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Nonlinear System Identification of an F-16 Aircraft Using the Acceleration Surface Method Tiln Dossogne Jean-Philippe Nol Gaetan Kerschen Workshop on Nonlinear System Identification Benchmarks Brussels, April 24 th 2017 Design Cycle of

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SLIDE 1

Nonlinear System Identification

  • f an F-16 Aircraft Using the Acceleration

Surface Method

Tilàn Dossogne Jean-Philippe Noël Gaetan Kerschen

Workshop on Nonlinear System Identification Benchmarks – Brussels, April 24th 2017

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Design Cycle of Nonlinear Engineering Structures

Measure Identify Design Understand Uncover Model

Computer-aided modelling (FEM) Structure (prototype

  • r current design)
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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

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4

Starting from Equations of Motion

From Newton’s second law in DOF 2: 𝑛2 ሷ 𝑦2+ 𝑙23. 𝑦2 − 𝑦3 + 𝑙12. 𝑦2 − 𝑦1 + 𝑑23. ሶ 𝑦2 − ሶ 𝑦3 + 𝑑12. ሶ 𝑦2 − ሶ 𝑦1 + 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

+ 𝑔

𝑜𝑚 𝑒𝑏𝑛𝑞

ሶ 𝑦2 − ሶ 𝑦1 = 0

1 2 3

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Isolating the Nonlinear Force

Assessment of the nonlinear stiffness term: 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

= − 𝑛2 ሷ 𝑦2 − 𝑙23. 𝑦2 − 𝑦3 − 𝑙12. 𝑦2 − 𝑦1 − 𝑑23. ሶ 𝑦2 − ሶ 𝑦3 − 𝑑12. ሶ 𝑦2 − ሶ 𝑦1 − 𝑔

𝑜𝑚 𝑒𝑏𝑛𝑞

ሶ 𝑦2 − ሶ 𝑦1

1 2 3

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Discarding All Unavailable Terms from Measurements Only

Assessment of the nonlinear stiffness term: 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

= − 𝑛2 ሷ 𝑦2 − 𝑙23. 𝑦2 − 𝑦3 − 𝑙12. 𝑦2 − 𝑦1 − 𝑑23. ሶ 𝑦2 − ሶ 𝑦3 − 𝑑12. ሶ 𝑦2 − ሶ 𝑦1 − 𝑔

𝑜𝑚 𝑒𝑏𝑛𝑞

ሶ 𝑦2 − ሶ 𝑦1

1 2 3

Only keep data points where ሶ 𝑦2 − ሶ 𝑦1 ≈ 0

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Discarding All Unavailable Terms from Measurements Only

Assessment of the nonlinear stiffness term: 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

≅ − 𝑛2 ሷ 𝑦2 − 𝑙23. 𝑦2 − 𝑦3 − 𝑙12. 𝑦2 − 𝑦1 − 𝑑23. ሶ 𝑦2 − ሶ 𝑦3

1 2 3

Contributions from other connections are assumed to be either small or linear

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Discarding All Unavailable Terms from Measurements Only

Assessment of the nonlinear stiffness term: 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

≅ − 𝑛2 ሷ 𝑦2 − 𝑙12. 𝑦2 − 𝑦1

1 2 3

Linear contribution (additional slope only)

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Discarding All Unavailable Terms from Measurements Only

Assessment of the nonlinear stiffness term: 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

≅ − 𝑛2 ሷ 𝑦2

1 2 3

Only slope affected

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A Simple Expression Approximates the Nonlinear Elastic Force

Assessment of the nonlinear stiffness term: 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

≅ − ሷ 𝑦2

Relative Velocity ሶ 𝑦2 − ሶ 𝑦1 Relative Displacement 𝑦2 − 𝑦1 − Acceleration − ሷ 𝑦2

Cross-section around ሶ 𝑦2 − ሶ 𝑦1 = 0

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The Shape of the Nonlinearity Can Be Found

Assessment of the nonlinear stiffness term: 𝑔

𝑜𝑚 𝑡𝑢𝑗𝑔𝑔 𝑦2 − 𝑦1

≅ − ሷ 𝑦2

− Acceleration − ሷ 𝑦2 Relative Displacement 𝑦2 − 𝑦1

Qualitative NL stiffness curve

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Same Machinery Applied for the Nonlinear Damping Term

Assessment of the nonlinear damping term: 𝑔

𝑜𝑚 𝑒𝑏𝑛𝑞

ሶ 𝑦2 − ሶ 𝑦1 ≅ − ሷ 𝑦2

Relative Velocity ሶ 𝑦2 − ሶ 𝑦1 Relative Displacement 𝑦2 − 𝑦1 − Acceleration − ሷ 𝑦2

