Bending Wavenumbers Research Articles

Assessing the behavior of highly damped multilayered structures with controllable patches using condensation models

This study presents the finite element modeling of an adaptive sandwich structure that can be configured to exhibit various behaviors depending on the arrangement of the temperature fields in its core, which is made of a shape memory polymer with outstanding damping capacity. The sandwich structure is divided into patches corresponding to homogeneous temperature fields. Each patch is modeled as an equivalent thin plate using a condensation model and the effective properties are determined by preserving the real part of bending, shear (transverse) and extension (quasi-longitudinal) wavenumbers. The damping loss factor is estimated independently using the forced response from an analytical discrete model of the sandwich structure and the Power Input Method. Two different configurations of the sandwich structure are presented: in the first, the modal behavior of the sandwich structure is slightly affected; In the second, vibrations are reduced throughout the frequency range. The use of a suitable condensation model can be particularly interesting for solving optimization problems and determining optimal configurations. The behavior of the sandwich structure can be evaluated with reduced computational cost whenever an update of the temperature fields is required.

Estimation of the low-wavenumber component of turbulent boundary layer pressure field using vibration data

In structures subjected to a high-speed flow, the convective region of the wall pressure field (WPF) beneath a turbulent boundary layer (TBL) plays a crucial role in their vibration behaviour. However, when it comes to underwater structures experiencing low-speed flow, they effectively filter out the convective domain of WPF and the bending wavenumber of structures align with the low-wavenumber domain of the WPF. As a result, the primary cause of vibration in this case is the low-wavenumber components of the WPF. Thus, accurate estimation of the WPF at low-wavenumber domain is crucial for predicting vibration responses of these structures. Existing models for WPF accurately predict the convective region but differ significantly in predicting the low-wavenumber levels. This numerical study aims to investigate the feasibility of estimating the low-wavenumber WPF by analysing measured vibration data from a flat plate excited by a TBL. The WPF's cross spectrum in the wavenumber domain can be linked to the cross spectrum of the plate's acceleration. By employing regularization techniques and solving an inverse problem, the low-wavenumber components of the WPF can be then estimated. Virtual experiments are performed to evaluate the accuracy of the studied process by comparing its prediction to the input WPF model.

Equivalent dynamic model of multilayered structures with imperfect interfaces: Application to a sandwich structured plate with sliding interfaces

This works aims to build an equivalent model of multilayered structures with imperfect interfaces for vibro-acoustic modeling and characterization. To take into account imperfections, new interface conditions including constitutive equations, that describe the imperfections, are implemented. Once the displacement field with imperfect interfaces is obtained, the dispersion relation of the structure is derived from the equivalent model. The bending wavenumbers are then used to compute the equivalent flexural rigidity and the damping of the sandwich panel. In this paper, the methodology is applied to a sandwich panel with sliding interfaces. The equivalent parameters are computed and compared to the reference case, i.e. perfect interfaces model. The main impact of the imperfection on the results is a shift towards the low frequency of the equivalent parameter curves.

Algebraic wavenumber identification method in presence of uncertainty

This paper presents an algebraic wavenumber identification method to identify wavenumbers under stochastic conditions. Stochastic condition results from the introduction of small perturbation which is referred to the uncertainty of measurements points’ coordinates caused by the operation faults or problems with experimental errors. The proposed method is compared with two popular alternatives, namely: inhomogeneous wave correlation method and inverse convolution method which are both capable of extracting the bending wavenumbers of a meta-structure. A good performance is observed for the identification of complex wavenumbers in presence of uncertainty. In addition, the proposed method needs to solve a linear problem, reducing the computational cost compared to the inhomogeneous wave correlation method.

Mindlin 판 이론을 적용한 단순지지 단면 접수평판의 음향방사효율 수치해석

Acoustic radiation efficiency is a crucial factor to estimate Underwater Radiated Noise (URN) of ships accurately. This paper describes a numerical method to analyse acoustic radiation efficiency of one side-wetted rectangular Mindlin plate with simply supported boundaries excited by a harmonic point force. Transverse displacements of plate and acoustic radiation pressures are evaluated by the mode superposition method. The acoustic radiation efficiencies analyzed by both Mindlin and thin plate theories show little differences at monopole and corner modes of low frequency regions but relatively large differences at edge and critical modes of high frequency regions. Especially, the critical frequency with the highest acoustic radiation efficiency evaluated by the Mindlin plate theory is higher than that of thin plate theory. In addition, the acoustic loading effect of fluid also increases bending wave-number of plate and its critical frequency. Finally, the acoustic radiation characteristics of plates with different aspect ratios and thicknesses through numerical analyses are investigated and discussed.

