New Flutter Analysis Technique for Time-Domain Computational Aeroelasticity

Document

New Flutter Analysis Technique

for Time - Domain Computational Aeroelasticity

Prepared For:

58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

Grapevine, Texas, January 9 - 13, 2017

Prepared By:

Chan - gi Pak and Shun - fat Lung

Structural Dynamics Group, Aerostructures Branch (Code RS)

NASA Armstrong Flight Research Center

Overview

 Theoretical background (slides 3 - 6)

 Computational validation ( s lides 7 - 18)

 Conclusions (slide 19)

Structural Dynamics Group Chan - gi Pak - 2 /19

What the technology does

Problem Statement

 The classical method of determining the flutter speed from CFD

results is using a time - consuming trial - and - error process .

q < q

 Previous technologies provide system damping factors and

t

frequencies at a single dynamic pressure with a single CFD run.

Objective

 Develop a simple efficient approach for flutter speed and

q > q

frequency prediction

t

CFD CFD CFD run #1 run #2 run #3

q = q

F q q q D1 D2 D3

t

Dynamic Damping g pressure q D Structural Dynamics Group Chan - gi Pak - 3 /19

Previous technologies

 Bennett , R. M., and Desmarais , R. N., “Curve Fitting of Aeroelastic Transient Response Data with Exponential 𝑛𝑚 Functions,” NASA - SP - 415, pp. 43 - 58, 1975 .

− 𝜎 𝑡 𝑖

𝑞 ( 𝑡 ) = 𝑞 + 𝑒 { 𝐴 𝑐𝑜𝑠 𝜔 𝑡 + 𝐵 𝑠𝑖𝑛 𝜔 𝑡 }

0 𝑖 𝑑𝑖 𝑖 𝑑𝑖

 Non - linear least squares fitting

𝑖 = 1

 Optimization problem; strongly depends on starting damping factor and frequency values

 Results are system damping factors and frequencies

 Pak , C. - G., and Friedmann , P. P., “New Time Domain Technique for Flutter Boundary Identification,” AIAA - 92 - 2102, AIAA Dynamics Specialist Conference, Washington, D.C., 1992 .

 Assume that an aeroelastic (structure + aerodynamic) system is unknown .

 Estimate aeroelastic system matrices using single - input single - output parameter estimation together with

ARMA model

CFD CFD CFD

 Compute aeroelastic system damping factors and frequencies

run #1 run #2 run #3

q(t)

t

q q q D1 D3 D2 Dynamic Damping g pressure q D Structural Dynamics Group Chan - gi Pak - 4 /19

Technical features of new technology

 Approach  S tructural model is assumed known .

Structural dynamic model (known) Finite element model: M & K  The unsteady CFD analysis is performed using an 𝜼 𝜼 𝑵 + 𝑵 𝑘 𝑘 + 1 = 𝚿 + 𝚯 estimated dynamic pressure , q .

D 𝜼 𝜼 𝑘 + 1 𝑘  Use a linear panel code or test data  Non - dimensionalize orthonormalized aerodynamic force Step 4: Compute critical q D vector. Frequencies w & mode shapes F i using q – g & q – f curves D D  Estimate unknown aerodynamic system matrices, 𝐀 , 𝑎 𝑞 𝑞 𝐷 𝐷 𝑰 − 𝚯 𝐃 0 − 𝚯 𝐂 𝐁 , 𝐂 , & 𝐃 , using a multi - input multi - output parameter 1 𝑎 1 𝑎 𝜼 𝑎 𝑎 𝑎 2 2 estimation.

𝑞 𝑞 𝜼 = Updates q = q 𝐷 𝐷 D CD − 𝚯 𝐃 𝑰 − 𝚯 𝐂 2 𝑎 2 𝑎 𝑿  Multi - input: orthonormalized deflection vector 2 2 𝑘 + 1 0 0 𝐈  Multi - output: orthonormalized aerodynamic force 𝑞 𝑞 𝐷 𝐷 𝚿 + 𝚯 𝐃 𝚿 𝚯 𝐂 vector 11 1 𝑎 12 1 𝑎 𝜼 2 2 𝑞 𝑞  Compute the critical dynamic pressure using the known 𝐷 𝐷 𝜼 𝚿 + 𝚯 𝐃 𝚿 𝚯 𝐂 21 2 𝑎 22 2 𝑎 structural model and the estimated aerodynamic model.

