Modal Flexibility Research Articles

Structural damage assessment with incomplete and noisy modal data using model reduction technique and LAPO algorithm

The present article proposes an efficient method for damage assessment of structures with incomplete and noisy modal data using model reduction method and Lightning Attachment Procedure Optimisation (LAPO) algorithm. The damage identification problem is formulated as an optimisation problem, in which the extent of damaged elements and a combination of both natural frequencies and modal flexibility changes are considered as the continuous design variables and a hybrid objective function, respectively. The LAPO algorithm, a recently developed algorithm, is for the first time extended to solve the formulated optimisation problem. In addition, due to incomplete modal data, an iterated improved reduction system (IIRS) technique is adopted to condense structural physical properties. The efficiency and robustness of the proposed method are investigated using three types of structures and the results are compared with those obtained by other well-known optimisation algorithms. It is shown that regardless of the effect of incomplete and noisy modal data, the proposed method not only successfully determines the location and extent of single and multiple damage cases in the structures, but also has lower computational cost than the other optimisers.

A Study on Simulation Based Validation of Optimized Design of High Precision Rotating Unit for Processing Machinery

With the rapid development of processing technology, the demand for precision, durability and NVH performance of processing machinery has also increased. This paper presents an optimally designed gear train composed of a set each of spiral bevel gears and spur gears to serve as a high precision rotating unit for processing machinery. The safety of each component used in the rotating unit gear train was verified using the geometry method and empirical formulae. The load contact pattern and PPTE value, which is an important indicator of the gear whine characteristic, were optimized using micro geometry modifications. In addition, the transfer efficiency of the entire gear train was determined and a novel dynamic analysis based on the comparison of the dynamic transmission error and the modal flexibility was performed to verify the correctness of the geometry design.

Reliability Assessment of Deflection Limit State of a Simply Supported Bridge using vibration data and Dynamic Bayesian Network Inference.

Structural health monitoring (SHM) has been widely used in all kinds of bridges. It is significant to accurately assess the serviceability and reliability of bridge subjected to severe conditions by SHM technique. Bridge deflection as an essential evaluation index can reflect structural condition perfectly. In this study, an approach for deflection calculation and reliability assessment of simply supported bridge is presented. Firstly, a bridge deflection calculation method is proposed based on modal flexibility and Kriging method improved by artificial bee colony algorithm. Secondly, a dynamic Bayesian network is employed to evaluate the deflection reliability combined with monitoring results which include modal frequency, mode shape, environmental temperature, and humidity. A linear regression model is established to analyze the relationship between modal parameters and environmental factors. Thirdly, a simply supported bridge is constructed and monitored to verify the effectiveness of the proposed method. The results reveal that the proposed method can precisely calculate the bridge deflection. Finally, the time-dependent reliabilities of two cases are computed and the effects of monitoring factors on bridge deflection reliability are analyzed by sensitivity parameter. It indicates that the reliability is negatively correlated with temperature and more sensitive to mode shape than other three factors.

Modal flexibility based damage detection for suspension bridge hangers: A numerical and experimental investigation

This paper addresses the problem of damage detection in suspension bridge hangers, with an emphasis on the modal flexibility method. It aims at evaluating the capability and the accuracy of the modal flexibility method to detect and locate single and multiple damages in suspension bridge hangers, with different level of severity and various locations. The study is conducted numerically and experimentally on a laboratory suspension bridge mock-up. First, the covariance-driven stochastic subspace identification is used to extract the modal parameters of the bridge from experimental data, using only output measurements data from ambient vibration. Then, the method is demonstrated for several damage scenarios and compared against other classical methods, such as: Coordinate Modal Assurance Criterion (COMAC), Enhanced Coordinate Modal Assurance Criterion (ECOMAC), Mode Shape Curvature (MSC) and Modal Strain Energy (MSE). The paper demonstrates the relative merits and shortcomings of these methods which play a significant role in the damage detection of suspension bridges.

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.

