On the modal density and damping of cylindrical pipes

Measurements of the resonant frequencies and quality factors of a series of long, small diameter cylindrical pipes are presented. The dependence of the modal densities, calculated from these measurements, on pipe length, wall thickness and pipe material is in close agreement with theoretical statistical predictions. Measurements of the damping of steel pipes for several different end conditions are also presented. Different, but always well-ordered, variation of modal quality factor with mode order is found in each case. For free-free ends the modal quality factors are large (>1000) and determined by internal material damping, except for modes for which the acoustic radiation damping is large; the effect of radiation damping is most important for those modes which have high wave speeds at low frequencies. The quality factors for rigid end conditions are similar to those for free ends, except for translational modes of low axial order which are damped by vibration of the end supports. For end conditions which allow relative motion between the pipe and the end supports, there is considerable additional damping, probably ascribable to gas pumping in the joints.     

Dynamic stability of a rotating beam with a constrained damping layer

The dynamic behavior and dynamic instability of the rotating sandwich beam with a constrained damping layer subjected to axial periodic loads are studied by the finite element method. The influences of rotating speed, thickness ratio, setting angle and hub radius ratio on the resonant frequencies and modal system loss factors are presented. The regions of instability for simple and combination resonant frequencies are determined from the Mathieu equation that is obtained from the parametric excitation of the rotating sandwich beam. The regions of dynamic instability for various parameters are presented.     

Damping of stiffened plates by multiple layer treatments

It is shown in this paper that the modal damping and resonant frequencies of a stiffened plate structure, with a multiple layer constrained damping treatment attached to the surface, can be predicted from a knowledge of the equivalent complex modulus properties of the treatment. The equations used represent a simple extension of the classical equations of Oberst for a free layer treatment applied to an unstiffened beam or plate, with terms accounting for the effect of the stiffeners. The equivalent complex modulus properties of the treatment depend on a shear parameter, a geometrical parameter, the stiffness of the constraining layer and the loss factor of the adhesive. Experimental results are discussed.     

Analysis of mode shapes of a rigidly backed cylindrical foam using three-dimensional finite element acoustical analysis

In this study, the three-dimensional finite element frequency domain acoustical analysis is used to determine the modal shapes of cylindrical foam with a rigid backing and subjected to a unit normal incidence impulsive sound pressure loading while placed in the impedance tube. The acoustic results predicted for the foam are validated by data from the two-microphone acoustic measurements, and good agreement between the measured and predicted acoustic results is observed. The mode shapes of the incident face of the foam at a low frequency, resonant and anti-resonant frequencies as well as the frequency that occurring the peak loss modulus are illustrated. It is found that the modal behaviors of the cylindrical foam are dominated by the fluid, although the acoustic properties of the cylindrical foam are also influenced by the circumferential edge constraints and the modal movements of the solid skeleton.     

HEALTH-MONITORING METHOD FOR BRIDGES UNDER ORDINARY TRAFFIC LOADINGS

A method for damage estimation of a bridge structure is presented using ambient vibration data caused by the traffic loadings. The procedure consists of identification of the operational modal properties and the assessment of damage locations and severities. An experimental study is carried out on a bridge model with a composite cross-section subjected to vehicle loadings. Vertical accelerations of the bridge deck are measured while vehicles are running. The modal parameters are identified from the free-decay signals extracted using the random decrement method. The damage assessment is carried out based on the estimated modal parameters using the neural networks technique. As input to the neural networks, the ratios of the resonant frequencies between before and after damages and the mode shapes after the damages are used to take into account the mass effect of the traffic on the bridge. The identified damage locations and severities agree reasonably well with the inflicted damages on the structure.     

Sound transmission loss of foam-filled honeycomb sandwich panels using statistical energy analysis and theoretical and measured dynamic properties

Comparisons between the experimental and predicted sound transmission loss values obtained from statistical energy analysis are presented for two foam-filled honeycomb sandwich panels. Statistical energy analysis (SEA) is a modeling procedure which uses energy flow relationships for the theoretical estimation of the sound transmission through structures in resonant motion. The accuracy of the prediction of the sound transmission loss using SEA greatly depends on accurate estimates of: (1) the modal density, (2) the internal loss factor, and (3) the coupling loss factor parameters of the structures. A theoretical expression for the modal density of sandwich panels is developed from a sixth-order governing equation. Measured modal density estimates of the two foam-filled honeycomb sandwich panels are obtained by using a three-channel spectral method with a spectral mass correction to allow for the mass loading of the impedance head. The effect of mass loading of the accelerometer is corrected in the estimations of both the total loss factor and radiation loss factor of the sandwich panels.     

