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毕业设计(论文)外文资料翻译院 系 专 业 学生姓名 班级学号 外文出处 2004 American Society of Agricultural Engineers ISSN 0001-2351指导教师评语：指导教师签名： 年 月 日无线控制固体肥料撒播机上肥料散播倾斜转子的分析与设计参数J.Duhovnik,J.Benedicic,R.Bernik摘要：肥料散布机的发展是农产品耕作领域中重要的一环。因此，肥料散布机的技术进展一定要预先使用各种科学研究和分析的理论。而重点首先就是要改进散布机的工作原理，使用所谓的广角散布器。通过对转子材质的分析结果显示了大部分与散播机相关的参数。而分析证明摩擦并不是最重要的参数，肥料最初的位置和转子的转速更为重要。转子和肥料间的摩擦系数对转子上肥料的切向速度只会产生微小的影响，也就是说只会对撒播范围产生微小影响。而肥料颗粒在转子上的初始位置则会对它的切向速度和行程产生较大影响。肥料颗粒从转子上分离的速度会受到转子的尺寸和转速的影响。此结论的可贵之处在于它构造了一种新的施肥方法，或者说它提高了现有的施肥理论水平。关键词：无线应用，肥料撒播机，数学模型，施肥施肥是农作物耕作地中重要的一环。固体肥料是养牛业的副产品，但它的基本作用是直接用肥于田。虽然有许多关于固态肥料撒播的理论，但是它们都有一些相同的参数 (Redelberger and Kuyhlmann,1989)。最重要的标准是在相关地势施肥田表面的最大可能施肥 能力；另一个标准就是要最大可能的沿着施肥线垂线方向均匀撒布。另外，田间的一些部分 施肥后要进行适当的处理，比如施肥的起点和终点，沿着施肥线方向的盲区，角落，以及田 间有较大坡度变化的地方，等等(Frick et al，2001 )。这些标准构成了评价施肥技术进展水平 的基本要求。一旦某个技术被准确地确定，那么这项技术系统中那些合乎基本秦件要求的各 单元也可以确定(VDI，1993)。因此，按照规则，技术系统中的各个过程都是为了适应技术的发展。在这种方式下，撒播设备各个部分的改念设计都允许采用不同的技术方法(Zavbi and Duhovnik,2001 ;Zavbi and Duhovnik，1996)。目前市场上流行这一些著名生产商生产的此类产品，通过对它们的研究向人们展示了多种肥料撒播理论，如下文所述(Frick et al，2001;Huschke，2001;Benedicic，2002)。固体肥料撒播中最通用的方法是使用垂直敲打器，比如德国Goldenstedt 州L.bergmann GmbH 公司的Bergmann 撒播机(Huschke，2001:Benedicic，2002)。水平敲打器的撒播机也有生产，比如澳大利亚Grieskirchen 州 Alois Pottinger Maschinenfabrik 公司的 Pottinger 撒播机(Huschke，2001;Benedidc，2002);而 且近来有越来越多的广角设计撒播机，如德国Stadtlohn州Maschinenfabrik Kemper GmbH公司产的Kemper撒播机和德国Goldenstedt州L.bergmann GmbH公司的Bergmann撒播机 (Huschke,2001 ;Benedicic，2002)。大部分撒播机设计都不允许在操作时调整撒播的宽度。另外，系数变化(CV)对撒播的整体宽度有很大的影响(Fricketal，2001)。peEN 13080标准规定CV测试只使用于重叠的情况(Final Draft European Standard)。因此，在小表面上撒播肥料（在不重叠的地方）不存在实际的数据。有效的撒播宽度取决于撒播机的型号。如果希望地表均匀的肥沃化，撒播的肥料 必须和邻边的撒播带重叠，因为沿着撒播带边缘的肥料数量非常少(Frick et al)。现代的撒播 机可以更加均匀地、精确地将有机肥料撒播在所要的表面上。这就要提高操作者的控制水平， 也会提高撒播过程的质量，而且也将明显地提高撒播利用率。根据我们的研究，肥料撒播机 最重要的参数是肥料颗粒从机器容器到田地的轨迹。提高肥料撒播机速率(kg/s)最好的方法 就是增加撒播宽度。通过提高车辆沿着撤播线的速度来提高撒播速度是有限的，它会受到撒 播地表面状况和拖拉机及联合机车的性能限制。因而，近来的撒播机更多的增加撒播宽度。 越来越多的生产商开始生产具有广角撒播技术的撒播机，此类机器的撒播宽度己经宽达24米。当技术方法确定好了，肥料撒播的宽度直接取决于撒播机的撒布装置(基本的一个肥料输出装置)。撒布装置是离心抛射动作的首要执行者(Meriam，1993)。如果是非常相似的物质流(在这种情况下仍是固体肥料)，那么离心撒布装置的方法可以划分得十分仔细。然而在这种情况下，物质流十分不同。因此，物质流中首先进入撒布装置的相同或相似的肥料颗粒是衡量撒播质量的一个重要条件。在如下分析中，重点将是肥料撒播的过程方法。我们的分析得出了一些影响技术方法的重要参数。所有的基本参数(即直径，转速，肥料的类型，需要施肥的区域及肥料的撒播范围)都进行分析并且将它们列成更清晰的图表形式。现在已经建立了一个肥料撒播的数字模拟系统，而且对施肥影响最大的那些参数也建立了同一函数。这些在预期条件下会生成最佳结果的参数就可以确定。这些参数包括施肥的均匀性以及这种方法对应的撒播距离范围。在对由撒播机施到田表的肥料块流的数据计算中，这种方法提供了可行的理论基础。在己知的液压驱动转子上的肥料模拟分析中，这并非唯一的物流控制例子(Hess and Keuper,2002)；但是，到目前为止，还没有发现一种固体肥料撒播的数字分析方法。Hess和 Keuper更多地注意考虑能量的损耗和物流的控制。我们在物流控制方面和使用这种方法控 制后的撒播质量方面己经取得领先的水平。肥料撒播方法：想要了解肥料的进展水平，必须对其方法作一个仔细的、有比较性的分类。可以明显地知道，其方法的分类可以依据Zavbi和Duhovnik在2001年发表的文章中的几个基本依据来划分。这些基本依据说明了需要考虑到哪些部分的功能，以期达到期望效果，即撒播的肥料。分开考虑各部分的条件可以帮助我们建立撒播的数字分析模型。通过对现有撒播机型号的详细分析可以知道，一些撒播机仅仅在工作原理的技巧方面 有分别。少数情况下，肥料撒播机的功能可以分为两大部分，即肥料的粉碎和撒播功能。1、肥料供给固体肥料首先要装入撒播机的料箱中，然后还要运输到粉碎装置。肥料的供给靠一个牵 引装置实现。这个装置应该是可以连续控制或按渐进方式进给的。到达撒播装置的肥料总数 取决于供给装置的速度。2、月巴料粉碎料箱中的肥料都是成块的，这些通过供给装置输送到粉碎装置的的肥料必须切割成小块。粉碎装置一般由垂直和水平的敲打器组成。这些东西将料箱中的肥料切割成小颗粒，然后母其均匀地分配到整个撒播所需的宽度。3、输送到撤播器输送肥料的作用是保证它们由撒播机的粉碎装置输送到撒播装置。将肥料输送到撒播装 置的制定区域是十分重要的。肥料颗粒不会直接落到撒播装置，因为它们的方向不确定性会 沿着金属挡板运动才达到制定区域。4、肥料撒播肥料撒播在所有功能中是最为重要的一环因为预期方法的功能要靠它来实现。撒播范 围和均匀性可以多样化。撒播工作必须使肥料颗粒加速，以期达到工作方法中的最大速度和 因此而达到的最大撒播距离，即范围。市场上通用的大多肥料撤播机具有如下部分的功能：粉碎、运输以及撒播肥料颗粒。大 多数撒播机的构造方式是用一个单独的单元来实现一个功能，但是近来许多制造商已经开始 更多地生产广角撒播机，即只由粉碎装置和撒播装置组成。粉碎装置保持不变，仍然用来粉碎的水平和垂直敲打器。然后这些肥料颗粒通过撒播装置离开撒播机。在广角撒播机中，这 些部分的功能可以通过逐个相连的单独部分来实现。肥料撒播方法的原理及特性在肥料撒播方法中，撒播是以离心抛撒的方法为基础的。进入离心抛撒的物流量相对较 大，因此这个过程是建立在肥料颗粒运动分析的基础上。输入的数据构建了分析的重要的一部分。在必要因素中的是肥料本身，即固态肥料。仔细分析肥料的力学和物理特性是十分重 要的。有机肥料的力学及物理特性所有的材料都有一些影响它们使用范围的特定属性。在数字分析中，肥料和与它发生滑 动表面间的摩擦系数（）是必须要求知道的。取决于肥料的表面、密度、湿度和类型。因此,的值是可以相应地变化的(从0.6到0.98),比如在仓储条件下(比如家禽粪肥)(Pezzi and Rondelli,2002)。受到许多因素的影响，我们讨论的是通过DIN 51131标准(DIN标准，1999) 测量的牛粪肥料的摩擦系数。我们的目的是查明肥料湿度对摩擦系数的影响。可以通过两种类型的肥料来测定：七组较高湿度肥料的测量数据；以及七组较低湿度肥料的测量数据。在所有数据组中去掉最大和最小值。然后去余下数据的平均值就可以作为摩擦系数的值用于 数据计算(如表1)。测量证明低湿度肥料的摩擦系数比高湿度的摩擦系数要大，密度越大，肥料的湿度越 大。相似的结果在其它文献中也有陈述(Redelberger and Kuyhlmann,1989;Dohler and| Biskuper,l993;Highnett,1985)。数学模型建立数学模型是计算过程中的重要步骤(Benedicic,2002;Hesse and Keuper,2001 ;Meriam, 1993)，同时计算结果的正确性以取决于它。因此当挤压叶片和底板固定时，可以建立转子的数学模型(如图2)。这种类型转子模型的建立可以对沿着转子表面运动的肥料颗粒进行计算分析。颗粒沿着两个方向运动，因而，数学模型也可以创建成二维的。转子表面肥料颗粒的运动依照动力学规则（Meriam,1993）。为了简化分析，可以建立一个中心坐标来解决。肥料颗粒的运动完全可以用与运动学相关的径向方向（）和切向方向（） 来描述。由于转子的作用力和颗粒运动的作用，肥料颗粒相对于转子沿着径向和切向运动。在径向（），肥料颗粒受到如下作用：肥料和底盘之间的摩擦（），肥料和叶片间的摩擦（），以及径向的离心力。在正切方向(图2中)，肥料颗粒受如下作用：切向加速度力（）和叶片的支持力，正是这两个力导致了的产生(如图3)；底版和肥料间的摩擦（）也要加以考虑。当确定了数学模型后，要知道哪些影响和特性要纳入计算，哪些应当忽略。上面已经提 到空气阻力是不纳入计算的。空气阻力只影响结果(速度、范围等)的打消而不影响运动方向 或对撒播质量有意义的参数。因此，不将空气阻力纳入的简化算法可以得出最重要的那些参 数；这些参数可以在设计新的散播机械时使用。作为个种类，有机肥料是一种不均勻的材 质，但是在散播用的是压碎的细顆粒。颗粒的类型也取决于库存的时间。库存时间相对于较 长的肥料大多小于12毫米（Pezzi and RondeHi, 2002)。小颗粒的肥料可以当作均匀对待， 因此所有颗粒的动力学和物理属性可以认为是相同的。这对于牛类肥料也同样适用。对一个肥料颗粒的动力分析可以按两步来：（1）肥料沿转子表明面和（2)肥料离开转子后的运动。肥料颗粒沿转子表面的运动一个肥料颗粒落到转子表明（图2）后，它将仍然呆在原位置直到叶片碰到它，然后它将随着叶片进行圆周运动。离心力将致使肥料沿着叶片和转子的边沿运动。这个运动在径向可以写成一个不用的等式：（1）在切向写成：（2）（1） 和（2）写成另外微分：（3）方程3的解在等式4中已经给出。它把径向运动又定义成时间函数r（t），肥料沿着转子表面的运动也是一个时间函数，它受到摩擦系数，肥料离叶片中心初始距离（），和转子的角速度三者的影响。方程3并不要求肥料的质量，它就是说通过方程1和2的合并已经把质量消去了。因此计算结果并不依赖肥料的类型。肥料块的类型是很难确定的，因为肥料是一种不均匀的材料而且它的性质取决于很多参数。重力加速度（g）的取值9.81m/。微分方程（eq.3）可以分解成齐次部分（系数1和2）和非齐次部分： （4）（5）方程5描述了肥料的速度随着时间的变化，也描述了肥料沿叶片的运动。这是径向速度 (图3中的Vr)它仅仅用于肥料沿叶片方向的运动。切向速度（）取决于它的半径（r) 和转子角速度（），肥料的最终速度叫系统速度（）： （6）肥料离开转子后的运动图4画出了肥料散出后转子的位置。散播是对称的，因而只要分析一个转子就可以了。当一个肥料块处于转子的边沿，肥料已经达到一个适当的速度。这应当归位一个矢量，因为它的 大小和方向（r，）。计算肥料的落点可以利用斜抛和抛物运动的原理（Meriam,1993） （7）其中Lt=抛物运动的距离 a=地面和抛射方向的角度(图18) =抛射的最大高度 t=肥料的飞行时间我们必须更改方程7，因为必须考虑肥料块的初始条件。方程7完全定义了斜抛运动的整个 轨迹。对我们来说，仅仅要考虑抛物运动的第二部分就是从抛物线最高点（）到地面（h=0)的过程。为了计算散播的范围转子离地的高度（图4中的h),肥料块 处于转子边缘时的位置，和肥料块的大小及方向必须要知道坐标系统。坐标系统的起点设立 在转子在叶片中轴线上的中心，而且作为中点来计算撒播宽度(只用于一个转子X如图4b). 当计算肥料颗粒在横向(x轴)的分布时，肥料x向的速度可以满足因为这对分析横向的肥 料撒播已经足够。转子的放置应当是与地面并且平行与重力友速度的方向垂直的水平位置，如此就可以使 方程7中的(t图18)。肥料的速度矢量（）等于与转子外圆速度相同的切向速度(Vt)和径向逨度(Vr)的和。为了计算撤播宽度，只需要知道横向速度()(如图4b)。这就是垂直于撒播方向的速度。等式8可以计算肥料从飞离转子到落地的时间(t)，等式9可以计算肥料 落地的横向距离（Lx）。这种方式下，每个肥料颗粒落到田表的位置都可以确定： （8） （9）空气阻力对肥料运动的影响也要进行理论分析。Fzr力取决于正面区域表面(A),速度和 空阻系数（图10中的）。空阻系数的值主要取决于肥料的形状球形体，棱柱体，板状体， 车身等物体的风阻系数是已知的。但要对更复杂的形状体进分析时，就必须进行试验来测定 风阻系数。肥料颗粒的形状是随机不确定的，所以我们要仔细分析那些已知形状的风阻，并 从中挑出最合适的一种来用于肥料颗粒的分析可以知道球形和棱柱形的肥料颗粒的风阻系 数是可以进行理论分析的。作为颗粒的基准，相对较大的家禽粪肥颗粒的尺寸己经制定了标 准(Pezzi and Rondelli,2002)。第二个例子的取值是第一个值村的4倍，因为要减小牛粪颗粒 的尺寸要比家禽肥料困难的多。分析两种尺寸对撒播范围的影响(如图5和6)。可以使用风 阻的二次方程式(方程10)，因为颗粒的速度相对较大而且会随时间变化。等式11包含颗粒的质量(mg)，它用于特定尺寸颗粒的计算： （10） （11）图5显示了|种形状肥料颗粒的不同撒播范围。计算中的球形和近似棱柱形颗粒的尺寸 是1cm。棱柱形颗粒(Cw=1.56)的撒播范围(Lt)比没用风阻时少了 30%。而球形颗粒(Cw=0.47) 相比于棱柱形则多了15%的宽度范围。分析肥料尺寸对空气阻力大小的影响，图6显示了对尺寸为4cm肥料的分析结果。通过 对图5和图6的比较说明，撒播范围L0)随着颗粒尺寸的增加而增加，原因如下文所述。当 一个球体的半径增加时，它的体积随着半径的立方增加而增加：而且假设密度不变，它的质 量也随之增加。颗粒的动能与质量成正比，所以动能也随半径立方的增加而增加。范围L(t) 与动能成正比，因此它也与半径的立方成正比。然而，空气阻力与颗粒的横截面积成正比， 所以对球体来说，空气阻力与半径的平方成正比。因而，当颗粒的半径增加时，空气阻力增 加，但不是和动能同等级得增加。所以半径增加导致范围L(t)的增加。一般来说，肥料的质 量随半径的立方增加，同时横截面积也随半径的平方增加。二者的不同导致撒播距离的不同。 相似地，对非球体颗粒来说，大致尺寸面积的增加导致撒播范围的增加。对于宽度4cm的棱柱形(Cw =1.56)，肥料的撒播范围L(t)比忽略风阻时少了 10%。撒播质量：不但对撒播肥料的选择很重要，而且如何来进行撒播也很重要。EN 13083标准规定了必 须估量的参数和撒播质量的检测理论。有机肥料的横向撒播范围和它沿着机车行驶方向的纵 向范围必须要进行测试。各种变化系数都要计算到。撒播材料的使用范围也很重耍。在曰常使用中，操作者会受到肥料纵向撒播很大的影响。肥料是不均匀地装入料箱中， 而且拖拉机或机车的行驶速度和肥料的撒播速度都可以由造作者来控制。这些因素会相当大 地提髙或降低肥料的纵向撒播质量。然而在现有的撒播机的撒播过程中，造作者并不能对侧 面的撒播质量产生任何有意义的影响。即时可能通过微呆改变转速来部分改变，横向撒播主 要取决于撒播方法的类别，使用频率，以及肥料的类型。由于使用了标准中特别规定的统一 测量，操作者对撒播质量的影响就大大降低了。当假设为无方向撒播时，撒播的变化系数不可以大于30%。市场上所有的肥料撒播机都 应肖满足这个条件(Final Draft European Standard,202).变化系数越小，肥料在撒播范围内擻 播得就越均匀。