Raman spectroscopy of CaM2+Ge2O6(M2+ = Mg, Mn, Fe, Co, Ni, Zn) clinopyroxenes

The Raman spectra of Ge-clinopyroxenes CaM2+Ge2O6 (M2+ = Mg, Mn, Fe, Co, Ni, Zn), general formula M2M1T2O6, are reported for the first time. Their spectral features are discussed in comparison with corresponding Si-pyroxenes. The vibrational wavenumbers of germanates may be roughly obtained by a scale factor of about ~0.8 by those of the corresponding silicates, due to the Ge-Si mass difference. The main peaks in the germanate Raman spectra at ~850 and ~540 cm−1 may be related to Ge-O tetrahedral stretching and chain bending, respectively; minor peaks between 200 and 400 cm−1 are ascribed to bending and stretching of the non-tetrahedral cations. Within Ge-pyroxenes, possible correlations between crystallographic parameters and the vibrational wavenumbers are investigated. The main stretching mode at ~850 cm−1 shows wavenumber changes with M2+ substitutions, but no simple correlation can be found with M2+ cation mass or size. On the other hand, the chain bending wavenumber linearly decreases with increasing ionic radius of the M2+ cation: the expansion of the M1 polyhedron reduces the chain kinking angle and the Ge-Ge distances correspondingly increase. Copyright © 2015 John Wiley & Sons, Ltd.

Vibroacoustic flexural properties of symmetric honeycomb sandwich panels with composite faces

The vibroacoustic bending properties of honeycomb sandwich panels with composite faces are studied from the wavenumber modulus to the mechanical impedance, passing through the modal density. Numerical results extracted from finite element software computations are compared with analytical results. In both cases, the homogenization method is used to calculate the global properties of the sandwich panel. Since faces are made of composite material, the classical laminate theory serves as reference. With particular conditions used in the application for symmetric panels, the original orthotropic mechanical properties can be reduced simply to three parameters commonly used in vibroacoustic characterizations. These three parameters are the mass per unit area, the bending rigidity and the out-of-plane shear rigidity. They simultaneously govern the wavenumber modulus, the modal frequencies, the modal density and the mechanical impedance. For all of these vibroacoustic characterizations, a special frequency called the transition frequency separates two domains. In the first domain, below the transition frequency or for low frequencies, the orthotropic sandwich panel has a classical isotropic plate behavior. In the second domain, above the transition frequency or for high frequencies, the out-of-plane shear rigidity is very significant and changes the behavior.However, the results discussed are only valid up to a certain frequency which is determined by the thickness and out-of-plane shear stiffness of the honeycomb core, the thickness and the bending stiffness of the laminated face sheets and then the mass per unit area and bending stiffness of the total sandwich structure. All these parameters influence the final choice of model and simplifications presented.Experimental measurements of the bending wavenumber modulus and modal frequencies for our own application were carried out. In the vibroacoustic domain, the critical frequency is also an important frequency. It again depends on the mass per unit area, the bending rigidity and the out-of-plane shear rigidity. The experimental and numerical results of the article are reasonably in agreement with the analytical formula. They all confirm the changes in frequency through different boundary conditions around the panel.The analytical modal frequencies of rectangular sandwich panels with transverse shear, under simply supported boundary conditions are well known, but under free boundary conditions it is more difficult to predict them. For experiments, however, these latter conditions are the most common. We present, in this paper, an analytical formula that we have developed for the modal frequencies of such a panel under free boundary conditions. All parameters being controlled, it is possible from dynamic measurements and with a special process to identify some honeycomb and composite mechanical properties.

The average specific forced radiation wave impedance of a finite rectangular panel.

