A Novel Multi–Scale One–Dimensional Convolutional Neural Network for Intelligent Fault Diagnosis of Centrifugal Pumps

Abstract

Centrifugal pumps are susceptible to various faults, particularly under challenging conditions such as high pressure. Swift and accurate fault diagnosis is crucial for enhancing the reliability and safety of mechanical equipment. However, monitoring data under fault conditions in centrifugal pumps are limited. This study employed an experimental approach to gather original monitoring data (vibration signal data) across various fault types. We introduce a multi–scale sensing Convolutional Neural Network (MS–1D–CNN) model for diagnosing faults in centrifugal pumps. The network structure is further optimized by examining the impact of various hyperparameters on its performance. Subsequently, the model’s efficacy in diagnosing centrifugal pump faults has been comprehensively validated using experimental data. The results demonstrate that, under both single and multiple operating conditions, the model not only reduces reliance on manual intervention but also improves the accuracy of fault diagnosis.

1. Introduction

Mechanical fault diagnosis and mechanical health detection have always been essential research areas in mechanical engineering [1]. Centrifugal pumps are common mechanical devices, widely used in various fields [2]. Due to the high–strength and high–corrosive working environment of centrifugal pumps, they may be more prone to failure [3]. These issues can accumulate over time. If maintenance issues are not detected and repairs implemented promptly, they will adversely impact safe operation. Therefore, it is necessary to develop and apply methods for timely diagnosis and assessment of centrifugal pump damage.

In general, personnel often rely on visual inspections to detect and assess damage in centrifugal pumps, a process that frequently demands significant human intervention and time [4]. Moreover, visual inspection is only suitable for small pump structures, and its diagnostic efficiency for complex structures is low. To address the limitations of visual inspection, extensive research has been conducted, resulting in the development of advanced fault–diagnosis methods [5]. Among these methods, utilizing mechanical vibration data and data analysis for fault diagnosis demonstrates robustness and promising prospects. Fault diagnosis based on mechanical vibration data can be categorized into two types [6]. One involves establishing mathematical models for simulation verification, and many successful model–based methods have been developed and applied [7]. These methods, compared to visual inspection, significantly save time and effort in fault diagnosis. However, they do have certain limitations. For instance, they are often computationally complex, requiring a substantial amount of historical data, samples, and multi–dimensional features during the training phase [8]. Moreover, labeling of the current state by experienced experts is necessary, but data in the industrial field often lack such labels.

In addition, constructing models and adjusting model parameters demands a high level of expertise, and they are static, lacking flexibility to adapt to new changes [9]. In contrast, signal–based machine learning methods offer more flexibility and effectiveness, exemplified by the widely utilized support vector machine (SVM) [10]. When combined with other intelligent algorithms, support vector machines can achieve precise fault identification even with limited fault samples. Huang et al. [11] proposed an SVM model based on an optimized genetic algorithm, successfully identifying faults such as loosening of base screws and failure of buffer springs. The machine–learning–based fault–diagnosis method mainly involves two stages: feature extraction and fault diagnosis. During feature extraction, the model extracts key features from the measured signal and learns the extracted feature information to predict the final output. In recent years, various machine learning methods combining feature extraction have been successively proposed. Lee et al. [12] introduced a motor–fault–detection method that combines a wavelet transform and a BP neural network. The BP neural network model is a multi–layer feedforward network trained using the error backpropagation algorithm, constructed with a layer–by–layer calculation and an error feedback correction approach. This established model exhibits greater stability, requiring fewer iterations and faster calculation times [13]. Li et al. [14] integrated the principal–component analysis method and SVM to establish a rolling bearing state life evaluation model. Principal–component analysis reduces the dimensions of input data, enhancing the efficiency of training and prediction while reducing the risk of overfitting, aiding SVM in handling high–dimensional data more effectively. However, when combining PCA and SVM, consideration should be given to the number of principal components [15]. Tang et al. [16] employed a combination of continuous wavelet transform and CNN model to accurately perform intelligent fault diagnosis of a hydraulic pump. To assess the performance of different training methods from the perspective of feature extraction, Moysidis, D.A. et al. [17] applied various pretreatment methods, including continuous wavelet transform. Continuous wavelet transform demonstrates certain robustness, enhancing feature extraction through filtering and thresholding. Through the aforementioned research, it is evident that the combination of feature extraction and machine learning has a positive impact on fault diagnosis. However, finding a suitable combination algorithm remains a challenging task, and feature extraction consumes considerable computing time. Consequently, one–dimensional convolutional neural networks are introduced, capable of directly extracting features from the original vibration data and internally predicting the diagnosis results. Ugli et al. [18] proposed a 1D–CNN model based on a genetic algorithm, optimizing hyperparameters through the use of a genetic algorithm, successfully completing the fault–detection task of an axial hydraulic piston pump. Han et al. [19] employed 1D–CNN and fused dilated convolutional layers to capture the deep feature relationships of the signal. Liang et al. [20] used a multi–scale similarity matrix to detect diesel–engine faults and improve fault–identification ability. Nevertheless, a single model cannot learn spatial relationships in the data. Currently, there are limited existing vibration data for centrifugal pumps, and there are fewer fault detection methods using neural network models for centrifugal pumps.

