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The work is devoted to the study of vibrations and buckling of a graphene sheet (GS) with the shape of a right triangle, and embedded in a nonlinear polymer matrix. The research is based on the nonlocal elasticity theory and the application of the Bubnov‐Galerkin method. Due to the influence of the polymer surrounding, the study of vibrations is reduced to the analysis of a nonlinear cubic ordinary differential equation considering small‐scale effects. The nonlocal buckling problem for the triangular GS under in‐plane uniform compression is investigated including the finding of critical buckling load and the post‐buckling behaviour determined by the nonlinear medium effect. [ABSTRACT FROM AUTHOR]

In this paper, oscillations of the electrically actuated microbeam are considered, described by strongly nonlinear differential equations. One of the essentially nonlinear effects is the pull‐in phenomenon, that is, the transition of the oscillatory regime to the attraction regime. This effect is considered in the paper on the basis of various approximate models. In particular, the error of replacing the original nonlinearity by its expansion in the Maclaurin series is estimated. It is shown that such an approximation can only be used for voltage values that are far from critical. Estimates are also given for initial displacements and velocities, which guarantee an oscillating character of solutions. The electrically activated oscillations of the microbeam are considered taking into account the geometric nonlinearity within the framework of the Kirchhoff model. The dependence of pull‐in value on geometric nonlinearity is investigated. [ABSTRACT FROM AUTHOR]

Free vibrations of the orthotropic micro/nanoplate with nonclassical shape are investigated. The considered model is based on the nonlocal elasticity theory. The developed method uses the Ritz method as well as R-function theory for the construction of the system of coordinate functions. The linear frequencies are obtained for a rectangular plate with two cutouts on opposite sides, while the boundary conditions are considered of several types, including simply supported and clamped edges. The small-scale effects for various sizes of cutouts are discussed. [ABSTRACT FROM AUTHOR]

Paper is devoted to the effective stationary heat conductivity for the fibre composite materials. We are aimed on getting on analytical expression for effective thermal conductivity coefficient. Asymptotic homogenization approach, based on the multiple scale perturbation method, is used. This allows to reduce the original boundary value problem in multiply connected domain to the sequence of boundary value problems in simply connected domains. These problems include: the local problem for the periodically repeated cell and homogenized problem with effective coefficient. It is shown that for densely packed and high contrast fibre composites, the cell problem can be solved analytically. For this aim, lubrication approach (asymptotics of thin layer) has been employed. We also generalise obtained solution to the case of medium‐sized inclusions in the framework of the Padé approximants. [ABSTRACT FROM AUTHOR]

In the present article, we prove some new generalizations of dynamic inequalities of Hardy-type by utilizing diamond- α dynamic integrals on time scales. Furthermore, new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are proved. Moreover, we present Hardy inequalities for several functions by using the diamond- α dynamic integrals on time scales. The results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales. Our main results extend existing results of the integral and discrete Hardy-type inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

We use two parameters for functions whose second (q, ω)-derivatives are bounded in order to prove several recent extensions of the Ostrowski inequality and its companion inequalities on (q, ω)-Hahn difference operator. Furthermore, we procure some q-integral and continuous inequalities as special cases of the main results as well these generalizations. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

The solitary wave solutions of the (2 + 1) -dimensional nonlinear Chiral Schrödinger ((2 + 1) -D CNLS) equation by making use of the modern extended direct algebraic approach are investigated in this paper. Such a nonlinear model's different categories of solitary waves solutions are found. The division of semi-dark solitons, singular dark-pitch solitons, single solitons of type 1 along with 2, intermixed hyperbolically, trigonometrically and rational solitons are developed and evaluated using software tools for numerical calculations. Detailed graphical interpretation of results with deliberation is also discussed. This paper effectively depicts 3D- and 2D-contour plots to provide further detail about the physical characteristics of the obtained solutions. We compare our solutions to previously reported findings along with study motivation and future directions. The fractional quantum hall effect's edge states are defined by the discussed model. [ABSTRACT FROM AUTHOR]

