The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

The effects of large vibration amplitudes on the first two axisymmetric mode shapes of clamped thin isotropic circular plates are examined. The theoretical model based on Hamilton's principle and spectral analysis developed previously by Benamar et al. for clamped-clamped beams and fully clamped rectangular plates is adapted to the case of circular plates using a basis of Bessel's functions. The model effectively reduces the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. Numerical results are given for the first and second axisymmetric non-linear mode shapes for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency are given. The non-linear frequencies associated to the fundamental non-linear mode shape predicted by the present model were compared with numerical results from the available published literature and a good agreement was found. The non-linear mode shapes exhibit higher bending stresses near to the clamped edge at large deflections, compared with those predicted by linear theory. In order to obtain explicit analytical solutions for the first two non-linear axisymmetric mode shapes of clamped circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. for beams and rectangular plates, has been adapted to the case of clamped circular plates, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values of 0.5 and 0.44 for the first and second axisymmetric non-linear mode shapes, respectively.     

Free vibration analysis of cooling towers with column supports

A cooling tower with column supports is a rotationally periodic structure. This property has been used to develop a finite element technique to analyze the free vibration of such structures whilst explicitly including the effects of column supports. The resonant frequencies and mode shapes of a model cooling tower have been measured and shown to be in good agreement with those predicted by the theoretical technique. As far as the authors are aware this agreement is better than heretofore reported in the literature.     

Individual and collective vibrational modes of nanostructures studied by picosecond ultrasonics

We report on picosecond ultrasonic measurements obtained on aluminum and platinum nanostructures with variable dot size and lateral periodicity which realized a 2D phononic crystal. Performing investigations at different resolution scales, we have identified individual modes of vibration depending on the dot size, and mode of vibration strongly correlated with the bi-dimensional organization. The platinum dots sputtered on an aluminum layer have shown a behavior of isolated oscillators without any coupling between neighbor elements in this phononic crystal. The frequency of such normal modes, extracted from time resolved measurements are in good agreement with 3D finite element simulations. In contrast, with aluminum dot systems where the coupling is more efficient we observe a complex spectrum of vibrational modes related to the band structure induced by the bi-dimensional patterning.     

Combustion oscillations in gas fired appliances: Eigen-frequencies and stability regimes

This paper presents a one-dimensional acoustic model for prediction of the frequencies of self-excited oscillation and acoustic mode shapes in combustion systems. The impedance of the combustion system is represented in terms of a frequency response function (FRF). Impedances of the settling and combustion chambers are predicted by using the acoustic model, taking into account the temperature distribution in the combustion chamber. Reasonably good agreement between measured and predicted acoustic resonance frequencies and mode shapes was achieved. Some data on stability regimes are discussed.     

Free vibrations of a vessel consisting of circular plates and a shell of revolution having varying meridional curvature

An analytical solution procedure is presented for the free vibration of vessels consisting of a shell of revolution having varying meridional curvature and circular plate lids. The Lagrangian of vibration of the combined system is obtained in quadratic forms of boundary values. The frequency equation and the relations among the boundary values are obtained from minimizing conditions of the Lagrangian with respect to the unknown boundary values. The natural frequencies and the mode shapes of vessels having elliptical and hyperbolical meridians have been obtained by carrying out numerical calculations. Effects of various parameters upon natural frequencies and mode shapes are illustrated in discussions of numerical results.     

The effect of trailing end geometry on the vibration of a circular cantilevered rod in nominally axial flow

The effect of “trailing end” geometry on the vibration of a circular cantilevered rod in nominally axial water flow is reported. Eleven different trailing end geometries were tested. For each end geometry, rms displacement response and damping were measured and are presented as functions of mean axial flow velocity. The various geometric end shapes are ordered according to their effectiveness in attenuating rod vibration. The results, which are in qualitative agreement with similar experimental results from tests on flat plates, provide the designer with empirical information that allows selecting a trailing end geometry that minimizes response to parallel flow excitation. The results have application in the design of nuclear reactor fuel pins and certain instrumentation probes.     

