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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
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[10]YeHD. Elliptical isopachous screening technology and its application. Sintering and Palletizing, 1999，5(3):30 – 33.
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變橢圓軌跡振動(dòng)篩的動(dòng)力學(xué)和篩選特性
何小梅，劉楚生 機(jī)械和電氣工程學(xué)院，中國(guó)礦業(yè)大學(xué)科技，江蘇省徐州市，中國(guó)
摘要：理想的運(yùn)動(dòng)特征的振動(dòng)篩是根據(jù)恒定床厚篩分過(guò)程的原理介紹。提出了一種新振動(dòng)篩具有可變橢圓軌跡。一個(gè)精確的機(jī)械模型，根據(jù)所要求的運(yùn)動(dòng)的結(jié)構(gòu)特征構(gòu)成。應(yīng)用多度的自由度振動(dòng)原理，使振動(dòng)篩的特點(diǎn)進(jìn)行了分析。獲得振動(dòng)篩的運(yùn)動(dòng)軌跡為直線，圓形或橢圓形的運(yùn)動(dòng)學(xué)參數(shù)。動(dòng)力學(xué)方程的穩(wěn)定的解決方案通過(guò)計(jì)算機(jī)模擬的方式給了振動(dòng)篩的運(yùn)動(dòng)。工藝參數(shù)，包括振幅，運(yùn)動(dòng)速度沿屏幕表面五項(xiàng)具體點(diǎn)和投擲指數(shù)，是通過(guò)理論計(jì)算獲得。實(shí)驗(yàn)結(jié)果表明，新設(shè)計(jì)的振動(dòng)篩的痕跡按照理想的篩選運(yùn)動(dòng)。篩分效率和處理能力可能因此被有效改善。
關(guān)鍵詞：變橢圓軌跡；固定床厚度的篩選過(guò)程；動(dòng)力學(xué)模型；運(yùn)動(dòng)特性；篩選特征
1、介紹：
篩選操作是煤炭加工的一個(gè)重要組成部分。振動(dòng)篩是一種最廣泛使用的篩選工具。振動(dòng)篩，如直線振動(dòng)篩，圓振動(dòng)篩、平動(dòng)橢圓振動(dòng)篩，有一個(gè)簡(jiǎn)單的平移運(yùn)動(dòng)。運(yùn)動(dòng)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(dǎo)致低的篩分效率。增強(qiáng)的拋擲指數(shù)提高打破了激振電機(jī)的處理能力，降低了工作強(qiáng)度。
在本文中，我們報(bào)告與變速運(yùn)動(dòng)的痕跡，是基于等厚篩分過(guò)程[ 3 - 4 ]的原則的一個(gè)新的振動(dòng)篩的設(shè)計(jì)。該振動(dòng)篩橢圓軌跡穿越不同的部位產(chǎn)生的運(yùn)動(dòng)程度與理想的運(yùn)動(dòng)。因此，屏幕處理能力和效率均可以提高。
2、振動(dòng)表面的理想運(yùn)動(dòng)和變橢圓軌跡振動(dòng)篩的建議
2.1常見(jiàn)的振動(dòng)篩篩分特性
振動(dòng)篩，通常在一個(gè)固定的振動(dòng)強(qiáng)度的工作。扔在屏幕表面移動(dòng)材料，滾動(dòng)或滑動(dòng)運(yùn)動(dòng)。常見(jiàn)的安檢人員，物料粒度分布在進(jìn)料端廣泛。能賦予材料粒子從振動(dòng)篩是嚴(yán)重消耗。因此，大量的粒子成為層壓只有很短的距離從進(jìn)料端。材料的穿透屏幕內(nèi)的第一個(gè)1 / 4至1 / 2的屏幕，從而影響篩選和降低處理能力[ 5 ]。細(xì)粒物質(zhì)的減少導(dǎo)致比。顆粒的尺寸接近，或大于，網(wǎng)格增加。因此，篩選效率急劇下降。物料粒度均勻，同時(shí)成為能量從振動(dòng)對(duì)材料受點(diǎn)損失。因此，粒子組成的物質(zhì)的幅度和速度的增加。這使物料床層深度的進(jìn)料端是厚而在放電結(jié)束它薄。這種運(yùn)動(dòng)導(dǎo)致沿篩面不對(duì)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(dòng)篩表面的理想運(yùn)動(dòng)和實(shí)施方案
篩面運(yùn)動(dòng)的理想描述如下，根據(jù)等厚篩分原理。屏幕的進(jìn)料端有一個(gè)較大的拋擲指數(shù)和較高的材料刪除應(yīng)用速度，使物料迅速滲透，導(dǎo)致迅速脫層壓。早期的層壓材料增加細(xì)顆粒材料通過(guò)網(wǎng)格的概率。屏幕上有一個(gè)適當(dāng)?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(xì)粒度的材料和滲透均勻地沿篩面長(zhǎng)度。一個(gè)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(lì)網(wǎng)更完整的滲透。目前有兩種方法可用來(lái)提高篩分效率[ 7 - 8 ]。首先是從多個(gè)進(jìn)料口材料添加到屏幕。
這在實(shí)際使用中是個(gè)麻煩，尤其是在控制的分布不同的粒狀材料。因此，在實(shí)際生產(chǎn)中，很少使用。第二種方法是采用新的篩選設(shè)備，1艾克，例如，一個(gè)恒定的厚度的屏幕。新的篩面運(yùn)動(dòng)導(dǎo)致材料保持不變，或增加，厚度。它達(dá)到更理想的運(yùn)動(dòng)。
