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Application of Probabilistic and Nonprobabilistic Hybrid Reliability Analysis Based on Dynamic Substructural Extremum Response Surface Decoupling Method for a Blisk of the Aeroengine

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  • School of Energy and Power Engineering

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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.
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Research Article
Application of Probabilistic and Nonprobabilistic
Hybrid Reliability Analysis Based on
Dynamic Substructural Extremum Response Surface
Decoupling Method for a Blisk of the Aeroengine
Bin Bai,1,2 Wei Zhang,1,2 Botong Li,1,2 Chao Li,3and Guangchen Bai3
1College of Mechanical Engineering & Applied Electronics Technology, Beijing University of Technology, Pingleyuan 100,
Chaoyang District, Beijing 100022, China
2Beijing Key Laboratory for the Nonlinear Vibration and Strength of Mechanical Structures, Beijing University of Technology,
Pingleyuan 100, Chaoyang District, Beijing 100022, China
3School of Energy and Power Engineering, Beihang University, Beijing 100191, China
Correspondence should be addressed to Bin Bai; baibin@.com and Wei Zhang; sandyzhang@yahoo.com
Received 23 September 2016; Accepted 9 January 2017; Published 28 March 2017
Academic Editor: Giacomo Frulla
Copyright ©  Bin Bai et al. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For the nondeterministic factors of an aeroengine blisk, including both factors with sucient and insucient 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. e model includes random variables and interval variables to determine the interval failure probability and the interval
reliability index. e extremum response surface function and its ow chart of mixed reliability analysis are given. e interval
analysis is embedded in the most likely failure point in the iterative process. e 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 eciency of this method is superior and more reasonable for analysing and designing a mistuned blisk. erefore,
this methodology has very important practical signicance.
1. Introduction
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
eective method for handling uncertainty. However, the
probabilistic model and fuzzy model require more data to
dene 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
[, ]. e probabilistic model has certain limitations for the
complexity structures, which has continued to increase with
the progress of science and technology. In addition, only
a certain range of uncertainties can be obtained instead of
their probabilistic distribution. Some scholars believe that
the probabilistic model cannot be accurately dened with
too little data. Currently, the interval model can be used to
describe uncertainties; the nonprobabilistic method is used
to analyse the structure. In , Ben-Haim [] proposed the
conceptofnonprobabilisticreliabilitybasedontheconvexset
Hindawi
International Journal of Aerospace Engineering
Volume 2017, Article ID 5839620, 11 pages
https://doi.org/10.1155/2017/5839620
International Journal of Aerospace Engineering
model and measured reliability through the maximum extent
of the uncertainty for a system. Ben-Haim and Elishako
[] proposed a measurable method for the nonprobabilistic
concept; he thought that nonprobabilistic reliability belonged
to a certain range of nondeterministic parameters; that is, the
reliability index is an interval rather than a specic variable.
e models mentioned in this paper can be used to
describe uncertainties. Some uncertain parameters have
sucient 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. erefore, 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 non-
probabilistichybridreliabilitymethodcantakefulladvantage
of the known information to achieve an eective analysis.
Many scholars have studied probabilistic and nonprobabilis-
tic 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 insucient. en 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 ve-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 reli-
ability model that contained randomness, fuzziness, and
nonprobabilistic uncertainty based on the structural fuzzy
random reliability and nonprobabilistic set-based models
was presented in []. Furthermore, based on a residual
strength model, the fatigue reliability was evaluated using
hybrid uncertain parameters in []. Furthermore, Fang et
al. converted the interval variables and fuzzy variables into
random variables based on the maximum entropy principle
and eectively analysed the reliability in the situation that
random variables, fuzzy variables, and interval variables
coexist in a problem. e branch-and-bound method for
the probabilistic reliability analysis of structural systems was
combined by Wang et al. with the nonprobabilistic branch-
and-bound method for determining the dominant failure
modes of an uncertain structural system; meanwhile, the
compatibility of the classical probabilistic model and the
interval-set model was discussed by them, verifying the
physical meaning of safety measures. In [], a modied fuzzy
interval perturbation method was proposed for predicting a
hybrid uncertain temperature eld involving both interval
and fuzzy parameters in material properties and boundary
conditions. In addition to this, An et al. introduced the
truncated probability reliability model, the fuzzy random
reliabilitymodel,andthenonprobabilisticintervalreliability
model to present a new hybrid reliability index to evaluate
structural hybrid reliability based on the random-fuzzy-
interval theory. Jiang et al. employed the probabilistic and
nonprobabilistic convex model methods to address two cases
for the uncertainty domain and the failure surface; they
analysed cracked structures, addressed the diculties in epis-
temic uncertainty modelling, adopted the scaled boundary
nite element method to calculate the stress intensity, and
developedtheresponsesurfacetosolvethehybridreliability
model. Furthermore, Han et al. adopted the response surface
technique to compute the interval of the failure probability
of the structure based on the probability-interval hybrid
uncertainty.
