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In this paper, a mathematical model of complex oscillations of a flexible micropolar cylindrical mesh structure is constructed. Equations are written in displacements. Geometric nonlinearity is taken into account according to the Theodore von Karman model. A non-classical continual model of a panel based on a Cosserat medium with constrained particle rotation (pseudocontinuum) is considered. It is assumed that the fields of displacements and rotations are not independent. An additional independent material parameter of length associated with a symmetric tensor by a rotation gradient is introduced into consideration. The equations of motion of a panel element, the boundary and initial conditions are obtained from the Ostrogradsky – Hamilton variational principle based on the Kirchhoff – Love’s kinematic hypotheses. It is assumed that the cylindrical panel consists of n families of edges of the same material, each of which is characterized by an inclination angle relative to the positive direction of the axis directed along the length of the panel and the distance between adjacent edges. The material is isotropic, elastic and obeys Hooke’s law. To homogenize the rib system over the panel surface, the G. I. Pshenichnov continuous model is used. The dissipative mechanical system is considered. The differential problem in partial derivatives is reduced to an ordinary differential problem with respect to spatial coordinates by the Bubnov – Galerkin method in higher approximations. The Cauchy problem is solved by the Runge – Kutta method of the 4th order of accuracy. Using the establishment method, a study of grid geometry influence and taking account of micropolar theory on the behavior of a grid plate consisting of two families of mutually perpendicular edges was conducted.

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.

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 theory of nonlinear dynamics of a flexible single-layer micropolar cylindrical shell of a network structure is constructed. The geometric nonlinearity is taken into account by the model of Theodor von Karman. We consider a nonclassical continuum shell model based on the Cosserat medium with constrained particle rotation (pseudocontinuum). It is assumed that the displacement and rotation fields are not independent. An additional independent material length parameter associated with the symmetric tensor of the rotation gradient is introduced into consideration. The equations of motion of the shell element, boundary and initial conditions are obtained from the variational principle of Ostrogradskii–Hamilton on the basis of kinematic hypotheses of the third approximation (Peleha–Sheremetyev–Reddy), allowing to take into account not only the rotation, but also the curvature of the normal after deformation. It is assumed that the cylindrical shell con- sists of n families of edges, each of which is characterized by an inclination angle with respect to the positive direction of the axis directed along the length of the shell and the distance between neighboring edges. The shell material is isotropic, elastic, and obeys Hooke’s law. A dissipative mechanical system is considered. As a special case, the system of equations of motion for Kirchhoff–Love’s micro-polar reticulated shell is presented. The theory constructed in this paper can be used, among other things, for studying the behavior of CNTs under the action of static and dynamic loads.

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.

Microelectromechanical systems (MEMS) are increasingly playing a significant role in the aviation industry and space exploration. Moreover, there is a need to study the neutron radiation effect on the MEMS structural members and the MEMS devices reliability in general. Experiments with MEMS structural members showed changes in their operation after exposure to neutron radiation. In this study, the neutron irradiation effect on the flexible MEMS resonators&rsquo, stability in the form of shallow rectangular shells is investigated. The theory of flexible rectangular shallow shells under the influence of both neutron irradiation and temperature field is developed. It consists of three components. First, the theory of flexible rectangular shallow shells under neutron radiation in temperature field was considered based on the Kirchhoff hypothesis and energetic Hamilton principle. Second, the theory of plasticity relaxation and cyclic loading were taken into account. Third, the Birger method of variable parameters was employed. The derived mathematical model was solved using both the finite difference method and the Bubnov&ndash, Galerkin method of higher approximations. It was established based on a few numeric examples that the irradiation direction of the MEMS structural members significantly affects the magnitude and shape of the plastic deformations&rsquo, distribution, as well as the forces magnitude in the shell middle surface, although qualitatively with the same deflection the diagrams of the main investigated functions were similar.

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.

In this work a generalization of the Pshenichnov theory is introduced which is devoted to study of rectangular in plane structures composed of meshed shells with two families of mutually perpendicular ribs. It is based on an account of higher order couple stresses while employing the classical modified couple stress theory. The considerations are based on the Kirchhoff-Love theory including von Karman geometric nonlinearity. The worked out algorithms are aimed on reduction of the governing PDEs to ODEs using the FDM (finite difference method) of the second order. The obtained Cauchy problem is solved with a help of the 4th order Runge-Kutta method, whereas the static problems are solved using the set-up method. Convergence of the proposed computational scheme is validated with regard to spatial and time coordinates. Computational examples are provided.

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.

