Jeffrey M. Brown

Confidence Interval Simulation for Systems of Random Variables

Bayesian network models are seen as important tools in probabilistic design assessment for complex systems. Such network models for system reliability analysis provide a single probability of failure value whether the experimental data used to model the random variables in the problem are perfectly known or derive from limited experimental data. The values of the probability of failure for each of those two cases are not the same, of course, but the point is that there is no way to derive a Bayesian type of confidence interval from such reliability network models. Bayesian confidence (or belief) intervals for a probability of failure are needed for complex system problems in order to extract information on which random variables are dominant, not just for the expected probability of failure but also for some upper bound, such as for a 95% confidence upper bound. We believe that such confidence bounds on the probability of failure will be needed for certifying turbine engine components and systems based on probabilistic design methods. This paper reports on a proposed use of a two-step Bayesian network modeling strategy that provides a full cumulative distribution function for the probability of failure, conditioned by the experimental evidence for the selected random variables. The example is based on a hypothetical high-cycle fatigue design problem for a transport aircraft engine application.

Reduced-Order Model Development for Airfoil Forced Response

Two new reduced-order models are developed to accurately and rapidly predict geometry deviation effects on airfoil forced response. Both models have significant application to improved mistuning analysis. The first developed model integrates a principal component analysis approach to reduce the number of defining geometric parameters, semianalytic eigensensitivity analysis, and first-order Taylor series approximation to allow rapid as-measured airfoil response analysis. A second developed model extends this approach and quantifies both random and bias errors between the reduced and full models. Adjusting for the bias significantly improves reduced-order model accuracy. The error model is developed from a regression analysis of the relationship between airfoil geometry parameters and reduced-order model error, leading to physics-based error quantification. Both models are demonstrated on an advanced fan airfoils frequency, modal force, and forced response.

Uncertainties of an Automated Optical 3D Geometry Measurement, Modeling, and Analysis Process for Mistuned Integrally Bladed Rotor Reverse Engineering

An automated reverse engineering process is developed that uses a structured light optical measurement system to collect dense point cloud geometry representations. The modeling process is automated through integration of software for point cloud processing, reverse engineering, solid model creation, grid generation, and structural solution. Process uncertainties are quantified on a calibration block and demonstrated on an academic transonic integrally bladed rotor. These uncertainties are propagated through physics-based models to assess impacts on predicted modal and mistuned forced response. Process details are discussed and recommendations made on reducing uncertainty. Reverse engineered parts averaged a deviation of 0.0002 in. (5 μm) which did not significantly impact low and midrange frequency responses. High frequency modes were found to be sensitive to these uncertainties demonstrating the need for future refinement of reverse engineering processes.

Probabilistic Mistuning Assessment Using Nominal and Geometry Based Mistuning Methods

Two deterministic mistuning models utilizing component mode synthesis methods are used in a Monte Carlo simulation to generate mistuned response distributions for a geometrically perturbed Integrally Bladed Rotor. The first method, a frequency-perturbation approach with a nominal mode approximation used widely in academia and industry, assumes airfoil geometric perturbations alter only the corresponding modal stiffnesses while its mode shapes remain unaffected. The mistuned response is then predicted by a summation of the nominal modes. The second method, a geometric method utilizing non-nominal modes, makes no simplifying assumptions of the dynamic response due to airfoil geometric perturbations, but requires recalculation of each airfoil eigen-problem. A comparison of the statistical moments of the mistuned response distributions and prediction error is given for three different frequency ranges and engine order excitations. Further, the response distributions are used for a variety of design and testing scenarios to highlight impacts of the frequency-based approach inaccuracy. Results indicate the frequency-based method typically provides conservative response levels.Copyright

Probabilistic Analysis of Geometric Uncertainty Effects on Blade Modal Response

This paper investigates the effect of manufacturing variations on the modal response of a transonic low aspect ratio fan. A simulated set of coordinate measurement machine measurements from a single rotor, representative of actual manufacturing variations, are used to investigate geometric effects. Principal component analysis is used to statistically model spatial geometry variations and reduce variable space dimensionality. Statistics from this analysis are used with Monte Carlo sampling to generate random blades realizations that are used to predict response distributions for a simulated fleet of 1000 blades. An existing approach to approximate blade frequency response is extended to include modal displacement and stress. These approximations are based on eigensensitivity analysis and first order Taylor series approximations. An approximation error analysis is conducted to quantify accuracy. The effect of small geometry variations on blade natural frequency, mode shape, and modal stress is investigated with results showing that small variations on the order of mils can cause significant variations in both scale and location of free and forced response.© 2003 ASME

Automated Finite Element Model Mesh Updating Scheme Applicable to Mistuning Analysis

