1School of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin 150080, China2Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong3School of Energy and Power Engineering, Beihang University, Beijing 100191, China
Correspondence should be addressed to Cheng-Wei Fei; [email protected]
Received 6 May 2015; Revised 30 June 2015; Accepted 8 July 2015
To study accurately the influence of the deformation, stress, and strain of turbine blisk on the performance of aeroengine, thecomprehensive reliability analysis of turbine blisk with multiple disciplines and multiple objects was performed based on multipleresponse surfacemethod (MRSM) and fluid-thermal-solid coupling technique. Firstly, the basic thought ofMRSMwas introduced.And then the mathematical model of MRSM was established with quadratic polynomial. Finally, the multiple reliability analysesof deformation, stress, and strain of turbine blisk were completed under multiphysical field coupling by the MRSM, and thecomprehensive performance of turbine blisk was evaluated. From the reliability analysis, it is demonstrated that the reliabilitydegrees of the deformation, stress, and strain for turbine blisk are 0.9942, 0.9935, 0.9954, and 0.9919, respectively, when the allowabledeformation, stress, and strain are 3.7× 10−3m, 1.07× 109 Pa, and 1.12× 10−2m/m, respectively; besides, the comprehensive reliabilitydegree of turbine blisk is 0.9919, which basically satisfies the engineering requirement of aeroengine.The efforts of this paper providea promising approach method for multidiscipline multiobject reliability analysis.
1. Introduction
An aeroengine as the power system of aircraft seriouslyinfluences the performance and reliability of air vehicle[1]. Bladed disk as one pivotal part of aeroengine is theimportant fault source of aeroengine for suffering from hightemperature, high pressure, and high rotation speed duringoperation [2].Of the faults of aeroengine, the rate of blisk faultis 25%.The performances of aeroengine safety, reliability, androbustness are to decline sharply once blisk fault occurs [3].Therefore, it is of great significance for improving the wholeperformance and reliability of aeroengine to investigate thereliability of blisk.
Recently, some efforts spring up on the improvementof aeroengine. Qi et al. studied the time history variationof the blade-tip clearance of aeroengine high turbine byfinite element method under considering the effects oftemperature, pressure, and rotation speed [4]. Pillidis and
Maccallum, focused on the change rule of aeroengine high-pressure blade-tip radial running clearance through calcu-lating the radial deformations of turbine disk, blade, andcasing by adopting thermal-solid coupling method under theinfluences of heat load and centrifugal force load [5].Wang etal. analyzed the stress of contact region on the blade and diskof aeroengine based on finite element method [6]. Meguidet al. discussed the effect of impact force by simulatingthe impact of bird against the blade of aeroengine usingfinite element method [7]. The above investigations onlyfocus on the deterministic analysis without considering therandomness of influencing parameter on aeroengine blisk,so that it is very difficult to gain reasonable results for bliskdesign and analysis.
To address this issue, one viable alternative to determin-istic analysis is probabilistic analysis, which does considerthe randomness of factors to describe the blisk deformationof aeroengine with acceptable accuracy. The probabilistic
Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2015, Article ID 649046, 10 pageshttp://dx.doi.org/10.1155/2015/649046
2 Advances in Materials Science and Engineering
analysis method has been widely applied in many fields. Forinstance, Lu and Low conducted the probabilistic analysisof underground rock excavations [8]; Kartal et al. imple-mented the probabilistic nonlinear analysis of CFR dams[9]; Fitzpatrick et al. applied probabilistic analysis method tomultivariate sensitivity evaluation patellofemoral mechanics[10]; Zona et al. studied design assessment of continuoussteel concrete composite girders with probabilistic analysis[11]. Meanwhile, some works were also done in the prob-abilistic analysis of typical aeroengine components. Hu etal. researched the probabilistic design for turbine disk athigh temperature [12]; Nakamura and Fujii analyzed thetransient heat of an atmospheric reentry vehicle structureusing probabilistic method [13]; Fei et al. finished the prob-abilistic analyses of turbine disk [14] and casing [15, 16].Additionally, some approaches of probabilistic analysis havebeen developed such as Monte Carlo method (MCM) [13],response surface method (RSM) [17, 18], extremum responsesurface method [14, 15], and support vector machine [16, 19],for the probabilistic analyses and reliability analyses of typicalaeroengine components and the blade-tip radial runningclearance of aeroengine high-pressure turbine. The effortsonly keep a watchful eye on the reliability analysis of singleobject which needs to build one model and do not considerthe influences of all factors. In fact, although the computa-tional precision and efficiency were greatly improved by theabove method, blisk reliability is determined by many failuremodels and multiple disciplines. Although multiple responsesurface model (MRSM) was developed for the reliabilityanalysis of aeroengine blade-tip clearance by establishingmany response surface models for different disciplines [18],two deficiencies yet exist: (1) only consider the influenceof centrifugal load and heat load without the effect due tofluid; (2) only finish the reliability analysis of single object bysingle response surface model without considering multipleoutput response (multiobject) by multiple response surfacemodels.
