Nonlinear System Identification of an F-16 Aircraft Using the Acceleration Surface Method Tilàn Dossogne Jean-Philippe Noël Gaetan Kerschen Workshop on Nonlinear System Identification Benchmarks – Brussels, April 24 th 2017
Design Cycle of Nonlinear Engineering Structures Structure (prototype or current design) Understand Measure Identify Design Model Uncover Computer-aided modelling (FEM) 2
Outline • The Acceleration Surface Method (ASM) Main assumptions Application to the F-16 aircraft • A modified Acceleration Surface Method for quantitative estimation Basic Principles Application to the F-16 aircraft 3
ሶ ሶ ሶ Starting from Equations of Motion From Newton’s second law in DOF 2 : 𝑛 2 ሷ 𝑦 2 + 𝑙 23 . 𝑦 2 − 𝑦 3 + 𝑙 12 . 𝑦 2 − 𝑦 1 + 𝑑 23 . 𝑦 2 − ሶ 𝑦 3 + 𝑑 12 . 𝑦 2 − ሶ 𝑦 1 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑒𝑏𝑛𝑞 + 𝑔 + 𝑔 𝑦 2 − ሶ 𝑦 1 = 0 𝑜𝑚 𝑜𝑚 1 2 3 4
ሶ ሶ ሶ Isolating the Nonlinear Force Assessment of the nonlinear stiffness term: 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑔 = − 𝑛 2 ሷ 𝑦 2 𝑜𝑚 − 𝑙 23 . 𝑦 2 − 𝑦 3 − 𝑙 12 . 𝑦 2 − 𝑦 1 − 𝑑 23 . 𝑦 2 − ሶ 𝑦 3 − 𝑑 12 . 𝑦 2 − ሶ 𝑦 1 𝑒𝑏𝑛𝑞 − 𝑔 𝑦 2 − ሶ 𝑦 1 𝑜𝑚 1 2 3 5
ሶ ሶ ሶ Discarding All Unavailable Terms from Measurements Only Assessment of the nonlinear stiffness term: 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑔 = − 𝑛 2 ሷ 𝑦 2 𝑜𝑚 − 𝑙 23 . 𝑦 2 − 𝑦 3 − 𝑙 12 . 𝑦 2 − 𝑦 1 − 𝑑 23 . 𝑦 2 − ሶ 𝑦 3 − 𝑑 12 . 𝑦 2 − ሶ 𝑦 1 𝑒𝑏𝑛𝑞 − 𝑔 𝑦 2 − ሶ 𝑦 1 𝑜𝑚 Only keep data points where ሶ 𝑦 2 − ሶ 𝑦 1 ≈ 0 1 2 3 6
ሶ Discarding All Unavailable Terms from Measurements Only Assessment of the nonlinear stiffness term: 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑔 ≅ − 𝑛 2 ሷ 𝑦 2 𝑜𝑚 − 𝑙 23 . 𝑦 2 − 𝑦 3 − 𝑙 12 . 𝑦 2 − 𝑦 1 − 𝑑 23 . 𝑦 2 − ሶ 𝑦 3 Contributions from other connections are assumed to be either small or linear 1 2 3 7
Discarding All Unavailable Terms from Measurements Only Assessment of the nonlinear stiffness term: 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑔 ≅ − 𝑛 2 ሷ 𝑦 2 − 𝑙 12 . 𝑦 2 − 𝑦 1 𝑜𝑚 Linear contribution (additional slope only) 1 2 3 8
Discarding All Unavailable Terms from Measurements Only Assessment of the nonlinear stiffness term: 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑔 ≅ − 𝑛 2 ሷ 𝑦 2 𝑜𝑚 Only slope affected 1 2 3 9
ሶ ሶ A Simple Expression Approximates the Nonlinear Elastic Force Assessment of the nonlinear stiffness term: 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑔 ≅ − ሷ 𝑦 2 𝑜𝑚 Cross-section around 𝑦 2 − ሶ 𝑦 1 = 0 − Acceleration − ሷ 𝑦 2 Relative Velocity Relative Displacement 𝑦 2 − ሶ 𝑦 1 𝑦 2 − 𝑦 1 10
The Shape of the Nonlinearity Can Be Found Assessment of the nonlinear stiffness term: 𝑡𝑢𝑗𝑔𝑔 𝑦 2 − 𝑦 1 𝑔 ≅ − ሷ 𝑦 2 𝑜𝑚 Qualitative NL stiffness curve − Acceleration − ሷ 𝑦 2 Relative Displacement 𝑦 2 − 𝑦 1 11
ሶ ሶ Same Machinery Applied for the Nonlinear Damping Term Assessment of the nonlinear damping term: 𝑒𝑏𝑛𝑞 𝑔 𝑦 2 − ሶ 𝑦 1 ≅ − ሷ 𝑦 2 𝑜𝑚 Cross-section around 𝑦 2 − 𝑦 1 = 0 − Acceleration − ሷ 𝑦 2 Relative Velocity Relative Displacement 𝑦 2 − ሶ 𝑦 1 𝑦 2 − 𝑦 1 12
Summary of the Acceleration Surface Method Provides qualitative views of nonlinear stiffness and damping. Fast method requiring only two measured output signals. Requires Sine-sweep excitation. Assumptions: • Linear or negligible contributions from the surrounding connections • Small damping forces around zero relative velocity • Only one mode responding 13
Application to the F-16 Aircraft Two sensors from both sides of the back wing-to-payload connection 14
The Reference Is Taken On The Payload Two sensors from both sides of the back wing-to-payload connection 15
Only One Mode Is Selected At A Time - Acceleration [m/s²] Relative velocity [m/s] Relative displacement [m] - Acceleration [m/s²] Portion of the time signal around 1 mode (under a 96N sine-sweep excitation) Sweep frequency [Hz] 16
Stiffness and Damping Are Decoupled Using Cross-sections - Acceleration [m/s²] Relative velocity [m/s] Relative displacement [m] 17
Qualitative Stiffness and Damping Curves Are Found - Acceleration [m/s²] 8 4 Stiffness curve Damping curve - Acceleration [m/s²] - Acceleration [m/s²] 0 0 -8 -4 -0.6 0 0.6 -80 0 80 Relative Velocity [mm/s] Relative Displacement [mm] 18