ere are other methods for HRA [–]; for exam-
ple,[]developedanewHRAtechniqueforstructures
with multisource uncertainties that contained randomness,
fuzziness, and nonprobabilistic boundedness. Hurtado and
Alvarez proposed a Monte Carlo method for probability and
interval approaches for reliability analysis; meanwhile, they
discussed the use of Monte Carlo methods for both reliability
and interval analysis from the point proposed representation.
In addition to this, a mixed perturbation Monte Carlo
method with a mixture of random and interval parameters
was presented by Gao et al., and, meanwhile, three theories
or methodologies, namely, the Taylor expansion, matrix
perturbation theory, and random interval moment method,
were combined to develop expressions for the mean value and
standard deviation of random interval structural responses.
Besides, a hybrid probabilistic model was presented in []
to solve the fatigue reliability problems of steel bridges.
Considering the parameter characteristics of the plot of stress
against the number of cycles to failure (-curve), Wu et al.
proposedahybridprobabilisticandintervalcomputational
scheme to robustly assess the stability of engineering struc-
tures. Also, based on -approach and fracture mechanics
approach, Xue et al. assessed a tension leg platform using
hybrid probabilistic and nonprobabilistic models. In addition
to this, the HRA was also used in the nuclear industry;
for example, Ib´
a˜
nez-Llano et al. presented a new approach
for estimating the exact probabilistic quantication results
by combining Monte Carlo simulation with the truncation
limits of the binary decision diagram approach in the nuclear
industry.
Other researchers optimized structures and studied their
sensitivity through probabilistic and nonprobabilistic hybrid
reliability methodology as well [–]. For example, Luo and
Zhang investigated an adhesive bonded steel-concrete com-
posite beam with probabilistic and nonprobabilistic uncer-
tainties and mathematically formulated the reliability-based
optimization, incorporating mixed reliability constraints as
a nested problem. Pedroni and Zio considered the model of
an aircra with a twin-jet engine, including  inputs and
 outputs; meanwhile, they propagated the aleatory uncer-
tainty described by probability distributions using MCS and
solved the numerous optimization problems related to the
propagation of epistemic uncertainty using interval analysis.
Besides, Liu et al. developed a hybrid uncertainty model and
used an ecient decoupling strategy to solve the nesting
optimization problem to obtain an equivalent single-layer
optimization model. Furthermore, based on active learning
International Journal of Aerospace Engineering
kriging, Yang et al. investigated HRA. e nonprobabilistic
set-theory convex model was combined with the classical
probabilistic approach to optimize structures exhibiting ran-
dom and uncertain-but-bounded mixed uncertainties in [].
In other applications as well, for example, [–], Xia
and Yu proposed the change-of-variable interval stochastic
perturbation method to predict the interval of the response
probability density function and the response condence
interval of a hybrid uncertain structural-acoustic system with
random and interval variables. Meanwhile, Chen et al. pre-
sented a hybrid stochastic interval perturbation method for
the unied energy ow analysis of coupled vibrating systems.
