A closed form mechanistic cutting force model for helical peripheral milling of ductile metallic alloys Abhijit Bhattacharyya a,1 , John K. Schueller a,Ã , Brian P. Mann b , John C. Ziegert c , Tony L. Schmitz a , Fred J. Taylor d , Norman G. Fitz-Coy a a Department of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA b Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA c Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA d Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611, USA article info Article history: Received 30 September 2009 Received in revised form 2 March 2010 Accepted 8 March 2010 Available online 15 March 2010 Keywords: Cutting force Cutting coefficients Helical milling Analytical model Runout Uncertainty abstract A closed form mechanistic model is developed for cutting forces in helical peripheral milling (endmilling) of ductile metallic alloys. This paper presents an alternative derivation, using the frontal chip area, to describe two series of cutting force expressions—one using a Heaviside unit step function and the other using a Fourier series expansion. A specific advantage of the present work is highlighted by deriving analytical expressions for sensitivity coefficients required to analytically propagate the uncertainty in the cutting-force model parameters. Another advantage is that even very small radial immersions can be used to derive cutting coefficients reliably, along with their variances. The aforementioned analytical investigations are applied to a series of experimental cutting tests to estimate the force-model cutting coefficients. Experimental investigations include the study of a tool having radial runout. Finally, confidence intervals are placed on predicted forces which experimentally verify the validity of the proposed force model. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Machining with helical endmills is one of the most widely used processes in the manufacture of prismatic parts. Predictive models of cutting forces in endmilling are required for various purposes including the estimation of power consumed, and the prediction of stability and surface placement. Applications based on real time cutting force sensing, including tool wear and tool breakage monitoring, have the potential to use predictive models. A mechanistic model presented by Sabberwal [15] related the tangential force to the chip area with the cutting coefficient being the constant of proportionality. The coefficient was called the specific pressure, and was experimentally demonstrated to be independent of the helix angle for a few work materials. In a review paper, Ehman et al. [6] discussed various formulations of cutting force models available in the literature. Tlusty [21] has described the milling process in great detail, including the modeling of cutting forces. Tlusty and MacNeil [22] were the first to present a closed form analytical solution for two components of the force, in the plane of cutter rotation, for helical peripheral milling. Four separate sets of equation were used to describe each force component. Altintas and Spence [2] presented another two dimensional solution in which different expressions were used depending on the position of the cutter in one rotation. A two dimensional, Fourier series based solution of Schmitz and Mann [18] includes the effect of the helix angle by dividing the cutter into thin axial disks and summing the effects to obtain the total force. A three dimensional integral model was developed by Mann et al. [13] using an equivalent complex Fourier series representation of forces to facilitate symbolic manipulation. Abrari and Elbestawi [1] published a three dimen- sional closed form solution where the forces were expressed as a linear combination of a set of basis functions, but the linearized cutting coefficients were replaced with a matrix which incorpo- rated the helix angle. Engin and Altintas [7] have offered a general solution for a mechanistic model in a set of integral expressions. The integrations have to be carried out by the user for any specific endmilling process. The three dimensional force model presented in this paper differs from prior work in several ways. Each force component has a single, closed form expression which is valid for the entire cutter rotation, but the simple structure of linearized cutting coefficients is retained. Any arbitrary value of radial immersion, and hence very small immersions, can be used to experimentally determine ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijmactool International Journal of Machine Tools & Manufacture 0890-6955/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2010.03.003 Ã Corresponding author. E-mail addresses: abmech@ufl.edu (A. Bhattacharyya), schuejk@ufl.edu (J.K. Schueller). 1 Present address: 21 SE 11th Ave, Apt 2, Fort Lauderdale, FL 33301, USA. International Journal of Machine Tools & Manufacture 50 (2010) 538–551