Search

We study the nonlinear modulation property of flexural-gravity waves on a water surface covered by a compressed ice sheet of given thickness and density in a basin of a constant depth. For weakly nonlinear perturbations, we derive the nonlinear Schrödinger equation and investigate the conditions when a quasi-sinusoidal wave becomes unstable with respect to amplitude modulation. The domains of instability are presented in the planes of governing physical parameters; the shapes of the domains exhibit fairly complicated patterns. It is shown that, under certain conditions, the modulational instability can develop from shorter groups and for fewer wave periods than in the situation of deep-water gravity waves on a free water surface. The modulational instability can occur at the conditions shallower than that known for the free water surface kh = 1.363, where k is the wavenumber and h is the water depth. Estimates of parameters of modulated waves are given for the typical physical conditions of an ice-covered sea. [ABSTRACT FROM AUTHOR]

This paper presents a computationally efficient beam‐column element to simulate the nonlinear flexural and shear behaviors of reinforced masonry shear walls subjected to bilateral seismic forces. A three‐field mixed formulation based on the Hu‐Washizu variational principle is adopted. This mixed element is free of shear locking, and allows a wall to be modeled with one element. The axial and flexural responses are evaluated with a fiber‐section model at each integration point along the element, while the shear response in each loading direction is represented by a phenomenological model. The element accounts for the influence of the axial load, wall aspect ratio, and wall flange, if any, on the shear response of a wall. To consider axial‐flexure‐shear interaction, the shear model takes into account the axial stress resultant from the fiber‐section model, and the compressive strength of masonry in the fiber‐section model decreases when severe shear damage developed. The model has been calibrated and validated with experiment data. It has been demonstrated that it is able to reproduce experimental results from quasi‐static cyclic loading tests of single walls as well as shake‐table tests of wall systems with good accuracy. Furthermore, the ability of the proposed model to simulate the response of reinforced masonry shear wall systems subjected to bilateral seismic forces has been examined by comparing with results from a detailed finite element model. [ABSTRACT FROM AUTHOR]

A novel elastically coupled double beam (ECDB) finite element with superconvergent properties is developed to primarily study wave propagation in the bonded region of an adhesively bonded single lap joint. An ECDB element consists of two axial-flexural-shear coupled beams coupled with continuously distributed linear elastic springs. A general formulation applicable to both metals and composites is proposed. An exact solution to the static part of six-coupled governing equations of motion is obtained, which is then used to formulate exact stiffness and mass matrices. Consequently, the ECDB element thus formulated has superconvergent properties and is inherently free of shear locking. It is shown that merely a single ECDB element is sufficient to accurately capture the tip deflections of a cantilever double beam type geometry subjected to static loading, following which eigenvalue analysis is performed, and comparisons are made with the commercial finite element software—Abaqus. Further, wave propagation studies are carried out to demonstrate the efficiency of the element in evaluating ultrasonic wave responses across various metallic and composite ECDBs, and comparisons are made with Abaqus and frequency-domain spectral finite element (SFEM) model described in Paunikar and Goapalakrishnan ("Wave propagation in adhesively bonded metallic and composite lap joints modelled through spectrally formulated elastically coupled double beam element," Compos. Struct., under review). Following this, wave propagation in symmetric metallic, geometrically asymmetric metallic, and symmetric laminated composite lap joints with strong and weak bonding is studied by employing two-noded Lagrangian frame elements to mesh the adherends and superconvergent elastically coupled double beam (SECDB) elements to mesh the bonded region. The results obtained from the superconvergent finite element (SCFE) simulations are compared with those given by Abaqus and SFEM. Lastly, the SCFE models for the cases of perfectly bonded aluminium and symmetric ply laminated composite are experimentally validated. We have shown that the SECDB element developed in this work may be used to efficiently carry out static and dynamic analysis in any symmetric or asymmetric bonded joint comprising of only metals or composites or a combination of the two. It is also demonstrated that various levels of adhesion in a bonded joint can be simulated by varying the coupling spring stiffness value. Additionally, the SECDB element may also be used to study other double beam like coupled structures, for instance, space platforms or dynamic vibration absorbers. [ABSTRACT FROM AUTHOR]

Glass fiber-reinforced polymer (GFRP) and carbon fiberreinforced polymer (CFRP) sheets bonded to concrete were investigated to evaluate bond behavior, including effective bond lengths under two conditions: sustained temperature (ST) or sustained load with increasing temperature (SL). Tests were conducted using either double shear or flexural shear. Effective bond lengths of 70 and 100 mm (2.8 and 3.9 in.) were inferred for GFRP and CFRP sheets, respectively, from the double-shear tests. The effective bond lengths were then used for elevated temperature specimens, which showed significant reductions in the bond strength beyond the glass-transition temperature (55°C [131°F]). For 60°C (140°F) ST condition, up to 73% and 52% bond strength reductions were observed in double-shear and flexure specimens, respectively. The elevated temperatures also shifted the failure mode from thin concrete shearing to adhesive failure. Due to adhesive failure, the SL specimens with 30% and 50% sustained loads failed around temperatures of 125 and 110°C (257 and 230°F), respectively, regardless of the fiber type. [ABSTRACT FROM AUTHOR]

Purpose: Flexural wave propagation in non-homogeneous plates has wide engineering backgrounds, such as the out-of-plane vibration of the decks in buildings, and so on, and the corresponding research will provide some useful suggestions for the vibration control of the associated structures. Method: Taking the out-of-plane vibration of the basal plate bonded with electronic elements encapsulated in the spacecraft induced by the impact series as the background, the flexural wave propagation in the square damping thin plate bonded with two cylinders symmetrically is investigated by the structure-preserving approach in this paper. Based on the variational principle, the dynamic equation describing the flexural wave propagation in the square damping non-homogeneous plate is deduced. Employing the structure-preserving method, the wave propagation in the plate excited by the impact series is simulated. Results and Conclusions: In the numerical simulation, the impact series is classified into two types and different sizes of the cylinders are considered. From the numerical results, we find that the flexural elastic wave forms are symmetric if the impact series is symmetric and the vibration induced by the impact acting on the bolts located at the center of the edge is more severe than that induced by the impact acting on the bolts located at the corner of the plate if the impact series is asymmetric. In addition, the effects of the size of the cylinders on the flexural wave propagation in the non-homogeneous plate are also investigated by the numerical simulation. [ABSTRACT FROM AUTHOR]

