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Mathematics and Computers in Simulation xxx (2009) xxx–xxx

Uncertainty treatment in forced response calculation of mistuned bladed disk Mohammad Rahimi, Saeed Ziaei-Rad ∗ Department of Mechanical Engineering, Isfahan University of Technology (IUT), Isfahan 84156-83111, Iran Received 29 September 2008; received in revised form 13 March 2009; accepted 17 July 2009

Abstract Mistuning, imperfections in cyclical symmetry of bladed disks is an inevitable and perilous occurrence due to many factors including manufacturing tolerances and in-service wear and tear. It can cause some unpredictable phenomena such as mode splitting, mode localization and dramatic difference in forced vibration response. In this paper first, a method is presented which calculates the forced vibration response of a mistuned system based on an exact relationship between tuned and mistuned systems. Then, the genetic algorithm is used for solving an optimization problem to find the worst-case response of bladed-disk assembly. The second part tries to find methods to reduce the system worst-case response. Intentional mistuning which breaks the nominal symmetry of a tuned bladed disk and rearranging the bladed-disk assembly are introduced and used to reduce the system worst-case response. Finally, a two degree of freedom per blade simplified model with 56 blades is used to demonstrate the capabilities of the techniques in reducing the worst response of the bladed-disk system. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Worst response; Mistuned bladed disk; Genetic algorithm; Intentional mistuning

1. Introduction Bladed disks may be used in several engineering systems such as fans, impeller pumps, turbine generators and jet engines. Ideally, these systems are tuned and all blades are identical but, in practice there always exist small, random differences among the blades due to manufacturing tolerances, in-operation wear, and so on that can make destroy the cyclic symmetry of bladed disk. It is well known that even a small amount of mistuning can induce a large forced response known as mode localization [24,13] and modal analysis shows that double eigenvalues appear in tuned cyclic symmetric structures which would be split in mistuned systems [14]. Afolabi’s investigation [1] concluded that the blade with the most mistuning is likely to be the blade with the largest amplitude. Rahimi and Ziaei-Rad [13] showed that the maximum forced response increases with increasing mistuning up to a certain level. The effects of coupling stiffness, damping and engine order excitation on mistuned bladed disk response is evaluated in [13]. Knowledge of increase the largest response level caused by the mistuning and of the arrangements of blades that are the most favorable, and those which are the most dangerous, is very important for design practice. The answers can ∗

Corresponding author. Tel: +98 311 3915244; fax: +98 311 3912628. E-mail address: [email protected] (S. Ziaei-Rad).

0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.07.002

Please cite this article in press as: M. Rahimi, S. Ziaei-Rad, Uncertainty treatment in forced response calculation of mistuned bladed disk, Math. Comput. Simul. (2009), doi:10.1016/j.matcom.2009.07.002

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M. Rahimi, S. Ziaei-Rad / Mathematics and Computers in Simulation xxx (2009) xxx–xxx

