Conservative finite difference time domain schemes for the prestressed Timoshenko, shear and Euler–Bernoulli beam equations

Abstract

This paper presents a number of finite difference time domain (FDTD) schemes to simulate the vibration of prestressed beams to various degrees of accuracy. The Timoshenko, shear and Euler–Bernoulli models are investigated, with a focus on the numerical modelling for the Timoshenko system. The conservation of a discrete Hamiltonian to machine accuracy ensures stability and convergence of the numerical schemes. The difference equations are in the form of theta schemes, which depend on a number of free parameters that can be tuned in order to reduce numerical dispersion. Although the schemes are built by means of second-order accurate finite difference operators only, fully fourth-order accurate schemes may be designed through modified equation techniques, and wideband-accurate schemes are also possible. The latter are schemes designed to maximise the resolving power at all wavelengths. Investigation of beams of cross section varying from slender to thick allows a thorough comparison between the various schemes, for the three beam models.