REFLECTION OF FLEXURAL WAVES IN GEOMETRICALLY PERIODIC BEAMS

2020

In this thesis, the definition and effects of quasi-periodicity in periodic structure are investigated. More importantly, the presence of irregularity in periodic structures and its significant impact in vibroacoustic responses of elastic systems are analyzed. In the extant literature, it has already shown that a sandwich panel, optimized for vibroacoustic performance with added random properties of the core, can exhibit stop band characteristics in some frequency ranges. Therefore, an additional target can exist in framing the abovementioned property under the Wave Finite Element Method (WFEM) for resulting in some design guideline. In this paper, (1) the numerical stud- ies of the vibrational analysis of 1D finite, periodic, and quasi-periodic beams are presented. The paper's content deals with the finite element models of beams focusing on spectral analysis and the damped forced responses. The quasi-periodicity is defined by invoking the Fibonacci sequence for building the as...

The International Conference on Applied Mechanics and Mechanical Engineering, 2008

Periodic Structures have been in the focus of research for their useful characteristics and ability to attenuate vibration in frequency bands called "stop-bands". In the previous studies, periodic changes in the geometry of beams were introduced to create the periodicity; then the vibration characteristics were studied accordingly. In this study, for the first time, we are analyzing the vibration characteristics of a sandwich beam with the periodic change in the sandwich material. The new technique preserves the external geometry of the beam structure and depends on changing the material of the sandwich material. The periodic analysis and the vibration response characteristics of the model are investigated using finite element model for a sandwich beam. The response to bending excitation has shown promising results for periodic sandwich beams, that may encourage further study of the problem with more practical configurations.

This paper is concerned with flexural wave propagation and vibration transmission in beams with periodically attached vibration absorbers. Such periodic systems feature unique wave filtering characteristics that can find applications in the control of wave propagation in flexural beam structures. The study is performed by using an exact analytical approach based on a combination of the spectral element method and periodic structure theory. Both infinite and finite periodic structures are considered. An explicit expression is provided for the calculation of propagation constants and thus the complex band structures, and it is further developed to examine the effects of various system parameters on the band-gap behavior, including the position, width and wave attenuation performance of all the band gaps. The band formation mechanisms of such periodic systems are explained via both derivations and physical models, yielding explicit equations to enable the prediction of all the band edge frequencies in an exact manner without the need to calculate propagation constants. Based on these equations, explicit formulas are further derived to determine the conditions for the transition and near-coupling between local resonance and Bragg scattering, each being a unique band-gap opening mechanism.

Periodic Structures have been in the focus of research for their useful characteristics and ability to attenuate vibration in frequency bands called "stop-bands". In the previous studies, periodic changes in the geometry of beams were introduced to create the periodicity; then the vibration characteristics were studied accordingly. In this study, for the first time, we are analyzing the vibration characteristics of a sandwich beam with the periodic change in the sandwich material. The new technique preserves the external geometry of the beam structure and depends on changing the material of the sandwich material. The periodic analysis and the vibration response characteristics of the model are investigated using finite element model for a sandwich beam. The response to bending excitation has shown promising results for periodic sandwich beams, that may encourage further study of the problem with more practical configurations.

Fibre Science and Technology, 1983

This paper deals with the propagation of waves through a nonhomogeneous elastic material, with a periodic structure of elastic constant and density variation, in terms of Floquet waves. Dispersion curves are evaluated and plotted for some particular examples.

Archives of Civil and Mechanical Engineering, 2018

Considered are free and forced transverse vibrations of slender periodic beams of finite length. It is assumed that the vibration amplitude is of the order of cross-section dimensions, still much smaller than the beam length. An averaged non-asymptotic model is proposed as a tool in analysis. The description is based on the tolerance approach to averaging of differential operators, using the concept of weakly slowly-varying function. The resulting differential equations with constant coefficients involve the effect of periodicity cell length. The model is verified by comparison of linear frequencies and mode shapes with Finite Element Method results, and then applied in analysis of free and forced vibrations of beam with variable cross-section. The method employed in obtaining the solution involves Galerkin orthogonalization and Runge-Kutta (RKF45) method. The results of nonlinear vibrations analysis are presented by backbone and amplitude-frequency response curves, time series, Poincare sections and bifurcation diagrams.

2020

In this thesis, the definition and effects of quasi-periodicity in periodic structure are investigated. More importantly, the presence of irregularity in periodic structures and its significant impact in vibroacoustic responses of elastic systems are analyzed. In the extant literature, it has already shown that a sandwich panel, optimized for vibroacoustic performance with added random properties of the core, can exhibit stop band characteristics in some frequency ranges. Therefore, an additional target can exist in framing the abovementioned property under the Wave Finite Element Method (WFEM) for resulting in some design guideline. In this paper, (1) the numerical stud- ies of the vibrational analysis of 1D finite, periodic, and quasi-periodic beams are presented. The paper's content deals with the finite element models of beams focusing on spectral analysis and the damped forced responses. The quasi-periodicity is defined by invoking the Fibonacci sequence for building the as...

Materials, 2021

This paper deals with the linear natural vibrations analysis of beams where the geometric and material properties vary periodically along the beam axis. In contrast with homogeneous prismatic beams, the frequency spectra of such beams are irregular as there exist enlarged intervals between some adjacent frequencies. Presented here are two averaged models of beams based on the tolerance modelling approach. The assumptions of classical Euler–Bernoulli and Timoshenko–Ehrenfest beam theories are adopted as the foundations. The resulting mathematical models are systems of differential equations with constant, weight-averaged coefficients. This makes it possible to apply any classical method of solution suitable for homogeneous beams, such as Galerkin orthogonalization. Here, emphasis is placed on the comparison of natural frequencies neighbouring the frequency band-gaps that are obtained from these two theories. Two basic cases of material and geometric property distribution in a periodi...

Nonlinear Dynamics, 2017

Nonlinear finite-DOF dynamical system is derived to describe the beam vibrations with transversal crack. The beam deflections are expanded by using the eigenmodes and contact parameter. The Galerkin method is applied to the partial differential equation, which describes the structure vibrations. Two-and three-DOF nonlinear dynamical systems with internal resonance are analyzed. The multiple scales method is used to investigate both the principle second resonance and the combination one. The resonance quasiperiodic and sub-harmonic motions are analyzed. The quasi-periodic motions are arisen due to the Neimark-Sacker bifurcation.

Vibrations in Physical Systems, 2018

Elastic periodic structures with variable material and geometrical properties exhibit dynamic characteristics that are investigated in this contribution. The paper is devoted to analysis of geometrically linear vibrations of Rayleigh and Timoshenko beams with cross-sections and material properties periodically varying along the longitudinal axis. The period of inhomogeneity is assumed to be sufficiently small when compared to the beam length. Equations of motion in both beam theories under consideration have highly-oscillating coefficients. In order to derive the averaged model equations with constant coefficients for vibrations, the tolerance averaging approach is applied. The method of averaging differential operators with rapidly varying coefficients is applied to obtain averaged governing equations with constant coefficients. An assumed tolerance and indiscernibility relations and the definition of slowly varying function found the applied technique. Numerical results from the tolerance Rayleigh and Timoshenko beam model equations are compared.