The sound power of an individual mode of a clamped-free annular plate

The main aim of this study is an analysis of sound radiation of some clamped-free and free-clamped annular plates. The plates are embedded in some infinite perfectly rigid baffles and vibrate with the free-field condition satisfied. The Kirchhoff-Love linear theory of a perfectly elastic plate is used to solve the plates’ equations of motion. The sound pressure at the plates’ surfaces is expressed by its Hankel transform. The closed-path integral technique known from the literature and the stationary-phase method are used to find the standardized active and reactive sound power of an individual mode in the form of some high-frequency asymptotic formulae useful for some highly efficient engineering computations. Their non-oscillating and oscillating parts have been separated. It is assumed that the vibration and sound radiation are axisymmetric and time harmonic. Low fluid loading is also assumed. The complex sound power of an individual mode forms the basis for computing the total sound power of some excited and damped vibrations in the fluid. The total sound power is not discussed in this study.     

Theoretical analysis of sound radiation of an elastically supported circular plate

This study focuses on the analysis of the active and reactive sound power of the axisymmetric modes of free vibrations of elastically supported circular plates embedded in a rigid baffle. Some linear and time-harmonic processes have been considered. It is assumed that the plate radiates some acoustic waves into a hemisphere filled with a lossless gaseous medium. The integral formulations for the active and reactive sound power have been derived and expressed in their Hankel representations. They have been used to derive some elementary formulations in the form of some high-frequency asymptotes valid for frequencies higher than the successive coincidence frequencies of the plate. Therefore, the discussion on some sample numerical results mostly covers the sound power radiated at those frequencies. The asymptotes are easy to express in a computer code and they do not need great processor capacity. They are therefore useful for engineering use.The main benefit of the analysis presented in this paper is that the sound power for all the possible boundary configurations of the boundary stiffnesses, i.e., classical clamped, guided, simply supported or completely free boundaries as well as all the intermediate situations, has been described using the same formulae. This is possible simply by changing the two values of stiffnesses associated with the boundary conditions, whose influence on the radiated sound power has been discussed. The solution of the problem of sound power radiated by a vibrating elastically supported circular plate presented herein is essentially more general than the solutions presented earlier for the classical boundary configurations, such as clamped, simply supported, guided or completely free circular plates.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.     

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.     

磁声参量振荡的理论

在本文中,我们从磁-弹性耦合的宏观表达式,通过经典场论的方法,求得弹性振动和磁振璗的耦合方程,用来分析了伴随波长约等于铁氧体样品的线度的声振动而存在的磁振璗(磁声模)。文中指出,Spencer和LeCraw所发现的磁声效应是磁声模和静磁模在注入场的激发下产生的参量振璗现象(也可以说是热声子的电磁讯号的放大)。我们引用Berk等人在讨论一种半静磁操作放大器的文章中给出的公式,算出Spencer-LeCraw实验所需要的功率,其结果与记录的数据相接近。我们提出了使任一静磁模配合磁声模产生振璗的调谐条件以及降低激发功率和观测几十到几百兆赫的声频的办法。通过磁声模和静磁模的交变场向量的空间对称性的分析,我们推导出磁声参量振璗的选择定则:对于球体三个主要弹性振动模(旋转模、向径模和椭球模),(1)静磁模(n,m,r)的Walker指标n是偶数者不产生磁声效应;(2)指标m是奇数者不与旋转模产生磁声效应,m是偶数者不与向径模或椭球模产生磁声效应。我们也举出第一类本征振动中有只可能和n是偶数、m是奇数的静磁模产生参量振璗的例子。Spencer-LeCraw局限于使静磁场调谐在(110)模上,所观察到的现象仅仅是本文所给出理论预见的一个特殊情况。他们发现了椭球模和向径模的频率显著地出现,但并无旋转模的频率,这是上述的选择定则的具体验证。最后,我们指出,热声子的参量放大可形成铁氧体微波放大器的噪声的来源。     

Acoustic characteristics inside cylindrical structure with end plates excited at different frequencies

In order to study coupling between vibration of a structure and a sound field in contact with the structure, a cavity surrounded by a rigid cylinder having thin elastic plates at both ends is adopted as an analytical model. When excitation forces of different frequencies are applied to the respective plates, the plate vibrations and the sound field inside the cavity become aperiodic, because of the coupling between the systems. In the present investigation, distribution of the sound pressure level inside the cavity is studied in detail in order to clarify the coupling behavior. The results show that when the respective plates vibrate at the same circumferential order, the vibration modes of the plates, which effect the coupling of the plate vibrations and the sound field, cause the aperiodic nature of each system to develop. In this case, since the dominant mode exists in formation of the sound field, it significantly influences the aperiodic nature of the coupling systems. In the case of vibration modes where the plates vibrate at different circumferential orders, the behaviors of the three systems, whose coupling has been restrained, approximate a steady state. Consequently, the dominant mode does not appear in the sound field.     