be obtained only for some specific bladed disk assemblies and under specific forcing conditions since it is very case specific. There are different predictions of ratio of worst response to tuned response, e.g., 1.82 in [3], 1.63 in [4], 2.05 in [6], and 2.1 in [15]. These different results may have been based on the different models and parameter values used in the studies. The problem of searching for the worst and best mistuning patterns has been formulated as an optimization problem in [12–14]. An attempt to use the direct optimization search for the worst-case response has also been made in [11]. Recently, Rotea and D’Amato used an upper bound and lower bound method and solved it by LFTLB algorithm [14]. Petrov and Ewins found worst response for a large finite element model as an optimization problem [12] and Sinha used the method of polynomial chaos to analytically compute the statistics of the forced response of a mistuned bladed-disk assembly [20]. Although random mistuning can lead to failure, intentional mistuning can provide benefits [10]. Most previous studies have only used intentional mistuning for pushing back flutter and increase stability of system where forced response had increased in contrast [18]. Recently, Slater and Minkiewicz introduced intentional mistuning into the design of bladed disks in order to reduce the maximum forced response [21]. You [25] also showed if the specific pattern arrangement of bladed-disk assembly changed, the response of new system would change. As a result it can be an ideal way to decrease forced vibration response. In this paper, first a method for exact calculation of the forced vibration response of mistuned bladed disk is represented which is based on a relationship between tuned and mistuned systems. The method is also used in this survey to reduce enormous amount of calculations required for stochastic optimization using genetic algorithm. The problem of finding worst specification is formulated as an optimization problem subjected to constraints such as the manufacturing tolerances. Next, two namely intentional mistuning and rearrangement techniques are used to decrease the worst response of bladed-disk assembly. Finally, a 56 blades simplified model with two degree of freedom per blade is illustrated to show the usual mistuning phenomena and the capabilities of the proposed techniques. 2. Forced response analysis The equation of motion for forced vibration of any linear system can be written in following form: ¨ + (C∗ + G)Q ˙ + (K + H)Q = f exp(iωt) MQ

(1)

where C∗ , damping matrix consists of viscous damping, C, and structural damping D (C∗ = C + 1/ωD). M, K, G and H are mass, stiffness, gyroscopic and circulatory matrices, respectively; Q is the response vector and f is the external √ load vector, ω is the excitation frequency and i = −1. The steady state solution of the above equation can be written as Q = q exp(iωt)

(2)

where q is a vector of complex response amplitudes. By substituting Eq. (2) into Eq. (1), one obtains the following algebraic relations: [−ω2 M + iω(C + G) + K + H + iD]q = [Z(ω)]q = f

(3)

where Z is the dynamic stiffness matrix. In mistuning case, it can be assumed that the dynamic stiffness matrix consists of a matrix corresponding to the tuned bladed disk, Z0 , and a mistuning matrix, Z which reflects the deviation from the tuned system. As a result Eq. (3) for a mistuned bladed disk can be written as [Z0 + Z(ω)]q = f

(4)

The response of the mistuned bladed disk is expressed as q = [Z0 + Z(ω)]−1 f

(5)

However, response calculation of the above equation by using direct inverse method needs enormous times and is costly especially when the system is large. Let us consider the so-called Sherman–Morrison–Woodbury [19] identity which relates the inverse of the sum of two matrices as follows: −1

(Z0 + Z)−1 = (Z0 + UVT )

−1

−1 T −1 = Z−1 0 − Z0 U(I + V Z0 U)

VT Z−1 0

(6)


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The above exact identity is valid for any invertible square matrix Z0(n×n) and any matrix Z that expressed by multiplication of two rectangular matrices U(n×m) and V(n×m) , if (I +VTU) is invertible. Here, I(m×m) is an identity matrix, n and m are equal to the degrees of freedom and the rank of matrix Z, respectively. Consequently the best use of this method is when that (I + VT Z−1 0 U) is only a scalar (i.e. the rank of Z is equal to one). In this paper, the mistuning matrix is represented by sum of individual mistuning matrices. Furthermore, each individual mistuning matrix is written as multiplication of two vectors as follows: Z =

m

zj ; zj = uj vTj

(7)

j

where uj is a unit vector with one nonzero component at its jth elements which is corresponds to the jth nonzero row of matrix Z and vj is the jth nonzero column of matrix ZT . Therefore, the influence of each individual mistuning matrix, zj , can be calculated from the following expression which is obtained by substituting Z from Eq. (7) to Eq. (6): (Z0 + zj )−1 = Z−1 0 −

a0j vTj 1 + vTj a0j

Z−1 0

(8)

where a0j refers to jth column of Z−1 0 . By substituting Eq. (8) into Eq. (5) and by taking into account all pairs of uj , vj , one can obtain an efficient formula for calculating the forced response levels of mistuned systems without any need to matrix inversion as following: q(j+1) = q(j) −

A

(j+1)

=A

(j)

