3.1. Machine Tool Structure and Model Description
Here, the proposed approach was applied to a four-axis machine tool model in uniaxial configuration, which is depicted in
Figure 4. For this system, a high-fidelity and well-parameterized model with a high conformity to the real machine tool structure exists [
6], which can be used to simulate reference data and validate the method presented here without measurement and modeling errors. Due to a beneficial combination of substructuring, MOR [
28], and the linearization of the involved nonlinear friction models [
21], the model is computationally efficient, parametric, and, at the same time, position-flexible. Note that some parameters of the original unreduced model, as, for example, the components’ Young’s moduli, are fixed after the MOR and cannot be identified here. However, this is not a drawback of the presented approach, but the examined model and could be overcome with more advanced MOR approaches. The model has already been subjected to a dimensionality reduction step [
20], resulting in the following 27 unknown machine tool parameters representing the lumped stiffness and damping properties of:
The three mounting elements (MEs) ( each)
The fixed bearing (FB) supporting the BSD ()
The coupling (CPL) between the motor shaft and the BSD ()
The BSD ();
The linear guiding system (LGS) ( used for all four shoes)
Here, indices indicate the direction of the stiffness (k) and viscous (b) and hysteretic (d) damping parameters.
Note that the considered machine tool system fulfills all assumptions stated in
Section 2.1: Modal parameters can be simulated with high accordance [
6], and the machine tool is lightly damped [
11,
21] (A1 and A2). Suitable bounds can, in this case, be easily chosen since the true model parameters are already known (A3). Here, the intervals:
are used. In real-world scenarios, where the true model parameters are unknown, these intervals are believed to be large enough to ensure the validity of assumption A3 even when only data sheet values provided by the machine tool component manufacturers or values from similar machine tools are known. Lastly, all nonlinear damping sources were linearized and replaced by spring–damper systems (A4) [
21].
For evaluating the machine tool’s modal parameters, 32 nodes (see
Table 1 and
Figure 4) were used in the simulations. Both the computational modal analysis and the EMA were conducted at four axis positions covering the full motion range of the
z-axis (
,
,
, and
). Note that there is no unconstrained DOF in the model since the motor brake of the
x-axis is applied and the
z-axis is constrained by the linear replacement stiffness resulting from the friction model linearization [
12].
3.2. Parameter Identification of an Ideal Machine Tool Model
To demonstrate the effectiveness of the proposed approach, an ideal test case was set up first (i.e., “ideal model vs. simulated data”). Reference data were simulated using the model described in
Section 3.1. Thus, in this case, the parameters to be identified are already known in advance, enabling a direct evaluation of the final identification results in addition to an indirect evaluation in terms of model evaluation criteria (e.g., frequency response assurance criterion (FRAC) [
29], cross-signature scale factor (CSF) [
30], MACXP, and NDD). The data comprise the first 20 modes covering the frequency range up to 311
and four axis positions, of which three are used in the identification (
,
, and
) and one is held back for validation purposes (
). Modes with higher eigenfrequencies are not related to the machine tool structure, but the workpiece or the tool [
31,
32] and are excluded from further consideration. Afterward, random values (within their bounds defined in Equation (
14)) are assigned to all 27 model parameters, providing the starting point for the identification procedure. All computations were performed on a state-of-the-art workstation with 40 Intel
® Xeon
® Gold 6148 CPU cores.
Following the approach in
Section 2.3, 120 GSAs (see Equation (
10)) were performed for each of the 20 modes at three axis positions for both the MACXP and NDD
. On the available workstation, this resulted in a computation time of 1
4
.
Figure 5 exemplarily shows two resulting parameter rankings for different position–mode combinations. It can be seen that instead of all 27 model parameters, in both cases, only a few parameters significantly affect the specific model outcomes (i.e., the fit criteria for the displayed modes at the chosen position). The GSA results can also be confirmed by looking at the mode shapes: For example, mode 8 at position
(see
Figure 5a) at
is the first bending mode of the machine tool bed in the
-plane. This mainly involves large vertical (
y-axis) movements of the back MEs (i.e., ME2 and ME3), making them the most significant parameters. Other ME parameters are significant as well due to smaller movements because of asymmetries in the machine tool model. More importantly, feed drive parameters (e.g., BSD, CPL, and FB) are missing here since there is no deformation of the feed drive, but only movement in the unconstrained screw DOF of the BSD. Note that there is, per the definition (see assumption A1 in
Section 2.1), no influence of the damping parameters on the MACXP.
