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附錄1
冷軋薄帶時，軋制參數(shù)對軋輥邊緣接觸的影響
在一些冷軋制造薄帶的過程中，我們已經(jīng)發(fā)現(xiàn)了工作輥邊部接觸并且使薄帶變形的問題。在工作輥邊緣接觸的問題上，形成了一個新的在滾動中變形的特性，這一特性已得到分析。在本文中，作者重點研究軋制參數(shù)對特定的力如軋制力，中間力，邊緣接觸力和薄帶鋼冷軋工作輥邊緣接觸時的影響。目前已研究出一個影響函數(shù)法來模擬此特殊軋制過程?；跀?shù)值模擬，得到了軋制參數(shù)對力學和變形影響的冷軋薄帶。數(shù)字模擬試驗，驗證了這個已較為成熟的方法的有效性。
冷軋薄帶被廣泛的應(yīng)用在電子和儀器行業(yè)當中。隨著科學和技術(shù)的迅速發(fā)展，薄帶鋼已經(jīng)越來越廣泛的應(yīng)用于工業(yè)當中。一般來說,這種薄帶是由一個冷連軋機組的一個非圓形的工作輥所制成。
Sutcliffe等人為薄帶的軋制研究了一種新的方法進行負載和帶鋼軋機斷面軋薄的測量。在薄帶鋼軋制中，一個比較估計軋輥轉(zhuǎn)矩和一個修正橫向擴散的方法也已經(jīng)被研究出來。Jiang等人計算薄帶的彈性變形，和在冷軋薄帶中的薄帶的形狀、輪廓和平整度。軋輥的彈性變形導致輪廓、外形和平整度的問題。鋼鐵制造商一直關(guān)心如何改進它的形狀，平整度和尺寸精度這一問題。研究員已經(jīng)從新的制造工廠中發(fā)現(xiàn)對這些問題的解決辦法通過引入軋輥連續(xù)變型(CVC)和軋輥交叉(PC)的軋機。有了這些軋制程序，能夠使相對較厚的薄帶被軋制的時候，工作輥彼此不接觸。
在一些冷軋過程中，例如當薄帶被軋制的時候，工作輥的端部接觸而且變形(見圖1)。在冷軋薄帶的分析中，我們不得不考慮工作輥端部接觸時可能導致毀滅性結(jié)構(gòu)的問題。這種情況，模擬變形模型的技巧不同于傳統(tǒng)的薄帶冷軋程序。當工作輥接觸邊緣地帶之外時，不僅改變了壓力分布，而且對變形模型的工作輥、摩擦界面都將帶來磨損。工作輥接觸邊緣地帶研究如何確定軋制力、中間力量、邊緣接觸力和剖面的地帶，以改善其質(zhì)量。作者這篇文章的重點在于冷軋過程中旋轉(zhuǎn)參數(shù)對特定的力量和軋件的描繪效果的研究。
當軋件超出軋輥接觸邊緣時，Edwards和Spooner根據(jù)一個分析方法也簡短地描述了冷軋薄帶毀壞兼容性的關(guān)系。但是到目前為止詳細的結(jié)果還沒有被報告出來?；跀?shù)字的模擬、旋轉(zhuǎn)參數(shù)和改變，冷軋過程中邊緣的損壞的效果得以演示。數(shù)字的模擬測試已經(jīng)證明此研究的可行性。
圖1冷軋薄帶工作輥的邊緣接觸
變形軋輥在工作輥和支撐輥之間，工作輥和軋件之間，是以工作輥之間的換置兼容性關(guān)系為基礎(chǔ)的。由于左邊和右邊的對稱，鑄坯在軋輥的中心線快速前進，一半軋輥當做一個研究目的被分離出來。分開區(qū)域在圖2被顯示出來。工作輥和支撐輥之間的軋制力在該區(qū)域是統(tǒng)一的，軋輥和軋件的毀壞在圖2也被表示出來。