To solve the above issues, accompanied with the heuristicidea of MRSM [18], the comprehensive reliability analysis ofaeroengine turbine blisk with multifailure models (multiob-ject) was completed by considering multidiscipline of heat,fluid, and structure.
In what follows, Section 2 introduces the basic thoughtof MRSM and establishes its mathematical model basedon quadratic polynomial. In Section 3, the fluid-thermal-structural analysis of turbine blisk is completed by consid-ering various parameters from different disciplines. Section 4focuses on the comprehensive reliability evaluation of turbineblisk from the reliability analyses of deformation, stress,and strain of blisk under the effect of fluid-thermal-solidinteraction. Section 5 summarizes the conclusions of thiswork.
2. Multiple Response Surface Method (MRSM)
2.1. Basic Principle. For structural reliability analysiswith multiple disciplines and multiple objects, MRSM is
Select input random variables and their distribution characteristics
Find the maximum output responses ofmultiple structural analyses randomly
Extract enough fitting samples with inputsamples and output samples
Establish multiresponse surface modelbased on the fitting samples
The linkage simulation of multiresponsesurface model by MCM
Structural comprehensive reliabilityanalysis
Output results
Begin
Figure 1: Flow chart of structural reliability analysis based onMRSM.
structured and applied based on response surface method.The basic thought of MRSM is summarized as follows:
(1) Select input random variables reasonably to completethe deterministic analyses of multiple objects forcomplex structure.
(2) Find the maximum values of multiobject analyticalresults as output responses to complete multiobjectreliability analysis.
(3) Structure the multiple response surface models byextracting the samples of input variables and calcu-lating the output response of each object based onsimulation methods like MCM [13].
(4) The reliability analysis of complex structure withmultidiscipline and multiobject is completed by sim-ulating themultiple response surface models by usingMCM.
The flow chart of structural reliability analysis based onMRSM is shown in Figure 1.
2.2. Mathematical Model of MRSM. Response surfacemethod (RSM) is used to fit a simple response surface by aseries of deterministic analyses replacing the real limit-statefunction [15]. When 𝑦 and X express the output responseand the vector of input random variables, respectively,
Advances in Materials Science and Engineering 3
the quadratic polynomial response surface function is struc-tured as follows:
𝑦 (X) = 𝐴 + BX + XTCX (1)
in which𝐴,B, andC are the coefficient of constant, the vectorof linear term coefficients, and the matrix of quadratic termcoefficients, respectively. B, C, and X are denoted by
B = [𝑏1
𝑏2
⋅ ⋅ ⋅ 𝑏𝑘
] ,
C =
[[[[[[[[
[
𝑐11
𝑐22
𝑐33
d
𝑐𝑘𝑘
]]]]]]]]
]
,
X = [𝑥1
𝑥2
⋅ ⋅ ⋅ 𝑥𝑘
]T,
(2)
where 𝑘 is the number of input random variables.In this paper, the mathematical model of MRSM was
established on the foundation of quadratic polynomialresponse surface function. Assuming that the reliability anal-ysis of a complex structure involves𝑚 (𝑚 ∈ 𝑍) output objects,the input random variable vector of the 𝑖th output object isX(𝑖), and the corresponding output variable is denoted by 𝑦(𝑖),the relationship of X(𝑖) and 𝑦
(𝑖) is
𝑦(𝑖)
= 𝑓 (X(𝑖)) (𝑖 = 1, 2, . . . , 𝑚) (3)
in which 𝑓(⋅) is the function of input random variables.In the light of quadratic polynomial response surface
function, (3) is rewritten as
𝑦(𝑖)
(X(𝑖)) = 𝐴(𝑖)
+ B(𝑖)X(𝑖) + (X(𝑖))TC(𝑖)X(𝑖), (4)
where 𝐴(𝑖) is the constant coefficient of the 𝑖th output object,B(𝑖) the coefficient vector of linear term of the 𝑖th outputobject, and C(𝑖) the coefficient matrix of quadratic term.
