利用CAD功能进行数控铣削中的物料去除量计算(
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利用CAD功能进行数控铣削中的物料去除量计算(,利用,CAD,功能,进行,数控,铣削,中的,物料,去除,计算
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西安工程大学毕业设计英文译文 原文题目：Removed material volume calculations in CNC milling by exploiting CAD functionality 译文题目：利用CAD功能进行数控铣削中的物料去除量计算 - 32 -摘要加工过程中的材料去除体积计算在各种铣削仿真应用中都很重要，包括材料去除率估算和加工力计算。在本文中，为此目的提出了两种不同的方法，即Z maps和具有实体模型的布尔运算。Z-map方法很简单，但会产生大文件，需要复杂的例程来提供可接受的准确性。在工具和工件的精确实体模型之间的布尔运算在容易获得的CAD系统应用程序编程接口上实现。除了与精度水平相关的计算负荷之外，通过工具的一次旋转需要足够数量的插值点才是值得信赖的。沿着刀具路径在特定的关注点使用是实用的。关键词：材料量；铣；Z地图；实体模型；刀具路径；计算机数字控制*英文原文摘自：Benardos P, Vosniakos G. Removed material volume calculations in CNC milling by exploiting CAD functionalityM. 2017. 1 介绍用几何建模的方法研究铣削等加工过程中去除的材料体积是必要的。其目的主要涉及物料去除率的计算、工件表面静止材料分布的映射、表面形貌、可能的碰撞确定以及切削力模型的开发。一般来说，可以区分两种主要的方法，即基于实体模型的方法和基于点的方法。可以使用实体造型来表示刀具和工件，使用构造性的实体几何(Spence和 Altintas，1994；Imani 和 Elbestawi，2001)或边界表示(El-Mounayri等人，1998年)。各种变化包括使用工具的等分切片(Merdol和Altintas，2008年)，被切削刃扫过的包络体积(Sun等人，2009年；Layegh等人，2012年)。这种方法在确定它们的公共(相交)体积时具有相当高的精度，但计算量很大，即O(N4)表示CSG和O(N1.5)表示B-REP，其中N是刀具运动段的个数(Bohez等人，2003年)。此外，体积表示对较低计算成本的变化包括：再分配(Karunakaran等人，2010年)和(Zhang等人，2009年)。基于点的方法特别包括Z-MAP和离散矢量方法。Z贴图由Anderson（1978）介绍了研磨，并进一步开发和一直持续到最近（Baek和Ko，2008）。点网络通常被创建,在工件的底部和与这些工件的任何一个相对应的工件高度上,点存储在阵列中。当切削刀具在点网络上移动时,只有在移除材质时，才会更新相应的高度。这很容易实现具有低的计算成本O(n)并且允许材料去除速率容易地估计。然而，精度在很大程度上取决于网络密度和工件上的工件。定向（Choi和Jerard，1998）。在类似上下文中，各种点和基于曲线的也使用了离散化方法(Bouzakis等，2003)。离散矢量方法代表工件表面在其不同点使用法向矢量(Chapel,1983年)。后者与切削刀具运动包络相交。最终长度如果向量对应于去除小于或大于预期的材料，则该向量对应于去除更少或更多的材料。分别在表面之上或之下延伸。该方法适用于5轴研磨并允许材料去除偏离理想，但不直接计算。材料去除速率。在类似的上下文中，已经使用了各种几何分析方法，使用过（Tsai和Liao，2010）。可以使用形式Ft = k * MRR / V的等式基于体积模型计算平均切削力，其中Ft是切削力的切向分量，k是特定切削力，MRR是材料去除率，V是 切割速度。 材料去除率是在工具的一个或多个转数上计算的，并未考虑精确的切削刃几何形状或沿切削刃的进入和退出角度的变化。 瞬时切削力可以通过将切削工具离散成切片并考虑切屑厚度以及长度（轴向）和宽度（周向）的机械模型来计算（参见例如Merdol和Altintas，2008）。在这项工作中，如第2至第4节所述，研究了刀具每转去除的材料体积的计算，以便用于力预测神经模型(Benardos 和 Vosniakos，2014年)。这就构成了一个挑战，特别是在工件上存在着由于先前的再加工道次而存在的情况下，这使得在一个系统中对工件表面的表示是不可缺少的。此外，移除的材料应在数控加工的任何点上识别，通常使用下一节简要描述的位置插值。结果载于第5节，结论载于第6节。2 分点计算对应于铣削路径的g码逐行读取。切割运动受到歧视，进一步按插补类型分类，见图1。目前所支持的是直线(直线型)和圆形型(等分和反直)，但任何对机器专用插补类型的扩展都是可能的，例如样条、等高线等。图1.并行点计算逻辑在每次切割运动中，记录两个连续点的坐标，如果连接它们的线的长度大于一个值，则执行相应的插值例程，以便计算形成连续等长段的适当数量的中间点(除了-可能是最后一个)。线性和圆形插值计算被认为是微不足道的，因此这里没有给出它们。相应的例程作为MATLAB应用程序编写。现有点和计算点的所有坐标都存储在文件中。图2使用z映射的材料去除计算逻辑3 用z-映射法计算移除材料z-map方法是根据图2所示的步骤使用API和MATLAB实现的。利用api程序，在平面上的网络点上生成原点的射线，得到其与工件一个或多个表面的交点坐标，计算出z图。图3(b)-3(d)给出了图3(a)所示部分的相关z-映射和点网络的不同密度的例子。值得注意的是，z映射方法的精度取决于点网络的密度，这一点对于曲面来说更加突出。这会导致非常大的文件，并对计算时间产生不利影响。对于图3(a)所示的对应于尺寸为10021034的盒的部分和所试验的三种网络密度，见图3(b)-3(d)，相应的文件大小为165，约为绝大部分，确认了文件大小为点网络距离的平方。因此，约为100m的距离(这是可接受的)将导致文件大小为67 MB，而距离为10m，这无疑是足够的，这将需要6.6GB的存储空间，这在任何标准下都是明显过大的。图3（a）样本部分（股骨植入）（b）点网络距离的Z-映射等于2毫米（上），1毫米（中）和0.5毫米（下）考虑的一种解决方案是只在局部计算z-映射，即只在刀具周围计算，以减少文件大小和计算量。然而，由于z映射不能代表工件的整体，因此在已经从其中移除材料的区域不能更新z映射。另一种原因是需要将刀具加工成有限厚的切片，以便详细计算每一种刀具的进出角，并由此计算出刀具与工件之间的“周边”啮合面积。表1 z图与实体模型的精度比较图4（a）带平面和球头立铣刀的扁平工件和（b）带球头立铣刀的自由形状粗加工工件（参见在线版本的颜色）请注意，Z贴图仅包含曲面上的边界点，而不包括内部点手上的固体。该缺失信息需要通过例程来添加,该例程以离散Z级的点填充有界体积。然而，即使是这样做问题，与填充点的Z步骤相关。因此，如果该步骤是大的精度，因为切片将不足够薄以保证它。相反，如果该步骤太小了将落在特定切片中的边界点的数目。可能太小，无法保证进入和退出角度计算的准确性。(或)使用Z图与计算面积偏差的指示使用实体建模功能进行的计算，如在下一节中详细说明的那样，如表1所示。相关部件/工具情况如图4所示。如果零件形状简单，精度很高，但在零件形状复杂的情况下，误差相当大。因此，出于效率和准确性的原因，Z-MAP方法被放弃以有利于下面描述的实体建模方法。4 用实体模型计算一移除材料这个方法是在连续的步骤中执行的，如图5所示。其主要思想是将工件在连续位置上对应于刀具的体积进行切割。图5使用实体模型的刀具每转移料量的计算逻辑因此，有必要对工件进行精确的描述，此外，对刀具进行尽可能精确的建模也是非常重要的。工件的初始形状可以在CAD程序中使用标准技术进行设计。但是，如果需要使用中间形式，例如在一个阶段之后的工件，则可以使用由CAM程序导出的表示形式，如CAM程序。