Cross-section around 𝑦2 − 𝑦1 = 0

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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
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Application to the F-16 Aircraft

Two sensors from both sides

  • f the back wing-to-payload

connection

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The Reference Is Taken On The Payload

Two sensors from both sides

  • f the back wing-to-payload

connection

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Only One Mode Is Selected At A Time

Relative velocity [m/s]

  • Acceleration [m/s²]

Relative displacement [m]

Sweep frequency [Hz]

  • Acceleration [m/s²]

Portion of the time signal around 1 mode (under a 96N sine-sweep excitation)

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Stiffness and Damping Are Decoupled Using Cross-sections

  • Acceleration [m/s²]

Relative velocity [m/s] Relative displacement [m]

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Qualitative Stiffness and Damping Curves Are Found

  • Acceleration [m/s²]

8

  • 8
  • 0.6

0.6 Relative Displacement [mm]

  • Acceleration [m/s²]

4

  • 4
  • 80

80 Relative Velocity [mm/s]

  • Acceleration [m/s²]

Stiffness curve Damping curve

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ASM Gives Insights About the Physical Nonlinear Mechanism

8

  • 8
  • 0.6

0.6 Relative Displacement [mm]

  • Acceleration [m/s²]

4

  • 4
  • 80

80 Relative Velocity [mm/s]

  • Acceleration [m/s²]

Stiffness curve Damping curve

Sliding connection between the wing and the payload ⇒ Opening of the connection (softening) ⇒ Impacts (hardening) ⇒ Coulomb friction

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Consistent Stiffness and Damping Curves

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Useful Information from ASM

What can be extracted from ASM curves:

  • Clearances
  • Mathematical form of the NL

(piecewise linear)

What cannot be extracted from ASM curves:

  • Nonlinear stiffness values (slopes)

8

  • 8

0.05 Relative Displacement [mm]

  • Acceleration [m/s²]

0.54

  • 0.04
  • 0.41

Can we use the ASM to extract quantitative information about the nonlinear parameters?

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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

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ASM Methodology for Experimental Structures

Prototype or actual structure Measurements ASM

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ASM Can Also Be Applied On Numerical Results

Linear FE model Nonlinear elements Prototype or actual structure Measurements Simulations ASM ASM

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Similar Models Must Give Similar ASM Curves

Linear FE model Nonlinear elements Prototype or actual structure Measurements Simulations ASM ASM Qualitative estimation of the nonlinearity BUT Quantitatively comparable between each other

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Quantitative Nonlinear Parameters Can Be Extracted

Linear FE model Nonlinear elements Prototype or actual structure Measurements Simulations ASM ASM Qualitative estimation of the nonlinearity BUT Quantitatively comparable between each other Related to actual nonlinear parameters

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Optimal Nonlinear Parameters Are Found Iteratively

Linear FE model Nonlinear elements Simulations Reference ASM ASM

Optimization on the nonlinear parameters

Cost function: difference between the 2 ASM curves

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Finite Element Modeling of the Right Wing

Shell and beam elements Clamping at the root chord Nonlinear spring

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Parameters of the Nonlinear Spring Are Not Known A Priori

Shell and beam elements Clamping at the root chord Nonlinear spring

? ? ? ?

Relative Displacement Restoring Force

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The Procedure Is Applied to the F-16 Case

Linear FE model Nonlinear element F-16 aircraft GVT measurements Simulations ASM ASM

error > 𝜁

Error computation

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The Experimental Reference Solution Is Improved

Curve fitting step to simplify the error computation and decrease the number of parameters.

8

  • 8
  • 0.04
  • 0.41

0.05 Relative Displacement [mm]

  • Acceleration [m/s²]
  • 4

4 0.54

Experimental data points Curve fit

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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 =

𝑧ASM num − 𝑧ASM ref 2

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Good Fit Is Obtained After Several Iterations

8

  • 8
  • 0.04
  • 0.41

0.05 Relative Displacement [mm]

  • Acceleration [m/s²]
  • 4

4 0.54

Simulation data points Reference ASM from experimental

The error is minimized between the experimental and the numerical ASM sitffness curve.

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Quantitative NL Parameters Are Found

Linear FE model Nonlinear element F-16 aircraft GVT measurements Simulations ASM ASM

NL parameters: [0.4 0.08 1 0.08 1.1] x 107 N/m

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Conclusions ASM provides a quick and intuitive way to characterize the nonlinear behavior. Using a reliable linear finite element model, the ASM can quantitavely estimate optimal nonlinear parameters. Caution must be taken regarding the selection of the mode and the reference sensors.