Computer-Aided Design 42 (2010) 903–910
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Global optimization of tool path for five-axis flank milling with a conical cutter LiMin Zhu a,∗ , Gang Zheng a , Han Ding b , YouLun Xiong b a
State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, PR China
b
State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, PR China
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Article history: Received 12 November 2009 Accepted 25 June 2010 Keywords: Conical cutter Flank milling Tool path Distance function Global optimization
abstract In this paper, optimum positioning of the conical cutter for five-axis flank milling of slender surfaces is addressed from the perspective of approximating the tool envelope surface to the data points on the design surface following the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation. Based on the observation that a conical surface can be treated as a canal surface, i.e. envelope surface of one-parameter family of spheres, the swept envelope of a conical cutter is represented as a sphere-swept surface. Then, an approach is presented to efficiently compute the signed distance between a point in space and the swept surface without constructing the swept surface itself. The first order differential increment of the signed point-to-surface distance with respect to the differential deformation of the tool axis trajectory surface is derived. By using the distance function, tool path optimizations for semi-finish and finish millings are formulated as two constrained optimization problems in a unified framework, and a sequential approximation algorithm along with a hierarchical algorithmic structure is developed for the optimization. Numerical examples are given to confirm the validity and efficiency of the proposed approach. Comparing with the existing approaches, the present one improves the machining accuracy greatly. The rationale developed applies to general rotary cutters. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Turbo-machinery components including centrifugal compressor impellers and turbine blades are often considered to be the most challenging components to manufacture. Generally, their tight tolerances and smooth surface finishes are obtained by applying high speed machining with point milling. The major advantages of point milling are that almost any surface can be machined and it is relatively easy to position the cutting tools. Form the manufacturer’s point of view, however, it is very time consuming. It may require more finish passes and each pass removes only a small amount of material. Another disadvantage is that the point milling process produces a scalloped surface finish, leaving a small cusp height between two consecutive passes. For slender parts, like turbine blades and impellers, free-form surfaces are usually approximated by piecewise ruled surfaces [1]. Flank milling offers a better choice for machining such slender surfaces. It is performed by employing the side of a cutter to touch the desired surface. Compared with point milling, flank milling has its unique advantages. It can increase the material removal rate, lower the cutting forces, eliminate necessary hand finish and ensure improved component accuracy. To this end, advanced tool
∗
Corresponding author. Tel.: +86 2134206545. E-mail address:
[email protected] (L. Zhu).
0010-4485/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2010.06.005
path generation algorithms should be developed. Cylindrical and conical cutters are the two most commonly used tools for flank milling. In the last decade, the problem of optimum positioning of a cylindrical cutter has been studied extensively [2–8]. Cylindrical cutters can satisfy most of the demand for flank milling. However, when an application requires flank milling within a confined space, a conical cutter may prove to be more suitable because it has a smaller tip and a stronger shank in comparison with a cylindrical cutter having a small diameter [9]. Some strategies proposed for cylindrical cutters are applicable to conical cutters, but they will result in high surface deviations. Recently, increasing attention has been drawn onto the problem of using a conical cutter for flank milling ruled surfaces. Monies et al. [10,11] first presented a conical tool based milling strategy, in which the conical tool has three tangent points: two are on the directrix curves and one is on the ruling line between the directrix curves. The only hypothesis is that the curvatures of the directrix curves in the zone considered are kept constant. This method leads to seven transcendental equations and the solution of these equations dictates the final tool position. In a subsequent work [12], they proposed an algorithm to calculate the interference error between the cutter and the workpiece so that one can determine the optimal cutter dimensions (cone radius and angle). Motivated by Bedi’s work [7], Li et al. [9] presented a three-stepoptimization approach, which needs to solve three transcendental equations numerically. The program is easy to implement and