圖2死滾軋機的力學模型
在工作輥和支撐輥之間，由于彎矩、剪切力和泊松比的影響，工作輥之間的干擾，通過計算軋輥歪斜得到不成形的工作卷物描繪，以上內(nèi)容在下面的段落中會詳細介紹。
采用輥撓度的計算理論對彎曲和剪切組件得到了廣泛的應(yīng)用，一個典型的軋輥歪斜模型如圖3所示。
圖3由于加載點的中性軸的偏轉(zhuǎn)
軋輥歪斜在彎曲力的效果之下在某一位置x能被描述為：
式中，E是彈性模量，I是橫截面積。
通過O'connor和Weinstein，軋輥變形可以調(diào)整為:
式中，A是橫截面積，J是剪切模量。
如果有彎曲，中間的歪斜軸能在圖4被顯示出來而且表示成：
式中，M是彎曲力矩，是在x位置的軋輥的半徑。
卷物中軸的歪斜由于在表面運動的泊松比是
式中，R是工作輥半徑。
基于假定長的接觸的柔性氣缸，適當大小的壓力，-軋輥壓力q(x)能通過下面的公式能夠被表達出來。
是在工作輥和支撐輥之間干擾的影響；寫在底下的數(shù)字W和B分別地提及工作輥和支撐輥。被下列的方程式?jīng)Q定：
式中，υ表示泊松比。
基于軋輥的等高線,工作輥和支撐輥之間的影響得計算：
式中，是軋輥干擾的中心地帶，是全體的支撐輥的撓度，是完全的支撐輥隆起包括平面隆起，熱的隆起和軋輥磨耗。是全體工作輥的撓度，而且是完全的工作輥隆起包括平面加冠，熱的隆起和軋輥磨耗。
軋輥在工作軋輥的接觸面積變平和薄帶能被描述為
B是薄帶寬度，而且由：
圖5超越邊緣地帶的工作輥之間的影響
式中，是旋轉(zhuǎn)的壓力，而且和是由實驗決定的常數(shù)。因為軟鋼(0.1-0.25%C)，和分別地被估計當做32.92和0.86mm/kN。當軋制洋鐵的時候，薄帶可能是非常薄的，而且工作軋輥的撓度能充分造成工作軋輥接觸超過薄帶的邊緣?，F(xiàn)在的彎輥力系統(tǒng)使用單獨的工作輥觸摸彼此之外的邊緣地帶。工作軋輥之間的影響，能依照下列各項被計算：
式中，是工作輥的寬度，是出口薄帶在薄帶中心的厚度。
由圖5可知，左邊和右手邊超過那被卷的薄帶的邊緣叫做軋輥邊緣接觸區(qū)域。和分別是接觸壓力在工作軋輥在左邊和右手接觸區(qū)域。
下面是被用在模擬冷軋方面的重要參數(shù)的價值：
工作輥的直徑：400mm；
支撐輥的直徑：1200mm；
工作輥的長度：1600mm；
支撐輥的長度：1600mm；
工作輥的初次隆起：0mm；
支撐輥的初次隆起：0mm；
中心距在螺旋之間：2700mm；
中心距在彎曲氣缸之間：2700mm；
工作的楊氏模數(shù)卷：220000N/mm；
支撐輥的楊氏模數(shù)：22000N/mm；
工作輥的浦松氏比：0.3；
支撐輥的浦松氏比：0.3；
板層厚度：2.02mm；
進入薄帶的厚度：0.45、0.40、0.35或0.32mm；
薄帶的出口厚度：0.3mm；
薄帶的寬度：1000mm；
特定的前面拉力：165N/mm；
特定的背部之里面拉力：160N/mm；
旋轉(zhuǎn)的速度：1000m/min；
磨擦系數(shù)：0.017；
在進入的薄帶的初次隆起：0mm；
定義來自邊緣的薄帶隆起的點：25mm；
工作軋輥彎曲力：0、50、100或150kN/chock.
