For the acquisition of a sufficient training set and, therewith, validation of the presented method, frequency response measurements were performed through impact hammer tests on two machine tools in different poses. On a five-axis machining center with swivel head kinematic Heller FT 4000, denoted as

${M}_{1}$ in the following, the positions of the axes influencing the pose of the tool,

X,

Y, and

C, were varied. Furthermore, on a five-axis machining center with fork head kinematic DMG HSC 75 linear, denoted as

${M}_{2}$, the

Y-,

Z-, and

B-axis positions were varied. For both of the machining centers, the axis positions were selected on the basis of an extended centrally composed experimental design with star points, in order to be able to investigate higher order influences as well as possible cross-correlations (cf.

Figure 1). A total number of

${P}_{1}=46$ and

${P}_{2}=49$ poses were investigated using

${M}_{1}$ and

${M}_{2}$, respectively.

The FRF measurements were conducted using a Fraisa X7400 spherical end mill with a diameter of $d=10\mathrm{mm}$ and a free length of ${l}_{\mathrm{l}}=30.4\mathrm{mm}$. Hereto, an impact hammer Kistler 8206 and an acceleration sensor 352C23 by PCB Piezotronics, mounted at the tool tip, were used. Fast Fourier transformations were performed up to ${f}_{\mathrm{max},}\phantom{\rule{4.pt}{0ex}}1=6400\mathrm{Hz}$ and ${f}_{\mathrm{max},}\phantom{\rule{4.pt}{0ex}}2=3200\mathrm{Hz}$ for ${M}_{1}$ and ${M}_{2}$, respectively. The finite length of the impulse response function measurements led to a frequency resolution of $\u2206{f}_{1}=0.25\mathrm{Hz}$ and $\u2206{f}_{2}=0.5\mathrm{Hz}$ for ${M}_{1}$ and ${M}_{2}$, respectively.

Slot milling experiments were conducted using the aforementioned milling tool, the machine tool

${M}_{2}$, and an increasing depth of cut

${a}_{\mathrm{p}}$, using a starting value of

${a}_{\mathrm{p},\mathrm{start}}$ and a depth of cut of

${a}_{\mathrm{p},\mathrm{end}}$ at the end of each slot, in order to validate the calculated SLDs. A high speed steel ASP 2012 hardened to approx. 58 HRC was machined. Different inclination angles and spindle speeds were investigated. There are different publications in literature, which investigated the detection of chatter vibrations applying wavelet analysis using various signals, e.g., cutting forces [

38] or tool accelerations [

39,

40,

41]. In this contribution, discrete acoustic emission signals were recorded and analyzed utilizing the continuous wavelet transform [

42]

for each experiment. In this context,

${\xi}^{\left(i\right)}$ is the

i-th sample of the original acoustic emission signal,

N is the total number of samples of the signal,

${\mathrm{\Psi}}^{*}$ is the complex conjugate of a mother wavelet

$\mathrm{\Psi}$,

${\delta}_{t}$ is the time difference between two samples of the signal, and

a and

b are the scaling and translation variables, respectively. Each scaled and translated mother wavelet corresponds to an investigated frequency at a time instant. By calculating the convolution between the original signal and the scaled and translated mother wavelet, the correlation between the signal and a frequency can be estimated for each time instant. For the mother wavelet, the complex Morlet wavelet [

43]

was used. The stability limit

${a}_{\mathrm{p},\mathrm{crit}}$ of each experiment was estimated as

whereby

$W\left(a\right)$ is the wavelet transform of the signal,

$\tau $ is a user-defined threshold value and

S is the set of investigated scales, which correspond to a set of natural frequencies of the FRF of the regarded pose. For the Morlet wavelet, the relationship between a scale

a and the Fourier period

$\lambda $ can be described as

whereby

${\omega}_{0}$ is the non-dimensional central frequency of the mother wavelet. Because natural frequencies measured for a non-rotating spindle may differ from natural frequencies of a rotating system, the function

$T({W}^{\left(i\right)}\left(a\right),{N}_{T})$ calculates a weighted average of the wavelet intensities, incorporating

${N}_{T}$ neighboring frequencies of the natural frequency, which corresponds to the scaling

a. For the weighting, the Blackman window function [

44]

with

was used, whereby

${W}^{\left(i\right)}\left(a\right)$ was weighted with

$w({N}_{T}/2)$. Equation (4) represents different values for the translation variable

b with the index

i, assuming that there is a set

$\left\{{W}^{\left(i\right)}\left(a\right)\phantom{\rule{4pt}{0ex}}\forall i\in \left\{1,\cdots ,N\right\}\right\}$, whereby each

${W}^{\left(i\right)}\left(a\right)$ corresponds to a sample

${\xi}^{\left(i\right)}$ of the original signal and

b is estimated according to the temporal location of

${\xi}^{\left(i\right)}$ within the time series. Thus, the stability limit is estimatednby identifying a critical index

${i}_{\mathrm{crit}}$, which is defined as the index, where each of the weighted averages of wavelet intensities of a frequency and its neighboring frequencies of a set of investigated natural frequencies exceeds

$\overline{W\left(a\right)}+\tau \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{W}_{\mathrm{s}}\left(a\right)$, whereby

are the mean and the standard deviation of a wavelet transform

$W\left(a\right)$, respectively.

Figure 2 visualizes the identification of the stability limit using a spindle speed of

n = 3025 min

^{−1} and an inclination angle of

$B=-45{}^{\circ}$.

Acoustic emission signals often are highly influenced by noise. As a result, a high amount of frequencies correlate with the corresponding scaled and translated wavelet transforms. The natural frequencies 1415 $\mathrm{Hz}$, 1500 $\mathrm{Hz}$ and 1580 $\mathrm{Hz}$ of the three dominant peaks of the FRF in X- and Y-direction were considered when analyzing the wavelet transform, since they were expected to have the most influence on the stability behavior of the process. Using the aforementioned approach, a stability limit of ${a}_{\mathrm{p},\mathrm{crit}}=0.5\mathrm{mm}$ was identified, since all og the wavelet transforms, which corresponded to the three natural frequencies, exceeded the defined threshold at the corresponding point in time. The stability limits, which are calculated using the presented method, have to be interpreted with caution. The resulting limits are highly dependent on the signal-to-noise ratio of the acoustic emission signal and the choice of $\tau $. As a result, the transferability to processes that differ from the considered configuration is limited.