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Mathematical models of planar physically nonlinear inhomogeneous plates with rectangular cuts are constructed based on the three-dimensional (3D) theory of elasticity, the Mises plasticity criterion, and Birger’s method of variable parameters. The theory is developed for arbitrary deformation diagrams, boundary conditions, transverse loads, and material inhomogeneities. Additionally, inhomogeneities in the form of holes of any size and shape are considered. The finite element method is employed to solve the problem, and the convergence of this method is examined. Finally, based on numerical experiments, the influence of various inhomogeneities in the plates on their stress–strain states under the action of static mechanical loads is presented and discussed. Results show that these imbalances existing with the plate’s structure lead to increased plastic deformation.

We propose applying topological optimization methods based on measuring temperature and heat fluxes to estimate the thermal conductivity of inhomogeneous thermally conductive plates and determine the shape and location of foreign inclusions. Examples of plates subjected to external heat fluxes and heat sources are considered. The solutions are obtained with the help of the finite element method combined with the method of moving asymptotes. The results show that identification accuracy depends on the defined boundary conditions, the source intensity value, heat fluxes, temperature distribution and the shape of the inclusions. For the problem of 18 circular inclusions under different heat fluxes, identification accuracy increases with increasing source intensity values.

In this work, mathematical models of physically nonlinear plates and beams made from multimodulus materials are constructed. Our considerations are based on the 3D deformation theory of plasticity, the von Mises plasticity criterion and the method of variable parameters of the theory of elasticity developed by Birger. The proposed theory and computational algorithm enable for solving problems of three types of boundary conditions, edge conditions and arbitrary lateral load distribution. The problem is solved by the finite element method (FEM), and its convergence and the reliability of the results are investigated. Based on numerical experiments, the influence of multimodulus characteristics of the material of the beam and the plate on their stress–strain states under the action of transverse loads is illustrated and discussed.

This review/research paper deals with the reduction of nonlinear partial differential equations governing the dynamic behavior of structural mechanical members with emphasis put on theoretical aspects of the applied methods and signal processing. Owing to the rapid development of technology, materials science and in particular micro/nano mechanical systems, there is a need not only to revise approaches to mathematical modeling of structural nonlinear vibrations, but also to choose/propose novel (extended) theoretically based methods and hence, motivating development of numerical algorithms, to get the authentic, reliable, validated and accurate solutions to complex mathematical models derived (nonlinear PDEs). The review introduces the reader to traditional approaches with a broad spectrum of the Fourier-type methods, Galerkin-type methods, Kantorovich–Vlasov methods, variational methods, variational iteration methods, as well as the methods of Vaindiner and Agranovskii–Baglai–Smirnov. While some of them are well known and applied by computational and engineering-oriented community, attention is paid to important (from our point of view) but not widely known and used classical approaches. In addition, the considerations are supported by the most popular and frequently employed algorithms and direct numerical schemes based on the finite element method (FEM) and finite difference method (FDM) to validate results obtained. In spite of a general aspect of the review paper, the traditional theoretical methods mentioned so far are quantified and compared with respect to applications to the novel branch of mechanics, i.e. vibrational behavior of nanostructures, which includes results of our own research presented throughout the paper. Namely, considerable effort has been devoted to investigate dynamic features of the Germain–Lagrange nanoplate (including physical nonlinearity and inhomogeneity of materials). Modified Germain–Lagrange equations are obtained using Kirchhoff’s hypothesis and relations based on the modified couple stress theory as well as Hamilton’s principle. A comparative analysis is carried out to identify the most effective methods for solving equations of mathematical physics taking as an example the modified Germain–Lagrange equation for a nanoplate. In numerical experiments with reducing the problem of PDEs to ODEs based on Fourier’s ideas (separation of variables), the Bubnov–Galerkin method of static problems and Faedo–Galerkin method of dynamic problems are employed and quantified. An exact solution governing the behavior of nanoplates served to quantify the efficiency of various reduction methods, including the Bubnov–Galerkin method, Kantorovich–Vlasov method, variational iterations and Vaindiner’s method (the last three methods include theorems regarding their numerical convergence). The numerical solutions have been compared with the solutions obtained by various combinations of the mentioned methods and with solutions obtained by FDM of the second order of accuracy and FEM for triangular and quadrangular finite elements. The studied methods of reduction to ordinary differential equations show high accuracy and feasibility to solve numerous problems of mathematical physics and mechanical systems with emphasis put on signal processing.

