Curvature Mode Research Articles

Damage detection in beam through change in measured frequency and undamaged curvature mode shape

ABSTRACTThis paper develops and presents a technique on the identification of the damage location and depth in a beam using a frequency contour method. Unlike the frequency contour methods presented in the literature, the one presented in this work does not require neither the solving of the governing differential equation analytically to obtain the contour lines, nor the calculation of the frequencies by varying all damage severities and locations in order to use an inverse problem to plot the contours. In other words, the detection method does not require the modelling of the damage. In this work, a contour line is plotted using only the value of the changes in the measured natural frequencies and the vectors of the curvature mode shapes of the intact structure which can be calculated numerically or can be derived from the measured mode shapes. In order to verify the practical applicability of the developed theoretical method, a numerical simulation and an experimental framework on cantilever beam have been designed and the results confirm the applicability of the developed method in the real applications.

Failure Analysis of Composite Mono Leaf Spring Using Modal Flexibility and Curvature Method

Post-failure analysis of composite structures by damage identification at pre-stage has gained considerable interest in recent years. In the paper, modal flexibility and modal curvature methods have been used for premature failure analysis of composite mono leaf spring through analytical and finite element approaches. Initially, finite element model of healthy and cracked spring is used to evaluate the eigenvalue and eigenvectors. Subsequently, change in flexibility and absolute curvature difference among both springs are evaluated after mass normalization. The presence, location and severity of crack in a spring are identified through differences in local flexibilities and change in absolute modal curvatures. The modal curvature method predicts location and severity of crack more precisely than the modal flexibility method.

The surface of implied firm’s asset volatility

Abstract This paper analyzes the surface of CDS implied firm’s asset volatility at the aggregate market level, using a sample of European investment-grade firms during the 2007–2014 period. The term structure of asset implied volatilities is backed-out from the term structure of CDS spreads, while the moneyness dimension is proxied by the ratio of the default barrier to asset value. We find both a downward sloping term structure and a negative skew. Principal component analysis on the entire volatility surface shows that the first four components interpreted as a level, a term structure, a skew and a moneyness-related curvature mode capture 86% of the daily variation in asset implied volatility. We also find that the term structure slope is related to market and funding illiquidity, investors’ risk aversion, informational frictions, demand/supply factors and momentum.

Damage Identification Based on Curvature Mode Shape using Cubic Polynomial Regression and Chebyshev Filters

Structure Health Monitoring (SHM) has been applied in various application such as aerospace, machinery and civil structures to maintain structure’s safety and integrity. Gapped smoothing method (GSM) is most popular non-destructive identification (NDI) method due to its simplicity and did not require baseline data for comparisons. However, GSM is less accurate to detect wide size of damage in structure and cause false detection. Objective of this study is to propose a method to detect damage in structure using curvature mode shape data estimated from damaged structure and did not require data from undamaged structure. Finite element analysis (FEA) on a free-free boundary condition steel beam was carried out to demonstrate the feasibility of the proposed method that estimate undamaged curvature mode shape data using cubic polynomial regression (CPR) and Chebyshev filters (CF) methods. The results shows proposed method that used Chebyshev filters has better accuracy damage detection on wide notch compared to GSM. Although application of an interpolation and Chebyshev filters showed results with a high potential for overcoming the issue of false detection due to different notch size, however the proposed method still need refinement to better detection of different damage cases.

Delamination monitoring in CFRP laminated plates under noisy conditions using complex-wavelet 2D curvature mode shapes

Delamination monitoring in carbon fiber reinforced polymer (CFRP) laminated plates is crucial to ensure the integrity and safety of the structures that accommodate the plates. To identify delaminations in CFRP laminated plates, the two-dimensional (2D) curvature mode shape method is a prevailing method that features instant and simultaneous determination of the presence and location of the delamination. However, this method has two noticeable deficiencies in characterizing incipient small-sized delaminations, namely lack of damage sensitivity and inadequate noise robustness. To this end, this study proposes a new dynamics feature of the complex-wavelet 2D curvature mode shape to discriminate small-sized delaminations. This feature is delicately formulated based on the integration of the 2D curvature mode shape with the complex wavelet. The complex-wavelet 2D curvature mode shape is superior to the 2D curvature mode shape by virtue of its stronger damage sensitivity and noise robustness. These merits can be attributed to the adjustable localization and the multi-scale properties of the second-order Gabor wavelet, respectively. Proof of concept of the complex-wavelet 2D curvature mode shape is numerically undertaken in a finite-element laminated CFRP plate with a small-sized delamination, with emphasis on sensitivity to damage and robustness against noise. The applicability of the feature is experimentally validated on a CFRP laminated plate with a small-sized delamination, whose mode shapes are acquired via the non-contact measurement using a scanning laser vibrometer. The numerical and experimental results show that the complex-wavelet 2D curvature mode shape can effectively designate the presence and location of the delaminations in CFRP laminated plates under noisy conditions.