Microwave transmission through a single subwavelength annular aperture in a metal plate

The resonant transmission of a small annular aperture, with a diameter much smaller than the radiation wavelength, in a thin metal plate is studied at microwave frequencies. It transpires that such an annular aperture supports several resonant guided modes, including those that are not quantized in the azimuthal direction. Such modes have resonant frequencies that are largely independent of the diameter of the annular aperture, thus being supported by annular apertures that tend to zero radius. The transmittance of such a structure at microwave frequencies is detailed and compared with the predictions of a finite element method model.     

Reverberation time measurements in non-diffuse acoustic field by the modal reverberation time

The increasing presence of low frequency sources and the lack of acoustic standard measurement procedures make the extension of reverberation time measurements to frequencies below 100 Hz necessary. In typical ordinary rooms with volumes between 30 m³ and 200 m³ the sound field is non-diffuse at such low frequencies, entailing inhomogeneities in space and frequency domains. Presence of standing waves is also the main cause of bad quality of listening in terms of clarity and rumble effects. Since standard measurements according to ISO 3382 fail to achieve accurate and precise values in third octave bands due to non-linear decays caused by room modes, a new approach based on reverberation time measurements of single resonant frequencies (the modal reverberation time) has been introduced. From background theory, due to the intrinsic relation between modal decays and half bandwidth of resonant frequencies, two measurement methods have been proposed together with proper measurement procedures: a direct method based on interrupted source signal method, and an indirect method based on half bandwidth measurements. With microphones placed at corners of rectangular rooms in order to detect all modes and maximize SNRs, different source signals were tested. Anti-resonant sine waves and sweep signal turned out to be the most suitable for direct and indirect measurement methods respectively. From spatial measurements in an empty rectangular test room, comparison between direct and indirect methods showed good and significant agreements. This is the first experimental validation of the relation between resonant half bandwidth and modal reverberation time. Furthermore, comparisons between means and standard deviations of modal reverberation times and standard reverberation times in third octave bands confirm the inadequacy of standard procedure to get accurate and precise values at low frequencies with respect to the modal approach. Modal reverberation time measurements applied to furnished ordinary rooms confirm previous results in the limit of modal sound field: for highly damped modes due to furniture or acoustic treatment, the indirect method is not applicable due to strong suppression of modes and the consequent deviation of the acoustic field from a non-diffuse condition to a damped modal condition, while standard reverberation times align with direct method values. In the future, further investigations will be necessary in different rooms to improve uncertainty evaluation.     

The role of the induction zone on the detonation–turbulence linear interaction

Detonation–turbulence linear interaction analysis extends the non-reactive shock–turbulence analog by considering geometrical scaling of the noise with respect to the half-reaction distance. The analysis emphasizes the effect of structure in energizing selective frequencies, and determining acoustic amplification in the farfield. Natural frequencies are determined as eigenvalues of the inviscid non-forced interaction problem. They modify postshock energy spectra by supporting resonant amplification, and cast light on the role of the activation energy on the detonation–turbulence interaction. Detonations with higher activation energies amplify smaller scales by resonant amplification. An analysis of the bifurcation parameters reveals a strong link between detonation overdrive and acoustic attenuation. The damping is correlated with the subcritical nature of the characteristic solutions for high overdrives. For detonation conditions on the stability boundary, a larger overdrive supports a weaker resonant peak in both the temperature and longitudinal velocity spectra. Postshock temperature variances feature a well-defined maximum within the reaction zone, which is found to be sensitive to changes in detonation structure.     

A parametric multi-body section model for modal interactions of cable-supported bridges

A parametric section model is formulated to synthetically describe the geometrically nonlinear dynamics of cable-stayed and suspended bridges through a planar elastic multi-body system. The four-degrees-of-freedom model accounts for both the flexo-torsional motion of the bridge deck and for the transversal motion of a pair of hangers or stay cables. After linearization around the pre-stressed static equilibrium configuration, the coupled equations of motion governing the global deck dynamics and the local cable motion are obtained. A multi-parameter perturbation method is employed to solve the modal problem of internally resonant systems. The perturbation-based modal solution furnishes, first, explicit formulae for the parameter combinations which realize the internal resonance conditions and, second, asymptotic approximations of the resonant frequencies and modes. Attention is focused on the triple internal resonance among a global torsional mode of the deck and two local modes of the cables, due to the relevant geometric coupling which maximizes the modal interaction. The asymptotic approximation of the modal solution is found to finely describe the multiple veering phenomenon which involves the three frequency loci under small variation of the most significant mechanical parameters, including terms of structural coupling or disorder. Moreover, the veering amplitude between any two of the three frequency loci can be expressed as an explicit parametric function. Finally, the disorder is recognized as the only parameter governing a complex phenomenon of triple modal hybridization involving all the resonant modes. The entire hybridization process is successfully described by an energy-based localization factor, presented in a new perturbation-based form, valid for internally resonant system.     

