Improvement of Discrete-Mechanics-Type Time Integration Schemes by Utilizing Balance Relations in Integral form Together with Picard-Type Iterations

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature.

On Nonlinear Motions of Two-Degree-of-Freedom Nonlinear Systems with Repeated Linearized Natural Frequencies

In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes’ rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms.

Integration of Advection and Mixing Using Fluid Mass-Momentum-Covering Relationships

Classical discretization of discrete fluids and parameterization of mixing bring modeling uncertainties for momentumuneven fluids by setting the grid spacing as the distance for fluids to move within one time step, using low resolutions, and introducing dozens of parameters dependent of scales and modeling processes. Low resolutions have linearized, in a rather extent, nonlinear motions and failed to forecast accelerations that can cause climate adjustments, especially during solving tide-associated dynamics. Here, I wrote Newton Law directly for discrete fluids under conservations of momentum and mass using “mass and momentum-covering relationships (MMCR)”, and combined advection and mixing terms. Mixing became a higher-order advection terms induced from speed gradients times coefficients that are functions of density, grid spacing, and the gradients of speed, density and momentum. Parameterization can be avoided and no uncertain parameters are involved in modeling. Preliminary modeling experiments showed skill for simulation of sea surface temperatures over tropical oceans where momentum was more uneven than as compared with higher latitudes.

Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order

This paper deals with the methods for investigating the nonlinear dynamics of discrete chaotic systems (DCS) applied to piecewise linear systems of the third order. The paper proposes an approach to the analysis of the systems under research and their improvement. Thus, effective and mathematically sound methods for the analysis of nonlinear motions in the models under consideration are proposed. It makes it possible to obtain simple calculated relations for determining the basic dynamic characteristics of systems. Based on these methods, the authors developed algorithms for calculating the dynamic characteristics of discrete systems, i.e. areas of the existence of steady-state motion, areas of stability, capture band, and parameters of transients. By virtue of the developed methods and algorithms, the dynamic modes of several models of discrete phase synchronization systems can be analyzed. They are as follows: Pulsed and digital different orders, dual-ring systems of various types, including combined ones, and systems with cyclic interruption of auto-tuning. The efficiency of various devices for information processing, generation and stabilization could be increased by using the mentioned discrete synchronization systems on the grounds of the results of the analysis. We are now developing original software for analyzing the dynamic characteristics of various classes of discrete phase synchronization systems, based on the developed methods and algorithms.

Oscillating Wave Surge Converter Forced Oscillation Tests

Hydrodynamic numerical models of Wave Energy Converters (WEC) contain hydrodynamic coefficients that are commonly obtained from numerical codes that solve linear potential flow problems using Boundary Element Methods (BEM codes). The assumptions made by the BEM codes in their calculation of the linear hydrodynamic coefficients are violated by the large and nonlinear motions that wave activation body class WECs often go through during operation. In this study, Forced Oscillation Tests were used to evaluate the hydrodynamic torque coefficients estimated for an Oscillating Wave Surge Converter (OWSC) WEC by two BEM codes; WAMIT and Nemoh. The paper describes the tests and the active Force Feedback Dynamometer test rig used to perform them. The results indicate good agreement between the BEM codes and experimental data for small angular displacement amplitude oscillations, as expected; up to 0.3 rad. The torque not predicted by the BEM codes is presented and shown to have an amplitude and phase that vary throughout the range of tests performed.