In the research on fault diagnosis of rotating machinery, the cited literature primarily focuses on the signal–noise–reduction capability of the diagnostic algorithm, the screening of an optimal feature’s set dimension, and the accuracy of fault recognition. However, it neglects the equipment’s ability to characterize multiple faults under different working conditions and measurement points. To address issues such as uncertain fault signals, limited representation of various working conditions, and inadequate classification and identification of multiple faults, this paper proposes a fault–diagnosis method using a one–dimensional convolutional neural network with a novel multi–scale sensing function. The method captures feature information across different scales through multi–scale sensing, retaining the most useful features while minimizing manual intervention, thereby achieving the goal of diagnosing centrifugal pump faults [21]. In the experiments, six typical test points, highly sensitive to fault characteristics, are individually tested to increase data volume and enhance robustness. Additionally, the optimal convolution kernel scale is determined to enhance the performance of the convolutional neural network model in classifying and identifying centrifugal pump faults. Finally, the model’s reliability is verified by establishing datasets for multiple working conditions, filling the gap in fault diagnosis under various working conditions.

2. Data Acquisition

2.1. Experimental Platform Construction

Currently, due to limited fault information and challenges in extracting relevant data [22], we have established a testing platform for centrifugal pump equipment. This platform enables us to collect data related to normal operations and performance during faults. The identified fault types fall into five distinct groups: impeller eccentricity (shaft misalignment), bearing inner ring fault, bearing outer ring fault, base loosening, and pump body leakage. We have achieved a fault–isolation rate of at least 85% for all six categories.

Figure 1 illustrates the key components of the test setup, including the base, motor, centrifugal pump, power supply, control cabinet, transducer set, and data acquisition equipment. The measurement point strategically captures vibration–energy transmission to the elastic foundation or other system parts. Measurements are conducted both parallel and perpendicular to the direction of maximum deflection, ensuring precise assessment of the maximum value. The acceleration sensor is positioned at six designated locations on the vertical centrifugal pump, as depicted in Figure 2. In the experiment as shown in Figure 3, the acceleration sensor is attached to the flange surfaces of the inlet and outlet ports, the impeller’s central position on the pump body shaft, the left and right electrical racks, and the central motor shaft position.

Acceleration sensors assess vibrations at crucial bearing seats, the mounting base of the centrifugal pump, inlet and outlet flange surfaces, and the volute. Power meters and tachometers measure the motor’s power output and rotational speed. Simultaneously, a data acquisition card is used to collect and process vibration signals with high precision.

2.2. Fault Simulation Experiment

(1)

Impeller eccentricity Experiment

In the experiment simulating imbalance failure of a centrifugal pump impeller, mounting holes (610 mm) were drilled in the impeller web. Counterweight bolts were then added to these holes, introducing an unbalanced weight of approximately 400–800 g, as depicted in Figure 4. This setup effectively simulates the imbalance failure in the impeller parameters of the centrifugal pump.

(2)

Inner and outer ring faults test

Bearing inner and outer ring faults are experimentally simulated using wire cutting. Faults, 24 mm wide and 12 mm deep, are intentionally introduced along the axial direction on the contact side of the rolling bearings’ inner and outer rings, as illustrated in Figure 5 and Figure 6:

(3)

Base loosening

The experiment replicates a loose–base failure by adjusting the tightness of centrifugal pump bolts, with two to four turns loosening in this instance, as illustrated in Figure 7.

(4)

Pump body leakage

The term “water seal” in a pump typically denotes the mechanical seal preventing liquid (e.g., water) from leaking out of the pump shaft. A water seal serves as a type of sealing device. In this experiment, the pump body’s leakage failure was induced by adjusting the rubber O–ring seal, as illustrated in Figure 8. The left side shows the water seal in normal condition, the right side shows the water seal in missing and slack condition.

The rotational speed was set to 600 r/min, 1000 r/min, and 1450 r/min for both normal working conditions and the aforementioned five fault conditions, resulting in 18 experiment sets in total.

In these experiments, 2500 sampling points were taken as samples, with vibration data collected every 1 s. The impeller completes 24 rotations per second, justifying the selection of a 2500 Hz sampling frequency for extracting centrifugal–pump–fault characteristics. Six measurement points were established, encompassing water inlet and outlet, the base perpendicular to the test bench center, the left and right sides of the motor base, and the motor shaft center. Each measurement point comprises five types of fault data and one type of normal data. The pump operates at three rotational speeds for 3 min, resulting in the collection of 100 data sets for each of the six measurement points, totaling 1800 sets of data and reaching 4.5 million data points.

