Reliability analysis on resonance for low-pressure compressor rotor blade based on least squares support vector machine with leave-one-out cross-validation

Abstract

This research article analyzes the resonant reliability at the rotating speed of 6150.0 r/min for low-pressure compressor rotor blade. The aim is to improve the computational efficiency of reliability analysis. This study applies least squares support vector machine to predict the natural frequencies of the low-pressure compressor rotor blade considered. To build a more stable and reliable least squares support vector machine model, leave-one-out cross-validation is introduced to search for the optimal parameters of least squares support vector machine. Least squares support vector machine with leave-one-out cross-validation is presented to analyze the resonant reliability. Additionally, the modal analysis at the rotating speed of 6150.0 r/min for the rotor blade is considered as a tandem system to simplify the analysis and design process, and the randomness of influence factors on frequencies, such as material properties, structural dimension, and operating condition, is taken into consideration. Back-propagation neural network is compared to verify the proposed approach based on the same training and testing sets as least squares support vector machine with leave-one-out cross-validation. Finally, the statistical results prove that the proposed approach is considered to be effective and feasible and can be applied to structural reliability analysis.

Keywords

Resonance reliability analysis natural frequency rotor blade least squares support vector machine leave-one-out cross-validation

Introduction

High reliability is usually required for the stabilities of aeronautic and astronautic structures. Especially, resonant reliability plays an important role in the safety operation of low-pressure compressor (LPC) rotor blade. Vibration is inevitable, but high resonant reliability is helpful to improve the structural security. Some research works^1–4 on vibration of rotor blade have been carried out based on deterministic analysis, but its resonant reliability was rarely involved. Therefore, it is extremely necessary to analyze the resonant reliability for LPC rotor blade by considering the influence of random factors on vibration characteristic.

In the reliability analysis, the Monte Carlo method (MCM)^5–9 is often applied to analyze and calculate structural reliability, but this method possesses low computational efficiency due to the large number of simulations required. Some classical algorithms, such as first-order reliability method (FORM) and second-order reliability method (SORM), possess lower calculation precision for the complicated or implicit limit state functions in the reliability problems.¹⁰ In addition, the classical response surface method (RSM) is the main approach to solve the reliability problem with explicit limit state functions.^11,12 However, it is formidable for RSM to fit highly nonlinear response surface for the reliability problem with multiple basic random variables.^13–15

In recent years, machine learning algorithms have been successfully applied to structural reliability field, mainly including back-propagation neural network (BPNN) and support vector machine (SVM). In 1995, Cortes and Vapnik¹⁶ proposed a novel machine learning algorithm called SVM, which follows the principle of structure risk minimization in the statistical learning theory.^17–19 SVM can acquire the globally optimal solution on the basis of available data. In addition, SVM enjoys the advantage of fitting highly nonlinear response surface and ensuring highly computational efficiency and precision. Accordingly, SVM has been widely adopted in academic research and industrial applications.^20–24 The essence of SVM is to solve a convex and quadratic programming problem, and it is time-consuming.²⁵ Based on the traditional SVM, this article proposes least squares support vector machine (LSSVM),^25,26 which converts the process of training a model to solving a linear equation. Consequently, LSSVM can avoid the complex computation for the quadratic programming problem. LSSVM simplifies the calculation process, greatly reduces the computational complexity, and improves the solution speed.^27–32

To construct a high-precision LSSVM model, the parameters of LSSVM should be carefully set.^30–32 Appropriate parameters ensure the precision and efficiency of the structural reliability analysis. In this article, leave-one-out cross-validation (LOOCV)^33–35 is applied to seek the optimal parameters to improve the approximation capability and generalization ability of the LSSVM model, which makes the LSSVM model effectively overcome the appearance of over-learning and under-learning and greatly improves computational accuracy. Then, the parameters and training set are associated to construct the least squares support vector machine with leave-one-out cross-validation (LSSVM-LOOCV) model, and the model is used to predict vibration frequencies. Based on the randomness of the influence factors on frequencies, reliability analysis on resonance for LPC rotor blade is finished. Finally, the proposed method is compared with BPNN and MCM to examine the predictive ability of LSSVM-LOOCV.

