Prediction of cutting forces in three and five-axis ball-end milling with tool indentation effect

Simulation of multi-axis ball-end milling of dies, molds and aerospace parts with free-form surfaces is highly desirable in order to optimize the machining processes in virtual environment ahead of costly trials. This paper presents a mechanics model that predicts the cutting forces in feed (x), normal (y) and axial (z) directions by modeling the chip thickness distribution, and cutting and indentation mechanics. The shearing forces are based on commonly known cutting mechanics models. The indentation of the cutting edge into the work material is modeled analytically by considering elasto-plastic deformation of the work material pressed by a rigid cutting tool edge with a positive or negative rake angle. The distribution of chip thickness and geometry of indentation zone are evaluated by considering five-axis motion of the tool along the toolpath. The proposed model has been experimentally validated in plunge indentation, as well as in three and five-axis ball-end milling of free-form surfaces. The prediction of axial (z) cutting forces is shown to be improved significantly when the proposed indentation model is integrated into the mechanics of ball-end milling.     

Prediction of ball-end milling forces from orthogonal cutting data

The mechanics of cutting with helical ball-end mills are presented. The fundamental cutting parameters, the yield shear stress, average friction coefficient on the rake face and shear angle are measured from a set of orthogonal cutting tests at various cutting speeds and feeds. The cutting forces are separated into edge or ploughing forces and shearing forces. The helical flutes are divided into small differential oblique cutting edge segments. The orthogonal cutting parameters are carried to oblique milling edge geometry using the classical oblique transformation method, where the chip flow angle is assumed to be equal to the local helix angle. The cutting force distribution on the helical ball-end mill flutes is accurately predicted by the proposed method, and the model is validated experimentally and statistically by conducting more than 60 ball-end milling experiments.     

An improved process simulation system for ball-end milling of sculptured surfaces

A simulation system is developed in this paper, which deals with the geometry and mechanics of machining with ball-end milling cutters. The geometry of the workpiece, the cutter, and the cutter/workpiece engagement is modeled using a geometric simulation system. This module uses a commercial solid modeler (ACIS) as a geometric engine and automatically extracts the critical geometric information required for the physical simulation system. To calculate the instantaneous cutting forces, a new mechanistic force model is developed. This force model takes into account the variations of the cutting coefficients along the cutting edge, and considers the variations of the rake angle and the chip flow direction on the rake face. The calibration of the developed model is performed for half-immersion ball-end milling operation. The applicability of the developed system is verified experimentally for various up-hill angles. It is shown that as the up-hill angle increases, the ball-nose tip engagement decreases which in turn significantly affects the magnitude of the resultant forces. Also, lower cutting forces and powers are experienced if cutting with the vicinity of the tool tip is avoided.     

The prediction of cutting forces in the ball-end milling process—II. Cut geometry analysis and model verification

This paper presents a model for the prediction of cutting forces in the ball-end milling process. The steps used in developing the force model are based on the mechanistic principles of metal cutting. The cutting forces are calculated on the basis of the engaged cut geometry, the underformed chip thickness distribution along the cutting edges, and the empirical relationships that relate the cutting forces to the undeformed chip geometry. A simplified cutter runout model, which characterizes the effect of cutter axis offset and tilt on the undeformed chip geometry, has been formulated. A model building procedure based on experimentally measured average forces and the associated runout data is developed to identify the numerical values of the empirical model parameters for the particular workpiece/cutter combination.     

Prediction of cutting force distribution and its influence on dimensional accuracy in peripheral milling

Cutting force has a significant influence on the dimensional accuracy due to tool and workpiece deflection in peripheral milling. In this paper, the authors present an improved theoretical dynamic cutting force model for peripheral milling, which includes the size effect of undeformed chip thickness, the influence of the effective rake angle and the chip flow angle. The cutting force coefficients in the model were calibrated with the cutting forces measured by Yucesan [18] in tests on a titanium alloy, and the model was proved to be more accurate than the previous models. Based on the model, a few case studies are presented to investigate the cutting force distribution in cutting tests of the titanium alloy. The simulation results indicate that the cutting force distribution in the cut-in process has a significant influence on the dimensional accuracy of the finished part. Suggestions about how to select the cutter and the cutting parameters were given to get an ideal cutting force distribution, so as to reduce the machining error, meanwhile keeping a high productivity.     