In this work, a mathematical model describing the nonlinear dynamics of a plate-beam system is proposed. The model takes account of coupling between temperature and deformation fields as well as external mechanical, temperature, and noise excitation. It considers a system of integro-differential equations of a hyperbolic-parabolic type and of different dimensions. Both the proof of existence and uniqueness of the solution to the problem and the proof of convergence of the Faedo–Galerkin method used for solving the problem are given. The obtained mathematical model governs the work of the members of the micromechanical system. Algorithms aimed at solving the plate-beam structures, constructed based on the Faedo–Galerkin method in higher approximations as well as the 2nd and 4th order finite difference method (FDM), are employed to reduce PDEs to the Cauchy problem. The last is solved by Runge–Kutta methods of different types. In order to define the character of vibrations of the plate-beam structure, the sign of the largest Lyapunov exponent (LLE) is analyzed with the help of Wolf, Kantz, and Rosenstein methods. This complex approach allows one to obtain reliable results and true chaotic orbits. In addition, a few computational examples are provided.

In this paper, we construct a new theory of nanobeams taking into account the dependence of the material properties on the stress state. The theory is based on the Euler–Bernoulli kinematic model in the first approximation. The beam material is isotropic but heterogeneous. For the first time, the physical nonlinearity and the dependence of the material properties on the temperature are taken into account in the study of nanobeams, and the theory is developed for arbitrary materials. It is based on the theory of small elasticplastic strains and on the the modified torque theory of elasticity. The stationary temperature field is determined by solving the three-dimensional Poisson equation with boundary conditions of orders 1–3. The initial equations are derived from the Hamilton–Ostrogradskii principle. The desired system of partial differential equations is reduced to the Cauchy problem by the finite difference method of the second order of accuracy, and the Cauchy problem is solved by the Runge–Kutta or Newmark method. At each time step, an iterative procedure is developed by the Birger method of variable elasticity parameters. The stationary solution follows from the dynamic solution of the problem obtained by the method of determination (the method of the parameter position). The convergence of the solution is investigated depending on the number of points of partition along the length and thickness of the beam in the finite difference method as well as on the method of solving the Cauchy problem and the size-dependent parameter, i.e., the solution of the problem is considered to have almost infinite number of degrees of freedom. Numerical examples are given for a beam rigidly clamped at the ends with the stress-strain diagram for aluminum. Accounting for the size-dependent parameter in the nanobeam theory significantly affects the load-carrying capacity of nanobeams.

In this study, an analysis of mechanical vibrations influenced by external additive white Gaussian noise and colored noise is conducted using the principal component analysis. The principal component analysis is widely employed for encoding images in image processing, biology, economics, sociology, and political science. However, it is hereby applied to analyze nonlinear dynamics of continuous mechanical systems for the first time. A rich class of objects, including straight beams, beams on Winkler foundations and spherical shells, is investigated in the present paper. The basic differential equations are obtained based on the Bernoulli–Euler hypothesis, and solutions of the linear PDEs are analyzed by means of the principal component analysis. Results obtained with the principal component analysis are compared with those for the method of empirical modal decomposition and the wavelet-packet decomposition.

A new mathematical model of oscillations of mesh micropolar geometrically nonlinear cylindrical panels under the action of a normal alternating distributed load has been constructed. The equations of motion for an element of a smooth panel equivalent to the mesh and the boundary and initial conditions are obtained from the Hamilton–Ostrogradsky energy principle taking into account the Kirchhoff–Love kinematic hypotheses and the von Karman theory. To take into account the size-dependent behavior, a non-classical continual model based on a Cosserat medium is used in the work, where, along with the usual stress field, momentary stresses are also taken into account. The panel consists of n sets of densely arranged ribs of the same material, which makes it possible to average the ribs on the panel surface using the Pshenichnov theory. To reduce the partial derivative problem to a system of ordinary differential equations in spatial coordinates, two fundamentally different methods: the finite difference method with the second-order approximation and the Bubnov–Galerkin method with higher approximation are used. The obtained Cauchy problem has been solved by the Runge–Kutta-type methods with different orders of accuracy. The results obtained by different numerical methods are compared. The nonlinear dynamics of the examined systems is investigated depending on the mesh geometry. The necessity of studying the propagation of longitudinal waves has been justified.