不同材料参数薄板振动中的热力耦合效应

 利用有限Hankel变换法,导出了周界等温-弹性支撑圆薄板在激光束辐照下的轴对称耦合热弹性弯曲振动近似解;针对具有不同弹性模量和热膨胀系数的薄板进行了热力耦合和非耦合弯曲振动的解析和有限元计算与分析。结果表明：热力耦合效应使薄板振动的振幅和周期都有所减小,其程度与材料的性能参数（如弹性模量和热膨胀系数等）密切相关,材料弹性模量和热膨胀系数越大,板振动中的热力耦合效应就越明显。     

Radiation efficiency of a baffled circular plate in flexural vibration

The radiation efficiency of an edge-clamped circular plate, which is vibrating flexurally in one of its natural modes and is mounted in an infinite baffle, is theoretically determined from the total power radiated to the far field. The vibrations of the plate are investigated both by the classical plate theory and by the improved plate theory (Mindlin plate theory). Approximation formulae for the low frequency region are derived, and curves covering the entire frequency range for the first fifteen modes are obtained through numerical calculation. Except for frequencies much higher than the critical frequency, there exist some differences in the radiation efficiencies between the results obtainedby the two theories. The difference increases with the thickness of the plate and with the mode numbers, especially for modes having many nodal diameters.     

Radiation efficiency of submerged rectangular plates

The average radiation efficiency of point-excited submerged rectangular plates is investigated in two methods, deterministic analysis and statistical approach, respectively. In the deterministic analysis, the effect of mutual impedance by water loading on the velocity of the plate is illustrated analytically by using a modal summation method. The cross-modal contributions to the average radiation efficiency are averaged to zero by averaging over all possible excitation positions on the plate. In the statistical approach, by analyzing the engineering formulae of the average radiation efficiency in air, this paper modifies the formulae to be applicable in water. The numerical comparisons show that the modified formulae reflect the average level in the frequency region controlled by corner modes and are accurate enough in the region controlled by monopole and edge modes. On this basis, approximate expressions for predicting the average radiation efficiency of the submerged beam-stiffened rectangular plates are proposed. The main differences between air loading and water loading are considered. Firstly, as dry modes taken in analysis instead of real vibration modes in water, the vibration of the stiffened plate is not only determined by the first mode but by several modes simultaneously at low frequencies. Secondly, the “corner mode region” becomes inappropriate as the plate is stiffened. The proposed formulae are validated numerically in different size, thickness, and stiffener amount of the stiffened plate.     

On the steering of sound energy through a supercritical plate by a near-field transducer array

The ability to direct sound energy through the flexural vibrations of a submerged plate at various angles of incidence using a near-field transducer array is investigated. An alumina bar is placed in front of a one-dimensional, eight-element transducer array, between the array and the water. Operating in a receive mode, data were taken as a function of angle of incidence and compared to data taken without the presence of the alumina bar. The array was also operated in transmit mode and results were compared to corresponding receive mode data, showing that reciprocity holds. Results show that in fact sound energy can be steered through a plate, and that the measurement method used provides a convenient method of measuring the angular dependence of transmission through a plate, including measurements at frequencies above the plate's critical frequency. Experimental results of sound transmission versus angle of incidence of finite sized plates agree qualitatively with theoretical results from an analysis of the transmission through an unbounded flexible partition.     