This research focuses on the nonlinear vibration control of a self-excited single-degree-of-freedom system. The integral resonant controller (IRC) is introduced to stabilize the unstable motion and suppress nonlinear oscillations of the considered system. The nonlinear dynamical equations that govern the vibratory behaviors of the proposed closed-loop control system are investigated using perturbation analysis, where loop delays have been included in the studied model. The system bifurcation behaviors have been visualized in both the two and three-dimensional spaces, and corresponding dynamical behaviors have been explored numerically using the bifurcation diagrams, Poincaré map, time-response, zero-one chaotic test algorithm, and frequency spectrum. The obtained analytical investigations revealed that the uncontrolled system can oscillate with one of four vibration modes depending on the excitation frequency, which are mono-stable periodic motion, bi-stable periodic motion, periodic and quasi-period motion, and quasi-periodic motion only. In addition, it is found that the existence of time delays in the control loop can either improve or degrade the control performance. Therefore, an objective function has been introduced to design the optimum control parameters. Based on the derived objective function, it is found that the performance of the proposed control strategy is proportional to the product of the control and feedback gains and inversely proportional to the internal loop feedback gain when the loop delays are neglected. Moreover, it is reported that the controller performance is a periodic function of the total sum of the loop delays. Accordingly, the optimal operating conditions of the time-delayed integral resonant controller have been explained. Finally, numerical validations for all obtained analytical results have been performed, where an excellent correspondence between the analytical and numerical investigations has been demonstrated. [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 work investigates a novel design of an electromagnetic vibration energy harvester array based on pendulum structures under harmonic base excitation. The design includes combined magnetic-solenoid pendulum harvesters with different mechanical coupling configurations. A comparison is made with standalone magnetic and solenoid pendulum based harvester arrays under similar excitation conditions. Additionally, the positions of harvesters are shuffled and analyzed. The effects on harvested power, power density and frequency bandwidth are evaluated by considering dimensionless coupling ratios, damping ratios, mass ratios and length ratios as parameters. Further, the efficiency of the harvester against these parameters has also been reported. In addition, experimental validation of the numerical formulation is also conducted. The study shows that the proposed magnetic-solenoid harvester array can harvest sustained energy in a wider bandwidth compared to standalone magnetic and solenoid pendulum harvester arrays. [ABSTRACT FROM AUTHOR]

This work investigates a parametrically coupled electromechanical system of van der Pol and Duffing oscillators. Such a model finds applications in the development of sensors and transducers for the purpose of measurements and actuation. First, the model was developed from the basic components of electrical and mechanical systems, and the governing equations were derived from an energy-based principle. The system was solved analytically by multiple scales method to validate the numerical solutions. Its stability was examined by Floquet theory, and further analysis reveals the presence of Hopf and Neimark–Sacker bifurcations. Other phenomena, such as period doubling and quasiperiodic routes to chaos, were discovered. Additionally, the system exhibits multistability due to the presence of coexisting attractors and synchronization at parametric resonances. Finally, the chaotic system was stabilized by designing a nonlinear feedback controller with time-varying coefficients, bringing its trajectories to the desired equilibrium at the origin for vibration suppression and the desired limit cycle for periodic solutions, respectively. The controlled outputs show a stable invariant closed curve with its Floquet multipliers falling within a unit circle. • Dynamic analysis of the system shows Hopf and Neimark–Sacker bifurcations. • The coupled system exhibits multistability, synchronization, and chaos. • The synchronization is caused by both self- and parametric excitations. • The nonlinear feedback control stabilized the chaotic system to desired states. [ABSTRACT FROM AUTHOR]