In addition to this, combining with a backpropagation neural
network, Peng et al. investigated reliability analysis based on
the hybrid uncertainty reliability mode, in the process of
their research, the random variables and interval variables
were used as the input layer of the neural network, and
the response variables were obtained through the output
layer. Furthermore, Fan and Zhang used a convex model to
simulate the uncertainties of isolated structural parameters
andusedarandommodeltosimulatetheuncertaintiesof
the seismic input. An interval analysis for parameter identi-
cations was presented by Zhang et al. to address both mea-
surement noise and model uncertainty. Xu et al. developed
a model called inexact two-stage fuzzy chance-constrained
programming for handling multiple uncertainties associated
with solid waste management systems. Meanwhile, they
structured the safety margin equation to analyse the fatigue
reliability of the system by adopting the “stress-strength”
theory. Also, Drugowitsch and Pouget used probabilistic
and nonprobabilistic approaches for the neurobiology of
perceptual decision-making.
For an aeroengine working in a complicated high tem-
perature, high pressure, and high rotational speed envi-
ronment, signicant information is available as uncertain
parameters, such as rotational speed and temperature. How-
ever, other uncertain parameters, such as the coecient of
thermal conductivity and the coecient of expansion, lack
sucient data. erefore, 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 [].
is new method is called the probabilistic and nonprob-
abilistichybridreliabilityanalysisbasedondynamicsub-
structural extremum response surface decoupling method
(P-NP-HRA-DS-ERSDM) and is used to analyse the blisk
of aeroengine. e reliability problems are better resolved
bythehybridmodelcontainingbothrandomandinterval
variables.
2. Basic Theory of P-NP-HRA-DS-ERSDM
For designing and analysing engineering structures, random
variables and interval variables need to be included simul-
taneously. erefore, a probabilistic and nonprobabilistic
X2
X1
Reliable domain
Failure domain
0

Yg(X, Y)=0

Yg(X, Y)=0
F : e region of extreme postural belt.
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 =
(1,2,...,𝑚)and the interval variable is =(
1,2,...,
𝑛). en, the expression of the performance function is
written as =(,), and the failure probability is dened
as 𝑓=Pr{(,) ≤ 0}.
Aer introducing the interval variable ,thelimitstate
surface function (,) = 0 is no longer the unique surface
in the space ,but it is a belt body that consists of two bound-
ary surfaces, that is, max𝑌(, = 0) and min𝑌(, = 0),
asshowninFigure.
erefore, the failure probability 𝑓has upper and lower
boundaries:
min
𝑓=Pr max
𝑌(,)≤0,
max
𝑓=Pr min
𝑌(,)≤0. ()
e reliability index is not a specic value but an inter-
val (i.e., ∈[
𝐿,𝑅]), where 𝐿and 𝑅are the maximum
and the minimum reliability indexes, respectively.
en the following two optimization problems can be
solved: the maximum and the minimum reliability indexes
of the limit state belt can be obtained.
𝑅=min
s.t.max
𝑌(,)=0, ()
𝐿=min
s.t.max
𝑌(,)=0, ()
where (⋅)is the performance function of transformed into
standard normal space and (⋅) is standard normal space.
International Journal of Aerospace Engineering
e failure probabilities of the maximum and the mini-
mum values are expressed as
min
𝑓
𝑅,
max
𝑓
𝐿()
In engineering applications, the maximum failure proba-
bility of the structure is oen the one of most concern, and it
has signicant reference value for engineering and technical
sta. erefore, 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 eciency is very low for some
mechanical parts such as a complex aeroengine working in a
poor environment. us, the decoupling method proposed in
[] is utilized in this paper, and the proposed methodology
is called P-NP-HRA-DS-ERSDM. e modal and vibration
response of a blisk are analysed using this method. Accord-
ing to the actual conditions, the cross terms are ignored,
and the response surface function can be represented as
follows:
(,)=
0+𝑚
𝑖=1𝑖𝑖+𝑛
𝑗=1𝑗𝑗+𝑚
𝑖=1𝑖2
𝑖+𝑛
𝑗=1𝑗2
𝑗,()
where 0,,,,andare (2 + 2 + 1) undetermined
coecients in the quadratic polynomial.
e sample centre point (,) is continually updated in
the solution process. For the rst time, the iteration of (,)
is located in (𝑋,𝑐),where𝑋is the mean of the random
variable ,𝑐is the median of the interval variable ,and
thecoordinatesaregivenin().eother(2 +2)sample
points are selected around the centre point, as shown in
Figure .
