Data-driven machine learning for pattern recognition and detection of loosening torque in bolted joints

Abstract

One of the primary challenges associated with bolts, frequently used in assembly structures and systems, is the issue of torque loosening. This problem can arise due to shock and vibration, potentially leading to significant damage and structural failure. The difficulty in identifying and monitoring torque loosening arises from the variability and nonlinear effects present in bolted joints. In this paper, we proposed a machine-learning algorithm architecture designed for pattern recognition, detection, and quantification of torque loosening in bolted joints. This approach combines unsupervised and supervised machine learning algorithms to address the challenges of assessing the bolt torque loosening issue. Our algorithm utilises a damage index, calculated from the frequency response of the jointed system using the Frequency Response Assurance Criterion, as input data for the unsupervised–supervised classification algorithm. This classification ML algorithm effectively identifies and categorises instances of torque loss by analysing indirect vibration measurements, even in situations where the bolted system’s state is unknown. Additionally, we introduce a regression algorithm to quantify torque loosening levels. The results obtained from our proposed machine learning algorithm to overcome torque loosening in bolted joints show that the inherent uncertainties of a data-driven approach intrinsically influence torque-related issues. This assessment is based on experimental raw data collected under diverse test conditions for the bolted structure. We employ a range of validation and cross-validation metrics to evaluate the effectiveness and accuracy of these ML algorithms in detecting and diagnosing torque-related issues. These metrics play a crucial role in assessing the algorithms’ efficiency and precision in determining the state of the bolted connection and the corresponding torque levels with associated uncertainty quantification.

Appendix A: Machine learning techniques

An appendix contains supplementary information that is not an essential part of the text itself but may help provide a more comprehensive understanding of the research problem, or it is information that is too cumbersome to be included in the body of the paper.

1.1 A.1: K-Means clustering

The K-Means clustering algorithm is an unsupervised ML in which data objects are distributed into a specified number of k clusters [53]. The K is a hyperparameter that specifies the number of clusters that should be created. It is a useful approach for clustering (labelling) or partitioning the data prior to feeding the labelled data as the output of a supervised ML algorithm. The aim is to find centroids that are a measure of the centre point of the cluster, such that the sum of the squared distances of each data sample to its nearest cluster centre is minimal. The nearest here is with respect to the Euclidean norm (L2 norm). Thus, the objective function is

$$\begin{aligned} J = \sum _{i=1}^{n} \textrm{min} _{k}\left( \left\| x_{i} - \mu _{k}\right\| ^2 \right) \end{aligned}$$

(A1)

where $x_{i}$ is referred to as the ith instance in cluster k and $\mu _{k}$ is referred to as the mean of the samples or “centroid” of cluster k.

The K-Means algorithm is widely used due to its simplicity of implementation and low computational complexity, but one of the biggest problems of K-Means clustering algorithms is the initial definition of the number of clusters that must be used. When dealing with highly complex problems where the cluster count is hard to define, the “elbow” method can provide insights into the potential number of required clusters. Another disadvantage of k-means is that it is very sensitive to outlier points, which can distort the centroids and the clusters [54].

1.2 A.2: K-Nearest-neighbour classifier

K-nearest neighbour is one of the simplest supervised learner methods [54, 55] and widely used for pattern recognition [56]. k-NN can be used for classification and regression, where data with discrete labels usually uses classification and data with continuous labels regression. The classification is calculated from a simple majority vote of the nearest neighbours of each point: a query point is assigned the data class with more representatives within the nearest neighbours of the point. For this, a metric between the points is used spaces [54].

The k-NN algorithm, in its simplest version, only considers exactly one nearest neighbour, which is the closest training data point to the point we want to predict. The prediction is then simply the known output for this training point. Depending on the value of ‘k’, each sample is compared to find similarity or closeness with ‘k’ surrounding samples. For example, when k = 5, the individual samples compare with the nearest five samples; hence, the unknown sample is classified accordingly [54]. The optimal choice of the value of ’k’ is highly data-dependent. In general, a larger suppresses the effects of noise but makes the classification boundaries less distinct.

1.3 A.3: Decision tree and random forest

Decision tree supervised algorithm can target categorical variables such as the classification of a damaged or undamaged statement and continuous variables as regression to compare the signal with the healthy state of the system [55]. Learning a decision tree means learning the sequence of if/else questions that gets us to the true answer most quickly. A tree contains a root node representing the input feature(s) and the internal nodes with significant data information. Each node (also called a leaf or terminal node) represents a question containing the answer. The interactive process is repeated until the last node (leaf node) is reached such that the node becomes impure [54]. The data get into the form of binary features in our application, and a classification procedure is performed.

Random Forest ML algorithm is an ensemble classifier that consists of many decision trees where the class output is the node composed of individual trees. The RF has high prediction accuracy, robust stability, good tolerance of noisy data, and the law of large numbers they do not overfit, and it has been used for structural damage detection. It has shown a better performance [57].

