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Dynamics and screening characteristics of a vibrating screen with variable elliptical trace HE Xiao-mei,LIU Chu-sheng School of Mechanical and Electrical Engineering, China University of Mining ＆Technology, Xuzhou, Jiangsu 221116，China Abstract: the ideal motion character sties for the vibrating screen was presented，according to the principle of screening process with constant bed thickness. A new vibrating screen with variable elliptical trace was proposed. An accurate mechanical model was constructed according to the required structural motion features. Applying multi degree of-freedom vibration theory, characteristics of the vibrating screen was analyzed. Kinematics parameters of the vibrating screen which motion traces were linear, circular or elliptical were obtained. The stable solutions of the dynamic equations gave the motions of the vibrating screen by means of computer simulations. Technological parameters, including amplitude, movement velocity and throwing index, of five specific points along the screen surface were gained by theoretical calculation . The results show that the traces of the new designed vibrating screen follow the ideal screening motion . The screening efficiency and processing capacity may thus be effectively improved. Keywords: variable elliptical trace; screening process with constant bed thickncss;dynamic model;motion characteristic;screening characteristics 1、Introduction Screening operations are an important part of coal processing. The vibrating screen is one of the most extensively used screening tools. Vibrating screens, such as linear vibrating screen, circular vibrating screen or elliptical vibrating screen, have a simple translational motion. The motion follows the same path everywhere on the screen and so the screen has constant transport velocity and throwing index, which leads to low screening efficiency. Augmenting the throwing index to improve breaks the exciting motors processing capacity lowers the working. In this paper，we report on the design of a new vibrating screen with variable motion traces that is based on the principle of screening process with constant bed thickness [3 - 4]. Different parts of the vibrating screen traverse different elliptical traces and the resulting motion grees well with the ideal motion . Thus the screen processing capacity and efficiency can both be improved. 2、Ideal motion for a screen surface and the proposal of a vibrating screen with variable elliptical trace 2. 1 Screening characteristics of common vibrating screens Vibrating screens commonly work at a fixed vibration intensity . Material on the screen surface moves by throwing, rolling or sliding motions . For common screeners , material granularity is widely distributed at the feed end . The energy imparted to the material particles from the vibrating screen is severely dissipated . Consequently，a large number of particles become laminated only a short distance from the feed end . The material penetrates the screen within the first 1/4 to 1/2 of the screen , which affects screening and lowers processing capacity [5]. The decrease of fine-grained material causes the ratio of. particles close in size to，or larger than，the mesh to increase . Thus，the screening efficiency declines dramatically . The material granularity simultaneously becomes uniform and the energy imparted from the vibrations to the material suffers little loss . Hence , the amplitude and velocity of the material particles increase . This causes the material bed depth at the feed end to be thick while at the discharge end it is Thin . This kind of motion leads to an asymmetrical penetration along the screen surface, which influences the screening efficiency and processing capability [6]. Common screening characteristics are shown in Fig. 1 2. 