Finite-element modeling of static surface errors in the peripheral milling of thin-walled workpieces

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Abstract

The present study develops a finite-element model along with an adequate end milling cutting-force model to analyze the surface dimensional errors in the peripheral milling of thin-walled workpieces. The helical fluted end mill is modeled with the pre-twisted Timoshenko beam element that can more accurately simulate the specific geometry and structural behavior of the cutter. The workpiece is modeled with a 3D isoparametric 12-node element that can take into account the geometry and thickness variations of the workpiece during peripheral milling. This study neglects the dynamic effect during milling and assumes that the tool and the workpiece deform to their static equilibrium positions at any milling instant. For a flexible cutting system, the effect of cutting system deflections on the cutting force distribution must be included. Hence the cutting force distribution and the cutting system deflections are solved iteratively by the modified Newton–Raphson method in this study. After the converged cutting system deflections are obtained at each cutting step, the surface dimensional error of the workpiece at the corresponding grid point can be computed easily. From the present study it is found that there will exist serious surface dimensional errors in the peripheral milling of very flexible components even the metal removal rate is very small. The present simulation model is verified experimentally and is helpful in determining the cutting parameters without performing real cutting experiments.

Introduction

The end mill is an important tool in the milling process. A typical example for the end mill is the milling of a pocket and slot in which a lot of material is removed from the workpiece. Therefore the proper selection of cutting parameters for end milling is one of the important factors affecting the cutting cost. Amongst the factors that influence the determination of cutting parameters, such as the material of the tool and then workpiece, the deflection of the tool and the workpiece is also an important factor, since it causes surface dimensional error. Hence many works on surface dimensional error in the peripheral milling of thin plates have been undertaken 1, 2, 3, 4, 5.

In general, the surface dimensional error is induced mainly by the deflection of the tool and the workpiece during milling, which does not make the removed material as planned. If the workpiece has high rigidity and is fixed firmly, the deformation of the workpiece is small and can be neglected in the contribution of surface dimensional error, but in the peripheral milling of a very flexible component (see Fig. 1), the deflection of the tool and the workpiece is large and can produce surface form errors. Hence the effect of the deformed tool and workpiece on the machined surface form must be considered when a process simulation model of the peripheral milling of a flexible plate is developed. In the following, the previous study on the peripheral milling of flexible structures is reviewed briefly. Kline et al. [1]developed a method to predict the surface form error in the milling of a flexible plate with a flexible end mill. He used a rectangular plate element to model the plate, and a uniform cantilevered beam for the end mill. The distributed cutting force is approximated by an equivalent concentrated force, which is calculated by using the rigid model, i.e., assuming that the chip thickness is not affected by the tool and workpiece deflections. The equivalent cutting force is then used to compute the deflections of the tool and the workpiece, which are summed to obtain the surface dimensional error. Sutherland and DeVor [2]considered the effect of tool deflection on the chip thickness and proposed a method to estimate the instantaneous uncut chip thickness that also allows for the run-out of tool. Although Sutherland's new model predicted very accurate cutting forces and surface form error, this model did not analyze the end milling of flexible thin plates. Elbestawi and Sagherian [3]developed a simulation model to predict the surface dimensional errors in the peripheral milling of thin-walled workpieces including the dynamic effects of the tool and the workpiece. In Ref. [3]the workpiece was modeled using a 3D isoparametric 8-node finite element, and the end mill was divided into several disk-like elements of which the modal parameters were obtained experimentally by mounting the tool at the end of the spindle and using an impact testing. Altintas et al. [4]presented a dynamic model to show the interaction of a very flexible plate structure and a rigid end mill during peripheral milling. Budak and Altintas [5]included the tool and workpiece deflections and used iteration to predict the cutting force distribution and the dimensional error of the machined surface. In Ref. [5]the end mill was represented by a cantilevered elastic beam with an equivalent diameter and the workpiece was modeled with the 3D isoparametric 8-node element. Ref. [5]also demonstrated the applicability of the proposed model in the prediction of surface form error in the peripheral milling of a flexible plate.

This paper proposes a model to predict the surface form error in the peripheral milling of a flexible plate, in which the end mill and the workpiece are modeled with structural elements different from those in the literature. The two-flute end mill is used widely and has a cross-section that is different from that of a circle (see Fig. 3). Hence in this study the two-flute end mill is represented by a pre-twisted beam and modeled with a pre-twisted Timoshenko beam element developed by the authors in a previous work [6]. Also in the present analysis, milling forces are calculated with the flexible model, which includes the effect of the tool and workpiece deflections, and the cutting force distribution is determined through iteration. The thin-walled workpiece is modeled with a three-dimensional 12-node isoparametric element (see Fig. 1, Fig. 5) that can better approximate the curved surface of the tool–workpiece contact zone and include the thickness change of the workpiece in the feed direction during milling.

Since the flexible tool–workpiece system is adopted here, in order to meet the static equilibrium condition, the modified Newton–Raphson iteration method is used to determine the chip load, the cutting forces and the surface form error. In the following, first the finite element modeling of the tool and workpiece structures is presented. Next the cutting force models, both rigid and flexible force models, are introduced. Then the surface error generation and determination are outlined. In Section 5the present model to simulate the surface form errors in peripheral milling of a flexible plate is proven experimentally. Finally, some suggestions are presented for future investigation.

Section snippets

Finite-element modeling of the end mill

In Fig. 2, the shank of a helical fluted end mill with two flutes is clamped in a collet that is installed in the spindle of a milling machine, L being the gauge length of the helical end mill from the collet end. The approximate cross-section of the two-fluted end mill is shown in Fig. 3, where XYZ is the fixed coordinate system, xyz is the coordinate system attached to and rotating with the cutter, and the x′, y′ and z′ axes are the principal axes of the two-fluted end mill at every

Cutting force model for a helical fluted end milling cutter

Since in this study the helical fluted end mill is considered, the chip thickness varies also along the axis of cutter. Therefore the helical fluted end mill (or the modeling beam element) is sliced into many equal axial segments, as shown in Fig. 2, Fig. 6. For each axial segment the helical fluted cutting edge is viewed as a straight edge [2]. Summing up the cutting forces for each axial segment engaged with the workpiece, the instantaneous cutting forces acting on the whole end mill can be

Surface error generation

To predict the machined surface form error, there is only need to consider the tool and workpiece deflections in the Y and Y directions, respectively. The differential cutting force components, DFY and DFZ, of the cutting edge of each axial segment are assumed as uniform line forces that are applied on the two nodes in the middle of the corresponding 3D element of the workpiece. Hence the differential cutting forces of each axial segment are assumed to be applied on B–B line of the workpiece,

Finite element models verification

In this study, the finite-element method is used to model the whole cutting system, including the end mill and the workpiece. As mentioned in Section 2.1, the helical fluted end mill is modeled with the pre-twisted Timoshenko beam element which was also used in the authors' previous work [6], where good results were obtained. The thin-walled workpiece is modeled with the 3D 12-node isoparametric hexahedral element as described in Section 2.2. If the thin plate structure is modeled with the 3D

Conclusions

This study develops a finite-element model to analyze the surface dimensional errors in the peripheral milling of thin-walled workpieces. The helical fluted end mill is modeled with the pre-twisted Timoshenko beam finite element, which can more accurately model the specific geometry and structural behavior of the cutter, compared with treating the tool as a cantilevered uniform beam with an equivalent diameter and using the equations of strength of materials. The workpiece is modeled with a 3D

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