Introduction

Milling is a widely used method to remove materials from the initial configuration of a workpiece for machining monolithic parts in aeronautic, aerospace and automobile manufacturing industries. Due to the characteristics of large size and weak rigidity, cutting deformations and chatter vibrations will be easily induced during the cutting process. As a result, machining accuracy and surface quality of workpieces are not easily achieved and useless products will be produced in the worst case. Traditionally, the usual approach to remedying the machining precision was to validate the NC program by expensive trial and error cutting. Recently, an alternative approach is to numerically simulate the milling process a priori. It is desired that a quasi net-shaping will be obtained practically with optimal cutting parameters in perhaps one pass without grinding and polishing. To this end, research on the mechanics and dynamics of milling process is of great significance in developing strategies to guarantee accuracy. Issues such as cutting force modeling, surface quality prediction, chatter stability analysis and clamping scheme design are the key to this aspect.

I.1. Cutting force modeling

Cutting force modeling is the basis of all simulation schemes. In early research [MER 44], the concept of specific cutting energy was employed in cutting force modeling where cutting forces were assumed to be entirely related to shearing and friction effects. Under this assumption, the lumped force model was proposed as a classical one [KOE 61]. It approximates the entire cutting process as an equivalent shearing mechanism. This means that the cutting forces are supposed to be proportional to the chip load and the proportion scale is named the cutting force coefficient.

Since then, research efforts have been focused on how to effectively determine the cutting force coefficients in the lumped force model. For instance, Kline et al. [KLI 82a] and Larue and Anselmetti [LAR 03] treat the coefficients as constants. The former calibrated them by means of measured average cutting forces, whereas measured cutter deflections were used by the latter. Endres et al. [END 90] used empirical relations that mapped three independent variables of interest, i.e. instantaneous uncut chip thickness, cutting speed and rake angle, to represent the cutting force coefficients. Altintas and Spence [ALT 91] assumed that the coefficients are a power function of the average chip thickness, determined based on a strict integral technique. Instantaneous models such as the Weilbull function proposed by Ko et al. [KO 02] were also used to characterize instantaneous influences of process geometry parameters upon the cutting force coefficients for the lumped cutting force model.

It was, however, recognized by Thomsen [THO 66] that the cutting forces do not converge to zero when the chip thickness approaches zero. This phenomenon is the so-called rubbing effect associated with the clearance face of the flank edge and responsible for cutting process damping [END 95]. Masuko [MAS 56] and Albrecht [ALB 60, ALB 61] proposed the dual-mechanism model to separate the chip removal and flank rubbing mechanisms for the machining process of constant chip thickness. With regard to the milling process of periodically changing chip thickness, Altintas [ALT 12] modeled the chip removal and the flank rubbing effects separately as functions of chip load and chip width, respectively. Meanwhile, Budak et al. [BUD 96] calibrated the cutting force coefficients using orthogonal cutting tests with oblique cutting analysis and transformation. Gonzalo et al. [GON 10] identified the constant coefficients by means of measured instantaneous cutting force data. Wang and Zheng [WAN 03] used the convolution integration method to identify the cutting force coefficients. However, it is interesting to remark that the methods proposed above were developed for each cutter type individually, e.g. the flat end mill and the ball end mill. To have a unified cutting force model of general end mills, Engin and Altintas [ENG 01] developed a generalized mechanics and dynamics model where cutting force coefficients are predicted from an orthogonal database. Alternatively, Gradisek et al. [GRA 04] calibrated the cutting force coefficients for a general end mill based on the average cutting forces measured.

With respect to the cutting force prediction, however, most work was based on the assumption that machine set-up errors such as cutter tilt and cutter offset runout were ignored. In contrast, a rigid mechanistic cutting force model including the cutter radial runout was proposed by Kline and DeVor [KLI 83]. This model was later extended by Sutherland and DeVor [SUT 86] and a regeneration model was developed to predict the cutting forces in flat end milling, accounting for the cutter flexibility and the cutter runout. In this context, we suppose that the cutter runout parameters are known a priori. Wang and Liang [WAN 96] developed an analytical model for the calculation of instantaneous uncut chip thickness and cutting forces.

