CHINESE JOURNAL OF MECHANICAL ENGINEERING
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Vol. 26, No. 6, 2013
DOI: 10.3901/CJME.2013.06.1160, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn
Distributed Collaborative Response Surface Method for Mechanical Dynamic Assembly Reliability Design BAI Guangchen and FEI Chengwei* School of Energy and Power Engineering, Beihang University, Beijing 100191, China Received August 30, 2012; revised March 13, 2013; accepted March 15, 2013
Abstract: Because of the randomness of many impact factors influencing the dynamic assembly relationship of complex machinery, the reliability analysis of dynamic assembly relationship needs to be accomplished considering the randomness from a probabilistic perspective. To improve the accuracy and efficiency of dynamic assembly relationship reliability analysis, the mechanical dynamic assembly reliability(MDAR) theory and a distributed collaborative response surface method(DCRSM) are proposed. The mathematic model of DCRSM is established based on the quadratic response surface function, and verified by the assembly relationship reliability analysis of aeroengine high pressure turbine(HPT) blade-tip radial running clearance(BTRRC). Through the comparison of the DCRSM, traditional response surface method(RSM) and Monte Carlo Method(MCM), the results show that the DCRSM is not able to accomplish the computational task which is impossible for the other methods when the number of simulation is more than 100 000 times, but also the computational precision for the DCRSM is basically consistent with the MCM and improved by 0.40~4.63% to the RSM, furthermore, the computational efficiency of DCRSM is up to about 188 times of the MCM and 55 times of the RSM under 10000 times simulations. The DCRSM is demonstrated to be a feasible and effective approach for markedly improving the computational efficiency and accuracy of MDAR analysis. Thus, the proposed research provides the promising theory and method for the MDAR design and optimization, and opens a novel research direction of probabilistic analysis for developing the high-performance and high-reliability of aeroengine. Key words: machinery dynamic assembly, reliability analysis, distributed collaborative response surface method, blade-tip radial running clearance
1
Introduction ∗
The multi-component assembly relationship of complex machinery directly influences the quality of mechanical product[1], thus, the assembly relationship design plays an important role in mechanical design. SEKHAN, et al, analyzed the statistical tolerance and clearance for assembly[2]. ZONG, et al, proposed the multi-objective stability control algorithm of heavy tractor semi-trailer based on differential braking[3]. HUANG, et al, studied the non-linear torsional vibration characteristics of an internal combustion engine crankshaft assembly[4]. These precision designs based on the tolerance match and reasonable assembly technologies can only ensure the static(steady or cold) mechanical assembly relationship without operation. However, for the complex machinery like an aeroengine under high temperature and high rotation operation, the change of components assembly relationship is always very significant in a working cycle[5–6]. Therefore, the assembly * Corresponding author. E-mail:
[email protected] This project is supported by National Natural Science Foundation of China(Grant Nos. 51175017, 51245027), Innovation Foundation of Beihang University for PhD Graduates, China(Grant No. YWF-12-RBYJ008), and Research Fund for the Doctoral Program of Higher Education of China(Grant No. 20111102110011) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2013
relationships need to be dynamically analyzed and designed to keep a rational assembly clearance under all work conditions. The blade-tip radial running clearance(BTRRC) analysis of aeroengine high pressure turbine(HPT) is the representative issue of mechanical dynamic assembly relationship design. LATTIME, et al, reviewed on the current practices and future directions of BTRRC and evaluated the BTRRC dynamically[7–8]. JIA, et al, defined the effect of rotor vibration on the variation and fast active control of BTRRC[6]. ANNETTE, et al, built the Model of BTRRC to validate the thermal effects on gas turbine transients[9]. KYPUROS, et al, established a reduced model to predict the thermal and rotational effects on BTRRC[10]. These efforts are based on a deterministic analysis for mechanical dynamic assembly relationship design, which hardly considers the random factors on the assembly relationship. According to engineering practice, many factors possess strong randomness on the variation of mechanical dynamic assembly relationship. For this reason, the randomness of parameters should be considered to research the reliability design of mechanical dynamic assembly relationship from a probabilistic perspective. Only in this way, can the dynamic assembly relationship be more accurately described for improving the rationality and reliability of mechanical assembly design.