In this paper a new approach is taken to find the exact solutions for generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate and is analyzed for ramped wall temperature and velocity with constant concentration. The vertical plate is suspended in a porous medium and encounters radiation effects. Solutions based on special functions are obtained using an integral transform for an unsteady MHD Maxwell fluid in the presence of ramped velocity, temperature and constant concentration. The relations for Nusselt number and skin-friction coefficient are efficiently computed to precisely estimate the rate of heat transfer at the boundary and the shear stress. Results are also discussed in detail and demonstrated graphically using software to comprehensively analyze the dynamics of the proposed problem, and the physical impact of several system parameters, such as magnetic field M, Prandtl number Pr, the relaxation time λ, dimensionless time τ, Schmidt number Sc, Mass and Thermal Grashof numbers Gm and Gr, respectively, is studied. Furthermore, solutions for some recently published work are compared with the current study that endorses the authenticity of our derived results and proves that those investigations are limiting or special cases of the current problem. [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]

This is an introductory paper of the Focus Issue Recent advances in modeling complex systems: Theory and applications, where papers presenting new advances and insights into chaotic dynamics, fractional dynamics, complex oscillations, complex traffic dynamics, and complex networks, as well as their applications, are collected. All these different problems share common ideas and methods and provide new perspectives for further progress in the modeling of complex systems. [ABSTRACT FROM AUTHOR]

The free vibration of plates and shallow shells with/without cutouts made of functionally graded materials (FGM) is investigated using variational FG shallow shells with temperature dependent mechanical characteristics of the constituent materials. First‐order shear deformation theory of shallow shells is employed. It is supposed that material properties vary through thickness according to a power‐law distribution of the constituent's volume fraction. They depend on both the temperature and the thickness. Temperature field is modeled by one‐dimensional heat transfer equation, since the temperature is varied only in thickness direction. Solution of this equation is determined by a polynomial power series expansion. Corresponding software was developed to implement the proposed approach. The vibration analysis was carried out for FG plates and shallow shells with cutout and various boundary conditions. [ABSTRACT FROM AUTHOR]

• Estimation of the regions of attraction using the quadratic Lyapunov functions is reported. • Recursive procedure for constructing a polynomial Lyapunov function is developed and described. • Effectiveness of the proposed approach is shown using various examples including systems with uncertain parameters. This paper deals with the problem of obtaining estimates of the domains of attraction and stability in nonlinear dynamical systems. It is based on a special recursive procedure for constructing polynomial Lyapunov functions. The procedure is tested on several examples used in previous studies by other authors, with a comparison of the results. The approach is also applied to estimate the domain of attraction for single DoF mechanical systems with uncertain parameters (Van der Pol and Duffing oscillators). [ABSTRACT FROM AUTHOR]

Our aim in this article is to solve the composite fractional relaxation differential equation by using different definitions of the non-integer order derivative operator D t α , more specifically we employ the definitions of Caputo, Caputo–Fabrizio and Atangana–Baleanu of non-integer order derivative operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley's generalised G function. Furthermore, the effects of the parameters involved in the model are graphically highlighted. [ABSTRACT FROM AUTHOR]

In this work the chaotic dynamics of flexible curvilinear Euler–Bernoulli micro‐beams embedded into a stationary temperature field is investigated. The temperature field is modelled based on a the Duhamel–Neumann theory and is free from the restrictions on the temperature field distribution along beam thickness. The von Kármán geometric strain–stress relations are employed. The governing nonlinear PDEs are yielded by the Hamilton principle with an account of the modified couple stress theory. The finite dimension problem is truncated to a finite system of nonlinear ODEs using the finite difference method (FDM) and then the Cauchy problem is solved with a help of the Runge–Kutta method. Action of the 2D thermal field is defined by solution to the heat transfer PDE which is also solved by FDM of the second order of accuracy. The so‐called charts of vibration regimes (amplitude‐frequency planes) are constructed. In particular, novel features of nonlinear (chaotic) dynamics versus the change of the magnitude of the size (length) dependent parameter are reported. The carried out numerical analysis is supported by the monitoring of frequency power spectra based on the fast Fourier transform (FFT), phase space projections, Poincaré maps and LLEs (largest Lyapunov exponents). We have also analyzed system chaotic dynamics of the classical and size‐dependent beam models versus series of values of the two control parameters, i.e. beam curvature and its size‐dependent length. Moreover, we have detected and illustrated the novel scenarios of transition form regular to chaotic vibrations of the studied beams, governed by non‐linear PDEs. [ABSTRACT FROM AUTHOR]

