Optimal Design of Intentional Mistuning for a Bladed Disk With Interval Uncertainty

2015 ◽  
Author(s):  
David Yoo ◽  
Jiong Tang
Keyword(s):  
Optimal Design ◽  
Design Optimization ◽  
Probability Of Failure ◽  
Forced Response ◽  
Response Amplitude ◽  
Interval Uncertainty ◽  
Probabilistic Constraints ◽  
Worst Case ◽  
Bladed Disk ◽  
Gradient Based

This paper presents a methodology for the optimal design of intentional mistuning for a mistuned bladed disk with interval uncertainty. For a bladed disk where blades are weakly coupled, presence of random mistuning can easily induce vibration localization. This phenomenon will lead to great amplification in response amplitude of certain blades. To achieve desired reliability of a bladed disk, amplified response must be reduced to certain level, which requires probabilistic or reliability analysis. In this study, it is considered that blades have random distribution and coupling between blades has interval uncertainty. To treat the interval uncertainty appropriately, the worst-case combination of interval couplings is searched first, then probability of failure is evaluated under the worst-case condition. To increase reliability of a bladed disk, intentional mistuning is used in this study. While applying the intentional mistuning, it is also wanted to minimize the degree of intentional mistuning to minimize the cost of implementation. To find optimal combination of intentional mistuning parameters to achieve dual goals, gradient-based design optimization approach is utilized, which is expected to guarantee efficient convergence. To carry out gradient-based design optimization, sensitivities of objective function and probabilistic constraints with respect to intentional mistuning parameters are derived. During the sensitivity analysis, distribution of forced response amplitude is identified through Gaussian fit and eigenvalue perturbation theory is referred to. Monte Carlo simulation is utilized to accurately calculate probability of failure and its sensitivity. The proposed method is demonstrated with numerical examples of two distinct bladed disks.

Sampling-Based Approach for Design Optimization in the Presence of Interval Variables

2013 ◽  
Author(s):  
David Yoo ◽  
Ikjin Lee
Keyword(s):  
Monte Carlo Simulation ◽  
Design Optimization ◽  
Random Variables ◽  
Delta Function ◽  
Probability Of Failure ◽  
Probabilistic Constraints ◽  
Worst Case ◽  
Performance Function ◽  
Dirac Delta ◽  
Interval Variables

This paper proposes a methodology for sampling-based design optimization in the presence of interval variables. Assuming that an accurate surrogate model is available, the proposed method first searches the worst combination of interval variables for constraints when only interval variables are present or for probabilistic constraints when both interval and random variables are present. Due to the fact that the worst combination of interval variables for probability of failure does not always coincide with that for a performance function, the proposed method directly uses the probability of failure to obtain the worst combination of interval variables when both interval and random variables are present. To calculate sensitivities of constraints and probabilistic constraints with respect to interval variables by the sampling-based method, the behavior of interval variables at the worst case is defined by utilizing the Dirac delta function. Then, Monte Carlo simulation is applied to calculate constraints and probabilistic constraints with the worst combination of interval variables, and their sensitivities. The important merit of the proposed method is that it does not require gradients of performance functions and transformation from X-space to U-space for reliability analysis after the worst combination of interval variables is obtained, thus there is no approximation or restriction in calculating the sensitivities of constraints or probabilistic constraints. Numerical results indicate that the proposed method can search the worst case probability of failure with both efficiency and accuracy and that it can perform design optimization with mixture of random and interval variables by utilizing the worst case probability of failure search.

Design Optimization Toward Alleviating Forced Response Variation in Cyclically Periodic Structure Using Gaussian Process

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
K. Zhou ◽  
A. Hegde ◽  
P. Cao ◽  
J. Tang
Keyword(s):  
Optimal Design ◽  
Gaussian Process ◽  
Design Optimization ◽  
Periodic Structure ◽  
Periodic Structures ◽  
Forced Response ◽  
Design Modification ◽  
Vibration Localization ◽  
Bladed Disk ◽  
Response Variation