This paper presents a series of experimental investigations into the flexural fatigue performance of rubberised concrete (RuC). A detailed account of fatigue response of reference conventional concrete materials (CCM) and RuC under flexural loading is presented. The experimental arrangement is described as well as the constituent materials, and the volumetric rubber replacement ratio of both fine and coarse aggregates varies between 0 % and 20 % in this study. The effect of three different stress ratios (R = 0.1, 0.3 and 0.5) on the flexural fatigue response are studied. It is found the static flexural strength of RuC with the rubber contents (ρ rv) of 10 % and 20 % show a reduction of 12 % and 24 % compared to their counterpart of CCM, respectively. A stress level of 0.9 is embedded in this study while despite the uncertainty of fatigue life of concrete materials, the increase of stress ratio from 0.1 to 0.5 lead to an increase of fatigue life of RuC in general. It is notable that the inclusion of rubber in concrete increases the fatigue life uncertainty whereas significantly increases the fatigue life of RuC at the same time, i.e. the average fatigue life of RuC with 20 % rubber content shows an increase of 170 % compared to its counterpart of CCM when R = 0.5. Based on the experimental results in this study, a series of three-parameter Weibull distribution models are proposed to describe the failure probability of RuC under fatigue flexural loadings, which consider the effects of ρ rv , stress ratio, stress level, and loading frequency on the probability of failure (PF) of RuC. This paper presents the attempt to investigate the possibility distribution of fatigue life of RuC, and all numerical calculation curves are in good agreement with the experimental results carried out in this study. • Flexural strength and fatigue behaviour of conventional and rubberised concrete were tested. • Cyclic flexural tests at different stress ratios were carried out. • The fatigue life probabilistic model of rubberised concrete is established. • The three-parameter Weibull distribution is introduced as a cumulative failure model. • Effect of stress ratio and rubber content is included. [ABSTRACT FROM AUTHOR]

The external clamp of hydraulic pipeline plays an important role in strengthening pipeline stiffness and improving vibration attenuation ability. In this study, wave propagation method is proposed to evaluate the translational and rotational stiffness and their loss factors of the clamp by analyzing the flexural wave propagation characteristics of pipeline. In order to validate the measured dynamic properties, experimental modal analysis was performed on the two-end elastically-supported pipeline. By modelling the pipeline with translational and rotational edge restraints, the analytical modal frequency parameters, wave numbers, and waveform coefficients were obtained. By comparing of the first-order flexural mode frequencies between experiment and analysis, the dynamic stiffness predicted by the wave propagation method was validated. The effects of the magnitude of torque, direction, clamping torque cycle, and clamp size on the complex dynamic properties of pipeline clamp were investigated. The proposed wave propagation method allows non-destructive continuous monitoring of a clamp stiffness, provides additional information about the vibration attenuation feature. It has advantages in clamp applications and is an effective methodology for measuring the dynamic properties of viscoelastic materials. [ABSTRACT FROM AUTHOR]

The propagation characteristics of F‐(flexural) wave and the reflection from vertical cracks in pipe‐piles subjected to an eccentric vertical excitation were numerically investigated using elastodynamic finite integration technique (EFIT). The accuracy of the self‐written code was validated by comparing computed results against the measured data. The snapshots of circumferential, radial and vertical velocities in intact and cracked pipe‐piles, as well as the velocity histories of the pile‐top were presented. It is found that not only L‐(longitudinal) wave (corresponding to axisymmetric mode) but also F‐wave (corresponding to non‐axisymmetric mode) can be excited in pipe‐piles subjected to an eccentric vertical excitation. F‐wave causes motion in all directions, while L‐wave only causes vertical motion. The three‐dimensional (3‐D) interference is weak at 90° away from the excitation position for circumferential velocity response, while at 45° or 135° for radial velocity response. The reflected F‐wave (RFW) from the pipe‐toe has multiple peaks due to the dispersion effect of F‐wave. Reflection occurs when F‐wave encounters vertical cracks, while L‐wave is not sensitive to vertical cracks. The intensity of RFW from vertical cracks varies with excitation‐and‐sensor modes. For shallow vertical cracks, the RFW strength varies with the sensor positions; while the intensity of the RFW caused by deep vertical cracks varies little with the sensor positions. With the increase of crack length, the strength of the first arrival RFW due to the vertical crack has no obvious change; however, the strength of RFW from the pile‐toe decreases. [ABSTRACT FROM AUTHOR]

Flexural elastic waves and sound in solids are of great interest in wide-ranging contexts such as ultrasound in plates, geophysics, ocean engineering, aerospace and automotive structures, and musical acoustics. Despite bending waves being the most important elastic waves for such surface structures, their propagation in the presence of the inevitable non-uniformity is poorly understood. Here we show the branching and focusing behaviour of highly dispersive flexural waves travelling in elastic plates of non-uniform thickness. The thickness profile has isotropically correlated spatial randomness. The correlation length is much larger than the wavelength. The location of wave focusing shows a scaling relationship with randomness, which is consistent with those previously reported in other random media. We show this analytically and numerically. This suggests a universality in the scaling between the location of wave focusing with randomness and the correlation length, regardless of the physics of the waves in question. Despite the relevance of flexural elastic waves to many diverse physical systems, modelling their propagation is non-trivial. Here, branching and focusing of highly dispersive flexural waves in heterogeneous elastic plates is described numerically and analytically. [ABSTRACT FROM AUTHOR]

Metasurfaces exhibiting spatially asymmetric inner structures have been shown to host unidirectional scattering effects, benefiting areas where directional control of waves is desired. In this work, we propose a non-Hermitian planar elastic metasurface to achieve unidirectional focusing of flexural waves. The unit cells are constructed by piezoelectric disks and metallic blocks that are asymmetrically loaded. A tunable material loss is then introduced by negative capacitance shunting. By suitably engineering the induced loss profile, a series of unit cells are designed, which can individually access the exceptional points manifested by unidirectional zero reflection. We then construct a planar metasurface by tuning the reflected phase to ensure constructive interference at one side of the metasurface. Unidirectional focusing of the incident waves is demonstrated, where the reflected wave energy is focused from one direction, and zero reflection is observed in the other direction. The proposed metasurface enriches the flexibility in asymmetric elastic wave manipulation as the loss and the reflected phase can be tailored independently in each unit cell. [ABSTRACT FROM AUTHOR]

