Approximate Analytical Solutions for Primary Chatter in the Non-Linear Metal Cutting Model

Abstract

This paper considers an accepted model of the metal cutting process dynamics in the context of an approximate analysis of the resulting non-linear differential equations of motion. The process model is based upon the established mechanics of orthogonal cutting and results in a pair of non-linear ordinary differential equations which are then restated in a form suitable for approximate analytical solution. The chosen solution technique is the perturbation method of multiple time scales and approximate closed-form solutions are generated for the most important non-resonant case. Numerical data are then substituted into the analytical solutions and key results are obtained and presented. Some comparisons between the exact numerical calculations for the forces involved and their reduced and simplified analytical counterparts are given. It is shown that there is almost no discernible difference between the two thus confirming the validity of the excitation functions adopted in the analysis for the data sets used, these being chosen to represent a real orthogonal cutting process. In an attempt to provide guidance for the selection of technological parameters for the avoidance of primary chatter, this paper determines for the first time the stability regions in terms of the depth of cut and the cutting speed co-ordinates. #

Response of a magneto‐rheological fluid damper subjected to periodic forcing in a high frequency limit

“…In fact the multiple time-scale method is an efficient and widely used treatment for nonlinear mechanical systems [8,9,10] to get an approximate analytic solution. In these cases the time-scales are introduced to account for nonlinear terms in the examined systems.…”

Nonlinear Dynamics and Chaos of a Nonsmooth Rotor-Stator System

“…Savi et al [4] discussed a one-degree-of-freedom (1DOF) system considering numerical and experimental approaches. Several engineering systems are related to nonsmooth nonlinearities including oil drilling [5][6][7][8], manufacturing [9], and rotor dynamics [10][11][12][13].…”

Dynamical changes during composite milling: recurrence and multiscale entropy analysis

“…Over the past years, its fast development gave way to a reliable high-speed cutting procedure. Consequently, elimina-tion and stabilization of the associated chatter oscillations have become a high interest in science and technology [1][2][3][4][5][6][7]. The plausible adaptive control concept, based on relatively short time series, is studied to gain deeper understanding.…”