Search

Mechanical periodic structures exhibit unusual dynamic behavior thanks to the periodicity of their structures, which can be attributed to their cellular arrangement. The source of this periodicity may result from periodic variations of material properties within their cells and/or variations in the cell geometry. The authors present the results of their studies on the optimization of physical parameters of a three-dimensional axisymetrical periodic beam in order to obtain the desired vibroacoustic properties. The aim of the optimization process of the unit cell shape was to obtain band gaps of a given width and position in the frequency spectrum. [ABSTRACT FROM AUTHOR]

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports. [ABSTRACT FROM AUTHOR]

The dynamic response of periodically restrained flexural structures, modeled as Euler–Bernoulli beams supported by flexible and dissipative supports at equally spaced points, to moving loads is investigated. Bloch–Floquet theorem and modal analysis are used to examine the dispersion relation and band structure of the infinitely long periodic beam and the frequency spectrum of the finite beam, respectively. It is shown that the natural frequencies of the finite beam fall within the propagation bands of the corresponding infinite beam and get closer to each other as the number of spans increases ultimately covering the propagation bands. The dynamic response to moving load is obtained via a novel method based on Floquet transformation in the frequency domain for infinite periodic beams and through modal summation for finite periodic beams. Through numerical examples, the effect of boundary conditions, e.g. support stiffness and damping, and the impact of moving load speed on the flexural response as well as the energy stored and dissipated within the supports are discussed. [ABSTRACT FROM AUTHOR]

• In the work dynamics of a 3-D periodic beam is investigated. • The beam is modelled by 6-node SFEs of smooth geometry optimised to predefined dynamic properties. • This results in the emergence of one common frequency band gap in the beam spectrum. • The common frequency band gap allows to use the beam a sound isolator or filter. • Numerical and optimisation results by TD-SFEM/FEM agree well with experimental ones by 1D-SLDV. This paper presents results of investigations on a three-dimensional (3-D) isotropic periodic beam. The beam can represent a vibroacoustic isolator of optimised dynamic characteristics in the case of its longitudinal, flexural and torsional behaviour. The optimisation process concerned both the widths as well as the positions of particular frequency band gaps that are present in the frequency spectrum of the beam. Since the dynamic behaviour of the beam is directly related to its geometry, through an optimisation process of the beam geometry, desired dynamic characteristics of the beam were successfully obtained. For the purpose of the optimisation process a new numerical model of the beam, based on the spectral finite element method in the time domain (TD-SFEM), was developed by the authors. This model enabled the authors to investigate the beam behaviour not only in a wide frequency spectrum, but also ensured a high accuracy of the model predictions. The accuracy of this modelling approach was checked against well-known analytical formulas. However, in the case of the optimised geometry of the beam for the verification of the correctness of the modelling approach a commercial finite element method (FEM) package was used. Finally, based on the results of numerical predictions and optimised geometry of the beam a sample for experimental verification was prepared. Experimental measurements were carried out by the authors by the application of one-dimensional (1-D) laser Doppler scanning vibrometry (LDSV). The results of experimental measurements obtained by the authors confirmed the correctness of the numerical predictions, showing a high degree of correspondence. [ABSTRACT FROM AUTHOR]

Abstract The subject of this article is analysis of transverse vibrations of beams which geometric and material properties vary periodically along the longitudinal axis. The aim is to present averaged models that take into account the shear deformation and geometric non-linearity, and to analyse transverse vibrations of such beams in moderately large deflection range. As the theoretical foundations, we use Timoshenko beam theory with von Kármán-type non-linearity. This results in obtaining new differential equations with constant coefficients, some of which explicitly depend on the beam inhomogeneity period size. Then, a reasonably simplified model is proposed to describe the vibrations of the considered beams in the low frequency range. The differential equations are transformed into a system of algebraic equations according to the Galerkin method. The response of the beam to transverse harmonic load is investigated by means of a pseudo arc-length continuation scheme. Non-linear coupling between vibration modes and the possibility of superharmonic resonance occurrence are taken into account. As an example of application, few special cases of beam geometry and boundary conditions are examined and compared. The results have the potential application to structural vibration control. [ABSTRACT FROM AUTHOR]