CHINESE JOURNAL OF MECHANICAL ENGINEERING The complex machinery assemblage invariably consists of many components. The assembly relationship design for each component always refers to multiply disciplines. Hence the dynamic reliability analysis of assembly relationship for complex machinery like the HPT BTRRC is a multi-object multi-disciplinary(MOMD) reliability analysis of dynamic assembly relationship. Although some work have been done for multi-object or multi-discipline by a deterministic analysis[11–13], the proposed approaches and ideas cannot be applied to the dynamic assembly relationship reliability design for no complete theory. Therefore, in order to conduct preferably the MOMD reliability analysis of dynamic assembly relationship, it is necessary to address two key problems as follows. One key issue is to develop a new theory. In fact, the essence of dynamic assembly relationship design is the dynamic assembly design between static component and movement component. Currently, there is no related theory for the dynamic assembly relationship reliability design. The other key issue is to propose an effective method due to the great computational consumption of MOMD reliability analysis of dynamic assembly relationship. The Response surface method(RSM) of structural dynamic reliability analysis experiences a rapid development to improve the calculation efficiency while preserving the computing precision. Many RSMs have been proposed such as kriging surrogate model[14], neural network RSMs[15–16], support vector machine RSMs[17–19], rough set model[20] and stochastic RSM[21]. Compared with the general structure dynamic reliability, these RSMs are unfeasible to be directly used for the dynamic assembly relationship reliability design of complex machinery. To overcome the two issues, this paper is to develop the mechanical dynamic assembly reliability(MDAR) theory around the realistic engineering background that a deterministic design is transformed into a probabilistic design for the dynamic assembly relationship of complex machinery like the BTRRC of aeroengine HPT, and conceives a distributed collaborative response surface method(DCRSM) based RSM according to the MDAR analysis features. The MDAR is defined as the ability to maintain the specified assembly relationship of assembly objects under the specified work status and during the specified time. The assembly failure caused by the failures of components[5–10] is the problem of structural reliability and the probabilistic design of tolerance match dimension chain is the static assembly reliability issue[22–23], which does not belong to the MDAR content. However, the MDAR is very complicated and no related achievement for the MDAR analysis is published so far. The purpose of this paper is to present MDAR theory and DCRSM, and takes the BTRRC MDAR analysis of aeroengine HPT as an example to verify the theory and method.
2
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Mechanical Dynamic Assembly Reliability Analysis features
The RSM has been studied continuously for structural reliability analysis and perfectly employed in engineering [14–21, 24–26] . The RSM is able to improve the analytical accuracy and efficiency of structural reliability and provides an enlightening insight for the MDAR design with MOMD. Nevertheless, compared with the single structural reliability analysis, the MDAR needs to conduct the cooperative response analysis of assembly objects from a probabilistic perspective, which greatly increases the analysis difficulty and complexity. Thus, how to apply RSM to MDAR analysis deserves further study. Firstly, due to the dynamic assembly relationships often determined by the coordinating responses of assembly objects, the MDAR analysis always require to establish the analytical model of each assembly object respectively. Secondly, the mechanical dynamic assembly relationship subjects to the influences of many parameters such as transient temperature field, centrifugal force, etc. Hence, different analytical models need to be structured for different disciplines(heat transfer theory, rotor dynamics, aerodynamics, vibration etc) when the mechanical dynamic assembly relationship is designed. The structure reliability analysis based on RSM only need one analytical model. Although the thousands of times simulations is required to perform for finite element model (FEM), the analysis has high computational efficiency yet because of easily realizing the automatic operations. However, for the MOMD collaborative response of MDAR analysis, no published method is able to engage multiply structure models in automatic operation when an assembly relationship feature(like a clearance) was directly taken as a objective to establish a single response surface model(like the single structure reliability analysis), If the calculation is controlled artificially repeatedly, the computational efficiency is unacceptable. In addition, despite of the establishment of the response model of assembly relationship feature, the increased number of variables cannot either ensure the precision of response surface model compared with the single response surface model of structural reliability analysis. Therefore, the key advantage of MDAR analysis is to cope with the reliability analysis of multiply models and keep the acceptable precision and efficiency, which is also an important distinction from structure reliability analysis.