这些数字分析的结果不可能完全地移植到EN 13080标准上，因为这个过程没有考虑机 车的运动。然而，横向撒播可以建立数字分析来建立它的统一性。目的是为了达到在最大撒 播范围中肥料覆兼的最大均匀效果。分析方法：方程4、7和9描述了肥料从离开转子到络到地面的运动过程。质点的为止计算是十分复杂的，所以可以用C语言建立一个程序来解决。这个程序可以给出肥料颗粒沿转子叶片的运动轨迹。图7给除了这个程序的框架图。这个程序也可以算出肥料离开转子的脱离点和肥料即将落到田表的位置。在肥料从转子到田间的过程中，颗粒的速度和方向是影响肥料撒播范围最重耍的因素。颗粒的轨迹和它落到田间的位置必须知道。可以用一个随机数来模拟随机落在转子上的肥料下降，所以在模拟过程中允许肥料落在转子的任何位置。落到转子的肥料数应该是预先设定的。为了防止计算和后面演算结果中的重大错误，必须有足够的肥料颗粒参(2000)与到数字模拟中。这是因为，使用较大数量的肥料可以减少每一个颗粒个体对最后结果的影响。作为边界条件，转子半径(r)，转子转速(n)，转子离地的垂直高度(h)，以及肥料和与它相 对滑动表面间的摩擦系数（）都要输入到程序中。结果将作为广角撒播机的分配模式。在此基础上，这些结果可以用来评估，而且可以改变边界条件。只有单个的一个肥料颗粒可以使用不同边界条件来模拟。结果介绍如下文。数字分析结果肥料颗粒在转子上的运动肥料颗粒依照运动学规律在转子上运动(Meriam，1993)。颗粒在整个过程中经历的时间取决于它落在转子上的位置、转子转速、摩擦系数等等。方程4可以描绘它在转子上的运动。 肥料在转子上的位置取决于摩擦系数，颗粒和转子的最初接触位置，转子转速和半径。图8 显示了肥料颗粒离转子中心距离关于时间的函数。图8也给出了计算中的边界条件。肥料在转子上的运动与转速的关系肥料的运动曲线因为转子的转速不同而不同，转速大都在100到700r/min之间。转速同 样影响到转子的圆周速度，所以肥料脱离转子的速度随着转速的增加而增加。为了了解肥料 颗粒在转子上运动轨迹的变化，模拟系统中速度分别取100，300, 500，和700r/min(图9)。 通过对曲线图(图9)比较可以发现，不管使用哪个转速，颗粒都经历170到345的轨迹路 线。在转速从300到700变化时，肥料颗粒离开转子的位置几乎没有改变，在我们的调査中 只有5%的差异，而肥料的脱离速度的方向有10%的变化。可以看出，当转速低于300时 肥料的轨迹范围比转速在300到700是有大幅度地减少。图9边上给出了分析过程中的常量 参数。肥料在转子上的运动与摩擦系数的关系不同的有机肥料在和与它接触表面滑动时有着不同的摩擦系数。转子与肥料的摩擦系数 对运动的影响是需要考虑的。图10中用了 4种值，从=0.35(Glancey and Hoffman, 1996)到=0.95(Pezzi and Rondelli,2002)。分析显不轨迹长度随着摩擦系数的增加而增加。当=0.35时，肥料绕转子轴心旋转了大约145；=0.55时，角度达到了166；=0.95 时，则到了214。摩擦系数因此会会对肥料在转子上的运动产生影响，也因而会对它离开转子的位置产生影响。肥料颗粒轨迹的变化主要因为摩擦力的大小（）。当转子转速不变时，它们是阻碍颗粒向外运动的力。当摩擦系数从=0.35到0.95变化时，轨迹增加了33%的长度。图10边上给出了分析中的常量参数。肥料在转子上的运动与它初始位置的关系肥料的运动也取决于它刚开始向转子边缘移动时的位置。肥料颗粒的初始位置姓不同的， 因此肥料在转子上的运动也要观测。肥料颗粒的初始位置是从（）到之间。图11中的曲线说明，当肥料的初始位置原理转子中心时，它在转子上的运动距离是如何减少的。如果初始位置在=0.05时，转子必须转过244才可以让肥料达到其边缘(图11)。当r=0.15m时，转子只需要旋转130，而r=0.35m时则只需55（图11）。如此的一个轨迹减少说明抛射力随着半径的增加而增加(方程1)。图11边上列出分析中的常量参数。肥料在转子上的运动与与转子类型的关系紧随着肥料，肥料轨迹过程计算中最重要的元素是转子本身。转子的类型决定所有参数。 如果转子较小，肥料到达其边缘的时间就较少。如果肥料出于较大的转子上，它与转子分离 的时间就更长，而且分离时的圆周速度也更大，这将导致撒播范围的增大。当转子直径增加 时，肥料在转子上的运动时间以及它的运动长度都会增加。这也将导致肥料在转子上时转子 旋转角度的增加(图12)。图12边上列出了分析中的常量参数。肥料在转子上的速度决定肥料撒播范围的参数是它的速度，肥料撒播速度随着速度的增加而增加。肥料颗粒 包括两部分的速度：（1）切向速度(Vt)和它沿着叶片的径向速度(Vr)。Vt取决于它在转子上的位置(半径r)和转子角速度；Vr取决于肥料在转子上的位置，角速度，以及肥料和叶片间的摩擦系数（）。两者合成就是系统速度(V)，V是随肥料离转子中心距离的增加而增加(图13a)。图13说明系统速度(V)与时间的平方成正比(图13)而且几乎与半径成线性关系（图13b）。 从转子上释放出来的系统速度只要取决于径向速度（Vr），切向和径向速度的方向是相互垂直的。肥料颗粒沿径向的加速度是相对不变的，而切向加速度随时间而增加的（图13a）。然后，随着肥料在转子上的移动，每个速度都会增加。切向速度曲线沿着它的整体线性变化，因为颗粒是随着叶片转动的。速度关于转子转速的函数为了确定转子速度对肥料颗粒速度的影响，转速取值从100到700r/min。如果转速降至 100以下，肥料颗粒将停留在起始半径处（）(图14)，因为离心力要比摩擦力小（）。径向速度随着转子速度的增加明显增加，但是曲线的形状仍然保持不变。从100r/min转速的转子上脱离的肥料速度是700r/min时的12%。图14列出了分析中的常量参数。速度与摩擦系数的关系摩擦系数对运动轨迹的长度和径向速度的大小都有影响。摩擦系数（）越大，径向速度(Vr)越小(图15)。摩擦系数变化的影响比转速变化的影响要小。当肥料在转子上处于半径0.4m 时，摩擦系数从0.35到0.95之间变化时，径向速度相应地从8.5m/s变化到14.5m/s(图15)，这个变化率高达40%。图15边上列出了分析中的常参数。速度陆初始位置的变化肥料颗粒的速度也与颗粒在转子上的运动速度有关。肥料轨迹的长度随着初始位置与转 子中心的距离大小的增加而增加（图11）。同时颗粒的速度也在增加（图16），但是没有轨迹长度的增加那么明显(图11)。如果颗粒的初始半径小于0.15m，它脱离转子时的径向速度保持不变。当从小于0.15m增加到0.35m时，径向速度（Vr）从11.5减少到8m/s.当从小于0.15m的间距时，离心力稍大于摩擦力，因此在如此小的半径内速度饿增加很小(从0.1到 0.5m/s)（图4）。在径向方向增加距离时，离心力随之增加，但是摩擦不变，因而会导致速度的变化。图16边上列出了分析中的常量参数。速度随转子尺寸的变化大的转子意味着最在大的离心力，因此肥料颗粒离开转子时会产生大的径向速度()。 图17列出了分析中的常量参数。肥料颗粒离开转子后的运动一旦一个肥料颗粒离开转子，它脱离转子的点和落到田间的位置就确定了。平抛和抛射 运动的物理特性可以用来计算运动轨迹（等式6）。等式7和8说明肥料的撒播范围主要取决于它的速度和脱离是从田表到脱离点的垂直髙度(图4中的h)。肥料颗粒从转子到田表的运动可以在X-Z坐标系统中观测（图18）；这样可以涵盖肥料的整个撒播宽度，因为撒播机是纵向行进的，撒播机是沿着y轴负方向前进的。分析两个转子的撒播机（广角撒播机）时可以使用先前分析转子运动的结论。为了更可能地接近真实情况，分析中使用的撒播肥料颗粒数目为2000。肥料落在转子上的初始位置是随机的(图19)。分析速度关于初始位置的函数(图16)说明，当肥料落在初始半径()小于 0.15m时，肥料脱离转子时的径向速度(Vr)几乎相同。正因如此，肥料颗粒被随机地散布在初始半径大于0.15m的转子部位。图19显示了肥料从转子到田表的过程的分析结果。综上所述，广角撒播机按照扇形方式散布肥料。这个扇形的尺寸大小取决于肥料脱离转子圆周时的速度和位置。在任一肥料颗粒处于转子上的时间段中，转子旋转了 0到200的角位移。图19的模型展示了一个相对均 匀的、宽17m的散布过程。图20列出了每单位横向距离上散布的肥料数量。边缘的肥料数量更多一些，因为在模型的中间部分没有较大的偏差出现。由于在进行模拟计算时并没有考虑空气对肥料颗粒的阻 力，所以边缘的数量更多。由于空气阻力的影响，肥料颗粒的飞行距离随速度的平方而变化。 对我们来说，这意味这颗粒的速度随它离开撒播机的距离的增加而减小，而且肥料的最大散 布范围也减小。在这种情形下，更多的肥料将落在均匀散布区，而边缘则要更少一些(图20)。讨论与结论在耕地上均匀地施肥仍然是一个重要的课题。由于环保问题和为了更高效率地使用肥料，均匀施肥的重要性将持续增加。在现有理论模型的基础上，可以得出如下结论：转子的转速必须在300到700m/inin之间，因为低于300时肥料在转子上的运动轨迹 长度会增加。例如，在100m/min转速时的轨迹长度是300m/min时的两倍。300m/min 以上时肥料在转子上的轨迹长度不会迅速增加。300和700m/min时的差别只有近似 5%。摩擦系数从=0.35增加到0.95时轨迹有33%的增加。肥料颗粒的初始半径必须大于0.15ra。因此，转子上必须安装特殊的分布装置。 分析肥料颗粒在转子上的速度可以得出如下结论：摩擦系数从=0.35增加到0.95时，径向速度(Vr)增加了 40%。肥料在转子转速为100r/min时的脱离速度是700r/min时的12%。将转子的半径增到大于0.5m时，肥料颗粒的速度会增加，但是这样作没有作用，因为颗粒在转子上的运动时间也会增加。广角撒播机通常有两个转子，这样的撒播机虽然也是0. 5m的转子，但是它可以壜加2m的散布宽度.将转子半径增加到0.6m以上时仅仅可以增加2. 5m的散布宽度，但是这样不方便使用，而且也不方便在小路上运输。综分析结果可以得出，肥料的初始位置半径必须大于0.15m，转子的半径最多0.5m，转子的最大转速应该为700r/min，最小应该为300r/min。摩擦对撒播的影响比预期的要小，摩擦系数的变化对肥料在转子上的运动或脱离转子的速度没有明显的影响。然而，我们必须考虑到空气阻力被忽略的事实。空气阻力的响应该通过实验来测量。现在已经得出了肥料颗粒在转子上运动和抛洒距离的理论模型。应该按照相关标准来建 立一个标准，测量肥料的轨道，并且分析肥料的分布情况（Final Draft European Standard,2002）。实际的撒播距离应当比理论要小，而且肥料的撒播应当比现有型号的固体肥料撒播机的效果要好。任务书题 目花生联合收获机的总体设计论文时间20*年2月24日至 20*年6月14日课题的依据、主要内容及要求 随着我国经济的不断发展，农户对农机具的需求日益高涨，尤其需要基本型农机具，其成本低，适用于一个农户家庭的收割作业。需要设计一个精密型花生联合收割机并与12马力手扶拖拉机相配套，有效利用拖拉机动机，并将动力配给皮带传输轮，碎土轮等，成本控制在约1万元左右。本设计是在已研制的机型上改进而来。结合花生的生长情况，参考花生联合收获机的工作原理，采用东风-12作为动力，采用皮带将其和收割机连接。对输送装置的动力来源则采用齿轮箱变速，提高传动效率；对分土轮采用链传动。整个动力传动系统结构紧凑，传动效率充分予以保证。该机将花生的藤苗、果实和土壤三者同时起耕，将藤和果、土一次分离，一次性完成收获。本设计主要以总体设计为重点。参考了各方面的资料，确定了几种方案，最后选用了这种双排输送的设计方案。另外也设计了一些部件的设计。课题的实施的方法、步骤及工作量要求1.查阅有关资料和设计手册，了解国家或行业对花生联合收获机的要求等；2.与12马力拖拉机的动力配套；3.绘制花生联合收获机总体设计5张图纸；4.完成外文翻译汉字3000字以上；5.完成毕业设计说明书（1万汉字以上）。指定参考文献 1陈乐萍，杨胜军.新型花生联合收获机J.农业装备技术，2003，1：20-302于明忠.YN-2型花生联合收获机J.农业知识，2004，16：10-203耿忠申，陈西军.双力“果神”花生联合收获机J.山东农机，2004，8：13-134网泽.华农-1花生联合收果机J.农业装备技术，2005，1：8-85陈书法，李耀明，孙星钊.花生联合收获机挖掘装置的设计研究J.中国农机化，2005，1：10-156闻邦椿主编.机械设计手册M.第一卷.第五版.北京：机械工业出版社，2010毕业设计(论文)进度计划(以周为单位) 第 1 周（20*年2月24日-20*年2月28日）：下达设计任务书，明确任务，熟悉课题，收集资料，上交外文翻译、参考文献和开题报告第 2 周第 3 周（20*年3月3日-20*年3月14日）：制定总体方案，绘制总装图草图。第 4 周第 5 周（20*年3月17日-20*年3月28日）：完成设计方案，拟定振动筛的设计草图。第 6 周第 7 周（20*年3月31日-20*年4月11日）：完成振动筛设计总图及有关零件设计图。第 8 周（20*年4月14日-20*年4月18日）：提交第1-8周的指导记录表和已做的毕业设计内容，由指导老师初审后上交学院第 9 周第 13 周（20*年4月21日-20*年5月23日）：在指导老师指导下修改并完成设计，完成相关设计图纸，同时撰写毕业设计说明书，并提交指导老师初审。第 14 周第 16 周（20*年5月26日-20*年6月14日）：修改毕业设计图纸及说明书，完成后参加毕业答辩。备注注：表格栏高不够可自行增加。此表由指导教师在毕业设计（论文）工作开始前填写，每位毕业生两份，一份发给学生，一份交院（系）留存。Transactions of the ASAEVol. 47(5): 13891404? 2004 American Society of Agricultural Engineers ISSN 000123511389ANALYSIS AND DESIGN PARAMETERS FOR INCLINED ROTORSUSED FOR MANURE DISPERSAL ON BROADCASTSPREADERS FOR SOLID MANUREJ. Duhovnik, J. Benedii, R. BernikABSTRACT. The process of manure spreading is an important step in cultivating land for agricultural production. For this rea-son, the technical process of manure spreading was studied using various methods of scientific research and analysis. Thefocus was primarily on improving the working principles of manure spreading, using socalled wideangle manure spreaders.Results of the analysis of material transport along the rotor revealed the most relevant parameters of manure spreading. Itwas found that friction is not the most important parameter. The initial position of pieces of manure and the rotational frequen-cy of the rotor are far more important. The coefficient of friction between the rotor and pieces of manure has a minimum influ-ence on the tangential speed of a piece on the rotor, which in turn also means a minimum influence on the range of spreading.The initial position of the piece on the rotor has an influence on the tangential speed and travel of the piece on the rotor. Thetangential speed of the piece as it departs from the rotor is influenced by the size of rotor and the rotational frequency of therotor. Included