The dependence of the quality factor of nonlinear microbeam resonators under thermoelastic damping for Timoshenko beams with regard to geometric nonlinearity has been studied. The constructed mathematical model is based on the modified couple stress theory which implies prediction of size-dependent effects in microbeam resonators. The Hamilton principle has yielded coupled nonlinear thermoelastic PDEs governing dynamics of the Timoshenko microbeams for both plane stresses and plane deformations. Nonlinear thermoelastic vibrations are studied analytically and numerically and quality factors of the resonators versus geometric and material microbeam properties are estimated. Results are presented for gold microbeams for different ambient temperatures and different beam thicknesses, and they are compared with results yielded by the classical theory of elasticity in linear/nonlinear cases.

A theory of flexible shallow nano-shells is developed based on the modified couple stress theory in higher approximation. The shell material is considered as isotropic, elastic and both von Karman and Kirchhoff-Love hypotheses are taken into account. Variational Hamilton's principle yields differential equations of motion of both shallow size-dependent nano-shells with rectangular planforms and axially symmetric nano-shells with circular planforms. The derived PDEs are reduced to the Cauchy problem by using the FDM (finite difference method), and then solved by the Runge-Kutta-type methods. Convergence of the numerical results is analysed with respect to the number of the shell radius partitions and time step. The Fourier power spectra, various wavelets-type spectra, phase and modal portraits, as well as signs of the LEs (Lyapunov exponents) are investigated. All results associated with the analysis of LEs are validated based on the case studies of non-linear dynamics. Two kinds of boundary conditions are employed: movable and fixed clamping along the shell edge. It is shown that the size-dependent parameter essentially influences shell vibrations (in particular, the chaotic vibrations become periodic).

Different kinematic mathematical models of nonlinear dynamics of a contact interaction of two microbeams are derived and studied. Dynamics of one of the microbeams is governed by kinematic hypotheses of the first, second, and third approximation orders. The second beam is excited through a contact interaction with the first beam and is described by the kinematic hypothesis of the second-order approximation in both geometric linear and nonlinear frameworks. The derived nonlinear partial differential equations (PDEs) are transformed to the counterpart system of nonlinear ordinary differential equations (ODEs) by the finite difference method. Nonlinear contact interaction dynamics of the microbeam structure is analyzed with the help of time series (signals), Fourier spectra, and wavelet spectra based on various mother wavelets, Morlet wavelet spectra employed to study synchronization phenomena, Poincaré maps, phase portraits, and the Lyapunov exponents estimated with the Wolf, Kantz, and Rosenstein algorithms. We have illustrated that neglecting the shear function (Euler–Bernoulli model) yields erroneous numerical results. We have shown that the geometric nonlinearity cannot be neglected in the analysis even for small two-layer microbeam deflection. In addition, we have detected that the contact between two microbeams takes place in the vicinity of x ≈ 0.2 x \approx 0.2 and x ≈ 0.8 x \approx 0.8 instead of the beams central points.

A mathematical model of the loss of dynamic stability of curvilinear size-dependent MEMS and NEMS elements embedded in a temperature field and subjected to large deflections was derived and studied. The fundamental governing dynamical equations of MEMS/NEMS members were yielded by Hamilton's principle. The investigations were based on combining the modified couple stress theory, the first-order approximation kinematic (Euler–Bernoulli) model, the von Karman geometric non-linearity, and the Duhamel–Neumann law regarding the temperature input (the beam material is elastic, isotropic and there are no constraints imposed on the temperature distribution). The temperature field was defined by solving a heat transfer equation. The computational algorithm was based on the finite difference method and the Runge–Kutta method. The numerical methods were validated by estimating the temporal and spatial convergence and reliability of the obtained solution was validated with the Lyapunov exponents obtained by qualitatively different methods. A few case studies related to the loss of stability, the magnitude of the size-dependent parameter, the type and intensity of the temperature input, and the parameters of uniformly distributed transverse load were investigated.

