An automatic mode pairing strategy using an enhanced modal assurance criterion based on modal strain energies

In the context of finite element model updating using output-only vibration test data, natural frequencies and mode shapes are used as validation criteria. Consequently, the correct pairing of experimentally obtained and numerically derived natural frequencies and mode shapes is important. In many cases, only limited spatial information is available and noise is present in the measurements. Therefore, the automatic selection of the most likely numerical mode shape corresponding to a particular experimentally identified mode shape can be a difficult task. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches.In this paper, the purely mathematical modal assurance criterion will be enhanced by additional physical information from the numerical model in terms of modal strain energies. A numerical example and a benchmark study with experimental data are presented to show the advantages of the proposed energy-based criterion in comparison to the traditional modal assurance criterion.     

On exact and approximated formulations for scaling-mode shapes in operational modal analysis by mass and stiffness change

When operational modal analysis (OMA) is used to estimate modal parameters, mode shapes cannot be mass normalized. In the past few years, some equations have been proposed to scale mode shapes using the mass-change method, which consists of repeating modal testing after changing the mass at different points of the structure where the mode shapes are known. In this paper, the structural-dynamic-modification theory is used to derive a set of equations, from which all the existing formulations can be derived. It is shown that the known equations can be divided into two types, the exact and the approximated equations, where the former type does in fact fulfill the equations derived from the theory of structural modification, whereas the remaining equations do not, mainly because the change of the mode shapes of the modified structure is not properly taken into account. By simulations, the paper illustrates the large difference in accuracy between the approximate and the exact formulations. The paper provides two new exact formulations for the scaling factors, one for the non-modified structure and – for the first time in the literature – one for the modified structure. The simulations indicate the influence of errors from the measured natural frequencies and mode shapes on the estimation of the scaling factors using the two exact formulations from the literature and the new exact formulation proposed in this paper. In addition, the paper illustrates statistics of the errors on mode-shape scaling. All simulations were carried out using a plate with closely spaced modes.     

Minute displacement and strain analysis using lensless Fourier transformed holographic interferometry

An optical system for lensless Fourier transformed holographic interferometry is constructed to enable the measurement of minute displacements from nanometers to micrometers scale and to obtain corresponding strain distributions using a CCD camera with poor spatial resolution. Since a Fourier spectrum of an object beam is recorded on a hologram in this technique, the image reconstruction is easily performed with a single pass of 2-D fast Fourier transformation. Then, the map of the phase difference over the whole field is obtained by comparing two images before and after deformation. A suitable and effective unwrapping process is, however, inevitably required since the phase difference distribution is wrapped from −π to π in this technique. For phase unwrapping, the maximum spanning tree method is adopted here, which seeks a spanning tree that maximizes overall edge weights given by the cross amplitude. In-plane and out-of-plane displacements are obtained separately from the phase difference distributions at one's request. Moreover, in-plain strain is easily calculated from the in-plane displacement distribution.     

光栅衍射多普勒效应位移测量的理论分析和实验结果

本文研究了一种利用光栅衍射的多普勒效应进行位移测量的新方法.文中进行了光路结构和测量原理的理论分析,得出测量公式,证明了用光栅拾取多路差拍信号的相位关系.实验结果表明,它具有高的信号质量,可有效地用于位移的精密测量.     

Numerical and experimental modal analysis of the reed and pipe of a clarinet

A modal computation of a complete clarinet is presented by the association of finite-element models of the reed and of part of the pipe with a lumped-element model of the rest of the pipe. In the first part, we compare modal computations of the reed and the air inside the mouthpiece and barrel with measurements performed by holographic interferometry. In the second part, the complete clarinet is modeled by adjoining a series of lumped elements for the remaining part of the pipe. The parameters of the lumped-resonator model are determined from acoustic impedance measurements. Computed eigenmodes of the whole system show that modal patterns of the reed differ significantly whether it is alone or coupled to air. Some modes exhibit mostly reed motion and a small contribution of the acoustic pressure inside the pipe. Resonance frequencies measured on a clarinet with the mouthpiece replaced by the cylinder of equal volume differ significantly from the computed eigenfrequencies of the clarinet taking the actual shape of the mouthpiece into account and from those including the (linear) dynamics of the reed. This suggests revisiting the customary quality index based on the alignment of the peaks of the input acoustical impedance curve.     

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.     

An experimental and theoretical investigation of the frequencies and mode shapes of axisymmetric shell models

The use of doubly curved isoparametric elements in a finite element analysis enabled successful prediction to be made of the natural frequencies and mode shapes of axisymmetric shell models. The models chosen were in the form of a natural draught cooling tower, part of a large hemispherical dome and part of a smaller hemispherical dome, and all were formed from thermoplastic materials. Several natural frequencies and mode shapes were determined experimentally for all the models. The theoretical predictions compared very favourably with the experimental results. For example in the case of the cooling tower model the lowest frequency mode, with four nodal diameters and one nodal circle, was predicted to occur at 19·99 H, and experimentally it was 19·4 H. The effect of numerical results of the choice of the number of elements and hence the number of degrees of freedom was demonstrated. The use of five elements on the cooling tower model gave a difference between experimental and theoretical results of less than 3%. In representing the domes by three, five and six elements the deployment of more degrees of freedom led to a significant improvement in the results. It was also shown that some functional modelling could be misleading owing to deficiencies occurring during the incorporation of geometric and boundary conditions at the formulation stage.     