𝑖±𝑥⋅
𝑋𝑖, =1,2,...,,
𝑗±𝑦⋅𝑟
𝑗, =1,2,...,, ()
where 𝑋𝑖is the standard deviation of the random variable
𝑖,𝑟
𝑗is the radius of the interval variable 𝑖,and𝑥,𝑦are
the coecients of the sample points for the random variable
and the interval variable, respectively.
e value of the original performance function is calcu-
lated at each sample point; then, undetermined coecients of
theresponsesurfacefunctionaresolved.
Combining the probabilistic and nonprobabilistic hybrid
reliability model with (), the approximate mixed reliability
model is constructed as follows:
𝐿=min
𝑈
s.t.max
𝑌
(,)=0, ()
where
is the approximate limit state function in the space
.
P-NP-HRA-DS-ERSDM is used to solve the probabilistic
andnonprobabilisticHRAoftheblisk,andthespecic
iterative process is as follows.
X2(Y2)
X2(Yr
2)
X1(Yr
1)
X(Y)
X1(Y1)
e initial sample points
New sample points
X(Yc)
F : Processing of the sample points selected.
Assume that (𝑘) and (𝑘) are obtained in the iteration
process of the th step, interval variable (𝑘) is xed in the
next step, and (𝑘+1) is calculated; namely,
(𝑘+1) =
(𝑘) +(𝑘),()
where the searching direction is expressed by
(𝑘) =
(𝑘),(𝑘)(𝑘)𝑇
(𝑘),(𝑘)
(𝑘),(𝑘)
2
⋅∇
(𝑘),(𝑘)−(𝑘),
()
where
is the gradient of
, which is determined by the
following function:
m(,)=1
2+
(,)
,()
where is a constant that satises  > /∇
((𝑘),(𝑘))
and =2
(𝑘)/∇
((𝑘),(𝑘)) + 10.is the iteration step
length, which is determined by
=
( = 0.5),
=max |
(𝑘) +(𝑘),(𝑘)−(𝑘),(𝑘)
<0.
()
en, (𝑘+1) iscalculatedusingintervalanalysisaer
obtaining (𝑘+1):
(𝑘+1) =min
𝑌
(𝑘+1),(𝑘)
s.t.
𝐿≤
(𝑘) ≤
𝑅()
International Journal of Aerospace Engineering
Setting the initial point: U(k) =
X,Y
(k) =Y
c
k=1
Fixation Y(k)
Searching U(k+1)
d(k) =
GU(k),Y
(k)U(k)T
GU(k),Y
(k)
GU(k),Y
(k)
2
GU(k),Y
(k)−U
(k)
Fixation U(k+1)
Searching Y(k+1)
Y(k+1) =Y
(k)
Convergence
k=k+1
Stop
s.t.
Y(k+1) =
YL≤Y
(k) ≤Y
R

Y
GU(k+1),Y
(k)
U(k+1) =U
(k) +d
(k)
F : Flow chart of decoupling method.
until it meets
(𝑘+1) −(𝑘)
(𝑘)
≤
1,
(𝑘+1),(𝑘+1)≤
2,
()
where 1and 2are arbitrary small numbers.
e design test point (
,
)is obtained from the ow
chart shown in Figure .
e design test point (
,
)is obtained by HRA and
() in each iteration step. en, the sample centre point
(,)that is closer to the failure surface is obtained through
interpolation, as shown in Figure .
e expressions of the connective points (𝑋,𝑐,
(𝑋,𝑐)) and (
,
,(
,
))are obtained by
(,)−
𝑋,𝑐
−
𝑋=
,
−(,)
− ,
(,)−
𝑋,𝑐
−𝑐=
,
−(,)
− .
()
When (,) = 0, the design points close to the real limit
state surface are acquired as follows:
=
𝑋+
−
𝑋𝑋,𝑐
𝑋,𝑐−
,
,
󸀠=
𝑐+
−𝑐𝑋,𝑐
𝑋,𝑐−
,
.