軋制力由福特-希爾公式計算
B是軋制前的薄帶的寬度，拉力因數(shù)，被描述的變形阻力，由下列方程得:
是一個常數(shù)，污染率，寫在底下的指示靜止的和
是靜止的變形阻力一個常數(shù)，在這一公式=740MPa，m和n是常數(shù)，m=0.01和n=0.23，是平均的整體還原被描述為
是板層厚度
是一個常數(shù)。(0.75)半徑是一將工作輥的半徑變平卷能被Hitchcock模型推論：
b是軋制、H薄帶的寬度，h薄帶的出口厚度，操作軋輥半徑，CHHitchcock系數(shù)和F軋制力。能被描述為
(17)
磨擦系數(shù)。
工作輥和支撐輥的撓度使用簡單梁理論計算彎曲和剪切。
基于影響力功能方法，模擬程序表在個人計算機上發(fā)展起來。,獲得為不同的軋制薄帶入口的厚度，彎曲力和工作的狀態(tài)或沒有邊緣接觸力；旋轉(zhuǎn)的力、中間的力，邊緣接觸力和薄帶的輪廓。
板層厚度是2.02毫米，薄帶的出口厚度是0.30毫米和彎曲力是零。進入厚度的效果在特定的力上的薄帶在圖6被顯示。它能被見到，旋轉(zhuǎn)的力增加當進入薄帶的厚度增加。因為還原增加當做薄帶的進入厚度增加，它也被見到那中間的力增加當進入厚度薄帶增加，而且它在邊有一個逐漸增加的趨勢由于邊緣接觸工作軋輥快速前進。當進入厚度是0.32毫米，邊緣接觸力是零，這方法沒有邊緣接觸。邊緣接觸力用薄帶(還原)的進入厚度的增大，那邊緣工作軋輥的接觸變得更重要當薄帶增大的進入厚度，有一重要的在中間的力方面的影響力。
圖7表演薄帶的出口厚度的分布對于不同進入厚度的薄帶。當進入厚度薄帶增大它能被見到那出口薄帶的隆起增加(也見表1)因此，即使工作軋輥連絡(luò)超過薄帶的邊當彎曲力是零，被卷的薄帶的輪廓變成具有進入厚度的增加。
圖6入口厚度在特定地帶的影響
圖7入口厚度在輪廓地帶的影響
板層厚度是2.02毫米，進入厚度0.40毫米，出口厚度0.30毫米，彎曲力是零。特定的力作用下的效果邊緣接觸如圖8所示。它能反應(yīng)當邊緣接觸的時候，在薄帶的邊附近的旋轉(zhuǎn)的力減少。因為邊緣接觸工作軋輥，邊緣接觸力增大和那中間的力超過薄帶的邊也增加。因此，旋轉(zhuǎn)的力減少。邊緣接觸的效果在薄帶的輪廓上在圖9顯示出來。很輕易能發(fā)現(xiàn)那出口薄帶的隆起的減少。工作軋輥接觸彼此的邊緣(見表2)，因此工作軋輥的邊緣接觸能改良輪廓。如果沒有在薄帶中被應(yīng)用的卷板機系統(tǒng)。
圖8特定力量對邊緣的影響
板層厚度是2.02毫米，進入厚度0.40毫米，出口厚度0.30毫米。彎曲特性方面的力的效果力在如圖10所示。它能反應(yīng)當彎曲力增大的時候在軋輥邊緣的力的減小。然而，當寬度里面的中間的力使薄帶減少，然后在邊緣附近增的力和那當彎曲應(yīng)力增加的時候，就能操作軋輥。因為邊緣接觸的效果，接近的中間力工作的邊緣卷桶稍微增加。當彎曲力增加，中間增大力時，使工作的邊緣卷桶變得更重要。我們能看到邊緣接觸力減少，彎曲力增加的時候，表示那邊緣接觸力可能是可以忽略的。這時彎曲應(yīng)力150kN/chock。當彎曲力增大時，薄帶的輪廓變成比較的彎曲。(見到圖11)因此，減少邊緣接觸力而有效的改良邊緣變形的方法就是增大彎曲應(yīng)力。
圖9邊緣接觸對薄帶邊緣的影響
圖10彎曲力的影響
圖11邊緣地帶彎曲力的影響
這是一個研究軋輥在軋制過程中通過模擬邊緣力和彎曲應(yīng)力而改善軋輥作用下薄帶邊緣變形的模型。結(jié)果表示那些特定的力，像是旋轉(zhuǎn)的力，中間的力而且對于薄帶軋制這種特殊的生產(chǎn)過程所造成的特別的影響。當薄帶的厚度增加的時候，那邊緣接觸力增大，工作的邊緣接觸軋輥變得非常重要，在中間施加作用力所產(chǎn)生的中還要得效果就是使，出口薄帶的形變成很小的。如果沒有彎曲應(yīng)力的作用，薄帶在出口處的隆起將會減小，工作輥邊緣的變形也隨之減小。因為邊緣接觸能改良薄帶的輪廓，因此各個生產(chǎn)廠家已經(jīng)引入了邊緣檢出應(yīng)力分析的裝置來提升薄帶生產(chǎn)的產(chǎn)品質(zhì)量。在這些裝置的作用下，薄帶的變形變的微乎其微。因此，增加彎曲和應(yīng)力能夠在成產(chǎn)過程中很大程度上減小薄帶邊緣在工作輥作用下的變形。
致謝
這項工作受到一個澳大利亞研究理事會的支持。
附錄2
Effect of rolling parameters on cold rolling of thin strip during work rolls edge contact
In some cold rolling mills, a problem has been found that the sides of work rolls touch and deform when thin strip is rolled. The problem of work roll contact at the edges, which forms a new deformation feature in rolling, is analyzed. In this paper, the authors focus on the research of the effects of rolling parameters on specific force such as rolling force, intermediate force, edge contact force and the profile of thin strip in cold rolling when the work roll edges contact. An influence function method is developed to simulate this special rolling process. Based on numerical simulation, the effects of the rolling parameters on the mechanics and deformation of the cold rolled thin strip are obtained. Numerical simulation tests have verified the validity of this developed method.
A cold rolled thin strip is widely used in the electronic and instrument industries. With the rapid development of science and technology, thin strip has been finding more and more applications in industry. In general, this kind of strip is produced by a tandem cold rolling mill where the work rolls are flattened to a non-circular deformed shape.
