The cutting area is evaluated with calculating the uncut chip thickness. Figure 30 shows simulation of the height in the cutting area when cutting with a 2 flutes ball end mill during a rotation of the cutter. Because the cutting edges were divided into small segments in the simulation, the cutting areas are designated by symbols. When the cutter axis is not inclined, each edge cuts the material alternatively. Therefore, the tool always contacts the workpiece, as shown in Figure 30(a). Figure 30(b) shows the simulation in cutting with the tool inclined at 45. The noncutting time, during which neither of the teeth cut the workpiece, appears and the actual cutting time of each tooth reduces during a rotation of the cutter. Figure 31 shows the cutting force of crown glass measured by a piezoelectric dynamometer. The force components loaded on the tool are shown in Figure 31(a). The noncutting is observed in the measured cutting force, as the simulated results shown in Figure 30(b).
Figure 30. Cutting area during a rotation of cutter: (a) inclination angle, 0 and (b) inclination angle, 45. Cutting conditions: spindle speed, 20 000 rpm; feed rate, 0.48 mm min1; depth of cut, 0.02. Tool: type, ball; diameter, 0.4 mm; flutes, 2; helix angle, 0.
Figure 31. Cutting force in glass milling with cutter axis inclined at 45: (a) cutter axis inclination and (b) cutting force. Cutting conditions: spindle speed, 20 000 rpm; feed rate, 0.48 mm min1; depth of cut, 0.02 mm; lubrication, water. Tool: type, ball; material, TiAlN-coated carbide; diameter, 0.4 mm; flutes, 2.
The actual cutting time during a rotation of the cutter decreases when increasing the inclination angle of the tool, as shown in Figure 32(a). Figure 32(b) shows the height of the cutting area removing the material with the inclination angle, where the symbols show the minimum and the maximum height. The cutting area spreads when increasing the inclination angle. The cutting velocities increase with the cutter heights corresponding to the rotation radii of the cutting area.
On the thermal effect, the time for the heat generation reduces with the actual cutting time. Because heat conduction requires time to heat, the temperature rise is expected to be small even if the cutting velocities increase in the cutting area with the tool inclination. On the other hand, the teeth are cooled well during the noncutting time by supplying water. Consequently, tool wear can be suppressed at low cutting temperatures when cutting with the inclined tool.
Figure 33(a) shows the range of the cutting velocity in the cutting area. The cutting velocity increases with the inclination angle. Then, the surface finish can be improved with increasing the cutting speed. Figure 33(b) illustrates the changing rates in the cutter radius at a lower cutting area and that of a higher cutting area. The changing rate is smaller at a higher cutting area, and the range of the cutting velocity in the cutting area reduces when increasing the inclination angle. Therefore, a uniform surface can be finished due to little change of the cutting velocity when the inclination angle is large.
The mechanistic force model has been formulated following those models presented by Imani  and Yucesan . The modeling approach taken was to first develop the cutting edge geometry, then to develop the cutting force expressions.
The cutting edge geometry for a ball end mill is shown in figure 2. The coordinate values of any point Pi on the cutting edge can be determined from the NURBS curve representation, if given the corresponding parameter ui. The parameter ui, normalizes points at equal spacing along the rotational axis (z axis) of the tool. Given features of the geometry in the locale of point Pi, the instantaneous helix angle z, lead angle 1. and angle of rotation i. is determined from basic geometric considerations. The instantaneous normal rake angle 1 is determined from the nominal rake angle, 0, the lead angle, and the instantaneous cutter helix angle,
In order to evaluate the proposed procedure, we develop a prototype system and conduct an experiment using NC program simulates finish machining. In the prototype system, we introduced P-Voxel representation method (Kaneko J. 2002) in order to accelerate the existence evaluation.
Figure 5 shows the estimated results by the developed system. The NC program is created by commercial CAM system. Workpice is sculptured by contour milling of 7600 steps with square end mill and profile milling of 10600 steps by ball end mill. Figure 5(a) shows the estimated result when helix angle of flutes on cutting tool is 30 degrees. And, Figure 5(b) shows the result when the helix angle is 0 degree. The difference of each result is caused by changes of loaded cutting force resulted from the helix angle of cutting edge and removal process of workpiece volume.