The average specific forced radiation wave impedance of a finite rectangular panel is of importance for the prediction of both sound insulation and sound absorption. In 1982, Thomasson published numerical calculations of the average specific forced radiation wave impedance of a square of side length 2e for wave number k in half octave steps of ke from 0.25 to 64. Thomasson's calculations were for the case when the forced bending wave number kb was less than or equal to k. Thomasson also published approximate formulas for values of ke above and below the published results. This paper combines Thomasson's high and low frequency formulas and compares this combined formula with Thomasson's numerical calculations. The real part of the approximate formula is between 0.7 dB higher and -1 dB lower than the numerical calculations. The imaginary part of the approximate formula is between 2.3 dB higher and -2.6 dB lower than the numerical calculations. This paper also gives approximate formulas for the case when kb is greater than or equal to k. The differences are between 0.8 and -1.2 dB for the imaginary part and between 6.2 and -2.4 dB for the real part.

The evolution of the structural, vibrational and electronic properties of the cyclic ethers – on ring size. An ab initio study

The molecular structures, vibrational spectra and atomic charges of the alicyclic ethers containing from two to five carbon atoms have been determined by means of ab initio calculations, at the level of second order Møller–Plesset perturbation theory and using Dunning’s augmented correlation-consistent polarized valence triple-zeta basis set. Two isomers of the oxetane, tetrahydrofuran and tetrahydropyran molecules have been identified and their relative energies determined. Structural properties, such as the COC bond angles and the CH bond lengths, are found to increase steadily with increasing ring size and with decreasing ionization energy. The mean CH2 stretching and bending wavenumbers exhibit the reverse behaviour, while the mean wavenumbers of the CH2 wagging and twisting modes follow the same trend as the structural features. The ring mode wavenumbers vary in a less regular way. The charges of the oxygen, α-carbon and axial and equatorial α- and β-hydrogen atoms also do not show systematic dependences on ring size or ionization energy. The trends in the values of these properties have been rationalized.

Asymptotic expansions for the structural wavenumbers of isotropic and orthotropic fluid-filled circular cylindrical shells in the intermediate frequency range

We consider wavenumbers in in vacuoand fluid-filled isotropic and orthotropic shells. Using the Donnell–Mushtari (DM) theory we find compact and elegant asymptotic expansions for the wavenumbers in the intermediate frequency range, i.e., around the ring frequency. This frequency range corresponds to the frequencies where there is a rapid change in the values of bending wavenumbers and is found to exist in isotropic and orthotropic shells (in vacuoand fluid-filled) for low circumferential orders n only. The same is first identified using the n=0 mode of an orthotropic shell. Following this, using the expression for the intermediate frequency, asymptotic expansions are found for other cases. Here, in order to get compact expansions we consider slight orthotropy (ϵ⪡1) and light fluid loading (μ⪡1). Thus, the orthotropy parameter ϵ and the fluid loading parameter μ are used as asymptotic parameters along with the non-dimensional thickness parameter β. The methodology can be extended to any order of ϵ, only the expansions become unwieldy. The expansions are matched with the numerical solutions of the corresponding dispersion relation. The match is found to be good.

On the measurement of angular-dependent, airborne sound transmission through finite supercritical bars: Further results

A method of measuring the angular dependence of sound transmission through supercritical bars in air is discussed. The coincidence effect occurs in a supercritical bar when the component of the acoustic wave number parallel to the bar matches the bending wave number in the bar. The transmission of sound is at a maximum at the angle where this trace wave number matching occurs. The theory of the coincidence effect is well-defined for unbounded thin plates using plane-wave excitation. An experimental setup has been developed in order to observe the coincidence effect using continuous-wave excitation and phased-array methods through finite bars. Experimental results through a 0.5 mm thick aluminum bar exhibit strong maxima at the predicted coincidence angles, showing that coincidence is observable using continuous waves. Measurements of the coincidence angle at frequencies spanning from the critical frequency up to nearly three times the critical frequency have been made. A curve fit to the frequency depende...

