New procedures for calibration of instantaneous cutting force coefficients and cutter runout parameters in peripheral milling

In this paper, new procedures are proposed to calibrate the instantaneous cutting force coefficients and the cutter runout parameters for peripheral milling. By combining with optimization algorithm, i.e., the Nelder–Mead simplex method, detailed calibration schemes are derived for a mechanistic cutting force model in which the cutting force coefficients are described as the exponential functions of the instantaneous uncut chip thickness. Three different cutter runout models are considered in the calculation of instantaneous uncut chip thickness. Only one or two tests are required to perform the calibration. Experimental verifications are also conducted to validate the proposed procedures, and the results show that they are efficient and reliable. To see the effect of different runout models on milling process, comparisons among the predicted results under a wide range of cutting parameters are made to study the consistency and limitations of different models. It is found that the radial cutter runout model is a recommendable one for cutting force modelling.     

Numerical simulation and prediction of cutting forces in five-axis milling processes with cutter run-out

Radial cutter runout is a common issue in milling processes and has a direct effect on milling stability due to variations of resulting chip load and forces. This paper presents a new method to effectively model and predict the instantaneous cutting forces in 5-axis milling processes with radial cutter runout based on tool motion analysis. First, the undeformed chip thickness model taking runout effect into account is established under continuous change of cutter axis orientation by means of the sweep traces of cutter edges. Second, the engaged cutting edge is determined and cutting coefficients are subsequently calibrated. Finally, the method of identifying runout parameters from the measured cutting forces is proposed, and mechanistic method is then applied to predict the cutting force. Since this method is completely based on the relative motion analysis of tool-part, it can reduce the prediction errors of cutting forces effectively and is suitable for generic rotation cutters. Several validation examples are given under different cutting conditions to prove its effectiveness and accuracy. The results reveal that the developed method can predict the cutter forces with a high accuracy and has the ability to be used in simulations and optimizations of five-axis machining.     

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.     

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.     

Prediction of chatter stability for multiple-delay milling system under different cutting force models

This paper systematically studies the stability lobe prediction methods for the milling process with multiple delays, which are often induced by cutter runout. Emphasis is put on how to effectively incorporate the instantaneous cutting force model into the prediction procedure of stability lobes. Two original methods are proposed based on the vibration time history of the cutter motion, which is numerically obtained by time domain simulation. A comparison study is made with the existing method taken from the literature and experimental verifications are also carried out to validate both methods.In this study, two different instantaneous cutting force models together with the constant cutting force model are considered in the calculation of the cutting force coefficients during the simulation. The effects of different cutting force models on stability lobes, their consistencies and limitations are highlighted. At the same time, cutting force coefficients calibrated from three tests with different feeds per tooth are also considered in order to show the influences of cutting force coefficient's accuracy. It is found that both types of cutting force models and the calibration accuracy of the cutting force coefficients have great influences on the reliability of the stability lobes.     

Systematic study on cutting force modelling methods for peripheral milling

This paper systematically studies the cutting force modelling methods in peripheral milling process in the presence of cutter runout. Emphasis is put on how to efficiently calibrate the cutting force coefficients and cutter runout. Mathematical derivations and implementation procedures are carried out based on the measured cutting force or its harmonics from Fourier transformation. Five methods are presented in detail. In the first three methods the cutting force coefficients are assumed to be constants whereas in the last two they are taken as functions of instantaneous uncut chip thickness. The first method and the fifth one are taken from literatures for comparison. The second, the third and the fourth methods are original contributions, which are carried out with optimization ideas. The second method proceeds using the first and Nkth harmonic forces as the source signal while the third and the fourth are derived based on the measured cutting forces and its first harmonics. The engagement of the cutter with the workpiece is considered in these three new calibration procedures without the requirement of a prior knowledge of the actual cutter runout. Comparisons among the calibrated results from different methods are made to study the limitations and consistency of the presented methods. Experiments are also conducted to show the prediction ability of all methods.     