Equation (4) is also reshaped as
𝑦(𝑖)
= 𝑎(𝑖)
+
𝑘
∑
𝑗=1
𝑏(𝑖)
𝑗
𝑥(𝑖)
𝑗
+
𝑘
∑
𝑗=1
𝑐(𝑖)
𝑗
(𝑥(𝑖)
𝑗
)
2
(𝑝 = 1, 2, . . . , 𝑘)
(5)
in which 𝑥(𝑖)
𝑗
denotes the 𝑗th component of the input variable𝑥 in the 𝑖th output response (object) and 𝑎
(𝑖), 𝑏(𝑖)
𝑗
, and𝑐(𝑖)
𝑗
denote the undetermined coefficients of constant term,linear term, and quadratic term, respectively. The number ofundetermined coefficients is 𝑚 (2𝑘 + 1). The undeterminedcoefficients are gained based on least square method whenthe number of samples is enough, because the vector D(𝑖) isformed by
D(𝑖) = [𝐴(𝑖)
𝑏(𝑖)
1
𝑏(𝑖)
2
⋅ ⋅ ⋅ 𝑏(𝑖)
𝑘
𝑐(𝑖)
11
𝑐(𝑖)
22
⋅ ⋅ ⋅ 𝑐(𝑖)
𝑘𝑘
] . (6)
Geometry
Figure 2: Structure model of blisk.
From (4),D(𝑖) may be deduced as
D(𝑖) = [(X(𝑖))TX(𝑖)]−1
(X(𝑖))T𝑦(𝑖)
. (7)
From (6), we can gain the undetermined coefficients of(5) and further the mathematical model of MRSM is
The working condition of aeroengine is so harsh that theblisk of turbine suffers from the high-temperature gas andlarge centrifugal force. To simulate the real work conditionof turbine blisk, the fluid-thermal-solid coupling analysis wasexecuted based on the discrete coupling analysis method[20, 21]. Therein, the structure of turbine blisk is shown inFigure 2 and TC4 alloy was selected as the material of turbineblisk. In this deterministic analysis, the inlet speed is 160m/s,the inlet pressure is 600 000 Pa, gas temperature is 1150K,and the rotation speed of blisk is 1168 rad/s [17–19]. Theinlet flow velocity V of air, inlet pressure 𝑝, material density𝜌, temperature 𝑡, and rotation speed 𝑤 were selected asrandom variables obeying normal distributions with mutualindependence.
4 Advances in Materials Science and Engineering
Figure 3: Flow field grid of blisk.
