The radiation impedance of a rectangular piston

The radiation impedance of a rectangular piston is expressed as the Fourier transform of its impulse response, which is obtained from the recent work of Lindermann [1]. The analytical evaluation of the transform is performed and new integral expressions are presented for both the radiation resistance and reactance. The integrals are readily evaluated in terms of elementary functions at both the low and high frequency limits. The integrals are also expressed as series of Bessel functions which are valid for all frequencies and aspect ratios. Numerical results are presented to illustrate the behavior of the radiation resistance and reactance as a function of the aspect ratio of the piston and a normalized frequency parameter. Additional numerical results are then presented to illustrate the accuracy of the analytical expressions for the radiation resistance and reactance at low and high frequencies. Finally, numerical results are presented to illustrate the application and accuracy of using standard FFT algorithms to evaluate the radiation resistance and reactance directly from the impulse responses.     

Vibration testing based on impulse response excited by pulsed-laser ablation: Measurement of frequency response function with detection-free input

We have developed a non-contact vibration-measurement system that is based on impulse excitation by laser ablation (i.e. laser excitation) to measure the high-frequency-vibration characteristics of objects. The proposed method makes it possible to analyse the frequency response function just by measuring the output (acceleration response) of a test object excited by pulsed-laser ablation. This technique does not require detection of the input force. Firstly, using a rigid block, the pulsed-laser-ablation force is calibrated via Newton's second law. Secondly, an experiment is conducted in which an object whose natural frequency lies in the high-frequency domain is excited by pulsed-laser ablation. The complex frequency spectrum obtained by Fourier transform of the measured response is then divided by the estimated pulsed-laser-ablation force. Finally, because of the error involved in the trigger position of the response with respect to the impulse arrival time, the phase of the complex Fourier transform is modified by accounting for the response dead time. The result is the frequency response function of the object. The effectiveness of the proposed method is demonstrated by a vibration test of an aluminium block.     

The measurement of the free field impulse response of microphones in a laboratory environment

A method is presented of measuring the free field frequency and impulse response of microphones by using a pulse technique in an ordinary laboratory environment. The pressure transient generated by exciting a loudspeaker with a narrow pulse is detected at some point in the loudspeaker's far field by a “reference” microphone whose response is assumed flat over the frequency range of interest. The microphone to be tested is then substituted in exactly the same position and its response to the transient measured. The outputs of the two microphones are accurately sampled and deconvolved by using a discrete Fourier transform technique to give the magnitude and phase parts of the “test” microphone's frequency response, and hence, via the discrete Fourier transform, its response to a delta function of pressure propagating in the free field. The computed impulse responses are presented and the accuracy of the method discussed.To illustrate the use of the method, the free field frequency response and free field correction curves of a one inch instrumentation microphone are measured. The method is then used to measure the pressure which occurs at the centre of the flat end-face of a long a cylinder when excited by an impulse of acoustic pressure propagating in the free field from various angles of incidence.     

Natural frequencies of orthotropic rectangular plates with stepped thickness

It is shown how one can derive an approximate formula for estimating a natural frequency of an orthotropic rectangular plate with stepped thickness by using several natural frequencies of the corresponding isotropic plate reduced from the orthotropic one. To justify the method, an orthotropic two-part rectangular plate with simply supported sides is discussed, and an approximate formula is proposed for estimating the fundamental natural frequency.     

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh-Ritz method using the ordinary and degenerated free-free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates.     

A reduction method for problems of vibration of orthotropic plates

It is shown that the problem of vibration of an orthotropic plate can be reduced to that of another orthotropic plate by a simple co-ordinate transformation, and reduction formulae are obtained. To justify the reduction formulae, fundamental natural frequencies of orthotropic rectangular plates with various boundary conditions and of a clamped orthotropic elliptical plate are discussed. As an example, an exact natural frequency of a simply supported generally orthotropic skew plate with special flexural rigidities is obtained from that of a simply supported isotropic rectangular plate.     

