Deterministic-Stochastic Subspace Identification of the Modal Parameters of a Machine Tool During Milling

Abstract

Modal analysis is a standard tool for evaluating the dynamic behaviour of machine tools. Since the dynamic behaviour can differ for operating and analysis conditions, the use of operational modal analysis for machine tools has been researched over the last decade. However, the operation of a machine tool is a special case with respect to the excitation, as the excitation is both deterministic and stochastic in nature. Therefore, a perfect broadband excitation, which can be assumed to be white noise, does not occur in machine tools. This fact must be taken into account for the identification and makes the application of classical techniques of operational modal analysis to milling machines insufficient. The approach for identification of the modal parameters of a milling machine during machining, presented in the paper, is based on the deterministic-stochastic subspace identification, which can consider both the deterministic and the stochastic character of the excitation in machine tools better than the stochastic subspace identification being often used in operational modal analysis. For this purpose, the formulation of the input vector for a milling machine is crucial. The approach is developed with the help of simulations and demonstrated on a real machine tool. The results show an improvement compared to the stochastic subspace identification, which is visible in more clear stability diagrams and in a higher number of identified resonance frequencies.

1. Introduction

The dynamic behaviour of a machine tool is defined as the response (vibration) of a machine tool to mechanical excitations [1]. The sources of the mechanical excitation can be external (e.g., neighbouring machines) or internal (e.g., cutting process, movements of feed axes and imbalances of rotating components). Since the vibration limits the working accuracy and the productivity of a machine tool, attention is paid to its avoiding, reducing or suppression. For this purpose, information about structural properties impacting the vibration like natural frequencies, mode shapes and modal damping (in other words modal parameters) is essential. Such information can be obtained by conducting the experimental modal analysis (EMA) [2]. Since the EMA is always conducted on a machine tool at a standstill (at special conditions customised to conduct EMA), there can be a significant difference between modal parameters identified within the EMA and those existing during operation, as referred in [3,4,5]. The difference can be caused by the process damping [6,7], changing inertial masses due to the workpiece and tool weight, excitation by moving masses [8,9], changed static pre-load in components with non-linear characteristics (like bearings) and gyroscopic moments [10].

For this reason, the implementation of the operational modal analysis (OMA) (see e.g., [11] or [12]) to machine tools has been investigated since the last decade, see e.g., [13,14]. In the case of the OMA, responses of the structure measured during operation are only evaluated. The excitation remains unknown. However, it is assumed that there is a process noise that acts as an excitation source and has the characteristics of a white noise signal. Furthermore, it is assumed that the process noise excites the structure so that a mode shape vector is uniformly excited. Moreover, the process noise is statistically distributed according to the normal distribution. Such assumptions can be valid for large structures like buildings, wind power plants, ships or aircraft. Here, the excitation results from ambient conditions such as wind or water flows. The principle of the OMA can be mathematically demonstrated in the frequency domain very easily. Assumed a linear time-invariant system, the vector of responses of a mechanical structure with l degrees of freedom (DOF) $\mathit{Y}\in {\mathbb{C}}^{l\times 1}$ can be expressed by multiplying the matrix of frequency response functions $\mathit{H}\in {\mathbb{C}}^{l\times l}$ by the vector of excitations $\mathit{F}\in {\mathbb{C}}^{l\times 1}$ in the frequency domain according to Equation (1)

$$\mathit{Y}\left(\omega \right)=\mathit{H}\left(\omega \right)\mathit{F}\left(\omega \right)=\left(\sum _{r=1}^R{\displaystyle \frac{g_r{\mathbf{\Psi }}_r{\mathbf{\Psi }}_r^{\mathrm{T}}}{\mathrm{j}\omega -p_r}}+{\displaystyle \frac{g_r^{*}{\mathbf{\Psi }}_r^{*}{\mathbf{\Psi }}_r^{\mathrm{H}}}{\mathrm{j}\omega -p_r^{*}}}\right)\mathit{F}\left(\omega \right).$$

(1)

Here, $\omega$ and $\mathrm{j}$ stand for angular frequency and complex operator, respectively. The frequency response functions in matrix $\mathit{H}$ can be expressed by the sum of R sets of the modal parameters. The r-th set of the modal parameters consists of the r-th mode shape vector ${\mathbf{\Psi }}_r\in {\mathbb{C}}^{l\times 1}$ and the r-th eigenvalue $p_r=-{\delta }_r+\mathrm{j}{\omega }_{d,r}$ (sometimes in the literature referred to as pole), where ${\mathit{\delta }}_r$ and ${\omega }_{d,r}$ denote the decay constant of the modal damping of the r-th mode shape and the r-th angular resonance frequency, respectively. Every mode shape vector is multiplied by a specific scale factor $g_r$. The value considers the chosen normalisation of the mode shapes (e.g., Unit modal mass or Unit A matrix). The superscripts ${(.)}^{\mathrm{H}}$, ${(.)}^{\mathrm{T}}$ and ${(.)}^{*}$ represent the Hermitian transpose, the transpose and the complex conjugate number, respectively. If power spectral density (PSD) of the response ${\mathit{G}}_{YY}\left(\omega \right)=\mathit{Y}\left(\omega \right){\mathit{Y}}^{\mathrm{H}}\left(\omega \right)=\mathit{H}\left(\omega \right)\mathit{F}\left(\omega \right){\mathit{F}}^{\mathrm{H}}\left(\omega \right){\mathit{H}}^{\mathrm{H}}\left(\omega \right)=\mathit{H}\left(\omega \right){\mathbf{G}}_{FF}\left(\omega \right){\mathit{H}}^{\mathrm{H}}\left(\omega \right)$ is computed, it can be seen that the excitation does not affect the estimation of the modal parameters for certain conditions. In case of excitation with white noise signals, the PSD matrix of excitation ${\mathit{G}}_{FF}$ is constant and the matrix ${\mathit{G}}_{YY}$ can be expressed by modal parameters according to Equation (2) [11]

$${\mathit{G}}_{YY}\left(\omega \right)=\sum _{r=1}^R{\displaystyle \frac{{\mathbf{\Psi }}_r{\mathbf{\Theta }}_r^{\mathrm{T}}}{\mathrm{j}\omega -{\mathrm{p}}_r}}+{\displaystyle \frac{{\mathbf{\Psi }}_r^{*}{\mathbf{\Theta }}_r^{\mathrm{H}}}{\mathrm{j}\omega -{\mathrm{p}}_r^{*}}}+{\displaystyle \frac{{\mathbf{\Theta }}_r{\mathbf{\Psi }}_r^{\mathrm{T}}}{-\mathrm{j}\omega -{\mathrm{p}}_r}}+{\displaystyle \frac{{\mathbf{\Theta }}_r^{*}{\mathbf{\Psi }}_r^{\mathrm{H}}}{-\mathrm{j}\omega -{\mathrm{p}}_r}}.$$

(2)

This implies that the PSD matrix of the responses ${\mathit{G}}_{YY}$ only depends on the modal parameters. However, the excitation influences the operational reference vector ${\mathbf{\Theta }}_r$ associated with the r-th mode shape in analogy to the modal participation vector ${\mathbf{\Xi }}_r=g_r{\mathbf{\Psi }}_r$. Unlike the vector ${\mathbf{\Theta }}_r$, the vector ${\mathbf{\Xi }}_r$ is not influenced by the excitation.

In the case of machine tools, there are many excitation sources during operation in general, like process forces, servo drives, unbalanced components, chip conveyors, lubrication units, coolant pumps, various ventilators, hydraulic and pneumatic devices, etc. However, these excitation sources act locally and cause harmonic or multiple harmonic excitation of machine tools. Such sources are not able to excite the mode shapes uniformly either over the frequency range or spatially. The first kind of uniformity, i.e., uniformly excitation over the frequency range, leads to a broadband excitation with a nearly constant magnitude over the investigated frequency range like a white noise signal. In general, a dominant harmonic excitation is linked with a peak in the frequency spectrum and this peak can be misinterpreted as a resonance frequency. Therefore, it is important to ensure a broadband excitation as far as possible and, further, to check for spurious frequency within the pretreatment of captured signals, as referred in [11]. The need for a broadband excitation of machine tools within OMA is reflected in [5,8,9,13,15,16]. In most cases, the process forces are considered to be dominant for the excitation of the machine. Thus, various modifications of the cutting process like varying cutting parameters (see [13,15,16] for more details) or machining a special workpiece (see [5,8,13]) are suggested to excite a machine tool in broadband manner. Besides the modification of the cutting process, a special movement of the feed axis has been suggested for the excitation of the machine tool as well, as referred to in [9]. The investigation of various excitation sources of a machine tool for the OMA is presented in [17]. Here, the modification of the cutting process seems to be a more suitable way for such excitation. However, it was not possible to excite the machine during milling constantly over the frequency range of interest (unlike the white noise) due to the finite servo drive dynamics. The second kind of non-uniformly excitation leads to distorted mode shapes since mode shapes are amplified near the local excitation. This becomes an issue if there exists locally acting excitation sources, which affect the operational reference vector ${\mathbf{\Theta }}_r$. Unfortunately, this is the case of every machine tool since all the excitation forces in a machine tool (like process forces, servo drives, unbalanced components, coolant pumps, hydraulic and pneumatic devices, etc.) act locally. This is an important aspect when a machine tool is investigated since the evaluation of mode shapes helps to indicate weak spots in the mechanical design. The local distorted mode shape can lead to misinterpretation of the weak spots.