At et al. developed a robust model for rolling of thin strip and foil and carried out the experimental measurements of load and strip profile during thin strip rolling. In thin strip rolling, a comparison of methods to estimate the roll torque and a modified method for lateral spread has also been conducted. et al. calculated the elastic deformation of strip, and the shape, profile and flatness of strip in cold rolling of thin strip. Elastic deformation of the rolls brings about problems of profile, shape and flatness. The problem on how to improve its shape and flatness, and the dimensional accuracy has always been of major interest to the steel manufacturers. Researchers have found solutions to these problems by introducing new types of mills, such as continuous variable crown (CVC) and pair cross (PC) mills equipped with roll shifting roll crossing and work roll bending. These are rolling processes where the work rolls do not contact each other when relatively thick strip is rolled.
In some cold rolling mills, for example, it has often been found that the edges of work rolls touch and deform (see Fig. 1) when the thin strip is rolled. The problem of work roll contact at the edges should be considered in an analysis of the cold rolling of thin strip, which forms a new deformation feature. In this case, the models of deformation and mechanics are different from the traditional cold rolling processes of strip. Not only the distribution of the roll pressure will change when the work rolls contact beyond the edges of the strip, but also the deformation model of work rolls, friction at the interface of the rolls and the strip and work roll wear. How to determine the rolling force, intermediate force, edge contact force and profile of the strip, to improve its quality when the work rolls contact beyond the edges of the strip is the main feature of this study. The authors focus on the research of the effect of rolling parameters on specific force and profile of thin strip in cold rolling, which is a highlight of this paper.
Edwards and Spooner also described briefly deformation compatibility relationship for the cold rolling of the thin strip when the work rolls contact beyond the edges of strip by an analysis method. But up to now detailed results have not been reported. In this study, an influence function method has been developed to simulate this special rolling process. Based on the numerical simulation, the effect of the rolling parameters on the mechanics and deformation of the cold rolling of thin strip are obtained. Numerical simulation tests have verified the validity of this developed method.
The calculation of the deformed rolls is based on the displacement compatibility relationships between the work roll and backup roll, work roll and thin strip, and the work rolls. Due to symmetry of the left and right sides of the rolls at the central line of the roll barrels, one-half of the roll barrels is selected as a research objective, and the equal divided zone is shown in Fig. 2. The rolling pressure and the pressure between the work roll and backup roll are uniform in zone. The deformations of the rolls and the strip are described in Fig. 2.
