Suppression of Regenerative Chatter via Impedance Modulation

Abstract

A new technique for the suppression of machine tool chatter in turning based on modulation of the system impedance is analyzed. The modulation frequency is selected as half the spindle frequency. Using the method of harmonic balance and a numerical root finding scheme on the transcendental characteristic equation, stability diagrams arc computed for a variety of modulation amplitudes. The stability boundaries are verified by using the Nyquist stability criterion and growth exponents computed from transient simulations of the vibratory response. It is shown in this way that the impedance modulation technique successfully scrambles the regenerative process that leads to chatter and widens the stability region (by increasing the maximum allowed cutting depth), particularly for low values of the spindle speed.

Semi‐discretization method for delayed systems

“…Another approach was developed by Insperger and Stà epà an [4] for milling process analysis, when the discrete time delay is approximated by special continuous ones, and the inÿnite-dimensional eigenvalue problem is transformed into an approximate ÿnite-dimensional one. An alternative of Hill's method was used by Seagalmann and Butcher [27] to determine stability properties of turning process with harmonic impedance modulation. Insperger and Stà epà an [28] also used Hill's method to determine the closed-form stability chart for the Mathieu equation with one discrete delay.…”

Stability of up-milling and down-milling, part 1: alternative analytical methods

“…After the extensive work of Tlusty et al [3], Tobias [4] and Kudinov [5,6], the so-called regenerative effect has become the most commonly accepted explanation for machine tool chatter [7][8][9][10]. This effect is related to the wavy workpiece surface generated by the previous cutting tooth passage.…”

Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials

“…For illustrative examples, the stability charts of delayed Mathieu equations are obtained for different cases of proportionality between parametric period and the delay period. Also, stability charts are produced for the dynamic model of impedance modulation in turning and compared with those obtained in a prior study [9]. The results show that the proposed techniques can provide accurate stability boundaries for systems modeled by time-periodic DDEs.…”

“…Equation (89) is the dimensionless form of dynamic model of machine tool vibration studied by Segalman and Butcher [9] which models turning with impedance modulation. The independent variable is the revolution angle co-ordinate, and the non-dimensional rotation speed is = 1/ .…”

“…Recently, however, a few numerical methods have been proposed to study the stability of this kind of system. These include time finite element analysis [3], numerical simulation [10], statistical signal variance [14], harmonic balance and infinite determinants [4,7,9,15], approximation of the delay by weighted integrals [6] and the semi-discretization method [16]. In each of these papers, the traditional stability charts related to different dynamic models are given.…”