Patent Publication Number: US-2004042860-A1

Title: Methods of simulating end-milling operations

Description:
TECHNICAL FIELD  
       [0001] This disclosure relates generally to end-milling. More specifically, this disclosure relates to methods of simulating end-milling operations.  
       BACKGROUND  
       [0002] Many different machining methods have been developed. Machining as used herein is the act of removing material from a piece of stock material. Carving of a piece of wood into a desired shape is a basic or primitive form of machining. More complex machining methods include lathing and end-milling of materials such as, but not limited to, metals, plastics, ceramics, other materials and combinations of any of the foregoing. In the lathing process, a piece of stock material is rotated while a sharpened edge of a stationary tool is selectively applied to the rotating material. The edge of the tool removes material from the piece of stock to provide the desired shape.  
       [0003] End-milling operations consist of sweeping a rotating tool across a piece of stock material. The rotating tool includes a plurality of sharpened teeth or flutes that remove material from the stock. This process finds extensive use in the automotive, aerospace, tool, and die industries, because of its inherent flexibility in terms of tool paths/trajectories, and its ability to provide complex part features.  
       [0004] In order to predict variables of the end-milling operations (e.g., tool path, tool passes, and tool speed), models that capture the process kinematics have been developed. However, these models are limited in their application due to the restrictions on the part shape, tool shape and the tool path. For example, these models can have limited use when the tool is an insert-type tool where the teeth of the tool are not a single entity along a helix, but rather are a series of discrete tooth inserts that can be randomly positioned along the circumference of the tool.  
       [0005] Further, these models are limited due to the methods with which the model calculates the material removal rate. These prior methods can be both computationally intensive and time-consuming. It has also been found that these prior methods can have a less than desired accuracy.  
       [0006] Accordingly, there is a continuing need for easier and more accurate methods for simulating end-milling operations.  
       SUMMARY  
       [0007] A method of simulating a milling operation is provided. The method includes approximating the milling operation as a first two-dimensional representation; approximating the milling operation as a second two-dimensional representation; and computing a chip length. The first two-dimensional representation is a first plane of intersection of a milling tool with a work piece. The second two-dimensional representation is a second plane of intersection of the milling tool with the work piece. The chip length is computed from the first and second planes of intersection.  
       [0008] A method of simulating an end-milling operation is provided. The method includes approximating a tool as a first polygon having two dimensions; approximating a work piece as a second polygon having two dimensions; intersecting the first polygon at a first position along a selected tool path of the endmilling operation with the second polygon to determine a first contact arc between the first and second polygons; representing a circumference of the tool as a two-dimensional area; defining a cutting window on the tool in the two-dimensional area, the cutting window having a first dimension equal to the first contact arc and a second dimension equal to a contact depth of the tool with the work piece; and determining a length of a chip by computing an intersection of one or more teeth of the tool with the work piece in the cutting window.  
       [0009] A method of simulating a milling operation is also provided which includes creating a first three-dimensional model of a tool and a second three-dimensional model of a work piece; defining a first plane through the first three-dimensional model of the tool to obtain a first polygon having two dimensions; defining the first plane through the second three-dimensional model of the work piece to obtain a second polygon having two dimensions; intersecting the first and second polygons with one another at a first position along a selected tool path; determining tool entry and exit points, the tool entry and exit points representing intersection points of the second polygon with the first polygon at the first position, the tool entry and exit points defining a first contact arc; approximating a circumference of the three-dimensional model of the tool as a two-dimensional area; defining a cutting window on the two-dimensional area, a first dimension of the cutting window being the first contact arc and a second dimension of the cutting window being a selected depth of the tool in the work piece; and plotting the tool entry and exit points in the cutting window to determine teeth entry and exit points, the teeth entry and exit points representing intersection points of teeth of the tool with the work piece at the first position, the tool entry and exit points defining a length of a chip.  
       [0010] The above-described and other features are appreciated and understood by those skilled in the art from the following detailed description, drawings, and appended claims. 