3.1. Fluid Analysis of Turbine Blisk. In fluid analysis, thestandard 𝑘-𝜀 turbulence model [22, 23] without gravity effectwas selected as follows:
𝜕
𝜕𝑡
(𝜌𝑘) +
𝜕
𝜕𝑥𝑖
(𝜌𝑘𝑢𝑖
)
=
𝜕
𝜕𝑥𝑗
[(𝜇 +
𝜇𝑡
𝜎𝑘
)
𝜕𝑘
𝜕𝑥𝑗
] + 𝐺𝑘
+ 𝐺𝑏
− 𝜌𝜀 − 𝑌𝑀
+ 𝑆𝑘
,
𝜕
𝜕𝑡
(𝜌𝜀) +
𝜕
𝜕𝑥𝑖
(𝜌𝜀𝑢𝑖
)
=
𝜕
𝜕𝑥𝑗
[(𝜇 +
𝜇𝑡
𝜎𝜀
)
𝜕𝜀
𝑥𝑗
] + 𝐶1𝜀
𝜀
𝑘
(𝐺𝑘
+ 𝐶3𝜀
𝐺𝑏
)
− 𝐶2𝜀
𝜌
𝜀2
𝑘
+ 𝑆𝜀
,
𝜇𝑡
= 𝜌𝐶𝜇
𝑘2
𝜀
,
( 9 )
where 𝑘 is the turbulence energy; 𝜀 the specific dissipationrate;𝜇
𝑡
the eddy viscosity;𝐺𝑘
the turbulence energy generatedfrom the mean velocity gradient; 𝐺
𝑏
the turbulence energygenerated from flotage; and 𝑌
𝑀
the effect of the fluctuatingexpansion of compressible speed turbulence on the totaldissipation rate. The coefficients 𝐶
𝜇
, 𝜎𝑘
, 𝜎𝜀
, 𝐶1𝜀
, 𝐶2𝜀
, and 𝐶3𝜀
were the constants of 0.09, 1, 𝜎𝜀
= 1.3, 𝐶1𝜀
= 1.44, and 𝐶2𝜀
=
1.92, respectively, and the 𝐺𝑏
and 𝑌𝑀
were not considered influid analysis. The flow field model of turbine blisk is builtwith the diameter 1.2m and the length 2m.The finite elementmodel of flow field was established as shown in Figure 3 withthe number of elements being 589 428 and the number ofnodes being 842 703. In line with the boundary condition ofblisk flow field, the numerical simulation analysis of flow fieldwas completed based on the finite element volume methodand 𝑘-𝜀 standard turbulence model. From the analysis, thestatic pressure distribution of turbine blisk is revealed inFigure 4.
P (P
a)
0
2.748 × 104
2.532 × 104
2.369 × 104
2.113 × 104
1.814 × 104
1.654 × 104
1.336 × 104
1.057 × 104
8.792 × 103
5.513 × 102
Figure 4: Static pressure distribution of blisk surface.
Figure 5: Grid of blisk.
3.2. Thermal Analysis of Turbine Blisk. The thermal analysisof turbine blisk was finished based on the following energyconservation equation [23]:
𝜕
𝜕𝑡
(𝜌𝐶𝑝
𝑇0
) +
𝜕
𝜕𝑥
(𝜌V𝑥
𝐶𝑝
𝑇0
) +
𝜕
𝜕𝑦
(𝜌V𝑦
𝐶𝑃
𝑇0
)
+
𝜕
𝜕𝑧
(𝜌V𝑧
𝐶𝑝
𝑇0
)
=
𝜕
𝜕𝑥
(𝐾
𝜕𝑇0
𝜕𝑥
) +
𝜕
𝜕𝑦
(𝐾
𝜕𝑇0
𝜕𝑦
) +
𝜕
𝜕𝑧
(𝐾
𝜕𝑇0
𝜕𝑧
)
+𝑊V + 𝐸𝑘
+ 𝑄V + Φ +
𝜕𝑃
𝜕𝑡
(10)
in which 𝐶𝑝
is the specific heat, 𝑇0
the total temperature, 𝐾the heat conductivity coefficient, 𝑊V the viscous dissipation,𝐸𝑘
the kinetic energy, 𝑄V the volume heat source, and Φ theitem of viscous heat.
The finite element model of turbine blisk was built asshown in Figure 5, which includes the number of elements,34 875, and the number of nodes, 68 678. The heat loads
Advances in Materials Science and Engineering 5
t (K)
1.150 × 103
1.069 × 103
9.885 × 102
9.075 × 102
8.265 × 102
7.455 × 102
6.645 × 102
5.836 × 102
5.026 × 102
4.216 × 102
3.406 × 102
Figure 6: Temperature distribution of blisk surface.
from high-temperature gas were exerted on the blisk ofturbine, where the temperature distribution of turbine bliskis demonstrated in Figure 6.
3.3. Structural Analysis of Turbine Blisk. Tetrahedron wasselected as the element of blisk’s finite element model. Thestructural analysis of turbine blisk was performed by trans-forming the analytical results of fluid analysis and thermalanalysis into the surface of turbine blisk based on finiteelement method. In order to more accurately express theresults of structure analysis, the shape function of tetrahedronelements (in (11)) and the displacement equations (in (12))[24] are applied to solve the node deformation of turbineblisk.The concentrated force andmoment of joints are equiv-alent to the distribution force based on the results of nodesdeformation and the relationship between displacement andstress (in (13)). Besides, the strain results of turbine blisk aregained by using (14).
