Identification of cutter offset in end milling without a prior knowledge of cutting coefficients

This paper presents a method for the identification of cutter offset through milling force without requiring the specific cutting coefficients to be known as priori. The total milling force in the presence of cutter offset is first obtained on the basis of dual cutting mechanisms, where the local force is comprised of a constant plowing force and a linear shearing force proportional to the chip load under the cutter offset. The total milling force is synthesized through convolution and is shown to be the sum of three force components: the nominal chip shearing force component, the plowing force components and the offset related shearing force component. Fourier analysis of these force components reveals the effects of offset geometry and yields formulas for the identification of offset geometry. The identification process requires only two cutting tests and the evaluation of two algebraic expressions; the shearing constants are found from the average forces of cutting tests and the offset geometry is identified from the force component at the spindle frequency. Through numerical simulation and experimental results, the efficacy of the identification method is demonstrated; the effects of feed per tooth and cutting depths on the accuracy of the model are investigated and criteria for the appropriate selection of these parameters are suggested.     

Identification of shearing and ploughing cutting constants from average forces in ball-end milling

This paper presents an analytical model for the direct identification of global shearing and ploughing cutting constants from measured average cutting forces in ball-end milling. This model is based on the linear decomposition of elemental local cutting forces into a shearing component and a ploughing component. Then, a convolution integral approach is used to obtain the average cutting forces leading to a concise and explicit expression for the global shearing and ploughing cutting constants in terms of axial depth of cut, cutter radius and average milling forces. The model is verified by comparisons with an existing force model of variable cutting coefficients. Cutting constants are identified through milling experiments and the prediction of cutting forces from identified cutting constants coincides with the experimental measurements. A model for identifying the lumped shearing constants is obtained as a subset of the presented dual mechanism model. Experimental results indicate that a model with dual-mechanism cutting constants predicts the ball-end milling forces with better accuracy than the lumped force model.     

An analytical force model with shearing and ploughing mechanisms for end milling

An analytical force model with both shearing and ploughing mechanisms is established for the end milling processes. The elemental forces are defined as the linear combination of shearing and ploughing forces in six cutting constants. The analytical model for the total milling forces in the angular and frequency domain are derived by convolution approach and Fourier transform respectively and are expressed as the superposition of the shearing force component and ploughing force component. This dual-mechanism model is analyzed and discussed in the frequency domain and compared with the lumped shear model. An expression is derived for identifying the cutting constants of the dual-mechanism model from the average milling forces. Explicit inclusion of ploughing force in the model is shown to result in better predictive accuracy and yields a linear force model with constant cutting coefficients. Experiments verify the accuracy and the frequency analysis of the dual-mechanism model and show that cutting constants for the dual-mechanism model are fairly independent of chip thickness.     

Cutting force prediction of sculptured surface ball-end milling using Z-map

The cutting force in ball-end milling of sculptured surfaces is calculated. In sculptured surface machining, a simple method to determine the cutter contact area is necessary since cutting geometry is complicated and cutter contact area changes continuously. In this study, the cutter contact area is determined from the Z-map of the surface geometry and current cutter location. To determine cutting edge element engagement, the cutting edge elements are projected onto the cutter plane normal to the Z-axis and compared with the cutter contact area obtained from the Z-map. Cutting forces acting on the engaged cutting edge elements are calculated using an empirical method. Empirical cutting mechanism parameters are set as functions of cutting edge element position angle in order to consider the cutting action variation along the cutting edge. The relationship between undeformed chip geometry and the cutter feed inclination angle is also analyzed. The resultant cutting force is calculated by numerical integration of cutting forces acting on the engaged cutting edge elements. A series of experiments were performed to verify the proposed cutting force estimation model. It is shown that the proposed method predicts cutting force effectively for any geometry including sculptured surfaces with cusp marks and a hole.     

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.     

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.     

