Vol. 6, No. 4 (2017): Regular Issue

This is the Special Issue dedicated to the 13th International Conference on "Dynamical Systems - Theory and Applications" (DSTA-2015) which was held on December 7-10, 2015 in Łódź Poland. The main aim of the conference was to provide a platform for researchers and engineers to present and discuss the current state and contemporary investigations in different disciplines of science, bioscience, and engineering. This issue presents eight selected manuscripts related to the modeling, analysis and control of nonlinear discrete and continuous mechanical structures met in mechatronic applications.

In this report we study the mathematical model for mutually penetrating continua. This model consists of the wave equation describing the carrying medium and the equation for oscillators forming the oscillating inclusion. Prescribing the constitutive equation of the carrying medium and kinetics of oscillator’s dynamics for model in question, the cubic nonlinearity is accounted. We are interested in the structure of wave solutions obeying the dynamical system of Hamiltonian type. This allows us to determine the peculiarities of the phase space of dynamical system, namely, the relation describing the homoclinic trajectories, the division of phase plane into the parts with equivalent orbits’ behavior, the conditions of bifurcations. To simulate the wave dynamics, we construct the three level finite-difference numerical scheme and study the evolution of solitary waves, their pair interactions and stability. Propagation of periodic waves is modeled as well.

The paper presents a model-based tracking controller design for a freefloating space robot for a mission scenario of intercepting an object. Such missions are of interest due to a growing number of objects needed to be removed from space. The free-floating mode requires spacecraft thrusters to be off and linear and angular momentum are conserved then. Momentum conservation generates holonomic and nonholonomic constraint equations, respectively. The free-floating mode implicates underactuation, so the robot becomes multi-constrained. Many control algorithms are designed for underactuated robots but they are specific mission and robot dependent. Motivations for the presented research come from the growing space exploration, which results in more space debris and requires sophisticated removal services. Service tasks and debris removal need to be performed by specialized robots. The debris interception scenario presented in the paper consists of estimation of target properties, a controller design to track and intercept the debris, and move it to the graveyard orbit. Simulation results of the theoretical control development for the robot intercepting a non-tumbling object are provided.

This paper examines the implementation of a standalone 1 MW Wave Energy Converter (WEC), based on a novel concept of a class of WECs, consisting in fully enclosed appropriate internal body configurations, which provide inertial reaction against the motion of an external vessel. Acting under the excitation of the waves, the external vessel is subjected to a simultaneous surge and pitch motion in all directions, ensuring maximum wave energy capture. The internal body is suspended from the external vessel body in such an appropriate geometrical configuration, that a symmetric four bar mechanism is essentially formed. The first advantage of this suspension geometry is that a linear trajectory results for the center of the mass of the suspended body with respect to the external vessel, enabling the introduction of a quite simple form of a Power Take-Off (PTO) design. The simplicity and symmetry of the suspension geometry and of the PTO, ensure a quite simple and robust technological implementation. Mass and inertia distribution of the internal body are optimized, ensuring maximal conversion and storage of wave energy. As a result, the internal body assembly is essentially, dynamically equivalent to a vertical physical pendulum. However, the resulting equivalent pendulum length and inertia can far exceed those that can be achieved by a simple horizontal or vertical pendulum, suspended or inverted, leading to a significant reduction of the suspended mass.

Zona Pelucida (ZP) is highly specialized glycosylated and sulfated polymer gel that surrounds mammalian oocyte, exhibits elastic or viscoelastic properties and change of diameter in different maturation stages. In process of in vitro fertilization, sperm cells affect the external ZP surface and transfer certain amount of energy to the ZP structure that goes through the deformation work. ZP thickness and number of motile sperm cells are important factors that may influence the fertility. Using the quasi-static approximate ZP model in the form of hollow sphere the numerical analysis of how specific deformation work depends of different variables like: ZP thickness, specific point in ZP, external pressure that comes from sperm cells were done. According to the model, sperm that make pressure upon ZP surface are in the form of homogeny discrete continuum distribution in radial directions. For specific ZP point in the model, analytical expressions of component stresses and strains were obtained as well as deformations and volume dilatation in ZP. The results are discussed from biomechanical aspect of fertilization. Limitations of the model are also discussed.

The current work is devoted to the problem of fatigue cracks detection and evaluation in the case of isotropic rectangular plates with circular holes. The cutout is located in the geometrical center of the plate. The structure is subjected to the cyclic tension. It causes the formation of the fatigue cracks in the vicinity of the hole. This type of damages can be effectively detected by the analyzing of elastic waves propagation. In the presented work the propagation of the elastic waves in the vicinity of a hole is simulated with the use of the finite element method. It is assumed that the elastic waves are excited and picked up by piezoelectric sensors. The most effective position of the piezoelectric actuator is looked for while the position of the array of sensors is fixed. Four different possibilities of the location of the actuator are studied. Moreover, the advanced algorithm of crack detection and evaluation is also discussed.

The topic of this study is the first mode of natural transverse vibrations of isotropic, homogeneous and elastic bars (columns or beams) with clamped bottom and free head. The columns of the first group are shaped as truncated solid cones or as tubes with linearly variable wall thickness and with different inclination of lateral faces, from cylinder to cone. The columns of the second group were shaped in similar way, but the generatrices of the solids of revolution were curvilinear ?in the shape of a parabola. The first frequency of free vibrations was determined using the Rayleigh method. The deflection line of the column axis during the vibration was assumed in form of the bending line of the column axis subjected to a uniform load. Resulting frequencies (or periods) were compared with these obtained with the use of FEM (ANSYS) and a good compliance of results was observed. As the expression for the energy of an elementary slice of material was integrated over the length of the rod, the formula for the frequency was obtained in form of an integral equation. In some cases an exact solution of integral equation was obtained, however in other cases only a numerical solution was possible.

In the paper, the known problem of vibration control have been studied for elements which have defined configurations and connections of piezostack actuator with external electric network and can exhibits negative values. Depending on the phase of the synthesis process, used for the design of the mechatronic systems that have to comply with some given requirements, negative elements have been identified and described. Subsequently with the study on selection of corresponding negative stiffness and damping, presented in the graph form, optimal values for the systems can be compared. Following examples with selected calculations, the goal of this paper is to present limits and constrains that may support the physical realization, as well as applications of the considered systems.

The paper deals with searching for the phenomenological model of a fibre. A multibody model of a simple weight-fibre-pulley-drive mechanical system is created. This mechanical system is focused on the investigation of a fibre behaviour and experimental laboratory measurements are performed on it. The carbon fibre, which is driven by one drive, is led over a pulley. On its other end there is a prism-shaped steel weight, which moves in a prismatic linkage on an inclined plane. Drive excitation signals can be of different shapes with the possibility of variation of a signal rate. Time histories of the weight position and of the force acting in the fibre are measured. At simulating with the multibody model of this system there is investigated the influence of dependences of the fibre spring-damper coefficients on the velocity of the weight motion in the computational model, on the coincidence of the simulation results and the experimental measurement results are evaluated. The aim of the simulations is to create a phenomenological model of the fibre, which will be utilizable in fibre modelling in the case of more complicated mechanical or mechatronic systems.