Time-dependent variance and covariance of responses of structures to non-stationary random excitations

In this paper techniques for the analysis of non-stationary random responses of linear structures, discretized by the finite element method so that they can be analyzed as multi-degree of freedom systems, subjected to non-stationary random excitation are developed. The non-stationary random excitation is represented as a product of (a) an exponentially decaying function and a white noise process, and (b) a modulating function in the form of an exponential envelope and a white noise process. Closed form expressions for the time-dependent variance and covariance of response of structures are presented. Application of these expressions is made for the analysis of non-stationary random responses of a physical model of a class of mast antenna structures subjected to base excitation. It is concluded that (a) the coupling terms do have a definite influence on the response; the magnitude of the influence is proportional to the amount of damping in the structure and proximity of the modes excited; (b) the non-stationary random excitations considered are general in that the modulating functions are not necessarily identical, and therefore the influence of various modulating functions of the excitations applied to different locations of the structure on responses can be examined quantivatively; and (c) for a given damping parameter the magnitudes of the modulating function parameters cannot be chosen arbitrarily though the shapes of normalized modulating functions can be selected to best fit the excitation realizations.     

Modal parameter identification through Gabor expansion of response signals

A new method of identifying modal parameters by decomposing response signals with Gabor transform is presented in this paper to estimate natural frequencies, damping ratios and mode shapes of linear time invariant systems. According to Gabor expansion theory, responses of a multi-degree-of-freedom system can be decomposed into uncoupled signal components, each vibrating at a single natural frequency. From these uncoupled signals, modal parameters are subsequently extracted with common methods. The proposed method can process stationary and non-stationary responses and requires no input signal except for the response signals generated by unknown excitation acting on a system. In the sense of less restriction on the in-out signals, the approach based on time-frequency decomposition is very general. A simulation study on a simply supported beam under non-stationary excitation has demonstrated that the proposed method is effective in parameter estimation.     

On the application of blind source separation for damping estimation of bridges under traffic loading

The accurate and reliable estimation of modal damping from output-only vibration measurements of structural systems is a continuing challenge in the fields of operational modal analysis (OMA) and system identification. In this paper a modified version of the blind source separation (BSS)-based Second-Order Blind Identification (SOBI) method was used to perform modal damping identification on a model bridge structure under varying loading conditions. The bridge model was created with finite elements and consisted of a series of stringer beams supported by a larger girder. The excitation was separated into two categories: ambient noise and traffic loads with noise modeled with random forcing vectors and traffic simulated with moving loads for cars and partially distributed moving masses for trains. The acceleration responses were treated as the mixed output signals for the BSS algorithm. The modified SOBI method used a windowing technique to maximize the amount of information used for blind identification from the responses. The modified SOBI method successfully found the mode shapes for both types of excitation with strong accuracy, but power spectral densities (PSDs) of the recovered modal responses showed signs of distortion for the traffic simulations. The distortion had an adverse affect on the damping ratio estimates for some of the modes but no correlation could be found between the accuracy of the damping estimates and the accuracy of the recovered mode shapes. The responses and their PSDs were compared to real-world collected data and patterns similar to distortion were observed implying that this issue likely affects real-world estimates.     

Modal identification based on continuous wavelet transform and ambient excitation tests

This paper deals with the use of the continuous wavelet transform for modal identification from ambient vibration tests. The wavelet analysis of the free responses of a linear mechanical system has been used to estimate its natural frequencies, viscous damping ratios, and mode shapes, using either the modulus or the phase of the wavelet transform with “ridge” and “skeleton” notions. This technique is extended in this paper, to the response from ambient excitation tests based on the equivalent formulation between the free response and the correlation function of a linear mechanical system under stationary stochastic excitation. The main novelties involve the new formulation of continuous wavelet transform and the use of the singular value decomposition algorithm for ridges and skeleton detection under the noisy conditions of ambient vibration tests. A complete procedure for modal identification of ambient response, including these new formulations, is presented and then applied to numerical examples and experimental test to illustrate its validity.     