2 2 𝑿 𝑘 𝐁 0 𝐀 𝑎 𝑎  Each iteration solves for the critical dynamic pressure , q , and uses this value in subsequent D h k Step 1: Run a CFD code @ Mach iterations Create Aerodynamic model (unknown) Single number M & dynamic pressure q complete a D 𝑿 = 𝐀 𝑿 + 𝐁 𝜼 𝑘 + 1 𝑎 𝑘 𝑎 𝑘 CFD run q - g & q - f D D N 𝑵 = 𝑞 𝐂 𝑿 + 𝑞 𝐃 𝜼 𝑘 𝐷 𝑎 𝑘 𝐷 𝑎 𝑘 k curves 𝐀 , 𝐁 , 𝐂 , & 𝐃 𝑎 𝑎 𝑎 𝑎 Step 2: Compute orthonormalized q q D1 CD Dynamic aerodynamic force vector N at Step 3: Estimate aerodynamic k pressure q D each time k system matrices using system ID Damping g N / q Use classical q - g curve k D D to find the critical q D Structural Dynamics Group Chan - gi Pak - 5 /19

Technical features of new technology (continued)

 Structural dynamic differential equations of motion in matrix form:

𝐌 𝒒 + 𝐂 𝒒 + 𝐊 𝒒 = 𝑸

 Generalized displacement vector 𝒒 :

𝜱 =mode shape

𝒒 ≡ 𝜱𝜼

𝜼 =orthonormalized coordinate vector

 Orthonormalized differential equations of motion: 𝑇

𝜼 + 2 𝛇𝛚 𝜼 + 𝛚 𝜼 = 𝑵

𝑵 = 𝜱 𝑸

 State differential equation in continuous time t :

𝜼

0 𝑰

= 𝑨 + 𝑩𝑵 𝑨 = 𝑩 =

𝜼

− 𝛚 − 2 𝛇𝛚

𝜼

𝑰

 State difference equation in discrete time k : 𝛥𝑇

𝜼 𝜼 𝑵 + 𝑵

𝑘 𝑘 + 1 ) 𝑨 𝛥𝑇 𝑨 ( 𝛥𝑇 − 𝜎

= 𝚿 + 𝚯 𝛥𝑇 = time step

𝚿 = e Q = 𝚪 B 𝚪 = e 𝑑𝜎

𝜼 𝜼

𝑘 + 1 𝑘 Structural Dynamics Group Chan - gi Pak - 6 /19

Computational Validation

Cantilevered rectangular wing model

 Configuration of a wind tunnel test article  Has aluminum insert (thickness = 0.065 in ) covered with 6% circular arc cross - sectional shape ( plastic foam )  lumped mass weight are computed based on 6% circular - arc cross sectional shape .

 Use structural dynamic model tuning technique  Chan - gi Pak and Samson Truong, “Creating a Test - Validated Finite - Element Model of the X - 56A Aircraft Structure,” Journal of Aircraft , Vol. 52, No. 5, pp. 1644 - 1667, 2015. doi : http:// arc.aiaa.org/doi/abs/10.2514/1.C033043  Modal analysis Measured and computed natural frequencies  NASTRAN sol. 103 Mode Measured (Hz) Computed (Hz) % Error 1 14.29 14.29 0.0 2 80.41 80.17 - 0.3 Y A 3 89.80 89.04 - 0.8 A - A 6% Circular arc 4.56 inch 0.065” aluminum insert Flexible plastic foam A 11.5 inch X Structural Dynamics Group Chan - gi Pak - 8 /19

CFL3D model & spline between CFL3D and NASTRAN

 CFL3D v.6 code is used.

 C ompute orthonormalized displacement and aerodynamic force vectors.

 The CFD grid is a multi - block (97 × 73 × 57) grid with H - H topology.

 The first three flexible modes are used.

 Splines between CFL3D and NASTRAN  Use interpolation element, RBE3, between FE grids and CFD grids.

 Include CFD grids in structural FE model  Structural FEM grids: master DOF  Surface CFD grids: slave DOF FEM grids: master DOF CFD model Finite element model Z X Flow direction CFD grids: slave DOF RBE3 elements between FEM Y and CFD Structural Dynamics Group Chan - gi Pak - 9 /19

FEM and CFD grids connection using RBE3 elements

RBE3 elements

CFD grids

FEM grids

Structural Dynamics Group Chan - gi Pak - 10 /19

Mode shapes of the cantilevered rectangular wing on structural and aerodynamic models

FEM Mode 3 Mode 2 Mode 1 CFD Structural Dynamics Group Chan - gi Pak - 11 /19

Local Mach number contour from steady CFD computations

0.714 0.795 0.851 1.017 0.956 0.913 . .

