For the nondeterministic factors of an aeroengine blisk, including both factors with sufficient and insufficient statistical data, based on the dynamic substructural method of determinate analysis, the extremum response surface method of probabilistic analysis, and the interval method of nonprobabilistic analysis, a methodology called the probabilistic and nonprobabilistic hybrid reliability analysis based on dynamic substructural extremum response surface decoupling method (P-NP-HRA-DS-ERSDM) is proposed. The model includes random variables and interval variables to determine the interval failure probability and the interval reliability index. The extremum response surface function and its flow chart of mixed reliability analysis are given. The interval analysis is embedded in the most likely failure point in the iterative process. The probabilistic analysis and nonprobabilistic analysis are investigated alternately. Tuned and mistuned blisks are studied in a complicated environment, and the results are compared with the Monte Carlo method (MCM) and the multilevel nested algorithm (MLNA) to verify that the hybrid model can better handle reliability problems concurrently containing random variables and interval variables; meanwhile, it manifests that the computational efficiency of this method is superior and more reasonable for analysing and designing a mistuned blisk. Therefore, this methodology has very important practical significance.
In practical engineering, many uncertainties need to be considered when analysing the reliability of a complex structure. Typically, probability theory and fuzzy set are used to address uncertainty in the traditional method, and the probabilistic model has become the most common and effective method for handling uncertainty. However, the probabilistic model and fuzzy model require more data to define the parameters of the probability distribution function. Furthermore, probabilistic reliability is very sensitive to the tail of the probability density function, which plays a key role in the calculation; therefore, a small error in the data may lead to a large error in the structural reliability calculation [
The models mentioned in this paper can be used to describe uncertainties. Some uncertain parameters have sufficient statistical data that can be used to establish the probabilistic distribution function, while other uncertainties can be described only by a range of their variables due to a lack of statistical information; only convex sets or interval variables can be used to describe the latter uncertainties. In this case, the probabilistic method is not suitable for the structure, and the existing statistical information cannot be used by the nonprobabilistic method. Therefore, a satisfactory result cannot be obtained with only one type of model. Nevertheless, for a problem that contains both probabilistic and nonprobabilistic variables, the probabilistic and nonprobabilistic hybrid reliability method can take full advantage of the known information to achieve an effective analysis. Many scholars have studied probabilistic and nonprobabilistic hybrid reliability analysis (HRA) [
For instance, Qiu et al. developed a hybrid of probabilistic and nonprobabilistic reliability theory, with the structural uncertain parameters as interval variables when statistical data are found insufficient. Then they proposed a new reliability model to improve the evaluation of probabilistic and nonprobabilistic hybrid structural systems. In addition, he presented a recognition method for the main failure modes using a five-bar statically indeterminate truss structure and an intermediate complexity wing structure to demonstrate that the new model was more suitable for analysis and design than the probabilistic model. A new hybrid reliability model that contained randomness, fuzziness, and nonprobabilistic uncertainty based on the structural fuzzy random reliability and nonprobabilistic set-based models was presented in [
There are other methods for HRA [
Other researchers optimized structures and studied their sensitivity through probabilistic and nonprobabilistic hybrid reliability methodology as well [
In other applications as well, for example, [
For an aeroengine working in a complicated high temperature, high pressure, and high rotational speed environment, significant information is available as uncertain parameters, such as rotational speed and temperature. However, other uncertain parameters, such as the coefficient of thermal conductivity and the coefficient of expansion, lack sufficient data. Therefore, a new type of probabilistic and nonprobabilistic HRA is proposed based on the traditional probabilistic model and nonprobabilistic model. Unlike the probabilistic and nonprobabilistic hybrid models investigated by the abovementioned scholars, the methodology presented in this paper is connected with the improved hybrid interface substructural component modal synthesis method in [
For designing and analysing engineering structures, random variables and interval variables need to be included simultaneously. Therefore, a probabilistic and nonprobabilistic hybrid reliability model needs to be established to fully describe the actual situation of the structures.
Assume that the random variable of the system is
After introducing the interval variable
The region of extreme postural belt.
Therefore, the failure probability
The reliability index
Then the following two optimization problems can be solved: the maximum and the minimum reliability indexes of the limit state belt can be obtained.
