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

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 from slender to thick allows a thorough comparison between the various schemes, for the three beam models.