Vibration Band Gaps Research Articles

Mechanical vibration bandgaps in surface-based lattices

In this paper, the phonon dispersion curves of several surface-based lattices are examined, and their energy transmission spectra, along with the corresponding bandgaps are identified. We demonstrate that these bandgaps may be controlled, or tuned, through the choice of cell type, cell size and volume fraction. Our results include two findings of high relevance to the designers of lattice structures: (i) network and matrix phase gyroid lattice structures develop bandgaps below 15 kHz while network diamond and matrix diamond lattices do not, and (ii) the bandwidth of a bandgap in the network phase gyroid lattice can be tuned by adjusting its volume fraction and cell size.

Traveling wave analytical solutions of vibration band gaps of composite periodic beams

This paper is concerned with elastic wave propagation and vibration band gaps in periodic beams formed by alternating combinations of various materials with the same geometric parameters. Such composite periodic systems have the similar feature of filtering characteristics of periodic beams. The study is performed by using an exact analytical approach based on a combination of the traveling wave analysis method and periodic structure theory. Based on the wave theory, a general mathematical model of composite periodic beam with uniform cross-section is established, the band gaps mechanism and vibration characteristics of this model are discussed, and the analytical solutions of the dynamic model and the explicit expression of band gaps are derived. Based on these equations, an explicit expression is provided for the calculation of intermediate frequency boundary frequency, all the band gaps information can be obtained in an exact and simple manner, and several illustrative examples are also provided.

Elastic wave and vibration bandgaps in two-dimensional acoustic metamaterials with resonators and disorders

The vibration properties of two-dimensional (2D) finite acoustic metamaterials (AMs) consisting of discrete masses and springs are investigated in this paper. In order to clarify the mechanisms of the bandgaps in AMs, the infinite symmetrical systems are also studied, especially the impact of the physical asymmetry caused by the discrepant stiffness coefficients of the springs connecting the unit-cells. Although the asymmetry cannot change the bandgap width, the change of the phase velocity and the generation of the shear mode have been proven analytically. To study the vibration properties of the finite AM model, the effective mass of each unit-cell is used. The effective mass properties of the unit-cell with single resonator and multi-resonators in 2D AMs subjected to a time-harmonic excitation are discussed. The effects of the number of the unit-cells, multi-resonators in each unit-cell, graded resonators in the finite AM model on the vibration suppression are thoroughly examined by the frequency response analysis. Numerical results show that the degree of the vibration attenuation is related to the size of the model and the number of the bandgaps changes corresponding to the disorders of the local resonators. Furthermore, imperfections or defects are introduced into the finite AM system. Wave propagation and guiding in a finite model for straight waveguide are investigated and discussed. The defect states induced by changing the central resonator of a supercell are demonstrated. Defect bands are obtained and their location is illustrated.

Analysis of vibration band gaps in an Euler–Bernoulli beam with periodic arrays of meander-shaped beams

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.

Complete vibrational bandgap in thin elastic metamaterial plates with periodically slot-embedded local resonators

This paper presents a metamaterial plate (metaplate) consisting of a periodic array of holes on a homogeneous thin plate with slot-embedded resonators. The study numerically proves that the proposed model can generate a complete vibrational bandgap in the low-frequency range. A simplified analytical model was proposed for feasibly and accurately capturing the dispersion behavior and first bandgap characteristics in the low-frequency range, which can be used for initial design and bandgap study of the metaplate. A realistic and practical unit metaplate was subsequently designed to verify the analytical model through finite element simulations. The metaplate not only generated a complete vibrational bandgap but also exhibited excellent agreement in both analytical and finite element models for predicting the bandgap characteristics. This study facilitates the design of opening and tuning bandgaps for potential applications such as low-frequency vibration isolation and stress wave mitigation.