are recommendations regarding the process of constructing a new spreading device or improving existingmethods of spreading manure.Keywords. Broadcast application, Manure spreaders, Mathematical models, Spreading.he spreading of manure is an important step in culti-vating land for agricultural production. Solid ma-nure is a side product of cattle raising, and its directuse for field fertilization is a basic requirement.Different technologies for spreading solid stable manure areknown, but all have certain characteristic parameters in com-mon (Redelberger and Kuyhlmann, 1989). The most impor-tant criterion is the maximum possible spreading capacitywith respect to the relief of the surface to be fertilized. Anoth-er criterion is the maximum possible uniformity of spreadingperpendicular to the line of spreading. In addition, certainparts of the field need to be covered properly, such as the be-ginning and end of manure spreading, blind angles along theline of spreading, corners, large variations in the slope of thefertilized surface, etc. (Frick et al., 2001). These criteriaconstitute the functional requirements for defining the tech-nical process of manure spreading.Once the technical process has been properly defined, theelements of the technical system that will fulfill the set offunctional requirements in their entirety or partially can bedetermined (VDI, 1993). Therefore, individual solutions forthe technical system are, as a rule, adjusted to the technicalprocess. The conceptual design process defined in thismanner for each piece of work equipment allows varioustechnical solutions to be used in the spreading process (?avbiArticle was submitted for review in December 2002; approved forpublication by Power & Machinery Division of ASAE in July 2004.The authors are Jo?e Duhovnik, Professor, and Janez Benedii, PhDStudent, Faculty of Mechanical Engineering, and Rajko Bernik, Professor,Biotechnical Faculty, University of Ljubljana, Slovenia. Correspondingauthor: Prof. Dr. Jo?e Duhovnik, Faculty of Mechanical Engineering,University of Ljubljana, Slovenia; phone: +38614771507; fax:+38612527232; email: joze.duhovniklecad.unilj.si.and Duhovnik, 2001; ?avbi and Duhovnik, 1996). A studyof the literature and the equipment supplied by wellknownmanufacturers that is currently available on the market hasshown several methods of manure spreading, as describedbelow (Frick et al., 2001; Huschke, 2001; Benedii, 2002).The most commonly used method for stable manurespreading uses vertical beaters (e.g., Bergmann spreader, L.Bergmann GmbH, Goldenstedt, Germany) (Huschke, 2001;Benedii, 2002). A manure spreader with horizontal beatersis also manufactured (Pttinger spreader, Alois PttingerMaschinenfabrik GmbH, Grieskirchen, Austria) (Huschke,2001; Benedii, 2002), and recently there has been anincreasing number of manure spreaders with wideanglespreading devices (e.g., Kemper spreader, MaschinenfabrikKemper GmbH, Stadtlohn, Germany, and Bergmann spread-er, L. Bergmann GmbH, Goldenstedt, Germany) (Huschke,2001; Benedii, 2002).Most spreading devices do not permit adjusting thespreading width during operation. In addition, the coefficientof variation (CV) is very large over the entire width ofspreading (Frick et al., 2001). The prEN 13080 standardstipulates CV measurement only for the case of overlapping(Final Draft European Standard, 2002). Thus, no real data areavailable on the spreading of manure over small surfaces(where there is no overlapping). The effective spreadingwidth depends on the type of the manure spreader. If thedesire is to have the surface uniformly fertilized, then manuremust be spread by overlapping the adjacent manure spreaderswaths, since the amount of manure along the edges of onespreading width is very small (Frick et al., 2001). Modernmanure spreaders should provide a more uniform and precisespreading of organic manure over the desired surface. Thiswould increase the operators control, increase the quality ofthe spreading process, and indirectly increase the utilizationof the spread manure.T1390TRANSACTIONS OF THE ASAEManure SpreadingUnspread BulkManureSpreadManureFeeding ofManureShredding ofManureTransport toSpreaderDispersalFigure 1. Division of the manure spreading function into partial functions.Based on our research, it was recognized that the mostimportant parameter of the manure spreading process is thepath that pieces of manure cover from the container of themanure spreader to the fertilized surface. The best method forincreasing the rate of manure application (kg/s) from amanure spreader is by increasing the spreading width.Increasing the rate of manure application by increasing thespeed of the vehicle along the manure spreading line islimited, both with respect to the relief of the ground and theperformance of the tractor and trailer combination. For thisreason, more recent development of manure spreaders hasbeen oriented toward increasing only the spreading width. Anincreasing number of manufacturers produce manure spread-ers with wideangle spreading devices, which achievespreading widths of up to 24 m.While defining the technical process, it was found that themanure spreading width directly depends on the dispersalassembly (essentially an ejection assembly) of the manurespreader. The dispersal assembly functions on the principleof centrifugal throwout and projectile motion (Meriam,1993). If very homogeneous material flow is achieved (ofstable manure, in this case), then the centrifugal dispersalprocess can be defined in great detail. However, in this case,material flow is highly nonhomogeneous. Therefore, theuniformity or maximum possible homogeneity of pieces ofmanure in the material flow prior to entering the dispersalassembly is an important condition for defining the quality ofspreading.In the analysis below, the focus will be on the process ofspreading manure pieces. Our analysis showed a fewimportant parameters that affect the technical solution. All ofthe essential ones (i.e., diameter, rotational frequency, size ofmanure pieces, manure feeding area, and manure spreadingarea) were analyzed and are shown in figures for clearerpresentation. A numerical simulation of the manure spread-ing process was also performed, and the parameters that mostaffect the quality of spreading were identified. Values ofthese parameters that would yield the best results forspreading under predefined conditions, including the unifor-mity of spreading and range, which is the distance traveledby a manure piece thrown by a spreading device, weredetermined.This article provides the theoretical basis for numericcalculation of the flow of pieces of manure from thespreading device onto the fertilized surface. It is not the onlyexample of flow control of artificial manure on hydraulicallydriven rotors that was found in the literature (Hesse andKeuper, 2002); however, a numeric analysis of stable manuredispersal has not been found yet. Hesse and Keuper paid moreattention to the energy consumption and mass flow control.In our case, the flow control and the quality of dispersal as aresult of the flow control of mass pieces were given thepriority (Benedii, 2002).MANURE SPREADING PROCESSTo understand the manure