Primena nelokalne teorije kontinuuma u analizi dinamičkog ponašanja i stabilnosti sistema spregnutih nano-struktura

This dissertation investigates vibration and stability behavior of complex nano-scale systems composed of single and multiple carbon nanotubes and graphene sheets. Based on the assumptions introduced through the nonlocal continuum theory, the nanotubes are modeled as nanobeams and graphene sheets are represented as nanoplates where the influence of inter-atomic forces and the discrete nature of nanomaterials are introduced as material parameters. To such mechanical models of nanostructures one can apply the second Newton’s law of motion or Hamilton’s principles to derive the governing equation of motion of the system. In order to obtain solutions of partial differential equations, the analytical and approximation methods will be employed. Special attention is devoted to determining the analytical solutions for natural frequencies and critical buckling load of systems with multiple nanostructures (nanorods, nanobeams and nanoplates) and special cases of such systems. Thus obtained analytical solutions are validated by using the numerical methods as well as the results from molecular dynamics simulations, where excellent agreement of the results is confirmed. In addition, the longitudinal vibration of systems with a single or multiple coupled nanorods will be analyzed using nonlocal elasticity and viscoelasticity theories. What should be noted are the effects of temperature changes and magnetic fields on the dynamic behavior of a cracked carbon nanotube embedded in an elastic medium. It is shown that the possibility of change in the overall system stiffness by changing the parameters of external physical fields leads to certain changes in natural frequencies without any change in other parameters of the model. The case of the free nonlinear vibration and dynamic stability of carbon nanotubes subjected to variable axial force and external magnetic field will be presented in the example of a single nanobeam embedded in a viscoelastic medium by considering the geometric nonlinearity. Analytical approximation results are determined for nonlinear frequencies, amplitude-frequency curve by using the multiple scales method. It is shown that it is possible to avoid resonant states as well as changes in stability and instability regions by changing the external magnetic field parameter without any change in other parameters of the system. A parametric study is performed for all presented systems, and effects of different physical and geometrical parameters on the dynamic behavior and stability are examined in detail

VIBRATION AND STABILITY OF A NONLINEAR NONLOCAL STRAIN-GRADIENT FG BEAM ON A VISCO-PASTERNAK FOUNDATION

This study investigates the stability of periodic solutions of a nonlinear nonlocal strain gradient functionally graded Euler–Bernoulli beam model resting on a visco-Pasternak foundation and subjected to external harmonic excitation. The nonlinearity of the beam arises from the von Kármán strain-displacement relation. Nonlocal stress gradient theory combined with the strain gradient theory is used to describe the stress-strain relation. Variations of material properties across the thickness direction are defined by the power-law model. The governing differential equation of motion is derived by using Hamilton's principle and discretized by the Galerkin approximation. The methodology for obtaining the steady-state amplitude-frequency responses via the incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the numerical integration method and stability analysis is performed by utilizing the Floquet theory

ENERGY ANALYSIS OF FREE TRANSVERSE VIBRATIONS OF THE VISCO-ELASTICALLY CONNECTED DOUBLE-MEMBRANE SYSTEM

The presented paper deals with the analysis of energy transfer in the visco-elastically connected circular double-membrane system for free transverse vibration of the membranes. The system motion is described by a set of two coupled non-homogeneous partial differential equations. The solutions are obtained by using the method of separation of variables. Once the problem is solved, natural frequencies and mode shape functions are found, and then the form of solution for small transverse deflections of membranes is derived. Using the obtained solutions, forms of reduced kinetic, potential and total energies, as functions of dissipation of the whole system and subsystems, are determined. The numerical examples are given as an illustration of the presented theoretical analysis as well as the possibilities to investigate the influence of different parameters and different initial conditions on the energies transfer in the system.

Nonlocal axial vibration of a fractional order viscoelastic nanorod

This communication presents a novel nonlocal fractional order viscoelastic model of nanorod. The main assumptions for the proposed model are discussed. Solution of the motion equation involving fractional order derivative is presented. Influences of a nonlocal parameter and fractional order derivative on the free damped vibration of nanorod are presented through numerical example

Nonlinear vibration of a nonlocal functionally graded beam on fractional visco-Pasternak foundation

This paper investigates the nonlinear dynamic behavior of a nonlocal functionally graded Euler-Bernoulli beam resting on a fractional visco-Pasternak foundation and subjected to harmonic loads. The proposed model captures both, nonlocal parameter considering the elastic stress gradient field and a material length scale parameter considering the strain gradient stress field. Additionally, the von Karman strain-displacement relation is used to describe the nonlinear geometrical beam behavior. The power-law model is utilized to represent the material variations across the thickness direction of the functionally graded beam. The following steps are conducted in this research study. At first, the governing equation of motion is derived using Hamilton's principle and then reduced to the nonlinear fractional-order differential equation through the single-mode Galerkin approximation. The methodology to determine steady-state amplitude-frequency responses via incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the perturbation multiple scales method for the weakly nonlinear case and numerical integration Newmark method in the case of strong nonlinearity. It has been shown that the application of the incremental harmonic balance method in the analysis of nonlocal strain gradient theory-based structures can lead to more reliable studies for strongly nonlinear systems. In the parametric study, it is shown that, on the one hand, parameters of the visco-Pasternak foundation and power-law index remarkable affect the amplitudes responses. On the contrary, the nonlocal and the length-scale parameters are having a small influence on the amplitude-frequency response. Finally, the effects of the fractional derivative order on the system's damping are displayed at time response diagrams and subsequently discussed.This is the peer reviewed version of the article: Nešić, N.; Cajić, M.; Karličić, D.; Obradović, A.; Simonović, J. Nonlinear Vibration of a Nonlocal Functionally Graded Beam on Fractional Visco-Pasternak Foundation. Nonlinear Dynamics 2022, 107 (3), 2003–2026. [https://doi.org/10.1007/s11071-021-07081-z

