# Damage Identification in Rotary Shafts via Vibration Monitoring and Physics-Informed Neural Networks

Vasiliki Panagiotopoulou, Konstantinos Vlachas, Claudio Sbarufatti, Eleni Chatzi

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Vasiliki Panagiotopoulou, Konstantinos Vlachas, Claudio Sbarufatti, Eleni Chatzi. Damage Identification in Rotary Shafts via Vibration Monitoring and Physics-Informed Neural Networks. 11th International Operational Modal Analysis Conference, Institut National de Recherche en Informatique et en Automatique [Inria], May 2025, Rennes, France. hal-05398782

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# Damage Identification in Rotary Shafts via Vibration Monitoring and Physics-Informed Neural Networks

Vasiliki Panagiotopoulou <fnref n="1" />, and Konstantinos Vlachas <fnref n="2" />, and Marco Giglio <fnref n="1" />, and Claudio Sbarufatti <fnref n="1" />, and Eleni Chatzi <fnref n="2" />

## ABSTRACT

Integrating machine learning (ML) with physics-based principles enables virtual models to approximate high-fidelity system behavior while supporting real-time evaluations. However, many data-driven models operate as black boxes, providing predictions that may lack physical interpretability or fail to align with underlying system dynamics. This work develops a physics-informed framework for damage identification in a helicopter transmission shaft, facilitating informed decision-making. It employs two parallel physics-informed neural networks (PINNs): one localizes damage by learning an adjustment vector, while the other estimates severity by inferring stiffness loss and increased excitation forces. Once deployed, the framework processes vibration data for online damage assessment. A numerical case study on a rotary helicopter shaft under ballistic impacts validates its accuracy, demonstrating reliable predictions using limited vibration measurements.

Keywords: damage identification, PINN models, vibration monitoring

## 1. INTRODUCTION

Identification of damage in rotary shafts is a significant challenge in mechanical engineering, particularly in critical systems like aeronautical applications. Vibration monitoring is commonly used to detect damage, as changes in vibration patterns can indicate wear, cracks, or imbalances. However, traditional methods often struggle to distinguish damage types under varying conditions. Machine learning (ML) offers a promising solution by learning patterns from large datasets, improving damage detection accuracy even in complex scenarios [1].

Conventional data-driven ML methods face challenges in damage identification, particularly due to their reliance on sparse sensor networks, which often lead to noisy or incomplete data [2]. Although accurate,

<footnote n="1">¹ Department of Mechanical Engineering, Politecnico di Milano, Via La Masa 1, 20156, Italy, vasiliki.panagiotopoulou@polimi.it</footnote>

<footnote n="2">² Department of Civil, Environmental and Geomatic Engineering, ETH Zurich, Switzerland, vlachas@ibk.baug.ethz.ch</footnote>
these models depend on correlations that lack physical interpretability, making them hard to apply in real-world scenarios where physical reasoning or domain knowledge is essential. Finally, black-box models may fail to generalize or provide predictions that align with physical behavior, leading to unreliable decision making [3].

A plausible solution to the limitations described above is the encoding of physical principles, such as partial differential equations and boundary conditions, into the underlying ML techniques to improve interpretability while reducing violations of physical laws [4]. In this context, Physics-Informed Neural Networks (PINNs) pose as an established alternative [5]. Unlike conventional models, PINNs ensure consistency of the inferred output with the physical laws governing the behavior of the monitored system. This physics-informed approach enables more accurate extrapolation beyond the available data, yielding reliable predictions even in scenarios with sparse data [6]. Recent studies have developed such models to extract valuable insights and applied them to simulate and predict system dynamics, as well as for system identification [7]. Furthermore, PINNs are increasingly being utilized to solve engineering problems related to structural integrity and damage detection [8]. Lastly, they offer a promising approach for failure prediction, leveraging both sensor data and physical principles [9].

This study develops a framework comprising a cluster of two PINNs trained in parallel to localize damage in a shaft and infer the system's state and parameter at the damaged location. The performed inference tasks rely on incoming vibration measurements of accelerations and displacements at the system's bearings. The first network identifies the damage location by learning an adjustment vector that aligns the system's mass, damping, and stiffness matrices with the measured state. The damage location is then determined by identifying the index of an informative metric with the minimum value, which quantifies the alignment between the updated system structure and the predicted adjustment vector. Subsequently, the second network predicts the acceleration and displacement responses at the damage location, as well as a set of parameters that infer the stiffness loss and load increase at the identified damage location.