−

vTj q(j)

j a j j

1 + vTj aj 1

j [a vT A(j) ]; j j j T 1 + v j aj

(9) j = 0, 1, . . . , m − 1

where A0 is equal to Z−1 0 and j indicates the number of the nonzero rows of the mistuning matrix which is being accounted for in the current recurrence update. As a result, q(m) is a vector of responses when the whole mistuning matrix, Z, is accounted for. As one can see from Eq. (9), in genetic algorithm calculations or in search for the worst mistuning specification, there is no need to invert matrices except tuned response for once. 3. Optimization strategy 3.1. Summary of genetic algorithm Generally, one can categorize the optimization methods into two major classes, namely gradient-based methods and global-based methods. However, in some engineering applications conventional gradient-based algorithms are ineffective due to the problem of local minima or the difficulty in calculating gradients. The genetic algorithm is one of the optimization methods that require no gradient and can achieve a global optimal solution (Holland [7], Davis [2]). GAs are called so because they attempt to use the supposition of evolution as a basic mechanism for improvement, that is, learning/survival of the fittest, in solving a problem. The GAs are computationally simple but powerful and not limited by assumptions about the search space. Following the terminology of true genetic researches, the computational GAs developed by Holland [7] and his students encode potential solutions into chromosome-like structures and then allow these structures to compete, reproduce and mutate to produce better solutions over time. GAs have been increasingly used in optimization studies over the past decade and have more recently been used in multidisciplinary optimization. Many facts control the way that a GA works. A potential solution first has to be encoded along with all of the other potential solutions that form a generation. Basic operators which create successive improved populations include selection, crossover, mutation, and interchange. Once a pair of parents is selected, the mating of the pair also involves a random process called crossover. Mutation is implemented by changing, at random and with small probability, the value of a gene and serves the purposes of avoiding premature loss of diversity in the designs. Please cite this article in press as: M. Rahimi, S. Ziaei-Rad, Uncertainty treatment in forced response calculation of mistuned bladed disk, Math. Comput. Simul. (2009), doi:10.1016/j.matcom.2009.07.002

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GA moves from generation to generation selecting and reproducing parents until a termination criterion which is the maximum number of generations is reached. Another termination strategy involves population convergence criteria. In general, GAs will force much of the entire population to converge to a single solution. When the sum of the derivations among individuals becomes smaller than some specified threshold, the algorithm can also be terminated due to lack of improvement in the best solution over a specified number of generations. GAs are global optimizers because of mutation and their general probabilistic non-gradient nature. However, in engineering applications, good answers are desired as fast as possible. Whether the answer is the absolute global optimum or not is less of a concern than whether a good answer can be found in the time allotted by management. Besides, for complicated problems, there is no analytical way to determine the global optimum and so arguments over whether the true optimum has been found are academic. 3.2. The worst case analysis The worst blade in a special arrangement of bladed disk is a blade that has amplitude larger than the rest in a specified frequency range. According to turbomachine design, one can never foretell the perturbation parameters prior to manufacture, therefore it is necessary to predict the probable worst case. All structures are made with particular tolerances. Even though bladed disk are manufactured by some tight tolerances, they may have large response due to high sensitivity of bladed disks assembly to perturbation. For this object, the worst mistuning pattern can be formulated as follows: q∗ (ω, {αj }) = max(qbj ); j = 1,..,N, αj ≤ μ, ωmin ≤ ω ≤ ωmax (10) where qbj is the jth blade response to a mistuning parameter αj and external excitation, ωj . μ, ωmin and ωmax are upper bound for perturbation parameter, minimum external excitation and maximum external excitation, respectively. Eq. (10) is a constrained optimization problem that its maximum is desired. In this paper, the optimization is carried out by use of genetic algorithm. 4. Methods for reducing the worst-case response The main goal of this part is to find out the best way for decreasing the worst response of bladed-disk assembly. It is clear that designing the turbomachines based on the worst pattern response can be very costly. Therefore, it is necessary to find an effective remedy for reducing the worst-case response. Here, two remedies are suggested. The first method is based on intentional mistuning. In this technique, blades can be designed different from each other. In other words, the method breaks cyclical symmetry of the structure deliberately. However, it can cause expensive manufacturing procedure because of nominal differences among blades. The second technique uses rearranging of blades on the assembly to decrease the worst response. This method is very cost effective and is shown to be more efficient than the intentional mistuning. 4.1. Intentional mistuning Intentional mistuning is known to reduce the sensitivity of the response to unintentional and inevitable perturbations. Intentional mistuning is carried out by introducing designed perturbations that break the nominal symmetry of a tuned bladed disk with the objective of reducing its negative effects and therefore reduce the worst-case response in the presence of unintentional perturbations. In this subsection, it is illustrated how to use the tools to determine an optimal intentional mistuning solution in which the worst-case response is minimized. First, a pattern of the following form is considered: αj = δj + μj