The topological optimization of composite structures is widely used while tailoring materials to achieve the required engineering physical properties. In this paper, the problem of topological optimization of the microstructure of a composite aimed at the construction of a material with most effective values of the bulk modulus of elasticity and thermal conductivity taking into account competing mechanical and thermal properties of the materials included in the composite is defined and solved. A two-phase composite consists of two base materials, one of which has a higher Young modulus but lower thermal conductivity, while the other has a lower Young modulus but higher thermal conductivity. A new class of problems for composites containing material pores or technological inclusions of different shapes is considered. Effective thermoelastic properties are obtained using the asymptotic homogenization method. A modified solid isotropic material with penalization (SIMP) model is used to regularize the problem. The problem of the isoperimetric constraints is solved by the method of moving asymptotes (MMA). The influence on optimal topology of the composite in the presence of two competing materials, and optimality criteria using linear weight functions are investigated. Pareto spaces that provide deep understanding of how these goals compete in achieving optimal topology are constructed.

A theory of contact interaction of a plate locally supported by a beam, under the influence of external lateral load and external additive color noise (pink, red, white) was constructed. Also described design is in a stationary temperature field. For the plate, the Kirchhoff kinematic model was adopted; for the beam, Euler - Bernoulli, the physical nonlinearity is taken into account according to the theory of small elastic-plastic deformations. The temperature field is taken into account according to the Duhamel - Neumann theory, and there are no restrictions on the temperature distribution over the plate thickness and the height of the beam. The temperature field is determined from the solution of the three-dimensional (plate) and two-dimensional (beam) heat conduction equations. The theory of B.Ya. Cantor. The heat conduction equations are solved by the finite difference method of the second and fourth order of accuracy. The system of differential equations is reduced to the Cauchy problem by the Bubnov - Galerkin methods in higher approximations and finite differences in spatial variables. Next, the Cauchy problem is solved by the fourth-order Runge - Kutta method and the Newmark method. At each time step, the iterative procedure of I. Birger was applied. The results of a numerical experiment are given. To analyze the results, the methods of nonlinear dynamics were used (construction of signals, phase portraits, Poincare sections, Fourier power spectra and Morlet wavelet spectra, analysis of the sign of Lyapunov indices by three methods: Wolf, Kantz, Rosenstein). The effect of color noise on the contact interaction between the plate and the beam has been studied. It has been established that red additive noise has the most significant effect on the oscillation pattern of the lamellar-beam structure compared to pink and white noise.

An attempt is made to clear vibrations of a beam resting on a viscoelastic support from noise effects. It is assumed that Bernoulli-Euler hypothesis holds. Effects of white, red, pink, purple and blue noise are considered. Noise is accounted for as a component of an alternating distributed load. Equations of motion of the beam areobtained as partial derivatives from Hamilton-Ostrogradski principle. Partial derivative equations are reduced to a Cauchy problem, using a second-order accuracy finite difference method, which is solved by Runge-Kutta-type methods. To clear vibrations of the beam from noise, the main component method was applied. This method was used to process the solutions of linear partial differential equations describing vibrations of rectangular beams resting on a viscoelastic support. Solutions of the equations were represented in the form of a 2D data array corresponding to deflections in the nodes of the beam at different times. The quality of clearing was assessed by comparing the Fourier power spectra obtained in the absence of noise effects with those that had noise effects, and after clearing. Problems for beams simply supported at both ends, fully fixed at both ends, simply supported at one end and fully fixed at the other one are considered. It was possible to clear the signals from four types of noise: white, pink, blue and purple.

In this work, a mathematical model of multilayer orthotropic shells with the account of both the 3rd-order generalized model (the so-called Grigoluk-Kulikov model) and a temperature field is presented. An asymptotically stable modified model is proposed. The reported conservative difference schemes associated with the considered models are developed based on the variational-difference method. The stability of a symmetric/non-symmetric pack of layers is addressed. In particular, the influence of the number of layers on the shell stability properties is illustrated and discussed.

