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

The Purpose of the Paper. Qualitative and quantitative analysis of selected parameters of mandible movements, electronically registered in patients with temporomandibular joint dysfunction and healthy ones. Material. Function test of the mandible movements was conducted in 175 patients. Gender distribution was 143 women and 32 men, aged 9 to 84. Methods. The studied population, after accurate clinical examination, was divided into age groups with the range of five years. All the patients had Zebris JMA computerized facebow examination done, according to the generally accepted principles and procedures. Results. Mean values of mouth opening calculated to 45.6 mm in healthy group and 37.6 mm in TMJ dysfunction group. Mean length of condylar path amounted to 39 ± 7% of the maximum value of mouth opening in the group of healthy people, 44 ± 11% in the case of muscle-based disorders, and 35 ± 11% with joint-based. The mean value of the condylar path inclination oscillated in the range of 25° to 45°. Conclusions. The ratio of length of the condylar path to the size of mouth opening may be a significant value characterising the type and degree of intensification of the TMJ dysfunctions.

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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.

The present study aimed to determine the numeric projection of the function of the mandible and muscle system during mastication. An experimental study was conducted on a healthy 47 year-old subject. On clinical examination no functional disorders were observed. To evaluate the activity of mastication during muscle functioning, bread cubes and hazelnuts were selected (2 cm2 and 1.2/1.3 cm in diameter, respectively) for condyloid processing. An assessment of the activity of mastication during muscle functioning was determined on the basis of numeric calculations conducted with a novel software programme, Kinematics 3D, designed specifically for this study. The efficacy of the model was verified by ensuring the experimentally recorded trajectories were concordant with those calculated numerically. Experimental measurements of the characteristic points of the mandible trajectory were recorded six times. Using the configuration coordinates that were calculated, the dominant componential harmonics of the amplitude-frequency spectrum were identified. The average value of the dominant frequency during mastication of the bread cubes was ~1.16±0.06 Hz, whereas in the case of the hazelnut, this value was nearly two-fold higher at 1.84±0.07 Hz. The most asymmetrical action during mastication was demonstrated to be carried out by the lateral pterygoid muscles, provided that their functioning was not influenced by food consistency. The consistency of the food products had a decisive impact on the frequency of mastication and the number of cycles necessary to grind the food. Model tests on the function of the masticatory organ serve as effective tools since they provide qualitative and quantitative novel information on the functioning of the human masticatory organ.

Purpose of the Paper. This paper is an attempt to mathematically describe the mastication organ muscle functioning, taking into consideration the impact of the central nervous system. Material. To conduct model tests, three types of craniums were prepared: short, normal, and long. The necessary numeric data, required to prepare the final calculation models of different craniofacial types, were used to identify muscle and occlusion forces generated by muscles in the area of incisors and molars. The mandible in model tests was treated as a nondeformable stiff form. Methods. The formal basis for the formulated research problem was reached using the laws and principles of mechanics and control theory. The proposed method treats muscles as “black boxes,” whose properties automatically adapt to the nature of the occlusion load. The identified values of occlusion forces referred to measurements made in clinical conditions. Results. The conducted verification demonstrated a very good consistency of model and clinical tests' results. The proposed method is an alternative approach to the so far applied methods of muscle force identification. Identification of muscle forces without taking into account the impact of the nervous system does not fully reflect the conditions of mastication organ muscle functioning.

This paper presents the results of computer simulations of a gear model, where the variable stiffness of the meshing and backlash are considered. The outcome of such assumptions is a non-linear mathematical model in which chaotic phenomena can occur. During model studies, attention was paid to the identification of areas limited by the physical parameters, for which the analysed system behaved chaotically. To determine the ranges of irregular gear behaviour, numerical procedures were used to plot the bifurcation diagram, the Lyapunov exponent, the amplitude-frequency distribution and the Poincaré cross section.