在任何CAD系统中，这都很容易转换成另一种格式，包括在这种情况下。刀具也采用标准实体造型技术。在这种情况下，两种切削工具都是中性的，即有四台平头铣刀和一台带有两台的球头铣刀，参见图6。参数化造型是指用直径、长度、凹槽或角、轴角和径向角等几个参数值来控制刀具的形状，从而避免从零开始对刀具进行完全建模。图6(a)平头铣刀和(b)球头铣刀模型要计算移除材料体积的位置是通过沿着指定零件的线性运动和相应的刀具的旋转运动而得出的。在每一步中，执行刀具与工件的相交。通过将对应于刀具完全旋转的位置数的所有零件的公用体积相加，计算出从工件到工件的总物料体积。例如，当线性进给f=150 mm/min，主轴转速s=5,000时，刀具在一次旋转中所覆盖的距离为l=f/s=0.03 mm/rev，该距离分为四段，步长为0.0075 mm。得到的结果是对工件表面的足够精确的表示。在实现方面，通过在全旋转范围内将工具在连续的位置上移动，使用布尔值和交点，以及API的指令，实现了并行计算。在旋转过程中去除的体积可以在任何一点上计算，但是在几百个或几千个点这样做是不实际的。相反，这种计算只在一小部分感兴趣的点进行，例如，预计移除材料的体积会发生突然变化。当然，随后可以在这些点的附近进行更详细的计算，即在它们之间的较小距离上使用更多的点。例如，从彼此之间最大距离为0.0075的点集合中，可以选择对应于单元长度的特定部分的那些点，以便执行体积去除计算。在其余点，工件只需要进行更新，以便反映研磨后的状态。在这些位置，由于速度和效率的原因，使用切割工具的简化模型来执行工件更新。在这种情况下，使用气缸来表示平轧机和具有半球形端部的气缸，以表示球磨机。再次，使用布尔运算符执行更新操作，即表示简化工具的连续位置的实体体积与来自工件实体的实体体积的结合，以反映工件的形状，直到要执行关于移除体积的详细计算的特定点之前的点为止。按照图7所示的步骤，这也是用SolidworksAPI执行的。图7使用实体模型的工件更新逻辑在实现方面，所有的Solidworks API例程都在大约500行的VisualBasic脚本中使用。它们与选择点的应用程序通信，参见第2节，使用文件而不是系统内存。5 结果与讨论空气动力叶片被检查为第一示例，特别是其压力侧，见图8(a)。该侧面用20mm直径的平端研磨机在填充策略之后粗糙化，即所执行的切割工具在XY平面上在曲折状图案中线性移动，通过的水平距离为工具直径的50，垂直距离为0.75mm。叶片的粗糙形状如图8(b)所示。用直径为6毫米的球头铣床沿着x轴进行精加工，每道次加工都是在垂直(平行)平面上进行的，见图8(C)。传球距离是可变的，以10m为限。进给量设定为200 mm/min，主轴转速设定为3000 mm/min。模拟精加工快照如图8(d)所示。图8空气动力，(A)外形(B)(C)精加工策略(D)光洁度(颜色见网络版)计算新点的插补步骤为16m，每周刀件的体积去除量每10 mm计算一次。与前1500毫米长度对应的结果(150点)如图9所示。图9每转150分的精加工量计算第二个例子涉及图10(a)所示的股骨头，如图3(A)所示。它是用直径为20 mm的平端磨机进行的，采用垂直孔型距离设置为1mm。采用直径为6mm的球头铣床，在垂直步长可变的水平平面上对零件进行仿形加工，使工件高度保持在10m以下，见图10(b)和10(c)。饲料和速度再次设定为200mm/min和3,000RPM。图10股骨头，（a）粗加工（b）精加工（c）精加工策略（见在线版本的颜色）材料去除的计算条件与前面的例子完全相同，但是对于更深的，而不是1500毫米的长度和结果如图11所示。图11股骨头精加工刀具路径330点的每次转速计算的体积在这两个例子中，很明显，每一次旋转的体积变化很大，这是因为表面的曲率是相当大的，同时也是由于在过渡阶段形成的。这些都是比较接近的情况下，在交叉的再加工，而在情况下，在股骨头的情况下，更接近的主要方向。计算的速度和精度几乎完全取决于表示工件模型的文件的尺寸和质量。它以STL格式表示，如同在此工作中的情况一样，这减少了构成模型的三角形数目。然而，后者可以用任何其它可用的格式表示，例如SAT、步骤等，利用部件的边界表面的闭合形式表示。然而，该格式被认为适合于表示零件的等高线形状，特别是因为大多数凸轮系统都支持它作为一种通用格式来表达零件从最初到最终形状的中间形式。以(T)为单位，记录了t=*n-，其中=0.361，=130.85的亚型(N)数与计算时间之间的线性关系。这些计算可进一步用于估算或估计适当校准后的切削力。根据该方法中的芯片尺寸，精确地计算了轴向和周边接合尺寸。然而，芯片厚度只是比较好的，但是当一个完全旋转的点数增加，通常是24到12个，对应于15到30的旋转时，近似得到了更好的效果。这在技术上是可能的，但只适用于少数部位。6 结论由于各种原因，需要计算铣削加工中刀具每转去除的材料量，以估算物料去除率和铣削作业的经济性，计算切屑尺寸，估算切削力等。Z-map方法原则上很简单，但会产生大文件，或者为了纠正这个问题，它需要变得更加复杂，从而失去其简单性。主要的当需要更新工件以反映移除的材料时，会出现困难在加工期间和由于刀具的基于曲线的表示。这是基于精确的刀具模型和工件之间的布尔运算的实体建模。除了与精度水平相结合的计算负荷外，还需要通过一次旋转获得足够数量的可信赖点。在特定的兴趣点上使用是可行的，但对后者的整体也适用。在任何情况下，数控加工都需要根据相关的g码计算，并随后使用插值方法。需要对移除材料进行详细体积计算的兴趣点，应根据与这些计算计划的进一步使用有关的适当标准手动或自动选择。致谢这项工作由欧洲联盟-欧洲社会基金和希腊国家发展部研究和技术总秘书处出资，项目编号为01131号，75%由欧洲联盟-欧洲社会基金供资。参考文献1 Anderson, R.O. (1978) 数控加工中的碰撞检测与消除，计算机辅助设计, Vol. 10, No. 4, pp.231237.2 Baek, D.K. and Ko, T.J. (2008) 基于数控验证模型的自由曲面调度“国际机床与制造杂志”, Vol. 48, No. 2,pp.163172.3 Benardos, P. and Vosniakos, G-C. (2014) 离线柔性进给和速度在数控加工中，利用专用切削力。第二部分：“工程制造学报”， Vol. 228,No. 6, pp.878892.4 Bohez, E.L.J., Minh, N.T.H., Kiatsrithanakorn, B., Natasukon, P., Ruei-Yun, H. and Son, L.T.(2003) 五轴刀具路径验证的缓冲区扫描平面算法，计算机辅助设计， Vol. 35, No. 12, pp.11291142.5 Bouzakis, K-D., Aichouh, P. and Efstathiou, K. (2003) 自由曲面铣削加工的切屑几何形状、切削力和粗糙度，球头铣刀，“国际机床与制造杂志”， Vol. 43, No. 5, pp.499514.6 Chapel, I.T. (1983) 利用矢量模拟数控铣削去除的材料，计算机辅助设计， Vol. 15, No. 3, pp.156158.7 Choi, B.K. and Jerard, R.B. (1998) 高级表面加工，理论与应用，学术研究，Kluwer Academic Publishers, London. 