Propagation of Uncertainty in Stress Estimation in Turbine Engine Blades Giorgio Calanni, Vitali Volovoi, and Massimo Ruzzene Georgia Institute of Technology Atlanta, Georgia and Charles Vining Naval Air Systems Command Peter Cento Aerospace Testing Alliance (ATA) Arnold Engineering Development Center Arnold Air Force Base, Tennessee May 11, 2006, Barcelona Spain Air Force Materiel Command Arnold Engineering Development Center Arnold Air Force Base, TN 37389
Outline Motivation and Problem Statement Background, Objectives, and Approach Stress Estimation Methodology Results Analysis on a beam-like blade Analysis on a bladed disk Concluding Remarks Future Work
Motivation and Problem Statement Satisfactory assessment of stress levels in turbine disks is key to engine reliability It relies on: high-fidelity analysis tools high measurement accuracy identification of uncertainties Development of probabilistic-based methodology to: improve stress prediction support analysis and interpretation of test results
Background and Objective Stress prediction in turbine disks in a testing environment is affected by: Limitations on number and locations of sensors Sensor type (e.g. Strain gauges vs. NSMS) Uncertainty due to variations and inaccuracies in the instrumentation Uncertainty due to manufacturing variations (e.g. blade mistuning) Errors associated with the stress estimation technique Mapping between field and test operating conditions. Objective of overall project : Quantification of uncertainty related to stress estimation in turbine engine blades Objective of present work: Estimate propagation of uncertainty in stress estimation due to a limited number of uncertain parameters
Approach Development of a simplified beam model: Model is developed in Matlab environment Model is based on 1D beam theory, and considers basic model for rotor flexibility Centrifugal effects are included Model allows single blade and entire rotor analyses Development of a detailed rotor blade model Model is generated in Ansys environment Purdue Transonic Multistage Research Compressor data Cyclic symmetry analysis is performed to reduce computational cost
Approach Models are used to simulate effects and propagation of uncertainty Model-based maximum stress evaluation process is affected by: - Variability inherent with test procedures (test conditions, component configuration) - Uncertainty on sensor location and sensitivity - Uncertainty on model input (i.e. blade geometry, material properties, boundary conditions) INPUT UNCERTAINTY and - Limited model fidelity, or capability to reproduce component s behavior MODELING UNCERTAINTY
Stress Estimation Methodology
Model-based Stress Inference Process Evaluation of maximum stresses relies on model information. Identification of the mode correlating to the maximum sensor amplitudes (forced response) defines the modal participation factors Experimental data and identified modal information are used to extrapolate maximum stress values Example: two-sensor case w Sensor S1 Sensor S2 a S1, max Frequency response read by sensor S1 as 2, max Frequency response read by sensor S2 as1 as 2 f max, 1 frequency f max, 2 frequency
Model-based Stress Inference Process
Model-based Stress Inference Process Given asj, max (i.e. the maximum amplitude measured by the j-th sensor), the maximum response in the structure can be extrapolated as follows: STEP 1: Minimization of the fitness parameter. Frequency of max response Experimental Response ratio i-th natural frequency i-th modal ratio STEP 2: Maximum-value estimate via scaling of modal responses Φ p and Φ q : a max = max a s1 ( f max, p ) max( φ ) φ s1, p p, a s2 ( f max, q ) max( φ q ) φs2, q
Modal Analysis on system model System Response Estimation under Uncertainty Modeling Uncertainty (via mode-shape perturbation) Prediction Scheme (two-sensor case) as j, max as j Sensor-based Uncertainty (normally distributed error added to nominal values) 1. Identification of modes Φ p and Φ q 2. Maximum-response estimation via modes Φ p and Φ q f max, S j Excitation frequency Data from (simulated) experiments on system (model)
Sources of input uncertainty Material and geometric uncertainties; Sensor related uncertainties: sensor location x: normal distribution sensor measurement uncertainty: L r = a system s reference dimension A = magnitude of sensor noise