3
DCRSM for Multiply Models Reliability Analysis
3.1 RSM The RSM employs a response surface fitted by a series of experiments to simulate a real limit state surface[18]. If quadratic response surface model(in Eq. (1)) is applied to
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describe the relationship between the system response Y and the random parameter X[X1, X2, , Xr], the quadratic response surface function would be fitted by n sample values of random variables and the corresponding response n Yi i1 to replaces the finite element model(FEM) for reliability analysis.
Y ( X ) a0 BX X T CX ,
c11 , C cr1 crr
Y ij f ( X ij ) ，i 1,2, , m, j 1, 2, , n.
(2)
Y ij A0ij B ij X ij X ijT C ij X ij .
function(DRSF) of SOSD. A0ij , B ij and C ij are the undetermined coefficients. B ij , C ij and X ij are
(3)
3.2.2 Mathematical model of DCRSM Based on a quadratic response surface model in Eq. (1), the mathematical model of DCRSM is built for MDAR. Assuming that an assemblage involves m(m∈Z) assembly objects and each object refers n(n ∈ Z) subjects, the MOMD problem of complex machinery may be decomposed into multiple SOSD problems. Assuming Xij as the input parameter of the jth discipline in the ith object
B ij b1ij , b2ij , ,bkij ,
(6)
cij 11 , C ij ij ij ck1 ckk
(7)
T
X ij x1ij x2ij xkij ,
DCRSM
3.2.1 Basic thought of DCRSM The MDAR analysis requires building many analysis models in light of assembly objects and disciplines. DCRSM is proposed to address the MOMD MDAR analysis. The basic thought of DCRSM is as follows. Firstly, the “big” model of complex machinery that involves many elements(sub-components) and multiple subjects is divided into many “small” models which are single-object single-disciplinary(SOSD). Secondly, distributive response surface models(DRSMs) are built, which are equivalent to structure the SOSD response surface models according to each response analysis features. Lastly, the DRSMs are employed in the collaborative response reliability analysis that many “small” models are reintegrated to deal with the collaborative relationship among MOMD responses in order to accomplish the collaborative reliability analysis of many SOSD responses by collaborative response surface model(CRSM) or specified rule. Obviously, the DCRSM idea is beneficial to improve expediently the computing precision and efficiency of complex mechanical reliability analysis. The flow chart of MDAR analysis based on DCRSM is shown in Fig. 1.
(5)
This relationship is called distributed response surface
where r is the number of input variables, C the lower triangular matrix. 3.2
(4)
The above equation is rewritten by the format of response surface function, i.e.,
(1)
where a0, B and C are the undetermined coefficients of constant term, linear term and quadratic term respectively. B and C are
B b1 , b2 , , br ,
and Yij is the output response, their relationship is
(8)
where k is the number of SOSD input random variables
Fig. 1.
Flow chart of MDAR analysis based on DCRSM
Taking the output responses {Y ij }nj1 of all disciplines in each object as the input variables X i of single-object multi-discipline(SOMD) response surface model is denoted by
X i {Y ij }nj1 .
(9)
When Y i is the output response of SOMD response surface model, there is
Y i f ( X i ) A0i B i X i X iT C i X i .
(10)
CHINESE JOURNAL OF MECHANICAL ENGINEERING This relationship is called SOMD distributed response surface function(DRSF), where
A0i ,
i
B, C
i
are the
coefficients. Similarly, the output responses {Y i }im1 of all objects are regarded as the input random variables X of CRSM expressed by
X {Y i }im1 .