8 El-Mounayri, H., Spence, A.D. and Elbestawi, M.A. (1998) 铣削过程模拟一种通用的基于固体的范例，制造科学与工程杂志，Vol. 120, No. 2, pp.213221.9 Imani, B.M. and Elbestawi, M.A. (2001) 球头铣削作业仿真，制造科学与工程杂志，Vol. 123, No. 5, pp.177184.10 Karunakaran, K.P., Shringi, R., Ramamurthi, D. and Hariharan, C. (2010) 基于八叉树的利用瞬时力模型优化磨削进给速度的数控仿真系统，先进制造技术国际报， Vol. 46, Nos. 58, pp.465490.11 Layegh, S.E.K., Erdim, H. and Lazoglu, I. (2012) “5轴研磨”中复合自由曲面的“离线力控制和进料速率调度”, Procedia CIRP, Vol. 1, pp.96101.12 Merdol, S.D. and Altintas, Y. (2008) 三轴铣削加工与优化，国际机床与制造杂志，Vol. 48, No. 10, pp.10631071.13 Spence, A.D. and Altintas, Y. (1994) 基于实体的铣削过程模拟与规划系统，“工业工程学报”，Vol. 116, No. 1, pp.6169.14 Sun, Y., Ren, F., Guo, D. and Jia, Z. (2009) 加工表面球头铣削中切削力的检验和实验验证，国际机床与制造杂志，Vol. 49, No. 15, pp.12381244.15 Tsai, C.L. and Liao, Y.S. (2010) 利用几何分析、先进制造技术国际期刊球端磨中切削力预测；Vol. 46, Nos. 58, pp.529541.16 Zhang, L., Feng, J., Wang, Y. and Chen, M. (2009) 基于几何和模型的自由曲面加工调度策略，国际先进制造技术杂志，Vol. 40, Nos. 1112, pp.11911201.Removed material volume calculations in CNC millingby exploiting CAD functionalityAbstractMaterial removal volume calculations in machining processes areimportant in a variety of milling simulation applications, including materialremoval rate estimation and machining force calculation. In this paper twodifferent approaches are presented to this end, i.e., Z-maps and Booleanoperations with solid models. The Z-map method is simple but results in largefiles and needs sophisticated routines to render acceptable accuracy. Booleanoperations between accurate solid models of the tool and the workpiece isimplemented on readily available CAD system application programminginterface. Beside the computational load which is bound to the accuracy level,it requires a sufficient number of interpolated points through one revolution ofthe tool to be trustworthy. It is practical to use at particular points of interestalong the toolpath. Keywords: material volume; milling; Z-map; solid model; toolpath;computer-numerical control.1 IntroductionThe volume of removed material in machining processes such as milling is necessary in studying these processes by geometric modelling approaches. The aims concern, among others, material removal rate calculation, mapping of rest material distribution on the workpiece surface, machined surface topography visualization, possible collision determination as well as cutting force model development.In general, two main classes of approaches may be distinguished, namely those based on solid modelling and on point-based discretization. Solid modelling may be used to represent both the cutting tool and the workpiece, using constructive solid geometry (Spence and Altintas, 1994; Imani and Elbestawi, 2001), or boundary representation (El-Mounayri et al. 1998). Variations include use of discretized slices of the tool (Merdol and Altintas, 2008), envelope volume swept by the cutting edges (Sun et al., 2009; Layegh et al. 2012). This method offers considerable accuracy in determining their common (intersection) volume but at considerable computational cost, namely O(N4) for CSG and O(N1.5) for B-rep where N is the number of cutting tool motion segments (Bohez et al., 2003). In addition, volume representation variations to lower computational cost include octrees (Karunakaran et al., 2010) and dexels (Zhang et al., 2009).Point-based methods include notably Z-map and discrete vector methods. Z-maps were introduced by Anderson (1978) regarding milling and were further developed and continually used until recently (Baek and Ko, 2008). A point network is created usually on the base of the workpiece and the workpiece height corresponding to any of these points is stored in an array. As the cutting tool moves over the point network the