We construct a mathematical model of non-linear vibration of a beam nanostructure with low shear stiffness subjected to uniformly distributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elastic material obey the Hooke law and are governed by the kinematic third-order approximation (Sheremetev–Pelekh–Reddy model). The von Karman geometric non-linear relation between deformations and displacements is taken into account. In order to describe the size-dependent coefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations, as well as the initial and boundary conditions. A solution to the dynamical problem is found via the finite difference method of the second order of accuracy, and next via the Runge–Kutta method of orders from two to eight, as well as the Newmark method. Investigations of the non-linear nanobeam vibrations are carried out with a help of signals (time histories), phase portraits, as well as through the Fourier and wavelet-based analyses. The strength of the nanobeam chaotic vibrations is quantified through the Lyapunov exponents computed based on the Sano–Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling procedure allowed us to achieve reliable results. In particular, we have detected chaotic and hyper-chaotic vibrations of the studied nanobeam, and our results are authentic, reliable, and accurate.

A theory of geometrically nonlinear dynamics of nanoplates has been constructed with allowance for the temperature and strain fields connection on the basis of the modified moment theory taking into account the transverse load and additive color noise. A research method based on the qualitative theory of differential equations has been developed. An example of the additive color noise influence on nonlinear vibrations for an elastic Euler-Bernoulli beam is given.

A methodology to detect true chaos (in terms of non-linear dynamics) is developed on an example of a structure composed of two beams with a small clearance. The Euler–Bernoulli hypothesis is employed, and the contact interaction between beams follows the Kantor model. The complex non-linearity results from the von Karman geometric non-linearity as well as the non-linearity implied by the contact interaction. The governing PDEs are reduced to ODEs by the second-order Finite Difference Method (FDM). The obtained system of equations is solved by Runge–Kutta methods of different accuracy. To purify the signal from errors introduced by numerical methods, the principal component analysis is employed and the sign of the first Lyapunov exponent is estimated by the Kantz, Wolf, Rosenstein methods and the method of neural networks. In the lattermost case, a spectrum of the Lyapunov exponents is estimated. It is illustrated how the number of nodes in the FDM influences numerical results regarding chaotic vibrations. It is also shown that an increase in the distance between beams implies stronger action of the geometric non-linearity. Convergence of the used numerical algorithm for FDM is demonstrated. The essential influence of initial conditions on the numerical results of the studied contact problem is presented and discussed.

This paper is devoted to the investigation of non-linear dynamics of non-homogeneous beams with a material optimally distributed along the height and length of the beam. The study was initiated by topological optimization for the given boundary and loading conditions, which yielded maximum stiffness of a beam microstructure. As a result, the beam with an optimized microstructure exhibiting non-homogeneity in two directions, i.e. along beam thickness and length, was obtained. In the second step, a beam model was derived based on the kinematic Euler–Bernoulli hypotheses and the modified couple stress theory including the von Karman geometric non-linearity and heat flow action obeying the Duhamel–Neumann law. Both static and dynamic behaviour of the optimized (non-homogeneous) and homogeneous beams were studied for different values of the material length-dependent parameter and temperature. Differences and peculiarities in static and dynamic problems were illustrated and discussed. In particular, the influence of the scale size parameter on chaotic beam dynamics was investigated. Also, scenarios of transition into deterministic chaos were detected and analysed for both homogeneous and optimized beams.

The hardware and software of an android assistant (AA) has to handle multiple tasks, where the student’s condition analysis subsystem has an important role to play. Condition analysis uses electroencephalogram (EEG) processing, identifies brain activity patterns, etc. The subsystem contains EEG preprocessing modules, one of which identifies brain activity patterns.This paper dwells upon the structure of this subsystem and illustrates its capabilities. The research effort involved an epileptic patient. The team analyzed EEG signals over 21 channels. The analytical approach was based on the fact that EEG signals are chaotic signals. The approach was successfully tested using tools of nonlinear dynamics, e.g. the spectrum of Lyapunov exponents. Abnormal brain activity was identified and confirmed by a professional analysis of MRI and EEG readings. THe paper shows the proposed Lapunov exponent-based brain activity analysis methods can be used to analyze EEG signals for early detection of electroencephalographic changes.