Precise prediction of forces in milling circular corners

Pocket corner is the most typical characters of aerospace structure components. To achieve high-quality product and stable machining operation, manufacturer constantly seek to control the cutting forces in pocket corner milling process. This paper presents the cutting force in corner milling considering the precision instantaneous achievements of tool engagement angle and undeformed chip thickness. To achieve the actual milling tool engagement angle in corner milling process, the details of tool–corner engagement relationship are analyzed considering the elements of tool trajectory, tool radius, and corner radius. The actual undeformed chip thicknesses in up and down milling operations are approached on account of the trochoid paths of adjacent teeth by a presented iteration algorithm. Error analysis shows that the presented models of tool engagement angle and undeformed chip thickness have higher precision comparing with the traditional models. Combined with the cutting force coefficients fitted by a series of slot milling tests, the predicted cutting force in milling titanium pocket with different corner structure and milling parameters are achieved, and the prediction accuracy of the model was validated experimentally and the obtained predict and the experiment results were found in good agreement.     

The prediction of dimensional error for sculptured surface productions using the ball-end milling process. Part 1: Chip geometry analysis and cutting force prediction

The study of machining errors caused by tool deflection in the balkend milling process involves four issues, namely the chip geometry, the cutting force, the tool deflection and the deflection sensitivity of the surface geometry. In this paper, chip geometry and cutting force are investigated. The study on chip geometry includes the undeformed radial chip thickness, the chip engagement surface and the relationship between feed boundary and feed angle. For cutting force prediction, a rigid force model and a flexible force model are developed. Instantaneous cutting forces of a machining experiment for two 2D sculptured surfaces produced by the ball-end milling process are simulated using these force models and are verified by force measurements. This information is used in Part 2 of this paper, together with a tool deflection model and the deflection sensitivity of the surface geometry, to predict the machining errors of the machined sculptured surfaces.     

Predictive cutting force model in complex-shaped end milling based on minimum cutting energy

A force model is presented to predict the cutting forces and the chip flow directions in cuttings with complex-shaped end mills such as ball end mills and roughing end mills. Three-dimensional chip flow in milling is interpreted as a piling up of the orthogonal cuttings in the planes containing the cutting velocities and the chip flow velocities. Because the cutting thickness changes with the rotation angle of the edge in the milling process, the surface profile machined by the previous edge inclines with respect to the cutting direction. The chip flow model is made using the orthogonal cutting data with taking into account the inclination of the pre-machined surface. The chip flow direction is determined so as to minimize the cutting energy, which is the sum of the shear energy on the shear plane and the friction energy on the rake face. Then, the cutting force is predicted for the chip flow model at the minimum cutting energy. The predicted chip flow direction changes not only with the local edge inclination but also with the cutting energy consumed in the shear plane cutting model. The cutting processes with a ball end mill and a roughing end mill are simulated to verify the predicted cutting forces in comparison with the measured cutting forces.     

A model for thread milling cutting forces

This paper presents a mechanistic model for prediction of the thread milling forces. The mechanics of cutting for thread milling is analyzed similar to the end milling process but with modified cutting edge geometry. The chip thickness and cutting force models are developed considering the unique geometry of the tool. The model has been calibrated for 6061 Aluminum and validated. The effects of tool and thread geometry have been studied using the model.     

Mechanics of boring processes—Part I

Mechanics of boring operations are presented in the paper. The distribution of chip thickness along the cutting edge is modeled as a function of tool inclination angle, nose radius, depth of cut and feed rate. The cutting mechanics of the process is modeled using both mechanistic and orthogonal to oblique cutting transformation approaches. The forces are separated into tangential and friction directions. The friction force is further projected into the radial and feed directions. The cutting forces are correlated to chip area using mechanistic cutting force coefficients which are expressed as a function of chip-tool edge contact length, chip area and cutting speed. For tools which have uniform rake face, the cutting coefficients are predicted using shear stress, shear angle and friction coefficient of the material. Both approaches are experimentally verified and the cutting forces in three Cartesian directions are predicted satisfactorily. The mechanics model presented in this paper is used in predicting the cutting forces generated by inserted boring heads with runouts and presented in Part II of the article [1].     