2.3. Establishment of a Database

Specific speed is a dimensionless parameter employed for standardizing pump performance, facilitating a more comprehensive comprehension of pump operational characteristics. Its concept derives from the interplay among actual speed, flow, and head, establishing a consistent criterion. The formula for specific rotational speed is as Equation (1):

$$n_s=\omega \frac{Q^{1/2}}{{(gH)}^{3/4}}$$

(1)

Here, $\omega$ represents the actual speed, $Q$ denotes the flow rate, and $H$ signifies the head.

Table 1. Specific speed at different operating conditions.

Parameter	Condition 1	Condition 2	Condition 3
$H$ (m)	18.00	14.95	11.58
$Q$ (m3/h)	50.00	34.48	20.69
$\omega$ (r/min)	1450	1000	600
$n_s$	211.83	139.44	78.49

displays the resulting specific speeds of the pump under various operating conditions.

The centrifugal pump vibration data were categorized into three sets based on distinct rotational speeds. Data set A comprises the vibration data in Condition 1, data set B combines data from Condition 1 and Condition 2, and data set C includes data from Condition 1, Condition 2, and Condition 3. Categorization was also completed based on the measurement points’ locations. Each sample consists of 2500 data points, resulting in a total of 1.5 million data points. The data were then split into training and test sets in a 7:3 ratio, with the training set containing 420 samples and the test set containing 180 samples. The data composition is detailed in

Table 2. Composition of training and test data.

Data Sets	A		B		C
Condition	1		2		3
Location of measuring point	train	test	train	test	train	test
0, 1, 2, 3, 4, 5	420	180	840	360	1260	540
Total	2520	1080	5040	2160	7560	3240

.

Using condition 1 as an example, 5000 vibration data points from measurement point 0 are chosen to create time–domain graphs for both normal operation and five fault states. The acceleration sensor collects 250 sample points during one impeller rotation cycle, and the time–domain graph is generated from 250 to 500 sample points. This enables the observation of the overall trend of the vibration signal. Additionally, detailed plots are used to observe distinct peaks in vibration signals and abrupt frequency changes.

Figure 9 reveals distinct differences in vibration waveforms and periods between the pump’s normal and fault states. Furthermore, unique vibration waveforms are observed for various pump faults, indicating the potential for feature extraction in subsequent stages.

3. Build Theoretical Frameworks for Centrifugal–Pump–Fault Diagnosis

3.1. One–Imensional Convolutional Neural Network

This paper utilizes the TensorFlow machine learning framework and employs the Python programming language for implementing the 1D–CNN. The 1D–CNN, a neural network variant, is primarily designed for processing one–dimensional signals [23], sequentially applying convolutional operations to the input signal using a convolutional kernel and generating corresponding feature information as the signal undergoes these operations [24].

Equation (2) illustrates the convolutional process of a one–dimensional convolutional neural network:

$$z^{\mathrm{l}}=f\left({\displaystyle \sum _{i=1}^{c^{l-1}}{w}_{i,c}^l\ast x_i^{l-1}+b_i^l}\right)$$

(2)

The pooling layer serves as a down–sampling operation for further feature extraction from the preceding layer [25]. Simultaneously, the pooling layer can mitigate the overfitting phenomenon to a certain extent. Given that pump vibration data contains noise, Max–Pooling is employed in this study to filter out irrelevant information.

The formula for calculating Max–Pooling is as Equation (3):

$$g^{n(i,j)}=\max \cdot pooling(x^{n(i,t)})=\underset{(j-1)L+1\le t\le jL}{\max }(x^{n(i,t)})$$

(3)

Table 3. Some variables used in Equations (2) and (3).

Variables	Description
$z^l$	The output of the $l$th layer
$b_i^l$	The $i$th convolution kernel bias
$x_i^{l-1}$	The $l$th layer input and the $l-1$th layer output
$w_{i,c}^l$	The weight of the $l$th layer
$c^{l-1}$	The $c$th channel of the $l-1$th layer
$f(\cdot )$	The activation function
$x^{n(i,j)}$	The value of the $t$th neuron in the $i$th channel of the $n$th layer
$L$	The size of the pooling kernel

provides a definition of the terms used in one–dimensional convolutional neural network.

The traditional 1D–CNN model for diagnosing centrifugal pump faults involves three primary processes. Initially, the original vibration time–domain signal during pump operation serves as the system input. Forward propagation through the convolution and pooling layers refines the time–domain signal into intricate feature expressions, enabling adaptive feature extraction for each pump–fault type. Subsequently, reverse propagation fine–tunes network parameters in each layer to establish mapping from the feature space to the fault–type space. Lastly, the Softmax classifier produces the classification results. This entire process integrates the self–learning of fault features and the recognition of fault types. Figure 10 illustrates the flowchart and internal structure of the 1D–CNN–based fault–classification algorithm, featuring a four–layer network comprising input, hidden, fully connected, and output layers.

