Damage detection in composite panels based on mode-converted Lamb waves sensed using 3D laser scanning vibrometer

This paper introduces damage identification approach based on guided ultrasonic waves and 3D laser Doppler vibrometry. The method is based on the fact that the symmetric and antisymmetric Lamb wave modes differ in amplitude of the in-plane and out-of-plane vibrations. Moreover, the modes differ also in group velocities and normally they are well separated in time. For a given time window both modes can occur simultaneously only close to the wave source or to a defect that leads to mode conversion. By making the comparison between the in-plane and out-of-plane wave vector components the detection of mode conversion is possible, allowing for superior and reliable damage detection. Experimental verification of the proposed damage identification procedure is performed on fuel tank elements of Reusable Launch Vehicles designed for space exploration. Lamb waves are excited using low-profile, surface-bonded piezoceramic transducers and 3D scanning laser Doppler vibrometer is used to characterize the Lamb wave propagation field. The paper presents theoretical background of the proposed damage identification technique as well as experimental arrangements and results.     

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.     

Detecting damage in vibrating structures with a scanning LDV

It has been demonstrated, through experiments on laboratory-scale structures, that structural defects such as cracks can be detected and located using a continuously scanning laser Doppler vibrometer (LDV) if vibration sufficient to flex the defect can be induced and if the defects are such as to produce localised mode shape discontinuities. This paper describes such a method of defect detection using a short linear scan at the crack location. Through-cracks are easily detected in thin metal plates whereas narrow slots in a solid cantilever beam have no easily identifiable effect unless they extend more than half-way through the thickness. Cracks in a reinforced-concrete beam introduced marked and identifiable discontinuities in mode shapes. Speckle noise affects the measurements, sometimes seriously. A simple low-pass filter may improve the signal quality.     

Non-linear forced vibrations of a beam carrying concentrated mass under gravity

The non-linear dynamic behavior of a simply supported beam, with ends restrained to remain a fixed distance apart, carrying a concentrated mass and subjected to a harmonic exciting force at an arbitrary point under the influence of gravity is analysed. By using the one mode approximation and applying Galerkin's method, the governing equation of motion is reduced to the well known Duffing type equation. The harmonic balance method is applied to solve the equation and the dynamic response of a concentrated mass is derived. The effects of the weight, the location, and the vibratory amplitude of the concentrated mass on the natural frequency are also discussed.     

Overview of the New OMAH Technique for Scaling OMA Mode Shapes

Methods for scaling mode shapes determined by operational modal analysis (OMA) have been extensively investigated in the last years. A recent addition to the range of methods for scaling OMA mode shapes is the so-called OMAH technique, which is based on exciting the structure by harmonic forces applied by an actuator. By applying harmonic forces in at least one degree-of-freedom (DOF), and measuring the response in at least one response DOF, while using at least as many frequencies as the number of mode shapes to be scaled, the mode shape scaling (modal mass) of all modes of interest may be determined. In previous publications on the method the authors have proven that the technique is easy and robust to apply to both small scale and large scale structures. Also, it has been shown that the technique is capable of scaling highly coupled modes by using an extended multiple reference formulation. The present paper summarizes the theory of the OMAH method and gives recommendations of how to implement the method for best results. It is pointed out, as has been shown in previous papers, that the accuracy of the mode scaling is increased by using more than one response DOF, and by selecting DOFs with high mode shape coefficients. To determine the harmonic force and responses, it is recommended to use the three-parameter sine fit method. It is shown that by using this method, the measurement time can be kept short by using high sampling frequency and bandpass filtering whereas spectrum based methods require long measurement times. This means that even for structures with low natural frequencies, the extra measurement time for scaling the mode shapes can be kept relatively short.     

一种新型声学超材料平板对机械波吸收性能的模拟与实验研究

基于质量-弹簧微结构通过光学振动模式对机械波进行选择性吸收的原理, 本文设计了一个由集中质量与弹性薄膜构成的二维声学超材料等效平板结构. 利用有限元法对等效平板代表性胞元的振动特性模拟结果表明, 二维超材料平板结构表现出与质量-弹簧微结构相同的振动模式. 基于模拟结果, 制作加工了一个正方形超材料平板, 并通过Polytec扫描式激光测振仪(PSV)对其振动特性进行了试验测试, 结果表明, 本文设计制作的超材料平板可以对频率157.5 Hz的机械波进行选择吸收.     