Stabilization of time domain acoustic boundary element method for the interior problem with impedance boundary conditions

The time domain boundary element method (BEM) is associated with numerical instability that typically stems from the time marching scheme. In this work, a formulation of time domain BEM is derived to deal with all types of boundary conditions adopting a multi-input, multi-output, infinite impulse response structure. The fitted frequency domain impedance data are converted into a time domain expression as a form of an infinite impulse response filter, which can also invoke a modeling error. In the calculation, the response at each time step is projected onto the wave vector space of natural radiation modes, which can be obtained from the eigensolutions of the single iterative matrix. To stabilize the computation, unstable oscillatory modes are nullified, and the same decay rate is used for two nonoscillatory modes. As a test example, a transient sound field within a partially lined, parallelepiped box is used, within which a point source is excited by an octave band impulse. In comparison with the results of the inverse Fourier transform of a frequency domain BEM, the average of relative difference norm in the stabilized time response is found to be 4.4%.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

一种频域合成房间频率响应的人工混响方法

给出了一种频域合成房间频率响应的方法用于卷积法人工混响,基于频域内房间频率响应的后期部分为高斯随机过程的假设,用自回归滑动平均模型为其自协方差函数和功率谱密度进行参数化描述,在对自回归滑动平均模型中的参数求解后,通过逆滤波得到了房间频率响应后期部分,与房间频率响应前期部分组合后经过傅里叶反变换得到完整的房间脉冲响应。仿真结果表明该方法的混响效果与镜像源法接近,明显优于反馈延迟网络法,但其计算复杂度比镜像源法低,便于实时应用。     

基于光学相干测振系统的微悬臂梁缺陷检测

提出了基于光学相干测振(optical coherence vibrometer,OCV)系统的微悬臂梁缺陷检测方法.自搭建的OCV系统最大振动位移量程、最大振动频率分别为2.574 mm和138.5 kHz,应用该系统对含缺陷微悬臂梁-附加质量块耦合结构进行振动测量获得其固有频率,并利用附加质量块对固有频率的影响特性...     

The effect of different combinations of boundary conditions on the average radiation efficiency of rectangular plates

The boundary conditions of a vibrating plate are known to have an influence on its sound radiation for frequencies below the critical frequency. To investigate this effect in a systematic way, the average radiation efficiency and radiated power are calculated for a rectangular plate set in an infinite baffle using a modal summation approach. Whereas analytical expressions exist for simply supported boundary conditions, a numerical approach is required for other cases. Nine combinations of boundary conditions are considered, consisting of simply supported, clamped and free edges on different plate edges. The structural vibration is approximated by using independent beam functions in orthogonal directions allowing simple approximate formulae for mode shapes and natural frequencies. This assumption is checked against a finite element model and shown to give reliable results. It is shown that a free plate has the lowest radiation efficiency and a clamped plate the highest for most frequencies between the fundamental panel natural frequency and the critical frequency. Other combinations of boundary condition give intermediate results according to the level of constraint introduced. The differences depend on frequency: excluding the extreme case of a fully free plate all the other boundary conditions give results within a range of 8 dB in the middle part of the short-circuiting region, decreasing towards the critical frequency. At low frequency the differences can be even greater, in some cases up to 20 dB. These conclusions are shown to hold for a range of plate thicknesses and dimensions.     

Investigation of the phase velocities of guided acoustic waves in soft porous layers

A new experimental method for measuring the phase velocities of guided acoustic waves in soft poroelastic or poroviscoelastic plates is proposed. The method is based on the generation of standing waves in the material and on the spatial Fourier transform of the displacement profile of the upper surface. The plate is glued on a rigid substrate so that it has a free upper surface and a nonmoving lower surface. The displacement is measured with a laser Doppler vibrometer along a line corresponding to the direction of propagation of plane surface waves. A continuous sine with varying frequencies was chosen as excitation signal to maximize the precision of the measurements. The spatial Fourier transform provides the wave numbers, and the phase velocities are obtained from the relationship between wave number and frequency. The phase velocities of several guided modes could be measured in a highly porous foam saturated by air. The modes were also studied theoretically and, from the theoretical results, the experimental results, and a fitting procedure, it was possible to determine the frequency behavior of the complex shear modulus and of the complex Poisson ratio from 200 Hz to 1.4 kHz, in a frequency range higher than the traditional methods.     

Calibration of ear canals for audiometry at high frequencies

A procedure is described for determining the absolute sound pressure at the inner end of the ear canal when a sound source is coupled to the ear, for frequencies in the range 8-20 kHz. The transducer that generates the sound is coupled to the ear canal through a lossy tube, yielding a source impedance that is approximately matched to the characteristic impedance of the ear canal. A small microphone is located in the coupling tube close to the entrance to the ear canal. Calibration is carried out by measuring the response at this microphone when an impulse is applied at the transducer. To estimate the sound pressure at the medial end of the ear canal, the Fourier transform of this impulse response is corrected by an all-pole function in which the poles are estimated from the minima in this Fourier transform. Data on individual ear canals are presented in terms of gain functions relating the sound pressure at the medial end of the ear canal to the sound pressure when the coupling tube is blocked. The average gain function for a group of adult ears increases from 2 to 12 dB over the frequency range 8-20 kHz, in rough agreement with data from ear-canal models. Possible sources of error in the calibration procedure are discussed.     