(11)
The whole response surface function Y is
. X T CX Y A0 BX
(12)
This relationship is called collaborative response surface function(CRSF) of MOMD where A , B , C are the 0
coefficients. From the above analysis, it is obvious that the quadratic response surface model Eq. (1) of MDAR analysis is decomposed into multiple DRSFs such as Eqs. (5), (10) and (12). This method is called DCRSM. It is also called the DCRSM of MDAR analysis when DCRSM is applied in the MDAR analysis. 3.2.3 DCRSM strengths As shown by the DCRSM principle of MDAR analysis, the DCRSM decomposes an MOMD problem of complex machinery, difficult and complicated to directly analyze by RSM, into the simple collaborative response problems of multiple objects, multiple disciplines and multiple models, which are easily analyzed possibly. Along the heuristic thought, the DCRSM is promising to conquer the MOMD problem of complex machinery by collaborative reliability analysis of multiple DRSMs, which is impossible for the traditional RSM. Therefore, it is valid for DCRSM to tackle various problems of increasing variable number, low computational efficiency and accuracy, etc. The reasons are as follows. (1) Compared with the “big” model(complicated mechanical model), the greatly reduced variable number of single (“small”) model is conducive to considerably decrease the computational task into easy analysis problem, and reduce the time of fitting response surface model to improve the fitting efficiency. In addition, the simulation efficiency can be ameliorated due to the relatively simple response surface model. (2) With the improvement of the simulation speed of response surface model, the precision of reliability analysis is likely to be enhanced by increasing the simulation times and even directly analyzing FEM using the random simulation methods. (3) The nonlinearity between input random variables and output response can be reasonably analyzed for “small” model to improve the calculation accuracy by increasing the simulation times since more factors can be considered comprehensively.
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(4) The automatic operation and the parallel computation for “small” model can be implemented simultaneously on different computers to significantly save the computing cost and improve the computational efficiency. (5) It is unique that the DRSMs can be built more reasonably by different response surface models (such as quadratic function, support vector machine and artificial neural network, etc) to improve the precision and validity of reliability analysis according to the response analysis properties of each object and discipline. Moreover, DCRSM can provide useful insight for the optimization design of MDAR. Through MDAR analysis, the main factors effecting assembly relationship are gained, which contributes to control each model and main factors to achieve the goals of optimization and improve the computing efficiency by the automatic operation and parallel computation in reliability design and optimization.
4
MDAR Analysis of BTRRC
The HPT BTRRC is an important assembly relationship of gas turbine, which seriously affects the performance and reliability of aeroengine. Large BTRRC directly reduces the performance of aeroengine[5]. According to the engineering experiences, the BTRRC reduction is beneficial to decrease the specific fuel consumption, extend air flight time, expand compressor surge margin and increase payload[6]. For example, when the ratio of blade clearance to blade length decreases by 0.01, it is promising to improve at 0.8%–1.2% in the efficiency of compressor or turbine, to decline in the specific fuel consumption by 2% in double rotor turbofan engine and by 1.5% in turboshaft engine[7]. In fact, the BTRRC is constantly changing under operating conditions. If the designed blade-tip clearance is too small, the friction fault between blade-tip and casing may occur and even the criticality failures(blade fracture, casing damage, etc) may come up. Hence, the BTRRC of gas turbine engine has to be dynamically analyzed and even actively controlled in order to make aeroengine keep the reasonable clearance in various working conditions. Previous efforts in the BTRRC design and analysis of aeroengine are based on the deterministic analysis method [5–10] , which adopts specific parameters to quantify the BTRRC. The deterministic method does not consider the randomness of various parameters impacting on the clearance, so the design and control of BTRRC is merely able to ensure no friction by leaving margin for the minimum clearance, which is similar to the safety factor method in strength design. Too small clearance may cause the friction during operation, while too large clearance has to sacrifice the efficiency of aeroengine. Therefore, the deterministic clearance design method possesses some blindness for determining the minimum margin of clearance so as not to quantitatively trade off the two aspects of the contradictory between reducing blade-tip clearance and no friction. The BTRRC design of
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aeroengine HPT contains many objects (disk, blade and casing) and many disciplines (heat transfer, rotor dynamics, etc)[5–10] and is also a typical dynamic assembly problem of components among rotor and stator. The reliability design of BTRRC is also a typical MDAR design issue. Consequently the MDAR analysis of aeroengine HPT BTRRC was selected as the object of study to verify the DCRSM in this paper. 4.1
Distributed Response Surface Model
4.1.1 Assembly object FEMs Based on the basic thought of DCRSM, the BTRRC analysis was divided into the radial deformation analyses of disk, blade and casing. Three assembly objects were reduced to build the FEMs as shown Fig. 2. The loads and constraint conditions on disk were assumed to be axisymmetric. The tenon of blade was configured on the disk model and the cooling was ignored. The bushing ring of casing is a sensitive element, the expansion and contraction of which always cause the radial deformation of casing and the change of blade-tip clearance[9]. Thus, the axial cross section of bushing ring is only analyzed in this paper.
the insides of A, B, C, D and the outside of casing respectively. Table 1 Object
Disk
Blade
Casing
Fig. 2.