ASM Gives Insights About the Physical Nonlinear Mechanism Sliding connection between the wing and the payload ⇒ Opening of the connection (softening) ⇒ Impacts (hardening) ⇒ Coulomb friction Stiffness curve Damping curve 8 4 - Acceleration [m/s²] - Acceleration [m/s²] 0 0 -8 -4 -0.6 0 0.6 -80 0 80 Relative Velocity [mm/s] Relative Displacement [mm] 19
Consistent Stiffness and Damping Curves 20
Consistent Stiffness and Damping Curves 21
Consistent Stiffness and Damping Curves 22
Consistent Stiffness and Damping Curves 23
Consistent Stiffness and Damping Curves 24
Consistent Stiffness and Damping Curves 25
Consistent Stiffness and Damping Curves 26
Consistent Stiffness and Damping Curves 27
Consistent Stiffness and Damping Curves 28
Consistent Stiffness and Damping Curves 29
Consistent Stiffness and Damping Curves 30
Useful Information from ASM What can be extracted from ASM - Acceleration [m/s²] curves: 8 • Clearances • Mathematical form of the NL (piecewise linear) 0 What cannot be extracted from ASM curves: -8 -0.41 -0.04 0.05 0.54 • Nonlinear stiffness values (slopes) Relative Displacement [mm] Can we use the ASM to extract quantitative information about the nonlinear parameters? 31
Outline • The Acceleration Surface Method Main assumptions Application to the F-16 aircraft • A modified Acceleration Surface Method for quantitative estimation Basic Principles Application to the F-16 aircraft 32
ASM Methodology for Experimental Structures Prototype or actual structure Measurements ASM 33
ASM Can Also Be Applied On Numerical Results Prototype or actual structure Linear FE model Nonlinear elements Measurements Simulations ASM ASM 34
Similar Models Must Give Similar ASM Curves Prototype or actual structure Linear FE model Nonlinear elements Measurements Simulations ASM ASM Qualitative estimation of the nonlinearity BUT Quantitatively comparable between each other 35
Quantitative Nonlinear Parameters Can Be Extracted Prototype or actual structure Linear FE model Nonlinear elements Related to Measurements Simulations actual nonlinear parameters ASM ASM Qualitative estimation of the nonlinearity BUT Quantitatively comparable between each other 36
Optimal Nonlinear Parameters Are Found Iteratively Optimization on the nonlinear parameters Linear FE model Nonlinear elements Cost function: Simulations difference between the 2 ASM curves Reference ASM ASM 37
Finite Element Modeling of the Right Wing Shell and Clamping at the beam elements root chord Nonlinear spring 38
Parameters of the Nonlinear Spring Are Not Known A Priori Shell and Clamping at the beam elements root chord Nonlinear spring Restoring Force ? ? ? ? Relative Displacement 39
The Procedure Is Applied to the F-16 Case F-16 aircraft Linear FE model Nonlinear element GVT measurements Simulations error > 𝜁 ASM ASM Error computation 40
The Experimental Reference Solution Is Improved Curve fitting step to simplify the error computation and decrease the number of parameters. 8 - Acceleration [m/s²] 4 Experimental 0 data points Curve fit -4 -8 -0.41 -0.04 0.05 0.54 Relative Displacement [mm] 41
Simulations Inside an Optimization Loop Are Performed Newmark time integration for a sine-sweep excitation • Level: 96N • Frequency bounds: from 8 to 6 Hz (restricted to 1 mode) • Sweep rate: 1Hz/min • Sampling frequency: 5000Hz (finer time step for the integration) Unconstrained Optimization • 5 parameters (slopes of the 5 portions) • Scaling • Cost function: err = 𝑧 ASM num − 𝑧 ASM ref 2 42
Good Fit Is Obtained After Several Iterations The error is minimized between the experimental and the numerical ASM sitffness curve. 8 - Acceleration [m/s²] 4 Simulation 0 data points Reference ASM from experimental -4 -8 -0.41 -0.04 0.05 0.54 Relative Displacement [mm] 43
Quantitative NL Parameters Are Found F-16 aircraft Linear FE model Nonlinear element NL parameters: [0.4 0.08 1 0.08 1.1] GVT measurements Simulations x 10 7 N/m ASM ASM 44
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