LSSVM

LSSVM, an improved SVM, was proposed by Suykens and colleagues^25,26 based on regularization theory. LSSVM adopts a least squares linear system to solve classification and estimation problems instead of the quadratic programming method used in the traditional SVM.

Suppose that there is a set of given data ${(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{l}, y_{l})} \in R^{n} \times R$ , where $x_{i}$ is the input vector and $y_{i}$ is the corresponding output value. The goal of modeling is to construct a regression function $y = f (x)$ to predict the outputs ${y_{i}}$ corresponding to a new set of samples and minimize the structure risk.^27–32 In the feature space, the linear regression function is expressed as follows

f (x) = ω^{T} \cdot φ (x) + b

(1)

where $ω$ is the weight vector; $φ (\cdot)$ denotes the nonlinear mapping function, which can map nonlinear regression in the original feature space into linear regression in the high-dimensional feature space; and $b$ is the bias term.

Furthermore, the linear regression problem can be expressed as an optimization problem with equality constraints according to the principle of structural risk minimization (synthetically considering the minimization of the Vapnik–Chervonenkis (VC) dimension and empirical risk). The objective function of the optimization problem is given by equation (2)

\begin{matrix} \min_{ω, b, e} \frac{1}{2} ‖ ω ‖^{2} + \frac{1}{2} C \sum_{i = 1}^{l} e_{i}^{2} \\ subject to e_{i} = y_{i} - ω^{T} \cdot φ (x_{i}) - b, i = 1, 2, \dots, l \end{matrix}

(2)

where $C$ is the regularization parameter to determine the trade-off between the training error minimization and smoothness of the estimation function and $e_{i}$ is the error between the actual output and the predictive output of the ith sample.³⁰

To solve the above optimization problem, the Lagrange multiplier $α_{i}$ is introduced. Then equation (2) is converted into the optimization problem without constraint given by equation (3)

\begin{matrix} L (ω, b, e_{i}, α_{i}) = \frac{1}{2} ‖ ω ‖^{2} + \frac{1}{2} C \sum_{i = 1}^{l} e_{i}^{2} \\ - \sum_{i = 1}^{l} α_{i} [e_{i} - y_{i} + ω^{T} \cdot φ (x_{i}) + b] \end{matrix}

(3)

Based on the Karush–Kuhn–Tucker (KKT) conditions, the following equations are obtained

{\begin{matrix} \frac{\partial L}{\partial ω} = 0 \to ω = \sum_{i = 1}^{l} α_{i} φ (x_{i}) \\ \frac{\partial L}{\partial b} = 0 \to \sum_{i = 1}^{l} α_{i} = 0 \\ \frac{\partial L}{\partial e_{i}} = 0 \to α_{i} = {Ce}_{i}, i = 1, 2, \dots, l \\ \frac{\partial L}{\partial α_{i}} = 0 \to ω^{T} \cdot φ (x_{i}) + b + e_{i} - y_{i} = 0, i = 1, 2, \dots, l \end{matrix}

(4)

Finally, the regression decision function is written as the following explicit expression

f (x) = \sum_{i = 1}^{l} α_{i} K (x, x_{i}) + b, i = 1, 2, \dots, l

(5)

where $α$ and $b$ follow from equation (4), and $K (x, x_{i})$ is a kernel function with Mercer condition.

Selection of LSSVM parameters

The LSSVM parameters, including the form of kernel function, regularization parameter, and kernel parameter, are the key influence factors on the generalization ability of LSSVM model and fully describe the spatial structure and characteristics of the model. In order to improve the generalization ability of LSSVM, selecting the optimal parameters is indispensable. Specifically, the type of kernel function implicitly confirms the mapped high-dimensional feature space. The regularization parameter can adjust the confidence interval of statistical learning machine, and the kernel parameter implicitly affects the complexity of the distribution of input subspace. In this article, the radial basis function ( $K (x, x_{i}) = \exp (- {(x - x_{i})}^{2} / σ^{2})$ ) is selected as the kernel function of LSSVM, and LOOCV is applied to acquire the tuning parameters.^16–19