The required time for estimation of case (a) was about 130 seconds, and was 133 seconds in case (b). The total number of estimation points on sculptured surface is about 44000. The prototype system can calculate the cutting error on finished workpiece surface in about 0.003 seconds per one estimation point. As a result, it is thought that the proposed procedure realizes the estimation of cutting error distribution with sufficient performance.
Exact modeling of the milling process remains elusive because of the uncertainty in predicting fundamental parameters (rake face friction, shear angle, etc.). For this reason, mechanistic models, which must be calibrated for each tool and part material combination, are widely used. Work towards determining new tool model parameters from orthogonal cutting tests has begun , but is early in development. Additional research into developing models for ball end mills  and more complex cutter shapes is needed. Modeling of thin part deflection under machining forces  is also required. All of these models are needed to further understanding of the machining process.
For tool path planning and control assistance simulations, less detailed models will suffice. Adaptive force control of sculptured surface machining, for example, requires only recognition of areas where part surface interruptions and sudden force increases will occur. The feedback capabilities of the controller will fine tune the process. By accepting more approximate models, there is potential to separate the geometric portion of the process model from the cutting mechanics portion. If this can be achieved with adequate model accuracy, then much of the computationally intensive simulation work can be completed and stored prior to actual part machining. This is necessary if online, time-critical cooperation between the machine tool controller and solid modeler is to be achieved.
The internal working faces of mills consist of renewable liners, which must withstand impact, be wear-resistant, and promote the most favorable motion of the charge. Rod mill ends have plain flat liners, slightly coned to encourage the self-centering and straight-line action of rods. They are made usually from manganese or chrome-molybdenum steels, having high impact strength. Ball mill ends usually have ribs to lift the charge with the mill rotation. The ribs prevent excessive slipping and increase liner life. They can be made from white cast iron, alloyed with nickel (Ni-hard), other wear-resistant materials, and rubber (Durman, 1988). Trunnion liners are designed for each application and can be conical, plain, with advancing or retarding spirals. They are manufactured from hard cast iron or cast alloy steel, a rubber lining often being bonded to the inner surface for increased life.
Shell liners have an endless variety of lifter shapes. Smooth linings result in much abrasion, and hence a fine grind, but with associated high metal wear. The liners are therefore generally shaped to provide lifting action and to add impact and crushing. From a survey (Wei and Craig, 2009), the most common shapes were wave, rib, step, and Osborn (Figure 7.11). The liners are attached to the mill shell and ends by forged steel countersunk liner bolts.
Rod mill liners are generally of alloyed steel or cast iron, and of the wave type, although Ni-hard step liners may be used with rods up to 4cm in diameter. Lorain liners consist of high carbon rolled steel plates held in place by manganese or hard alloy steel lifter bars. Ball mill liners may be made of hard cast iron when balls of up to 5cm in diameter are used, but otherwise cast manganese steel, cast chromium steel, or Ni-hard are used.
Efforts to prolong liner life are constantly being made. With the lost production cost associated with shut-downs for replacing liners, the trend is toward selecting liners that have the best service life and least relining down-time (Orford and Strah, 2006).
Rubber liners and lifters have supplanted steel at some operations, particularly in ball mills. They have been found to be longer lasting, easier and faster to install, and their use results in a significant reduction of noise level. In primary grinding applications with severe grinding forces, the higher wear rate of rubber tends to inhibit its use. Rubber lining may have drawbacks in processes requiring the addition of flotation reagents directly into the mill, or temperatures exceeding 80C. They are also thicker than their steel counterparts, which reduces mill capacity, a potentially important factor in small mills. There are also important differences in design aspects between steel and rubber linings (Moller and Brough, 1989). A combination of rubber lifter bars with steel inserts embedded in the face, the steel providing the wear resistance and the rubber backing cushioning the impacts, is a compromise design (Moller, 1990).
A different concept is the magnetic liner. Magnets keep the lining in contact with the steel shell and the end plates without using bolts, while the ball scats in the charge and any magnetic minerals are attracted to the liner to form a 3040mm protective layer, which is continuously renewed as it wears.
