On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics

This paper focuses on the development of an explicit finite difference numerical method for approximating the solution of the inhomogeneous fourth-order Euler–Bernoulli beam bending equation with velocity-dependent damping and second moment of area, mass and elastic modulus distribution varying with...

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Autores principales:	Dominique Fleischmann, László Könözsy
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	inhomogeneous Euler–Bernoulli equation stability analysis high-order finite difference schemes aeroelasticity comparisons with experimental data flexible aircraft Motor vehicles. Aeronautics. Astronautics TL1-4050
Acceso en línea:	https://doaj.org/article/741cc4e2c1ce48a887b3995f4dbc2c79
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spelling	oai:doaj.org-article:741cc4e2c1ce48a887b3995f4dbc2c792021-11-25T15:57:53ZOn a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics10.3390/aerospace81103562226-4310https://doaj.org/article/741cc4e2c1ce48a887b3995f4dbc2c792021-11-01T00:00:00Zhttps://www.mdpi.com/2226-4310/8/11/356https://doaj.org/toc/2226-4310This paper focuses on the development of an explicit finite difference numerical method for approximating the solution of the inhomogeneous fourth-order Euler–Bernoulli beam bending equation with velocity-dependent damping and second moment of area, mass and elastic modulus distribution varying with distance along the beam. We verify the method by comparing its predictions with an exact analytical solution of the homogeneous equation, we use the generalised Richardson extrapolation to show that the method is grid convergent and we extend the application of the Lax–Richtmyer stability criteria to higher-order schemes to ensure that it is numerically stable. Finally, we present three sets of computational experiments. The first set simulates the behaviour of the un-loaded beam and is validated against the analytic solution. The second set simulates the time-dependent dynamic behaviour of a damped beam of varying stiffness and mass distributions under arbitrary externally applied loading in an aeroelastic analysis setting by approximating the inhomogeneous equation using the finite difference method derived here. We compare the third set of simulations of the steady-state deflection with the results of static beam bending experiments conducted at Cranfield University. Overall, we developed an accurate, stable and convergent numerical framework for solving the inhomogeneous Euler–Bernoulli equation over a wide range of boundary conditions. Aircraft manufacturers are starting to consider configurations with increased wing aspect ratios and reduced structural weight which lead to more slender and flexible designs. Aeroelastic analysis now plays a central role in the design process. Efficient computational tools for the prediction of the deformation of wings under external loads are in demand and this has motivated the work carried out in this paper.Dominique FleischmannLászló KönözsyMDPI AGarticleinhomogeneous Euler–Bernoulli equationstability analysishigh-order finite difference schemesaeroelasticitycomparisons with experimental dataflexible aircraftMotor vehicles. Aeronautics. AstronauticsTL1-4050ENAerospace, Vol 8, Iss 356, p 356 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	inhomogeneous Euler–Bernoulli equation stability analysis high-order finite difference schemes aeroelasticity comparisons with experimental data flexible aircraft Motor vehicles. Aeronautics. Astronautics TL1-4050
spellingShingle	inhomogeneous Euler–Bernoulli equation stability analysis high-order finite difference schemes aeroelasticity comparisons with experimental data flexible aircraft Motor vehicles. Aeronautics. Astronautics TL1-4050 Dominique Fleischmann László Könözsy On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics
description	This paper focuses on the development of an explicit finite difference numerical method for approximating the solution of the inhomogeneous fourth-order Euler–Bernoulli beam bending equation with velocity-dependent damping and second moment of area, mass and elastic modulus distribution varying with distance along the beam. We verify the method by comparing its predictions with an exact analytical solution of the homogeneous equation, we use the generalised Richardson extrapolation to show that the method is grid convergent and we extend the application of the Lax–Richtmyer stability criteria to higher-order schemes to ensure that it is numerically stable. Finally, we present three sets of computational experiments. The first set simulates the behaviour of the un-loaded beam and is validated against the analytic solution. The second set simulates the time-dependent dynamic behaviour of a damped beam of varying stiffness and mass distributions under arbitrary externally applied loading in an aeroelastic analysis setting by approximating the inhomogeneous equation using the finite difference method derived here. We compare the third set of simulations of the steady-state deflection with the results of static beam bending experiments conducted at Cranfield University. Overall, we developed an accurate, stable and convergent numerical framework for solving the inhomogeneous Euler–Bernoulli equation over a wide range of boundary conditions. Aircraft manufacturers are starting to consider configurations with increased wing aspect ratios and reduced structural weight which lead to more slender and flexible designs. Aeroelastic analysis now plays a central role in the design process. Efficient computational tools for the prediction of the deformation of wings under external loads are in demand and this has motivated the work carried out in this paper.
format	article
author	Dominique Fleischmann László Könözsy
author_facet	Dominique Fleischmann László Könözsy
author_sort	Dominique Fleischmann
title	On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics
title_short	On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics
title_full	On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics
title_fullStr	On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics
title_full_unstemmed	On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics
title_sort	on a novel approximate solution to the inhomogeneous euler–bernoulli equation with an application to aeroelastics
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/741cc4e2c1ce48a887b3995f4dbc2c79
work_keys_str_mv	AT dominiquefleischmann onanovelapproximatesolutiontotheinhomogeneouseulerbernoulliequationwithanapplicationtoaeroelastics AT laszlokonozsy onanovelapproximatesolutiontotheinhomogeneouseulerbernoulliequationwithanapplicationtoaeroelastics
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On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics

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