Determination of Cutting-Condition-Independent Coefficients and Runout Parameters in Ball-End Milling

This paper presents a systematic and practical method for determining cutting-condition-independent coefficients in ball-end milling. An approach for estimating the runout offset and its location angle is also described based on a single cutting force measurement. An in-depth analysis of the characteristics of these cutting coefficients, which can be determined from only a few test cuts, is provided. The size effect is also modelled, including the characteristics of the ball-end mill geometry, and incorporated into the cutting force model. This method of estimating the 3D ball-end milling force coefficients was tested experimentally for various cutting conditions and gave excellent cutting force predictions. Also, the estimated values of the runout offset agreed well with the measurements.     

A force-model-based approach to estimating cutter axis offset in ball end milling

Cutter runout due to cutter axis offset is quite common in a milling process, yet it is difficult to directly measure the runout geometry of a ball end cutter during the cutting process. This paper presents an analytical method for the estimation of cutter radial offset via forces in ball end milling. Closed form expression for the total milling force in the presence of cutter offset is first obtained. Fourier series coefficients for the offset related force component are shown to be expressed explicitly in terms of the offset geometry and serve as the basis for the identification of the offset geometry from the measured cutting forces. The offset geometry including its magnitude and the phase angle are directly calculated from the measured force component at the spindle frequency through two algebraic expressions. The identification method is finally validated by milling experiments.     

A separate-edge force coefficients’ calibration method using specific condition for cutters with variable helix and pitch angles combining the runout effect

Classical ways of computing cutting force coefficients cannot be used by the cutters with non-uniform helix and pitch angles. So, this paper presents a novel separate-edge-forecast method to compute cutting force coefficients for any kind of flank-end cutter, especially for cutters with non-uniform helix and pitch angles. Using this method, the cutter runout can be combined into the cutting force coefficients without computing the cutter runout parameters. Simultaneously, the method predicts the cutting force coefficients for every cutter edge. Firstly, a series of three-axis machining experiments, which must satisfy the specific condition that only one cutter edge is removing materials at any time, is conducted. Then, the cutting force-curves are divided into N force lobes. Each lobe is assigned to the corresponding cutter edge using an algorithm. Subsequently, the cutter edge and the corresponding cutting force lobe are used to determine the cutting force coefficients. This means N cutter edges have N groups of cutting force coefficients, correspondingly. Finally, in order to verify the validity and correctness of the proposed method, a cutter with non-uniform helix and pitch angle is utilized to predict cutting force coefficients based on which the cutting forces are also computed. The results demonstrate that the cutting forces predicted agree well with the data measured. Simultaneously, it can be observed that the method can predict the coefficients considering the cutter runout effect.     

New mathematical method for the determination of cutter runout parameters in flat-end milling

The cutting force prediction is essential to optimize the process parameters of machining such as feed rate optimization, etc. Due to the significant influences of the runout effect on cutting force variation in milling process, it is necessary to incorporate the cutter runout parameters into the prediction model of cutting forces. However, the determination of cutter runout parameters is still a challenge task until now. In this paper, cutting process geometry models, such as uncut chip thickness and pitch angle, are established based on the true trajectory of the cutting edge considering the cutter runout effect. A new algorithm is then presented to compute the cutter runout parameters for flat-end mill utilizing the sampled data of cutting forces and derived process geometry parameters. Further, three-axis and five-axis milling experiments were conducted on a machining centre, and resulting cutting forces were sampled by a three-component dynamometer. After computing the corresponding cutter runout parameters, cutter forces are simulated embracing the cutter runout parameters obtained from the proposed algorithm. The predicted cutting forces show good agreements with the sampled data both in magnitude and shape, which validates the feasibility and effectivity of the proposed new algorithm of determining cutter runout parameters and the new way to accurately predict cutting forces. The proposed method for computing the cutter runout parameters provides the significant references for the cutting force prediction in the cutting process.     

Improved dynamic cutting force model in ball-end milling. Part I: theoretical modelling and experimental calibration

An accurate cutting force model of ball-end milling is essential for precision prediction and compensation of tool deflection that dominantly determines the dimensional accuracy of the machined surface. This paper presents an improved theoretical dynamic cutting force model for ball-end milling. The three-dimensional instantaneous cutting forces acting on a single flute of a helical ball-end mill are integrated from the differential cutting force components on sliced elements of the flute along the cutter-axis direction. The size effect of undeformed chip thickness and the influence of the effective rake angle are considered in the formulation of the differential cutting forces based on the theory of oblique cutting. A set of half immersion slot milling tests is performed with a one-tooth solid carbide helical ball-end mill for the calibration of the cutting force coefficients. The recorded dynamic cutting forces are averaged to fit the theoretical model and yield the cutting force coefficients. The measured and simulated dynamic cutting forces are compared using the experimental calibrated cutting force coefficients, and there is a reasonable agreement. A further experimental verification of the dynamic cutting force model will be presented in a follow-up paper.     

