)的動(dòng)壓力。Fw的影

23、響因素：CD和A。A由于乘坐空間的制約變化不大,CD變化較大。19501970年為0.40.6,1990年為0.250.40,概念車(chē)約為0.2,當(dāng)前 Passat 0.28。1. 前部低 2. 過(guò)渡平滑3. 后部加擾流板4. 掠背式(艙背式)5. 底部導(dǎo)流、平整化，向后應(yīng)逐步升高6. 整車(chē)俯視形狀為腰鼓式7. 改進(jìn)氣流進(jìn)、出口位置8. 商用車(chē)頂(側(cè))部安裝導(dǎo)流罩CD降低的要點(diǎn)降低汽車(chē)空氣阻力的基本思路Basic thought for decreasing aerodynamic drag MV坡道阻力grade resistanceRoad resistance coefficientGra

24、de angle 汽車(chē)行駛方程式MV motion formulaCorrelation of drive force and driving resistance隔離法：平面受力圖driven tireInsulation method: force diagram with two dimensionsLongitudinal forceNormal forcetangential reactioncircumferential force從動(dòng)輪受力分析 reaction analysis of the moved tireVertical forceNormal reaction從動(dòng)輪受

25、力分析（續(xù)）analysis of forces on the moved tire隔離法：受力圖Forces reacting on drive tireInsulation method: force diagram with two dimensions驅(qū)動(dòng)輪受力分析Forces acting on the tracition tire除以r驅(qū)動(dòng)輪受力分析(續(xù))Forces on the tracitive tire加速時(shí)車(chē)身受力圖Force diagram on the body while acceleration車(chē)體受力分析 forces on the body飛輪慣性矩Flywh

26、eel inertial車(chē)體受力分析 (續(xù)) forces on the body車(chē)體受力分析 (續(xù)) forces on the body車(chē)體受力分析 (續(xù)) forces pressed on the body汽車(chē)行駛方程式_總結(jié) 只是表示各個(gè)物理量之間的數(shù)量關(guān)系。 有些項(xiàng)并不是外力。Ft不是作用于車(chē)輪上的切向反作用力，僅為了計(jì)算方便才定義為驅(qū)動(dòng)力。 滾動(dòng)阻力也不是作用于汽車(chē)上的阻力，而是以滾動(dòng)阻力偶矩的形式作用于車(chē)輪上。 作用在汽車(chē)上的慣性力是mdu/dt，而不是mdu/dt。飛輪慣性力矩作用在汽車(chē)的橫截面上，而不作用于車(chē)輪上。 Fj 代表慣性力和慣性力矩的總效應(yīng)。1.3 汽車(chē)行駛驅(qū)動(dòng)-

27、附著條件及汽車(chē)附著力1.汽車(chē)行駛驅(qū)動(dòng)附著條件Driving traction, adhesive condition and its forceDriving tractionadhesive condition驅(qū)動(dòng)（行駛）的必要條件:Driving essential condition:the adhesive condition between the tire and ground tire shows the phenomenon of slide revolution on the road 后驅(qū)動(dòng)汽車(chē)Fz2Reaction of ground on traction tire,

28、i.e. It is controlling the second condition that MV drives. generally is the necessary and full conditions that the MV drives.Rear drive MV2. 汽車(chē)附著力 MV adhesive force在硬路面上， 主要取決路面的種類(lèi)和狀況，也與 有關(guān)。On the hard road, it depends mainly on the category and conditions of road surface, also is relevant with the

29、 velocity路面類(lèi)型 混凝土（干） 混凝土（濕）附著系數(shù) 0.70.8 0.50.6路面類(lèi)型 碎石 土路（干） 土路（濕）附著系數(shù) 0.60.7 0.50.6 0.20.4L=wheel baseB=wheel track前輪驅(qū)動(dòng)和后輪驅(qū)動(dòng)時(shí)的地面反力對(duì)后輪取矩，得到Torque received on rear tire , givenRoad Reactions on front/rear drive tire對(duì)前輪取矩，得到Torque received on front tire, given軸距增加，重心高度小，動(dòng)載荷減小。良好路面Static load on axleDyna

30、mic load on axle前輪靜載荷 static load front tire 后輪靜載荷static load rear tire Getting from above formulae由此得到For rear drive對(duì)于后驅(qū)動(dòng)后驅(qū)動(dòng)地面附著力adhesive force of rear driveFor rear drive對(duì)于前驅(qū)動(dòng)前驅(qū)動(dòng)地面附著力adhesive force of front driveFor 4WD(AWD)Prior conditionThen it can make use of well前輪、后輪驅(qū)動(dòng)附著利用率For RWDFor FWDUtili

31、zation of MV adhesives while front or rear axle driveUsually for front drive MV , the static load of front axle larger then that of rear axle. The position of the mass center can be determined by means of weighing.附著利用率RWDFWD4WDAdhesive factorAdhesive utilizationAdhesive utilization depending on dri

32、ve type1.4 汽車(chē)驅(qū)動(dòng)力-行駛阻力平衡圖便可分析在附著良好的典型路面上汽車(chē)的行駛能力。即在油門(mén)全開(kāi)時(shí)，汽車(chē)可能達(dá)到最大速度、加速能力和爬坡能力。汽車(chē)行駛方程式反映了汽車(chē)行駛時(shí)，驅(qū)動(dòng)力和外界阻力之間的普遍情況；若已知1.4 balance Between Traction & driving MV resistanceThe driving equation shows the correlation between the traction and driving resistance when the MV drives. Ifare known, can analyze then

