张力微分方程的研究

修改稿No.1冷连轧动态变规格张力微分方程TandemcoldrollingFGCtensiondifferentialequation摘要：介绍了冷连轧动态变规格概念及轧制工艺特点。以冷连轧机组机架间带钢受张力拉伸为研究对象，根据动态变规格过程中冷轧带钢受力和变形特点，通过理论推导建立起冷连轧动态变规格变形微分方程和张力微分方程。分析了动态变规格过程相关因素对张力变化的影响，证明了稳态轧制条件下张力微分方程是动态变规格张力微分方程的特例。关键词：冷连轧；动态变规格；张力；微分方程；变形；速度差中图分类号：TG335.11文献标识码：AAbstract：IntroducestheconceptofthetandemcoldrollingFGCandrollingprocesscharacteristics.Usingstriptensionbetweenthetandemcoldrollingmillunitsofthestandsastheresearchobject,accordingtocoldstripforceanddeformationfeaturesintheFGCprocessthroughtheoreticalderivationsetupcoldrollingFGCdeformationdifferentialequationandtensiondifferentialequation.AnalyzestheeffectsoftheFGCprocessrelatedfactorsontensionchangeandprovethesteady-staterollingconditionstensiondifferentialequationisaspecialcaseinFGCtensiondifferentialequation.Keywords:tandemcoldrolling;FGC,Tension,Differentialequations;Deformation;VelocitydifferenceChineselibraryclassificationnumber：TG335.11Literatureidentificationcode：A动态变规格FGC（FlyingGaugeChange），是在轧制过程中进行带钢的规格变化，即在连轧机组不停机的条件下，通过对辊缝、速度、张力等参数的动态调整，实现相邻两卷带钢的钢种、厚度、宽度等规格的变换[1,2]。冷连轧机组实现动态变规格全连续轧制后，消除了穿带、甩尾过程，缩短了加、减速过程的时间，从而可以提高轧机生产率，改善带钢的质量，特别是带钢的头、尾部的厚度偏差，同时板形质量得到较好控制，进而减少了带钢的切损，提高了成材率[3~7]。FGCisthespecificationchangeinrollingprocessofstripsteel,namelyincontinuousrollinglinenon-stopconditions,throughthedynamicadjustmentoftherollgap,speed,tensionandotherparameters,realizethespecificationsofcommutationontheadjacenttworollsofstripsteelgrade,thicknessandwidth.TandemcoldrollingmillunitsrealizeFGCfullcontinuousrolling,eliminatesweartake,swingtailprocess,shortenthetimeofacceleration,decelerationprocess,whichcanimprovetheproductivity,improvestripsteelquality,especiallythethicknessdeviationofheadandtail,atthesametimethicknessgetgoodcontrolofthestripshapequality,andthenreducethestripcuttingloss,improvetheyield.动态变规格复杂之处在于，在极短的时间内由前一卷带钢的轧制规程切换到下一卷带修改稿No.2钢的轧制规程。在这一变化过程，辊缝和辊速需要进行多次、大幅度调整。因此动态变规格必须按照一定的规律进行，为了研究不同动态变规格控制模型的控制效果，通常需对冷连轧动态变规格过程进行动态仿真研究，这需要建立起包括张力微分方程在内的动态变规格控制模型[8~11]。ThecomplexofFGCliesinthat,therollingscheduleofthepreviouscoilswitchestothenextcoilinaveryshorttime.Inthisprocess,rollgapandrollspeedneedtocarryonthemultipleandlargeadjustment.ThereforeFGCmustaccordingtocertainrules,inordertostudythecontroleffectbetweendifferentcontrolmodelofFGC,usuallyneedtododynamicsimulationsearchontheprocessofcoldrollingFGC,thisneedstoestablishtheFGCcontrolmodelwhichincludestensiondifferentialequation.1基本假设1、Basicassumptions图1表示在某一时刻动态变规格点（用iL1、1i2L描述）正位于第i机架和第i+1机架之间的某一位置。