In this paper, model tests were carried out, which mainly focused on the numerical mapping of the characteristics of the gear backlash. In particular, the effect of the approximation function on the value of the largest Lyapunov exponent was investigated. The generated multicoloured maps served as a criterion for verifying the results of the model tests. The analysis involved polynomial functions of the third degree, its modified structure, and the logarithmic equation. As a pattern to which the results of model tests were derived, the mathematical model of the gear was used, in which the characteristics of the backlash were modelled with a non-continuous function describing the so-called dead zone. We show that the dependencies described by polynomials imprecisely describe the dynamics of a single-stage gear transmission mechanism. Additionally, the value of the logarithmic coefficient, which approximates the backlash characteristics, for which the Poincare cross section corresponds with its model counterpart, is determined. The coefficient of the logarithmic function was optimized on the basis of
This paper presents the results of numerical simulations of a non-linear, tristable system for harvesting energy from vibrating mechanical devices. Detailed model tests were carried out in relation to the system consisting of a beam and three permanent magnets. Based on the derived mathematical model and assuming a range of control parameter variability, a three-dimensional image of the distribution of the largest Lyapunov exponent was plotted. On its basis, the regions of chaotic and predictable movement of the considered system exist have been established. With reference to selected plane of the largest Lyapunov exponent cross-sections, possible co-existing solutions were identified. To identify multiple solutions, a diagram of solutions (DS) diagram was used to illustrate the number of existing solutions and their periodicity. The proposed calculation tool is based on the so-called fixed points of Poincaré cross-section. In relation to selected values of the control parameter w, coexisting periodic solutions were identified for which phase trajectories and basins of attraction were presented. Based on the model tests carried out, it was found that in order to efficiently harvest energy, appropriate transducer adjustment is required. Calibration of the transducer is necessary to obtain the greatest amplitude of vibration of the beam, which corresponds to the phase trajectory limited by external energy potential barriers. As expected, the average voltage induced on the electrodes of the piezoelectric transducer and the average electrical power recorded on the resistive element are directly proportional to the amplitude and average kinetic energy of the beam.
The aim of this work is to model the dynamics of flexible couplings. On the basis of a non-linear mathematical model solved by bond graph, the ranges of excitation frequency were determined, in which the movement of the couplings is chaotic. For three couplings, the 3D distributions of the largest Lyapunov exponent and correlation dimension diagram (CDD) were plotted. The proposed diagram (CDD) illustrates how the geometric structure of the attractor changes when the conditions of excitation change. The classic Poincare cross-section, completed by us with the density of points distribution, significantly enhances information about geometrical structures of strange attractors. It has been shown that in relation to large ranges of changes in the control parameter, the geometric structure of the strange attractor is stretched and curved. The areas with the highest densification of the Poincaré cross section are most often located in places where the chaotic attractor is curved. Untersuchung der nichtlinearen Dynamik flexibler Kopplungen mit Hilfe von Bondgraphen ZusammenfassungZiel dieser Arbeit ist es, die Dynamik von elastischen Kupplungen zu modellieren. Anhand eines durch einen Bindungsgraphen gelösten nichtlinearen mathematischen Modells wurden die Bereiche der Anregungsfrequenz bestimmt, in denen die Bewegung der Kopplungen chaotisch ist. Für drei Kopplungen wurden die 3D-Verteilungen des größten Lyapunov-Exponenten und das Korrelationsdimensionsdiagramm (CDD) aufgezeichnet. Das vorgeschlagene Diagramm (CDD) zeigt, wie sich die geometrische Struktur des Attraktors ändert, wenn sich die Anregungsbedingungen ändern. Der von uns vervollständigte klassische Poincaré-Querschnitt mit der Dichte der Punkteverteilung verbessert die Informationen über geometrische Strukturen fremder Attraktoren erheblich. Es hat sich gezeigt, dass in Bezug auf große Änderungsbereiche der Steuerparameter die geometrische Struktur des seltsamen Attraktors gedehnt und gebogen wird. Die Bereiche mit der höchsten Verdichtung des Poincaré-Querschnitts befinden sich am häufigsten an Stellen, an denen der chaotische Attraktor gekrümmt ist.
Summary. The article presents the results of numerical analysis whose aim was to compare the basic dynamic and strength parameters of lightweight load-carrying crane structures made of aluminium alloys and steel. The analysis covered the typical construction of workshop cranes with a span of 3 to 5 meters, girders in the form of an Ibeam and maximum load capacities amounting to 5 tons. The values of stresses, deflections and natural frequencies were compared and then matched with the masses of the various structures. In the simulation a girder model was used and computed by the finite element method. BADANIA MODELOWE WYKORZYSTANIA STOPÓW ALUMINIUM W USTROJACH NOŚNYCH LEKKICH SUWNICStreszczenie. W artykule przedstawiono wyniki analizy numerycznej, której celem było porównanie podstawowych parametrów dynamicznych i wytrzymałościowych ustrojów nośnych lekkich suwnic wykonanych z aluminium i stali. Analizie poddano typowe konstrukcje suwnic warsztatowych o rozpiętości od 3 do 5 metrów, o dźwigarach w postaci dwuteownika i maksymalnych udźwigach wynoszących 5 ton. Porównywano wartości naprężeń, ugięć i częstotliwości drgań własnych, a następnie zestawiono je z masami poszczególnych ustrojów. W symulacji wykorzystano model dźwigara obliczany z wykorzystaniem metody elementów skończonych.
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