3.2. Multi–Scale One–Dimensional Convolutional Neural Network

During centrifugal pump troubleshooting, relying solely on single kernels in the convolutional layer may lead to the neglect of finer features. This results in incomplete information in the extracted feature expression, impacting the effectiveness of feature learning [26]. Considering the perspective of learning local sensory field feature information, convolution kernels with different scales can extract features with varying accuracy [27]. Smaller–scale convolution kernels detect more detailed information, but they may also lead to the loss of upper and lower information connections when processing a broader range of information [28]. Larger–scale convolutional kernels access a larger receptive field, enabling the capturing of more global information during signal processing [29]. Therefore, this paper introduces a fault–diagnosis system based on a convolutional neural network (MS–1D–CNN), which integrates multiple scales of convolutional kernel sizes for pump–fault diagnosis. By employing different sizes of convolutional kernels for convolutional operations, the combination enhances the expression of feature information, preventing the oversight of different accuracy features that may occur when using a single convolutional kernel.

The convolutional structure of the multi–scale one–dimensional convolutional neural network is illustrated in Figure 11. The convolutional layer in this system comprises parallel convolutional layers, each incorporating various scales of convolutional kernel sizes. Different scales of convolution are applied to the one–dimensional input signals to extract distinct features. The features extracted from each convolutional layer are subsequently combined and forwarded to the next layer. Figure 12 depicts the workflow of the intelligent fault–diagnosis method for centrifugal pumps utilizing the multi–scale one–dimensional convolutional neural network.

3.3. Evaluation Criteria

3.3.1. Average–Validation–Loss Function

When employing a deep algorithmic model on a provided dataset, to assess the model’s performance, we introduce a metric known as loss, quantifying the error produced by the model [30]. The closer the average validation loss is to zero, the better the model learns. The formulation of the average–validation–loss function is presented in Equation (4):

$$L\mathrm{oss}(x)=-{\displaystyle \sum _{i=1}^{output \mathrm{size}}{\mathrm{y}}_i}\cdot \log \stackrel{\wedge }{y_i}$$

(4)

where ${\mathrm{y}}_i$ is the true value label corresponding to the $i$th category.

3.3.2. Accuracy Rate

Accuracy represents the ratio of correctly predicted samples by the model to the total number of samples and is a common metric for evaluating the performance of a classification model [31]. It can be calculated using Equation (5):

$$A\mathrm{cc}uracy=\frac{TP}{(TP+FP)}$$

(5)

TP represents the count of correctly classified samples, and FP represents the count of incorrectly classified samples.

4. Parameter Settings for MS–1D–CNN Model

4.1. Basic Model Parameter Selection

Various convolutional neural network structures serve distinct recognition functions, and the network’s performance is determined by its hyperparameter settings [32]. To identify the optimal network structure, this study explores the impact of activation function, convolution kernel size, batch size, epoch number, optimizer selection, and the number of convolutional layers on the model [33]. Figure 13 presents the statistical investigation results for six different run parameters of the 1D–CNN model with varying hyperparameters.

This paper investigates the impact of five common activation functions on model performance. Figure 13a illustrates that, in comparison to other activation functions, the 1D–CNN model exhibits superior performance when employing the Relu activation function. A more intuitive observation can be taken from Appendix A.

Table A1. Mean validation loss and accuracy for different activation modes.

Activation Functions	Tanh	Softsign	Selu	Relu	Elu
Val–accuracy	0.8444	0.8759	0.8759	0.9593	0.8444
Val–loss	0.3129	0.3231	0.2994	0.1720	0.3495

reveals that the model achieves an accuracy rate as high as 0.9593 and the lowest average verification loss (0.1720) when utilizing the Relu activation function.

When assessing the impact of convolution kernel size on model performance and choosing the convolution kernel, it is advisable to opt for odd numbers. This preference arises from the unique central position of convolution kernels with odd side lengths, maintaining symmetry in the convolution kernel direction. This choice mitigates the introduction of irrelevant bias and balances the edge effect of data. In line with modeling experience, odd numbers like 3, 5, 7, 9, 11, 13, and 15 are commonly favored. In this study, while keeping other parameters constant, we systematically test the convolution kernel sizes mentioned above to investigate their influence. Figure 13b illustrates that the model achieves optimal performance with a convolution kernel size of 11, and

Table 4. The average validation loss and accuracy for various kernel sizes.

Kernel Sizes	3	5	7	9	11	13	15
Val–accuracy	0.8315	0.8944	0.9019	0.9593	0.9611	0.9019	0.8722
Val–loss	0.3993	0.2999	0.3009	0.1720	0.1005	0.1796	0.23

further demonstrates that with a kernel size of 11, the model achieves an accuracy as high as 0.9611.

Figure 13c illustrates the variation in model performance for different batch sizes: 128, 256, 512, 1024, and 2048. The figure indicates that with a batch size of 128, the training model exhibits the smallest average verification loss and the fastest convergence. This could be attributed to the better classification results achieved with a smaller batch size. However, excessively small batches may lead to oscillations in the model’s average validation loss curve, preventing convergence to a stable result.

Table 5. The average validation loss and accuracy for various lot sizes.

Batch Size	128	256	512	1024	2048
Val–accuracy	0.9463	0.9000	0.8593	0.8630	0.8167
Val–loss	0.1279	0.1779	0.2580	0.3318	0.4817

further reveals that with a batch size of 128, the model achieves an accuracy rate of 0.9463, meeting the accuracy requirement.