Parametric non-linear vibrations of flexible cylindrical panels subjected to additive white noise are studied. The governing Marguerre equations are investigated using the finite difference method (FDM) of the second-order accuracy and the Runge-Kutta method. The considered mechanical structural member is treated as a system of many/infinite number of degrees of freedom (DoF). The dependence of chaotic vibrations on the number of DoFs is investigated. Reliability of results is guaranteed by comparing the results obtained using two qualitatively different methods to reduce the problem of PDEs (partial differential equations) to ODEs (ordinary differential equations), i.e. the Faedo-Galerkin method in higher approximations and the 4th and 6th order FDM. The Cauchy problem obtained by the FDM is eventually solved using the 4th-order Runge-Kutta methods. The numerical experiment yielded, for a certain set of parameters, the non-symmetric vibration modes/forms with and without white noise. In particular, it has been illustrated and discussed that action of white noise on chaotic vibrations implies quasi-periodicity, whereas the previously non-symmetric vibration modes are closer to symmetric ones.

Mathematical models of multi-layer orthotropic shells were reconsidered based on the Timoshenko hypothesis. A new mathematical model with $$\varepsilon $$ -regularisation was proposed, and the theorem regarding the existence of a generalised solution to the model was formulated and proved. The algorithms of numerical investigation of models studied with the aid of the variational-difference method were developed. The associated stability problem was also addressed. A comparison of the results yielded by the considered models was carried out and discussed for numerous factors and parameters.

In this work, chaotic dynamics of flexible spherical axially symmetric shallow shells subjected to sinusoidal transverse load is studied with emphasis put on the vibration modes. Chaos reliability is verified and validated by solving the implemented mathematical model by partial nonlinear equations governing the dynamics of flexible spherical shells and by estimating the signs of the largest Lyapunov exponents with the help of qualitatively different approaches. It is shown how the scenario of transition of the investigated shells from regular to chaotic vibrations depends on the boundary condition. The following cases are considered: (1) movable and fixed simple supports along the shell contours, taking into account shell stiffness (Feigenbaum scenario) and shell damping (Ruelle–Takens–Newhouse scenario), and (2) movable clamping (regular shell vibrations). The presence of dents, the location and character of which essentially depend on the shell geometric parameters, boundary conditions, and the external load parameters, is detected in some regions of the shell surface and discussed.

A mathematical model of a contact interaction between two plates made from materials with different elasticity modulus is derived taking into account physical and design nonlinearities. In order to study the stress–strain state of this complex mechanical structure, the method of variational iteration has been employed allowing for reduction of partial differential equations to ordinary differential equations (ODEs). The theorem regarding convergence of this method is formulated for the class of similar-like problems. The convergence of the proposed iterational procedure used for obtaining a solution to contact problems of two plates is proved. In the studied case, the physical nonlinearity is introduced with the help of variable parameters associated with plate stiffness. The work is supplemented with a few numerical examples. Both Fourier and Morlet power spectra are employed to detect and analyse regular and chaotic vibrations of two interacting plates.

A mathematical model of flexible physically non-linear micro-shells is presented in this paper, taking into account the coupling of temperature and deformation fields. The geometric non-linearity is introduced by means of the von Karman shell theory and the shells are assumed to be shallow. The Kirchhoff-Love hypothesis is employed, whereas the physical non-linearity is yielded by the theory of plastic deformations. The coupling of fields is governed by the variational Biot principle. The derived partial differential equations are reduced to ordinary differential equations by means of both the finite difference method of the second order and the Faedo-Galerkin method. The Cauchy problem is solved with methods of the Runge-Kutta type, i.e. the Runge-Kutta methods of the 4th (RK4) and the 2nd (RK2) order, the Runge-Kutta-Fehlberg method of the 4th order (rkf45), the Cash-Karp method of the 4th order (RKCK), the Runge-Kutta-Dormand-Prince (RKDP) method of the 8th order (rk8pd), the implicit 2nd-order (rk2imp) and 4th-order (rk4imp) methods. Each of the employed approaches is investigated with respect to time and spatial coordinates. Analysis of stability and nature (type) of vibrations is carried out with the help of the Largest Lyapunov Exponent (LLE) using the Wolf, Rosenstein and Kantz methods as well as the modified method of neural networks. The existence of a solution of the Faedo-Galerkin method for geometrically non-linear problems of thermoelasticity is formulated and proved. A priori estimates of the convergence of the Faedo-Galerkin method are reported. Examples of calculation of vibrations and loss of stability of square shells are illustrated and discussed.