The need for broadband excitation of the white noise character results from the application of classical techniques of operational modal analysis to machine tools. Such techniques like stochastic subspace identification are, however, not able to consider the deterministic character of excitation in machine tools properly. The state-of-the-art shows that it is not possible to provoke a stochastic excitation in machine tools with sufficient results, see e.g., [17,18]. This leads to some difficulties related to non-uniformly excited mode shapes or scattered stability diagrams being used for the selection of resonance frequencies when the operational modal analysis is performed during machining. Therefore, this paper presents an approach for the identification of modal parameters of a machine tool during milling that takes into account the deterministic character of the machine excitation originating from the milling process through the utilisation of deterministic-stochastic subspace identification. The deterministic-stochastic subspace identification requires an input vector that can consider the deterministic excitation by the milling process in a machine tool. The approach is developed theoretically and applied to a machine tool to prove its benefit in comparison to stochastic subspace identification, which belongs to classical techniques for operational modal analysis. Since the milling process can be assumed to be the most significant excitation of a machine tool, the principal drawback of performing operational modal analysis on machine tools during milling can be significantly reduced. The other excitation sources, which may have a deterministic character, are considered by the process noise. This leads to the fact that they are treated like stochastic signals.

The deterministic-stochastic subspace identification is applied to the computation of the modal parameters of large civil structures like bridges and buildings from only output signals [19,20,21]. In general, the OMA is used for the health monitoring of such structures. The deterministic-stochastic subspace identification improves the identification results of the OMA significantly.

The approach presented in this paper contributes to the application of OMA to machine tools. The OMA is a suitable way to perform modal analysis on large machine tools due to realisation of excitation with sufficient levels without the need for shakers. Furthermore, the downtime needed for dynamic investigations can be significantly reduced. The identification of modal parameters during operation can also be utilised for enhanced condition and/or process monitoring due to the assessment of system-related parameters. The knowledge of modal parameters being relevant to a milling process can contribute to a customised process design for high productivity and/or precision.

This paper starts with an explanation of the mathematical background of stochastic subspace identification (SSI) and deterministic-stochastic subspace (DSSI) identification. In this way, the difference between both methods is clearly visible. Since the DSSI requires an input vector, the formulation of an input vector for the face milling process is presented. The challenge consists in preserving the OMA principle, i.e., the measurement of excitation (in our case process forces) with special measuring devices must be avoided. The investigations of the DSSI follow in the next sections. Here, the basic assumptions regarding the kind of excitation, the influence of the process and measuring noise, the signal length and the sampling frequency are theoretically and experimentally investigated. The realisation of the DSSI for a machine tool during a face milling process is also presented and the results are evaluated in comparison to SSI. The discussion wraps up the paper.

2. Methods

2.1. Stochastic Subspace Identification

The most advanced methods for the identification of modal parameters within the OMA are based on the stochastic state-space model [11]. This model has an order n and it assumes that the structure is excited by the process noise with the characteristic of the white noise, which is denoted by the process noise vector $\mathit{w}\left(t\right)$. The state of the system is described by the state vector $\mathit{x}\left(t\right)$. Since measured response signals $\mathit{y}\left(t\right)$ are sampled for the evaluation within OMA, the discrete stochastic state-space model is used for the identification

$$\begin{array}{c}\hfill {\mathit{x}}_{k+1}=\mathit{A}{x}_k+{\mathit{w}}_k\\ \hfill {\mathit{y}}_k=\mathit{C}{\mathit{x}}_k+{\mathit{v}}_k.\end{array}$$

(3)

Here, the matrices $\mathit{A}\in {\mathbb{R}}^{n\times n}$ and $\mathit{C}\in {\mathbb{R}}^{l\times n}$ stand for the discrete system matrix and the output matrix, respectively. l represents the number of DOF being measured. The measured response signals (output) captured at time instance k are summarised in the vector ${\mathit{y}}_k\in {\mathbb{R}}^{l\times 1}$. Every measured response can generally be affected by the measuring noise, which is considered by the vector ${\mathit{v}}_k\in {\mathbb{R}}^{l\times 1}$. The measuring noise includes disturbance in the signal resulting from the sensor coupling, the sensor itself, the cable, the connectors and the electronics of the measuring hardware. The vectors ${\mathit{w}}_k\in {\mathbb{R}}^{n\times 1}$ and ${\mathit{v}}_k$ are zero mean. It is assumed that the model is stationary, i.e., $\mathbf{E}\left[{\mathit{x}}_k\right]=0$ ($\mathbf{E}\left[.\right]$ stands for the Expected value). The matrix $\mathit{A}$ is stable; thus, the state covariance matrix $\mathbf{E}\left[{\mathit{x}}_k{\mathit{x}}_k^{\mathrm{T}}\right]:=\mathrm{\Sigma }=\mathrm{const}$. Since the process noise and measuring noise are white noise sequences independent of ${\mathit{x}}_k$, it can be further assumed $\mathbf{E}\left[{\mathit{x}}_k{\mathit{w}}_k^{\mathrm{T}}\right]=0$ and $\mathbf{E}\left[{\mathit{x}}_k{\mathbf{v}}_k^{\mathrm{T}}\right]=0$. The OMA for machine tools requires the computation of the System matrix $\mathit{A}$, from which the eigenvalues ${\mu }_m$ and eigenvectors ${\mathit{\chi }}_m$ are extracted through the eigendecomposition. These eigenvalues and eigenvectors must be transformed to eigenvalues ${\lambda }_m$ and eigenvectors ${\mathit{\psi }}_m$ for continuous system according to: ${\lambda }_m=ln\left({\mathit{\mu }}_m\right)/\Delta t$ and ${\mathit{\psi }}_m={\mathit{\chi }}_m$. It must be remarked that eigenvalues $\lambda$ correspond to poles p from equation (1). The computation of the matrix $\mathit{A}$ is based on orthogonal projection of the measured responses from future (denoted by the subscript f) into past (denoted by the subscript p) according to ${\mathbf{\Pi }}_i:={\mathit{Y}}_f/{\mathit{Y}}_p={\mathit{Y}}_f{\mathit{Y}}_p^{\mathrm{T}}{\left({\mathit{Y}}_p{\mathit{Y}}_p^{\mathrm{T}}\right)}^{+}{\mathit{Y}}_p$. The ${(.)}^{+}$ denotes the Moore–Penrose inverse. The concept of past and future is just intellectual and it can be visualised by using the Hankel matrix as follows

$$\left({\displaystyle \frac{\begin{array}{ccc}\hfill {\mathit{y}}_0& \hfill \dots & \hfill {\mathit{y}}_{j-1}\\ \hfill ⋮& \hfill \ddots & \hfill ⋮\\ \hfill {\mathit{y}}_{i-1}& \hfill \dots & \hfill {\mathit{y}}_{i+j-2}\end{array}}{\begin{array}{ccc}\hfill {\mathit{y}}_i& \hfill \dots & \hfill {\mathit{y}}_{i+j-1}\\ \hfill ⋮& \hfill \ddots & \hfill ⋮\\ \hfill {\mathit{y}}_{2i-1}& \hfill \dots & \hfill {\mathit{y}}_{2i+j-2}\end{array}}}\right):=\left({\displaystyle \frac{{\mathit{Y}}_p}{{\mathit{Y}}_f}}\right),$$

(4)

where the matrices ${\mathit{Y}}_p$ and ${\mathit{Y}}_f$ denote the responses for past and future, respectively. The choice of the number i should consider $i>n$ and j should be as large as possible since the mathematical proof assumes $j\to \infty$. Therefore, the number of observations s should be $s>2i+j-2$. The matrix ${\mathbf{\Pi }}_i$ can be supplemented by two weighting matrices ${\mathit{W}}_1\in {\mathbb{R}}^{li\times li}$ and ${\mathit{W}}_2\in {\mathbb{R}}^{j\times j}$ so that singular value decomposition is performed according to

$${\mathit{W}}_1{\mathbf{\Pi }}_i{\mathit{W}}_2=\mathit{U}\mathit{S}{\mathit{V}}^{\mathrm{T}}.$$

(5)

The choice of the matrices ${\mathit{W}}_1$ and ${\mathit{W}}_2$ determines the algorithm:

• Principal component algorithm (PC): ${\mathit{W}}_1=\mathit{I}$ and ${\mathit{W}}_2={\mathit{Y}}_p^{\mathrm{T}}{\mathbf{\Phi }}_{Y_p,{\mathit{Y}}_p}^{-1/2}{\mathit{Y}}_p$;
• Unweighted principal component algorithm (UPC): ${\mathit{W}}_1=\mathit{I}$ and ${\mathit{W}}_2=\mathit{I}$;
• Canonical variate algorithm (CVA): ${\mathit{W}}_1={\mathbf{\Phi }}_{Y_f,Y_f}^{-1/2}$ and ${\mathit{W}}_2=\mathit{I}$.

by using ${\mathbf{\Phi }}_{A,B}$, the covariance of two matrices A, B. The matrices from the singular value decomposition are further used to compute the extended observability matrix according to

$${\mathbf{\Gamma }}_i={\mathit{W}}_1^{-1}{\mathit{U}}_1{\mathit{S}}_1^{1/2}.$$

(6)

Matrices ${\mathit{U}}_1$ and ${\mathit{S}}_1$ denote sub-matrices of $\mathit{U}$ and $\mathit{S}$, respectively, that contain the selected singular values. The Kalman’s estimation of the states at time instance k and $k+1$ (${\tilde{\mathit{x}}}_k$ and ${\tilde{\mathit{x}}}_{k+1}$) are computed according to: ${\tilde{\mathit{x}}}_k={\mathbf{\Gamma }}_i^{+}{\mathbf{\Pi }}_i$ and ${\tilde{\mathit{x}}}_{k+1}={\underline{{\mathbf{\Gamma }}_i}}^{+}{\mathbf{\Pi }}_{i-1}$. Here, $\underline{{\mathbf{\Gamma }}_i}$ is obtained by removing the last l rows in ${\mathbf{\Gamma }}_i$ and ${\mathbf{\Pi }}_{i-1}$ is the projection matrix of reduced response in future and extended response in past, i.e., the middle line in the Hankel matrix, Equation (4), is shifted down by l rows, ${\mathbf{\Pi }}_{i-1}={\mathit{Y}}_{f-}/{\mathit{Y}}_{p+}$. Finally, the matrix $\mathit{A}$ is computed according to $\mathit{A}={\mathit{x}}_{k+1}{\mathit{x}}_k^{+}$ (referred to as 1. approach in this paper). Alternatively, the matrix $\mathit{A}$ can be computed by using the extended observability matrix as follows $\mathit{A}={\underline{{\mathbf{\Gamma }}_i}}^{+}\overline{{\mathbf{\Gamma }}_i}$ (referred to as the 2. approach in this paper), where $\overline{{\mathbf{\Gamma }}_i}$ is obtained from ${\mathbf{\Gamma }}_i$ by removing the first l rows. These two approaches, each with the three algorithms (PC, UPC and CVA), can be summarised as data-driven stochastic subspace identification (DD SSI) techniques. The theory is explained more in detail in [22].

