An investigation of cutting forces in machining with worn ball-end mill

Reliable tool wear monitoring technique is one of the important aspects for achieving an integrated and self-adjusting manufacturing system. In this study, an analytical model is proposed to estimate the cutting forces, the tool geometry, and the chip geometry in relation to the flank wear, when milling with a ball-end mill. Modeling is based on thermomechanical modelling of oblique cutting. The worn tool geometry is decomposed into a series of axial elementary cutting edges. At any active tooth element, the flank wear geometry is calculated and the chip formation is obtained from an oblique cutting process characterised by local undeformed chip section and local cutting angles. Coated carbide ball-end tool, and a titanium workpiece material have been considered in this paper. The results found by using developed models have shown good agreement with experimental results.     

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.     

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.     

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.     

A chip formation based analytic force model for oblique cutting

An oblique cutting force model has been developed using an analytic orthogonal force model. The force model uses a thermo-visco-plastic material constitutive law to represent the shear stress during deformation of the material. The strains and strain rates used for defining the shear stress were obtained from chip formation and morphology derived from orthogonal cutting tests and has been extended to oblique cutting. A time domain simulation using the in-cut chip geometry to define the chip load area has been developed. The oblique force model was used to predict the cutting forces during ball milling of hardened AISI D2 tool steel. The predicted forces were verified experimentally and showed good correlation.     

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.     

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.     

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.     

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.     

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.     

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.     

Cutting force estimation in sculptured surface milling

Cutting force milling models developed up to now are mostly used for planar milling using end-mills. Only a reduced number of models applying ball-end mills have been developed. Furthermore these models usually only consider horizontal surface machining, even though the main application of ball-end mills is sculptured surface machining. This article proposes a model that is able to estimate the cutting forces in inclined surfaces machined both up-milling and down-milling. For this purpose a semi-mechanistic model has been developed that calculates the cutting forces based on a set of coefficients which depend on the material, the tool, the cutting conditions, the machining direction and the slope of the surface.A coordinate transformation has been included in order to consider the slope milling case with different cutting directions.The model has been tested on two materials, an aluminum alloy Al7075-T6 and a 52 HRC tool steel AISI H13. Validation tests have been carried out on inclined planes using different slopes and different machining directions.The results provide errors below 10% in most of the cases and both the value and shape of the predicted forces adjust the measured cutting force.     

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.     

复杂型面数控加工中的切削力预测

球头铣刀加工水平面的切削力的预测模型已经非常成熟,而对于球头铣刀加工复杂型面切削力预测的研究却很少。文章提出一种模型来预测复杂型面加工过程中的切削力。此模型通过坐标转换．使零件坐标系统与刀具坐标系统一致,并根据具体的材料,刀具,切削条件,加工方向以及表面斜率的一套参数来预测切削力。这个模型将为复杂型面加工条件的选择提供参考,并提高加工效率。     

A chatter free calibration method for determining cutter runout and cutting force coefficients in ball-end milling

The accuracy of cutting force coefficients plays an important role in predicting reliable cutting force, stability lobes as well as surface location error in ball-end milling. In order to avoid chatter risk of the traditional calibration test with an entire-ball-immersed cutting depth, a cylindrical surface milling method is proposed to calibrate the cutting force coefficients with the characteristics of low cutting depth and varying lead angle. A dual-cubic-polynomial function is also presented to describe the non-uniform cutting force coefficients of the ball part cutting edge and the nonlinear chip size effect on cutting force. The variation of the maximum chip thickness versus the lead angle is established with the consideration of cutter runout. According to the dependence of chip thickness on lead angle, a runout identification method is introduced by seeking the critical lead angle at which one of the cutter flutes is just thoroughly out of cut. Then, a lumped equivalent method is adopted for the low cutting depth condition so that the dual-cubic-polynomial model can be calibrated for the chip size effect and the cutting force coefficients respectively. The accuracy of the proposed calibration method has been validated experimentally with a series of milling tests. The stability examinations indicate that the proposed method has an evident chatter-free advantage, compared with that of varying cutting depth method.     

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.     

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.     

机器人铣削加工让刀误差建模与分析

在机器人切削加工工艺过程中,让刀误差是影响切削加工精度的重要因素之一。以球头铣刀铣削加工为研究对象,视其为准静态运动过程,根据机器人静弹性力学模型和球头铣刀切削力模型,建立了机器人切削过程的让刀误差数学模型。同时,提出了一种基于基因遗传算法的刀具姿态优化方法,以减小切削过程的让刀误差。最后,通过仿真分析了切削参数、刀具姿态和机器人刚度等因素对让刀误差的影响,验证了刀具姿态优化方法的可行性。     

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.