The prediction of dimensional error for sculptured surface productions using the ball-end milling process. Part 2: Surface generation model and experimental verification

This paper presents a surface generation model for sculptured surface productions using the ball-end milling process. In this model, machining errors caused by tool deflections are studied. As shown in Part 1 of this paper, instantaneous horizontal cutting forces can be evaluated from the cutting geometries using mechanistic force models. In this paper, a tool deflection model is developed to calculate the corresponding horizontal tool deflection at the surface generation points on the cutter. The sensitivity of the machining errors to tool deflections, both in magnitude and direction, has been analyzed via the deflection sensitivity of the surface geometry. Machining errors are then determined from the tool deflection and the deflection sensitivity of the designed surface. The ability of this model in predicting dimensional errors for sculptured surfaces produced by the ball-end milling process has been verified by a machining experiment. In addition to providing a means to predict dimensional accuracy prior to actual cutting, this surface generation model can also be used as a tool for quality control and machining planning.     

The prediction of cutting force in ball-end milling

Due to the development of CNC machining centers and automatic programming software, the ball-end milling have become the most widely used machining process for sculptured surfaces. In this study, the ball-end milling process has been analysed, and its cutting force model has been developed to predict the instantaneous cutting force on given machining conditions. The development of the model is based on the analysis of cutting geometry of the ball-end mill with plane rake faces. A cutting edge of the ball-end mill was considered as a series of infinitesimal elements, and the geometry of a cutting edge element was analysed to calculate the necessary parameters for its oblique cutting process assuming that each cutting edge was straight. The oblique cutting process in the small cutting edge element has been analysed as an orthogonal cutting process in the plane containing the cutting velocity and chip flow vectors. And with the orthogonal cutting data obtained from end turning tests on thin-walled tubes over wide range of cutting and tooling conditions, the cutting forces of ball-end milling could be predicted using the model. The predicted cutting forces have shown a fairly good agreement with test results in various machining modes.     

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.     

Simplified and efficient calibration of a mechanistic cutting force model for ball-end milling

Accurate evaluation of the empirical coefficients of a mechanistic cutting force model is critical to the reliability of the predicted cutting forces. This paper presents a simplified and efficient method to determine the cutting force coefficients of a ball-end milling model. The unique feature of this new method is that only a single half-slot cut is to be performed to calibrate the empirical force coefficients that are valid over a wide range of cutting conditions. The instantaneous cutting forces are used with the established helical cutting edge profile on the ball-end mill. The half-slot calibration cut enables successive determination of the lumped discrete values of the varying cutting mechanics parameters along the cutter axis whereas the size effect parameters are determined from the known variation of undeformed chip thickness with cutter rotation. The effectiveness of the present method in determining the cutting force coefficients has been demonstrated experimentally with a series of verification test cuts.     

Development of a reference cutting force model for rough milling feedrate scheduling using FEM analysis

Recently developed feedrate scheduling systems regulate cutting forces at the desired level by changing the feedrate to reduce the machining time and to avoid undesirable situations. For effective scheduling, an optimized criterion is required to adjust the feedrate. In this study, a method to obtain the most appropriate reference cutting force for rough milling was developed. The reference cutting force was determined by considering the transverse rupture strength of the tool material and the area of the rupture surface. A finite element method analysis was performed to accurately calculate the area of the rupture surface. Using the analyzed results, the effect of various cutting parameters on the chipping phenomenon was determined. The calculation method for the reference cutting force considered the area of the rupture surface, the effect of the rake angle, and the axial depth of the cut. The reference cutting force calculated using the developed model was applied to feedrate scheduling for pocket machining. The experimental results clearly show that the reference cutting force obtained from the proposed method met the desired constraints that guarantee higher productivity without tool failure.     

The use of cutting force and acoustic emission signals for the monitoring of tool insert geometry during rough face milling

The use of acoustic emission (AE) and cutting force has been applied to the detection of changes in milling tool insert geometries. These geometries, produced by precision grinding, were intended to simulate different forms of naturally occurring wear such as crater, notch and flank wear, local changes in rake angle and edge breakdown. The results indicate that both cutting force and AE can be used as indicators of changes in cutting tool geometry with consequent implications for diagnostic, geometry-specific, wear detection.     

