Stability of milling using Chebyshev collocation method

Abstract

The dynamic stability of the milling process is investigated through single and two degree-of-freedom mechanical models by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems like milling are modeled by linear delay-differential equations (DDEs) with time-periodic coefficients. A new approximation technique for studying the stability properties of such systems is presented in this thesis. The approach is based on the properties of Chebyshev polynomials and a collocation expansion of the solution at their extremum points, the Chebyshev collocation points. The stability properties are determined by the eigenvalues of the monodromy matrix which maps collocation points from one interval to the next and which is a finite dimensional approximation to which the exact infinite dimensional Floquet transition matrix (monodromy operator). We check the results for convergence by varying the number of Chebyshev collocation points and by simulation of the transient response via the DDE23 MATLAB routine. Stability charts and chatter frequency diagrams are produced for up-milling and down-milling cases of 1, 2, 4 and 8 cutting teeth and 0 to 100 % immersion levels. The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations are found which agree with the results found by other techniques in the previous literature. An in-depth investigation in the vicinity of the critical immersion ratio for down-milling (where the average cutting force changes from negative to positive) and its implication for stability is presented.