Cyclically periodic structures, such as bladed disk assemblies in turbomachinery, are widely used in engineering systems. It is well known that small uncertainties exist among their substructures, which in certain situations may cause drastic change in the dynamic responses, a phenomenon known as vibration localization. Previous studies have suggested that the introduction of small, prespecified design modification, i.e., intentional mistuning, may alleviate vibration localization and reduce response variation. However, there has been no systematic methodology to facilitate the optimal design of intentional mistuning. The most significant challenge is the computational cost involved. The finite-element model of a bladed disk usually requires a very large number of degrees-of-freedom (DOFs). When uncertainties occur in a cyclically periodic structure, the response may no longer be considered as simple perturbation to that of the nominal structure. In this research, a suite of interrelated algorithms is proposed to enable the efficient design optimization of cyclically periodic structures toward alleviating their forced response variation. We first integrate model order reduction with a perturbation scheme to reduce the scale of analysis of a single run. Then, as the core of the new methodology, we incorporate Gaussian process (GP) emulation to conduct the rapid sampling-based evaluation of the design objective, which is a metric of response variation under uncertainties, in the parametric space. The optimal design modification can thus be directly identified to minimize the response variation. The efficiency and effectiveness of the proposed methodology are demonstrated by systematic case studies.

Experimental Investigation of Mode Localization and Forced Response Amplitude Magnification for a Mistuned Bladed Disk

2000 ◽  
Author(s):  
John Judge ◽  
Christophe Pierre ◽  
Oral Mehmed
Keyword(s):  
Experimental Investigation ◽  
Traveling Wave ◽  
Forced Response ◽  
Response Amplitude ◽  
Mode Shapes ◽  
Bladed Disks ◽  
Element Analysis ◽  
Mode Localization ◽  
Bladed Disk ◽  
Out Of Plane

The results of an experimental investigation on the effects of random blade mistuning on the forced dynamic response of bladed disks are reported. The primary aim of the experiment is to gain understanding of the phenomena of mode localization and forced response blade amplitude magnification in bladed disks. A stationary, nominally periodic, twelve-bladed disk with simple geometry is subjected to a traveling-wave, out-of-plane, “engine order” excitation delivered via phase-shifted control signals sent to piezo-electric actuators mounted on the blades. The bladed disk is then mistuned by the addition of small, unequal weights to the blade tips, and it is again subjected to a traveling wave excitation. The experimental data is used to verify analytical predictions about the occurrence of localized mode shapes, increases in forced response amplitude, and changes in resonant frequency due to the presence of mistuning. Very good agreement between experimental measurements and finite element analysis is obtained. The out-of-plane response is compared and contrasted with the previously reported in-plane mode localization behavior of the same test specimen. This work also represents an important extension of previous experimental study by investigating a frequency regime in which modal density is lower but disk-blade interaction is significantly greater.

Combinatorial Optimization of Mistuned Blade Rearrangement Based on Reduced-Order FEA Model

2017 ◽  
Author(s):  
Tianyuan Liu ◽  
Ding Guo ◽  
Di Zhang ◽  
Yonghui Xie
Keyword(s):  
Combinatorial Optimization ◽  
Ant Colony Algorithm ◽  
Forced Response ◽  
Response Amplitude ◽  
Reduced Order Model ◽  
Ant Colony ◽  
Order Model ◽  
3D Fem ◽  
Reduced Order ◽  
Bladed Disk

This paper is focused on the optimization of mistuned blades assembling rearrangement under the forced response. First, in order to avoid the greatly increase of the calculation greatly by the whole circle bladed-disk finite element model, a reduced-order model is developed based on the component mode synthesis. CPU+GPU heterogeneous architecture parallel computation is used to accelerate modal analysis of the disk and blade sectors substructures. Second, a modified ant colony algorithm is applied to the combinatorial optimization to find the optimal rearrangement pattern of bladed-disk assembly. Different from classical algorithm, the individual mistuned information is used to construct heuristic function based on intentional mistuning pattern, which can avoid slow convergence of ant colony algorithm and increase the search speed efficiently. At last, a high-fidelity 3D FEM model with 43 mistuned blades is used to demonstrate the capabilities of the techniques in reducing the maximum displacement resonance response of the bladed-disk system. The numerical simulation showed that this program based on the reduced-order model proposed in this article gained 4.3 speedup compared with ANSYS full model under the scale of 500k nodes. The displacement response amplitude of the blades decreased by 32% with 60 steps (1200 times FEM calculation) by the new optimization method. The physical mechanism of reducing the bladed-disk response is explained by comparing the optimized and worst arrangement patterns. The results clearly demonstrate that the optimized rearrangement pattern of mistuned blades is able to reduce the response amplitude of the forced vibration significantly, and the algorithm proposed in this article is practical and effective.

