Chapter 5. Reliability-Oriented Optimal Design of Intentional Mistuning for Bladed Disk with
5.3 Reliability Analysis of a Bladed Disk with Interval and Random Uncertainties
5.3.1 Interval Analysis under Disk Connection Uncertainty
It starts from analyzing response under interval disk connection uncertainty only. Since the upper bound of performance function occurs at the maximum response, the worst-case combination of interval uncertainties that maximizes the vibration response needs to be found. Traditionally, this has been achieved by considering all possible combinations of interval uncertainties (Mourelatos et al. 2005; Yoo and Lee 2014). The steps of executing this method involve dividing all interval uncertainties into a discrete number of sub-intervals and calculating vibration responses for all possible combinations of these sub-intervals. The worst-case combination can then be determined by finding the actual maximum
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responses. While intuitive, for this particular problem of bladed disk analysis, the computational cost would be impractical. For instance, let there be 12 numbers of blade in the assembly, and let each interval disk connection between the adjacent blades be divided into 10 sub-intervals. There would be a total of 1012 response calculations.
Here it is proposed to resort to an efficient, sampling-based technique, known as the Metropolis- Hastings (M-H) algorithm (Cai et al. 2008), to solve the above-mentioned challenge. The M-H algorithm is a Markov chain Monte Carlo method, and its primary application is to identify multi-dimensional output distribution by generating input random samples (Roberts and Rosenthal 2001). In the research in this chapter the M-H algorithm is modified such that the stopping criterion is deterministic instead of probabilistic, since the distribution information is unknown inside the uncertainty interval. The proposed method with appropriate choice of convergence parameters will efficiently narrow the search down into one possible scenario and then further converge to true global minimum/maximum. Other sampling-based methods such as Latin Hypercube sampling or Genetic Algorithm will be relatively inefficient choices for this problem, since they essentially will search through all possible scenarios (Helton and Davis 2003;
Sivanandam and Deepa 2007).
The flowchart of the proposed method that searches for the worst-case interval coupling stiffness combination, denoted by kcworst, with Nws number of iterations, is shown in Figure 5.3. The procedure can be summarized as:
Step 1. Decide the number of samples Nws. (Based on case studies in this research, the recommended value of Nws is between 500 and 1000. However it can be further adjusted if convergence is not achieved.) Set the iteration counter j = 0. Initialize kc 0 by drawing kc 0 from uniform distribution between kcL and kcU where the superscripts L and U indicate, respectively, the lower and upper bounds.
Step 2. Propose a value for kc j , which is denoted by kc*, by drawing kc* from Nkc j ,kc j where
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j U L j
c c c
k k k (5.16)
where and are adjustable parameters to expedite the convergence (Roberts and Rosenthal 2001).
Repeat Step 2, until kcLkc*kcU. Please note that value of is usually close to 1.
Step 3. Check the stopping criterion that is given as
*
max max
minx / x j ,1 1 (5.17)
where *
xmax and
max
x j are the maximum blade responses solved over the interested frequency range
under coupling stiffness combinations kc* and kc j , respectively. If the stopping criterion in Equation (5.17) is satisfied, kc j kc*. The next iteration counter is set as j = j + 1.
Step 4. If jNws, go to Step 2. If jNws, kcworst= kc j .
Figure 5.3 Flowchart of proposed M-H based method for worst-case interval coupling stiffness search.
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To demonstrate the accuracy as well as the efficiency of the proposed M-H based method for the potential usage in bladed disk analysis, relatively simple examples are carried out first on two weakly- coupled bladed disks with 4 (even) and 5 (odd) blades, respectively. All other parameters are the same as those used in Section 5.5 on more complex structures and will be explained in detail later. For these two bladed disks with smaller number of blades and with interval uncertainty only, comprehensive worst-case analysis can be indeed carried out by using the traditional approach, i.e., discretizing each interval coupling into multiple sub-intervals and evaluating the responses under all possible combinations. Figure 5.4 show the responses for all possible combinations, based on which the worst-case response can be determined.
Meanwhile, the results obtained by the traditional approach and the proposed M-H based method are compared in Figure 5.5 where responses are normalized with respect to the nominal response that is extracted without considering interval uncertainty. In Figure 5.5, the worst-case responses obtained by the proposed method for bladed disks with 4 and 5 blades are 109.06% and 133.32% of the nominal response, respectively, while those obtained by the traditional method are 108.90% and 132.63%. It can be observed that the M-H based method converges quickly to the worst-case response obtained by the traditional approach within just a few hundred iterations, which clearly demonstrates its efficiency. In terms of accuracy, the worst-case responses obtained by the proposed method are slightly higher than the ones obtained by the traditional approach (0.16% and 0.69%). This is due to the fact that the proposed method can search for the worst-case combination continuously within the intervals, while the traditional approach considers only the discrete combinations of sub-intervals. Since the proposed method can find slightly higher responses, it may be concluded that it gives better result in the context of searching for the worst-case responses. The worst-case combinations of interval couplings obtained by the proposed method are shown in Figure 5.6.
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(a) (b)
Figure 5.4 Responses for all possible combinations of interval couplings for bladed disks.
(a) 4 blades; (b) 5 blades.
(a) (b)
Figure 5.5 Worst-case amplitude searches by the proposed M-H based method for bladed disks. (a) 4 blades; (b) 5 blades.
(a) (b)
Figure 5.6 Worst-case combinations of interval couplings obtained by the proposed M-H based method for bladed disks. (a) 4 blades; (b) 5 blades.
0 2000 4000 6000 8000 10000
98 100 102 104 106 108 110
Interval Coupling Combination
Normalized Maximum Response (%)
0 2 4 6 8 10
x 104 90
100 110 120 130 140
Interval Coupling Combination
Normalized Maximum Response (%)
0 200 400 600 800 1000
100 102 104 106 108 110
Iteration
Normalized Maximum Response (%)
Proposed M-H Based Method Traditional Method
0 200 400 600 800 1000
100 110 120 130 140
Iteration
Normalized Maximum Response (%)
Proposed M-H Based Method Traditional Method
1 1.5 2 2.5 3 3.5 4
1.9 2 2.1 2.2 2.3 2.4x 104
Coupling kc (N/m)
1 2 3 4 5
1.9 2 2.1 2.2 2.3 2.4x 104
Coupling kc (N/m)