Prediction of bolt missing fault for multistage rotor by experimental test and analysis

The high-pressure rotor of aero-engine is assembled by numerous bolts under high manufacture precision. The connected structure is subjected to both axial force and transverse vibration during service, which may result in individual bolt loosen. In this study, the influence of bolt missing on the dynamic characteristics is analyzed by numerical simulation. A test rig capable of impact and frequency sweeping experiment under axial tension was constructed. The vibration response features in the simulation were then extracted. The loss function of the mean absolute error and the decision method of extreme gradient boosting were used to predict the bolt missing position. The results show that the proposed model can reach a prediction precision of more than 90%. Moreover, the coefficient of determination evaluation index of the prediction effect reaches 0.9, which is significantly higher than those of other conventional models such as multivariate linear regression and multivariate adaptive regression spliness.


Introduction
The high-pressure rotor is the core component of an aeroengine. It is precisely assembled using numerous bolts and an interference fit. However, such a bolted structure will be subjected to both axial force and transverse vibration during service, resulting in individual bolt loosen. Thus, the detection capabilities are particularly important before periodic maintenance. The assembly quality, such as the connection performance, can be correlated with the complex excitation environment in the operation state, which is an important basis for evaluating the product quality of large and complex rotating machinery.
Many studies on precision assembly technologies, such as on geometric deformation and error transmission in the assembly process of components, have concentrated on the final assembly quality, which can be achieved by controlling the fitting accuracy [1][2][3]. Mu et al. comprehensively considered the manufacturing error of high-pressure rotor parts and the influence of deformation in the assembly process on assembly accuracy [4]. An assembly accuracy evaluation model with a rough surface fit was established using a homogeneous coordinate transformation method. Ding et al. proposed a three-dimensional deviation model and combined a stack concentricity control strategy based on the improved Jacobian spinor theory for determining the rotor rotation characteristics and multistage assembly characteristics [5]. Other researchers also considered the assembly requirements of rigid unbalanced masses and proposed several evaluation methods for rotor assembly quality [6][7][8]. However, the assembly quality of a rotor cannot be completely reflected only by the geometric and mass deviations. In particular, it is difficult to reflect the dynamic characteristics of a rotor based only on the above research content.
Many researchers have gradually realized that the bolt connection state has a significant impact on the assembly quality [9][10][11]. For example, Liu et al. found that the external loads and connecting structures could change the nonlinear contact characteristics of bolts, which affect the rotating state of the rotor [12]. Another study combined bolt fastening with rotor dynamic response analysis to establish the effective dynamic characteristics [13][14][15]. Marc proposed a nonlinear dynamic model of the local contact and bolt connection quality for a bolted flange structure of an aero-engine [16]. Yu established a nonlinear dynamic model considering the bending force of bolts under the action of an axial force, indicating that the critical speed distribution of the rotor is affected by the bolt connection [17]. Although the aforementioned studies have gradually expanded from structural performance to comprehensive manufacturing testing, many deficiencies still exist in the research of high-pressure rotor bolt assembly technology for service performance. The fundamental problem is that the bolt connection performance of the multistage rotor and multistage mating surface is not integrated with the composite load environment oriented to service performance. Furthermore, no special test equipment and identification method for bolt assembly quality have been established.
XGBoost is an effective machine learning algorithm for multiple nonlinear problems [18][19][20][21]. In addition, this algorithm implements a variety of transformations that can be applied for data analysis and fault detection of different engineering objects [22,23]. Lin et al. combined the XGBoost model with finite element simulation to predict the strength of self-punching riveting joints [24]. They achieved an error in the strength prediction results of less than 7.6% compared with experimental values. Phan et al. proposed a hybrid prediction model considering different numerical weather forecasts and the XGBoost model for short-term wind power generation [25]. The prediction results showed that their model has good prediction performance and accuracy. Patnaik et al. applied the XGBoost model to perform grid fault detection and classification [26]. Each level decomposed by the maximum overlapping discrete wavelet transform was converted into differential energy, which was used as the input of the XGBoost model to realize early detection and classification of power grid faults. Choi developed an XGBoost prediction model for an aggregate material database to accurately reflect the properties of materials. The prediction accuracy index R2 reached 0.985 [27]. Although XGBoost can be applied to all types of regression prediction problems, it is not suitable for evaluating the quality of multi-bolt assembly in rotation machinery under composite loads.
In this study, a high-pressure aero-engine rotor was taken as the research object. A dynamic model considering a multi-bolt connection was established to analyze the influence of single bolt missing on the first three natural frequencies. To eliminate the influence of over fitting caused by fluctuation of experimental value, a prediction model was developed using XGBoost based on the mean absolute error (MAE-XGBoost). Single bolt missing at different junction surface (JS) were investigated by a multi-excitation test rig, and the vibration response features were obtained according to the dynamic characteristics of the multistage rotor. The proposed prediction model was used to evaluate the precision and effectiveness of the bolt loss position.