Modeling uncertainty Modeling uncertainty is introduced by considering perturbed mode shapes Φ (p) for stress estimation: weights i-th mode obtained from the model. The Modal Assurance Criterion (MAC) is used to gage the mismatch between model modes and the perturbed modes used for stress prediction:
Results: Analysis on Beam-like Blade
System Characteristics Length: 22.5 cm Thickness: 2 cm Chord: 15.3 cm Density: 4430 Kg/m 3 Young s Modulus: 114GPa N Beam-like Blade 1 k (N) c k (i) r k (i) Tr R h k (i+1) c k (i-1) c k (i) c i+1 i Types of uncertainty: Analysis Input data uncertainty geometry, material properties. Sensor-induced uncertainty: placement and measurement inaccuracy. Modeling uncertainty. S 2 S 1 x x 1 = 0.25 L o x 2 = 0.75 L o
Geometric related Uncertainties Theoretical Normal distribution N(t o,σt o ) Effect of a variation in blade chord c o and thickness t o (±2% perturbation): 2 c = c [1 + N(0, σ )] σ = max maxσ Ω, ω a alternating, axial 0 2 t = t0[1 + N(0, σ )] 3σ =.02 c = c t = t 0 0 3σ =.02 2 [1 + N(0, σ )] 2 [1 + N(0, σ )] Theoretical Normal distribution N(c o,σc o ) MonteCarlo (3000 samples) Goodman line Some input combinations may cause the structure to critically approach or exceed the fatigue limit
Sensor related Uncertainty Sensors at 25% & 75% of beam length Sensor results simulated via harmonic analysis on nominal system Sensor noise A defined as a percentage of the maximum amplitude response for the selected mode Standard deviation equal to 0.01 Variability due to uncertainty in sensor readings Variability due to uncertainty in sensor location Absolute Error in Maximum Longitudinal Stress Estimate [Pa] Variability due to uncertainty in both sensor locations Absolute Error in Maximum Longitudinal Stress Estimate [Pa] Absolute Error in Maximum Longitudinal Stress Estimate [Pa]
Mode n. (p1) Φ 5 4 3 2 MAC - Configuration 1 Mode n. - Φ (p2) Modeling Uncertainty 5 4 3 2 MAC - Configuration 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 1 1 2 3 4 5 Mode n. -Φ (m) 5 1 1 2 3 4 5 MAC - Configuration 3 Mode n. -Φ (m) 0.1 0 MAC matrices for two perturbed modes and original unperturbed mode Mode n. Φ (p) - 4 3 2 Perfect correlation 1 1 2 3 4 5 Mode n. Φ - (m)
Modeling Uncertainty Reducing modeling uncertainty reduces uncertainty on stress estimation.18 Maximum Stress (output).16.14 Unperturbed modes Configuration 3 Probability density.12.1.08.06.04 Configuration 2 Perturbed modes.02 Configuration 1 0-0.02-0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 (max) (max) x 10 σ [ ] [Pa] 3 x μ σ x
Modeling Uncertainty Reducing modeling uncertainty reduces uncertainty on stress estimation.18 Maximum Stress (output).16.14 Improved (perfect) model Probability density.12.1.08.06 Initial Model.04.02 uncertainty Input uncertainty Modeling uncertainty 0-0.02-0.015-0.01-0.005 0 0.005 0.01 0.015 0.02 (max) (max) x 10 σ [ ] [Pa] 3 x μ σ x
Results: Analysis on a Bladed Disk
Bladed Disk Characteristics The disk is composed of 19 sectors. Disk dimensions extrapolated from photographs of a real system. Blade geometry representing Purdue Multistage Transonic Compressor [1]. Dimensions Blade height: 5.08 cm Blade chord: 12.7 cm Hub outer radius: 10.16 cm Material Properties (Ti-6-4) Density: 4430 Kg/m 3 Young s Modulus: 114 GPa Poisson s Ratio: 0.33 Hub inner radius: 1.27 cm Disk core thickness: 1.52 cm [1] Sanders, A., Fleeter, S. (2000). Experimental Investigation of Rotor-Inlet Guide Vane Interactions in Transonic Axial-Flow Compressor. Journal of Propulsion and Power, Vol. 16, Issue 3, May-June 2000.
Bladed Disk Finite Element Model Blades and disk modeled as one integral component (i.e. a blisk) Discretization via SOLID95 elements in the FE commercial package Ansys Externally applied boundary conditions: zero displacement at disk-shaft interface Modeling approach accounted for the system s cyclic-symmetry nature, via internal compatibility constraints
Cyclically Symmetric Structures Cyclically symmetric structures analyzed by the modeling of only the fundamental unit (i.e. a sector) Equivalent compatibility conditions and loading functions applied to the modeled sector to account for the missing part of the structure Ansys employs the Duplicate Sector Approach: Fourier transformations used to convert loads applied on any part of the entire structure to equivalent loads applied only on the modeled structure; Existing Ansys routine has been modified to enable forced harmonic analysis.