     
    
    
     DESCRIPTION OF THE DRAWINGS  
     [0011]FIG. 1 is a schematic depiction of an end-milling operation;  
     [0012]FIG. 2 illustrates an exemplary embodiment of a method of simulating end-milling operations;  
     [0013] FIGS.  3 - 4  illustrate various exemplary embodiments of two-dimensional polygonal representations of a tool;  
     [0014]FIG. 5 illustrates an exemplary embodiment of a three-dimensional work piece;  
     [0015]FIG. 6 illustrates an exemplary embodiment of a two-dimensional polygonal representation of the work piece of FIG. 5;  
     [0016]FIG. 7 illustrates an exemplary embodiment of a first two-dimensional representation of an end-milling operation;  
     [0017]FIG. 8 illustrates a side view of an end-milling operation;  
     [0018]FIG. 9 illustrates an exemplary embodiment of a tool having its circumference approximated as a two-dimensional area;  
     [0019]FIG. 10 illustrates the forces predicted by the simulating method of FIG. 2 during an end-milling operation; and  
     [0020]FIG. 11 illustrates the forces measured during the end-milling operation simulated by FIG. 10. 
    
    
     DETAILED DESCRIPTION  
     [0021] Referring now to the figures and in particular to FIG. 1, an end-milling operation  10  is illustrated. The end-milling operation  10  comprises sweeping a rotating end-mill tool  12  across a work piece  14 . The tool  12  comprises a plurality of sharpened teeth or flutes  16  defined on a shaft or shank  18 . The shank  18  is secured to a spindle  20  of a end-milling machine (not shown).  
     [0022] The milling machine is configured to rotate the tool  12  in a selected direction  22  with a selected speed, referred to herein as the spindle rate. The milling machine is also configured to move the tool  12  along a selected tool path  24  at a selected speed, referred to herein as the feed rate.  
     [0023] The work piece  14  is held in place on the milling machine while the tool  12  is moved along the tool path  24  such that the teeth  16  of the tool remove material, in the form of chips, from the work piece to form the desired part.  
     [0024] A method  26  of simulating an end-milling operations  10  is illustrated schematically in FIG. 2. Method  26  provides a simple means for simulating end-milling operations having two-dimensional tool paths (e.g., tool paths in directions normal to the axis of rotation of the tool), three-dimensional parts, and complex tool shapes (e.g., tools having cylindrical, spherical, fluted, insert-type tools, and other shapes). Specifically, the method uses a two-stage two-dimensional computation technique that uses a polygonal intersection computation to ensure accuracy with high computational speeds, and ease of processing and maintaining data.  
     [0025] Method  26  uses Boolean alegbra in conjunction with the two-stage two-dimensional computation to simulate the milling operation  10 . Boolean algebra is defined as a set with two binary operations plus a unary operation. The binary operations are cumulative, associative, and each operation distributes over the other. The binary operations also require two identity elements that satisfy specific equations for all elements in the set. The two commutative binary operations can be described by any of various systems of postulates all of which can be deduced from the postulates that an identity element exists for each operation, that each operation is distributive over the other, and that for every element in the set there is another element which when combined with the first under one of the operations yields the identity element of the other operation.  
     [0026] The first stage  28  computes the region of contact (e.g., entry and exit) of the tool  12  into the work piece  14 . The second stage  30  computes the length of the chip by determining the entry and exit of each tooth  16  of the tool  12  into the work piece.  
     [0027] These two stages  28  and  30  provide an accurate simulation of the resulting chip length. The chip shape, thickness, and area can be determined at node  32  from the length as will be described in detail herein. Once the area of the chip has been determined, the forces on the tool can be computed at nodes  34  and  36 . Specifically, it has been deteremined that the forces on the tool are proportional to the chip area.  
     [0028] The forces on the tool can be used to determine the deflections in the tool during the end-milling process  10 . By repeating the above computational method at every angular increment of the tool with respect to the work piece (illustrated as feedback loop  35  in FIG. 2), the forces and deflections can be determined for the entire tool path  24 .  
     [0029] The quality of the part, such as the tolerance or variation from desired dimensions, that is formed by the end-milling operation  10  can be determined from these forces and deflections. Such simulations can also be useful in determining the amount of time necessary to produce a part from a particular end-milling operation.  