Once deployed, the PINNs operate as a damage identification model, performing parallel training and delivering predictions of both the damage location and the state-parameter. To enforce physical consistency, dynamic equations derived from a high-fidelity physics-based model are incorporated into the loss functions of the networks. This high-fidelity model simulates the dynamic behavior of the rotary shaft using interconnected finite beam elements and accounts for ballistic impact damage through a dedicated damage model. In this study, the framework is validated using purely numerical data generated by the high-fidelity model, which defines the system's structural response under a range of ballistic impact scenarios. The results confirm the framework's capability to deliver rapid and accurate predictions, even when relying on limited vibration measurements.

## 2. FORWARD MODEL W FOR DYNAMIC ANALYSIS

The structural dynamics of damaged rotary shafts form the basis of the high-fidelity model used here. The rotary shaft of interest can be discretized into $N$ finite elements of length $l_e$ and two nodes per element by applying Timoshenko's beam theory and interconnected finite beam elements. The four degrees of freedom of each of the corresponding $N + 1$ nodes indicated by the subscript $i = 1, 2, ..., N + 1$ include the rotations along each axis, represented by $\zeta_i$, $\psi_i$ for node $i$, and the vertical and lateral translations, represented by $z_i$ and $y_i$, respectively. For the entire (damaged) shaft system, the corresponding spatial discretization used produces the following set of governing equations:

$$M\ddot{x}(t) + (C + \omega G)\dot{x}(t) + Kx(t) = q(t) + w \quad (1)$$

where the nodal response vector is $x(t) = [x_1^T \ x_2^T \ ... \ x_{N+1}^T]^T \in \mathbb{R}^{n \times 1}$, and $x_i(t) = [z_i \ y_i \ \zeta_i \ \psi_i]$ represents the response vector of node $i$, which includes the translational and rotational displacements about the respective stationary axes. The mass is represented by $M \in \mathbb{R}^{n \times n}$, the stiffness matrix by $K \in \mathbb{R}^{n \times n}$, and the gyroscopic and damping matrices of the shaft system by $C, G \in \mathbb{R}^{n \times n}$. The dimensionality of the problem is represented by the variable $n = 4 * (N + 1)$, which indicates the
number of degrees of freedom in the system; t represents the time. A Rayleigh damping formulation is used for every element with $C = \alpha K$, where $a$ is the weighted average factor. The inter-element excitation is applied through vectors $q \in \mathbb{R}^{n \times 1}$ and $w \in \mathbb{R}^{n \times 1}$ that denote the gravity and imbalance force vector respectively. The above mathematical formulation is largely based in [10, 11].

The shaft experiences eccentric stresses due to mass loss and stiffness asymmetry after an impact event, which increases the vibration amplitude and harmonic resonance [12]. This study uses the breathing crack method, which is described in [10], to model the damage. The damaged stiffness matrix is expressed as $K(t) = \tilde{K} - f(t) \cdot K_c$, where $f(t)$ is a cosine function representing the cyclic opening and closing of the crack during rotation, and $\tilde{K}$ is the healthy stiffness matrix. The matrix $K_c \in \mathbb{R}^{n \times n}$ is calculated at the crack location using standard finite element methods, following the analytical derivations in [10]. Following Eq. (1), the equation of motion for the single element $j$ with two nodes at $i = j$ and $i = j + 1$ can be obtained as:

$$m_j \ddot{x}_j(t) + (c_j + \omega g_j) \dot{x}_j(t) + k_j x_j(t) = q_j(t) + w_j \quad (2)$$

where $x_j = [z_i y_i \zeta_i \psi_i z_{i+1} y_{i+1} \zeta_{i+1} \psi_{i+1}]^T \in \mathbb{R}^{8 \times 1}$, $q_j, w_j \in \mathbb{R}^{8 \times 1}$ are the unbalance force and gravity force vectors at the nodes of element $j$, $m_j$ is the element mass, $k_j$ is the element's stiffness matrix, $c_j$ is the damping matrix and $g_j$ the element's gyroscopic matrix. Lastly, the unbalance and gravity forces referring to the degrees of freedom vector $[z_i y_i \zeta_i \psi_i]$ of node $i$ for the damaged (cracked) shaft system are given in [13] as:

