An analysis of the sensitivity of cutting force coefficients and their influence on the variability of stability diagrams

In order to systematically analyze the variability and validity of force parameter values, an extensive database of measurements of different scenarios defined by the radial and axial depth of cut as well as the milling strategy (up- and down-milling) was generated. For this purpose, a full factorial design was applied in terms of the defined parameter range of the axial and radial engagement listed in Table 1. To investigate the effect of model calibration, three different materials (low-alloyed steel 42CrMo4 (AISI 4140) in a heat-treated state, the titanium alloy TiAl6V4 (Titan Gr. 5, AMS 4928) and the aluminum alloy (Al7075 T6) were examined. An overview of the selected tools consisting of cemented carbide can be found in Table 2. These experiments were conducted on a five-axis machining center (DMG Mori, type DMU 50). The specimen were mounted on a dynamometer (Kistler, type 9257B). The experiments consisted of stable cutting processes which is essential for the calibration of conventional force models [15]. Each scenario was carried out using an up- and down-milling strategy. The spindle speeds for machining 42CrMo4 and TiAl6V4 were chosen within the range recommended by the tool manufacturer. The spindle speed for machining Al7075 was selected based on the dynamic transfer function of the dynamometer, which was measured using an impact hammer (Brüel & Kjær, type 8206-002) in order to avoid severe dynamically affected force signals (cf. Fig. 1). Force components that occur at the spindle frequency \(f_\text {s}\), its double, i.e. the tooth engagement frequency, or its triple are close to a transfer factor of one. During the machining of 42CrMo4 and TiAl6V4, the tool was monitored to avoid significant wear-related influence on the cutting forces by comparing the process forces with a reference process in the initial state of the tool.

In order to determine an appropriate section of the measured force signal for the calibration of the model coefficients, the quality of the signals was evaluated. Due to non-stationary engagement situations, the beginning and end of the measured time series of the tool engagements were unsuitable for the calibration of the force model. Variations of process forces can also occur during stationary engagement situations as a result of superimposed vibrations due to the initial excitation of the tool. To account for this and assess the local signal quality, the measurements (Fig. 2a) were evaluated using a window-based analysis. For each measured tooth engagement, the variance \(\sigma\) and covariance \(\rho\) for a window of consecutive tooth feeds were determined for x- and y-direction, where a window size of 30 tooth engagements was sufficient. For calibration, a tooth engagement with a high covariance and a low variance value was selected (Fig. 2b, orange line). To account for slight variations due to the runout of the tools, a set of tooth feeds for one tool rotation was extracted for each measurement (Fig. 2c).

Based on the selected measurement data (cf. Sect. 2.2), the coefficients (\(k_i, m_i\)) of the force model in Eq. (1) were determined using regression analysis. For this purpose, an iterative optimization approach using the least squares fitting method with randomized starting values was applied [6]. One set of coefficients was determined for each individual calibration experiment. As depicted in Fig. 3a for the use case of aluminum, these coefficients can deviate from one another, which highlights the ambiguity of the parameters. In order to evaluate the variability of the parametrization of the force model, a regression analysis was also carried out using 455 sets for aluminum, which consisted of different scenarios with varying process parameter values (cf. Sect. 2.1). Thus, force model coefficients were determined for a varying number of measurements based on a randomized selection of the measured force signals. As can be seen in the decrease of the envelope area defined by the blue lines (Fig. 3), the model shows less variation with an increasing number of scenarios. The calibration experiments were evaluated in the same way for the materials mentioned. As can be seen in Figs. 4 and 5, there existed an initial high variation, which converged to a smaller envelope area (blue line) as the number of scenarios increased. Here it should be noted that for each material, even with the highest number of scenarios considered, a variation in the resulting model parameter values can still be observed. In general, the shape of the envelope area determined using eight scenarios differed for the three different materials. In particular, the force models parameterized for machining Al7075 exhibited rather high values of \(k_\mathrm {c}\) and \(m_\mathrm {n}\) and small values of \(k_\mathrm {t}\) and \(m_\mathrm {c}\) compared to those calibrated for machining 42CrMo4 and TiAl6V4. This may be attributed to the shape of the milling tools used, especially the corner radius present. A detailed evaluation of the determined parameter values and their influence on the calculation of stability diagrams is described in the following for the aluminum use case.

In order to evaluate the quality of the determined coefficients, the predicted process forces for the aluminum use case were compared to the measurements of all conducted experimental process configurations using the mean squared error (MSE). These experimental measurements correspond to the time points determined in Sect. 2.2. The sets of parameter values with their corresponding process configurations are illustrated in Fig. 6 and sorted according to their respective error values (Fig. 6e). The configuration of each set can be identified based on the depicted entries for each process characteristic (\(a_\mathrm {e}\), \(a_\mathrm {p}\), strategy). For a set consisting of one scenario (cf. Fig. 6a, orange “x”), each of Fig. 6b and c shows one process configuration. For sets consisting of, e.g., eight scenarios (cf. Fig. 6, blue triangle), the corresponding column in Fig. 6b and c contains eight entries. The “strategy ratio” in Fig. 6d indicates the ratio between the number of scenarios with up- and down-milling in one set. Based on this information, recommendations can be established for the considered use case regarding the amount and configuration of the calibration experiments. For instance, force predictions based on model coefficients that were parameterized using only single scenarios generally showed a higher MSE value compared to sets consisting of multiple scenarios (cf. Fig. 6, gray shaded area). Only a few of these scenarios based on a single measurement are located in the medium MSE range. This is especially the case for those scenarios where the radial depth of cut \(a_\mathrm {e}\) is larger than the tool radius (\(r = 5\,\hbox {mm}\)). In general, an increase in set size results in an enhanced over-all prediction of process forces, where sets consisting of four to five scenarios can already yield very low MSE values. In addition to the consideration of larger radial depth of cut, an important characteristic of these sets is a balanced strategy ratio towards an equal number of up- and down-milling processes. In the following, two sets were selected as examples for a more detailed discussion: The set with the best MSE value (Fig. 6, set A, Table 3), consisting of eight scenarios (Table 4) and a set with a higher MSE value (Fig. 6, set B, Table 3), corresponding to a typical process configuration for force calibrations (\(a_\mathrm {e} = 2\,\hbox {mm}\), \(a_\mathrm {p} = 3\,\hbox {mm}\), down-milling). In order to analyze the quality of these two sets, they were used to predict the cutting forces for a series of machining processes with varying process parameter values and compared to the corresponding force measurements. In Fig. 7, the resulting forces of an exemplary tooth engagement (\(a_\mathrm {p} = 3\,\hbox {mm}\), up-milling) with different radial depths of cut are depicted. As expected, the prediction of higher depths of cut using parameter set B is insufficient since this range of the undeformed chip thicknesses was not considered in the calibration step. This will have a particular effect on the determination of the stability limit since deflections of milling tools can lead to a large variation of the chip thicknesses (cf. Sect. 3.2).

