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 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.     

Modal Overlap Factor of a beam with an acoustic black hole termination

An Acoustic Black Hole (ABH) effect is a passive vibration reduction technique which takes advantage of properties of wave propagation in thin structures of varying thickness. A practical implementation of ABH on a uniform beam consists in an extremity whose thickness follows a power-law profile covered by a very thin layer of additional damping material. A modal analysis based on a High Resolution technique shows that the ABH significantly increases the Modal Overlap Factor (MOF) of the beam, thus reducing the resonant behaviour of the structure. Further investigations, including a two-dimensional numerical model of the structure based on the finite difference method, show that this MOF can be explained by an increase of the modal density and a high damping of a number of modes of the structure due to the ABH.     

Effect of geometrical and material imperfections on damping flexural vibrations in plates with attached wedges of power law profile

In the present paper, an efficient method of damping structural vibrations using the acoustic black hole effect is further investigated experimentally. This method is based on some specific properties of flexural wave propagation in tapered plates (wedges) of power-law profile that have to be partially covered by narrow thin strips of absorbing layers. Ideally, if the power-law exponent of the profile is equal or larger than two, the flexural wave never reaches the sharp edge and therefore never reflects back, which constitutes the acoustic black hole effect. It has been previously established theoretically and confirmed experimentally that this method of damping structural vibrations is very efficient even in the presence of edge truncations. The present work describes the results of the experimental studies of the effects of manufacturing intolerances on damping flexural vibrations in wedge-like structures of power-law profile. In particular, the effect of mechanical damage resulting from the use of cutting tools to wedge tips is investigated, including tip curling and early truncation, as well as the placement of absorbing layers on different wedge surfaces. Also, the effects of welded and glued bonding of wedge attachments to basic rectangular plates (strips) are investigated. The results show that, although the above-mentioned geometrical and material imperfections reduce the damping efficiency by varying degrees, the method of damping structural vibrations using the acoustic black hole effect is robust enough and can be used widely without the need of high precision manufacturing.     

Vibration and stability of a circular plate of unidirectionally varying thickness

An analysis is presented for the vibration and stability of an elastically restrained circular plate of unidirectionally varying thickness subjected to an in-plane force. For this purpose, the transverse deflection of a circular plate of variable thickness is written in a series of the deflection functions of a uniform circular plate without the action of a force. The dynamical energies of the plate are evaluated analytically, and the frequency equation of the plate is derived by the Ritz method. The analysis is applied to circular plates of unidirectionally tapered or stepped thickness; the natural frequencies and the divergence loads are calculated numerically, and the effects of the varying thickness and edge conditions on the vibration and stability are studied.     

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.     

Free vibration analysis of piezoelectric coupled circular plate with open circuit

A vibration analysis of a circular steel substrate surface bonded by a piezoelectric layer with open circuit is presented. A solution for the electrical potential along the thickness direction of the piezoelectric layer satisfying the open circuit electric boundary condition is developed for the first time. The mechanical model and solutions for the vibration analysis of the piezoelectric coupled circular plate are then established based on the developed electrical potential, the Kirchhoff plate model, and the Maxwell equation. The first four mode shapes and the corresponding resonant frequencies of the plate with two standard boundary conditions are presented in numerical simulations and compared with those of a piezoelectric coupled plate with the closed circuit condition. The simulations show that the resonant frequencies of the open circuit piezoelectric coupled plate are higher than those of the closed circuit piezoelectric coupled plate. Corresponding discussions are thus provided for the higher piezoelectric effect from the open circuit piezoelectric layer.     

变厚度圆板、环板振动分析的传递矩阵法*

变厚度圆板和环板是在工程设计中经常遇到的一类构件,与等厚度板相比,通过适当的沿径向厚度的变化,这种变厚度板在振动、失稳、弯曲等方面能起到更好的效果。将沿径向任意变厚度圆板、环板划分为一系列等厚度环板单元,基于Mindlin中厚板理论采用逆向推导的方式推导了其传递矩阵,建立起变厚度圆板、环板的频率方程。通过计算线性变厚度环板自由振动时的频率,并与ANSYS模态分析结果相比较,验证了计算模型的精确性。逆向推导法避免了高阶数传递矩阵推导复杂的问题,是对传递矩阵法的很好推广。     

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.     

Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs

Using a periodic expansion by means of the Bloch theorem, the flexural vibration band gaps are studied in a thin plate with two-dimensional ternary locally resonant structures, i.e. a thin epoxy plate containing a periodic square array of lead discs hemmed around by rubber. The full band gaps of flexural vibration in the thin plate are obtained within which sound and vibration will be forbidden. The numerical results are used to show how the width of the first full band gap depends on the radius ratio of lead disc to hemmed disc, filling fraction, lattice constant (distance between the centers of the nearest lead discs) and thickness of the thin plate. It is observed that the gap width can be changed a lot by modulating these physical parameters.     

Optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates

This paper concerns the optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates. The system loss factor is expressed in terms of the mechanical properties of the plate and damping layer and the layer/plate thickness ratio. Optimum distributions of the thickness ratio that maximize the system loss factor are obtained through sequential unconstrained minimization techniques. Results are presented for both simply-supported and edge-fixed rectangular plates with aspect ratios of 1·0 to 4·0. These results indicate that the system loss factor can be increased by as much as 100%, or more, by optimizing the thickness distribution of the damping treatment. Also revealed are the regions of the plate where added damping treatments are most effective.     

上海光源储存环主支撑减振研究

 上海光源是一台在建的第三代同步辐射光源,对束流轨道稳定性要求很高。由磁铁和支架组成的支撑组件的机械稳定性是影响束流轨道稳定性的重要因素。对主支撑组件样机的测试结果表明,其最低共振频率处放大倍数为50左右,超过要求5倍。因此,需要研究相应的减振措施。利用阻尼减振原理设计了一种约束阻尼结构。在样机上的测试结果表明,安装该装置后,支撑组件的共振放大倍数最大可以降低91.8%,对应的功率谱密度的峰值可以降低25 dB。因此,该装置可以用来增加支撑的稳定性。     

Vibration of tapered Mindlin plates in terms of static Timoshenko beam functions

In this paper, the free vibrations of rectangular Mindlin plates with variable thickness in one or two directions are investigated. The thickness variation of the plate is continuous and can be represented by a power function of the rectangular co-ordinates. A wide range of tapered rectangular plates can be described by giving various index values to the power function. Two sets of new admissible functions are developed, respectively, to approximate the flexural displacement and the angle of rotation due to bending of the plate. The eigenfrequency equation is obtained by using the Rayleigh-Ritz method. The complete solutions of displacement and angle of rotation due to bending for a tapered Timoshenko beam (a strip taken from the tapered Mindlin plate in some direction) under a Taylor series of static load have been derived, which are used as the admissible functions of the rectangular Mindlin plates with taper thickness in one or two directions. Unlike conventional admissible functions which are independent of the thickness variation of the plate, the static Timoshenko beam functions presented in this paper are closely connected with the thickness variation of the plate so that higher accuracy and more rapid convergence can be expected. Some numerical results are furnished for both truncated Mindlin plates and sharp-ended Mindlin plates. On the basis of convergence study and comparison with available results in literature, it is shown that the first few eigenfrequencies can be obtained with quite satisfactory accuracy by using only a small number of terms of the static Timoshenko beam functions.     

Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness

Free axisymmetric vibrations of an isotropic, elastic, non-homogeneous circular plate of linearly varying thickness have been studied on the classical theory of plates. The non-homogeneity of the material of the plate is assumed to arise due to the variation of Young's modulus and density with the radius vector whereas Poisson's ratio is assumed to remain constant. The governing differential equation of motion is solved by the method of Frobenius. The transverse displacement of the plate has been expressed as a power series in terms of the radial co-ordinate. The frequency parameters corresponding to the first two modes of vibration have been computed for different values of the non-homogeneity parameter and taper constant and for clamped and simply supported edge conditions of the plate. A comparison between the numerical results for homogeneous and non-homogeneous material of the plate is also made.     