Identification of embedded horizontal cracks in beams using measured mode shapes

Mode shapes (MSs) have been extensively used to detect structural damage. This paper presents two new non-model-based methods that use measured MSs to identify embedded horizontal cracks in beams. The proposed methods do not require any a priori information of associated undamaged beams, if the beams are geometrically smooth and made of materials that have no stiffness discontinuities. Curvatures and continuous wavelet transforms (CWTs) of differences between a measured MS of a damaged beam and that from a polynomial that fits the MS of the damaged beam are processed to yield a curvature damage index (CDI) and a CWT damage index (CWTDI), respectively, at each measurement point. It is shown that the MS from the polynomial fit can well approximate the measured MS and associated curvature MS of the undamaged beam, provided that the measured MS of the damaged beam is extended beyond boundaries of the beam and the order of the polynomial is properly chosen. The proposed CDIs of a measured MS are presented with multiple resolutions to alleviate adverse effects caused by measurement noise, and cracks can be identified by locating their tips near regions with high values of the CDIs. It is shown that the CWT of a measured MS with the n-th-order Gaussian wavelet function has a shape resembling that of the n-th-order derivative of the MS. The crack tips can also be located using the CWTs of the aforementioned MS differences with second- and third-order Gaussian wavelet functions near peaks and valleys of the resulting CWTDIs, respectively, which are presented with multiple scales. A uniform acrylonitrile butadiene styrene (ABS) cantilever beam with an embedded horizontal crack was constructed to experimentally validate the proposed methods. The elastic modulus of the ABS was determined using experimental modal analysis and model updating. Non-contact operational modal analysis using acoustic excitations and measurements by two laser vibrometers was performed to measure the natural frequencies and MSs of the ABS cantilever beam, and the results compare well with those from the finite element method. Numerical and experimental crack identification can successfully identify the crack by locating its tips.     

Three-dimensional myocardial strain analysis based on short- and long-axis magnetic resonance tagged images using a 1D displacement field

A robust algorithm to estimate three-dimensional strain in the left-ventricular heart wall, based on magnetic resonance (MR) grid-tagging in two sets of orthogonal image planes, is presented. Starting-point of this study was to minimize global interpolation and smoothing. Only the longitudinal displacement was interpolated between long-axis images. Homogeneous strain analysis was performed using small tetrahedrons. The method was tested using a stack of short-axis images and three long-axis images in six healthy volunteers. In addition, the method was subjected to an analytical test case, in which the effect of noise in tag point position on the observed strains was explored for normally distributed noise (0.5 mm RMS). In volunteers, the error in the longitudinal displacement due to interpolation between the long-axis image planes was -0.10 +/- 0. 48 mm (mean +/- SD). The resulting error in the longitudinal strain epsilon(l) was -0.003 +/- 0.02. The analytical test case was used to quantify the effects of three sources of errors on the observed strain. The SD of the difference between homogeneous strain and true strain was 0.06 for epsilon(r.) The error due to the 3-D reconstruction was 0.004 for epsilon(r.) The error in epsilon(r) resulting from simulated noise in the tag point position was 0.10. Equivalent results were obtained for all other strain parameters; thus, the error resulting from noise in the tag point position dominates the error introduced by approximations in the method. Because the proposed method uses a minimum of global interpolation and smoothing, it offers the prospect to detect small regions of aberrant contraction.     

Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure

Fpr a two-degree-of-freedom structure, an experimental and theoretical investigation has been made of the primary resonances of the system, which occur when the frequency of excitation is near one of the natural frequencies, ω₁ and ω₂. The additional constraint of $\text{ω}{}_2\text{2}\text{?}\text{ω}{}_1$ has been included and its profound influence on the response studied. The investigation has revealed some characteristics of the response that are the result of both non-linearity and more than one degree of freedom, the principal one of these being saturation. As far as is known this is the first time that saturation has beeb observed and studied in detail in a physical structure.     

Experimental analysis of thermal displacement and strain distributions in a small outline J-leaded electronic package by using wedged-glass phase-shifting moiré interferometry

To evaluate the influence of a printed wiring board (PWB) with a high coefficients of thermal expansion (CTE) on the thermal deformation of a small outline J-leaded electronic package (SOJ), a newly developed phase-shifting method was applied to moiré interferometry. This phase-shifting moiré interferometry method uses a wedged glass plate as a phase shifter to obtain displacement fields with a sensitivity of 100 nm/line. This technique also enabled the quantitative determination of strain distributions in all observation areas. Thermal loading was applied from room temperature (25 °C) to an elevated temperature (100 °C), and then the thermal strains of SOJ with and without the PWB were compared. The results showed that the concentrations of the longitudinal strains ε_xx and ε_yy became increasingly prominent when mounted on the PWB, and the shear strains γ_xy were concentrated at the corners of the silicon chip. The values of these strains increased by about 50% when the SOJ was mounted on the PWB.     

