Displacement Mode Shapes Research Articles

Analysis and mitigation of uncertainties in damage identification by modal-curvature based methods

The objective of this paper is to address and reduce the uncertainties associated with measurement noise and discretization in damage identification methods based on modal curvature analysis. The experimental case study considers the local reduction in the bending stiffness of a slender beam under free-free and simply supported boundary conditions. First, the analysis of error sources and their propagation is theoretically set up. Second, the mitigation of uncertainties in damage localization is pursued using a two-stage approach based on multiple hypothesis testing relative to the normalized indices and the definition of a combined macro-index. Finally, the Monte Carlo method is exploited to obtain the statistical error distribution of the experimental damage position and severity predictions by randomizing the numerical displacement mode shapes with the identified noise. The present analysis allows us to find the optimal number of sensors that minimizes the combination of bias and truncation errors, to highlight how sensor spacing and data noise affect damage localization, and to determine the uncertainty bounds of the predicted damage severity. The two-stage approach, enhanced by selecting thresholds related to real noise levels and tuned on SHM objectives, appears to improve identification accuracy compared to the separate use of damage indices based on absolute confidence levels.

A modal-based method for blast-induced damage assessment of reinforced concrete columns: Numerical and experimental validation

Reinforced concrete (RC) columns play a crucial role in the overall performance of modern RC frame buildings under blast loading. The RC columns under blast loads might experience various irreversible consequences such as concrete cracks, crushing and spalling, changing the status of both modal properties and load-carrying capacity. In this paper, the blast-induced damage assessment of the RC columns involving the measurement of modal parameters was introduced and validated. Firstly, a modal-based method for blast-induced damage assessment of RC columns was proposed, in which the damage is characterized as the reduction of axial bearing capacity and further expressed as the explicit function of the vibration frequencies and the mass-normalized displacement mode shapes. Then, the finite-element analysis program LS-DYNA was employed to establish the numerical model of RC columns under axial compression loads and lateral blast loads. The residual load-bearing capacities and modal properties of the RC columns were illustrated. By using the numerical results collected in the present study, the validity of the proposed damage assessment method was discussed and proved numerically. Further, field explosion tests and experimental modal tests were conducted, and the results showed that the modal-based damage assessment method exhibited remarkable agreement with the blast damage determined by residual bearing capacities. Based on the numerical and experimental results, the proposed modal-based damage assessment method is applicable for the non-destructive evaluation of blast-induced damage of RC columns.

A modified mode shape data-based method for beams structural damage detection

Damage detection methods based on changing modal parameters have received considerable attention in engineering applications due to the satisfactory results and the low associated cost compared with other techniques. The Mode Shape Data Based Indicator (MSDBI) is a damage indicator available in the literature, being used to identify damage in beam structures from the mode shape, mode shape slope and mode shape curvature, in the undamaged and damaged configurations of the element under study. However, in some situations, the configuration of the displacement mode shape of the ith mode, of the undamaged structure compared to the damaged one, presents mirroring. The damage identification algorithm could be a better indicator when these situations occur. Them, this paper presents a proposal to modify this method, called MSDBIM. The proposed modified method (MSDBIM) and the traditional method (MSDBI) were applied in two numerical examples that were elaborated in commercial software of finite elements, namely a simply supported concrete beam and a fixed-end steel beam in different single and multiple damage scenarios with sensitivity studies. A new discretization for the fixed-end beam was performed to assess whether there is a direct influence on the damage identification method. The results show that the proposed method (MSDBIM) performs better than the traditional method (MSDBI).