For the determination of the modal parameter values of the production system, the dynamic compliance at the tool center point (TCP) was evaluated. For this purpose, an automated impact hammer (Maul-Theet, type vImpact-62), which allowed for a reproducible excitation of the relevant eigenfrequencies, and an accelerometer (Brüel & Kjær, type 4374) for measuring its response was used. To account for the influence of the rotating spindle on the dynamic compliance [17], additional FRFs were determined. Since the displacements could not be measured at the rotating cutting edges at the TCP, the measurements were conducted at the shaft (approx. \(30\,\hbox {mm}\) above the TCP) of the rotating tool using a laser sensor (Blum-Novotest, type PSC), which determined the displacement based on the shading of the beam after excitations with the automated impact hammer. It should be noted that only modes within a frequency range of up to \(4500\,\hbox {Hz}\) were identified in the measurements of the rotating tool and correlated to the FRF of the non-rotating tool at the TCP. The dynamic compliance was analyzed for spindle speeds between \(n = [10{,}000; 15{,}000]\,{\mathrm {RPM}}\) with an increment of 1000 RPM. The modal parameters of each identified mode were adjusted according to the determined correlation between these two measurement locations at the TCP and shaft to represent the compliance behavior for each measured rotational speed. In order to take the speed dependency of the dynamic compliance for the prediction of stability limits into account, the modal parameter values were linearly interpolated between the selected speed increments within the considered spindle speed range. Modes above 4500 Hz were adopted from the FRF at the TCP of the non-rotating tool. For validation, experiments were conducted in which only the peripheral cutting edges were engaged in order to avoid the influence of the minor cutting edges on the process dynamics. For this purpose, the same process conditions were applied (tool, machine tool, workpiece material) as described in Sect. 2.1. To determine the stability limits, the radial depth of cut \(a_\mathrm {e}\) was varied, while the axial depth of cut was set to \(a_\mathrm {p} = 3\,\hbox {mm}\). For stability evaluation, the acoustic emissions of the process were recorded (PCB, type 130F20) and analyzed in the frequency domain to allow for an assignment of the eigenmodes involved. To minimize the influence of run-in effects of the milling tools used on the process dynamics [18], the tools were replaced on average after seven experiments with a cutting length of 50 mm each. Considering the spindle speed dependency of the modal parameters, the calculated stability limit (cf. Fig. 8a, “Sim. interpolated”) showed a higher correspondence to the experiments, particularly for spindle speeds between \(n = [12{,}250; 14{,}000]\,{\mathrm {RPM}}\), than the limit “Sim. \(n = 0\,{\mathrm {RPM}}\)”, which was calculated based on the FRF measured at the TCP of the non-rotating tool. An overestimation of the process stability was present at spindle speeds of \(n = 10{,}500\), 12,000 and \(14{,}000\,{\mathrm {RPM}}\). For these processes (red and purple dot), the modes were adopted from the FRF measured at the TCP since they were not present in the measurements taken during tool rotation. This may be the reason for the deviations of the stability limits. Processes that could not be assigned to a unique mode were labeled “inconclusive” (grey dots).

Based on the results of the evaluated process stability described in Sect. 3.1, the influence of the force parametrization on the accuracy of the predicted stability limits was analyzed. For this purpose, a selection of sets with different MSE values were used in order to initialize the force model of the geometric physically-based simulation. The variability of the predicted stability limits is depicted in Fig. 9 for different parameter sets for spindle speeds between \(n = [11{,}000; 14{,}000]\,{\mathrm {RPM}}\). While the stability limits that were calculated using force parameter sets with smaller MSE values converged towards the limit calculated using set A with the best MSE value (black line), the predicted stability limit resulting from sets with higher MSE values diverged significantly. The largest deviation of the predicted limits of more than 3.5 mm was present at a spindle speed of \(n = 13{,}500\, {\mathrm {RPM}}\). Generally, the experimentally determined stability limit was underestimated when using parameter sets with higher MSE values. For parameter set B, which represents a conceivable scenario for force calibration, the calculated limit (red line) underestimated the experiments in particular between \(n = [12{,}250; 14{,}000]\,{\mathrm {RPM}}\). In addition to a shift of the predicted lobes, e.g., at approx. \(n = 12{,}500\,{\mathrm {RPM}}\), a different formation of the shapes of individual lobes was also observed, especially at spindle speeds of \(n = 12{,}000\,{\mathrm {RPM}}\) and 13,500 RPM. This effect could be caused by complex interactions of different modes and their respective response to varying force excitations.