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.     

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.     

Experimental research on cutting force variation during regenerative chatter vibration in a plain milling operation

A plain milling operation is characterized by a transient and intermittent cutting process, in which undeformed chip thickness varies continuously. The reverse is the case in variations of undeformed chip thickness in the processes of up- and down-milling. In the present study, the property of regenerative chatter vibration in a plain milling operation is investigated from the viewpoint of cutting force variation. With primary chatter vibration, the vibration energy supply is closely related to the collision of the cutting tool flank against the workpiece surface during vibration, which is induced by the bending vibration or the torsional vibration of the arbor. In addition to this factor, the regenerative effect is considered to be one of the main causes of the chatter excitation in regenerative chatter vibration. The simulation result of the cutting force variation during regenerative chatter vibration agrees well with the experimental result, when considering these factors. It is shown that the regenerative chatter vibration in the-down-milling process occurs more easily than in the up-milling process.     

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.     

Stability lobes for general turning operations with slender tools in the tangential direction

During the turning of large and heavy cases and rings, the tool must be cantilevered over long distances, and chatter due to modes in the tangential direction may occur. This paper proposes a numerical method based on a nonlinear cutting force model, which includes the effects of the cutting parameters to construct precise stability charts for this special machining case. This method has been successfully applied to general cutting geometries in rough and medium longitudinal turning operations, assuming a rigid workpiece and a flexible tool. This study proposes a dynamic model to implement the effects of the tangential mode on chip regeneration in the regenerative plane based on an experimentally obtained dynamic displacement factor. From the simulations and experimental results, the model provides a reliable approach to obtain chatter-free turning conditions.     

Experimental research on cutting force variation during primary chatter vibration occuring in plain milling operation

Plain milling operation is characterized by a transient and intermittent cutting process, in which undeformed chip thickness varies continuously. The undeformed chip thickness variation is opposite in the up milling and down milling processes. First, the property of primary chatter vibration in plain milling operation is investigated. In the up milling process, the transient vibration generated in the initial stage of the cutting operation develops into primary chatter vibration, along with the chip thickness increase. On the other hand, a large amount of vibration energy is supplied during the initial collision of the cutting edge with the workpiece at a large undeformed chip thickness in the down milling process; and immediately after this collision, the primary chatter vibration of almost stable amplitude continues. Secondly, the vibration energy supply during the primary chatter vibration of plain milling operations is investigated on the basis of the experimental results. The exciting mechanism can be explained by considering the interference between the tool flank and the workpiece surface accompanying the arbor vibration. An unusual phenomenon is also discussed, in which the normal cutting force component has two maxima during one period of vibration in up milling. From the above results, the cutting edge shapes (effective relief angle and cutting edge radius), and the torsional rigidity of the milling arbor must be carefully determined, to prevent the primary chatter vibration in plain milling operation.     

Linear analysis of chatter vibration and stability for orthogonal cutting in turning

The productivity of high speed milling operations is limited by the onset of self-excited vibrations known as chatter. Unless avoided, chatter vibrations may cause large dynamic loads damaging the machine spindle, cutting tool, or workpiece and leave behind a poor surface finish. The cutting force magnitude is proportional to the thickness of the chip removed from the workpiece. Many researchers focused on the development of analytical and numerical methods for the prediction of chatter. However, the applicability of these methods in industrial conditions is limited, since they require accurate modelling of machining system dynamics and of cutting forces. In this study, chatter prediction was investigated for orthogonal cutting in turning operations. Therefore, the linear analysis of the single degree of freedom (SDOF) model was performed by applying oriented transfer function (OTF) and \tau decomposition form to Nyquist criteria. Machine chatter frequency predictions obtained from both forms were compared with modal analysis and cutting tests.     