Response of plates with unconstrained layer damping treatment to random acoustic excitation,part I: Damping and frequency evaluations

For theoretical evaluation of the response of a structure under random acoustic excitation a complete understanding is required of the various modes of vibration and the modal damping associated with each mode. In order to evaluate these parameters for plates with unconstrained layer damping treatment, some of the theoretical approaches applicable are used. Experimentally observed modal frequencies and associated loss factors are compared with those estimated by different theories for all edges simply supported and all edges clamped boundaries, after accounting for the damping at supports. The modes of vibration used in the theoretical analysis for these boundaries are compared with those observed in the experiments. These results are made use of in Part II for the evaluation of response under random acoustic excitation.     

Non resonant transmission modelling with statistical modal energy distribution analysis

Statistical modal Energy distribution Analysis (SmEdA) can be used as an alternative to Statistical Energy Analysis for describing subsystems with low modal overlap. In its original form, SmEdA predicts the power flow exchanged between the resonant modes of different subsystems. In the case of sound transmission through a thin structure, it is well-known that the non resonant response of the structure plays a significant role in transmission below the critical frequency. In this paper, we present an extension of SmEdA that takes into account the contributions of the non resonant modes of a thin structure. The dual modal formulation (DMF) is used to describe the behaviour of two acoustic cavities separated by a thin structure, with prior knowledge of the modal basis of each subsystem. Condensation in the DMF equations is achieved on the amplitudes of the non resonant modes and a new coupling scheme between the resonant modes of the three subsystems is obtained after several simplifications. We show that the contribution of the non resonant panel mode results in coupling the cavity modes of stiffness type, characterised by the mode shapes of both the cavities and the structure. Comparisons with reference results demonstrate that the present approach can take into account the non resonant contributions of the structure in the evaluation of the transmission loss.     

Insertion loss of an acoustic enclosure

Acoustical enclosures are the common arrangements in reducing airborne noise from shipboard machinery such as engines and generators. In this paper the theoretical models, established based on statistical energy analysis, are presented for predicting the insertion loss of acoustical enclosures in different frequency ranges. In addition to the consideration of resonant modal coupling between internal sound field and enclosure structural vibration, the nonresonant transmission though and the interaction between enclosure walls in the models are also included. It is shown that the insertion loss of enclosures is mainly controlled by the nonresonant modes in the intermediate frequency range. At high frequencies, the insertion loss of enclosures can be improved by increasing the sound absorption at the internal boundaries of enclosures. Experiments were carried out on two enclosures made of different materials. The measured results are compared with the predicted values and the good agreement between them is the initial demonstration of the validity and feasibility of the theoretical models.     

Mechanical properties of silicon nanobeams with an undercut evaluated by combining the dynamic resonance test and finite element analysis

Mechanical properties of silicon nanobeams are of prime importance in nanoelectromechanical system applications.A numerical experimental method of determining resonant frequencies and Young’s modulus of nanobeams by combining finite element analysis and frequency response tests based on an electrostatic excitation and visual detection by using a laser Doppler vibrometer is presented in this paper.Silicon nanobeam test structures are fabricated from silicon-oninsulator wafers by using a standard lithography and anisotropic wet etching release process,which inevitably generates the undercut of the nanobeam clamping.In conjunction with three-dimensional finite element numerical simulations incorporating the geometric undercut,dynamic resonance tests reveal that the undercut significantly reduces resonant frequencies of nanobeams due to the fact that it effectively increases the nanobeam length by a correct value △L,which is a key parameter that is correlated with deviations in the resonant frequencies predicted from the ideal Euler-Bernoulli beam theory and experimentally measured data.By using a least-square fit expression including △L,we finally extract Young’s modulus from the measured resonance frequency versus effective length dependency and find that Young’s modulus of a silicon nanobeam with 200-nm thickness is close to that of bulk silicon.This result supports that the finite size effect due to the surface effect does not play a role in the mechanical elastic behaviour of silicon nanobeams with thickness larger than 200 nm.     