Model and theory
In this section, variations in the structural state parameters caused by the bolt flange are introduced into the rotor dynamic model. In addition, a mathematical evaluation method using MAE-XGBoost is established to analyze the deviation of the mounting peripheral bolts. This method can provide a theoretical model for the subsequent analysis and evaluation of the experimental results.

Equivalent bending stiffness of bolt deviation
The bolts were applied stage-by-stage in a two-component assembly. The connection provides both the axial and radial forces, which relate to the rotor bending response. The distribution in the equivalent stiffness loss of a single bolt is determined as where R and r are the radii of the outer and inner circles, respectively, and is the semicircular central angle corresponding to the equivalent crack. Angle is the azimuth of the bolt deviation, which is directly determined based on the structure and material [28,29]. According to a previous study [17], the angle for a bolt connection region can be expressed as follows Based on Eq. (1), the cross-sectional moment of inertia of the equivalent crack section with respect to axes x and y can be obtained: where S denotes the region affected by the equivalent crack,A represents the sectorial area of r , while e x and e y are the geometric eccentricities along the x and y directions, respectively. The equivalent bending stiffness loss of the bolt deviation in the corresponding zone is calculated as The equivalent elastic modulus E * can be obtained from the material parameters and contact theory [30]: where E 1 and E 2 are the Young's moduli of the front and rear fitting bodies, while 1 and 2 are the corresponding Poisson's ratios.
The displacement vector bolt of the connecting flange of the bolt-connected element can be expressed as

Dynamic model of rotor system
The section area A s and the section moment of inertia I s in the variable section at s is shown in Fig. 3, which can be expressed as where 1 , 1 , 2 , 2 , 2 , 2 , are given by With Eqs. (6)- (13), the motion equation of the axis element of variable section can be obtained as where the mass matrix shaft , gyroscopic matrix shaft , and stiffness matrix K shaft of the shaft element are where the mass matrix d and gyroscopic matrix d are from disk element. sys is the damping matrix and sys is the stiffness matrix of rotor system. is the rotational speed. bolt is the bolt missing matrix.̈(t),̇ (t) and (t) are the acceleration, velocity, and displacement vectors of the rotor system.
In the absence of bolt missing, the first three natural frequencies of the rotor system are 76.79 Hz, 138.14 Hz, and 885.64 Hz, respectively. When the single bolt at JS2 is missed, the first three critical speed is almost the same.

Formulation of MAE-XGBoost model
XGBoost is an integrated regression algorithm whose objective function is [20] where l i ,̂i denotes the mean squared error (MSE) loss function for measuring the difference between the predicted value ̂i and target value i based on the second-order Taylor expansion. Ω f t stands for the penalty term, which is used to control the complexity of the model. The iterative process minimizes the gap between i and ̂i by constantly adding f t to the objective function, which is treated as an iterative optimization process.
The difference between the target and predicted values is squared to the MSE loss, and the MAE loss is linear. The MAE loss reflects the true error in the modeling process of the XGBoost model, and can reduce the excessive loss of the model.
However, the second-order Taylor expansion of the MAE is zero, which is inapplicable to the requirements of the objective function. Therefore, an approximate function L for the MAE loss function is proposed as follows: where c is a real number that can be adjusted according to the distribution of the actual data. The objective function of the improved model is Through the above transformation, an MAE-XGBoost prediction model is established.