Model- and Non-model-based Damage Detection Methods Using Vibration Data

Modal parameters of a structure, including natural frequencies, modal damping ratios and mode shapes, are directly related to its material properties, such as mass, damping and stiffness. Changes in material properties of astructure due to occurrence of damage can result in those in its modal properties. In this work, model- and non-modelbased structural damage detection methods by use of modal parameters are introduced. The model-based method uses changes in natural frequencies of a structure; it requires an accurate physics-based model of the structure and a minimum amount of vibration measurement. The nonmodel-based methods use measured mode shapes and curvature mode shapes, and they do not require use of structural models. Experimental damage detection on a space frame, a beam and a plate are carried out to validate the model- and non-model-based methods.

Free vibration of FGM sandwich doubly-curved shallow shell based on a new shear deformation theory with stretching effects

Free vibration of the functionally graded material (FGM) sandwich doubly-curved shallow shells under simply supported conditions is presented. Three common types of the shell, namely, the shell with homogeneous hardcore and FGM facesheet (type A), the shell with homogeneous softcore and FGM facesheet (type B) and the shell with FGM core and homogeneous facesheet (type C), are discussed. The dynamics model is derived according to a new shear deformation theory with stretching effects. The dimensionless natural frequency is obtained by Navier method. Numerical examples of the FGM sandwich doubly-curved shallow shells are provided for different gradient material parameters, core thicknesses, radii of curvature, aspect ratios, side-to-thickness ratios and vibration modes to demonstrate the accuracy of the present theory.

Development of a mechanical strain amplifying transducer with Bragg grating sensor for low-amplitude strain sensing

Vibration-based damage identification is a well-known method to support health monitoring of civil engineering structures. Damage in such structures can be identified by measuring changes of the natural frequencies, damping factors or modal displacements of the structure. However, this approach suffers from the low sensitivity of these natural frequencies and modal displacements to certain types of damage. Modal strains and curvatures can be more sensitive to local damage, but direct monitoring of these quantities with sufficient spatial resolution is not possible with current measurement techniques due to the very small strain levels (sub-microstrain) caused by ambient or operational excitation. To deal with this issue, we propose a novel mechanical transducer equipped with an optical fiber Bragg grating (FBG) sensor that enhances the sensitivity to strain with a factor larger than 30. The principle of operation of the transducer exploits a symmetric cantilever structure that enlarges the strain experienced by the FBG sensor compared to the strain applied to the transducer itself. We carried out dynamic and static tests to verify the ability of the strain-amplifying transducers to measure small-amplitude strain levels and to evidence the potential for carrying out FBG based modal strain measurements on concrete civil engineering structures.

Nondestructive identification of composite beams damage based on the curvature mode difference

The nondestructive identification method for composite beams damage was explored based on the curvature mode difference (CMD). By combining hammer experiment tests with finite element simulation, modal analysis were performed on intact and damaged composite beams with single, multiple or different degree of damage, respectively. The first three-order modal parameters were obtained by using dynamic signal test system and finite element software ANSYS. From the obtained mode shapes, the CMD were calculated by the central difference operation to identify the damage in composite beams. Experimental and simulation results revealed that damage in the composite beams would cause mutation of CMD. The presence, location and size of damage, as well as the multiple damage can be accurately detected by the mutation position. In addition, when the number of delamination at the same damage location was increased from 1 to 3, the CMD of damage unit obtained from the finite element simulation results were increased from 5.45 to 54.63, with the corresponding experimental results increased from 3.79 to 40.54, which were significantly higher than those from the undamaged units. It can be concluded that the CMD can be used to quantitatively determine the damage degree at the same damage location.