     [0030] Method  26  simulates the end-milling of three-dimensional parts using a plurality of readily available inputs. These inputs can include the shape of the tool  12 , the shape of the work piece  14 , the tool path  24 , the material properties of the tool, the material properties of the work piece, and the end-milling parameters (e.g., the feed rate of tool, the spindle speed, etc.).  
     [0031] The three-dimensional shape of the tool  12  and the work piece  14  are developed at a node  38 . Next, two-dimensional approximations of these components are determined at a node  40 .  
     [0032] The portion of the tool  12  having the teeth  16  is approximated as a two-dimensional shape in FIGS.  3 - 4 . This portion of the tool  12  can be approximated as a circular cross section, illustrated in FIG. 3. Alternately, this portion of the tool  12  can be represented as an n-sided polygon  42 . Here, the number of sides “n” are a number sufficient to retain the accuracy of the computation. In the embodiment illustrated in FIG. 4, the tool  12  is approximated as a nine-sided polygon.  
     [0033] The three-dimensional shape of the work piece  14  is approximated as a two-dimensional shape by defining a cross-sectional plane  44  through the work piece  14  as illustrated in FIG. 5. In this manner, the workpiece  14  can also be represented as an n-sided polygon, where the number of sides “n” are a number sufficient to retain the accuracy of the computation. For example, the work piece  14  (FIG. 5) is approximated as a closed polygon  46  (FIG. 6).  
     [0034] As illustrated, the work piece  14  includes a preformed feature  48 , which can include features molded or precut into the work piece prior to the milling operation  10 .  
     [0035] It should be recognized that the work piece  14  is described above by way of example only as having one preformed feature  48  and as being taken along one plane  44 . Of course, a number “m” of planes can be defined through the work piece at incremental depths along the axis of the tool to define multiple polygons. Here, the model is reduced to repeating the steps for the polygon  46  “m” number times (e.g., one for each plane) and combining the results. Additionally, more or less than one preformed feature can be defined in the work piece.  
     [0036] Thus, the three-dimensional tool and work piece are be simplified into two-dimensional polygons  42  and  46 . The three-dimensional tool and work piece and the two-dimensional approximations can be provided by many commercial computer aided drafting or “CAD” software packages. For example, many CAD software packages, such as UniGraphics, IDEAS, ProE, KATIA, and others, can produce faceted representations of any three-dimensional model (e.g., an STL format). A cross section of this three-dimensional model can then be used to generate the required two-dimensional polygons. The two-dimensional approximation of the tool  12  (e.g., polygon  42 ) and the work piece  14  (e.g., polygon  46 ) are provided as an input to method  26  at node  40 .  
     [0037] The tool  12  removes material in the form of chips from the work piece  14 . The shape of a chip for any given instant of time can be determined by knowing which portions of the tool are interacting with the work piece at that given instant. The cross section of the chip can depend on the length of contact of the tool with the work piece, and the thickness or depth of the tool into the part.  
     [0038] The first stage  28  determines region of contact between the tool and the work piece. The first stage  28  uses the inputs from nodes  38  and  40  to compute the entry and exit of the tool into the work piece. Specifically and with reference to FIG. 7, the method  26  simulates the movement of polygon  42  into the polygon  46  along the tool path  24 . The intersection of the polygons at each position can be used to approximate a contact arc  50  of the tool with the work piece. Thus, the contact arc  50  is defined by the intersection of the polygons  42  and  46 . This arc  50  can be used to determine the area of a chip  52  that is removed from the work piece.  
     [0039] It should be noted that the contact arc  50  is the intersection between the tool and the work piece. Of course, since the tool and the work piece are represented by polygons having “n” number of sides, the intersection may be a series of lines. Specifically, contact arc  50  may be a polyline defining the intersection of these polygons.  
     [0040] The first stage  28 , by using Boolean alegbra, reduces the complex problem of determining the entry and exit points of the tool into the work piece to a simple problem of finding the intersections of a circle (e.g., the rotating polygon  42 ), and the polygon  46  to provide the contact arc  50 . Here, the end-milling operation  10  is represented as a first two-dimensional representation defined as a first plane of intersection of the tool  12  and the work piece  14 . The first plane is taken through plane  44 .  