Dynamic bucking criteria for spherical shells of a rectangular form under sinusoidal lateral load are proposed and developed taking into consideration geometric and physical non-linearity. A mathematical model of thin shallow shells is constructed on the basis of the Kirchoff-Love hypothesis and the von Karman geometric non-linearity, whereas the physical non-linearity follows the Ilyushin theory of plastic deformations. Reliability of the results is proved by comparing them with the results obtained by means of higher-order approximations of the Faedo-Galerkin method. Three scenarios (Feigenbaum, Ruelle-Takens-Newhouse and Pomeau-Manneville) are detected while transiting from regular to quasi-periodic/chaotic vibrations.

In this work, mathematical models of physically nonlinear beams (and plates) are constructed in a three-dimensional and one-dimensional formulation based on the kinematic models of Euler–Bernoulli and Timoshenko. The modeling includes achievements of the deformation theory of plasticity, the von Mises plasticity criterion and the method of variable parameters of the Birger theory of elasticity. The theory is built for arbitrary boundary conditions, transverse loads, and stress-strain diagrams. The issue of solving perforated structures is also addressed. The numerical investigations are based on the finite element method and the method of variable elasticity parameters. Convergence of the method is also investigated.

In this work mathematical models of temporal part of chaos at chosen spatial locations of a plate locally reinforced by ribs taking into account an interplay of their interactions are derived and studied numerically for the most relevant dynamical parameters. In addition, an influence of the additive external noise on chaotic vibrations of multi-layer beam–plate structures coupled only by boundary conditions is investigated. We illustrate and discuss novel nonlinear phenomena of the temporal regular and chaotic contact/no-contact dynamics with the help of Morlet wavelets and Fourier analysis. We show how the additive white noise cancels deterministic chaos close to the boundary of chaotic region in the space of parameters, and we present windows of on/off switching of the frequencies during the contact dynamics between structural members. In order to solve the mentioned design type nonlinear problem we apply methods of qualitative theory of differential equations, the Bubnov–Galerkin method in higher approximations, the Runge–Kutta methods of 4th, 6th and 8th order, as well as the computation and analysis of the largest Lyapunov exponent (Benettin׳s and Wolf׳s algorithms are used). The agreement of outcomes of all applied qualitatively different numerical approaches validate our simulation results. In particular, we have illustrated that the Fourier analysis of the studied mechanical structures may yield erroneous results, and hence the wavelet-based analysis is used to investigate chaotic dynamics in the system parameter space.

The increasing complexity of the constructive forms and shell elements structure leads to the need to develop both the theory and methods for solving static and dynamic problems for non-homogeneous (heterogeneous) shells. By the shell heterogeneity, we mean heterogeneity in a broad sense: these are inclusions in the shell body of the different rigidity elements and, as a special case, these are holes; the material inhomogeneity caused by a change in the stress–strain state under the influence of both static and dynamic loads; stiffeners; and by taking into account the physical nonlinearity and different modulus of the shell material. This work is devoted to the mathematical model creation of the statics and dynamics for non-homogeneous shells in the above sense and consists of two parts. In the first part, a mathematical model of statics and dynamics for rectangular shells described by the Kirchhoff–Love kinematic model is constructed. Geometric nonlinearity is taken into account on the basis of T. von Karman’s geometric model, physical nonlinearity — according to the deformation theory of plasticity, based on the elasticity variable parameters method. Stiffness heterogeneity is taken into account using the Heaviside function. The original equations were obtained from Hamilton’s variational principle. A numerical experiment is performed using the Faedo–Galerkin method in higher approximations. The convergence of this method is investigated. In the second part, the nonlinear dynamics of axisymmetric elastic variable thickness shells is investigated. In contrast to the first part of this work, where the Faedo–Galerkin method in higher approximations was used to reduce partial differential equations to the Cauchy problem, in this part the Ritz method in higher approximations is employed. For differential operators used in two parts of this work, according to the research of S.G. Mikhlin, the Ritz and Faedo–Galerkin methods are equivalent. The convergence of the applied numerical method is investigated. An approach is proposed for the zones study of oscillation types for variable thickness shells using dynamic modes maps.