In the present study, 45 patients with schizophrenia syndromes and 39 healthy subjects are studied with electroencephalogram (EEG) signals. The study groups were of different genders. For each of the two groups, the signals were analyzed using 16 EEG channels. Multiscale entropy, Lempel-Ziv complexity and Lyapunov exponent were used to study the chaotic signals. The data were compared for two groups of subjects. Entropy was compared for each of the 16 channels for all subjects. As a result, topographic images of brain areas were obtained, illustrating the entropy and complexity of Lempel-Ziv. Lempel-Ziv complexity was found to be more representative of the classification problem. The results will be useful for further development of EEG signal classification algorithms for machine learning. This study shows that EEG signals can be an effective tool for classifying participants with symptoms of schizophrenia and control group. It is suggested that this analysis may be an additional tool to help psychiatrists diagnose patients with schizophrenia.

On the basis of the kinematic hypotheses of the Kirchhoff-Love built a mathematical model of micropolar cylindrical meshed panels vibrations under the action of a normal distributed load. In order to take into account the size-dependent behavior, the panel material is considered as a Cosser’s pseudocontinuum with constrained particle rotation. The mesh structure is taken into account by the phenomenological continuum model of G. I. Pshenichnov. For a cylindrical panel consisting of two systems of mutually perpendicular edges, a scenario of transition of oscillations from harmonic to chaotic is constructed. It is shown that in the study of the behavior of cylindrical micropolar meshed panels it is necessary to study the nature of the oscillations of longitudinal waves.

This paper analyzes a case with the patient having focal structural epilepsy by processing electroencephalogram (EEG) fragments containing the “sharp wave” pattern of brain activity. EEG signals were recorded using 21 channels. Based on the fact that EEG signals are time series, an approach has been developed for their analysis using nonlinear dynamics tools: calculating the Lyapunov exponent’s spectrum, multiscale entropy, and Lempel–Ziv complexity. The calculation of the first Lyapunov exponent is carried out by three methods: Wolf, Rosenstein, and Sano–Sawada, to obtain reliable results. The seven Lyapunov exponent spectra are calculated by the Sano–Sawada method. For the observed patient, studies showed that with medical treatment, his condition did not improve, and as a result, it was recommended to switch from conservative treatment to surgical. The obtained results of the patient’s EEG study using the indicated nonlinear dynamics methods are in good agreement with the medical report and MRI data. The approach developed for the analysis of EEG signals by nonlinear dynamics methods can be applied for early detection of structural 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.

Mathematical models of continuous structural members (beams, plates and shells) subjected to an external additive white noise are studied. The structural members are considered as systems with infinite number of degrees of freedom. We show that in mechanical structural systems external noise can not only lead to quantitative changes in the system dynamics (that is obvious), but also cause the qualitative, and sometimes surprising changes in the vibration regimes. Furthermore, we show that scenarios of the transition from regular to chaotic regimes quantified by Fast Fourier Transform (FFT) can lead to erroneous conclusions, and a support of the wavelet analysis is needed. We have detected and illustrated the modifications of classical three scenarios of transition from regular vibrations to deterministic chaos. The carried out numerical experiment shows that the white noise lowers the threshold for transition into spatio-temporal chaotic dynamics. A transition into chaos via the proposed modified scenarios developed in this work is sensitive to small noise and significantly reduces occurrence of periodic vibrations. Increase of noise intensity yields decrease of the duration of the laminar signal range, i.e., time between two successive turbulent bursts decreases. Scenario of transition into chaos of the studied mechanical structures essentially depends on the control parameters, and it can be different in different zones of the constructed charts (control parameter planes). Furthermore, we found an interesting phenomenon, when increase of the noise intensity yields surprisingly the vibrational characteristics with a lack of noisy effect (chaos is destroyed by noise and windows of periodicity appear).

A mathematical model of the vibrations of nanoelectromechanical sensors components as a nanobeam connected to an electrode at small distance was built. Nano beam was in a stationary temperature field, under impact Casimir forces, transverse alternating load and additive color noise. Geometric nonlinearity was taken into account according to the theory of Karman. In the particular case of the dependence of the Casimir force - deflection for elastic preheated beams have been studied.

The aim of the work is to use the variational iterations method to study the three-dimensional equations of mathematical physics and visualize the solutions obtained on its basis and the 3DsMAX software package. An analytical solution of the three-dimensional Poisson equations is obtained for the first time. The method is based on the Fourier idea of variables separation with the subsequent application of the Bubnov-Galerkin method for reducing partial differential equations to ordinary differential equations, which in the Western scientific literature has become known as the generalized Kantorovich method, and in the Eastern European literature has known as the variational iterations method. This solution is compared with the numerical solution of the three-dimensional Poisson equation by the finite differences method of the second accuracy order and the finite element method for two finite element types: tetrahedra and cubic elements, which is a generalized Kantorovich method, based on the solution of the three-dimensional stationary differential heat equation. As the method study, a set of numerical methods was used. For the results reliability, the convergence of the solutions by the partition step is checked. The results comparison with a change in the geometric parameters of the heat equation is made and a conclusion is drawn on the solutions reliability obtained. Solutions visualization using the 3Ds max program is also implemented.