Non-linear structural dynamics characterization using a scanning laser vibrometer

This paper presents the use of a scanning laser vibrometer and a signal decomposition method to characterize non-linear dynamics of highly flexible structures. A Polytec PI PSV-200 scanning laser vibrometer is used to measure transverse velocities of points on a structure subjected to a harmonic excitation. Velocity profiles at different times are constructed using the measured velocities, and then each velocity profile is decomposed using the first four linear mode shapes and a least-squares curve-fitting method. From the variations of the obtained modal velocities with time we search for possible non-linear phenomena. A cantilevered titanium alloy beam subjected to harmonic base-excitations around the second, third, and fourth natural frequencies are examined in detail. Influences of the fixture mass, gravity, mass centers of mode shapes, and non-linearities are evaluated. Geometrically exact equations governing the planar, harmonic large-amplitude vibrations of beams are solved for operational deflection shapes using the multiple shooting method. Experimental results show the existence of 1:3 and 1:2:3 external and internal resonances, energy transfer from high-frequency modes to the first mode, and amplitude- and phase-modulation among several modes. Moreover, the existence of non-linear normal modes is found to be questionable.     

HEALTH-MONITORING METHOD FOR BRIDGES UNDER ORDINARY TRAFFIC LOADINGS

A method for damage estimation of a bridge structure is presented using ambient vibration data caused by the traffic loadings. The procedure consists of identification of the operational modal properties and the assessment of damage locations and severities. An experimental study is carried out on a bridge model with a composite cross-section subjected to vehicle loadings. Vertical accelerations of the bridge deck are measured while vehicles are running. The modal parameters are identified from the free-decay signals extracted using the random decrement method. The damage assessment is carried out based on the estimated modal parameters using the neural networks technique. As input to the neural networks, the ratios of the resonant frequencies between before and after damages and the mode shapes after the damages are used to take into account the mass effect of the traffic on the bridge. The identified damage locations and severities agree reasonably well with the inflicted damages on the structure.     

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.     

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.     

Analysis of dynamics of excitation and dephasing of plasmon resonance modes in nanoparticles

A novel theoretical approach to the dynamics analysis of excitation and dephasing of plasmon modes in nanoparticles is presented. This approach is based on the biorthogonal plasmon mode expansion, and it leads to the predictions of time dynamics of excitation of specific plasmon modes as well as their steady state amplitude and their decay. Temporal characteristics of plasmon modes in nanoparticles are expressed in terms of their shapes, permittivity dispersion relations, and excitation conditions. In the case of the Drude model, analytical expressions for time-dynamics of plasmon modes are obtained.     

Pulsed TV-holography recording for vibration analysis applications

TV-holography is a well-known tool for vibration analysis. Using the so-called time-average method, this technique allows to record interferograms showing the mode shapes of a structure submitted to vibration excitation and is currently used for modal identification.Within the frames of a BRITE-EURAM program called vibration intensity processing using full-field multi-pulse laser technique (VIP), a TV-holography equipment has been developed, working with a 25 Hz pulsed laser and allowing easy on-site measurements. A measurement procedure has been defined and a specific data processing has been developed for the determination of structural intensity fields, which give transfer path of the vibration energy within a structure.The measurement of these quantities is possible the using classical means (accelerometers, stress gauges, etc.) but the data processing is complex and require a lot of accurate sensors because it is based on spatial derivatives of high order. The optical techniques (laser vibrometry, holography, etc.) are more suited for that purpose because of the high density of measuring points and because of the well-known advantages of these methods: reduced measurement time and no modification of the mass parameters of the structure as it is the case when using contact sensors.Different kinds of output data are then given: operational deflection shape, amplitude and phase, structural intensity field and its divergence through a further step of data processing. The complete procedure with the associated data processing has been tested (for various configurations of excitation and damping) first on a clamped plate, then on a cylinder and at the end on several industrial components.This paper describes the general measurement procedure and the equipment used. The data processing is also presented and various measurement results are shown. The conclusion gives the main advantages and limitations of the method and evaluates the application possibilities.     