2 Ideal motion for screen surface and implementing scheme The ideal motion for screen surface is described below, according to the principle of screening process with constant bed thickness . The feed end of the screen has a bigger throwing index and a higher material del ivery velocity，which makes bulk material quickly penetrate and causes rapid de-laminating. Earlier lamination of material increases the probability of fine-grained material passing through the mesh . The screen has an appropriate throwing index and a little higher material delivery velocity in its middle part .This is of benefit for stabilizing fine-grained materials and for penetrating uniformly along the screen length . A lower throwing index and material delivery velocity near the discharge end causes the material to stay longer on the screen and encourages more complete penetration of the mesh. Two methods are currently used to improve screening efficiency [7 - 8]. The first is to add material to the screen from multiple feed ports. This is troublesome in practical use especially in terms of controlling the distribution of differently granulated materials. Hence it is rarely used in practical production. The second way is to adopt new screening equipment 1 ike, for example, a constant thickness screen. The motion of the new screen surface causes material to maintain the same, or an increased, thickness .It achieves a rather more ideal motion. The main problem with the constant thickness screen is that it covers a bigger area and that the structure is complicated and hard to maintain . A simple structure with good screening efficiency is still a necessity. We have designed a new vibration screen with a variable elliptical trace that is based upon an ideal screen motion for use in raw coal classification. The size of the vibrating screen is 3. 6 mX7. 5 m, the feed granularity is 0 to 50 inin and the classification granularity is 6mnu Elliptically vibrating screens combine the basic advantages of both circular and linear vibrating screens [9 - 10]. The long axis of the ellipse determines material delivery and the short axis influences material loosening, to be exact. 3、Dynamics model analysis of vibrating screen with variable elliptical trace We made the exciting force deviate from the center of gravity, to change the motion pattern of the now vibrating screen. The stiffness matrix of the vibration isolation spring was not zero under these circumstances and the vibrating system had multiple degrees of freedom. Minor transverse wagging was neglected to simplify the research. The motion was considered to be a linear vibration of a rigid beam in the longitudinal ly symmetrical plane. At each point the vibration is a comhination of the translation of the center of gravity and the screen pitching about the center of gravity. Previous studies neglected the influence of elastic forces in the horizontal and vertical direction on the swing of the vibrating screen [3, 11]. An accurate dynamic model consisting of three differential equations that include coupling of