(11)

where μj and δj are intentional parameter and perturbation parameter, respectively. If μj = 0, it means there is no intentional mistuning. By substituting Eq. (11) into Eq. (10) one can formulate the best intentional mistuning pattern as follows: qw (ηj ) = max(max(qbj ); j = 1, . . . , N, δj ≤ μ, ωmin ≤ ω ≤ ωmax ), μ ≤ ηj ≤ μ∗ (12) Please cite this article in press as: M. Rahimi, S. Ziaei-Rad, Uncertainty treatment in forced response calculation of mistuned bladed disk, Math. Comput. Simul. (2009), doi:10.1016/j.matcom.2009.07.002

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Fig. 1. Simplified model of the bladed disk.

where qw is the peak of amplitude of frequency response for the worst case and μ∗ is upper bound for intentional mistuning parameter. Eq. (12) is a constrained optimization problem that its minimum is desired. In a special case, one can assume an alternating pattern for blades of the form AB . . . ABAB. In other words, it means that only two intentional mistuning parameters can be used in manufacturing of blades. Therefore Eq. (12) can be simplified as qw (ηj ) = max(max(qbj ); j = 1, . . . , N, δj ≤ μ, ωmin ≤ ω ≤ ωmax ), ηj = η1 if j = Odd,ηj = η2 if j = Even,

μ ≤ |η1 , η2 | ≤ μ∗

(13)

Here, all odd blades have an intentional mistuning parameter η1 while the even blades have an intentional mistuning parameter η2 . 4.2. Rearranging the bladed-disk assembly It can be shown that by changing the arrangement of a mistuned bladed-disk system, the maximum response will be changed. Therefore, there exists a best arrangement for the bladed-disk assembly in which maximum response will be decreased effectively. Based on the aforementioned explanations, the optimization problem for finding the best arrangement can be formulated as follows: qˆ (ω, {αj }) = max(max(qbj ), Subjected to : ωmin ≤ ω ≤ ωmax ),

αj ∈ a∗ ,

j = 1, . . . , N

(14)

where a∗ is the worst specification vector, i.e. answer of Eq. (10) and is the maximum amplitude over the interest frequency range. The above equation is a discrete minimization problem that can be solved by use of genetic algorithm. 5. Simpliﬁed model for mistuning analysis A simplified model of a 56 bladed-disk assembly as shown in Fig. 1 is considered in this example. Each sector consists of two lumped masses for blade, mb and disk, md , respectively. Note that all disk masses are coupled with each other by lumped spring kt , and also connected to the ground by another spring, kd . Blade mass is also coupled with corresponding disk sector mass by spring kb , linear viscous damping force is assumed to act on each mass. Each mass is also excited by external harmonic force with magnitude f, frequency ω, and the phase at the jth blade ϕj , which is given by 2(j − 1)πE (15) N Note that the phase differs from blade to blade. In Eq. (15) E is the engine order of the excitation and N is number of blades. The equation of motion in the frequency domain, in the absence of structural damping, gyroscopic and ϕj =


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circulatory matrices can be derived as follows: [−ω2 M + iωC + K]q = f where

⎡

mb

⎢ ⎢ 0 ⎢ M=⎢ ⎢ 0 ⎢ ⎣ . 0 f = [1

0

0 md

0

.