恒定厚度的屏幕的主要問(wèn)題是，它包括一個(gè)面積大，結(jié)構(gòu)復(fù)雜，維護(hù)困難。一種篩分效率好簡(jiǎn)單的結(jié)構(gòu)仍然是必要的。我們?cè)O(shè)計(jì)了一個(gè)新的具有可變橢圓軌跡，是基于一個(gè)理想的屏幕運(yùn)動(dòng)中使用的原煤分級(jí)篩。
振動(dòng)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(dòng)篩結(jié)合圓形和直線振動(dòng)篩的[ 9 - 10 ]的基本優(yōu)點(diǎn)。橢圓的長(zhǎng)軸和短軸確定物料輸送的影響物質(zhì)的松動(dòng)，是準(zhǔn)確的。
3、變橢圓軌跡振動(dòng)篩的動(dòng)力學(xué)模型分析
我們做的激振力偏離重心，改變了振動(dòng)篩的運(yùn)動(dòng)模式。多自由度振動(dòng)系統(tǒng)在隔振彈簧的剛度矩陣不為零的情況下，簡(jiǎn)化研究忽略了小橫搖的情形。運(yùn)動(dòng)被認(rèn)為是在縱向上對(duì)稱平面剛性梁的線性振動(dòng)。在每個(gè)點(diǎn)的振動(dòng)是一個(gè)三的重心和屏幕俯仰重心有關(guān)翻譯。以往的研究忽略了在水平和垂直方向的彈性力對(duì)振動(dòng)篩的擺動(dòng)的影響[ 3，11 ]。一個(gè)準(zhǔn)確的動(dòng)態(tài)模型由三個(gè)微分方程，包括在垂直方向上的自由度耦合，橫向和擺動(dòng)的方向。
?該振動(dòng)篩數(shù)學(xué)模型如圖2所示。重力的中心，作為在靜態(tài)平衡一個(gè)矩形的坐標(biāo)系的原點(diǎn)，在剛體運(yùn)動(dòng)按照飛機(jī)上[ 12 ]。在廣義坐標(biāo)使用重心坐標(biāo)中心的微分方程組，（x，y），和擺偏角θ，可以寫(xiě)成：
其中m是振動(dòng)篩的質(zhì)量，目前M相對(duì)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(gè)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(zhì)量和令人興奮的角頻率，L1和L2距離半徑；每個(gè)支撐彈簧和重力的中心之間的距離的旋轉(zhuǎn)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(jì)。則式（1）可簡(jiǎn)化為式（2）：
4、變橢圓軌跡振動(dòng)篩的運(yùn)動(dòng)和篩選的效果分析
4.1分析多自由度運(yùn)動(dòng)參數(shù)振動(dòng)理論多用來(lái)解決受迫振動(dòng)[13]的一個(gè)穩(wěn)定解，如下：
替代參數(shù)在式（3）到（2）式。允許一個(gè)穩(wěn)定的解決方案被發(fā)現(xiàn)
假設(shè)一個(gè)點(diǎn)的屏幕坐標(biāo)為D（Dξ，Dψ）運(yùn)動(dòng)方程如下：
當(dāng)E2S2+C2H2+2ESCH=0時(shí)，D點(diǎn)的軌跡是一條直線。當(dāng)E=∑，X=H∑時(shí)，D點(diǎn)的軌跡是圓。一般來(lái)說(shuō)，（6）式表示方程的橢圓面直角坐標(biāo)系。XOY坐標(biāo)系以γ角速度逆時(shí)針旋轉(zhuǎn)從而給定一個(gè)新的坐標(biāo)系x'oy'。一個(gè)標(biāo)準(zhǔn)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(diǎn)在一條線或一個(gè)圓圈屏幕上移動(dòng)而移動(dòng)的橢圓，只要對(duì)旋轉(zhuǎn)中心的偏心塊和重心的相對(duì)位置進(jìn)行適當(dāng)調(diào)整的，屏幕會(huì)得到變運(yùn)動(dòng)的橢圓。這提供了一個(gè)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(yùn)動(dòng)軌跡和篩分效率分析
穩(wěn)定振動(dòng)系統(tǒng)解決方案，就振動(dòng)篩而言的，可以給出
在振動(dòng)篩上任一點(diǎn)的運(yùn)動(dòng)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(dòng)存在三個(gè)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(jié)論
11）新振動(dòng)篩變橢圓根據(jù)原則提出了運(yùn)動(dòng)軌跡常數(shù)床厚度的篩選過(guò)程。振動(dòng)篩跟蹤不同的不同點(diǎn)橢圓路徑。運(yùn)動(dòng)規(guī)律也同意配合篩面的理想運(yùn)動(dòng)特性。因此，篩選能力和處理效率會(huì)增加。
22）振動(dòng)的理論運(yùn)動(dòng)學(xué)分析屏幕做是為了研究如何變不同參數(shù)會(huì)影響屏幕的意義。抽搐振動(dòng)篩參數(shù)的議案線性跟蹤，獲得圓或橢圓。
33）?總振動(dòng)篩運(yùn)動(dòng)的痕跡通過(guò)計(jì)算機(jī)模擬獲得。篩選技術(shù)參數(shù)，包括振幅、速度和引發(fā)指數(shù)五個(gè)的特定點(diǎn)沿屏幕表面計(jì)算。這些參數(shù)是與篩分效率。結(jié)果顯示模式設(shè)計(jì)的議案振動(dòng)篩符合理想的篩選議案，設(shè)計(jì)能夠有效提高篩分效率。
44）?勵(lì)磁機(jī)軸中心的地位，相對(duì)振動(dòng)篩的重心，是篩選高效的極為重要的。因此，我們可以設(shè)計(jì)一個(gè)振動(dòng)篩具有更高的處理的能力而又不會(huì)增加功耗調(diào)整軸中心的相對(duì)位置。這是一個(gè)點(diǎn)，需要進(jìn)一步研究。
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