Mechanistic modeling of five-axis machining with a general end mill considering cutter runout

The accurate and fast prediction of cutting forces in five-axis milling of free-form surfaces remains a challenge due to difficulties in determining the varying cutter-workpiece engagement (CWE) boundaries and the instantaneous uncut chip thickness (IUCT) along the tool path. This paper proposes an approach to predict the cutting forces in five-axis milling process with a general end mill considering the cutter runout effect that is inevitable in the practical machining operations. Based on the analytical model of cutting edge combined with runout parameters, the expression of the rotary surface formed by each cutting edge undergoing general spatial motion is firstly derived. Then by extracting the feasible contact arc along the tool axis, a new arc-surface intersection method is developed to determine the CWE boundaries fast and precisely. Next, the circular tooth trajectory (CTT) model is developed for the calculation of the IUCT with a slight sacrifice of accuracy. In comparison with the true IUCT calculated by the trochoidal tooth trajectory model, the approximation error introduced by the circular assumption is negligible while the computational efficiency improves a lot. Finally, combining with the calibrated cutting coefficients and runout parameters, comprehensive formulation of the cutting force system is set up. Simulations and experimental validations of a five-axis flank milling process show that the novel CTT model possesses obvious advantages in computing efficiency and accuracy over the existing approaches. Rough machining of a turbo impeller is further carried out to test the practicability and effectiveness of the proposed mechanistic model.     

Sculpture surface machining: a generalized model of ball-end milling force system

A new mechanistic model is presented for the prediction of a cutting force system in ball-end milling of sculpture surfaces. The model has the ability to calculate the workpiece/cutter intersection domain automatically for a given cutter location (CL) file, cutter and workpiece geometries. Furthermore, an analytical approach is used to determine the instantaneous chip load (with and without runout) and cutting forces. In addition to predicting the cutting forces, the model also employs a Boolean approach for a given cutter, workpiece geometries, and CL file to determine the surface topography and scallop height variations along the workpiece surface which can be visualized in 3-D. The results of model validation experiments on machining Ti-6A1-4V are also reported. Comparisons of the predicted and measured forces as well as surface topography show good agreement.     

A cutting force model for finishing processes using helical end mills with significant runout

Analytical cutting force models play an important role in a wide array of simulation approaches of milling processes. The accuracy of the simulated processes directly depends on the predictive power of the applied cutting force model, which may vary under specific circumstances. End milling processes with small radial cutting depths, e.g. finishing processes, are particularly problematic. In this case, the tool runout, which is usually neglected in established cutting force models, can become quite significant. Within this article, well-known cutting force models are implemented for runout-prone finishing processes and modified by integrating additional parameters. A method is presented for how these additional runout parameters can be efficiently determined alongside commonly used cutting coefficients. For this purpose, a large number of milling experiments have been performed where the cutting forces were directly measured using a stationary dynamometer. The measured cutting forces were compared with the simulated cutting forces to verify and assess the modified model. By using the presented model and calibration method, cutting forces can be accurately predicted even for small radial cutting depths and significant tool runout.     

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.     

Efficient calibration of instantaneous cutting force coefficients and runout parameters for general end mills

This paper aims at developing a unified approach to identify the cutting force coefficients together with the cutter runout parameters for general end mills such as cylindrical, ball, bull nose ones, etc. The cutting forces that are modeled using the instantaneous cutting force coefficients are analyzed and separated into two terms: a nominal component independent of the runout and a perturbation component induced by the runout. The nominal component enables the calibration of the instantaneous cutting force coefficients whereas the runout parameters are determined from the perturbation component. The validity of the present method is demonstrated with simulation and experimented data.     

Prediction of cutting forces in milling of circular corner profiles

This paper proposes an approach to predict the cutting forces in peripheral milling of circular corner profiles in which varying radial depth of cut is encountered. The geometric relationship between an end mill and the corner profile is investigated and a mathematical model is presented to describe the different phases of the cutter/workpiece contact. The milling process for circular corner is discretized into a series of steady-state cutting processes, each with different radial depth of cut determined by the instantaneous position of the end mill relative to the workpiece. A time domain analytical model of cutting forces for the steady-state machining conditions is introduced to each segmented process for the cutting force prediction. The predicted cutting forces can be calculated in terms of tool/workpiece geometry, cutting parameters and workpirece material property, as well as the relative position of the tool to workpiece. Experiments are conducted and the measured forces are compared to the predictions for the verification of the proposed method.     

Tool deflection compensation in peripheral milling of curved geometries

This paper presents compensation of surface error due to cutting force-induced tool deflections in a peripheral milling process. Previous research attempts on this topic deal with error compensation in machining of straight geometries only. This paper is concerned with peripheral milling of variable curvature geometries where the workpiece curvature changes continuously along the path of cut. In the case of curved geometries, both process geometry and the cutting forces have shown to have strong dependence on workpiece curvature and hence variation of surface error along the path of cut. This calls for a different error compensation strategy than the one which is normally used for machining straight geometries. The present work is an attempt to improve accuracy in machining of curved geometries by use of CNC tool path compensation. Mechanistic model for cutting force estimation and cantilever beam model for cutter deflection estimation are used. The results based on machining experiments performed on a variety of geometries show that the dimensional accuracy can be improved significantly in peripheral milling of curved geometries.     