$$q_i = \rho A_r l_e e \omega^2 \begin{bmatrix} -\cos(\omega t + \beta_i) & \sin(\omega t + \beta_i) & 0 & 0 \end{bmatrix}^T \quad (3)$$

$$w_i = \rho A_r l_e \begin{bmatrix} 0 & -1/2 & l_e/12 & 0 \end{bmatrix}^T \quad (4)$$

where $e$ is the imbalance quantity, $\rho$ is the density, and $\omega$ is the shaft's rotational speed, and $A_r = A - A_c$ is the residual area of the damaged element, and $A_c$ indicates the damaged area. Finally, the eccentric angle is $\beta_i$. The expressions in Eqs. (3) and (4) define the components of the excitation force vector $f$ acting on the system. The harmonic balance approach can be used to discretize the system's unknown response functions in time through their Fourier components, assuming periodic excitation of the shaft. Specifically, the required solution $x$ of the system can be approximated by a finite Fourier series, which, when substituted into Eq. (5), allows for the transformation of the differential equations into a set of algebraic equations in the frequency domain.

$$M \ddot{x}(t) + (C + \omega G) \dot{x}(t) + K(p_1) x(t) = f(p_2, t) \quad (5)$$

where $K$ and $f$ depend on the parameters $p_1$ and $p_2$, respectively. The parameter $p_1$ is the multiplicative coefficient modeling the stiffness loss due to impact damage, and $p_2$ is the multiplicative coefficient that captures the discrepancy in the amplitude of the excitation force.

### 3. INVERSE MODEL FOR DAMAGE IDENTIFICATION

The damage identification problem can be formulated as an inverse problem with two phases: the first phase, damage localization, focuses on identifying the damaged element of the rotary shaft, while the second phase, damage quantification, aims to assess the extent of the damage, described by the stiffness loss and the load increase. Two feed-forward multi-layer neural networks, $N_s$ and $N_f$, are employed to approximate each phase of the inverse problem. Both networks receive as input the measured acceleration response $\ddot{x}_b$ and displacement response $x_b$ at the two supports -denoted with the superscript b- of the shaft's boundary $\partial\Omega$. The proposed method employs dimensionless outputs to simplify the problem and eliminate the influence of varying unit scales.

First, the network $N_s$ approximates the adjustment vector, denoted as $s \in \mathbb{R}^{N_d \times 1}$, from which it infers the damage position $l_t$, assuming fixed damage extent $\mu$. The respective mathematical formulation of the model $N_s$ is expressed in Eq. (6), where $N_d$ is the total number of potential damaged elements and
$\Theta_s$ consists of the network learnable weights and biases. Initially, the network invokes the model $\mathcal{W}$, which defines the system's structure through the matrices $M_l, C_l$, and $K_l$, as well as the velocity vector $\dot{x}_l$ and the input load $f_l$, corresponding to any potentially damaged element at position $l = 1, \dots, N_d$. At each time step, the adjustment vector $\hat{s}_l$ updates the mass, damping, and stiffness matrices, enabling the evaluation of the equation of motion and the computation of its residual for every possible damage position $l$.

$$ \hat{s}_l = \mathcal{N}_s(\ddot{x}_b, x_b; \Theta_s) \in \mathbb{R}^{N_d \times 1}, \forall l = 1, \dots, N_d \quad (6) $$

An informative metric, denoted as $\hat{J}_l$, is derived from the predicted adjustment vector $\hat{s}_l$ to evaluate its alignment with components of a physical system governed by a set of partial differential equations (PDEs). It quantifies the required structural adjustment—either an increase or decrease—based on state measurements, to closely satisfy the equation of motion at each damage position $l = 1, \dots, N_d$. This metric can be expressed as

$$ \hat{J}_l = \frac{\overline{M_l}}{M_l \cdot \hat{s}_l} + \frac{\overline{C_l}}{C_l \cdot \hat{s}_l} + \frac{\overline{K_l}}{K_l \cdot \hat{s}_l}, \forall l = 1, \dots, N_d \quad (7) $$

where $\overline{M_l}, \overline{C_l}, \overline{K_l}$ represent the mass, damping, and stiffness properties averaged over time, respectively, associated with a candidate damage position $l$. The alignment terms $\overline{M_l} \cdot \hat{s}_l, \overline{C_l} \cdot \hat{s}_l, \overline{K_l} \cdot \hat{s}_l$ measure how well these components of the system align with the predicted adjustment vector.

