Predictive cutting force model in complex-shaped end milling based on minimum cutting energy

A force model is presented to predict the cutting forces and the chip flow directions in cuttings with complex-shaped end mills such as ball end mills and roughing end mills. Three-dimensional chip flow in milling is interpreted as a piling up of the orthogonal cuttings in the planes containing the cutting velocities and the chip flow velocities. Because the cutting thickness changes with the rotation angle of the edge in the milling process, the surface profile machined by the previous edge inclines with respect to the cutting direction. The chip flow model is made using the orthogonal cutting data with taking into account the inclination of the pre-machined surface. The chip flow direction is determined so as to minimize the cutting energy, which is the sum of the shear energy on the shear plane and the friction energy on the rake face. Then, the cutting force is predicted for the chip flow model at the minimum cutting energy. The predicted chip flow direction changes not only with the local edge inclination but also with the cutting energy consumed in the shear plane cutting model. The cutting processes with a ball end mill and a roughing end mill are simulated to verify the predicted cutting forces in comparison with the measured cutting forces.     

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

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

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.     

Modelling the cutting edge radius size effect for force prediction in micro milling

This paper presents a theoretical model for cutting force prediction in micro milling, taking into account the cutting edge radius size effect, the tool run out and the deviation of the chip flow angle from the inclination angle. A parameterization according to the uncut chip thickness to cutting edge radius ratio is used for the parameters involved in the force calculation. The model was verified by means of cutting force measurements in micro milling. The results show good agreement between predicted and measured forces. It is also demonstrated that the use of the Stabler's rule is a reasonable approximation and that micro end mill run out is effectively compensated by the deflections induced by the cutting forces.     

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

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

A three-dimensional analytical cutting force model for micro end milling operation

In the present day manufacturing arena one of the most important fields of interest lies in the manufacturing of miniaturized components. End milling with fine-grained carbide micro end mills could be an efficient and economical means for medium and small lot production of micro components. Analysis of the cutting force in micro end milling plays a vital role in characterizing the cutting process, in estimating the tool life and in optimizing the process. A new approach to analytical three-dimensional cutting force modeling has been introduced in this paper. The model determines the theoretical chip area at any specific angular position of the tool cutting edge by considering the geometry of the path of the cutting edge and relates this with tangential cutting force. A greater proportion of the helix face of the cutter participating in the cutting process differs the cutting force profile in micro end milling operations a bit from that in conventional end milling operations. This is because of the reason that the depth-of-cut to tool diameter ratio is much higher in micro end milling than the conventional one. The analytical cutting force expressions developed in this model have been simulated for a set of cutting conditions and are found to be well in harmony with experimental results.     

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.     

Chatter Stability of Plunge Milling

Plunge milling operations are used to remove excess material in boring cylinders, roughing pockets, dies and mold cavities. This paper presents a frequency domain, chatter stability prediction theory for plunge milling. The regenerative chip thickness is modeled as a function of lateral, axial and torsional vibrations. The stability of the plunge milling is formulated as a fourth order eigenvalue problem by relating the regenerative chip thickness, cutting forces and torque, and the structural modes of the cutter. The stability lobes are predicted analytically from the eigenvalue solution. The stability lobes are experimentally proven by conducting over one hundred plunge milling tests.     

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。     

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.     

Cutting force modeling for flat end milling including bottom edge cutting effect

This study proposes a novel mechanistic cutting force model for flat end milling. The prominent feature of this model lies in that the overall cutting forces contributed by both the flank edge and the bottom edge cuttings are simultaneously taken into consideration. In the model formulation, to reflect the size effect in flank cutting, the flank cutting force coefficients are treated as an exponent function of instantaneous uncut chip thickness and are identified by nonlinear least-square algorithm. With the help of the calibrated flank cutting force coefficients, the bottom cutting force coefficients are instantaneously calibrated by the force component that is obtained by subtracting the flank force component from the total measured force. It is shown that the bottom cutting force coefficients can be treated as constants. The validity of the proposed cutting force model is also experimentally demonstrated over a relatively wide range of cutting conditions. It turns out that the bottom edge cutting has a remarkable effect on total cutting forces, when the axial depth of cut is relatively small.     

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.     

Mechanistic identification of specific force coefficients for a general end mill

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

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.     