On the sound radiation of a rolling tyre

The sound radiation from rolling tyres is still not very well understood. Although details such as horn effect or directivity during rolling have been investigated, it is not clear which vibrational modes of the tyre structure are responsible for the radiated sound power. In this work an advanced tyre model based on Wave Guide Finite Elements is used in connection with a contact model validated in previous work. With these tools the tyre vibrations during rolling on an ISO surface are simulated. Starting from the calculated contact forces in time the amplitudes of the modes excited during rolling are determined as function of frequency. A boundary element model also validated in previous work is applied to predict the sound pressure level on a reference surface around a tyre placed on rigid ground as function of the modal composition of the tyre vibrations. Taking into account different modes when calculating the vibrational field as input into the boundary element calculations, it is possible to identify individual modes or groups of modes of special relevance for the radiated sound power. The results show that mainly low-order modes with relative low amplitudes but high radiation efficiency in the frequency range around 1 kHz are responsible for the radiated sound power at these frequencies, while those modes which are most strongly excited in that frequency range during rolling are irrelevant for the radiated sound power. This fact is very essential when focusing on the design of quieter tyres.     

Generalized thermoelastic extensional and flexural wave motions in homogenous isotropic plate by using asymptotic method

In this paper the asymptotic method has been applied to investigate propagation of generalized thermoelastic waves in an infinite homogenous isotropic plate. The governing equations for the extensional, transversal and flexural motions are derived from the system of three-dimensional dynamical equations of linear theories of generalized thermoelasticity. The asymptotic operator plate model for extensional and flexural free vibrations in a homogenous thermoelastic plate leads to sixth and fifth degree polynomial secular equations, respectively. These secular equations govern frequency and phase velocity of various possible modes of wave propagation at all wavelengths. The velocity dispersion equations for extensional and flexural wave motion are deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate. The approximation for long and short waves along with expression for group velocity has also been obtained. The Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations and its equivalence established with that of asymptotic method. The numeric values for phase velocity, group velocity and attenuation coefficients has also been obtained using MATHCAD software and are shown graphically for extensional and flexural waves in generalized theories of thermoelastic plate for solid helium material.     

Violin bow vibrations

The modal frequencies and bending mode shapes of a freely supported tapered violin bow are investigated by finite element analysis and direct measurement, with and without tensioned bow hair. Such computations are used with analytic models to model the admittance presented to the stretched bow hairs at the ends of the bow and to the string at the point of contact with the bow. Finite element computations are also used to demonstrate the influence of the lowest stick mode vibrations on the low frequency bouncing modes, when the hand-held bow is pressed against the string. The possible influence of the dynamic stick modes on the sound of the bowed instrument is briefly discussed.     

弯管对末端带弹性障板充液管路辐射声能量的影响

基于声固耦合有限元方法建立了末端带弹性障板的充液管路数值模型,重点分析了不同激励下弯管对管口辐射声能量的影响.结果表明:弯管引入的高阶周向模式耦合使结构振动和流体声传播都发生明显改变,以致系统辐射声能量及主要能量贡献源也发生转移,并随激励方式和频率而不同.对本文管路模型,平面波激励下弯管系统在低频的结构辐射声能量明显增...     

The low frequency axisymmetric modal sound radiation efficiency of an elastically supported annular plate

This paper deals with the sound radiation efficiency of a vibrating, thin, elastically supported annular plate embedded into a flat rigid baffle. The free axisymmetric time harmonic vibrations have been considered for a single mode. It has been assumed that the influence of the air column above the plate on the plate's vibrations is negligible. First, the sound radiation efficiency has been formulated as an integral. Further, rigorous mathematical manipulations have been carried out based on the theory of summation of multiple expansion series containing the hypergeometric functions. As a result, the formulations have been expressed as some fast convergent expansion series containing only the Bessel and Struve functions of integer order and the spherical Bessel functions. The presented formulations of sound radiation efficiency of an elastically supported annular plate are useful for numerical calculations within the low frequency range what is important for practical reasons. The formulations are valid for axisymmetric boundary conditions and they enable changing the values of boundary stiffness constants. Consequently, the analysis of influence of the plate's edge attachment on the sound radiation efficiency has been performed. The limiting transitions have also been performed from formulations valid for the elastically supported annular plates to the formulations valid for annular plates with classical boundary conditions (clamped, simply supported and free) at one edge or at both edges.     