Figure 13d illustrates the impact of the number of iterations on model performance. In conjunction with

Table 6. The average validation loss and accuracy with different number of epochs.

Epoch	30	60	90	120
Val–accuracy	0.8944	0.9463	0.9315	0.9130
Val–loss	0.2409	0.1279	0.2278	0.1544

, it becomes evident that as the number of iterations increases, the model becomes unstable and is susceptible to overfitting issues. The model achieves its highest stability and least average validation loss when the number of iterations is 60.

Furthermore, the choice of different optimizers significantly impacts the model’s performance. Analyzing the average validation loss curve and model prediction accuracy in Figure 13e and

Table 7. The average validation loss and accuracy of different optimizers.

Optimizer	Adagrad	Adam	Adamax	Nadam	RMSProp	SGD
Val–accuracy	0.4167	0.9500	0.8722	0.9611	0.6704	0.3315
Val–loss	1.3724	0.1405	0.2357	0.1255	0.6479	1.1825

reveals that training the model with Adam and Nadam yields better results, with Nadam exhibiting a higher correctness rate compared to Adam. Therefore, in this study, Nadam is selected for backpropagation calculations to compute the gradient of each parameter. Subsequently, the models’ weights and biases are updated to minimize training error.

In the final analysis, we investigate the impact of convolutional layers on model performance. As illustrated in Figure 13f and

Table 8. The average validation loss and accuracy for different numbers of convolution layers.

Number of Convolution Layers	1	2	3
Val–accuracy	0.8315	0.9611	0.8926
Val–loss	0.4233	0.1255	0.1867

, the two–layer convolutional neural network exhibits the highest and most consistent accuracy in pump–fault diagnosis, effectively addressing the issue of overfitting. The three–layer convolutional neural network, while less effective than the two–layer counterpart, could be attributed to the limited vibration data available for the centrifugal pump. The three–layer convolutional neural network is susceptible to gradient explosion and overfitting, resulting in reduced diagnostic accuracy and an unstable average validation loss curve. The single–layer convolutional neural network performs the poorest in terms of accuracy due to its insufficient feature extraction, a consequence of its limited number of convolutional layers, making it prone to underfitting.

The individual parameters of the optimized 1D–CNN model are presented in

Table 9. Optimized 1D CNN model parameter table.

Layer	Hidden Layer Type	Number of Feature Maps	Kernel Size	Activation	Stride
1	Conv1D	16	11	Relu	1
2	MaxPooling1D	2	2	-	1
3	Conv1D	32	11	Relu	1
4	MaxPooling1D	2	2	-	1
5	Dropout	-	-	-	-
6	Flatten	6	-	-	-

.

4.2. Convolutional Kernel Scale Selection

The optimal parameters for the convolutional neural network were determined using control variables in Section 3.2. Subsequently, the construction of the multi–scale convolutional neural network was performed based on the optimal one–dimensional convolutional neural network. The convolution kernel sizes were primarily selected from 1 × 7, 1 × 9, 1 × 11, 1 × 13, 1 × 32, 1 × 64, aiming to capture signal features with varying degrees of fineness from high to low.

The two–layer multi–scale convolutional neural network is chosen based on the following design principles: the first layer utilizes convolution kernels with a broad scale range to extract features over a wide range with lower refinement; the second layer employs convolution kernels with a smaller scale range to convolve the previously extracted features with varying precision, enhancing feature extraction to achieve a higher level of feature expression. Following these design principles, the scale selection for the convolution kernels in the two layers is experimentally analyzed, and the number of convolution kernels in the chosen two–layer multi–scale convolutional layer is presented in

Table 10. Number of convolution kernels.

First Layer	k1	k2	k3
Convolution kernels	1 × 7, 1 × 32	1 × 7, 1 × 32,1 × 64	1 × 7, 1 × 11, 1 × 32, 1 × 64
Second layer	k4	k5	k6
Convolution kernels	1 × 9, 1 × 11	1 × 7, 1 × 9, 1 × 11	1 × 7, 1 × 9, 1 × 11, 1 × 13

.

4.2.1. Validity Verification of MS–1D–CNN Model

Input Sample Settings

The input data consists of one–dimensional time–domain vibration signals, typically necessitating a substantial volume of data for effective learning by a one–dimensional convolutional neural network. Simultaneously, for optimal utilization of experimental data, we select dataset C with the most complicated rotational speed at measurement point 0 in Table 1 for the comparative analysis. This dataset encompasses 1800 sample points, with each point comprising 2500 data points, totaling 4.5 million data points. Subsequently, the entire dataset is divided into a training set and a test set for input into the network model during the learning test.