degrees of freedom in the vertical,horizontal and swing directions is proposed. The mathematical model of the vibrating screen is shown in Fig. 2. The center of gravity, is taken as the origin of a rectangular coordinate system at static equilibrium, in accordance with rigid motion on the plane [12]. Simultaneous differential equations in generalized coordinates using center of gravity coordinates, (x, y), and the swing declination angle θ , may be written as where M is the mass of the vibrating screen, s the moment of inertia of M relative to the center of gravity, 0;x and y the displacements in the x and y0 directions;x and y the velocities in the x and y directions’and y the accelerations in the x and y directions; is the swing angular displacement; a the installation angle;fx, f yond father damping coefficients in the x,y and directions; x k and k the stiffness coefficients of the supporting spring along the x and y directions；AO the amplitude of the exciting force, given hy2 0 A =mrc , where r is the radius of eccentricity the mass of the eccentric block and the exciting angular frequency; L1 and L2 the distances between each supporting spring and the center of gravity' s the distance between the rotating center of the eccentric block and the center of gravity; and, P the included angle between the 1 and x directions. The damping force is rather small and can be neglected. Then Eq. (1) can be simplified to Eq. (2) 4、Motion and screening effect analysis of a vibrating screen with variable elliptical trace 4.1 Analysis of the motion parameters Multiple degree of freedom vibration theory was used to find a stable solution for the forced vibration [13],as follows: Screen coordinates,?assuming a?point to D?(D?ξ,?D?ψ) equations of motion are as follow： When E2S2+C2H2+2 ESCH=0, the trace of point D is a line. When E =Sand C =H，the trace of point D is a circle. In general. (6) expresses the equation of an ellipse. The xoy coordinate was rotated Y degrees anticlockwise to give a new set of x ' oy' coordinates. A standard elliptical equation was then obtained after eliminating D D x y in Eq. (7) From this we know that some points on the screen move in a line or a circle while others move in an ellipse .As long as the relative position of the rotating center of the eccentric block and the center of gravity are properly adjusted, variable elliptical motion of the screen will be obtained . This provides a reasonable throwing index and material delivery velocity and improves screening efficiency. 4. 2 Analysis of motion trace and screening efficiency. The stable solution of a vibrating system, in terms of the vibrating screen, can be given by The equations of motion for any point on the vibrating screen are Eq. (8) shows that the center of gravity traces an approximate circle and that the amplitude in the horizontal and vertical directions is between 3. 5 mm and 5 mm. Fig. 3 shows how the center of gravity moves in three degrees of freedom. Fig. 3 gives the angular phase difference between the horizontal and vertical directions as well as the amplitude of the swing angle. 