Periodic Beam Structure Research Articles

Research on Vibration Absorption and Isolation Characteristics of Periodic Acoustic Black Hole Beam Resonators and their Enhancement

An acoustic black hole (ABH) resonator is regarded as an efficient approach for controlling vibration caused by flexural wave energy. In this paper, the beam models with periodic ABH beam resonators are designed. Both the vibration absorption and isolation performances are investigated. Theoretical models based on the Transfer Matrix Method are presented to evaluate the reflection coefficient, which is validated both by the semi-analytic method combined with the Finite Element Method (FEM) and the Impedance Matrix Method. Meanwhile, FEM models of periodic ABH beam resonators acting as the beam terminator and isolator are established and analyzed. The results show that the periodic ABH beam resonators are of a better vibration reduction performance in lower frequency and have wider bandgaps for lower reflection coefficient and higher transmission loss than the single wedge. Moreover, with the increasing number of periods, the advantages of the periodic ABH beam resonators in reducing vibration become more obvious. Through the complex plane and dynamic analyses, it shows that multimode coupling and meta-damping effect lead to superior performance since the enriched modal content is introduced by the periodic ABH beam structure. This effect is also verified by the experimental result. Besides, the study also reveals the paradoxical relationship between vibration absorption and isolation performances. Additionally, parametric studies are conducted to disclose the effects of structural parameters. Based on the analyses, two approaches are proposed to enhance the vibration reduction performances, including the composite beam resonators and compound beam resonators. This paper illustrates a promising vision for applying the periodic ABH beam resonators to various vibration control fields.

Vibration control of periodically supported pipes employing optimally designed dampers

Vibration in periodic piping structures requires efficient attenuation measures. Structures composed of periodically arranged identical units show distinct bandgap characteristics, wherein waves and vibrations of particular frequencies cannot pass through. Along this vein, a pipe supported periodically on a rack structure is analyzed to determine the flexural wave propagation characteristics and thereby to design vibration control strategies. To understand the influence of stiffness of the rack on wave propagation, two models are examined; (i) pipe supported on a rack and (ii) pipe without rack. The relevant dispersion relations for both the cases are derived by means of Floquet-Bloch theory and the resulting bandgaps are validated using FE models. The results show that resonance and Bragg type bandgaps emerge in the pipe due to the existence of local resonance and spatial periodicity. The dependence of the propagation characteristics on the number of unit-cells is then investigated. It is found that attenuation within the bandgaps is significantly enhanced with an increase in the number of unit-cells. Further, to control two specific frequency ranges within a particular passband, two single-degree-of-freedom (SDoF) localized mass dampers (LMDs) are attached to the center of each unit-cell of the pipe. The LMDs are provided with identical mass ratios and are designed using a genetic algorithm (GA) based optimization for both the models. As an alternative to the LMDs, a novel possibility of using already existing pipes of smaller cross-sections in the rack as distributed mass dampers (DMDs) are evaluated. For identical mass ratios, the performance of LMDs was found to be higher than the DMDs. However, the latter represents a new, cost-effective and promising technique for vibration mitigation. Further, the effectiveness of both the LMDs and DMDs is verified by applying a Gaussian white noise as input. The findings of this paper are not only limited to pipelines, but can be applied when similar periodic beam structures are present in the system such as railway tracks, bridges resting on piers, etc.

Multi-Objective Collaborative Flexibility Optimization Design of the Strengthening Layer of a Flexible Riser

Abstract Marine flexible risers are widely used in ocean oil and gas extraction, and need to withstand environment loads (wave and current) and the large offset of the floater. Therefore, the flexible riser is subjected to tension, bending, and torsion loads, which are mainly resisted by the key strengthening layer. Small bending stiffness of a cross section of the strengthening layer with larger tension and torsion stiffness are required to be compliant with the ocean environment. The traditional design of the key strengthening layer is partially rigid with larger cross-sectional stiffnesses. Therefore, the innovative configurations of the strengthening layer are imperative to make sure that the flexible riser is reliable and safe during the installation and operation. The strengthening layer of the flexible riser is treated as the cylindrical shell composed of periodic unit-cell beam structures, which is a hypothetical model. The optimization design is conducted through the novel implementation of the asymptotic homogenization (NIAH) method. The multi-objective collaborative flexibility optimization formulation of cylindrical shell structure is proposed, considering the ratio of cross-sectional tensile torsion stiffness to bending stiffness of the strengthening layer as the objectives. The optimal configuration results, the helically wound structures, are obtained, which are the alternative strengthening components of flexible risers. Finally, the optimal structures are compared with the commonly used marine flexible riser, which gives a great verification of the methodology feasibility and explains why the strengthening layer is designed as the type of helically wound structure.

