Damping of flexural vibrations in turbofan blades using the acoustic black hole effect

The results of the experimental study into the damping of flexural vibrations in turbofan blades with trailing edges tapered according to a power-law profile are reported. Trailing edges of power-law profile (wedges), with small pieces of attached absorbing layers, materialise one-dimensional acoustic black holes for flexural waves that can absorb a large proportion of the incident flexural wave energy. The experiments were carried out on four model blades made of aluminium. Two of them were twisted, so that a more realistic fan blade could be considered. All model blades, the ones with tapered trailing edges and the ones of traditional form, were excited by an electromagnetic shaker, and the corresponding frequency response functions have been measured. The results show that the resonant peaks are reduced substantially once a power-law tapering is introduced to the blade. An initial study into the aerodynamic implications of this method has been carried out as well, using measurements in a closed circuit wind tunnel. In particular, the effects of the trailing edge of power-law profile on the airflow-excited vibrations of the fan blades have been investigated. It has been demonstrated that trailing edges of power-law profile with appropriate damping layers are efficient in reduction of the airflow-excited vibrations of the fan blades. The obtained results demonstrate that power-law tapering of trailing edges of turbofan blades can be a viable method of reduction of blade vibrations.     

Damping of flexural vibrations in rectangular plates using the acoustic black hole effect

The reduction of flexural vibration in plate structures has been investigated using the recently reported acoustic black hole effect for flexural wave reflection in plates with the local thickness varying according to h(x)=εx^m and m≥2. Since sharp edges of such plates (wedges) are always truncated before x=0, the real reflection coefficients are relatively high, therefore the application of a small amount of damping is required to achieve large reductions in vibration amplitude. This paper presents a numerical model of a plate incorporating an acoustic black hole wedge, with predictions for vibration amplitudes. These are compared to equivalent experimental measurements for a range of applied damping layers. It is concluded that the above-mentioned power-law wedges can be used as effective vibration dampers in plate structures over a wide frequency range of interest.     

Point mobility of a cylindrical plate incorporating a tapered hole of power-law profile

The paper describes the results of experimental measurements of point mobility carried out on circular plates containing tapered holes of quadratic power-law profile with attached damping layers. The obtained results are compared to the developed numerical model, as a means of validation. The profiles of the tapered hole in the plates are designed to replicate near zero reflection of quasi-plane waves from a tapered hole in geometrical acoustics approximation, also known as acoustic black hole effect. The driving point mobility measurements are provided, showing a comparison of the results for a constant thickness circular plate, a constant thickness plate with a layer of damping film applied and a plate with a quadratic power-law profile machined into the center, which is tested with a thin layer of elastic damping material attached. The results indicate a substantial suppression of resonant peaks, agreeing with a numerical model, which is based on the analytical solution available for the vibration of a plate with a central quadratic power-law profile. The paper contains results for the case of free boundary conditions on all edges of the plates, with emphasis placed on the predictions of resonant frequencies and the amplitudes of vibration and loss factor.     

Damping of flexural vibrations in circular plates with tapered central holes

The paper develops a numerical approach to the calculation of mobilities for a circular plate with a tapered central hole of power-law profile. The exact solution of the corresponding flexural wave equation that exists for m=2 has been used in the process of the numerical solution of the corresponding boundary problem. Note that this value of m belongs to the power-law range m≥2 associated with zero reflection of quasi-plane waves from a tapered hole in geometrical acoustics approximation. Two cases of added damping in the central hole area have been considered: a thin absorbing layer and a constrained layer. Cross and point mobilities have been calculated for both these cases. The obtained results for point and cross mobilities show a substantial suppression of resonant peaks (up to 17 dB), in comparison with the cases of a plate with an uncovered hole of the same power-law profile and of a reference circular plate of constant thickness covered or uncovered by a thin absorbing layer. Further theoretical and experimental research is needed to examine applications of the obtained numerical results to more practical situations, e.g. to rectangular plates or other structures with arbitrary locations of tapered holes.     

Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect

Flexural waves in beams and plates slow down if their thickness decreases. Such property was used in the past for establishing the theory of acoustic black holes (ABH). The aim of the present paper is to establish reliable numerical and experimental approaches for designing, modelling and manufacturing an effective passive vibration damper using the ABH effect. The effectiveness of such vibration absorbers increases with frequency. Initially, the dynamic behaviour of an Euler-Bernoulli beam is expressed using the Impedance Method, which in turn leads to a Riccati equation for the beam impedance. This equation is numerically integrated using an adaptive Runge-Kutta-Fehlberg method, yielding the frequency- and spatially-dependent impedance matrix of the beam, from which the reflection matrix is obtained. Moreover, the mathematical model can be extended to incorporate an absorbing film that assists for reducing reflected waves from the truncated edge. Therefore, the influence of the geometrical and material characteristics of the absorbing film is then studied and an optimal configuration of these parameters is proposed. An experiment consisting of an elliptical plate with a pit of power-law profile placed in one of its foci is presented. The elliptical shape of the plate induces a complete focalisation of the waves towards ABH in case they are generated in the other focus. Consequently, the derived 1-D method for an Euler-Bernoulli beam can be used as a phenomenological model assisting for better understanding the complex processes in 2-D elliptical structure. Finally, both, numerical simulations and experimental measurements show significant reduction of vibration levels.     

Artificial cochlea and acoustic black hole travelling waves observation: Model and experimental results

An inhomogeneous fluid structure waveguide reproducing passive behaviour of the inner ear is modelled with the help of the Wentzel–Kramers–Brillouin method. A physical setup is designed and built. Experimental results are compared with a good correlation to theoretical ones. The experimental setup is a varying width plate immersed in fluid and terminated with an acoustic black hole. The varying width plate provides a spatial repartition of the vibration depending on the excitation frequency. The acoustic black hole is made by decreasing the plate?s thickness with a quadratic profile and by covering this region with a thin film of viscoelastic material. Such a termination attenuates the flexural wave reflection at the end of the waveguide, turning standing waves into travelling waves.     

An approximate solution for the bending vibrations of a combination of rectangular thin plates

The appropriate method often used for calculating the bending vibration of a single rectangular plate is extended to calculate the bending vibrations of a global system of combinations of rectangular plates with elastically supported and damped non-coupled edges. Two examples, a series of T-combinations and an L-combination of rectangular thin isotropic plates, are considered and the input and transfer mobilities due to point excitation derived. Numerical results are presented for the case of combinations of concrete plates and the effects of varying the material damping of the plates and edge damping are investigated.The eigenfrequencies of an L-combination of plates with one plate of very high bending stiffness are calculated and results compare well with the eigenfrequencies of a single plate calculated by means of the classical Ritz-Rayleigh method.     

On the quality of plate reverberation

Metallic plates are used in recording studios to create artificial reverberation. In this paper, the flexural vibrations of such plates are analysed in order to better understand the influence of geometry and materials on the quality of plate reverberation. A comparison is made with acoustic reverberation, both in the time and frequency domain. This comparison is supported by objective criteria such as modal density and density of reflections. From a physical point of view, the internal and external damping mechanisms are examined in detail since they determine the frequency dependence of the reverberation time. Theoretical results are validated through comparison with experiments performed on an EMT140 reverberation plate. The underlying model is aimed at developing numerical simulations of plates (virtual reverberators) allowing a large variety of reverberations based on variations of geometry and materials as input parameters.     

Acoustic streaming induced by ultrasonic flexural vibrations and associated enhancement of convective heat transfer

Acoustic streaming induced by ultrasonic flexural vibrations and the associated convection enhancement are investigated. Acoustic streaming pattern, streaming velocity, and associated heat transfer characteristics are experimentally observed. Moreover, analytical analysis based on Nyborg's formulation is performed along with computational fluid dynamics (CFD) simulation using a numerical solver CFX 4.3. Two distinctive acoustic streaming patterns in half-wavelength of the flexural vibrations are observed, which agree well with the theory. However, acoustic streaming velocities obtained from CFD simulation, based on the incompressible flow assumption, exceed the theoretically estimated velocity by a factor ranging from 10 to 100, depending upon the location along the beam. Both CFD simulation and analytical analysis reveal that the acoustic streaming velocity is proportional to the square of the vibration amplitude and the wavelength of the vibrating beam that decreases with the excitation frequency. It is observed that the streaming velocity decreases with the excitation frequency. Also, with an open-ended channel, a substantial increase in streaming velocity is observed from CFD simulations. Using acoustic streaming, a temperature drop of 40 degrees C with a vibration amplitude of 25 microm at 28.4 kHz is experimentally achieved.     