Stationary response of nonlinear magneto-piezoelectric energy harvester systems under stochastic excitation

Recent years have shown increasing interest of researchers in energy harvesting systems designed to generate electrical energy from ambient energy sources, such as mechanical excitations. In a lot of cases excitation patterns of such systems exhibit random rather than deterministic behaviour with broad-band frequency spectra. In this paper, we study the efficiency of vibration energy harvesting systems with stochastic ambient excitations by solving corresponding Fokker-Planck equations. In the system under consideration, mechanical energy is transformed by a piezoelectric transducer in the presence of mechanical potential functions which are governed by magnetic fields applied to the device. Depending on the magnet positions and orientations the vibrating piezo beam system is subject to characteristic potential functions, including single and double well shapes. Considering random excitation, the probability density function (pdf) of the state variables can be calculated by solving the corresponding Fokker-Planck equation. For this purpose, the pdf is expanded into orthogonal polynomials specially adapted to the problem and the residual is minimized by a Galerkin procedure. The power output has been estimated as a function of basic potential function parameters determining the characteristic pdf shape.     

Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods

This study provides two non-contact optical techniques to investigate the transverse vibration characteristics of piezoceramic rectangular plates in resonance. These methods, including the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), are full-field measurement for AF-ESPI and point-wise displacement measurement for LDV, respectively. The edges of these piezoceramic rectangular plates may either be fixed or free. Both resonant frequencies and mode shapes of vibrating piezoceramic plates can be obtained simultaneously by AF-ESPI. Excellent quality of the interferometric fringe patterns for the mode shapes is obtained. In the LDV system, a built-in dynamic signal analyzer (DSA) composed of DSA software and a plug-in waveform generator board can provide the piezoceramic plates with the swept-sine excitation signal, whose gain at corresponding frequencies is analyzed by the DSA software. The peaks appeared in the frequency response curve are resonant frequencies. In addition to these optical methods, the numerical computation based on the finite element analysis is used to verify the experimental results. Good agreements of the mode shapes and resonant frequencies are obtained for experimental and numerical results.     

Non-stationary random response of a finite cable

Numerous problems of current concern involve the designs of aerodynamic systems which either travel at high speeds or contain structural elements which are excited by moving pressure fluctuations. In a number of recent papers responses of dynamic systems to random excitation have been considered. The appropriate theory for calculating the mean square response of linear systems to both stationary and non-stationary random excitation is well known [1–7]. In this paper, the mean square response of a finite cable to non-stationary random excitation is considered. The non-stationary random excitation is of the form s(t) = e(t)α(t), where e(t) is a well defined envelope function and α (t) is the Guassian, narrow band, stationary part of the excitation which has zero mean. Both the unit step and rectangular step functions are used for the envelope function, and both white noise and noise with an exponentially decaying harmonic correlation function are used to prescribe the statistical property of the excitation. The results obtained are shown to be a complete expression for the mean square response when checked for accuracy by reduction to expressions previously obtained by Lyon [4]. It is felt that these results will aid the design of both linear and two-dimensional aerodynamic systems excited by random pressure fluctuations.     

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.     

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.     

Equivalent linear stochastic seismic response of isolated bridges

Stochastic response of bridges seismically isolated by lead-rubber bearings (LRB) is investigated. The earthquake excitation is modeled as a non-stationary random process (i.e. uniformly modulated broad-band excitation). The stochastic response of isolated bridge is obtained using the time-dependent equivalent linearization technique as the force-deformation behavior of the LRB is highly nonlinear. The non-stationary response of isolated bridge is compared with the corresponding stationary response in order to study the effects of non-stationary characteristics of the earthquake input motion. For a given isolated bridge system and excitation, it was observed that there exists an optimum value of the yield strength of LRB for which the root mean square (rms) absolute acceleration of bridge deck attains the minimum value. The optimum yield strength of LRB is investigated under important parametric variations such as isolation period and damping ratio of the LRB and the frequency content and intensity of earthquake excitation. It is shown that the above parameters have significant effects on the optimum yield strength of LRB. Finally, closed-form expressions for the optimum yield strength of LRB and corresponding response of the isolated bridge system are proposed. These expressions were derived based on the model of bridge with rigid deck and pier condition subjected to stationary white-noise excitation. It was observed that there is a very good comparison between the proposed closed-form expressions and actual optimum parameters and response of the isolated bridge system. These expressions can be used for initial optimal design of seismic isolation system for the bridges.     

