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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 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]

• 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]

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]

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]

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]

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]

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]

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]

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]

This Special Issue of Applied Mathematical Modelling journal consists of selected quality papers originally presented at 15th International Conference "Dynamical Systems - Theory and Applications" (DSTA) which took place on December 2-5, 2019 in Lodz, Poland. B Dzyubak et al. b analysed a non-linear multi-scale diffusion cancer invasion model that describes the interactions of the tumour cells, matrix-metalloproteinases, matrix-degradative enzymes and oxygen. Chaotic cancer attractors are quantified and multi-parametric evolution of conditions leading to cancer invasion is defined. [Extracted from the article]

Abstract: There are many examples of mechanical systems with non-point friction contacts (billiard ball, Thompson top, wobblestone, electric polishing machine, the wobblestone, the Celtic stone), where the assumption of one-dimensional dry friction model do not necessarily lead to satisfactory accuracy of the numerical simulation. Moreover the rolling resistance often plays an important role in such systems. The paper is devoted to the problem of developing an approximate coupled model of resulting dry friction force and moment as well as rolling resistance, suitable for fast numerical simulation of rigid bodies with friction contacts, i.e. allowing to avoid the space discretization. An integral model of dry friction components is built under assumption of classical Coulomb friction law and fully developed sliding on the contact area of general shape and arbitrary contact pressure distribution. Then the special model of stress distribution over the elliptic contact area is developed, being a kind of generalization of Hertzian normal stress distribution, with addition of special distortion related to the rolling resistance. Finally some original approximate models of friction force and moment are proposed, based on Padé approximants and their generalizations as well as in the form of piecewise polynomial functions. [Copyright &y& Elsevier]

Abstract: Non-linear dynamics of a high-speed rotor symmetrically supported on the magneto-hydrodynamic bearing is analyzed. In the case of soft magnetic materials the method of multiple scales is applied. Non-resonant and primary resonance cases with and without internal resonance are illustrated and discussed. In the case of rigid magnetic materials the hysteretic properties are taken into consideration by means of the Bouc–Wen hysteretic model. In order to characterize the hysteretic close-loop control of the rotor motion, instability regions as well as the amplitude level contours of vertical and horizontal vibrations of the rotor are obtained in the form suitable for engineering applications, i.e. in the ‘frequency-amplitude’ parametric plane. [ABSTRACT FROM AUTHOR]

Abstract: Nonlinear equations governing chaotic dynamics exhibited by a class of lumped mechanical systems with dry friction are analysed. It is shown that the obtained corresponding Melnikov''s function can be simplified in some cases to yield analytical conditions for chaos occurrence. The analytical predictions are verified numerically. [Copyright &y& Elsevier]

Abstract: By using Tichonov theorem for singularly perturbed differential equations, we study the relationship between dynamics of discontinuous differential equations and their continuous approximations along periodic solutions. [Copyright &y& Elsevier]

Abstract: The paper focuses on 1D continuous models derived from a discrete micro-structure. A new continualization procedure that takes into account the nonlocal interaction between variables of the discrete media is proposed. The proposed procedure mainly contains an application of two-point Padé approximations and allows obtaining continuous models suitable to analyze 1D lattice vibrations with arbitrary frequencies. [Copyright &y& Elsevier]

• Simplified beam equations with inertial terms and nonlinear curvatures are derived. • They can be treated as the first approximation to the Kirchhoff nonlinear equations. • Simplified equations versus boundary conditions are illustrated and discussed. • Nonlinear modes of beam vibrations are constructed for various boundary conditions. This paper is devoted to the construction of asymptotically correct simplified models of nonlinear beam equations for various boundary conditions. V.V. Bolotin mentioned that in some cases (e.g., if compressed load is near the buckling value), the so-called "nonlinear inertia" must be taken into account. The effect of nonlinear inertia on the oscillations of the clamped-free beam is investigated in many papers. Bolotin used some physical assumption and did not compare the order of nonlinear terms in original equations. Below we propose our method for deriving those, which we will named "Bolotin's equations". This approach is based on fractional analysis of original boundary value problems. [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]

• Non-linear multi-scale diffusion cancer invasion model is developed. • Interactions of tumor cells, matrix-metalloproteinases/enzymes and oxygen are analyzed. • Multi-parametric evolution of conditions leading to cancer invasion is carried out. • Chaotic versus regular attractors and phase spaces are illustrated and discussed. • Amplitude level contours and oxygen concentrations are obtained. In this study we performed a simulation of carcinogenesis evolution based on varying system parameters in the multi-parametric space. A non-linear multi-scale diffusion cancer invasion model that describes the interactions of the tumor cells, matrix-metalloproteinases (MM), matrix-degradative enzymes (MDE) and oxygen was employed for simulation. To quantify chaotic cancer attractors, the technique based on the wandering trajectories analysis was applied. Multi-parametric evolution of conditions leading to cancer invasion was defined and illustrated in numerous figures. Chaotic attractors and phase spaces of the regular oscillations of basic indicators of a biological system with parameters that correspond to the chaotic and regular regions, respectively, have been presented. Amplitude level contours of MM, MDE and oxygen concentrations have been obtained and juxtaposed with the corresponding parametric planes. In all cases the carcinogenesis is accompanied by significant increase in chemical oscillation amplitudes of MM, MDE and oxygen concentrations. The obtained results derived from all control parameter planes could be used for evaluations of conditions resulting in cancer invasion as well as the modes to inhibit and/or stabilize carcinogenesis. [ABSTRACT FROM AUTHOR]

• Nonlinear unimorph and bimorph models for cantilever-based FGMEEM energy harvesters are presented. • Analytical perturbation solutions for the free vibration and primary resonance cases are provided. • Several comparisons are made to ensure the accuracy of the introduced models. • The supremacies of the present harvesters especially the bimorph case compared to the conventional FGPM systems are addressed. This paper nonlinearly models cantilever-based functionally graded magneto-electro-elastic energy harvesters (FGMEEEH) for the first time. The coupled magneto-electro-mechanical model is obtained on the basis of the Euler-Bernoulli beam theory. A hybrid procedure including Ritz's method is then utilized to generate reduced order models for both asymmetric unimorph and symmetric bimorph configurations. The resulting sets of initial value problems, whose convergence will be examined, are then analytically solved using the method of multiple time scales for both the free vibrations and primary resonance cases. The analytical time-histories of the system are compared by those obtained numerically and excellent agreements between them are observed. In addition, simplifying the frequency response function of the system in the primary resonance case, the present findings are validated by those available in the literature for linear unimorph systems. The influences of the harvester configurations, base acceleration amplitude, the value of the tip mass, the material gradation index as well as the resistances of the piezoelectric and magnetic circuits on the nonlinear response of the system are also studied in detail. It is observed that FGMEEEEHs enjoy much more efficiency in comparison to piezoelectric-based systems. [ABSTRACT FROM AUTHOR]