Finite element models of assembly objects
4.1.2 Random variables selection In order to verify the DCRSM for BTRRC reliability analysis, some parameters was reasonably selected as random variables as shown in Table 1, such as rotor speed ω , material density ρ , temperatures at different locations and surface coefficients of heat transfer. These parameters were assumed to be mutually independent and obey a normal distribution. In Table 1, T is temperature and α represents the surface coefficients of heat transfer. In the variables of disk, the subscripts of T indicate the temperature points, and the subscripts d1, d2 and d3 of α are the locations of B1, B2 and B3 respectively. In blade variables, the subscripts b1, b2, b3 and b4 of T and α are the positions in the blade (respectively blade-tip, upper, lower and root). In casing variables, Ti and To are the inside and outside temperature of bushing ring respectively. The subscripts c1, c2, c3, c4 and the subscript o of α denote
Random variables of BTRRC reliability analysis Variable Ta1 ℃ Ta2 ℃ Ta3 ℃ Tb1 ℃ Tb2 ℃ α d1 (W·m–2·K–1) α d2 (W·m–2·K–1) α d3 (W·m–2·K–1) ω (rad·s–1) ρ (kg·m–3) T1 ℃ T2 ℃ T3 ℃ T4 ℃ α b1 (W·m–2·K–1) α b2 (W·m–2·K–1) α b3 (W·m–2·K–1) α b4 (W·m–2·K–1) ω (rad·s–1) ρ (kg·m–3) Ti ℃ To ℃ α c1 (W·m–2·K–1) α c2 (W·m–2·K–1) α c3 (W·m–2·K–1) α c4 (W·m–2·K–1) α o (W·m–2·K–1)
Mean 540 210 200 245 320 1 527 1 082 864 1 168 8 210 1 030 980 820 540 11 756 8 253 6 547 3 130 1 168 8 210 1 050 320 6 000 5 400 4 800 4 200 2 600
Standard deviation 16.2 6.3 6.0 7.35 9.6 45.81 32.46 25.92 35.04 0.123 31 29.4 24.6 16.2 352.68 247.59 196.41 93.9 23.36 246.3 31.5 9.6 180 162 144 126 78
4.1.3 Deterministic analysis for assembly objects The flight profile and computing range of aeroengine were chosen from the start-idle-take off-climb -cruise of aeroengine[7, 9–10], in which 12 critical points were selected as the computing points as shown in Fig. 3(a). Considering the nonlinearity of thermal conductivities and expansion coefficients of disk, blade and casing and the dynamics of rotor speed and gas temperature[10], the radial deformations of objects were calculated respectively by the thermalstructural coupled method based on the means of variables in Table 1 and the FEMs in Fig. 2. The change curves of assembly object radial deformations with time are obtained and shown in Fig. 3(b). If the blade-tip static assembly clearance δ is 2 mm, the variation of Y (t ) with time is gained and shown in Fig. 3(b). As shown in Fig. 3(b), the minimum of BTRRC Y is gained synchronously at t180 s. In addition, it can be seen that the BTRRC decreases before the cruise of aeroengine and reaches at the minimum during accelerating climb, while slightly increases into the cruise. Therefore, the dangerous point may be considered in the climbing process with the highest gas temperature and rotor speed at t180 s. Thus, the dangerous point should be regarded as the calculating point of BTRRC reliability analysis because the safety at t180 s can ensure overall flight security.
CHINESE JOURNAL OF MECHANICAL ENGINEERING
Fig. 3.
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Load spectrum of aeroengine and change curves
4.1.4 Distributed response surface model Inputting the statistical characteristics of random variables in Table 1 into the FEMs, respectively, 161, 161 and 57 times simulations are fulfilled for three FEMs and gain the test samples by the Box-Behnken Matrix method[26]. These samples are used respectively to fit three response surface functions, which are shown in Eqs. (13), (14) and (15) (ignoring the coefficients of less than 10–3): Yd 1.236 6.37910 –2 ω 2.302102 Ta1 9.456103 Ta3 9.338103 Tb1 1.099102 Tb2 1.704103 α d1 1.483103 ω 2 ,
Yb 1.342 6.407 103 ω 1.903102 T3 ,
(13) (14) Fig. 4.