Mould and dies are ball-end milled in high speed milling centres to achieve precise shapes, good surface finishing and a good economical performance. Several aspects must be taken into account: the use of three- or five-axis milling machines, powerful CAM systems, high-tech milling tools, and skilled programmers and machinists. Milling of free form surfaces using ball-end milling is the main operation in this application, known as sculptured surface milling (SSM) (Choi and Jerard, 1999). Ball-end milling is also known as copying milling because of the old practice of copy forms from jigs 40 years ago before the introduction of CNCs in milling machines. In Table3.4, some applications of SSM, along with their requirements are presented.
Sub-micrograin grade tungsten carbide ball-end milling tools coated by AlTiN monolayer are used (see Figures3.32a and 3.38b), applying 200400m/min for finishing. In some deep parts, the more rigid bull-nose or conical shank ballend mills are available. Polycrystalline cubic boron nitride (PCBN) sintered tools are also available with higher cutting speeds than carbides, up to 800m/min can be achieved. However, its feasibility depends a lot on the CNC program depuration, because unexpected material left by previous milling operations could break cutting edges. Machine operators prefer the minor risk of carbide tools in spite of the improved speed of the very expensive PCBN tools.
Ball-end milling must remove a 0.3mm allowance in the finishing pass, using ball-end mills with diameters less than 20mm to reproduce the free-form small features. Slopes found in sculptured forms are from 0 to 90 and the fillet radius can in some applications be very small.
Effective cutting speeds must be between 200 and 400m/ min, the current recommendation for AlTiN coated carbide tools. In the view of Eq.[3.1], introducing the small ap, the recommended cutting speed, the small tool diameter, and surface slope, results in a spindle rotational speed over 15,000rpm at least (Siller et al., 2006). Therefore a highspeed spindle is required (Flores et al., 2007); the maximum rotational speed of todays industrial high-speed machines is around 20,000-25,000rpm, with power ranging from 14 to 20KW. For this rotational speed and recommended feed per tooth of around 0.070.1mm/z, the machine working linear feed results in 1015m/min. The control of interpolated curves at this feed rate is not a problem for current CNCs.
The most typical milling machine is a three-axis vertical spindle, a bridge structure with moving table, with a workspace 1mX1mX1m for mould making, and for big stamping dies those briefly presented in Section 3.4.1.
Since 2000, five-axis milling centres have been derived from adding two additional orientation axes that are being introduced for mould making, allowing the milling of very complex parts that cannot be machined using three-axis machines (Tsutsumi and Saito, 2004). Regarding sculptured surface milling, the cutting speed is zero at the tool tip, making tool cutting very unfavourable and causing typical breakage of the tool tip. With five axes, milling can be performed, thus avoiding the tool tip cutting. Nevertheless, using five-axis milling the tool overhang can be reduced and thus tool stiffness is greater, which increases machining precision.
In Figure3.33 a small plastic injection mould, manufactured in a five-axis machine is shown. Table3.5 presents the milling data for this mould. The initial block was tempered steel hardened to 35HRC. The roughing process was performed using high-feed milling, characterized for the very high feed per tooth of 1.2mm/z and the small axial depth of cut. The main advantage of this milling type was that no big stairs were produced in roughing, so the semi-finishing operation was eliminated. Total machining time decreased by more than 40% with respect to other milling approaches. Finishing of cavities was carried out by the simultaneous five-axis, keeping a tool tilt angle of approximately 20 and a not too long tool overhang.
In most cases, the material used for making the flexible bodies is a room-temperature vulcanizing (RTV) silicone rubber (e.g. Silastic E RTV made by Dow Corning). It is supplied in a two-component kit, and the two fluids, one of which is the catalyst, are mixed in the prescribed ratio (typically 10:1) just before manufacture. The mixture is poured into a mould, cured, and then extracted, as described in what follows. Another essential item is a liquid agent supplied by the manufacturer for coating surfaces to which the silicone rubber should adhere; e.g. the edges of the metal strip sometimes embedded in pipes (Figure D.1), which are thus constrained to oscillate in 2-D. A releasing agent is also available, for coating surfaces on which the silicone rubber should not adhere at all, e.g. the inner surface of the mould and the middle length of the metal strip; a cheap substitute is PAM no-stick solution spray, used in cooking. In what follows, we shall continue using the moulding of a pipe as an example; a few words on other structures are given in Section D.2.