Experimental program
This section describes the experimental object, multiaxis loading test device, and experimental process.

Experimental subject
The multistage rotor system has the same structural composition, namely, assembled by a front shaft, high pressure compressor (HPC) stage-by-stage disks, turbine disk, rear shaft, and other key components. The position and direction of the multiaxis excitation are shown in Fig. 4. The experiment mainly uses the JS2 and JS4 as the bolt missing flange, which is assembled by 36 and 24 bolts, respectively. The tightening torque of each bolt is 8 N·m.

Multi-excitation test rig
The multi-excitation loading device consists of two major parts, as shown in Fig. 5. The excitation system comprises a vibration system (DC-600 produced by Suzhou Sushi Testing Group Co., Ltd) and an axial tension control system, which simulates the real loading state of transverse vibration and axial force for the high-pressure rotor during operation. The main function of the (22) vibration system is to provide a certain frequency and amplitude. It has two modes: random excitation and sweep excitation. The excitation frequency range is 0-3000 Hz and the excitation direction is indicated in Fig. 5. The tension control system can produce an axial force ranging from 0 to 150 kN, with an error fluctuation of ± 30 N around the set tension value. The second part is the digital signal acquisition system, which includes a digital signal acquisition instrument (INV3062A, produced by China Orient Institute of Noise and Vibration) and an acceleration sensor (CAYD115 type ICP accelerometer, produced by Niell-Tech). With these devices, the vibration responses of the measuring points under different bolt-deviation positions and specified multiaxial loading conditions can be obtained.

Experimental scheme
To determine the relationship between the overall vibration characteristics and bolt connection state of the flange, it is necessary to remove the bolts at a specified matching surface and apply a certain axial tension, as shown in Fig. 6. The vibration characteristics of multistage rotor system were collected in real time through the accelerometer. Accordingly, each bolt at JS2 and JS4 were removed in turn, and their vibration responses were recorded. Based on the tests, the disturbance effect of a single experiment was eliminated. The experimental process is illustrated in Fig. 5. Both impact method and frequency sweeping method were selected as the transverse vibration excitation. Frequency response analysis of 0-1000 Hz was carried out by impact excitation. However, an obvious resonance between the test board and rotor occurred when the sweeping excitation frequency was less than 300 Hz. Therefore, the sweeping excitation range was defined as 350-1000 Hz. The axial tension was set as 30 kN to ensure that the rotor has stable and enough placement response.

Analysis of experimental data
This section takes the change of bolt missing location on JS2 as an example to illustrate the data feature extraction and distribution analysis, while bolt missing location on JS4 has the same process.

Signal conversion and feature selection
The vibration signal in the time domain is converted into the frequency domain by Fourier transformation. This transformation expands an arbitrary function x(t) that satisfies the condition of absolute integrability into a weighted summation of a standard function: As shown in Fig. 7(a), the correctness of the first three natural frequencies of the dynamic model in Sect. 2 were verified. However, the impact excitation not only had an influence on the excessive axial force fluctuation, but also affected the frequency feature identification of single bolt missing. Therefore, sweep frequency within 350-1000 Hz was used for feature extraction to avoiding above influence, showed in Fig. 7(b).
To extracting enough vibration features, two feature selection criteria were established according to the natural frequencies: a) the extracted frequency peak and its amplitude are the frequency doubling of the simulation result, and b) the extracted features have significant differences, at least in terms of frequency or amplitude.
According to the above criteria of feature selection, the selected frequency values, the corresponding frequency multiples of the first two natural frequencies are presented in Table 1. The amplitude corresponding to the extracted frequency is also added to the eigenvalue cluster. Thus, a dataset of 37 samples, with the extracted features and the corresponding bolt missing positions, is established, including a sample of the fully connected state.