corresponding height is updated only if material is removed. This is easy to implement has low computational cost O(N) and allows material removal rate to be readily estimated. However, accuracy depends heavily on the network density and on workpiece orientation (Choi and Jerard, 1998). In a similar context, various point and curve-based discretization methods have been used, too (Bouzakis et al., 2003). Discrete vector method represents workpiece surface at its various points using normal vectors (Chapel, 1983). The latter are intersected with the cutting tool motion envelope. The final length of the vector corresponds to removal of less or more material than anticipated, if the vector extends above or below the surface, respectively. This method is suited to 5-axis milling and allows material removal deviations from the ideal, but not direct calculation of material removal rate. In a similar context, various geometric analysis methods have been used, too (Tsai and Liao, 2010).Mean cutting forces may be calculated based on volumetric models using an equation of the form Ft = k * MRR / V, where Ft is the tangential component of the cutting force, k is the specific cutting force, MRR is the material removal rate and V is the cutting speed.Material Removal Rate is calculated over one or more revolutions of the tool and does not take into account either the exact cutting edge geometry or the variation of entry and exit angle along the cutting edges. Instantaneous cutting forces may be calculated by mechanistic models discretizing the cutting tool into slices and taking into account chip thickness as well as length (axial direction) and width (peripheral direction) (see e.g. Merdol and Altintas, 2008).In this work, calculations of the material volume removed per revolution of the cutting tool are investigated, as reported in sections 2 to 4 in order to be used in force predictive neural models (Benardos and Vosniakos, 2014). This constitutes a challenge, especially in the presence of scallops on the workpiece owing to previous roughing passes, which make representation of the workpiece surface in a CAD system indispensable. In addition, the material removed should be identified at any point along the CNC toolpath, typically using position interpolation as briefly described in the next section. Results are presented in Section 5 and Conclusions in Section 6.2 Calculation of toolpath pointsThe G-code corresponding to the milling path is read off an ASCII file line by line. Cutting movements are discriminated and further categorized by type of interpolation see Figure 1. Currently, straight lines (G01) and circular lines (G02 and G03 for clockwise and anti-clockwise) are supported, but any extension to encapsulate machine specific interpolation types is possible, e.g. spline, ellipse, hyperbolae etc.Figure 1 Toolpath point calculation logicWithin each cutting movement the coordinates of two consecutive points are registered and if the length of the line connecting them is larger than a predefined value, then the corresponding interpolation routine is executed in order to calculate the appropriate number of intermediate points that form consecutive segments of equal length (except for - possibly - the last one). Linear and circular interpolation calculations are considered to be trivial and for this reason they are not presented here. The corresponding routines were written as a Matlab application. All