Suppression of lateral vibration in rectangular ultrasonic plastic welding tool based on phononic crystal structure

Ultrasonic welding is one of the main applications of high-power ultrasound and is used in the automotive industry and aerospace. Transducers and tool are important parts of the ultrasonic welding system. Different tools are required for different welding objects. For larger plastic welded parts, it is necessary to weld them with large-sized welding tools. Due to the large size of the welding tool, under the excitation of the transducer, the tool will produce a coupling effect of longitudinal vibration and lateral vibration. Lateral vibration will cause the radiation surface of the tool to be non-uniformly displaced, and the working efficiency and welding results of the welding system will also be affected. So, in this paper, the phononic crystal bandgap theory and coupling vibration theory are used to study the coupled vibration of large-sized rectangular plastic ultrasonic welding tools. In order to improve the work efficiency and radiation surface's displacement uniformity of the tool, the phononic crystal structure is used to suppress the lateral vibration of the large-sized plastic ultrasonic welding tool, and the lateral vibration band gap of the phononic crystal structure is calculated. The longitudinal resonance frequency of the system is designed in the band gap range of the lateral vibration of the tool. So the lateral vibration of the tool can be effectively suppressed. The longitudinal vibration displacements on the radiation surface of the rectangular tool before and after vibration suppression are analyzed and compared with each other. The vibration mode of the ultrasonic welding system is simulated by the Comsol Multiphysics finite element software. The large-scaled tool with phononic crystal structure has a radiation surface displacement compared with the tool without phononic crystal structure, and the results show that the radiation surface displacement with phononic crystal structure will increase and tend to be uniform, greatly optimize the welding effect, improve the working efficiency of the welding system, and meet the needs of practical engineering. It is concluded that the longitudinal resonance frequency of the ultrasonic plastic welding system within the lateral vibration bandgap on the phononic crystal structure can not only suppress the lateral vibration, but also make the longitudinal displacement of the radiation surface more uniform and larger. Therefore, the work efficiency is greatly improved.

Galloping Reduction of Transmission Lines by Using Phononic Crystal

Considering the combination of the transmission lines and phononic crystals (PCs), we propose a new method to solve the problem of the galloping of overhead transmission lines. The method has two key points: attaching the suitable mass-spring system on each spacer, and periodically arranging the modified spacers along a transmission line. Based on the Bloch’s theorem, the PC transmission lines could generate vibration band gaps (BGs), which would reduce galloping. In order to implement our point, we establish the two-dimensional model of the PC transmission lines and derive the transfer matrix method to calculate the frequency dispersion relation of the vertical transverse vibration. Then, the extremely low frequency BG, in the range of galloping frequency, is obtained and verified based on an example of single conductor. To widen the BG range, we also study the effects of the spacer and the attached mass-spring system on the BG. The wide BG, which even covers the range of 0.338–0.909 Hz, could be given just by using the suitable setting of the spacer and mass-spring system.

An Analysis of Structural-Acoustic Coupling Band Gaps in a Fluid-Filled Periodic Pipe**Supported by the National Natural Science Foundation of China under Grant No 11372346.

A periodic pipe system composed of steel pipes and rubber hoses with the same inner radius is designed based on the theory of phononic crystals. Using the transfer matrix method, the band structure of the periodic pipe is calculated considering the structural-acoustic coupling. The results show that longitudinal vibration band gaps and acoustic band gaps can coexist in the fluid-filled periodic pipe. The formation of the band gap mechanism is further analyzed. The band gaps are validated by the sound transmission loss and vibration-frequency response functions calculated using the finite element method. The effect of the damp on the band gap is analyzed by calculating the complex band structure. The periodic pipe system can be used not only in the field of vibration reduction but also for noise elimination.

Three-component pillared phononic crystal plate containing viscoelastic material

Three-component pillared phononic crystal plates (PPCPs) usually contain viscoelastic materials. The influence of viscoelasticity on the band structures cannot be ignored. Based on the finite element method (FEM) and the standard linear solid model, this paper proposes a method to obtain the band structures of three-component PPCP containing viscoelastic rubber. The influence of the rubber storage modulus shifting along with frequency on the vibration band gaps is investigated, and the band structures of loss factor under different relaxation time are obtained.