1.4 A.4: Support vector machine

Support Vector Machines are supervised machine learning techniques developed from the statistical learning theory that can be used for classifying and regressing clustered data. In the case of linear classification, with two classes, let $\{(x_{i}, y_{i}),...,(x_{n}, y_{n})\}$, a training dataset with n observations, where $x_{i}$ represents the set of input vectors and $y_{i}(+1, -1)$ is the class label of $x_{i}$, the hyperplane is a straight line that separates the two classes with a marginal distance (as seen in Fig. 11). The purpose of an SVM is to construct a hyperplane using a margin, defined as the distance between the hyperplane and the nearest points that lie along the marginal line termed as support vectors [58].

Fig. 11

SVM algorithm operation

One can define the hyperplane by Eq. (A2), where we have the dot product between x and w added to the term b:

$$\begin{aligned} D(x)=w^T.x + b = c \quad for\quad -1< c < 1 \end{aligned}$$

(A2)

where x represents the points within the hyperplane, w is the weights that determine the orientation of the hyperplane, and b is the bias or displacement of the hyperplane. When $c = 0$, the separating hyperplane is in the middle of the two hyperplanes with $c= 1$ and ${-}1$. An SVM aims to maximise the data separation margin by minimising w. This optimisation problem can be obtained as the quadratic programming problem given by

$$\begin{aligned} \textrm{min}\quad \frac{w^2}{2} \quad s.t \quad y_{i}(w^T.x_{i}+b)\ge 1 \quad \textrm{for} \quad i=1,2,...,n \end{aligned}$$

(A3)

where w is the Euclidean norm.

SVM algorithm encompasses not only linear and nonlinear classification but also linear and nonlinear regression. The main idea of the algorithm consists of fitting as many instances as possible a “tube” while limiting margin violations. Therefore, SVR wants to find a hyperplane that minimises the distance from all data to this hyperplane. The width of the “tube” is controlled by a hyperparameter, which has an error “insensitive” area, defined by $\epsilon$, as illustrated by Fig. 11. The larger the $\epsilon$, the larger the diameter of this tube, and the less sensitive the model is in predicting points within it. In contrast, the smaller $\epsilon$, the smaller the diameter of the tube, the greater the chances of points being on the edges of the tube, making the model more robust. The samples that fall into the $\epsilon$-margin do not incur any loss. Points outside the tube are examined and considered with respect to the $\epsilon$-insensitive region. Compared to a previously defined error called slack variables ($\xi$). This approach is similar to the “soft margin” concept in SVM classification because the slack variables allow regression errors to exist up to the value of $\xi$ and $\xi _{i}^{*}$, yet still satisfy the required conditions. Including slack variables leads to the objective function given by Eq. (A4).

$$\begin{aligned} \begin{aligned}&\textrm{Minimise}:\quad \frac{1}{2}\left\| w \right\| ^2 + C\sum _{i=1}^{n}\left( \xi _{i} +\xi _{i}^{*} \right) \\ \textrm{Constraints}:\quad&\\&y_{i} - w^T.x_{i} - b \le \epsilon + \xi _{i}\\&w^T.x_{i} + b - y_{i} \le \epsilon + \xi _{i}^{*}\\&\xi _{i}, \xi _{i}^{*} \ge 0, \quad i=1,...,n \end{aligned} \end{aligned}$$

(A4)

1.5 A.5: Naïve bayes

Naïve s classification method based on Bayes theorem with the assumption of independence between features, considered a simple technique for constructing classifiers with models that assign class labels to problem instances, represented as vectors of feature values, where the class labels are drawn from some finite set. There are three classes in sk-learn, the Gaussian-NB, Multinomial-NB, and Bernoulli-NB. The first assumes a Gaussian distribution, the second is for discrete occurrence counters, and the third is for discrete boolean attributes [59].

Naive Bayes classifiers are highly scalable, requiring several parameters linear in the number of variables in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression. In other words, one can work with the naive Bayes model without accepting Bayesian probability or using any Bayesian methods. An advantage of naive Bayes is to train a model with few samples [60].

1.6 A.6: Extreme gradient boosting (XGBoost)

XGBoost (short for Extreme Gradient Boosting) is an efficient implementation of Gradient Boosting Machines (GBM), developed by Tianqi Chen [61], widely recognised for its superior performance in supervised learning. This versatile algorithm is also considered to be an ensemble tree technique that can be used for both regression and classification tasks.

XGBoost follows the concept of weak-learner, where each predictor could be improved by sequentially training new trees to the model [62]. In other words, the XGBoost makes predictions by creating numerous smaller decision trees, also known as subtrees. Each subtree makes predictions for the data, and their individual predictions are combined to form the final prediction for the given input. This ensemble approach helps improve the accuracy and generalisation ability of the predictive model. The process involves iterative training of these subtrees to correct the errors made by the previous subtrees, gradually refining the overall prediction as more trees are added.

Another feature related to XGBoost is that it uses L1 and L2 regularisation, which helps with model generalisation and overfitting reduction. It uses an optimisation strategy that produces better weights as it calculates the weights of the component models. It also uses slightly less tiny component models.