coordinates of both existing and calculated points are stored in a file.Figure 2 Material removal calculation logic using Z-maps3 Removed material calculation by Z-mapsThe Z-map method was implemented using Solidworks application programming interface (API) and Matlab according to the steps depicted in Figure 2.The Z-map was calculated by exploiting the RayIntersection routine of Solidworks API, creating rays with their origin on network points on XY plane and obtaining their intersection coordinates with one or more surfaces of the workpiece. Examples of relevant Z-maps for the part shown in Figure 3(a) and different densities of the point network are given in Figures 3(b)-3(d). It is conspicuous that accuracy of the Z-map approach depends on the density of the point network, this need being more pronounced for sculptured surfaces. This results in very large files and has an adverse effect in computation time. Indicatively, for the part shown in Figure 3(a) corresponding to a bounding box of size 100 x 210 x 34 mm3 and the three network densities that were tried, see Figures 3(b)-3(d), the corresponding file sizes were 165 kB, 651 kB and 2,680 kB, confirming proportionality of the file size to the square of the point network distance. Thus, a distance of the order of 100 which is marginally acceptable results in a file size of 67 MB and a distance of 10 m, which would be most certainly adequate would need 6.6 GB storage space, which is clearly excessive by any standards.Figure 3(a) Sample part (femoral implant) (b) Z-maps for point network distance equal to 2 mm (up), 1 mm (middle) and 0.5 mm (down)A solution that was considered was to calculate Z-maps only locally, i.e. only around the cutting tool in order to reduce file size and computational load. However, in this way the Z-map could not be updated at regions from which material had already been removed, since the Z-map could not represent the whole of the workpiece.A further hindrance was due to the need to discretize the tool into finite thickness slices so as to calculate in detail the entry and exit angles for each one of them and from there the peripheral area of engagement between tool and workpiece.Table 1 Accuracy of Z-map compared to solid modellingFigure 4(a) Flat workpieces with flat and ball end mill and (b) freeform roughed workpiece with ball end mill (see online version for colours)Note that the Z-map includes only boundary points on surfaces and no internal points of the solids at hand. This missing information would need to be added via a routine that would fill the bounded volume with points at discrete Z levels. However, even this was problematic, related to the Z-step of filling points. Thus, if the step was large accuracy was reduced since the slices would not be thin enough to warrant it. On the contrary, if the step was small the number of boundary points that would fall in the particular slice might be too small to guarantee accuracy of entry and exit angle calculation. An indication of the deviations of the calculated area using Z-maps compared to the calculations performed using solid modelling functions, as detailed in the next section, is given in Table 1. The relevant part/tool cases are depicted in Figure 4.If part shape is simple accuracy attained is high, but in cases where part shape is complex the error is considerable. Thus, for reasons of both efficiency and accuracy the Z-map approach is