Vibration studies on some integral rib-stiffened plates

Results from a theoretical and experimental comparison study of the vibrations of four integrally machined rib-stiffened plates are presented. Two plates had one rib-stiffener and two plates had two rib-stiffeners. The finite element method was used for the theoretical predictions and real-time laser holography was used for the experimental verifications. The first twenty-four vibration modes were predicted and measured for each plate, and detailed comparisons are presented. Reasonably good quantitative agreement was obtained in all cases for both frequencies and mode shapes.     

Vibrations of a square plate with parabolically varying thickness

Vibration results for a square plate with a parabolically varying thickness distribution and built-in edges are presented. Frequency and mode shape predictions obtained from a finite element analysis are compared with measurements made with real-time laser holography. In general, the agreement between the two is very good except for a few of the lower modes where the predicted frequencies are about 15% high. So far, a satisfactory explanation for this has not been found. The vibration mode shapes for the plate exhibit a striking blend of radial and square symmetries that results from the axisymmetric thickness distribution and the square symmetry of the boundary frame.     

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.     

La3Ga5SiO14晶体的介电性质、弹性与压电性质

研究了La3Ga5SiO14晶体的介电性质、弹性和压电性质，其中弹性和压电性质是利用高频振动模式通过谐振法测定的. 关键词： 介电性质 弹性 压电性质     

Equivalent circuits and directivity patterns of air-coupled ultrasonic transducers

Air-coupled transducers for producing ultrasonic radiation in gases are studied. The transducer consists of a circular thin plate in flexural vibration and a sandwich longitudinal electromechanical vibrator that is attached to the center of the plate. The lowest-order axially symmetric flexural vibrational mode of a circular thin plate is analyzed. The equivalent circuits of the circular plate in flexural vibration and the compound transducer are presented and the frequency equation is derived. The radiated ultrasonic field of the circular thin plate in flexural vibration is calculated and the directivity pattern is obtained theoretically. Some transducers of this type are designed according to the frequency equation, and their resonance frequencies are measured. The measured resonance frequencies are in good agreement with the theoretical results, and the calculated radiation ultrasonic field is also in good agreement with the measured results of a previous work.     

First- and second-harmonic radiations from a near-millimeter-wave gyromonotron

First- and second-harmonic radiations in frequency regions centered around 120 and 240 GHz, respectively, have been observed from a gyromonotron employing magnetic fields between 34 and 54 kG and a magnetron injection gun preducing a 30-kV, 1- to 2-A electron beam. Extensive redesign of the cavity, electron beam tunnel holders, and vacuum envelope resulted in greatly improved tube performance over that obtained previously. Nine first-harmonic modes and three second-harmonic modes were identified with specific TE cavity modes. These identifications were based on agreement between the measured frequency of a radiation and the calculated resonant frequency of a TE mode, and one or both of the following: (a) the agreement between the magnetic field at which the radiation occurred and that predicted by linear theory calculations of threshold current versus magnetic field for that mode; and (b) the similarity between the far-field antenna pattern measured with calibrated, waveguide-mounted detectors and that predicted for that mode. Output power was measured with a Scientech 362 Calorimeter and also estimated by integrating the far-field antenna patterns measured with the calibrated detectors. The power ranged from a fraction of a watt to 200 W for second-harmonic modes, and from tens of watts to 2 kW for first-harmonic modes.     