Measurement of Lamb wave polarization using a one-dimensional scanning laser vibrometer (L)

A single head scanning laser Doppler vibrometer is used for the estimation of the polarization of the first symmetric (S(0)) and antisymmetric (A(0)) Lamb wave modes. The measurements at two known incidence angles are performed in order to resolve the two components of motion. Filtering in the frequency/wavenumber domain of the response recorded along a scan line separates the contributions from each mode and allows the evaluation of the corresponding elliptical trajectories of particle motion. Comparison between measured and analytically estimated trajectories validates the measurement technique and suggests its application for the development of material characterization and diagnostics tools.     

A SEMI-ANALYTICAL APPROACH TO THE NON-LINEAR DYNAMIC RESPONSE PROBLEM OF BEAMS AT LARGE VIBRATION AMPLITUDES, PART II: MULTIMODE APPROACH TO THE STEADY STATE FORCED PERIODIC RESPONSE

The semi-analytical approach to the non-linear dynamic response of beams based on multimode analysis has been presented in Part I of this series of papers (Azrar et al., 1999 Journal of Sound and Vibration224, 183-207 [1]). The mathematical formulation of the problem and single mode analysis have been studied. The objective of this paper is to take advantage of applying this semi-analytical approach to the large amplitude forced vibrations of beams. Various types of excitation forces such as harmonic distributed and concentrated loads are considered. The governing equation of motion is obtained and can be considered as a multi-dimensional form of the Duffing equation. Using the harmonic balance method, the equation of motion is converted into non-linear algebraic form. Techniques of solution based on iterative-incremental procedures are presented. The non-linear frequency and the non-linear modes are determined at large amplitudes of vibration. The basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated. The percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution. Also, the percentage contributions of the higher modes to the bending moment near to the clamps are given, in order to determine accurately the error introduced in the non-linear bending stress estimated by different approximations. Solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency. The non-linear deflection shapes in various regions of the solution, the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes. The numerical results obtained here for the non-linear forced response are compared with those from the linear theory, with available non-linear results, based on various approaches, and with the single mode analysis.     

THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN ELASTIC SHELLS. PART II: A NEW APPROACH FOR FREE TRANSVERSE CONSTRAINED VIBRATION OF CYLINDRICAL SHELLS

The non-linear dynamic behaviour of infinitely long circular cylindrical shells in the case of plane strains is examined and results are compared with previous studies. A theoretical model based on Hamilton's principle and spectral analysis previously developed for non-linear vibration of thin straight structures (beams and plates) is extended here to shell-type structures, reducing the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. In the present work, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions corresponding to the constrained vibrations, which exclude the axisymmetric displacements. The non-linear strain energy is expressed by taking into account the non-linear terms due to the considerable stretching of the shell middle surface induced by large deflections. It has been shown that the model presented here gives new results for infinitely long circular cylindrical shells and can lead to a good approximation for determining the fundamental longitudinal mode shape and the associated higher circumferencial mode shapes (n>3) of simply supported circular cylindrical shells of finite length. The non-linear results at small vibration amplitudes are compared with linear experimental and theoretical results obtained by several authors for simply supported shells. Numerical results (non-linear frequencies, vibration amplitudes and basic function contributions) of infinite shells associated to the first four mode shapes of free vibrations, are obtained, using a multi-mode approach and are summarized in tables. Good agreement is found with results from previous studies for both small and large amplitudes of vibration. The non-linear mode shapes are plotted and discussed for different thickness to radius ratios. The distributions of the bending stresses associated with the mode shapes are given and compared with those obtained via the linear theory.     

Flow characterization using a laser Doppler vibrometer

This paper shows how a scanning laser Doppler vibrometer (LDV), an instrument designed to measure vibrations of structures or objects, can be used in a non-traditional fashion to measure the flow around a cylinder. In particular the von Karmann vortex street which appears in the cylinder wake will be visualized. This is achieved by measuring the changes in the optical path induced by local fluctuation of air refraction index to which the laser vibrometer is sensitive.The measurements obtained with the LDV will be compared visually to measurements done with particle image velocimetry and also with CFD computations for one test case. The specific frequency of the Von Karmann street predicted by the vibrometer, will be compared numerically to the other techniques for two different sized cylinders at three different velocities.     