Radiation efficiency of a guitar top plate linked with edge or corner modes and intercell cancellation

This paper was based on a theoretical framework to determine strong and weak radiation by a guitar top plate, vibrating through deflections hard to analyze: multipolar mode shapes. The air-structure interaction was examined in terms of edge modes or corner modes, and considering even or odd number modes. A numerical model was implemented and experimentally calibrated, exhibiting several advantages exploring the coupling between vibratory and acoustic waves in a top plate. Two analyses were applied detecting high or low radiation efficiency for the structure. First, the addition of volume velocity for odd numbers of poles and cancellation for even numbers were examined. In fact, both behaviors can happen at the same time, as it was shown for a corner radiator case used as an example. Second, the ratio between bending and acoustic wavenumbers was explored. To illustrate the importance of this ratio, some theoretical features of a more efficient radiator than the corner mode were exposed in an edge mode example. Labeling multipolar mode shapes as efficient or inefficient radiators showed to be a useful alternative analyzing the top plate behavior. It can be applied knowing the nodal lines of the vibration pattern and estimating the bending and acoustic wavelengths.

Verdigris pigment: a mixture of compounds. Input from Raman spectroscopy

AbstractThe reproduction of an old recipe to obtain the pigment known as verdigris leads to a complex mixture of hydrated copper(II) acetates containing neutral verdigris and two varieties of basic verdigris. Since the characterization of such mixture is complicated by the lack of reference standards, we complemented our Raman measurements with Fourier transform infrared (FTIR) and X‐ray diffraction (XRD) experiments to perform our analysis, which has been compared with available studies. The spectral analysis allowed us to distinguish the different verdigris varieties, with stretching and bending wavenumbers assigned to the following chemical groups: OH groups (OHhydroxyand OHH2O) (3600–3050 cm−1), CuO bonds (CuO and CuOH) (1000–450 cm−1) and those corresponding to the acetate group (CH3COO−), i.e. (CH3: 3000–2850 cm−1) and (COO−: 1600–1400 cm−1and 700–600 cm−1). Both wavenumbers and Raman intensities were used to quantify the relative compositions of the mixture components. Copyright © 2010 John Wiley & Sons, Ltd.

Sound radiation characteristics of a box-type structure

The finite element and boundary element methods are employed in this study to investigate the sound radiation characteristics of a box-type structure. It has been shown [T.R. Lin, J. Pan, Vibration characteristics of a box-type structure, Journal of Vibration and Acoustics, Transactions of ASME 131 (2009) 031004-1–031004-9] that modes of natural vibration of a box-type structure can be classified into six groups according to the symmetry properties of the three panel pairs forming the box. In this paper, we demonstrate that such properties also reveal information about sound radiation effectiveness of each group of modes. The changes of radiation efficiencies and directivity patterns with the wavenumber ratio (the ratio between the acoustic and the plate bending wavenumbers) are examined for typical modes from each group. Similar characteristics of modal radiation efficiencies between a box structure and a corresponding simply supported panel are observed. The change of sound radiation patterns as a function of the wavenumber ratio is also illustrated. It is found that the sound radiation directivity of each box mode can be correlated to that of elementary sound sources (monopole, dipole, etc.) at frequencies well below the critical frequency of the plates of the box. The sound radiation pattern on the box surface also closely related to the vibration amplitude distribution of the box structure at frequencies above the critical frequency. In the medium frequency range, the radiated sound field is dominated by the edge vibration pattern of the box. The radiation efficiency of all box modes reaches a peak at frequencies above the critical frequency, and gradually approaches unity at higher frequencies.

Hydrodynamic and hydroelastic analyses of a plate excited by the turbulent boundary layer

Recent studies have demonstrated that the characterisation of wall-pressure fluctuations for surface ships is of great interest not only for military applications but also for civil marine vehicles. A ship model towed in a towing tank is used to perform pressure and structural measurements at high Reynolds numbers. This facility provides ideal flow conditions because background turbulence and noise are almost absent. Free surface effects are naturally included in the analysis, although in the particular section chosen for the present study do not have significant consequences on pressure spectra. Scaling laws for the power spectral density are identified providing the possibility to estimate pressure spectra for different flow conditions and in particular for full-scale applications. The range of validity of some theoretical models for the cross-spectral density representation is analysed by direct comparison with experimental data of wall-pressure fluctuations measured in streamwise and spanwise direction. In a second phase, an indirect validation is performed by comparing the measured vibrational response of an elastic plate inserted in the catamaran hull with that obtained numerically using, as a forcing function, the modelled pressure load. In general, marine structures are able to accept energy mainly from the sub-convective components of the pressure field because the typical bending wavenumber values are usually lower than the convective one; thus, a model that gives an accurate description of the phenomenon at low wavenumbers is needed. In this work, it is shown that the use of the Chase model for the description of the pressure field provides a satisfactory agreement between the numerical and the experimental response of the hull plate. These experimental data, although acquired at model scale, represent a significant test case also for the real ship problem.