This knowledge can be exploited by setting up optimization problems searching for only the significant parameters while using fixed arbitrary values for the rest. Here, significant parameters were distinguished from non-significant ones by a threshold of 1% regarding their total effect. To circumvent still-existing local minima [
20], 60 (i.e.,
) optimization problems were set up for all position–mode combinations with the MACXP and repeated
times, resulting in total in 6000 independent optimization problems. As the number of unknown parameters is small for each individual run, gradient-based algorithms can be efficiently used. The sequential least-squares programming (SLSQP) approach [
33] was implemented in this work, resulting in an overall computation time of 1
8
. Each optimization run ended up at different final values for the significant (search) parameters because local minima are present, and the non-significant parameters and the initial values for the search parameters were chosen randomly. The final identified stiffness parameter values were determined as the mean value of all repetitions from the position–mode combination with the smallest standard deviations (see
Section 2.3).
Figure 6a shows the relative deviation of the selected stiffness parameter values from their, in this case, true known values. It can be seen that all parameters but two were identified almost perfectly with deviations of less than 5%. Only the stiffnesses of the first and second MEs in the
z-direction (ME1 k z and ME2 k z) were identified poorly with deviations of
% and
%, respectively. However, the high conformity with the reference frequency response function (FRF) shown in
Figure 7 indicates that the influence of these model parameters is small. Minimal deviations toward higher stiffnesses of the other model parameters seem to be able to compensate the underestimated MEs’ stiffnesses in the
z-direction.
In the next step, the model’s damping parameters can be identified using an LS approach, which was described in
Section 2.3. In principle, an LS problem could be set up with all position–mode combinations with NDD
. To avoid the propagation of errors, only the 20 position–mode combinations with the lowest influence of all non-search parameters on the NDD
were selected. This information can be directly taken from the GSAs conducted before. To reduce numerical errors, the number of equations was further reduced by removing all rows from Equation (
13) with a residual damping
smaller than 0.5%, resulting in 18 equations for 11 unknown damping parameters. The relative deviations of the identified damping parameters from their, in this case, known values can be found in
Figure 6b. It can be seen that approximately half of the parameters can be identified with a still reasonably good accuracy of
%. The rest is spread all over the allowed interval (see Equation (
14)) with some parameters even at their limits as, for example, the rotary hysteretic damping of the coupling (CPL d rz) and the viscous damping of the second ME in the
x-direction (ME2 b x) with deviations of
% and
%, respectively. It is important to note that this is not a shortcoming of the damping parameter identification approach. The targeted parameters were found almost perfectly with maximum deviations of
% when it was assumed that the perfect stiffness parameters were found before. This can be explained by again looking at
Figure 5b, which shows that the model’s stiffness parameters influence the NDD
value and, with it, the modal damping. From the latter, the right-hand side of the LS problem for the damping estimation was constructed (see
Section 2.3.3), leading to the displayed deviations.
Table 2 demonstrates the accuracy of the identified model in terms of common model evaluation criteria (i.e., FRAC, CSF, MACXP, and NDD) for the evaluated modes at the three axis positions considered in the identification (
,
, and
). In general, FRAC and CSF values above 80% [
11] or even above 70% [
21] are considered a good match. Thus, it can be stated that the model’s conformity is very high, with worst-case FRAC, CSF and MACXP values of
% and mean values close to 100%. Note, that the worst-case NDD indicates a relative and not absolute deviation of
%, which is also very low.
Table 3 shows the same data considering only the validation position
. The FRAC and CSF values are a bit lower, but in the same range. However, the modal fitness values are very similar with worst-case MACXP and NDD values of
% and
%, respectively.