Microstructural topology optimization of periodic beam structures based on the relaxed Saint-Venant solution

Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

On the solutions to the Saint–Venant problem of heterogeneous beam-like structures with periodic microstructures

This paper presents displacement solutions of the Saint-Venant problem for linear elastic heterogeneous beam structures with arbitrarily shaped periodic microstructures. The solutions are generated by introducing unknown axially periodic displacement functions in place of the axially invariant warping functions in Ieşan's rational scheme, due to the periodic characteristics of the beam structure, while retaining the displacement fields integrated from rigid-body displacements, which represent basic kinematics in beam theories. The governing equations of the unknown functions, defined on a base cell, are then developed from equilibrium equations and boundary conditions of the periodic beam structure. Moreover, an improved FE formulation of the governing equations is presented and its efficient numerical implementation approach, which can be readily realized using FE software as a black box, is proposed with the aid of NIAH (Novel Numerical Implementation of Asymptotic homogenization) approach. To illustrate the validity of the Saint-Venant solution and the effectiveness of the numerical implementation approach, several numerical examples are presented and compared with detailed three-dimensional FE analysis for the heterogeneous beam structures in terms of stress components.

Generating Lieb and super-honeycomb lattices by employing the fractional Talbot effect

We demonstrate a novel method for producing optically-induced Lieb and super-honeycomb lattices, by employing the fractional Talbot effect of specific periodic beam structures. Our numerical and analytical results display the generation of Lieb and super-honeycomb lattices at fractional Talbot lengths effectively and with high beam quality. By adjusting the initial phase shifts of the interfering beams, the incident periodic beam structures, as well as the lattices with broken inversion symmetry, can be constructed in situ. This research suggests not only a possible practical utilization of the Talbot effect in the production of novel optically-induced lattices but also in the studies of related optical topological phenomena.

Elastic wave propagation in a periodic composite beam structure: A new model for band gaps incorporating surface energy, transverse shear and rotational inertia effects

A new model for determining band gaps for elastic wave propagation in a periodic composite beam structure is developed using a non-classical Timoshenko beam model that incorporates the surface energy, transverse shear and rotational inertia effects. The Bloch theorem and transfer matrix method for periodic structures are employed in the formulation. The new model reduces to the classical elasticity-based model when the surface energy effect is not considered. It is shown that the band gaps predicted by the current model depend on the surface elastic constants of each constituent material, beam thickness, unit cell size, and volume fraction. The numerical results reveal that the band gap based on the current non-classical model is always larger than that given by the classical model when the beam thickness is very small, but the difference is diminishing as the thickness becomes large. Also, it is found that the first frequency for producing the band gap and the band gap size decrease with the increase of the unit cell length according to both the current and classical models. In addition, it is observed that the volume fraction has a significant effect on the band gap size, and large band gaps can be obtained by tailoring the volume fraction and material parameters.

