Analytical stability lobes including nonlinear process damping effect on machining chatter
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, Acr, 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 Acr or greater; or in a state of “finite amplitude stability”, where the amplitude of vibration stabilizes at a value between zero and Acr. 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 Acr. 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.
Section snippets
System dynamic model
The vibratory model is shown in Fig. 1. It is a single degree of freedom in the normal direction Y and the system is assumed to be rigid in the other directions. The forces acting on the system are Fy in the feed or the radial direction Y and Fx the tangential force acting in the velocity or the tangential direction X. Each force is composed of shear and ploughing components according towhere s designates shearing while p denotes ploughing. The shear components are obtained
Analytical representation of process damping
In this section the geometry of the indentation area will be discussed along with the analytical computation of this area. This is utilized in formulating the equivalent viscous damping that will be employed in establishing the stability lobes. An analytical formula to compute the equivalent damping will also be presented.
Establishing stability lobes including process damping
In establishing the lobes in this work, other sources of process damping such as changes in the magnitude and direction of the shear force are neglected in comparison with the indentation of material underneath the flank face of the tool. It is a reasonable assumption based on the study conducted by Huang and Wang [20] where the authors showed that the energy dissipated by the other sources are very small compared to that dissipated by the indentation.
Starting with the equation of motion
Comparisons with time domain simulations and empirical model
Altintas et al. [2] conducted plunge turning experiments to develop an empirical model for the process damping. They represented the process damping effect by an additional dashpot acting on the vibratory system. Accordingly, they expressed the process damper for 1045 steel in the form
The authors obtained Cp by harmonic excitation at 0.035 mm amplitude. The parameters of the vibratory model in Ref. [2] were as follows: K=6.48×106 N/m, M=0.56 kg, and Cs=145 N s/m. The cutting force
Experimental verifications
Plunge turning experiments of 1018 steel, 100 mm diameter workpieces were conducted in this work at three feedrate: 0.035, 0.05, and 0.075 mm/rev. Because of equipment availability, the experiments were conducted on a 3-axes milling machine rather than a lathe. As shown in Fig. 11, the workpiece was clamped in a tool holder that was mounted in the spindle. The tool was clamped in a fixture mounted on a Kistler 9255 table dynamometer, which in turn was clamped to the machine table. In effect, the
Conclusions
An iterative procedure was developed to establish the stability lobes analytically including process damping. Upper and lower bound stability lobes were established instead of the traditional stability lobes that represent a single boundary between stable and unstable regions. Having two boundaries of stability allowed having a region in between where the process is in a state of finite amplitude stability. In this way the effect of process damping nonlinearity was preserved. It was also shown
Acknowledgment
This work was funded by the Natural Sciences and Engineering Research Council of Canada, NSERC.
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