An improved version of mode shape based indicator for structural damage identification

Damage detection is crucial for the operation and maintenance of an existing bridge or building. A vibration-based method is a popular approach that can be used to identify failures based on changes in the dynamic properties. However, this approach often requires information on both healthy and unhealthy states. In this current study, the structural failure determination is carried out through a damage index based solely on the unhealthy state. A combination of the gapped smooth and modal curvature methods is the core of calculating the proposed indicator. In particular, damage scenarios by means of assumed stiffness reduction, and cuts are investigated. Three categories of beam-like structures consisting of a simply supported beam, a cantilever beam, and a free-free beam are used to validate the applicability of the proposed approach. For comparison purposes, the traditional gapped smooth method is implemented. The location of damage can be identified using only displacement mode shapes. Then, damage quantification at the identified location is estimated using a stochastic optimization process. The promising findings indicate that the proposed approach can enhance the effectiveness of damage identification based on vibration data in the unhealthy state of the structures

Shear horizontal wave propagation on a piezoelectric semiconductor substrate under slight ridge or thin metal strip gratings

In the present work, analytical solutions for shear horizontal (SH) waves propagating in an n-type piezoelectric semiconductor (PSC) substrate under periodic slight ridge or thin metal strip gratings are obtained. The solutions of the various physical fields can be expressed as a summation of infinite harmonic waves by using the Bloch–Floquet theorem due to the existence of the periodic ridge or metal grating. Based on the numerical calculations, the effects of the width, height and shape of the ridge, the doping density of the PSC substrate and the material properties of the metal grating on the dispersion and attenuation curves of the SH waves are investigated. The results show that both the periodic ridge and metal grating have evident effects on the propagation characteristics and displacement mode shapes of the SH waves and the effects are closely related with the ratio of the wavelength to the structure period. In addition, the metal gratings can better control the propagation characteristics of SH waves than the ridges, namely, the effects of the metal gratings are more obvious and flexible than that of the ridges.

Shear wave modes in an equilateral triangular prismatic bar.

Shear wave excitation of an equilateral triangular solid bar shows the existence of horizontally polarized (SH) and vertically polarized (SV) shear waves. The SH wave modes are Lamé solutions of the Neumann problem for displacement and Dirichlet problem for stress in the equilateral triangle. It is shown that Lamé's solutions are obtained from superposition of three equal plane waves and their reflections. The Neumann SH modes have cutoff wavenumber 4mπ/(3a), where m=1,2,3,…, a being the side length of triangular cross section. The first Dirichlet SH mode has a higher cutoff wavenumber 4π7/(3a). The SV modes are related to the 30°-60°-90° and 30°-120°-30° sub triangles. Their cutoff wavenumbers lie between those of the first (m = 1) and second (m = 2) Neumann SH modes. The standing wave condition in each sub triangle is related to three unequal plane waves which decompose via symmetrical component analysis into two different sets of equal plane waves. A nonlinear product solution based on the new plane wave set leads to modal SV wave solutions. Dispersion curves calculated using COMSOL confirm the correct cutoff wavenumbers of both SH and SV waves. Normal displacement mode shapes are calculated analytically and verified experimentally using laser vibrometer with good agreement for all modes.

Multiple damage prediction in tubular rectangular beam model using frequency response-based mode shape curvature with back-propagation neural network

In the civil infrastructure, the structures made up of tubular sections played an important role due to an economic point of view, lower self-weight, and stability against functional loads. The periodical maintenance or inspection of the structure is mandatory nowadays to achieve the potential utilization of the structural system. In this study, an experimental, numerical, and analytical study is carried out to investigate the structural fault and its severity in a tubular rectangular beam made up of structural steel. Modal parameters are extracted with the help of a dynamic data logger (B&K) by exciting an impact hammer on model structure and extracting a data with a set of unidirectional accelerometers. First, two displacement mode shapes are extracted using obtained modal parameters. The modal parameters are expected to contain environmental noise during experimentation, so, de-noising is must to obtain noise-free data. Artificial neural network training is utilized to reduce the noise from experimental modal parameters. Using the modified modal parameters, the mode shape curvature is obtained, and so-called modified mode shape curvature (MMSC) is used to calculate the curvature damage index. The curvature damage index is appropriate to investigate multiple fault locations with different fault levels in tubular rectangular beam structures.