Besides the DD SSI methods, there is an alternative approach for the identification of the modal parameters being based on covariance matrices, as referred in [11]. This approach is called covariance-driven stochastic subspace identification (CD SSI). For this purpose, the Toeplitz matrix of covariances is used: ${\mathit{T}}_i={\mathbf{\Phi }}_{Y_f,Y_p}$. The singular value decomposition of the Toeplitz matrix leads to: ${\mathit{T}}_i={\mathit{U}}_c{\mathit{S}}_c{\mathbf{V}}_c^{\mathrm{T}}$ and further to the extended observability matrix ${\mathbf{\Gamma }}_i={\mathit{U}}_{c1}{\mathit{S}}_{c1}^{1/2}$ in Analogy to Equation (6). The system matrix results from $\mathit{A}={\underline{{\mathbf{\Gamma }}_i}}^{+}\overline{{\mathbf{\Gamma }}_i}$, where the nomenclature is analogical to the second approach of DD SSI.

2.2. Deterministic-Stochastic Subspace Identification

In order to take into account the situation with the excitation of machine tools, the forward stochastic deterministic model is used for the identification of modal parameters in this paper

$$\begin{array}{c}\hfill {\mathit{x}}_{k+1}=\mathit{A}{\mathit{x}}_k+\mathit{B}{\mathit{u}}_k+{\mathit{w}}_k\\ \hfill {\mathit{y}}_k=\mathit{C}{\mathit{x}}_k+\mathit{D}{\mathit{u}}_k+{\mathbf{v}}_k,\end{array}$$

(7)

where $\mathit{B}$, $\mathit{D}$ and ${\mathit{U}}_k$ denote the input matrix, the feedforward matrix and the input vector at time instance k, respectively. The process noise ${\mathit{w}}_k$ contains a part of the state vector that cannot be considered by the term $\mathit{A}{\mathit{x}}_k+\mathit{B}{\mathit{u}}_k$, i.e., vibrations caused by excitation sources being not considered by the input vector. The unit of the process noise is the same as for the state vector. In the case of the deterministic-stochastic subspace identification (DSSI), the procedure is very similar to that for SSI being presented above. However, in this case, the oblique projection of the output future matrix ${\mathit{Y}}_f$ onto the Hankel matrix of past inputs and past outputs ${\mathit{W}}_p$

$${\mathit{W}}_p=\left(\begin{array}{c}\hfill {\mathit{U}}_p\\ \hfill {\mathit{Y}}_p\end{array}\right)$$

(8)

along the input future matrix ${\mathit{U}}_f$ is employed here. Thus, the projection matrix results from ${\mathit{O}}_i:=\left[{\mathit{Y}}_f{\mathit{U}}_f^{\mathrm{T}}{\left({\mathit{U}}_f{\mathit{U}}_f^{\mathrm{T}}\right)}^{+}{\mathit{U}}_f\right]{\left[{\mathit{W}}_p{\mathit{U}}_f^{\mathrm{T}}{\left({\mathit{U}}_f{\mathit{U}}_f^{\mathrm{T}}\right)}^{+}{\mathit{U}}_f\right]}^{+}{\mathit{W}}_p$. Moreover, the second projection matrix ${\mathit{Z}}_i$ is defined as orthogonal projection according to

$${\mathit{Z}}_i:={\mathit{Y}}_f{\left(\begin{array}{c}\hfill {\mathit{W}}_p\\ \hfill {\mathit{U}}_f\end{array}\right)}^{\mathrm{T}}{\left(\left(\begin{array}{c}\hfill {\mathit{W}}_p\\ \hfill {\mathit{U}}_f\end{array}\right){\left(\begin{array}{c}\hfill {\mathit{W}}_p\\ \hfill {\mathit{U}}_f\end{array}\right)}^{\mathrm{T}}\right)}^{+}\left(\begin{array}{c}\hfill {\mathit{W}}_p\\ \hfill {\mathit{U}}_f\end{array}\right)$$

(9)

In analogy to ${\mathbf{\Pi }}_{i-1}$, the matrix ${\mathit{Z}}_{i+1}$ is computed with reduced future matrices ${\mathit{Y}}_{f-}$ and ${\mathit{U}}_{f-}$ and extended past matrix ${\mathit{W}}_{p+}$ each by l rows. The system matrix $\mathit{A}$ is computed through solving the following linear equation

$$\left(\begin{array}{c}\underline{{\mathbf{\Gamma }}_i}{\mathit{Z}}_{i+1}\\ {\mathit{Y}}_{i|i}\end{array}\right)=\left(\begin{array}{c}\mathit{A}\\ \mathit{C}\end{array}\right){\mathbf{\Gamma }}_i^{+}{\mathit{Z}}_i+\kappa {\mathit{U}}_f+\mathit{\rho }.$$

(10)

Here, ${\mathbf{\Gamma }}_i$ is the extended observability matrix. However, this matrix is computed through the singular value decomposition of the oblique projection matrix ${\mathit{O}}_i$ and matrices ${\mathit{W}}_1\in {\mathbb{R}}^{li\times li}$ and ${\mathit{W}}_2\in {\mathbb{R}}^{j\times j}$ in Analogy to Equations (5) and (6). Again, the choice of the matrices ${\mathit{W}}_1$ and ${\mathit{W}}_2$ determines the algorithm for DSSI:

• Numerical algorithms for Subspace State Space System IDentification (N4SID): ${\mathit{W}}_1=\mathit{I}$ and ${\mathit{W}}_2=\mathit{I}$;
• Multivariable Output-Error State sPace (MOESP): ${\mathit{W}}_1=\mathit{I}$
  and ${\mathit{W}}_2=\mathit{I}-{\mathit{U}}_f^{\mathrm{T}}{\left({\mathit{U}}_f{\mathit{U}}_f^{\mathrm{T}}\right)}^{+}{\mathit{U}}_f$;
• Canonical variate algorithm (CVA): ${\mathit{W}}_1={\mathbf{\Phi }}_{{\mathbf{\Pi }}_{YU},{\mathbf{\Pi }}_{YU}}^{-1/2}$
  and ${\mathit{W}}_2=\mathit{I}-{\mathit{U}}_f^{\mathrm{T}}{\left({\mathit{U}}_f{\mathit{U}}_f^{\mathrm{T}}\right)}^{+}{\mathit{U}}_f$.

Here, ${\mathbf{\Pi }}_{YU}={\mathit{Y}}_f-{\mathit{Y}}_f{\mathit{U}}_f^{\mathrm{T}}{\left({\mathit{U}}_f{\mathit{U}}_f^{\mathrm{T}}\right)}^{+}{\mathit{U}}_f$. The matrix $\underline{{\mathbf{\Gamma }}_i}$ is obtained by removing the last l rows in ${\mathbf{\Gamma }}_i$ like in case of SSI and ${\mathit{Y}}_{i|i}$ is the matrix with the first l rows of the matrix ${\mathit{Y}}_f$. The computation of the system matrix $\mathit{A}$ from Equation (10) determines the first approach of DSSI. The second approach of DSSI employs shifted oblique projection matrix ${\mathit{O}}_{i+1}:=\left[{\mathit{Y}}_{f-}-{\mathit{Y}}_{f-}{\mathit{U}}_{f-}^{\mathrm{T}}{\left({\mathit{U}}_{f-}{\mathit{U}}_{f-}^{\mathrm{T}}\right)}^{+}{\mathit{U}}_{f-}\right]\left[{\mathit{W}}_{p+}-{\mathit{W}}_{p+}{\mathit{U}}_{f-}^{\mathrm{T}}{\left({\mathit{U}}_{f-}{\mathit{U}}_{f-}^{\mathrm{T}}\right)}^{+}{\mathit{U}}_{f-}\right]{\mathit{W}}_{p+}$. In analogy to the first approach of DSSI, the matrix $\mathit{A}$ is computed through solving a linear equation

$$\left(\begin{array}{c}{\widehat{\mathit{X}}}_{i+1}\\ {\mathit{Y}}_{i|i}\end{array}\right)=\left(\begin{array}{cc}\mathit{A}& \mathit{B}\\ \mathit{C}& \mathit{D}\end{array}\right)\left(\begin{array}{c}{\widehat{\mathit{X}}}_i\\ {\mathit{U}}_{i|i}\end{array}\right)+\mathit{\rho }.$$

(11)

In Equation (11), matrices with the Kalman’s estimate sequences result from ${\widehat{\mathit{x}}}_i={\mathbf{\Gamma }}_i^{+}{\mathit{O}}_i$ and ${\widehat{\mathit{x}}}_{i+1}={\underline{{\mathbf{\Gamma }}_i}}^{+}{\mathit{O}}_{i+1}$.

2.3. Realisation of the Input Vector for Machine Tools

The idea behind the deployment of DSSI for OMA on machine tools is based on the assumption that the cutting process is the dominant excitation source and vibrations caused by the other excitation sources are a part of the process noise ${\mathit{w}}_k$. Furthermore, the process forces remain unknown, i.e., they are not measured. The aim of the input vector is therefore to consider the deterministic part of the excitation resulting from the milling process. For this purpose, only the geometric relations are needed, as they reflect the dynamics of the process forces. Thus, the input vector ${\mathit{u}}_k$ can reproduce the dynamics of the cutting process.