In fact, the problem also arises of how to figure out the runout parameters and the cutting force coefficients simultaneously based upon the measured cutting forces. For cylindrical end milling, a numerical scheme was proposed by Armarego and Despande [ARM 89], who estimated the runout parameters through a best-fit procedure. Liang, Zheng and Wang et al. [LIA 94, ZHE 97, WAN 03] analyzed the influence of the cutter runout on the cutting forces using the convolution integration method. Cutting force coefficients and cutter runout parameters were identified by means of the Fourier series. An alternative approach was suggested by Cho et al. [YUN 00, YUN 01] who calibrated the cutting force coefficients and the cutter runout for cylindrical end mills based on the instantaneous cutting forces rather than the average ones. Attention was also received in ball end milling. Feng and Menq [FEN 94a, FEN 94b] calibrated the cutting force coefficients and the runout using the mechanistic approach for the modelling of complicated ball end milling process. Ko and Cho [KO 05] calibrated the instantaneous cutting force coefficients and the runout parameters for ball end milling with the synchronization procedure.

I.2. Surface quality simulation

In a milling process without chatter, the static surface form error caused mainly by elastic deflections of the cutter and of the workpiece is often the dominant defect when milling a thin-walled workpiece made up of titanium or aluminum alloys at a low spindle speed [BUD 95, TSA 99]. The surface form error is mainly made up of the force-induced deflection, which results in a deviation of the depth of cut. Many research attempts have been focused on this problem. Kline [KLI 82b], Larue [LAR 03], Budak [BUD 94], Shirase [SHI 96], Ryu [RYU 03] and Paksiri [PAK 04] used the cutter deflection to predict the surface form errors; whereas Ratchev et al. [RAT 04a, RAT 04b, RAT 06] used the workpiece deflections to calculate surface form error of a flexible workpiece. For example, Kline et al. [KLI 82b] studied the prediction of surface form errors in the peripheral milling of a clamped-clamped-clamped-free rectanglular plate. The cutter is modeled as a continuous cantilevered beam and the plate is discretized by the FEM. In the meantime, cutting forces are assumed to be concentrated forces in the calculation of the cutter and workpiece deflections. Budak and Altintas [BUD 94] and Shirase et al. [SHI 96] studied the surface form errors uniquely caused by the deflection of the cutter that is modeled as an assemblage of discrete elements with equal length. Thus, cutting forces are discretized onto the element nodes to calculate the deflection of the cutter. This approximation is valid when the workpiece has relatively a large rigidity. Zhang et al. [ZHA 01a] determined the surface form errors by evaluating the deflections of both the cutter and the workpiece without considering the coupling effect between cutter and workpiece. To consider this coupling effect in a flexible milling process, many researchers used iteration schemes to predict the cutting forces and the surface form errors [BUD 95, TSA 99]. Budak and Altintas [BUD 95] and Tsai and Liao [TSA 99] developed iteration schemes to retain the coupling effect of deflections between the cutter and workpiece, as well as the rigidity diminution of the workpiece due to material removal. Meanwhile, the workpiece is meshed by one layer of 8-node and 12-node isoparametric volume elements along the thickness direction, respectively. Nevertheless, the generated mesh of the workpiece has to coincide, element to element, with that of the cutter. In addition, the stiffness reduction of the workpiece due to material removal can be simulated by changing nodal coordinates of such a one-layer element. This requirement becomes a major limitation in the modeling step, especially in the modeling process of complex workpieces. In the above work, it can be generally said that the surface form errors were predicted either by means of an analytical/finite element method [LAR 03, BUD 95, RYU 03] or by means of neural networks [PAK 04].

Based on the obtained values of surface form errors, compensation techniques have been widely used to reduce the resulting errors without sacrificing the machining productivity. Depince [DEP 06], Rao [RAO 06], Landon [LAN 03] and Law [LAW 99, LAW 03] studied the cutting-force-induced tool deflection compensation in peripheral milling by the mirror method. Based on the closed loop volumetric error relations, Bohez [BOH 02] proposed a...