LSSVM-LOOCV model

In the process of establishing the LSSVM-LOOCV model, the main issue is to seek an optimal hyper-parameter combination, the regularization parameter $C$ , and kernel parameter $σ^{2}$ , corresponding to the actual problem. But no acknowledged and consistent method can be applied to select these parameters. This article utilizes LOOCV to search for the hyper-parameters and ensure their global optimality. LOOCV, an extreme case of k-fold cross-validation with k = n,^36,37 is widely applied in many fields³⁸ to get a reliable and stable computational model. This method possesses two obvious advantages: one is almost all the given sample points are used to train a model, and the predictive result is reliable; another is the experimental process which is repeatedly performed, because there is no random factor affecting experimental data.^33–35

Suppose that the given training set is $A = {(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})}$ with n points, and that they are distributed independently. For each iteration, n– 1 sample points are taken as the training subset to fit a model, and the remaining points are used as the testing subset to estimate the performance of the model.³³ This process is repeated for each sample point. Finally, all the estimates of the performance are combined, and the mean of the estimates is used as the global performance index.^34,35 Then, the optimal hyper-parameter combination is acquired. The parameters and relevant variables are associated to construct the LSSVM-LOOCV model. Figure 1 illustrates the process of establishing the LSSVM-LOOCV model.²⁴

Figure 1.

The process of establishing the least squares support vector machine with leave-one-out cross-validation (LSSVM-LOOCV) model.

Selection random variables

Modal analysis

This article considered an LPC rotor blade as research object. Figure 2 shows the finite element model with 2445 elements and 13,634 nodes. The mechanical properties of the blade material, Titanium alloy, such as the elastic modulus, the density, and Poisson’s ratio are set to 118 GPa, 4500 kg/m³, and 0.34, respectively. Both the flanks of the dovetail-shaped root were fixed to execute the modal analysis of the rotor blade. Figure 3 shows its profile diagram, where $w$ is the profile width, $t$ is the maximum thickness of the profile, and $γ$ is the torsion angle of the profile relative to the rotation shaft.

Figure 2.

Blade finite element model.

Figure 3.

Blade profile diagram.

To accomplish vibration characteristics analysis, the modal analysis was primarily implemented to calculate the natural frequencies and vibration modes of the rotor blade. In this section, the frequency distribution was calculated under the static state, crawling state, cruising state, and designing state with the rotating speeds of 0, 3737.0, 6150.0, 9916.2, and 11,000.0 r/min. The first four natural frequencies are shown in Table 1.

Table 1.

The first four natural frequencies of the blade.

Rotating speed (r/min)	Frequency (Hz)
Rotating speed (r/min)	1	2	3	4
0	196.73	632.04	956.35	1421.4
3737.0	219.96	650.16	960.56	1433.5
6150.0	254.51	679.25	967.92	1452.8
9916.2	324.99	743.78	987.63	1494.1
11,000.0	347.30	764.83	995.43	1507.0

The Campbell chart is shown in Figure 4, where the horizontal axis represents the engine rotating speed $n$ , the vertical axis denotes the natural frequency $ω_{i}$ , the dynamic frequency curves indicate the relationship between the natural frequencies and the rotating speeds, and each ray shows the exciting frequency $ς$ . Each intersection between the ray and the dynamic frequency curve is a resonance point.

Figure 4.

The Campbell diagram of the blade.

Extensive facts illustrate that resonance caused in the first three modes under the crawling state is the main failure form of the rotor blade. Therefore, the harmonic response analysis at the rotating speed of 6150.0 r/min was performed to test whether the first three modes were in a state of resonance (Figure 5).

Figure 5.

The harmonic response curve of the blade at the rotating speed of 6150.0 r/min.

The curve illustrates that resonance occurs in the first and third modes at the rotating speed of 6150.0 r/min, which is in good agreement with the actual situation.

Random variables

The material parameters, structural dimension, and operating condition possess inherent randomness, which possess enormous impact on vibration frequencies. Therefore, $x = [E ρ μ w_{1} w_{2} w_{3} t_{1} t_{2} t_{3} γ_{1} γ_{2} γ_{3} H n]$ was selected as the input vector, and the natural frequency was considered as the output $y$ . The statistical characteristics of random variables are shown in Table 2.

Table 2.

The statistical characteristics of random variables.