Advancement of optical geometric measurement hardware has enabled the construction of accurate 3D tessellated models for a wide range of turbomachinery components. These tessellated models can be reverse-engineered into computer-aided design (CAD) models and input into grid generation software for finite element analyses. However, generating a CAD model from scan data is a time consuming and cumbersome process requiring significant user-involvement for even a single model. While it is possible to generate finite element models (FEMs) directly from tessellated data, current direct-grid methods produce unstructured grids that can introduce fictitious, numerical mistuning in these models, obscuring geometric mistuning. Nonetheless, as-measured scan data captured in a structured grid is essential for accurate geometric mistuning analyses, provided the tessellated scan data can be rapidly and accurately transformed into a FEM. This paper outlines and demonstrates an approach for rapidly generating structured FEMs for a population of integrally bladed rotors (IBRs) without requiring the arduous task of generating a CAD model for each as-measured IBR. This is accomplished by morphing the structured mesh of a nominal model to the tessellated data set collected from an optical scanner. It is shown that the fidelity and structure of these FEMs can be utilized for accurate mistuning analyses.© 2014 ASME

Component-Mode Reduced-Order Models for Geometric Mistuning of Integrally Bladed Rotors

Two methods that explicitly model airfoil geometry surface deviations for mistuning prediction in integrally bladed rotors are developed by performing a modal analysis on different degrees of freedom of a parent reduced-order model. The parent reduced-order model is formulated with Craig–Bampton component-mode synthesis in cyclic symmetry coordinates for an integrally bladed rotor with a tuned disk and airfoil geometric deviations. The first method performs an eigenanalysis on the constraint-mode degrees of freedom that provides a truncated set of interface modes, whereas the second method includes the disk fixed-interface normal mode in the eigenanalysis to yield a truncated set of ancillary modes. Both methods can use tuned or mistuned modes, where the tuned modes have the computational benefit of being computed in cyclic symmetry coordinates. Furthermore, the tuned modes only need to be calculated once, which offers significant computational savings for subsequent mistuning studies. Each geometric mist...

Probabilistic Analysis of Geometric Uncertainty Effects on Blade-Alone Forced Response

This paper investigates the effect of manufacturing variations on the blade-alone forced response of a transonic low aspect ratio fan. A simulated set of coordinate measurement machine measurements from a single rotor, representative of actual manufacturing variations, are used to investigate geometric effects. A reduced order model is developed to rapidly solve for the forced response and is based on eigensensitivity analysis and dynamic response mode superposition. An approximation error analysis is conducted to quantify accuracy of the new tool and errors between approximate and full finite element analysis solutions are shown to be small for low order modes with some high order modes having moderate error. A study of the simulated measured blade results show a significant amount of forced response variation along the leading edge of the airfoil. Statistics from this simulated measured rotor are used with Monte Carlo sampling to generate random blades realizations that are solved with the reduced order model. This procedure allows the prediction of the variation across an entire fleet of blades from a small sample of blades. The large variations predicted, up to 40%, could have a significant impact of the blade design process including the procedures to account for foreign object damage damage tolerance, how non-intrusive stress measurement systems are used, and how mistuning prediction algorithms are validated.Copyright

Probabilistic Gradient Kriging to Efficiently Predict Failure Probability Confidence Intervals

Probabilistic methods predict response variations caused by a system’s random design parameters. While it is generally assumed that the parameter statistics are known absolutely, they are usually an estimate taken from a small sample. A sampling distribution is associated with the statistical estimate that quantifies its uncertainty and from which confidence intervals can be determined. The sampling distributions also lead to a distribution and confidence intervals for probabilistically predicted responses such as failure probability. Calculating the failure probability distribution is expensive because each sample can require a full Monte Carlo simulation. This work develops a failure probability approximation method to efficiently quantify the effect of design parameter sampling distributions on failure probability distributions and confidence bounds. First, Response Surface and Kriging approximations are used and then a new failure probability approximation approach is developed. The method is labeled Probabilistic Gradient Kriging (PGK), and is based on an augmented Kriging approach which uses both the function values and analytically calculated probabilistic sensitivities. Fatigue crack growth and a dynamic oscillator demonstration problem are used to compare the traditional response surface method, the Kriging approach, and the new PGK method. Results show that the PGK method can fit surfaces extremely well, reduce approximation error by an order of magnitude, and enable efficient design parameter sample statistic uncertainty propagation. It is also shown that the statistical estimates from small samples can cause large variations in failure probabilites that should be accounted for to ensure system reliability.

Probabilistic High Cycle Fatigue Assessment Process for Integrally Bladed Rotors

This paper defines a probabilistic High Cycle Fatigue (HCF) assessment process for a fan or compressor Integrally Bladed Rotor (IBR). It identifies key design variables, how they are statistically modeled, the probabilistic integration technique, and the physics-based modeling process. It defines how previous eigensensitivity based reduced order models cannot be used for IBR assessment and validates an alternate approach. An autoregressive model accounts for correlation between IBR blade-to-blade variabilities. An approach is also defined to combine sector tuned stress variation and mistuning amplifications. Predicted stress variations integrate with a probabilistic Goodman Diagram to allow an IBR risk assessment. The paper concludes by summarizing several remaining areas that are necessary for a practical assessment process. These areas are probabilistic fluid dynamic prediction, probabilistic mission analysis, propagating model error, and the need for an effective validation strategy.Copyright