spreading process, it isnecessary to make a detailed and comprehensive descriptionof the process. It is evident from the literature (?avbi andDuhovnik, 2001) that the spreading process can be describedby the functional structure, which shows what partialfunctions need to be performed in order to obtain the desiredresult, i.e., dispersed manure. Division into partial functionshelps us to build the model of numeric analysis of dispersal.A detailed review of the available manure spreader typesshowed that some spreaders differ only in the method ofexecution of the working principle, while in a few, the manurespreading function is divided into the partial functions ofshredding and dispersal of pieces of manure (fig. 1).1. Feeding of ManureStable manure is loaded into the manure spreaderscontainer and needs to be transported to the shreddingassembly. The feeding of manure is performed by a dragapron. This can be driven continuously or in a stepwisemanner. The amount of manure reaching the dispersalassembly depends on the speed of the feeding assembly.2. Shredding of ManureThe pile of manure from the container, which is trans-ported by the feeding assembly to the shredding assembly,needs to be cut into small pieces. The shredding assemblytypically consists of vertically or horizontally positionedbeaters. These mash and cut the manure from the containerinto small pieces and then distribute it evenly over the entirewidth of the spreading device.3. Transport to SpreaderThe partial function of transporting pieces of manureensures that these are moved from the shredding assembly tothe dispersal assembly of the manure spreader. It is importantto guide pieces of manure from the shredding assembly to aspecific area of the dispersal assembly. The pieces that do notfall directly onto the dispersal assembly because of theirinappropriate direction will travel along the metal guard andthen reach this area.4. Dispersal of ManureThe partial function of manure dispersal is the mostimportant one because the functionality of the entire devicedepends on it. The range and uniformity of spreading varies.The dispersal working principle must accelerate pieces ofmanure so that they exit from the spreading device at thehighest possible speed and thus achieve the maximumdispersal length, i.e., range.Most manure spreaders available on the market have thefollowing partial functions: shredding, transport, and dis-persal of manure pieces. The construction of most manurespreaders on the market enables the performance of partial1391Vol. 47(5): 13891404Table 1. Static coefficient of friction of manure on unpainted steel.Coefficient of Friction(dimensionless)Moisture Content(% wet basis)Moist Manure(m)Dry Manure(d)MoistManureDryManure0.670.777970functions with a single unit, but lately manufacturers have in-creasingly been offering manure spreaders (“wideangle”manure spreaders) that consist of only a shredding assemblyand a dispersal assembly. The shredding assembly has re-mained unchanged and has vertical or horizontal beaters,which crush the manure into pieces. These pieces then leavethe manure spreader via the dispersal assembly. In the caseof wideangle manure spreaders, these partial functions areperformed by separate devices that follow one another.THEORY AND PROPERTIES OF THE MANURESPREADING PROCESSIn the manure spreading process, dispersal is based on theprinciple of centrifugal throwout. The mass flow into thecentrifugal ejector is relatively large, so the simulation wasbased on the analysis of the movement of manure pieces.Input data constitute an important part of the analysis.Among the essential factors is the material itself, i.e., stablemanure. It is important to analyze the mechanical andphysical properties of the manure in detail.MECHANICAL AND PHYSICAL PROPERTIES OF ORGANICMANUREAll materials have certain properties that affect theirusability. For numerical simulations, the coefficient offriction (?) between manure and the surface on which it slidesis required. The coefficient of friction (?) depends on thesurface and on the density, moisture content, and type of themanure. Therefore, values for the coefficient of friction (?)vary considerably (from 0.6 to 0.98), for example withstorage conditions (e.g., poultry manure) (Pezzi and Rondel-li, 2002). The coefficient of friction is influenced by manyfactors. In our case, the coefficient of friction was measuredon cattle manure according to the DIN 51131 standard (DINStandards, 1999). The objective was to verify the influenceof manure moisture content on the coefficient of friction (?).Measurements were performed on two types of manure:seven measurements on manure with a high moisture content,and seven on manure with a low moisture content. In bothgroups of measurements, the lowest and highest values wereeliminated. The average value of the coefficient of friction(?) for the remaining measurements was used in thenumerical calculations (table 1).Measurements indicated that cattle manure with a lowmoisture content has a higher coefficient of friction thancattle manure with a high moisture content. The higher thedensity, the greater the moisture content. Similar results havealso been stated in other literature (Redelberger and Kuyhl-mann, 1989; Dohler and Biskupek, 1993; Highnett, 1985).MATHEMATICAL MODELThe selection of the mathematical model is an importantstep in developing the procedure for performing the calcula-tions of a numerical simulation (Benedii, 2002; Hesse andKeuper, 2001; Meriam, 1993), as the correctness of simula-tion results considerably depends on it. A mathematicalmodel was thus made for the case of a rotor in which theblades rotate and the base is stationary (fig. 2). The selectionof this type of rotor enabled numerical simulation of theFigure 2. Top view of blades and rotor surface showing the mathematical model.1392TRANSACTIONS OF THE ASAEmovement of pieces of manure along the rotor surface. Thepieces move in two dimensions; therefore, the mathematicalmodel was also made for a twodimensional space.A manure piece on the rotor surface moves according tothe laws of dynamics (Meriam, 1993). To simplify theanalysis, the problem was solved in a polar coordinatesystem. The motion of a manure piece is fully described byequations in the radial direction (re) in relation to themovement and in the tangential direction (e) in relation tothe movement. Due to the forces acting on the piece as aresult of the rotor rotation and the movement of the piece, themanure piece moves in radial and tangential directionsrelative to the rotor.In the radial direction (re), forces acting on the manurepiece are the following: the force of friction (1TF) betweenthe manure piece and the stationary surface, the force offriction (2TF) between the blade and the manure piece, andthe radial force (ram). In the tangential direction (?e infig. 