Theoretical and experimental research on a disk-type non-contact ultrasonic motor

We developed a disk-type non-contact ultrasonic motor based on B22 vibration mode. The rotors of SU-8 photoresist are fabricated by the UV-LIGA process to control their shapes and thicknesses. So the structures of them are optimized by the experiments. It is found that the revolution speed of disk-type non-contact ultrasonic motor not only depends on the vibration amplitude of the stator, but also the weight and construction of the rotors. The maximum revolution speed of the optimal rotor is 3569 rpm at the input voltage of 20 V and the driving frequency of 45.6 kHz. The exciting principle of traveling wave is presented with theoretical equations. The electric signals applied to the piezoelectric ceramic are designed by the principle. The natural frequency and corresponding vibration mode are calculated and analyzed using finite element method. It is shown that experimental results are in good agreement with simulation, which verifies the effectiveness of the finite element model. Moreover, the levitation distance between the stator and rotor is measured by a CCD laser displacement transducer.     

Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

In Parts I and II of this series of papers, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written on the modal basis, in the neighbourhood of each resonance, has been developed for beams and fully clamped rectangular plates.¹ Simple explicit formulae have been derived, which allowed, via the so-called first formulation, direct calculation of the basic function contributions to the first three non-linear mode shapes of clamped-clamped and clamped-simply supported beams, and the two first non-linear mode shapes of FCRP. Also, in Part I of this series of papers, this approach has been successively extended, in order to determine the amplitude-dependent deflection shapes associated with the non-linear steady state periodic forced response² of clamped-clamped beams, excited by a concentrated or a distributed harmonic force in the neighbourhood of the first resonance.This new approach has been applied in the present work to obtain the NLSSPFR formulation for FCRP, SSRP, and CCCSSRP, leading in each case to a non-linear system of coupled differential equations, which may be considered as a multi-dimensional form of the well-known Duffing equation. The single-mode assumption, and the harmonic balance method, have been used for both harmonic concentrated and distributed excitation forces, leading to one-dimensional non-linear frequency response functions of the plates considered. Comparisons have been made between the curves based on these functions, and the results available in the literature, showing a reasonable agreement, for finite but relatively small vibration amplitudes. A more accurate estimation of the FCRP non-linear frequency response functions has been obtained by the extension of the improved version of the semi-analytical model developed in Part I for the NLSSPFR of beams, to the case of FCRP, leading to explicit analytical expressions for the “multi-dimensional non-linear frequency response function”, depending on the forcing level, and the amplitude of the response induced in the range considered for the excitation frequency.     

Analysis of the sandwich piezoelectric ultrasonic transducer in coupled vibration

The coupled vibration of the sandwich piezoelectric transducer with a large cross-section is analyzed using an approximate analytic method. The resonance frequency equations of the transducer are derived and the effect of the geometrical dimensions on the resonance frequency is studied. It is illustrated that when the radial vibration in the transducer is considered, the vibration of the sandwich transducer becomes more complex. Apart from the longitudinal resonance frequency, the radial resonance frequency can also be obtained. For comparison, numerical methods are also used to simulate the coupled vibration; the resonance frequency and the vibrational displacement distribution are computed. Compared with one-dimensional longitudinal theory, the radial dimensions of the transducer are no longer limited because the coupled vibration is considered. Compared with numerical methods, the physical meaning of the analytic method is concise. It is illustrated that the resonance frequencies obtained from the coupled resonance frequency equations are in good agreement with those from numerical methods, and they are in better agreement with the measured results than those from one-dimensional theory. Since the radial and the coupled vibration are considered in the analysis, more resonance frequencies can be obtained. Therefore, using the coupled resonance frequency equations, the sandwich transducer with multifrequency or wide frequency bandwidth can be designed and used in ultrasonic cleaning, ultrasonic sonochemistry and other applications.     