Figure D.1. Cut-away view of a pipe with an embedded metal strip typically of 0.005 in (0.127mm) feeler gauge. The holes are for equalizing the pressure in the two channels during flow testing, in case of small asymmetries.
A schematic of a mould for a pipe with an embedded metal strip is shown in Figure D.2(a). It is basically composed of (a) a split outer mould and (b) a split cylindrical core (or, when no metal strip exists, a whole cylinder). The outer mould is made of two solid-block halves; after the interfaces are ground flat, semicircular grooves are carefully milled in each with a ball end-mill, so as to produce a fine finish. Similar care should be taken to make sure that the split core when sandwiching the metal strip is cylindrical and of the required diameter. The mould can be made of plexiglas to allow viewing while casting or, for better dimensional tolerances and robustness, of brass. The alignment of the two halves of the mould and of the components of the core is crucial, since it controls the quality of the final product: axial symmetry, straightness, central positioning of the metal strip, and so on. Hence, tight tolerances should be imposed, and dowel pins used to ensure correct assemblage every time. Long plexiglas moulds should be reinforced with metal reinforcing plates. The end-supports serve (a) to support the central core and (b) to connect to the injector or collect some overflow (since the silicone rubber contracts a little during curing). All surfaces must be thoroughly cleaned and then treated with a thin film of either adhering or releasing agent, just before manufacture of the pipe.
Figure D.2. (a) Schematic of the mould; (b) schematic of the injection syringe. A: lower end-support for connection with injector outlet; B: split cylinder core; C: split outer mould; D: reinforcing plate; E: upper end-support for holding overflow; F: injector outlet; G: transparent-wall injector; H: injector piston with O-rings; I: threaded rod; J: injector handle.
The injector, Figure D.2(b), is an elephant-size syringe typically 10cm in diameter and 30cm long. The two components of the silicone rubber are mixed in a beaker with the aid of an electric drill, and then poured into the syringe, typically filling 110-18 th of its volume. Then the top of the injector is connected to a vacuum pump, capable of generating a pressure of 0.1 atm approximately, to boil off trapped air in the viscous mixture (of the consistency of bread dough), but not low enough to reach the boiling point of the silicone rubber itself; hence the piston in the injector must be leak-proof. Air is trapped not only by the folding of the mixture during pouring, but also in the form of small bubbles trapped during mixing, which cannot rise to the surface fast enough. The vacuum is applied and held long enough for the mixture to expand, filling half or two-thirds of the injector volume, allowing the larger bubbles to burst and the mixture to collapse. This cycle of (a) application of the vacuum, (b) holding it, and (c) releasing it gradually has to be repeated perhaps up to 50 times until application of the vacuum results in no noticeable change in volume.
The working time available before the mixture begins to set varies from one silicone rubber to another, but it is typically 12h. Room-temperature curing takes about 72h, but in a temperature-controlled oven at 160C this can be accelerated to 1h.
Once the mixture is deaerated, it is injected into the mould slowly, so as to rise in it at no more than 0.5mm/s. The mould and injector are arranged in a vertical configuration and, usually, remain so connected during curing.
Extracting the casting from the mould is perhaps the most challenging aspect of the manufacturing process. Even with the mould-release agent, the casting just does not slide off the central core, because of the vacuum that needs to be broken between the surfaces. An effective way is to put the pipe, with the core in it, on a long V-block and then apply compressed air (at no more than 140 psi or 1MPa) at one end, to slightly expand and lift the pipe off the core; a little water lubrication helps to then draw the core out from the other end. In the case of a split core, the first half is removed in this way, but the second one has to be painstakingly eased out mechanically, by tapping it with a smaller rod carefully, so as not to damage the bond between the metal strip and the pipe.
Lower-quality, but easier to manufacture pipes and cylinders may be cast in glass tubes which, after curing, are broken and the core removed in the manner just described. The weakness here is the imperfect uniformity and straightness of the glass tubes.
For cantilevered pipe experiments, it is best to make the free end square to the long pipe-axis at manufacture. If cutting a piece of the free end becomes necessary, however, it should be done with great care. A good way is to sandwich the pipe between a close-fitting rod inside and a shorter pipe outside with a square-cut end, then to slice the elastomer pipe with a sharp razor, slowly and with minimum local deformation.