Analysis of a non-linear structure by considering two non-linear formulations

In recent years, modal synthesis methods have been extended for solving non-linear dynamic problems subjected to harmonic excitation. These methods are based on the notion of non-linear or linearized modes and exploited in the case of structures affected by localized non-linearity. Actually, the experimental tests executed on non-linear structures are time consuming, particularly when repeated experimental tests are needed. It is often preferable to consider new non-linear methods with a view to decrease significantly the number of attempts during prototype tests and improving the accuracy of the dynamic behaviour.This article describes two fundamental non-linear formulations based on two different strategies. The first formulation exploits the eigensolutions of the associated linear system and the dynamics characteristics of each localized non-linearity. The second formulation is based on the exploitation of the linearized eigensolutions obtained using an iterative process. This article contains a numerical and an experimental study which examines the non-linear behaviour of the structure affected by localized non-linearities. The study is intended to validate the numerical algorithm and to evaluate the problems arising from the introduction of non-linearities. The complex responses are evaluated using the iterative Newton-Raphson method and for a series of discrete frequencies. The theory has been applied to a bi-dimensional structure and consists of evaluating the harmonic responses obtained using the proposed formulations by comparing measured and calculated transfer functions.     

Assessment of acoustic pressure holograms from membrane velocity measurements

The aim of this paper is to demonstrate experimentally the possibility of acquiring acoustic pressure holograms using a light membrane and a scanning laser vibrometer. The velocity of a light membrane placed in an acoustic field can be measured without contact by means of a laser vibrometer. The ideal membrane must be optically reflective, acoustically transparent (having as little mass as possible), impermeable, and mounted without tension. The measured velocity is equal for continuity reasons to the normal acoustic velocity, but differs from the acoustic velocity without the membrane because the membrane is never completely transparent to acoustic waves. The effect of the mass of the membrane can be taken into account to correct this difference. Then, acoustic pressure holograms can be deduced from velocity holograms using the 2D Discrete Fourier Transform. An experimental validation is carried out; acoustic pressures derived from laser measurements are compared with microphone measurements, with a very satisfying match over a wide frequency range.     

Damage detection in membrane structures using non-contact laser excitation and wavelet transformation

In this paper, a vibration testing and health monitoring system based on an impulse response excited by laser is proposed to detect damage in membrane structures. A high power Nd: YAG pulse laser is used to supply an ideal impulse to a membrane structure by generating shock waves via laser-induced breakdown in air. A health monitoring apparatus is developed with this vibration testing system and a damage detecting algorithm which only requires the vibration mode shape of the damaged membrane. Artificial damage is induced in membrane structure by cutting and tearing the membrane. The vibration mode shapes of the membrane structure extracted from vibration testing by using the laser-induced breakdown and laser Doppler vibrometer are then analyzed by 2-D continuous wavelet transformation. The location of damage is determined by the dominant peak of the wavelet coefficient which can be seen clearly by applying a boundary treatment and the concept of an iso-surface to the 2-D wavelet coefficient. The applicability of the present approach is verified by finite element analysis and experimental results, demonstrating the ability of the method to detect and identify the positions of damage induced on the membrane structure.     

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.     

Experimental study and signal analysis in the Lamb wave conversion at a bevelled edge

This paper deals with a quantitative study of the conversion of a Lamb wave at the bevelled edge of a plate. A harmonic wave (successively the A1 and S0 Lamb modes) is generated using a wedge. The normal displacements at the surface of the plate are measured with a laser vibrometer and used to compute an energy evaluation. In order to determine a good mean value of incident and reflected wave amplitudes, the signals are isolated by a Fourier treatment. Then, the energy flow repartition among the converted modes is computed in both cases and for different values of the bevel angle.     

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.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

Mode effects in second harmonic generation and their uses for laser stabilization

The influence of laser modes on second harmonic generation has been investigated using a three mode He-Ne laser operating at 633 nm as the source of the fundamental radiation. Mode locking which occurs when the laser modes are symmetrically placed relative to the centre of the gain profile enhances the second harmonic signal. This effect has been used to stabilize the frequency of the He-Ne laser to ± 1.5 MHz.     

Mode shapes analysis of a cracked beam and its application for crack detection

In this paper, mode shapes of a cracked beam with a rectangular cross section beam are analysed using finite element method. The 3D beam element is applied for this finite element analysis. The influence of the coupling mechanism between horizontal bending and vertical bending vibrations due to the crack on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam change from plane curves to space curves. Thus, the existence of the crack can be detected based on the mode shapes: when the mode shapes are space curves there is a crack in the beam. Also, when there is a crack, the mode shapes have distortions or sharp changes at the crack position. Thus, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. While in previous studies using 2D beam element, distortions in the mode shapes caused by a small crack could not be detected, these distortions in the case using the 3D beam element can be amplified and inspected clearly by using the projections of the mode shapes on appropriate planes. The quantitative analysis is also implemented to relate the size and position of the crack with the observed coupled modes. These results can be applied for crack detection of a beam. In this paper, the stiffness matrix of a cracked element obtained from fracture mechanics is presented and numerical simulations of three case studies are provided.     

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.