Analytical stability lobes including nonlinear process damping effect on machining chatter

Abstract

Indentation of the tool edge and flank face into workpiece surface undulations has been recognized in the literature as the main source of process damping. This damping affects the process stability at low cutting speed greatly. Numerical simulations have allowed integrating the nonlinear indentation force into machining chatter models. It is shown in this paper that the indentation force requires very high discretization resolution for accurate numerical simulation. The objective of the current work is to develop the stability lobes analytically taking into account the effect of nonlinear process damping. The developed lobes could be established for different amplitudes of vibration. This is a departure from the traditional notion that the stability lobes represent a single boundary between fully stable and fully unstable cutting conditions. Plunge turning is utilized in the current work to illustrate the procedure of establishing the lobes analytically. Experimental cutting tests were conducted at three feedrates for sharp and worn tools and the results agreed well with the analytically established lobes.

Introduction

Stability lobes have been proven to be of great value in selecting cutting conditions at which machining chatter could be avoided. These lobes show the limit width of cut versus cutting speed. At low cutting speeds the stability is greatly affected by process damping generated at the tool/workpiece interface. Linear process damping models have been developed to establish the lobes analytically. These models were established either empirically [1], [2] using sinusoidal excitations at particular amplitudes or by adopting Wu’s indentation force model [3] with the assumption of small amplitude vibration [4]. The empirical models are valid at the utilized excitation amplitudes and they lead to errors in estimating stability boundaries associated with other amplitudes. The assumption of small amplitude vibration was utilized [4] to linearize the nonlinear indentation force model of Wu. As such, the model in Ref. [4] yields a lower bound of process stability where the vibration amplitude is close to zero and leads to errors in establishing process stability associated with higher amplitudes. The nonlinear indentation force model was implemented in numerical simulations of turning operations [5], [6] and the results showed that at a range of widths of the cut the process falls into a state of “finite amplitude stability”. In this state the vibration amplitude stabilizes at a value greater than zero and smaller than critical amplitude, A_cr, at which the tool disengages from the workpiece periodically. In other words, the process could be “fully stable”, where the vibration dies down to zero amplitude; “fully unstable”, where the vibration amplitude stabilizes at A_cr or greater; or in a state of “finite amplitude stability”, where the amplitude of vibration stabilizes at a value between zero and A_cr. The case of “finite amplitude stability” was investigated and confirmed experimentally by the current authors [7]. Although more accurate in predicting the stability status of the cut, the numerical implementation of the nonlinear process damping requires high discretization resolution of the indentation pulse, which translates into long simulation time at each width of cut. Establishing the stability lobes numerically over a typical speed range would thus require a very long simulation time. The objective of the current work is to develop a linearized model of the process damping while preserving the vibration-amplitude dependence of the indentation pulse on the resulting stability. In this way the linearized model will reflect the nonlinear effect of the indentation force and will allow discerning the status of the cut. It will be used in establishing different stability lobes in a turning operation, each associated with a particular amplitude of vibration in the range zero to A_cr. Associating the stability lobes with different vibration amplitudes is a departure from the existing notion that the stability lobes represent a single boundary between “fully stable” and “fully unstable” conditions. While the existing notion is correct at high speeds, it does not agree with theoretical [5], [6] and experimental [7], [8] evidence at low speed due to the nonlinearity associated with process damping. The task of establishing a “band” of stability will be carried out in the current paper.

The turning simulation model is briefly described next to establish the basic variables involved including the indentation force pulse associated with process damping. The effect of discretization resolution is also investigated in that section to demonstrate the need to utilize very high resolution in order to represent the indentation pulse accurately. This will be followed by analytical computations of the process damping that involve computations of the indentation area and equivalent viscous damping. A faster method to the computations of this equivalent damping will also be presented, which will help speed up establishing the stability lobes greatly. The subsequent section will present the approach of establishing the lobes at particular amplitudes of vibration as well as boundary lobes analytically. In these lobes the nonlinear effect of process damping will be preserved by maintaining the relationship with vibration amplitude. The analytically established lobes will be tested in the subsequent section against time domain simulations, as well as against the lobes presented by Altintas et al. [2] using an empirical damping model. This will be followed by experimental verifications of the proposed approach using plunge turning of steel and employing sharp and worn tools. It will be shown that the analytically established lobes yield practically the same results obtained from time domain simulations and they are in good agreement with results obtained from cutting tests.