Structures carrying moving loads present rich vibration phenomena and colorful spectrum characteristics. This study analytically investigates the moving-inertial-load-pumped frequency comb free vibration behavior of a single-span rectangular linear elastic Kirchhoff thin plate in the low-speed region. Multiple-time-scales expansion is made and the asymptotic solution describing the non-resonance modal oscillation is derived. Case studies on a single-span rectangular isotropic plate with the clamped opposite boundaries dominated by the first two transverse flexural modes are presented. Numerical computation based on finite element method structure modeling and response numerical integration verifies the modal frequency comb characteristics as predicted by the proposed theoretical model. The results also present the weak modal interaction for the studied single-span structure in the observed light-mass low moving speed region. The outcomes of this study will benefit the related structural dynamic system design and intervention. [ABSTRACT FROM AUTHOR]

Nonreciprocal mechanical devices are of great interest for directional elastic wave manipulation. In this Letter, we introduce a design of a compact low-frequency nonreciprocal metamaterial for flexural waves, whose dimension is less than 1 / 3 of the operating wavelength. This structure is made of two well-placed coil-cantilever-magnet resonators, where the electromagnetic forces can be temporally modulated, which enables time varying of the effective stiffness of the resonators. A phase shift is introduced between the stiffness modulations of these two resonators, which breaks the time-reversal symmetry and enables nonreciprocal wave propagation at the resonance frequency of the structure. A semi-analytical method based on harmonic wave decomposition is developed to describe the system, leading to results that match well with numerical predictions from a finite element method. We also experimentally demonstrate nonreciprocal flexural wave propagation with good agreement with the predictions made. Our system could inspire the design of compact nonreciprocal devices for flexural waves. [ABSTRACT FROM AUTHOR]

The intraplate Hawaiian‐Emperor Seamount Chain has long been considered a hotspot track generated by the motion of the Pacific plate over a deep mantle plume, and an ideal feature therefore for studies of volcanic structure, magma supply, plume‐crust interaction, flexural loading, and upper mantle rheology. Despite their importance as a major component of the chain, the Emperor Seamounts have been relatively little studied. In this paper, we present the results of an active‐source wide‐angle reflection and refraction experiment conducted along an ocean‐bottom‐seismograph (OBS) line oriented perpendicular to the seamount chain, crossing Jimmu guyot. The tomographic P wave velocity model, using ∼20,000 travel times from 26 OBSs, suggests that there is a high‐velocity (>6.0 km/s) intrusive core within the edifice, and the extrusive‐to‐intrusive ratio is estimated to be ∼2.5, indicating that Jimmu was built mainly by extrusive processes. The total volume for magmatic material above the top of the oceanic crust is ∼5.3 × 104 km3, and the related volume flux is ∼0.96 m3/s during the formation of Jimmu. Under volcanic loading, the ∼5.3‐km‐thick oceanic crust is depressed by ∼3.8 km over a broad region. Using the standard relationships between Vp and density, the velocity model is verified by gravity modeling, and plate flexure modeling indicates an effective elastic thickness (Te) of ∼14 km. Finally, we find no evidence for large‐scale magmatic underplating beneath the pre‐existing crust. Plain Language Summary: As lithospheric plates move across mantle hotspots, melts will rise and pass through the plate to form a series of volcanos. The Hawaiian‐Emperor Seamount Chain is one of the Earth's most well‐known hotspot tracks and an ideal laboratory therefore for studying the nature and characteristics of volcano building and the extent to which oceanic plates bend under the weight of these volcanos, which is an indication of oceanic plate strength. As a major component of the chain, the Emperor Seamounts have been relatively little studied. To address these issues, we carried out an active‐source seismic imaging experiment across the Emperor Seamounts at Jimmu guyot. The seismic velocity image reveals that Jimmu has a core made from magmas that froze within its edifice but was built mainly by volcanic processes. Due to volcanic loading, the oceanic crust is depressed by ∼3.8 km. The density structure derived from the tomographic image well‐predicts observed gravity, while plate flexure modeling indicates an effective elastic thickness of ∼14 km, suggesting that the oceanic plate on which the Emperor chain was built was weaker at the time of volcano formation than the Hawaiian Ridge. Key Points: Seismic tomography experiment reveals crust and upper mantle structure along a 495‐km‐long east‐west line crossing the Emperor SeamountsJimmu guyot was built mainly by extrusive processes, and the underlying ∼5.3‐km‐thick oceanic crust was depressed by ∼3.8 kmThere is no evidence for large‐scale magmatic underplating beneath the edifice [ABSTRACT FROM AUTHOR]

To effectively and efficiently control vibrations in steel bridge decks, the flexural wave band gap and vibration attenuation mechanism are studied. The band structures and displacement fields of the eigenmodes at the band gap edges of infinite periodic U-rib stiffened plates are calculated. The calculation method is combining the finite element method with Bloch periodic boundary conditions. A scaled specimen test of a finite periodic U-rib stiffened plate is used to validate the numerical results. A comparison of results confirms that introducing the U-rib allows the periodic U-rib stiffened plates to yield several flexural wave band gaps. In the band gaps, flexural wave propagation is stopped and clear flexural vibration suppression is achieved. Furthermore, the effects of geometric parameters and structural schemes on the flexural wave band gaps are analyzed in detail. Finally, by analyzing a specific case, it is demonstrated that the flexural wave band gaps and vibration attenuation properties can be controlled. This provides a new approach to vibration and noise control in steel bridges. [ABSTRACT FROM AUTHOR]

We provide an analytical formulation to model the propagation of elastic waves in a homogeneous half-space supporting an array of thin plates. The technique provides the displacement field obtained from the interaction between an incident wave generated by a harmonic source and the scattered fields induced by the flexural motion of the plates. The scattered field generated by each plate is calculated using an ad-hoc set of Green's functions. The interaction between the incident field and the scattered fields is modelled through a multiple scattering formulation. Owing to the introduction of the multiple scattering formalism, the proposed technique can handle a generic set of plates arbitrarily arranged on the half-space surface. The method is validated via comparison with finite element simulations considering Rayleigh waves interacting with a single and a collection of thin plates. Our framework can be used to investigate the interaction of vertically polarized surface waves and flexural resonators in different engineering contexts, from the design of novel surface acoustic wave devices to the interpretation of urban vibration problems. [ABSTRACT FROM AUTHOR]