Mechanics and dynamics of general milling cutters.: Part I: helical end mills

A variety of helical end mill geometry is used in the industry. Helical cylindrical, helical ball, taper helical ball, bull nosed and special purpose end mills are widely used in aerospace, automotive and die machining industry. While the geometry of each cutter may be different, the mechanics and dynamics of the milling process at each cutting edge point are common. This paper presents a generalized mathematical model of most helical end mills used in the industry. The end mill geometry is modeled by helical flutes wrapped around a parametric envelope. The coordinates of a cutting edge point along the parametric helical flute are mathematically expressed. The chip thickness at each cutting point is evaluated by using the true kinematics of milling including the structural vibrations of both cutter and workpiece. By integrating the process along each cutting edge, which is in contact with the workpiece, the cutting forces, vibrations, dimensional surface finish and chatter stability lobes for an arbitrary end mill can be predicted. The predicted and measured cutting forces, surface roughness and stability lobes for ball, helical tapered ball, and bull nosed end mills are provided to illustrate the viability of the proposed generalized end mill analysis.     

Cutting process in glass peripheral milling

Peripheral glass milling for trimmings of several devices and touch panels is studied with measuring cutting forces and observing surface damages. Peripheral millings were performed to cut the end faces of 1 mm thick glass plates. In order to discuss the typical cutting force in glass milling, the cutting forces were compared with those of 0.45% carbon steel (AISI 1045) at high feed rates in a large radial depth of cut. The differences of the cutting force in glass milling from that of metal milling are: (1) the change in the cutting force does not correspond to the uncut chip thickness; and (2) the maximum cutting force does not change with the feed rate. A model is proposed to predict the cutting forces in glass millings, which are performed in ductile, ductile/brittle complex and brittle modes. The cutting force depends on the uncut chip thickness in a ductile mode. In a brittle mode, the mean value of the cutting force does not change though the vibration component becomes large. Because the uncut chip thickness changes with the dynamic displacement of the cutting edge, the cutting process is performed in a ductile/brittle complex mode when the cutting mode changes in ductile–brittle transition. The critical uncut chip thickness at the transition from a ductile to a ductile/brittle complex mode and that of the transition from a ductile/brittle complex to brittle mode are determined in the rate of the cutting force change. The force model is verified by the cutting forces in up- and down-cutting milling operations. Then, the surface finishing and crack propagation in up- and down-cutting millings were analyzed to define the cutter path in glass trimming. Cracks propagate to the surface to be finally finished in down-cutting; while cracks propagate to the chip to be removed in up-cutting. The cutter path in up-cutting milling should be selected to finish the fine surfaces.     

Influence of tool inclination on brittle fracture in glass cutting with ball end mills

Glass milling is discussed with influences of tool inclination on brittle fracture. Cutting tests are performed to observe surfaces in the up-cut and the down-cut processes with a ball end mill inclined in the feed direction of the cutter. Brittle fracture occurs in the down-cut process at high feed rates. Then the machined surfaces in cutting with the ball end mills tilted in the vertical plane with respect to the feed direction are associated with those of the up-cut and the down-cut processes. The cutting forces are also measured to discuss brittle fracture with the change of the undeformed chip thickness. The scratches on the surface finished with the tilted ball end mill are shown in an analytical model with a notched edge shape. The maximum feed rates at which brittle fracture does not occur are shown with the tool inclination in the cutting tests.     

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.     

A study on instantaneous cutting force coefficients in face milling

In this paper, the characteristics of instantaneous cutting force coefficients in face milling are studied. In order to estimate instantaneous cutting force coefficients in face milling, the relationships between instantaneous cutting force coefficients and measured cutting force signals are derived. A series of experiments are then conducted to study the natures of instantaneous cutting force coefficients. The relationships between instantaneous cutting force coefficients and other cutting parameters are also established. It is found that the normal force coefficient is mainly affected by chip thickness and cutting speed; the vertical force coefficient is mainly affected by chip thickness, cutting edge length and cutting speed; and that the horizontal force coefficient is not only affected by chip thickness, cutting speed and length of cut, but also the variation rate of chip thickness.     

Mechanics of micro-milling with round edge tools

This paper presents analytical prediction of micro-milling forces from constitutive model of the material and friction coefficient. The chip formation process is predicted with a slip-line field model which considers the strain hardening, strain-rate and temperature effects on the flow stress of the material. The cutting force coefficients are identified from series of slip-line field simulations at a range of cutting edge radii and chip loads. The predicted cutting force coefficients are used to simulate micro-milling forces. The proposed chain of predictive micro-milling model is experimentally proven by conducting brass cutting tests with a 200 μm diameter helical end mill.