Nonlinear vibrations of double-layer nanoplate system are studied by numerical simulation. The governing equations are derived on a base of the nonlocal elasticity theory, von Kármán nonlinear theory, and Kirchhoff's hypotheses. Both layers are assumed to be with the same material properties, while interaction between them is taken by the Winkler model. Discretization of the system of governing differential equations is performed by the Bubnov-Galerkin method with the two-mode approximation model, and the obtained system of ordinary differential equations is analysed by the Runge–Kutta method. The study of a linearized system allows the construction of regions of instability. The application of nonlinear dynamics methods made it possible to determine the nature of oscillations when stability is lost, and when small-scale effects are manifested. • Nonlinear vibrations of double-layer graphene system under in-plane load are studied; • the nonlocal governing system based on von Kármán nonlinear relations is applied; • the proposed approach uses the Bubnov-Galerkin method with the two-mode model; • the Floquet theory is used to define instability zones; • it is concluded that small-scale effects on the stability zones are not significant; • small-scale effects can influence the nature of vibrations after loss of stability. [ABSTRACT FROM AUTHOR]

The effective properties of the fiber-reinforced composite materials with fibers of circular cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For analytical solution of the periodically repeated cell problem, the Schwarz alternating process is employed. The principal term of the refined formula coincides with the classical Maxwell formula. On the other hand, the refined formula can be used far beyond the area of applicability of the Maxwell formula. It can be used for dilute and non-dilute composites. It is confirmed by comparison with known numerical and asymptotic results. [ABSTRACT FROM AUTHOR]

The effective properties of the fiber-reinforced composite materials with fibers of circle cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For an analytical solution of the periodically repeated cell problem the Schwarz alternating process (SAP) was employed. Convergence of this method was proved by S. Mikhlin, S. Sobolev, V. Mityushev. Unfortunately, the rate of the convergence is often slow, especially for nondilute high-contrast composite materials. For improving this drawback we used Padé approximations for various forms of SAP solutions with the following additive matching of obtained expressions. As a result, the solutions in our paper are obtained in a fairly simple and convenient form. They can be used even for a volume fraction of inclusion very near the physically possible maximum value as well as for high-contrast composite constituents. The results are confirmed by comparison with known numerical and asymptotic results. [ABSTRACT FROM AUTHOR]

Biological models are usually described using difference equations. As a result, we are - in this work - interested in studying a general difference model which includes two biological models as special cases. In detail, we study the qualitative behaviors (local and global stability, boundedness and periodicity character) of a general difference model. Furthermore, we apply our general results to the population model with two age classes and the flour beetle model. [ABSTRACT FROM AUTHOR]

The paper presents a new method for the identification of joint friction, reductor self-locking and gear backlash parameters. The method was developed using an existing hexapod robot. During the process, the motor driving currents given by the model and measured on the real device were observed and compared. Initially, the current curves showed specific differences. After the analysis of these deviations, their causes were identified as joint friction, reductor self-locking and gear backlash. The parameters of these phenomena were determined using swarm optimization. As a result of this process, a validated model was obtained. The mentioned mechanical properties were identified using a step-by-step method, in which one property aided in the discovery of another. The robot model was created in a MATLAB/Simulink environment, using the Simscape Multibody Toolbox. [ABSTRACT FROM AUTHOR]