In this study, whirl characteristics and stability of an axially functionally graded (AFG) liquid-filled rotor are investigated. The rotor is modeled based on the spinning Timoshenko beam theory. The governing equations for flexural vibration are derived via Hamilton's principle. For pinned–pinned AFG liquid-filled rotor, the analytical solutions are derived for both the exact whirl frequency equation and the stability model. To validate the present formulations, comparative studies by numerical solutions available in the literature are conducted. Some numerical examples are performed to investigate the effects of gradient parameter, mass ratio, cavity ratio, rotary inertia, and shear deformation on the whirl speed, the critical spinning speed, and the stability of the AFG liquid-filled rotor system. The results show that these parameters have noticeable influences on dynamic behavior and stability of the rotor system. In particular, the rotary inertia and shear deformation play an important role in the stability analysis for different length rotors. [ABSTRACT FROM AUTHOR]

The solution to the problem of flexural-gravity waves in an ice-covered channel is constructed and investigated when the ice thickness varies linearly from one to another channel wall. The channel cross-section is rectangular. It is assumed that the ice-plate thickness is small as compared with the channel depth and the plate edges are frozen to the channel walls. The ice cover deflection is described within the framework of the linear theory of elastic plates, fluid flow beneath ice is assumed to be potential. The problem is solved using expansions of the plate deflection in the modes of vibration of the plate whose thickness varies linearly across the the channel width. The dispersion relations, the profiles of flexural-gravity waves across the channel, and the strain distributions in the ice cover are determined. The group and phase velocities of hydroelastic waves are found. The hydroelastic characteristics are investigated as functions of the linear coefficient of the ice thickness. The results obtained are compared with the results for the uniform ice cover. [ABSTRACT FROM AUTHOR]

For the release of a slab avalanche, crack propagation within a weak snowpack layer below a cohesive snow slab is required. As crack speed measurements can give insight into underlying processes, we analysed three crack propagation events that occurred in similar snowpacks and covered all scales relevant for avalanche release. For the largest scale, up to 400 m, we estimated crack speed from an avalanche movie; for scales between 5 and 25 m, we used accelerometers placed on the snow surface and for scales below 5 m, we performed a propagation saw test. The mean crack speeds ranged from 36 ± 6 to 49 ± 5 m s−1, and did not exhibit scale dependence. Using the discrete element method and the material point method, we reproduced the measured crack speeds reasonably well, in particular the terminal crack speed observed at smaller scales. Finally, we used a finite element model to assess the speed of different elastic waves in a layered snowpack. Results suggest that the observed cracks propagated as mixed mode closing cracks and that the flexural wave of the slab is responsible for the energy transfer to the crack tip. [ABSTRACT FROM AUTHOR]

The influence of wave blocking on the scattering of a flexural-gravity wave by a linear crack in a thin ice cover resting over a two-layer ocean having a rigid flat seafloor is investigated. The wave dispersion curve reveals the existence of multiple propagating wave modes within the blocking frequencies, either in the surface or interface mode. The Sommerfeld radiation condition depends on multiple propagating wave modes within the blocking frequency and contributes to wave energy propagation. The solution process involves the appropriate transition of wave modes within the blocking frequencies, which is obtained with the help of the dispersion curve. The reflection and transmission coefficients are generalized in the case of multiple propagating wave modes, and the associated energy balance relation is derived using Green's integral theorem. The scattering matrix is generated to describe all the possible transmitted and incident wave modes. The role of lateral compressive force and the density ratio on the scattering process, ice deflection, and interface elevation are shown graphically. This study reveals the occurrence of removable and jump discontinuities in the reflection and transmission coefficients at the saddle point as well as blocking frequencies and at the frequency for which incident wave mode changes. Irregular plate deflection and interface elevation patterns are found due to the superposition of multiple propagating wave modes within the primary and secondary blocking frequencies. [ABSTRACT FROM AUTHOR]

An acoustic black hole (ABH) has the ability to concentrate and manipulate flexural waves, which can be used for structural vibration suppression and noise attenuation. In this paper, a 2D ABH rectangular plate is designed and a 2D ABH plate-cavity coupling system is constructed using the 2D ABH plate and five rectangular elastic plates of uniform thickness. Series of vibro-acoustic FEM models of the plate-cavity parallelepiped coupling system embedded with 2D ABHs are established, and the vibro-acoustic coupling simulation is conducted to analyze the effects of ABHs on the coupling modes, vibro-acoustic coupling characteristics, and mechanism of ABHs and the damping layer. It is shown that at most frequencies in the range of 3600~5000 Hz, the damping 2D ABH plate-cavity parallelepiped coupling system can significantly suppress the sound pressure and greatly reduce the peak values. It is also found that the significant reduction of the participation factor of the acoustic modes within the 100th order is the main mechanism for the obvious suppression of the sound pressure in the damping ABH plate-cavity coupling system at 3701 Hz. Finally, an experimental platform of vibro-acoustic measurement of the 2D ABH coupling system is constructed, and the accuracy of the vibro-acoustic FEM models of the 2D ABH coupling systems established in this paper and the numerical simulation calculation are verified by the vibro-acoustic measurement experiment. [ABSTRACT FROM AUTHOR]

Herein, a thermally induced flexural vibration problem of inhomogeneous rectangular plates in which the material constants are given as a power of the coordinate in the thickness direction is considered. Assuming that the top surface of the plate uniformly receives cyclic heat supply and that the bottom surface is kept at zero temperature change, a problem of 1D heat conduction in an unsteady state in the inhomogeneous plate is analyzed theoretically. Analytical solutions of temperature change, deflection, and stresses in the plate with simply supported edges in which power exponents in the material constants are free from any constraints are derived. Performing numerical calculations, an effect of inhomogeneity in the specific heat capacity on amplitude and phase shift in temperature change inside of plate is clarified. To evaluate the influence of inhomogeneity in the material constants on the inertia effect in the thermoelastic response of the plate, amplification factors are defined as ratios of maximum amplitudes in dynamic solutions of deflection and stresses to those in quasistatic solutions. It is clarified that amplification factors in deflection and stresses of the plate are significantly affected by inhomogeneity in the Young's modulus of elasticity and the mass density. [ABSTRACT FROM AUTHOR]

A micromachined resonator immersed in liquid provides valuable resonance parameters for determining the fluidic parameters. However, the liquid operating environment poses a challenge to maintaining a fine sensing performance, particularly through electrical characterization. This paper presents a piezoelectric micromachined cantilever with a stepped shape for liquid monitoring purposes. Multiple modes of the proposed cantilever are available with full electrical characterization for realizing self-actuated and self-sensing capabilities. The focus is on higher flexural resonances, which nonconventionally feature two-dimensional vibration modes. Modal analyses are conducted for the developed cantilever under flexural vibrations at different orders. Modeling explains not only the basic length-dominant mode but also higher modes that simultaneously depend on the length and width of the cantilever. This study determines that the analytical predictions for resonant frequency in liquid media exhibit good agreement with the experimental results. Furthermore, the experiments on cantilever resonators are performed in various test liquids, demonstrating that higher-order flexural modes allow for the decoupled measurements of density and viscosity. The measurement differences achieve 0.39% in density and 3.50% in viscosity, and the frequency instability is below 0.05‰. On the basis of these results, design guidelines for piezoelectric higher-mode resonators are proposed for liquid sensing. [ABSTRACT FROM AUTHOR]