When $\hat{J}_l$ is evaluated for different scenarios, where damage occurs at various spatial locations $l$, it reflects the reciprocal efficiency of the system's mass, damping, and stiffness properties, which are aligned by the adjustment vector $\hat{s}_l$. A smaller value of $\hat{J}_l$ indicates that the adjustment vector $\hat{s}_l$ is effectively fine-tuning the system properties to satisfy the equation of motion with good alignment. This implies that the system requires minimal adjustment in each of these physical properties, and so the respective damage position $\hat{l}$ is obtained. In contrast, a larger value of $\hat{J}_l$ penalizes systems where the mass, damping, and stiffness components do not effectively participate in the dynamic behavior.

Then, the network $\mathcal{N}_f$ approximates the acceleration response, denoted as $\hat{\dot{x}}_e$, and the displacement response denoted as $\hat{x}_e$, at the damaged element. Additionally, the network infers damage by predicting the excitation force $\hat{f}_e$, and the predicted reduced stiffness value $\hat{k}_e$ at the identified damaged element $e$. This is achieved by updating the following set of parameters $\Theta = \{w, b, [p_1, p_2]\}$, which consist of the network learnable weights $w$ and biases $b$, as well as the unknown coefficients $p_1$ and $p_2$. These multiplicative coefficients update the initially assumed stiffness value $k_e^w$ and force vector $f_e^w$ imported from the model $\mathcal{W}$. The mathematical formulation corresponding to the second phase of the PINN model is as follows

$$ \{\hat{\dot{x}}_e, \hat{x}_e\} = \mathcal{N}_f(\ddot{x}_b, x_b; \Theta) \in \mathbb{R}^{1 \times N_t} \quad (8) $$

where $N_t$ is the total number of time steps. The evaluation of $N_f$ requires the stiffness matrix and excitation vector for the predicted damage position $\hat{l}$. This is because the damage effect is represented by the predicted multiplicative parameters $p_1$ and $p_2$, as outlined in Eq. (5). The updated stiffness matrix and excitation vector at the element level are then obtained using the formulations $\hat{k}_e = \hat{p}_1 \cdot k_e^w \in \mathbb{R}^{4 \cdot (N_d+1) \times 4 \cdot (N_d+1)}$ and $\hat{f}_e = \hat{p}_2 \cdot f_e^w \in \mathbb{R}^{1 \times N_t}$, respectively. Here, $k_e^w$ and $f_e^w$ are the initial estimates of the stiffness matrix and excitation vector, respectively, retrieved from the model $\mathcal{W}$ based on the identified damaged element $e$. Finally, the stiffness matrix and excitation vector, based on the global system formulation, can be expressed as $\hat{\mathbf{K}}(t) = g_k(\hat{k}_e(t))$ and $\hat{\mathbf{f}}(t) = g_f(\hat{f}_e)$, where $g_k$ and $g_f$ are nonlinear functions.

### 3.1. $\mathcal{N}_s$ model for damage localization

The network $\mathcal{N}_s$ infers the damage location $\hat{l}$. The adjustment vector $\hat{s}_l$ updates the system's structure at each time step for every potentially damaged element, facilitating the calculation of the partial differential
equation (PDE) residuals at each step. The accuracy of the derived approximations is evaluated using the relative $L_2$ norm, based on a loss function denoted as $L_s$, which consists of two components. The first term, $L_1$, quantifies the error in the PDE residuals evaluated at the potential damaged positions for each time step. The second term, $L_2$, assesses the predicted adjustment vector $\hat{s}_l$ by minimizing the predicted metric $\hat{J}_l$. The $L_s$ is expressed as

$$L_s = \frac{1}{N_t} \sum_{j=1}^{N_t} \frac{1}{N_d} \sum_{i=1}^{N_d} \|R_{phy,l}^s(i, j)\|^2 + \frac{1}{N_t} \sum_{j=1}^{N_t} \|\hat{J}_l(j)\|^2 \quad (9)$$

where the metric $\hat{J}_l$ is described in Eq. (7), while the residual equation is described as:

$$R_{phy,l}^s := M_l \hat{s} \cdot \ddot{x} + (C_l + \omega G) \hat{s} \cdot \dot{x}_l + K_l(t; \Theta_s) \hat{s} \cdot x - f_l(t; \Theta_s) \quad \forall l = 1, \dots, N_d. \quad (10)$$