Sr3TaGa3Si2O14(STGS)单晶的晶格振动

根据空间群理论,预测了Sr3TaGa3Si2O14(STGS)单晶的振动模式,并分别计算了非极性和极性振动模式的Raman散射强度。测量了STGS晶体的Raman光谱,并对其振动模式进行了指认。实验结果表明STGS晶体具有6个A1对称类特征振动模式:波数为126cm-1的振动峰可指认为SiO4团簇、Sr原子和TaO6团簇间的相对平动;245cm-1的振动峰是SiO4团簇的扭曲振动和Sr—TaO6—Sr伸缩振动耦合的结果;特征峰557和604cm-1分别来源于O—Ta—O和O—Ga—O的伸缩振动;896cm-1谱带对应着两个SiO4四面体的O—Si—O伸缩振动;991cm-1的谱带对应着两个SiO4四面体的Si—O伸缩振动。实验结果和理论计算结果均确证了STGS晶体的层状结构,其弱的各项异性和压电模量归因于十面体结构单元的微弱形变。     

Experimental analysis of the vibrational characteristics of the human skull

Analytical models of the human skull structure have generally been constructed so as to characterize the gross geometric features and material properties; however, a model should also have accurate frequency response characteristics since these are essential for collision and head injury analyses. An experimental investigation was conducted to identify the dynamic characteristics of freely vibrating human skulls. Resonant frequencies and associated mode shapes in the frequency band from 20 Hz to 5000 Hz were delineated for two dry human skulls. Osteometrically, one skull corresponds to a 50th percentile male (skull 1) and the second is representative of a 5th percentile female skull (skull 2). Digital Fourier analysis techniques were used to identify the resonant frequencies and corresponding mode shapes of each skull. Eleven resonant frequencies were identified for skull 1, with the lowest being 1385 Hz. In contrast, skull 2 exhibited only 6 resonant frequencies with the first being 1641 Hz. Nine mode shapes were identified for skull 1, but only 5 modes were recognized for skull 2. The vibrational pattern of the human skull, as indicated by its mode shapes in this limited study, seems to be a unique property of a particular skull. Skull satures did not appear to influence the modal pattern.     

Effect of crystal and collimator misalignments on Bragg angle measurements

A mathematical analysis has been made of the combined effect of crystal and collimator misalignments on Bragg angles measured from zero-layer Weissenberg photographs. The results obtained here have been compared with those available for Bond’s method of measuring Bragg angles with diffractometers. The comparison shows that, for a given amount of crystal and collimator misalignments, Bond’s method and the 2 θ-method do not give identical results for small and intermediate Bragg angles.     

Coupled bending-bending vibrations of pretwisted tapered cantilever beams treated by the Reissner method

Theoretical natural frequencies and mode shapes of the first four coupled modes of a uniform pretwisted cantilever blade and the first five coupled flexural frequencies of pretwisted tapered blading are determined by using the Reissner method. The shape functions for the bending moments and deflections are developed in series form and with these used in the dynamic Reissner functional, the frequency equation is obtained by minimizing it through the Ritz process. A convergence study made in the case of the pretwisted uniform blade indicates that there appears to be a quicker convergence of the natural frequencies and that a five-term solution yields a set of results that are in good agreement with the theoretical and experimental values of other authors, available in the literature. The mode shapes obtained from the present analysis are compared with those from an earlier investigation and the effect of ignoring the shear deflection and rotary inertia in the analysis is discussed. The effects of breadth taper and depth taper on the vibration characteristics of pretwisted cantilever blading are discussed from the results obtained in the present limited study and it is observed that an extensive investigation appears to be necessary to draw positive conclusions covering wide ranges of pretwisted blade parameters.     

Force generation characteristics of standard heavyweight impact sources used in the sound generation of building floors

Several heavyweight impact sources have been used in applying standardized excitation forces to evaluate the noise generation characteristics of building floors. In this study, heavyweight sources (a bang machine and an impact ball) were filmed with a high-speed camera on the period of deformation and restitution of the impact. The generated impact force was measured, and the effects of the taken deforming shapes were investigated. Experimental modal analysis was performed to determine the natural frequencies and mode shapes of the impact sources. With respect to deforming shapes during impacts, the influence of the mode shapes was clearly demonstrated, especially for the impact ball. Using the measured modal properties, the impact force was predicted and compared to the measured excitation force. By comparing the transient variation of the excitation force and its frequency spectrum, the influence of the modal characteristics of the source on the generated impact force was investigated.