Curvature Mode Research Articles

Primordial black holes and local non-Gaussianity in canonical inflation

Primordial black holes (PBHs) cannot be produced abundantly enough to be the dark matter in canonical single-field inflation under slow roll. This conclusion is robust to local non-Gaussian correlations between long- and short-wavelength curvature modes, which we show have no effect in slow roll on local primordial black hole abundances. For the prototypical model which evades this no go, ultra-slow roll (USR), these squeezed non-Gaussian correlations have at most an order unity effect on the variance of PBH-producing curvature fluctuations for models that would otherwise fail to form sufficient PBHs. Moreover, the transition out of USR, which is necessary for a successful model, suppresses even this small enhancement unless it causes a large increase in the inflaton kinetic energy in a fraction of an e-fold, which we call a large and fast transition. Along the way we apply the in-in formalism, the delta N formalism, and gauge transformations to compute non-Gaussianities and illuminate different aspects of the physical origin of these results. Local non-Gaussianity in the squeezed limit does not weaken the Gaussian conclusion that PBHs as dark matter in canonical single-field inflation require a complicated and fine-tuned potential shape with an epoch where slow roll is transiently violated.

Curvature and topology dependency of the cosmological spectra

In this article we investigate dependency of the adiabatic and entropy spectral indices of the cosmological perturbations on the geometry and topology of the background universe. Our discussion includes the post-inflationary universe i.e. radiation-dust mixture era. For this purpose, we first extract an explicit equation describing evolution of the comoving curvature perturbation in the FLRW universe with arbitrary spatial sectional curvature. We may percept when $K\neq 0$, curvature scale would be as significant as the perturbations scales to recognize the behavior of the spectral indices. We also focus on the entropy perturbation in order to extract behavior of the isocurvature spectral index in terms of the curvature index and time. Our analysis shows that spectra of curvature and entropy perturbations in sub-horizon scales could be function of topology. Moreover, an accurate analysis makes clear that time-average of isocurvature index in case $K=0$ is about zero,so that imprint of entropy perturbation in time duration may be negligible. We also consider evolution of the cosmological indices for super-curvature modes in the case $K=-1$. In the most results dependency to curvature, initial conditions and scale modes are thoroughly vivid.

Operational modal analysis using lifted continuously scanning laser Doppler vibrometer measurements and its application to baseline-free structural damage identification

A continuously scanning laser Doppler vibrometer (CSLDV) system is capable of efficient and spatially dense vibration measurements by sweeping its laser spot along a scan path assigned on a structure. This paper proposes a new operational modal analysis (OMA) method based on a data processing method for CSLDV measurements of a structure, called the lifting method, under white-noise excitation and applies a baseline-free method to identify structural damage using estimated mode shapes from the OMA method. The lifting method enables transformation of raw CSLDV measurements into measurements at individual virtual measurement points, as if the latter were made by use of an ordinary scanning laser Doppler vibrometer in a step-wise manner. It is shown that a correlation function with nonnegative time delays between lifted CSLDV measurements at two virtual measurement points on a structure under white-noise excitation and its power spectrum contain modal parameters of the structure, that is, natural frequencies, modal damping ratios, and mode shapes. The modal parameters can be estimated by using a standard OMA algorithm. A major advantage of the proposed OMA method is that curvature mode shapes associated with mode shapes estimated by the method can reflect local anomaly caused by small-sized structural damage, while those estimated by other existing OMA methods that use CSLDV measurements cannot. Numerical and experimental investigations are conducted to study the OMA method and baseline-free structural damage identification method. In the experimental investigation, effects of the scan frequency of a CSLDV system on the two methods were studied. It is shown in both the numerical and experimental investigations that modal parameters can be accurately estimated by the OMA method and structural damage can be successfully identified in neighborhoods with consistently high values of curvature damage indices.

Damage Identification of Plate Structure Based on the Method of Modal Flexibility Curvature Difference

According to the higher order modes of the structure are difficult to extract and the lower order are easy to obtain in practice, it is put forward that only uses the parameters of lower or first-order modal to constitute the modal flexibility curvature difference (MFCD) as the index of damage identification, which based on the flexibility matrix of structure being sensitive to the structure modal characteristics. Numerical analysis is made on different damage conditions of a plate structure, and the result shows that the MFCD can not only accurately identify the single-damage and multi-damage position of plate structure, but also reflect the size of damage degree. It has a great significance to apply the index into the range of damage identification of actual structures.