Abstract Wave propagation control in piezoelectric meta-structures has been developed in recent years, which focuses on coupling proper electric media via piezoelectric materials for elastic wave filtering properties within a low frequency range. In order to develop broadband band gaps and diversified wave propagation characteristics of mechanical meta-structures for different complex practical applications, the research proposes an innovative piezoelectric beam meta-structure using electronic negative capacitance dual-adjacent/staggered electrical connections. In the proposed meta-structures, a unit periodic cell composed of four adjacent primitive periodic cells (including four piezoelectric patches with different polarization directions) are connected to negative capacitance circuit shunts using electrical dual-adjacent/staggered patterns. Based on the Timoshenko beam theory and wave propagation theory, a theoretical modeling of the proposed meta-structure is established for evaluating wave propagation properties. Furthermore, wave attenuation performance of the proposed structure is investigated and compared with the traditional meta-structure with negative capacitance independent electronic networks in which negative capacitance shunts are independently applied to single piezoelectric patch or bimorph piezoelectric patches within a unit primitive periodic cell. Results show that the proposed innovative meta-structure utilizes less negative capacitance shunts for achieving better wave attenuation performance, and has more stable negative capacitance control system in practical applications. [ABSTRACT FROM AUTHOR]

In this study, an Euler–Bernoulli beam carrying periodic arrays of meander-shaped beams is introduced. Each meander-shaped beam consists of several connected beam elements. Two models with different number of beam elements are considered. In each model, the effects of geometrical parameters on the lower and upper edges of the first three band gaps are investigated using the Adomian decomposition method. Results show that the wide band gaps at low frequency ranges can be obtained by changing the geometrical parameters. Furthermore, the band gaps are very close to each other for specific values of the geometrical parameters. Another advantage of this periodic beam is that its length is shorter than other types of periodic beams. These features make this periodic beam very useful in different applications of the band gap phenomenon such as vibration absorption. The finite element simulation (ANSYS software) is used to validate the analytical results and good agreement is found. [ABSTRACT FROM AUTHOR]

In this study, a new periodic beam model is introduced. This beam consists of the concentrated rigid masses and tapered beam elements with linearly variable width. The theoretical equations are derived by employing the Euler-Bernoulli beam and the Bloch–Floquet theorem and then solved using the generalized differential quadrature rule method to calculate the first two band gaps. The effects of the mass, mass moment of inertia and taper ratio on the widths and central frequencies of the first two band gaps are investigated. Results show that the wide band gaps at low frequency ranges can be obtained by changing the geometrical parameters. This is of interest for different applications of the band gap phenomenon such as broadband piezoelectric energy harvesting. Finally, the finite element simulation (ANSYS software) is used to validate the analytical method and good agreement is found. [ABSTRACT FROM AUTHOR]

A topology optimization method is developed for microstructure design of heterogeneous periodic beam structure aiming at its extreme or specified effective stiffness. The effective stiffness is calculated using a FEM formulation of asymptotic homogenization method for heterogeneous periodic beam. Sensitivity of stiffness to the density design variable is derived analytically based on the solution of unit cell problems under corresponding generalized strain fields. Implementation of optimization procedure is generalized to take full advantage of commercial FEM software capabilities, with several examples presented to demonstrate its effectiveness. It is shown here the proposed method is extendible to periodic truss beam design. [ABSTRACT FROM AUTHOR]

The flexural wave propagation in a periodic beam is investigated using the method of multiple reflections. The effects of the clamped boundary condition and disturbance on the flexural wave propagation in the periodic beam are considered. A propagating disturbance is incident upon a discontinuity and gives rise to transmitted and reflected waves, and all of the transmitted and reflected waves of given flexural wave incident upon the beam at some location are found and superposed. The method is extended to the case of incident fading wave. The relation between the wave field of incident waves and the wave field of resulting waves on any segment is expressed. Much attention is devoted to the response in the frequency ranges with gaps in the band structure for the corresponding periodic beam. As an example, the application of the results to the analysis of a finite periodic beam with clamped-free boundary condition and a propagating disturbance is then demonstrated. [ABSTRACT FROM AUTHOR]

