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Purpose: In the work there are presented results of the synthesis and additional validation of previously developed mathematical models of two different mechanical oscillators with 1 degree of freedom and harmonic excitation: (i) with magnetically modified elasticity generating a double symmetrical minimum of potential; (ii) with linear mechanical springs and with a one-sided limiter of motion. Methods: In the first case, original mathematical models of non-linear magnetic springs were developed, allowing for effective and fast numerical simulations of the bifurcation dynamics of a real mechanical oscillator with Duffing type stiffness. In the second system, various models of impact were proposed and tested: continuous models based on the generalized Hunt–Crossley model and original discontinuous versions of this model based on the restitution coefficient and with a finite duration of the collision. In the frame of the present work, a system consisting of magnetic springs used in the first system and obstacles from the second oscillator was built and investigated. The system was built as a new configuration of a special universal stand used in the earlier studies mentioned here. Results and Conclusion: In the current study, the parameters of the models identified in previous studies on two different systems were used, the synthesis of which is the current work. A very good agreement was obtained between numerical simulations and experimental data, thus demonstrating the correctness and effectiveness of the adopted mathematical models. [ABSTRACT FROM AUTHOR]

The application of fuzzy theory is vital in all scientific disciplines. The construction of mathematical models with fuzziness is little studied in the literature. With this in mind and for a better understanding of the disease, an SEIR model of malaria transmission with fuzziness is examined in this study by extending a classicalmodel of malaria transmission. The parameters ß and d, being function of the malaria virus load, are considered fuzzy numbers. Three steady states and the reproduction number of the model are analyzed in fuzzy senses. A numerical technique is developed in a fuzzy environment to solve the studied model, which retains essential properties such as positivity and dynamic consistency. Moreover, numerical simulations are carried out to illustrate the analytical results of the developed technique. Unlike most of the classical methods in the literature, the proposed approach converges unconditionally and can be considered a reliable tool for studying malaria disease dynamics. [ABSTRACT FROM AUTHOR]

This manuscript analyses a multi-frequency based coupled piezo-electromagnetic hybrid vibration energy harvester. Most of the energy harvesting devices discussed in the literature are based on the assumption that the base excitation is harmonic. However, in general, energy harvesting devices are deployed in an ambient environment, which is random in nature. Hence, in this regard, the manuscript discusses the power harvesting capabilities of the hybrid energy harvester under random base excitation using analytical and experimental studies. Acceleration with band limited Gaussian noise has been provided as an input to the base of the harvester. Comparative studies are also made to understand the performance of the hybrid harvester over conventional standalone piezoelectric and electromagnetic harvesters under the same support excitation. Thus, the novelty of the present work is to analyse a novel design of a hybrid harvester to improve harvesting performance across a wide range of frequencies under random base excitation compared to conventional harvesters. The results demonstrate that the proposed hybrid energy harvester can yield sufficient power over a wide range of frequencies. In addition, parametric studies have also been performed to characterize the system parameters in order to enhance the performance of the harvester. Studies show that the harvested power can be enhanced by maximizing the values of mass ratio, frequency ratio, non-dimensional electromechanical coupling coefficient and minimizing the values of structural damping ratios. Finally, practical applications of the proposed device have also been reported. [Display omitted] • Hybrid energy harvester (HH) is studied to improve harvesting performance. • Simplified mathematical model is developed and validated through experimentation. • Performance of HH is compared to conventional harvesters under random excitation. • In comparison, HH widens the operating bandwidth and produces high current/voltage. • Few applications of the hybrid harvester are demonstrated. [ABSTRACT FROM AUTHOR]

This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells K (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate K of cancerous cells depend on the cells concentration. [ABSTRACT FROM AUTHOR]

It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and nonslippage assumptions for Oldroyd-B fluid with magnetized permeability. An innovative definition of Caputo-Fabrizio time fractional derivative is implemented to hypothesize the constitutive energy and momentum equations. The exact solutions of presented problem, are determined by using mathematical techniques, namely Laplace transform with slipping boundary conditions have been invoked to tackle governing equations of velocity and temperature. The Nusselt number and limiting solutions have also been persuaded to estimate the heat emission rate through physical interpretation. In order to provide the validation of the problem, the absence of retardation time parameter led the investigated solutions with good agreement in literature. Additionally, comprehensively scrutinize the dynamics of the considered problem with parametric analysis is accomplished, the graphical illustration is depicted for slipping and non-slipping solutions for temperature and velocity. A comparative studies between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. [ABSTRACT FROM AUTHOR]

In this part of the paper, the theory of nonlinear dynamics of flexible Euler-Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied. [ABSTRACT FROM AUTHOR]

A new dynamic two-dimensional friction model is developed that is based on the bristle theory. Actually, it is the Reset Integrator Model converted into a two-dimensional space. Usually, two-dimensional friction models are in fact one-dimensional models that are rotated into a slip velocity direction. However, this common approach cannot be applied to the bristle model. That is why the idea of a two-dimensional bristle is presented. The bristle's deformation is described using polar coordinates. The carried-out numerical simulation of a planar oscillator has proved that the new model correctly captures the mechanism of smoothing dry friction by dither applied in both a perpendicular and co-linear way regarding body velocity. Furthermore, the introduced mathematical model captures two-dimensional stick-slip behaviour. Cartesian slip velocity components are the only inputs to the model. In addition, our proposed model allows one to describe friction anisotropy using bristle parameters. The paper contains the results of an experimental verification of the new friction model, conducted with a special laboratory rig employed to investigate a two-dimensional motion in the presence of dither as well as to validate our numerical results. [ABSTRACT FROM AUTHOR]

In the process of modelling of the Celtic stone rotating and rolling on a plane surface, different versions of simplified models of contact forces between two contacting bodies are used. The considered contact models take into account coupled dry friction force and torque, as well as rolling resistance. They were developed using Padé approximations, their modifications, and polynomial functions. Before the use of these models, some kinds of regularisations have been made, allowing to avoid singularities in the differential equations. The models were tested both numerically and experimentally, giving some practical guesses of the most essential elements of contact modelling in the Celtic stone numerical simulations. Since the tested contact models do not require the space discretisation, they can find application in relatively fast numerical simulations of rigid bodies with friction contacts. [ABSTRACT FROM AUTHOR]

We consider a mathematical model of two-layer beams coupled by boundary conditions in a stationary temperature field taking into account geometric nonlinearity. The stationary temperature field is defined by a 2D heat transfer equation with boundary conditions of the first kind. The geometric nonlinearity is introduced via von Kármán's relations for both beams. Equations of beam deflection are derived due to the Euler–Bernoulli hypothesis. The contact interaction is described using Winkler's model. Scenarios of a transition from regular to chaotic regimes are studied. Phase synchronization of beam vibrations versus both character and intensity of the applied temperature field is investigated. [ABSTRACT FROM AUTHOR]

Mathematical modeling, theoretical/numerical analysis, and experimental verification of wear processes occurring on the contact surface of friction linings of a mechanical friction clutch are studied. In contrast to many earlier papers we take into consideration wear properties and flexibility of friction materials being in friction contact. During mathematical modeling and numerical simulations we consider a general nonlinear differential model of wear (differential wear model) and a model of wear in the integral form (integral wear model). Equations governing contact pressure and wear distributions of individual friction linings, decrease of distance between clutch shields, and friction torque transmitted by the clutch are derived and compared with experimental data. Both analytical and numerical analyses are carried out with the qualitative and quantitative theories of differential and integral equations, including the Laplace transform approach to ODEs. We show that theoretical results and numerical simulations agree with the experimental data. Finally, a numerical analysis of the proposed mathematical models was carried out in a wider range of parameters of the considered system. [ABSTRACT FROM AUTHOR]

The quadrocopter, an unmanned aerial vehicle (UAV), is a type of aerodynamic object using thrust generated by the propellers revolving around the rotor mast. The main areas of focus for complete design and engineering research should be on its stabilization and realization of tasks to be performed during flight. To accomplish these tasks, it is necessary to consider the following stages: modeling of motion dynamics, computer-aided design, programming of the control unit, precise implementation, carefully selected motors and sensors, i.e. a gyroscope, accelerometer, camera, communication module, GPS module, and others. The presented results of research and experiments carried out at Lodz University of Technology in the Department of Automation and Biomechanics take into consideration most of these issues. [ABSTRACT FROM AUTHOR]

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor's series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect. [ABSTRACT FROM AUTHOR]

In this paper, for the first time, the complete set of Tangens hyperbolicus approximations of model of dry friction coupled with rolling resistance for circular contact area between interacting bodies is proposed. The developed approximations are compared with corresponding Padé approximants of the first and second order well known from the literature and with the numerical solution of the exact integral model as well. It is shown that Tangens hyperbolicus approximants are closest to the exact solution. Then the approximated models are applied to the celtic stone dynamics, however with the significant simplifying assumption of circular contact between stone and the table, presenting differences between them again. Certain specific approximations and regularizations of the friction and rolling resistance models enabling and facilitating their application to the real problem are shown. The analysis of the response dependence on initial conditions is performed by the use of a special kind of diagram. [ABSTRACT FROM AUTHOR]

Nonlinear dynamics of a real plane and periodically forced triple pendulum is investigated experimentally and numerically. Mathematical modeling includes details, taking into account some characteristic features (for example, real characteristics of joints built by the use of roller bearings) as well as some imperfections (asymmetry of the forcing) of the real system. Parameters of the model are obtained by a combination of the estimation from experimental data and direct measurements of the system's geometric and physical parameters. A few versions of the model of resistance in the joints are tested in the identification process. Good agreement between both numerical simulation results and experimental measurements have been obtained and presented. Some novel features of our real system chaotic dynamics have also been reported, and a novel approach of the rolling bearings friction modeling is proposed, among other. [ABSTRACT FROM AUTHOR]

The paper presents an experimental and simulative study on anisotropic dry friction. Friction force components tangential and normal to sliding velocity are measured for the corrugated cardboard-felt frictional pair. Six different curves with two-fold symmetry are analysed in terms of modelling anisotropic friction. Next, the parameters of the analysed curves are identified based on the experimental results. Friction hodograph is best described by the Gielis' curve and the superellipse, whereas the ellipse seems to be the worst choice. The sliding potential is described best by the oval. It is shown that modelling of the sliding potential can be crucial in capturing the dynamics of the system and the superelliptical sliding rule can lead to qualitatively different results. • Anisotropic friction forces tangential and normal to sliding velocity are measured. • Sliding rule is modelled with an ellipse, a superellipse, a cosine curve and an oval. • Friction hodograph is best described by the Gielis' curve and the superellipse. • Best description for the sliding potential is the oval. • Free sliding and anisotropic friction-induced oscillations are analysed numerically. [ABSTRACT FROM AUTHOR]

• Experimental and mathematical models of a one-degree-of-freedom impact oscillator are presented. • A new discontinuous impact model with finite collision duration is developed. • The impact model is validated during numerical and experimental bifurcation analysis of the investigated system. In the work it is proposed a new discontinuous impact model based on the restitution coefficient and the finite duration of the collision, which are functions of the velocity before the collision. The model was validated during modelling, experimental and simulation studies of the bifurcation dynamics of an oscillator of one-degree-of-freedom with a one-sided limiter. The simulation results using the new discontinuous impact model were compared with numerical solutions obtained with the use of the continuous five-parameter Hunt-Crossley model assuming the obstacle compliance. It was shown that the new discontinuous model produces results that are as realistic and consistent with the experimental data as the compliant impact model. Moreover, the significance of the dependence of the restitution coefficient on the speed before the collision and the contact time during the collision in obtaining a good agreement of the simulation results with the experimental data were shown. [ABSTRACT FROM AUTHOR]

The paper deals with the studies of the forced impacting oscillator when taking into account the dry and viscous resistance as well as the generalized Hertz contact law during an impact. Numerical investigations of the mathematical model are accompanied by validations obtained with an experimental rig. To study the solutions of the mathematical model, the sequences of impacts, when the system evolves in periodic and chaotic modes, is used. The statistical properties of chaotic impact events are considered in detail. In particular, successive iterations of the impact map, autocorrelation function and coefficient of variation for the impact train, histograms for the inter-impact intervals and values of obstacle penetrations are analyzed. It is revealed that the impact sequence is stationary but non-Poissonian and contains temporal scales that do not relate to the external stimulus. This sequence can be described by a bimodal distribution. The findings are confirmed by the analysis of experimental data. • Mathematical model and experimental stand of a forced impacting oscillator are studied. • Impact event sequences are considered numerically and experimentally. • Statistical properties of event sequences are examined. • Agreement between properties of numerical and experimental model is shown. [ABSTRACT FROM AUTHOR]

Chaotic and regular behavior of the system consisting of a double physical pendulum with two repulsive permanent magnets is studied. We are focused on mathematical modelling, numerical simulations, experimental measurements and in particular on a novel magnetic interaction modelling. System parameters are identified by matching the output signals from experiments and numerical solutions to the developed mathematical model governed by a strongly non-linear set of two second order ODEs including the friction and the magnetic interaction torques. Considered system shows chaotic and periodic dynamics. Few chaotic zones have been detected numerically and confirmed experimentally. Scenarios of transition from regular to chaotic motion and vice versa, as well as the bifurcation diagrams are illustrated and discussed. Good agreement between the numerical simulation and experimental measurement is achieved. [ABSTRACT FROM AUTHOR]

The aim of the work is to develop and test simple mathematical models suitable for fast and realistic simulations of bifurcation dynamics of systems with a double potential well generated by a pair of repulsive magnets positioned transversely to the direction of motion, where there are motion resistances typical for rolling guides found in industry. Mathematical modelling of a harmonically forced one-degree-of-freedom oscillator with magnetically modified elasticity generating a double symmetrical minimum of potential was carried out. The system is composed of a cart moving along a linear rolling bearing with inertial excitation by means of a stepper motor with an unbalanced disc. The stiffness is realized by a pair of neodymium magnets of axes perpendicular to the direction of motion, in a position corresponding to the static equilibrium position of the oscillator, with additional mechanical linear springs, which generate a system similar to the Duffing system. Different models of the interaction force between cylindrical neodymium magnets were developed. The model parameters are estimated based on experimental data. Then the model is validated through additional experimental and numerical bifurcation analysis of the system. A very good agreement between numerical simulations and experimental data is obtained. • Experimental and mathematical models of 1DOF oscillator with magnetically modified stiffness. • Good agreement between numerical simulations and experimental data is obtained. • Bifurcation dynamics is investigated by experimental and numerical methods. [ABSTRACT FROM AUTHOR]

• Experimental and mathematical models of 2DOF oscillator with magnetic springs are presented. • Good agreement between numerical simulations and experimental data is obtained. • Regular and chaotic dynamics is investigated by numerical methods. In this paper, oscillations in the physical model consisting of two carts mounted on a guide are considered. The movement of the carts is restricted by the repelling permanent magnets. The interaction between carts is provided by a linear spring. The harmonic external loading is applied to one of the carts. According to the experimental studies, the period doubling cascade leading to the creation of a chaotic attractor occurs with an increase in the forcing frequency in the system. Chaotic attractor crises, quasi-periodic regimes, and finite periodic doubling cascade are revealed as well. To describe system's dynamics, the nonlinear mathematical model is developed. Based on the experimental data, the procedure of model validation is introduced. The numerical and qualitative analysis of the mathematical model allows one to identify bifurcation types, reveal unstable modes, and discover phenomena existing beyond the main period doubling cascade. [ABSTRACT FROM AUTHOR]