The shape function of tetrahedron elements is
𝑁𝑖
=
1
6𝑉
(𝑎𝑖
+ 𝑏𝑖
𝑥 + 𝑐𝑖
𝑦 + 𝑑𝑖
𝑧) (𝑖 = 1, 2, 3, 4) ; (11)
here 𝑉 is the volume of tetrahedron and 𝑎𝑖
, 𝑏𝑖
, 𝑐𝑖
, and 𝑑𝑖
arethe related coefficients of node geometry.
The displacement equations of element node on threedirections are
𝑢 (𝑥, 𝑦, 𝑧) = 𝑎0
+ 𝑎1
𝑥 + 𝑎2
𝑦 + 𝑎3
𝑧,
V (𝑥, 𝑦, 𝑧) = 𝑏0
+ 𝑏1
𝑥 + 𝑏2
𝑦 + 𝑏3
𝑧,
𝑤 (𝑥, 𝑦, 𝑧) = 𝑐0
+ 𝑐1
𝑥 + 𝑐2
𝑦 + 𝑐3
𝑧.
(12)
The relationship between displacement and stress on theelement of turbine blisk is denoted by
𝜎𝑥
=
𝜕𝑢
𝜕𝑥
;
𝜎𝑦
=
𝜕V𝜕𝑦
;
Table 1: Input random variables of MRSM reliability analysis.
Input random variables Distribution Mean Standarddeviation
Inlet speed, V/(m/s) Normal 160 3.2Inlet pressure, 𝑝/Pa Normal 600 000 18 000Temperature, 𝑡/K Normal 1 150 15.56Material density, 𝜌/(kg/m3) Normal 4 620 92.4Rotation speed, 𝑤/(rad/s) Normal 1 168 23.36
𝜎𝑧
=
𝜕𝑤
𝜕𝑧
,
𝛾𝑥𝑦
=
𝜕𝑢
𝜕𝑦
+
𝜕V𝜕𝑥
;
𝛾𝑦𝑧
=
𝜕V𝜕𝑧
+
𝜕𝑤
𝜕𝑦
;
𝛾𝑧𝑥
=
𝜕𝑢
𝜕𝑧
+
𝜕𝑤
𝜕𝑥
.
(13)
The relationship between stress and strain on the elementof turbine blisk is expressed by
{𝜎} = [D] {𝜀el} , (14)
where 𝜎𝑥
, 𝜎𝑦
, and 𝜎𝑧
and 𝛾𝑥𝑦
, 𝛾𝑦𝑧
, and 𝛾𝑧𝑥
are the normalstresses and shear stresses on 𝑥, 𝑦, and 𝑧 directions, respec-tively; {𝜎} = [𝜎
𝑥
, 𝜎𝑦
, 𝜎𝑧
, 𝛾𝑥𝑦
, 𝛾𝑦𝑧
, 𝛾𝑧𝑥
]T are the components of
stress; [D] is the elastic matrix (or elastic stiffness matrix orthe stress and strainmatrix); {𝜀el} is the vector of elastic strain.
Under the effect of fluid pressure, heat stress, and cen-trifugal force, the deformation, stress, and strain of turbineblisk were analyzed. From this analysis, the nephogramsof deformation, stress, and strain are listed in Figure 7. InFigure 7, 𝑢, 𝜎, and 𝜀 indicate the deformation, stress, andstrain of turbine blisk (similarly hereinafter). The changingcurves of deformation, stress, and strain are shown inFigure 8. Figures 7 and 8 reveal that the maximum defor-mation locates on the blade-tip of turbine blisk, while themaximum stress and strain locate on root of turbine blisk.
As shown in Figures 7 and 8, the highest temperaturelocates on the top of blisk; meanwhile, the temperaturegradually reduces from the top of blisk to the root of blisk.However, the maximum stress and strain are on the root ofblisk, and the stress and strain decrease from the root ofblisk to the top of blisk in which the variations of the stressand strain of turbine blisk hold close relationship with thegeometrical shape of turbine blisk.The above conclusions areconsistent with practical engineering.