Band gap property analysis of periodic plate structures under general boundary conditions using spectral-dynamic stiffness method

Vibration band gap properties of periodic rectangular plate structures with general boundary conditions are studied using the spectral-dynamic stiffness method (S-DSM). Material and dimensional properties are assumed to vary periodically in the whole plate structure. In the S-DSM, the general solution governing each unit plate is firstly derived with the aid of spectral method and then the dynamic stiffness matrix is established by substituting the general solution into corresponding boundary equations. Finally, the global spectral dynamic matrix for the whole periodic plate structure is assembled in a similar way to the finite element method (FEM). The harmonic responses are calculated to illustrate the band gap properties of the periodic structures. By comparing the results obtained by the FEM, it can be verified that the S-DSM is of superior accuracy and convergence rate. A parametric study is also conducted to investigate the effects of the material properties, geometrical parameters and structural damping on the vibration band gap behaviors of the periodic structures.

The two-degree-of-freedom local resonance elastic metamaterial plate with broadband low-frequency bandgaps

A theoretical model of a new type of two-degree-of-freedom local resonance mass-spring system with the resonator considering vertical and rotational vibration (the moment of inertia) is proposed to generate a negative effective mass over specific frequency ranges. Based on the theoretical analysis, a novel elastic metamaterial plate is designed. The flexural vibration band structures as well as the transmission spectrum of the metamaterial plate are investigated by using the finite element method. Subsequently, the formation mechanism of the band gaps is analyzed by studying the displacement field of the eigenmodes at the band gap edges. At last, the effects of the geometrical parameters on the flexural vibration band gaps (FVBGs) are studied and discussed in detail. The related results well confirm that the two-degree-of-freedom metamaterial shows negative mass density in two distinctive asymptotic regions, and the proposed elastic metamaterial plate has two FVBGs with the total width of 94.45 Hz below 200 Hz. The magnitude of torques introduced into the local resonance system can obviously affect the locations of the FVBGs. For the elastic metamaterial plate with thick local resonance lead plate and weak rubber is appropriate to obtain a lower gap, but the total width of the FVBGs becomes narrow. However, it does just the opposite for the condition with thin local resonance lead plate and strong rubber. For the double-side stubbed plate, it can enhance the coupling of the flexural traveling wave and the local resonance modes, which can greatly enlarge the bandwidths. The new working mechanism and the related calculation results of the designed structures would provide an effective way for elastic metamaterial plate to broaden the FVBGs at low frequencies, which has potential applications in low-frequency vibration and noise attenuation.

Flexural vibration bandgaps in local resonance beam with a novel two-degree-of-freedom local resonance system

In this paper, an elastic metamaterial beam with a novel two-degree-of-freedom local resonator is investigated theoretically, and the dispersion relation is calculated by using transfer matrix (TM). In order to confirm the existence of band gaps, the transmission spectrum of flexural wave are also studied by using finite element method. The formation mechanism of the flexural vibration bandgaps (FVBGs) are further analyzed by studying the displacement fields of the eigenmodes at the band-gap edges. At last, the evolution of the dispersion relations with the increasing of the distance from the one side rubber to the center of the local resonance mass are discussed in detail, and the effects of the outside diameter of the Cu ring and the equivalent stiffness k of the rubbers on the FVBGs are also investigated. Through the above analysis, we can draw the following conclusions, due to the unequal of the torques provided by the two rubbers, two different rotational vibrations of local resonance mass with two different local resonance frequencies are introduced in the local resonance system, thus the elastic metamaterial beam shows two FVBGs at low frequencies. The theoretical results are in good agreement with the numerical results. The magnitude of torques introduced in the local resonance system can obviously affect the locations of the FVBGs. With the asymmetry decreasing, the frequency region of the first FVBG moves to the higher value, while that of the second FVBG tends to the lower value, and when the two torques are equal, the two FVBGs coupled into one wider gap. For the elastic metamaterial beam with heavy resonance mass and weak rubbers is appropriate to obtain a lower band gap, and the total width of the FVBGs becomes wider. However, it does just the opposite under the condition of the case with light Cu ring and strong rubbers, but the total width of the band gaps also becomes wider. The propagation properties of the flexural wave in the designed local resonance beam can potentially be used to control and insulate vibration at low-frequency range and the unequal distances between the two rubbers to the center of the local resonance mass provide possibilities to expand the application of the beams for the vibration control in the needed frequency ranges.