Mechanics of boring processes—Part II—multi-insert boring heads

Mechanics of boring operations are presented in the paper. The distribution of chip thickness along the cutting edge is modeled as a function of tool inclination angle, nose radius, depth of cut and feed rate. The cutting mechanics of the process is modeled using both mechanistic and orthogonal to oblique cutting transformation approaches. The forces are separated into tangential and friction directions. The friction force is further projected into the radial and feed directions. The cutting forces are correlated to chip area using mechanistic cutting force coefficients which are expressed as a function of chip-tool edge contact length, chip area and cutting speed. For tools which have uniform rake face, the cutting coefficients are predicted using shear stress, shear angle and friction coefficient of the material. Both approaches are experimentally verified and the cutting forces in three Cartesian directions are predicted satisfactorily. The mechanics model presented in this paper is used in predicting the cutting forces generated by inserted boring heads with runouts and presented in Part II of the article [1].     

Dynamic response analysis in end milling using pre-twisted beam finite elements

This paper develops an analytical model for estimating the dynamic responses in end milling, i.e. dynamic milling cutter deflections and cutting forces, by using the finite-element method along with an adequate end milling-cutting force model. The whole cutting system includes the spindle, the bearings and the cutter. The spindle is modelled structurally with the Timoshenko-beam element, the milling cutter with the pre-twisted Timoshenko-beam element due to its special geometry, and the bearings with lumped springs and dampers. Because the damping matrix in the resulting finite-element equation of motion for the whole cutting system is not one of proportional damping due to the presence of bearing damping, the state-vector approach and the convolution integral is used to find the solution of the equation of motion. To assure the accuracy of prediction of the dynamic response, the associated cutting force model should be sufficiently precise. Since the dynamic cutting force is proportional to the chip thickness, a quite accurate alogorithm for the calculation of the variation of the chip thickness due to geometry, run-out and spindle-tool viration is developed. A number of dynamic cutting forces and tool deflections obtained from the present model for various cutting conditions are compared with the experimental and analytical results available in the literature, good agreement being demonstrated for these comparisons. The present model is useful, therefore, for the prediction of end milling instability. Also, the tool deflections obtained using the pre-twisted beam element are found to be smaller than those obtained using the straight beam element without pre-twist angle. Hence neglecting the pre-twist angle in the structural model of the milling cutter may overestimate the tool deflections.     

A method for the identification of the specific force coefficients for mechanistic milling simulation

This paper presents a new method to obtain the specific cutting coefficients needed to predict the milling forces using a mechanistic model of the process. The specific coefficients depend on the tool–material couple, the cutting conditions and the geometry of the tool, being usually calculated applying the force model in an inverse way. The most used inverse method is based on the calculation of the average cutting force per revolution values measured in a series of slot machining tests at different feed rates. In this research work, the inverse method is applied using the instantaneous cutting force values, solving the equations system by a constrained least squares fitting method. Furthermore, the cutting force and specific cutting coefficients relation with rake angle and chip thickness is analysed. The results are validated by the comparison of the simulations and experiments in orthogonal cutting test, showing the advantages of using the new method.     

Modelling the cutting edge radius size effect for force prediction in micro milling

This paper presents a theoretical model for cutting force prediction in micro milling, taking into account the cutting edge radius size effect, the tool run out and the deviation of the chip flow angle from the inclination angle. A parameterization according to the uncut chip thickness to cutting edge radius ratio is used for the parameters involved in the force calculation. The model was verified by means of cutting force measurements in micro milling. The results show good agreement between predicted and measured forces. It is also demonstrated that the use of the Stabler's rule is a reasonable approximation and that micro end mill run out is effectively compensated by the deflections induced by the cutting forces.     

Generalized dynamic model of metal cutting operations

This paper presents a unified mathematical model which allows the prediction of chatter stability for multiple machining operations with defined cutting edges. The normal and friction forces on the rake face are transformed to edge coordinates of the tool. The dynamic forces that contain vibrations between the tool and workpiece are transformed to machine tool coordinates with parameters that are set differently for each cutting operation and tool geometry. It is shown that the chatter stability can be predicted simultaneously for multiple cutting operations. The application of the model to single-point turning and multi-point milling is demonstrated with experimental results.     

Simulation of cutting process in peripheral milling by predictive cutting force model based on minimum cutting energy

The cutting force and the chip flow direction in peripheral milling are predicted by a predictive force model based on the minimum cutting energy. The chip flow model in milling is made by piling up the orthogonal cuttings in the planes containing the cutting velocities and the chip flow velocities. The cutting edges are divided into discrete segments and the shear plane cutting models are made on the segments in the chip flow model. In the peripheral milling, the shear plane in the cutting model cannot be completely made when the cutting point is near the workpiece surface. When the shear plane is restricted by the workpiece surface, the cutting energy is estimated taking into account the restricted length of the shear plane. The chip flow angle is determined so as to minimize the cutting energy. Then, the cutting force is predicted in the determined chip flow model corresponding to the workpiece shape. The cutting processes in the traverse and the contour millings are simulated as practical operations and the predicted cutting forces verified in comparison with the measured ones. Because the presented model determines the chip flow angle based on the cutting energy, the change in the chip flow angle can be predicted with the cutting model.     

