# Assessment of harmonic suppression methods within OMA of an axial piston pump under constant speed

Vijayasankar Irissappane, Christian Svendsen, Luis David Avendaño-Valencia

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Vijayasankar Irissappane, Christian Svendsen, Luis David Avendaño-Valencia. Assessment of harmonic suppression methods within OMA of an axial piston pump under constant speed. 11th International Operational Modal Analysis Conference, Institut National de Recherche en Informatique et en Automatique [Inria], May 2025, Rennes, France. hal-05398796

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# Assessment of harmonic suppression methods within OMA of an axial piston pump under constant speed

Vijayasankar Irissappane<fnref n="1" />, Christian Svendsen<fnref n="2" /> and Luis David Avendaño-Valencia<fnref n="1" />

## ABSTRACT

Operational Modal Analysis (OMA) is a widely used technique for modal parameter identification from in-operation data. OMA on rotary machines like Axial Piston Pumps (APPs) is challenging due to the presence of strong harmonic components. In this study, OMA is applied to an APP running at constant speed and constant pressure conditions. Time-domain vibration data from the pump are preprocessed using *cepstrum editing* and *time synchronous averaging* (TSA) to suppress deterministic harmonics from rotation and pumping. *Frequency Domain Decomposition (FDD)* and *Stochastic Subspace Identification (SSI)* algorithms are applied to extract modal parameters. The results demonstrate that although FDD-based methods are computationally efficient, they require domain knowledge and manual interpretation of peaks, thereby complicating the process. The SSI algorithm, while generally considered more robust, had challenges in identifying the modal parameters as well. The preprocessing techniques improve the performance of OMA algorithms by suppressing the effects of the harmonics present in the measured signals, but it does not remove them completely. This paper concludes with discussions on the implications of these techniques for structural health monitoring (SHM) and future work.

Keywords: Operational Modal Analysis, Frequency Domain Decomposition, Stochastic Subspace Identification, Axial Piston Pumps

## 1. INTRODUCTION

Operational Modal Analysis (OMA) is an output-only technique for determining the dynamic characteristics of structures using vibration responses recorded under ambient or operational excitations [1]. A key assumption in most OMA techniques is that the excitation is a broadband white noise. Since the input forces are unknown, it is assumed that the nature of the loads entering the system is white Gaussian [1],

<footnote n="1">¹ Department of Electrical and Mechanical Engineering, University of Southern, Denmark, Campusvej 55, Odense, 5230, Denmark, viris@sdu.dk, ldav@sdu.dk</footnote>

<footnote n="2">² HPP, Danfoss A/S, Nordborgvej, Nordborg, 6430, Denmark, c.svendsen@danfoss.com</footnote>
so that all the structural modes in a frequency band of interest are equally excited and the corresponding modal parameters can be identified.

In practice, however, many structures are subjected to periodic deterministic forces that violate this assumption. Rotary machines like *Axial Piston Pumps* (APPs) experience strong harmonic components [2, 3] due to the operational forces that dominate the output measurement. The harmonic components, thus conceal the inherent structural response of the system. Therefore, it is necessary to develop OMA procedures that can deal with this harmonic content. Otherwise, harmonics may mask modal content or be mistakenly identified as lowly damped structural resonances [4].

Among the signal processing techniques available to detect and remove harmonics in measured responses, *cepstrum editing* [5] and *time synchronous averaging* (TSA) [6] stand out. Cepstrum editing allows for the removal of repeating tonal components by removing periodic components detected in cepstral domain, while TSA aims at identifying and subtracting periodic components in the response associated with single rotations of the shaft/rotor. Order tracking methods such as computed order tracking and Vold-Kalman filtering are particularly effective under variable speed conditions [7]. Cyclostationary analysis has shown strong capabilities in revealing modulated fault signatures otherwise masked by harmonics [8]. Recent advances in adaptive filtering [9], blind source separation [10], and time-frequency techniques such as wavelet packet decomposition [11] have also been used to isolate harmonics and improve fault detection.

Although OMA has previously been applied to structures with periodic excitations, these typically involve simple harmonic content and remain limited in rotating machinery [12]. In addition to signal processing techniques to remove harmonics, there have been some applications of direct OMA in rotating machinery, such as a full-scale ferry [13], wind turbines [14], and axial compressors [12], which showcase the real-world application of OMA in rotating machinery. The objective of this work is to assess the potential of OMA combined with preprocessing techniques to extract structural dynamic characteristics of an APP operating at constant speed. The study can be summarized as follows: (i) Detection and suppression of periodic signal components on raw APP vibration data with cepstrum editing and TSA. (ii) Application of FDD and SSI algorithms to extract modal parameters of the APP based on original and edited vibration signals.

## 2. THEORY OVERVIEW

A brief summary of the OMA identification algorithms and the preprocessing techniques related to harmonic reduction are described in this section.

### 2.1 OMA identification algorithms

OMA algorithms are broadly classified into frequency and time domain methods. Frequency domain techniques [15] (e.g., *Frequency-Domain Decomposition* (FDD)) analyze *Power Spectral Densities* (PSDs) of measured responses to extract modal parameters. Time-domain methods (e.g., *Ibrahim Time Domain (ITD)* method, *Auto-Regressive Moving Average* (ARMA) models, or *Stochastic Subspace Identification (SSI)*), make use of time-series to construct parameterized linear system approximations which can be used to extract modal parameters. In this section, a brief overview of FDD and SSI is presented.

#### 2.11. *Frequency Domain Decomposition*

FDD leverages the *Singular Value Decomposition* (SVD) of the response PSD matrix to extract modal characteristics of the underlying system. For a system consisting of $n$ unknown inputs $x(t) \in \mathbb{R}^n$ and $r$ measured responses $y(t) \in \mathbb{R}^r$, the response PSD matrix $G_{yy}(j\omega) \in \mathbb{C}^{r \times r}$ can be decomposed as [15]:

$$G_{yy}(j\omega) = H(j\omega) \cdot G_{xx}(j\omega) \cdot H(j\omega)^H = (U \cdot [h_k(j\omega)] \cdot V^H) \cdot G_{xx}(j\omega) \cdot (V \cdot [h_k^*(j\omega)] \cdot U^H) \quad (1)$$

where $G_{xx}(j\omega) \in \mathbb{C}^{n \times n}$ is the excitation PSD matrix, $H(j\omega) \in \mathbb{C}^{r \times n}$ *Frequency Response Function* (FRF), which accepts the SVD based on the left and right singular vectors $U \in \mathbb{C}^{r \times n}$ and $V \in \mathbb{C}^{n \times n}$,
respectively, and $[\mathbf{h}_k(j\omega)] \in \mathbb{C}^{n \times n}$ is a diagonal matrix with the FRFs of each modal coordinate. For white Gaussian excitation equally exciting all the system's DOFs, Eq. (1) simplifies to:

$$G_{yy}(j\omega) = U \cdot S(j\omega) \cdot U^H(j\omega) \qquad (2)$$

where $S(j\omega) := \sigma_x^2 \cdot |[\mathbf{h}_k(j\omega)]|^2$ is the diagonal singular value matrix with the PSDs of each modal coordinate. The singular value matrix is dominated by individual modes at their resonance frequencies. The FDD algorithm exploits this property, by computing the SVD of an estimate of the PSD matrix $\hat{G}_{yy}(j\omega)$ at each frequency $\omega$. Hence, resonances are identified as peaks in the largest entries of the singular value matrix. Natural frequencies and damping ratios can be extracted from the peak, whereas mode shapes are extracted from the corresponding entry of the (left) singular vector matrix $U$ [15].

## 2.12. Stochastic Subspace Identification

The dynamics of a linear time-invariant structural dynamics system with unknown excitation can be represented by the discrete-time stochastic state-space model as:

$$\mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{w}_k \qquad \mathbf{w}_k \sim \text{NID}(\mathbf{0}, \mathbf{Q}) \qquad (3a)$$

$$\mathbf{y}_k = \mathbf{C}\mathbf{x}_k + \mathbf{v}_k \qquad \mathbf{v}_k \sim \text{NID}(\mathbf{0}, \mathbf{R}) \qquad (3b)$$

where $\mathbf{x}_k \in \mathbb{R}^n$ is the state vector, $\mathbf{y}_k \in \mathbb{R}^r$ is the vector of (discrete) measured responses, $\mathbf{A} \in \mathbb{R}^{n \times n}$ and $\mathbf{C} \in \mathbb{R}^{r \times n}$ are the state-transition and observation matrices, respectively, and $\mathbf{w}_k \in \mathbb{R}^n$ and $\mathbf{v}_k \in \mathbb{R}^r$ are *Normally and Independently Distributed* (NID) sequences representing the process and measurement noises, respectively.

SSI attempts at estimating the system matrices $(\mathbf{A}, \mathbf{C})$, up to an equivalence transformation, based on analysis of the data block Hankel matrix:

$$\mathbf{Y} = \begin{bmatrix} \mathbf{y}_1 & \mathbf{y}_2 & \cdots & \mathbf{y}_{N-2p} \\ \mathbf{y}_2 & \mathbf{y}_3 & \cdots & \mathbf{y}_{N-2p+1} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{y}_{2p} & \mathbf{y}_{2p+1} & \cdots & \mathbf{y}_{N-1} \end{bmatrix} = \begin{bmatrix} \mathbf{Y}^+ \\ \mathbf{Y}^- \end{bmatrix} \qquad \mathbf{Y} \in \mathbb{R}^{2rp \times (N-2p)} \qquad (4)$$

where $p$ is the subspace dimensionality, and $\mathbf{Y}^- \in \mathbb{R}^{rp \times (N-2p)}$ and $\mathbf{Y}^+ \in \mathbb{R}^{rp \times (N-2p)}$ are the top and bottom halves of the Hankel matrix, splitting it into *past* and *future* responses. The SSI algorithm operates on the *projection matrix* $\mathcal{O} \in \mathbb{R}^{rp \times (N-2p)}$ [16]:

$$\mathcal{O} := E\{\mathbf{Y}^+|\mathbf{Y}^-\} = \mathbf{Y}^+(\mathbf{Y}^-)^T(\mathbf{Y}^-(\mathbf{Y}^-)^T)^{-1}\mathbf{Y}^- \qquad (5)$$

which can be factorized into the *observability matrix* $\hat{\mathbf{\Gamma}} \in \mathbb{R}^{rp \times rp}$ and the *Kalman state matrix* $\mathbf{X}^p \in \mathbb{R}^{rp \times (N-2p)}$ with the help of the SVD, as follows:

$$\mathcal{O} = \hat{\mathbf{\Gamma}}\mathbf{X}^p = U\mathbf{\Sigma}\mathbf{V}^T \qquad \Rightarrow \qquad \hat{\mathbf{\Gamma}} = U\mathbf{\Sigma}^{1/2}, \mathbf{X}^p = \mathbf{\Sigma}^{1/2}\mathbf{V}^T \qquad (6)$$

where $U \in \mathbb{R}^{rp \times rp}$ and $V \in \mathbb{R}^{rp \times (N-2p)}$ are orthogonal matrices comprised by the left and right singular vectors, and $\mathbf{\Sigma} \in \mathbb{R}^{rp \times rp}$ is a diagonal matrix of singular values. Estimates of the state and measurement matrices of the associated discrete-time state space representation $\hat{\mathbf{A}} \in \mathbb{R}^{rp \times rp}$ and $\hat{\mathbf{C}} \in \mathbb{R}^{r \times rp}$ can be extracted from the observability matrix. In turn, respective modal properties are calculated from the eigenvalue decomposition of the estimated state matrix.

The identification procedure may be repeated for different subspace dimensionalities $p$, until a maximum dimension is achieved. In this form, it is possible to analyze how modal parameter estimates stabilize with increasing subspace dimension, in the so-called *Frequency Stabilization Diagrams* (FSD). While this can be performed by changing the size of the Hankel matrix, a more computationally efficient approach consists of gradually increasing the number of singular values in the calculation of the observability matrix. Further details can be found at [17].
## 2.2 Preprocessing of vibration responses for harmonic suppression

A fundamental assumption of both FDD and SSI algorithms is that the excitation sources must be white Gaussian. However, measured vibration responses from rotating machines are dominated by deterministic components arising from periodic excitations. In a typical APP, these periodic excitations are related to both the pump's rotation and piston compression cycles. These create countless harmonics, observed as peaks at multiples of the piston passing frequency. These strong harmonics challenge structural identification. To overcome these challenges, various signal preprocessing techniques have been proposed to reduce the impact of these harmonics before conducting modal analysis. Here, we provide a brief overview of two widely used methods in OMA: Cepstrum Editing and Time Synchronous Averaging.

### 2.21. Time Synchronous Averaging (TSA)

TSA is a signal processing technique used to extract periodic components from a signal. In the context of OMA of rotating machinery, it can be used to remove periodic components and enhance the identifiability of structural modes [6]. By averaging vibration signals over multiple rotations of a reference component, TSA isolates the strictly periodic signal component. The procedure can be facilitated with the use of a tachometer signal, marking the period T of a reference shaft. Hence, an estimate of the periodic signal component is given by:

$$y_{\text{TSA}}(t) = \frac{1}{N} \sum_{k=1}^{N} y(t + (k - 1)T), \quad 0 \le t < T, \qquad (7)$$

where N is the total number of revolutions in the signal. The averaged signal $y_{\text{TSA}}(t)$ represents the deterministic periodic content, and the residual

$$y_{\text{residual}}(t) = y(t) - y_{\text{TSA}}(t) \qquad (8)$$

contains mainly the stochastic component suitable for OMA. TSA is effective at suppressing harmonics when the rotational speed is stable. However, even feeble speed variations may require additional angular resampling or order-tracking techniques to improve the results of this procedure [18].

### 2.22. Cepstrum Editing

Cepstrum editing is based on manipulation of the response signal cepstrum, where harmonics in the spectrum are manifested as periodic peaks in the cepstrum. The real cepstrum $c_y(\tau)$ of a signal $y(t)$ is defined as:

$$c_y(\tau) = \mathcal{F}^{-1}\{\ln|Y(\omega)|\}(\tau), \qquad (9)$$

where $Y(\omega)$ is the Fourier transform of $y(t)$ and $\tau$ is the quef frequency. Peaks in $c_y(\tau)$ indicate periodic components with a period $T_h = 1/f_h$. By applying a lifter function $L(\tau)$ that zeroes out these peaks, the liftered cepstrum becomes

$$\tilde{c}_y(\tau) = c_y(\tau) \cdot L(\tau). \qquad (10)$$

The edited signal $\tilde{y}(t)$ is then reconstructed by combining the modified cepstrum with the phase information of the original signal. This method effectively reduces the deterministic harmonic content, resulting in a signal that better satisfies the random excitation assumption [5].

## 3. EXPERIMENTAL METHODOLOGY AND ANALYSIS TECHNIQUES

This section presents the experimental setup on the APP under study, as well as a brief description of the preprocessing and modal parameter identification settings.
## 3.1 Structure, Instrumentation and Data Acquisition

The APP under study (model APP-38 manufactured by Danfoss A/S) was tested on an endurance test rig at standard lab conditions. The pump is driven by an electric motor through a variable-frequency drive. Inlet and outlet pressure are controlled by a pump and a hydraulic motor, respectively. The pump is controlled via an instrument panel, where all the pressure, RPM, and the torque are monitored, which are not logged during the test.

A total of 20 single-axis accelerometers (Dytran 3097A2T, sensitivity 100 mV/g) are mounted on the pump housing and end cover to capture the vibration response of the pump, as can be observed in Fig. 1. Additionally, two single-axis reference accelerometers (Wilcoxon 793-10, sensitivity 10 mV/g) are used — one in the axial direction and one in the radial direction — to serve as fixed references during data processing and subsequent modal parameter extraction if needed. The DAQ used in this setup was NI-cDAQ-9188 chassis, with multiple input modules to accommodate the required number of channels.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Schematic of the APP under study and accelerometer distribution, including the two reference sensors.</figcaption></figure>

Three case studies are conducted to emulate the varying operational conditions of a typical APP, as described in Table 1. Two cases of fixed pump speed are tested, in which the electric motor runs at 1500 RPM and the pump operates at an outlet pressure of 60 and 80 bar. A third case considers the pump running at a varying speed between 1000 and 1500 RPM in 3 different settings (3.1 to 3.3), tabulated as Test Case 3.1-3.3 in Table 1. In all sub-cases, the outlet pressure is set to 60 bar at the start. The tachometer signal is recorded to track the changing speed during each ramp. A slight pressure increase is observed as the speed increases, but to maintain consistency of the system settings, only the RPM is modified in the control system. As the APP reaches the target RPM as per the test conditions, it is allowed to stabilize for 10 minutes before starting the measurement process. Data is acquired at a sampling rate of 10 240 Hz for 300 s. This process is repeated several times to consider experimental variability. In the analysis presented here, only the cases of fixed RPM are under consideration.

<table><caption>**Table 1:** Running conditions of APP for various test scenarios</caption><thead><tr><th>Test Case</th><th>Start Speed [RPM]</th><th>End Speed [RPM]</th><th>Start Pressure (Set) [bar]</th><th>Ending Pressure (Observed) [bar]</th><th>Duration of Test [Seconds]</th><th>Tacho Signal</th></tr></thead><tbody><tr><td>1</td><td>1500</td><td>1500</td><td>80</td><td>80</td><td>300</td><td>No</td></tr><tr><td>2</td><td>1500</td><td>1500</td><td>60</td><td>60</td><td>300</td><td>Yes</td></tr><tr><td>3.1</td><td>1000</td><td>1500</td><td>60</td><td>65</td><td>300</td><td>Yes</td></tr><tr><td>3.2</td><td>1200</td><td>1500</td><td>60</td><td>64</td><td>600</td><td>Yes</td></tr><tr><td>3.3</td><td>1400</td><td>1500</td><td>60</td><td>62</td><td>600</td><td>Yes</td></tr></tbody></table>
## 3.2 Preprocessing

Preprocessing comprises initially angular resampling with the help of a tachometer signal. Considering the number of pistons of the APP (9), raw vibration signals are resampled to $2^7 \times 9 = 1152$ samples per revolution. This results in an adjusted sampling frequency of $f_s = 14440(2^6 \times 9 \times 25 \text{ Hz})$. Subsequently, cepstrum editing and TSA are used to extract periodic components from raw accelerometer data before modal parameter extraction. The **cepstrum editing** procedure operates on the cepstrum of angular-resampled with signals, based on FFTs of the complete data length. Peaks in the obtained cepstrum at quefrency values corresponding to multiples of the rotor period are located and edited. The editing procedure involves replacing the values around the peak by a segment of the same length from the neighborhood. The edited signals are transformed back to time domain to be used for modal parameter extraction. For **TSA**, the mean signal at each channel is calculated using Eq. (7) and subsequently removed from the resampled signals. Modal parameter extraction operates on the residual signals.

## 3.3 Modal Parameter Identification

After preprocessing, the data of all the channels from the APP are used as input for modal parameter extraction using the FDD and SSI. For comparison, the methods are also applied to the raw signals. FDD operates on Welch estimates of the PSD matrix, calculated using a Hanning window with blocksize $N_f = 2^{16}$, and 75% samples of overlap. Candidate modes are identified as peaks in the first singular value of the PSD matrix. These are obtained using the MATLAB's *findpeaks* function, considering a minimum prominence of 5% relative to the global maximum, and a minimum frequency separation of 20 Hz. In parallel, **SSI-data** is used on the original and filtered signals. The maximum model order is selected as $p = 75$, while 10000 samples are used to construct the data matrices. For automated modal parameter extraction, the algorithm is set to find 10 modes in the frequency range $f_{Range} = [0 \text{ 3000]}$, damping ratio range $\zeta_{Range} = [0 \text{ 0.02]}$ and minimum MAC value $MAC = 0.90$. The mode sorting procedure is based on histograms calculated on frequency bins of size $\Delta f = 2 \text{ Hz}$, as described in [19].