Turning Telecommunication Fiber-Optic Cables into Distributed Acoustic Sensors for Vibration-Based Bridge Health Monitoring

We introduce a nondedicated bridge health monitoring (BHM) system that turns pre-existing telecommunication fiber-optic cables into distributed acoustic sensors to collect bridge dynamic strain responses. Due to extensively installed telecommunication fiber cables in the cities, our telecommunication cable-based system enables efficient and low-cost BHM without the requirement of on-site sensor installation and maintenance; however, it is challenging to extract bridge damage-sensitive information (e.g., natural frequencies and mode shapes) from this nondedicated strain data as it has large measurement noise and error propagation. To overcome the challenge, we develop a physics-guided system identification method that models strain mode shapes based on physics-guided parametric mode shape functions derived from bridge dynamics. We then estimate the displacement mode shape function by analytically double-integrating the modeled strain mode shape. Our method improves the accuracy of estimating bridge damage-sensitive features and reduces error propagation by constraining strain and displacement mode shapes with bridge dynamics. We evaluated our system on a concrete continuous three-span bridge in San Jose, California. Our system successfully identified the first three natural frequencies and reconstructed strain and displacement mode shapes in a meter-scale resolution.

Three-Dimensional Digital Image Correlation Based on Speckle Pattern Projection for Non-Invasive Vibrational Analysis.

Non-contact vibration measurements are relevant for non-invasively characterizing the mechanical behavior of structures. This paper presents a novel methodology for full-field vibrational analysis at high frequencies using the three-dimensional digital image correlation technique combined with the projection of a speckle pattern. The method includes stereo calibration and image processing routines for accurate three-dimensional data acquisition. Quantitative analysis allows the extraction of several deformation parameters, such as the cross-correlation coefficients, shape and intensity, as well as the out-of-plane displacement fields and mode shapes. The potential of the methodology is demonstrated on an Unmanned Aerial Vehicle wing made of composite material, followed by experimental validation with reference accelerometers. The results obtained with the projected three-dimensional digital image correlation show a percentage of error below 5% compared with the measures of accelerometers, achieving, therefore, high sensitivity to detect the dynamic modes in structures made of composite material.

Experimental Investigation: Vibration Measurement of a Rotating Blade with Digital Image Correlation and Blade Tip-Timing.

High cycle fatigue has been known as an important form of aeroengine blade failure. This study aims to achieve a method of investigation for a rotating blade vibration measurement, combining the two non-contact optical techniques of digital image correlation (DIC) and blade tip-timing (BTT). Dynamic parameters of a thin-blade were obtained on a stationary vibration platform with stereo-DIC system. Meanwhile, the finite element analysis (FEA) of this thin-blade was performed within different rotating speeds. Then, the set of thin-blades was mounted in a simulated compressor test rig equipped with BTT and a wireless strain gauge (SG) system. A rotor speed sweep experiment was carried out and the blade synchronous resonance parameters were extracted. Results show that the displacement mode shapes match well between DIC and FEA, and that MAC values of the first six order modes are over than 0.88. The predicting strain from the FE model and SG agreed to within 32.41% in the worst case, and the predicting strain from the DIC model corresponds to 28.53% in the worst case. This is an effective non-contact, high-precision full-field deformation measurement method that is worth exploring for structural design and dynamic strain assessment of vibrating components.

Influence of frost and local stiffness variations on the strain mode shapes of a steel arch bridge

Natural frequencies are probably the most widely used modal characteristics in vibration-based monitoring. However, they can be highly influenced by temperature and this influence can completely mask the effect of even severe damage. This translates into a necessity for time-consuming data-normalization techniques to remove the influence of temperature and identify damage. Displacement mode shapes of homogeneous material structures are less influenced by temperature, but obtaining them in a dense grid, which is required for damage localization, is cumbersome due to the large number of sensors needed. Strain mode shapes on the other hand can be insensitive to temperature variations, while obtaining them in a dense grid is possible when fiber-optic sensors such as fiber-Bragg gratings (FBG) are used. This work presents the results of the continuous monitoring of a steel railway bridge for a period of almost two years, where modal data were collected for a wide temperature range. The bridge is instrumented with eighty FBG strain sensors, multiplexed in four fibers. The natural frequencies and strain mode shapes of ten modes have been automatically identified from operational strain time histories, on an hourly basis. A clear influence of temperature on the natural frequency of most modes is identified, especially during frost periods. On the contrary, the strain mode shapes are mostly insensitive to temperature changes and only these of some higher-order modes are slightly and uniformly influenced when frost occurs. This behavior is confirmed also by a finite element model (FE) of the bridge. Furthermore, the FE model is used to investigate the influence of local stiffness changes on the modal characteristics. A clear and local change of the modal strain amplitude is observed at the location of the reduced stiffness, especially when information from all modes is combined in a sensitive damage index.