2), the following forces act on a manure piece: thetangential force (am) and the force of the blade (F), whichcauses the force of friction (2TF) between the blade and themanure piece (fig. 3). The force of friction (3TF) between themanure piece and the stationary surface on which it movesshould also be taken into account.When determining the mathematical model, it is veryimportant to know which influences and properties to takeinto account and which to neglect. It has already beenmentioned that the influence of air resistance will not betaken into account in the numerical simulation. Air resistanceaffects only the magnitude of the results (speed, range, etc.)but not the direction of movement or the selection ordetermination of the parameters having a significant effect onthe quality of manure spreading. Thus, a simplified numeri-cal simulation, which does not take into account the influenceof air resistance, can be used to determine the most importantparameters; these parameters can later be used whendesigning new manure spreading mechanisms. As a whole,organic manure is a nonhomogeneous material, but crushingduring the manure spreading process yields small pieces. Thesizes of pieces also depend on the storage time. Manurestored for relatively long times has a majority of piecessmaller than 12 mm (Pezzi and Rondelli, 2002). It waspresumed that manure with small pieces can be treated ashomogeneous, so the mechanical and physical properties ofall pieces can be considered nearly the same. This is also truefor the cattle manure.Simulation of the movement of a manure pieces wasperformed in two steps: (1) movement of manure piecesalong the rotor surface, and (2) movement of manure piecesafter leaving the rotor.Movement of Manure Pieces Along the Rotor SurfaceA manure piece falls onto the rotor surface (fig. 2), staysin the same location on the rotor until a blade reaches it, andthen starts a circular movement together with the blade. Thecentrifugal force causes the manure piece to move toward theFigure 3. Top view showing movement of pieces of manure along the rotor surface.1393Vol. 47(5): 13891404edge of the blade and rotor. This movement is described bya differential equation in the radial direction (re):rTTamFF=21 (1)and for the tangential direction (e):=+amFFT3 (2)Equations 1 and 2 are solved as differential equations:)(222ggrrr+=+?. (3)The solution of equation 3 is given in equation 4, whichdefines the radius of motion as a function of time, r(t). Themovement of a manure piece along the rotor surface is alsoa function of time, and is influenced by the coefficient offriction (?) of the manure piece on the surface on which itmoves, the initial distance of the manure piece from thecenter of rotation (r0), and the rotor angular velocity (?).Equation 3 is not dependent on the mass of the manure piece,which means that by solving equations 1 and 2, the effect ofthe mass has been eliminated. The results of the calculationsare thus independent of the size of the manure piece. Sizes ofmanure pieces are difficult to determine, as manure is anonhomogeneous material and its properties depend on manyother parameters. A value of 9.81 m/s2 has been used for gra-vitational acceleration (g).The differential equation (eq. 3) is solved for its homoge-neous part (coefficients ?1 and ?2) and nonhomogeneous part(D):112221+=+=?22+=ggD?where g is gravitational acceleration. DeDreDrtrtt+=2112012201)()(? (4)tteDreDr+=211202122011)(r.?(t)v?r (5)Equation 5 describes the variation in speed of a manurepiece with time, in the direction of its movement along theblade. This is the radial speed (v?r in fig. 3), which only appliesFigure 4a. Rotor with respect to the fertilized surface, viewed from front of spreader.Figure 4b. Top view of rotors with respect to the fertilized surface.1394TRANSACTIONS OF THE ASAEto the direction of movement of the manure piece along theblade. There is also the tangential speed (v?t), which dependsonly on the radius (r) and the angular velocity of the rotor (?).The resultant is called the system speed (?), which is thespeed of the manure piece:22rt+=v?v?v? (6)Movement of Manure Pieces After Leaving the RotorFigure 4 shows the position of the rotors when manure isspread rearwards. The spreading device is symmetrical, soonly one rotor will be analyzed. Once a manure piece is onthe rotor edge, the piece has achieved a certain speed. Thisis treated as a vector, as it has both magnitude and direction(r, ?). The position of the point where the piece will fall onthe ground is calculated using the theory of oblique throw andprojectile motion (Meriam, 1993):2221)sin()cos(tgvhtvLtt= (7)whereLt = distance of projectile motion? = angle between the ground and direction of thethrow (fig. 18)ht = maximal height of the throwt = time of the piece in the air.In our case, we have to modify equation 7 because theinitial conditions of the piece are taken into consideration.Equation 7 is defined for the whole trajectory of the inclinedthrow. In our case, only the second part of the inclined throwis taken into consideration, i.e., from the maximum trajectoryheight (? = 0, h = ht) to the ground (h = 0). To calculate therange of the manure piece, the vertical distance of the rotorfrom the ground (h in fig. 4a), the position of the manurepiece manure on the rotor edge, and the magnitude anddirection of the speed of the piece must be known. Thecoordinate system was used as the starting point. The originof the coordinate system is placed in the center of the rotorsFigure 5. Variation of magnitude of the range with the shape for 1 cm ma-nure pieces.axis of rotation and is also used as the center point for calcu-lating the spreading width (for a single rotor) (fig. 4b). Whencalculating the distribution of pieces of manure in the trans-verse direction (x axis), the speed of the manure piece in thex direction will suffice, as this is enough for the analysis ofmanure spreading in the transverse direction.The rotor is positioned horizontally with respect to theground and the direction of acceleration of gravity (g) so thatin equation 7, ? = 0 (fig. 18). The speed vector (v) of themanure piece is the sum of the tangential speed (tv ), whichis equal to the rotor peripheral speed, and the radial speed(rv). For calculating the manure spreading width, only thetransverse speed (xv) of the manure piece is required(fig. 4b). This is the speed component perpendicular to thedirection of travel of the manure spreader. Equation 8calculates the time (t) required for a manure piece to fly offthe rotor and fall onto the fertilized surface, and equation 9calculates the distance (Lx) from the rotor to the point in thetransverse direction where the piece will fall on the ground.In this manner, the position of each manure piece on thefertilized surface can be determined:ght=2 (8)xxvtL= (9)The effect of the force of air resistance on the movementof manure pieces was also theoretically analyzed. Force Fzrdepends on the frontal surface area (A), the speed, and thecoefficient of air resistance (Cw in eq. 10). The value of thecoefficient of air resistance depends primarily on shape. Forbodies such as a sphere, prism, plate, car, etc. coefficients ofair resistance are known, while for more complex shapes theyneed to be determined experimentally. Shapes of manurepieces are unpredictable and random, so we consideredshapes for which the coefficients of air resistance (Cw) areknown, and selected one that would best fit the shapes ofmanure pieces. It was presumed that the air resistance ofFigure 6. Variation