Flexural edge waves in semi-infinite elastic plates

This paper presents a detailed analysis of the dispersion for flexural edge waves in semi-infinite isotropic elastic plates. A solution to the dynamic equations of motion is constructed by the superposition of two partial solutions, each providing zero shear stresses at the plate faces. A dispersion equation is expressed via the determinant of an infinite system of linear algebraic equations. The system is reduced to a finite one by taking into account the asymptotic behaviour of unknown coefficients. The accuracy of the solution is confirmed by a good agreement with the available experimental data and by a proper satisfaction of the prescribed boundary conditions.A detailed analysis of dispersion properties for the edge wave and corresponding displacements at various frequencies is carried out. In addition to the well-known results it is shown that the plate height does not influence the existence of the edge wave at high frequencies and, as the frequency increases, the phase velocity of the edge wave in a semi-infinite plate asymptotically approaches the velocity of an edge wave in a right-angled wedge. The performed analysis allows evaluating the plate theories such as the Kirchhoff theory or other refined plate theories developed for modeling edge waves in semi-infinite elastic plates at low frequencies.     

Reliability assessment of different plate theories for elastic wave propagation analysis in functionally graded plates

The importance of elastic wave propagation problem in plates arises from the application of ultrasonic elastic waves in non-destructive evaluation of plate-like structures. However, precise study and analysis of acoustic guided waves especially in non-homogeneous waveguides such as functionally graded plates are so complicated that exact elastodynamic methods are rarely employed in practical applications. Thus, the simple approximate plate theories have attracted much interest for the calculation of wave fields in FGM plates. Therefore, in the current research, the classical plate theory (CPT), first-order shear deformation theory (FSDT) and third-order shear deformation theory (TSDT) are used to obtain the transient responses of flexural waves in FGM plates subjected to transverse impulsive loadings. Moreover, comparing the results with those based on a well recognized hybrid numerical method (HNM), we examine the accuracy of the plate theories for several plates of various thicknesses under excitations of different frequencies. The material properties of the plate are assumed to vary across the plate thickness according to a simple power-law distribution in terms of volume fractions of constituents. In all analyses, spatial Fourier transform together with modal analysis are applied to compute displacement responses of the plates. A comparison of the results demonstrates the reliability ranges of the approximate plate theories for elastic wave propagation analysis in FGM plates. Furthermore, based on various examples, it is shown that whenever the plate theories are used within the appropriate ranges of plate thickness and frequency content, solution process in wave number-time domain based on modal analysis approach is not only sufficient but also efficient for finding the transient waveforms in FGM plates.     

Flexural vibration band gaps in thin plates with two-dimensional binary locally resonant structures

The complete flexural vibration band gaps are studied in the thin plates with two-dimensional binary locally resonant structures, i.e. the composite plate consisting of soft rubber cylindrical inclusions periodically placed in a host material. Numerical simulations show that the low-frequency gaps of flexural wave exist in the thin plates. The width of the first gap decreases monotonically as the matrix density increases. The frequency response of the finite periodic thin plates is simulated by the finite element method, which provides attenuations of over 20dB in the frequency range of the band gaps. The findings will be significant in the application of phononic crystals.     

Pitch change during reverberant decay

The natural frequencies and mode shapes of a number of box beams are calculated by using the finite element displacement method. The structures are considered as assemblages of plates, and in general it is necessary to consider both the in-plane and transverse motion of the plates. A method of representing these two types of motion in the analysis of the vibrations of box beams is presented. A number of box beams of varying sectional parameters are analysed as systems of plates and the results compared with the predictions of Euler and Timoshenko beam theories. The comparisons show that for short beams constructed of thin plates, the new method can successfully represent the localized plate deformations, which cannot be described by beam theory.     