()
e interval variable of the most recently obtained 󸀠may
overow the boundary of the interval in the process of each
iteration, and, in this case, canbeexpressedas
=min 󸀠,𝑅, if 󸀠>
𝑅,
=max 󸀠,𝐿, if 󸀠<
𝑅.()
Basedontheaboveanalysis,theprobabilisticandnon-
probabilistic HRA process are as follows:
() Set the initial iteration point (𝑡) =(
𝑋1,𝑋2,...,
𝑋𝑚),() = (𝑐
1,𝑐
2,...,𝑐
𝑛).
International Journal of Aerospace Engineering
X2(Y2)
X(Y)
X1(Y1)
X,
Y,g
X,
Y
X,Y
c,gX,Y
c
0
g(X, Y)=0
F : Obtaining a new sample centre point.
Initialize random variables and
interval variables
Convergence
e maximum failure
probability
Establish performance function
Sampling
Solve approximate
solution of the reliability
model
Build extreme response
surface model
Update the sample
space
Perform simulation based on the response
surface
Ye s
No
F : Flow chart of hybrid reliability analysis based on the extreme response surface.
() Establish a quadratic polynomial extremum response
surface function. e sample centre point is set to
(𝑋,𝑐)in the initial iteration; then, calculate the
function values of (2 + 2 + 1) points, and obtain
the undetermined response surface coecients.
() Solve the approximate mixed reliability problems
using (), and obtain the test point (
,
).
() Solve the new sample centre point ((𝑡+1),(𝑡+1)).
() Judge convergence. If (𝑡+1) − (𝑡)/(𝑡)≤
1,go
to step (6); otherwise, set =+1,andgotostep(2).
()Simulatetheconstructedresponsesurfacefunction,
andcalculatethemaximumfailureprobability.
e maximum failure probability of the probabilistic and
nonprobabilistic HRA can be obtained, as shown in Figure .
To contain both random variable and interval variable anal-
ysis, MLNA is usually adopted in the conventional research,
which is time-consuming and has a low computational e-
ciency. Nevertheless, the P-NP-HRA-DS-ERSDM is utilized
and the interval analysis is embedded in the most likely
failure point in the iterative process. Probabilistic analysis
and nonprobabilistic analysis are investigated alternately, so
International Journal of Aerospace Engineering
T : Distribution of nondeterministic variables of tuned blisk.
Nondeterministic variables Parameter  Parameter  Distribution pattern
/rads−1  . Normal distribution
/C  . Normal distribution
/×−5/C−1 . . Normal distribution
/W(mC)−1 . . Normal distribution
/mm . . Normal distribution
/×11/Pa . . Interval variable
pr . . Interval variable
den/kgm−3   Interval variable
Note. In the normal distribution, parameter  and parameter  represent the mean and the standard deviation of a random variable. In the interval variable,
parameter  and parameter  represent the lower bound and upper bound. (e meaning in Table  is the same as in Table .)
T : Analysis results of modal and vibration response of P-NP-HRA-DS-ERSDM for the tuned blisk.
(a)
Random variable Interval variable
. ×11 .
. den .
. pr .
. —
×−5 . —
——
(b)
Response e maximum failure probability
— MCM MLNA P-NP-HRA-DS-ERSDM
max
𝑓MCS/% /(h) max
𝑓/% /(h) Er/% MC/% max
𝑓/% /(h) Er/% MC/% ML/%
.
.
.
.
.
.
.
.
.
. .
. HZ
sum . . . . .
.    
strs (×13). . . . .
.
str_e(×7). . . . .
.
𝑦. . . . . . . . . . .
.
Note.𝑃max
𝑓MCS: the maximum failure probability using MCS, 𝑡: computing time, Er: relative error to the MCM, 𝜂MC: relative computational eciency to MCM,
and 𝜂ML: relative computational eciency to MLNA. (e meaning in Table  is the same as in Table .)
the intermediate values can call each other in turn, and the
computational eciency is improved, observably.
3. P-NP-HRA of Blisk
P-NP-HRA-DS-ERSDM is used to analyse a blisk and is com-
pared with MLNA and MCS to verify its scientic rationality.