A linearized model is presented for the scattering problem of flexural gravity waves by the bottom with a finite number of rectangular steps in the ocean. The matched eigenfunction expansion method is used to solve the boundary-value problem, and the results are expressed in terms of a set of integral equations solved by the multiterm Galerkin's approximation technique. Numerical values of the reflection and transmission coefficients are calculated and graphed to display the influence of various parameters of the problem. The accuracy of the present method is verified by using the energy balance relation for the reflection and transmission coefficients. The results are compared with those available in some earlier works. [ABSTRACT FROM AUTHOR]

Wedge members with variable thickness are widely used in ship structures, aerospace structures, and building structures. Considering their application scenarios, their high-frequency vibration characteristics have important research value. Energy finite element analysis (EFEA) is a powerful tool to predict high-frequency vibration response. EFEA is essentially applying finite element analysis to solve the energy density governing equation. However, the equation for wedge beam is missing. Energy flow analysis aims to obtain energy density governing equation. The energy flow model of wedge beam with variable thickness is established in this paper. Firstly, according to the geometric acoustic approximation method, the general displacement solution of the bending deformation equation for a wedge beam is deduced. Furthermore, the wave dispersion relation of bending deformation of wedge beam is derived. Then, the relationship between the time-averaged vibration energy density and energy flow of the wedge beam is also derived. Also, the governing equation taking energy density as a variable of the wedge beam structure is derived through the energy balance relationship in the microelement body. The governing equation is numerically solved, and the energy density distribution on the wedge beam structure is calculated. Finally, the energy density distribution of uniform beam and wedge beam under the same excitation is analyzed and summarized; the influence of different excitation frequencies and power exponent of thickness change on the performance of wedge beam structure is also summarized. Geometrically, the wedge beam satisfies the one-dimensional acoustic black hole structure. The calculation results show that the energy density on the beam structure increases with the decrease of thickness and reaches its maximum near the tip. The wedge acoustic black hole beam structure has a good energy absorption effect with a frequency between 250 Hz and 2000 Hz. With the further increase of excitation frequency, the energy absorption effect of the wedge beam structure has dropped significantly. [ABSTRACT FROM AUTHOR]

Motivated by the importance of lattice structures in multiple fields, we numerically investigate the propagation of flexural waves in a thin reticulated plate augmented with two classes of metastructures for wave mitigation and guiding, namely metabarriers and metalenses. The cellular architecture of this plate invokes the well-known octet topology, while the metadevices rely on novel customized octets either comprising spherical masses added to the midpoint of their struts or variable node thickness. We numerically determine the dispersion curves of a doubly-periodic array of octets, which produce a broad bandgap whose underlying physics is elucidated and leveraged as a design paradigm, allowing the construction of a metabarrier effective for inhibiting the transmission of waves. More sophisticated effects emerge upon parametric analyses of the added masses and node thickness, leading to graded designs that spatially filter waves through an enlarged bandgap via rainbow trapping. Additionally, Luneburg and Maxwell metalenses are realized using the spatial modulation of the tuning parameters and numerically tested. Wavefronts impinging on these structures are progressively curved within the inhomogeneous media and steered toward a focal point. Our results yield new perspectives for the use of octet-like lattices, paving the way for promising applications in vibration isolation and energy focusing. [ABSTRACT FROM AUTHOR]

Fused filament fabrication (FFF) is a widely used extrusion-based additive manufacturing process due to its low operating cost and available material options. Polylactic acid (PLA) is a bio-based compostable polymer extensively used as a feedstock material in FFF. The application of PLA in engineering and medical applications is limited due to the low thermal stability associated with the polymer. The present work aims to study the effect of thermal annealing on FFF-printed PLA in static and dynamic flexural loading. Response surface methodology (RSM) is chosen as a design of experiment method to study the effect of annealing time and temperature on flexural properties of printed PLA. Annealing is found to be advantageous in increasing the dynamic flexural properties. Improved glass transition temperature, stiffness, thermal stability, and crystallinity are also evident with annealing. However, annealing is found to be detrimental in improving static flexural properties. Considering the desirability function (DF) approach, annealing time and temperature are optimised for measured static and dynamic flexural properties. Samples are further investigated in scanning electron microscope for failure mode and differential scanning calorimetry for crystalline behaviour of annealed PLA. Empirical models are proposed for predicting the flexural behaviour of printed PLA. The current study has shown that the FFF-printed PLA can be tailored to specific application requirements due to its stable characteristics for thermal loading with thermal annealing. [ABSTRACT FROM AUTHOR]

This Letter reports the feasibility of a structure specifically designed for the control of flexural waves propagating in thin perforated plates. The structure, here denominated as a redirector device, consists of a square array of free holes that splits the impinging beam and transmits sideways their vibrational energy. This behavior is known as a Poisson-like effect, and it was theoretically described in different acoustic structures. This effect is experimentally demonstrated for flexural waves excited in an aluminum perforated plate, and it is explained in terms of a physical mechanism different to that reported for acoustic waves interacting with thin hollow cylinders embedded in water. In addition, a collimator device based also in free holes is designed and validated with the purpose of providing the beam impinging the redirector device. The measurements indicate that the amount of redirected energy is strongly enhanced when a barrier of two-beam resonators is added at the rear side of the redirector. All the designs are validated by an experimental setup employing 1 mm thick aluminum plates. [ABSTRACT FROM AUTHOR]

In order to meet the needs of vibration reduction in complex vibration environments, one-dimensional phononic crystals constituted with periodic tunnels on elastic foundations are proposed. The governing equations of the flexural wave in a periodic structure are established based on Euler beam theory and its band structure is calculated by the plane wave expansion method. The theoretical and numerical results show that there are distinct forbidden bands and pass bands when the flexural wave propagates in the periodic tunnel structure. The elastic foundation stiffness, elastic modulus ratio, and axial prestress have significant effects on the location and width of the band gap. The results of the paper could be used for the vibration mitigation of the tunnel structure. [ABSTRACT FROM AUTHOR]