In general, the cutting forces can be modelled by using the linear cutting force model, as referred in [23]: $F_t=K_{tc}bh+K_{te}b$, $F_r=K_{rc}bh+K_{re}b$, $F_a=K_{ac}bh+K_{ae}b$, where b, h, $K_c$ and $K_e$ are the width of cut, the uncut chip thickness, cutting constants, and edge coefficients, respectively. The indices t, r, and a denote the tangential, radial and axial directions, respectively. Since the stationary process is assumed, i.e., $\mathbf{E}\left[\mathit{u}\left(t\right)\right]=0$, the constant term of cutting forces (the term with edge coefficients) can be ignored. In the case of face milling with a face milling tool without helix and with the side cutting edge angle of 0° and considering notation from Figure 1, the process forces in x, y and z directions can be computed as

$$\begin{array}{c}{F}_x=\sum _{q=1}^Q-K_{tc}b{h}_{q}cos{\mathit{\varphi }}_q-K_{rc}b{h}_{q}sin{\mathit{\varphi }}_q=-b{K}_{tc}\sum _{q=1}^Q(f_t+r_q)\left(sin{\mathit{\varphi }}_{q}cos{\mathit{\varphi }}_q+q_{rt}{sin}^2{\mathit{\varphi }}_q\right)\hfill \\ F_y=\sum _{q=1}^Q{K}_{tc}bhsin{\mathit{\varphi }}_q-K_{rc}bhcos{\mathit{\varphi }}_q=b{K}_{tc}\sum _{q=1}^Q(f_t+r_q)\left({sin}^2{\mathit{\varphi }}_q-q_{rt}sin{\mathit{\varphi }}_{q}cos{\mathit{\varphi }}_q\right)\hfill \\ F_z=K_{ac}bh=K_{ac}b\sum _{q=1}^Q(f_t+r_q)sin{\mathit{\varphi }}_q\hfill \end{array}$$

(12)

respectively, where $f_t$ and $q_{rt}$ denote the feed rate per tooth and the ratio of the radial to the tangential cutting constant, respectively. The index q stands for the q-th cutting insert of a face milling tool with Q inserts. The process forces can only exist if an insert is in cut, i.e., ${\mathit{\varphi }}_q\in 〈{\mathit{\varphi }}_{st},{\mathit{\varphi }}_{ex}〉$. The run-out error r can influence the dynamic of the milling process significantly. However, the run-out error must be measured, which may not always be possible. Thus, it is not considered when computing the input vector. As mentioned above, the input vector ${\mathit{u}}_k$ only contains the dynamic influencing term of Equation (12)

$${\mathit{u}}_k=b{f}_t\sum _{q=1}^Q\left(\begin{array}{c}-sin{\mathit{\varphi }}_{q,k}cos{\mathit{\varphi }}_{q,k}-q_{rt}{sin}^2{\mathit{\varphi }}_{q,k}\hfill \\ {sin}^2{\mathit{\varphi }}_{q,k}-q_{rt}sin{\mathit{\varphi }}_{q,k}cos{\mathit{\varphi }}_{q,k}\hfill \\ sin{\mathit{\varphi }}_{q,k}\hfill \end{array}\right)$$

(13)

The cutting coefficients ($K_{tc}$ and $K_{ac}$), which are unknown, are a part of the input matrix $\mathit{B}$ being identified within OMA. The instantaneous angle of immersion of the q-th insert can be computed as ${\mathit{\varphi }}_q\left(t\right)={\mathit{\varphi }}_0+{\mathit{\varphi }}_{p,q}+\int 2\pi n_T\left(t\right)\mathrm{d}t$. The tool resp. spindle speed $n_T\left(t\right)$ is traced by the control system during cutting. Thus, the input vector can be easily computed. The biggest challenge consists of finding the constant angle ${\mathit{\varphi }}_0$. This will be found out during the identification process as explained further in this paper. The angle ${\mathit{\varphi }}_{p,q}$ is the position angle of the q-th insert on the face milling tool, e.g., ${\mathit{\varphi }}_{p,q}=(q-1){\mathit{\varphi }}_p$ for a constant pitch angle of the tool ${\mathit{\varphi }}_p$.

This approach (DSSI in combination with the input vector being based on geometric relations, which are derived from the computation of process forces) can theoretically handle every process for which the geometric relations can be formulated. The part of the deterministic excitation that is not considered by the input vector is automatically treated as process noise.

2.4. Theoretical Study with Harmonic Oscillator

In the next step, the methods for identification of modal parameters during milling presented above are investigated. The impact of basic assumptions on the results of the methods is theoretically investigated in the example of a harmonic oscillator with three DOF. The harmonic oscillator is defined by the inertial matrix $\mathit{M}\in {\mathbb{R}}^{3\times 3}$, the damping matrix $\mathit{B}$ and the stiffness matrix $\mathit{K}$ as follows

$$\begin{array}{ccc}\mathit{M}=\mathit{I}\mathrm{kg},& \mathit{B}=\left(\begin{array}{ccc}{b}_1+b_2& -b_2& 0\\ -b_2& b_2+b_3& -b_3\\ 0& -b_3& b_3\end{array}\right),& \mathit{K}=\left(\begin{array}{ccc}{k}_1+k_2& -k_2& 0\\ -k_2& k_2+k_3& -k_3\\ 0& -k_3& k_3\end{array}\right),\end{array}$$

(14)

where $b_1=0.85$ Ns/m, $b_2=2b_1$, and $b_3=2b_2$, $k_1=45e3$ N/m, and $k_2=2k_1$, $k_3=2k_2$. Various scenarios related to the excitation and measurement noise are implemented according to

Table 1. Investigated scenarios with the harmonic oscillator.

Simulation	Description
No.1	Correlated white noise in all DOF
No.2	Correlated white noise in all DOF and measurement noise 1%
No.3	Correlated white noise in all DOF and measurement noise 10%
No.4	Uncorrelated white noise in all DOF
No.5	Uncorrelated white noise in all DOF and measurement noise 10%
No.6	Uncorrelated white noise in all DOF of various level (DOF 1—100%, DOF 2—60%, DOF 3—30%)
No.7	White noise at point 1
No.8	White noise at point 1 and measurement noise 10%
No.9	White noise at point 2
No.10	White noise at point 2 and measurement noise 10%
No.11	White noise at point 3
No.12	White noise at point 3 and measurement noise 10%
No.13	Process forces at point 3
No.14	Broadband process forces at point 3
No.15	Broadband process forces at point 3 and measurement noise 10%
No.16	Broadband process forces at point 3, measurement noise 10%, and $j=5000$
No.17	Broadband process forces at point 3, measurement noise 10%, and $f{s}^{*}=400$ Hz
No.18	Broadband process forces at point 3, measurement noise 10%, and process noise

* Sampling frequency.

. The investigations put focus on the identified resonance frequencies and mode shapes by using SSI and DSSI in comparison to the analytical results. For this purpose, stability diagrams of eigenvalues and MAC (Modal Assurance Criterion) values for mode shapes are deployed (for more details see [24]).

Here, the white noise excitation has the power spectral density of 0.1 Hz^−1/2. The process forces resulting from a face milling process (see Figure 1) are computed according to Equation (12) for following parameters: ${\mathit{\varphi }}_{st}=162.54$°, ${\mathit{\varphi }}_{ex}=360$°, $b=1$ mm, $n_T=1000$ rpm, $f_t=0.075$ mm, $q_{rt}=0.2$, $K_{tc}=4224$ N/mm², $K_{ac}=650$ N/mm². The tool is a face milling cutter of 40 mm diameter with four inserts ($Q=4$) and a constant pitch angle of ${\mathit{\varphi }}_p=90$°. The process force is computed for the spindle speed of 3000 rpm. The broadband process forces result from a modification of the described face milling process through spindle speed variation according to $n_T=3000+2400sin\left(2\pi 0.18t\right)$ rpm. This modification is performed with the constant feed rate of 300 mm/min so that the spindle speed variation leads to the variation of the chip thickness, and thus, to the variation of process forces. Each load case is simulated without measuring noise, i.e., ${\mathit{v}}_k=0\forall k$, as well as with measuring noise modelled by white noise sequence, which is uncorrelated for DOF, of level 10% and 1% related to simulated responses without measuring noise. It must be noticed that such a level of measuring noise is actually very high as the accelerometers being commonly used for EMA feature much lower noise than the level of 10%. In this context, the influence of the signal length on the results is investigated, since j is not infinite in the praxis. For this purpose, j will be varied from high ($j=40\times {10}^3$, default value in this paper) to short signal length ($j=612$). The maximal signal length is limited by the computing power due to the singular value decomposition of projection matrices, which are of the dimension $i\times j$. Furthermore, the sampling frequency $fs$ is the subject of investigation since too high a sampling frequency leads to a short captured time period if the number of samples j cannot be exceeded. In such cases, the high-frequency information resulting from the high sampling frequency might not be relevant for the identification due to a low-frequency range being often of interest when machine tools are investigated. Therefore, it can be meaningful to reduce the sampling frequency to achieve a longer captured time period for more information at low frequencies. Considering the highest natural frequency of the harmonic oscillator of 104.1 Hz, the sampling frequency being investigated is set to 400 Hz and 1000 Hz (default value in this paper) for the constant number of samples $j=40\times {10}^3$.

2.5. Theoretical Study with Machine Model

In the next step, a model of higher order is used to investigate the impact of higher complexity in the dynamic behaviour on the results of the suggested method. Furthermore, the impact of the process noise is assessed, since the process noise plays an important role when EMA during milling is conducted, as referred in [18]. For this purpose, the model of a real machine tool is used. The model is based on modal parameters, which were identified within EMA

$$\begin{array}{c}\left(\begin{array}{c}\dot{\mathit{q}}\\ \ddot{\mathit{q}}\end{array}\right)=\left(\begin{array}{cc}\mathbf{0}& \mathit{I}\\ -\mathbf{\Lambda }& -2\delta \end{array}\right)\left(\begin{array}{c}\mathit{q}\\ \dot{\mathit{q}}\end{array}\right)+\left(\begin{array}{c}\mathbf{0}\\ {\mathit{P}}^{\mathrm{T}}\end{array}\right)\mathit{f}\hfill \\ \mathit{y}=\left(\begin{array}{cc}\mathit{P}& \mathbf{0}\end{array}\right)\left(\begin{array}{c}\mathit{q}\\ \dot{\mathit{q}}\end{array}\right).\hfill \end{array}$$

(15)

Here, the output vector $\mathit{y}\left(t\right)\in {\mathbb{R}}^{l\times 1}$ stands for measured displacements. The matrix $\mathit{P}\in {\mathbb{R}}^{l\times R}$ is the matrix of undamped mode shapes scaled according to unit modal mass. The number of DOF of the presented model amounts to 3, i.e., $l=3$, and 19 mode shapes were identified, i.e., $R=19$. The diagonal matrices $\mathbf{\Lambda }\in {\mathbb{R}}^{R\times R}$ and $\mathbf{\delta }\in {\mathbb{R}}^{R\times R}$ contain the natural frequencies and decay constants of modal damping, respectively. Considering the number of identified mode shapes R, the model order amounts to 38. The excitation is in the form of forces for all measured DOF, which are included by the input vector $\mathit{f}\in {\mathbb{R}}^{l\times 1}$. In this model, the 3 DOF correspond to displacements in x, y, and z coordinates at the tool tip (TCP). The vector $\mathit{q}\in {\mathbb{R}}^{R\times 1}$ consists of measured displacements in the modal coordinate system.

In analogy to the investigation with the harmonic oscillator, there are various scenarios applied in this model and the modal parameters are identified with the methods of SSI and DSSI presented in this paper. The evaluation of the results follows through a comparison of the identified eigenvalues and mode shapes to those from the EMA. In such a way, the capability of the investigated methods to identify modal parameters of the system with realistic model order can be assessed. The overview of the investigated scenarios with the machine model is given in

Table 2. Investigated scenarios with the machine model.

Simulation	Description
No.19	Uncorrelated white noise excitation at TCP in all three coordinates
No.20	Broadband process forces at TCP explicitly considered by the input vector
No.21	Broadband process forces excitation at TCP implicitly considered by the input vector
No.22	Broadband process forces excitation at TCP implicitly considered by the input vector, and process noise as white noise signal
No.23	Broadband process forces excitation at TCP implicitly considered by the input vector, and process noise as harmonic signal
No.24	Broadband process forces excitation at TCP implicitly considered by the input vector, and process noise as impulse sequence

.

The investigation is similar to those for the harmonic oscillator, i.e., the subject of investigations with this model is the influence of the signal length and of the sampling frequency on the identification results. The measuring noise is set to 10% for all investigations. This value represents the worst case in the investigation with the oscillator. As the machine model covers the frequency range from 0 to 400 Hz, the sampling frequency is set to 1 kHz, 2 kHz and 10 kHz. Additionally, the investigation of the process noise is included here. As the machine model is defined in modal coordinates, the process noise is applied in the model as an additional force excitation. In this way, the physical meaning of the process noise (excitation sources being out of the scope of the input vector) can be kept during the investigation. The term $\mathit{f}$ becomes $(\mathit{f}+\mathit{w})$. The DSSI only considers $\mathit{f}$ in the input vector, i.e., ${\mathit{u}}_k\widehat{=}{\mathit{f}}_k$. The process noise being investigated is white noise sequence, harmonic function with frequency of 672.3 rad/s, and impulse sequence with the period between impulses of 1 s. The level of the process noise is always 10% and 1% of process forces.

2.6. Identification with Measured Signals

2.6.1. Pseudo-OMA

All experimental investigations were carried out on a machine tool of medium size whose modal parameters were used for modelling in the previous section. First, investigations for excitation with a shaker are performed. The shaker is installed on the table of the machine tool and excites the spindle by using a tool dummy (see Figure 2). The oblique orientation of the excitation force ensures that the machine is excited in the three coordinates. The response of the machine to the excitation is measured with triaxial accelerometers at various locations on the machine (Headstock, Machine table, and Machine slide of the y nc axis). The evaluation is conducted according to the SSI and DSSI methods, presented in this paper. Therefore, these investigations are called Pseudo-OMA in this paper, since the excitation does not result from the operation; however, methods being used for OMA are employed for the evaluation. Thus, the capability of the DSSI and SSI methods can be assessed through a direct comparison of identified modal parameters to those from EMA. In such a way, the machine can be excited by the real white noise signal, which matches the assumption of OMA. Furthermore, the process noise is partly controllable and can be implemented in various way. There is still a process noise originating from the surroundings of the machine. This kind of process noise cannot be controlled. For the quantification of this portion of process noise, responses on the machine are measured without any excitation. Afterwards, the shaker excitation with a white noise signal is conducted. In the next steps, various kinds of additional excitation sources, which cause process noise, are implemented in the machine. In such a way, the influence of the process noise on the identification results can be assessed. All measurements being taken within the Pseudo-OMA are listed in the

Table 3. Investigated scenarios for Pseudo-OMA.

Experiment	Description
No.1	Process noise from the surroundings of the machine, i.e., machine is switched off
No.2	Shaker excitation with white noise + machine off
No.3	Shaker excitation with white noise + machine in stand by
No.4	Shaker excitation with white noise + process noise due to pulse sequence at the headstock
No.5	Shaker excitation with white noise + process noise due to pulse sequence at the machine table
No.6	Shaker excitation with white noise + process noise due to pulse sequence at the machine slide
No.7	Shaker excitation with white noise + process noise due to harmonic excitation with second shaker at the headstock
No.8	Shaker excitation with white noise + process noise due to harmonic excitation with second shaker at the machine table
No.9	Shaker excitation with white noise + process noise due to harmonic excitation with second shaker at the machine slide
No.10	Shaker excitation with white noise + process noise due to sinus-swept excitation with second shaker at the headstock
No.11	Shaker excitation with white noise + process noise due to sinus-swept excitation with second shaker at the machine table
No.12	Shaker excitation with white noise + process noise due to sinus-swept excitation with second shaker at the machine slide

.

Additional excitation sources for provoking process noise were implemented by using additional devices (impulse hammer for the impulse sequence and second shaker for the harmonic and sinus-swept signal) near each triaxial accelerometer. The sinus-swept signal was chosen to replace a white noise signal since the second shaker was not able to excite the machine with a white noise signal of a sufficient level. In such a way, performing a broadband signal disturbing the measurement was achieved.

2.6.2. Face Milling

In the last experimental investigation, a face milling process is performed and responses to this kind of excitation are measured with the same three triaxial accelerometers as for Pseudo-OMA. Here, a workpiece with the dimensions 300 mm × 200 mm × 350 mm (L × W × H) is processed, as shown in Figure 3. The excitation caused by the process forces is measured with a rotating dynamometer RCD 9170A by Kistler (further referred to as RD). The face milling process is the same as described in Section 2.4. Therefore, the time period per one milling path amounts to 60 s if the workpiece length of 300 mm and the constant feed rate of 300 mm/min are considered. This time period defines the signal length available for the signal processing per one milling path.

3. Results

3.1. Theoretical Study with Harmonic Oscillator

The results of the simulations listed in Table 1 are presented here. For this purpose, the stability diagram and the MAC diagram are employed to assess the quality of the identification of modal parameters by using the presented methods. A stability diagram, which is a standard tool for selecting resonance frequencies in modal analysis, shows identified eigenvalues for continuous system ${\lambda }_m$ (as explained in Section 2.1). The m-th eigenvalue contains the m-th modal damping in the form of the decay coefficient ${\delta }_m$ and the m-th circular resonance frequency ${\omega }_{d,m}$, so that ${\lambda }_m=-{\delta }_m+\mathrm{j}{\omega }_d,m$. The number of identified eigenvalues depends on the model order n being used for the eigendecomposition. With increasing model order, which is depicted on the vertical axis of the stability diagram, the number of identified eigenvalues also increases. The circular resonance frequency of the identified eigenvalue is depicted on the horizontal axis of the stability diagram. Eigenvalues that are related to the physics of the investigated system are always present independent of the model order. Therefore, vertical lines in the stability diagram indicate the existence of an eigenvalue related to the physics of the system. In this paper, the stability diagram contains markers of various colours. The grey colour depicts all eigenvalues being identified. The black colour indicates all eigenvalues with physical meaning, i.e., there is dissipation in the system (real part of an eigenvalue is negative), the eigenvalues are complex conjugate, and the damping is not higher than 100%. The red colour flags eigenvalues that are stable in frequency, i.e., the frequency difference between neighbouring model order is less than $1/\left(2\pi \right)rad/s$, and the blue colours correspond to eigenvalues being stable in frequency and damping, i.e., the damping difference is less than 1%. The ideal case would be lines of blue markers overlaying the yellow lines since the yellow lines depict the reference eigenvalues of the oscillator.

The MAC diagram compares the identified mode shapes (referred to as OMA) to the reference mode shapes of the oscillator (referred to as ref.). The MAC value lies in the range from 0 to 1 at which 0 means no correlation and 1 indicates the perfect correlation between the mode shapes. The colours are just used for better distinguishing between the bars. The ideal case would look like bars with the height of 1 lying on the diagonal of the grid, as referred in [24].

In Figure 4, the stability diagrams for scenario No. 1 (Correlated white noise excitation in all three DOF) are shown. It is obvious that DD SSI only shows very good results with the UPC algorithm. In this case, the reference eigenvalues can be identified. The CD SSI could identify eigenvalues at reference lines; however, more stable eigenvalues are identified which could cause some confusion in the practical use since the reference lines are unknown. The DSSI methods lead to the best results at which N4SID and MOESP algorithms in the 2. approach result in some stable eigenvalues in the very high-frequency range, which do not correspond to the reference lines. The MAC diagrams show a very good correlation for all methods, approaches, and algorithms with stable eigenvalues, as depicted in Figure 5. The very good results for DSSI are expectable since the input vector contains the white noise signal explicitly. The MAC values are for each result of simulation with the oscillator clearly interpretable, i.e., either the diagonal is always dominating or the results are very poor. Therefore, the MAC values are further presented in

Table 4. MAC values for stochastic excitation of the oscillator (Mode shape pairs depicted as 1-1, 2-2, 3-3).