abandoned in favor of the solid modelling approach which is described next.4 Removed material calculation by solid modelsThis method is conducted in successive steps as shown in Figure 5. The main idea is to subtract the volume corresponding to the cutting tool from that of the workpiece at successive positions.Figure 5 Calculation logic of material removal per revolution of the cutting tool using solid modelsTherefore, it is necessary to have an accurate representation of the workpiece and, in addition, it is important to model the cutting tool as precisely as possible. The initial shape of the workpiece may be designed in the CAD program using standard techniques. However, if an intermediate form needs to be used, such as the workpiece after a roughing stage, an STL representation may be used, as derived by a CAM program, such as SolidcamTM. This is easily converted into a Brep format in any CAD system, including SolidworksTM in this case.The cutting tool is modelled using standard solid modelling techniques, too. Two cutting tools were parametrically modelled in this case, namely a flat end mill with four flutes and a ball end tool with two flutes, see Figure 6. Parametric modelling consists in controlling the shape of the tool by just a few parameter values, i.e. diameter, length, flute or helix angle, axial and radial rake angle, thus avoiding complete re-modelling of the tool from scratch.Figure 6(a) Flat end mill and (b) ball end mill modelsThe positions at which removed material volume is to be calculated are derived by following the linear movement along the designated toolpath and corresponding rotary stepwise movement of the tool. At each step an intersection of the tool and the workpiece is performed. By adding up the common volume of all intersections for the number of positions that correspond to a full rotation of the tool, the total material volume to subtract from the workpiece is calculated.For example, for linear feed f = 150 mm/min and spindle speed S = 5,000 RPM, the distance covered by the tool in one revolution is l = f / S = 0.03 mm/rev. Discretising this distance into four segments yields a step distance of 0.0075 mm.The result is an accurate enough representation of the workpiece surface. In terms of implementation the calculation is implemented in SolidWorks by moving the tool in the successive positions within a full rotation and by using Boolean subtraction and intersection as well as the MassProperty. Volume commands of the Solidworks API.The volume removed during a revolution can be calculated at any point of the toolpath, but it is not practical to do so at hundreds or thousands of points. On the contrary, this calculation is performed only at a small set of points of interest, e.g. where abrupt changes in volume of removed material are expected. Of course, it is possible to subsequently perform the calculation in the neighbourhood of such points with greater detail, i.e., using more points at respectively smaller distance between them. For example, from the set of points at a maximum distance of 0.0075 from each other it is possible to choose those corresponding to a specific part of the toolpath of 10mm length in order to perform the volume removal calculations.At the rest of the points the workpiece needs to only be updated in order to reflect the state it is at after milling is performed. At these positions, workpiece updating is performed using a