The failure probabilities of the maximum and the minimum values are expressed as
In engineering applications, the maximum failure probability of the structure is often the one of most concern, and it has significant reference value for engineering and technical staff. Therefore, the maximum failure probability is taken as the measurement for the structural reliability in this paper.
When using the traditional multilevel nested algorithm (MLNA), the computation efficiency is very low for some mechanical parts such as a complex aeroengine working in a poor environment. Thus, the decoupling method proposed in [
The sample centre point
Processing of the sample points selected.
The value of the original performance function is calculated at each sample point; then, undetermined coefficients of the response surface function are solved.
Combining the probabilistic and nonprobabilistic hybrid reliability model with (
P-NP-HRA-DS-ERSDM is used to solve the probabilistic and nonprobabilistic HRA of the blisk, and the specific iterative process is as follows.
Assume that
Then,
The design test point
Flow chart of decoupling method.
The design test point
Obtaining a new sample centre point.
The expressions of the connective points
When
The interval variable of the most recently obtained
Based on the above analysis, the probabilistic and nonprobabilistic HRA process are as follows: Set the initial iteration point Establish a quadratic polynomial extremum response surface function. The sample centre point is set to Solve the approximate mixed reliability problems using ( Solve the new sample centre point Judge convergence. If Simulate the constructed response surface function, and calculate the maximum failure probability.
The maximum failure probability of the probabilistic and nonprobabilistic HRA can be obtained, as shown in Figure
Flow chart of hybrid reliability analysis based on the extreme response surface.
P-NP-HRA-DS-ERSDM is used to analyse a blisk and is compared with MLNA and MCS to verify its scientific rationality.
First, the reliability of the tuned blisk is investigated, and the natural frequency, modal shape, and vibration response are studied using P-NP-HRA-DS-ERSDM.
In the research, the rotational speed
Distribution of nondeterministic variables of tuned blisk.
Nondeterministic variables | Parameter 1 | Parameter 2 | Distribution pattern |
---|---|---|---|
|
1046 | 31.38 | Normal distribution |
|
1050 | 31.50 | Normal distribution |
|
1.216 | 0.03648 | Normal distribution |
|
27.21 | 0.8163 | Normal distribution |
|
3.0 | 0.09 | Normal distribution |
|
1.75 | 1.94 | Interval variable |
pr | 0.2915 | 0.3325 | Interval variable |
den/kg·m−3 | 8090 | 9005 | Interval variable |
Analysis results of modal and vibration response of P-NP-HRA-DS-ERSDM for the tuned blisk.
Random variable | Interval variable | ||
---|---|---|---|
|
1129.3 |
|
1.931 |
|
943.2 | den | 8072.9 |
|
29.14 | pr | 0.297 |
|
2.827 | — | — |
|
1.339 | — | — |
— | — |
Response | The maximum failure probability | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
— | MCM | MLNA | P-NP-HRA-DS-ERSDM | ||||||||
— |
|
|
|
|
Er/% |
|
|
|
Er/% |
|
|
|
98.32 | 289.41 | 99.07 | 65.56 | 0.76 | 77.34 | 99.15 | 51.98 | 0.84 | 82.03 | 20.71 |
2611.8 HZ | |||||||||||
|
97.68 | 99.17 | 1.52 | 99.26 | 1.61 | ||||||
7.2114 | |||||||||||
|
98.17 | 99.32 | 1.17 | 99.33 | 1.18 | ||||||
1.4218 | |||||||||||
str_ |
98.54 | 98.94 | 0.41 | 99.12 | 0.59 | ||||||
7.3512 | |||||||||||
|
|||||||||||
|
98.16 | 297.86 | 99.12 | 77.23 | 0.97 | 74.07 | 99.76 | 62.54 | 1.62 | 79.01 | 19.02 |
4.12 |
Distribution of nondeterministic variables of mistuned blisk.
Nondeterministic variables | Parameter 1 | Parameter 2 | Distribution pattern |
---|---|---|---|
|
1046 | 31.38 | Normal distribution |
|
1050 | 31.50 | Normal distribution |
|
3.0 | 0.09 | Normal distribution |
|
0.3181 | 0.009542 | Normal distribution |
|
1.268 | 0.03806 | Normal distribution |
|
29.72 | 0.8915 | Normal distribution |
|
8010 | 8943 | Interval variable |
|
1.689 | 1.932 | Interval variable |
|
0.3143 | 0.009429 | Normal distribution |
|
1.216 | 0.03648 | Normal distribution |
|
27.21 | 0.8163 | Normal distribution |
|
8170 | 9280 | Interval variable |
|
1.714 | 1.996 | Interval variable |
Analysis results of modal and vibration response of P-NP-HRA-DS-ERSDM for the mistuned blisk.