On the scalar modal analysis of optical waveguides using approximate methods

We have discussed the approximate methods which are used for obtaining scalar guided modes of optical waveguides. The methods include the perturbation method, the variational method including the Rayleigh-Ritz method, and the Galerkin and the collocation method. The main purpose of this paper is to discuss the inter-relationships and equivalences of these methods, and to bring out the fact that these relationships have, in fact, not been recognized in the guided wave optics literature, although in the numerical electromagnetics and applied mathematics literature some of these relationships are well known. We have also pointed out specific examples where, due to this lack of recognition of relationships, there are repetitions in the literature. In particular, we have noted that the Rayleigh-Ritz method and the Galerkin method have been used using the same set of basis functions for the same kind of waveguides without recognizing the existing literature. We have also reported for the first time an explicit relationship between the Galerkin method and the collocation method. This relationship also points out in which cases one method is more accurate and/or numerically efficient than the other. Another interesting relationship explored is that between the perturbation method and the variational method.     

Numerical and experimental analysis of uncertainty on modal parameters estimated with the stochastic subspace method

Modal parameters of structures are often used as inputs for finite element model updating, vibration control, structural design or structural health monitoring (SHM). In order to test the robustness of these methods, it is a common practice to introduce uncertainty on the eigenfrequencies and modal damping coefficients under the form of a Gaussian perturbation, while the uncertainty on the mode shapes is modeled in the form of independent Gaussian noise at each measured location. A more rigorous approach consists however in adding uncorrelated noise on the time domain responses at each sensor before proceeding to an operational modal analysis. In this paper, we study in detail the resulting uncertainty when modal analysis is performed using the stochastic subspace identification method. A Monte-Carlo simulation is performed on a simply supported beam, and the uncertainty on a set of 5000 modal parameters identified with the stochastic subspace identification method is discussed. Next, 4000 experimental modal identifications of a small clamped–free steel plate equipped with 8 piezoelectric patches are performed in order to confirm the conclusions drawn in the numerical case study. In particular, the results point out that the uncertainty on eigenfrequencies and modal damping coefficients may exhibit a non-normal distribution, and that there is a non-negligible spatial correlation between the uncertainty on mode shapes at sensors of different locations.     

Noncontact modal analysis of a pipe organ reed using airborne ultrasound stimulated vibrometry

The goal of this study was to excite and measure, in a noncontact manner, the vibrational modes of the reed from a reed organ pipe. To perform ultrasound stimulated excitation, the audio-range difference frequency between a pair of ultrasound beams produced a radiation force that induced vibrations. The resulting vibrational deflection shapes were measured with a scanning laser vibrometer. The resonances of any relatively small object can be studied in air using this technique. For a 36 mm x 6 mm brass reed, displacements and velocities in excess of 5 microm and 4 mm/s could be imparted at the fundamental frequency of 145 Hz. Using the same ultrasound transducer, excitation across the entire range of audio frequencies was obtained. Since the beam was focused on the reed, ultrasound stimulated excitation eliminated background effects observed during mechanical shaker excitation, such as vibrations of clamps and supports. The results obtained using single, dual and confocal ultrasound transducers in AM and two-beam modes, along with results obtained using a mechanical shaker and audio excitation using a speaker are discussed.     

On the undamped natural frequencies and mode shapes of a finite-element model of the cat eardrum

This paper presents a three-dimensional finite-element model of the cat eardrum which includes inertial effects. The model is implemented using a hierarchical modeling scheme which permits the mesh resolution to be varied. The static behavior of the model is calculated as a function of mesh resolution in order to check the validity of an earlier model. The first six undamped natural frequencies, and the corresponding modal vibration patterns, are then calculated. They are found to lie between about 1.8 and 3.2 kHz for the standard values chosen for the model parameters. The effects on the natural frequencies of varying seven parameters of the model are described.     

基于封装光纤Bragg光栅传感器的混凝土应变监测试验研究

研究了光纤Bragg光栅封装工艺,提出并实现了两种封装光纤Bragg光栅传感器:工字型钢管封装光纤Bragg光栅应变传感器及钢管封装光纤Bragg温度传感器。研究了传感器的埋设工艺,采用金属丝固定法、钢管抽出法两种方法进行了埋设。将它们埋入到钢筋混凝土梁中,实现对混凝土的拉、压应变及温度的测量。结果显示,在试验梁弹性阶段,可以较为准确的监测混凝土应变。     

零平均折射率带隙中离散模的实验研究

通过实验研究了零平均折射率带隙中离散模的传输特性.零平均折射率带隙存在于左手材料与右手材料交替形成的一维光晶体中.在实验中,左手材料与右手材料分别由左/右手复合传输线和右手复合传输线实现.实验结果表明,左手材料与右手材料的色散特性使离散模并没有按照理论上期待的"单点频"传输,而是彻底地覆盖了零平均折射率带隙.由于左手材料在本质上是色散的,因此该实验结论普遍适用于零平均折射率带隙中的离散模,即离散模的出现会使零平均折射率带隙消失.     

An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia

In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal.