Modal identification of machine tool spindle units by output only operational modal analysis

Abstract

Accurate tracking of modal characteristics is a valuable diagnostic tool for condition monitoring of machine tool spindle units. While experimental modal analysis (EMA) is the conventional method used for machine tool modal identification, it is often impractical to implement in production settings due to the invasive and manual nature of the impact hammer test. In this study, a new technique for operational modal analysis (OMA) based on output-only vibration measurements obtained during a milling operation with variable spindle speed is proposed. Modal identification is performed using two OMA standard methods, namely stochastic subspace identification (SSI) and frequency domain decomposition (FDD). The modal characteristics are compared to values obtained from conventional EMA from impulse hammer testing on the static spindle, and from the operational spindle during cutting using force measurements collected by a table dynamometer. The percentage difference between the natural frequencies identified by the proposed OMA method and frequencies identified by conventional impulse hammer testing was less than 10%, and for the operational spindle during cutting tests, the difference was less than 3%. These results demonstrate the validity of a new modal identification method that can be practically implemented in production.

Introduction

Understanding a machine tool’s dynamic behavior is essential for optimizing machining performance, preventing failures, and enabling condition-based maintenance (CBM). Modal identification is the process of measuring the modal characteristics of a mechanical system. Modal characteristics describe the oscillatory nature of machine, tool, and workpiece vibrations during cutting, which are critically important to the manufacturing process. For example, self-excited machine tool vibrations known as chatter are known to cause problems in manufacturing processes such as poor workpiece surface finish, tool breakage, and increased wear on machine components [1]. Beyond chatter, shifts in modal characteristics have been linked to spindle failures and repairs [2], and to long-term degradation of spindle bearings [3]. Utilizing modal characteristics in CBM programs could therefore detect changes in a machine’s dynamic behavior before problems escalate and cause expensive downtime. However, modal identification for machine tool systems is difficult to implement in practice.

Modal identification in machine tool applications is conventionally done through experimental modal analysis (EMA) using controlled artificial excitation. The input excitation and output response are both measured, and the ratio of the output response spectrum to the input excitation spectrum describes the dynamic frequency response function (FRF) for the system. Different types of devices have been proposed to provide and measure the input excitation, including mechanical shakers [4], electromagnetic forcing devices [5, 6], and specialized mechanical impulse devices [7, 8]. However, this type of artificial excitation is costly to implement both in hardware expenses and in the time required to set-up and perform the testing procedures [9]. The most widely used method for modal identification in machine tools involves the use of an instrumented impulse hammer to strike the tool with a known input force and to measure the vibration response of the machine. Procedures for impact hammer testing of machine tool systems are well-detailed in [10, 11]. The impact hammer has a compact form factor, and impact testing is fast compared to more complicated excitation devices. However, due to the manual nature of the impact hammer EMA, production must be stopped to perform the test. This interruption to production schedules makes impact hammer EMA infeasible for condition monitoring applications since the cost of lost production is not justifiable to the manufacturing enterprise. Furthermore, impact hammer EMA under static spindle conditions may not always provide accurate modal characteristics, since they are known to change with operational speed and load conditions [12, 13]. Modal identification methods that can be performed under ambient conditions of the machine tool during nominal operation are more desirable for these applications.

Operational modal analysis (OMA) describes methods for the modal identification of structures subject to ambient operational excitations. Under the assumption of white-noise excitation, identification can be carried out without knowledge of the input forces [14]. However, the trade-off is that it is not possible to determine modal mass, dynamic stiffness, or system residues as these parameters are dependent on knowledge of the unknown input excitations. Ambient force excitation in machine tool spindle units comes from two main sources, namely inertial forces generated by high-speed rotation of the spindle and cutting forces. In general, neither of these excitation sources fulfills the assumption of white-noise excitation, since both inertial and cutting forces have strong spectral components related to the spindle rotation frequency and harmonics [15]. The harmonic content presents a problem for OMA as strong harmonics introduce bias in the identified modal characteristics near these frequencies [16, 17].

Specialized workpieces have been constructed to broaden cutting force spectra during machining. Poddar et al. used a pseudo-random workpiece that contained a series of non-uniformly spaced slots to generate different cutting forces [18]. Mohammadi and Ahmadi machined a porous aluminum foam workpiece to generate random force components for modal identification in a robotic milling application [19]. Despite the specialized workpiece designs, strong content was still observed at spindle harmonics, which needed to be suppressed by subsequent signal processing steps. Additionally, the specialized workpiece designs are undesirable because they increase the cost of performing modal analysis. Other researchers have varied the spindle speed during machining to produce the random cutting forces. In [20, 21], researchers pre-programmed the spindle to follow a Brownian walk speed profile to generate a broad force spectrum while cutting a specialized single-toothed workpiece. Recently, Franco et al. proposed an EMA modal identification method based on face milling with a linearly varying spindle speed [22]. However, this work relied on obtaining cutting force measurements using a table dynamometer, which is often not available in production environments.

For modal identification to become useful for CBM programs in manufacturing environments, the current methods must be improved with a focus on practical implementation. The presence of strong harmonics in the operational cutting force input excitation, the need for expensive force plate dynamometers, and the need for specialized workpiece geometries are all factors that limit the practicality of operational modal identification. To address these challenges, the present study introduces a low-cost output-only OMA technique for identifying spindle-unit modal identification during machining. The proposed technique improves upon the state-of-the-art by eliminating the need for cutting force measurement and the need for specialized workpiece geometries. Both of which reduce the total cost of performing the test. Modal characteristics (i.e., natural frequency and damping ratio) are identified using two independent identification algorithms applied to vibration data collected from the spindle housing. The results are validated against conventional impact hammer EMA for the static spindle and EMA during dynamic operation, demonstrating accurate, non-intrusive modal monitoring suitable for CBM integration.

OMA methods

Frequency domain decomposition

The FDD method used in this study was formulated based on the works of Brincker, originally used for OMA in buildings and civil structures [23, 24]. Consider the output vibration response of a mechanical system that is modelled as a linear combination of modal coordinates:

$$\begin{array}{cc}\hfill \mathbf{y}(t)=& \sum _{n=1}^{2N}{\phi }_n{q}_n(t)\hfill \\ \hfill =& \mathrm{\Phi }\mathbf{q}(t)\hfill \end{array}$$ 1

where the mode shape matrix $\mathrm{\Phi }$ describes the orthogonal vibratory mode shapes of the system, and the general modal coordinates $\mathbf{q}(t)$ describe the contribution of each shape at any instance in time. The time-domain covariance matrix of the responses is defined as:

$$\begin{array}{cc}\hfill {\mathbf{R}}_y(\tau )=& \text{E}[\mathbf{y}(t+\tau )\mathbf{y}{(t)}^{\text{T}}]\hfill \\ \hfill =& \text{E}[\mathrm{\Phi }\mathbf{q}(t+\tau )\mathbf{q}{(t)}^{\text{T}}{\mathrm{\Phi }}^{\text{T}}]\hfill \\ \hfill =& \mathrm{\Phi }{\mathbf{R}}_q(\tau ){\mathrm{\Phi }}^{\text{T}}\hfill \end{array}$$ 2

where ${\mathbf{R}}_q(\tau )$ is the covariance matrix of the modal coordinates. By taking the Fourier transform of Eq. 2, one obtains the following spectral density matrix:

$$\begin{array}{cc}\hfill {\mathbf{G}}_y(\omega )=& \mathrm{\Phi }{\mathbf{G}}_q(\omega )\mathrm{\Phi }_{}^{\ast }\hfill \\ \hfill =& \mathrm{\Phi }[{\mathbf{g}}_q^2(\omega )]\mathrm{\Phi }_{}^{\ast }\hfill \end{array}$$ 3

where the superscript $(_{}^{\ast })$ indicates the complex-transpose operation. If a decomposition as Eq. 3 can be obtained, then the diagonal elements $[{\mathbf{g}}_q^2(\omega )]$ of ${\mathbf{G}}_q(\omega )$ should be interpreted as the auto-spectral densities of the modal coordinates. This line of reasoning is the core motivation behind the following procedures for FDD. First, an estimated PSD matrix at each discrete frequency ${\widehat{\mathbf{G}}}_y(\omega )$ is computed for the output vibration data. Then, singular value decomposition of the PSD matrix is performed at each frequency:

$$\begin{array}{cc}\hfill {\widehat{\mathbf{G}}}_y(\omega )=& \mathbf{U}\mathbf{S}\mathbf{U}_{}^{\ast }\hfill \\ \hfill =& \mathbf{U}[s_n^2]\mathbf{U}_{}^{\ast }\hfill \end{array}$$ 4

where $\mathbf{S}$ is a diagonal matrix with singular values $[s_n^2]$ along the diagonal. The usage of $\mathbf{U}_{}^{\ast }$ in the decomposition rather than the typical $\mathbf{V}_{}^{\ast }$ is due to the fact that ${\widehat{\mathbf{G}}}_y(\omega )$ is symmetric positive definite by definition of the cross-PSD matrix. The singular values $[s_n^2]$ obtained from this procedure are interpreted as the auto-spectral densities of each modal coordinate $[{\mathbf{g}}_q^2(\omega )]$. A peak-picking procedure is used to select the modal coordinates from the auto-spectral densities [25], and the inverse Fourier transform is used to convert the selected auto-spectrum data back into the time domain. This time-domain signal corresponds to the autocorrelation function of a free decay of a single vibratory mode of the mechanical system. The extrema and zero-crossings of the autocorrelation function are examined to estimate the damping ratio and natural frequency of the mode as in [25]. The period between zero-crossings is determined, and the inverse of the period gives the natural frequency of the mode. The values of the extrema in the free decay are used to determine the damping ratio by the method of logarithmic decrement. The mode shape is given by the value of the first singular vector ${\mathbf{u}}_1(\omega )$ in $\mathbf{U}$.

Stochastic subspace identification

The second method used for OMA in this study was the stochastic subspace identification (SSI) formulation originally presented by Van Overschee and DeMoore [26]. Consider the stochastic state space formulation for a vibrating mechanical system:

$$\begin{array}{cc}\hfill {\mathbf{x}}_{k+1}=& \mathbf{A}{\mathbf{x}}_k+{\mathbf{w}}_k\hfill \\ \hfill {\mathbf{y}}_k=& \mathbf{C}{\mathbf{x}}_k+{\mathbf{v}}_k\hfill \end{array}$$ 5

where ${\mathbf{y}}_k$ is the measured output response vector, ${\mathbf{x}}_k$ is the discrete state vector, ${\mathbf{w}}_k$ is the process noise, and ${\mathbf{v}}_k$ is the measurement noise. We assume both ${\mathbf{w}}_k$ and ${\mathbf{v}}_k$ are zero-mean white noise. $\mathbf{A}$ is the state transition matrix that characterizes the system dynamics by its eigenvalues, and $\mathbf{C}$ is the output matrix that specifies how the internal states are transformed to the outside world. In the case of output-only OMA, the state vector ${\mathbf{x}}_{k+1}$ is unknown and only the system output ${\mathbf{y}}_k$ is available. The correlation function of the output vector can be estimated as:

$$\begin{array}{c}\hfill {\widehat{R}}_i=\text{E}\left[{\mathbf{y}}_{k+1},({\mathbf{y}}_k^{\text{T}})\right]\end{array}$$ 6

By calculating the correlation function at different time delays, one may construct the following Hankel matrix from the output response:

$$\begin{array}{c}\hfill {\mathbf{H}}_i=\left[\begin{array}{cccc}{\mathbf{R}}_i& {\mathbf{R}}_{i-1}& \cdots & {\mathbf{R}}_1\\ {\mathbf{R}}_{i+1}& {\mathbf{R}}_i& \cdots & {\mathbf{R}}_2\\ ⋮& ⋮& \ddots & ⋮\\ {\mathbf{R}}_{2i-1}& {\mathbf{R}}_{2i-2}& \cdots & {\mathbf{R}}_i\\ \end{array}\right]=\left[\begin{array}{c}{\mathbf{H}}_p\\ {\mathbf{H}}_f\end{array}\right]\end{array}$$ 7

Using a particular selection of correlation functions with different time delays, one may obtain the following correlation function matrix:

$$\begin{array}{c}\hfill {\mathbf{O}}_i=\mathbf{E}\left[\frac{{\mathbf{H}}_f}{{\mathbf{H}}_p}\right]\end{array}$$ 8

Van Overschee and De Moore showed that this correlation function matrix is equal to the product of the controllability matrix of the system ${\mathrm{\Gamma }}_i$ and the state matrix ${\widehat{X}}_i$ [26].

$$\begin{array}{c}\hfill {\mathbf{O}}_i={\mathrm{\Gamma }}_i{\widehat{X}}_i\end{array}$$ 9

By performing singular value decomposition on the projection matrix ${\mathbf{O}}_i$ one obtains:

$$\begin{array}{c}\hfill {\mathbf{O}}_i=\left(\begin{array}{cc}{U}_1& U_2\end{array}\right)\left(\begin{array}{cc}{S}_1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}{V}_1^T\\ V_2^T\end{array}\right)=U_1{S}_1{V}_1^T\end{array}$$ 10

The order of the system can be obtained from the number of non-zero singular values from $S_1$. Then the corresponding observability matrix ${\mathrm{\Gamma }}_i$ and state sequence ${\widehat{X}}_i$ can be determined:

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_i=& U_1\sqrt{S_1}\hfill \end{array}$$ 11

$$\begin{array}{cc}\hfill {\widehat{X}}_i=& \sqrt{S_1}{V}_1^{\text{T}}\hfill \end{array}$$ 12

From this point, it is possible to recover the system matrices $\mathbf{A}$ and $\mathbf{C}$ by solving the least-squares problem.

$$\begin{array}{c}\hfill \left(\begin{array}{c}\mathbf{A}\\ \mathbf{C}\end{array}\right)=\left(\begin{array}{c}{\widehat{X}}_{i+1}\\ Y_{i|i}\end{array}\right)\widehat{X}_i^{\ast }\end{array}$$ 13

Finally, the modal characteristics of the system can be derived from the eigenvalues of the state matrix $\mathbf{A}$. For a continuous-time eigenvalue of $\mathbf{A}$ denoted by ${\lambda }_{\mathit{ck}}$, the corresponding discrete-time eigenvalue is:

$$\begin{array}{c}\hfill {\lambda }_k=\frac{ln({\lambda }_{\mathit{ck}})}{\mathrm{\Delta }T}\end{array}$$ 14

The natural frequency ${\omega }_k$ and damping ratio ${\zeta }_k$ corresponding to the eigenvalue are:

$$\begin{array}{cc}\hfill {\omega }_k=& \frac{|{\lambda }_k|}{2\pi }\hfill \end{array}$$ 15

$$\begin{array}{cc}\hfill {\zeta }_k=& \frac{-\text{Re}({\lambda }_k)}{|{\lambda }_k|}\hfill \end{array}$$ 16

The eigenvector of $\mathbf{A}$ corresponding to ${\lambda }_k$ represents the mode shape vector.

Variable spindle speed experiment

The modal identification experiment was conducted in two stages. In Stage 1, conventional EMA testing was performed on the static spindle by striking the tool tip with an impulse hammer (PCB Piezotronics 086C01) and recording the output vibration responses. Impulse tests were conducted by striking in both the X and Y directions. In Stage 2, dynamic testing was conducted using a pre-programmed swept spindle speed profile over the operating range of the machine. Cutting tests were performed on a three-axis vertical machining center (FANUC Robodrill D14SiA) equipped with a 5.5 kW spindle and BT30 spindle taper. Figure 1 displays a photograph of the experimental set-up.

Fig. 1.

Annotated photograph of the experimental set-up showing (1) tool, (2) workpiece, (3) force plate dynamometer, and accelerometer locations x1, x2, x3, and y1, y2, y3

Variations in spindle speed were used to achieve broad-spectrum input excitation. The speed was increased from 100 to 10,000 RPM throughout the first half of the cut, then decreased back down to 100 RPM in the second half. Spindle speed was changed linearly with respect to cutting length, and the feed rate was a constant 0.02 mm per tooth. As a result, the spindle speed profile takes on a parabolic curve with respect to time, which can be observed in the cutting force spectrogram shown in Fig. 2.

Fig. 2.

Spectrogram of X direction cutting force from swept spindle profile test

Two types of tooling were used in the spindle: a two-flute carbide endmill (Sowa 103857), and a carbon-steel wire brush. A 25 mm square bar of Al-6061 was used as the workpiece. For the endmill, the axial depth of cut was 19 mm and the radial engagement was 0.5 mm. For the wire brush, the axial engagement was 3 mm and the radial engagement was 6 mm. A discussion of these cutting parameters is found in Sect. 4.

Input force excitations for the dynamic experiment were recorded using the table dynamometer (Kistler 9255B), and the output vibration responses were recorded from the spindle housing, as with the static case. Dynamic tests were conducted for both climb and conventional milling. Output vibrations on the spindle housing were captured with six accelerometers (PCB Piezotronics 352C03), with three each measuring in the X and Y directions, as seen in Fig. 1. All data channels were recorded at a sampling rate of 12,800 Hz through a multi-channel data acquisition unit (National Instruments NI-9234).

Results

Characterization of cutting force excitation

The input excitation force was examined to determine if the cutting forces produced with the swept spindle profile were similar to random excitation. Figure 3 shows a comparison of the endmill force spectrum produced by the swept profile and an example cutting test performed at a constant spindle speed of 6000 RPM. Spikes in cutting force energy are seen for the 6000 RPM test at 100 Hz increments, corresponding to the rotation frequency of the spindle and harmonics. It was also observed that even-numbered harmonics showed higher energy due to the two-flute geometry of the endmill. In contrast, the swept spindle profile displayed a broad spectrum more closely resembling white-noise. These results indicated that the swept spindle speed profile was an effective strategy to broaden cutting force inputs over a wide range of frequencies. Post-processing techniques to identify and remove harmonic content were not required using the proposed method of spindle speed variation.

Fig. 3.

X direction force spectrum for swept spindle profile and harmonics produced at constant spindle speed of 6000 RPM

Figure 4 displays the X component force spectrum produced by the swept profile for the endmill and the wire brush, and the force produced by the impact hammer. Broad spectrum excitation was observed for the wire brush as well; however, the overall excitation magnitude was lower compared to the endmill cutting forces. Due to the low levels of input excitation produced by the wire brush, the modal identification results were relatively poor. The remainder of this article examines only the results obtained from the endmill cutting tests. In the range beyond 1500 Hz, both the force spectrum for the endmill and brush show deviation from the broad pattern, while also decreasing in energy. The source of this anomaly was the natural frequencies of the dynamometer located near 1600 and 2200 Hz. The presence of these natural frequencies in the dynamometer limited the usable range of the force data used for validation. In the remainder of this study, the analysis region was restricted to frequencies between 0 and 700 Hz to ensure bias from the dynamometer did not compromise the results.

Fig. 4.

X direction force spectrum for the static impulse hammer and dynamic endmill and wire brush tests

Acceleration output responses

The signal-noise-ratio (SNR) was calculated to determine that the output vibration responses were sufficient for reliable OMA. The reference signal was collected during machine operation using the spindle sweep profile using the same tool path but with a vertical offset so the endmill did not contact the workpiece. Figure 5 displays a comparison of the vibration levels during cutting against the reference levels for the accelerometer located at position x1.

Fig. 5.

Comparison of x1 acceleration levels during cutting test against reference levels in time and frequency domain

The SNR was calculated as:

$$\begin{array}{c}\hfill \text{SNR}(\omega )=20log\left(\frac{P(\omega )}{P_{\mathit{ref}}(\omega )}\right)\end{array}$$ 17

where P is the power spectral density (PSD) of the accelerometer channel during cutting, while $P_{\mathit{ref}}$ is PSD of the reference signal. Best practice for OMA recommends that the SNR should be greater than 30 dB for reliable detection [14]. The SNR values for each accelerometer channel for the endmill test are shown in Fig. 6 with the 30 dB guideline annotated. Some regions displayed SNR values less than 30 dB for the different sensor channels. Results in this range should be interpreted with caution, as sufficient excitation does not exist in this range to identify modes with a high level of confidence. However, significant portions of the signal exceeded the 30 dB guideline, and it will be shown in Sect. 4.3 that these regions are where natural frequencies were identified.

Fig. 6.

Example SNR for all six accelerometer channels from dynamic swept milling test with threshold guideline annotated at 30 dB

Identification by conventional EMA

The FRFs obtained by EMA for the endmill are shown in Fig. 7 for the X and Y directions for both the static and dynamic cases. The X and Y directions displayed different FRF shapes due to the asymmetry of the machine tool system.

Fig. 7.

Receptance FRF magnitude for static and dynamic EMA of the two-flute endmill

The X direction FRF displayed two notable peaks located near 175 and 230 Hz, while the Y direction displayed three peaks near 240, 300, and 400 Hz. The dynamic FRFs in both directions displayed an increased receptance magnitude compared to the static case. The modal characteristics for each FRF were identified by a least-squares complex exponential fit of the modal FRF model to the experimental data [27]. The fitting was repeated for different model orders ranging between 1 and 50, and the natural frequency and damping information were obtained from the poles of each fitted model. A stabilization diagram was generated using the different model orders to identify which modal fit parameters were consistent across the fitted models. In this diagram, the stability criteria were selected as 5% for natural frequency and 10% for damping ratio. An example stabilization diagram for the endmill static FRF in the Y direction is shown in Fig. 8. The vertical lines of stable points annotated in red indicate modal characteristics, which consistently appeared in the model fit for many different model orders. Stable modal identification at all model orders is suggestive that the mode is physically present in the observed system response. While other modes were evident in Fig. 8, the three most prominent were located near 240, 300, and 400 Hz.

Fig. 8.

Stabilization diagram for the static endmill EMA in the Y direction, with three notable modal characteristics annotated in red

Identification by proposed OMA method

The output-only OMA methods were used to determine the modal characteristics of the system without any knowledge of the input excitation. The PSD matrix was computed for all six accelerometer channels, then SVD was performed on the matrix at each discrete frequency. The six singular values resulting from this decomposition were plotted at each frequency line as seen in Fig. 9. Only the first two singular values SV1 and SV2 showed significant spectral content. These values corresponded to vibratory modes in the X and Y direction. The remaining four singular values contained little spectral information that was distinguishable from noise, which indicated that the number and placement of accelerometers were sufficient to resolve two orthogonal vibration signatures by SVD. It was also noted that the overall shapes of SV1 and SV2 corresponded to the shapes of the X and Y FRFs captured in Fig. 7 using the conventional input–output EMA method. Peaks were identified in SV1 at 175 and 230 Hz, corresponding to the X direction FRF, and in SV2 at 240, 300, and 400 Hz, corresponding to the Y direction.

Fig. 9.

Magnitude of the complex mode indicator function (CMIF) for six singular values of the output vibrations from the dynamic endmill with region of interest annotated for SV1 in red

FDD was conducted for the SV1 and SV2 functions near these prominent peaks, which were identified by manual inspection. The region of interest near 175 Hz shown in Fig. 9 indicates the portion of SV1 that was considered to be the autospectrum of the first mode. This autospectrum was transformed into the time domain by inverse Fourier transform and is shown in Fig. 10. Examination of the period between zero-crossings of this signal was used to estimate the natural frequency of the mode. Similarly, the damping ratio was determined from the extrema of the autocorrelation function by the method of logarithmic decrement. Modal identification by FDD was repeated for all five peaks using this approach. The manual identification of peaks is the most notable limitation of the proposed OMA methods for practical long-term condition monitoring of machine tool systems, as some subjective bias can be introduced by the analyst. In this work, the peaks were evident in the singular value plots. Future work, however, should examine methods to automate the identification process for practical implementation.

Fig. 10.

Reconstructed autocorrelation function by FDD corresponding to 175 Hz mode. Critical features used to determine natural frequency and damping ratios are indicated on the wave

SSI was the fourth method for modal identification. System orders ranging between 1 and 50 were considered, and the state matrices were constructed from the extended observability and controllability matrices as described in Sect. 2.2. Stabilization diagrams were constructed using the same stability criteria as in the EMA case. The example stabilization diagram generated by SSI is shown in Fig. 11, with the modal characteristics of prominent stable modes annotated in red. The first five modes identified by SSI appear near model order 10 at 175, 230, 240, 300, and 400 Hz. As the model order increased, it was observed that stable detection of these five modes persisted to higher orders, indicating that they were physically present in the system response.

Fig. 11.

Stabilization diagram for the first five modes for the dynamic endmill using SSI

The SSI method was simple to implement as the modal characteristics were directly extracted from the six accelerometer time-domain signals. However, due to the close spacing of the modes at 230 and 240 Hz it may be difficult to determine if these modes correspond to different vibrations. To demonstrate that the closely spaced modes do in fact correspond to different vibratory behavior of the system, the modal assurance criterion (MAC) was computed to compare each of the five mode shapes for linear dependency. A MAC value of unity indicates that two mode shape vectors are perfectly correlated with one another, while a MAC value of zero indicates uncorrelated mode shapes. Often, a high MAC value (ex., 0.98) is used as a threshold to determine if two mode shapes are indistinguishable [28]. Figure 12 displays the MAC values for the FDD method and Fig. 13 for the SSI method. Strong agreement was found between Mode 1 (175 Hz) and Mode 2 (230 Hz). Similarly, Mode 3 (240 Hz), Mode 4 (300 Hz), and Mode 5 (400 Hz) were observed to have a moderate correlation to each other. However, the MAC value between Mode 2 and Mode 3 was moderate to low, providing clear evidence to distinguish these vibrations despite the closeness in natural frequency.

Fig. 12.

MAC values for mode shape vectors identified by FDD

Fig. 13.

MAC values for mode shape vectors identified by SSI

In this experiment, only two significant mode shapes were identified, which largely corresponded to motion in the X and Y directions. A normalized mode shape plot for the five modes is shown in Fig. 14, where the similarity for X direction modes (1 and 2) and Y direction modes (3, 4, and 5) was evident. This corresponds to the presence of only two significant non-zero singular values seen in the CMIF magnitude plots seen in Figs. 9 and 11. Resolving the mode shape vectors for the spindle is one way OMA methods may be used to localize vibrations to physical locations on the machine tool. Such localization aids the maintenance engineer in diagnostic tasks related to assigning faulty behavior to a specific component of the machine, such as spindle bearings. The placement of more accelerometers along the vertical direction of the spindle headstock would improve the spatial resolution of the mode shape vectors and allow for the differentiation of more mode shapes. However, increasing the number of sensors and data acquisition channels also increases the cost of the vibration monitoring system. In long-term CBM applications at scale, it may only be justifiable to use a small number of vibration sensors in an effort to minimize cost. Examination of the MAC values and mode shapes provides a means to justify the minimum number and locations of sensors in a particular CBM application.

Fig. 14.

Visualization of five normalized mode shapes identified by OMA

Modal characteristics

The bar chart in Fig. 15 displays natural frequency values corresponding to modes in the X direction for each of the four identification methods. A similar bar chart for the Y direction modes is shown in Fig. 16. For the Y direction modes, and to a lesser extent the X modes, the natural frequency values were lower for the three dynamic tests compared to the static EMA. A moderate decrease in natural frequency for the dynamic case is consistent with “speed softening” effects, which are known to present in high-speed spindles. Spindle natural frequencies have been shown to decrease at high speeds since significant centrifugal and gyroscopic forces develop that cause reduced stiffness in the spindle bearings [29, 30]. The observed decrease in natural frequency for these modes is evidence that these natural frequencies are indeed related to the spindle.

Fig. 15.

Natural frequency values of X direction modes identified by four methods

Fig. 16.

Natural frequency values of Y direction modes identified by four methods

When considering only the dynamic cutting conditions, the mean value for natural frequency identifications for the dynamic EMA, SSI, and FDD methods agreed within 3% error, which is comparable to the standard deviation of the estimates within the grouping of each individual method. No measurable difference was evident in the mean values of natural frequencies identified by dynamic EMA, SSI, and FDD. In the context of CBM activities, this level of precision is sufficient to determine changes in spindle health. In a modal identification case study conducted by the authors at an automotive manufacturing facility reported in [8], it was observed that spindle natural frequencies changed on the order of 30% between a spindle failure and after corrective maintenance action, suggesting that the proposed method would be able to distinguish such a change. It is noted that despite the agreement in mean values, both OMA algorithms produced a larger variance in the identified natural frequency. For example, the variance of the Mode X2 natural frequency calculated using the FDD algorithm is an order of magnitude larger than the corresponding dynamic EMA value, as shown in Fig. 15. In this work, the original peak-picking method proposed in [25] was applied to identify the modal characteristics; however, it is known that the original identification approach can be sensitive to noise and interferences present in real-world systems [31]. Improvements to the original FDD formulation have been proposed, such as recently in [32], which provides a least-squares approach to enhance the precision of the identification. While the results demonstrated in the present study demonstrate a level of precision that is sufficient to characterize the natural frequencies of the spindle, improvement of the identification algorithms for enhanced precision provides an opportunity for future work.