of magnitude of the range with the shape for 4 cm ma-nure pieces.1395Vol. 47(5): 13891404manure pieces that have the shape of a prism and a sphere canbe theoretically calculated. As the benchmark for piece size,the size of relatively large poultry manure pieces was as-sumed (Pezzi and Rondelli, 2002). The value for the secondexample was four times as large, as it is more difficult to re-duce the size of cattle manure pieces than it is for poultry ma-nure. The influence of the two shapes on the range of amanure piece was analyzed (figs. 5 and 6). The quadratic lawof air resistance (eq. 10) was used because the speeds of thepieces are relatively large and vary considerably with time.Equation 11 contains the piece mass (mg), which was calcu-lated for a certain piece size and shape:221vCAFwzr= (10)1ln(1)1ln(1)(2vCvtCCtLmCACgw+= (11)Figure 5 shows the difference in ranges for varioustheoretical shapes of manure pieces. The diameter of thesphere and major dimension of the prism used in thecalculations was 1 cm. A prismshaped piece (Cw = 1.56) hasa 30% lower range, L(t), than one with no air resistance. Asphereshaped manure piece (Cw = 0.47) has a 15% greaterrange than one shaped like a prism.The influence of manure piece size on the magnitude ofair resistance was theoretically analyzed, and figure 6 showsthe results for 4 cm pieces. A comparison of figures 5 and 6shows that an increase in piece size increases the range, L(t),and the reason for this follows. As the radius of a sphereincreases, its volume increases with the cube of the radius,and assuming constant density, the mass would also increasewith the cube of the radius. The kinetic energy of the pieceis proportional to the mass, so the kinetic energy wouldincrease with the cube of the radius. The range, L(t), tends tobe proportional to the kinetic energy, so it tends to beproportional to the cube of the radius. The air resistance,however, is proportional to the crosssectional area of thepiece, so for a sphere, it is proportional to the square of theradius. Therefore, as the radius of the piece increases, the airresistance increases, but not as fast as the kinetic energy in-creases, so an increase in radius causes the range, L(t), to in-crease. In general, the mass of a manure piece increases withthe cube of its diameter, whereas crosssectional area in-creases with the square of the diameter. The differences be-tween the two functions cause differences in the throwingdistances of manure pieces. Similarly, for nonsphericalpieces, an increase in the major dimension of the piece causesan increase in the range. For a prism (Cw = 1.56) with a widthof 4 cm, the range of the piece, L(t), was 10% lower than fora piece in which the influence of air resistance was neglected.QUALITY OF SPREADINGIt is not only important that organic manure is selected forspreading, but also how it is spread. Standard EN 13080stipulates the parameters that need to be measured and themeasurement methods prescribed for checking the quality ofmanure spreading with the use of spreaders. The transversedistribution of dispersed organic manure and its longitudinaldistribution with respect to the direction of vehicle travelneed to be measured. The coefficient of variation iscalculated for both cases. The working width of materialspreading is also important.In everyday use, the operator has a considerable influenceon the longitudinal distribution of manure spreading. Themanure is loaded into the container unevenly, and the speedof the tractor or transporter and the travel speed can both bedetermined by the operator. These factors can significantlyimprove or worsen the quality of spreading in the longitudi-nal direction. With existing manure spreaders, however, theoperator cannot have any significant effect on the quality ofdistribution in the lateral direction during the spreadingprocess. Even though this is partially possible by smallchanges in the rotational speed of the spreading device, thetransverse distribution primarily depends on the type ofspreading device, the application rate, and type of manure.Measurements are carried out in accordance with specificrules set in the standard, so the operators influence on thequality of dispersal in much lower.The coefficient of variation for spreading should be nogreater than 30% when simulating unidirectional distribu-tion, or “to and from” distribution. This requirement shouldbe fulfilled by all manure spreaders available on the market(Final Draft European Standard, 2002). The lower thecoefficient of variation, the more uniformly manure will bespread across the spreading width.The results of such numerical simulation cannot beentirely evaluated on the basis of standard EN 13080 becausethe travel of the manure spreader is not simulated. However,numerical simulation can be done for transverse spreading toestablish its uniformity. The objective is to achieve themaximum uniformity of manure coverage across the maxi-mum spreading width.ENTRY OFBOUNDARYCONDITIONSCONTROLOFPARTICLEPOSITIONCHANGES INBOUNDARYCONDITIONSDIAGRAM OFPARTICLE MOVEMENTALONG THE ROTORUANALYSISOF RESULTSDIAGRAM OFPARTICLEPATHFigure 7. Algorithm of the procedure.1396TRANSACTIONS OF THE ASAEFigure 8. Movement of a manure piece as a function of time.SIMULATION METHODOLOGYEquations 4, 7, and 9 describe the movement of manurepieces from their first contact with the rotor until they landon the fertilized surface. Calculations of particle position arecomplex, so a program was developed in the C programminglanguage. The program enables the tracking of the movementof manure pieces along the rotor blade. The algorithm of thisprocedure is shown in figure 7. The program also calculatesthe point at which the piece will leave the rotor and thelocation on the fertilized surface onto which it will land.During the transit of pieces of manure from the rotor to thefertilized surface, the speed and direction of the piece are thefactors that have the greatest influence on the magnitude ofthe range of the piece. Paths of individual pieces of manureand positions of the pieces on the fertilized surface need to beknown. A random number generator was used to simulate thefalling of pieces of manure onto random locations on therotor, so the simulation allowed pieces to fall anywhere on therotor. The number of pieces falling on the rotor was preset.It is important to have a sufficient number of pieces in thenumerical simulation (2000) to prevent major errors in thecalculation and for subsequent evaluation of the results. Thisis because use of a relatively large total number of piecescauses each individual piece to have a small effect on the finalresult.For the boundary conditions, the rotor radius (r), the rotorrotational frequency (n), the rotor vertical distance from theground (h), and the coefficient of friction between the manureand the surface on which it slides (?) are entered in theprogram. The result is a distribution pattern for a wideanglemanure spreader. On the basis of this distribution pattern,results can be evaluated and the boundary conditionschanged. Only one piece was simulated using differentboundary conditions. The results are presented below.Figure 9. Movement of a manure piece