Yc 0.763 5.624102 Ti 9.285103 To 3.020103 α c1 2.675103 α c2 2.647 103 α c3 2.554103 α c4 1.351102 α o
1.940103 Ti 2 2.112103 Tiα o .
(15)
Three response surface models replacing the FEMs were simulated respectively 10 000 times for assembly objects reliability analysis by MC method. The output response histograms of assembly objects are obtained in Fig. 4. As shown by Fig. 4, all output responses obey a normal distribution, where the mean values of Yd, Yb and Yc are respectively 1.236 1 mm, 1.342 7 mm and 0.763 mm and the standard deviations are respectively 0.030 025 mm, 0.008 631 5 mm and 0.025 249 mm.
Radical deformation histograms of assembly objects
4.2 MDAR Model for BTRRC Assembly Relationship Assuming that the deformations of disk, blade and casing at t are severally Yd(t), Yb(t) and Yc(t), the variation τ(t) of blade-tip radial clearance[4–7, 12] is
τ (t ) Yd (t ) Yb (t ) Yc (t ) .
(16)
When the static (steady) blade-tip clearance of HPT is δ, the BTRRC Y(t) (limit state function)[7, 12] is
Y (t ) δ τ (t ) δ Yd (t ) Yb (t ) Yc (t ) .
(17)
As know by Eq. (17), Y＞0 is safety for the blade-tip assembly of aeroengine HPT, otherwise failure. If Y obeys a normal distribution, the variables are independent each other and their mean and variance matrix are μ(μd, μb, μc)
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and σ(σd, σb, σc), the mean function E[Y] and variable function D[Y][26] are
E[Y ] µY ( µd , µb , µc , σ d , σ b , σ c ) ,
(18)
D[Y ] DY ( µd , µb , µc , σ d , σ b , σ c ) .
(19)
Thus, the reliability index β and probability R of output response are[26]
β
µY DY
, R Φ ( β ).
(20)
4.3 BTRRC MDAR Analysis The DRSM and numerical characteristics of output responses were obtained through the radial deformation reliability analyses of disk, blade and casing. Based on the basic principle of DCRSM, these analysis results were taken as the input random variables of CRSF. Eqs. (16) and (17) were selected as the CRSFs for BTRRC reliability analysis. The CRSFs were simulated 10 000 times. The results show that the distribution features of variation τ are the mean 1.815 8 mm, the standard variance 0.013 4 mm and obey a normal distribution. For the distribution characteristics of Y, the failure number is 38, the failure probability 0.003 8, the reliability 0.996 2, the reliability index 3.291 9 and the computing time 2.581 82 s when δ1.86 mm. The simulation history and histogram of τ are shown in Fig. 5.
4.4 DCRSM Validation To support the validity and feasibility of DCRSM, the Monte Carlo method(MCM), RSM and DCRSM were selected for the reliability analysis of BTRRC. In this analysis process, the DCRSM was based on the automatic parallel computation of disk, blade and casing in three computers, where the reliability analysis of assembly objects(disk, blade and casing) was accomplished firstly and then the output responses were regarded as the input random variables of BTRRC to conduct the MDAR analysis. Nevertheless, the RSM and MCM were applied to the reliability analysis of whole assembled BTRRC that integrated the disk, blade and casing using the same variables, computer configuration and computing condition. The analysis results of three methods were compared under different sampling number. Three method computing time is shown in Table 2 and the BTRRC reliabilities at δ 1.85 mm are listed in Table 3 (ignoring the computing time which is greater than 120 h(432 000 s)). Table 2.
Fitting times
Method
MCM – RSM 114 696 DCRSM 2 088
Table 3. Sampling number 102 103 104 105 106
Fig. 5.