Modeling cutting forces in milling operations is important for predicting cutting forces that would be generated, power and torque required, and machine tool vibration for a given set of machining parameters prior to actual machining, proper selection of tools and fixtures for successful and optimum milling operation. High cutting force will induce large deformation of the workpiece and cutting tool leading to bad quality of machined components, severe tool wear, and breakage. Real time measurement of cutting forces is time consuming and very costly. Fast and low cost method is needed for their prediction. This section presents progress of research conducted by the author in cutting force modeling and prediction to be integrated with virtual machine tool for machining simulation and learning in the web.
Milling processes are used extensively in automotive, aerospace, and tool and die manufacturing industries to make prismatic components. It is categorized as peripheral and face/end milling. For peripheral milling process the cuter axis is parallel to the milled surface, while in face/end milling, the cutter axis is perpendicular to milled surface. Methods reported in the literature for modeling cutting forces in milling include: empirical, analytical, and mechanistic [9,10]. Empirical method correlates experimentally measured cutting response variables like cutting force components and tool wear with machining input variables using empirical functions, which is time consuming and very costly. Analytical method modulates physical mechanics of cutting using numerical algorithms, which does not characterize completely the situation at the flank and rake face of the cutting edges, like high strain rates and temperature gradients . Mechanistic modeling method, derives total cutting forces, that is, the summation of the cutting force that is proportional to differential cross-sectional area of undeformed chip, and the edge force that is proportional to differential cutting edge length. Cutting force is responsible for the effect due to shearing, whereas the edge force is responsible for the effect due to ploughing . The proportionality constants associated with shearing effect are referred to as cutting force coefficients in the tangential, radial, and axial directions of the cutter diameter (ktc, krc, kac, respectively), and the proportionality constants associated with ploughing effect are referred to as edge force coefficients in the tangential, radial, and axial directions (kte, kre, kae, respectively). They account the end mill geometry and mechanical properties of the machined workpiece [10,11].
Several investigations have been conducted on mechanistic cutting force prediction , including early study by Martellotti in 1941 [15,16] on kinematics of end milling, chip formation, and surface finish. Zheng et al.  developed a mechanistic model to predict cutting forces in peripheral milling of Aluminum 7075-T6 using helical end-mill, which only accounts for shearing effect and dry machining. Budak et al.  mechanistic cutting force model added edge force coefficients to account for the ploughing effect and shearing effect using orthogonal cutting data and oblique cutting analysis, which they claim could replace experiments for the determination of cutting and edge force coefficients. Altintas and Lee  analyzed Budak et al.s model by adding differential edge length to the ploughing effect and used it predicts cutting forces in end milling with helical ball end mill. Engin and Altintas [20,21] developed a generalized geometric model that represents different shapes of end mill, which accounts for shearing and ploughing effects. Their model considered only end mill envelope geometry but not cutting edge geometry. Zhang et al.  developed mechanistic cutting force model for serrated-edge end mills, accounted only shearing effect. Merdol and Altintas  investigated the effects of serrated end mill profile on chip load, cutting force, power, and vibration.
Aderoro and Wen  predicted cutting force coefficients, using arbitrary Lagrangian formulation and finite element analysis. It was also reported that the determination of force coefficients does not require experiments . Gradisek et al.  used semi empirical mechanistic cutting force model for the identification of force coefficients for general end mills from radial immersion cutting tests of varying sizes. Similarly, during high speed end milling of titanium alloy, a mechanistic identification of cutting and edge force coefficients for simulation of cutting forces was also done .
In this section, the results of mechanistic cutting force model development for wavy edge bull-nose helical end mill (WEBNHE) are presented. WEBNHE was conceived to reduce self-excited vibrations and improve machining dynamics. Thus a need has been established for mechanistic cutting force model that can accounts for the effects of cooling and lubrication as well.
Okafor and Sultan  developed mechanistic cutting force model for WEBNHE for milling Inconel 718. Sultan and Okafor  used the developed and experimentally validated cutting force model to investigate the effect of end mill geometric parameters on cutting forces in end milling Inconel 718, using MQL cooling Strategy. WEBNHE developed by the author incorporates the effects of emulsion and MQL cooling, and it was used to predict cutting forces in end milling of Inconel-718.