     [0041] In real end-milling operations, the input parameters to define the first plane can constantly change as the tool is moved across the part. In the exemplary embodiment, method  26  provides a means to determine first plane  44  as a function of time and distance. This allows method  26  to compensate for the dynamic nature of real world end-milling operations.  
     [0042] In the illustrated example, the polygon  42  enters the polygon  46  at point  54 , but exits at point  56  where the arc  50  intersects the feature  48 . The polygon  42  re-enters the polygon  46  at point  58 , but exits at point  60 . Thus, the arc  50  has two portions. The first portion of the arc is defined between points  54  and  56 , and the seond portion is defined between points  58  and  60 .  
     [0043] The first stage  28  computes the contact arc between the tool into the work piece by approximating the tool and work piece as two-dimensional polygons laid over one another. In the first stage  28 , plane  44  simulates the three-dimensional end-milling process as a first two-dimensional representaion.  
     [0044] The area of the chip  52  also depends on the angular position of the teeth  16  with respect to the work piece  14 . Thus, the second stage  30  computes the entry and exit points of the teeth into the work piece. Specifically, the point of entry of the teeth and the point of exit of the teeth is needed to determine the length of the chip  52 . Description of the second stage  30  is made with reference to FIGS. 8 and 9.  
     [0045] The tool  12  is illustrated in FIG. 8 engaged with the work piece  14 . During the end-milling operation  10 , the tool  12  is positioned a selected depth  62  into the work piece  14 . The arc  50  and the depth  62  define a cutting window  64  of the tool.  
     [0046] The cutting window  64  is best seen in FIG. 9, where the tool  12  is illustrated opened out along its circumference. Namely, the circumference of the tool is approximated as a two-dimensional rectangular area. The cutting window  64  is defined in this two-dimensional area to represent a second plane of intersection between the tool  12  and the work piece  14 . The cutting window  64  therefore simulates the three-dimensional end-milling process in a second two-dimensional plane.  
     [0047] The tool  12  comprises four teeth  66 ,  68 ,  70 , and  72 , respectively, helically cut into the tool. The helix angle on the teeth  66 - 72  allow the teeth to come in contact and leave the work piece  14  as the tool  12  is rotated during the end-milling operation. Only the portions of each tooth  66 - 72  that lie within the cutting window  64  actually participate in the cutting and chip creation process. Accordingly, one or more of the teeth and/or one or more portions of each tooth may be entering or exiting the work piece in the cutting window at any given instant. When multiple portions of the teeth are in the cutting window at any given instant, each portion forms a segment of the chip  52 . The chip  52  can therefore be an amalgamation of multiple chip segments, where multiple portions of the teeth are involved in creating the chip at the particlar instant.  
     [0048] The second stage  30  uses the entry and exit points of the tool  12  to determine the entry and exit points of the teeth  66 - 72 . Namely, the second stage  30  plots the entry and exit points of the tool  12  determined by the the first stage  28  in the cutting window  64 . Boolean alegbra is then used to determine the entry/exit points of the teeth  66 - 72 .  
     [0049] In the illustrated example, the arc  50  covers about 75 degrees of the tool. Here, the first portion of the arc  50  covers from about zero degrees to about 20 degrees and the second portion covers from about 65 degrees to about 75 degrees. Tooth  54  is the only tooth that is engaged with the polygon  46  in the cutting window  64 . Specifically, tooth  54  enters polygon  46  at point  74  and exits at point  76 . Thus, tooth  54  is engaged with the polygon  46  from about 0 degrees to about 20 degrees. In this example, chip  52  is has only one chip segment due to the fact that only one portion of one tooth is engaged in the cutting window.  
     [0050] The length  78  of the chip is equal to the portion of the teeth that is in contact with the work piece. Thus in the illustrated example, the length  78  is the distance between points  74  and  76 .  
     [0051] The shape, thickness, and area of chip  52  can be determined at node  32 . Node  32  computes the chip  52  by comparing the polygon  42  in a first position  80  (in illustrated phantom) with respect to the polygon  46  and the polygon  42  in a second position  82  with respect to the polygon  46 . The shape of the chip  52  is the area obtained by the intersection of polygon  46  with polygon  42  at the first position  80  as compared to the second position  82 .  
