Virtual compensation of deflection errors in ball end milling of flexible blades
Introduction
The current research trend is to digitally model the machining operations, and optimize them in virtual environment before conducting costly physical trials. A comprehensive review of the virtual machining research, which consists of modelling cutter–workpiece engagement (CWE), cutting forces, machine and machining dynamics, simulation and optimization of machining operations can be found in CIRP key note paper [1]. This paper focuses specifically on the accurate ball-end milling of integrally bladed rotors, which is mostly affected by the quasi-static deflections of highly flexible thin-walled blades and the slender tools. Currently, the dimensional errors are measured after a machining trial, and the tool path is modified to compensate the errors iteratively until the tolerance of the blade is satisfied. It is desired to develop the mathematical model of the process leading to prediction and compensation of deflection errors digitally to eliminate the costly machining tests.
Tool deflection errors are either controlled indirectly by constraining the cutting forces through feedrate scheduling [2], [3], or by modifying the tool path [4]. Varying flexibility of the part along the tool path and changing tool–workpiece engagements due to the deflections of the tool and the part have not been considered. There have been studies to predict the deflections of the thin-walled parts in milling, where the static stiffness of the blade changes with the material removal. Biermann and Kersting [5], [6] compared Finite Elements (FE), particle and oscillator based methods to predict the flexible workpiece deflections. These models are applicable for the analysis of a small portion of the workpiece due to the required complex mesh modifications and re-meshing. Budak et al. [7] combined FE with a structural modification technique to predict the structural dynamics of flexible parts. Koike et al. [8] updated the static stiffness of the in-process workpiece, and minimized the static deflections of a cantilevered beam by orienting the cutting forces in the direction having the highest stiffness. Instead of using computationally prohibitive re-meshing based methods, Tuysuz and Altintas [9] modelled the initial workpiece in FE only once, and analytically updated its static and dynamic stiffness as the material is removed by developing a substructuring method.
In this paper, a comprehensive virtual model of dimensional form errors and their compensation in ball-end milling of thin-walled blades are presented as outlined in Fig. 1. The article is organized as follows. The cutting force and cutter–workpiece engagement models for a rigid blade machining system are briefly overviewed in Section 2. The stiffness of the blade is updated analytically as the material is removed along the tool path (Section 2.1). The cutter–workpiece engagement boundaries are updated by considering both the tool and workpiece deflections induced by cutting forces (Section 2.2). The deflection errors are predicted and compensated by modifying the tool path in Section 3. The proposed form error reduction has been experimentally demonstrated in ball-end milling of a sample blade in Section 4, and the paper is concluded in Section 5.
Section snippets
Modelling of dimensional surface form errors
NC program for blade milling is assumed to be chatter free, and the objective is to estimate and compensate the static deflection errors imprinted on the blade surface as (Fig. 1). The time invariant static stiffness of the slender end mill is modelled as a cantilevered elastic beam divided into disk elements along its axis. The blade is meshed into discrete tetrahedral elements having the same nominal height of tool’s discrete disk elements, and used to predict the time varying stiffness
Toolpath compensation
The calculated form error in the flexible milling system is iteratively compensated at cutter–blade contact (CC) points along the toolpath. The goal is to force the surface error towards zero while keeping the new CC points as close as to the original blade dimensions. Assume that the deflection vector at the CC point is calculated as d1 (Fig. 6). The nominal CC point on the blade moves to CC1 in the surface normal direction. CC1 has a surface error vector of E1 = d1. The compensation algorithm
Experimental verification
The form error prediction and compensation model has been verified in 3-axis ball-end milling of a blade made of AL7050-T7451. The workpiece is rigidly mounted on the machine table (Fig. 8(a)) to have cantilevered boundary condition on the blade. The workpiece is cut with 0.1 mm/tooth feed at 1000 rev/min spindle speed under chatter-free conditions. Sandvik R216.44-12030-AK26N 4-fluted solid carbide ball-end mill with 12 mm diameter was used. The cutting forces were measured with Kistler 9123C
Conclusion
Traditionally, the deflection errors left on the highly flexible machined blades are measured, tool path is modified with offsets equivalent to the deflections, and the process is iteratively experimented on the machine until tolerance of the blade is met. This paper proposes physics based digital prediction and compensation of deflection errors on the surface of ball-end milled, highly flexible blades. The critical challenges in the modelling are iterative estimation of cutter–workpiece
Acknowledgment
This research is supported by NSERC — Pratt & Whitney Canada Industrial Research Chair Grant on Virtual Machining (IRCPJ 260683-12).
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