Laser Raman and infrared spectral studies of dl-phenylalaninium nitrate

The vibrational band assignments of dl-phenylalaninium nitrate in the crystalline state are made by recording the infrared and Raman spectra at room temperature. The presence of carbonyl (C O) group has been identified. The prominent marker bands of the aromatic amino acid phenylalanine have been observed and the various modes of vibration have been assigned. The extensive intermolecular hydrogen bonding in the crystal has been identified by the shifting of bands due to the stretching and bending modes of the various functional groups. The nitrate group forms the anion. The stretching and bending wavenumbers of the NO 3 − anion are different from those observed for free ion state and the degenerating mode of vibrations is also lifted. These reveal that the crystalline field has influenced the symmetry of the nitrate ion.

Vibrational spectral studies of l-citrullinium perchlorate

Infrared and Raman spectra of l-citrullinium perchlorate crystals have been recorded at room temperature. The vibrational assignments of the observed wavenumbers are proposed on the basis of group theoretical analysis. The presence of carbonyl group indicates that the molecule exists in the ionic form. The shifting of stretching and bending wavenumbers indicates the presence of extensive hydrogen bonding in the crystal. The anion fundamentals however continue to be degenerated. This suggests that its symmetry is not affected in the crystal.

Infrared and laser Raman spectra of bis(DL-methioninium) sulfate

The infrared and laser-excited Raman spectra of bis(DL-methioninium) sulfate crystal were recorded and analyzed at room temperature. The vibrational assignments of the observed wavenumbers were made based on group theoretical analysis. The presence of CO group is confirmed. Factor group analysis was carried out and the number of normal vibrational modes was calculated. The shifting of several stretching and bending wavenumbers suggests that the extensive intermolecular hydrogen bonding causes both linear and angular distortions of several groups. The splitting of the threefold degenerate stretching and bending modes of the sulfate group confirms the distortion of its Td symmetry due to hydrogen bonding. Copyright © 2005 John Wiley & Sons, Ltd.

Laser vibrometry technique for measurement of contained stress in railroad rail

Based on the known sensitivity of a beam's effective flexural rigidity to its static axial load, we experimentally investigate whether measurements of lateral bending wavenumber, at a specified imposed frequency, can be used to nondestructively determine that load. Initial estimates are that at an imposed frequency in the vicinity of 200 Hz, where wavelengths are of the order of 2 m, the sensitivity should be adequate if the wavenumber can be extracted with sufficient precision. Scanned laser vibrometry, followed by digital lock-in, and referencing with a fixed accelerometer, is found to accurately measure steady-state vibration distributions. Nonlinear least-square fits to theoretical forms consisting of a sum of guided vibration modes then give a best-fit value for that wavenumber, a value that correlates well with the known levels of load in the experiments. The proposed technique appears viable.

Estimating in-situ Material Properties of a Wood Joist Floor: Part 1 — Measurements of the Real Part of Bending Wavenumber

This paper outlines and evaluates a non-destructive experimental technique used to obtain in-situ measures of the real part of the bending wavenumber in the two principal directions of a wood joist floor. In-situ measured wavenumbers are compared to those obtained from beam samples cut from the floor sheathing to identify the frequency range when the joists significantly affect vibration response in the floor sheathing. Wavenumber measurements confirm the highly orthotropic nature. Measurements indicate that the wavenumbers in the direction parallel to the joists are a function of location between the joists. Lower wavenumbers in this direction can be expected closer to the joists. Some discussion is given regarding the rates of attenuation with distance that can be expected parallel and perpendicular to the joists given the variation in wavenumbers, however this will be the topic of a separate paper, Part 2.