Search

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.

Here, the transfer matrix method along with the Bloch theorem are employed to study the frequency spectra in various cases of Timoshenko’s beam periodicity, such as structural, material and boundary periodicity. We additionally compare the location and size of band gaps, and the shape of the band structure in general for different types of foundation layers and relevant parameter values.

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.

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.

This paper shows the transverse vibration analysis of complex multiple coupled nanobeams system. Observed system is composed of an arbitrary number of aligned nanobeam structures embedded in viscoelastic medium where damping features of nanobeams and viscoelastic medium are represented by the fractional viscoelastic models while small-scale effects are considered using the Eringen’s nonlocal elasticity. In addition, non-homogeneities of the system such as different nanobeams cross-sections or density are observed within the paper. Governing equations are derived using using the d’Alembert’s principle, differential form of nonlocal stress relation and local stress-strain viscoelastic constitutive equation of fractional Kelvin-Voigt type. Semi-analyitical solutions for the transient response of simply supported nanobeams in the system are obtained using the separation of variables method, Laplace and Mellin-Fourier integral transforms, residue theory and modal expansion method. The problem of decoupling the governing equations of a non-homogenous system is solved by adopting the methodology from the literature. Several numerical examples are given to show the effects of different physical parameters on the transient response of such a system. Presented dynamic analysis and results could be used in future theoretical studies with additional physical effects included into the model. There is also a potential for use of this type of analysis in design procedures of modern nanodevices for calculation of their dynamic behavior that is much faster compared to atomistic based models.

The investigation into the dynamics of robotic and complex mechanical systems has been an active topic of research for many years. The modelling complex rigid multibody systems (RBS) using symbolic equations can provide many advantages over the more widely-used numerical methods of modeling these systems. In this contribution, we propose using procedure for recursive symbolic form computation of the complete dynamics of robotic systems with the open kinematic chain structures using Rodriquez approach for matrices of coordinate transformations. Dynamic equations are given as Lagrange equations of the second kind in the covariant form with external generalized forces of the gravity, motor-torque, viscous and spring. On the other side, the use of adaptive (hereditary/actuator: viscoelastic element with an actuator, piezo-viscoelastic,thermo-viscoelastic and magneto-rheologic) elements in complex RBS can be significant for the additional control of these systems and for reducing undesirable vibrations. Recently, fractional calculus (FC) has attracted an increased attention of scientific society. The fractional integro-differential operators are a generalization of integration and derivation to fractional operators where fractional derivatives (FD) are often used to describe viscoelastic, rheological properties of advanced materials and dissipative forces in structural dynamics. Here, modeling of dynamics of multibody systems involving generalized forces of a spring/spring-pot/actuator (SSPA) and MR elements modeled with fractional order derivatives, including recently new obtained definitions of FD, is studied. The system is defined as a discrete material system where in the force–displacement relation, mass of the element is neglected. Generalized forces of an element are derived by using the principle of virtual work and force–displacement relation of the fractional order Kelvin–Voigt/Zener model. Finally, the results obtained for generalized forces are compared for d

Предмет ове докторске дисертације је истраживање у области моделирања и анализе динамичког понашања система крутих тела повезаних хибридним елементима фракционог типа и сложених система наноструктура чија су вискоеластична својства описана изводима фракционог реда..., The research subject of this doctoral dissertation is in the field of modeling and analysis of dynamic behavior of rigid multibody systems, connected with fractional order hybrid elements, and complex nanostructure based systems with fractional order viscoelastic properties...

In this paper a closed-loop PD alpha-type iterative learning control (ILC) of fractional order singular time-delay system is considered. In particular, we discuss fractional order linear singular timed-delay systems in state space form. The sufficient conditions for the convergence in time domain of the proposed PD-alpha type ILC for a class of fractional order singular system are given by the corresponding theorem together with its proof. Finally, the validity of the proposed PD alpha 1LC scheme for a class of fractional order singular time-delay system is verified by a numerical example.

Nanostructures with similar shapes as macro structures such as beams, plates or shells are grown using various technological processes. Nano-scale structures mechanical behaviour is similar to macro structures with the main difference that influences of various size-effects cannot be neglected. Particulary, dynamic behaviour of nanostructures is significant for the development of new types of devices such as nanoactuators, nanosensors or resonators. Nonlinear oscillations and dissipation effects from different sources are important for the investigation of nano-scale resonators. Application of experimental and atomistic methods is limited only to specific and less complex nanostructure systems. Modified continuum based theoretical models have attracted a great attention of researchers in recent years. Here, we observe combined parametric and external resonance of a geometrically nonlinear nanobeam 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 systems, accounting size effects and dissipation effects from external medium. 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 in combined parametric and external resonance case for variable axial and transverse loads. Influences of different system parameters on frequency-amplitude response are examined on several numerical examples.

In this communication, we observe the interaction of fundamental parametric resonances with subharmonic resonances of order one-half of a geometrically nonlinear nonlocal nanobeam model resting on a fractional Pasternak-type viscoelastic foundation. Euler-Bernoulli beam theory and nonlinear von Karman strain-displacement relation 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. Combined parametric resonance from axial load and subharmonic resonance under external excitation are examined for different parameters of the model. Validation of the multiple scales solution against numerical solution in the phase plane and Poincare map will be provided

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.

In this paper stability problem of double inverted pendulum controlled by a fractional differential compensator is investigated. Pendubot is an underactuated mechanical system, i.e. it has only one control input and two degrees of freedom. Detailed mathematical model of Pendubot is derived using the Rodriguez method and then fractional order lead compensator is introduced in order to stabilize it around unstable upright position. D-decomposition method is used to solve the problem of asymptotic stability of closed loop system. Stability regions in control parameters space are calculated using this technique, which allows tuning of the fractional differential compensator to be carried out.