			No. 1	No. 2	No. 3	No. 4	No. 5	No. 6	No. 7	No. 8	No. 9	No. 10	No. 11	No. 12
DD SSI 1. approach	UPC	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	0.96	0.98	0.75	1	1	1	1	1	1	1	1	1
	PC	1-1	NaN	1	1	1	1	1	NaN	1	NaN	1	NaN	1
		2-2	NaN	1	1	1	1	1	NaN	1	NaN	1	NaN	1
		3-3	NaN	0.81	NaN	1	1	1	1	1	NaN	1	1	1
	CVA	1-1	NaN	1	1	1	1	1	NaN	1	NaN	1	NaN	1
		2-2	NaN	0.98	1	1	0.99	1	NaN	1	1	1	NaN	1
		3-3	NaN	0.32	NaN	1	1	1	NaN	1	0.99	1	0.99	0.99
DD SSI 2. approach	UPC	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	0.96	0.98	0.75	1	1	1	1	1	1	1	1	1
	PC	1-1	NaN	1	1	1	1	1	NaN	1	1	1	NaN	1
		2-2	NaN	1	1	1	1	1	NaN	1	1	1	NaN	1
		3-3	NaN	0.96	NaN	1	1	1	NaN	0.99	NaN	1	NaN	1
	CVA	1-1	NaN	0.99	1	0.98	1	0.73	NaN	1	NaN	1	NaN	0.99
		2-2	NaN	0.81	0.98	0.81	1	0.83	NaN	0.99	NaN	0.96	NaN	0.97
		3-3	NaN	0.99	0.82	0.74	0.99	0.67	NaN	0.99	NaN	0.99	NaN	0.99
CD SSI	-	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	0.92	0.96	NaN	1	1	1	1	0.99	1	1	1	1
DSSI 1. approach	N4SID	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	1	0.96	0.72	1	1	1	1	1	1	1	1	1
	MOESP	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	1	0.96	0.72	1	1	1	1	1	1	1	1	1
	CVA	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	1	0.99	0.74	1	1	1	1	1	1	1	1	1
DSSI 2. approach	N4SID	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	1	0.94	0.78	1	1	1	1	1	1	1	1	1
	MOESP	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	1	0.94	0.87	1	1	1	1	1	1	1	1	1
	CVA	1-1	1	1	1	1	1	1	1	1	1	1	1	1
		2-2	1	1	1	1	1	1	1	1	1	1	1	1
		3-3	1	0.99	0.86	1	1	1	1	1	1	1	1	1

and

Table 5. MAC values for excitation of the oscillator by process force.

			No. 13	No. 14	No. 15	No. 16	No. 17	No. 18
DD SSI 1. Algorithm	UPC	1-1	1	1	1	1	1	1
		2-2	0.99	1	1	1	1	1
		3-3	1	0.99	1	1	0.99	1
	PC	1-1	NaN	NaN	1	1	1	1
		2-2	NaN	NaN	1	1	1	1
		3-3	0.97	NaN	0.99	1	1	0.99
	CVA	1-1	NaN	NaN	1	1	1	1
		2-2	NaN	NaN	1	0.99	1	0.99
		3-3	0.97	NaN	0.92	1	1	0.99
DD SSI 2. Algorithm	UPC	1-1	1	1	1	1	1	1
		2-2	0.98	1	1	1	1	1
		3-3	1	0.99	1	1	0.99	1
	PC	1-1	NaN	NaN	1	1	1	1
		2-2	NaN	NaN	1	1	1	1
		3-3	NaN	NaN	1	1	0.99	1
	CVA	1-1	1	NaN	1	0.99	1	1
		2-2	NaN	NaN	0.96	0.95	0.92	0.96
		3-3	0.97	NaN	0.96	0.86	0.99	0.96
CD SSI	-	1-1	1	1	1	1	1	1
		2-2	NaN	1	1	1	1	1
		3-3	0.97	1	1	1	0.99	1
DSSI 1. Algorithm	N4SID	1-1	1	1	1	1	1	1
		2-2	0.97	1	1	1	1	1
		3-3	NaN	1	1	1	1	1
	MOES.	1-1	1	1	1	1	NaN	1
		2-2	0.98	1	1	1	1	1
		3-3	1	1	1	1	1	1
	CVA	1-1	1	1	NaN	1	1	1
		2-2	0	1	1	1	1	1
		3-3	0	1	1	1	1	1
DSSI 2. Algorithm	N4SID	1-1	1	NaN	NaN	NaN	1	1
		2-2	0.98	1	1	1	1	1
		3-3	NaN	1	1	1	0.99	1
	MOES.	1-1	1	NaN	NaN	NaN	NaN	1
		2-2	NaN	1	1	1	1	1
		3-3	NaN	1	1	1	0.9	1
	CVA	1-1	1	NaN	1	NaN	1	1
		2-2	NaN	1	1	1	1	1
		3-3	NaN	1	1	1	0.99	1

for the simulations with the oscillator.

It is obvious that the measurement noise of 1% leads to better results of DD SSI than in case of no measurement noise, see Figure 6. Here, every approach and algorithm yields stable eigenvalues, which match the reference eigenvalues. In the case of all methods, the measurement noise of 1% causes additional eigenvalues (mathematical eigenvalues), which makes the identification of the physical eigenvalues more difficult. The most clear results come from the DD SSI second approach with the CVA algorithm and DSSI methods. It is interesting to see that the measurement noise actually improved the results of DD SSI with the CVA and PC algorithms. This originates from the definition of matrices ${\mathit{W}}_1$ and ${\mathit{W}}_2$ for these two algorithms. In the case of no noise and correlated excitation in all DOF, these matrices are very badly conditioned which leads to poor stability in the eigenvalues, as shown in Figure 4.

The next set of results, which is further presented, contains the previous simulation extended by the measurement noise of 1% and 10%, as explained in Section 2.4.

The 10% level of the measurement noise limits the identification of the third eigenvalue of the oscillator (see Figure 7) in general. Here, no method is able to find stable eigenvalues being overlaid to the third reference line. The bias between the third reference eigenvalue and the next stable eigenvalue is clearly visible in the MAC values, see Table 4. From this table, the best results are provided by the DSSI for the second approach and MOESP algorithm.

The next simulation (No. 4) is related to uncorrelated white noise excitation in all three DOF without measurement noise. In this case, all methods show very good results with the exception of DD SSI with the CVA algorithm (see Figure 8). Here, the highest oscillator resonance frequency is hard to recognise. The MAC values are ideal for all methods excepting DD SSI 2. approach and CVA algorithm, see Table 4.

After the addition of the measurement noise, all methods can identify all three resonance frequencies of the oscillator, see Figure 9. The DSSI method shows more mathematical eigenvalues than the SSI method. The uncorrelated white noise excitation fulfils assumptions of SSI in the best way, which explains the very good results. The difference between the measurement noise levels (1% and 10%) is very low. The MAC values are very good for all methods. Again, the presence of the measurement noise improves the numerical conditioning of weighting matrices ${\mathit{W}}_1$ and ${\mathit{W}}_2$ (see Equation (5)) of the CVA algorithm.

The sixth simulation run considers the uncorrelated white noise excitation in all three DOF; however, with different power spectral density, i.e., 100% for DOF 1, 60% for DOF 2, 30% for DOF 3. In such a case, the eigenvalues are clearly recognisable in the stability diagrams for all methods except the second approach of DD SSI with CVA algorithm (see Figure 10) which is reflected by the lower MAC values, see Table 4.

The addition of the measurement noise leads to the same effects as in the previous simulation results, i.e., the high level of measuring noise causes more mathematical eigenvalues. In the presence of noise, the second approach of DD SSI with the CVA algorithm is able to identify the resonance frequencies and mode shapes with better results.

The excitation in only one DOF mostly corresponds to real excitation in a machine tool, since the cutting process excites a machine tool very locally. Simulation No. 7 considers white noise excitation in DOF 1 (see Figure 11). In simulation No. 8, the measurement noise of level 10% is added to the excitation (see Figure 12). In the similar manner, simulation No. 9 and 10 consider the excitation in DOF 2 (see Figure 13 and Figure 14) as well as simulation No. 11 and 12 the excitation in DOF 3 (see Figure 15 and Figure 16). It can be seen that DSSI provides good results in all cases. DD SSI needs a measurement noise to work properly except the UPC algorithm, which works very well without measurement noise.

The MAC values from simulations No. 1 to 12 are listed in Table 4. The table entry ‘NaN’ means that a mode shape could not be identified, which would correspond to the reference mode shape. The table shows that DSSI methods lead to high MAC values in all simulation runs except the correlated white noise excitation with measurement noise, which, however, has little practical relevance.

Starting with simulation No. 13, the excitation is not stochastic but deterministic in nature, which corresponds to a dominant excitation of a machine tool caused by the cutting process with a constant spindle speed of 3000 rpm. For this purpose, the excitation is solely implemented in DOF 3. Simulation No. 13 considers a process force of a face milling process $F_x$ (see Equation (12)). In this case, all methods fail (see Figure 17). In the stability diagrams, the harmonics of the frequency 314.16 rad/s are often stated to be stable. The algorithms UPC and CD SSI and some algorithms of the DSSI can only identify the first eigenvalue. Therefore, the investigation of the influence of measuring noise is omitted here.

The modification of the process force for broadband excitation according to Section 2.3 leads to acceptable results for DSSI, CD SSI and the UPC algorithm of DD SSI. However, there are eigenvalues being stated as stable but they do not correspond to any physical eigenvalue. These eigenvalues are referred to as mathematical eigenvalues. In this context, the most clear results provide the N4SID and MOESP algorithms of the DSSI while CD SSI identifies many mathematical eigenvalues, see Figure 18. It is worth noticing that the algorithms PC and CVA within SSI do not work properly. The reason is the poor numerical conditioned matrices, as explained above for white noise excitation. After the addition of the measurement noise, the identification of the first eigenvalue is a big challenge for DSSI in general, as depicted in Figure 19. Here, only the first approach of DSSI with the N4SID and MOESP algorithm is able to compute partial stable results for the first eigenvalue. The CVA algorithm in combination with the first approach of DSSI leads to stable eigenvalues in frequency. On the contrary, the measurement noise causes better results of SSI methods. Here, stable eigenvalues are identified but it is difficult to distinguish between physical and mathematical eigenvalues.

