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

Simulation of the Geometrically Exact Nonlinear String via Energy Quadratisation

“…On the other hand, since k ∝ h 2 for small h, the error has a slope of 1 when plotted against the time step. Note that these results are consistent with the error curves presented in [37].…”

“…Thus, a check on the eigenvalues of the linear part allows to conclude that a necessary and sufficient condition for the non-negativity of the total energy is given by (see e.g. [8,37])…”

“…[46,47]. For this reason, it may be preferable to reduce dispersion over the entire range of frequencies [48,37], rather than increasing the formal order of accuracy. The linear parts of (1a) and (1b) will now be parameterised.…”

A numerical algorithm for the nonlinear Timoshenko beam system

“…In Reference [8], the mixed finite element approximation was proposed by Franca and Loula. Ducceschi and Bilbao [9] present a number of finite difference time domain schemes to simulate the vibration of prestressed beams to various degrees of accuracy. Fu-le Li and Zhi-Zhong Sun [10] and Almeida Júnior [11] investigated some difference schemes.…”

“…where p = 1 or p = 2. Let us consider the value s 0 depended on the functions from initial conditions in (9) and function v(x, t)…”