The deformed work roll profile is obtained by calculating the roll deflections due to bending, shear and effect of Poisson’s ratio, bending moment, interference between the work roll and the backup roll, and work roll flattening, which are described in the following paragraphs.
Beam theory for the bending and shear components has been widely employed to calculate the roll deflections. A typical roll deflection model under the effect of point load is shown in Fig. 3.
The roll deflection of the beam under the effect of bending at a position x can be described as follows:
Where E is the Young’s modulus, I the second moment of areaand the point loads
According to O’Connor , the deflection of the neutral axis for short stubby beams due to shear is given by
Where is the cross-sectional area and G the shear modulus of the beam?
If there is a bending moment, the deflection of the neutral axis can be illustrated in Fig. 4 and expressed as follows:
Where is the Poisson’s ratio, M the bending moment, and R(x) the radius of the roll at x position.
The deflection of the roll neutral axis due to the effect of Poisson’s ratio on the movement of the surfaces is
Given by
Where is the work roll radius?
Based on the assumption of two infinitely long elastic cylinders in contact, the interference under inter-roll pressure can be described as follows
Where is the interference between the work roll and backup roll; subscripts and refer to the work roll and backup roll, respectively. And are determined
By the following equation:
Where is the Poisson’s ratio? Based on the contours of the rolls, the interference between the work roll and backup roll can be calculated as follows
Where is the total roll interference at the strip , the total backup roll-axis deflection, and CB(x) the total backup roll crown including ground crown, thermal crown and roll wear. The total work roll-axis deflection, and CW(x) the total work roll crown including ground crown, thermal crown and roll wear.
Work roll flattening at the contact area of work roll and strip can be described as follows
Where B is the strip width, and is given by
Where p(x) is the rolling pressure, and b1 and b2 are constants determined by experiments. For mild steel (0.1–0.25% C), b1 and b2 are estimated as 32.92 and 0.86 mm/, respectively. When rolling tinplate, the strip can be very thin and the deflection of work rolls can be sufficient to result in work roll contact beyond the edges of the strip. Nowadays the roll bending systems are employed to separate work rolls from touching each other beyond the edges of the strip. The interference between work rolls can then be calculated as follows:
Where is the width of work roll barrel, the exit strip thickness at the strip ?
Given in Fig. 5, the left and right hand sides beyond the edges of the strip being rolled are named roll edge contact region. p__(x) and p_(x) are contact pressures between the work rolls at the left and right hand contact regions, respectively.
Given below are values of the important parameters used in the simulation for cold rolling:
? Diameter of the work roll: 400 mm;
? Diameter of the backup roll: 1200 mm;
? Length of the work roll barrel: 1600 mm;
? Length of the backup roll barrel: 1600 mm;
? Initial crown of the work roll: 0.0 mm;
? Initial crown of the backup roll: 0.0 mm;
? Center distance between housing screw: 2700 mm;
? Center distance between bending cylinder: 2700 mm;
? Young’s modulus of the work roll: 220 000 N/mm2;
? Young’s modulus of the backup roll: 22 000 N/mm2;
? Poisson’s ratio of the work roll: 0.3;
? Poisson’s ratio of the backup roll: 0.3;
? Slab thickness: 2.02 mm;
? Entry thickness of strip: 0.45, 0.40, 0.35 or 0.32 mm;
? Exit thickness of strip: 0.3 mm;
? Width of strip: 1000 mm;
? Specific front tension: 165 N/mm2;
? Specific back tension: 160 N/mm2;
? Rolling speed: 1000 m/min;
? Friction coefficient: 0.017;
? Initial crown of strip at entry: 0.0 mm;
? Defining point of strip crown from edge: 25 mm;
? Work roll bending force: 0, 50, 100 or 150 /chock.