Performance evaluation of piezo sensors with respect to accelerometers for 3D modal analysis of structures

Strain modal analysis, as a new domain in the health monitoring field, needs to be studied in depth for experimental modal testing. Towards this purpose, this paper experimentally investigates the efficacy of piezo sensors for structural identification for structural health monitoring under different excitations applicable to large-scale structures. The piezo sensors are evaluated against industry-standard accelerometers by experimental modal testing of a scaled-down model of a pedestrian foot over bridge. The model is excited under the impact hammer, electro-dynamic shaker-based sweep and random excitations, and pedestrian motion (PM)-based low-amplitude excitations. Piezo sensors are found to be capable of capturing the modal parameters (modal frequencies, damping ratios and mode shape vector) under all the excitations with excellent correlation with respect to accelerometer-based parameters. However, some modes are missed under the shaker and PM-based excitations compared to the impact hammer-based excitations for both accelerometers and piezo sensors. Modal parameters of lower modes are successfully extracted under low-level pedestrian excitations, the most efficient type of excitation acting in operational conditions. High modal assurance criteria values between the strain and the displacement mode shapes establish the piezo sensors as effective for strain-based vibration testing and structural identification.

Periyodik yığılı kütleli kirişlerde eğilme dalgalarının yayılımı

In this study, dispersion analysis is carried out in the Euler-Bernoulli beam with periodic lumped mass, and periodicity effects are investigated. First, the dispersion relation is derived using the propagator matrix method for an infinitely long periodic beam with lumped mass. The banded frequency spectrum is given depending on the mass ratio. Then, in the case of a finite number of periodic lumped masses, the effect on wave propagation was investigated and the transmission function was obtained. Finally, the displacement mode shapes of the barrier consisting of these lumped masses were obtained for the pass and stop band frequency values. The results show that it is possible to use designs made with periodic lumped masses as wave barriers.

Radar Interferometric Experimental Reconstruction of Three-Dimensional Displacement Vectors and Mode Shapes for Masonry Constructions

Abstract Radar interferometry is an innovative measurement technique capable of remotely measuring vibrations in terms of displacement history. It apparently allows overcoming some major drawbacks of the plodding and time-consuming conventional accelerometric experimental setup for vibration testing. Indeed, radar interferometry does not require any access to the structure; moreover, its use is simpler and faster. Therefore, this technique appears very appealing for ambient vibration testing on structures, and in particular on architectural heritage masonry constructions. However, radar interferometry is still affected by some relevant limitations, that currently outweigh the advantages. One of the most important limitations is that displacements are measured only along the line-of-sight; therefore, it is not possible to determine the whole three-dimensional displacement vector for moving structural points and, consequently, to reconstruct three-dimensional mode shapes of structures, whose knowledge is crucial for characterizing the dynamic behavior of structures and for detecting eventual damages. In this paper, a theoretical and experimental approach for reconstructing both three-dimensional displacement vectors and mode shapes of masonry constructions is proposed. This approach is based on the simultaneous use of two synchronized radar interferometers, and on the application of a theoretical model assuming a specific kinematical constraint, which is generally plausible in the case of masonry constructions. The proposed approach has been validated through in-situ experimental tests on a masonry bell tower.