Three-dimensional cutting force analysis based on the lower boundary of the shear zone. Part 1: Single edge oblique cutting

Traditional models of cutting based on Merchant's shear plane idealization are incapable of predicting any of the cutting force components without a priori knowledge of chip-tool friction. However, Rubenstein's work on orthogonal cutting has shown that this limitation can be avoided by utilizing the stress distributions on the lower boundary of the shear zone. The present work aims to extend this approach to oblique cutting with. single and two edged tools. This paper focuses on single edge oblique cutting whereas Part 2 analyses two edge cutting. It is assumed that the progressive deformation of the work material into chip material occurs within the effective plane. The resulting stress distributions on the lower boundary are integrated to yield expressions for estimating cutting forces from given tool and chip geometries. This provides a mechanism for predicting the power and lateral components of the cutting force in single edge oblique cutting. The predictions are verified against new and previously published experimental data.     

Uncertainty analysis and variation reduction of three dimensional coordinate metrology. Part 1: geometric error decomposition

In this paper our study focuses on the uncertainty analysis and variation reduction of coordinate system estimation using discrete measurement data and is associated with the applications that deal with parts produced by end-milling processes and having complex geometry. This paper consists of three parts. Since the uncertainty of the estimated coordinate transformation arises from the geometric errors on a part surface, Part 1 is devoted to the study of surface geometric errors. In this study, according to the characteristics of end-milling processes the sampled geometric error is divided into two components, and a decomposition procedure is developed for geometric error decomposition. The results of surface geometric error decomposition will be used in Part 2 for uncertainty analysis and in Part 3 for variation reduction.     

The harmonic fitting method for the assessment of the substitute geometry estimate error. Part II: statistical approach, machining process analysis and inspection plan optimisation

The Harmonic Fitting Method (HFM), presented in the first part of this work, is here used to describe the estimate error in statistical terms. In fact, the estimate error can be considered as a random variable that depends on three different random processes: the machining, the part positioning, and the measurement processes. With the HFM it is possible to determine the moments of the estimate error (systematic value and covariance matrix) as a function of the inspection plan. It will be demonstrated that the systematic part of the estimate error derives from the systematic part of the machined surface deviations, and that the random part derives from the variability of the machining and measurement processes. In particular, the causes of the machining process variability will be analysed in terms of harmonics, thus establishing a direct relationship between the machining process and the substitute geometry estimate error. Moreover, the possibility of obtaining the probability density function of the estimate error from the HFM will be analysed, together with the problem of the inspection plan optimisation. The HFM will be used to design the optimal inspection plan for a circular geometric feature, and it will be demonstrated that, when the part positioning is subjected to a rotation uncertainty, the estimate errors of the diameter and of the eccentricity are likely to follow a Gaussian and a Rayleigh distribution, respectively. A real case of turned shafts will be considered, and the HFM optimisation of the inspection plan will be discussed.     

Three-dimensional cutting force analysis based on the lower boundary of the shear zone. Part 2: Two edge oblique cutting

The problem of partitioning the overall cutting force between the two active cutting edges in two edge oblique cutting, although of crucial importance in modelling the temperature and wear rates prevailing in form cutting, has so far eluded solution. This paper presents two solutions to the problem: one utilizing the Merchant shear plane (MSP) approach and the other utilizing the solution developed in Part 1 for single edge oblique cutting based on the lower boundary (LB) of the shear zone. Both approaches avoid the anomalies encountered in previous attempts at solving the partitioning problem. This was done through the simultaneous application of the principles of chip equilibrium, force-velocity collinearity and chip interaction. Arguments are presented to indicate that the LB approach is more reliable than that of the MSP approach. Further, the LB approach is able to predict the power component of the overall cutting force in two edge cutting just as in the case of single edge cutting.     

An in-depth analysis of the synchronization between the measured and predicted cutting forces for developing instantaneous milling force model

This paper proposes an analytical approach to synchronize the measured and predicted cutting forces for calibrating instantaneous cutting force coefficients that vary with the instantaneous uncut chip thickness in general end milling. Essential issues such as the synchronization criterion, phase determination of measured cutting forces, specification of calibration experiments and related cutting parameters are highlighted both theoretically and numerically to ensure the calibration accuracy. A closed-form criterion is established to select cutting parameters ensuring the single tooth engagement. Numerical cutting simulations and experimental test results are compared to validate the proposed approach.     