This indicates very high conformity of the model, which is also confirmed by the WPT FRF in the
x-direction shown in
Figure 7. Here, hardly any deviations from the reference FRF can be seen at all. Note that this holds for the whole frequency range up to 500
, even though only modes up to 311
were considered in the identification, indicating that the global optimum of the model parameters has been found. Additionally,
Figure 7 stresses the importance of the damping parameter identification by also showing an FRF of a model with the same damping parameter deviations, but randomly reassigned to other (damping) parameters. For example, the deviation of the rotary hysteretic damping of the coupling (CPL d rz) from its true reference value of
% could be used to calculate a value for the viscous damping of the first ME in the
z-direction (ME3 b z), and so on. This leads to a poor qualitative match with the reference FRF, especially in comparison to the actual identification results.
Summarizing the above, it can be said that the shown approach provided stiffness parameters very close to their global optimum (see
Figure 6a). Note that this can only be stated since, in this case, simulated reference data from a model whose parameters are known were used as a starting point for the identification. In general, one would validate the model against reference data not used in the identification, as shown in
Table 3 and
Figure 7. However, this might lead to a misleading conclusion in the case of the damping parameters: the conformity of the model is very high (see
Table 3), as is the overall deviation of the parameters from their true known reference (see
Figure 6b). It is assumed that this conflict originates from the high influence of the stiffness parameters on the damping parameters (see
Figure 5b), which would mean that the found damping parameters still represent the global optimum of the model. This is supported by the fact that the damping parameters’ deviations almost vanish when it is assumed that the true stiffness parameters have been found. However, it cannot be completely ruled out that this conflict originates from a validation position too close to the identification positions or very different significance values of the damping parameters, resulting in only those with high sensitivity being calculated correctly. This would indicate local and, thus, non-transferable solutions for the damping parameters. More insights into this will be targeted in further research. For the machine tool model considered here (see
Section 3.1), the final parameter identification results were found to be comparably insensitive to the sensitivity threshold, the number of repetitions of each optimization problem
, and the number of involved position–mode combinations in the damping parameter identification. However, the importance of these hyperparameters will be reinvestigated in further research.
3.3. Parameter Identification of a Disturbed Machine Tool Model
In this section, the effectiveness of the proposed approach is shown by using reference data from a similar, but not matching model for the identification. Here, the model to be identified is disturbed by setting different values for 44 non-sensitive parameters [
20]. Note that, in contrast to
Section 3.2, assumption A2 from
Section 2.1 is now only approximately true. As the same simulated reference data as in
Section 3.2 were used, the identification results, that is the modal parameters, can again be evaluated directly by comparing them to the reference model’s parameters.
Based on the results of the GSAs for the disturbed model, the stiffness parameters of the model were identified first (see
Section 2.3 and
Section 3.2). The results can be found in
Figure 8a. It can be seen that most of the stiffness parameters were identified well with deviations of even less than 3%. Similar to
Section 3.2, the stiffness in the
x-direction of the first ME (ME1 k x) shows a larger deviation of
%. The identified and the reference FRFs in
Figure 9 match very well. Additionally, both the modal and the frequency-based conformity measures for the position considered in the identification process in
Table 4 are very high, suggesting that the global optimum was approximated well.
This is also supported by
Table 5, which, apart from the worst-case and 5% percentile MACXP (and NDD) values, also shows very high conformity for the validation position. The reason for this exception is that there are three poorly identified modes in the range 265
to 311
, in which the FRF has a low amplitude (see
Figure 9), leading to only a minor influence on the FRAC (and the NDD).
Similar to
Section 3.2, an LS problem was set up and solved for the yet-unknown damping parameters. Again, only the 20 position–mode combinations with the lowest interaction of the non-search parameters on the NDD
were selected. The chosen residual damping threshold of
% did not lead to any further reduction of the number of equations in this case. The final deviations depicted in
Figure 8b are slightly higher than in the “ideal model vs. simulated data” case (see
Figure 6b). However, it is believed that the identified damping parameters still represent the true and global optimum since the overall conformity shown in
Table 4 and
Table 5 is, except for three modes in the range 265
to 311
, very high. Furthermore, the identified model’s FRF in
Figure 9 matches well with the reference data. Again, the comparison with similar, but randomly assigned damping deviations stresses the importance of the damping parameter identification.