Online Monitoring of Flexural Damage Index of a Cable‐Stayed Bridge

This work introduces a recent application of the online nondestructive damage assessment system into a cable‐stayed bridge. A set of ambient modal parameters are automatically extracted every 20 minutes using real‐time signal data collected from a total of 26 accelerometers attached on the deck plate of the bridge. Then, a set of modal flexibilities are reconstructed by the combination of the extracted modal parameters with the approximated modal mass of the girder. Next, the curvature of the modal flexibility is approximated by a central difference formula. Finally, the set of flexural damage index equations is constructed by comparing the modal curvature of the damaged state to that of the undamaged state. Solving the overdetermined flexural damage index equations, the desired damage index is finally quantified. The resulting index clearly indicates the location and severity of the potential structural damage on the girder. Based on the overall performance of the implemented health monitoring system, the bridge operator’s damage index control criteria are set to ±20% of the undamaged state.

Application of spline interpolation to speckle shearography measurements for damage identification

This paper aims to explore a new technique for structural damage identification using cubic spline interpolation. The method is based on the interpolation of modal rotations measured with speckle shearography. In order to locate the damaged areas, we make use of the analytical derivative of the cubic spline function to compute the modal curvature, which is known to be very sensitive to damage. A comprehensive parametric study of the spatial sampling interval is carried out to find its influence on noise filtering. Furthermore, the identification quality dependency on the mode shape and respective noise is also examined. The results obtained with the proposed method show the consistency of the localizations. Additionally, the challenging tasks of identifying small and multiple damage are tackled, yielding a good performance.

Effect of inclusion lagrange interpolation method in mode shape curvature based damage detection

Mode shape curvature based damage detection capable to detect damage in structure with high sensitivity. Sparse and high density mode shape displacement data obtained experimentally pose difficulties for mode shape curvature algorithm to quantify damage size accurately. The objective of this study is to compare damage detection sensitivity of different mode shape curvature algorithm with the inclusion of Lagrange interpolation to enhance the algorithm damage detection sensitivity for sparse and high density curvature mode shape displacement data. Finite element analysis (FEA) model with free-free boundary condition of an aluminum beam has been carried out to investigate the feasibility of the proposed method. Undamaged curvature mode shape data from the damaged structure was estimated using Gapped Smoothing Method (GSM) and Savitzky-Golay (SG) filters with two different grid points of 149 and 74. Structural Irregularity Index (SII) and Damage Estimate Reliability (DER) were used to evaluate the effectiveness of the proposed algorithm. Numerical results show inclusion of Lagrange interpolation in mode shape curvature algorithm with Savitzky-Golay filter has better performance on estimate damage size by 2.81% of DER value for less dense (74 grid points) compared to GSM. The present method shows the inclusion of Lagrange interpolation has increased the sensitivity of mode shape curvature algorithm to identify damage size in beam-type structures compared to the previous method.

A novel mode shape reconstruction method for damage diagnosis of cracked beam

Abstract This paper presents a novel mode shape reconstruction method for damage diagnosis of cracked beam. The proposed method uses output-only data of damaged structure to identify and reconstruct the mode shape of healthy structure. Following that, the curvature mode shapes (CMS) is employed as a damage sensitive feature to locate the damage. In order to identify the mode shape of a target structure with measurement noises, EMD with intermittency criteria is applied to separate dynamic responses into individual modal responses of all grid points on the target structure. A statistical analysis is proposed to evaluate mode shapes from obtained modal responses. Then the Fourier series fitting is employed to mathematically represent the mode shape of the healthy structure. After that, the curvature mode shapes (CMS) which has been investigated as a promising feature for damage identification are acquired by a central difference approximation. Numerical studies are conducted for methodology illustration and validation. Effects of broader damage, multiple damage and measurement noise level are investigated in detail. The accuracy and effectiveness of the proposed method are demonstrated and verified by comparison with conventional method. Finally, a two-span continuous steel I-girder bridge is used to demonstrate the proposed method for more practical applications.