The analysis of the influence of the signal length on the results of DSSI shows that the results are satisfying for signal length at least $j=5000$, see Figure 20. It is interesting to see that the shorter signal length leads to slightly better stability diagrams for the first approach of DSSI. The first eigenvalue can be identified more clearly with the N4SID and MOESP algorithm than in the case of the initial signal length (j = 40,000). The minimal signal length ($j=612$), which is possible for the identification, leads to very poor results. In particular, the third eigenvalue cannot be identified.

The reduction in the sampling frequency from the initial value of 1000 Hz towards to 400 Hz make the identification more difficult, since there are many mathematical values close to the third resonance frequency in the case of SSI, see Figure 21. In the case of DSSI, there is no significant impact. Since DSSI requires a definition of the input vector, the input vector must theoretically contain all excitation sources. This is often not possible in practical use cases since many excitation sources are unknown. Such excitation sources lead to process noise when DSSI is computed. In order to investigate the influence of the process noise on the DSSI results, additional excitation as a white noise signal, which is not considered by the input vector, is implemented in the model. The white noise signal is defined with the power of 0.1 Hz^−1/2 in all DOF. It can be seen that the process noise improves the identification, see Figure 22. It is obvious that an even excitation of eigenvalues by process noise is better for DSSI than poor excitation without process noise. Furthermore, it is worth noting that the identification also works very well for the signal length of $j=5000$. For the sake of completeness, it must be remarked that the results of SSI are improved by the process noise as well as what is to be expected.

The MAC values from simulations No. 13 to 18 are listed in Table 5. They reflect the conclusions for stability diagrams. Considering the mathematical eigenvalues, which make the identification difficult, the first approach of DSSI seems to be appropriate for the identification of modal parameters for systems with local excitation of a deterministic nature.

3.2. Theoretical Study with Machine Model

The purpose of the simulation with a machine model is to show how far the DSSI and SSI methods can handle complex dynamic behaviour (i.e., a high number of mode shapes). Firstly, the model is excited at TCP with an uncorrelated white noise signal in all three coordinates in analogy to the investigation with the oscillator (simulation No. 19, see Table 2). Since the measurement noise makes the identification of eigenvalues more difficult due to the presence of mathematical eigenvalues (see investigations with the oscillator), the machine model always contains the measurement noise of 10% (see Figure 23). It is obvious that all methods of DSSI and SSI can identify all the eigenvalues clearly. However, there are some mathematical eigenvalues, which seem to be stale. This issue concerns DSSI methods more than SSI. The mathematical eigenvalues are mostly located in the higher frequency range.

If the modified process forces are used for the excitation and these forces are considered by the input vector explicitly, the stability diagrams yield acceptable results for the first approach of DSSI with MOESP and CVA algorithm, see Figure 24. In the case of SSI methods, the highest eigenvalue cannot be found by any approach and algorithm. The first approach with the CVA algorithm seems to be most capable of achieving the best results. The others cannot identify at least two eigenvalues in the frequency range around 1500 rad/s.

If the input vector considers the process force implicitly, i.e., the input vector is defined according to Equation (13) (further referred to as implicit input vector), the results do not change very much (see Figure 25). It is obvious that all the eigenvalues can be identified by using the first approach of DSSI with MOESP and CVA algorithms very well like in the previous case. Here, the DSSI method is only commented on as the definition of the input vector has no relevance for SSI.

Furthermore, the robustness of the DSSI with the implicit input vector is investigated in the presence of various kinds of process noise. For this purpose, the white noise signals, harmonic signals, and impulse sequences are separately added to the excitation but not considered by the input vector. Figure 26, Figure 27 and Figure 28 show that the identification for the first approach with MOESP and CVA algorithms is robust, i.e., the eigenvalues can be identified clearly despite the process noise. The process noise actually improves SSI methods, as shown above. However, the results are very similar to those for only broadband force, i.e., without process noise. The CVA algorithm in combination with the first approach of DD SSI achieves the best results among SSI methods.

The harmonic signal as process noise is visible as a stable eigenvalue in the stability diagrams of all methods.

There is no impact of the impulse sequence as a process noise on the stability diagrams. An impulse causes a broadband excitation and a sequence of impulses leads to a broadband excitation as well, however, with harmonics of the impulse frequency.

3.3. Identification with Measured Signals

3.3.1. Process Noise Analysis

Before the identification results for measured signals are presented in this section, a short analysis of the process noise is given here. The process noise plays an important role when EMA during machining is conducted, as referred to in [18]. In the previous section, some results show that the presence of the process noise can improve the identification results. The implemented process noise is assumed to be a white noise signal that fits the assumption of the methods perfectly. The real process noise can, however, differ from this ideal case. Thus, it would be worth knowing more about the level and the structure of the process noise. The machine being used for investigations in this paper is located on a shop floor with very rare production. Thus, the process noise originating from the environment is very low. The measurement is simultaneously conducted by three triaxial accelerometers being located at the table, headstock and slider of the y nc axis for three operating modes of the machine, in particular, machine off (referred to as environment), machine on (all auxiliary aggregates are switched on), stand by (the servo drives of nc axes are in the control loop). The measuring noise of the used sensors together with the acquisition device amounts to $0.002\mathrm{m}/{\mathrm{s}}^2\left(\mathrm{rms}\right)$ or $0.005\mathsf{\mu }\mathrm{m}\left(\mathrm{rms}\right)$. This value was experimentally determined. Figure 29 shows the time series of the captured displacement as a part of the process noise vector for the three operating modes of the machine. Here, the highest rms value amounts to $0.25\mathsf{\mu }\mathrm{m}\left(\mathrm{rms}\right)$ for the stand by mode. It can be seen that the process noise level depends on the operating mode of the machine. The character of the process noise can be seen in the frequency spectrum, as shown in Figure 30. Here, stochastic and harmonic parts of the signal can be recognised. It must be mentioned that the measured signals are double integrated to evaluate the displacement since the process noise vector has the same physical meaning as the state vector (see Equation (7)). The process noise is more like pink noise than white noise. Besides the stochastic part, the signal contains harmonic parts, especially for the mode machine. The harmonic parts come from the running aggregates. The operating mode ’stand by’ causes peaks that correspond to mechanical responses being excited by the servo drives. This is clearly visible in a frequency range around 1000 rad/s (wide peaks are recognisable).

The second experiment corresponds to an experimental setup that is typical for the EMA. The machine is switched off and excited by an electro-magnetic shaker. There is still a portion of process noise, which is depicted as ’Environment’ in Figure 29. However, the dominant excitation originates from the shaker. The minimal acceleration at headstock is greater than $1\mathrm{m}/{\mathrm{s}}^2\left(\mathrm{rms}\right)$, which is 500 times greater than the acceleration measured for excitation from the environment. The input vector for the DSSI is the force signal measured by the force transducer (see Figure 31). It must be noticed that the force signal is not a pure white noise sequence despite the voltage setting for the shaker, since the force is influenced by the vibration of the shaker, the shaker holder, the tool dummy, and the machine table. In the case of the OMA, the vibrations of the shaker and the shaker holder can affect the results by identifying them as resonance frequencies of the machine unlike EMA, in which these vibrations are considered by the measured force signal.

Figure 32 shows the stability diagrams for the second experiment. Here, the signals of the length $j=40,000$ and sampled with a frequency of 1024 Hz are evaluated. At the first look, the results of DSSI and SSI are very similar. If we look more in detail, the best results are achieved by the MOESP algorithm in the first approach. In the case of SSI, the PC algorithm seems to be most appropriate for such evaluation. In the stability diagram of the MOESP algorithm 13 eigenvalues being stated as stable correlates with the reference lines. The eigenvalues, which are not identified, correspond to the 3^rd, 10^th, 13^th, 16^th, 17^th, and 19^th reference lines. It is obvious that reference lines of couple modes (lines being close to each other) could not be identified. Setting the signal length to a higher value $j=60,000$ does not make a difference in results.

In the third experiment, the impact of the process noise existing in the ’Stand by’ modus of the machine is investigated. The stability diagrams are shown in Figure 33. Comparing the results to the previous ones (with a very low level of process noise), it can be seen that there is nearly no difference. This confirms the theoretical investigations, in which the process noise rather improves the identification.

The next experiments aim to investigate the process noise of a certain kind like impulse sequence, harmonic excitation, broadband excitation (sinus swept). In the case of the impulse sequence, the impact is very marginal (compare Figure 34 to Figure 32). The location of the impulse sequence does not play an important role.

The process noise as harmonic excitation is clearly visible as a stable eigenvalue at a frequency of 697 rad/s (see Figure 35). This eigenvalue is also visible for excitation at the machine bed (see Figure 36) and at the slide of the y-axis (see Figure 37).

The broadband excitation realised with sinus swept signal does not affect the results of the identification, as exemplarily depicted for the excitation at the headstock in Figure 38.

3.3.2. Face Milling

This section presents the results of the experiments with the face milling, as described in Section 2.6.2. Before the identification results are presented, the process forces measured by RD and the input vector being established by using the traced data of the spindle speed, as described in Section 2.3, are analysed. For this purpose, Figure 39 depicts the frequency spectra of the process forces and the components of the input vector. It can be seen that the frequency spectra of process forces feature some peaks that are not present in the frequency spectra of the input vector. These peaks result from the relative vibration between the tool and workpiece, i.e., from the dynamic behaviour. However, the peaks at frequencies ≈ 63, 189 and 252 rad/s correspond to the lowest spindle speed and its third and fourth harmonics. The fourth harmonic is the highest one and results from the tooth pass frequency of the tool at the lowest spindle speed.