Modeling subharmonic response from contrast microbubbles as a function of ambient static pressure

Variation of subharmonic response from contrast microbubbles with ambient pressure is numerically investigated for non-invasive monitoring of organ-level blood pressure. Previously, several contrast microbubbles both in vitro and in vivo registered approximately linear (5-15 dB) subharmonic response reduction with 188 mm Hg change in ambient pressure. In contrast, simulated subharmonic response from a single microbubble is seen here to either increase or decrease with ambient pressure. This is shown using the code BUBBLESIM for encapsulated microbubbles, and then the underlying dynamics is investigated using a free bubble model. The ratio of the excitation frequency to the natural frequency of the bubble is the determining parameter--increasing ambient pressure increases natural frequency thereby changing this ratio. For frequency ratio below a lower critical value, increasing ambient pressure monotonically decreases subharmonic response. Above an upper critical value of the same ratio, increasing ambient pressure increases subharmonic response; in between, the subharmonic variation is non-monotonic. The precise values of frequency ratio for these three different trends depend on bubble radius and excitation amplitude. The modeled increase or decrease of subharmonic with ambient pressure, when one happens, is approximately linear only for certain range of excitation levels. Possible reasons for discrepancies between model and previous experiments are discussed.     

Piezoelectric transducer parameter selection for exciting a single mode from multiple modes of Lamb waves

Excitation and propagation of Lamb waves by using rectangular and circular piezoelectric transducers surface-bonded to an isotropic plate are investigated in this work. Analytical stain wave solutions are derived for the two transducer shapes, giving the responses of these transducers in Lamb wave fields. The analytical study is supported by a numerical simulation using the finite element method. Symmetric and antisymmetric components in the wave propagation responses are inspected in detail with respect to test parameters such as the transducer geometry, the length and the excitation frequency. By placing only one piezoelectric transducer on the top or the bottom surface of the plate and weakening the strength of one mode while enhancing the strength of the other modes to find the centre frequency, with which the peak wave amplitude ratio between the S0 and A0 modes is maximum, a single mode excitation from the multiple modes of the Lamb waves can be achieved approximately. Experimental data are presented to show the validity of the analyses. The results are used to optimize the Lamb wave detection system.     

Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains

In predictions of railway-induced vibrations, a distinction is generally made between the quasi-static and dynamic excitation. The quasi-static excitation is related to the static component of the axle loads. The dynamic excitation is due to dynamic train–track interaction, which is generated by a large number of excitation mechanisms, such as the spatial variation of the support stiffness and the wheel and track unevenness. In the present paper, the quasi-static excitation and the dynamic excitation due to random track unevenness are evaluated by means of numerical predictions. A solution strategy is presented that allows for the evaluation of the second-order statistics of the response due to dynamic excitation based on the power spectral density function of the track unevenness. Due to the motion of the train, the second-order statistics of the response at a fixed point in the free field are non-stationary and an appropriate solution procedure is required. The quasi-static and dynamic contribution to the track and free-field response are analysed for the case of InterCity and high-speed trains running at a subcritical train speed. It is shown how the train speed affects the quasi-static and dynamic contribution. Finally, results of numerical predictions for different train speeds are compared with field measurements that have been performed at a site along the high-speed line L2 Brussels–Köln within the frame of homologation tests.     

Resonance fluorescence spectrum of a three-level atom. II. Numerical results

Numerical computations are presented for the excitation spectra arising from the interaction between a three-level atom in the cascade configuration and a strong electromagnetic field whose frequency mode is initially populated. The excitation spectra are considered when the laser field is at resonance with the equally spaced levels of the atom as well as a function of the detunings. The physical process of optical amplification occurs without population inversion and it is more pronounced when the laser field is detuned than when it is at resonance. The shapes of the spectral lines for a number of side-bands are of the absorption-amplification type rather than that of the absorption one. In the presence of detunings as well as in the cooperative two-photon cascade process, the resulting spectra are far more complicated than those occuring at resonance. Results of numerical calculations for a wide range of Rabi frequencies and detunings are presented graphically.