In this paper, we studied both kinematic and dynamic models of a mammal-like octopod robot. To control robot legs, we employed a simple gait generator based on a sine function and we proposed modified generators of gait. The introduced generators allowed us to obtain better kinematic and dynamic properties of motion of the whole robot during walking. By changing the length and the height of a single robot step, it is possible to use one model to control the initial, regular, and final phases of the robot gait without the need of generating additional control signals. The simulation parameters were estimated based on the prototype designed in Inventor software. In turn, for numerical research, we used a simulation model implemented in Mathematica. The obtained results were presented in the form of time histories of basic kinematic and dynamic quantities of the robot gait as well as the configuration of the robot legs at the chosen time moments. The presented method allows us to precisely control the robot position in the vertical direction, which is also presented in this paper. As a result, we obtained better stability of the whole robot during walking and performing tasks, also on slippery terrains, i.e. on terrains with relatively low coefficient of friction between the ground and the robot feet. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is to define and solve the problem of an optimized structural topology of the simply supported beam made from functionally graded material (FGM) enabling achievement of a maximum buckling load.Design/methodology/approach Two kinds of inclusions are considered: regular distribution of inclusions of different rigidities and non-uniform distribution of identical inclusions. It is shown that the optimal conditions are similar for both structural designs. The optimization problems are solved by using the homogenization method, and the target functions belong to the class of piece-wise continuous functions. Both optimized structures exhibit border zones free of any inclusions, and the largest amount of inclusions is localized in the central zone of the beams.Findings It has been shown that the final result of the carried out optimization of the internal structure for both studied types of FGM are similar. The relative increase in the buckling force of the FG beam with the optimized internal structure is on amount of 20 per cent while comparing it with the regular structure beam.Originality/value In contrary to a standard approach, this paper is aimed to detect and study a scenario of transition from heterogeneous to its counterpart homogeneous beam structure based on the consideration of the FGM inclusions. In addition, the problem of inversed transition from the optimized homogeneous structure to the optimal heterogeneous one is solved. [ABSTRACT FROM AUTHOR]

Abstract Linear and geometrically nonlinear vibrations of the three-layered functionally graded shallow shells with a complex form of the base are studied. It is assumed that outer and inner layers are made of functionally graded materials (FGM) or an isotropic material (metal or ceramic). The first-order shear deformation theory of shallow shells (FSDT) is employed. Effective material properties of layers are varied along the thickness according to a power law. To calculate different mechanical characteristics for different types of lamination schemes, analytical expressions are obtained. The linear problem is solved by combining the Ritz and the R-functions method. Linearization of the nonlinear problem is carried out by a novel original approach. Namely, the initial problem is reduced to solving a sequence of linear problems including those vibrations related to linear, special type of the elasticity problem and the nonlinear system of ordinary differential equations. Validation of the proposed method and the developed software has been examined on test problems for shallow shells with rectangular plan form and different boundary conditions. In order to demonstrate possibilities of the proposed approach, new results for laminated FGM shallow shells of the complex form of the base are presented. Effects of different material distributions, lamination schemes, curvatures, boundary conditions, and geometrical parameters on natural frequencies and backbone curves are reported and analyzed. Highlights • Free vibrations of sandwich shells are studied by R-function and Ritz methods. • Different types of boundary conditions are used. • Novel solutions are presented for shells with cutout of the complex shape. • Effects of shell parameters and boundary condition are reported. [ABSTRACT FROM AUTHOR]

The change in mechanical properties of the femoral bone tissue surrounding hip endoprosthesis stems during the post-operative period is one of the causes of implant instability, and the mathematical description of this phenomenon is the subject of much research. In the present study, a model of bone adaptation, based on isotropic Stanford theory, is created for further computer investigation. The results of implementation of such a mathematical model are presented regarding the choice of cement mantle rational thickness in cemented hip arthroplasties. The results show that for cement mantle thicknesses ranging from 1–1.5 mm, a peak stress value in the proximal part of the mantle exceeds the limit of durability of bone cement. Moreover, results show that high reduction in the bone density of distal and proximal regions was observed in cases of cement mantle thicknesses varying from 1–3 mm. No significant changes in bone density of the abovementioned regions were obtained for 4 mm and 5 mm. The outcome of numerical investigations can be treated as valuable and will lead to the improvement of cemented hip replacement surgery results. [ABSTRACT FROM AUTHOR]