     [0052] The thickness of the chip  52  is the distance between polygon  42  in the first and second positions. The position of the tool (both radially and along the tool path) in the first and second positions can be determined from the feed of the tool and the speed of the tool for a selected time-period. For example, the distance between the first position  80  and the second position  82  can be equal to the feed of the tool for the tooth. Feed per tooth of the cutter where the teeth are equispaced is given by feed per revolution divided by the number of teeth on the cutter.  
     [0053] The thickness of the chip  52  can vary along its length due to the helical configuration of the teeth. For example, when the teeth have a helix angle equal to zero, chip  52  has one chip segment. However, when the teeth have a helix angle other than zero, chip  52  comprises has multiple chip segments. In the example where chip  52  comprises one chip segment, the chip is substantially rectangular. In the example where chip  52  comprises multiple chip segments, each segment can be assumed to be substantially rectangular.  
     [0054] The rectangular chip shape allows node  32  to compute the area of the chip  52  as the product of the length  74  and the thickness of the chip. In this manner, node  32  can easily compute the chip area using two two-dimensional approximations of the tool  12  and the work piece  14 . Here, the first of the two-dimensional approximations is represented by plane  44  (determined at the first stage  28 ) and the second of the two-dimensional approximations is represented by cutting window  64  (determined at the second stage  30 ).  
     [0055] The method  26  repeats nodes  28 - 32  for each incremental movement of polygon  42  with respect to polygon  46  along the tool path  24 . By repeating nodes  28 - 32  of the computational method  26  at every angular increment of the tool along the tool path  24 , the forces and deflections can be determined for the entire tool path at nodes  34  and  36 . In alternate embodiments, method  26  can be repeated for “m” number times (e.g., one for each plane) and combining the results.  
     [0056] Node  34  determines the forces on each chip by assuming that the force is proportional to the chip area of that chip. Node  36  then sums the forces computed at node  34  to determine the resultant forces on the tool  12 .  
     [0057] Turning now to FIGS. 10 and 11, a comparision of the forces predicted by method  26  as compared to forces measure for the same end-milling operation is illustrated. In this example, the work piece is formed of gray cast iron, and the tool is an untreated carbide tool. The tool has a 12.7 mm diameter, a 30 degree helix angle, and four teeth with zero rake angle. The tool was has a feed of 0.08 mm/rev (millimeters per revolution), was rotated at 1500 rpm (revolutions per minute). The x-direction is along the direction of the feed, the y-direction is normal to the feed, and the z-direction is through the axis of rotation of the tool. As can be seen in FIGS.  10 - 11 , method  26  acurately and preceisely predicts the forces experienced during the end-milling operation.  
     [0058] Method  26  therefore computes the chip area, as well as the resulting force, for end-milling operations  10  of any three-dimensional workpiece and tool using a stack of multiple two-dimensional polygons, along a two-dimensional tool path.  
     [0059] Method  26  computes a chip area from a three-dimensional end-milling operation in two stages. The first stage uses polygonal intersections to compute the intersection between the tool and work piece. The first stage also computes the resultant part shape due to the removal of the chip. The second stage computes tooth entry exit angles into the work piece to determine the length of the chip. The result of the two stages is the chip shape and area. Boolean algebra is used to determine the polygon intersections without loss of accuracy or computational efficiency.  
     [0060] Method  26  allows workpieces and tools having any shape to be approximated as two-dimensional polygons, and uses Boolean intersections of these two polygons to compute the chip shape.  
     [0061] It should be recognized that the tool  12  is illustrated therein by way of example only. Of course, a tool having more or less than four teeth, a tool with or without helical teeth, a tool having an arc that covers more or less than 75 degrees, and others are contemplated to be with in the scope of the present disclsoure.  
     [0062] It should also be noted that the terms “first”, “second”, and “third” may be used herein to modify elements performing similar and/or analogous functions. These modifiers do not imply a spatial, sequential, or hierarchical order to the modified elements, unless otherwise indicated.  
     [0063] While the invention has been described with reference to one or more an exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims.