# Separation of vibratory components in complex systems for condition monitoring

Fadi Karkafi, Dany Abboud, Quentin Leclère, Jérôme Antoni, Mohammed El Badaoui

► To cite this version:

Fadi Karkafi, Dany Abboud, Quentin Leclère, Jérôme Antoni, Mohammed El Badaoui. Separation of vibratory components in complex systems for condition monitoring. International Journal of COMA-DEM, 2024, 27 (2), pp.21-29. hal-04619794

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Submitted on 21 Jun 2024

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# Separation of vibratory components in complex systems for condition monitoring

Fadi Karkafi, Dany Abboud, Quentin Leclère, Jérôme Antoni, Mohammed El Badaoui

<figure><img src="image_1.png" /></figure>

## To cite this version:

Fadi Karkafi, Dany Abboud, Quentin Leclère, Jérôme Antoni, Mohammed El Badaoui. Separation of vibratory components in complex systems for condition monitoring. International Journal of COMA-DEM, 2024, 27 (2), pp.21-29. hal-04619794

HAL Id: hal-04619794

https://hal.science/hal-04619794v1

Submitted on 21 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

Public Domain
<figure><img src="image_1.png" /></figure>

<figure><img src="image_2.png" /></figure>

# Separation of vibratory components in complex systems for condition monitoring

Fadi Karkafi<sup>a,b,*</sup>, Dany Abboud<sup>b</sup>, Quentin Leclère<sup>a</sup>, Jérôme Antoni<sup>a</sup>, and Mohammed El Badaoui<sup>b,c</sup>

## ABSTRACT

Mechanical signals are often a mixture of multiple components produced by different sources within multiple subsystems. The separation of these components is beneficial for differential diagnosis, fault severity assessment and prognosis. The synchronous average (SA) is a powerful and widely used technique for this purpose, providing optimal estimation performance and simplicity. However, the complexity of such systems often implies the existence of several spectral interferences, which jeopardize the straightforward application of the SA. This paper addresses this issue by proposing a general strategy based on the synchronous average and the set theory, to isolate each vibratory contribution related to a source of interest, while accounting for spectral interferences. This approach enables a comprehensive understanding of the system's health status and the ability to locate specific issues within a subsystem. The proposed methodology is demonstrated through an example of an accessory gearbox vibration signal acquired during a ground test campaign on a CFM56 turbojet engine.

Keywords: Synchronous average; Mechanical contribution separation; Condition monitoring; Vibrations analysis.

Article history:

## 1. Introduction

The study of mechanical systems has long been a crucial area of research in various industries [1]. Effective monitoring and diagnosis of mechanical systems is essential for maintaining smooth operations. Vibration and acoustic analysis is a widely used method for monitoring the health of mechanical systems. This approach is non-intrusive and provides valuable information about the internal forces in the system, offering a reliable tool for fault detection and diagnosis [2]. However, these systems are often complex, involving the conversion of multiple speed and torque ratios to achieve different tasks. Therefore, this often results in a large number of rotating parts, each of which contributes differently to the overall vibration signal. This makes it difficult to identify and diagnose specific components, thus the importance of signal separation in [3]. Furthermore, the presence of multiple subsystems and rotating parts can lead to some interferences between different sources. This problem is often addressed through frequency analysis, which assumes that the sources do not share their frequency content. However, in many real-world applications, frequency content overlap is unavoidable, which results in a spectral interference due to having the information of a certain line contribution masking the other. This superimposition can compromise the accuracy of diagnosis in terms of fault detection and localization by tracking the faulty component over the healthy one and vice versa, therefore, a false health state assumption. Consequently, to diagnose each mechanical component independently, it is essential to extract the exclusive contribution of each one by

eliminating the overlapped frequencies caused by other components.

Some generic signal processing techniques were proposed for signal decomposition and mechanical component contribution extraction using vibration analysis. Starting with a basic Synchronous Average (SA); this method involves averaging the signal over one or several periods to extract the sinusoidal component that refers to a certain frequency and its multiples. The advantage of this method is its simplicity making it possible to implement without any estimation. However, when considering a set of lines, the sum of all contributions would result in a replication of the interference mentioned before. This concept was addressed in [4] by exploiting the corresponding operator of the synchronous average and this replication problem when extracting the whole subsystem. However, the remaining spectral overlap for each contribution still exists which triggers again the problem stated before. Next, [5] involves the use of a band pass filter to extract the frequency content of the components of interest. This method is simple to implement and can provide good results for simple mechanical systems. However, it can be sensitive to the choice of filter parameters and may include some interferences within the chosen bandwidth around the centered frequency. In addition, the problem of frequency overlap is still considered having several sources sharing the same frequency. Otherwise, the Optimal Subspace-Kurtosis and Subspace-Extended Kalman Filter involves combining the advantages of both approaches to achieve optimal source separation [6]. It uses the subspace-kurtosis method to

<footnote><sup>a</sup>Univ Lyon, INSA Lyon, LVA, EA677, 69621 Villeurbanne, France</footnote>

<footnote><sup>b</sup>Safran Tech, Rue des Jeunes Bois - Châteaufort 78772 Magny-les-Hameaux, France</footnote>

<footnote><sup>c</sup>Univ Lyon, UJM-St-Etienne, LASPI, EA3059, F-42023, Saint-Etienne, France</footnote>

<footnote>* Corresponding author. Tel.: +33-7-65865954; email: fadi.karkafi@insa-lyon.fr</footnote>
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

# Separation of vibratory components in complex systems for condition monitoring

Fadi Karkafi<sup>a,b,*</sup>, Dany Abboud<sup>b</sup>, Quentin Leclère<sup>a</sup>, Jérôme Antoni<sup>a</sup>, and Mohammed El Badaoui<sup>b,c</sup>

## ABSTRACT

Mechanical signals are often a mixture of multiple components produced by different sources within multiple subsystems. The separation of these components is beneficial for differential diagnosis, fault severity assessment and prognosis. The synchronous average (SA) is a powerful and widely used technique for this purpose, providing optimal estimation performance and simplicity. However, the complexity of such systems often implies the existence of several spectral interferences, which jeopardize the straightforward application of the SA. This paper addresses this issue by proposing a general strategy based on the synchronous average and the set theory, to isolate each vibratory contribution related to a source of interest, while accounting for spectral interferences. This approach enables a comprehensive understanding of the system's health status and the ability to locate specific issues within a subsystem. The proposed methodology is demonstrated through an example of an accessory gearbox vibration signal acquired during a ground test campaign on a CFM56 turbojet engine.

Keywords: Synchronous average; Mechanical contribution separation; Condition monitoring; Vibrations analysis.

Article history:

## 1. Introduction

The study of mechanical systems has long been a crucial area of research in various industries [1]. Effective monitoring and diagnosis of mechanical systems is essential for maintaining smooth operations. Vibration and acoustic analysis is a widely used method for monitoring the health of mechanical systems. This approach is non-intrusive and provides valuable information about the internal forces in the system, offering a reliable tool for fault detection and diagnosis [2]. However, these systems are often complex, involving the conversion of multiple speed and torque ratios to achieve different tasks. Therefore, this often results in a large number of rotating parts, each of which contributes differently to the overall vibration signal. This makes it difficult to identify and diagnose specific components, thus the importance of signal separation in [3]. Furthermore, the presence of multiple subsystems and rotating parts can lead to some interferences between different sources. This problem is often addressed through frequency analysis, which assumes that the sources do not share their frequency content. However, in many real-world applications, frequency content overlap is unavoidable, which results in a spectral interference due to having the information of a certain line contribution masking the other. This superimposition can compromise the accuracy of diagnosis in terms of fault detection and localization by tracking the faulty component over the healthy one and vice versa, therefore, a false health state assumption. Consequently, to diagnose each mechanical component independently, it is essential to extract the exclusive contribution of each one by

eliminating the overlapped frequencies caused by other components.

Some generic signal processing techniques were proposed for signal decomposition and mechanical component contribution extraction using vibration analysis. Starting with a basic Synchronous Average (SA); this method involves averaging the signal over one or several periods to extract the sinusoidal component that refers to a certain frequency and its multiples. The advantage of this method is its simplicity making it possible to implement without any estimation. However, when considering a set of lines, the sum of all contributions would result in a replication of the interference mentioned before. This concept was addressed in [4] by exploiting the corresponding operator of the synchronous average and this replication problem when extracting the whole subsystem. However, the remaining spectral overlap for each contribution still exists which triggers again the problem stated before. Next, [5] involves the use of a band pass filter to extract the frequency content of the components of interest. This method is simple to implement and can provide good results for simple mechanical systems. However, it can be sensitive to the choice of filter parameters and may include some interferences within the chosen bandwidth around the centered frequency. In addition, the problem of frequency overlap is still considered having several sources sharing the same frequency. Otherwise, the Optimal Subspace-Kurtosis and Subspace-Extended Kalman Filter involves combining the advantages of both approaches to achieve optimal source separation [6]. It uses the subspace-kurtosis method to

<footnote><sup>a</sup>Univ Lyon, INSA Lyon, LVA, EA677, 69621 Villeurbanne, France</footnote>

<footnote><sup>b</sup>Safran Tech, Rue des Jeunes Bois - Châteaufort 78772 Magny-les-Hameaux, France</footnote>

<footnote><sup>c</sup>Univ Lyon, UJM-St-Etienne, LASPI, EA3059, F-42023, Saint-Etienne, France</footnote>

<footnote>* Corresponding author. Tel.: +33-7-65865954; email: fadi.karkafi@insa-lyon.fr</footnote>
extract periodic signals of interest, while the extended Kalman filter is used to estimate the corresponding contributions. The advantage of this method is that it provides a comprehensive solution to the problem of interferences and source separation, taking into account both the presence of periodic signals and the correction of interferences. Other works use Kalman filters to improve frequency analysis and order tracking methods [7, 8, 9, 10, 11] and recent techniques aim to improve the Vold Kalman Filtering for resampling techniques, high frequency resolution in Fourier analysis and crossing orders detection [12, 13, 14, 15, 16]. However, when many harmonics are considered in typical complex mechanical systems, it can be challenging for these approaches to accurately track all of them, which can lead to an increased computation time and cost.

In this paper, the proposed method addresses the challenge of interferences and contribution separation in complex mechanical systems. In Section 2, the problem statement defines the periodicity identification in such systems and its replication when extracting several sources. Section 3 presents the methodology of the proposed method systematically to solve the defined problem. Section 4 demonstrates the application of the method on an aircraft accessory gearbox. Finally, in Section 5, a conclusion wraps this paper up and highlights the most important details.

## 2. Problem Statement

Vibratory signals generated by rotating machines are periodic in nature. This periodicity is directly linked to the angle of rotation of the machine. In fact, when machines operate under a perfectly constant speed, this periodicity remains consistent over time. In practice, machines are subject to speed fluctuations, which can introduce errors when analyzing time domain signals. One widely used solution in mechanical signal processing is angular description [17, 18]. This can be achieved by resampling the time domain signal in the angular domain using an encoder signal indicating the position of the angle of rotation.

### 2.1. The synchronous average (SA)

In this context, the synchronous average (SA) is a powerful and widely used tool in vibration analysis, which particularly deals with the extraction of periodic waveforms (such as produced by imbalances, misalignments, anisotropic rotors, flexible coupling, gear meshing and other phenomena) from other competing but random sources, assuming that its period is known a priori [19, 20, 21]. The basic idea of the SA is to average the signal over a number of samples called a cycle. This is done by dividing the signal into segments, each corresponding to one cycle of the signal, and then averaging the values within each segment. Mathematically, let $x[\theta]$ be an angular signal of cycle $\Theta$ (for instance $\Theta = 2\pi$); its SA is defined as follows:

$$SA_{\alpha}(\bar{\theta}) = \left\langle x\left(\bar{\theta} + m\frac{2\pi}{\alpha}\right) \right\rangle_m, \quad m \in \mathbb{N} \quad (1)$$

where the operator $\langle f(m) \rangle_m$ denotes the empirical average of the function $f(m)$, $\alpha$ being the fundamental cyclic order (so that the averaging period equals $2\pi/\alpha$) and $\bar{\theta} \in [0, \frac{2\pi}{\alpha}]$ for the angular location in the cycle (i.e. $\bar{\theta} = \theta - k \frac{2\pi}{\alpha}$, with k the greatest integer smaller than $\theta/\alpha$).

The obtained SA signal is defined over a corresponding period with respect to a cyclic order $\alpha$ and then replicated to match with initial signal length as follows:

$$m_{\alpha}(\theta) \leftrightarrow SA_{\alpha}(\bar{\theta}) \quad (2)$$

The SA is typically used to extract the vibratory component associated with a rotating shaft, resulting in a Dirac comb in the frequency domain at the line order and its multiple harmonics.

The order of each shaft is defined as the ratio between the frequency of the shaft and the reference frequency, with the reference shaft having an order of 1.

### 2.2. Frequency Interferences

When approaching the problem of a complex mechanical system decomposition, given a general model illustrated in Figure 1 with the corresponding interferences in red, it is essential to firstly isolate the subsystem of interest ($a_i$, such that $j=1,...,N$). Some examples of subsystems in mechanical systems are: an accessory gearbox (AGB), a gas generator, a combustion chamber, etc...

<figure><img src="image_3.png" /><figcaption>Figure 1. General model of a rotating machine considering different interferences (red arrows) between different lines ($a_{11}, a_{12},...$) in each subsystem ($a_1, a_2,...$).</figcaption></figure>

Once the isolation of the potentially defective subsystem has been carried out, it is important in a second phase to proceed with the separation of the various components linked to the rotating parts of this subsystem (i.e. $a_{i1}$, such that $i=1,...,M$, from the subsystem $a_1$). This separation considers the exclusive contribution of each component (without interferences) to accurately locate the fault in the isolated subsystem.

In the following, and without loss of generality, the problem addressed in this paper is explained through a simplistic example. Considering a gearbox with an input shaft rotating at order 1 (reference shaft) and an output shaft rotating at 1.2, the associated frequency set with the SA of each shaft would lead to the following order set:

Input Shaft:

$$S_i : \left\{ \alpha_i = 1; 2; 3; 4; 5; 6; \dots, \text{ such that } \alpha_i \le \frac{N_S}{2} \in \mathbb{N} \right\}$$

Output Shaft:

$$S_o : \left\{ 1.2 \times k = 1.2; 2.4; 3.6; 4.8; 6; \dots, \text{ such that } 1.2 \times k \le \frac{N_S}{2} / k \in \mathbb{N} \right\}$$
extract periodic signals of interest, while the extended Kalman filter is used to estimate the corresponding contributions. The advantage of this method is that it provides a comprehensive solution to the problem of interferences and source separation, taking into account both the presence of periodic signals and the correction of interferences. Other works use Kalman filters to improve frequency analysis and order tracking methods [7, 8, 9, 10, 11] and recent techniques aim to improve the Vold Kalman Filtering for resampling techniques, high frequency resolution in Fourier analysis and crossing orders detection [12, 13, 14, 15, 16]. However, when many harmonics are considered in typical complex mechanical systems, it can be challenging for these approaches to accurately track all of them, which can lead to an increased computation time and cost.

In this paper, the proposed method addresses the challenge of interferences and contribution separation in complex mechanical systems. In Section 2, the problem statement defines the periodicity identification in such systems and its replication when extracting several sources. Section 3 presents the methodology of the proposed method systematically to solve the defined problem. Section 4 demonstrates the application of the method on an aircraft accessory gearbox. Finally, in Section 5, a conclusion wraps this paper up and highlights the most important details.

## 2. Problem Statement

Vibratory signals generated by rotating machines are periodic in nature. This periodicity is directly linked to the angle of rotation of the machine. In fact, when machines operate under a perfectly constant speed, this periodicity remains consistent over time. In practice, machines are subject to speed fluctuations, which can introduce errors when analyzing time domain signals. One widely used solution in mechanical signal processing is angular description [17, 18]. This can be achieved by resampling the time domain signal in the angular domain using an encoder signal indicating the position of the angle of rotation.

### 2.1. The synchronous average (SA)

In this context, the synchronous average (SA) is a powerful and widely used tool in vibration analysis, which particularly deals with the extraction of periodic waveforms (such as produced by imbalances, misalignments, anisotropic rotors, flexible coupling, gear meshing and other phenomena) from other competing but random sources, assuming that its period is known a priori [19, 20, 21]. The basic idea of the SA is to average the signal over a number of samples called a cycle. This is done by dividing the signal into segments, each corresponding to one cycle of the signal, and then averaging the values within each segment. Mathematically, let $x[\theta]$ be an angular signal of cycle $\Theta$ (for instance $\Theta = 2\pi$); its SA is defined as follows:

$$SA_{\alpha}(\bar{\theta}) = \left\langle x\left(\bar{\theta} + m\frac{2\pi}{\alpha}\right) \right\rangle_m, \quad m \in \mathbb{N} \quad (1)$$

where the operator $\langle f(m) \rangle_m$ denotes the empirical average of the function $f(m)$, $\alpha$ being the fundamental cyclic order (so that the averaging period equals $2\pi/\alpha$) and $\bar{\theta} \in [0, \frac{2\pi}{\alpha}]$ for the angular location in the cycle (i.e. $\bar{\theta} = \theta - k \frac{2\pi}{\alpha}$, with k the greatest integer smaller than $\theta/\alpha$).

The obtained SA signal is defined over a corresponding period with respect to a cyclic order $\alpha$ and then replicated to match with initial signal length as follows:

$$m_{\alpha}(\theta) \leftrightarrow SA_{\alpha}(\bar{\theta}) \qquad (2)$$

The SA is typically used to extract the vibratory component associated with a rotating shaft, resulting in a Dirac comb in the frequency domain at the line order and its multiple harmonics.

The order of each shaft is defined as the ratio between the frequency of the shaft and the reference frequency, with the reference shaft having an order of 1.

### 2.2. Frequency Interferences

When approaching the problem of a complex mechanical system decomposition, given a general model illustrated in Figure 1 with the corresponding interferences in red, it is essential to firstly isolate the subsystem of interest ($a_i$, such that $j=1,...,N$). Some examples of subsystems in mechanical systems are: an accessory gearbox (AGB), a gas generator, a combustion chamber, etc...

<figure><img src="image_4.png" /><figcaption>Figure 1. General model of a rotating machine considering different interferences (red arrows) between different lines ($a_{11}, a_{12},...$) in each subsystem ($a_1, a_2,...$).</figcaption></figure>

Once the isolation of the potentially defective subsystem has been carried out, it is important in a second phase to proceed with the separation of the various components linked to the rotating parts of this subsystem (i.e. $a_{i1}$, such that $i=1,...,M$, from the subsystem $a_1$). This separation considers the exclusive contribution of each component (without interferences) to accurately locate the fault in the isolated subsystem.

In the following, and without loss of generality, the problem addressed in this paper is explained through a simplistic example. Considering a gearbox with an input shaft rotating at order 1 (reference shaft) and an output shaft rotating at 1.2, the associated frequency set with the SA of each shaft would lead to the following order set:

Input Shaft:

$$S_i : \left\{ \alpha_i = 1; 2; 3; 4; 5; 6; \dots, \text{ such that } \alpha_i \le \frac{N_S}{2} \in \mathbb{N} \right\}$$

Output Shaft:

$$S_o : \left\{ 1.2 \times k = 1.2; 2.4; 3.6; 4.8; 6; \dots, \text{ such that } 1.2 \times k \le \frac{N_S}{2} / k \in \mathbb{N} \right\}$$
with $N_S$ being the number of samples per reference cycle assumed $2\pi$ in this paper.

The vibratory contribution of this gearbox (the subsystem) cannot be directly obtained by summing up the SA of both shafts. In fact, as seen in the Figures 2a – 2b, there are several orders in common between the shafts (i.e. 6 and its multiples 12, 18, 24, etc.). These common orders represent the frequency interferences of the subsystem components. The sum of both SA leads to incorrect estimation, which manifests through higher energy at the shared frequencies in the spectrum (see in Figures 2c – 2d). To be noted that this interference is not meant to be completely removed for the subsystem but rather its replication because it can provide important information about its global health state. In fact, these shared harmonics, representing the meshing harmonics and their energies, are widely used as efficient diagnostic indicators in gearboxes.

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Order Spectra for: a- (Top left) SA of input shaft, b- (Top right) SA of output shaft, c- (Bottom left) Sum of both SAs, d- (Bottom right) Comparison between the sum of both SAs and the real subsystem SA.</figcaption></figure>

Another interesting issue addressed in this paper is the localization of a fault. This needs to be achieved by considering the exclusive contribution of each shaft so the faulty one does not affect the others. As seen in Figures 2a – 2b, the SAs of the two shafts present an interference at the sixth order and its multiples, which could mislead the identification of the faulty shaft. Therefore, it is essential to eliminate the remaining interferences between each source to evaluate the health indicator applied on the exclusive contribution of each one. This paper goes beyond this simple example to the case of more complex systems, where multiple shafts with different rotating frequencies having multiple common frequencies in play.

### 3. Proposed methodology

The proposed methodology in this paper is designed to isolate the desired vibratory contribution of a given subsystem in a complex mechanical system using the SA, as well as the exclusive vibration response of each of the subset component. On the one hand, this is achieved by using the set theory to identify the common frequencies that represent the interferences between each line and eliminate its replication as seen in [4]. On the other hand, it can be implemented by the computation of the least common multiple (LCM) of each pair of orders.

Figure 3 illustrates this theory, where multiple sets are considered, having the set “$a_1$” representing the reference shaft. Each set corresponds to the orders ensemble of each shaft’s contribution resulted from its SA. By comparing the orders of shafts $a_1$ and $a_2$, it can be seen that some interferences exist (hatched part) between these two sets and others that are jointly common with shaft $a_3$ (green triangles).

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Illustration of the sets of SA orders with the different interferences.</figcaption></figure>

#### 3.1. Subsystems Isolation

Generally, one can consider N shafts with the corresponding set of fundamental orders $A = \{\alpha_i \in \mathbb{R} / i = 1, 2, \dots, N\}$ constituting a certain subsystem. The goal is to find the signal associated with this specific subsystem, which encompasses the total set of constituent lines excluding the replications of the interference between each considered line. This leads to the following equation:

$$ \sum_{\alpha_i \in A} m_{\alpha_i}(\theta) = s(\theta) + \Delta(\theta) \quad (2) $$

With $m_{\alpha_i}(\theta)$ being the SA of the angular signal $x(\theta)$ with respect to the order $\alpha_i$, $s(\theta)$ being the subsystem deterministic signal and $\Delta(\theta)$ being the replication contribution caused by the interfering components.

Taking the replication to be the unique interference between each two sets (i.e. $a_1 \cap a_2$ without $a_2 \cap a_1$; the interference is not totally eliminated between these two sets, but rather the replication of the interference.

The replication set requires a more generic definition because common orders between physically unlinked shafts can still exist. In addition, this interference will repeat for integer multiples of the common order, leading to two major problems that prompted the proposal of two solutions:

* Calculate the LCM between each pair of shafts to find the fundamental common order.

* Compute the synchronous average relative to the LCM order found to take into consideration all its multiples.

In order to calculate the corresponding LCM of all common orders for each shaft, a matrix of common orders (MCO) is proposed. To be able to free the replication of the interference between each two shafts without the elimination of the latter, this matrix must be an upper triangular matrix whose diagonal corresponds to the common order between two consecutive shafts. By taking the LCM between each pair of shafts, only the lowest common order is considered, eliminating the redundant frequencies when isolating the subsystem. Thus, the computation of the following matrix of common orders (MCO):

$$ M(i, j) = \begin{cases} \text{LCM}(\alpha_i, \alpha_{j+1}), & \text{if } i \le j \\ 0, & \text{if } i > j \end{cases} \quad (3) $$
with $N_S$ being the number of samples per reference cycle assumed $2\pi$ in this paper.

The vibratory contribution of this gearbox (the subsystem) cannot be directly obtained by summing up the SA of both shafts. In fact, as seen in the Figures 2a – 2b, there are several orders in common between the shafts (i.e. 6 and its multiples 12, 18, 24, etc.). These common orders represent the frequency interferences of the subsystem components. The sum of both SA leads to incorrect estimation, which manifests through higher energy at the shared frequencies in the spectrum (see in Figures 2c – 2d). To be noted that this interference is not meant to be completely removed for the subsystem but rather its replication because it can provide important information about its global health state. In fact, these shared harmonics, representing the meshing harmonics and their energies, are widely used as efficient diagnostic indicators in gearboxes.

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Order Spectra for: a- (Top left) SA of input shaft, b- (Top right) SA of output shaft, c- (Bottom left) Sum of both SAs, d- (Bottom right) Comparison between the sum of both SAs and the real subsystem SA.</figcaption></figure>

Another interesting issue addressed in this paper is the localization of a fault. This needs to be achieved by considering the exclusive contribution of each shaft so the faulty one does not affect the others. As seen in Figures 2a – 2b, the SAs of the two shafts present an interference at the sixth order and its multiples, which could mislead the identification of the faulty shaft. Therefore, it is essential to eliminate the remaining interferences between each source to evaluate the health indicator applied on the exclusive contribution of each one. This paper goes beyond this simple example to the case of more complex systems, where multiple shafts with different rotating frequencies having multiple common frequencies in play.

### 3. Proposed methodology

The proposed methodology in this paper is designed to isolate the desired vibratory contribution of a given subsystem in a complex mechanical system using the SA, as well as the exclusive vibration response of each of the subset component. On the one hand, this is achieved by using the set theory to identify the common frequencies that represent the interferences between each line and eliminate its replication as seen in [4]. On the other hand, it can be implemented by the computation of the least common multiple (LCM) of each pair of orders.

Figure 3 illustrates this theory, where multiple sets are considered, having the set “$a_1$” representing the reference shaft. Each set corresponds to the orders ensemble of each shaft’s contribution resulted from its SA. By comparing the orders of shafts $a_1$ and $a_2$, it can be seen that some interferences exist (hatched part) between these two sets and others that are jointly common with shaft $a_3$ (green triangles).

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Illustration of the sets of SA orders with the different interferences.</figcaption></figure>

#### 3.1. Subsystems Isolation

Generally, one can consider N shafts with the corresponding set of fundamental orders $A = \{\alpha_i \in \mathbb{R} / i = 1, 2, \dots, N\}$ constituting a certain subsystem. The goal is to find the signal associated with this specific subsystem, which encompasses the total set of constituent lines excluding the replications of the interference between each considered line. This leads to the following equation:

$$ \sum_{\alpha_i \in A} m_{\alpha_i}(\theta) = s(\theta) + \Delta(\theta) \quad (2) $$

With $m_{\alpha_i}(\theta)$ being the SA of the angular signal $x(\theta)$ with respect to the order $\alpha_i$, $s(\theta)$ being the subsystem deterministic signal and $\Delta(\theta)$ being the replication contribution caused by the interfering components.

Taking the replication to be the unique interference between each two sets (i.e. $a_1 \cap a_2$ without $a_2 \cap a_1$; the interference is not totally eliminated between these two sets, but rather the replication of the interference.

The replication set requires a more generic definition because common orders between physically unlinked shafts can still exist. In addition, this interference will repeat for integer multiples of the common order, leading to two major problems that prompted the proposal of two solutions:

* Calculate the LCM between each pair of shafts to find the fundamental common order.

* Compute the synchronous average relative to the LCM order found to take into consideration all its multiples.

In order to calculate the corresponding LCM of all common orders for each shaft, a matrix of common orders (MCO) is proposed. To be able to free the replication of the interference between each two shafts without the elimination of the latter, this matrix must be an upper triangular matrix whose diagonal corresponds to the common order between two consecutive shafts. By taking the LCM between each pair of shafts, only the lowest common order is considered, eliminating the redundant frequencies when isolating the subsystem. Thus, the computation of the following matrix of common orders (MCO):

$$ M(i, j) = \begin{cases} \text{LCM}(\alpha_i, \alpha_{j+1}), & \text{if } i \le j \\ 0, & \text{if } i > j \end{cases} \quad (3) $$
with LCM(a,b) = $\frac{a \times b}{GCD(a,b)}$, i and j iterating from 1 → N-1,
therefore M would be of size (N-1,N-1).

The greatest common divisor (GCD) can be computed using
several methods, i.e. the Euclidean algorithm that considers a k^th^
step that performs a division with remainder to find the quotient
q_k and remainder r_k so that: r_k-2 = q_k r_k-1 + r_k with
r_k-1 > r_k ≥ 0. In the initial step k = 0, the remainders are set to
r_2 = a and r_1 = b, thus the algorithm must eventually produce
a zero remainder r_N = 0. Therefore, the final nonzero remainder
is the greatest common divisor of a and b : GCD(a,b) = r_{N-1}.

Once the MCO is calculated, it is used to find the SAs with
respect to each found order and subtract it from the total sum of
SA of all considered shafts. This ensures that the final result
contains a single interference between each shaft, without
modifying the structure of the total set. In conclusion, the
formula (3) becomes:

$$
s(\theta) = \sum_{\alpha_i \in A} m_{\alpha_i}(\theta) - \sum_{i} \sum_{j} m_{M(i,j)}(\theta) \quad (5)
$$

with $m_{M(i,j)}(\theta)$ being the SA of the angular signal $x(\theta)$ with
respect to the order found at element $M(i,j)$.

3.2. Exclusive contributions extraction

Furthermore, to extract the exclusive contributions of each
shaft from the isolated subsystem, the same set theory and the
matrix MCO are used to find the following:

$$
s(\theta) = \sum_{\alpha_i \in A} \left( c_{\alpha_i}(\theta) + \varepsilon_{\alpha_i}(\theta) \right) \quad (6)
$$

with $c_{\alpha_i}(\theta)$ being the exclusive contribution of shaft of order $\alpha_i$.
The remaining interference $\varepsilon_{\alpha_i}(\theta)$ represents all common
contributions between the shaft of order $\alpha_i$ and all remaining
ones as seen in the following:

$$
\epsilon_{\alpha_i}(\theta) = \sum_j s_{M(i,j)}(\theta) \quad (7)
$$

with $s_{M(i,j)}(\theta)$ being the SA of the subsystem deterministic
signal $s(\theta)$ with respect to the order found at element $M(i,j)$
and $i$ being the index of $\alpha_i$ in the set $A$. One can identify the
following:

$$
c_{\alpha_i}(\theta) = s_{\alpha_i}(\theta) - \epsilon_{\alpha_i}(\theta) \quad (8)
$$

with $s_{\alpha_i}(\theta)$ being the SA of the subsystem deterministic signal
$s(\theta)$ with respect to order $\alpha_i$.

In conclusion, each shaft’s exclusive contribution is extracted
as follows:

$$
c_{\alpha_i}(\theta) = s_{\alpha_i}(\theta) - \sum_j s_{M(i,j)}(\theta) \quad (9)
$$

In summary, the proposed methodology in this paper uses the
set theory and the calculation of MCO to isolate each subsystem
within a complex mechanical system. By eliminating the
redundant frequencies present in the synchronous averages of the
shafts' orders, one can finally obtain the exclusive contribution of
each component.

4. Application

4.1. Data description

In this section, an experimental example is considered to
understand how to apply such method on aircraft subsystems. It
concerns a vibration signal acquired during a ground test
campaign on the CFM56 turbojet engine (Figure 4), which is a
civil aircraft engine with two main shafts low and high pressure,
and an accessory gearbox linked to the high-pressure shaft. The
acquired signal is obtained from the accelerometer called Acc2
with a sampling frequency of 50,000 Hertz and a time duration of
200 seconds. The acquisition was performed during a steady state
at idle followed by a slow acceleration up to full power. For this
application, the desired subsystem to isolate is the accessory
gearbox.

The kinematics of the gearbox and the rotating speeds of its
shafts are described in Figure 5. The corresponding signal
provided by the Acc2 accelerometer (located near the shafts L4
and L5 of the AGB) and the rotating frequency of shaft L1 is
depicted on Figure 6. This signal has been initially provided for a
contest during the International Conference Surveillance 8,
October 20 and 21, 2015, at the Roanne Institute of technology,
France. More details about the signal can be found in [22].

Accordingly, the orders are structured in a well-designed table
to clearly distinguish the order of each shaft. A table was
provided during the contest for the same specific reason by
listing all characteristic frequencies of the gearbox components.
In this embodiment, 12 frequencies are given and associated with
the rotation of the 12 shaft lines of the CFM56 AGB. Table 1
summarizes the dynamics of the AGB with the 12 frequencies of
the set f reference of shaft frequencies referenced by N2. It is
specified that these frequencies are obtained with a rotational
speed of shaft N2 of 10,000 rpm. The orders considered in this
kinematic correspond to the Tooth line F1(N2). Therefore, the
goal is to find all the interferences between the HP shaft and all
the others, then RDS and all the others, etc. In addition, it is
necessary to analyze the different sources that interfere jointly
(an interference might exist for several sources even if they are
not physically linked).

Consequently, the next step would be to calculate the MCO
with the corresponding orders. Figure 7 shows the resulting
matrix after including all orders provided by the Table 1. It is
essential to note that a non-integral number is mostly anticipated
for each order (aside the reference one). However, the
synchronous average takes an integer number of samples as the
period to average the signal. As a result, it is possible to find a
common order between each one. It can be identified in the
matrix above that the diagonal presents the meshes of the gears
located on the consecutive shafts or a divisor of this mesh in case
a LCM smaller than the mesh was found.

Afterwards, using the given instantaneous angular speed, the
signal undergoes resampling into its angular domain having the
corresponding trajectory with its order spectrum in Figure 8a.
The first phase of the proposed method extracts the contribution
of the AGB by utilizing MCO to eliminate the replication of
interferences among the subsystem lines. As a result, Figures 8b
and 8c depict the signals of the AGB (deterministic component)
and the residual, along with each corresponding order spectra.
This deterministic signal corresponds to the periodic part of all
the shafts of the gearbox including the common parts (the
interferences) between the different lines without any replication.
with $LCM(a,b) = \frac{a \times b}{GCD(a,b)}$, i and j iterating from $1 \to N-1$,
therefore $M$ would be of size $(N-1,N-1)$.

The greatest common divisor (GCD) can be computed using several methods, i.e. the Euclidean algorithm that considers a $k^{th}$ step that performs a division with remainder to find the quotient $q_k$ and remainder $r_k$ so that: $r_{k-2} = q_k r_{k-1} + r_k$ with $r_{k-1} > r_k \ge 0$. In the initial step $k=0$, the remainders are set to $r_{-2} = a$ and $r_{-1} = b$, thus the algorithm must eventually produce a zero remainder $r_N = 0$. Therefore, the final nonzero remainder is the greatest common divisor of $a$ and $b$: $GCD(a,b) = r_{N-1}$.

Once the MCO is calculated, it is used to find the SAs with respect to each found order and subtract it from the total sum of SA of all considered shafts. This ensures that the final result contains a single interference between each shaft, without modifying the structure of the total set. In conclusion, the formula (3) becomes:

$$s(\theta) = \sum_{\alpha_i \in A} m_{\alpha_i}(\theta) - \sum_{i} \sum_{j} m_{M(i, j)}(\theta) \quad (5)$$

with $m_{M(i, j)}(\theta)$ being the SA of the angular signal $x(\theta)$ with respect to the order found at element $M(i, j)$.

## 3.2. Exclusive contributions extraction

Furthermore, to extract the exclusive contributions of each shaft from the isolated subsystem, the same set theory and the matrix MCO are used to find the following:

$$s(\theta) = \sum_{\alpha_i \in A} (c_{\alpha_i}(\theta) + \varepsilon_{\alpha_i}(\theta)) \quad (6)$$

with $c_{\alpha_i}(\theta)$ being the exclusive contribution of shaft of order $\alpha_i$. The remaining interference $\varepsilon_{\alpha_i}(\theta)$ represents all common contributions between the shaft of order $\alpha_i$ and all remaining ones as seen in the following:

$$\varepsilon_{\alpha_i}(\theta) = \sum_j s_{M(i, j)}(\theta) \quad (7)$$

with $s_{M(i, j)}(\theta)$ being the SA of the subsystem deterministic signal $s(\theta)$ with respect to the order found at element $M(i, j)$ and $i$ being the index of $\alpha_i$ in the set $A$. One can identify the following:

$$c_{\alpha_i}(\theta) = s_{\alpha_i}(\theta) - \varepsilon_{\alpha_i}(\theta) \quad (8)$$

with $s_{\alpha_i}(\theta)$ being the SA of the subsystem deterministic signal $s(\theta)$ with respect to order $\alpha_i$.

In conclusion, each shaft's exclusive contribution is extracted as follows:

$$c_{\alpha_i}(\theta) = s_{\alpha_i}(\theta) - \sum_j s_{M(i, j)}(\theta) \quad (9)$$

In summary, the proposed methodology in this paper uses the set theory and the calculation of MCO to isolate each subsystem within a complex mechanical system. By eliminating the redundant frequencies present in the synchronous averages of the shafts' orders, one can finally obtain the exclusive contribution of each component.

# 4. Application

## 4.1. Data description

In this section, an experimental example is considered to understand how to apply such method on aircraft subsystems. It concerns a vibration signal acquired during a ground test campaign on the CFM56 turbojet engine (Figure 4), which is a civil aircraft engine with two main shafts low and high pressure, and an accessory gearbox linked to the high-pressure shaft. The acquired signal is obtained from the accelerometer called Acc2 with a sampling frequency of 50,000 Hertz and a time duration of 200 seconds. The acquisition was performed during a steady state at idle followed by a slow acceleration up to full power. For this application, the desired subsystem to isolate is the accessory gearbox.

The kinematics of the gearbox and the rotating speeds of its shafts are described in Figure 5. The corresponding signal provided by the Acc2 accelerometer (located near the shafts L4 and L5 of the AGB) and the rotating frequency of shaft L1 is depicted on Figure 6. This signal has been initially provided for a contest during the International Conference Surveillance 8, October 20 and 21, 2015, at the Roanne Institute of technology, France. More details about the signal can be found in [22].

Accordingly, the orders are structured in a well-designed table to clearly distinguish the order of each shaft. A table was provided during the contest for the same specific reason by listing all characteristic frequencies of the gearbox components. In this embodiment, 12 frequencies are given and associated with the rotation of the 12 shaft lines of the CFM56 AGB. Table 1 summarizes the dynamics of the AGB with the 12 frequencies of the set $f_{reference}$ of shaft frequencies referenced by $N_2$. It is specified that these frequencies are obtained with a rotational speed of shaft $N_2$ of 10,000 rpm. The orders considered in this kinematic correspond to the Tooth line $F_1(N_2)$. Therefore, the goal is to find all the interferences between the HP shaft and all the others, then RDS and all the others, etc. In addition, it is necessary to analyze the different sources that interfere jointly (an interference might exist for several sources even if they are not physically linked).

Consequently, the next step would be to calculate the MCO with the corresponding orders. Figure 7 shows the resulting matrix after including all orders provided by the Table 1. It is essential to note that a non-integral number is mostly anticipated for each order (aside the reference one). However, the synchronous average takes an integer number of samples as the period to average the signal. As a result, it is possible to find a common order between each one. It can be identified in the matrix above that the diagonal presents the meshes of the gears located on the consecutive shafts or a divisor of this mesh in case a LCM smaller than the mesh was found.

Afterwards, using the given instantaneous angular speed, the signal undergoes resampling into its angular domain having the corresponding trajectory with its order spectrum in Figure 8a. The first phase of the proposed method extracts the contribution of the AGB by utilizing MCO to eliminate the replication of interferences among the subsystem lines. As a result, Figures 8b and 8c depict the signals of the AGB (deterministic component) and the residual, along with each corresponding order spectra. This deterministic signal corresponds to the periodic part of all the shafts of the gearbox including the common parts (the interferences) between the different lines without any replication.
<figure><img src="image_6.png" /><figcaption>Figure 4. General overview of the engine and the accessory gearbox.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5. Kinematics of the gearbox.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6. (Top) Evolution of the gas generator shaft speed N1 in Hertz and (bottom) the raw vibration signal measured on the gas generator casing.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 4.** General overview of the engine and the accessory gearbox.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Kinematics of the gearbox.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** (Top) Evolution of the gas generator shaft speed N1 in Hertz and (bottom) the raw vibration signal measured on the gas generator casing.</figcaption></figure>
<table><caption>**Table 1.** Characteristic frequencies of the gearbox.</caption><thead><tr><th colspan="2">N2 speed (rpm)</th><th colspan="2">Line 1</th><th colspan="2">Line 2</th><th colspan="2">Line 3</th><th colspan="2">N2 tacho: 44</th><th colspan="2">Line 4</th><th colspan="2">Line 5</th><th colspan="2">Line 6</th><th colspan="2">Line 7</th><th colspan="2">Line 8</th><th colspan="2">Line 9</th><th colspan="2">Line 10</th><th colspan="2">Line 11</th></tr><tr><th colspan="2">10000</th><th>HP Shaft</th><th>RDS</th><th>Horizontal Drive Shaft</th><th>EEC Alternator</th><th>Handcranking</th><th>Starter</th><th>Fuel Pump</th><th>Intermediate</th><th>IDG</th><th>Intermediate</th><th>Hydro Pump</th><th>Lubrication Unit</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th><th>aval</th></tr></thead><tbody><tr><th>Number of teeth</th><td>47</td><td>35</td><td>31</td><td>32</td><td>47</td><td>47</td><td>47</td><td>47</td><td>62</td><td>62</td><td>61</td><td>31</td><td>73</td><td>73</td><td>41</td><td>41</td><td>55</td><td>55</td><td>73</td><td>44</td><td>73</td><td>73</td><td>44</td><td>44</td><td>44</td></tr><tr><th>Rotating Speed (rpm)</th><td>10000</td><td>13429</td><td>13009</td><td>13009</td><td>13009</td><td>9862</td><td>10023</td><td>4256</td><td>7579</td><td>5649</td><td>4256</td><td>2566</td><td>4256</td><td>13429</td><td>13429</td><td>13009</td><td>13009</td><td>13009</td><td>13009</td><td>13009</td><td>9862</td><td>9862</td><td>10023</td><td>10023</td><td>10023</td></tr><tr><th>F1 tooth (Hz)</th><td>167</td><td>224</td><td>224</td><td>217</td><td>217</td><td>164</td><td>164</td><td>167</td><td>71</td><td>126</td><td>126</td><td>94</td><td>94</td><td>71</td><td>71</td><td>43</td><td>43</td><td>71</td><td>71</td><td>43</td><td>43</td><td>71</td><td>71</td><td>71</td></tr><tr><th>F1 tooth (N2)</th><td>1.0000</td><td>1.3429</td><td>1.3429</td><td>1.3009</td><td>1.3009</td><td>1.3009</td><td>1.3009</td><td>0.9862</td><td>0.9862</td><td>0.4256</td><td>0.4256</td><td>0.7579</td><td>0.7579</td><td>0.5649</td><td>0.5649</td><td>0.4256</td><td>0.4256</td><td>0.2566</td><td>0.2566</td><td>0.4256</td><td>0.4256</td><td>0.2566</td><td>0.2566</td><td>0.4256</td></tr><tr><th>Gear Mesh Frequency (Hz)</th><td>7833</td><td>6938</td><td>10190</td><td>10190</td><td>10190</td><td>10190</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>3121</td><td>3121</td><td>47.0000</td><td>41.6286</td><td>61.1420</td><td>61.1420</td><td>61.1420</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>18.7284</td><td>18.7284</td><td>18.7284</td></tr><tr><th>Gear Mesh (mult N2)</th><td>4.8</td><td>7.0</td><td>216.8</td><td>3.5</td><td>2.7</td><td>2.3</td><td>1.7</td><td>2.3</td><td>1.3</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td></tr><tr><th>F coincidence (Hz)</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Figure 7.** Resulting MCO values (in black) with respect to each order (in blue).</caption><thead><tr><th>α<sub>1</sub></th><th>α<sub>2</sub>+1</th><th>1.3429</th><th>1.3009</th><th>1.3009</th><th>0.9862</th><th>1.0023</th><th>0.4256</th><th>0.7579</th><th>0.5649</th><th>0.4256</th><th>0.2566</th><th>0.4256</th></tr></thead><tbody><tr><th>1.0000</th><td>47.0000</td><td>147.0000</td><td>147.0000</td><td>285.0000</td><td>1.0000</td><td>63.0000</td><td>72.0000</td><td>274.0000</td><td>63.0000</td><td>137.0000</td><td>63.0000</td></tr><tr><th>1.3429</th><td>0</td><td>41.6286</td><td>41.6286</td><td>63.1143</td><td>209.4857</td><td>77.8857</td><td>76.5429</td><td>256.4857</td><td>77.8857</td><td>22.8286</td><td>77.8857</td></tr><tr><th>1.3009</th><td>0</td><td>0</td><td>1.3009</td><td>61.1420</td><td>61.1420</td><td>300.5062</td><td>165.2134</td><td>98.8679</td><td>300.5062</td><td>92.3634</td><td>300.5062</td></tr><tr><th>1.3009</th><td>0</td><td>0</td><td>0</td><td>61.1420</td><td>61.1420</td><td>300.5062</td><td>165.2134</td><td>98.8679</td><td>300.5062</td><td>92.3634</td><td>300.5062</td></tr><tr><th>0.9862</th><td>0</td><td>0</td><td>0</td><td>0</td><td>61.1420</td><td>80.8652</td><td>553.2362</td><td>54.2388</td><td>80.8652</td><td>233.7201</td><td>80.8652</td></tr><tr><th>1.0023</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>11.0256</td><td>31.0721</td></tr><tr><th>0.4256</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>31.0721</td><td>0.4256</td><td>18.7284</td><td>0.4256</td></tr><tr><th>0.7579</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>31.0721</td><td>65.9336</td><td>31.0721</td></tr><tr><th>0.5649</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>109.0350</td><td>31.0721</td></tr><tr><th>0.4256</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>18.7284</td><td>0.4256</td></tr><tr><th>0.2566</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>18.7284</td></tr></tbody></table>

Hence obtaining the contribution of each shaft without any remaining interference using the second phase of the proposed approach with the same MCO computed before. Regarding the

12 lines constituting the AGB, the different SAs are visualized in Figure 9 and their order spectra in Figure 10.

<figure><img src="image_9.png" /><figcaption>**Figure 8.**</figcaption></figure>

a - (Top Center) CFM56 angular signal and its order spectra

b - (Bottom Left) Exclusive CFM56 AGB deterministic Signal and its order spectra

c - (Bottom Right) CFM56 signal without AGB and its order spectra

## 4.2. Results and discussion

The proposed methodology successfully extracts the AGB's deterministic signal and its constituent lines while eliminating the interferences and its replication for subsystem contribution. However, as it is seen in the MCO (Figure 7) and in the extracted