4. Reliability Analysis of Turbine Blisk
In the light of the results of fluid-thermal-solid couplinganalysis, the points corresponding to the maximum valuesof blisk’s deformation, stress, and strain are regarded as thecomputational point of reliability analysis of turbine blisk.In accordance with the input random variables in Table 1
6 Advances in Materials Science and Engineering
u (m
)
3.740 × 10−3
3.428 × 10−3
3.116 × 10−3
2.805 × 10−3
2.493 × 10−3
2.181 × 10−3
1.870 × 10−3
1.558 × 10−3
1.247 × 10−3
9.349 × 10−4
6.233 × 10−4
3.116 × 10−4
0
(a) Deformation distribution on blisk
𝜎 (P
a)
9.557 × 108
8.760 × 108
7.964 × 108
7.168 × 108
6.371 × 108
5.575 × 108
4.779 × 108
3.983 × 108
3.186 × 108
2.390 × 108
1.594 × 108
7.975 × 107
1.734 × 105
(b) Stress distribution on blisk𝜀 (
m/m
)
9.774 × 10−3
9.143 × 10−3
8.312 × 10−3
7.481 × 10−3
6.649 × 10−3
5.819 × 10−3
4.988 × 10−3
4.157 × 10−3
3.326 × 10−3
2.459 × 10−3
1.633 × 10−3
8.823 × 10−4
1.223 × 10−6
(c) Strain distribution on blisk
Figure 7: Deformation, stress, and strain distribution on blisk.
and the thought of MRSM, the multiresponse surface modelturbine blisk was established as follows:
𝑦1
= −1.43 × 10−2
+ 9.8308 × 10−5V + 2.0023
× 10−9
𝑝 + 6.1827 × 10−6
𝑡 + 1.1185 × 10−6
𝜌
+ 1.6421 × 10−6
𝑤 − 3.1758 × 10−7V2
− 1.6927 × 10−15
𝑝2
− 2.6173 × 10−9
𝑡2
− 8.1784 × 10−11
𝜌2
+ 7.0349 × 10−10
𝑤2
,
𝑦2
= 3.5147 × 108
− 8.179 × 106V + 1.6905 × 10
3
𝑝
+ 1.7985 × 106
𝑡 + 9.3053 × 103
𝜌 − 1.3004
× 106
𝑤 + 2.5955 × 104V2 − 1.386 × 10
−3
𝑝2
− 19.2157𝑡2
− 0.6358𝜌2
+ 6.1378 × 102
𝑤2
,
Table 2: Single-object sampling statistics of blisk.
Object variables Maximum allowable values Sampling number𝑦1
/m ≤3.7 × 10−3 9942𝑦2
/Pa ≤1.07 × 109 9935𝑦3
/m/m ≤1.12 × 10−2 9954
𝑦3
= 0.0045 − 8.7321 × 10−5V + 1.822 × 10
−8
𝑝
+ 1.6927 × 10−5
𝑡 + 1.0621 × 10−7
𝜌 + 1.3835
× 10−5
𝑤 + 2.7717 × 10−7V2 − 1.494
× 10−14
𝑝2
+ 9.7086 × 10−10
𝑡2
− 7.515
× 10−12
𝜌2
+ 6.5152 × 10−9
𝑤2
.
(15)
When the maximum allowable deformation, stress, andstrain of turbine blisk are 𝑢
𝑎
= 3.7 × 10−3m, 𝜎
𝑎
= 1.07 ×
109 Pa, and 𝜀
𝑎
= 1.12 × 10−2m/m, the built multiresponse
surface model was simulated by 10 000 times using MCM.The analytical results are listed in Table 2.The histograms and
Figure 8: The curves of turbine blisk deformation, stress, and strain along radial direction.
Table 3: Multiobject sampling statistics of blisk.
𝑦1
/m, 𝑦2
/Pa, 𝑦3
/m/m Number ofsamples
𝑦1
≤ 3.7 × 10−3
∩ 𝑦2
≤ 1.07 × 109
∩ 𝑦3
≤ 1.12 × 10−2 9919
𝑦1
> 3.7 × 10−3
| 𝑦2
> 1.07 × 109
| 𝑦3
> 1.12 × 10−2 81
simulation history curves of maximum deformation 𝑢max,maximum stress 𝜎max, and maximum strain 𝜀max are shownin Figures 9 and 10, respectively. Through the comprehensiveperformance evaluation, the results were summarized inTable 3.