Sound radiation from a thin infinite plate in contact with a layered inhomogeneous fluid

In this paper the sound radiation from a thin infinite plate in contact with a layered inhomogeneous fluid subjected to single point excitation is studied. Sources contributing to the inhomogeneity are discussed, and the fluid is analytically expressed as a layered inhomogeneous field. Using Hankel transform, the equations governing the fluid-structure interaction are solved. The asymptotic form of the radiation pressure at far field is obtained using the method of stationary phase. Numerical examples show that the sound radiation patterns from a plate in contact with an inhomogeneous fluid is much more complicated than that in a homogeneous fluid. Three different radiation patterns (bell, disc, and their combination) were observed in the numerical examples, and the radiation patterns are sensitive to the field parameters. Depending on frequency, the sound pressure at far field is either magnified or reduced compared with that in a homogeneous fluid. And beyond some frequency the sound pressures are almost zero.     

Free vibrations of an infinite plate of parabolically varying thickness on elastic foundation

A simple model is presented for use by research workers in seismology, soil mechanics and earthquake vibrations in problems where the resistance offered by the soil foundation is very close to that of Winkler's assumption. In earth vibrations one earth layer can be treated as an infinite plate of variable thickness while layers below it may approximately serve as an elastic foundation. Here, the transverse displacement of a plate of parabolically varying thickness on an elastic foundation is expressed as a power series and the frequencies, deflections and moments corresponding to the first two modes of vibration are computed for various values of foundation modulus and taper constant, and for two combinations of boundary conditions. A problem discussed in previous work [1] can be considered as a particular case of the present problem if the foundation modulus is taken to be zero.     

Axisymmetric vibrations of a circular plate of linearly varying thickness on an elastic foundation according to Mindlin theory

Free axisymmetric vibrations of an elastic circular plate of linearly varying thickness on an elastic foundation have been studied on the basis of shear theory [1,2]. The transverse displacement and local rotation are expressed as an infinite series. The frequencies corresponding to the first two modes of vibrations are obtained for a circular plate with clamped and simply supported edge conditions for various values of the taper constant and the foundation modulus. The results have been compared with those of reference [3].     

圆薄板大撓度问题

Equations for the large deflection of thin plate established by Th. von Karman has been well known for many years. But so far there have been only a few iproblems studied with numerical certainty. S. Way was the first to apply these equations to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy found the solution of the simply supported rectangular plate under uniform load by the method of double trigonometric series. Both methods are too labourious to be applicable to other more important cases. Lately, Chien Wei-zang treated Way's problem again by means of the perturbation method and obtained excellent results. By the method as given by Chien Wei-zang, Yeh Kai-yuan worked out the problem of circular plate with a central hole under central concentrated load.In this paper, more results are given for various circular plates under various edge conditions. These include uniformly loaded circular plate under various edge conditions (section 2) and central concentrated loaded circular plate under various edge conditions (section 3). Such edge conditions are: (1) simply supported, (2) simply hinged, (3) rigidly clamped, (4) clamped but free to slip, (5) edge clamped but with possible slipping in horizontal direction, (6) edge simply supported but elastically fastened, and (7) edge clamped in elastic wall.All these results are presented in such a form that direct application in design problem is possible. In particular cases, under edge conditions (1) to (4), as σ=0.3, design formulae and curves for central deflection, radial tensile stress and radial bending stress are presented.     

FORCED AND FREE VIBRATIONS OF RECTANGULAR SANDWICH PLATES WITH PARAMETRIC STIFFNESS MODULATION

An active control of the resonant vibrations of a rectangular sandwich plate performed by the parametric stiffness modulation is analyzed. The controlled vibrations are those of the dominantly flexural type excited by the transverse force acting at the first resonant frequency of dominantly flexural vibrations. The stiffness modulation is performed at a comparatively high frequency identified by the resonance of a mode of the dominantly shear type. The method of direct partition of motions is used that predicts an existence of the modal interaction between these two modes of vibrations due to the parametric stiffness modulation. It is shown that such a parametric control can provide a significant shift of the first eigenfrequency of a controlled plate (the one subjected to the stiffness modulation) from its nominal value for an uncontrolled plate. Heavy fluid loading conditions are accounted for as well as the energy dissipation in the material of a plate. It is demonstrated that although heavy fluid loading reduces resonant frequencies of forced vibrations, the suggested mechanism of control remains valid in these cases. Dynamics of an elementary two-degree-of-freedom model mechanical system is considered to illustrate the mechanism of modal interaction, which is involved in the suggested way of an active control of vibrations of sandwich plates.