Towards a Non-linear Full Aircraft

Model for Large Passenger Aircraft

Loads CalculationsJavon Farao1

Prof Arnaud G. Malan1

Francesco Gambioli2

1UCT Industrial Computational Fluid Dynamics Research Group2AIRBUS – ‘Loads and Aeroelastics’ Department

IFASD Conference 2017June 2017

Full Aircraft Model

Incorporate various ROMs to form loads analysis tool

• Sloshing Effects

• High-fidelity CFD modelling via Elemental

• Equivalent mechanical models

• Amplification Factor slosh map

• Aerodynamics

• Discrete gust condition

• Structure

• Linear vs nonlinear

• Full order vs ROMs

Structure

Aerodynamics

Slosh

Various approaches to geometrically nonlinear beam structure

• Non-linear finite element analysis

• Updated/total Lagrangian formulation

• Computationally expensive approach

• Intrinsic beam

• Velocity and strains employed as degrees-of-freedom

• Co-rotational formulation

• Geometrically exact nonlinear beam

• Quadratic mode shape analysis

• Pragmatic approach to include quadratic nonlinearities

Brief Background

Quadratic Mode Shape Analysis

Sensible approach to implement current structural data

• Structure deforms in a curved path

• Includes non-linear kinematics not present in linear analysis

• Reduced number of dofs while encompassing non-linearities

• Extend normal modes to approximate parabolic motion

• Implemented in aeroelastic modelling with marked success [1]

Geometrically linear

Geometrically non-linear

[1] “Hybrid Finite-Volume Reduced-Order Model for Non-linear Aero-elastic modelling”. Mowat, Malan, et al (2014)

Structural Reduced Order Model (ROM)

Finite element structure to equivalent beam

• Static condensation

• Produce stiffness matrix

• Produce element properties

Modal analysis

• Large, sparse matrices

Temporal solution

• Newmark method

Quadratic mode shape [2]

[2] “A Method for Calculating the Dynamics of Rotating Flexible Structures, Part 1: Derivation”. Segalmann, Dohrmann (1995)

FEM Structure

Governing Equation

• Timoshenko Beam Theory

• Shear Deformation

• Rotary Inertia

12 d.o.f beam element

Modal Analysis

Large, sparse matrices

• Aircraft wing experiences excitation in lowest frequency modes

• Compute primary frequencies and associated modes

Solve generalised eigen-problem

Cholesky Factorization of mass matrix

• Simplify to standard eigen-problem

Modal Analysis

Arnoldi Method

• Produce upper Hessenberg matrix

• Contains similar primary eigen-pairs as overall vector

Iterative QR decomposition

• Compute eigen-values𝐻𝑚 = 𝑄𝑅, 𝑅 ∗ 𝑄 = 𝐻𝑚 𝜆𝑖 on diagonal

Shifted Inverse Power Method

• Compute associated eigen-vectors𝜆𝑖 , 𝒌𝑖 eigen-pairs of Hessenberg matrix

Transform into original coordinates𝒚𝑖 = 𝑽𝑚 ∗ 𝒌𝑖

𝒙𝑖 = 𝑳−𝑇 ∗ 𝒚𝑖

𝑉𝑚 𝑉𝑚 𝐻𝑚𝐷

Quadratic Mode Shape Analysis

Derivation [3]

• Compensate for extension due to rotation

• Consider truss element

• Elastic energy only stored via compression or expansion

• Introduce quadratic modal term to compensate higher-order extension

𝑹𝒆 = 𝒍 × (𝑢2 – 𝑢1)/𝒍2

𝑒𝑥𝑡 = 𝑞𝑖(𝑢2 – 𝑢1). 𝒍/ 𝒍2+1

2𝑞𝑖2𝑹𝒆𝟐𝒍 +𝑞𝑖

2 (𝒈2𝒊𝒋−𝒈1𝒊𝒋). 𝒍/ 𝒍2

• Solve quadratic mode shape to minimise the coefficient of 𝑞𝑖4 for the elastic

potential energy

𝑈 = 𝑞𝑖 . . + 𝑞𝑖2 … + 𝑞𝑖

3 … + 𝑞𝑖4

𝑒

𝐴𝐸

𝟐𝒍

1

2𝑹𝒆𝟐𝒍 +𝒈2𝒊𝒋−𝒈1𝒊𝒋

. 𝒍

𝒍2

2

• Differentiate coefficient w.r.t each component of the quad mode𝑲 𝑲𝒖𝒖𝑇𝑲 0

𝒈𝑖𝑗

𝜆=𝑭0

[3] “Quadratic Mode Shapes from Linear Finite Element Analysis”. Louw, Mathews (2012)

1 2Initial orientation

12 Modal orientation

𝑢2𝑢1

𝒍

Verification and Validation

Analytical Solution [4]

• Quadratic mode accounts for element stretch

• Compare truss network and beam structure

[4] “Applied Dynamic Programming for Optimization of Dynamical Systems”. Robinett, et al (2005)

𝐿 = 20 𝑚Truss elements = 81 Beam elements = 10

Verification and Validation

Analytical comparison

Verification and Validation

Non-linear dynamic beam analysis

• Nastran vs Quadratic Modal Analysis

𝐹

𝐿

𝑥

𝑦

𝑧

Verification and Validation

Geometrically non-linear beam inextensibility

Linear Cantilever Animation

Non-Linear Cantilever Animation – Quadratic Mode Shapes

Error Analysis

Verification and Validation

𝜀 = 6%

Application to CRM wing

CRM wing FEM model condensed to beam elements [5]

Compare linear model to non-linear QMS approach• Half-gust load applied at wing tip to elicit large deformations

• 2% structural damping applied

[5] Condensation performed by Dr R. Cook from University of Bristol

Application to CRM wing

Full Aircraft Platform

Modal Analysis

• Primary modal pairs extracted within ElementalTM

Construction of QMS

• Pragmatic approach to incorporate geometrically nonlinear deformation

Results

• Axial acceleration component improves accuracy of fuel loads integration

• Seamless integration into current loads analysis procedures

• Negligible impact on computational effort

• Valid structural response for the purpose of our study

Summary

Acknowledgements

National Aerospace Centre

South African Research Chair Initiative

Airbus ‘Loads and Aeroelasticity’ Department – Bristol University of

Bristol – Dr R. Cook and Prof J. Cooper

Part of the research leading to this work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 636053.