simplified model of the cutting tool for reasons of speed and efficiency. In this case a cylinder is used represent the flat endmill and a cylinder with hemispherical end to represent the ball endmill.Again, updating operations are performed using Boolean operators, i.e. a union of the solid volumes representing the consecutive positions of the simplified tool and subtraction from the workpiece solid, in order to reflect the shape of the workpiece up to the point of the toolpath preceding the specific point at which the detailed calculation concerning removed volume is to be performed. This was performed with Solidworks API, too, according to the steps depicted in Figure 7.Figure 7 Workpiece updating logic using solid modelsIn terms of implementation, all Solidworks API routines were used in Visual Basic scripts of about 500 lines. These communicate with the toolpath point selection application, see Section 2, using files rather than system memory.5 Results and discussionAn aerodynamic vane is examined as a first example, in particular its pressure side, see Figure 8(a). This side was roughed using a flat end mill of 20 mm diameter following a hatch strategy, i.e. the cutting tool performed linear moves in a meander-like pattern on the XY plane, the horizontal distance of passes being 50% of the tool diameter and the vertical distance being 0.75 mm. The roughed shape of the vane is shown in Figure 8(b).Finish machining of the vane was performed with a ball end mill of 6 mm diameter following a profiling strategy along X axis, each pass being performed on the vertical (YZ) plane, see Figure 8(c). The distance of passes was variable so as to limit scallop height to 10 m. Feed was set to 200 mm/min and spindle speed was set to 3,000 RPM. A simulated finish machining snapshot is shown in Figure 8(d).Figure 8 Aerodynamic vane, (a) shape (b) roughed (c) finish machining strategy(d) finish machined (see online version for colors)Interpolation step for calculating new points along the toolpath was 16 m and the volume removed per revolution of the tool was calculated every 10 mm of toolpath length. The results corresponding to the first 1,500 mm of toolpath length (150 points) are shown in Figure 9.Figure 9 Volume per revolution calculations at 150 points of the toolpath for vane finishingA second example concerns the hemi-spherical head shown in Figure 10(a) of a femoral prosthesis shown Figure 3(a). Its roughing was performed with a flat end mill of 20 mm diameter using hatching with vertical distance of passes being set at 1 mm.Finish machining was conducted with a ball end mill of diameter 6 mm following profiling of the part on horizontal planes with a variable vertical step distance so as to keep scallop height below 10 m, see Figures 10(b) and 10(c). Feed and speed were again set to 200 mm/min and 3,000 RPM.Figure 10 Femoral head, (a) roughed (b) finish machined (c) finish machining strategy (see online version for colors)Figure 11 Volume per revolution calculations at 330 points of the toolpath for femoral head finishingIn both examples it is obvious that volume per revolution varies widely, due to both the curvature of the roughed surface being machined and to the scallops being formed in the roughing stage. These scallops are machined in the case of the vane across the roughing toolpath, whereas in the case of the femoral head along the roughing toolpath main direction.Speed and accuracy