Determining the dynamic amplification factor of multi-span continuous box girder bridges in highways using vehicle-bridge interaction analyses

The impact factors of multi-span continuous box girder bridges in highways are influenced by many factors, including road roughness, vehicle-bridge interactions (VBIs), and travelling velocity. Currently, the empirical formulas specified by bridge design codes are based on single factors (bridge length or fundamental frequency). These formulas yield inconsistent results that can differ widely. In this paper, the regularity of the dynamic amplification factors (DAFs) of continuous beam bridges is investigated by selecting 15 continuous beam bridges and conducting VBI analyses. The results indicate that the DAFs of the continuous beam bridges increase dramatically when resonance phenomena occur. The vehicle-bridge resonance is closely related to vehicle frequency, bridge frequency and the disturbance frequency caused by vehicle movement. The travelling velocity exerts an important influence on the resonance. In most cases, the DAF of a continuous beam bridge peaks at the velocity range of 40–60 km/h. Vehicle-bridge resonance may be caused by the first-order or second-order mode of a bridge. Second-order curvature modes exert an especially crucial influence on the DAF of the negative moment at the interior supports. The empirical formulas used in current bridge design codes fail to account for the influence of resonance and travelling velocity on the DAFs of bridges. The maximum DAF at the interior supports of the continuous beam bridges may exceed 1.7.

A robust locating multi-optima approach for damage identification of plate-like structures

This paper proposes a robust optimization approach for multiple damage identification of plate-like structures. Different from traditional particle swarm optimizations (PSOs), a combined PSO and niche technique (NPSO) is proposed to solve multimodal optimization problems, with the full consideration of subswarm creation, merging and absorbing mechanism. As a hypersensitive parameter to damage, the curvature mode shape is adopted to construct the objective function. Case studies are conducted to investigate the effectiveness and robustness of the algorithm on multi-damage identification. Simulation results show that the proposed algorithm exhibits robust search performance on identifying damage locations accurately with good convergence behavior. It is hoped that this study can provide guidance on robust damage detection, especially when the structure is subject to multiple damages and external disturbances.

Modal curvature-based damage localization in weakly damaged continuous beams

Abstract Modal curvatures have been claimed to contain local information on damage and to be less sensitive to environmental variables than natural frequencies. However, simply using the difference between modal curvatures in the undamaged and damaged states can result into localization errors, due to the complex pattern that this quantity presents when considering broad damages or higher order modes. In this paper, we consider weakly damaged continuous beam and we exploit a perturbative solution of the beam equation of motion to obtain an analytical expression of the modal curvature variations in terms of damage distribution. The solution is then used to introduce a filtering technique of the modal curvature variation and to set up the inverse problem of damage localization based on modal curvatures only. Using numerical examples and experimental tests, we show that modal curvatures can be used for precise damage localization, once properly filtered.

Damage and retrofitting monitoring in reinforced concrete structures along with long-term strength and fatigue monitoring using embedded Lead Zirconate Titanate patches

This article aims at developing a generic system for the damage and retrofitting monitoring along with long-term strength and first-stage fatigue monitoring of reinforced concrete structures using embedded Lead Zirconate Titanate sensors in the form of concrete vibration sensors. The concrete vibration sensor is a ready-to-use sensor, and its unique packaging renders it very compatible for embedment in reinforced concrete structures. In addition to cost-effectiveness, the concrete vibration sensors are also characterized by excellent structure-compatibility and durability. In this article, both finite element method and experimental investigations have been employed to establish the feasibility of using curvature (second-order derivative) and other higher order derivatives of displacement mode shapes for damage detection and retrofitting assessment. The experiments are conducted on a real-life-sized reinforced concrete beam. The concrete vibration sensors embedded on the outer faces of the reinforced concrete beam are coupled to obtain the curvature and higher order mode shapes of the beam in pristine, damaged and retrofitted conditions. It is found that the curvature mode shape–based response of concrete vibration sensors can successfully identify the location of damage both numerically and experimentally. However, the third-order mode shape is unable to correctly identify the location of damage. Before introducing damage in the beam, the effect of long-term dynamic loading from Day 6 to Day 108 after casting of the reinforced concrete beam is also monitored. Both the global monitoring technique (in which flexural rigidity of the beam is monitored) and the local electro-mechanical impedance technique (where the equivalent stiffness identified by concrete vibration sensors is monitored) successfully detected the decreasing fatigue strength of the reinforced concrete beam. Degradation of the strength of reinforced concrete beam results due to the development of micro-cracks in the concrete because of the continuous vibrations (9.3 million load cycles) experienced by it via shaker. This is the first-of-its-kind proof-of-concept application of equivalent stiffness concept for monitoring curing of a large-sized reinforced concrete structure. It is also the first study on first-stage fatigue monitoring carried out before the ‘retrofitting-stage’ of the structure. Complete experimental investigations after the ‘retrofitting-stage’ covering all three stages of fatigue have been covered by the authors in their related publication.