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles. The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method. The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method. Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.

Chaotic dynamics of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and von Karman geometric nonlinearity. We assume that the beam properties are graded along the thickness direction. The influence of size-dependent and functionally graded coefficients on the vibration characteristics, scenarios of transition from regular to chaotic vibrations as well as a series of static problems with an emphasis put on the load-deflection behavior are studied. Our theoretical/numerical analysis is supported by methods of nonlinear dynamics and the qualitative theory of differential equations supplemented by Fourier and wavelet spectra, phase portraits, and Lyapunov exponents spectra estimated by different algorithms, including Wolf’s, Rosenstein’s, Kantz’s, and neural networks. We have also detected and numerically validated a general scenario governing transition into chaotic vibrations, which follows the classical Ruelle-Takens-Newhouse scenario for the considered values of the size-dependent and grading parameters.

In this paper, composites made of periodically repeating micro structures are investigated. The study aims at identifying the optimal spatial distribution of constituents within a composite material to obtain the material of desired/improved functional properties. To find the relationship between micro- and macro-structural properties of the composite material, the method of homogenization is used. The problem of finding optimal microstructures of various materials, with the aim of obtaining maximum rigidity, i.e., maximum volume and shear modules for the base cell of a composite that contains the original installation of technological holes and/or inclusions was first investigated. For illustration and validation of the proposed approach, numerical examples are provided.

In the first part of the paper we employ the Sheremetev-Pelekh-Reddy-Levinson hypotheses, which yield a non-linear mathematical model of a beam taking into account geometric and physical non-linearity as well as transverse shear based on the modified couple stress theory. The general model includes both Bernoulli-Euler and Timoshenko models with/without geometric/physical non-linearity, and the size-dependent beam behaviour. In addition, we present results of the development of the relaxation method for solution to numerous static problems. The influence of the size-dependent coefficient on the load-deflection and stress-strain states of the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson mathematical models has been also studied.

In the present part of the paper various problems of non-linear dynamics of nano-beams within the modified couple stress theory as well as the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson models are studied taking into account the geometric non-linearity. Different characteristics of the vibrational process, including Fourier spectra, wavelet spectra, phase portraits, Poincare maps as well as the largest Lyapunov exponents, are studied for the same physical-geometric parameter with and without consideration of the size-dependent behaviour. Vibration graphs are constructed and analysed, and scenarios of transition from regular to chaotic vibrations are illustrated and discussed.

This paper studies the chaotic dynamics of two cylindrical shells nested into each other with a gap and their reinforcing beam, also with a gap, which is subjected to a distributed alternating load. The problem is solved using methods of nonlinear dynamics and the qualitative theory of differential equations. The Novozhilov equations for geometrically nonlinear structures are used as the governing equations. Contact pressure is determined by Kantor’s method. Using finite elements in spatial variables, the partial differential equations for the beam and shells are reduced to the Cauchy problem, which is solved by explicit integration (Euler’s method). The chaotic synchronization of this system is studied.

We study networks of coupled oscillators governed by ODEs and yielded by physically validated sets of a few PDEs governing dynamics of structural members (plate and beams), chaos and phase synchronization and contact/no-contact non-linear dynamics of structural members coupled via boundary conditions. We have detected, illustrated and discussed a few novel kinds of hybrid states of the studied plate-beam(s) contact/no-contact interactions as well as novel scenarios of transition into chaos exhibited by the interplay of continuous objects. Classical (time histories, phase portraits, Poincare maps, FFT, Lyapunov exponents) and non-classical (2D Morlet wavelets) approaches are used while monitoring non-linear dynamics of the interacting spatial structural members. Our results include examples from structural mechanics and the studied objects are modelled by validated mechanical hypotheses and assumptions. Novel non-linear phenomena including switching to different vibration regimes and phase chaotic synchronization are illustrated and discussed.