As shown in Figure 10, the output responses (deforma-tion 𝑦
1
, stress 𝑦2
, and strain 𝑦3
) for turbine blisk obeynormal distributions with the corresponding mean values of
3.452 × 10−3m, 9.974 × 108 Pa, and 1.040 × 10−2m/m and thecorresponding standard deviation of 8.476 × 10−9m, 7.672 ×
102 Pa, and 8.385 × 10−8m/m, respectively. From Table 2, itis illustrated that the reliability degrees for the deformation,stress, and strain of turbine blisk are 0.9942, 0.9935, and0.9954, respectively. The above conclusions pledge the reli-ability and security of turbine blisk design.
As revealed in Table 3, the comprehensive reliabilitydegree of turbine blisk is obtained as 0.9919 through joint reli-ability analysis, which basicallymeets the design requirementof aeroengine turbine blisk.
5. Conclusions
The goal of this effort is to apply the high accuracy and highefficiency MRSM to the comprehensive reliability evaluation
8 Advances in Materials Science and Engineering
0 2000 4000 6000 8000 10000Number of samples
3
3.8
3.6
3.4
3.2
×10−3u
max
(m)
(a) Simulation history of deformation
0 2000 4000 6000 8000 10000Number of samples
0.85
0.9
0.95
1
1.05
1.1
1.15×109
𝜎m
ax(P
a)
(b) Simulation history of stress
0 2000 4000 6000 8000 10000Number of samples
0.9
0.95
1
1.05
1.1
1.15
1.2×10−2
𝜀 max
(m/m
)
(c) Simulation history of strain
Figure 9: Simulation samples of turbine blisk.
of aeroengine turbine blisk through themultiobject reliabilityanalyses of the deformation, stress, and stress of turbineblisk based on fluid-thermal-structural coupling analysis.The present study establishes the mathematical model ofMRSM with the quadratic response surface function. Someconclusions are drawn as follows:
(1) The maximum deformation, maximum stress, andmaximum strain of blisk are 3.74 × 10−3m, 9.557 ×
108 Pa, and 9.974 × 10−3m/m, respectively. Besides,the distributions of blisk’s deformation, stress, andstrain are gained.
(2) The reliability degrees of blisk’s deformation, stress,and strain are 0.9942, 0.9935, and 0.9954, respectively,when the allowable deformation, stress, and strain
are 𝑢𝑎
= 3.7 × 10−3m, 𝜎𝑎
= 1.07 × 109 Pa, and 𝜀𝑎
=1.12 × 10−2m/m, respectively. Based on the conclu-sions, the comprehensive reliability degree of blisk is0.9919.
(3) The fluid-thermal-structural coupling analysis meth-od is adopted for the reliability analysis of aeroengineturbine blisk, which is promising to improve compu-tational accuracy.
(4) The efforts of this paper demonstrate that MRSMcan be adopted to solve the comprehensive reliabilityanalysis withmultiple disciplines andmultiple objectsbesides single-object reliability analysis, which pro-vide a promising approach for complex structuralreliability analysis.
Advances in Materials Science and Engineering 9
3 3.83.63.43.20
200
400
600
800Re
lativ
e fre
quen
cy
×10−3umax (m)
(a) Frequency distribution of blisk deformation
0
200
400
600
800
Rela
tive f
requ
ency
0.85 0.9 0.95 1 1.05 1.1 1.15
×109𝜎max (Pa)
(b) Frequency distribution of blisk stress
0
200
400
600
800
Rela
tive f
requ
ency
0.9 0.95 1 1.05 1.1 1.15 1.2×10−2𝜀max (m/m)
(c) Frequency distribution of blisk strain
Figure 10: Frequency distribution of turbine blisk.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This paper is cosupported by the National Natural ScienceFoundation of China (Grants nos. 51275138 and 51475025),the Science Foundation of Heilongjiang Provincial Depart-ment of Education (Grant no. 12531109), and Hong KongScholars Program (Grant no. XJ2015002). The authors wouldlike to thank them.
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