33、the driveability of MV on the good road to face, namely at full acceleration pedal, the MV may attain the max. speed, acceleration ability and grade ability. 驅(qū)動(dòng)力與行駛阻力平衡圖定義：為了清晰地描述汽車(chē)行駛時(shí)受力情況及其平衡關(guān)系，通常將平衡方程式用圖解方式描述，即將驅(qū)動(dòng)力Ft和常見(jiàn)行駛阻力 Fw 和 Ff 繪在同一張圖上。1. 最高速度和部分負(fù)荷時(shí)的平衡2. uamax3. 最大爬坡度diagram of the equilibrium

34、 between drive force and driving resistanceDefinition: For describing the automobile to drive clearly be subjected to the reaction and its balance relations, usually draw equilibrium equation to illustrate the way to carry on the description, ie draw the drive force Ft and the driving resistance Fw

35、and Ff in the same diagram.1. max speed and force balance for part load (see next page)2.uamax3. max grade climbing汽車(chē)驅(qū)動(dòng)力行駛阻力平衡圖FtSpeed, ua (km/h)equilibrium between drive force and driving resistancedrive force2. 加速能力，可用aj評(píng)價(jià),， 但不方便，常用加速時(shí)間或加速距離來(lái)評(píng)價(jià)。加速度倒數(shù)：Acceleration reciprocalAcceleration timeAcceler

36、ation travel汽車(chē)加速度速度圖Speed, ua,km/hDiagram of acceleration vs velocity MV ua is different from uAcceleration aj,m/s2加速度倒數(shù)曲線圖1/ajua,Diagram of acceleration reciprocalacceleration reciprocaltSpeed, km/h將驅(qū)動(dòng)力、滾動(dòng)阻力和空氣阻力計(jì)算出，就可按公式計(jì)算加速度及其倒數(shù)，從而求得加速時(shí)間或者加速距離。After computing drive force, roll resistance and air

37、resistances, it can be computed the acceleration and its reciprocal according to the formula and get the acceleration time or travel.手工計(jì)算時(shí)一般忽略原地起步過(guò)程的離合器打滑過(guò)程，即假設(shè)在最初時(shí)刻，汽車(chē)已具備起步換檔所需的最低車(chē)速。換檔時(shí)刻的確定：若I-II加速度曲線相交，規(guī)定在交點(diǎn)處換檔；若I-II的加速度曲線不相交，則規(guī)定在發(fā)動(dòng)機(jī)最高轉(zhuǎn)速處換檔；換檔時(shí)間一般忽略不計(jì)（正態(tài)分布t=0.2-0.4s)。計(jì)算加速時(shí)間的用途：確定汽車(chē)加速能力；傳動(dòng)系匹配情況；合理選

38、擇發(fā)動(dòng)機(jī)的排量。注意NoticeGenerally when handicraft computes, at first ,the slide of clutch at the start process is neglected, then the assumption is in the first time, the MV has already had the lowest speed that start need when shift gear.Time of shift gear to determine: If the gear 1st- 2nd acceleration cu

39、rves is intersected, then shift gear 1st to 2nd; If the acceleration curves of the gear 1st and 2nd is not intersected, then at max engine speed shift the gear 1st to 2nd; neglect shift gear spending time. (normal distribution t=0.2-0.4 s)Use of computing acceleration time: to determine MV accelerat

40、ion ability, trade-off power-train; reasonably choose displacement of the engine.3. 利用驅(qū)動(dòng)力行駛阻力平衡圖確定汽車(chē)爬坡能力Its prior condition is a road surface good, except overcoming the Fw+ Ff ,all drive force is used for overcoming a grade resistance, namely.Diagram of equilibrium of drive force - driving resistan

41、ce to determine grade ability of MV其前提條件是路面良好，汽車(chē)克服Fw+Ff后的全部力都用于克服坡道阻力，即坡度通常用i表示i represents grade usuallyMV gradient iSpeed, km/himax is max sloping gradient for 1st gear.i0max is max sloping gradient for top gear.Objective: decreasing the number of shift on normal slope, which makes the speed of MV

42、 slowly down. 二、動(dòng)力特性圖評(píng)價(jià)汽車(chē)動(dòng)力性u(píng)nit drive force to evaluate MV tractive performanceNormalized drive forcedrive factorCharacteristic of normalized drive force and its use汽車(chē)動(dòng)力特性圖及其用途To solve imax ,let While MV acceleration, the road must be flatness. i.e.1.5 汽車(chē)功率平衡 MV power balance 汽車(chē)在行駛時(shí)驅(qū)動(dòng)力與阻力平衡 發(fā)動(dòng)機(jī)輸出功率也與行駛阻力功率平衡定義：用縱坐標(biāo)表示功率，橫坐標(biāo)表示車(chē)速，將發(fā)動(dòng)機(jī)功率與經(jīng)常遇到的阻力功率對(duì)車(chē)速的關(guān)系繪制在直角坐標(biāo)圖上，就得到功率平衡圖。The traction and resistance balances when the MV drivesThe engine power output also balances with the driving resistance power. Definition: let vertical coordinate as the power, horizontal coordinate as MV speed, draw diag