of the calculation depend almost entirely on the size and quality of the file representing the workpiece model. It this is expressed in stl format, as was the case in this work, this reduces to the number of triangles constituting the model. However, the latter can be expressed in any other format available, e.g. sat, step etc, making use of a closed form representation of the boundary surfaces of the part.Nevertheless, the stl format is considered suitable for expressing the roughed shape of the part not especially since most CAM systems support it as a universal format for expressing intermediate forms of the part from its initial to its final shape. Indicatively, a linear relationship between computation time in msec (t) and number of triangles (n) was recorded of the type: t = a * n - p, where a = 0.361 and p = 130.85 on an Intel i5TM processor.These calculations may be further used for MRR estimation or for estimation of the cutting forces after suitable calibration. In terms of chip dimensions within this method the axial and peripheral engagement dimensions are accurately calculated. However, chip thickness is only approximated, but the approximation gets better as the number of discretization points in a complete revolution is increased, typically 24 to 12, corresponding to a rotation of 15 to 30. This is technically possible but is practical only for a few positions.6 ConclusionsThe material removed per revolution by the cutting tool in end milling operations needs to be calculated for various reasons, among others, to estimate material removal rate and the economics of a milling operation, calculate the dimensions of the chip, estimate the cutting forces etc.The Z-map method is simple in principle but results in large files or, in order to rectify this, it needs to become more sophisticated, thereby losing its simplicity. The main difficulty arises when the workpiece needs to be updated to reflect the material removed during machining and is due to the curve-based representation of the tool.This is circumvented by solid modelling based on Boolean operations between accurate models of the tool and the workpiece. Beside the computational load which is bound to the accuracy level, it requires a sufficient number of interpolated points through one revolution of the tool to be trustworthy. It is practical to use at particular points of interest along the toolpath but it is impractical to cover the whole of the latter.In any case, the CNC toolpath needs to be calculated based on the relevant G-code and subsequently discretized using interpolation methods. The points of interest at which the detailed volume calculation of the removed material needs to be performed are to be selected either manually or automatically based on suitable criteria pertinent to the kind of further use that these calculations are planned for.AcknowledgementsThe work was funded as project number 01EA131 of the PENED2001 action at 75% by the European Union - European Social Fund and 25% the Greek State - Ministry of Development, General Secretariat for Research and Technology.References1Anderson, R.O. (1978) Detecting and eliminating collisions in NC machining5, Computer-.Aided Design, Vol. 10, No. 4, pp.231-237.2Baek, D.K. and Ko, T.J. (2008) Feedrate scheduling for free-form surface using an NC verification model， International Journal of Machine Tools
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