T-matrix method formulation applied to the study of flexural waves scattering from a through obstacle in a plate

Elastic wave scattering in a flat thin plate hosting a through obstacle of arbitrary closed form is examined using a numerical technique based on the T-matrix approach, which is applied to describe of flexural waves in plates. The limiting cases of a hole and a rigid obstacle are considered. The vibrations of the plate are described by the Kirchhoff model. The far field backscattered amplitude as a function of wave frequency for inclusions of elliptic, triangular and square form with rounded corners is analysed numerically. Comparison of present results for circular obstacles with the analytical solutions obtained by other authors show excellent agreement.     

A study of modal characteristics and the control mechanism of finite periodic and irregular ribbed plates

An analytical solution is presented in this paper to investigate the control mechanism and modal characteristics of finite periodic and irregular ribbed plates. Peak responses of a finite periodic ribbed plate were examined where they were grouped into two sets of propagation zones according to the coupling mechanism at beam/plate interfaces. Details of modal characteristics in pass bands of the periodic ribbed plate were elucidated and the control mechanism was discussed. Modes in each pass band that are governed by shear force couplings were characterized by one of the beam flexural modes whose modal responses could be represented approximately by those of the corresponding orthotropic plate modes. Modes in the second set of pass bands were found to retain the resonance frequencies of the corresponding modes of the unribbed base plate. Higher order orthotropic plate modes were also identified, which could not be grouped into any pass bands defined by the classical periodic theory. The control mechanism leading to vibration confinement in disordered and irregular ribbed plates was also discussed. It was found that beam spacing irregularity attributes to localization of the group of modes associated with flexural wave couplings but not the group of modes associated with moment couplings.     

A New Higher Order Shear Deformation Model for Static Behavior of Functionally Graded Plates

In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concerned flexural behavior of FG plates with Metal-Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static behavior of functionally graded plates.     

Numerical and experimental analysis of the vibration and band-gap properties of elastic beams with periodically variable cross sections

The band-gap properties of non-uniform periodic beams are analyzed using numerical and experimental methods. The flexural wave equations are established based on the Euler–Bernoulli and Timoshenko beam theories. The beams with periodically variable cross sections are investigated. The transfer matrix method is used to explore the dynamic behaviors of the periodic beams, that is, the natural frequencies of the finite periodic beams with different cross-section ratios between the adjacent sub-cells and the band-gaps of the infinite periodic beams based on the Bloch theory. The validity and accuracy of the band-gaps acquired by the present method are verified by comparing the results with those obtained from the finite element method and the vibration experiments. The effects of the different lengths of adjacent sub-cells on the band-gap properties are then investigated. The research results and conclusions should be useful in the study of vibration control applications.     

Active control of the plate energy transmission in a semi-infinite ribbed plate

Active control of the plate flexural wave transmission through the beam in a semi-infinite beam-reinforced plate is analytically investigated. The ribbed plate is modeled as a continuous system, using equations of motion to describe the plate in flexure and the beam in both flexure and torsion. The maximum transmission of the plate flexural waves through the reinforcing beam is found to occur at resonance frequencies corresponding to the optimal coupling between the plate flexural waves and the flexural and torsional waves in the beam. A single control force is applied to the beam, and a cost function is developed to attenuate the far-field flexural energy transmission. It can be observed that the transmission peaks corresponding to the flexural resonances in the beam are reduced. Similarly, the transmission peaks corresponding to the torsional resonance conditions in the beam can be attenuated using a single control moment applied to the beam. Significant attenuation of all the resonance peaks in the flexural wave transmission can also be achieved with the application of a single force and a single moment collocated on the beam. In this paper, the feasibility of attenuating the flexural wave transmission due to both the flexural and torsional resonance conditions by using a single point force and point moment collocated on the beam is demonstrated.     

