PREDICTION OF SURFACE LOCATION ERROR BY TIME FINITE ELEMENT ANALYSIS AND EULER INTEGRATION Tony L. Schmitz a , Philip V. Bayly b , Johannes A. Soons a , and Brian Dutterer a a National Institute of Standards and Technology, Gaithersburg, MD b Washington University in St. Louis, St. Louis, MO 1.0 Introduction Surface location error (SLE), which occurs when the machined surface does not lie in the commanded location due to the cutting process dynamics, is not traditionally considered when diagnosing part errors, even though the error magnitude can be as large or larger than those caused by the geometric or thermal errors of the machine tool. SLE can be visualized by considering a square that is produced by peripheral milling. Ideally, the tool will remove material to produce a square of the desired dimensions. However, due to the relative phasing between the cutter vibrations and the time at which the cutter produces the final surface (e.g., as the tool exits the cut in down milling), either more material may be removed than commanded (overcutting, which gives a smaller square) or less than commanded (undercutting, which produces a larger square). This relative phasing is defined by the system frequency response function and selected forcing frequency, i.e., spindle speed [1-2]. The expertise that has been gained in modeling cutter motions for chatter avoidance [e.g., 3-5] can be drawn upon to predict and compensate SLE and, more importantly, allow the pre-process selection of spindle speeds with significantly reduced errors. Surface location error is especially important in high-speed machining applications for three reasons: 1) due to the low damping in most systems, the surface location error changes dramatically for a small variation in spindle speed near the intentionally selected speeds; 2) the error magnitude grows near the system resonance which typically occurs at the high tooth passing frequencies preferred in high-speed machining; and 3) high-speed machining is often used to produce monolithic parts with thin-walled sections and the mechanical integrity of such features depends strongly on correct wall thickness. In this study, time domain simulation tools are applied to predict SLE in milling and comparisons are made with high-speed machined part dimensions. The simulations include the time finite element analysis (TFEA) method [6] and the well-known time marching Euler integration scheme [1]. The cutting tests included slot machining, low radial immersion (5 %) cuts, and thin rib production. These tests were carried out over a number of spindle speeds, the control parameter in this case, using two 12.7 mm diameter tools. 2.0 Simulations descriptions For both simulation methods in this study, a rigid workpiece is assumed. In the TFEA method, an exact solution for the X-Y displacement vector of the cutter is found using the system state transition matrix when the cutter is not in contact with the workpiece. When in contact, the time in the cut is broken up into multiple elements and the vector displacement during a single element is approximated as a linear combination of polynomial trial functions. The position and velocity at the beginning of the current time element are matched to the position and velocity at the end of the previous element. Using this formulation, a discrete version of the system (i.e., a linear map) is obtained that relates coefficients of the solution to coefficients one tooth passage earlier, which is required for the surface regeneration found in milling. The eigenvalues of the linear map determine its stability. The fixed points of the map represent the steady state solution, from which the displacement of the tool as it leaves the cut for down milling or enters the cut in up milling can be extracted. This displacement approximates the SLE of the finished surface. A complete description of the TFEA method can be found in reference [6]. The numerical time domain simulations used in this study are based on Tlusty’s Euler integration scheme [1]. In this method, the cutter angle is sequentially incremented in a time-based manner and, if the current tooth is within the angle bounded by the prescribed radial immersion, the thrust and radial forces, F t and F r , respectively, are calculated using Eq. 1 (the same force model was used for the TFEA simulations) [7]. If multiple teeth are engaged in the cut, the force is summed over all active teeth. The cutter deflection is then calculated using this force and the cutter velocity and deflection values from the previous time step. If the deflection is large enough so that the tool leaves the cut, a nonlinearity occurs. It is accounted for by setting the forces equal to zero in this situation. In Eq. 1, b is the commanded axial depth of cut and h is the instantaneous chip thickness that depends on the selected chip load or feed per tooth, the current tool deflection, and the previously cut surface. This dependence of the actual chip thickness, and therefore the cutting force, on previous cutter deflections is referred to as regeneration of waviness