A new method for cutting force prediction in peripheral milling of complex curved surface

The instantaneous uncut chip thickness and entry/exit angle of tool/workpiece engagement vary with tool path, workpiece geometry and cutting parameters in peripheral milling of complex curved surface, leading to the strong time-varying characteristic for instantaneous cutting forces. A new method for cutting force prediction in peripheral milling of complex curved surface is proposed in this paper. Considering the tool path, cutter runout, tool type(constant/nonconstant pitch cutter) and tool actual motion, a representation model of instantaneous uncut chip thickness and entry/exit angle of tool/ workpiece engagement is established firstly, which can reach better accuracy than the traditional models. Then, an approach for identifying of cutter runout parameters and calibrating of specific cutting force coefficients is presented. Finally, peripheral milling experiments are carried out with two types of tool, and the results indicate that the predicted cutting forces are highly consistent with the experimental values in the aspect of variation tendency and amplitude.     

CORRELATION-BASED ESTIMATION OF CUTTING FORCE COEFFICIENTS FOR BALL-END MILLING

This article presents a methodology to estimate cutting force coefficients based on the least squares approximation using correlation factor between the estimated and measured cutting forces in order to determine the corresponding tool angular position. This method can be applied on measured cutting force data over any small interval of time that need not contain information of the time instant when the cutting tool enters the workpiece, which has been the main requirement in the conventional method. This allows a quick estimation of the cutting force coefficients regardless of the chosen cutting conditions and tool-workpiece material, which is often the case in industrial machining processes. This proposed method has been validated by comparison of cutting force coefficients obtained using conventional estimation technique for a slot ball-end milling test. Besides being useful for predictive evaluation of forces, such estimation of cutting force coefficients of the cutting force model can be useful for understanding variations in cutting process over the tool life and can assist in online monitoring and process optimization.     

An experimental investigation of oblique cutting mechanics

In this study, an experimental investigation of oblique cutting process is presented for titanium alloy Ti-6Al-4V, AISI 4340, and Al 7075. Important process parameters such as shear angle, friction angle, shear stress, and chip flow angle are analyzed. Transformation of the data from the orthogonal cutting test results to oblique cutting process is applied, and the results are compared with actual oblique cutting tests. Effects of hone radius on cutting forces and flank contact length are also investigated. It is observed that the shear angle, friction angle, and shear stress in oblique cutting have the same trend with the ones obtained from the orthogonal cutting tests. The transformed oblique force coefficients from orthogonal tests have about 10% discrepancy in the feed and tangential directions. For the chip flow angle, the predictions based on kinematic and force balance results yield better results than Stabler's chip flow law. Finally, it is shown that the method of oblique transformation applied on the orthogonal cutting data yields more accurate results using the predicted chip flow angles compared to the ones obtained by the Stabler's rule.     

A study on calibration of coefficients in end milling forces model

This paper presents an improved approach to calibrate the cutting coefficients in an end-milling model. In order to predict end-milling forces, lots of simulative models are established. In order to use them, coefficients in the models, for example, cutting pressure constants etc., must firstly be calibrated experimentally, and simulative precision and applicability of the models are influenced by them. For simplicity, using average forces to calibrate cutting parameters are widely adopted by lots of researchers. However, the existence of an instruments zero-drift, noise, etc., will have effect on the precision of experimental data, so, it is difficult to directly obtain exact average-cutting forces through experimental data. Aiming at the above problem, the paper investigates milling forces in the frequency domain, discusses the impact of experimental data at different frequencies on cutting force coefficients and the influence of sensitivity of error on experimental data at different frequencies on coefficients is studied. Based on the research, an improved method to calibrate the cutting coefficients is provided. Based on a series of experiments and numerical simulations, the validity of the method is confirmed. At the end of the paper, some useful conclusions are drawn.     

Theoretical modelling of cutting forces in helical end milling with cutter runout

A theoretical cutting force model for helical end milling with cutter runout is developed using a predictive machining theory, which predicts cutting forces from the input data of workpiece material properties, tool geometry and cutting conditions. In the model, a helical end milling cutter is discretized into a number of slices along the cutter axis to account for the helix angle effect. The cutting action for a tooth segment in the first slice is modelled as oblique cutting with end cutting edge effect and tool nose radius effect, whereas the cutting actions of other slices are modelled as oblique cutting without end cutting edge effect and tool nose radius effect. The influence of cutter runout on chip load is considered based on the true tooth trajectories. The total cutting force is the sum of the forces at all the cutting slices of the cutter. The model is verified with experimental milling tests.     