Flexural wave motion in beam-type disordered periodic systems: Coincidence phenomenon and sound radiation

This paper deals with flexural wave motion in uniform beam-type periodic systems whose repeating units are identical finite beams with multiple beam-length disorders. A general expression derived for the propagation constants has been employed to study its variation with frequency for a beam system having 4-span disordered repeating units. This is helpful in understanding flexural wave motion in disordered periodic beams. Free flexural waves have been studied as wave groups consisting of a large number of harmonic components of different wavelengths, phase velocities and directions. Phase velocities have been computed and plotted for different frequencies in the propagation zones in which the free waves progress without attenuation. This has been found to be useful in understanding and predicting the coincidence phenomenon in disordered periodic beams under convected pressure field loading. The excitation of wave groups in disordered periodic beam-type systems by a slow (subsonic) convecting pressure field can include fast (supersonic) moving flexural wave components which can radiate sound. It has been pointed out that sound radiation from a disordered periodic beam (or plate) can be quite different as compared to that from a periodic beam under similar convected pressure field loading.     

一种多频局域共振型声子晶体板的低频带隙与减振特性

设计了一种多频局域共振型声子晶体板结构, 该结构由一薄板上附加周期性排列的多个双悬臂梁式子结构而构成. 由于多个双悬臂梁式子结构的低频振动与薄板振动的相互耦合作用, 这种局域共振型板结构可产生多个低频弯曲波带隙(禁带); 带隙频率范围内的板弯曲波会被禁止传播, 利用带隙可以实现对薄板的多个目标频率处低频减振. 本文针对这种局域共振型板结构进行了简化, 并基于平面波展开法建立了其弯曲波带隙计算理论模型; 基于该模型, 结合具体算例进行了带隙特性理论分析. 设计、制备了一种存在两个低频弯曲波带隙的局域共振型板结构样件, 通过激光扫描测振仪测试证实该结构存在两个低频带隙, 在带隙频率范围的板弯曲振动被显著衰减.     

Flexural and transversal wave motion in homogeneous isotropic thermoelastic plates by using asymptotic method

The present investigation is concerned with the flexural and transversal wave motion in an infinite, transversely isotropic, thermoelastic plate by asymptotic method. The governing equations for the flexural and transversal motions have been derived from the system of three-dimensional dynamical equations of linear theory of coupled thermoelasticity. The asymptotic operator plate model for free vibrations; both flexural and transversal, in a homogenous thermoelastic plate leads to fifth degree and cubic polynomial secular equations, respectively, that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. All the coefficients of differential operator have been expressed as explicit functions of the material parameters. The velocity dispersion equations for the flexural and transversal wave motion have been deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity have also been derived. The thermoelastic Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established with that of asymptotic method. The dispersion curves for phase velocity, group velocity and attenuation coefficient of various flexural and transversal wave modes are shown graphically for aluminum-epoxy material elastic and thermoelastic plates.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid

In this study, a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics (natural frequencies and associated mode shapes) of rectangular Kirchhoff plates. For the interaction of the Kirchhoff plate–Pasternak foundation, a mixed-type finite element formulation is employed by using the Gâteaux differential. The plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes, and at each node four degrees of freedom are present (one transverse displacement, two bending moments and one torsional moment). Therefore, a total number of 16 degrees-of-freedom are assigned to each element. A consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis. The plate structure considered is assumed clamped or simply supported along its edges and resting on a Pasternak foundation. Furthermore, the plate is fully or partially in contact with fresh water on its one side. For the calculation of the fluid–structure interaction effects (generalized fluid–structure interaction forces), a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid's free surface. It is assumed that the fluid is ideal, i.e., inviscid, incompressible, and its motion is irrotational. It is also assumed that the plate–elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. At the fluid–structure interface, continuity considerations require that the normal velocity of the fluid is equal to that of the structure. The normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements, obtained from the finite element analysis. By using the boundary integral equation method the fluid pressure is eliminated from the problem, and the fluid–structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients (due to the inertial effect of fluid). To asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure, the natural frequencies and associated mode shapes are presented. Furthermore, the influence of the submerging depth on the dynamic behavior is also investigated.     