A new thermomechanical model of cutting applied to turning operations. Part I. Theory

In this paper, an analytical approach is used to model the thermomechanical process of chip formation in a turning operation. In order to study the effects of the cutting edge geometry, it is important to analyse its global and local effects such as the chip flow direction, the cutting forces and the temperature distribution at the rake face. To take into account the real cutting edge geometry, the engaged part in cutting of the rounded nose is decomposed into a set of cutting edge elements. Thus each elementary chip produced by a straight cutting edge element, is obtained from an oblique cutting process. The fact that the local chip flow is imposed by the global chip movement is accounted for by considering appropriate interactions between adjacent chip elements. Consequently, a modified version of the oblique cutting model of Moufki et al. [Int. J. Mech. Sci. 42 (2000) 1205; Int. J. Mach. Tools Manufact. 44 (9) (2004) 971] is developed and applied to each cutting edge element in order to obtain the cutting forces and the temperature distributions along the rake face. The material characteristics such as strain rate sensitivity, strain hardening and thermal softening, the thermomechanical coupling and the inertia effects are taken into account in the modelling. The model can be used to predict the cutting forces, the global chip flow direction, the surface contact between chip and tool and the temperature distribution at the rake face which affects strongly the tool wear. Part II of this work consists in a parametric study where the effects of cutting conditions, cutting edge geometry, and friction at the tool–chip interface are investigated. The tendencies predicted by the model are also compared qualitatively with the experimental trends founded in the literature.     

Effect of cutter runout on process geometry and forces in peripheral milling of curved surfaces with variable curvature

This paper presents a novel method for cutting force modeling related to peripheral milling of curved surfaces including the effect of cutter runout, which often changes the rotation radii of cutting points. Emphasis is put on how to efficiently incorporate the continuously changing workpiece geometry along the tool path into the calculation procedure of tool position, feed direction, instantaneous uncut chip thickness (IUCT) and entry/exit angles, which are required in the calculation of cutting force. Mathematical models are derived in detail to calculate these process parameters in occurrence of cutter runout. On the basis of developed models, some key techniques related to the prediction of the instantaneous cutting forces in peripheral milling of curved surfaces are suggested together with a whole simulation procedure. Experiments are performed to verify the predicted cutting forces; meanwhile, the efficiency of the proposed method is highlighted by a comparative study of the existing method taken from the literature.     

Development of a thermomechanical cutting process model for machining process simulations

A thermomechanical model for cutting processes is presented. The deformation in the shear zone is represented using Johnson-Cook material model. The rake contact is modeled using sticking and sliding zones, and their lengths are also predicted. The parameters of the material model and the friction coefficient on the rake are directly identified from a few number of orthogonal cutting tests. The model can predict cutting forces, shear angle and stress, pressure distribution and contact lengths on the rake face and temperature distribution. The application of the model to common operations such as turning and multi-axis milling is also presented with experimental verification, and satisfactory results are obtained.     

Process geometry modeling with cutter runout for milling of curved surfaces

Prediction of cutting forces and machined surface error in peripheral milling of curved geometries is non-trivial due to varying workpiece curvature along tool path. The complexity in this case, arises due to continuously changing process geometry as workpiece curvature varies along tool path. In the presence of cutter runout, the situation is further complicated owing to changing radii of cutting points. The present work attempts to model process geometry in machining of curved geometries and in the presence of cutter runout. A mathematical model computing process geometry parameters which include cutter/workpiece engagements and instantaneous uncut chip thickness in the presence of cutter runout is presented. The developed model is more realistic as it accounts for interaction of cutting tooth trajectory with that of preceding teeth trajectories in computing process geometry. Computer simulation studies carried for this purpose has shown that it is essential to account for teeth trajectory interactions for accurate prediction of process geometry parameters. This aspect is further confirmed with machining experiments, which were conducted to validate this aspect. From the outcomes of present work, it is clearly seen that the computation of process geometry during machining of curved geometries and in presence of cutter runout is not straightforward and requires a systematic approach as presented in this paper.