GENERALIZED MODELING OF MILLING MECHANICS AND DYNAMICS: PART II - INSERTED CUTTERS Serafettin Engin Yusuf Altintas Graduate Student Professor and ASME Fellow <engin@mech.ubc.ca> <altintas@mech.ubc.ca> The University of British Columbia Department of Mechanical Engineering 2324 Main Mall, Vanocuver, B.C. , V6T 1Z4, CANADA tel: (604) 822 21 82, fax: (604) 822 24 03 ABSTRACT Inserted cutters are widely used in roughing and finishing of parts. The insert geometry and distribution of inserts on the cutter body vary significantly in industry depending on the application. This paper presents a generalized mathematical model of inserted cutters for the purpose of predicting cutting forces, vibrations, dimensional surface finish and stability lobes in milling. In this paper, the edge geometry is defined in the local coordinate system of each insert, and placed and oriented on the cutter body using cutter’s global coordinate system. The cutting edge locations are defined mathematically, and used in predicting the chip thickness distribution along the cutting zone. Each insert may have a different geometry, such as rectangular, convex triangular or a mathematically definable edge. Each insert can be placed on the cutter body mathematically by providing the coordinates of insert center with respect to the cutter body center. The inserts can be oriented by rotating them around the cutter body, thus each insert may be assigned to have different lead and axial rake angles. By solving the mechanics and dynamics of cutting at each edge point, and integrating them over the contact zone, it is shown that the milling process can be predicted for any inserted cutter. A sample of inserted cutter modeling and analysis examples are provided with experimental verifications. NOMENCLATURE: X, Y, Z : Cutter coordinate system O , O’ : Cutter and insert center points a , b : Rectangular insert width and height ( 29 z h , φ : Instantaneous chip thickness u, v, w : Insert local coordinate system β , δ , ϕ : Insert rotation angle around X, Y, Z axes M T : Total transformation matrix for the insert R , s θ , e θ , f o : Convex triangular insert dimensions t dF , r dF , a dF : Differential tangential, radial and axial force components acting on a chip element, respectively tc K , rc K , ac K : Cutting force coefficients in tangential, radial and axial directions, respectively te K , re K , ae K : Cutting coefficients in tangential, radial and axial directions, respectively x F , y F , z F : Cutting forces in X, Y, Z directions, respectively r I , z I : Insert center radial and axial distance from cutter tip, respectively V IC : Insert center position vector from cutter center V CE : Cutting edge position vector from insert center V CER : Cutting edge position vector from insert center after rotations V p r : Cutting edge position vector from cutter center ) ( R β x , ) ( R δ y , ) ( R ϕ z :Insert rotation matrixes for X, Y, Z, respectively 1. INTRODUCTION Inserted face and end milling cutters are widely used in industry. Face milling cutters have evenly or unevenly spaced inserts, and used in removing excess material from the face of parts such as transmission box, engine block or machine tool columns and beds. Inserts are distributed in both radial and axial directions on indexed end mills. Indexed cutters are used in removing massive amount of material from the periphery of parts such as aircraft spars, airframe support structures and rough pocketing of dies and molds. The distribution of inserts may be regular or arbitrary depending on the application. In order to improve the surface finish, chatter stability and cutting force balancing on the cutter body, the inserts with various shapes can be placed on the milling cutter body. The industry requires a generalized mathematical model, which allows the analysis of all inserted milling cutters used in practice (Wertheim et al., 1994). However, the research in the past mainly focused on the modeling of mechanics and dynamics of special cutter geometry. Fu et al. (1984) provided a comprehensive modeling of