lines with their spectra (Figures 9-10), it is possible for some shafts to share the same order. To overcome this issue, multiple input sensors can be employed to identify the exclusive contribution of each shaft while using the same proposed methodology.
<table><caption>**Table 1.** Characteristic frequencies of the gearbox.</caption><thead><tr><th colspan="10">N2 speed (rpm)</th></tr><tr><th colspan="10">10000</th></tr><tr><th rowspan="3">Number of teeth</th><th colspan="2">Line 1</th><th colspan="2">Line 2</th><th colspan="2">Line 3</th><th colspan="2">N2 tacho: 44</th><th colspan="2">Line 4</th><th colspan="2">Line 5</th><th colspan="2">Line 6</th><th colspan="2">Line 7</th><th colspan="2">Line 8</th><th colspan="2">Line 9</th><th colspan="2">Line 10</th><th colspan="2">Line 11</th></tr><tr><th colspan="2">HP Shaft</th><th colspan="2">RDS</th><th colspan="2">Horizontal Drive Shaft</th><th colspan="2">EEC Alternator</th><th colspan="2">Handcranking</th><th colspan="2">Starter</th><th colspan="2">Fuel Pump</th><th colspan="2">Intermediate</th><th colspan="2">IDG</th><th colspan="2">Intermediate</th><th colspan="2">Hydro Pump</th><th colspan="2">Lubrication Unit</th></tr><tr><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th><th>aval</th><th>amont</th></tr></thead><tbody><tr><td>10000</td><td>47</td><td>35</td><td>31</td><td>32</td><td>47</td><td>47</td><td>47</td><td>62</td><td>62</td><td>61</td><td>31</td><td>73</td><td>73</td><td>41</td><td>41</td><td>55</td><td>55</td><td>73</td><td>44</td><td>73</td><td>73</td><td>44</td></tr><tr><td>Rotating Speed (rpm)</td><td>13429</td><td>13429</td><td>13429</td><td>13429</td><td>13429</td><td>13429</td><td>13429</td><td>9862</td><td>9862</td><td>10023</td><td>10023</td><td>4256</td><td>4256</td><td>7579</td><td>7579</td><td>5649</td><td>5649</td><td>4256</td><td>4256</td><td>2566</td><td>2566</td><td>4256</td></tr><tr><td>F1 tooth (Hz)</td><td>167</td><td>224</td><td>224</td><td>217</td><td>217</td><td>217</td><td>217</td><td>164</td><td>164</td><td>167</td><td>167</td><td>71</td><td>71</td><td>126</td><td>126</td><td>94</td><td>94</td><td>71</td><td>71</td><td>43</td><td>43</td><td>71</td></tr><tr><td>F1 tooth (N2)</td><td>1.0000</td><td>1.3429</td><td>1.3429</td><td>1.3429</td><td>1.3429</td><td>1.3429</td><td>1.3429</td><td>0.9862</td><td>0.9862</td><td>1.0023</td><td>1.0023</td><td>0.4256</td><td>0.4256</td><td>0.7579</td><td>0.7579</td><td>0.5649</td><td>0.5649</td><td>0.4256</td><td>0.4256</td><td>0.2566</td><td>0.2566</td><td>0.4256</td></tr><tr><td>Gear Mesh Frequency (Hz)</td><td>7833</td><td>6938</td><td>10190</td><td>10190</td><td>10190</td><td>10190</td><td>10190</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>5179</td><td>3121</td><td>3121</td><td>3121</td><td>3121</td><td>3121</td><td>3121</td></tr><tr><td>Gear Mesh (mult N2)</td><td>47.0000</td><td>41.6286</td><td>61.1420</td><td>61.1420</td><td>61.1420</td><td>61.1420</td><td>61.1420</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td></tr><tr><td>F coincidence (Hz)</td><td>4.8</td><td>7.0</td><td>216.8</td><td>3.5</td><td>2.7</td><td>2.3</td><td>1.7</td><td>2.3</td><td>1.3</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td></tr></tbody></table>

<table><caption>**Figure 7.** Resulting MCO values (in black) with respect to each order (in blue).</caption><thead><tr><th>α<sub>1</sub></th><th>α<sub>2</sub>+1</th><th>1.3429</th><th>1.3009</th><th>1.3009</th><th>0.9862</th><th>1.0023</th><th>0.4256</th><th>0.7579</th><th>0.5649</th><th>0.4256</th><th>0.2566</th><th>0.4256</th></tr></thead><tbody><tr><td>1.0000</td><td>47.0000</td><td>147.0000</td><td>147.0000</td><td>285.0000</td><td>1.0000</td><td>63.0000</td><td>72.0000</td><td>274.0000</td><td>63.0000</td><td>137.0000</td><td>63.0000</td></tr><tr><td>1.3429</td><td>0</td><td>41.6286</td><td>41.6286</td><td>63.1143</td><td>209.4857</td><td>77.8857</td><td>76.5429</td><td>256.4857</td><td>77.8857</td><td>22.8286</td><td>77.8857</td></tr><tr><td>1.3009</td><td>0</td><td>0</td><td>1.3009</td><td>61.1420</td><td>61.1420</td><td>300.5062</td><td>165.2134</td><td>98.8679</td><td>300.5062</td><td>92.3634</td><td>300.5062</td></tr><tr><td>1.3009</td><td>0</td><td>0</td><td>0</td><td>61.1420</td><td>61.1420</td><td>300.5062</td><td>165.2134</td><td>98.8679</td><td>300.5062</td><td>92.3634</td><td>300.5062</td></tr><tr><td>0.9862</td><td>0</td><td>0</td><td>0</td><td>0</td><td>61.1420</td><td>80.8652</td><td>553.2362</td><td>54.2388</td><td>80.8652</td><td>233.7201</td><td>80.8652</td></tr><tr><td>1.0023</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>31.0721</td><td>11.0256</td><td>31.0721</td></tr><tr><td>0.4256</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>31.0721</td><td>0.4256</td><td>18.7284</td><td>0.4256</td></tr><tr><td>0.7579</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>31.0721</td><td>65.9336</td><td>31.0721</td></tr><tr><td>0.5649</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>31.0721</td><td>109.0350</td><td>31.0721</td></tr><tr><td>0.4256</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>18.7284</td><td>0.4256</td></tr><tr><td>0.2566</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>18.7284</td></tr></tbody></table>

Hence obtaining the contribution of each shaft without any remaining interference using the second phase of the proposed approach with the same MCO computed before. Regarding the

12 lines constituting the AGB, the different SAs are visualized in Figure 9 and their order spectra in Figure 10.

<figure><img src="image_10.png" /><figcaption>**Figure 8.**</figcaption></figure>

a - (Top Center) CFM56 angular signal and its order spectra

b - (Bottom Left) Exclusive CFM56 AGB deterministic Signal and its order spectra

c - (Bottom Right) CFM56 signal without AGB and its order spectra

## 4.2. Results and discussion

The proposed methodology successfully extracts the AGB's deterministic signal and its constituent lines while eliminating the interferences and its replication for subsystem contribution. However, as it is seen in the MCO (Figure 7) and in the extracted

lines with their spectra (Figures 9-10), it is possible for some shafts to share the same order. To overcome this issue, multiple input sensors can be employed to identify the exclusive contribution of each shaft while using the same proposed methodology.
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Figure 9. The signals of the 12 sources extracted from the AGB of the CFM56.</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Figure 9. The signals of the 12 sources extracted from the AGB of the CFM56.</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>
<figure><img src="image_14.png" /><figcaption>**Figure 10.** The order spectra of the 12 lines.</figcaption></figure>

**5. Conclusion**

This paper presents a new approach for isolating the subsystems in a complex mechanical system, specifically an aircraft accessory gearbox. The method leverages synchronous averaging, set theory, and the matrix of common orders to eliminate redundant frequencies and uncover the unique contributions of each subsystem. This allows for a cleaner analysis for having each contribution well isolated from interferences, as well as the ability to pinpoint specific issues within a subsystem. Overall, the proposed method can improve the maintenance and global performance of mechanical systems while providing a good understanding of the behavior of complex mechanical systems.

**Acknowledgments**

F. Karkafi gratefully acknowledges the European Commission for its support of the Marie Skłodowska Curie program through the ETN MOIRA project (GA 955681).

**6. References**

1. R. B. Randall and J. Antoni (2011). Rolling element bearing diagnostics—A tutorial. Mech. Syst. Signal Process., vol. 25, no. 2, pp. 485–520.

2. S. Braun, (1986), Mechanical signature analysis: theory and applications.

3. J. Antoni and R. B. Randall (2002). Differential diagnosis of gear and bearing faults. J Vib Acoust, vol. 124, no. 2, pp. 165–171.

4. J. Antoni, F. Bonnardot, A. Raad, M. El Badaoui (2004). Cyclostationary modelling of rotating machine vibration signals, Mechanical Systems and Signal Processing 18 1285–1314.

5. J. Antoni (2009). Cyclostationarity by examples. Mechanical Systems and Signal Processing, 23 987–1036.

6. J. Dion, C. Stephan, G. Chevallier, H. Festjens (2013). Tracking and removing modulated sinusoidal components: A solution based on the kurtosis and the Extended Kalman Filter. Mechanical Systems and Signal Processing, 38 (2), pp.428-439.

7. M. Bai, J. Huang, M. Hong, F. Su. (2005). Fault diagnosis of rotating machinery using an intelligent order tracking system. J. Sound Vib. 280, 699–718.