Non-model-based damage identification of plates using principal, mean and Gaussian curvature mode shapes

Mode shapes have been extensively used to identify structural damage. This paper presents a new non-model-based method that uses principal, mean and Gaussian curvature mode shapes (CMSs) to identify damage in plates; the method is applicable to mode shapes associated with low and high elastic modes on dense and coarse measurement grids and robust against measurement noise. A multi-scale discrete differential-geometry scheme is proposed to calculate principal, mean and Gaussian CMSs associated with a mode shape of a plate, which can alleviate adverse effects of measurement noise on calculating the CMSs. Principal, mean and Gaussian CMSs of a damaged plate and those of an undamaged one are used to yield four curvature damage indices (CDIs), including Maximum-CDIs, Minimum-CDIs, Mean-CDIs and Gaussian-CDIs. Damage can be identified near regions with consistently higher values of the CDIs. It is shown that a mode shape of an undamaged plate can be well approximated using a polynomial of a properly determined order that fits a mode shape of a damaged one, provided that the undamaged plate has a smooth geometry and is made of material that has no stiffness and mass discontinuities. Fitting and convergence indices are introduced to quantify the level of approximation of a mode shape from a polynomial fit to that of a damaged plate and to determine the proper order of the polynomial fit, respectively. A weight function is applied to the proposed CDIs to alleviate adverse effects of measurement noise on the CDIs and manifest existence of damage in the CDIs. A mode shape of an aluminum plate with damage in the form of a machined thickness reduction area was measured to experimentally investigate effectiveness of the proposed CDIs in damage identification; the damage on the plate was successfully identified. The experimental damage identification results were numerically verified by applying the proposed method to the mode shape associated with the same mode as that of the measured one from a finite element model of the damaged plate.

Numerical and experimental studies on vibration based methods for detection of damage in composite beams

In this paper a new damage index for detection and location of damage is introduced. It is improved version of the earlier developed Poincaré map method based on forced response of the structure. The proposed method is compared with the most popular modal based methods for damage detection: the modal displacements, modal curvatures and the strain energy methods. A finite element model of intact and damaged composite beams is used in order to validate correctness and efficiency of the method. Moreover, the modal based and forced response are tested experimentally by using a scanning laser vibrometer. The advantages of the offered method and its application in SHM are demonstrated by numerical simulations and experimental tests.

Two-dimensional Chebyshev pseudo spectral modal curvature and its application in damage detection for composite plates

To overcome the unstable problem in the classical modal curvature method, the two-dimensional Fourier spectral modal curvature method has been proposed in the former work of authors. The existences of the wrap-around effect and the Gibbs phenomenon in the Fourier spectrum-based method, however, necessitate the use of periodic extensions, which affects the efficiency of the modal curvature algorithm. To address the above problems, the two-dimensional Chebyshev pseudo spectral modal curvature is proposed in this work. The presented method inherits the spatial filtering capability and the spectral accuracy of the Fourier spectral modal curvature, and features an efficient modal curvature algorithm in absence of the periodic extension. Numerical and experimental validations are performed on composite structures. Validations show that the proposed method is accurate and effective in the damage detection for the two-dimensional composite structures.

Damage detection in beam-like composite structures via Chebyshev pseudo spectral modal curvature

This paper presents the Chebyshev pseudo spectral modal curvature formulation for damage detection in beam-like composite structures. The proposed method is the extension of authors' former work which is based on the Fourier spectral method. The Chebyshev pseudo spectral modal curvature overcomes the severe shortcomings of Fourier transform based spectral formulation which has serious problems associated with the wrap-around effect while analyzing the finite length signal. Moreover, the proposed method inherits the spatial filtering capability and spectral accuracy of the Fourier spectrum-based modal curvature, which ensures the robustness of the modal curvature calculation in noisy conditions. The validations of the presented method and comparisons with the Fourier spectrum based method are numerically and experimentally demonstrated for composite structures.