Sustained flight at hypersonic speeds is characterized by high pressure and aerothermal loads imposed on the structure of the aerodynamic vehicle. A consequence of lightening the structural design permits fluid–structure interaction phenomena that can significantly alter the flow and initiate unsteady structural responses. We investigate the coupling between high-speed laminar boundary layer flows over a mechanically compliant panel and analyze the dynamic system response of the coupled system to boundary layer instabilities by means of local convective linear stability analysis. The resulting non-dimensional interaction parameters describing the compliant system are shown to affect the boundary layer instabilities in the infinitely thin panel limit, and the transition from the rigid limit is described by two distinctly different responses: (a) a piston-like, one-dimensional panel deflection, or (b) a synchronization with flexural waves. Compliance is shown to non-monotonically change convective wave growth rates and induce uncertainty in the integrated N-factors. [ABSTRACT FROM AUTHOR]

An analytical study is presented here to investigate the scattering of oblique flexural gravity waves by a pair of totally submerged vertically placed porous barriers, located at some distance from each other, for a homogenous fluid flowing over a porous sea-bed. A thin ice-sheet, replacing the usual free surface, is considered as the upper surface where it is treated as a thin elastic plate by following Euler-Bernoulli beam equation. The complete analytical solution, under the assumption of small-amplitude theory and structural response, is acquired by employing eigenfunction expansion and least square method for the problem of flexural gravity waves interacting with the submerged porous barriers. Subsequently, computation for the reflection and transmission coefficients, energy loss and wave forces are carried out and discussed for different parameter values corresponding to the ice-sheet, porous sea-bed, and porous barriers. This study establishes that the oscillatory behavior exhibited by the reflection of the waves. It further shows that when the inertial effect of the porous-effect parameter of the barriers is increased, the minima in wave reflection occur. The vertical porous barriers are found to dissipate a significant portion of the wave energy when an increase in the inertial effect of the porous barriers is affected. The hydrodynamic force on the barriers also follows an oscillatory pattern, and it increases when the length of the barrier is increased. Furthermore, wave transmission decreases significantly due to the energy dissipation by the porous sea-bed. It is demonstrated that corresponding to various structural parameters, almost no reflection and full transmission take place for an impermeable sea-bed and also when only real porosity parameter of the porous sea-bed is considered. The effect of the ice-sheet on the propagation of the flexural waves is also examined by obtaining a number of results for variation of various parameters. Variation in the elastic parameter of the floating ice-sheet is observed to command a considerable influence when the wave impinges upon the submerged vertical porous barriers. [ABSTRACT FROM AUTHOR]

When cracks start to surface in the surrounding areas of the structure, they create a local softness zone and influences on the dynamic response of the structure. The beams are more susceptible to shear and flexural cracks because of being subjected to shear and bending stress. In this study, the dynamic response of the single-span and multi-span damped beam under moving load with multi-crack and elastic boundary condition is studied based on Timoshenko's theory. The Green's function method is used to calculate the dynamic response of the cracked beam. In addition, the Green's function method provides a solution for the differential equations. Moreover, the effects of the crack on the essential characteristics of the multi-span beams, especially the natural frequencies, are investigated. In this study, crack by itself is modeled in different situations and its effect on the behavior of the beam is analyzed. Also, the elastically restrained beam is modeled and its effect on the behavior of the beam is assessed. Because of the fact that the Euler–Bernoulli theory is also used in most beams, in this study, the results of the numerical examples are compared with the Euler–Bernoulli theory. Several examples are analyzed for a better understanding of the Timoshenko cracked beam. [ABSTRACT FROM AUTHOR]

Purpose: For the non-destructive interrogation purpose, the present work studies the flexural wave propagation characteristics in submerged pipes conveying viscous flowing fluid. The internal fluid-pipe and external fluid-pipe coupling effects are both considered. Methods: The internal fluid is assumed to be viscous, irrotational, and isotropic; and the external fluid is treated as ideal and irrotational. The pressures exerted on the pipe by the internal and external fluid can be derived via the Navier–Stokes equation. The pipe is modeled based on the first-order shear deformation shell theory. The motion equations of the submerged fluid-conveying pipe are given by the Hamilton's Principle. Results: The coupled equations of wave motion of the pipes are obtained based on the first-order shear deformation shell theory. The wave propagation relations of submerged pipes conveying viscous flowing fluid are given. Conclusion: Increasing internal fluid velocity, external fluid velocity, or reducing internal fluid mass density can result in an increase in phase velocity of the first flexural mode. In addition, the phase velocities of the first three flexural modes are almost constant under a large axial wave number. [ABSTRACT FROM AUTHOR]

The coupled flexural-torsional free vibration of circular horizontally curved beams made of an axially functionally graded (AFG) material was investigated. Beams with rectangular and elliptical cross-sections were designed to obey quadratic functions of Young's modulus and the mass density in the axial direction. Using the Timoshenko and St. Venant beam theories, the governing differential equations of motion were derived. Based on the trial eigenvalue method together with the numerical integration method, the differential equations were solved to obtain the natural frequencies. For validation purposes, the frequencies computed in this study and ADINA were compared. Parametric studies were also performed to clarify how the natural frequency of the flexural-torsion coupling depends on modular ratio, cross-sectional shape, aspect ratio, slenderness ratio, and opening angle of the beams. [ABSTRACT FROM AUTHOR]

The submerged floating tunnel (SFT) is a newly developed traffic structure for crossing the long waterway. On the basis of the vehicle–tunnel coupled vibration, the vortex-induced effect of the SFT in the water flow field is considered through a wake oscillator model. The tunnel is modeled as an Euler–Bernoulli beam supported by the continuous elastic foundation. The vehicle is regarded as a spring–mass block moving on the tube with a damping term. The finite-difference method is carried out to calculate the fluid–vehicle–tunnel coupled system vibration response evolution. The effects of the buoyancy weight ratio (BWR), the structural span, the vehicle weight, and the distributed flow field on the fluid–vehicle–tunnel coupled vibration are discussed. The results show that the vehicle moving in the SFT will aggravate the fluid–tunnel coupled vibration. The reverse displacement of the vehicle is driven by the coupled vibration. With an increase in the flow velocity, the short-span SFT will vibrate more intensively than the long-span SFT. The heavy-weight vehicle moves more stably during the fluid–vehicle–tunnel coupled vibration. The characteristics of the distributed flow field could be indirectly reflected by the vehicle vibration response. [ABSTRACT FROM AUTHOR]