Analysis of cutting force coefficients in high-speed ball end milling at varying rotational speeds

In high-speed ball end milling, cutting forces influence machinability, dimensional accuracy, tool failure, tool deflection, machine tool chatter, vibration, etc. Thus, an accurate prediction of cutting forces before actual machining is essential for a good insight into the process to produce good quality machined parts. In this article, an attempt has been made to determine specific cutting force coefficients in ball end milling based on a linear mechanistic model at a higher range of rotational speeds. The force coefficients have been determined based on average cutting force. Cutting force in one revolution of the cutter was recorded to avoid the cutter run-out condition (radial). Milling experiments have been conducted on aluminum alloy of grade Al2014-T6 at different spindle speeds and feeds. Thus, the dependence of specific cutting force coefficients on cutting speeds has been studied and analyzed. It is found that specific cutting force coefficients change with change in rotational speed while keeping other cutting parameters unchanged. Hence, simulated cutting forces at higher range of rotational speed might have considerable errors if specific cutting force coefficients evaluated at lower rotational speed are used. The specific cutting force coefficients obtained analytically have been validated through experiments.     

Identification of instantaneous cutting force coefficients using surface error

Cutting force coefficients are the key factors for efficient and accurate prediction of instantaneous milling force. To calibrate the coefficients, this paper presents an instantaneous milling force model including runout and cutter deformation. Also, forming of surface error is analyzed, and a surface error model considering runout is proposed. Using surface errors of two experiments completed with the same cutting conditions but different axial depth only, cutter deformation is obtained. Then, a new approach for the determination of instantaneous cutting force coefficients is provided. The method can eliminate influences of the other factors except cutter deformation and runout. A series of experiments are designed, and the results are used to identify the parameters. With the evaluated coefficients and runout parameters, the instantaneous milling force and surface error are predicted. A good agreement between predicted results and experimental results is achieved, which shows that the method is efficient, and effect of runout on surface error is not negligible.     

In-process estimation of radial immersion ratio in face milling using cutting force

In order to prevent tool breakage in milling, maximum total cutting force is regulated at a specific constant level, or threshold, through feed rate control. Since the threshold is a function of the immersion ratio, an estimation of the immersion ratio is necessary to flexibly determine the threshold. In this paper, a method of in-process estimation of the radial immersion ratio in face milling is presented. When an insert finishes sweeping, a sudden drop in cutting forces occurs. These force drops are equal to the cutting forces that act upon a single insert at the swept cutting angle and they can be acquired from cutting force signals in the feed and cross-feed directions. Average cutting forces per tooth period can also be calculated from the cutting force signals in two directions. The ratio of cutting forces acting upon a single insert at the swept angle of cut and the ratio of average cutting forces per tooth period are functions of the swept angle of cut and the ratio of radial to tangential cutting force. Using these parameters, the radial immersion ratio is estimated. Various experiments are performed to verify the proposed method. The results show that the radial immersion ratio can be estimated by this method regardless of other cutting conditions.Nomenclature F_T, F_R tangential and radial forces - F_X, F_Y cutting forces in feed direction and cross feed direction - dF_X, dF_Y cutting force differences before and after the immersion angle in X and Y direction - K_s specific cutting pressure - a depth of cut - r ratio between tangential force and radial force - s_t feed per tooth - instantaneous angle of cut - _s swept angle of cut - _T tooth spacing angle - w radial width of cut - R cutter radius - z number of inserts     

Improved Dynamic Cutting Force Model in Peripheral Milling. Part I: Theoretical Model and Simulation

The accurate prediction of cutting forces is important in controlling the tool deflection and the machining accuracy. In this paper, the authors present an improved theoretical dynamic cutting-force model for peripheral milling with helical end-mills. The theoretical model is based on the oblique cutting principle and includes the size effect of undeformed chip thickness and the influence of the effective rake angle. A set of closed-form analytical expressions is presented. Using the cutting forces measured by Yucesan [1] in tests on a titanium alloy, the cutting-force coefficients are estimated and the cutting- force model verified by simulation. The simulation results indicate that the improved dynamic cutting-force model does predict the cutting forces in peripheral milling accurately. Simulation results for a number of particular examples are presented. ID="A1" Correspondence and offprint requests to: Prof K. Cheng, School of Engineering, Leeds Metropolitan University, City Campus, Calverley Street, Leeds LS1 3HE, UK. E-mail: k.cheng@lmu.ac.uk     

Envelope surface formed by cutting edge under runout error in five-axis flank milling