Random variables	Mean	Standard deviation
Elastic modulus $E$ (MPa)	118,000	2360
Density $ρ$ (t/mm³)	4.5e–009	9e–011
Poisson’s ratio $μ$	0.34	0.0068
Blade-tip profile width $w_{1}$ (mm)	118	2.36
Blade-inner profile width $w_{2}$ (mm)	102.1	2.042
Blade-root profile width $w_{3}$ (mm)	86.5	1.73
Maximal thickness of blade-tip profile $t_{1}$ (mm)	2.785	0.0557
Maximal thickness of blade-inner profile $t_{2}$ (mm)	5.8	0.116
Maximal thickness of blade-root profile $t_{3}$ (mm)	7.1	0.142
Torsion angle of blade-tip profile $γ_{1}$ (rad)	0.9119	0.018238
Torsion angle of blade-inner profile $γ_{2}$ (rad)	0.5142	0.010284
Torsion angle of blade-root profile $γ_{3}$ (rad)	−0.1251	0.002502
Blade height $H$ (mm)	191.7	3.834
Rotating speed $n$ (rad/s)	644.03	12.8806

Algorithm models

In this section, algorithm models were constructed to predict natural frequencies. This article chose BPNN to compare the feasibility of the proposed method.²⁴ Some statistical metrics were used to evaluate the estimating performance of the model, such as the mean square error (MSE), mean absolute percentage error (MAPE), and correlation coefficient (R)

MSE = \frac{1}{l} \sum_{i = 1}^{l} (y_{i} - f (x_{i} {))}^{2}

(6)

MAPE = \frac{1}{l} \sum_{i = 1}^{l} \frac{| y_{i} - f (x_{i}) |}{y_{i}} \times 100 %

(7)

R = \frac{\sum_{i = 1}^{l} y_{i} f (x_{i})}{\sqrt{\sum_{i = 1}^{l} y_{i}^{2}} \sqrt{\sum_{i = 1}^{l} f {(x_{i})}^{2}}}

(8)

where $y_{i}$ and $f (x_{i})$ are the actual output and predictive output, respectively. MSE and MAPE evaluate the degree of data change, and R reflects the degree of the linear correlation between actual output and predictive output.

LSSVM-LOOCV model

Through some deterministic experiment, 57 groups of sample points and 285 groups of sample pointes were successively obtained, and were respectively considered as training set and testing set. The former was used to acquire the optimal hyper-parameter combination and train the LSSVM-LOOCV model, while the latter was responsible for estimating the generalization ability of the model. According to the principle of LOOCV, the estimates corresponding to the minimal objective value were considered as the optimal hyper-parameters, as shown in Table 3. Then, these parameters were applied to train the support values and the bias term to construct the LSSVM-LOOCV model.

Table 3.

Optimal hyper-parameter combinations of different modal orders.

Modal order	Regularization parameter $C^{*}$	Kernel parameter $σ^{* 2}$	Objective value
1	439,210.2198	3477.5323	3.008557e–003
3	217,364.6551	2204.6267	8.054864e–002

BPNN model

BPNN³⁹ mainly consists of the input layer, hidden layer, and output layer. The key issue is to select a proper hidden layer. But so far, there is no good analytic expression to determine the number of hidden nodes. It is generally believed that the number of hidden nodes is directly related to the number of input and output cells. Referring to previous experience, a simple formula is established⁴⁰

n = \sqrt{n_{i} + n_{o}} + a

(9)

where $n$ , $n_{i}$ , and $n_{o}$ are the number of hidden nodes, input cells, and output cells, respectively. $a$ is a constant, and $a \in [1, 10]$ . In conclusion, a standard three-layer BPNN structure was designed, which included 14 input cells, 1 output cell, and hidden layer with $n$ nodes, and $n \in [5, 14]$ . Additionally, in order to get a better BPNN model, the learning rate was set to 0.01, 0.04, 0.07, or 0.1, and the hidden layer and output layer used the sigmoid and pure-line transfer functions, respectively.²⁴ The training set and testing set used for BPNN were the same as for LSSVM-LOOCV. Finally, the network with the minimal testing MSE was thought to be the optimal BPNN model.

Table 4 shows that the {14-5-1} network structure with learning rate of 0.1 and {14-9-1} network structure with learning rate of 0.04 corresponding to the first- and third-order modes possess the best predictive ability, respectively, where ${\cdot}$ represents ${n_{i} - n - n_{o}}$ .

Table 4.

Analysis results of back-propagation neural network (BPNN) with different hidden layer structure.