3.1. P-NP-HRA of a Tuned Blisk. 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 ,gastemperature
, blade thickness , expansion coecient ,andthermal
conductivity coecient are regarded as random variables,
andallofthemareassumedtoobeythenormaldistribution
and to be independent of each other. e elastic modulus ,
density den, and Poissons ratio (pr) are regarded as interval
variables. eir distribution types and parameters are shown
in Table , and the results of the calculation are shown in
Table .
e maximum failure probabilities of the natural fre-
quency, modal shape, and vibration response for the tuned
blisk are investigated using MCM, MLNA, and P-NP-
HRA-DS-ERSDM, respectively. e relative errors and the
International Journal of Aerospace Engineering
T : Distribution of nondeterministic variables of mistuned blisk.
Nondeterministic variables Parameter  Parameter  Distribution pattern
/rads−1  . Normal distribution
/C  . Normal distribution
/mm . . Normal distribution
pr𝑏. . Normal distribution
𝑏/×−5/C−1 . . Normal distribution
𝑏/W(mC)−1 . . Normal distribution
den𝑏  Interval variable
𝑏×11/Pa . . Interval variable
pr𝑑. . Normal distribution
𝑑/×−5/C−1 . . Normal distribution
𝑑/W(mC)−1 . . Normal distribution
den𝑑  Interval variable
𝑑×11/Pa ×11/Pa . . Interval variable
computational eciency of all three methods are analysed,
as shown in Table . For the natural frequency, modal
displacement, modal stress, modal strain energy, and vibra-
tion response, the relative errors of MLNA to MCM are
.%, .%, .%, .%, and .%, respectively, and the
relative errors of P-NP-HRA-DS-ERSDM to MCM are .%,
.%, .%, .%, and .%, respectively; these values
meet the engineering requirements. For the calculation of
modal and vibration response, the computational eciency
of MLNA relative to MCM increases by .% and .%,
respectively, and the computational eciency of P-NP-HRA-
DS-ERSDM relative to MCM increases by .% and .%,
respectively. us, the computational eciency of P-NP-
HRA-DS-ERSDM relative to MLNA increased by .% and
.%, respectively. erefore, the computational accuracy
of using P-NP-HRA-DS-ERSDM and MLNA approximately
equals that of using MCM. However, the computational
eciency of P-NP-HRA-DS-ERSDM is higher than that of
MLNA.
3.2. P-NP-HRA of a Mistuned Blisk. For a mistuned blisk,
the rotational speed ,gastemperature, blade thickness
, Poisson’s ratio of the blade pr𝑏, expansion coecient of
the blade 𝑏, thermal conductivity coecient of the blade
𝑏, Poisson’s ratio of the disk pr𝑑, expansion coecient of
the disk 𝑑,andthermalconductivitycoecientofthedisk
𝑑are regarded as random variables, and all of them are
hypothesized to obey the normal distribution and to be
independent of each other. e elastic modulus of the blade
𝑏, density of the blade den𝑏,elasticmodulusofthedisk𝑑,
and density of the disk den𝑑are regarded as interval variables.
eir distribution types and parameters are shown in Table ,
and the calculation results are shown in Table .
e maximum failure probability of the natural fre-
quency, modal shape, and vibration response for the mis-
tunedbliskarecalculatedusingMCM,MLNA,andP-NP-
HRA-DS-ERSDM.erelativeerrorsandthecomputational
eciency of the three methods are analysed, as shown in
Table . When the blisk is mistuned, the computational
time is very long using MCM, and convergence may not
be achieved. e computational accuracy of P-NP-HRA-DS-
ERSDM approximately equals that of MLNA, while the com-
putational eciency of P-NP-HRA-DS-ERSDM is .%
and .% higher than that of MLNA. e increase in com-
putational eciency for P-NP-HRA-DS-ERSDM compared
to MLNA is higher for the mistuned blisk than for the tuned
blisk. Furthermore, the superiority of this methodology is
more obvious for the mistuned blisk than for the tuned blisk,
verifying that this method is feasible.