Analysis of Calculation Results

Figure 14 illustrates the iteration trend of the test set and training set with the first layer of the model’s convolution kernel scale set to 1 × 7, 1 × 32, and the second layer set to 1 × 9, 1 × 11. Figure 15 displays the outcomes when the first layer’s convolution kernel scale is set to 1 × 7, 1 × 32, and the second layer is set to 1 × 7, 1 × 9, 1 × 11. In Figure 16, the first layer’s convolution kernel scale is 1 × 7, 1 × 32, and the second layer’s is 1 × 7, 1 × 9, 1 × 11, 1 × 13. For Figure 17, the first layer’s convolution kernel scale is 1 × 7, 1 × 32, 1 × 64, and the second layer’s is 1 × 9, 1 × 11. Continuing, in Figure 18, the first layer’s convolution kernel scale is 1 × 7, 1 × 32, 1 × 64, and the second layer’s is 1 × 7, 1 × 9, 1 × 11. In Figure 19, the first layer’s convolution kernel scale is 1 × 7, 1 × 32, 1 × 64, and the second layer’s is 1 × 7, 1 × 9, 1 × 11, 1 × 13. In Figure 20, the first layer’s convolution kernel scale is 1 × 7, 1 × 11, 1 × 32, 1 × 64, and the second layer’s is 1 × 9, 1 × 11. In Figure 21, the first layer’s convolution kernel scale is 1 × 7, 1 × 11, 1 × 32, 1 × 64, and the second layer’s is 1 × 7, 1 × 9, 1 × 11. In Figure 22, the first layer’s convolution kernel scale is 1 × 7, 1 × 11, 1 × 32, 1 × 64, and the second layer’s is 1 × 7, 1 × 9, 1 × 11, 1 × 13. The model achieves high accuracies close to 1 in both the training and test sets, with loss values approaching 0. The models converge successfully with various convolutional kernels, demonstrating the overall good performance of the model.

4.2.2. Optimal Convolutional Kernel Scale

As depicted in Figure 23 and

Table 11. Convolution kernel comparison results.

Number of Kernel	k1,k4	k1,k5	k1,k6	k2,k4	k2,k5
Accuracy	0.9796	0.9667	0.9815	0.9667	0.9778
Loss	0.1158	0.1413	0.1043	0.1315	0.1091
Number of Kernel	k2,k6	k3,k4	k3,k5	k3,k6
Accuracy	0.9815	0.9778	0.9519	0.9704
Loss	0.1040	0.1088	0.1364	0.1007

, the one–dimensional convolutional neural network, comprising two multi–scale convolutional layers, exhibits superior fault identification capabilities, achieving an accuracy of 0.9815. In comparison with the conventional one–dimensional convolutional neural network, the multi–scale convolutional neural network excels in identifying deep–level feature signals, enhancing effectiveness in pump–fault–classification detection. The two–layer multi–scale convolutional kernel numbers are denoted as k₁ and k₃; the convolutional neural network at k₂ and k₆ attains 0.9815 accuracy post training. Acknowledging that an increase in network size and training parameters consumes excessive computational and storage resources, leading to prolonged training times and reduced rotational speed, and that augmenting the number of convolutional kernels yields diminishing changes in accuracy gap; convolution kernel parameters are selected for k₁ and k₃, respectively. The first layer of the multi–scale convolutional neural network has convolution kernel sizes of 1 × 7 and 1 × 32, while the second layer has sizes of 1 × 7, 1 × 9, 1 × 11, 1 × 13. A uniform padding method is employed to maintain equal lengths of input and output sequences, ensuring a smoother feature extraction process, preventing information loss, extracting signal features with maximum precision, and enhancing accuracy.

5. Result and Discussion

In Section 4 of this paper, we select the optimal hyperparameters and the number of multi–scale convolutional kernels to define the parameter settings for the MS–1D–CNN model. Leveraging the experimental data from Section 2, this section computes the classification results of the models under each operating condition. It confirms that the accuracy of the MS–1D–CNN model surpasses that of the 1D–CNN model in both single and multiple conditions of centrifugal pump operation, thereby substantiating the model’s reliability.

5.1. Fault–Diagnosis Results at Single Operating Condition

Choosing dataset A from Table 2 for the experiment, data from six distinct measurement points of the pump are fed into the model, and the resulting fault–classification comparative results are presented in

Table 12. Comparative analysis of outcomes between MS–1D–CNN and 1D–CNN at single operating condition.

Measuring Point (Dataset A)	Models	Val–Accuracy	Val–Loss
0	1D–CNN	0.9222	0.2307
	MS–1D–CNN	0.9517	0.1112
1	1D–CNN	0.7667	0.4308
	MS–1D–CNN	0.8967	0.1995
2	1D–CNN	0.8556	0.2957
	MS–1D–CNN	0.9217	0.1675
3	1D–CNN	0.6556	0.6026
	MS–1D–CNN	0.7222	0.4696
4	1D–CNN	0.9333	0.1735
	MS–1D–CNN	0.9450	0.1175
5	1D–CNN	0.9167	0.1930
	MS–1D–CNN	0.9467	0.1171

below:

In a single operating condition, the MS–1D–CNN model, proposed in this paper, exhibits superior accuracy and lower average validation loss at various measurement points compared to the 1D–CNN model.