Reliability-Based Multidisciplinary Design Optimization Using Probabilistic Gradient-Based Transformation Method

2012 ◽  
Vol 135 (2) ◽  
Author(s):  
Po Ting Lin ◽  
Hae Chang Gea
Keyword(s):  
Design Optimization ◽  
Multidisciplinary Design Optimization ◽  
Transformation Method ◽  
Performance Measure ◽  
Probabilistic Constraints ◽  
Multidisciplinary Design ◽  
Sensitivity Analyses ◽  
Chance Constrained Programming ◽  
Complex Design ◽  
Gradient Based

Recently, solving the complex design optimization problems with design uncertainties has become an important but very challenging task in the communities of reliability-based design optimization (RBDO) and multidisciplinary design optimization (MDO). The MDO algorithms decompose the complex design problem into the hierarchical or nonhierarchical optimization structure and distribute the workloads to each discipline (or subproblem) in the decomposed structure. The coordination of the local responses is crucial for the success of finding the optimal design point. The problem complexity increases dramatically when the existence of the design uncertainties is not negligible. The RBDO algorithms perform the reliability analyses to evaluate the probabilities that the random variables violate the constraints. However, the required reliability analyses build up the degree of complexity. In this paper, the gradient-based transformation method (GTM) is utilized to reduce the complexity of the MDO problems by transforming the design space to multiple single-variate monotonic coordinates along the directions of the constraint gradients. The subsystem responses are found using the monotonicity principles (MP) and then coordinated for the new design points based on two general principles. To consider the design uncertainties, the probabilistic gradient-based transformation method (PGTM) is proposed to adapt the first-order probabilistic constraints from three different RBDO algorithms, including the chance constrained programming (CCP), reliability index approach (RIA), and performance measure approach (PMA), to the framework of the GTM. PGTM is efficient because only the sensitivity analyses and the reliability analyses require function evaluations (FE). The optimization processes of monotonicity analyses and the coordination procedures are free of function evaluations. Several mathematical and engineering examples show the PGTM is capable of finding the optimal solutions with desirable reliability levels.

Tolerances and Robustness in Engineering Design Optimization

1990 ◽  
Author(s):  
Alan Parkinson ◽  
Carl Sorensen ◽  
Joseph Free ◽  
Bradley Canfield
Keyword(s):  
Statistical Analysis ◽  
Optimal Design ◽  
Design Optimization ◽  
Engineering Design ◽  
Tolerance Allocation ◽  
Worst Case ◽  
Engineering Design Optimization ◽  
New Strategy ◽  
Transmitted Variation

Abstract This paper describes a new strategy for incorporating tolerances as part of engineering design optimization. The tolerance problem is defined in a general way that allows for tolerances on all model variables and parameters and includes worst case and statistical analysis. Two methods are described for taking tolerances into account during optimization such that an optimal design will remain feasible despite fluctuations in variables or parameters. Methods are also presented for directly controlling transmitted variation, and a framework for optimal tolerance allocation is proposed.

A General Approach for Robust Optimal Design

1993 ◽  
Vol 115 (1) ◽  
pp. 74-80 ◽  
Author(s):  
A. Parkinson ◽  
C. Sorensen ◽  
N. Pourhassan
Keyword(s):  
Statistical Analysis ◽  
Optimal Design ◽  
Design Optimization ◽  
Design Problem ◽  
Optimization Process ◽  
Worst Case ◽  
Rigorous Approach ◽  
Design Variables ◽  
Robust Optimal Design ◽  
And Control

This paper describes a general, rigorous approach for robust optimal design. The method allows a designer to explicitly consider and control, as an integrated part of the optimization process, the effects of variability in design variables and parameters on a design. Variability is defined in terms of tolerances which bracket the variation of fluctuating quantities. A designer can apply tolerances to any model input and can analyze how the tolerances affect the design using either a worst case or statistical analysis. As part of design optimization, the designer can apply the method to find an optimum that will remain feasible when subject to variation, and/or the designer can minimize or constrain the effects of tolerances as one of the objectives or constraints of the design problem.