A numerical method to determine the displacement spectrum of micro-plates in viscous fluids

In this paper, we present a semi-numerical method for determining the dynamics of micro-resonators with finite width immersed in incompressible viscous fluids. The micro-resonator is modeled using Kirchhoff plate theory, and the hydrodynamic force acting on the plate is determined from a boundary integral formulation of the Stokes equations. The resulting equation of motion is solved with a continuous/discontinuous finite element method in which an interior penalty term imposes C1-continuity to the plate’s deflection. Numerical investigations show the method to be convergent with an exponent of the convergence rate equals 2. Examples demonstrate that the proposed method overcomes the limitations of existing semi-analytic methods, only applicable to beam geometries, considering arbitrary plate modes in the structure’s dynamics and their effects on the fluid flow. Different resonator geometries are investigated for which displacement spectrum, mode shapes, and quality factors are not determinable with existent semi-analytic methods. Moreover, we find excellent agreement between simulated and experimental data, which has not been achieved even with purely numerical methods. The present method will allow the understanding of high quality factors of wide micro-resonators in viscous fluids and facilitate new applications in liquid atomic force microscopy and gas sensing in ambient and low-pressure conditions.

An efficient stochastic-based coupled model for damage identification in plate structures

Mode shape-based method has shown its dominance in failure identification of beams. However, it is a challenge for fault detection in a plate structure. It requires combined information in two directions to determine the damage location. In this study, a damage index, namely mode shape derivative based damage identification (MSDBDI), is applied to localize damage in fixed-free plate structures. Two-dimensional (2D) displacement mode shapes and their derivatives are used to identify the MSDBDI index. It is a fact that this indicator is stuck in indicating the damage severity. Hence, a coupled model between an artificial neural network (ANN) and antlion optimizer (ALO), so-called ALOANN is used to overcome this drawback. In this method, ALO instead of a backpropagation algorithm is used to look for the best initial values of learnable parameters of ANN i.e. weights and biases through mean squared error (MSE). These obtained parameters are added to ANN for damage identification. The efficiency of the proposed approach is tested with two numerical studies of the plate structures with single and multiple damage scenarios. In the first application, damage scenarios in a plate-like structure are detected. In the second application, ALO first is used to build a FE model of a composite structure based on a vibration experiment. Then the slab of the updated model is assumed to be suffered several damage scenarios. In both applications, failures are localized by using damage index. Then, the proposed approach is used to quantify the corresponding extent by means of changes in frequencies and displacement mode shapes. Values of damage index are achieved from modal properties of one or three out of the first five modes of the two considered structures. A conventional ANN also is investigated for comparison. Results of damage identification indicate that the damage indicator coupled with ALOANN show better performance compared with using ANN alone even when a noise level is assigned to modal properties.

Deflection estimation of beam structures based on the measured strain mode shape

Deflection is of great significance for evaluating the safety performance of long-span bridges, while it is difficult or expensive to be directly measured in actual projects despite numerous displacement sensors have been developed. This paper proposes a displacement reconstruction framework to obtain the dynamic deflection of beam structures using strain data. The framework defines the judgment criteria of curvature symbols and standard side first, and then stochastic subspace identification algorithm is used to calculate the strain mode shapes involved in the vibration, which solves the unclear calculation process and the need for considerable measurement points of the traditional mode superposition method. Furthermore, the displacement mode shapes and corresponding modal coordinates are obtained, and then the dynamic deflections are derived. Numerical simulations of cantilever and simply supported beams illustrate the effectiveness of the proposed framework, and the results show that the error is only 0.69% with only four strain measurement points. The two corresponding model experiments have also been carried out, and the results demonstrate that the deflection estimation error is only 0.14% with five measurement points, which further proves that the developed displacement reconstruction framework can accurately reconstruct the dynamic deflection of the beam structures.