Five types of damage and a classification for a normal pump condition were considered. The corresponding fault labels are provided in

Table 13. The corresponding labels for the fault types are shown in the table below.

Type of Fault	Corresponding Labels	Binary Coding
Base loosening	0	000001
Bearing inner ring fault	1	000010
Pump body leakage	2	000100
Bearing outer ring fault	3	001000
Impeller eccentricity (shaft misalignment)	4	010000
Normal operation	5	100000

below.

The confusion matrix is a common tool for assessing the quality of classifier output. Figure 24a–f display fault–classification results for six distinct measurement points, utilizing the confusion matrix. The diagonal elements represent the probability of the predicted label matching the true label. The off–diagonal elements signify the probability of mislabeling by the classifier, and the sum of probabilities in each row is 1. A more favorable trajectory in the confusion matrix (i.e., a larger sum of elements on the main diagonal) indicates higher prediction accuracy [34].

In Figure 24a, under condition 1, the fault–classification accuracy for label one in the confusion matrix, corresponding to measurement point 0, is 0.71. The probability of misclassifying label one as label six is 0.29. The prediction accuracy for all other labels is 1. The classification accuracy for measurement point 0, with a pump rotational speed of 1450 r/min, is calculated as 0.9517 from the six diagonal elements. Better classification results were obtained.

Figure 24b–d display the fault–classification results obtained using the MS–1D–CNN model for measurement points 1, 2, and 3. At measurement point 1, the recognition rates are 0.61 for base loosening, 0.71 for the pump’s normal working condition, and all others are 1. For measurement point 2, the recognition rates are 0.9 for base loosening, 0.63 for the pump’s normal working condition, and all others are 1. The overall fault–classification accuracy for measurement point 2 is 0.9217, and the overall accuracy for the fault classification of measurement point 3 is 0.72.

Additionally, Figure 24e,f illustrates the fault–classification accuracies of the MS–1D–CNN model for measurement points 4 and 5, respectively. The computational results indicate that the accuracy of fault classification for measurement point 4 is 0.945, and for measurement point 5 is 0.9467.

The results indicate that MS–1D–CNN can identify the fault types of centrifugal pumps under single operating conditions, achieving fault–classification accuracy exceeding 0.85, with the highest reaching 1. Notably, the accuracy at measurement point 3 is comparatively lower, possibly attributed to excessive noise during the actual measurement of centrifugal pump vibration data. In addition, it can be observed that fault identification is best at measurement point 0.

5.2. Fault–Diagnosis Results at Double Operating Condition

Dataset B from Table 2 is chosen and input into the two models for comparison, with the results presented in

Table 14. Comparative analysis of outcomes between MS–1D–CNN and 1D–CNN at double operating condition.

Measuring Point (Dataset B)	Models	Val–Accuracy	Val–Loss
0	1D–CNN	0.9667	0.1036
	MS–1D–CNN	0.9767	0.0835
1	1D–CNN	0.9250	0.2009
	MS–1D–CNN	0.9400	0.1375
2	1D–CNN	0.9556	0.1700
	MS–1D–CNN	0.9650	0.1007
3	1D–CNN	0.7083	0.4959
	MS–1D–CNN	0.8767	0.3008
4	1D–CNN	0.8528	0.2497
	MS–1D–CNN	0.8817	0.2041
5	1D–CNN	0.9222	0.2048
	MS–1D–CNN	0.9517	0.1196

.

In scenarios involving a combination of dual operating conditions, the MS–1D–CNN model, as proposed in this paper, exhibits superior accuracy and lower average validation loss at various measurement points compared to the 1D–CNN model.

The fault–classification results for the six measurement points, as represented by the confusion matrix, are depicted in Figure 25a–f.

Figure 25a illustrates the results of fault classification at measurement point 0 when condition 1 and condition 2 are mixed, using the confusion matrix. The accuracy of fault classification for label one is 0.98, with a probability of mistakenly identifying label one as label six being 0.02. The recognition rate for the pump’s normal state is 0.9, for base loosening and bearing inner ring fault it is 0.98, and for all other faults it is 1. The fault–classification accuracy at measurement point 0 is calculated to be 0.9767 from the six diagonal elements.

Figure 25b–d present the fault–classification results of the MS–1D–CNN model for measurement points 1, 2, and 3, respectively. The fault–classification accuracy is 0.94 for measurement point 1, 0.965 for measurement point 2, and 0.8767 for measurement point 3.

Additionally, Figure 24e and Figure 23f illustrate the fault–classification accuracies of the MS–1D–CNN model for measurement points 4 and 5, respectively. The computational results indicate that the accuracy of fault classification is 0.8817 for measurement point 4 and 0.9517 for measurement point 5.

The results indicate that the MS–1D–CNN can accurately identify fault types under the mixed conditions of dual operating scenarios, achieving a fault–classification accuracy exceeding 0.85 in all instances, with the potential to reach up to 1. Experimentally, it can be seen that the fault classification of measurement point 0 is the best.