Vibration band gap analysis of a new periodic beam model using GDQR method

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.

Flexural vibration band gaps for a phononic crystal beam with X-shaped local resonance metadamping structure

Structural vibration is commonly seen in engineering, which can cause resonance and fatigue damage in structure. Therefore, it is very desirable in vibration control techniques to achieve structure with low-frequency and broadband damping feature. In this paper, we design a phononic crystal (PC) beam with X-shaped locally resonant metadamping (X-LRMD) structures. Based on the PC theory, the flexural wave propagation in X-LRMD beam is studied. The equivalent dynamic properties of the X LRMD structure are analyzed by Lagrange equation. It is shown that due to its geometric nonlinearity, the X LRMD can effectively increase the damping of the system, which is validated by the transfer matrix method. The influence of structural parameters of X LRMD on band gap characteristics of the PC beam is then discussed in detail by using the finite element method with COMSOL multiphysics software in conjunction with Matlab, where the PC beam with X LRMD is modeled with the multi-body dynamic module within COMSOL and the band gap characteristics are calculated. The damping properties of the system are studied also using the finite element method. It is shown that compared with the equivalent structures, the PC beam with X LRMD can magnify the damping of the structure system, demonstrating a meta-damping phenomenon. The X LRMD in the PC beam can not only generate lower frequency and wider range band gaps but also suppress the vibration in passband ranges. This can bring a new design for reducing the vibration of structural systems.

Periodic metamaterial plates with smart tunable local resonators

Vibration band gaps and elastic wave propagation are examined in the metamaterial plates manufactured with periodic locally resonant membranes arranged in a square array. Periodic metamaterials exhibit unique dynamic characteristics stemming from their ability to act as mechanical filters for wave propagation. As a result, waves propagate along the periodic cells only within specific frequency bands called the pass bands, while being blocked within other frequency bands called the stop bands. The proposed metamaterial plates are equipped with sources of local resonances which act as local absorbers of mechanical vibrations. The macroscopic dynamical properties of the resulting periodic structures depend on the resonant properties of substructures which contribute to the rise of interesting effects such as broad stop band characteristics that extend to lower frequencies. Externally excited piezoelectric polyvinylidene difluoride membranes are used to support the local resonators. The stiffness of the piezo-membranes is tunable by means of an external voltage allowing us to control the location and bandwidth of the local resonance frequencies. The predicted band structures are validated by investigating the frequency response of the plate to external mechanical excitations using a comprehensive finite element model of the entire structure. By examining the proposed metamaterial plate, it is shown that it would be possible to actively control the wave propagation both in the spectral and spatial domains in an attempt to steer, stop, and/or confine the propagation of undesirable external disturbances.

Research on the vibration band gaps of isolators applied to ship hydraulic pipe supports based on the theory of phononic crystals

According to the theory of phononic crystals, a new isolator applied to ship hydraulic pipe-support with a one-dimensional periodic composite structure is designed, which is composed of metal and rubber. The vibration of the ship hydraulic pipeline can be suppressed by the band gaps (BGs) of the isolator. The band structure and frequency response function of the isolator is figured out by the transfer matrix method and the finite element method respectively. The frequency ranges and width of the BGs can be modulated to obtain the best structure of the isolator by changing the geometrical parameters. The experimental results provide an attenuation of over 20 dB in the frequency range of the BGs, and the results show good agreement with those of the numeric calculations. The research provides an effective way to control the vibration of ship hydraulic pipelines.