Number of hidden nodes	Learning rate	First order		Third order
Number of hidden nodes	Learning rate	Iterations	Testing MSE	Iterations	Testing MSE
5	0.01	38	0.0125	121	0.0443
	0.04	9	0.0401	148	0.0502
	0.07	35	0.0144	230	0.0462
	0.1	51	0.0068	44	0.0558
6	0.01	16	0.0093	34	0.0530
	0.04	20	0.0243	29	0.0517
	0.07	14	0.0195	149	0.0465
	0.1	10	0.0471	53	0.0601
7	0.01	9	0.0438	74	0.0488
	0.04	17	0.0469	14	0.0421
	0.07	15	0.0684	18	0.0528
	0.1	10	0.0376	13	0.0606
8	0.01	8	0.0886	20	0.0442
	0.04	14	0.0505	37	0.0387
	0.07	9	0.0581	37	0.0551
	0.1	7	0.0691	97	0.0486
9	0.01	8	0.0496	26	0.0404
	0.04	5	0.0848	77	0.0337
	0.07	4	0.0715	83	0.0401
	0.1	9	0.0458	39	0.0492
10	0.01	17	0.0517	37	0.0535
	0.04	4	0.0601	65	0.0407
	0.07	6	0.0415	19	0.0428
	0.1	7	0.0305	35	0.0532
11	0.01	19	0.0651	35	0.0709
	0.04	8	0.0415	90	0.0462
	0.07	6	0.0514	37	0.0352
	0.1	10	0.0326	63	0.0424
12	0.01	9	0.0651	38	0.0518
	0.04	17	0.0721	112	0.0496
	0.07	9	0.0463	28	0.0508
	0.1	20	0.0507	67	0.0386
13	0.01	8	0.0794	17	0.0413
	0.04	16	0.0538	34	0.0437
	0.07	7	0.0648	38	0.0530
	0.1	15	0.0792	96	0.0607
14	0.01	6	0.0586	85	0.0525
	0.04	10	0.0861	43	0.0401
	0.07	5	0.0647	74	0.0456
	0.1	4	0.0659	31	0.0509

MSE: mean square error. The bold value represents the minimum and closest to 0 of all iterations.

Reliability analyses on resonance for LPC rotor blade

Basic theories

When the resonance of a stimulated system occurs at a rotating speed, the system is in failure state or critical failure state. According to the resonant reliability theory, the limit state function can be defined as

\begin{matrix} g_{i} (ς, ω_{i}) = | ς - ω_{i} | - δ \cdot ς \\ = {\begin{matrix} g_{i 1} = ω_{i} - (1 + δ) \cdot ς ς \leq ω_{i} \\ g_{i 2} = - ω_{i} + (1 - δ) \cdot ς ς \geq ω_{i} \end{matrix} 0 < δ < 0.3 \end{matrix}

(10)

where $δ$ is the reliability index, $ς$ is the exciting frequency, and $ω_{i}$ is the $i th$ natural frequency, which can be expressed by equation (5). $g_{i} (ς, ω_{i}) < 0$ indicates that the system is in a state of failure or critical failure. $g_{i} (ς, ω_{i}) > 0$ denotes that the system is normally working.⁴¹

We assume that the vibration analysis of the rotor blade is a tandem mode,⁴¹ and different modes are mutually independent. Therefore, the security and reliability of the system is based upon each subsystem in the normal working state. The reliability index $β$ and failure probability of the $i th$ mode $P_{i}$ are determined by

β = \frac{μ_{g}}{\sqrt{D_{g}}}, P_{i} = 1 - Φ (β) = 1 - \frac{1}{\sqrt{2 π}} \cdot \int_{- \infty}^{β} e^{- \frac{t^{2}}{2}} dt

(11)

So, the resonant reliability of the system is represented as

R = 1 - \sum_{i = 1}^{n} P_{i}

(12)

Numerical analyses

Based on the above algorithm models, the predicting simulation was performed against the testing set. Figures 6 and 7 show the distribution histogram for the first and third predictive frequencies. Table 5 shows the comparison of the simulation results for different methods. Tables 6 and 7 list the comparison of the resonant failure probability of the first and third modes under different reliability indices $δ$ for different methods. Table 8 shows the comparison of resonant reliability at the rotating speed of 6150.0 r/min for LPC rotor blade under different reliability indices $δ$ for different methods. In Tables 5 –8, the MCM is considered as the reference to which the LSSVM-LOOCV and BPNN can be compared.