球头铣刀铣削斜面的三维有限元仿真研究

在能源动力、汽车、航空航天、模具制造等关键零部件的加工过程中,球头铣刀因其特有的刀具几何结构,常作为零件加工的最终成型刀具。考虑到在球头铣刀立铣加工中不同的刀具与工件相对姿态会对切削过程产生不同的影响,本文研究切屑形成和不同走刀方式下切削过程中各物理量(切削力、切削温度等)的变化情况,结合有限元仿真技术在切削加工中的应用,建立硬质合金球头铣刀铣削斜面的有限元模型,模拟相同切削参数下,八种不同走刀方式的球头铣削过程,分析刀具切入切出工件时切屑的形成过程,探究切削力和切削温度的变化规律。仿真结果表明:不同的走刀方式,平均切削合力各不相同,同时切屑和工件的最大切削温度也出现较大差异,而斜坡上坡逆铣的走刀方式所对应的平均切削合力和最大切削温度均最优。     

立铣刀断屑槽对切削性能影响规律试验研究

采用析因试验法研究断屑槽结构的制备对整体式立铣刀切削性能的影响。选取2把断屑槽分布不同的刀具,以铣刀主轴转速、径向切削深度、每齿进给量为主要因素,建立正交铣削试验。结果表明：断屑槽分布密集的铣刀适合用于加工精度要求不高的零件;断屑槽分布稀释的铣刀,适合用于半精加工要求的工件;对比分析切屑形貌和表面质量,得到用断屑槽分布稀疏的刀具铣削工件时可获得较好的表面质量。     

Identification of dynamic cutting force coefficients and chatter stability with process damping

This paper presents a cutting force model which has three dynamic cutting force coefficients related to regenerative chip thickness, velocity and acceleration terms, respectively. The dynamic cutting force coefficients are identified from controlled orthogonal cutting tests with a fast tool servo oscillated at the desired frequency to vary the phase between inner and outer modulations. It is shown that the process damping coefficient increases as the tool is worn, which increases the chatter stability limit in cutting. The chatter stability of the dynamic cutting process is solved using Nyquist law, and compared favourably against experimental results at low cutting speeds.     

Peripheral milling conditions for improved dimensional accuracy

Peripheral milling with flexible helical cutters is analyzed and modelled. The end mill is modelled as a cantilevered beam clamped to the collet with linear springs. Cutting forces and resulting tool deflection marks on the surface are analytically expressed. It is shown that by proper selection of cutting conditions, the material removal rate can be increased significantly without sacrificing the dimensional accuracy of the finished product. A method of identifying optimal feedrate and width of cut for given cutter dimensions and cutting constants is presented with experimental verification.     

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.     

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.     

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.     

Time domain simulation of torsional–axial vibrations in drilling

A time domain model of the torsional–axial chatter vibrations in drilling is presented. The model considers the exact kinematics of rigid body, and coupled torsional and axial vibrations of the drill. The tool is modeled as a pretwisted beam that exhibits axial and torsional deflections due to torque and thrust loading. A mechanistic cutting force model is used to accurately predict the cutting torque and thrust as a function of feedrate, radial depth of cut, and drill geometry. The drill rotates and feeds axially into the workpiece while the structural vibrations are excited by the cutting torque and thrust. The location of the drill edge is predicted using the kinematics model, and the generated surface is digitized at discrete time intervals. The distribution of chip thickness, which is affected by both rigid body motion and structural vibrations, is evaluated by subtracting the presently generated surface from the previous one. The model considers nonlinearities in cutting coefficients, tool jumping out of cut and overlapping of multiple regeneration waves. Force, torque, power and dimensional form errors left on the surface are predicted using the dynamic chip thickness obtained from the exact kinematics model. The stability of the drilling process is also evaluated using the time domain simulation model, and compared with extensive experiments. This paper provides details of the mathematical model, experimental verification and simulation capabilities. Although the surface finish from unstable cutting can be predicted realistically, the actual drilling stability cannot be determined without including process damping.     

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.     

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 new dynamic model for drilling and reaming processes

This paper presents a new computer simulation model for drilling and reaming processes. The model is made of four parts: the force model for the cutting lips, the force model for the chisel edge, the dynamic model for the machine tool (including the cutter) and the regenerative correlation between the force and machine tool vibration. The models for the forces and the machine tool are similar to the existing models. The key to the model is the regeneration correlation between the cutting forces and the machine tool vibration. It uses a new 3D chip formation model to describe the interaction between the cutter and the workpiece. The model can predict the dynamic forces and chatter limit. It also reveals several interesting phenomena, such as how the feed and the point angle of the drill affect the chatter limit. The model is implemented using C++ language with an interface to I-DEAS™ CAE software system. The simulation results are validated experimentally by both drilling and reaming under various cutting conditions. The experiment results show that the simulation is accurate with average error about 10%. A number of research issues are also proposed for the future work.     

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.     

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.