Rolling force is calculated by using Bland–Ford–Hill model
where B is the width of strip before rolling, κ the tension factor, kp the deformation resistance which can be described by the following equation:
where α is a constant, ˙ε the stain rate, subscript s indicates static and
where is the static deformation resistance, which is determined under a constant stain rate 10?3 s?1, k0 a constant, in this simulation k0 = 740MPa, m and n are constants, m = 0.01 and n = 0.23, εm is average integral reduction which can be described as
Where H1 is slab thickness and
where β is a constant (0.75). R_ is a flatten radius of work roll which can be deduced by Hitchcock model:
Where is the width of strip after rolling, H, h the entry and exit thickness of strip, respectively, R the radius of the work roll, CH the Hitchcock coefficient [9], and F the rolling force. DP can be described as
(17)
where ε is the reduction and μ the friction coefficient.
Deflections of the work roll and backup roll are calculated by using simple beam theory for bending and shear.
Based on the influence function method, a simulation program was developed and performed on a PC. For different entry thickness of strip before rolling, bending force and the status of the work roll with or without edge contact, the rolling force, intermediate force, edge contact force and strip profile are obtained.
The slab thickness is 2.02mm, exit thickness of strip is 0.30mm and bending force is zero. The effect of entry thickness of strip Hen on specific force is shown in Fig. 6. It can be seen that the rolling force increases when the entry thickness of strip increases. Because the reduction increases as the entry thickness of strip increases. It is also seen that the intermediate force increases when the entry thickness of strip increases, and it has an increasing trend at the side of work roll barrel due to the edge contact. When the entry thickness is 0.32 mm, the edge contact force is zero; this means that there is no edge contact. The edge contact force increases with the entry thickness of strip (reduction). The edge contact of work rolls becomes more significant when the entry thickness of strip increases, which has a significant influence on the intermediate force. Fig. 7 shows the distribution of the exit thickness of strip for different entry thickness of strip Hen. It can be seen that the crown of exit strip increases when the entry thickness of strip increases (also see Table 1). Therefore, when the bending force is zero, the profile of the rolled strip becomes poor with the increasing of entry thickness even if the work rolls contact beyond the side of strip.
Because of the edge contact of the work rolls, the edge contact force increases and the intermediate force beyond the side of strip also increases. Therefore, the rolling force reduces. The effect of edge contact on the profile of strip is shown in Fig. 9. It is found that the crown of exit strip reduces significantly when the edges of work rolls contact each other (see Table 2). So the edge contact of work rolls can improve the profile of strip if there is no bending roll systems applied in a strip rolling mill
.
The slab thickness is 2.02 mm, entry thickness 0.40 mm, exit thickness 0.30mm and bending force is zero. The effect of edge contact on specific force is shown in Fig. 8. It can be seen that the rolling force near the side of strip reduces when the edge contact.
The slab thickness is 2.02 mm, entry thickness 0.40 mm, exit thickness 0.30 mm. The effect of bending force on specific force is shown in Fig. 10. It can be seen that the rolling force near the side of strip reduces when the bending force increases. However, the intermediate force within the width of strip decreases and then increases near the side of the work roll barrel when the bending force increases. Because of the effect of edge contact, the intermediate force near the edge of the work roll barrel increases a little. When the bending force increases, the increase of the intermediate force at the edge of the work roll barrel becomes more significant.
It is also seen that the edge contact force reduces when the bending force increases. This result shows that the edge contact can be negligible when the bending force is 150 /chock. The profile of strip becomes relative uniform when the bending force increases (see Fig. 11). Therefore, it is useful to reduce the edge contact force and improve the shape and profile of strip by introducing the bending force in the cold rolling of thin strip.
A model for the mechanics and profile of thin strip in cold rolling when the work rolls touch and deform beyond the side of strip was developed in this study. The results show that the specific force such as rolling force, intermediate force and the profile of the rolled strip for this special rolling is significantly different with the traditional strip cold rolling process. When the entry thickness of strip increases, the edge contact force increases, and the edge contact of work rolls becomes more significant, which has a significant effect on the intermediate force, and the shape of exit strip becomes poor. The crown of exit strip reduces significantly when the edge of work rolls contact if there is no bending force. So the edge contact can improve the profile of strip if there is no bending roll systems applied in a strip rolling mill. The profile of strip becomes relatively uniform when the bending force increases. Therefore, it is useful to decrease the edge contact force and improve the shape and profile of strip by introducing the bending force in the cold rolling of thin strip.
This work is supported by an Australian Research Council (ARC) Discovery-Project grant, Australia
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