8. J.R. Blough (2003). Development and analysis of time variant discrete Fourier transform order tracking. Mechanical Systems and Signal Processing 17, 1185–1199.

9. Y. Guo, K.K. Tan (2009). Order-crossing removal in Gabor order tracking by independent component analysis. J. Sound Vib. 325, 471–488.

10. M.-C. Pan, C.-C. Chiu (2006). Investigation on improved Gabor order tracking technique and its applications. J. Sound Vib. 295, 810–826.

11. M.-C. Pan, S.-W. Liao, C.-C. Chiu (2007). Improvement on Gabor order tracking and objective comparison with Vold-Kalman filtering order tracking. Mechanical Systems and Signal Processing 21, 653–667.

12. H. Vold, J. Leuridan (1993). High Resolution Order Tracking at Extreme Slew Rates, Using Kalman Tracking Filters. SAE Paper No. 931288.

13. H. Vold, H. Herlufsen (1997). Multi axle order tracking with the Vold-Kalman tracking filter. Sound Vib. 31 (5), 30–34.

14. Y. Guo, K.K. Tan (2010). High efficient crossing-order decoupling in Vold-Kalman filtering order tracking based on independent component analysis. Mechanical Systems and Signal Processing 24, 1756–1766.

15. M.-C. Pan, C.-X. Wu (2007). Adaptive Vold-Kalman filtering order tracking. Mechanical Systems and Signal Processing 21, 2957–2969.

16. K.S. Wang, P.S. Heyns (2011). The combined use of order tracking techniques for enhanced Fourier analysis of order components. Mechanical Systems and Signal Processing 25, 803–811.

17. P. D. McFadden (1989). Interpolation techniques for time domain averaging of gear vibration. Mechanical Systems and Signal Processing vol. 3, no. 1, pp. 87–97.
<figure><img src="image_15.png" /><figcaption>Figure 10. The order spectra of the 12 lines.</figcaption></figure>

**5. Conclusion**

This paper presents a new approach for isolating the subsystems in a complex mechanical system, specifically an aircraft accessory gearbox. The method leverages synchronous averaging, set theory, and the matrix of common orders to eliminate redundant frequencies and uncover the unique contributions of each subsystem. This allows for a cleaner analysis for having each contribution well isolated from interferences, as well as the ability to pinpoint specific issues within a subsystem. Overall, the proposed method can improve the maintenance and global performance of mechanical systems while providing a good understanding of the behavior of complex mechanical systems.

**Acknowledgments**

F. Karkafi gratefully acknowledges the European Commission for its support of the Marie Skłodowska Curie program through the ETN MOIRA project (GA 955681).

**6. References**

1. R. B. Randall and J. Antoni (2011). Rolling element bearing diagnostics—A tutorial. Mech. Syst. Signal Process., vol. 25, no. 2, pp. 485–520.

2. S. Braun, (1986), Mechanical signature analysis: theory and applications.

3. J. Antoni and R. B. Randall (2002). Differential diagnosis of gear and bearing faults. J Vib Acoust, vol. 124, no. 2, pp. 165–171.

4. J. Antoni, F. Bonnardot, A. Raad, M. El Badaoui (2004). Cyclostationary modelling of rotating machine vibration signals, Mechanical Systems and Signal Processing 18 1285–1314.

5. J. Antoni (2009). Cyclostationarity by examples. Mechanical Systems and Signal Processing, 23 987–1036.

6. J. Dion, C. Stephan, G. Chevallier, H. Festjens (2013). Tracking and removing modulated sinusoidal components: A solution based on the kurtosis and the Extended Kalman Filter. Mechanical Systems and Signal Processing, 38 (2), pp.428-439.

7. M. Bai, J. Huang, M. Hong, F. Su. (2005). Fault diagnosis of rotating machinery using an intelligent order tracking system. J. Sound Vib. 280, 699–718.

8. J.R. Blough (2003). Development and analysis of time variant discrete Fourier transform order tracking. Mechanical Systems and Signal Processing 17, 1185–1199.

9. Y. Guo, K.K. Tan (2009). Order-crossing removal in Gabor order tracking by independent component analysis. J. Sound Vib. 325, 471–488.

10. M.-C. Pan, C.-C. Chiu (2006). Investigation on improved Gabor order tracking technique and its applications. J. Sound Vib. 295, 810–826.

11. M.-C. Pan, S.-W. Liao, C.-C. Chiu (2007). Improvement on Gabor order tracking and objective comparison with Vold-Kalman filtering order tracking. Mechanical Systems and Signal Processing 21, 653–667.

12. H. Vold, J. Leuridan (1993). High Resolution Order Tracking at Extreme Slew Rates, Using Kalman Tracking Filters. SAE Paper No. 931288.

13. H. Vold, H. Herlufsen (1997). Multi axle order tracking with the Vold-Kalman tracking filter. Sound Vib. 31 (5), 30–34.

14. Y. Guo, K.K. Tan (2010). High efficient crossing-order decoupling in Vold-Kalman filtering order tracking based on independent component analysis. Mechanical Systems and Signal Processing 24, 1756–1766.

15. M.-C. Pan, C.-X. Wu (2007). Adaptive Vold-Kalman filtering order tracking. Mechanical Systems and Signal Processing 21, 2957–2969.

16. K.S. Wang, P.S. Heyns (2011). The combined use of order tracking techniques for enhanced Fourier analysis of order components. Mechanical Systems and Signal Processing 25, 803–811.

17. P. D. McFadden (1989). Interpolation techniques for time domain averaging of gear vibration. Mechanical Systems and Signal Processing vol. 3, no. 1, pp. 87–97.
18. F. Bonnardot, M. El Badaoui, R. B. Randall, J. Daniere, and F. Guillet (2005). Use of the acceleration signal of a gearbox in order to perform angular resampling (with limited speed fluctuation). *Mechanical Systems and Signal Processing* vol. 19, no. 4, pp. 766–785.

19. S. Braun (1975). The extraction of periodic waveforms by time domain averaging. *Acta Acust. United Acust.*, vol. 32, no. 2, pp. 69–77.

20. P. D. McFadden (1987). A revised model for the extraction of periodic waveforms by time domain averaging. *Mechanical Systems and Signal Processing* vol. 1, no. 1, pp. 83–95.

21. E. Bechhoefer, M. Kingsley (2009). A Review of Time Synchronous Average Algorithms. *Annual Conference of the Prognostics and Health Management Society*.

22. J. Antoni, J. Griffaton, H. André, L. David Avendaño-Valencia, F. Bonnardot, O. Cardona-Morales, G. Castellanos-Dominguez, A. Paolo Daga, Q. Leclère, C. Molina Vicuña, D. Quezada Acuña, A. Partogi Ompusunggu, E. F. Sierra-Alonso (2017). Feedback on the Surveillance 8 challenge: Vibration-based diagnosis of a Safran aircraft engine. *Mechanical Systems and Signal Processing* 97, 112–144.
18. F. Bonnardot, M. El Badaoui, R. B. Randall, J. Daniere, and F. Guillet (2005). Use of the acceleration signal of a gearbox in order to perform angular resampling (with limited speed fluctuation). *Mechanical Systems and Signal Processing* vol. 19, no. 4, pp. 766–785.

19. S. Braun (1975). The extraction of periodic waveforms by time domain averaging. *Acta Acust. United Acust.*, vol. 32, no. 2, pp. 69–77.

20. P. D. McFadden (1987). A revised model for the extraction of periodic waveforms by time domain averaging. *Mechanical Systems and Signal Processing* vol. 1, no. 1, pp. 83–95.

21. E. Bechhoefer, M. Kingsley (2009). A Review of Time Synchronous Average Algorithms. *Annual Conference of the Prognostics and Health Management Society*.

22. J. Antoni, J. Griffaton, H. André, L. David Avendaño-Valencia, F. Bonnardot, O. Cardona-Morales, G. Castellanos-Dominguez, A. Paolo Daga, Q. Leclère, C. Molina Vicuña, D. Quezada Acuña, A. Partogi Ompusunggu, E. F. Sierra-Alonso (2017). Feedback on the Surveillance 8 challenge: Vibration-based diagnosis of a Safran aircraft engine. *Mechanical Systems and Signal Processing* 97, 112–144.