Determination of the chip geometry, cutting force and roughness in free form surfaces finishing milling, with ball end tools

CAD/CAM systems offer various possibilities for finishing milling of parts such as dies and moulds, turbine blades and other high quality components, but most of them do not take into account the surface topomorphy expected, which is significantly affected, among others, by the milling kinematics and the contact conditions between the tool and the workpiece. In order to predict the workpiece roughness in multiaxis finishing milling with ball end tools, the computer supported milling simulation algorithm ‘ ’ was developed. By means of this algorithm, considering the individual movements of the cutting tool and of the workpiece due to the milling kinematics, the undeformed chip geometry, the cutting force components, the tool deflections and the final surface topomorphy expected are determined. Numerous investigations concerning the parameters mentioned above, with various workpiece materials have been carried out in order to determine the correlation of the experimental results with the corresponding calculated ones with the aid of the algorithm. Moreover the algorithm validity was extensively evaluated in milling of free form surfaces of large hydroturbine blades. The convergence between the experimental and the related calculated surface topomorphies by means of the computer program was found out to be satisfactory. Thus, the prediction of appropriate cutting conditions and milling kinematics to fulfill surface topomorphy requirements was enabled.     

Influence of the cutting edge rounding on the chip formation process: Part 1. Investigation of material flow, process forces, and cutting temperature

In order to meet the ever-increasing demand for high quality and low cost products, machining processes with geometrically defined cutting edges such as high speed cutting, or hard turning are being used. Due to the fact that cutting is accomplished through a physical interaction between the cutting edge and the workpiece, the characteristics of the cutting edge itself play a key role in influencing the machining process, which determines the product quality and the tool life. As a result, cutting edge design has attracted the focus of many researchers, and crucial improvements have been achieved. Rounded cutting edges have been found to improve the tool life, and the product quality. However, to better understand the impact of prepared cutting edges on the aspects of the machining processes, and to produce tailored cutting edges for specific load profiles, further investigation on the influence of the cutting edge design on the machining processes needs to be carried out. In this study, the effect of symmetrically and asymmetrically rounded cutting edge on the material in the vicinity of the cutting edge has been investigated using finite element simulation techniques. The results obtained from this investigation show that process forces and material flow under the flank face are mainly influenced by the micro-geometry S_??. However, the magnitude and the location of maximum nodal temperature are influenced by S_?? as well as S_??.     

The harmonic fitting method for the assessment of the substitute geometry estimate error. Part I: 2D and 3D theory

The dimensional inspection of a part consists of the measurement of several points on the manufactured surface and the fitting of a substitute geometry. The estimated parameters of the substitute geometry, which are related to the position and dimensions of the measured surface, are then compared with the specifications, leading to the acceptance or rejection of the manufactured part. Consequently, the reliability of the dimensional inspection depends on the precision of the estimated substitute geometry. The error that affects the estimate (estimate error) is a function of several factors (number and distribution of sampled points, measurement error, machining deviations, etc.), but the analytic expression of this relationship is still unknown.The first part of this work presents the Harmonic Fitting Method (HFM) for the assessment of the estimate error. It will be demonstrated that with the HFM the closed analytic relationship between the estimate error and the most important factors can be determined. This relationship is extremely simple when using the matrix notation, since the estimate error can be expressed as the product of different matrices, each one accounting for a single factor. In the first part of the work the general theory of the HFM will be presented and discussed for 2D features (line and circle) and 3D features (plane, cylinder and cone). In the second part the statistic approach and the machining process analysis will be presented, which are the basis for the practical use of the HFM for the inspection plan optimisation. The HFM will be applied to real data to optimise the inspection of circular geometric elements, which are particularly relevant for the mechanical industry.     

Model development for the prediction of surface topography generated by ball-end mills taking into account the tool parallel axis offset. Experimental validation

This paper presents a model for the topography prediction of ball-end milled surfaces, considering the tool parallel axis offset. First, the equations of cutting edge trajectories and the envelope equation of the material swept by the tool are derived. Later, the trajectories are cut by planes perpendicular to the feed direction obtaining a set of transcendental equations that are solved by transforming them to polynomial equations through Chebyshev expansions. This procedure presents the advantage on previous models in literature of not requiring any starting point to achieve the solution. Finally, experimental results are presented and compared to the model predictions.     