# 4. RESULTS AND DISCUSSION

The results of the analyses previously discussed are presented in the following sections.

## 4.1 Preprocessing of Input Signal

The input signals are processed using cepstrum editing and TSA techniques, as described in Sec. 2.2. Welch PSD estimates are calculated on the original and processed signals using the settings defined in Sec. 3.3. Resulting PSDs are shown in Fig. 2. The original signal consists of numerous harmonics corresponding to the multiples of the pump operating speed and the piston passing frequencies. As expected, the piston harmonics dominate the measured response and all the other structural responses are overwhelmed by the harmonics. Despite the TSA technique's success in removing certain harmonics from the original signals, as denoted by the green arrows, it is unable to completely remove them, particularly at higher frequencies, as shown by the pink arrows. On the other hand, cepstrum editing provides a better harmonic rejection capability, as evident in the red line, where there no significant harmonics are present.

## 4.2 Results with Frequency Domain Decomposition

Fig. 3 presents the results obtained with FDD on the original signal (top frame), and after TSA and cepstral editing (middle and bottom frames, respectively). Each frame shows the first three singular values and the ten dominant modes identified via automated peak picking. Identified modal frequencies for each preprocessing method are listed in Tab. 2. In the original signal, the automated peak picking primarily detects harmonics of the pump's running speed (multiples of 25 Hz), making it difficult to distinguish structural modes from harmonic content. Modes 1–4, 6, and 10 align with pump harmonics, while modes 5 and 7–9 correspond to piston passing harmonics (225 Hz).
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Welch's PSD estimate of the original, cepstrum edited and TSA residual signals of sensor 1</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3:** Singular values and salient peaks obtained with FDD in the original (top), TSA (middle) and cepstrum edited (bottom) signals.</figcaption></figure>

TSA facilitates peak identification by reducing the prominence of pump harmonics - particularly those around piston passing frequency of 225 Hz — though residual harmonics remain, as seen in the middle subplot of Figure 3. These harmonics are linked to an attached hydraulic motor, running at a speed close to that of the pump but not exactly the same. In this case, the hydraulic motor was observed to run at 24.2 Hz. Hence, peaks 1,3,5 belong to the hydraulic motor's piston passing frequency harmonics (24.2 × 9 = 217.8 Hz). In this case, FDD was able to identify 6 modes that do not belong either to the pump or motor harmonics. Cepstral editing provides the most effective harmonic suppression. In the bottom frame of Fig. 3, pump and motor harmonics are significantly attenuated, particularly sidebands around dominant peaks. While some harmonics from the hydraulic motor remain (e.g., peaks 5, 9, and 10), the remaining detected modes likely reflect the structural dynamics of the pump and its components.
<table><caption>**Table 2:** Identified natural frequencies using FDD and SSI methods after different preprocessing methods.</caption><thead><tr><th rowspan="2">Mode</th><th colspan="3">FDD</th><th colspan="3">SSI</th></tr><tr><th>Original<br>(Hz)</th><th>TSA<br>(Hz)</th><th>Cepstrum<br>(Hz)</th><th>Original<br>(Hz)</th><th>TSA<br>(Hz)</th><th>Cepstrum<br>(Hz)</th></tr></thead><tbody><tr><td>1</td><td>24.97</td><td>217.88</td><td>89.88</td><td>224.77</td><td>218.11</td><td>89.87</td></tr><tr><td>2</td><td>74.94</td><td>317.78</td><td>121.03</td><td>437.02</td><td>436.39</td><td>91.133</td></tr><tr><td>3</td><td>99.91</td><td>435.75</td><td>169.47</td><td>674.68</td><td>655.57</td><td>218.03</td></tr><tr><td>4</td><td>199.81</td><td>542.56</td><td>193.69</td><td>899.79</td><td>897.99</td><td>434.01</td></tr><tr><td>5</td><td>224.78</td><td>653.63</td><td>217.88</td><td>1125.70</td><td>1125.50</td><td>1122.70</td></tr><tr><td>6</td><td>324.69</td><td>681.25</td><td>266.28</td><td>1348.80</td><td>1349.00</td><td>1308.9</td></tr><tr><td>7</td><td>449.56</td><td>756.19</td><td>290.50</td><td>1798.10</td><td>1572.70</td><td>1570.00</td></tr><tr><td>8</td><td>674.34</td><td>817.82</td><td>327.19</td><td>2023.50</td><td>2023.30</td><td>—</td></tr><tr><td>9</td><td>899.13</td><td>892.22</td><td>435.75</td><td>2249.20</td><td>2249.00</td><td>—</td></tr><tr><td>10</td><td>924.09</td><td>931.00</td><td>653.59</td><td>2472.60</td><td>—</td><td>—</td></tr></tbody></table>