Effects of elastic supports and flexural cracking on low and high order modal properties of a reinforced concrete girder

A number of nondestructive evaluation (NDE) methods are available to detect cracks and damage in reinforced concrete structures. Methods such as ultrasonic, impact echo, or X-ray testing have high resolution but are time consuming to perform and therefore only used locally. Since the early 1970s, researchers have also studied the modal properties of structures for damage detection and identification. In this paper, we discuss the effects of elastic supports and flexural cracking on the dynamic response of a large-scale laboratory reinforced concrete girder. For this purpose, an instrumented hammer used for impulse response testing was employed to initiate vibrations and to subsequently estimate the natural frequencies and modes of a large-scale laboratory concrete girder. Natural frequencies up to the 13th mode were successfully extracted from the frequency response function (FRF). In addition, the mode shapes were extracted up to the 6th mode due to the limitation of the linearity of the FRF response. These modal properties were determined from the measured accelerations of two points on the girder using a roving hammer. The test procedure was performed on the uncracked beam and repeated after the beam had been cracked. The experimental measurements were verified with a finite element (FE) model consisting of one-dimensional beam elements that consider both flexural and shear deformations. Support flexibility was modeled as linear-elastic springs, and was found critical in order to explain experimentally-obtained measurements. The effects of cracking were found to be most obvious in the higher modes. Additionally, utilizing the modal flexibility method on the first mode (natural vibration frequency and mode shape) was an accurate predictor of flexural crack locations.

Damage identification in offshore jacket structures based on modal flexibility

A modal flexibility-based method using finite element model updating technique for damage detection purposes is presented. The motivation for choosing modal flexibility is explained and the objective function for an iterative optimization process is formulated. The simulated platform FE model is established with single and triple damage cases where noise contamination and model uncertainty are considered. Monte Carlo simulation is used for random parameter study. A substructure optimization algorithm is proposed together with a threshold to lower the number of design variables. When the flexibility-based method employs only the first order modal parameter and translational degree of freedoms on merely 4 nodes, satisfactory results can still be derived under high level of contaminated noise and random errors on elemental elasticity modulus and mass density. The influence of random parameters, spatial-incompleteness and tiny damage cases are discussed with several important conclusions summarized. The proposed substructure optimization algorithm and crucial problems during optimization process are described. The algorithm is applied to two numerical examples including a simply-supported beam and a jacket-platform, the study based on equivalent model reveals better accuracy and efficiency of the proposed algorithm, even with 3% measurement noise and model uncertainty.

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.

A Virtual Load Method for Damage Identification of Beam Structures

Background: Damage identification at the earliest possible stage is very important for beam structures to ensure safety, extend serviceability, and reduce maintenance costs according to recent patents. Methods: The goal of this work is to develop a virtual load method for beam damage identification by using only the vibration modes of the current structure. Central to the damage identification approach is the construction of the virtual load according to the support conditions of the structure. A virtual quasistatic deformation of the structure can be obtained by multiplying its modal flexibility matrix with the virtual force vector. And then the possible damage location in the structure can be determined by sudden increases in curvatures of the virtual quasi-static deformation. Subsequently the damage extent can be simply computed if necessary. Results: Three types of beams are used as examples to demonstrate the efficiency of the presented virtual load method in structural damage identification. For comparison, damage localization was also conducted by the existing uniform load surface (ULS) curvature method. The results showed that the proposed virtual load method is applicable to all types of beams but the ULS curvature method is only applicable to the simple supported beam. Conclusion: It was found that the proposed method can successfully determine damage location and extent only using the partial modes of the current structure. Keywords: Damage identification, virtual load, vibration mode, beam structures, curvature, flexibility matrix.

Multi-damage identification of large-scale truss structures using a two-step approach

In this paper, a two-step damage detection approach is presented to identify the location and severity of multiple damage cases in large-scale truss structures. In the first step, a damage locating vector (DLV) method based on EDS (exponential decreased stress) considering modal flexibility differences is employed to localize the damaged elements of a structure. Next, a crow search algorithm (CSA) is applied to estimate the severity of reported damaged elements using the results of the first step. Numerical examples including three large-scale trusses are considered to evaluate the efficiency of the proposed approach. In order to assess the effect of noise on the process of damage detection, a statistical study on each example is investigated, and selection of an appropriate threshold is studied in this article. The results indicate that the combination of the DLV method using EDS and CSA can reliably determine the location and severity of multiple structural damage cases even though there are a high number of elements and a low number of considered modes.