Random variable | Interval variable | ||
---|---|---|---|
|
1129.3 |
|
8072.9 |
|
943.2 |
|
7977.8 |
|
29.14 |
|
1.931 |
|
27.63 |
|
1.889 |
|
0.297 | — | — |
|
0.294 | ||
|
2.827 | ||
|
1.251 | ||
|
1.339 |
Response | The maximum failure probability | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
— | MCM | MLNA | P-NP-HRA-DS-ERSDM | ||||||||
— |
|
|
|
|
Er/% |
|
|
|
Er/% |
|
|
|
— | — | 99.17 | 289.21 | — | — | 99.87 | 223.09 | — | — | 22.86 |
2611.8 HZ | |||||||||||
|
99.37 | 99.31 | |||||||||
20.943 | |||||||||||
strs(×1013) | 99.46 | 99.32 | |||||||||
3.6745 | |||||||||||
str |
98.89 | 99.24 | |||||||||
372.84 | |||||||||||
|
|||||||||||
|
— | — | 99.21 | 312.56 | — | — | 99.86 | 231.59 | — | — | 25.18 |
6.56 |
The maximum failure probabilities of the natural frequency, modal shape, and vibration response for the tuned blisk are investigated using MCM, MLNA, and P-NP-HRA-DS-ERSDM, respectively. The relative errors and the computational efficiency of all three methods are analysed, as shown in Table
For a mistuned blisk, the rotational speed
The maximum failure probability of the natural frequency, modal shape, and vibration response for the mistuned blisk are calculated using MCM, MLNA, and P-NP-HRA-DS-ERSDM. The relative errors and the computational efficiency of the three methods are analysed, as shown in Table
A nondeterministic analysis method with high efficiency and high accuracy, P-NP-HRA-DS-ERSDM, is investigated for a blisk. In the analysis process, for the hybrid nondeterministic problem containing both random variables and interval variables, the interval failure probability and interval reliability index are obtained. A flow chart is presented for solving the reliability analysis and the extremum response surface HRA using P-NP-HRA-DS-ERSDM.
Probabilistic and nonprobabilistic HRA is used to analyse the blisk. The variables mainly affecting the output response are regarded as interval variables, and the other variables are regarded as random variables. In the tuned blisk,
The maximum failure probabilities of the natural frequency, modal displacement, modal stress, modal strain energy, and vibration response are calculated for blisk, comparing the computational time and the relative error of P-NP-HRA-DS-ERSDM, MCM, and MLNA. For the tuned blisk, the relative errors of the natural frequency, modal displacement, modal stress, modal strain energy, and vibration response using MLNA to MCM are 0.76%, 1.52%, 1.17%, 0.41%, and 0.97%, respectively, and those of using P-NP-HRA-DS-ERSDM are 0.84%, 1.61%, 1.18%, 0.59%, and 1.62%; these values meet the engineering requirements. The computational efficiency of MLNA relative to MCM increases by 77.34% and 74.07%, respectively, but that of P-NP-HRA-DS-ERSDM increases by 82.03% and 79.01%, respectively. Thus, the computational efficiency of P-NP-HRA-DS-ERSDM relative to MLNA increased by 20.71% and 19.02%. However, for the mistuned blisk, the computational efficiency of P-NP-HRA-DS-ERSDM increases by 22.86% and 25.18% higher than that of MLNA, and the computational time is very long using MCM, and the convergence may not be achieved, which manifests that the superiority of this methodology is more obvious for the mistuned blisk than for the tuned blisk. The scientific rationality and validity for researching a blisk using P-NP-HRA-DS-ERSDM is verified, and this method is shown to be particularly superior to MLNA for a mistuned blisk.
The author declares that there are no conflicts of interest regarding the publication of this paper.
This work has been supported by the National Natural Science Foundation of China (Grant no. 51375032) and Project supported by Beijing Postdoctoral Research Foundation (Grant no. 2016ZZ-12).