Impedance of infinite Kirchhoff and Mindlin plates with a rigid circular massless plug

Point force impedance expressions have been previously developed for infinite Kirchhoff and Mindlin plates. The present work develops impedance expressions for the more general case of an infinite plate with a circular, massless, rigid plug using both Kirchhoff and Mindlin plate theories. The models have been developed to analyze vibration propagation in buildings. The plate with the rigid plug provides a more reasonable model of the kinematic constraint at the column/floor interface. The models are used to investigate the potential benefits of using thick floors to block the transmission of structure-borne vibration in buildings.     

Experimental investigation of vibration power flow in thin technical orthotropic plates by the method of vibration intensity

Vibration intensity technique is used to measure vibration power transmission in thin single layer technical orthotropic plates for flexural waves. Measurement of flexural wave power is carried out in far-field conditions. All measurements are undertaken in the frequency domain using the cross-spectra of acceleration signals, facilitating the use of FFT analyzer. The two-transducer technique applicable to these plates is used for these measurements. Technical orthotropic (rectangular corrugation) plates of steel are used for the measurements. One isotropic plate of steel is also considered for comparison. Method of elastic equivalence technique is used. Both input power and vibration power transmission through the plates are estimated. Far-field power is normalized with the input power for flexural wave. Influence of flexural rigidity on vibration energy transfer is also investigated.     

Wave reflection and transmission in beams containing delamination and inhomogeneity

This paper presents an analytical approach using higher order plate theories to determine wave reflections from and transmissions through a damaged region in a beam. The damaged region is either treated as two split beams or as an inhomogeneity. The reflection ratios and transmission ratios are found to depend strongly on the frequency of the incident flexural waves, as well as the size of the damage, which gives rise to strong stop/pass band behaviour. Using the spectral analysis method, the transient wave propagation in a beam with a part-through delamination is predicted and compared with experimental results, indicating a good agreement in the phases and amplitudes of both the reflected and transmitted waves.     

Resonances of the Rayleigh waves in an elastic semi-infinite strip

Kirchhoff’s theory of plates is used to study forced harmonic vibrations of a semi-infinite strip when the latter is in the generalized stressed state or experiences flexural deformation. The forced vibrations are excited by a load applied to the strip end. Cross-boundary conditions are imposed on the strip’s sides, which allows one to obtain a closed solution. The presence of an infinite real frequency spectrum corresponding to the edge resonances is revealed. The relation of these resonances to the Rayleigh planar and flexural waves is established.     

Bending wave propagation of carbon nanotubes in a bi-parameter elastic matrix

This article studies transverse waves propagating in carbon nanotubes (CNTs) embedded in a surrounding medium. The CNTs are modeled as a nonlocal elastic beam, whereas the surrounding medium is modeled as a bi-parameter elastic medium. When taking into account the effect of rotary inertia of cross-section, a governing equation is acquired. A comparison of wave speeds using the Rayleigh and Euler-Bernoulli theories of beams with the results of molecular dynamics simulation indicates that the nonlocal Rayleigh beam model is more adequate to describe flexural waves in CNTs than the nonlocal Euler-Bernoulli model. The influences of the surrounding medium and rotary inertia on the phase speed for single-walled and double-walled CNTs are analyzed. Obtained results turn out that the surrounding medium plays a dominant role for lower wave numbers, while rotary inertia strongly affects the phase speed for higher wave numbers.     

Localized bending vibrations of piezoelectric plates

This paper investigates the existence and propagation of electro-elastic bending waves localized at the free edge of a piezoelectric plate. The problem is considered within the framework of the high-order refined plate theory introduced by Ambartsumian. The condition for existence of a localized bending wave is obtained, and the dispersion equation solved with respect to a dimensionless frequency. It is shown that the piezoelectric effect can increase the attenuation coefficient for a localized wave by up to 70% compared with that for a purely elastic plate, thus significantly decreasing the depth of penetration. The problem is also solved within the classical Kirchhoff theory. A comparison of results is carried out between two theories.