Non-Gradient Based Parameter Sensitivity Estimation for Robust Design Optimization

2003 ◽  
Author(s):  
S. Gunawan ◽  
S. Azarm
Keyword(s):  
Design Optimization ◽  
Robust Design ◽  
Parameter Sensitivity ◽  
Robust Design Optimization ◽  
Optimization Approach ◽  
Case Scenario ◽  
Worst Case ◽  
Design Alternative ◽  
Sensitivity Estimation ◽  
Gradient Based

We present a method for estimating the parameter sensitivity of a design alternative for use in robust design optimization. The method is non-gradient based: it is applicable even when the objective function of an optimization problem is non-differentiable and/or discontinuous with respect to the parameters. Also, the method does not require a presumed probability distribution for parameters, and is still valid when parameter variations are large. The sensitivity estimate is developed based on the concept that associated with each design alternative there is a region in the parameter variation space whose properties can be used to predict that design’s sensitivity. Our method estimates such a region using a worst-case scenario analysis and uses that estimate in a bi-level robust optimization approach. We present a numerical and an engineering example to demonstrate the applications of our method.

Reliability-Based Design Optimization Concerning Objective Variation Under Mixed Probabilistic and Interval Uncertainties

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Jianhua Zhou ◽  
Min Xu ◽  
Mian Li
Keyword(s):  
Design Optimization ◽  
Performance Measure ◽  
Interval Uncertainty ◽  
Reliability Based Design Optimization ◽  
Worst Case ◽  
Reliability Based Design ◽  
Matrix Operations ◽  
Interval Uncertainties ◽  
Treat All ◽  
Objective Robustness

Uncertainties, inevitable in nature, can be classified as probability based and interval based uncertainties in terms of its representations. Corresponding optimization strategies have been proposed to deal with these two types of uncertainties. It is more likely that both types of uncertainty can occur in one single problem, and thus, it is trivial to treat all uncertainties the same. A novel formulation for reliability-based design optimization (RBDO) under mixed probability and interval uncertainties is proposed in this paper, in which the objective variation is concerned. Furthermore, it is proposed to efficiently solve the worst-case parameter resulted from the interval uncertainty by utilizing the Utopian solution presented in a single-looped robust optimization (RO) approach where the inner optimization can be solved by matrix operations. The remaining problem can be solved utilizing any existing RBDO method. This work applies the performance measure approach to search for the most probable failure point (MPFP) and sequential quadratic programing (SQP) to solve the entire problem. One engineering example is given to demonstrate the applicability of the proposed approach and to illustrate the necessity to consider the objective robustness under certain circumstances.

Minimizing Maximum Modal Force in Mistuned Bladed Disk Forced Response

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
Keith W. Jones
Keyword(s):  
Lower Boundary ◽  
Forced Response ◽  
Response Amplitude ◽  
Absolute Minimum ◽  
Turbine Engine ◽  
Bladed Disk ◽  
Engine Order ◽  
Modal Force ◽  
Low Amplitude ◽  
Infinite Linear

Amplitude magnification is defined as the maximum forced response amplitude of any blade on a mistuned bladed disk divided by the maximum response amplitude of any blade on a tuned bladed disk over a range of engine order excitation frequencies. This paper shows that amplitude magnification can be approximated as the maximum ratio of modal force divided by modal vector magnitude in an isolated family of turbine engine bladed disk modes. An infinite linear mistuning pattern, defined by a constant interblade stiffness increment between an infinite number of blades, is found to minimize the maximum modal force when subjected to engine order N/4 excitation. Linear mistuning, an approximation of the infinite linear mistuning pattern, approximately minimizes the maximum modal force for bladed disks with a finite number of blades when subjected to engine order N/4 excitation. From this theory, 2/N is proposed to be a lower boundary for amplitude magnification. The linear mistuning method is demonstrated to produce very low amplitude magnifications numerically and experimentally. The numerical examples suggest that linear mistuning may produce amplitude magnifications near the absolute minimum possible.

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