for various rotor rotational frequencies.1397Vol. 47(5): 13891404Figure 10. Movement of a manure piece for various coefficients of friction.RESULTS OF NUMERICAL ANALYSISMOVEMENT OF PIECES OF MANURE ON THE ROTORManure pieces travel on the rotor according to the laws ofdynamics (Meriam, 1993). The transit time of a piecedepends on the place on the rotor onto which it falls, the rotorrotational frequency, the coefficient of friction, etc. Themovement of pieces on the rotor is described by equation 4.The position of a piece on the rotor varies depending on thecoefficient of friction, the point at which the piece firstcontacts the rotor, and the rotor rotational frequency andradius. Figure 8 shows the distance of the manure piece fromthe rotor center as a function of time. Boundary conditionsused in the calculations are given in figure 8.Movement of a Manure Piece on the Rotor Based on theRotor Rotational FrequencyThe curve along which a manure piece travels also varieswith respect to the variation of the rotor rotational frequency,which was varied from 100 to 700 min1. Variations of therotor rotational frequency also affect the rotor peripheralspeed, so the speed with which a piece leaves the rotorincreases as the rotor rotational frequency increases. Toobserve changes in the path of a manure piece on the rotor,a simulation of piece travel on the rotor was performed at 100,300, 500, and 700 min1 (fig. 9). A comparison of the graphs(fig. 9) shows that at any rotational frequency, the piece willcover a path ranging from 170 to 345. The point at whichthe piece will depart from the rotor will remain almostunchanged, varying by only about 5% in our investigation,for rotor rotational frequencies from 300 to 700 min1. Thedirection of the velocity of a piece upon release varies byFigure 11. Movement of a manure piece for various initial positions on the rotor.1398TRANSACTIONS OF THE ASAEFigure 12. Movement of a manure piece on the rotor for various rotor sizes.10% for rotational speeds from 300 to 700 min1. It wasfound that the path distances for rotational frequencies below300 min1 were considerably less that those for rotational fre-quencies from 300 to 700 min1. Parameters that were heldconstant in this simulation are listed next to the graph in fig-ure 9.Movement of a Manure Piece on the Rotor Based on theCoefficient of FrictionDifferent types of organic manure have different coeffi-cients of friction between the manure and the surface onwhich the piece slides. The influence of the coefficient offriction on the movement of manure pieces on the rotor wasobserved. Four values of the coefficient of friction were used(fig. 10), from ? = 0.35 (Glancey and Hoffman, 1996) to ? =0.95 (Pezzi and Rondelli, 2002). Simulation showed that thepath length of travel increases with an increase in thecoefficient of friction. At ? = 0.35, the manure piece rotatesaround the axis of rotor rotation for about 145, at ? = 0.55the angle is 166, and at ? = 0.95 it is 214. The coefficientof friction thus exerts an influence on the distance a manurepiece travels on the rotor and consequentially on the pointfrom which it leaves the rotor. The manure piece pathchanges primarily because of a greater force of friction (1TFand 2TF), which opposes movement of the piece outward,while the radial force remains unchanged (eq. 1). An increasein the coefficient of friction from ? = 0.35 to ? = 0.95 resultsin a 33% longer path. Parameters that were held constant inthis simulation are listed next to the graph in figure 10.Movement of a Manure Piece on the Rotor Based onChanges in its Initial PositionThe movement of a manure piece also depends on theplace from which the piece begins moving towards the rotoredge. The initial position of the piece was varied, and themovement of the manure piece on the rotor was observed.The initial position was moved from the rotors axis ofrotation (rinitial = 0 m) outward torinitial = 0.3 m. The graphsin figure 11 show how the distance traveled on the rotor bya manure piece decreases when the initial position is movedaway from the rotors axis of rotation. If the initial positionof the manure piece is at rinitial = 0.05 m, then the rotor mustturn 244 for the piece to reach the rotor edge (fig. 11). Atrinitial = 0.15 m, the piece will need the rotor to rotate over anangle of only 130 to reach the rotor edge, and only 55 atrinitial = 0.35 m (fig. 11). Such a reduction in the path lengthtraveled is a result of an increase in the radial force with anincrease in the radius of the initial position, rinitial, (eq. 1). PaRotational frequencyn = 500min 1Coefficient of friction= 0.6Rotor radiusr = 0.4 mRadius of initialpiece positionrinitial = 0.1 m(a)(b)Figure 13. Speed (?) of a manure piece on the rotor: (a) as a function of time (t), and (b) as a function of radius (r).1399Vol. 47(5): 13891404Figure 14. Radial speed for various rotational frequencies.rameters that were held constant in this simulation are listednext to the graph in figure 11.Movement of a Manure Piece on the Rotor Based on theRotor SizeFollowing the manure, the most important element of thesimulation of manure piece transit on the rotor is the rotoritself. The rotor size affects all parameters. If the rotor issmall, the manure piece will reach the rotor edge in a shorttime. For a manure piece on a large rotor, however, more timeis needed for the piece to depart from the rotor, but theperipheral speed is greater, which yields a longer range. Asthe rotor diameter increases, the manure piece transit time onthe rotor and the path length traveled by a manure piece onthe rotor increase. This causes the angle through which therotor rotates while the piece is on the rotor to increase(fig. 12). Parameters that were held constant in this simula-tion are listed next to the graph in figure 12.SPEED OF MANURE PIECES ON THE ROTORThe parameter that determines the maximum range of amanure piece is its speed. The manure piece range increases asthe manure piece speed increases. The piece has two compo-nents of speed: tangential speed (?t), which depends on theposition on the rotor (radius r) and on the angular velocity of therotor (?), and the transit speed of the piece along the blade,which is the radial speed (v?r) and depends on the position on therotor, the angular velocity, and the coefficient of friction (?)between the manure piece and the rotor and blade surface. Theresultant is the system speed (?), which increases with thedistance from the axis of rotation (fig. 13).Figure 15. Radial speed for various coefficients of friction.1400TRANSACTIONS OF THE ASAEFigure 16. Radial speed for various initial positions.Figure 13 shows that system speed (?) increases approxi-mately with the square of the time (fig. 13a) and almostlinearly with respect to the radius (fig. 13b). The magnitudeof the system speed at release from the rotor primarilydepends on the magnitude of the radial speed (v?r). Thedirections of the radial and tangential speeds are perpendicu-lar to one another. The manure piece acceleration in the radialdirection was relatively constant, while the acceleration inthe tangential direction increased with time (fig. 13a). Later,both speeds increase together