## 4.3 Results with Stochastic Subspace Identification

As in FDD, the original signal and, TSA and cepstrum edited signals are fed to the SSI algorithm, using the parameters mentioned in Sec. 3.2. The respective results displayed in Fig. 4. Detected modal frequencies are also listed in Tab. 2. In the original signal, SSI fails to detect any of the structural modes. All the 10 modes identified by the algorithm correspond to either the pump or the motor harmonics. In case of TSA, some of the identified modes correspond to the motor harmonics which remained in the signal. But at higher frequencies, over 1000 Hz, the pump frequencies dominate the detected peaks. Also, the algorithm can detect only nine modes successfully within the given frequency range. Lastly, as in FDD, the cepstrum edited signals demonstrate better results, following from the most effective harmonic reduction. Though there are some influence of the harmonics, the SSI method successfully identifies modes that are not belonging to either of the pump or motor harmonics.

<figure><img src="image_5.png" /><figcaption>**Figure 4:** Frequency stabilization diagrams obtained with SSI for the APP operating at 1500 RPM and 60 bar.</figcaption></figure>
## 5. CONCLUSIONS

This study explored the application of OMA techniques to identify the dynamic characteristics of an APP operating under constant operating conditions. The challenges posed by deterministic harmonics inherent to rotary machinery were addressed through pre-processing techniques namely, cepstrum editing and TSA. Both techniques demonstrated efficacy in suppressing harmonic content. While TSA was able to reduce the harmonics at lower frequencies, higher frequencies were still challenging to be addressed solely by TSA. Cepstrum editing was found to remove a broader range of harmonics, including some at higher frequencies, though the influence of the hydraulic motor running at a different speed was still present. Modal parameter identification was studied using FDD and SSI. While FDD provided computational efficiency, it required manual intervention for accurate peak identification, particularly in the presence of residual harmonics. Conversely, SSI facilitates automated detection but involves higher computation time. Residual harmonics also hampered the detection of structural modes.

The results confirm that pre-processing significantly enhances the clarity and reliability of modal estimates, though some residual harmonic influences persist. In future work, the modal parameters extracted from OMA will be validated against either a finite element model or by comparison with Experimental Modal Analysis results. In addition, there are plans to investigate alternative signal processing techniques and to explore OMA techniques that can directly handle harmonic excitation. Finally, an automated method for extracting modal parameters and tracking modes across experiments and APPs is under development, enabling the creation of a digital twin of the APP, that can be deployed for SHM. These advances will improve the fidelity of OMA in complex operational environments, contributing to more reliable condition monitoring and structural health assessment of APPs.

## ACKNOWLEDGMENTS

The authors acknowledge the financial support of Danfoss A/S, High Pressure Pumps division, and the Innovation Fund Denmark via the Industrial PhD Grant (2040-00009B) for the project "Digital Twins of Axial Piston Pumps (APPs) for Machine Learning-based Condition Monitoring".

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