Mechanical properties of silicon nanobeams with undercut evaluated by combining dynamic resonance test and finite element analysis

Mechanical properties of silicon nanobeams are of prime importance in nanoelectromechanical system applications. A numerical experimental method of determining resonant frequencies and Young's modulus of nanobeams by combining finite element analysis and frequency response tests based on an electrostatic excitation and visual detection by laser Doppler vibrometer is presented in this paper. Silicon nanobeams test structures are fabricated from silicon-on-insulator wafers by using a standard lithography and anisotropic wet etching release process, which inevitably generates the undercut of the nanobeam clamping. In conjunction with three-dimensional finite element numerical simulations incorporating the geometric undercut, dynamic resonance tests reveal that the undercut significantly reduces resonant frequencies of nanobeams due to the fact that it effectively increases the nanobeam length by a correct value Δ L, which is a key parameter that is correlated with deviations in the resonant frequencies predicted from the ideal Euler-Bernoulli beam theory and experimentally measured data. By using a least-square fit expression including Δ L, we finally extract Young's modulus from the measured resonance frequency versus effective length dependency and find that Young's modulus of silicon nanobeam with 200-nm thickness is close to that of bulk silicon. This result supports that the finite size effect due to surface effect does not play a role in mechanical elastic behaviour of silicon nanobeams with the thickness larger than 200 nm.     

In VivoLongitudinally Detected ESR Measurements at Microwave Regions of 300, 700, and 900 MHz in Rats Treated with a Nitroxide Radical

A signal detector of longitudinally detected ESR (LODESR) is independent of the resonant frequency. We developed anin vivoLODESR spectrometer operating in the regions of 300, 700, and 900 MHz. Using this apparatus, we estimated signal intensities at different operating frequencies obtained from non- or high-dielectric loss phantoms that contained nitroxide radical solutions and from live rats that had received a nitroxide radical. Our result, higher signal intensities in the high-dielectric loss samples (such as physiological saline solution and animals) at a lower frequency, shows that the influence of a decrease in dielectric loss dominates over the signal reduction caused by smaller Zeeman splitting. We believe that this finding strongly supports anin vivoESR resonant frequency that tends to be low.     

稳态损耗因子的衰减法识别研究

根据稳态损耗因子的定义, 推导了含多阶模态的频带稳态损耗因子公式, 得到结论: 稳态损耗因子不一定介于各阶模态损耗因子之间, 而是与各阶模态对振动响应的贡献程度有关. 提出了过程损耗因子的概念, 并给出了利用频带内各模态固有频率、损耗因子和振幅计算过程损耗因子的方法. 当时间趋于无穷时, 过程损耗因子趋于只由最小模态损耗因子贡献的稳态损耗因子. 传统衰减法测试稳态损耗因子在频带内仅有单个模态或模态密集的情况下精度较高, 但对于含有多阶模态且模态不密集的中频带, 采用传统衰减法准确获取稳态损耗因子存在困难. 根据过程损耗因子的特点, 提出了利用时域衰减曲线逐步分离频带内不同衰减特性分量及其响应幅度从而获取稳态损耗因子的方法. 仿真和实验均表明: 提出的利用时域衰减数据获取稳态损耗因子的方法具有很高精度, 可以弥补传统衰减法在中频段损耗因子实验确定中的不足.     

Calculation methods for the transmission loss of single,double and triple partitions

Methods of calculating the transmission loss for single and double walls are presented. These methods are developed using a statistical energy analysis (SEA). In principle the methods are based on separate calculations of resonant and non-resonant transmission for frequencies less than the critical frequency, $\text{f}{}_{\mathrm{c}}$, of the panels and calculation of only resonant transmission for frequencies equal to, and greater than, $\text{f}{}_{\mathrm{c}}$.Comparisons between calculated and measured results show good agreement.A calculation method is also presented for the transmission loss of triple panels for frequencies greater than the cut-off frequency for the cavities. For frequencies less than the cut-off frequency for the smallest cavity depth it is shown that, in most cases, the effect of the middle panel is very slight.     

Vibration and damping analysis of annular plates with constrained damping layer treatments

The natural frequencies and modal loss factors of annular plates with fully and partially constrained damping treatments are considered. The equations of free vibration of the plate including the transverse shear effects are derived by a discrete layer annular finite element method. The extensional and shear moduli of the viscoelastic material layer are described by the complex quantities. Complex eigenvalues are then found numerically, and from these, both frequencies and loss factors are extracted. The effects of viscoelastic layer stiffness and thickness, constraining layer stiffness and thickness, and treatment size on natural frequencies and modal loss factors are presented. Numerical results also show that the longer constrained damping treatment in radial length does not always provide better damping than the shorter ones.     

Analysis of the sound transmission loss of panels with resonant systems on the basis of equivalent representations

Effects of increasing the sound transmission loss of panels with the help of resonant systems on the basis of equivalent representations are analyzed. Emphasis is placed on the least-studied resonant systems, the inertial bodies of which simultaneously interact with media on each side of the panels. A universal expression for the sound transmission loss of panels with an arbitrary system of resonant elements with one degree of freedom is presented. It includes the parameters common to all types of resonators (the total mass, compressibility, quality factor, and characteristic frequencies). The expression can be directly used to compare the efficiency of different types of resonant systems mounted on a panel and to determine their optimum parameters.