Output-only damage detection in buildings using proportional modal flexibility-based deflections in unknown mass scenarios

Methods based on the evaluation of structural deflections from experimental modal flexibility matrices are important tools in vibration-based damage detection, especially for building structures. Modal flexibility matrices, however, can not be directly estimated when dealing with output-only vibration data (e.g. in the important case of structures tested under ambient vibrations). One can use an a-priori estimate of the mass matrix of the structure or the added mass method to obtain the modal normalization constants needed to assemble the flexibility matrix, but these operations can be challenging for real-life civil structures. To address this problem, a modal flexibility-based approach for output-only damage detection and localization in building structures that can be applied with minimal or no a-priori information on the structural masses is proposed in this paper. This approach is also able to deal with the general case in which mass modifications, e.g. due to operational variability in structures, are present before and after damage. The approach is based on the estimation (directly from output-only data) of modal flexibility-based deflections of building structures that are proportional with respect to the corresponding true deflections. From analytical investigations it was found that the missing scaling factor between the two deflections can be made theoretically equal to the total mass of the structure, as proposed in the paper. Then, interstory drifts evaluated from the proportional deflections are used for damage detection and localization according to two proposed strategies. The first strategy can be applied with minimal a-priori information on the masses (i.e. using a parameter that quantifies an eventual relative modification of the total mass of the structure before and after damage). The second one is a more advantageous strategy that can be applied from output-only data without any a-priori information on the masses, even in the case in which the masses are varied before and after damage. The effectiveness of the proposed approach was demonstrated using both numerical simulations and experimental vibration tests on frame building structures.

Damage assessment in plate-like structures using a two-stage method based on modal strain energy change and Jaya algorithm

ABSTRACTThe paper presents an effective two-stage approach based on modal strain energy change and Jaya algorithm for damage assessment in plate-like structures. In the first stage, a newly developed damage indicator, named as normalized modal strain energy-based damage index (nMSEBI), is proposed to help locate potential damage elements more effectively. After eliminating most of the healthy elements, the actual damaged sites and its extent in plate structure are determined in the second stage by minimizing an objective function, which is solved using Jaya algorithm. For finding a suitable objective function used in optimization process, two different objective functions are considered to examine their effects on the performance of the utilized optimization algorithm. The efficiency and accuracy of the proposed two-stage damage detection method are investigated by two numerical examples comprising a concrete plate and a four-layer (0°/90°/90°/0°) laminated composite plate with multiple damage locations. All of the obtained results indicate that even under measurement noise, the proposed method can identify the actual damage sites and estimate the extent of damage with high precision. In addition, the numerical results also show that the computational cost of the optimization process using the objective function based on modal flexibility change is much lower than that using the objective function based on mode shapes change.

A generalized flexibility matrix-based model updating method for damage detection of plane truss and frame structures

Nowadays, many non-destructive damage detection methods for determining the location and severity of damage in the field of health monitoring are considered in order to reduce the cost of maintenance and improve safety and reliability of structure. In this paper, damage specification is obtained by sensitivity-based updating approach. By applying changes on sensitivity matrix and using measured flexibility data, it is concluded that the results of proposed method are more accurate and efficient than the old modal flexibility methods. The mass modeling error and measurement error of flexibility and natural frequency are calculated in order to ensure the accuracy and robustness of proposed method for 2-D finite element truss and frame model. Close index, measuring the performance of the method, and the coefficient of variation, which represents the distribution of response, are used. Compared with Wang method, the proposed method is capable of accurately localizing and quantifying damage in all scenarios.

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.