Disk Modal Data Harmonic Index = 1; ω n =1070.6 Hz Harmonic Index = 6 ω n = 3366.4 Hz
Bladed Disk: Harmonic Analysis Setup Sensor S 1 Sensor locations: span-wise upper part of blade, suction and pressure surfaces on the same blade Force applied on blade tip, and of the form: Sensor S 2 F j = F o e i ωt+ 2π k ( j 1) N ω = excitation frequency k = harmonic index j = sector number
Bladed Disk: Sensor-induced Uncertainty Assessment of the impact of sensor measurement uncertainty. aˆ Excited mode ω n = 1070.6 Hz (k=0). S ( f ) = a S ( f ) + A N(0, σ 2 ) Variability due to uncertainty in sensor S 1 reading Comparison between sensors
Bladed Disk: Effect of Modeling Uncertainty Sensor uncertainties in the presence of modeling perturbations Excited mode: ω n = 1070.6 Hz (k=0). MAC Configuration 2 Comparison between different MAC configurations increasing modeling uncertainty Perfect model MAC Configuration 3 μ MAC i ( σ μ VM ) μ MAC 1 ( σ MAC 1 VM ) ( σ VM ) 5.73%, i = 2 = 16.35%, i = 3
Bladed Disk: Sensor-induced & Modeling Uncertainty Excited mode: @ ω n = 837.45 Hz (k=1). Coincident modes Two sensor measurements needed for stress inference. Forced response near the natural frequency ω n can be approximated as: (1) ω (2) ω a = αφ + βφ where α, β ( R, R) n ( a s 1, as2 ) n Two sensor measurements are needed to compute the coefficients α and β: a a s1 s2 = φs φs (1) 1, ωn (1) 2, ω n φ φ (2) s1, ωn (2) s2, ω n α β α β = φs φs (1) 1, ωn (1) 2, ω n φ φ (2) s1, ωn (2) s2, ω n 1 a a s1 s2
Bladed Disk: Sensor-induced Uncertainty (1/2) Assessment of the impact of sensor measurement uncertainty. Excited mode: double mode @ ω n = 3366.4 Hz (k=6). a a s1 s2 = φs φs (1) 1, ωn (1) 2, ω n φ φ (2) s1, ωn (2) s2, ω n α β Impact of measurement uncertainty on maximum-stress estimate Impact of measurement uncertainty on maximum-displacement estimate reading a s1 of S 1 only perturbed reading a s2 of S 2 only perturbed readings of S 1 & S 2 perturbed
Bladed Disk: Effect of Modeling Uncertainty (2/2) MAC Configuration 2 MAC Configuration 3 Combination of modeling and sensor-induced uncertainties. Excited mode: double mode @ ω n = 3366.4 Hz (k=6). Both sensor readings perturbed. Comparison between different MAC configurations Comparison between different MAC configurations MAC 2 MAC 1 MAC 3 Perfect Model μ MAC i ( σ ) μ MAC 1 ( σ μ VM MAC 1 VM ) ( σ VM ) - 2.37%, = - 4.67%, i = 2 i = 3
Focus of the work: Concluding Remarks Modeling of simplified sources of uncertainty: Sensor related and analysis input uncertainties Modeling uncertainties Knowledge gained: Response estimates affected by instrument settings even in simple cases Modeling uncertainties further deteriorates the accuracy of results Challenges: How to separate the effects of various sources of uncertainty, or quantify their correlations, in unknown settings Accomplish optimization for a given uncertainty in the absence of cause-effect relationships Extrapolation of results to not-modeled conditions/systems in the presence of notmodeled uncertainty
Future Work Expand the scope of the work: Include additional sources of uncertainty at the testing and system modeling level Utilize available experimental results to quantify uncertainty Application of statistical techniques to integrate various sources of knowledge/information within a unified estimation scheme
Acknowledgments Work is supported by the Air Force Office of Scientific Research s (AFOSR) Test & Evaluation Program (Grant # FA9550-05-1-0149)
BACKUP SLIDES
Modeling Uncertainty Model-based stress prediction is affected by limited accuracy of the model Lack of model accuracy is simulated by perturbing the modes used for maximum stress evaluation Modes used for stress inference weight Model modes Correlation between modes is quantified through the MAC: Perfect correlation corresponds to a unitary MAC
Cyclic Symmetry: Compatibility Constraints Compatibility constraints: r u r u A B Side r cos kα sin kα u = sin k cos k r u 2 α α for these constraints to be valid, the nodal degrees of freedom u are to be expressed in nodal local cylindrical coordinate systems A B Side 1 α Nodal cylindrical reference frame Side 2 harmonic index N 0, 1,..., N even k = 2 N 1 0, 1,..., N odd 2 Side 1 basic and duplicate sectors share the same physical location N = number of sectors α = sector angle = 360 /N
Cyclic Symmetry: External Loads Let F j be an external load applied on the j-th sector (node-wise) The equivalent loads applied on basic and duplicate sectors (separate contributions for each of the harmonic indices) are: k = ( F ) 0 1 N N k = 0 = A j= 1 ( F ) = 0 k = 0 B F j 0 < k N < 2 N 2 ( Fk ) = F j cos[ ( j 1) kα ] A N 2 ( Fk ) = Fj sin[ ( j 1) kα ] B For each k, cyclic components of the solution are obtained for basic and duplicate sectors: N N (N even or odd) j= 1 j= 1 [( U ) ( U ) ] k A k B k = N (N even only) 2 N 1 ( j ( F 2 ) = ( 1) A N k = N / j= 1 ( F ) = 0 k = N / 2 Sector numbering for loads B 1) j = 2 F j j = 1 j = N
Cyclic Symmetry: Solution Composition [( U k ) ( U ) ] Given then cyclic k-components of the solution A (e.g. displacements or stresses), the overall solution is obtained via superposition: k B U max( k = ) ( U ) cos[ ( j 1) kα ] ( U ) sin[ ( j 1) kα ] j =,..., N j k A k 1 B k = 0 The solution U j as given above is valid only in the nodal local cylindrical reference frames. j-th sector Nodal local cylindrical frame α