Mechanistic model for the reaming process with emphasis on process faults

A mechanistic model is developed to predict the cutting forces for an arbitrary reamer geometry. Inputs to the model include: tool geometry, feed, speed, initial hole geometry, and process faults including parallel offset runout, spindle tilt, their respective locating angles, and tool/hole axis misalignment. Given these input parameters, the model predicts torque, thrust, and radial forces. The cutting edges of the reamer are divided into elements and the elemental forces are determined from a fundamental oblique cutting model. The model is calibrated over a range of feed, speed, and varying tool geometry. Model validation tests were conducted and model predictions match experimental data well. The effects of process faults on cutting forces are examined through model-based studies. It was found that parallel offset runout, spindle tilt, spindle tilt locating angle, and tool/hole axis misalignment have significant effects on the radial forces. These radial forces are shown to be correlated to hole quality.     

Angle and frequency domain force models for a roughing end mill with a sinusoidal edge profile

This paper presents analytical force models for a cylindrical roughing end mill with a sinusoidal edge profile in both the angle and frequency domains. Starting from a general expression for the chip thickness model, it is shown that under normal feed conditions, there exists only one cutting point at any axial position for an N-flute roughing end mill with its chip thickness N times that of a regular end mill, while the effective axial depth of cut is only 1/Nth that of a regular end mill. Based on the chip load model, the analytical force model is subsequently established through convolution integration of the elemental cutting function with the cutting edge geometry function in the angular domain, followed by Fourier analysis to obtain the frequency domain force model. Distinctive features of the milling forces for a roughing end mill are illustrated and compared with a regular end mill in the frequency as well as in the angular domain. The effects of the geometric parameters of a roughing end mill on the chip load distribution and on the features of milling force are discussed. The force models in both the frequency and angular domains are finally verified through 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.     

Compensation of progressive radial run-out in face-milling by spindle speed variation

This paper investigates the use of spindle speed variation to compensate for progressive radial run-out in the face-milling process. Machine setup errors like spindle axis tilt, cutter axis tilt and cutter center offset result in radial and axial run-out on the inserts. Cutter center offset, in particular, results in progressive radial run-out on the inserts which can be compensated by continuously varying the spindle speed in a sinusoidal pattern. The amplitude and phase of the sinusoidal trajectory significantly influence the capability of this technique for run-out compensation. A method is proposed for selection of optimal values of the speed variation amplitude and phase based on the idea of matching the chip load pattern under variable speed machining conditions in the presence of progressive radial run-out with that of constant speed machining in the absence of any run-out. Experiments performed with optimal values of the speed variation parameters show a significant reduction in the cutting force component at the spindle frequency and a shift in the frequency content of cutting force from the harmonics of the spindle frequency to those of the tooth passing frequency. An industrial implementation strategy of the proposed scheme is also presented that integrates run-out estimation with its subsequent compensation.     

Phase width analysis of cutting forces considering bottom edge cutting and cutter runout calibration in flat end milling of titanium alloy

This paper is concerned with the combined cutting effects of both flank and bottom edges based on a systematic study of the cutting force in flat end milling of the titanium alloy. Besides the flank edge, the bottom edge of the cutter is also found to be an important factor influencing the cutting force distributions and can lead to uniform phase widths for non-zero cutting forces even under considerable cutter runout. One such phenomenon of uniform phase width induced by the bottom edge for the cutting force is deeply revealed. To do this, the models for characterizing the cutting force coefficients related to both edges are established based on the measured instantaneous cutting forces, and cutter runout is considered in the computation of process geometry parameters such as cutter/workpiece engagements and instantaneous uncut chip geometry parameters. Novel algorithms for identifying the cutter runout parameters and the bottom uncut chip width are also developed. Results definitely show that the flank cutting force coefficients can be treated as constants and that size effect obviously exists in the bottom cutting force coefficients that can be characterized by a power function of the bottom uncut chip width.The proposed model is validated through a comparative study with the existing model and experiments. From the outcomes of the current work, it is clearly seen that the prediction of cutting forces for titanium alloy can resort to the proposed model instead of traditional ones.     

