# PCA Based Stress Monitoring of Cylindrical Specimens Using PZTs and Guided Waves

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## Abstract

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## 1. Introduction

## 2. Review of Ultrasonic Stress Monitoring Techniques

## 3. Theoretical Framework

#### 3.1. Acoustoelasticity Effect

#### 3.2. Principal Components Analysis (PCA)

**T**is the projection of the original data over the direction of vector ${p}_{j}$ (jth principal component). The projected data in the new space are uncorrelated and have maximal variance, thus it can be potentially the best representation of the process features. Since eigenvectors are ordered according to variance, it is possible to reduce the dimensionality of the data set

**X**by choosing only a reduced number, $\varrho <K$, of eigenvectors related to the $\varrho $ highest eigenvalues. In this way, given the reduced matrix $\widehat{\mathbf{P}}\in {\mathcal{M}}_{K\times \varrho}\left(\mathbb{R}\right)$, the score matrix is defined as

- PCA Based indexOne well-known PCA statistical index used to distinguish abnormal behavior in a process is the Q-statistic or Square Prediction Error (SPE)-statistic.This index uses the residual error matrix $\mathbf{E}$ to represent the variability of the data projected on the residual subspace. The Q-statistic is based on the assumption that the underlying process follows approximately a multivariate normal distribution, where the first moment vector is zero. Therefore, this index denotes that events are unexplained by the reduced model. In other words, it is a measurement of the difference, or residual, between a sample and its retrieved version by using the reduced model. The Q-statistic of the ith experimental trial is defined as the sum of the squared residuals of each variable as follows:$$\begin{array}{}\mathrm{(13)}& \hfill {Q}_{i}& =\parallel {e}_{i}{\parallel}^{2}={e}_{i}^{T}{e}_{i}=\sum _{\ell =1}^{K}{e}_{i,\ell}^{2}\hfill \mathrm{(14)}& & ={x}_{i}^{T}\left(\mathbf{I}-\mathbf{P}{\mathbf{P}}^{T}\right){\left(\mathbf{I}-\mathbf{P}{\mathbf{P}}^{T}\right)}^{T}{x}_{i}\hfill \mathrm{(15)}& & ={x}_{i}^{T}\left(\mathbf{I}-\mathbf{P}{\mathbf{P}}^{T}-\mathbf{P}{\mathbf{P}}^{T}+\mathbf{P}{\mathbf{P}}^{T}\right){x}_{i}\hfill \mathrm{(16)}& & ={x}_{i}^{T}\left(\mathbf{I}-\mathbf{P}{\mathbf{P}}^{T}\right){x}_{i}\hfill \end{array}$$

## 4. PCA Based Stress Monitoring Approach

#### 4.1. Modeling

- A set of I experiments are conducted on the specimen at nominal condition (residual or initial stress). The experiment consists of exciting the specimen by a PZT, via a modulated pulse at a single probe position and capturing the guided wave by a PZT, at a point distant from the excitation, such that the interest zone is covered. This measurement is repeated several times (experimental trials). The collected data are arranged as follows:$$X=\left[\begin{array}{cccccc}{x}_{11}& {x}_{12}& \cdots & {x}_{1k}& \cdots & {x}_{1K}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {x}_{i1}& {x}_{i2}& \cdots & {x}_{ik}& \cdots & {x}_{iK}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {x}_{I1}& {x}_{I2}& \cdots & {x}_{Ik}& \cdots & {x}_{IK}\end{array}\right]=\left[\begin{array}{c}{x}_{1}\\ \cdots \\ {x}_{i}\\ \cdots \\ {x}_{I}\end{array}\right]=({v}_{1}|{v}_{2}|\cdots |{v}_{k}|\cdots |{v}_{k})$$This $\mathbf{X}\in {\mathcal{M}}_{I\times K}\left(\mathbb{R}\right)$ is the vector space of $I\times K$ matrices over $\mathbb{R}$, which contains information from K discretization instant times and I experimental trials. Each row vector $\left({x}_{i}\right)$ represents measurements from the sensor at a specific ith trial. In the same way, each column vector $\left({v}_{k}\right)$ represents measurements at the specific kth discretization instant time in the whole set of experiments trials.
- Cross correlation analysis is applied between the acting and sensing signals of the I experiments to eliminate noisy data trends.The cross-correlation function between two signals X(t) and Y(t) is defined by Equation (18).$${R}_{XY}(t,t+\tau )=\underset{k\to \infty}{lim}\frac{1}{K}\sum _{k=1}^{K}{X}_{k}\left(t\right){Y}_{k}(t+\tau )$$
- The correlated signals are arranged in the matrix $\tilde{X}$ for I experiments of 2K-1 samples, conducted on the same scenario in order to consider noise and variance due to the stochastic nature of the technique.
- The matrix $\tilde{X}$ is normalized by considering each column as a measured variable and normalized to mean zero and variance equal to one for the I experiments. This step minimizes bias and scale variance effects. The Equations (19)–(21) are used for the mentioned preprocessing.$${\mu}_{j}=\frac{1}{I}\sum _{i=1}^{I}{x}_{ij},j=1,\dots ,(2K-1)$$$$\mu =\frac{1}{I(2K-1)}\sum _{i=1}^{I}\sum _{j=1}^{2K-1}{x}_{ij}$$$$\sigma =\sqrt{\frac{1}{I(2K-1)}{\displaystyle \sum _{i=1}^{I}}{\displaystyle \sum _{j=1}^{(2K-1)}}{({x}_{ij}-\mu )}^{2}}$$

**P**) extracted from the singular value decomposition of the covariance matrix $\tilde{x}$, according to that described in the previous section.

#### 4.2. Monitoring

## 5. Experimental Setup

#### 5.1. Steel Rod

#### 5.2. Hollow Cylinder

#### 5.3. Influence of the Transducer Configuration on the Guided Wave Propagation

## 6. Results

#### 6.1. Rod

#### 6.2. Hollow Cylinder

## 7. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Quiroga, J.; Mujica, L.; Villamizar, R.; Ruiz, M.; Camacho, J.
PCA Based Stress Monitoring of Cylindrical Specimens Using PZTs and Guided Waves. *Sensors* **2017**, *17*, 2788.
https://doi.org/10.3390/s17122788

**AMA Style**

Quiroga J, Mujica L, Villamizar R, Ruiz M, Camacho J.
PCA Based Stress Monitoring of Cylindrical Specimens Using PZTs and Guided Waves. *Sensors*. 2017; 17(12):2788.
https://doi.org/10.3390/s17122788

**Chicago/Turabian Style**

Quiroga, Jabid, Luis Mujica, Rodolfo Villamizar, Magda Ruiz, and Jhonatan Camacho.
2017. "PCA Based Stress Monitoring of Cylindrical Specimens Using PZTs and Guided Waves" *Sensors* 17, no. 12: 2788.
https://doi.org/10.3390/s17122788