Performance evaluation of damage detection algorithms for identification of debond in stiffened metallic plates using a scanning laser vibrometer

While several studies have focused on the detection and localization of delamination in composite plates, few comprehensive studies have been performed for the identification of debond in stiffened metallic plates using vibration-based approaches. Therefore, this study is motivated by the need to evaluate the qualitative performance of existing damage detection algorithms, namely modal curvature, the gapped smoothing method (GSM), the generalized fractal dimension (GFD) and the wavelet transform coefficient (WTC), in detecting debond in stiffened metallic plates. Extensive experimental investigation is performed using laser Doppler vibrometer as a noncontact sensing device and LDS Permanent Magnetic Shaker as an actuator. The obtained results show high susceptibility to noise and lesser accuracy in locating the debond zone, except the WTC and GFD. However, the WTC fails to provide good results for higher debond lengths, and the GFD shows prominent false alarms at the free edges of the plates. To circumvent these difficulties, two different modifications of the fractal dimension algorithm, namely the modified GFD (MGFD) and the GFD with GSM (GFD-GSM), have been proposed. Extensive numerical simulations are further carried out using commercially available finite element package ANSYS 14.0 in order to examine the experimental findings. In contrast to most previous work, the signal-to-noise ratio (SNR) in the experimental data has been appropriately quantified and noise of the same SNR level has been synthetically generated and applied on the modal data obtained from numerical simulations. Comprehensive studies for different debond locations and lengths suggests a similar trend as that obtained from the experimental investigations. Finally, a study on damage severity has been performed using the WTC and proposed modifications of the GFD. It is found that the proposed modifications of the fractal dimension perform outstandingly well in all circumstances, and can be used as an excellent tool for debond localization and quantification.

Effective filtering of modal curvatures for damage identification in beams

In this work, we investigate the effectiveness of a damage identification technique recently proposed in [1] and assess how it is affected by the number and position of the sensors used. Mode shapes and curvatures have been claimed to contain local information on damage and to be less sensitive to environmental variables than natural frequencies. It is known that notch-type damage produces a localized and sharp change in the curvature that unfortunately could be difficult to detect experimentally without the use of an adequate number of sensors. However, we have recently shown that even a coarse description of the modal curvature can still be employed to identify the damage, provided that it is used in combination with other modal quantities. Here, by exploiting the perturbative solution of the Euler-Bernoulli equation, we consider the inverse problem of damage localisation based on modal curvatures only and we ascertain the feasibility of their sole use for recostructing the damage shape. To do so, we set up a filtering procedure acting on modal curvatures which are expressed in a discrete form enabling further investigation on the effect of using a reduced number of measurement points. The sensitivity of the procedure to damage extension is further assessed.

Damage detection methodology on beam-like structures based on combined modal Wavelet Transform strategy

Different approaches on the detection of damages based on dynamic measurement of structures have appeared in the last decades. They were based, amongst others, on changes in natural frequencies, modal curvatures, strain energy or flexibility. Wavelet analysis has also been used to detect the abnormalities on modal shapes induced by damages. However the majority of previous work was made with non-corrupted by noise signals. Moreover, the damage influence for each mode shape was studied separately. This paper proposes a new methodology based on combined modal wavelet transform strategy to cope with noisy signals, while at the same time, able to extract the relevant information from each mode shape. The proposed methodology will be then compared with the most frequently used and wide-studied methods from the bibliography. To evaluate the performance of each method, their capacity to detect and localize damage will be analyzed in different cases. The comparison will be done by simulating the oscillations of a cantilever steel beam with and without defect as a numerical case. The proposed methodology proved to outperform classical methods in terms of noisy signals.

Numerical study on damage identification using shearography with different shearing amounts

Shearography is a technique appropriate for the analysis of damage, namely by allowing the direct measurement of the gradient or derivative of the displacement field, i.e. the rotation field. This technique is based on the subtraction of the phase of laser light coming from the reflection at two neighboring points in the surface. The distance between these two points define what is called shearing amount. The measurement sensitivity can be adjusted by varying the shearing amount, such that we obtain large sensitivities with high values of this parameter. However, high values of shearing amount lead to the smoothing of the results. In this paper, a study on damage identification in beams using shearography with different shearing amounts is reported. Damage indicators based on differences in modal curvatures are applied to data coming from simulation of shearography. These damage indicators show a great dependency on the shearing amount, such that, for instance, with large values of this parameter the damage is spread across a larger area than the actual one. The peak values of the damage indicators present an attenuation and the damaged area also increases with the shearing amount. According to the findings presented in this paper, it is advisable to take precautions when localizing damage with measurements obtained with shearography.