Output response results of BTRRC reliability analysis
Computing time of three method for BTRRC reliability analysis Computing time of different simulationss 102 103 104 105 106 2 956 35 028 392 400 – – 6.48 59.76 0.212 12 528 – 0.102 0.225 0.367 22.32 1728
BTRRC reliability analysis results with three methods (δ1.85mm) Methods MCM
0.989 4 0.981 2 0.979 8 – –
RSM 0.943 0.972 0.976 0.98 –
Precision%
Improved precision DCRSM RSM DCRSM % 0.99 95.31 99.94 4.63 0.979 99.05 99.78 0.73 0.979 8 99.6 100 0.4 0.978 9 – – – 0.979 7 – – –
4.5 Discussion As shown by Table 2, it is obvious from the fitting time of response surface functions that the time of DCRSM is far less than that of RSM. The time is equivalent to 1.82% that of RSM. From the simulation speed, it no doubt that the calculation speed of DCRSM is the fastest, which is about 188 times of MCM and 55 times of RSM under 104 times simulation. Videlicet, the computing time of DCRSM is far less than those of RSM and MCM. In addition, the strengths (fast speed and high efficiency) of DCRSM are towardly to become more obvious along with the increasing number of simulation. As demonstrated by Table 3, the computing precision of DCRSM is higher than that of RSM and almost consistent with that of MCM. Especially, the computational accuracy of DCRSM is improved by 0.4 to that of RSM and absolutely equal to that of MCM under 104 times simulations.
CHINESE JOURNAL OF MECHANICAL ENGINEERING In addition, according to Table 2 and Table 3, the DCRSM can complete the calculation which the RSM and MCM almost unlikely achieve when the simulation times are more than 106. By the above conclusions, it is fully supported that the DCRSM is promising to not only resolve the previous issues that are impossibly addressed by the traditional methods (MCM and RSM), but also greatly save the computing time and improve the calculation speed and efficiency while keeping the high computation precision. Therefore, DCRSM is verified to be an effective and feasible methodology in MDAR analysis.
5
Conclusions
(1) The MOMD dynamic assembly reliability design of complex machinery, difficult to be analyzed by the traditional reliability design methods(RSM and MCM), could be promisingly resolved by the MDAR theory is initiated in this paper. The original MDAR theory provides a promising idea for the dynamic assembly optimization design of complex machinery as well as enriches and develops the mechanical reliability theory further. (2) The MDAR design of aeroengine HPT BTRRC can be accomplished by the proposed DCRSM. In addition, the calculation precision and efficiency of BTRRC MDAR design have also been greatly improved. Therefore, the DCRSM is an effective and feasible approach with high precision and efficiency in MOMD MDAR design. (3) Based on the MDAR theory and DCRSM, the reliability analysis of aeroengine HPT BTRRC is originally realized, and opens up a novel research direction for aeroengine HPT BTRRC design from a probabilistic perspective, which contributes to develop the highperformance and high-reliability of aeroengine. (4) According to the reliability analysis of BTRRC in the presented study, the results show that the static blade-tip clearance δ 1.85 mm is advisable for aeroengine HPT design because the reliability degree 0.979 8 basically satisfies the requirement of engineering. It is expectant to verify further the DCRSM in the MDAR design and optimization. In addition, since the MDAR is a new research direction, the theory and methodology of MDAR need to be continuously improved, especially, to explore more reasonable and efficient MDAR design method.
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Biographical notes
BAI Guangchen, born in 1962, is currently a professor at Beihang University, China. He received his PhD degree from Harbin Institute of Technology, China, in 1993. His main research interests include aerospace reliability engineering, mechanism and structure reliability design, aeroengine multidisciplinary reliability analysis and optimization design, etc. He has published more than one hundred technical papers in mechanical or aeronautical journals. Tel: +86-10-82317418; E-mail:
[email protected]
FEI Chengwei, born in 1983, is currently a PhD candidate at School of Energy and Power Engineering, Beihang University, China. He has received his master’s degree from Shenyang Aerospace University, China, in 2010. His main research interest focuses on aerospace reliability engineering, structure reliability design, aeroengine multidisciplinary reliability analysis and optimization design, aeroengine vibration fault analysis and diagnosis, etc. He has already published more than 20 papers in mechanical or aeronautical journals. He was granted the national graduate scholarship of China, the “Guanghua” scholarship, the Innovation Foundation of Beihang University for PhD Graduates and the Academic Scholarship for PhD Candidates of Beihang University for his researching efforts. Tel: +86-15201127536; E-mail:
[email protected]