Process visualization of static stability loss in mechanics is shown by the micropolar meshed cylindrical panel example with two families of mutually perpendicular ribs. The mathematical model of the panel's behavior is based on the Kirchhoff-Love hypotheses. The micropolar theory is applied to ac-count for scale effects. Geometric nonlinearity is taken into account according to the theory of Theodor von Karman. The mesh structure is taken into account based on the Pshenichnov I. G. continuum model. Visualization of numerical results using Autodesk 3ds Max software made it possible to more clearly assess the phenomenon of static buckling of the shell in question. Visualization of the results using 3D made it possible to establish that an in-crease in the distance between the edges of the mesh panel and an increase in the parameter depending on the size does not change the bending shape of the panel, as well as the diagrams of moments and forces at subcritical and supercritical loads.

This paper describes the influence of intensity of external additive white noise on the nonlinear dynamics of rectangular plates as mechanical systems with an infinite number of degrees of freedom. A new scenario is discovered, which is a combination of the classic Feigenbaum and Pomeau–Manneville scenarios. The classical methods of nonlinear dynamics and wavelet transforms were used to reveal the peculiarities of a modified scenario. The noise-induced transitions are investigated, and it is shown that the noise exposure is accompanied with the transition to chaotic fluctuations with a lower amplitude of the driving load. It is determined that the presence of external fluctuations does not affect the scenario of transition from harmonic to chaotic fluctuations.

In this paper regular and chaotic vibrations of flexible curvilinear beams with (and without) the action of temperature and electric fields are studied. Results obtained are based on the reduction of PDEs governing non-linear dynamics of straight and curvilinear beams to large sets of non-linear ODEs putting emphasis on reliability and validation of the results. In spite of the applied classical approaches to study bifurcational and chaotic dynamics, we have employed 2D and 3D Morlet wavelets and we have computed first four Lyapunov exponents. Numerous results are reported regarding scenarios of the transition from regular to chaotic vibrations including the occurrence of hyper-hyper chaos and deep chaos. Snap-through phenomena have been detected and analyzed, and the influence of boundary conditions of three types of the considered fields (mechanical, thermal and electrical) as well as of temperature on non-linear dynamics of the beam have been reported.

A mathematical model of behavior a spherical size-dependent shell and plate taking into account contact interaction is constructed. The effect of the color noise field on the mechanical system is taken into account. The following hypotheses were taken as initial hypotheses: the shell and plate material is isotropic, the Cosserat elastic pseudo-continuum with constrained particle rotation, and the Kirchhoff – Love kinematic model. The equations of motion are obtained on the basis of the following theories: Theodore von Karman (geometric non-linearity), B.Ya.Cantor theory (accounting for contact interaction).

Based on the micro polar and the Kirchhoff-Love theories, the mathematical model of the cylindrical mesh panel’s oscillations is constructed. The panels are consisting of two families of mutually perpendicular edges. The scenarios of the transition of panel oscillations to chaos are investigated. To justify the reliability of the results obtained, the numerical implementation was carried out by fundamentally different numerical methods. The conclusion is drawn about the optimal combinations of methods for the numerical implementation of the task.

The theory of nonlinear dynamics of the circular sensing element of a nanoelectromechanical sensor in the form of flexible elastic axisymmetric nano plates is constructed. The developed theory is general. It is based on the kinematic model of the third approximation (Sheremetev-Pelekh-Reddy). Two other theories follow from it as a special case: the theory of nonlinear dynamics and flexible nano-plates, obtained on the basis of the kinematic model of the first approximation (Kirchhoff), the second approximation (Timoshenko). The general theory obtained follows from the variational principle of Hamilton. For each of the kinematic hypotheses, a system of nonlinear partial differential equations is obtained. Obtaining a “true” solution is guaranteed using the methodology outlined in [1]. As an example, the model of the first approximation of the nonlinear dynamics of flexible elastic axisymmetric nano-plates is studied. In a numerical experiment, the required equations are solved by different methods, their convergence is investigated. It is shown that taking into account the size-dependent parameter significantly affects the character of plate oscillation and changes their character.