Vibration characteristics of a sandwich plate with viscoelastic periodic cores

The vibration characteristics of a sandwich plate with viscoelastic periodic cores are examined analytically and experimentally, which extends the previous research of corresponding periodic sandwich beam structure. Closed-form solutions for forced response and band structure of periodic sandwich plate are theoretically derived, providing computational support on the attenuation analysis. In the theoretical model, a new admissible displacement function for sandwich plate is proposed. Although it is used in free boundary condition in this paper, it is also suitable for clamped, simply supported, slipping, and elastic boundary conditions. The formation of the band gap is carefully studied, showing that the overall band gap is proved to be the intersection of all the cross-stream modal band gaps, which is quite different from a corresponding periodic beam structure. The parametric analysis shows further that the overall band gap could disappear when length ratio, element width, or core thickness exceeds a cut-off value, which provides a guidance in the band-gap design. The attenuation in a sandwich plate with viscoelastic periodic cores is mainly dominated by Bragg scattering mechanism in the band gap and by energy dissipation out of the band gap. Owing to the combined effect of both mechanisms, the sandwich plate with viscoelastic periodic cores provides better attenuation performance than that with a uniform viscoelastic core. This research could possibly provide useful guidance for the researches and engineers on the design of plate-type damping structures.

Wave component solutions of free vibration and mode damping loss factor of finite length periodic beam structure with damping material

Free vibration and mode damping loss factor of the finite periodic beam structure with viscoelastic damping material are investigated in this paper based on wave component method. A simple iterative operation for the natural vibration modes of the beam structure is firstly proposed by considering that the mode shape can be formed through superposition of the wave shapes of the wave components of the beam structure at the corresponding natural frequency. And then, wave component solution of the mode damping loss factor of the beam structure is obtained and analyzed based on the modal stain energy method. It is found that the kinetic energy of the unit cell of propagative wave component is always the same with the strain energy, and for non-propagative wave component these two quantities will never be equal. In numerical examples, the proposed iterative operation is validated.

Analysis and design of periodic beams for vibration attenuation

This paper presents a new finite-element-based design procedure for geometrically periodic beam structures to target a specific desired attenuation frequency band. Using the “forward approach,” the effects of the cell geometric parameters on the location of the stop bands are first studied. Then using the “reverse approach,” explicit expressions for the start and end frequencies of all stop bands are derived as functions of the cell geometric parameters using a symbolic solver. These expressions are used in the design procedure to determine the cell geometric parameters that will result in vibration attenuation in the desired attenuation band. The design procedure has been validated by comparison with the results of the forward approach in different problems targeting the first or second stop bands. It was found that, depending on the frequency range of the desired attenuation band, and the stop band selected to target the desired attenuation band in the design, one, two or no solution set for the cell geometric parameters can be obtained. For the case of two available solutions, the designer will have two design options that satisfy the design requirements to choose from.

Band gaps for elastic wave propagation in a periodic composite beam structure incorporating microstructure and surface energy effects

A new model for determining band gaps for elastic wave propagation in a periodic composite beam structure is developed using a non-classical Bernoulli–Euler beam model that incorporates the microstructure, surface energy and rotational inertia effects. The Bloch theorem and transfer matrix method for periodic structures are employed in the formulation. The new model reduces to the classical elasticity-based model when both the microstructure and surface energy effects are not considered. The band gaps predicted by the new model depend on the microstructure and surface elasticity of each constituent material, the unit cell size, the rotational inertia, and the volume fraction. To quantitatively illustrate the effects of these factors, a parametric study is conducted. The numerical results reveal that the band gap predicted by the current non-classical model is always larger than that predicted by the classical model when the beam thickness is very small, but the difference is diminishing as the thickness becomes large. Also, it is found that the first frequency for producing the band gap and the band gap size decrease with the increase of the unit cell length according to both the current and classical models. In addition, it is observed that the effect of the rotational inertia is larger when the exciting frequency is higher and the unit cell length is smaller. Furthermore, it is seen that the volume fraction has a significant effect on the band gap size, and large band gaps can be obtained by tailoring the volume fraction and material parameters.

Broadband locally resonant band gaps in periodic beam structures with embedded acoustic black holes

The Acoustic Black Hole (ABH) effect can be used to effectively reduce structural vibrations by trapping flexural waves in a thin-walled structure with a power-law thickness variation. In the present study, we used a wavelet-decomposed energy method to investigate an Euler-Bernoulli beam embedded with multiple ABHs. Broadband transmission attenuation bands at relatively low frequencies are observed in a beam containing only a few ABH elements. To explain the underlying phenomena, an infinite structure with periodic ABH elements is analyzed. Numerical results show that the periodic boundary conditions in terms of displacement and rotational slope of a unit cell, based on the finite model, are sufficient to describe the band structures, without requiring full treatment of the entire infinite structure. This provides an efficient and flexible means to predict, and eventually optimize, the band structure based on a single element. Meanwhile, the ABH-induced locally resonant band gaps coincide with the attenuation bands observed in the finite beams. Because of the unique ABH feature, the proposed beam requires only a small number of elements to obtain broad attenuation bands, which offers great potential for vibrational isolation applications and wave filter designs in beam structures.

Active tuning of vibration and wave propagation in elastic beams with periodically placed piezoelectric actuator/sensor pairs

A novel strategy is proposed to actively tune the vibration and wave propagation properties in elastic beams. By periodically placing the piezoelectric actuator/sensor pairs along the beam axis, an active periodic beam structure which exhibits special vibration and wave propagation properties such as the frequency pass-bands and stop-bands (or band-gaps) is developed. Hamilton's principle is applied to establish the equations of motion of the sub-beam elements i.e. the unit-cells, bonded by the piezoelectric patches. A negative proportional feedback control strategy is employed to design the controllers which can provide a positive active stiffness to the beam for a positive feedback control gain, which can increase the stability of the structural system. By means of the added positive active stiffness, the periodicity or the band-gap property of the beam with periodically placed piezoelectric patches can be actively tuned. From the investigation, it is shown that better band-gap characteristics can be achieved by using the negative proportional feedback control. The band-gaps can be obviously broadened by properly increasing the control gain, and they can also be greatly enlarged by appropriately designing the structural sizes of the controllers. The control voltages applied on the piezoelectric actuators are in reasonable and controllable ranges, especially, they are very low in the band-gaps. Thus, the vibration and wave propagation behaviors of the elastic beam can be actively controlled by the periodically placed piezoelectric patches.

Vibration reduction for smart periodic structures via periodic piezoelectric arrays with nonlinear interleaved-switched electronic networks

Smart periodic structures covered by periodically distributed piezoelectric patches have drawn more and more attention in recent years for wave propagation attenuation and corresponding structural vibration suppression. Since piezoelectric materials are special type of energy conversion materials that link mechanical characteristics with electrical characteristics, shunt circuits coupled with such materials play a key role in the wave propagation and/or vibration control performance in smart periodic structures. Conventional shunt circuit designs utilize resistive shunt (R-shunt) and resonant shunt (RL-shunt). More recently, semi-passive nonlinear approaches have also been developed for efficiently controlling the vibrations of such structures. In this paper, an innovative smart periodic beam structure with nonlinear interleaved-switched electric networks based on synchronized switching damping on inductor (SSDI) is proposed and investigated for vibration reduction and wave propagation attenuation. Different from locally resonant band gap mechanism forming narrow band gaps around the desired resonant frequencies, the proposed interleaved electrical networks can induce new broadly low-frequency stop bands and broaden primitive Bragg stop bands by virtue of unique interleaved electrical configurations and the SSDI technique which has the unique feature of realizing automatic impedance adaptation with a small inductance. Finite element modeling of a Timoshenko electromechanical beam structure is also presented for validating dispersion properties of the structure. Both theoretical and experimental results demonstrate that the proposed beam structure not only shows better vibration and wave propagation attenuation than the smart beam structure with independent switched networks, but also has technical simplicity of requiring only half of the number of switches than the independent switched network needs.

Stiffness design of heterogeneous periodic beam by topology optimization with integration of commercial software

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.

Bandgap of flexural wave in periodic bi-layer beam

A periodic bi-layer beam structure is proposed and the bandgap characteristic of flexural wave is studied in this paper. The single cell is made up of two bi-layer beams with four components. For the infinite structure, the flexural wave bandgap frequency algorithm is theoretically derived through Timoshenko beam theory, Hamilton principle, Bloch-Floquet theory and transfer matrix method. An analytical example is presented to illustrate the bandgap characteristic and FEA software simulation is conducted to demonstrate the validation of the algorithm. For the finite structure, the vibration transmission characteristic is studied with FEA software to show the flexural wave attenuation behavior of the periodic bi-layer beam. The results reveal that, the flexural wave is attenuated gradually in the stopband along the direction of wave propagation, while in the passband, it will propagate without attenuation. Comparisons with periodic single layer beam are studied to verify the convenience and flexibility of bi-layer beam. Finally, parametric influences on bandgaps are discussed, which will help the designers to make a better design for vibration reduction.

