Application of a Novel Picard-Type Time-Integration Technique to the Linear and Non-Linear Dynamics of Mechanical Structures: An Exemplary Study

Applications of a novel time-integration technique to the non-linear and linear dynamics of mechanical structures are presented, using an extended Picard-type iteration. Explicit discrete-mechanics approximations are taken as starting guess for the iteration. Iteration and necessary symbolic operations need to be performed only before time-stepping procedure starts. In a previous investigation, we demonstrated computational advantages for free vibrations of a hanging pendulum. In the present paper, we first study forced non-linear vibrations of a tower-like mechanical structure, modeled by a standing pendulum with a non-linear restoring moment, due to harmonic excitation in primary parametric vertical resonance, and due to excitation recordings from a real earthquake. Our technique is realized in the symbolic computer languages Mathematica and Maple, and outcomes are successfully compared against the numerical time-integration tool NDSolve of Mathematica. For out method, substantially smaller computation times, smaller also than the real observation time, are found on a standard computer. We finally present the application to free vibrations of a hanging double pendulum. Excellent accuracy with respect to the exact solution is found for comparatively large observation periods.

Dynamic analysis of thick beams with functionally graded porous layers and viscoelastic support

This study presents dynamic responses of a composite thick beam with a functionally graded porous layer under dynamic sine pulse load. The boundary conditions of the composite beam are considered as viscoelastic supports. Three layers are considered, and face sheet layers have porous functionally graded materials in which the distribution of material gradation through the graded layer is described by the power law function, and the porosity is depicted by three different distributions (i.e., symmetric distribution, X distribution, and ◊ distribution). The layered composite thick beam is modeled as a two-dimensional plane stress problem. The equation of motion is obtained by Lagrange’s equations. In formation of the problem, the finite element method is used with a 12-node 2D plane element. In the solution process of the dynamic problem, a numerical time integration method of the Newmark method is used. In numerical analyses, influences of stiffness and damping coefficients of viscoelastic supports, material gradation index, porosity parameter, and porosity models on the dynamic response of thick functionally graded porous beam are investigated under the pulse load.

Separate Time Integration Based on the Hilber, Hughes, Taylor Scheme for Flexible Bodies With a Large Number of Modes

In the current development of flexible multibody dynamics, the efficient and accurate consideration of distributed and nonlinear forces is an active area of research. Examples are, forces due to body-body contact or due to elastohydrodynamics (EHD). This leads to many additional modes for representing the local deformations in the areas on which those forces act. Recent publications show that these can be several hundred to several thousand additional modes. A conventional, monolithic numerical time integration scheme would lead to unacceptable computing times. This paper presents a method for an efficient time integration of such systems. The core idea is to treat the equations associated with modes representing local deformations separately. Using the Newmark formulas, a fixed point iteration is proposed for these separated equations, which can always be stabilized with decreasing step size. The concluding examples underline this property, as well as the fact that the proposed method massively outperforms the conventional, monolithic time integration with increasing number of modes.

Combined Rayleigh–Taylor and Kelvin–Helmholtz instabilities on an annular liquid sheet

This paper describes the three-dimensional destabilization characteristics of an annular liquid sheet when subjected to the combined action of Rayleigh–Taylor (RT) and Kelvin–Helmholtz (KH) instability mechanisms. The stability characteristics are studied using temporal linear stability analysis and by assuming that the fluids are incompressible, immiscible and inviscid. Surface tension is also taken into account at both the interfaces. Linearized equations governing the growth of instability amplitude have been derived. These equations involve time-varying coefficients and have been analysed using two approaches – direct numerical time integration and frozen-flow approximation. From the direct numerical time integration, we show that the time-varying coefficients evolve on a slow time scale in comparison with the amplitude growth. Therefore, we justify the use of the frozen-flow approximation and derive a closed-form dispersion relation from the appropriate governing equations and boundary conditions. The effect of flow conditions and fluid properties is investigated by introducing dimensionless numbers such as Bond number ($Bo$), inner and outer Weber numbers ($We_{i}$, $We_{o}$) and inner and outer density ratios ($Q_{i}$, $Q_{o}$). We show that four instability modes are possible – Taylor, sinuous, flute and helical. It is observed that the choice of instability mode is influenced by a combination of both $Bo$ as well as $We_{i}$ and $We_{o}$. However, the instability length scale calculated from the most unstable wavenumbers is primarily a function of $Bo$. We show a regime map in the $Bo,We_{i},We_{o}$ parameter space to identify regions where the system is susceptible to three-dimensional helical modes. Finally, we show an optimal partitioning of a given total energy ($\unicode[STIX]{x1D701}$) into acceleration-induced and shear-induced instability mechanisms in order to achieve a minimum instability length scale (${\mathcal{L}}_{m}^{\ast }$). We show that it is beneficial to introduce at least 90 % of the total energy into acceleration induced RT instability mechanism. In addition, we show that when the RT mechanism is invoked to destabilize an annular liquid sheet, ${\mathcal{L}}_{m}^{\ast }\sim \unicode[STIX]{x1D701}^{-3/5}$.

Exact Time Integration for Dynamic Interaction of High-Speed Train and Railway Structure Including Derailment During an Earthquake

A robust and efficient computational method to solve the dynamic interaction of a high-speed train and railway structure including derailment during an earthquake is given. Mechanical models to express contact–impact behaviors during and after derailment are described. A modal reduction has been developed to solve nonlinear equations of motions of the train and railway structure effectively. The exact time integration in the modal coordinate is given that is free from the round-off error normally appeared in the numerical time integration for very small time increments to solve the interaction including derailment during an earthquake. Some examples are demonstrated.