设变规格前后带钢的厚度为H、h，宽度为W、w，机架间长度为iL。带钢在第i机架的出口速度为HiV，在第i+1机架的入口速度为)1i(hV。Hi)1i(hVV，即存在速度差，因此带钢处于张力轧制状态。张力微分方程表征了张力与速度差间的微分关系。以i机架出口到i+1机架入口的带钢张力轧制状态为研究对象，见图2。做如下假设：(1)带钢在张力作用下的变形是弹性变形，服从虎克定律；(2)变规格前后的带钢横断面上变形、应力是均布的；(3)带钢轧制过程中为无宽展的平面变形；Figure1saidatacertainhourFGCpoint(useiL1、1i2Ltodescribe)islocatedjustinonepositionbetweentheistandsandi+1stands.AssumingthatbeforeFGCstripthicknessforH,h,widthisW,w,thelengthbetweenrackisiL.TheinletvelocityofstripintheistandsisHiV,theentranceofspeedinthei+1standsis)1i(hV.Hi)1i(hVV,i.e.therearevelocitydifference,sostriprollingisintensionrollingcondition..Tensiondifferentialequationsrepresentthedifferentialrelationshipbetweentensionandspeeddifference.Usingstriptensionrollingconditionofexportofistandstoofi+1standsasobjectofresearch,asfigure2shown.Meanwhiledotheassumptions:(1)Thestriptensionundertheactionofdeformationiselasticdeformation,obeyhooke'slaw；(2)Thedeformation,stressonstripcrosssectionisuniformbeforeandafterFGC；(3)Thereisnobroadsidingwhentheplanedeformationisinstriprollingprocess；修改稿No.3图1动态变规格轧制状态Fig.1RollingsituationofFGC图2机架间带钢规格变化Fig.2Gaugechangeofstripbetweenstands图2中T—带钢所受总张力；L1，L2—弹性变形后变规格点前后带钢长度；L1’，L2’—弹性变形前变规格点前后带钢长度；e1，e2—变规格点前后带钢的绝对伸长；W,w—变规格点前后带钢宽度；T-totaltensionofstrip;L1,L2-striplengthafterelasticdeformationwhichisbeforeandafterFGCpoint;L1',L2'-striplengthbeforeelasticdeformationwhichisbeforeandafterFGCpoint;e1,e2-theabsoluteelongationofstripwhichisbeforeandafterFGCpoint;W,w-stripwidthwhichisbeforeandafterFGCpoint;2变形微分方程Deformationdifferentialequation在图2所示的张力作用下，变规格点前后带钢的绝对伸长量为：Undertheinfluenceofthetensionshowninfigure2,beforeandafterFGCpointtheabsolutestretchingquantityofstripis:i机架i+1机架LiL2iL1iHh)1i(hvHivL2L1TVHTVhHhe2e1L2'L1'修改稿No.42L2L2e1L1L1e(1)相对伸长为Relativeelongationis2e2L2e2L2e21e1L1e1L1e1(2)上式可变化为Theaboveformulacanbechangedinto212L22e111L11e(3)于是带钢绝对总伸长可用相对伸长表示为Thenstripofabsolutetotalelongationcanbeexpressedinrelativeelongation212L2111L12e1ee(4)将绝对总伸长e对时间t取导数，得到用相对伸长表示的带钢拉伸速度：Willabsolutelytotalelongationeoftimetaketimetderivative,thestrippullingspeedsarerelativeelongation:dt2dL212dt1dL111dt2d212Ldt1d111Ldt2dedt1dedtde22（5）另一方面，带钢拉伸速度又可用速度差来表示。设在dt时间内带钢在i机架的出口处位移量为U，在i+1机架入口处的位移量为u。其出入口的位移差u－U即为带钢的绝对伸长e，即：e=u－U。将e对t取导数后可得：Ontheotherhand,strippullingspeedscanberepresentedbyusingvelocitydifference.AssumingthatinthetimeofdtthedisplacementstripattheoutletoftheistandsisU,thedisplacementattheentranceofthei+1standsisu.Thedisplacementsbetweenu–Uname