Modeling of process geometry in peripheral milling of curved surfaces

This paper presents an approach for modeling of the process geometry in peripheral milling of curved surfaces. The modeled process geometry involves feed direction, equivalent feedrate and cutter entry/exit angle. The equivalent feedrate, which is defined at the centroid of cutting cross-section, is proposed to measure the actual machining feedrate. The milling process of curved surface is discretized at intervals of feed per tooth for describing the variation of process geometry. The mathematical models for calculating the process geometry of each segmented cutting process are presented. Two same curved surfaces are machined. The tool paths are carried out with NURBS interpolation and traditional consecutive small line segments interpolation, respectively. The simulated and measured results are presented for the verification of the proposed method.     

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.     

Feed rate scheduling model considering transverse rupture strength of a tool for 3D ball-end milling

This paper presents an analytical model of off-line feed rate scheduling to determine desired feed rates for 3D ball-end milling. Off-line feed rate scheduling is presented as the advanced technology to regulate cutting forces through change of feed per tooth, which directly affects variation of uncut chip thickness. In this paper, the uncut chip thickness is calculated by following the movement of the position of the cutter center, which is determined by runout and cutter deflection. Also, since the developed cutting force model uses the cutting-condition-independent coefficients, off-line feed rate scheduling can be effectively performed regardless of continuous change of cutting conditions. Transverse rupture strength of the tool is used to determine the reference cutting force at which resultant cutting forces are regulated through feed rate scheduling. Experiments validated that the presented feed rate scheduling model reduced machining time drastically and regulated cutting forces at the reference cutting force.     

Development of a virtual machining system, part 1: approximation of the size effect for cutting force prediction

In this three-part paper, components of a virtual machining system for evaluating and optimizing cutting performance in -axis NC machining are presented. Part 1 describes a new method of calculating cutting-condition-independent coefficient and its application to the prediction of cutting forces over a wide range of cutting conditions. The prediction of the surface form error and transient cutting simulations, described in Parts 2 and 3, respectively, can be effectively performed based on the cutting force model with the improved size effect model that is presented in Part 1.

The relationship between the instantaneous uncut chip thickness and the cutting coefficients is calculated by following the movement of the center position of the cutter, which varies with nominal feed, cutter deflection and runout. The salient feature of the presented method is that it determines the cutting-condition-independent coefficients using experimental data processed for one cutting condition. The direct application of instantaneous cutting coefficient with size effects provides more accurate predictions of the cutting forces. A systematic comparison of the predicted and measured cutting forces over a wide range of cutting conditions confirms the validity of the proposed mechanistic cutting force and size effect models.     

Estimation and experimental validation of cutting forces in ball-end milling of sculptured surfaces

Chip thickness calculation has a key important effect on the prediction accuracy of accompanied cutting forces in milling process. This paper presents a mechanistic method for estimating cutting force in ball-end milling of sculptured surfaces for any cases of toolpaths and varying feedrate by incorporation into a new chip thickness model. Based on the given cutter location path and feedrate scheduling strategy, the trace modeling of the cutting edge used to determine the undeformed chip area is resulted from the relative part-tool motion in milling. Issues, such as the selection of the tooth tip and the computation of the preceding cutting path for the tooth tip, are also discussed in detail to ensure the accuracy of chip thickness calculation. Under different chip thicknesses cutting coefficients are regressed with good agreements to calibrated values. Validation tests are carried out on a sculptured surface with curved toolpaths under practical cutting conditions. Comparisons of simulated and experimental results show the effectiveness of the proposed method.     

Effect of direction of parameterization on cutting forces and surface error in machining curved geometries

The present paper investigates the effect of two variables, namely direction of parameterization and cutter diameter on process geometry, cutting forces, and surface error in peripheral milling of curved geometries. In machining of curved geometries where the curvature varies continuously along tool path, the process geometry variables, namely feed per tooth, engagement angle, and maximum undeformed chip thickness too vary along tool path. These variations will be different when a given geometry is machined from different parametric directions and with different cutter diameters. This difference in process geometry variations result in changed cutting forces and surface error along machined path. This aspect has been studied for variable curvature geometries by machining from both parametric directions and using cutters of different diameter. The computer simulation studies carried out show considerable amount of shift in the location of peak cutting forces with the change in cutting direction and cutter diameter, particularly in concave regions of workpiece geometry. A new parameter γ that relates the instantaneous curvature of workpiece with cutter radius is defined. The larger value of γ is an indicator of greater shift in the location of peak forces from the point of maximum curvature on the workpiece. The simulation results are validated by carrying out machining experiments with curved workpiece geometry and are found to be in good agreement.     

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.