Combining dynamic XFEM with machine learning for detection of multiple flaws

AbstractA novel methodology for multiple flaw detection is presented in this study. It combines the dynamic extended finite element method (XFEM) with machine learning for the first time. The extreme learning machine (ELM) is chosen as a learning rule for modeling and prediction. The XFEM is employed to overcome the issues associated with the large quantity of input data required for ELM network training, whereas the ELM itself is used to bypass the time‐consuming repeated analyses ordinarily required for the detection of multiple flaws. A large amount of potential flaw data for a structure is quasi‐randomly generated by a Sobol sequence. For each effective flaw datum, the dynamic XFEM with circular/elliptical void enrichment is used to compute the structural dynamic characteristics (i.e., frequencies and displacement mode shapes), which is possible because re‐meshing is not required for each flaw. The available data generated from the results of XFEM analyses are used for ELM network training. According to the measured dynamic characteristics, the trained ELM is then utilized to predict the size and location of flaws. The results show that the proposed novel methodology can identify accurately the location and size of circular or elliptical structural flaws. The method also is more efficient than previously proposed approaches, as it can avoid time‐consuming iterative analyses, and it is robust against noise.

Spatiotemporal compressive sensing of full-field Lagrangian continuous displacement response from optical flow of edge: Identification of full-field dynamic modes

The present state of condition monitoring of civil infrastructure involves the application of a large number of contact-based vibration sensors at different locations of the structures. The traditional vibration sensors, such as accelerometers or strain gauges, require considerable effort in laying out the connection wires. Also, their cost of maintenance is quite high considering the harsh environment they are exposed to. In recent times, there is a growing urge in developing contactless, vision-based vibration sensors that potentially alleviate the previously mentioned drawbacks. In a previous paper, the authors proposed a framework of acquiring full-field spatiotemporal Lagrangian displacement response of a continuous vibrating edge from its video by computing the optical-flow of the edge using d’Alembertian of Gaussian filter. Such spatially dense measurements are required to compute full-field mode shapes. But, from the perspective of condition monitoring of large infrastructures, such spatially dense measurements record a large amount of high dimensional data over the whole period of acquisition time. It poses a considerable challenge in the form of an increase in storage and data transmission capacity. In this paper, such drawbacks are addressed by suggesting a computational strategy of spatiotemporal compressive sensing of the full-field Lagrangian displacement response from the video of the vibrating structure with unknown geometric properties and boundary conditions. The non-uniformly sampled data, simultaneously both in the spatial and temporal dimension, still retains the low order dynamics of the system from which the full-field and high dimensional displacement responses are reconstructed. Subsequently, the modal parameters and full-field mode shapes are estimated from the reconstructed full-field Lagrangian displacement response. The experimental validation of the proposed method satisfactorily reconstructs the dense full-field displacement response along with estimating full-field dynamic modes from the low-dimensional spatiotemporal sub-sampled measurement data for a three-story steel frame and an aluminum cantilever beam for different compression ratios. The applicability of the proposed method is demonstrated on a simulated wind turbine tower model with unknown geometry or boundary conditions. The full-field displacement response and mode shapes of the non-prismatic wind turbine tower with a concentrated mass at the top are successfully reconstructed from the noise-induced sub-sampled data.

Efficient flexibility identification method using structured target rank approximation and extended Prony's method

An accurate estimation of the structural flexibility is crucial in applications such as deflection prediction, damage detection, and finite element model updating. By introducing a unit impulsive response function (UIRF) matrix estimated from noisy input and output signals collected through impact testing, this paper proposes a new method for estimating the structural flexibility matrix from a noisy UIRF. The advantages lie in the way the noise and decoupling modes of the impulse response signal are handled in the time domain. The proposed method consists of three steps: (1) Constructing an enhanced unit impulse response function (EUIRF) from the UIRF by utilizing the orthogonality of the displacement mode shapes for mode decoupling, (2) Noise removal from the constructed EUIRF to obtain a filtered one by implementing the improved Cadzow's algorithm for the structured target rank approximation (STRA) of the Hankel matrix constructed from the EUIRF, and (3) Structural flexibility identification from the EUIRF using the extended Prony's method to identify the basic modal parameters and modal scaling factors for a single-mode impulse response function with one degree of freedom. To validate the applicability of the proposed method, numerical and laboratory case studies were performed. A noisy impact test was simulated for a three-span continuous beam bridge, and the identified modal parameters and structural flexibility matrix were found to be accurate. Moreover, an impact test was conducted on a simply supported beam in a laboratory, and the results showed that the deformation predicted from the identified flexibility matrix and the one measured from the static load test are in good agreement.