An efficient approach for optimal sensor placement and damage identification in laminated composite structures

This paper proposed an efficient approach for optimal sensor placement and damage identification in laminated composite structures. This approach first utilized a model reduction technique, namely iterated improved reduced system (IIRS) method, to develop a reduced order model for optimal sensor placement (OSP), and then the OSP strategy using Jaya algorithm is conducted by formulating and solving an optimization problem for finding the best sensor locations. The objective function of the optimization problem is defined based on the correlation between the flexibility matrix obtained from an original finite element model and the corresponding one calculated from IIRS method. Next, the approach uses the measured incomplete modal data from optimized sensor locations for detecting and assessing any stiffness reduction induced by damage. In order to do this, the damage identification problem is formulated as an optimization problem where the damage extent of elements and the modal flexibility change are taken as the continuous design variables and the objective function, respectively. The Jaya algorithm is again adopted to solve the optimization problem for determining the actual damage sites and extents. Numerical simulations of a three cross-ply (0°/90°/0°) beam and a four-layer (0°/90°/90°/0°) laminated composite plate are carried out to demonstrate the applicability and efficiency of the proposed approach.

Damage detection in asymmetric buildings using vibration-based techniques

In recent times, esthetic and functionality requirements have caused many buildings to be asymmetric. An asymmetric building can be defined as one in which there is either geometric, stiffness, or mass eccentricity. Such buildings exhibit complex vibrations as there is coupling between the lateral and torsional components of vibration and are referred to as torsionally coupled buildings. These buildings require 3-dimensional modeling and analysis. Despite recent research and successful applications of damage detection techniques in civil structures, assessing damage in asymmetric buildings remains a challenging task for structural engineers. There has been considerably less investigation on the methodologies for detecting and locating damage specific to torsionally coupled asymmetric buildings. This paper develops a multicriteria approach using vibration-based damage indices for detecting and locating damage in asymmetric building structures. These vibration indices are based on the modified versions of the modal flexibility and the modal strain energy methods. The proposed procedure is first validated through experimental testing of a laboratory scale asymmetric building model. Numerically simulated modal data of a larger scale asymmetric building obtained from finite element analysis of the intact and damaged asymmetric building models are then applied into the modified modal flexibility and modal strain energy algorithms for detecting and locating the damage. Results show that the proposed method is capable of detecting both single and multiple damages in the beams and columns of asymmetric building structures.

Dynamic Optimization Design of Machine Tool Based on Modal Flexibility and Energy Distribution

Dynamic Optimization Design of Machine Tool Based on Modal Flexibility and Energy Distribution

Structural Damage Detection with Different Objective Functions in Noisy Conditions Using an Evolutionary Algorithm

Dynamic properties such as natural frequencies and mode shapes are directly affected by damage in structures. In this paper, changes in natural frequencies and mode shapes were used as the input to various objective functions for damage detection. Objective functions related to natural frequencies, mode shapes, modal flexibility and modal strain energy have been used, and their performances have been analyzed in varying noise conditions. Three beams were analyzed: two of which were simulated beams with single and multiple damage scenarios and one was an experimental beam. In order to do this, SAP 2000 (v14, Computers and Structures Inc., Berkeley, CA, United States, 2009) is linked with MATLAB (r2015, The MathWorks, Inc., Natick, MA, United States, 2015). The genetic algorithm (GA), an evolutionary algorithm (EA), was used to update the damaged structure for damage detection. Due to the degradation of the performance of objective functions in varying noisy conditions, a modified objective function based on the concept of regularization has been proposed, which can be effectively used in combination with EA. All three beams were used to validate the proposed procedure. It has been found that the modified objective function gives better results even in noisy and actual experimental conditions.

A High-Sensitivity Fusion Index for Delamination Identification of Composite Laminate Structures

A significant obstacle that hinders the extensive application of large composite structures in the aerospace industry is the lack of valid identification methods for composite damage, where the delamination damage is the most common one. In this paper, a high-sensitivity fusion index (HSFI) is proposed for detecting, locating, and quantifying damages in aerospace polymer composite structures. The proposed fusion index is developed based on modern information fusion of three vibration-based indexes: modal strain, modal flexibility and static strain indexes. And its advantage is high sensitivity and strong noise immunity. By detecting the delamination damage in a composite laminate plate, it is verified that this fusion index is highly sensitive for delamination position and size in both cases of little and large damage in the presence of 2% level noise.