Feedrate scheduling for indexable end milling process based on an improved cutting force model

In CNC machining, an optimal process plan is needed for higher productivity and machining performance. This paper proposes a mechanistic cutting force model to perform feedrate scheduling that is useful in process planning for indexable end milling. Indexable end mills, which consist of inserts and a cutter body, have been widely used in the roughing of parts in the mold industry. The geometry and distribution of inserts compose a discontinuous cutting edge on the cutter body, and tool geometry of indexable end mill varies with axial position due to the geometry and distribution of inserts. Thus, an algorithm that calculates tool geometry data at an arbitrary axial position was developed. The developed cutting force model uses cutting-condition-independent cutting force coefficients and considers run out, cutter deflection, geometry variation and size effect for accurate cutting force prediction. Through feedrate scheduling, NC code is optimized to regulate cutting forces at given reference force. Experiments with general NC codes show the effectiveness of feedrate scheduling in process planning.     

Milling force prediction using a dynamic shear length model

Modelling of cutting forces in milling is often needed in machining automation. In this paper, a new method for the determination of the cutting forces in face milling is presented, which applies a predictive machining theory originally developed for orthogonal cutting to milling operations, with a dynamic shear length model developed and incorporated. The proposed dynamic shear length model is developed based on the analysis for the true tooth trajectories of a milling cutter, taking into account of the characteristic wavy surface effects in milling. The prediction for the cutting forces is carried out at each step of the angular increment of cutter rotation from input data of fundamental workpiece material properties, tool geometry and cutting conditions. Cutting forces at a cutter tooth can be predicted once the shear angle, shear length, shear plane area, and the shear flow stress along the shear length have been determined. The milling force prediction using the dynamic shear length model is verified through milling experimental tests. The sensitivity of the difference between the static and dynamic shear length models with respect to the feed per tooth and the cutter diameter is discussed.     

Identification of the specific cutting force for geometrically defined cutting edges and varying cutting conditions

Cutting force modeling is a major discipline in the research of cutting processes. The exact prediction of cutting forces is crucial for process characterization and optimization. Semi-empirical and mechanistic force models have been established, but the identification of the specific cutting force for a pair of tool and workpiece material is still challenging. Existing approaches are depending on geometrical idealizations and on an extensive calibration process, which make practical and industrial application difficult. For nonstandard tools and five axis kinematics there does not exist a reasonable solution for the identification problem.In this paper a co-operative force model for the identification of the specific cutting forces and prediction of integral forces is presented. The model is coupled bidirectionally with a multi-dexel based material removal model that provides geometrical contact zone information. The nonlinear specific forces are modeled as polynomials of uncut chip thickness. The presented force model is not subjected to principal restrictions on tool shape or kinematics, the specific force and phase shift are identified with help of least square minimization. The benefit of this technique is that no special calibration experiments are needed anymore, which qualifies the method to determine the specific forces simultaneously during the machining process. In this paper, experiments with different cutting conditions are analyzed and systematically rated. Finally, the method is validated by experiments using different cutting conditions.     

Cutting force prediction in ball end milling of sculptured surface with Z-level contouring tool path

This paper presents an approach to predict cutting force in 3-axis ball end milling of sculptured surface with Z-level contouring tool path. The variable feed turning angle is proposed to denote the angular position of feed direction within tool axis perpendicular plane. In order to precisely describe the variation of feed turning angle and cutter engagement, the whole process of sculptured surface milling is discretized at intervals of feed per tooth along tool path. Each segmented process is considered as a small steady-state cutting. For each segmented cutting, the feed turning angle is determined according to the position of its start/end points, and the cutter engagement is obtained using a new efficient Z-map method. Both the chip thickness model and cutting force model for steady-state machining are improved for involving the effect of varying feed turning angle and cutter engagement in sculptured surface machining. In validation experiment, a practical 3-axis ball end milling of sculptured surface with Z-level contouring tool path is operated. Comparisons of the predicted cutting forces and the measurements show the reliability of the proposed approach.     

