张力微分方程研究.doc

修改稿 No.8冷连轧动态变规格张力微分方程Tandem cold rolling FGC tension differential equation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：Introduces the concept of the tandem cold rolling FGC and rolling process characteristics. Using strip tension between the tandem cold rolling mill units of the stands as the research object, according to cold strip force and deformation features in the FGC process through theoretical derivation set up cold rolling FGC deformation differential equation and tension differential equation. Analyzes the effects of the FGC process related factors on tension change and prove the steady-state rolling conditions tension differential equation is a special case in FGC tension differential equation .Keywords: tandem cold rolling; FGC, Tension, Differential equations; Deformation; Velocity differenceChinese library classification number：TG335.11 Literature identification code：A动态变规格FGC（Flying Gauge Change），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37。FGC is the specification change in rolling process of strip steel, namely in continuous rolling line non-stop conditions, through the dynamic adjustment of the roll gap, speed, tension and other parameters, realize the specifications of commutation on the adjacent two rolls of strip steel grade, thickness and width. Tandem cold rolling mill units realize FGC full continuous rolling, eliminates wear take, swing tail process, shorten the time of acceleration, deceleration process, which can improve the productivity, improve strip steel quality, especially the thickness deviation of head and tail, at the same time thickness get good control of the strip shape quality, and then reduce the strip cutting loss, improve the yield.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811。The complex of FGC lies in that, the rolling schedule of the previous coil switches to the next coil in a very short time. In this process, roll gap and roll speed need to carry on the multiple and large adjustment. Therefore FGC must according to certain rules, in order to study the control effect between different control model of FGC, usually need to do dynamic simulation search on the process of cold rolling FGC, this needs to establish the FGC control model which includes tension differential equation.1 基本假设1、Basic assumptions图1表示在某一时刻动态变规格点（用、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。带钢在第i机架的出口速度为，在第 i+1机架的入口速度为。，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；Figure 1 said at a certain hour FGC point (use、to describe) is located just in one position between the i stands and i + 1 stands. Assuming that before FGC strip thickness for H, h, width is W, w, the length between rack is . The inlet velocity of strip in the i stands is, the entrance of speed in the i + 1 stands is . , i.e. there are velocity difference, so strip rolling is in tension rolling condition.Tension differential equations represent the differential relationship between tension and speed difference. Using strip tension rolling condition of export of i stands to of i + 1 stands as object of research, as figure 2 shown. Meanwhile do the assumptions:(1) The strip tension under the action of deformation is elastic deformation, obey hookes law；(2) The deformation, stress on strip cross section is uniform before and after FGC；(3) There is no broadsiding when the plane deformation is in strip rolling process； i机架 i+1机架LiL2iL1iH h 图1 动态变规格轧制状态Fig. 1 Rolling situation of FGCL2L1TVHTVhHhe2e1L2L1图2 机架间带钢规格变化Fig .2 Gauge change of strip between stands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 - total tension of strip; L1 , L2 - strip length after elastic deformation which is before and after FGC point;L1 , L2 - strip length before elastic deformation which is before and after FGC point;e1, e2 - the absolute elongation of strip which is before and after FGC point;W, w - strip width which is before and after FGC point;2 变形微分方程Deformation differential equation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：Under the influence of the tension shown in figure 2, before and after FGC point the absolute stretching quantity of strip is: (1)相对伸长为Relative elongation is (2)上式可变化为The above formula can be changed into (3)于是带钢绝对总伸长可用相对伸长表示为Then strip of absolute total elongation can be expressed in relative elongation (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：Will absolutely total elongation e of time take time t derivative, the strip pulling speeds are relative elongation: （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uU即为带钢的绝对伸长e，即：e= uU。将e对t取导数后可得：On the other hand, strip pulling speeds can be represented by using velocity difference. Assuming that in the time of dt the displacement strip at the outlet of the i stands is U, the displacement at the entrance of the i+ 1 stands is u. The displacements between u U namely for e which is the absolute elongation of strip: i.e. e = u -U. Will e on t take derivative after available: (6)由式(5)和式( 6)得到动态变规格变形微分方程式：By type (5) and type (6),we can get FGC deformation differential equations: (7)3 张力微分方程Tension differential equation由假设可知：且 ，和都是小量，故式(7)中后二项为高阶小量，可以忽略不计。为弹性模量，由虎克定律，式(7)可简化为We can know from the hypothesis: and, and are all small, so the subsequent two items in (7) are high-order small, which can be neglected. is modulus of elasticity, formula (7) can be simplified by hookes law as (8)张应力、与张力T的关系为：The relationship of tension T between tensile stress、 is (9)将式(9)代入式(8)得Take formula(9) into formula(8),we can get (10)将式(10)写成一般形式Make formula(10) into general form (11)4 讨论Discussion式(11)为带钢在变规格轧制时的简化张力微分方程式。该式表明，动态变规格过程张力的变化不仅与前后机架的出入口速度差有关，而且与前后卷带钢的宽度、厚度等规格有关，还与变规格点在机架间所处的位置有关。这正是动态变规格轧制与稳态轧制时张力微分方程的区别所在。Formula (11) is the simplified tension differential equations when strip rolling FGC. This formula shows that the tension changes in dynamic FGC process not only related to the velocity difference in front and rear frame with entrance and inlet, but also related to the strip width and thickness in front and rear frame of such specifications, at the same time related to the location of the FGC point between frames. This is the difference between dynamic FGC rolling and tension differential equation in steady-state rolling time. 当或=0时，式(11)就变成稳态轧制时常用的张力微分方程12。When or = 0, formula (11) became commonly used tension differential equation when steady rolling. (12)上式即为Formula before namely for (13)所以，稳态张力微分方程是变规格张力微分方程的特殊形式。So, the steady-state tension differential equation is the special form of FGC tension of differential equation.5 结论conclusion变规格轧制时的张力微分方程，不同于稳定轧制时的张力微分方程，需要重新建立。稳态轧制时机架间张力的变化与前后机架出入口速度差有关。动态变规格过程中，机架间张力变化不仅与速度差有关，还与前后卷带钢规格及变规格点所处机架间的位置有关。 When rolling FGC, tension differential equation is different from the tension differential equation when regular rolling, and needs to be re-established. tension changes between frames when steady rolling related to the velocity difference in front and rear frame with entrance and inlet. In the process of dynamic FGC, tension changes between frame not only related to the specifications in front and rear frame and location of the FGC point between frames.参考文献1 Foster MA , Marshell SA. Modelling of a tandem cold-strip rolling millA , Mathematical process models in iron and steel makingC. Amsterdam, 1973 ,103-112.2 Yoshikazu. 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