This paper investigates the analytical envelope surface model formed by specially designed cutting edge under cutter runout error, including axis offset error and tilt error, in five-axis flank milling. This model, which is independent of the machine tool type, is determined by the tangency condition in envelope theory. First, the cutter runout is defined by four parameters, namely offset distance, offset direction angle, locating angle, and tilt angle. Then, the cutting edge represented by cubic B-spline curve is used as the generator of cutter rotation surface to formulate the closed-form envelope surface model. In particular, the runout error and feedrate are both integrated into the model. In addition, we study special cases of the analytical model and the runout effect on envelope surface. Finally, computer example validates the feasibility of the proposed model with runout. We find that envelope surface formed by cutter edge is different from each other at the existence of runout, and envelope surface dedicated to final machined surface is generated by the composition of some segments of cutter edges. The results can be applied to tool path optimization in five-axis flank milling and NC simulation with cutter runout.     

螺旋棒铣刀铣削力建模与仿真

在对螺旋棒铣刀铣削力建模中考虑了切削厚度变化对铣削力影响的指数关系、铣刀偏心对实际切削厚度、切入与切出角、铣削力波动的影响,并提出采用实测各刀齿铣削最大值比求解铣刀偏心和识别铣削力系数的方法。在考虑铣刀偏心因素的情况下仿真与实测的铣削力达到非常好的一致性。提出的铣削力仿真方法充分反映了铣削力的实际状态,提高了铣削力仿真精度。     

Implementation of analytical boundary simulation method for cutting force prediction model in five-axis milling

A simulation system was developed that deals with cut geometry and machining forces when a toroidal cutter is used during semifinishing in five-axis milling. The cut geometry was calculated using an analytical method called analytical boundary simulation (ABS). ABS was implemented to calculate the cut geometry when the machining used an inclination angle and a screw angle. The effect of tool orientation on the cut geometry was analyzed. The accuracy of the proposed method was verified by comparing the cut lengths calculated using ABS with cuts obtained experimentally. The result indicated that the method was accurate. ABS was subsequently applied to support a cutting force prediction model. A validation test showed that there was a good agreement with the cutting force generated experimentally.     

Modelling the cutting forces in micro-end-milling using a hybrid approach

This paper presents the development of a cutting force model for the micro-end-milling processes under various cutting conditions using a hybrid approach. Firstly, a finite element (FE) model of orthogonal micro-cutting with a round cutting edge is developed for medium-carbon steel. A number of finite element analyses (FEA) are performed at different uncut chip thicknesses and velocities. Based on the FEA results, the cutting force coefficients are extracted through a nonlinear algorithm to establish a relationship with the uncut chip thickness and cutting speed. Then, the cutting force coefficients are integrated into a mechanistic cutting force model, which can predict cutting forces under different cutting conditions. In order to account for the cutting edge effect, an effective rake angle is employed for the determination of the cutting force. A comparison of the prediction and experimental measured cutting forces has shown that the developed method provides accurate results.     

基于改进粒子群优化算法的变截面涡旋盘瞬时铣削力模型参数求解

针对变截面涡旋盘瞬时铣削力预测存在的多元非线性难题,从涡旋盘实际铣削过程出发,建立了考虑刀具跳动的瞬时铣削力数学模型,提出了一种基于改进粒子群优化算法（PSO）对铣削力模型参数进行求解的方法,以提高瞬时铣削力预测模型精度。通过4组不同铣削参数下的瞬时铣削力实验对该方法进行验证,结果表明：该方法求解得到的变截面涡旋盘瞬时铣削力与实验测得的瞬时铣削力在形状和峰值处有较高的吻合度,4组实验的峰值误差在15%以内;采用自适应惯性权重和随机扰动因子的改进PSO算法能够有效地提高变截面涡旋盘瞬时铣削力系数辨识的收敛速度和收敛效果,还能提高算法整体搜索能力。该方法只需较少的实验次数就能辨识出较高精度的模型参数,比平均铣削力求解方法的实验成本更低,对涡旋盘的加工具有重要参考价值。     

基于斜角切削理论的钛合金螺旋铣孔切削力建模

通过分析螺旋铣孔的加工原理和计算加工过程中的运动向量,结合侧刃和底刃对切削力的影响,建立了螺旋铣孔过程的切削力解析模型。提出了基于斜角切削的切削力系数辨识方法,并根据斜角切削过程几何关系推导出摩擦角、剪切角、剪切应力的约束方程。开展切削力系数辨识试验和钛合金螺旋铣孔试验对仿真值进行验证,结果表明,切削力的仿真值与试验值误差较小,平均误差为9.55%,从而验证了斜角切削系数辨识方法的有效性和切削力模型的正确性。