First and second derivatives of internal combustion engine cylinder pressure diagrams

Approximate formulae are proposed for estimating natural frequencies of isotropic and specially orthotropic skew plates with clamped sides. It has been shown previously that one can estimate a natural frequency of a generally orthotropic skew plate with clamped sides by using an approximate formula for the isotropic plate which one can relate to the orthotropic one by applying a previously described reduction method. The accuracy of the proposed approximate formulae is demonstrated by comparing numerical and experimental results for several typical cases.     

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

Fourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.     

Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions

In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed.     

Time domain identification of standing wave parameters in gas piping systems

A method for experimentally determining the natural frequencies and modal pressures of an air or gas piping system is presented. Such information is of interest in installations where pressure pulsations caused by pumps or compressors are of importance. In the method a time domain based technique is used which was originally developed as an alternative to frequency response methods for determining the vibration parameters (natural frequencies, modes, damping factors) of structures, to avoid difficulties often encountered in interpreting complex and non-conclusive frequency response data such as arises from systems having numerous modes, some of which may be highly damped or closely spaced in frequency. In this application, a straight steel pipe with a sound source at one end and closed at the other end was used. Two microphones were used to measure the pressure at two locations in the pipe. The free pressure response following a rapidly swept sinewave input was recorded, digitized and then used in a computational procedure based on a lumped parameter representation of the system. The natural frequencies and the corresponding modal pressure ratios at the two stations, thus obtained, are compared with mention here that although in the experiment reported here an external frequency sweep excitation was used, the technique works as well with free decay response after a system shut-off, impulse response or random responses from normal system operation.     

Structural waveguide behaviour of a beam–plate system

Structures which have a constant cross-section normal to a longitudinal axis can be considered as waveguides in which vibration can propagate in the form of various waves in the longitudinal direction. The dynamic behaviour of such systems can be found by using a Fourier transform approach in terms of wavenumbers in the longitudinal direction. Analytical solutions are available for simple, infinitely long waveguides, whereas for more complex waveguides numerical approaches have been developed using finite element techniques to describe the cross-section. In the present paper an analytical Fourier transform approach is used to find the dynamic behaviour of a system consisting of two parallel beams coupled by a plate, when a point force is applied to one of the beams. Multiple waves occur in the longitudinal direction, the number of waves depending on the number of modes of the equivalent cross-section. However, the motion of the driven beam is shown to be dominated by the contribution from only one or two waves at each frequency, these having wavenumbers closest to that of the uncoupled beam. The motion of the plate is also shown to be dominated by these wavenumbers for excitation on the beam. Experimental results are obtained on beam–plate–beam systems with identical and non-identical beams, which show good agreement with the predictions. In particular, these confirm that the plate response is dominated by waves with wavenumbers in the beam direction that follow those of the excited beam.     

Sound radiation from shear deformable stiffened laminated plates

Sound radiation from shear deformable stiffened laminated plates is studied theoretically. The equations of motion for the composite laminated plate are derived on the basis of the first-order shear deformation plate theory. Two sets of parallel stiffeners interact with the laminated plate only through the normal line forces. By using the Fourier wavenumber transform and the stationary phase method, the far-field sound pressure is described analytically. Sound pressure given by the first-order shear deformation plate theory and the classical thin plate theory is compared, and the differences of sound pressure are shown in the high frequency range for an isotropic plate. Sound pressure and the transverse displacement spectra are presented to illustrate the effects of force location, stiffeners and angle-ply layers. Sound radiation from symmetric and antisymmetric composite plates with multiple loadings is also investigated.     

分数傅里叶变换啁啾滤波

针对常规傅里叶变换所不能解决的啁啾噪声滤除问题,利用Wigner分布函数分析分数傅里叶变换的空域和频域特性,提出在分数傅里叶变换域进行啁啾滤波的方法。并将该方法应用到图像处理中,对分数傅里叶变换滤除一维和二维图像的啁啾噪声进行了计算机仿真,获得了满意的效果,结果表明该方法在图像处理中的有效性。