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.     

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.     

通用立铣刀真实切削轨迹下五轴铣削力计算

针对五轴铣削中刀具位姿变化和刀具类型差异所导致的铣削力预测难的问题，提出通用立铣刀五轴铣削力计算方法。基于通用立铣刀结构形式，建立通用立铣刀几何模型；综合考虑刀齿真实运动轨迹和刀具姿态变化，构建刀具瞬时切屑厚度模型；将刀具沿轴线方向等分成若干切削刃微元，并根据线性切削力假设建立刀具微元铣削力；将微元铣削力从刀具坐标系转换至工件坐标系下，并沿刀具轴向铣削深度进行积分，获得通用立铣刀的五轴铣削力模型；最后，在混联五轴数控加工实验平台上开展了铣削力测试。实测结果表明:所提铣削力计算方法正确有效，可作为后续五轴铣削工艺参数优选的理论依据。     

Machining of free-form surfaces. Part I: Analytical chip load

In the machining simulations of 3D free-form surfaces with ball-end mill, determinations of instantaneously changing tool-workpiece engagement regions and chip load play very critical roles in force and surface quality predictions. Cutting force models for ball-end milling are based on the undeformed chip load. Undeformed chip load can be constructed from the boundaries of instantaneous engagement region between the ball-end mill and workpiece. In order to predict the cutting forces accurately, precise determination of the varying engagement regions is important. In the literature, there are two main engagement region constructing methods; one is the Z-mapping and the other one is using solid modeler based on Boolean operation method. Both methods construct the engagement region within accuracy limits, on the other hand the computational time for these methods are long such that it is not possible to calculate the forces at the same time of CL-point construction. This paper presents development of a new analytical method for fast and accurate determination of the instantaneous engagement regions in the 3D machining of monotonic free-form surfaces. The analytical tool is grid size independent, thus it is much faster than discretization and Boolean methods. In addition to that, the analytical tool gives the most precise and exact engagement regions.     

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 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.     

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.     

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.     

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.     

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.     

Five-axis milling mechanics for complex free form surfaces

Accurate and fast prediction of machining forces is important in high performance cutting of free form surfaces that are commonly used in aerospace, automotive, biomedical and die/mold industries. This paper presents a novel and generalized approach for prediction of cutting forces in five-axis machining of parts with complex free form surfaces. Engagement simulations between cutter and part are performed precisely along the tool path by a recently developed boundary representation method. Moreover, mathematical model for five-axis milling mechanics is developed for any given solid model of parts with complex free form surfaces. Theoretical simulations and experimental validations show that cutting forces are predicted fast and precisely for five-axis machining of complex free form surfaces.     

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.     

Time domain model of plunge milling operation

Plunge milling operations are used to remove excess material rapidly in roughing operations. The cutter is fed in the direction of spindle axis which has the highest structural rigidity. This paper presents time domain modeling of mechanics and dynamics of plunge milling process. The cutter is assumed to be flexible in lateral, axial, and torsional directions. The rigid body feed motion of the cutter and structural vibrations of the tool are combined to evaluate time varying dynamic chip load distribution along the cutting edge. The cutting forces in lateral and axial directions and torque are predicted by considering the feed, radial engagement, tool geometry, spindle speed, and the regeneration of the chip load due to vibrations. The mathematical model is experimentally validated by comparing simulated forces and vibrations against measurements collected from plunge milling tests. The study shows that the lateral forces and vibrations exist only if the inserts are not symmetric, and the primary source of chatter is the torsional–axial vibrations of the plunge mill. The chatter vibrations can be reduced by increasing the torsional stiffness with strengthened flute cavities.     

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.     

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.     

Generalized modeling of chip geometry and cutting forces in multi-point thread turning

A generalized mechanics model of multi-point thread turning operations is presented. The cross section of the chip is determined from the thread profiles of the current and previous teeth as well as the infeed settings of the tool. The chip is discretized along the cutting edge, and the cutting force coefficients are evaluated for each element considering the varying effective oblique cutting angles and chip thickness. The nonlinear Kienzle force model is used to account for the effect of edge radius at low chip thickness values. Total cutting forces are obtained by resolving the elemental forces in the insert coordinate system, and integrating them along the engaged teeth. The experimentally validated generalized mechanics model can be used to predict the chip and cutting load distributions for multi-point inserts with custom thread profiles and infeed plans. The model can be used for both process planning and insert design.     

Predictive force model for ball-end milling and experimental validation with a wavelike form machining test

This paper presents a predictive force model for ball-end milling based on thermomechanical modelling of oblique cutting. The tool geometry is decomposed into a series of axial elementary cutting edges. At any active tooth element, the chip formation is obtained from an oblique cutting process characterised by local undeformed chip section and local cutting angles. This method predicts accurately the cutting force distribution on the helical ball-end mill flutes from the tool geometry, the pre-form surface, the tool path, the cutting conditions, the material behaviour and the friction at the tool-chip interface. The model is applied for a complex surface which is a wavelike form used as a validation machining test. The results are compared with experimental data obtained from ball-end milling tests performed on a 3-axis CNC equipped with a Kistler dynamometer.     

Feedrate scheduling strategies for free-form surfaces

Free-form machining is one of the commonly used manufacturing processes for several industries such as automobile, aerospace, die and mold industries. In 3D complicated free-form surfaces, it is critical, but often difficult, to select applicable cutting conditions to achieve high productivity while maintaining high quality of parts. It is essential to optimize the feedrate in order to improve the machining efficiency of the ball-end milling. Conservative constant feedrate values have been mostly used up to now since there was a lock of physical models and optimization tools for the machining processes.The common approach used in feedrate scheduling is material removal rate (MRR) model. In the MRR based approach, feedrate is inversely proportional to either average or instantaneous volumetric removal rate. Commonly used CAM programs and NC code generators based on only the geometric and volumetric analysis, but they do not concern the physics of the free-form machining process yet. The new approach that is also introduced in this paper is based on the mechanics of the process. In other words, the force-based models in which feedrate is set to values which keep either average or instantaneous machining forces to prescribed values. In this study, both feedrate scheduling strategies are compared theoretically and experimentally for 3D ball-end milling of free-form surfaces. It is shown that MRR based feedrate strategy outputs higher feedrate values compared to force based feedrate strategy. High feedrate values of the MRR strategy increase the cutting forces extensively which can be damaging to the part quality and to the CNC Machine.When the new force based feedrate-scheduling strategy introduced in this paper is used, it is shown that the machining time can be decreased significantly along the tool path. The force-based feedrate scheduling strategy is tested under various cutting conditions and some of the results are presented in the paper.     

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.     

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.