Previous studies [11,12,17,20,21,2426] show that cutting forces in end milling are proportional to undeformed chip instantaneous cross-sectional area that is time dependent. Differential tangential, radial, and axial cutting force components (dFt, dFr, and dFa) acting at point Pij along the cutting edge are calculated using the following equation:
Eq. (6.1) neglects dynamic and run out of the end mill. The six constants of proportionality as shown in Eq. (6.1) are Ktc, Krc, Kac, Kte, Kre, and Kae. As mentioned previously, they incorporate the effect of tool geometry, and properties of the tool and workpiece material [10,12,17,20,21,25,26]. These coefficients were determined mechanistically from end milling test by the authors MS student Ameen (2014) .
The developed MATLAB code was used to simulate WEBNHE geometric model. The geometric simulation involves: (1) the definition of geometric parameters, creating primary points; (2) the rotation of primary points to create knots; (3) the definition of other geometric and cutting parameters; (4) the calculation of spline constants; (5) cubic spline approximation of wavy edge profile; (6) the transformation of linear distances to polar coordinates; and (7) the creation of all cutting edges. The detail procedure is given in Okafor and Sultan .
The geometric model is defined first using the identified six cutting and edge force coefficients, and table feed (f). Then the feed per tooth (ft) is calculated using specified table feed and tooth pitch in polar coordinates before cutting force simulation and prediction.
The goal of the virtual simulation is to aid the producer to choose the adequate machining parameters before the realization of the parts on the machine tools. These parameters are chosen according to the imposed quality criteria. The most important criterion is the topography of the machined surface; it is influenced by the flexion and vibrations phenomena resulting from the cutting forces.
The cutting forces modeling gains more importance in order to prevent excessive cutter deflection and surface errors in milling process. Its prediction is not easy because of the difficulty to control the cutting geometries during machining. In fact, it is important to determinate the instantaneous flute enter and exit angles, this information help to know the exact value of the instantaneous chip thickness and the cutting length which are the two fundamental parameters in the determination of the cutting forces from mechanistic models.
In the previous works, some researchers have been interested in the calculation of the (CWE) region. Ju et al.  proposed a discrete boundary representation based on the exact Boolean method to calculate the (CWE) at every cutter location. Zhiyang  proposed a method based on the tessellated format. The CAD surface model is stored in tessellated format as an STL file which is transformed in a very large number of triangular facets. The algorithm finds the intersection curves form one or multiple closed regions on the ball end mill which forms the (CWE) regions. Boz et al.  compared two models for (CWE) calculations. The first method is a discrete model which uses three-orthogonal dexelfield and the second method is a solid modeler based model using parasolid boundary representation kernel. The second method was proved more accurate. In the study of Kiswanto et al. , the (CWE) region is determined by finding the length of each cut at every engagement angle between the lower engagement (LE) point and the upper engagement (UE) point and the swept envelope of the removed volume. Ozturk and Lazoglu  proposed a numerical method to determine the chip load of a ball-end mill during free-form machining. The chip load was obtained by defining three engagement boundaries: the tool entry boundary, the exit boundary and the workpiece surface boundary.
Iwabe et al.  calculated the chip area of the inclined surface machined by ball end mill cutter by a contour path method using 3D-CAD. The chip area is calculated by the interference of the rake surface and the chip volume. Sato et al.  calculated the (CWE) region based on the geometric tool-workpiece intersection in order to predict the topography and surface roughness as a function of cutting parameters. The geometric model is presented in the case of a flat surface machined by a ball end mill on one way strategy. Another phenomenon which influences the (CWE) region and the chip thickness and is not considered in this study is the motion errors of feed drive systems. Then, Nishio et al.  investigated the influence of dynamic motion errors of feed drive systems onto the surface machined by a square end mill.
In the study of Erdim et al. , the (CWE) is calculated from the output Cutter Location data file by an analytical simulation. Gong et al.  determined the (CWE) using a triangle mesh model and the intersection calculations of the cutter swept volume. Ozturk et al.  and Erdim and Sullivan  calculated the (CWE) by subtracting it from the swept volume of tool motion. The tool is divided into elementary disks and the engagement points are extracted, then the start and exit angles are calculated.