In this paper is proposed the methodology for determining the true chaos from the position of nonlinear dynamics for distributed mechanical systems in the form of a beam structure of two beams described by the kinematic firstapproximation hypothesis (Euler-Bernoulli). There is a small gap between the beams. The lower beam (beam 2) will be considered as an elastic base for the upper beam (beam 1). The transverse distributed alternating load acts on beam 1. The contact interaction of the beams was taken into account by the Cantor model. This problem has a great nonlinearity due to the account of the geometric nonlinearity of beams according to T. von Karman and the constructive nonlinearity of the structure due to the contact interaction. Differential equations in partial derivatives reduce to the ODE system by the finite differences method (FDM) of the second order accuracy. The resulting system is solved by Runge-Kutta methods of various accuracy orders. In general case, the problem solution essentially depends on the methods of reducing partial differential equations to ODEs and methods for solving the Cauchy problem, boundary and initial conditions. When solving the problem by the method of finite differences with approximation O(h2), the solution will depend on the number of points of the integration interval partition and the time step in solving the Cauchy problem. The analysis of the obtained results is carried out by the nonlinear dynamics methods and the qualitative theory of differential equations. The results are compared for geometrically linear and nonlinear beams, taking into account the contact interaction. The mechanical structure is considered as a system with an infinite number of freedom degrees. A complete coincidence of solutions is achieved, depending on the partitions number in the spatial coordinate in the chaos. The sign of the first Lyapunov exponent is determined by the Kantz, Wolf and Rosenstein methods.

A relaxation method is applied to estimate and predict a critical set of parameters responsible for stability loss (buckling) of spherical circle axially symmetric shells. The buckling phenomenon under static loading was investigated by solving the Cauchy problem for a set of ordinary differential equations and the Hausdorff metrics was applied while quantifying the data obtained within the novel approach.

In this work the mathematical modeling and analysis of the chaotic dynamics of flexible Euler-Bernoulli beams is carried out. Algorithms reducing the studied objects associated with the boundary value problems are reduced to the Cauchy problem through both the Finite Difference Method (FDM) with approximation of O ( c 2 ) and the Finite Element Method (FEM). The constructed Cauchy problem is solved via the fourth- and sixth-order Runge-Kutta methods. The validity and reliability of the obtained results is rigorously discussed. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincare and pseudo-Poincare maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. In particular, we study a transition from symmetric to asymmetric vibrations and we explain this phenomenon. Vibration-type charts are reported regarding two control parameters: amplitude q 0 and frequency ω p of the uniformly distributed periodic excitation. Furthermore, we have detected and illustrated chaotic vibrations of the Euler-Bernoulli beams for different boundary conditions and different beams thickness. In addition, we study chaotic dynamics and synchronization of multi-layer beams coupled only via boundary conditions. Computational examples of the theoretical investigations are given, where geometric, physical and design non-linearities are taken into account.

The chaotic synchronization of the vibrations of contacting multilayer beam–plate–shell structures under external loading is studied. There are gaps between their layers. Such systems are called structurally nonlinear. Their behavior is studied comprehensively, and the influence parameters that characterize the safe and critical modes are determined. A method to study the chaotic phase synchronization of various nature is developed. Vibration synchronization for the following systems is analyzed: (i) a stack of two-layer plates (each equation is linear; however, there is structural nonlinearity if the plates contact), (ii) a plate reinforced with a beam, (iii) a stack of two-layer beams (each component is both physically and geometrically nonlinear; their contact interaction causes structural nonlinearity), (iv) a stack of two-layer shells (each component is both physically and geometrically nonlinear; their contact interaction causes structural nonlinearity)

In this work we study PDEs governing beam dynamics under the Timoshenko hypotheses as well as the initial and boundary conditions which are yielded by Hamilton’s variational principle. The analysed beam is subjected to both uniform transversal harmonic load and additive white Gaussian noise. The PDEs are reduced to ODEs by means of the finite difference method employing the finite differences of the second-order accuracy, and then they are solved using the 4th and 6th order Runge-Kutta methods. The numerical results are validated with the applied nodes of the beam partition. The so-called charts of the beam vibration types are constructed versus the amplitude and frequency of harmonic excitation as well as the white noise intensity.

The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA).

The Euler–Bernoulli kinematic model as well as the von Karman geometric non-linearity are used to derive the PDEs governing flexible beam vibrations. The beam is embedded into a 2D temperature field, and its surface is subjected to action of the electric potential. We report how an increase of the exciting load amplitude yields the beam turbulent behavior, and how the temperature changes a scenario from a regular/laminar to spatio-temporal/turbulent dynamics. Both classical Fourier analysis and Morlet wavelets are used to monitor a strong influence of temperature on regular and chaotic beam dynamics.