By evaluating the metric for $\forall l = 1, \dots, N_d$ and identifying the index with the minimum value, the predicted damage position $\hat{l}$ can be accurately determined. Finally, the matrices $M', C', K'$ at this position are input to the loss function of the $N_f$ model. For brevity, this formulation follows [10] for detailed matrix representations.

## 3.2. $N_f$ model for damage quantification

The $N_f$ model addresses the second task of damage quantification, which involves predicting the system's state and parameters at the damaged element. The validation loss is calculated based on the relative $L_2$ norm between the actual $\{\ddot{x}^w, x^w\}$ and the predicted $\{\hat{x}, \hat{x}\}$ values at the predicted damage position $\hat{l}$ for every time step $N_t$. The residual of the PDEs can be obtained by substituting the predicted vectors $\hat{f}_e, \hat{k}_e$ and $\{\hat{x}_e, \hat{x}_e\}$ into the Eq. (14). The system's structure, i.e., the $M', C'$ and $G$ matrices, is provided by the model $N_s$ and encoded into the residual equation. In this study, the velocity response is assumed to be available from the model $W$. Additionally, it is assumed that the virtual model of the system (model $W$) offers measurements for the response $\{\ddot{x}^w, x^w\}$ for every element, including the damaged one. The PINN incorporates the residual of the PDEs into the tailored loss function to enhance the information content of the data and constrain the learning process. Consequently, the loss function is composed of three terms as follows

$$L_d = \frac{1}{N_t} \sum_{j=1}^{N_t} \|\hat{\dot{x}}_e(j) - \ddot{x}_e^w(j)\|_2^2 + \frac{1}{N_t} \sum_{j=1}^{N_t} \|\hat{x}_e(j) - x_e^w(j)\|_2^2, \quad (11)$$

$$L_i = \|\hat{\dot{x}}_e(0) - \ddot{x}_e^w(0)\|_2^2 + \|\hat{x}_e(0) - x_e^w(0)\|_2^2, \quad (12)$$

$$L_p = \frac{1}{N_t} \sum_{j=1}^{N_t} \frac{1}{N_d} \sum_{i=1}^{N_d} \|R_{phy}(i, j)\|^2. \quad (13)$$

where the residual equation is described as

$$R_{phy} := M' \hat{x} + (C' + \omega G) \dot{x} + \hat{K}(t; \Theta) \hat{x} - \hat{f}(t; \Theta). \quad (14)$$

The first term $L_d$ represents the difference between the measured response data and the prediction of PINN for $N_d$ number of elements and $N_t$ time steps. The second term $L_i$ ensures that the response of the damaged element $e$ meets the initial conditions at $t_0$. Finally, the term $L_p$ captures the PDE residuals evaluated at the damaged element of the shaft and for each time step. Evaluating the third term requires both the field response of the damaged element and the element's system matrices from the high-fidelity model. However, full-field response data are unnecessary for training the PINN in this inverse problem, where damage parameters are estimated using sparse observations and physics-based loss functions. Instead, the PINN leverages boundary conditions, known physics, and partial observations to infer unmeasured states. This approach may limit the generalization performance of the
PINN, which can be improved by evaluating PDE residuals over the full field response and incorporating system matrices from a high-fidelity model. In such cases, reduced-order modeling can approximate the underlying dynamics, lowering computational costs, and enabling real-time applications. The final loss function can be expressed as

$$L(\Theta; N_t) = L_s(\Theta_s; \{N_d, N_t\}) + \lambda_d L_d(\Theta; \{N_t, N_d\}) + \lambda_i L_i(\Theta; N_i) + \lambda_p L_p(\Theta; \{N_t, N_d\}) \quad (15)$$

where $\lambda = \{\lambda_s, \lambda_d, \lambda_i, \lambda_p\}$ is the loss weighting vector and $L_s, L_d, L_i, L_p$ are the loss function terms, defined in Eq. (9) and (11)-(13). The applied weighing vector $\lambda$, introduced to optimally balance the contribution of the data-fitting objective and physics-informed criterion. The networks' learnable parameters $\Theta_s$ and $\Theta$ are obtained by minimizing their loss functions.

