Reliability analysis of turbine blades based on fuzzy response surface method

1 Introduction

Under complex loads and harsh running conditions, the failure of aero-engine blades due to damage and breakage usually appeared in the operating procession. The basic reason is that the effects of uncertain factors have not yet been fully investigated. There are two major types of uncertainty in engineering: randomness and fuzziness. Randomness is a type of uncertainty due to insufficiency of causality, which is described as unpredictability caused by the defect of causality. Fuzziness refers to ambiguity in the process of intermediary transition, which is described as the unclear boundary of things caused by the defects of the excluded middle law. The blade function involves much uncertainty because of the particularities of the research object. If the influence of uncertainty is neglected, then many safety states that become failure states in actual operation, will be inferred according to the fixed value method. Therefore, uncertain factors must be considered in reliability analysis of aero-engine blades.

At present, the so-called stochastic reliability analysis model, which considers the impact of stochastic uncertainty, has been more fully-studied and widely-used in water conservancy, civil engineering, geology and other fields [1, 15]. In recent years, the random reliability analysis [8, 22] based on the Response Surface Method (RSM) has been widely-used in the fields of risk assessment and mechanical structure sensitivity and instability analysis [2, 21]. The RSM [6, 7] plays an important role in reliability analysis by combining mathematical methods and statistical methods, allowing it to deal with the complex structures of reliability analysis which contains many uncertain variables. When combined with multi-body dynamics and the modal method, C.Y. Zhang andG.C. Bai [4] established the Extremum Response Surface Method (ERSM) by using multiple dynamic parameters as random input variables. In this method, the ERSM is demonstrated as a highly accurate and highly efficient method in the reliability analysis of a two-link flexible robot manipulator. C.W. Fei and G.C. Bai [5] applied ERSM to the multiple input variable reliability analysis on aero-engine blades, and the primary factors influencing the radial displacement was obtained. However, the above reliability analysis of stochastic uncertainty is based on probability, and a double state hypothesis and fuzziness were not considered, which is not reasonable for engineering practice. For example, engineering experience and prior knowledge cannot be incorporated into random reliability analysis, which is based on probability theory. With a small sample, the confidence degree of statistical parameters obtained by the probability theory and mathematical statistical methods will be very low, i.e., the variables have ambiguity. There is no absolute boundary between the safe operation and failure of most non-brittle structures, i.e., the limit states have fuzzy features. Therefore, a fuzzy reliability analysis model which considers fuzzy uncertainty should be established. At present, fuzzy reliability has been gradually applied to mechanical engineering. According to the stress strength interference theory, M.H. Xu, et al. [12] have studied the method, which considers the fuzziness of basic variables, the fuzziness of strength and random of stress, the random of strength and fuzziness of stress, the fuzziness of strength and stress, respectively. Based on the fuzziness of the limit state, M.H. Xu, et al. [12, 20] proposed the computational method of failure probability, which included the moment method and the subset sampling method. Although fuzzy reliability analysis in the reference literature [9, 20] investigates the influence of fuzzy factors, its analytical integral solution is obtained under the condition that the probability density function is explicit. It is difficult to obtain results in cases with many variables in reliability analysis. Meanwhile, previous studies have studied the fuzzy input variables and state variables separately, rather than considering their common impact to the entire system.

The response surface combined with high precision and high efficiency, designated the Fuzzy Response Surface Method (FRSM) of reliability analysis on turbine blades, is proposed in this paper, which takes the fuzziness of input variables and state variables into account. A certain type of aircraft gas turbine engine blade is chosen as the target of study in a reliabilitysimulation in which its running temperature, speed, material density and gravity are taken as the input variables, and the maximum radial blade deformation and maximum stress are taken as the output responses. At the end of this paper, a comparison between FRSM and RSM is conducted.

2 Fuzzy response surface method

The paper selects an aero-engine turbine blade as the research object; its material is TC4 alloy. The allowable stress of the blade is [σ]= 390MPa and allowable deformation of the blade is [δ]= 0.195 mm. The blade model is simplified to include simple cooling holes and tenons. A total of 1240 hexahedral units and 2000 nodes are manually generated by Marc software, and the simplified model is imported into analysis for simulation. The finite element mesh model of a blade is as shown in Fig. 2.

There are six working conditions in the operating process of an aircraft, which follows this trajectory: start, slow, maximum, continuous, take off, climb cruise. In order to simplify the calculation, the climbing phase, during which the blade stress and deformation are greatest, is selected as the calculation point [4].

To study the influence of the fuzzy limit state on calculation results, the relationship between positional parameter Z _a of the membership function and the mean square deviation σ _z is defined as follows: | z a | = k z σ z ( k z = 1 3 ˜ )(19) where k_z is the coefficient of fuzzy zone, which indicates the fuzziness of the limit state.

To study the influence of fuzzy input variables on the calculation result, the distance value d _{x i a} and the distance value d _{x i b} are assumed to be equal. The relationship between the two is defined as follows: d x i a = d x i b = w x i x im(20) where w _{x i} is the coefficient of the fuzzy zone, which indicates the fuzziness of input variables (w _{x i}  = 0.1 ∼ 0.5); the value d _{x i a} is the distance from the middle value x _im to the minimum value; and d _{x i b} is the distance from the middle value x _im to the maximum value.