As the process forces are measured by RD, the identification of the modal parameters can be conducted in two ways. Firstly, the input vector explicitly contains the measured process forces. Alternatively, the implicit input vector is considered. Since SSI does not require any input vector the results are presented only once. Figure 40 depicts the stability diagrams for SSI and DSSI with the explicit input vector. The SSI results are clearly visible in the diagram for the CVA algorithm in the first approach. The same stable lines are actually visible for the other algorithms and approaches. However, in the case of Approach 1 and the CVA algorithm, the eigenvalues outside any lines (spread eigenvalues) are not in such a high number as in the other diagrams. The first stable line is approx. at 65 rad/s corresponds to the lowest spindle speed. This is visible as the first peak in the frequency spectrum of the process forces (see Figure 39). The tooth pass frequency for the lowest spindle speed (255 rad/s—the highest peak in the frequency spectrum of $F_x$ in Figure 39) is also visible, however, it is not as a clear, stable line. It can be stated that the 1^st, 5^th, 7^th, 9^th, 11^th, 16^th, 17^th and 18^th eigenvalues could not be identified. If the frequency spectra of process forces are considered it is obvious that the excitation over 2000 rad/s is very small. Furthermore, the response is very low as can be seen in Figure 39 as well as the results for Pseudo-OMA show. Some of the identified eigenvalues are slightly shifted from the eigenvalues identified within EMA (yellow dot lines). The results of the DSSI are very similar to those of SSI. Here, it is obvious that the first approach leads to better results, especially in the lower frequency range (less than 600 rad/s). If the algorithm CVA used in the first approach is evaluated and compared to the equivalent of SSI, small differences are visible. For the DSSI, the first eigenvalues can be indicated by a stable line, which is however not continuous over the model order. The 9. and 10. as well as the 16., 17. and 18. eigenvalues could not be identified. Recognising the eigenvalues is not very clear.

The next results set is identified with DSSI and the implicit input vector. When implicit input vector is used, the constant phase angle ${\mathit{\varphi }}_0$ introduced in Section 2.3 must be quantified. For this purpose, stability diagrams for various values of this angle are evaluated and the best stability diagram is chosen for the identification of the eigenvalues. Figure 41 depicts a set of stability diagrams for DSSI of the two approaches and all algorithms (N4SID, MOESP and CVA). Here, the stability diagrams are established for a phase angle that corresponds to a phase shift on the time axis of 1.267 s. This means that the time axis of the traced spindle speed is shifted against the time axis of measured vibrations by this value. It can be seen that the results are very poor. Such results can be found in a range of the phase shift from 1.21 to 1.315 s.

The sensitivity of the results to the phase shift is in a certain range of the phase shift relatively low. Figure 43 shows stability diagrams for 3.5 s. It can be seen that the results are comparable to those for a phase shift of 4.018 s, see Figure 42.

4. Discussion

The aim of this work is to consider the deterministic character in the excitation of a machine tool during milling. The methods of the state-of-the-art such as the SSI methods consider the excitation as if it were stochastic. It can be assumed that the milling process is the strongest source of excitation in a machine tool during milling. This excitation source is largely deterministic. Therefore, it should be considered to be deterministic when OMA is conducted during milling. For this purpose, the use of DSSI is proposed in this paper. The DSSI, however, requires an input vector. One benefit of the OMA is that the input vector remains unknown. In this paper, a formulation for the input vector is presented that takes into account the milling process without the need for the quantification of the process forces; thus, the principle of the OMA remains unchanged. This paper presents investigations of how far DSSI methods can take into account the specific excitation situation in machine tools better than SSI methods. For this purpose, four kinds of investigation are presented namely theoretical investigation with an oscillator, theoretical investigation with the machine model, which is created by using modal parameters from the EMA, experimental investigations with a shaker and experimental investigation during milling.

The investigations with the oscillator show that white noise excitation uncorrelated for each DOF is necessary for a proper function of SSI. DSSI methods can handle every kind of excitation. However, the high level of measurement noise can lead to more mathematical eigenvalues. The level (amplitude) of the excitation depending on the DOF does not have an influence on the stability diagrams at all. Even the MAC values are very high as well. If the excitation acts locally, only the UPC algorithm of SSI works properly. The algorithms CVA and PC need a measurement noise for proper function otherwise the weighting matrices of these two algorithms are poorly numerically conditioned. It is interesting that the MAC values are satisfying if the eigenvalues are correctly identified despite the fact that the structure is spatially not excited uniformly. As expected, the DSSI method works properly every time. After the investigations with white noise sequence, which mostly fits the assumptions of SSI methods, the excitation by the milling process is investigated next. The identification of eigenvalues for the process force of face milling fails with all methods due to the insufficient broadband character of the excitation. The excitation with modified process forces leads to better results of the identification with SSI methods. In the case of DSSI methods, the identification of the first eigenvalue is a big challenge. Shortening the signal length from $j=40,000$ to $j=5000$ shows light improvement in this matter. However, the shorter the better is not valid in general, since the results are the worst for short signal length $j=612$. Reducing the sampling frequency from 1000 Hz to 400 Hz could eliminate mathematical eigenvalues in the upper frequency range of the stability diagram. The MAC values are satisfying every time the eigenvalues are correctly identified. The worst MAC values were achieved with the CVA algorithm of SSI.

The purpose of the simulation with the machine model was to show the capability of the methods for the investigation of structures with a high number of resonance frequencies. The model, which is created by using modal parameters originating from the EMA, features 19 resonance frequencies in the frequency range from 0 to 400 Hz. In the first step, uncorrelated white noise sequences (in three directions, i.e., x, y, and z) were used for the excitation at TCP. Here, every method yields very good results in the stability diagrams. The DSSI methods produced more mathematical eigenvalues in a high-frequency range that could be confusing for identification. After the implementation of the modified process forces, the best results were achieved with MOESP and CVA algorithms for the first approach of DSSI. SSI methods could not identify all eigenvalues and the N4SID algorithm as well as the second approach of DSSI in general were not clear in the stability diagram. The input vector containing the process forces was denoted to explicit since the physical input is explicitly considered. The paper suggests an implicit input vector to ensure using DSSI within the OMA of machine tools and the process forces remain unknown during milling. DSSI with the implicit input vector leads to very similar results as for the explicit vector, which confirms the usability of the implicit input vector. The use of the input vector for the identification brings up the question of the influence of the process noise. Therefore, process noise in the form of impulse sequences, a harmonic signal and white noise sequences were additionally implemented. The process noise has no impact on the results except the harmonic signal, which was identified as a stable eigenvalue by all methods. Concluding, the DSSI methods with the implicit input vector are theoretically capable for identification of eigenvalues of a machine tool during milling. The evaluation of MAC values was omitted since the vibration of one sensor with three DOF was evaluated here due to the computing power being needed for SVD.

After simulative investigations, measured signals are used for the identification. Before the identification is conducted, the real process noise is quantified and analysed. It was shown that the process noise contains harmonic signals in the operating mode ‘Stand by’, which is relevant for the OMA. However, they do not affect the identification at all due to their low level. The first idea for using OMA in machine tools is just to use measured signals when the machine is switched off and the excitation comes from its environment. However, this excitation is not sufficient enough to identify a relevant number of eigenvalues. The second experiment corresponds to a setup that is usual for the EMA, i.e., the machine is switched off and excited by a shaker with a white noise sequence. The results show that not all eigenvalues could be identified. The best results were achieved with the MOESP algorithm in the first algorithm of DSSI. SSI methods are also capable of identifying similar sets of eigenvalues. This confirms the investigations with the machine model. Nevertheless, the second approach of DSSI leads to acceptable results, unlike the theoretical investigation. The process noise in various levels and shapes does not affect the identification substantially except for the process noise as a harmonic signal, which is identified as a spurious eigenvalue like in theoretical investigations. The last experiment is the realisation of the face milling process. Here, the process forces were measured by RD. Furthermore, the vibration at the headstock was captured by a triaxial accelerometer and the real spindle speed was traced by the control system. In such a way, there are three kinds of signals being ready for evaluation. Firstly, the identification is conducted with acceleration signals and the process forces in the case of DSSI. It can be seen that the results for SSI and DSSI with the explicit input vector are very similar. There are some eigenvalues that could not be identified and the recognition of stable lines is not clear. If the implicit input vector, i.e., computed signal by using the traced spindle speed and equation introduced in this paper, is used for the DSSI, stable lines become more clear and all eigenvalues in the frequency range lower than 1300 rad/s could be identified. The frequency range higher than 2000 rad/s is not excited very well by the modified face milling process, which leads to the fact that no eigenvalues were identified. Furthermore, three eigenvalues in the frequency range from 1300 to 1500 rad/s could not be identified as well, since acceleration signals indicate very low vibrations without peaks in this range which could indicate a vibration node at the captured location. There are identified some spurious eigenvalues. Nevertheless, these result from the excitation by the modified face milling process and can be distinguished from the physical eigenvalues as the frequency is related to the process, e.g., the lowest spindle speed and the tooth pass frequency at the lowest spindle speed.

5. Conclusions

This paper presents an approach for conducting an operational modal analysis of a machine tool during milling. It has been suggested to use the deterministic-stochastic subspace identification (DSSI) to take the deterministic character of the excitation related to the milling process adequately into account, in contrast to the stochastic subspace identification (SSI), which is often used in the state of the art. However, DSSI requires a definition of the input vector. The machine tools are special in this context since the input vector can be computed just by using traced spindle speed and geometric relations for the milling process. Here, no material parameters are required for the definition of the input vector since these are a part of the input matrix being a subject of the identification. In such a way, very good results are achieved after the identification. The paper deals with many influencing factors and research issues through theoretical investigations with an oscillator of three degrees of freedom and a mathematical model of a machine tool. In the next steps, experiments in a machine tool are presented and discussed. It is shown that the DSSI leads to better results in some aspects like more clear stable lines for eigenvalues in stability diagrams and identified resonance frequencies in a low frequency range. It was also shown that the suggested input vector is useful for the identification within DSSI and the stable diagrams become more clearer than in the case of the input vector with process forces.