Vibration-based damage detection in beam structures with multiple cracks: modal curvature vs. modal flexibility methods

ABSTRACTThis study presents damage localisation in steel cantilever beams including multiple cracks based on changes in the modal curvature and flexibility. Natural frequencies and corresponding mode shapes of circular-hollow-sectional and box-sectional steel cantilever beams are obtained by analytical and experimental models under six damage scenarios. Based on these data, the modal curvature and modal flexibility methods are employed to localise the cracks within the beams. Comparisons are made to show effectiveness of the methods in determination of crack locations.

Structural Damage Recognition Based on Perturbations of Curvature Mode Shape and Frequency

A new method for structural damage identification is presented based on perturbations of curvature mode shape and frequency. Firstly, the structure’s mass and stiffness matrices are expressed as functions of its elements’ physical parameters, which reflect their damage states. According to differences in the curvature mode shape of the structure in undamaged and damaged states, possible damage equations and determined damage equations are established and used to solve for the elements’ damage parameters. Then, to ensure the accuracy of the recognition result, a reasonable solution is substituted into a damage-checking equation based on the change of frequency of the damaged structure. Numerical examples are used to show that, to identify the damage, only one order mode needs to be tested. When the degree of damage is low, first-order perturbation equations can be used to recognize the damage with sufficient accuracy. If the extent of the damage is high, second-order perturbation equations can be used to provide more accurate identification results.

Axial quasinormal modes of scalarized neutron stars with realistic equations of state

We compute the axial quasi-normal modes of static neutron stars in scalar tensor theory. In particular, we employ various realistic equations of state including nuclear, hyperonic and hybrid matter. We investigate the fundamental curvature mode and compare the results with those of General Relativity. We find that the frequency of the modes and the damping time are reduced for the scalarized neutron stars. In addition, we confirm and extend the universal relations for quasi-normal modes known in General Relativity to this wide range of realistic equations of state for scalarized neutron stars and confirm the universality of the scaled frequency and damping time in terms of the scaled moment of inertia as well as compactness for neutron stars with and without scalarization.

Modal frequency and shape curvature as a measure of implant fixation: A computer study on the acetabular cup

Modal parameters are often investigated in order to assess the initial fixation of an implant. Most of studies are focused on the natural frequencies and frequency response function. Usually the femoral stem is tested although the acetabular cup fixation is important as well. The results of implant stability assessment are inconsistent and seem to suggest that frequency as a stability indicator is not sufficiently sensitive. In this study the sensitivity of the modal properties to changes in the bone-implant interface was investigated with the help of the finite element method (FEM). A novel fixation index based on modal shape curvature was investigated as a potential measure of the implant fixation. Modal frequencies are sensitive to interface changes in some manner, but suffer from insensitivity to local changes at bone-implant interface. The sensitivity up to 44% of natural frequencies to stiffness change due insertion steps was observed. The tested damage indicators are able to detect localized small changes in peripheral stiffness (5% stiffness reduction) with 95% confidence under the noise up to 1%. The modal shapes and their curvatures have a great potential to be a robust fixation indicator.

Research on Damage Identification of a Beam Based on Curvature Mode

To study the bridge damage location and degree identification ability based on curvature mode, with a three-span continuous steel beam as the research object, compared with the damage location identification of curvature modal difference and displacement modal difference, analyzing the influence of damage location identification on the curvature modal difference index considering the two factors of the measuring point number reduce and the noise, identifying the damage degree by the method of relative variation of the curvature modal shapes. The results show that the location damage identification effect of curvature modal difference is more obvious than that of displacement modal difference, the identification effect has higher requirements for the density of the measuring points and has low anti-noise ability; when the damage degree is recognized, the larger the damage degree is, the smaller the identification result error is. The method of curvature calculation can be incorporated into other dynamic fingerprint methods, and then the ability of damage identification will be further improved.