On the dynamics of ball end milling: modeling of cutting forces and stability analysis

This paper presents a dynamic force model and a stability analysis for ball end milling. The concept of the equivalent orthogonal cutting conditions, applied to modeling of the mechanics of ball end milling, is extended to include the dynamics of cutting forces. The tool is divided into very thin slices and the cutting force applied to each slice is calculated and summed for all the teeth engaged. To calculate the instantaneous chip thickness of each tooth slice, the method of regenerative chip load calculation which accounts for the effects of both the surface undulations and the instantaneous deflection is used. To include the effect of the interference of the flank face of the tool with the finished surface of the work, the plowing force is also considered in the developed model. Experimental cutting forces are obtained using a five-axis milling machine with a rotary dynamometer. The developed dynamic model is capable of generating force and torque patterns with very good agreement with the experimental data. Stability of the ball end milling in the semi-finishing operation of die cavities is also studied in this paper. The tangential and radial forces predicted by the method of equivalent orthogonal condition are fitted by the equations F_t = K_t(Z)bh_av and F_r = K_r(Z)F_t, where b is the depth of cut and h_av is the average chip thickness along the cutting edge and Z is the tool axis coordinate. The polynomial functions K_t(Z) and K_r(Z) are the cutting force constants. The interdependency of the axial and radial depths of cut in ball end milling results in an iterative solution of the characteristic equation for the critical width of cut and spindle speed. In addition, due to different cutting characteristics of the cutting edge at different heights of the ball nose, stability lobes are represented by surfaces. Comparison of the time domain simulation for the shoulder removal process in die cavity machining with the analytical predictions shows that the proposed method is capable of accurate prediction of the stability lobes.     

Development of a virtual machining system, part 2: prediction and analysis of a machined surface error

In part 2 of this three-part paper, a newly developed method that predicts the three-dimensional machined surface errors generated during the peripheral end milling process is presented. From the cutting force prediction system of Part 1, since the uncut chip thickness is calculated by tracing the movement of the cutter, the positions at which the cutting edges pass over the workpiece surface can readily be obtained. In this part of the paper these positions are used to construct surface error maps. In addition, by using the estimated locations of the peak and valley values of the cutting force component normal to the machined surface, a quantitative analysis of the machined surface error is given and followed by theoretical explanations. A series of machining tests on aluminum workpieces were conducted to validate the effectiveness of the model. The predicted cutting forces and surface errors were found in good agreement with their measured counterparts.     

Bandsawing. Part I: cutting force model including effects of positional errors, tool dynamics and wear

This article presents a mechanical cutting force model for bandsawing. The model describes the variation in cutting force between individual teeth and relates it to initial positional errors, tool dynamics and edge wear. Bandsawing is a multi-tooth cutting process, and the terminology of the cutting action is discussed and compared with other cutting processes. It will also be shown that the setting pattern and the preset feed govern the cutting data.     

Evaluation of the total cutting force in drilling of CFRP: a novel experimental method for the analysis of the cutting mechanism

The CFRP drilling process has not yet been fully mastered, which is due to the fact that it is not possible to measure all the cutting force components. In order to gain new insights into the drilling process, a novel experimental setup is being developed in order to record all cutting force components (cutting force, feed force, passive force). The results show that the force components are strongly dependent on the fiber cutting angle θ and the wear condition. Thereby, it will be possible to draw conclusions about the cutting mechanics in the drilling of unidirectional CFRP based on the transformation of the forces perpendicular and parallel to the fiber.     

The effects of process faults and misalignments on the cutting force system and hole quality in reaming

A chip thickness and cutting force model that considers the deflection of the tool and the regenerative effect resulting from the presence of process faults and misalignments has been developed for the reaming process. Through a series of experiments, the model has been calibrated and validated. The model predicts tool displacement, torque, thrust, X and Y forces, and the average radius of the reamed hole. The developed model is also shown to be capable of being used as a basis for the on-line detection of process faults present in the system.