覆黏弹层加肋双层板的声反射

为评估覆盖层的吸声性能,研究黏弹层在无限大周期加肋双层板背衬下的声反射。首先通过弯曲波与Lamb波的比较,揭示了板近似理论不适用于黏弹性板的原因。为此采用(黏)弹性理论处理(黏)弹性板,而肋板的处理则采用板近似理论,并用Hull提出的方法解决肋板和平板的耦合问题,得到反射声场的解。与板近似理论计算加肋双层板声反射的结果进行比较验证了本文方法的正确性,对反射系数进行计算分析了背衬对声反射的影响,定义平均反射系数比较了不同覆盖层的性能。结果表明,频率较低时覆盖层无法有效抑制背衬的影响,板间流体层与双层板的耦合作用引起共振散射,肋板与双层板的耦合作用降低声反射。      

复杂结构冲击声合成及实验验证

以复杂结构受击振动响应的时域计算为目的, 讨论了结构阻尼的计算方法, 给出一种用于冲击声合成的综合数值方法, 并进行了实验验证. 首先, 考虑到阻尼是影响瞬态振动时变特性的重要因素, 详细讨论了两种模态阻尼的计算方法; 其次, 对阻尼板的受击振动和声辐射进行了时域仿真, 并与时域有限差分法的计算结果进行对比, 显示出两种声音合成方法的计算结果具有高度的一致性; 最后, 针对有限长圆柱壳的受击振动, 将合成声与实验录音进行了对比研究. 结果表明, 合成声与实际录音的时域包络、频谱结构以及衰减趋势基本一致, 证明了采用数值方法进行冲击声合成的有效性. 关键词： 声音合成 模态阻尼 冲击声 数值方法     

Pulsed Sounding of Cracks with the Use of the Modulation of Ultrasound by Vibrations

The possibility of using the effect of the modulation of ultrasound by vibrations due to the presence of cracks for the nonlinear acoustic detection of cracks is demonstrated. The method is based on a pulsed ultrasonic sounding with gating the received signal and simultaneously exciting low-frequency vibrations in the sample. The presence of a crack is characterized by the modulation of the ultrasonic wave reflected from the crack. The visualization of the crack position in a model object (a metal rod) is performed. The possibility of selecting a crack on the background of an intense signal reflected from a cavity is experimentally demonstrated. The manifestation of the nonlinear properties of a crack is studied as a function of the polarization of the flexural vibrations of the rod.     

Improving the air coupling of bulk piezoelectric transducers with wedges of power-law profiles: A numerical study

An air-coupled ultrasonic transducer is created by bonding a bulk piezoelectric element onto the surface of a thick plate with a wedge of power-law profile. The wedge is used to improve the ultrasonic radiation efficiency. The power-law profile provides a smooth, impedance-matching transition for the mechanical energy to be transferred from the thick plate to the air, through the large-amplitude flexural waves observed in the thinnest region of the wedge. The performance of the proposed transducer is examined numerically and compared to that of a design where the piezoelectric element is isolated and where it is affixed to a thin plate of uniform thickness. The numerical analysis is first focused on the free-field radiation of the transducers. Then, time-reversal experiments are simulated by placing the transducers inside a cavity of arbitrary shape with some perfectly reflecting boundaries. In addition to time-reversal mirrors, the proposed concept could be integrated in the design of phased arrays and parametric arrays.     

Thermoelastic damping in torsion microresonators with coupling effect between torsion and bending

Predicting thermoelastic damping (TED) is crucial in the design of high Q MEMS resonators. In the past, there have been few works on analytical modeling of thermoelastic damping in torsion microresonators. This could be related to the assumption of pure torsional mode for the supporting beams in the torsion devices. The pure torsional modes of rectangular supporting beams involve no local volume change, and therefore, they do not suffer any thermoelastic loss. However, the coupled motion of torsion and bending usually exists in the torsion microresonator when it is not excited by pure torque. The bending component of the coupled motion causes flexural vibrations of supporting beams which may result in significant thermoelastic damping for the microresonator. This paper presents an analytical model for thermoelastic damping in torsion microresonators with the coupling effect between torsion and bending. The theory derives a dynamic model for torsion microresonators considering the coupling effect, and approximates the thermoelastic damping by assuming the energy loss to occur only in supporting beams of flexural vibrations. The thermoelastic damping obtained by the present model is compared to the measured internal friction of single paddle oscillators. It is found that thermoelastic damping contributes significantly to internal friction for the case of the higher modes at room temperature. The present model is validated by comparing its results with the finite-element method (FEM) solutions. The effects of structural dimensions and other parameters on thermoelastic damping are investigated for the representative case of torsion microresonators.     