5.3. Fault–Diagnosis Results at Triple Operating Condition

Utilizing dataset C from Table 2, subsequently, the time–domain vibration data of the pump is fed into the two models for comparative analysis. The results of this comparison are presented in

Table 15. Comparative analysis of outcomes between MS–1D–CNN and 1D–CNN at triple operating condition.

Measuring Point (Dataset C)	Models	Val–Accuracy	Val–Loss
0	1D–CNN	0.8815	0.2081
	MS–1D–CNN	0.9700	0.1506
1	1D–CNN	0.7963	0.5109
	MS–1–DCNN	0.9083	0.3018
2	1D–CNN	0.8093	0.4722
	MS–1D–CNN	0.9133	0.3307
3	1D–CNN	0.7481	0.6712
	MS–1D–CNN	0.8733	0.2970
4	1D–CNN	0.8444	0.4085
	MS–1D–CNN	0.9150	0.2070
5	1D–CNN	0.8167	0.3490
	MS–1D–CNN	0.9250	0.2152

.

In scenarios involving three operating conditions, the MS–1D–CNN model, as proposed in this paper, exhibits superior accuracy and a slightly lower average validation loss when compared to the 1D–CNN model across various measurement points.

The fault–classification results, represented by the confusion matrix, for the six measurement points are illustrated in Figure 26a–f.

Figure 26a illustrates the fault–classification results using a confusion matrix at measurement point 0 under a mix of condition 1, condition 2, and condition 3. The accuracy for label one is 0.99, with a 0.01 probability of misclassifying it as label six. The fault identification rate for base loosening is 0.99, for bearing inner ring faults it is 0.98, and for the normal condition of the centrifugal pump it is 0.86. The overall accuracy at measurement point 0 is calculated as 0.97 from the six diagonal elements.

Figure 26b–d present the fault–classification results of the MS–1D–CNN model for measurement points 1, 2, and 3. Figure 24b shows that the recognition rate for base loosening fault is 0.9, for bearing inner ring faults it is 0.8, for impeller eccentricity it is 0.95, for pump normal condition it is 0.8, and for all other faults, it is 1. The accuracy for measurement point 1 is 0.9083, for point 2 is 0.9133, and for point 3 is 0.9517.

Figure 26e,f illustrate the fault–classification accuracies of the MS–1D–CNN model for measurement points 4 and 5. The computational results indicate an accuracy of 0.915 for point 4 and 0.925 for point 5. Among them, the recognition rate for base loosening at measurement point 4 is 0.94, for bearing inner ring faults it is 0.92, and for other faults it is 1. At measurement point 5, the recognition rate for base loosening is 0.89; for bearing inner ring faults it is 0.96. For bearing outer ring faults it is 0.99, and for pump body leakage it is 1.

The results indicate that the MS–1D–CNN can accurately classify faults in the mixed case of triple operating conditions, achieving an accuracy of over 0.87 in all cases and up to 1. It can be seen from the experiments that the fault classification of measurement point 0 is the best.

6. Conclusions

In this study on centrifugal pump mechanical–fault diagnosis, the analysis of measured vibration experimental data reveals distinct multiplier differences in vibration acceleration signals corresponding to various mechanical faults. Specifically, the signals for impeller eccentricity are approximately four times the baseline, for bearing outer ring faults about seven times, for bearing inner ring faults about three times, for base loosening about twice the baseline, and for pump body leakage about sixty–three times. These substantial deviations from the baseline provide essential characteristic information for fault diagnosis.

Then, this paper successfully develops efficient fault–diagnosis models for centrifugal pumps. The newly proposed MS–1D–CNN model, when compared to the traditional 1D–CNN model, demonstrates a significant improvement in fault–diagnosis performance. The MS–1D–CNN model introduces a novel multi–scale perception function, enhancing its ability to comprehensively capture contextual feature information in vibration signals. This innovation broadens the scope of data information learning, leading to improved fault diagnosis accuracy. In a comparison experiment at single operating condition, the MS–1D–CNN model shows an average accuracy improvement of about 6% compared to the traditional model. Furthermore, by incorporating the novel multi–scale perception function, the MS–1D–CNN model eliminates steps like data feature extraction, simplifying the data processing process. This not only enhances the model’s efficiency but also enables more direct input of measured data, increasing the model’s practicality.

Experimental results under various operating conditions demonstrate the strong performance of the MS–1D–CNN model. It achieves an average fault diagnosis accuracy of 89%, 91%, and 93% for single operating condition, double operating condition, and triple operating condition, respectively. This indicates that the MS–1D–CNN model is suitable not only for the straightforward scenario of single operating condition but can also handle intricate conditions such as multiple operating conditions at the same time, meeting practical application requirements. Additionally, the experiment reveals that measurement point 0 is more representative in fault diagnosis compared to other measurement points.

In conclusion, our study presents an advanced, efficient, and practical model for the mechanical–fault diagnosis of centrifugal pumps. This holds significant importance in enhancing the accuracy of pump–fault diagnosis and enabling predictive maintenance. Future research avenues may focus on expanding the dataset size, optimizing the model structure, and achieving real–time monitoring in depth.