0 .

0 . .

. . mb

.

0

0

exp(ϕ2 )

0

(16)

0

⎤

⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 0 ⎦ md 0 .

⎡

kb

⎢ ⎢ −kb ⎢ K=⎢ ⎢ 0 ⎢ ⎣ .

. . . exp(ϕN )

0 T

0] ,

−kb

0

0

.

0

kb + kd + 2kt

0

−kt

.

−kt

0 .

kb .

. .

. kb

. −kb

−kt

0

.

−kb

kb + kd + 2kt

...

qbN

q = [ qb1

qd1

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

T

qdN ]

The amplitude of force vector, f, is equal to one. The viscous damping matrix C is determined by assuming a damping ratio ζ for all modes of system. In other words C is given by C = 2ζ ∗ −1

where and are the modal matrix and natural frequency matrix. The symbol ∗ represents the complex conjugate of the modal matrix, . j j In this paper, only stiffness matrix is mistuned by blade stiffness parameter, i.e. kb = kb0 (1 + αj ), where αj , kb and 0 kb are mistuning parameter, jth blade stiffness and nominal blade stiffness, respectively. 6. Numerical results 6.1. System description In this section, a model of bladed-disk assembly with 56 blades with the following nominal parameters is considered (Fig. 1). kb0 = 1, kt = 493, kd = 1.1, mb = 1, md = 426 Fig. 2 shows amplitudes of forced frequency responses of all blades for a tuned bladed disk for engine order excitation, E = 9 and damping ratio, ζ = 0.005. Note that all blades have the same response amplitude. However, their responses have a phase difference according to Eq. (15), equal to ϕj , with each other.

Fig. 2. Frequency response of tuned system for all blades.


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Fig. 3. Fitness value used in the genetic algorithm.

6.2. Worst-case response Next, it is decided to solve the optimization problem of Eq. (10). The engine order excitation is E = 9 and the damping factor for all blades is assumed to be ζ = 0.005. The upper bound for perturbation parameter, μ, the upper and lower limits for frequencies, ωmin and ωmin are set to 0.02, 0.97 and 1.02, respectively. The optimization problem should find the worst pattern for 57 variables, i.e. 56 perturbation parameters and one excitation frequency. This case is solved using genetic algorithm. The GA is first tuned using heuristic crossover and multi-nonuniform mutation [9,26,27]. The number of population in each generation of GA is set to 200. The crossover rate is set to 90% and the mutation rate is 1%. The convergence rate for the GA to reach the optimum of the defined objective function is depicted in Fig. 3. The value of the objective function becomes flat after 40 generations. The number of generations for other runs was set to 40 as by increasing the number of generations almost no improvement was observed in the best solution. Increasing the number of generations in the GA algorithm may further reduce this value. However, this needs more CPU time and computational efforts with little gain in the accuracy of the pressure distribution. Since genetic algorithm convergence towards final solution is usually slow, it is decided to use a two-stage optimization here. After the genetic algorithm terminates, another deterministic optimization technique, sequential quadratic programming [17], is run in order to capture the final solution more accurately. In this case, the worst bladed disk response calculated from GA algorithm is around 235. After feeding this result into SQP, the final solution for the worst response is around 248. It is worth mentioning that without using the second optimization procedure, i.e. pure GA, in order to have the same worst response, the number of population in each generation should be increased up to 400. This of course needs huge CPU time. On the other hand, using pure SQP method, the solution depends on the initial guess and the algorithm may not able to find the global maximum response. The worst calculated pattern for perturbation parameters is shown in Fig. 4. Fig. 5 shows the tuned and the worst response of system which is mistuned by the pattern shown in Fig. 4. In the same figure, the results for this problem which are reported in [14] are also plotted. It is noteworthy that the maximum amplitude for the worst response is 247.85 which is higher than results reported by [14] (i.e. 180). It is also noticeable

Fig. 4. Perturbation parameter at each blade for the worst case.