as the position of the piece onthe rotor changes. The curve of tangential speed (v?r) is linearin its entirety (fig. 13b) since the piece moves with the blade.Speed as a Function of Rotor Rotational FrequencyTo determine the effect of the rotor rotational frequency onthe speed of a manure piece, the rotational frequency was variedfrom 100 to 700 min1. If the rotational frequency is decreasedbelow 100 min1, the manure piece would be stationary at theinitial radius (rinitial 0.1 m) (fig. 14), as the centrifugal forceis then weaker than the force of friction (rTTamFF21).The radial speed increases evenly with an increase in rotorrotational frequency, but the shape of the curve remainsunchanged. The speed of a piece of manure released at a rotationfrequency of 100 min1 is 12% of the speed of a piece releasedat 700 min1. Parameters that were held constant in thissimulation are listed next to the graph in figure 14.Figure 17. Radial speed for various rotor sizes. The 0.3 m and 0.5 m arrows denote the radius of the rotor.1401Vol. 47(5): 13891404Figure 18. Movement of pieces of manure.Speed versus Coefficient of FrictionThe coefficient of friction affects both the length of thepath traveled and the magnitude of radial speed. The higherthe coefficient of friction (?), the lower the radial speed (v?r)(fig. 15). The influence of changes in the coefficient offriction on the radial speed is smaller than the influence ofchanges in the rotational frequency. When a manure piece onthe rotor is at a radius of 0.4 m, a change in the coefficient offriction from 0.35 to 0.95 corresponds to a change in radialspeed from 8.5 to 14.5 m/s (fig. 15), which is greater than40%. Parameters that were held constant in this simulationare listed next to the graph in figure 15.Figure 19. Spreading of manure pieces. Triangles and diamonds represent pieces applied by rotors 1 and 2, respectively.1402TRANSACTIONS OF THE ASAEFigure 20. Number of manure pieces as a function of the transversal position across the spreading width.Variation of Speed with Variation of Initial PositionThe speed of a manure piece also depends on the time apiece requires for movement on the rotor. The length of themanure piece path decreases with an increase in the distancebetween the initial position of the piece and the rotor center(fig. 11). At the same time, the speed of the piece increases(fig. 16), but not as markedly as the length of the pathdecreases (fig. 11). If the radius of the initial position of themanure piece is smaller than 0.15 m, then the radial speed ofthe piece when it is released from the rotor remainsunchanged. With an increase in rinitial from less than 0.15 mto 0.35 m, the radial speed (v?r) decreases from 11.5 to 8 m/s.Over the interval of rinitial 0.15 m, the centrifugal force isnot significantly greater than the force of friction, so theincrease in the speed of the piece over such a small radius issmall (0.1 to 0.5 m/s) (fig. 14). With an increase in the radius,the centrifugal force increases, but the force of frictionremains unchanged, resulting in a difference in speed.Parameters that were held constant in this simulation arelisted next to the graph in figure 16.Variation of Speed with Variation of Rotor SizeGreater rotor radius means a greater centrifugal force onthe piece, and thus a higher radial speed (v?r) when the pieceleaves the rotor. Parameters that were held constant in thissimulation are listed next to the graph in figure 17.MOVEMENT OF PIECES OF MANURE AFTER LEAVING THEROTOROnce a manure piece has left the rotor, its point of releasefrom the rotor and the location on the ground are known. Thephysical principle of oblique throw and projectile motion wasused for calculating the path (eq. 6). Equations 7 and 8 showthat the range of the manure piece depends primarily on itsspeed and the vertical distance between the place of releasefrom the rotor and the fertilized surface (h in fig. 4).Movement of a manure piece from the rotor to the fertilizedsurface is observed in the xz plane of the coordinate system(fig. 18); this will suffice to cover the entire manure spreadingwidth, since the spreading device moves longitudinally. Thepositive y axis is opposite the direction of travel of the manurespreader.Simulation of manure spreading for two rotors (a wideangle manure spreader) was made using findings fromprevious analyses of manure piece movement on the rotor. Toapproximate the real situation as closely as possible, thenumber of pieces to be spread in the simulation was set at2000. The initial location of a manure piece on the rotor wasrandom (fig. 19). Analysis of speed as a function of the initialposition of the manure piece (fig. 16) showed that the piecesfalling onto points with radii (rinitial) less than 0.15 m have analmost equal radial speed (v?r) when they depart from therotor. Because of this, the pieces were fed randomly onto therotor at locations having radii greater than 0.15 m.Figure 19 shows the results of the simulation of movementof manure pieces from the rotor to the fertilized surface.Viewed from above, the wideangle manure spreaderdisperses pieces of manure in the form of an arc. The size ofthis arc depends on the speed of manure pieces when theyreach the rotor circumference and on the point at which theyleave the rotor. During the time interval for which any givenmanure piece was on the rotor, the rotor rotated through anangle from 0 to 200. The pattern shown in figure 19provides a relatively uniform distribution across the 17 mspreading width.1403Vol. 47(5): 13891404Figure 20 shows the number of pieces applied per unit oflateral distance. The number of pieces is greater along theedges, while no significant deviation occurs in the centralpart of the pattern. The greater number of pieces along theedges is a consequence of not taking account of air resistanceon the manure pieces when calculations were made by thesimulation. The distance traveled by a piece of manure varieswith the square of velocity due to the influence of airresistance. In our case, this means that the velocity of a piecedecreases with the pieces distance from the spreader, and thepieces maximum range also decreases. In this way, morepieces are obtained in the area with a uniform distribution,and fewer ones along the edges (fig. 20).DISCUSSION AND CONCLUSIONSApplication of accurate amounts of fertilizers on culti-vated land continues to be an important issue. For reasons ofenvironment protection and better manure use efficiency, theimportance of application accuracy will continue to grow.Based on the presented theoretical model, the followingconclusions were drawn:? The rotor rotational speed must be between 300 and700 min1 because at values below 300 min1 the pathlength of the piece on the rotor increases. For example,the path is twice as long at 100 min1 as at 300 min1.The path of the piece on a rotor with a rotational fre-quency over 300 min1 does not decrease as rapidly.The difference between 300 and 700 min1 is only mar-ginally over 5%.? An increase in the coefficient of friction from ? = 0.35to ? = 0.95 results in a 33% longer path.? The radius of the pieces initial position must be greaterthan 0.15 m. For this reason, special distri