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.     

Elastic characterization of closed cell foams from impedance tube absorption tests

A method is presented to determine the bulk elastic properties of isotropic elastic closed-cell foams from impedance tube sound absorption tests. For such foams, a resonant sound absorption is generally observed, where acoustic energy is transformed into mechanical vibration, which in turn is dissipated into heat due to structural damping. This article shows how the bulk Young's modulus, Poisson's ratio, and damping loss factor can be deduced from the resonant absorption. Also, an optimal damping loss factor yielding 100% of absorption at the first resonance is defined from the developed theory. The method is introduced for a sliding edge condition which is an ideal condition. Then, the method is extended to a bonded edge condition which is more easily achievable and additionally enables the identification of the Poisson's ratio. The method is experimentally tested on expanding closed-cell foams to find their elastic properties in both cases. Using the found properties, sound absorption predictions using an equivalent solid model with and without surface absorption are compared to measurements. Good correlations are obtained when considering surface absorption.     

Free vibrations of curved box girders

The problem of coupled free vibrations of curved thin-walled girders of non-deformable asymmetric cross-section is examined in this paper. The general governing differential equations are derived for quadruple coupling between the two flexural, tangential and torsional vibrations. An approximate solution for the case of triple coupling between the two flexural and the torsional vibrations is given for a simply supported girder, uniform specific gravity of the material of the box being assumed. Section warping is considered but axial forces, rotary inertia and structural damping are neglected. A parametric study is conducted to investigate the effect of relevant parameters on natural frequencies. Eigenfunctions satisfying the orthogonality condition are given. The solution derived herein for the general case is also shown to cover a variety of special cases of straight and curved girders with doubly symmetric or singly symmetric cross-sections.     

Distribution of eigenfrequencies for vibrating plates

Acoustic spectra of free plates with a chaotic billiard shape have been measured, and all resonance frequencies in the range 0-500 kHz have been identified. The spectral fluctuations are analyzed and compared to predictions of the Gaussian Orthogonal Ensemble (GOE) of random matrices. The best agreement is found with a superposition of two independent GOE spectra with equal density which indicates that two types of eigenmodes contribute to the same extent. To explain and predict these results a detailed theoretical analysis is carried out below the first cut-off frequency where only flexural and in-plane vibrations are possible. Using three-dimensional plate dispersion relations and two-dimensional models for flexural and in-plane vibrations we obtained two first terms of the asymptotic expansion of the counting function of these eigenmodes. The contribution of edge modes is also discussed. The results are in a very good agreement with the experimentally measured number of modes. The analysis shows that the two types of modes have almost equal level density in the measured frequency interval, and this explains the observed spectral statistics. For a plate with broken symmetry in the up-down direction (where flexural and in-plane modes are strongly coupled) experimentally observed spectral fluctuations correspond to a single GOE spectrum. Above the first cut-off frequency a greater complexity of the spectral fluctuations is expected since a larger number of types of modes will contribute to the spectrum. Received 5 January 1999 and Received in final form 5 September 1999     

Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates

Large amplitude flexural vibrations of slender beams, and thin circular and rectangular plates have been studied when a compatible longitudinal or inplane mode is coupled with the fundamental flexural mode. It is shown that the effect of longitudinal or inplane deformation and inertia is to reduce the non-linearity in the flexural frequency-amplitude relationship. Further, for slender beams and thin plates, the effect of longitudinal or inplane inertia is negligible.     

Transient vibrations of a beam/mass system fixed to a rotating body

Transient flexural vibrations of a beam/mass system fixed to a rotating body are investigated. The rotating body is driven so as to have a velocity profile of trapezoidal shape. The governing ordinary differential equations of the beam mass system are derived by use of the extended Galerkin method, and the transient response is obtained by the Laplace transformation. Then the effects of the flexibility of the beam and the rotational period of the rotating body upon the flexural vibrations are investigated.