5、Conclusions (1)A new vibrating screen with variable elliptical motion trace was proposed according to the principle of screening process with constant bed thickness. Different points on the vibrating screen trace differentelliptical paths. The motion pattern agrees well with the ideal motion characteristic for a screening surface. Thus, screening capacity and process efficiency can be increased. (2)A theoretical kinematic analysis of the vibrating screen was done to study how varying different parameters affects the motion of the screen. Kinema tics parameters of the vibrating screen that motion traces are 1 inear, circular or el 1iptical are obtained. （3）Motion traces of total vibrating screen were gained through computer simulations. Screening technological parameters, including amplitude, velocity and throwing index, of five specific points along the screen surface were calculated. These parameters are related to screening efficiency. The results show that the motion pattern of the designed vibrating screen conforms to an ideal screening motion and that the design is able to effectively improve screening efficiency. （4）The position of the exciter axle center, relative to the center of gravity of the vibrating screen, is extreme!y important for screening efficient. Thus, we can design a vibrating screen with higher processing capacity without increasing power consumption by adjusting the relative position of the axle center. This is a point that requires further study. References [1]Wen 13 C, Liu F Q. Theory and Application of Vibration Machines. Bei jing:China Machine Press, 1982. [2]Gu Q B, Zhang E G. Study on complex-locus vibration screen. Mining&Processing Equipment, 1998(1) :42 – 46 [3]Hao F Y. Coal Preparation Manual:Technology and Equipment. Beijing:China Coal industry Publishing House, 1993. [4]Yan F. Screening Machines. Bei jing:China Coal Indus-try Publishing House, 1995. [5]Liu C S, Zhao Y M. Study on nonlinear characteristics of single particle on screening surface. Mining&Processing Equipment, 1999(1):45 - 48 [6]Tao Y J, Luo Z F, Zhao Y M. Experimental research on dosulfurization of fine coal using an enhanced centri- fugal gravity separator. Journal of China University of Mining&Technology, 2006，16(4): 399 - 403. [7]Zhang E G. Screening, Crushing and Dcwatering Equipments. Beijing: China Coal Industry Publishing House, 1991. [8]Khoury D L. Coal Cleaning Technology . USA:Noyes Data Corporation, 1981. [9]Shang N X, Na J F. 2TYA1842 elliptical vibration screen. \Iining&Processing Equipment, 1990(2): 20 - 24. [10]YeHD. Elliptical isopachous screening technology and its application. Sintering and Palletizing, 1999，5(3):30 – 33. [11]Wen B C, Liu S Y, He Q. Theory and Dynamic Design Method of Vibration Machines. Beijing:China Machine Press, 2001. [12]Wang F, Wang H.Screening. Machines. Bei jing:China Machine Press, 2001. [13]Ni Z H. Vibration Mechanics. Xi' an:Xi' an Jiaotong University Press,1989. [14]Zhu W B. Working principle and computer simulation of vibrating screen with complicated motion trace. Mining &Processing Equipment, 2004(10):34 - 36. [15]Peeler M. The mogensen E-series—a new screening oncept. Mineral Processing, 1996, 7(37):311 -315. [16]Wen B C. Synchronization theory of self-synchronous vibrating machines with ellipse motion locus. Boston: American Society of Mechanical Engineers, 1987:495 - 500  變橢圓軌跡振動篩的動力學和篩選特性 何小梅，劉楚生 機械和電氣工程學院，中國礦業(yè)大學科技，江蘇省徐州市，中國 摘要：理想的運動特征的振動篩是根據(jù)恒定床厚篩分過程的原理介紹。提出了一種新振動篩具有可變橢圓軌跡。一個精確的機械模型，根據(jù)所要求的運動的結構特征構成。應用多度的自由度振動原理，使振動篩的特點進行了分析。獲得振動篩的運動軌跡為直線，圓形或橢圓形的運動學參數(shù)。動力學方程的穩(wěn)定的解決方案通過計算機模擬的方式給了振動篩的運動。工藝參數(shù)，包括振幅，運動速度沿屏幕表面五項具體點和投擲指數(shù)，是通過理論計算獲得。實驗結果表明，新設計的振動篩的痕跡按照理想的篩選運動。篩分效率和處理能力可能因此被有效改善。 關鍵詞：變橢圓軌跡；固定床厚度的篩選過程；動力學模型；運動特性；篩選特征 1、介紹： 篩選操作是煤炭加工的一個重要組成部分。振動篩是一種最廣泛使用的篩選工具。振動篩，如直線振動篩，圓振動篩、平動橢圓振動篩，有一個簡單的平移運動。