The book is written for students of universities and postgraduate students specializing in the field of nonlinear dynamics, as well as specialists in various fields of mechanical engineering. It is devoted to the development of original methods, and outlines the results of analytical studies of dynamical chaos, synchronization, and dynamic structures in lattices of coupled rotators. It presents its findings within the context of the phase space of models and by involving methods of the qualitative theory of differential equations, the theory of bifurcations, and qualitative numerical methods.@en

Consider a model shown in Fig. 3.1. An asynchronous motor is coupled to the elastic shaft with an unbalanced disk (rotator) and balancing sleeve (oscillator) elastically coupled to the disk that can oscillate along the shaft.

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This paper outlines the results of an experimental study of the influence of high hydrostatic pressure on the abrasive wear of hard-alloy materials based on tungsten carbide (~90% WC ± 10% Co), as well as alloys based on iron with high contents of chromium. A specially developed setup has been described in the paper that makes it possible to test materials under the hydrostatic pressure of up to 250 MPa at different friction speeds. An investigation of the surfaces of samples using the Scanning Electron Microscopy method has revealed that the main damage of alloy surface occurs due to the delamination and spalling of hard particles. It has been revealed that the hydrostatic pressure significantly influences the wear rate of the investigated materials. When the pressure increases to 200 MPa, the wear of materials with high contents of chromium increases seven times, while for the material based on tungsten carbide, it increases twice.

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This paper outlines the results of experimental study of the dynamic rock failure based on the comparison of dry and saturated limestone samples obtained during the dynamic compression and split tests. The tests were performed using the Kolsky method and its modifications for dynamic splitting. The mechanical data (e.g. strength, time and energy characteristics) of this material at high strain rates are obtained. It is shown that these characteristics are sensitive to the strain rate. A unified interpretation of these rate effects, based on the structural–temporal approach, is hereby presented. It is demonstrated that the temporal dependence of the dynamic compressive and split tensile strengths of dry and saturated limestone samples can be predicted by the incubation time criterion. Previously discovered possibilities to optimize (minimize) the energy input for the failure process is discussed in connection with industrial rock failure processes. It is shown that the optimal energy input value associated with critical load, which is required to initialize failure in the rock media, strongly depends on the incubation time and the impact duration. The optimal load shapes, which minimize the momentum for a single failure impact, are demonstrated. Through this investigation, a possible approach to reduce the specific energy required for rock cutting by means of high-frequency vibrations is also discussed.@en

This paper outlines results of the experimental study of abrasive wear of materials at hyperbaric pressures. Two materials have been treated: steel and tungsten carbide. The tests have been performed in a custom made pressure vessel. SEM microscopy has been used to reveal the physical mechanisms of wear. Obtained results might be interesting for offshore mining and oil & gas industries.

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Successful mining of deep sea deposits strongly depends on the proper choice of the right equipment. The most probable concept for a deep sea mining system would consist of the three major sub-components: Seafloor Mining Tool, Vertical Transport System and Mining Support Vessel. In this paper, emphasis is placed on the Vertical Transport System. We analyse the pros and cons of the different concepts such as hydraulic transport using centrifugal or positive-displacement slurry pumps, conventional and unconventional airlift systems, vertical offshore mining systems and vortex slurry transportation systems. All these systems are considered for their applicability at different water depths (from the relatively shallow to the relatively deep) for the different types of materials (from the relatively fine to the relatively coarse) and various production rates in terms of the efficiency, reliability and state of the art of technology.@en

Extractable deposits of silver, gold, copper, zinc, lead, gas hydrates and other valuable materials can be found at the ocean floor. The most valuable minerals are found at large depths, starting from 1000 m and deeper. Therefore, the leading offshore companies are currently designing systems and tools for deep sea mining. The mining at the desired depth of a few kilometers is a great challenge though as it has never been attempted before at the commercial scale. One of the fundamental design challenges lays in the understanding of and preventing from the problems associated with the dynamics of a subsea Vertical Transport System (VTS). The function of the VTS is to transport slurry (a thin mixture of water and finely divided minerals) from the seafloor to the mining support vessel. The VTS consists of a vertically hanging submerged pipe through which the slurry is transported upwards and a number of booster stations which maintain the pressure in the pipe that enables the desired slurry flow. The VTS system is subject to a number of the dynamic excitations such as the vessel motion, the slurry flow in the pipe, the sea current and a propulsion device that is envisaged to control the position of the lower end of the pipe at the desired location. To design a reliable VTS system the effect of all the above-mentioned excitation mechanisms has to be accounted for.@en

A mass moving uniformly along a Bernoulli-Euler beam with a periodically nonuniform cross section lying on an elastic foundation has been examined. It is shown analytically that a parametric instability takes place in the system. The instable zones for different values of nonuniformity have been obtained.

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This paper studies the conditions for existence and stability of stationary cluster structures in lattices of diffusively coupled dynamical systems within the framework of a new interpretation of cluster synchronization as classical synchronization of cluster oscillators (C-oscillators). The study of existence of cluster attractors is based on the linear chains of cluster oscillators, defining possible types of cluster structures in chains. First, we present interval estimates for the range of coupling strengths in which cluster attractors can exist. Then we formulate and prove the basic theorems about the local stability of the various cluster structures. The presented methodology can be extended to study cluster structures on lattices of different geometry and forms such as linear cluster structures in two-dimensional lattices, layered cluster structures in three-dimensional lattices and cluster structures in ring-shaped systems. Crown@en

A "flexible-rotor/limited-power-excitation-source" dynamical system, which is a model of the vibrations of shafts driven by a motor of limited power is considered. The types of rotation characteristics of the motor and resonance characteristics of the shaft in the resonance parameter zone for different values of the viscosity coefficient of the medium are investigated. These characteristics completely describe the dynamics of the system when its parameters vary.@en

This study discusses the development of a technique for analysis of the dynamical regimes of complex mechanical systems consisting of a rotor motor coupled to a system with multi-degrees-of-freedom. To understand the possible qualitatively different dynamical regimes in such systems, a simple mechanical system is considered of the "rotator-oscillator" type with a finite power source. This system has four degrees-of-freedom and is defined in four-dimensional cylindrical phase space with 12 parameters. Near the main resonance the original system is reduced to the Lorenz system with four parameters defined in a three-dimensional Cartesian phase space. This is done with the help of a special change of variables, parameters, and employing an averaging method. Studying the latter system, the existence of one of the chaotic attractors, namely of Lorenz attractor is established. Also established is the Feigenbaum attractor and the alternation. Chaotic limit sets define chaotic behavior of the instantaneous frequency of rotation of the asynchronous motor. The Poincare mappings are presented to show the correspondence of the original 4 dof and averaged 3 dof systems. The qualitative rotational characteristics for different values of the system parameters are obtained. In particular, the system can possess normal Sommerfeld effect, doubled Sommerfeld effect and a so-called scattering of the torque curve. The scattering of the torque curve (which is a known effect in micro-electronics) is likely to be a new effect in mechanics. In contrast to the Sommerfeld effect, when frequency or amplitude jumps occur instantaneously (once the unstable point of the characteristic is reached), the jump to a next stable point may take a certain time, even infinite one. Such chaotic mistuning of the motor frequency would result in random vibrations leading to system wear and damage.

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The effect of the dead weight on stability of a vehicle moving along a periodically inhomogeneous elastic guide has been studied. The account of the dead weight has been shown not to affect the system stability but to lead to a linear resonance.@en