Modal strain identification using sub-microstrain FBG data from a pre-stressed concrete beam during progressive damage testing

Vibration-based damage identification can constitute a successful approach for Structural Health Monitoring (SHM) of civil structures. It is a non-destructive condition assessment method, dependent on the identification of changes in the modal characteristics of a structure that are related to damage. However, the damage identification from the modal characteristics of existing structures currently suffers from a low sensitivity of eigenfrequencies and mode shapes to certain types of damage. Furthermore, the sensitivity of eigenfrequencies to environmental influences may be sufficiently high to completely mask the effect even of severe damage. Modal strains and curvatures are more sensitive to local damage, but the direct monitoring of these quantities is challenging when the strain level is very low. In the present work, the identification of the modal strains of a pre-stressed concrete beam, subjected to a progressive damage test, is performed. Dynamic measurements are conducted on the beam at the beginning of each cycle and its response is recorded with multiplexed Fiber-optic Bragg Grating (FBG) strain sensors. Bending, lateral and torsional modes are accurately identified from dynamic strains with a typical measured root mean square (RMS) strain value of about 0.35 micro-strains. The evolution of the modal characteristics of the beam after each loading cycle is investigated. Changes of the eigenfrequency values, the amplitude and the curvature of the strain mode shapes are observed. The changes in the strain mode shapes appear at the locations where the damage is induced, and are already identified from an early damaged state.

A method of curvature mode difference and wavelet transform to damage detection on space grid structure

A method combined curvature mode difference and wavelet transformation was used to determine the damage location of space gird structure. This method considered the curvature mode difference of pre and post damage as the wavelet transform analysis signal. Moreover, these signals were analyzed by wavelet transform using wavelet function db3. As the wavelet transform coefficient occur sudden change, the damage locations of links will be obtained. In this paper, square pyramid space grid structure, orthogonal space truss structure, three way latticed grid and honeycomb triangular pyramid grid structure will be analyzed using this method to identify single damage and multi-damage location. The solutions from numerical simulations show that this method can take effective identification of single damage location and multi-damage location of square pyramid space grid structure by the first mode condition. And this method has versatility to detect damages from different space grid structures.

The Behavior Semi-precast Slab Under Dynamic Load

Driven by increase in trend of construction two storey house in the community, more bamboo and wood are needed to construct formwork in the traditional concreting method. The present of waste from the discarded wooden formwork make the construction method is an inefficient and environmentally unfriendly. A semi-precast method offers a more economical method of construction owing to its less use of bamboo/wood in slab concreting process. This study is intended to characterize the dynamic behavior of semi-precast slab. There were 5 full scale slab specimens cast for this research. One specimen was monolith slab that acts as a control specimen. While the other 4 specimens were made with various shear connector arrangements. On each slab specimens, the dynamic testing was firstly proceeded. The dynamic testing was conducted to determine the dynamic characteristics, displacement mode, mode curvature and absolute difference of curvature mode of slab. The static test was carried out when the dynamic was completed to determine the ultimate capacity as well as collapse pattern of slab being tested. The results showed that there were differences in natural frequency for each slabs, either monolithic or semi-precast slabs. Monolith slab has natural frequency of 14.84Hz, while semi-precast slab without shear connector has natural frequency of 13.45Hz. Semi-precast slab with shear connector spacing of L/4(100) - L/2(150) - L/4(100), L/3(100) - L/3(150) - L/3(100) and 100mm have natural frequency of 13.62Hz, 13.46Hz and 15.97Hz. Displacement mode of monolith slab is smaller than that of semi-precast slab. These test results show that semi-precast slab has a slightly lower stiffness than monolith slab, but it has the same strength characteristics. An indication of similarity was also found between the characteristic of absolute difference of curvature mode and the crack pattern occur in slab that suggest the potential use of absolute difference of curvature mode to predict crack pattern in concrete structural members. In general, the results showed that semi-precast slab has similar structural characteristics to monolithic plates.