運動遵循相同的路徑都在屏幕上，所以屏幕具有恒定的傳輸速度和拋擲指數(shù)，這導致低的篩分效率。增強的拋擲指數(shù)提高打破了激振電機的處理能力，降低了工作強度。 在本文中，我們報告與變速運動的痕跡，是基于等厚篩分過程[ 3 - 4 ]的原則的一個新的振動篩的設計。該振動篩橢圓軌跡穿越不同的部位產生的運動程度與理想的運動。因此，屏幕處理能力和效率均可以提高。 2、振動表面的理想運動和變橢圓軌跡振動篩的建議 2.1常見的振動篩篩分特性 振動篩，通常在一個固定的振動強度的工作。扔在屏幕表面移動材料，滾動或滑動運動。常見的安檢人員，物料粒度分布在進料端廣泛。能賦予材料粒子從振動篩是嚴重消耗。因此，大量的粒子成為層壓只有很短的距離從進料端。材料的穿透屏幕內的第一個1 / 4至1 / 2的屏幕，從而影響篩選和降低處理能力[ 5 ]。細粒物質的減少導致比。顆粒的尺寸接近，或大于，網格增加。因此，篩選效率急劇下降。物料粒度均勻，同時成為能量從振動對材料受點損失。因此，粒子組成的物質的幅度和速度的增加。這使物料床層深度的進料端是厚而在放電結束它薄。這種運動導致沿篩面不對稱的滲透，從而影響篩分效率和處理能力[ 6 ]。常用的篩選特性如圖1所示 2.2振動篩表面的理想運動和實施方案 篩面運動的理想描述如下，根據(jù)等厚篩分原理。屏幕的進料端有一個較大的拋擲指數(shù)和較高的材料刪除應用速度，使物料迅速滲透，導致迅速脫層壓。早期的層壓材料增加細顆粒材料通過網格的概率。屏幕上有一個適當?shù)膾仈S指數(shù)和更高的物料輸送速度在它的中間部分。這有助于穩(wěn)定細粒度的材料和滲透均勻地沿篩面長度。一個較低的拋擲指數(shù)和交貨速度附近的放電端使物料停留在屏幕上，鼓勵網更完整的滲透。目前有兩種方法可用來提高篩分效率[ 7 - 8 ]。首先是從多個進料口材料添加到屏幕。 這在實際使用中是個麻煩，尤其是在控制的分布不同的粒狀材料。因此，在實際生產中，很少使用。第二種方法是采用新的篩選設備，1艾克，例如，一個恒定的厚度的屏幕。新的篩面運動導致材料保持不變，或增加，厚度。它達到更理想的運動。 恒定厚度的屏幕的主要問題是，它包括一個面積大，結構復雜，維護困難。一種篩分效率好簡單的結構仍然是必要的。我們設計了一個新的具有可變橢圓軌跡，是基于一個理想的屏幕運動中使用的原煤分級篩。 振動篩的尺寸為3。6 mx7。5米，排料粒度大小0至50在和分類的粒度是6mnu橢圓振動篩結合圓形和直線振動篩的[ 9 - 10 ]的基本優(yōu)點。橢圓的長軸和短軸確定物料輸送的影響物質的松動，是準確的。 3、變橢圓軌跡振動篩的動力學模型分析 我們做的激振力偏離重心，改變了振動篩的運動模式。多自由度振動系統(tǒng)在隔振彈簧的剛度矩陣不為零的情況下，簡化研究忽略了小橫搖的情形。運動被認為是在縱向上對稱平面剛性梁的線性振動。在每個點的振動是一個三的重心和屏幕俯仰重心有關翻譯。以往的研究忽略了在水平和垂直方向的彈性力對振動篩的擺動的影響[ 3，11 ]。一個準確的動態(tài)模型由三個微分方程，包括在垂直方向上的自由度耦合，橫向和擺動的方向。 ?該振動篩數(shù)學模型如圖2所示。重力的中心，作為在靜態(tài)平衡一個矩形的坐標系的原點，在剛體運動按照飛機上[ 12 ]。在廣義坐標使用重心坐標中心的微分方程組，（x，y），和擺偏角θ，可以寫成： 其中m是振動篩的質量，目前M相對于重力，0中心慣量；X和Y在X和y 0 角位移；X和Y在X和Y方向上的X和Y directions'and Y加速度是速度；擺角位移；一個安裝角在阻尼系數(shù)F在X，Y方向；X和K支撐彈簧的剛度系數(shù)沿X和Y方向；AO的激振力的振幅，給出2 0 A=mrw，其中R是偏心的偏心塊質量和令人興奮的角頻率，L1和L2距離半徑；每個支撐彈簧和重力的中心之間的距離的旋轉中心的偏心塊和重心之間；和，P包括1和X方向之間的夾角。阻尼力很小，可以忽略不計。則式（1）可簡化為式（2）： 4、變橢圓軌跡振動篩的運動和篩選的效果分析 4.1分析多自由度運動參數(shù)振動理論多用來解決受迫振動[13]的一個穩(wěn)定解，如下： 替代參數(shù)在式（3）到（2）式。允許一個穩(wěn)定的解決方案被發(fā)現(xiàn) 假設一個點的屏幕坐標為D（Dξ，Dψ）運動方程如下： 當E2S2+C2H2+2ESCH=0時，D點的軌跡是一條直線。當E=∑，X=H∑時，D點的軌跡是圓。一般來說，（6）式表示方程的橢圓面直角坐標系。XOY坐標系以γ角速度逆時針旋轉從而給定一個新的坐標系x'oy'。一個標準的橢圓公式在消除XDYD后可得公式（7）。 從這我們可以知道一些點在一條線或一個圓圈屏幕上移動而移動的橢圓，只要對旋轉中心的偏心塊和重心的相對位置進行適當調整的，屏幕會得到變運動的橢圓。這提供了一個合理的拋擲指數(shù)和交貨速度以及提高了篩分效率。 4.2運動軌跡和篩分效率分析 穩(wěn)定振動系統(tǒng)解決方案，就振動篩而言的，可以給出 在振動篩上任一點的運動方程為 公式（8）表示重心的痕跡近似圓形，水平和垂直方向的振幅在3.5mm和5mm之間。圖3表示如重心的移動存在三個自由度。圖3水平和垂直方向的相位差和擺角的振幅一樣。 5、結論 11）新振動篩變橢圓根據(jù)原則提出了運動軌跡常數(shù)床厚度的篩選過程。振動篩跟蹤不同的不同點橢圓路徑。運動規(guī)律也同意配合篩面的理想運動特性。因此，篩選能力和處理效率會增加。 22）振動的理論運動學分析屏幕做是為了研究如何變不同參數(shù)會影響屏幕的意義。抽搐振動篩參數(shù)的議案線性跟蹤，獲得圓或橢圓。 33）?總振動篩運動的痕跡通過計算機模擬獲得。篩選技術參數(shù)，包括振幅、速度和引發(fā)指數(shù)五個的特定點沿屏幕表面計算。這些參數(shù)是與篩分效率。結果顯示模式設計的議案振動篩符合理想的篩選議案，設計能夠有效提高篩分效率。 44）?勵磁機軸中心的地位，相對振動篩的重心，是篩選高效的極為重要的。因此，我們可以設計一個振動篩具有更高的處理的能力而又不會增加功耗調整軸中心的相對位置。這是一個點，需要進一步研究。 參考文獻 [1]Wen B C，Liu F Q.振動的理論與應用機器.北京：中國機械出版，1982 [2]Gu Q B, Zhang E G.復雜軌跡振動篩研究.1998(1) :42 - 46 [3]Hao F Y.煤制備手冊：技術設備.北京：中國煤炭行業(yè)出版社，1993 [4]Yan F.篩分機械.北京：中國煤炭行業(yè)出版社.1995 [5]Liu C S, Zhao Y M.篩面上的非線性特性的研究.礦物處理設備，1999(1) :45 – 48。 [6]Tao Y J, Luo Z F, Zhao Y M..在重力強力分離器使JO方面的煤脫硫的實驗研究.中國礦業(yè)與技 術大學.2006, 16(4) :399 – 403。 [7]Zhang E G.篩分，粉碎，脫水設備.北京：中國煤工業(yè)出版社，1991。 [8]Khoury D L.煤淸潔工藝.美國：諾易斯數(shù)據(jù)公司，1981 [9]Shang N X，Na J F. 2TYA1842 圓振動篩.礦業(yè)與加工設備,1990(2): 20 – 24。 [10]Ye H D.等厚圓篩分及其應用.燒結和碼垛，1999, 5 (3): 30 – 33。 [11]Wen B C, Liu S Y, He Q.振動機械理論和動力學設計方法.北京:中國機械出版社，200。 [12]Wang F, Wang H.篩分機械.北京:中國機械出版社，2001 [13]Ni ZH?振動力學?西安:西安交通大學出版社，1989 [14]Zhu W B.復雜運動軌跡振動篩的工作原理和計算機仿真.礦業(yè)與加工設,2004(10): 34 – 36。 [15]Peder M.莫根深系列——種新型的篩分概念.礦物加工，1996,7(37) :311 – 315。 [16]Wen B C.圓振動篩的自動同步理論.波士頓：美國機械工程協(xié)會，1987:495 – 500。