Sun et al.  calculated the (CWE) using a Z-map model. The workpiece is meshed into small grids whose projection onto the xy plane is a square. The engaged cutter element can be achieved according to the difference between the cutter element and the projection of the instantaneous workpiece height into the cutter element.
Boz et al.  used a B-rep modeler to calculate the (CWE). In the study of Mamedov and Lazoglu  a method based on solid modelers was employed in which the (CWE) was calculated at each cutter location point. From the cutter location file, the swept volume of the tool was calculated and subtracted from a blank workpiece. After subtraction, the start and exit angles of each discretized cutting disc were calculated.
The approximation proposed by Zeroudi et al. , calculates the (CWE) as a region delimited by three boundaries. The first was the relative position of the workpiece uncut surfaces. The second was the previous tool path considered without any tool deflection and with perfect surface finishes and has a cylindrical form. The third was the path of the previous tooth considered as a circular path. An analytical model was proposed by Sai et al. , where the cutter runout error was neglected.
The (CWE) zone between the tool and the machined part represents the integration limits used in the calculation of the cutting forces. The instantaneous cutting chip thickness can also be modeled correctly by knowing the area of engagement between the part and the tool. The validation of these models requires the modeling of the cutting forces. In the mechanistic method, cutting forces are assumed proportional to the uncut chip thickness or cutter swept volume.
Yucesan and Altnta  proposed a ball-end mill forces model based on a mechanistic relationship between cutting forces and chip load. This model discretizes the ball-end mill into a series of disks and describes in fairly rigorous detail the ball-end mill geometry and chip thickness calculation. In the study of Ko and Cho  the dynamic effect is considered in the chip thickness model. For more accuracy in the prediction of the cutting forces, Altintas and Lee  proposed a mechanistic cutting forces model for 3D ball-end milling using instantaneous cutting forces coefficients. These coefficients are independent of the cutting conditions and are set as a function of the instantaneous uncut chip thickness only. They consider the size effect produced near the tool tip at the low values of undeformed chip thickness.
Liu et al.  proposed a theoretical dynamic cutting forces model for ball-end milling using the integrated method. The elementary cutting forces components are integrated using the slices elements of the flute along the cutter-axis direction. The size effect of undeformed chip thickness and the influence of the effective rake angle are considered in the formulation of the differential cutting forces based on the theory of oblique cutting.
In this paper, the models of the instantaneous chip thickness and the (CWE) zone is developed for the calculation of the cutting forces for flat surfaces perpendicular to the tool axis and considering the runout error. The content is presented in the following section as follows: First, the geometrical description of the trajectory of an arbitrary point P in the (CWE) is presented in the local coordinate system attached to the center of the tool then in the local coordinate system attached to the spindle. Second, these coordinates are transformed to the global coordinate system attached to the workpiece. Third, based on the tooth trajectory and the geometry of the milled surface, the cutter workpiece engagement (CWE) is analyzed and the entrance and exit angles are extracted, then the instantaneous undeformed chip thickness is calculated. Finally, a simulation using Matlab software was proposed to predict the cutting forces based on a mechanistic approach.
This study deals with the effect of toolsurface inclination on cutting forces in ball-end milling. The following paper presents the determination of these cutting forces by using a predictive milling force model based on a thermomechanical modelling of oblique cutting. In this analytical model, the tool is supposed to be rigid and the working cutting edges are decomposed into a series of axial elementary cutting edges. At any active tooth element, the local chip formation is obtained from an oblique cutting process characterised by a local undeformed chip section and local cutting angles. This method is efficient to predict accurately the cutting force distribution on the helical ball-end mill flutes from tool geometry, pre-form surface, tool path, cutting conditions, material behaviour and friction at the toolchip interface. The radial run-out of the tool is also taken into account in the presented work. The model is applied to ball-end milling with straight tool paths but with various toolsurface inclinations. All the ramping and contouring, up- and down-cutting configurations are tested and the results are compared with data obtained from ball-end milling experiments performed on a three-axis CNC equipped with a Kistler dynamometer.