4. Conclusions
A nondeterministic analysis method with high eciency 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 vari-
ables, the interval failure probability and interval reliability
index are obtained. A ow 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
theblisk.evariablesmainlyaectingtheoutputresponse
are regarded as interval variables, and the other variables
are regarded as random variables. In the tuned blisk, ,den,
and pr are regarded as interval variables, while ,,,,
and are regarded as random variables. However, in the
mistuned blisk, den𝑏,𝑏, den𝑑,and𝑑are regarded as interval
variables, while ,,,pr𝑏,𝑏,𝑏,pr𝑑,𝑑,and𝑑are regarded
as random variables.
e maximum failure probabilities of the natural fre-
quency, 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 .%,
.%, .%, .%, and .%, respectively, and those of
using P-NP-HRA-DS-ERSDM are .%, .%, .%, .%,
and .%; these values meet the engineering requirements.
International Journal of Aerospace Engineering
T:Analysisresultsofmodalandvibrationresponse of P-NP-HRA-DS-ERSDM for the mistuned blisk.
(a)
Random variable Interval variable
. den𝑑.
. den𝑏.
𝑑. 𝑑×11 .
𝑏. 𝑏×11 .
pr𝑑.
——
pr𝑏.
.
𝑏×−5 .
𝑑×−5 .
(b)
Response e maximum failure probability
— MCM MLNA P-NP-HRA-DS-ERSDM
max
𝑓MCS/% /()max
𝑓/% /()Er/%MC/% max
𝑓/% /()Er/%MC /% ML /%
——
.
. —
.
. — .
. HZ
dsum . .
.
strs(×13). .
.
str_e (×7). .
.
𝑦 . . — — . . — — .
.
e computational eciency of MLNA relative to MCM
increases by .% and .%, respectively, but that of
P-NP-HRA-DS-ERSDM increases by .% and .%,
respectively. us, the computational eciency of P-NP-
HRA-DS-ERSDM relative to MLNA increased by .% and
.%.However,forthemistunedblisk,thecomputational
eciency of P-NP-HRA-DS-ERSDM increases by .%
and .% 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. e scientic rationality and validity for
researching a blisk using P-NP-HRA-DS-ERSDM is veried,
and this method is shown to be particularly superior to
MLNA for a mistuned blisk.
Conflicts of Interest
e author declares that there are no conicts of interest
regarding the publication of this paper.
Acknowledgments
is work has been supported by the National Natural
Science Foundation of China (Grant no. ) and
Project supported by Beijing Postdoctoral Research Founda-
tion (Grant no. ZZ-).
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... Dev.) and distribution types, are determined by the extremum selection method [44]. In respect of engineering practices [19], the random variables are considered to be normal distribution, for reducing the complexity of analysis [45,46]. The distribution features of all random inputs are listed in Table I. [47,48]. ...
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A new hybrid probabilistic and interval computational scheme is proposed to robustly assess the stability of engineering structures involving mixture of random and interval variables. Such hybrid approach possesses noticeable flexibility by directly implementing primitive information on uncertain system parameters, thus the validity of the structural safety assessment against uncertainties can be improved. By implementing the different types of uncertainties, the presented approach is able to separately investigate the effects of random and interval variables acting upon the overall structural stability. A unified interval stochastic sampling (UISS) approach is proposed to calculate the statistical characteristics (i.e., mean and standard deviation) of the lower and upper bounds of the linear bifurcation buckling load of engineering structure involving hybrid uncertain system parameters. Consequently, the stability profile of engineering structure against various uncertainties can be parametrically established, such that the bounds on the maximum structural buckling load at any particular percentile of probability can be effectively determined. Both academic and practically motivated engineering structures have been thoroughly investigated by rigorously establishing the corresponding structural stability profiles, so the applicability and accuracy of the proposed method can be critically justified.
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For determining the energy flow in coupled vibrating systems with limited information, two hybrid uncertain models have been introduced to model the uncertain parameters. One is the hybrid probability and interval model in which the probability and interval variables exist simultaneously. The other one is the hybrid interval probability model in which some distribution parameters of the probability variables are given as interval variables. A hybrid stochastic interval perturbation method (HSIPM) is proposed for the unified energy flow analysis in coupled vibrating systems under two hybrid uncertain models. In the proposed method, by temporarily neglecting the uncertainties of interval variables, the first-order stochastic perturbation method is adopted to calculate the expectation and variance of the energy vector. Afterwards, the interval of the expectation and variance of the energy vector can be obtained by the first-order interval perturbation method. Two numerical examples are given to illustrate the feasibility and effectiveness of the proposed method.