Accurate 3-D cutting force prediction using cutting condition independent coefficients in end milling

The instantaneous uncut chip thickness and specific cutting forces have a significant effect on predictions of cutting force. This paper presents a systematic method for determining the coefficients in a three-dimensional mechanistic cutting force model—the cutting force coefficients (two specific cutting forces, chip flow angle) and runout parameters. Some existing models have taken the approach that the cutting force coefficients vary as a function of cutting conditions or cutter rotation angle. This paper, however, considers that the coefficients are affected only by the uncut chip thickness. The instantaneous uncut chip thickness is estimated by following the movement of the position of the center of a cutter. To consider the size effect, the present method derives the relationship between the re-scaled uncut chip thickness and the normal specific cutting force, K_n with respect to the cutter rotation angle, while the other two coefficients—frictional specific cutting force, K_f and chip flow angle, θ_c—remain constant. Subsequently, all the coefficients can be obtained, irrespective of cutting conditions. The proposed method was verified experimentally for a wide range of cutting conditions, and gave significantly better predictions of cutting forces.     

Estimation of cutter deflection and form error in ball-end milling processes

This paper presents a method to analyze the 3-dimensional form error of a ball-end milled surface due to the elastic compliance of the cutting tool. In order to estimate the form error in various cutting modes, the cutting force and the cutter deflection models including the effect of the surface inclination were established. The cutting forces were calculated by using the cutter contact area determined from the Z-map of the surface geometry and the current cutter location. The tool deflection responding to the cutting force was then calculated by considering the cutter and the holder stiffness. The cutter was modeled as a cantilever beam consisting of the shank and the flute. The stiffness of the holder was measured experimentally. Various experimental works have been performed to verify the validity of the proposed model. It is shown that the proposed method is capable of accurate prediction of cutting forces and the surface form error.     

Mechanistic identification of specific force coefficients for a general end mill

The paper presents expressions for semi-empirical mechanistic identification of specific cutting and edge force coefficients for a general helical end mill from milling tests at an arbitrary radial immersion. The expressions are derived for a mechanistic force model in which the total cutting force is described as a sum of the cutting and edge forces. Outer geometry of the end mill is described by a generalized mathematical model valid for a variety of end mill shapes, such as cylindrical, taper, ball, bull nose, etc. The derivations follow a procedure originally proposed for a cylindrical end mill. The procedure itself is improved by including the helix angle in evaluation of the average edge forces. The resulting expressions for the specific force coefficients are verified by simulations and experiments.     

Modeling of velocity-dependent chip flow angle and experimental analysis when machining 304L austenitic stainless steel with groove coated-carbide tools

This paper presents an analysis of experimental cutting forces and the study of the chip flow angle when machining 304L austenitic steel with a groove coated tool under dry condition. Experiments were conducted on a wide range of cutting conditions with a particular attention to ensure a great confidence in the obtained results. A detailed analysis of experimental cutting forces and the identification of empirical cutting force equations similar to that usually used for flat tools are proposed. The main focus of this work is on the study of chip flow angle deduced here from experimental cutting forces, considering that the chip flow direction is collinear to the friction force. From a comparison between experiments and two often used approaches, it appears that the experimental chip flow angle estimation, based on neglecting the complex tool geometry and adopting a zero rake angle, is bounded by the two considered modelings that renders useful for the proposed study. From experiments it is also observed an increase of the chip flow angle as the cutting velocity is increased. A velocity-dependent modeling with two distinct strategies of identification is then proposed in order to capture the cutting velocity effect on the chip flow angle.     

Modelling and simulation of chatter in milling using a predictive force model

A predictive time domain chatter model is presented for the simulation and analysis of chatter in milling processes. The model is developed using a predictive milling force model, which represents the action of milling cutter by the simultaneous operations of a number of single-point cutting tools and predicts the milling forces from the fundamental workpiece material properties, tool geometry and cutting conditions. The instantaneous undeformed chip thickness is modelled to include the dynamic modulations caused by the tool vibrations so that the dynamic regeneration effect is taken into account. Runge–Kutta method is employed to solve the differential equations governing the dynamics of the milling system for accurate solutions. A Windows-based simulation system for chatter in milling is developed using the predictive model, which predicts chatter vibrations represented by the tool-work displacements and cutting force variations against cutter revolution in both numerical and graphic formats, from input of tool and workpiece material properties, cutter parameters, machine tool characteristics and cutting conditions. The system is verified with experimental results and good agreement is shown.