Data distribution analysis
The distribution relationship of the extracted frequency peak, frequency amplitude, and bolt missing angle from sensor 1 is illustrated in Fig. 8, where the number 0 indicates a fully connected state. The radar graphs show that the removed bolts at different angles affect both the frequency and amplitude. The fluctuation range of the frequency is approximately from 1 to 4 Hz. There are some obvious abnormal fluctuation values, such as the loss positions of bolt numbers 4, 9, 18, 21, and 27 in Fig. 8(a); 21 and 28 in Fig. 8(e). The change in the frequency amplitude is small, but there are also some obvious changeable values, as depicted in Fig. 8(b) and (f).
The fluctuations in the above eigenvalues may be caused by the uncontrollable environmental noise, stability disturbance during axial tension loading, or influence of residual stress during the testing process. Owing to the above variable factors, it is difficult to observe a clear relationship between the vibration response and bolt missing position directly.

Results and discussions
This study utilized the MAE-XGBoost model to systematically analyze the experimental datasets. The evaluation method of the bolt missing position was established from three aspects: main feature extraction, model parameter optimization, and prediction effect comparison. Some common prediction models were also used for comparison.

Input and output of the model
From 37 datasets collected in the experiment, the datasets of the extracted frequency and corresponding amplitude were used as the input parameter, while the bolt missing position angle (expressed in radians) was employed as the output parameter. Thus, the relationship between the input and output data was established. In addition, all data samples were distributed randomly, in which 80% were used for model training while the remaining 20% were utilized to identify the forecasting effect of the prediction model. Three commonly used evaluation metrics, namely, R 2 , MSE, and MAE, were used to evaluate the prediction accuracy. The data division and model verification processes are illustrated in Fig. 9.

Comparison of model input datasets
The collected vibration features were divided into three types of datasets: peak frequency only, frequency amplitude only, and both peak frequency and corresponding amplitude. A comparison of the prediction accuracy of each dataset using R 2 is displayed in Fig. 10. Compared with other models, MAE-XGBoost has the best evaluation results under all listed conditions [31]. The assessment result also indicates that the dataset containing the vibration amplitude has a better prediction effect, which shows that the amplitude of the peak frequency has the strongest correlation with the bolt deviation position. After comprehensively considering the prediction effect, the frequency value and frequency amplitude datasets were used as the inputs for all models.

Parameter optimization strategy
To improve the prediction accuracy of the MAE-XGBoost model, two important parameters need to be adjusted: depth D and number N of the underlying regression tree. These two parameters directly affect the prediction accuracy of the model. Taking the data samples obtained by sensor 1 as an example, the process of adjusting the parameters is as follows: (1) adjusting N until the number of regression trees reaches 17-the increase in the number of regression trees no longer affects the prediction accuracy, as depicted in Fig. 11(a); (2) adjusting D until the depth of the regression tree reaches 5-the prediction accuracy of the model changes slightly, as illustrated in Fig. 11(b). The specific parameters of the modeling process of JS2 is shown in Table 2.

Accuracy of the prediction model
After confirming the dataset division and parameter optimization, the prediction performances of different intelligent forecasting models were evaluated, as displayed in Fig. 12 for the bolt loss at JS4. Similarly, Fig. 13 shows the prediction of bolt missing at JS4. The prediction curves indicate that the results by MAE-XGBoost are closer to the actual values than XGBoost. Moreover, the prediction precisions for sensors 1 to 4 are over than 90%.
To quantitatively evaluate the performance, Table 3 lists the specific prediction accuracy. All evaluation indicators reveal that the proposed prediction model has the

Conclusions
The influence of bolt deviation on the natural frequencies of a high-pressure rotor was investigated numerically. In the experiments, multi-excitations under bolt missing were evaluated, and the vibration features were extracted. The MAE-XGBoost model was proposed to eliminate the experimental data and predict the bolt deviation position of the rotor accurately. The following conclusions were drawn from this study:   (1) Numerical dynamic model of drum multistage rotor with single bolt missing is deduced and established. It is found that the missing single bolt only has little influence on the first three critical speed of the rotating system. (2) A multi-axial loading system is introduced. The accuracy of the rotor model is verified by impact experiment. The frequency response characteristics within 350-1000 Hz were obtained by frequency sweeping experiment, which shows clearly that the difference of vibration characteristics caused by bolts missing at different positions.

Data availability
The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to their containing information that could compromise the privacy of research participant.

Declarations
Ethical approval Not applicable.

Consent to participate Not applicable.
Consent for publication Not applicable.