0 0 Structural Dynamics Group Chan - gi Pak - 12 /19

( 𝒒 − g) and ( 𝒒 − f) plots for initial q = 1.0 psi

𝑫 𝑫

D

10 10 1.321 0 0 1.174 -10 -10 -20 -20 -30 -30 -40 -40 -50 -50 Damping Damping -60 -60 -70 -70 -80 -80 -90 -90 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Dynamic pressure Dynamic pressure 100 100 90 90 80 80 70 70 60 60 50 50 42.02 38.90 40 40 n4sid q =1.00 psi SOCIT q =1.00 psi D Frequency Frequency D 30 30 20 20 10 10 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Dynamic pressure Dynamic pressure Structural Dynamics Group Chan - gi Pak - 13 /19

( 𝒒 − g) and ( 𝒒 − f) plots for SOCIT ( q = 1.20 psi) and n4sid ( q = 1.30 psi)

𝑫 𝑫 D D 10 10 1.390 1.457 0 0 -10 -10 -20 -20 -30 -30 -40 -40 -50 -50 Damping Damping -60 -60 -70 -70 -80 -80 -90 -90 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Dynamic pressure Dynamic pressure 100 100 90 90 80 80 70 70 60 60 50 50 SOCIT q =1.20 psi n4sid q =1.30 psi 40 40 35.49 35.33 D D Damping Frequency 30 30 20 20 10 10 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Dynamic pressure Dynamic pressure Structural Dynamics Group Chan - gi Pak - 14 /19

( 𝒒 − g) and ( 𝒒 − f) plots for SOCIT ( q = 1.40 psi) and n4sid ( q = 1.45 psi)

𝑫 𝑫 D D 10 10 1.465 1.471 0 0 -10 -10 -20 -20 -30 -30 -40 -40 -50 -50 Damping Damping -60 -60 -70 -70 -80 -80 -90 -90 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Dynamic pressure Dynamic pressure 100 100 90 90 80 80 70 70 60 60 50 50 40 40 SOCIT q =1.40 psi n4sid q =1.45 psi D D 33.37 Frequency Frequency 30 30 33.47 20 20 10 10 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Dynamic pressure Dynamic pressure Structural Dynamics Group Chan - gi Pak - 15 /19

Time histories of orthonormalized displacement with dynamic pressures of 1.45 and 1.46 psi

0.020 0.020 0.015 0.015 0.010 0.010 0.005 0.005 0.000 0.000 -0.005 -0.005 -0.010 -0.010 -0.015 Orthonormal displacement -0.015 Orthonormalized displacement -0.020 -0.020 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Time (sec) Time (sec) (a) q =1.45 psi D (b) q =1.46 psi D Structural Dynamics Group Chan - gi Pak - 16 /19

Flutter boundary of the cantilevered rectangular wing

1.8 1.6 1.4 35 1.2 0.8 0.6 Flutter frequency (Hz) 0.4 Dynamic pressure (psi) 10 : Test data : Test data 0.2 : Current method : Current method 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Mach number Mach number (a) Dynamic pressure (b) Flutter frequency 0.714 0.795 0.851 1.017 0.956 0.913 Structural Dynamics Group Chan - gi Pak - 17 /19

Time histories & PSDs of the first three orthonormal displacements

st :1 orthonormalized displacement 0.008 nd :2 orthonormalized displacement 0.006 rd :3 orthonormalized displacement 0.004  25Hz , 35Hz, 46Hz , & 90Hz  CFL3D with Euler option could not provide the correct orthonormalized 0.002 displacement and force vectors with the first three structural dynamic 0.000 modes.

-0.002 -0.004 -0.006 CFL3D results Orthonormalized displacement -0.008 0 0.05 0.1 0.15 0.2 Time (sec) 10 10 30 9 9 second third first 8 8 orthonormalized orthonormalized orthonormalized 7 7 displacement displacement displacement 6 6 5 5 15 4 4 Magnitude Magnitude Magnitude 3 3 2 2 1 1 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Frequency (Hz) Frequency (Hz) Frequency (Hz) Structural Dynamics Group Chan - gi Pak - 18 /19

Conclusions

 A new time - domain technique for computing flutter speed and frequency based on computational fluid dynamics (CFD) results was presented .

 The CFL3D v.6 code with the Euler option was used for solving the 3 - D flows on the structured grid .

 The full aeroelastic model is created by coupling the estimated aerodynamics model with the known structure dynamic model.

 The proposed approach is successfully implemented to identify the flutter boundaries of a cantilevered rectangular wing model .

 Computed flutter speeds and frequencies are in good match with measured quantities , however, the CFL3D code with the Euler option could not provide the correct orthonormalized displacement and force vectors with the first three structural dynamic modes in transonic speed regimes .

 S urface grids of the CFD model are included in the structural FE model.

 These surface CFD grids are connected to the nearest structural finite element method grids using interpolation ( RBE3 ) elements.

 This proposed fitting technique between structural finite element and CFD models is successful .

 The most critical technology for the success of the proposed approach is the robust MIMO parameter estimator .

Structural Dynamics Group Chan - gi Pak - 19 /19

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