Abstract
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.
Highlights
Pressures to minimise the environmental impact of air travel have led to aircraft manufacturers becoming focused on reducing the fuel consumption of the generation of civil transport aircraft
This paper addresses the problem of finding approximate solutions to the inhomogeneous fourth-order Euler–Bernoulli partial differential Equation (PDE) using the finite difference method for application to the analysis of flexible aircraft at the conceptual design stage
The finite difference method provides an alternative approach to the established mode shape analysis for the approximation of both the homogeneous and inhomogeneous versions of the Euler–Bernoulli equation
Summary
Pressures to minimise the environmental impact of air travel have led to aircraft manufacturers becoming focused on reducing the fuel consumption of the generation of civil transport aircraft. We applied the method to the construction of an explicit finite difference solution scheme for the Euler–Bernoulli equation for the application to the analysis of beams in the presence of time-varying external loads with a velocity-dependent damping term and taking into account mass, the second moment of area, and elastic modulus beam structural properties, that arbitrarily vary with the distance along the beam. The second set of computer experiments simulated the time-domain behaviour of a beam with properties chosen to represent the slender flexible wing structure of a generic aircraft This consisted of the stiffness and mass distributions varying along the length of the beam, while subjected to inertial and fluid-dynamic loading during a gust encounter. This paper concludes by summarising the main findings and giving suggestions for further work to expand the applicability of the methods presented
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