Figure 6.

The predictive output of least squares support vector machine with leave-one-out cross-validation (LSSVM-LOOCV).

Figure 7.

The predictive output of back-propagation neural network (BPNN).

Table 5.

Comparison of the simulation results for different methods.

Order	Method	Iterations	Running time (s)	Mean (Hz)	Standard deviation	MSE	MAPE (%)	R
1	MCM	10,000	10,074	254.5407	2.9568	–	–	–
	LSSVM-LOOCV	11	4.9062	254.5315	3.0071	0.0015	0.009612	1
	BPNN	51	5.3348	254.5222	3.0203	0.0068	0.022458	1
3	MCM	10,000	10,958	968.0200	13.4328	–	–	–
	LSSVM-LOOCV	14	5.3282	967.9954	13.6162	0.0325	0.012934	1
	BPNN	77	5.3468	968.2039	14.0089	0.0337	0.035268	0.9998

MSE: mean square error; MAPE: mean absolute percentage error; MCM: Monte Carlo method; LSSVM-LOOCV: least squares support vector machine with leave-one-out cross-validation; BPNN: back-propagation neural network.

Table 6.

Comparison of the resonant failure probability of the first mode for different methods.

Exciting frequency (Hz)	$δ$	MCM	LSSVM-LOOCV		BPNN
Exciting frequency (Hz)	$δ$	$P_{1}$ (%)	$P_{1}$ (%)	Precision (%)	$P_{1}$ (%)	Precision (%)
205	0.19	0.01633	0	–	0	–
	0.20	0.22510	0.35088	44.1226	0.35058	44.2559
	0.21	1.40238	1.75438	74.8998	2.10526	49.8795
	0.22	6.54903	7.36842	87.4884	7.36842	87.4884
	0.23	20.8221	19.2982	92.6813	19.6491	94.3665
	0.24	45.5798	46.6667	97.6154	46.6667	97.6154
	0.25	71.9560	73.3333	98.0859	73.3333	98.0859
	0.26	89.9334	89.1228	99.0987	89.4737	99.4888
	0.27	97.4201	97.1930	99.9976	96.8421	99.4067
	0.28	99.6709	100	99.6698	100	99.6698
	0.29	99.9500	100	99.9499	100	99.9499
	0.30	100	100	100	100	100
307.5	0.14	0.05004	0	–	0	–
	0.15	1.15560	1.05263	91.0895	0.70175	60.7260
	0.16	10.0666	10.8772	91.9476	10.5263	95.4334
	0.17	40.3623	41.7544	96.5510	41.4035	97.4204
	0.18	79.1779	80.7018	98.0753	80.3508	98.5187
	0.19	96.9477	96.1404	99.1673	95.7894	98.8052
	0.20	99.7749	99.6491	99.8739	99.6491	99.8739
	0.21	100	100	100	100	100

MCM: Monte Carlo method; LSSVM-LOOCV: least squares support vector machine with leave-one-out cross-validation; BPNN: back-propagation neural network.

Table 7.

Comparison of the resonant failure probability of the third mode for different methods.

Exciting frequency (Hz)	$δ$	MCM	LSSVM-LOOCV		BPNN
Exciting frequency (Hz)	$δ$	$P_{3}$ (%)	$P_{3}$ (%)	Precision (%)	$P_{3}$ (%)	Precision (%)
922.5	0.01	0.22409	0	–	0	–
	0.02	2.11682	2.10538	99.4596	2.80701	67.3950
	0.03	8.96519	10.8772	78.6730	11.5789	70.8460
	0.04	26.5288	27.3684	96.8351	28.4211	92.8669
	0.05	51.8688	50.1754	96.7352	50.1754	96.7352
	0.06	76.6903	75.4386	98.3679	75.0877	97.9103
	0.07	92.3573	92.9825	99.3231	93.6842	98.5633
	0.08	98.1383	98.2456	99.8907	96.8421	98.6792
	0.09	99.7114	99.6491	99.9375	99.6491	99.9375
	0.10	99.9744	100	99.9744	100	99.9744
	0.11	99.9927	100	99.9927	100	99.9927
	0.12	100	100	100	100	100
1025	0.01	0.02564	0	–	0	–
	0.02	0.39581	0.35088	88.6486	0.35087	88.6461
	0.03	2.67131	3.50877	68.6498	3.50877	68.6498
	0.04	11.6359	12.2807	94.4585	11.5789	99.5101
	0.05	33.2916	34.7368	95.6590	33.6842	98.8207
	0.06	62.7913	62.8070	99.9750	61.7544	98.3486
	0.07	86.6495	85.9649	99.2099	85.2632	98.4001
	0.08	97.0046	97.1930	99.8058	95.7895	98.7474
	0.09	99.6989	100	99.6980	100	99.6980
	0.10	99.9745	100	99.9745	100	99.9745
	0.11	100	100	100	100	100