Vibration band gaps for elastic metamaterial rods using wave finite element method

Band gaps in elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators are investigated. New techniques to analyze metamaterial systems are using a combination of analytical or numerical method with wave propagation. One of them, called here wave spectral element method (WSEM), consists of combining the spectral element method (SEM) with Floquet–Bloch׳s theorem. A modern methodology called wave finite element method (WFEM), developed to calculate dynamic behavior in periodic acoustic and structural systems, utilizes a similar approach where SEM is substituted by the conventional finite element method (FEM). In this paper, it is proposed to use WFEM to calculate band gaps in elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators of multi-degree-of-freedom (M-DOF). Simulated examples with band gaps generated by Bragg scattering and local resonators are calculated by WFEM and verified with WSEM, which is used as a reference method. Results are presented in the form of attenuation constant, vibration transmittance and frequency response function (FRF). For all cases, WFEM and WSEM results are in agreement, provided that the number of elements used in WFEM is sufficient to convergence. An experimental test was conducted with a real elastic metamaterial rod, manufactured with plastic in a 3D printer, without local resonance-type effect. The experimental results for the metamaterial rod with band gaps generated by Bragg scattering are compared with the simulated ones. Both numerical methods (WSEM and WFEM) can localize the band gap position and width very close to the experimental results. A hybrid approach combining WFEM with the commercial finite element software ANSYS is proposed to model complex metamaterial systems. Two examples illustrating its efficiency and accuracy to model an elastic metamaterial rod unit-cell using 1D simple rod element and 3D solid element are demonstrated and the results present good approximation to the experimental data.

Numerical and experimental investigations on the vibration band-gap properties of periodic rigid frame structures

The spectral element method (SEM) is extended to study the vibration band-gap characteristics of periodic rigid frame structures composed of Timoshenko beams. The tensional and bending elements are taken into account for deducing the element stiffness matrices which are assembled to establish the spectral equations of motion of the whole rigid frame structure. The validity and accuracy of the natural frequencies and frequency responses obtained by the SEM are verified by comparing the present numerical results with the results of the experiments and the finite element method (FEM). The results indicate that the SEM is very suitable for analyzing the dynamic response and vibration band-gap properties of the periodic frame structures, and the SEM is more effective and accurate than the FEM, especially for the high frequency responses. The influences of the structural and material parameters on the vibration band-gaps are analyzed, and some new configurations of the periodic rigid frame structures are designed to obtain more and wider frequency band-gaps and consequently to improve the structural vibration isolation capability.

Vibration band gaps in double-vibrator pillared phononic crystal plate

This paper proposes a double-vibrator three-component pillared phononic crystal plate and theoretically studies the properties of vibration band gaps of this plate. The band structures and the displacement fields of the eigenmodes are calculated by the finite element method. Comparing the transmission power spectrums of the vibrations in the plate, the flexural vibration gap is proved more useful than the longitudinal vibration gap. The influence of the lattice constant, the height, and diameter of the pillars on the flexural vibration gaps are investigated. A supercell composed of the uni-vibrator and the double-vibrator unit cells is also investigated, and the analysis shows that the starting frequencies of the gaps in this supercell structure depend on the features of its pillars. This research can be used in the low frequency vibration insulation of plate structures.

Research on the Longitudinal Vibration Band Gaps of Isolator Applied to Ship Hydraulic Pipe-Support Based on the Theory of Phononic Crystals

Research on the Longitudinal Vibration Band Gaps of Isolator Applied to Ship Hydraulic Pipe-Support Based on the Theory of Phononic Crystals