In this work we consider an optimal design of corrugated cylindrical shells. The functionally graded (FG) corrugation with amplitude following a given shape is defined. The problem is solved as a linear one assuming the axially symmetric loading. The method of estimation of the stress-strain state and, consequently, the shell optimal design based on homogenization approach is proposed. The illustrative examples of the optimal FG corrugation shell subjected to hydrostatic load have demonstrated high efficiency of the employed method. [ABSTRACT FROM AUTHOR]

The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations. [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 damped parametric pendulum with friction is identified twice by means of its precise and imprecise mathematical model. A laboratory test stand designed for experimental investigations of nonlinear effects determined by a viscous resistance and the stick-slip phenomenon serves as the model mechanical system. An influence of accurateness of mathematical modeling on the time variability of the nonlinear damping coefficient of the oscillator is proved. A free decay response of a precisely and imprecisely modeled physical pendulum is dependent on two different time-varying coefficients of damping. The coefficients of the analyzed parametric oscillator are identified with the use of a new semi-empirical method based on a coupled oscillators approach, utilizing the fractional order derivative of the discrete measurement series treated as an input to the numerical model. Results of application of the proposed method of identification of the nonlinear coefficients of the damped parametric oscillator have been illustrated and extensively discussed. [ABSTRACT FROM AUTHOR]

Background The change in mechanical properties of femoral cortical bone tissue surrounding the stem of the hip endoprosthesis is one of the causes of implant instability. We present an analysis used to determine the best conditions for long-term functioning of the bone–implant system, which will lead to improvement of treatment results. Methods In the present paper, a finite element method coupled with a bone remodelling model is used to evaluate how different three-dimensional prosthesis models influence distribution of the density of bone tissue. The remodelling process begins after the density field is obtained from a computed tomography scan. Then, an isotropic Stanford model is employed to solve the bone remodelling process and verify bone tissue adaptation in relation to different prosthesis models. Findings The study results show that the long-stem models tend not to transmit loads to proximal regions of bone, which causes the stress-shielding effect. Short stems or application in the calcar region provide a favourable environment for transfer of loads to the proximal region, which allows for maintenance of bone density and, in some cases, for a positive variation, which causes absence of the aseptic loosening of an implant. In the case of hip resurfacing, bone mineral density changes slightly and is closest to an intact femur. Interpretation Installation of an implant modifies density distribution and stress field in the bone. Thus, bone tissue is stimulated in a different way than before total hip replacement, which evidences Wolff's law, according to which bone tissue adapts itself to the loads imposed on it. The results suggest that potential stress shielding in the proximal femur and cortical hypertrophy in the distal femur may, in part, be reduced through the use of shorter stems, instead of long ones, provided stem fixation is adequate. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the investigations reported in the Part I. Based on the asymptotic approach and the lubrication theory, the composite with rhombic inclusions are studied. Models of composites with curvilinear rhombic inclusions and thin interfaces on phase boundaries are constructed with the help of non-smooth argument substitution and asymptotically equivalent functions. The effective conductivity is derived for absolutely conducting and non-conducting rhombic inclusions, taking into account thin interface effects. Generalisation of the classical Dykhne formula is proposed. Finally, asymptotic solutions are obtained for the effective conductivity of a hexagonal lattice with circular large-size and absolutely conducting/non-conducting inclusions. [ABSTRACT FROM AUTHOR]

Homogenisation theory and lubrication approach are adopted to study transport problems for systems of densely packed, high-contrast fibre composites. The effective conductivity properties of composites with inclusions of various shapes are derived. Contact conditions are analysed for boundaries between the composite phases. In addition, asymptotic properties are investigated. The following problems are studied: composites with inclusions of large circular size and conductivity, located in a square lattice; composites with non-conducting inclusions, matrix-inclusions contact conditions as well as composites with thin interfaces at boundaries between different phases. Numerous results are obtained with the use of methods of the non-smooth argument substitution and asymptotically equivalent functions. [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]