MCM: Monte Carlo method; LSSVM-LOOCV: least squares support vector machine with leave-one-out cross-validation; BPNN: back-propagation neural network.

Table 8.

Comparison of resonant reliability at the rotating speed of 6150.0 r/min for low-pressure compressor (LPC) rotor blade with different methods.

$δ$	MCM	LSSVM-LOOCV		BPNN
$δ$	$R$ (%)	$R$ (%)	Precision (%)	$R$ (%)	Precision (%)
0.01	99.7503	100	99.7496	100	99.7496
0.02	97.4874	97.5437	99.9420	96.8421	99.3381
0.03	88.3635	85.6140	96.8884	84.9123	96.0943
0.04	61.8353	60.3509	97.5994	60	97.0319
0.05	14.8396	15.0878	98.3274	16.1404	91.2343

MCM: Monte Carlo method; LSSVM-LOOCV: least squares support vector machine with leave-one-out cross-validation; BPNN: back-propagation neural network.

Figures 6, 7, and Table 5 illustrate that the predictive frequencies follow Gaussian distribution. The above statistics analyses prove that the resonant failure probability of the first and third modes increases, and the resonant reliability at the rotating speed of 6150.0 r/min for LPC rotor blade decreases with the increasing reliability index δ. Compared with MCM, an acceptable estimating performance on few iterations and shorter running time can be obtained by the LSSVM-LOOCV and BPNN models, while the LSSVM-LOOCV model possesses better generalization ability and precision to analyze the reliability on resonance for LPC rotor blade than BPNN.

Conclusion

This study applies LSSVM to analyze the resonant reliability at the rotating speed of 6150.0 r/min for LPC rotor blade. LOOCV is applied to the training set to obtain the optimal hyper-parameter combination so that the better LSSVM-LOOCV model is constructed. Then, considering the randomness of the influence factors on frequencies, the resonant reliability analysis for the rotor blade is achieved with the proposed approach. Additionally, BPNN is introduced to compare the feasibility of the proposed method. The experimental results suggest that the LSSVM-LOOCV model has an excellent nonlinearity approximation capability and better generalization ability than BPNN in cases of fewer sample points, and LSSVM-LOOCV is an excellent algorithm to improve the efficiency of reliability analysis on resonance at the rotating speed of 6150.0 r/min for LPC rotor blade. Some details are explained as follows:

From the solving process, the LSSVM model has lower computational complexity than SVM, because only the linear equations need to be solved instead of the quadratic programming problem. However, due to the loss of sparsity, when the LSSVM model handles massive amounts of data, the amount of computations is huge. Therefore, this research carefully selects 57 groups of experimental data to train the algorithm model. The statistical results prove that the established LSSVM-LOOCV model requires less running time and iteration times to obtain an acceptable result than MCM.

Because the sum of MSE is considered as fitting error in the objective function of the LSSVM model, the solution will be optimal when the errors follow normal distribution. From the above analyses, Figure 8 can be obtained (Figure 8).

Figure 8.

The predictive error distribution for least squares support vector machine with leave-one-out cross-validation (LSSVM-LOOCV).

From the distribution figures of the predictive errors, it is seen that the errors basically follow normal distribution. So, the LSSVM-LOOCV model is optimal to improve the computational efficiency of reliability analysis.

Footnotes

Academic Editor: Jia-Jang Wu

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was supported by the National Natural Science Foundation of China (Grant no. 51335003) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20111102110011).

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