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Fig. 5. Comparison between tuned and the worst response.

that amplitude for the tuned system is 84.1. This means that the worst response of mistuned system is 2.95 times greater than the response of the tuned system. The results indicate the efficiency of implemented method. 6.3. Intentional mistuning Fig. 6 shows the efficiency of intentional mistuning in decreasing the worst response of system. The best intentional mistuning parameters are obtained by solving the optimization problem mentioned in Eq. (13) and found to be η1 = 0.11, η2 = −0.04. The upper limit for the intentional mistuning parameter μ∗ is set to be 0.12 while the engine order excitation is set to 9 and the damping factor is assumed to be 0.005. The lower and upper bounds of excitation frequency are assumed to be 0.97 and 1.02, respectively. The system worst response with the obtained intentional mistuning parameters are depicted in Fig. 6 and compared with the worst response without intentional mistuning. The graph shows that the worst amplitude for mistuned system is decreased from 247.85 to 191.08. However, as one can see in Fig. 6, the peak response is shifted to the left and the modes are scattered in a larger interval than the case without intentional mistuning. 6.4. Rearrangement of bladed disk Optimization problems can be classified as combinatorial or continuous. The former case concerns discrete variables and involves a finite number of candidate solutions. The search space is finite and thus, intrinsically constrained. In the latter case, there is an undefined number of solutions, since continuous variables cannot be fully characterized with a finite number of levels.

Fig. 6. Efficiency of intentional mistuning method.


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Fig. 7. Perturbation parameter of each blade for the best arrangement.

Another important difference is the lack of notion of gradient in the combinatorial domain. Since the direction of the improvement is meaningless, local search methods in discrete variable problems are less obvious than in the continuous domain, and require a deep knowledge about the problem nature. All these differences are so significant that force a particular study and implementation of a hybrid genetic algorithm code. There are some reported examples on combinatorial minimization problems successfully solved by genetic local search algorithms [16,22,23,26,27]. The intentional mistuning is a classical nonlinear continuous optimization problem and therefore, both the realcoded GA and gradient optimization method, like the SQP, can be applied for its solution. The second problem, i.e. the rearrangement of bladed disks, is a classical combinatorial problem, and therefore any heuristics or metaheuristics like a GA is a logical answer [5]. For solving the intentional mistuning problem, the real-coded GA was used. The results indicate that convergence steps towards the final answer are very slow. Therefore, in order to accelerate the solution, it was used in conjunction with a SQP algorithm. For finding the best arrangement of the system in the problem of bladed disks rearrangement, an integer from 1 to 56 is allocated for each blade. The GA proposes 56 values between 1 and 56. In order to solve this integer optimization problem, once again GA with the heuristic crossover and multi-nonuniform mutation is used [8]. The crossover and mutation rates are set as the previous case. However, the number of population is decreased to 70, because using the bigger population does not improve the response. Fig. 7 shows perturbation parameters calculated for the best arrangement. After solving Eq. (14) as a discrete optimization problem by using genetic algorithm, the final arrangement is obtained. The perturbation parameters are chosen from the worst pattern calculated in Fig. 4 (i.e. a∗ ). It is noticeable that there are more than 7.1e+74 arrangements for this case. In other words, the search space dimension is very large. However, the developed algorithm is able to find the solution in a relatively short time. Using a standard desktop computer, the final solution for this case is achieved in less than an hour.

Fig. 8. Frequency response of mistuned system for the worst case after rearranging.



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Fig. 9. Efficiency of rearranging method.