Geometrically nonlinear vibrations of functionally graded shallow shells of complex planform are studied. The paper deals with a power-law distribution of the volume fraction of ceramics and metal through the thickness. The analysis is performed with the use of the R-functions theory and variational Ritz method. Moreover, the Bubnov-Galerkin and the Runge-Kutta methods are employed. A novel approach of discretization of the equation of motion with respect to time is proposed. According to the developed approach, the eigenfunctions of the linear vibration problem and some auxiliary functions are appropriately matched to fit unknown functions of the input nonlinear problem. Application of the R-functions theory on every step has allowed the extension of the proposed approach to study shallow shells with an arbitrary shape and different kinds of boundary conditions. Numerical realization of the proposed method is performed only for one-mode approximation with respect to time. Simultaneously, the developed method is validated by investigating test problems for shallow shells with rectangular and elliptical planforms, and then applied to new kinds of dynamic problems for shallow shells having complex planforms. [ABSTRACT FROM AUTHOR]

This paper introduces a study on the horizontal and vertical deflections of the cross section of a thin-walled rotating beam. These deflections are governed by a system of two ordinary differential equations in order to describe their Cartesian directions. Based on multiple time-scales analysis, truncated asymptotic expansions are assumed to be approximate solutions to the given problem. Furthermore, an extracted autonomous system of differential equations governs the change rate of the amplitudes and phases of the beam deflections. The beam's rotation speed is adjusted to be in the neighborhood of both of the natural frequencies of the deflections such that the beam is subjected to 1 : 1 : 1 simultaneous resonance. A stability test is conducted according to the first method of Lyapunov in order to determine whether the equilibrium point is asymptotically stable or not. The beam's deflections turn unstable once its speed is in the neighborhood of its modal natural frequencies. There exists a multistable solution at some values of the beam's speed depending on the hysteresis manner of the model according to forward or backward sweeping of this speed. Furthermore, a range of centrifugal forces of the rotating hub can make the beam's deflections exhibit quasiperiodic responses which are confirmed by time response, orbital map, and amplitude spectrum. Eventually, some remarks are recommended for the external excitation frequency in order that the beam stays in the periodic behavior. [ABSTRACT FROM AUTHOR]

The (G′/G)-expansion scheme is used to execute many wave results for the partial differential equation, namely, the nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation which plays a significant role in mathematical physics, and this equation is accomplished in quantum electron–positron–ion magneto plasmas. The Lie approach is used to find the infinitesimal generators, and group invariant solutions, and help us to reduce the considered PDEs into ODEs. Then a planer dynamical system approach is used to see the existence of closed-form solutions. All possible phase portraits are obtained and the existence of electrostatic wave potentials is reported. Meantime, periodic and super-nonlinear electrostatic wave potentials are found for different initial value conditions with the help of the Runge–Kutta method. Moreover, new traveling wave patterns of the considered models are constructed and presented graphically. Then, the conserved vectors of the given physical model with the use of the multiplier scheme are presented. [ABSTRACT FROM AUTHOR]

Nonlinear chains of atoms (NCA) are complex systems with rich dynamics, that influence various scientific disciplines. The lie symmetry approach is considered to analyze the NCA. The Lie symmetry method is a powerful mathematical tool for analyzing and solving differential equations with symmetries, facilitating the reduction of complexity and obtaining solutions. After getting the entire vector field by using the Lie scheme, we find the optimal system of symmetries. We have converted assumed PDE into nonlinear ODE by using the optimal system. The new auxiliary scheme is used to find the Travelling wave solutions, while graphical behaviour visually represents relationships and patterns in data or mathematical models. The multiplier method enables the identification of conservation laws, and fundamental principles in physics that assert certain quantities remain constant over time. [ABSTRACT FROM AUTHOR]

The work presents the effectiveness of a certain class of approximate models of the resulting friction force found during simulations of a simplified model of clutch dynamics. The friction models are based on integral expressions assuming fully developed sliding and Coulomb's classical law of friction on each element of the planar contact. Special approximations of the integral model of the friction force and the moment are proposed, which are based on Padé approximants and their generalizations. The system of clutch dynamics is simplified to a friction disk on a rotating master disk. Two different configurations are investigated, including the coaxial and non-coaxial arrangement of the disks. The model based on the generalization of Padé approximants is compared with the corresponding approximants based on the Taylor series expansion and with the model using the integral expressions for the resultant friction force and torque components. [ABSTRACT FROM AUTHOR]

Some special cases of a larger class of smooth models of the resultant friction force occurring on a planar contact area are developed and presented. These models are constructed under assumptions of classical Coulomb's law of friction on each element of contact and instantaneous transition between two modes: fully developed sliding and rigid stick state without local slips and deformations of the contact area. They are able to model cases of different values of static and kinetic friction factors. The considered models are very effective tools for fast and reliable numerical simulations of mechanical systems with frictional contacts. They are applied in modelling and numerical analysis of a special kind of mechanical system, i.e. a kind of a pendulum with special frictional driving. The rotational joint of the pendulum is elastically suspended in the motion plane. The pendulum is driven by circular frictional contact with a rotating disk. Examples of self-excited bifurcation and chaotic dynamics as well as stick-slip behaviour of the pendulum are presented. [ABSTRACT FROM AUTHOR]

This work is devoted to strain analysis and optimal design of a Functionally Graded (FG) rods and beams with small inclusions. The homogenization procedure plays a key role in our investigations. The method is illustrated using an example of the rod longitudinal deformation and bending of a beam. We consider the cases of FG inclusion sizes and FG steps between inclusions separately. Particular problems of optimal design are discussed in some details. The mathematical model of the bending beam, which adapts to the external load action, is proposed and an illustrative example of the adaptation process is given. [ABSTRACT FROM AUTHOR]

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge–Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement. [ABSTRACT FROM AUTHOR]

A method to study dynamical instability and non-linear parametric vibrations of symmetrically laminated plates of complex shapes and having different cutouts is proposed. The first-order shear deformation theory (FSDT) and the classical plate theory (CPT) are used to formulate a mathematical statement of the given problem. The presence of cutouts essentially complicates the solution of buckling problem, since the stress field is non-uniform. At first, a plane stress analysis is carried out using the variational Ritz method and the R-functions theory. The obtained results are applied to investigate buckling and parametric vibrations of laminated plates. The developed method uses the R-functions theory, and it may be directly employed to study laminated plates of arbitrary forms and different boundary conditions. Besides, the proposed method is numerical-analytical, what greatly facilitates a solution of similar-like non-linear problems. In order to show the advantage of the developed approach, instability zones and response curves for the layered cross- and angle-ply plates with external cutouts are constructed and discussed. [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]

A novel numerical/analytical approach to study geometrically nonlinear vibrations of shells with variable thickness of layers is proposed. It enables investigation of shallow shells with complex forms and different boundary conditions. The proposed method combines application of the R-functions theory, variational Ritz’s method, as well as hybrid Bubnov–Galerkin method and the fourth-order Runge–Kutta method. Mainly two approaches, classical and first-order shear deformation theories of shells are used. An original scheme of discretization regarding time reduces the initial problem to the solution of a sequence of linear problems including those related to linear vibrations with a special type of elasticity, as well as problems governed by non-linear system of ordinary differential equations. The proposed method is validated by the investigation of test problems for shallow shells with rectangular planform and applied to new vibration problems for shallow shells with complex planforms and variable thickness of layers. [ABSTRACT FROM AUTHOR]