FEM formulation of homogenization method for effective properties of periodic heterogeneous beam and size effect of basic cell in thickness direction

This paper applies the asymptotic homogenization method to predict effective properties of periodic heterogeneous beam structures. A improved FEM formulation and algorithm of the unit cell problems is developed. The effective properties are rewritten in terms of the nodal quantities of the FEM model of the unit cell. With this approach, periodic beam structure of complicated microstructure can be modeled by various elements and modeling techniques and solved by using the commercial software as a black box. Numerical examples illustrate the versatility and efficiency of the method. Finally, the size effect of basic cell in thickness direction is studied.

A Study of Power Transmission through a Finite Periodic Dual-Layer Beam Structure with Transverse Connection

A study of the structure-borne sound power transmission through a finite periodic dual-layer beam structure with connection branches was conducted. The results suggest that the power transmitted through the beam structure into the supporting structure depends not only on the characteristics of the source and the supporting structure, but also on the attenuation of the periodic beam elements with the coupling branches and the exciting position of the source. The results also reveal the model under the mono-coupling condition to be similar to that under the multi-coupling condition in the relatively low frequency region. The transmitted power was strongly attenuated in a number of frequency regions, and in a few regions the power was even transmitted into the second layer of the supporting structure. The analysis indicated that this study is useful in understanding the structure-borne sound transmission from the first layer into the second layer of the beam structure.

Wave propagation in a periodic elastic-piezoelectric axial-bending coupled beam

The wave propagation in a periodic elastic-piezoelectric axial-bending coupled beam is investigated in this paper by considering the mechanical–electrical coupling behavior. The strain energy and kinetic energy of each sub-cell are first formulated to extract the dynamic stiffness matrices, and then the compatibility and continuity conditions at the interface between the adjacent cells are utilized to derive the transfer matrix that governs the propagation of the wave along the periodic piezoelectric beam. By employing the Lyapunov exponent method, the dynamic behaviors of the periodic beam structure are evaluated with different base beam materials, dimension ratios, piezoelectric constants and elastic stiffness. The results indicate that regardless of the length ratio, there exist certain frequency intervals, where the width and magnitude of the prominent stop band of the aluminum beam with periodic piezoelectric patches are always broader and larger than those of the steel base system. In addition, as the thickness ratio decreases, the location of the stop band tends to move toward a higher frequency. Numerical studies also demonstrate that different piezoelectric constants and elastic stiffness affect the characteristics of wave propagation in completely different fashions. The investigation in the present study provides basic guidelines to design periodic elastic-piezoelectric laminate structures in order to achieve desired filtering characteristics.

Experimental study of coupled vibration in a finite periodic dual-layered structure with transverse connection

The analytical equations of the transfer matrix method are further derived for the multi-coupled vibration of flexural and longitudinal waves in a periodic dual-layered beam structure with connection branches, with full consideration given to the flexural and longitudinal motions that are tri-coupled at each connection. Measurements of mobilities at the junctions on the uni-layered beam and the cross-layered beam are made. The numerical results agree well with the experimental results at all frequencies from 10 to 2000 Hz, which verifies the theoretical methodology for the multi-coupled vibration in a finite dual-layered beam. The cross-layer energy transmission is calculated, which reveals that the transmitted longitudinal energy is enhanced not only at the longitudinal resonant modes but also at the flexural resonant modes of the connection branches due to the structural wave coupling. The flexural energy is excited by wave coupling and becomes stronger at the longitudinal resonant modes and the flexural resonant modes of the connection branches. The cross-layer vibration motions from coupled waves in the branches can be effectively controlled by the attached cantilevers with mass at the resonance modes. This method can be used to control the structure-borne sound transmission in multi-layer beam structures.