Fig. 8 shows amplitudes of all blades forced frequency responses for the worst case mistuned bladed disk after rearrangement. It can be observed that for this special arrangement, the energy transmitted by the external harmonic force to the system is distributed uniformly between all blades. Even many blades have the same maximum amplitude at the given frequency interval. In Fig. 9, the efficiency of rearranging method is shown by plotting the worst response of the bladed disk before and after rearrangement. The figure shows that the maximum amplitude for the worst response is decreased from 247.85 to 139.84. The results clearly demonstrate that the rearranging method is able to effectively decrease the worst response. Also, one can conclude that the rearrangement is more effective than intentional mistuning. In addition, this method does not impose any extra expenses to the turbomachine manufacturers. The upper bound for perturbation parameter which is an indication of manufacturing tolerance is set to 0.02 in all previous calculations. Since this is an important parameter, it is decided to investigate the effect of this parameter on the rearrangement method. Fig. 10 depicts the maximum amplitude of the worst response versus different values of the upper bound perturbation parameter, μ. The results show that for perturbation values larger than 0.02, the worst response and the response of the system after rearrangement are slightly increasing. However, for values lower than 0.02, the effect of this parameter is significant.

Fig. 10. Maximum amplitude of the worst response versus maximum admissible perturbation parameter (a: ζ = 0.01 and b: ζ = 0.005).


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The effect of damping factor on the worst and rearranged responses are also studied. Fig. 10 shows the worst response and the rearranged responses of the bladed disk for two values of damping factor, namely ζ = 0.005 and ζ = 0.01. The figure shows that the sensitivity to mistuning increases by decreasing damping parameter. 7. Conclusions This paper explains the fundamental techniques of mistuning in some details. Although a simple lumped model is used for numerical calculations, the techniques can be easily utilized for more detailed and complicated models such as finite element. It is found that the GA is well suited in finding the worst-case response of the bladed-disk assembly. Unlike gradient-based optimization approaches, GAs find good designs by learning their own design lessons, without having to be given a starting solution or sensitivity derivatives. Also, it is concluded that GA can solve discrete variables problem and work very well without divergence. It is also found that by using a two-stage optimization algorithm, the final solution can be obtained in shorter time with less CPU efforts. The results indicate that rearrangement is an efficient way to decrease the worst-case response of bladed-disk system without any extra cost in turbomachine design. The results also show that the rearrangement technique is more effective than intentional mistuning. For the reported case study, rearrangement reduces the worst-case response by a factor of 0.56 while this factor is equal to 0.77 for intentional mistuning. References [1] D. Afolabi, The frequency response of mistuned bladed disk assemblies, in: Vibration of Blades and Bladed Disk Assemblies, Mechanical Vibration and Noise, ASME, Cincinnati, OH/New York, 1985. [2] L. Davis, Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, 1991. [3] R.C.F. Dye, T.A. Henry, Vibration amplitudes of compressor blades resulting from scatter in blade natural frequencies, ASME J. Eng. Power 91 (1969) 182–187. [4] D.J. Ewins, Z.C. Han, Resonant vibration levels of a mistuned bladed disk, ASME J. Vib. Acoust. 106 (1984) 211–217. [5] V.B. Gantovnik, C.M. Anderson-Cook, Z. Gurdal, L.T. Watson, A genetic algorithm with memory for mixed discrete-continuous design optimization, Comput. Struct. 81 (2003) 2003–2009. [6] H.J. Griffin, T.M. Hoosac, Model development and statistical investigation of turbine blade mistuning, ASME J. Vib. Acoust. 106 (1984) 204–210. [7] J.H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence, University of Michigan Press, Ann Arbor, MI, 1975. [8] J.A. Joines, C.T. Culbreth, R.E. King, Manufacturing cell design: an integer programming model employing genetic algorithms, IIE Trans. 28 (1996) 69–85. [9] K. Matous, M. Leps, J. Zeman, M. Sejnoha, Applying genetic algorithms to selected topics commonly encountered in engineering practice, Comput. 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