The identification of damping ratios was more challenging than natural frequency. Figure 17 displays a box plot comparison of the damping ratios for the X direction modes, and a matching plot is shown in Fig. 18 for the Y direction. The damping ratios obtained from impact hammer EMA on the static spindle ranged between 2 and 8% for all five modes. However, a significantly wider set of results was obtained from the dynamic tests, ranging from less than 1% to beyond 40%. The variance in damping ratios observed from the OMA may suggest that other process-dependent effects are obscuring the damping values. For instance, the so-called ploughing effect caused by friction between the machined workpiece material and the clearance face of the cutting tool has been reported to contribute significant process damping at low cutting speeds [33]. Due to the variable spindle speed used in the experiment, cutting conditions are changing throughout the test, which may contribute to the variance in damping ratios. Other effects, such as tool wear, workpiece material properties, workpiece geometry, coolant and lubrication conditions, and the presence of chatter, were not examined in this experiment and may also influence damping ratio measurements. It has been suggested that the damping ratio may provide valuable diagnostic information for spindle maintenance programs, such as quality of the bearing rolling contact surface and bearing lubrication levels [2]; however, the challenge of obtaining these values in practice is expected to limit the usefulness to CBM applications. Further developments in modelling and measurement of damping ratios during cutting may be required for CBM programs to benefit fully from the information obtained from damping ratios.

Fig. 17.

Damping ratio values of X direction modes identified by four methods

Fig. 18.

Damping ratio values of Y direction modes identified by four methods

Finally, it is noted that these results were produced using only a single tool workpiece combination for only one depth of cut condition. In other studies, optimal cutting parameters were obtained from cutting force models, resulting in a depth of cut that was well suited to the OMA problem [22]. However, in the intended CBM application, the cutting conditions will be dictated by the manufacturing process, and this optimal depth of cut may not be possible to achieve. Similarly, it may not be practical to use a spindle sweep profile that covers the entire operational range of the spindle. Future works should investigate combinations of spindle speed variability and cutting parameters which produce acceptable OMA results while still achieving process requirements dictated by the manufacturing process.

Conclusions

The contribution of this work is a new technique for OMA of machine tool spindle units based on output-only vibration measurements obtained during side milling with variable spindle speeds. The proposed technique improves upon the state-of-the-art by eliminating the need for direct measurement of cutting forces, specialized workpiece or tool geometries, or complicated Brownian-walk spindle speed profiles. In this study, the proposed technique was implemented using two OMA algorithms, namely the time-domain SSI and frequency-domain FDD. Broad frequency excitation was achieved over the range of 0–700 Hz using the swept spindle profile, and the cutting force excitations were shown to be appropriate for OMA while avoiding discrete harmonic content. The output-only vibration spectrum obtained by OMA during cutting was shown to correspond to the FRFs of the spindle unit obtained using conventional EMA methods. The natural frequency values identified by OMA agreed with conventional EMA methods within 10% overall and within 3% when comparing the three dynamic tests to one another. The damping ratios determined from OMA methods were somewhat similar to the values determined from EMA. Larger variance in the damping ratios was observed, however, and this may limit the usefulness of the damping ratio as a reliable health indicator for spindle CBM programs.

While the OMA method showed good performance at identifying natural frequencies from the spindle, future work should address the robustness of the method under operating conditions more similar to a production environment. In particular, three areas of future study are recommended:

1. Investigate more narrow bands of swept spindle speed ranges to more closely match production cutting conditions.

2. Evaluate sensitivity to changing process conditions such as different workpiece materials, depths of cut, tool wear, and the presence of coolant and lubricants.

3. Implement fully autonomous OMA identification methods that do not require manual interpretation of the vibration data.

Author Contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Patrick Chin. The first draft of the manuscript was written by Patrick Chin, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Funding

This work was supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2024-06690), the Federal Economic Development Agency for Southern Ontario iHub Project (#814996).

Declarations

Competing interests

The authors declare no competing interests.