The temperature, speed, density, and gravity acceleration of the blade are as assumed to be fuzzy variables and are expected to obey normal distribution. The input variables are shown in Table 1. The blade stress and radial deformation vector diagram of deterministic analysis obtained by software are shown in Figs. 3and 4.

The parameterized model of finite elements can be obtained from the above deterministic analysis. The mean value μ x eq and the mean square deviation σ x eq can be obtained from the transformation of fuzzy variables. The coefficient of the fuzzy response surface function can be obtained by MCM linkage sampling of the finite element model, and the expression is as described by Equations (21) and (22).

The distribution histograms of the blade stress and deformation can be obtained after 10,000 samplings on the fuzzy response surface function, as shown in Figs. 5 and 6. y 1 ˜ = 2.607 × 10 8 - 5.510 × 10 4 ρ - 4.312 × 10 5 ω - 5.87 × 10 5 T + 3.188 × 10 6 g + 9.703 × 10 2 ρ ω + 7.432 × 10 2 ρ T + 3.733 × 10 2 ρ g + 8.97 × 10 2 g T + 6.15 × 10 2 g ω + 9.47 × 10 2 ω T + 6.99 × 10 - 2 ρ 2 + 1.94 × 10 2 ω 2 + 2.87 × 10 2 T 2 + 2.04 × 10 5 g 2(21)

y 2 ˜ = 1.839 × 10 - 4 - 3.824 × 10 - 8 ρ - 3.163 × 10 - 7 ω - 2.302 × 10 - 7 T - 4.807 × 10 - 8 g - 1.442 × 10 - 10 ρ g + 6.602 × 10 - 11 ρ ω + 5 . 024 × 10 - 11 ρ T + 4.128 × 10 - 11 g ω + 3.143 × 10 - 11 ω T + 1.248 × 10 - 11 gT + 3.071 × 10 - 15 ρ 2 + 1.343 × 10 - 10 ω 2 - 1.049 × 10 - 9 g 2 + + 4.537 × 10 - 10 T 2(22)

The stress and deformation of the blade obey a normal distribution. The mean value of stress is 3.5611×10⁸ Pa and the square deviation of stress is 1.9296×10³ Pa; the mean value and the standard deviation of deformation is 1.7076 × 10^-4m and 7.5756 × 10^-11m, respectively. Therefore, the limit state functions of stress and the deformation of blades are obtained, shown respectively in Equations (23) and (24). Z 1 = [ σ ] - y 1(23)Z 2 = [ δ ] - y 2(24)

The reliability index β can be expressed as follows: β = - Φ - 1 ( P f )(25) with different width coefficients of the fuzzy zone and ultimate state, w _x and k, the reliability index β is obtained, as shown in Fig. 7.

The selected blade speed, material density, acceleration of gravity and temperature are selected as the random input variables. The type distribution, mean and variance of random parameter of the maximum displacement and maximum stress point are shown in Table 2.

Without consideration of the fuzziness of the random input and output variables, the random response surface function model is built solely on a theoretical basis (10). The stress and deformation response surface function are shown respectively in Equations (26) and (27). y 1 = 2.38 × 10 8 - 4.15 × 10 4 ρ - 5.51 × 10 5 ω - 5.21 × 10 5 T + 4.74 × 10 6 g + 2.67 × 10 2 ρ g + 8.65 × 10 2 ρ ω + 8.35 × 10 2 ρ T + 7.55 × 10 2 g ω + 9.25 × 10 2 gT + 3.57 × 10 2 T ω + 1.05 × 10 2 T 2 + 5.53 × 10 2 ω 2 + 5.35 × 10 - 2 ρ 2 + 1.26 × 10 5 g 2(26)y 2 = 1.76 × 10 - 4 - 6.43 × 10 - 8 ρ - 8.58 × 10 - 7 ω - 1.26 × 10 - 7 T - 5.13 × 10 - 8 g - 2.56 × 10 - 10 ρ g + 2.58 × 10 - 11 ρ ω + 6.85 × 10 - 11 ρ T + 3.83 × 10 - 11 g ω + 3.67 × 10 - 11 T ω + 1.56 × 10 - 11 gT + 3.55 × 10 - 10 T 2 + 5.83 × 10 - 10 ω 2 + 8.55 × 10 - 15 ρ 2 - 8.93 × 10 - 8 g 2(27)

A mathematical model has been established for the 10,000 time linkage sampling, which was used to achieve reliability analysis. The number of stress failures and displacement failures of the blade and the corresponding reliability probabilities are shown in Tables 3 and 4.

The failure probability of stress and deformation can be obtained by Equation (18), and the reliability is shown in Table 5.

In this paper, the reliability of aero-engine blades was analyzed by simultaneously considering the fuzziness of the input variables and their limit states. Considering the fuzziness effect of basic variables, the fuzzy variables were converted into stochastic variables according to the entropy equivalence method. The failure probability was calculated based on the fuzzy probability theory, and the reliability index was obtained. The example analysis indicates that:

The fuzziness of basic variables and limit states lead to remarkable effects on the structural fuzzy random reliability index. The reliability index decreases when the coefficient of the fuzzy zone of the input variable increases simultaneously with the coefficient of the limit state of the mushy zone. The fuzziness of the limit state function is not considered when k = 0.