The flexural wave propagation in a periodic beam with a propagating disturbance is studied by the use of the multi-reflection method. A propagating wave is incident upon a discontinuity and gives rise to transmitted and reflected waves. Here all of the transmitted and reflected waves of given flexural wave incident upon the beam at some specified location are found and superposed, and the method is extended to the case of incident evanescent wave. The results of incident waves at some location between discontinuities in a periodic beam are concerned. The relation between the wave-field of incident waves and the wave-field of resulting waves on any segments is expressed. As an example, the application of the results to the analysis of a finite periodic beam with a propagating disturbance is then demonstrated. The influences of the number of cells on the energy associated with propagating waves are considered. [ABSTRACT FROM AUTHOR]

Abstract: To determine the flexural vibration band gaps in periodic beams, the theoretical equations are derived by employing the Bloch–Floquet theorem and then solved by the use of the differential quadrature method. Moreover, a comprehensive parametric study is also conducted to highlight the influences of shear deformation, geometrical parameters and material parameters on the gaps. The results show that the method is efficient and accurate and that the bandwidth can be enlarged by changing the geometrical or material parameters. The existence of vibration gaps lends a new insight into vibration isolation applications in areas such as mechanical and civil engineering. [Copyright &y& Elsevier]

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 periodicity cell are studied, and the natural frequencies and mode shapes are obtained for a simply supported beam. The results are supported by a comparison with the finite element method and partially exact solutions. [ABSTRACT FROM AUTHOR]

In this paper a stability analysis of microperiodic beams resting on the periodic inhomogeneous foundation is carried out. The main issue of this considerations, which is the analytical solution to the governing equations characterised by periodic, highly oscillating and non-continuous coefficients, is overwhelmed by the application of the tolerance averaging technique. As a result of such application, the governing equation is transformed into a form with constant coefficients which can be solved using well-known mathematical methods. In several calculation examples, the convergence of the results of two derived averaged models is examined, as well as the convergence of the lowest value of the critical force parameter derived from the averaged models with the FEM model. The results prove the superiority of the presented analytical solution over the FEM analysis in the optimisation process. [ABSTRACT FROM AUTHOR]

In this paper mathematical modelling of vibrations and stability problems for slender visco-elastic periodic beams is considered. In order to take into account the effect of microstructure a certain non-asymptotic approach is applied, called the tolerance modelling method. This technique allows to replace the equation with non-continuous, highly oscillating, periodic coefficients by a system of differential equations with constant coefficients. Moreover, the derived equations describe the effect of microstructure on the overall behaviour of the beams under consideration. In the framework of the tolerance modelling governing equations of two different tolerance models can be obtained – the standard and the general, under weakened assumptions. To evaluate the proposed general tolerance model obtained results are compared with results calculated within the known asymptotic homogenization, i.e. by the asymptotic model. In this note mathematical models describing investigated beams are only presented and illustrated by a simple example. [ABSTRACT FROM AUTHOR]

The mathematical modelling of certain problems of vibrations and stability for periodic slender visco-elastic beams is presented in this note. To consider these problems and take into account the effect of the microstructure, the tolerance modelling approach is proposed. Using this technique, the equation with non-continuous, periodic, highly oscillating coefficients is replaced by a system of differential equations with constant coefficients. Moreover, these governing equations describe the effect of the microstructure on the overall behavior of the beams under consideration. The tolerance modelling can lead to equations of two different tolerance models—the standard and the general, under weakened assumptions. This averaging tolerance method was assessed by comparison with the asymptotic homogenization, the governing equations of which omit this effect. My considerations were limited to proposing and presenting only mathematical models describing investigated beams. In a simple analytical example, the application of the presented average models is shown. [ABSTRACT FROM AUTHOR]