Axial quasinormal modes of static neutron stars in the nonminimal derivative coupling sector of Horndeski gravity: Spectrum and universal relations for realistic equations of state

We study axial quasinormal modes of static neutron stars in the nonminimal derivative coupling sector of Horndeski theory. We focus on the fundamental curvature mode, which we analyse for ten different equations of state with different matter content. A comparison with the results obtained in pure General Relativity reveals that, apart from modifying the spectrum of the frequencies and the damping times of the stars, this theory modifies several universal relations between the modes and physical parameters of the stars, that are otherwise matter-independent.

A Study on Efficacy of Wavelet Transform for Damage Identification in Reinforced Concrete Buildings

Structural damage and its extent can be detected by vibration-based techniques to avoid failure or to minimize maintenance. Among different damage identification techniques, modal curvature approaches are widely researched and applied one. On the contrary, wavelet transformation (WT) methods are gaining popularity in damage identification of real life buildings because of their suitability for non-stationary signals and non-dependency on baseline data. This paper presents a novel approach utilizing complex continuous wavelet to effectively locate change in physical properties of reinforced concrete (RC) buildings by virtue of variation in frequency and mode shapes due to small change in mass and stiffness. In this paper, the effect of variation of mass and stiffness of a building on the modal parameters is established analytically using theequation of motion for a multi-degree freedom system under forced vibration condition. A 3-D finite element model was developed for predicting the modal frequencies and mode shapes of the scaled down six storey RC building and the effect of addition of mass on a particular level of structure on the modal parameters was studied. Further, acceleration time histories were recorded with variation in mass on 3rd story of building using wireless tri-axial accelerometers and the time histories were processed to arrive at Curvature Damage Factor and wavelet coefficients for identification of the additional load on the particular floor. Vibration responses from all floors of RC building in ambient and loaded conditions were analyzed for frequency response spectra (FRS). Mode shapes were drawn for unloaded case and loaded cases. It was observed that the modal frequency of building decreases with the increase in mass at floors. It is observed that CDF approach could detect the change in mass in both numerical and experimental results. However, CDF algorithm could not detect the addition of load in case 1, 2 and 3, i.e. when load was less than 25 kg, i.e. only 2.6% of floor mass (960 kg). The acquired data for the above stated load cases were analyzed using complex Gaussian ‘cgau5’ wavelet in MATLAB toolbox to determine the singularity in the signal in terms of wavelet coefficient modulus. It is observed that the WT approach is able to precisely locate the change in physical parameters of the RC model building. However, it is seen that additional load could not be detected in case where only 9 kg, i.e. 0.93% of the total floor mass, was placed on 3rd floor. From the research work, it is observed that CDF technique is inefficient in damage detection and always demand prior baseline information, which is usually difficult to obtain in practice. However, the wavelet transform-based approach more accurately detects the location of change without relying on intact state vibration data.

Detecting Changes in Fiber Orientation in a Simulated Chopped Fiber Plate Using Curvature Mode Shapes

The use of chopped fibers in the manufacturing of carbon fiber composite parts is becoming more popular in order to reduce production costs, especially in the automotive, wind, and gas storage industries. The orientation of the fibers in a chopped fiber part is important because the material properties of the part depend upon it. Phenomena such as shear alignment can result in undesired material properties, and therefore, a method for detecting the presence of undesired fiber orientations is needed. In this paper, a metric based on a part's curvature mode shapes is developed to identify the presence and location of fibers whose orientation is different from that of a desired alignment. A proof-of-concept experimental analysis shows the effectiveness of the metric at locating a region in a carbon fiber laminate plate that has been modified by rotating the fibers 90 deg. A finite element model is also developed to validate the experimental results and explore other modification scenarios. In each case, the metric is effective in identifying areas in which fiber alignment changed relative to a baseline model. In one case, a change as small as 3 deg was identified.