Significance Statement
Mathematical models were used in predicting metal cutting process performance for each tool-operation combinations. However, developing dedicated model for each metal cutting application limits the use of science-based process simulation and optimization method in industry. Hence, a unified and generalized model is required to adapt any tool geometry and metal cutting operations that involve chip shearing process.
Despite great knowledge provided by previous researchers concerning geometric modelling and mechanics of individual operations, tool geometries and machining operation in a unified mathematical model have not been considered.
Dr. Kilic and Professor Altintas from the Department of Mechanical Engineering at University of British Columbia in Canada presented a unified geometric modelling of solid and indexable tools which was later used in generalized prediction of cutting forces, vibrations and dimensional surface errors generated on the part. The work published in International Journal of Machine Tools & Manufacture, using the same model starts by defining the tangent and rake face vectors at discrete elements along the cutting edge.
Generalized geometric modeling of cutting edge location and its rake face orientation vectors when considering indexable cutter had insert digitized along its cutting edges by series of points. Tangent vectors (t, n and e) define the orientation of each differential edge segment attracted to each point. The insert’s cutting edge was now completely defined as a whole body by an array of points PI(p1I, p2I,…pcI)(4x(4C)) with their coordinates and orientation vectors.
The insert is a given Euler angular rotations which defines cutting edge (ҡr), axial rake (ϒp) and radial rake (ϒf) angles at the insert pocket of tool body with orientations carried out in order of negative-ϒf rotation around ZI-axis, negative-ϒp rotation around XI-axis and negative-ҡr around YI-axis of insert.
Solid tool was defined directly from Frame I which is placed at the mid-point of each of its element’s cutting edge (AB). Tangent vectors (t, n and e) corresponds to axes of the Frame I (XI, YI and ZI). 0r1 was defined as position vector describing the distance between origins of tool body (Frame 0) and differential edge (Frame I). R10 is the rotation matrix orienting the differential cutting edge element along the tools edge in Frame O. Order of rotations include rotation around Zt1-axis by π/2-ψ, rotation around Xr1-axis by helix angle (β), rotation around Zr3-axis by negative rake angle (-ϒ) and rotation around Yr3-axis by negative axial immersion angle (-ҡ).
The coordinates and tangential vectors that define oblique and rake angles at points along the cutting edges are now defined in Frame 0. Arbitrary tool geometry for any cutting operation was defined by 15 parameters such as location of Cartesian coordinates (x0,y0 and z0), axial location, axial runout, angular immersion, pitch angles, angular orientation of cutting edge element, discretization of cutting edges, coordination of cutting edge on the tool, cutting edge-work material contact conditions, radius of cutting edge, tool runout and angular location of cutting edge.
For modelling of tool angles, Frame E was introduced as cutting edge coordinate frame which has cutting edge tangent along its XE axis, normal rake face vector along its YE axis and rake face along its ZE axis. Parameters defined which depends on the reference system includes cutting velocity, feed vector, resultant cutting vector, tool-in-use planes, tool reference plane, assumed working plane, working plane, rake face, cutting edge normal plane, cutting edge plane, cutting edge angle, inclination angle and normal rake angle.
When several tools were modeled to illustrate application of the proposed generalized geometry model, the inserts seat center at the origin of insert frame (Frame I) had each point pl of the geometry array PI(p1I, p2I,…pcI)(4x(4C)) of the rake face tangent and normal vectors are constructed. For a sample face milling tool, insert is of ISO-W type with 3 symmetric sides, 0.8mm nose radius and 6.5mm thickness and radial runout of the inserts are measured when the cutter was mounted on the spindle. In case of a drilling tool, position vectors and rotation angles are different for peripheral (j=1) and central (j=2) inserts of the drill.
A sample indexable end mill with two inserts, translational feed motion of cutter is aligned with positive X0-direction and tool rotation axis is the positive Z0-direction. There are C=500 points i.e. PI(p1I, p2I,…pcI)(4×2000) which are digitized from the inserts solid model. Insert frame of the first insert (Frame I-1) is located and oriented using translational and rotational parameter of [0rxI, 0ryI, orzI]insert-1= [7.09, 2.60, 5.09]mm and [ϒf, ϒp, ҡr]insert-1=[9.00, 9.90, 0.200] deg.
A sample three-fluted serrated end-mill is modeled and sampling distance between interpolation points axis is dz=cos.ds. Radial variation along flute j is mapped onto tool axis with radius, R(j,k) at each segment along tool axis. Axial immersion angle information is mapped onto tool axis to obtain K(j,k). The radial runout is obtained by normalizing each axial segment ҡ to its minimum using the radius, R(j,k). Axial segment thickness is set to dz=0.1mm as geometry is interpolated at the midpoint of each segment. Maximum depth of cut is set to 39.2mm which makes K=39.2/0.1=392.
For the sample multi-functional tool, cutting diameter of inserted section is 46mm. Rake angle of insert is zero and origin of the insert coordinate frame is set at bottom tip of the insert while other three inserts are located on the cutter in a similar way. Information of each insert (j=[1,2,3,4]) is stored in PI(p1I, p2I,…pcI)(4x(4C)). Axial location and angular location are plotted using the identified 3D geometry as axial runout is derived by shifting the minimum axial location value to zero at each segment-k. Pitch angles additionally depends on axial runout and feed rate.
This study developed geometric model that will help designers to decide suitability of tool for the operation and prediction of tool performance.
Journal Reference
Kilic, Z.M. Altintas, Y. Generalized Modelling of Cutting Tool Geometries for Unified Process Simulation. International Journal of Machine Tools and Manufacture, Volume 104, May 2016, Pages 14–25.
The University of British Columbia, Department of Mechanical Engineering, Manufacturing Automation Laboratory, 2054-6250 Applied Science Lane, Vancouver, Canada BC V6T 1Z4.
Go To International Journal of Machine Tools and Manufacture