## 4. PREDICTION MODE

A monitoring dataset $\ddot{x}_b, x_b$ acquired from the two virtual sensors at the corresponding damage scenario, described by the parameters $l$ and $\mu$, which represent the damage location and extent, respectively, are used as input to the two models. The proposed PINN scheme acts as an instant model, performing training in real-time and providing predictions immediately afterward. In this study, purely numerical data is implemented, generated by evaluating the model $W$, which provides the system's structure and response. The framework's prediction mode, which incorporates the real-time training of the network, is illustrated in Figure 1.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration depicting the damage identification scheme based on PINNs.</figcaption></figure>

The proposed model was developed and trained in MATLAB using the Adam optimization algorithm. The $N_s$ model features 4 inputs and 5 outputs, while the $N_f$ model has 4 inputs and 2 outputs. Both models consist of 5 hidden layers with 50 neurons per layer, employing the sigmoid activation function. The parallel training utilized a time-series dataset spanning 0.1 seconds, containing 1,356 data points, processed as a complete spatiotemporal domain divided into successive time segments. Validation was performed on an independent time-series segment of 0.9 seconds, comprising 5,424 data points excluded from the training phase.

## 5. RESULTS AND CONCLUSIONS

The proposed framework for damage identification in rotary shafts is based on the implementation of a PINN model designed to capture the dynamic behavior of the system under damage conditions. This PINN model serves as a predictive tool, aiming to processing measurements and generating near real-time predictions. Consequently, the model is evaluated at each time step, during which the structural integrity of the system may be altered, resulting in changes to the model's properties due to the presence
of damage. Nevertheless, each PINN model evaluation, from initialization to prediction, takes 25 to 30 seconds, making it a promising candidate for online damage monitoring.

The framework incorporates two neural networks trained concurrently: one for identifying the location of the damage and the other for estimating its severity. The severity of the damage is assessed through predictions of stiffness loss and excitation load at the identified damage location. As an intrusive method, the PINN model leverages a high-fidelity analytical model to solve the inverse problem for a non-dimensional 10-degree-of-freedom shaft system subjected to impact damage.

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Localization: Detected damage location $\hat{l}$ based on the predicted metric $\hat{J}$, as provided by the $N_s$ model. Quantification: Predicted state vectors {$\hat{\mathbf{x}}_e, \hat{\mathbf{x}}_e$}, and inferred stiffness $\hat{\mathbf{f}}_e$ and load $\hat{\mathbf{k}}_e$ at the identified damage location determined based on the predicted parameters $p_1$ and $p_2$, as provided by the $N_f$ model. (gray area corresponds to the training dataset)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3:** Validation accuracy of the PINN scheme evaluated for various predicted damage locations.</figcaption></figure>

During localization, the minimum value of the predicted metric $\hat{J}$ identifies the damage location among five possible positions. For example, Figure 2 illustrates the probabilistic predicted metric, derived from
the output of the $N_s$ model, when the PINN model receives state measurements from the shaft's support, subjected to damage characterized by damage parameters $\mu = 0.1$ and $\hat{l} = 1$. It is evident that the true damage location is predicted with a probability of 100%, as the adjustment vector has successfully aligned the measurements with the correct system properties.

Additionally, the proposed model effectively infers stiffness loss and excitation increase. However, state prediction accuracy at the damage location ranges between 60 – 80%, as reported in Figure 3. This relatively lower accuracy may be attributed to the parallel training of the two networks, which could limit their ability to fully converge on optimal predictions simultaneously.

Future work will focus on improving state prediction accuracy by refining the training process, potentially decoupling the networks or introducing sequential training. Efforts will also prioritize reducing the dependency on training points and handling noisy data.

## ACKNOWLEDGMENTS

Part of this work has been developed within the SAMAS 2 project, a Cat.-B project (B PRJ-RT 1074) coordinated by the European Defence Agency (EDA) and involving two nations, Italy and Poland.

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