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

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.