The State Space Model of a Single-Link Flexible Robot with a Fractional Order Viscoelastic Element in the Joint

This paper extends some results from the area of vibration damping of single-link flexible robots. Instead of the integer derivative model, the fractional order derivative model of a viscoelastic element is proposed. Euler-Bernoulli beam theory for a single-link flexible robot is used. An analytical solution for a set of decoupled ordinary differential equations is found and the state-space model with fractional order derivatives is forme

Nonlocal elasticity and fractional viscoelasticity models of nanobeams and nanoplates

It is well known that nonlocal elasticity models are successfully applied to various nanostructure based systems to study their stability or vibration behavior [1]. Such modified continuum approach shows to be reliable and much more efficient way to study complex nano-scale systems and structures compared to atomistic methods based on discrete nature of nanostructures. Nonlocal elasticity introduces the scale effects into the model via single material parameter also called nonlocal parameter. Dissipation of mechanical energy in nanostructures is important feature of nano-scale system that significantly affects their dynamic or stability behavior. Various rheological models can be applied to describe such effects. However, well known comparison of fractional derivative rheological models compared to classical integer order one, candidates them for this application [2]. Finally, combination of nonlocal elasticity and fractional order viscoelasticity constitutive relations yield hybrid models that due to their nonlocal nature can describe nonlocality in space domain as well as relaxation/retardation processes in time domain. Here, we apply nonlocal elastic and fractional viscoelastic models to study vibration behavior of nanoplate and nanobeam like structures [3]. Euler-Bernoulli beam theory and Kirchhoff-Love plate theory are used for nanobeams and nanoplates, respectively. Several fractional derivative rheological models are shown and some of them applied to given nanostructure models. Governing equations are derived using D’Alambert’s principle and solutions for the simply supported boundary conditions are found using separation of variables, Laplace transform and Mellin-Fourier inverse transform methods as well as residue theory. Complex poles of unknown functions are determined by finding the roots of the characteristic equation using technique that is available in the literature. In order to show the effects of fractional derivative parameters, damping coefficients and nonlocal parameter on complex roots i.e. damped frequencies and damping ratios as well as on transient response of the systems, several numerical examples are given

Combined sub-harmonic resonances of nanobeam on fractional visco-Pasternak type foundation

In this work, we observe combined parametric and external sub-harmonic resonances of order one-third of a geometrically nonlinear nonlocal nanobeam model resting on a fractional visco- Pasternak type foundation. Euler-Bernoulli beam theory, nonlinear strain-displacement relation and nonlocal elasticity constitutive equation are employed to obtained fractional order governing equation for the transverse vibration of a system. Under the assumption of small fractional damping, we used the perturbation multiple-scales method to obtain an approximated analytical solution for the frequency-amplitude response for variable axial and transverse external loads. Several numerical examples are given to show the effects of different parameters on frequencyamplitude response

Parametrically excited unidirectional wave propagation in thin beam phononics

Wave attenuation, filtering and guiding is an ongoing topic of scientific research, as there are many opportunities for improvement of existing solutions in modern industry. One of the recent advancements has been made with the use of non-reciprocal metamaterials.Certain properties of metamaterials have made them suitable for use in various engineering fields. In this study, we investigate non-reciprocal wave propagation behavior in coupled thin beams phononics, due to time-modulation of material properties and axial loads. We compare the results for the beams which are interconnected with Winkler’s type of elastic layers and elastic or viscoelastic Pasternak layers. An analytic approach is used to discover directional band gaps and investigate wave propagation through these systems of beams, at relevant excitation frequencies. The proposed framework can be exploited in further analysis of phononic systems based on multiple beams coupled through different mediums and structural elements modeled with higher-order beam theories

Non-reciprocal wave propagation in periodically structured Timoshenko beams

In this paper we will investigate non-reciprocal wave propagation in Timoshenko beams, due to space and time modulation of its elastic properties. To that end, an analytical approach is used: the Bloch theorem is applied when choosing the solution form for displacement components and the angle of rotation, which figure in the equations of motion along with the elastic properties. Also, the Fourier expansion is used to express the periodic nature of the modulation. By solving the eigenvalue problem for different modulation parameters, we obtain the band diagrams which can be used to analyze the directionality of wave propagation. These diagrams clearly represent the breakage of symmetry as a consequence of modulation. Thus, a modulated beam behaves as a kind of metamaterial, in which one-way propagation of elastic waves is possible. When shear and rotational effects are neglected, these results converge to the results for the Euler-Bernoulli beam, which are already present in scientific literature