A major obstacle to obtaining cost-effective experimental data on the fatigue life of sandwich panels is the prohibitive amount of time and cost required to carry out millions of cycles. On the other hand, vibration techniques applied to sandwich geometries fail to match the stress patterns that are obtained from standard flexural fatigue tests. To overcome such limitations, a vibration-based fatigue technique is proposed, which entails the use of sandwich specimens whose geometries are optimized to reproduce the stress distribution observed during three point bend loading while vibrating at the first resonant frequency. The proposed vibration technique was experimentally validated. The results, compared with the average number of cycles to failure at different stress ratios obtained via the Three-Point Bending test, showed high levels of accuracy. The proposed method is robust and time effective and indicates the possibility of attaining fatigue lifetime prediction of a wide class of composite elements, such as sandwich panels. [ABSTRACT FROM AUTHOR]

Waves, mathematical modelling, mechanics of solids and fluids, structured media Keywords: waves; structured media; mathematical modelling; mechanics of solids and fluids EN waves structured media mathematical modelling mechanics of solids and fluids 1 3 3 10/21/22 20221128 NES 221128 This is part 2 of the theme issue on 'Wave generation and transmission in multi-scale complex media and structured metamaterials'. It is focused on the analysis of wave scattering and wave localization in complex media, including acoustic and elastic waves, vibrations in viscoelastic media, and pulsating flows associated with waves in fluids, with an interesting range of industrial applications as well as ideas linked to biomechanics and the mechanics of metamaterials. [Extracted from the article]

The Late Cretaceous intraplate shortening event in central western Europe is associated with a number of marine basins of relatively high amplitude and short wavelength (2–3 km depth and 20–100 km width). In particular, the Harz Mountains, a basement uplift on a single, relatively steeply dipping basement thrust, have filled the adjacent Subhercynian Cretaceous Basin with their erosive product, proving that the two were related and synchronous. The problem of generating subsidence of this general style and geometry in an intraplate setting is dealt with here by using an elastic flexural model conditioned to take account of basement thrusts as weak zones in the lithosphere. Using a relatively simple configuration of this kind, we reproduce many of the basic features of the Subhercynian Cretaceous Basin and related basement thrusts. As a result, we suggest that overall, it shares many characteristics with larger-scale foreland basins associated with collisional orogens on plate boundaries. [ABSTRACT FROM AUTHOR]

An elastic metasurface with a gradient index of functional units, namely gradient folding metasurface (GFMS), is proposed to simultaneously manipulate the transmission phase and amplitude of flexural waves. By varying the gradient index common ratio (CR) between each adjacent unit, the GFMS was found to cover an entire 2π span of phase shift. At the same time, the amplitude of the transmitted flexural wave can also be modulated. Based on the generalized Snell's law, two different cases of the wave focusing effect are realized, which illustrate CR dependent wave propagation mode from symmetric to asymmetric and verify the ability of the proposed metasurface to manipulate complete elastic wave through simultaneous phase and amplitude modulation. This study expands phase-amplitude modulation from acoustics to elastic waves and provides more degrees of freedom in elastic metasurface designing. [ABSTRACT FROM AUTHOR]

The dynamic characteristics studies of piezoelectric structures usually focus on the single vibration modes such as the longitudinal or the flexural mode, and the losses that cause heat generation and energy waste are generally neglected, which leads to discrepancies compared with experiments. In the present paper, a beam-type piezoelectric structure with four kinds of losses under coupled longitudinal-flexural vibration mode is investigated. In this approach, first, impedance matrix of the structure is obtained based on the motion equation and the electrical-mechanical boundary conditions. Secondly, admittance curves of this piezoelectric structure under different boundary conditions are simulated by the equivalent circuit methods that contain four losses—piezoelectric and dielectric losses, as well as two elastic losses. The simulation results by equivalent circuit methods has good agreement with COMSOL results, which demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The subject of the paper are homogeneous beams of symmetrically variable depth and bisymmetrical cross sections. Free flexural vibrations of these beams are analytically and numerically studied. Based on Hamilton's principle, the differential equations of motion of these beams are obtained. The equations of motion are analytically solved with consideration of the bending lines of these beams subjected to their own weight. The fundamental natural frequency for exemplary beams is derived and presented in Tables and Figures. [ABSTRACT FROM AUTHOR]

A new model for predicting elastic flexural wave band gaps in periodic composite plate structures containing interconnected, star-shaped, transversely isotropic, magneto-electro-elastic (MEE) inclusions is developed using a microstructure-dependent Mindlin plate model. The Floquet–Bloch theorem and the plane wave expansion method for periodic media are employed to solve the non-classical wave equations and to determine the band gaps incorporating the microstructure effect. The current non-classical model recovers its classical elasticity-based counterpart as a special case if the microstructure effect is suppressed. A parametric study is performed to demonstrate the newly developed model. The numerical results of the study show that the microstructure effect on band gaps is large when the plate is very thin. In addition, the unit cell edge length, inclusion geometry and MEE coupling have significant effects on band gap sizes, and large band gaps can be generated by tailoring the controlling parameters. [ABSTRACT FROM AUTHOR]

The flexural behavior of functionally graded composite panels with single and multiple rectangular perforations is examined subjected to complex loading conditions, including uniform/sinusoidal pressure and uniform/linear/nonlinear thermal field. The constituent materials are assumed to be temperature-dependent, whereas the instantaneous material properties are evaluated using the power-law function via Voigt's homogenization scheme. Here, the displacement field is based on the equivalent single-layer higher-order kinematics with nine degrees of freedom. The equations of motion are obtained using minimum total potential energy principle and isoperimetric finite-element approximations via Lagrange elements. The mesh convergence test is carried out to show the stable and appropriate mesh density; however, to check the correctness, the present solution is compared with the benchmark solutions. In addition, new numerical examples are presented to demonstrate the effect of different parameters, such as the number of perforations, power-law coefficients, aspect ratios, loading, and edge conditions on the deflection responses of perforated functionally graded composite panels. [ABSTRACT FROM AUTHOR]

This study is concerned with the distribution of flexural vibrations in plates excited with a Gaussian white noise. The distribution of energy can be characterized by its homogeneity and isotropy. Some particular geometries like the Bunimovitch stadium generate a field that is both homogeneous and isotropic. But other geometries produce a field that is homogeneous but not isotropic (like the rectangle panel) or non-isotropic and not homogeneous (like the circular panel). It is known that these features drive the establishment of diffuse field. However, in the present work, we show that even at high frequency and for these three particular geometries, the diffuse field cannot be reached rigorously. Owing to symmetries, the vibrational response is always enhanced on some particular lines and points by the effect of coherence between rays. The enhancement factors are predicted theoretically with the image-source method. The presence of energy enhancement is also shown experimentally by measuring the vibrational energy density in the 20 Hz-4 kHz frequency range for these three plates excited with a random white noise. Measurement of enhancement factors shows a good agreement with their theoretical predictions. [ABSTRACT FROM AUTHOR]