水下多层均匀材料的声特性

本文研究水和空气中由不同厚度的两层弹性板及消声橡胶共同组成的分层介质在斜入射时的声特性,并采用数值方法求解多层结构声场的线性方程组。得到反射,透射系数,当板的厚度远小于波长时,采用弹性薄壳振动模型和弹性波模型进行计算,它们的结果基本相同,但是薄壳振动模型更便于计算和理论分析;对系统反射系数和透射系数的计算结果表明,系统增加消声橡胶层后,极大地改变后射系数和透射系数的频率响应;频率升高,反射系数减小     

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.     

FREE VIBRATION OF ORTHOTROPIC SQUARE PLATES WITH A SQUARE HOLE

An approximate method is extended for analyzing the free vibration problem of orthotropic square plate with a square hole. In this paper, a square plate with a square hole is transformed into an equivalent square plate with non-uniform thickness by considering the hole as an extremely thin part of the equivalent plate. Therefore, the dynamic characteristics of a plate with a hole can be obtained by analyzing the equivalent plate. The Green function, which is the discrete solution for the deflection of the equivalent plate, is used to obtain the characteristic equation of the free vibration. The effects of the side to thickness ratio, hole side to plate side ratio and the variation of the thickness on the frequencies are considered. Some numerical analyses are carried out for the simply supported orthotropic square plate with a square hole. The efficiency and accuracy of the numerical solutions by the present method are investigated.     

Vibration analysis of scanning thermal microscope probe nanomachining using Timoshenko beam theory

In this paper, the effect of thermal vibration on the resonant frequency of transverse vibration of scanning thermal microscope (SThM) cantilever probe is analyzed using the Timoshenko beam theory, including the effects of rotary inertia and shear deformation. The thermal vibration effect can be considered as an axial force and is dependent of temperature distribution of the probe. In this analysis, the temperature is assumed to be distributed in accordance with the constant, linear, and quadratic models along the probe length. The Rayleigh–Ritz method is used to solve the vibration problem of the probe. The numerical results show that the frequency obtained with the constant model is the highest, while it is the lowest for the quadratic model. The frequency of vibration modes of the probe increases with increasing the temperature of the probe. As the ratio of probe length to its thickness increases, the frequency of vibration modes decreases. In addition, the effects of rotary inertia and shear deformation on the frequency are significant, especially in higher order modes and smaller values of the ratio of the probe length to its thickness.     

Experimental and numerical investigations of resonant vibration characteristics for piezoceramic plates.

Electronic speckle pattern interferometry (ESPI) is a full field, non-contact technique for measuring the surface displacement of a structure subjected to static loading or, especially, to dynamic vibration. In this article we employ an optical system called the amplitude-fluctuation ESPI with out-of-plane and in-plane measurements to investigate the vibration characteristics of piezoceramic plates. Two different configurations of piezoceramic plates, namely the rectangular and the circular plates, are discussed in detail. As compared with the film recording and optical reconstruction procedures used for holographic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Because the clear fringe patterns will be shown only at resonant frequencies, both the resonant frequencies and the corresponding mode shapes are obtained experimentally at the same time by the proposed AF-ESPI method. Excellent quality of the interferometric fringe patterns for both the in-plane and out-of-plane vibration mode shapes is demonstrated. The resonant frequencies of the piezoceramic plates are also measured by the conventional impedance analysis. From experimental results, we find that the out-of-plane vibration modes (type A) with lower resonant frequencies cannot be measured by the impedance analysis and only the in-plane vibration modes (type B) will be shown. However, both the out-of-plane (bending) and in-plane (extensional) vibration modes of piezoceramic plates are obtained by the AF-ESPI method. Finally, the numerical finite element calculations are also performed, and the results are compared with the experimental measurements. It is shown that the numerical calculations and the experimental results agree fairly well for both the resonant frequencies and the mode shapes.     

Axisymmetric vibrations of linearly tapered annular plates under an in-plane force

Analysis and numerical results are presented for the axisymmetric vibrations of circular annular plates with linear variation in thickness under the action of a hydrostatic in-plane force on the basis of the classical theory of plates. The governing differential equation with variable coefficients has been solved by Chebyshev collocation technique. The effect of inplane force on the natural frequencies of vibration has been investigated for two different boundary conditions and for different radii ratio and taper constant. Transverse displacements, moments and the critical buckling loads in compression with thickness variation have also been computed for the first two modes.