The flexural wave propagation in a microbeam is studied based upon the nonlocal strain gradient model with the spatial and time fractional order differentials in the present work. To capture the dispersive behaviour induced by the inherent nanoscale heterogeneity, the stress gradient elasticity and the strain gradient elasticity are often used to model the mechanical behaviour. The present model incorporates the two models and introduces the fractional order derivatives which can be understood as a generalization of integral order nonlocal strain gradient model. The Laplacian operator in the constitutive equation is replaced with the symmetric Caputo fractional differential in the present model. To illustrate the flexibility of the present model, the flexural wave propagation in a microbeam is studied. The fractional order in the present model as a new material parameter can be adjusted appropriately to describe the dispersive properties of the flexural waves. The numerical results based on the new nonlocal strain gradient elasticity with fractional order derivatives are provided for both Euler–Bernoulli beam and Timoshenko beam. The comparisons with the integer order nonlocal strain gradient model and the molecular dynamic simulation are performed to validate the flexibility of the fractional order nonlocal strain gradient model. [ABSTRACT FROM AUTHOR]

Mutual forces can be induced between coupled structures when the structures are stimulated by external acoustic waves. In this work, we propose a concept for asymmetric interaction between two coupled plate-like structures generated by plane waves that are incident in opposite directions. In addition to the striking contrast between the interaction strengths produced by the different incident waves, the mutual force induced by one of these incident pairs can be tuned extremely strongly because of the resonant excitation of the flexural plate modes. The highly asymmetric interaction with enhanced strength on a single side is potentially useful in applications such as the design of ultrasound instruments and sensors. [ABSTRACT FROM AUTHOR]

Our first objective is to identify two-dimensional equations that model the displacement of a linearly elastic flexural shell subjected to the action of an external heat source. To this end, we embed the shell into a family of linearly elastic flexural shells, all sharing the same middle surface θ (ω ‾) , where ω is a domain in R 2 and θ : ω ‾ → E 3 is a smooth enough immersion and whose thickness 2 ε > 0 is considered as a "small" parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as ε approaches zero, the corresponding "limit" two-dimensional variational problem. Our second objective is to identify and justify a set of two-dimensional equations that are meant to approximate the original three-dimensional model in the case where the shell under consideration is either an elliptic membrane shell or a flexural shell. [ABSTRACT FROM AUTHOR]

In a previous study (Marigo et al. in J. Mech. Phys. Solids 143:104029, 2020) we have studied the effect of a periodic array of subwavelength plates or beams over a semi-infinite elastic ground on the propagation of waves hitting the interface. The study was restricted to the low frequency regime where only flexural resonances take place. Here, we present a generalization to higher frequencies which allows us to account for both flexural and longitudinal resonances and to evaluate their interplay. An effective model is obtained using asymptotic analysis and homogenization techniques, which can be expressed in terms of the ground alone with an effective dynamic (frequency-dependent) boundary conditions of the Robin's type. For an in-plane wave at oblique incidence, the scattered displacement fields and the reflection coefficients are obtained in closed forms and their effectiveness to reproduce the actual scattering is inspected by comparison with direct numerics in a two-dimensional setting. [ABSTRACT FROM AUTHOR]

An efficient comprehensive vibration analysis method for a helicopter rotor–fuselage coupling system is presented. This loose computational fluid dynamics (CFD)/computational structural dynamics (CSD) coupling approach with a free wake–panel coupled model is used for system vibration response analysis. The CSD model of the helicopter consists of a fuselage model using a refined three dimensional (3 D) finite element model (FEM) and a rotor model consisting of nonlinear moderate deflection beam elements with 15 degrees of freedom. The unsteady Euler CFD solver is used for the flow field analysis of the entire vehicle. The induced inflow of the quasi-steady aerodynamic force is calculated with the free wake–panel coupled model, which is used to simulate rotor–fuselage aerodynamic interference. Using a full-scale helicopter as an example, the vibration responses of the typical fuselage position in hovering and level flights are analysed. When compared with the literature results and flight test data, the predictions of the proposed method are closer to the test data than those of the traditional method in hovering and low forward ratio flights, and the difference between the two methods is minimal in high forward ratio flight. [ABSTRACT FROM AUTHOR]

The article presents an analysis of the dynamic behaviour of discrete flexural systems composed of Euler–Bernoulli beams. The canonical object of study is the discrete Green's function, from which information regarding the dynamic response of the lattice under point loading by forces and moments can be obtained. Special attention is devoted to the interaction between flexural and torsional waves in a square lattice of Euler–Bernoulli beams, which is shown to yield a range of novel effects, including extreme dynamic anisotropy, asymmetric wave propagation, wave-guiding, filtering and the ability to create localised defect modes, all without the need for additional resonant elements or interfaces. The analytical study is complimented by numerical computations and finite element simulations, both of which are used to illustrate the effects predicted. A general algorithm is provided for constructing Green's functions as well as defect modes. This algorithm allows the tuning of the lattice to produce pass bands, band gaps, resonant modes, wave-guides and defect modes, over any desired frequency range. [ABSTRACT FROM AUTHOR]

We present a multiple scattering analysis of robust interface states for flexural waves in thin elastic plates. We show that finite clusters of linear arrays of scatterers built on a quasi-periodic arrangement support bounded modes in the two-dimensional space of the plate. The spectrum of these modes plotted against the modulation defining the quasi-periodicity has the shape of a Hofstadter butterfly, which as suggested by previous works might support topologically protected modes. Some interface states appear inside the gaps of the butterfly, which are enhanced when one linear cluster is merged with its mirror reflected version. The robustness of these modes is verified by numerical experiments in which different degrees of disorder are introduced in the scatterers, showing that neither the frequency nor the shape of the modes is altered. Since the modes are at the interface between two one-dimensional arrays of scatterers deposited on a two-dimensional space, these modes are not fully surrounded by bulk gaped materials so that they are more suitable for their excitation by propagating waves. The generality of these results goes beyond flexural waves since similar results are expected for acoustic or electromagnetic waves. [ABSTRACT FROM AUTHOR]