Structural Damage Detection Using Generalized Flexibility Matrix and Changes in Natural Frequencies

Damage can be identified using generalized flexibility matrix based methods, by using the first natural frequency and the corresponding mode shape. However, the first mode is not always appropriate to be used in damage detection. The contact interface of rod-fastened-rotor may be partially separated under bending moment which decreases the flexural stiffness of the rotor. The bending moment on the interface varies as rotating speed changes, so that the first- and second-modal parameters obtained are corresponding to different damage scenarios. In this paper, a structural damage detection method requiring single nonfirst mode is proposed. Firstly, the system is updated via restricting the first few mode shapes. The mass matrix, stiffness matrix, and modal parameters of the updated system are derived. Then, the generalized flexibility matrix of the updated system is obtained, and its changes and sensitivity to damage are derived. The changes and sensitivity are used to calculate the location and severity of damage. Finally, this method is tested through numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the second mode is available. The results indicate that the proposed method can effectively identify single, double, and multiple damage using single nonfirst mode.

In this paper, we discuss the dynamics of beams with multiple delaminations. The objective of the study is to use the dynamic response of the delaminated beams in health monitoring of composite structures. As a first step, we present experimental results on the vibrations of a cantilever beam with multiple delaminations. These experiments are conducted by exciting the beam with piezoelectric actuators. The response is detected by using a high-speed video camera to study the existence of delamination opening modes in a beam with multiple delaminations. Then, an analysis of natural frequencies and mode shapes, for a delaminated beam, is presented by using analytical techniques. The measured dynamic responses of the beam with multiple delaminations are also presented. The experimentally detected nonlinear response is discussed. Introduction Delaminations or interlayer cracks and impact damages are typical defects observed in composite structures. The presence of these defects changes the dynamic characteristics of the structure . Some of the specific changes in the observed dynamic characteristics include changes in natural frequencies, changes in the mode shapes and changes in damping ratios. These changes can be used to identify the existence, location and magnitude of a delamination or an impact damage, before they can grow to their critical sizes. Studies, to date, are concerned with the dynamics of beams and plates with a single delamination. In this paper, we specifically discuss the structural dynamics of a beam with multiple delaminations. This information can be used to develop health-monitoring techniques for composite beams with multiple delaminations. In an earlier paper, Luo and Hanagud studied the effects of defects, like saw-cuts, impact damage and delamination, on the structural dynamic characteristics of composite beams. They have shown that perturbations in natural frequencies and modes of a beam, due to defects, can be used for health monitoring of structures. In particular, it has been shown that an integral equation that considers perturbations in curvature modes and natural frequencies can be very effectively used to detect local defects. To distinguish delaminations from other types of monitored (identified) defects Luo and Hanagud have discussed nonlinear effects like superharmonic response in the dynamic response of a harmonically excited delaminated beam. They considered only a single delamination. In this paper we study beams with multiple delaminations. More specifically, we discuss dynamic response and the changes in natural frequencies, mode shapes due to multiple delaminations in a composite beam. One of the earliest models for vibration analysis of composite beams with delaminations was proposed by Ramkumar et al. This model simply used four Timoshenko beams connected at the delamination edges to model a composite beam with one throughwidth delamination. The predicted frequencies, based on this model, were consistently lower than the results of experimental measurements. Wang et al. improved the analytical solution by including coupling between flexural and axial vibrations of the delaminated sublaminates. Using an isotropic beam with splits and the classical beam model, they found that the calculated natural frequencies were closer to experimental results. With similar considerations, Nagesh Gummadi and Hanagud formulated a finite element solution for arbitrary composite beams. In the finite element models, they considered a classical composite beam model as well as the beam model with high order shear deformations. Mujumdar and Suryanarayan have proposed a model which imposed a constraint between the delaminated sublaminates to force them to have the same flexural displacement. This model was unable to * Ph.D. Student, School of Aerospace Engineering † Professor, School of Aerospace Engineering, AIAA Member Copyright © 1999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved

Global dynamic technique based on frequencies change is used to detect the damages/ cracks in structures. This technique is based on changes in natural frequencies. However, it is found that changes in frequencies are negligible even in major damage cases. Several researchers have used modes shape instead of frequencies to detect the damage/ crack in structure. It is true that mode shapes approach is better compared to frequencies based analysis for damage detection; however, it fetches excellent result, if real mode shape is used. Researchers are using theoretical/ numerical/ interpolated modes shape, which produce erroneous results. In this paper, a method is proposed to extract the real mode shape of structure experimentally using only one accelerometer and damage was detected successfully in the steel beam using this mode shape. An artificial damages were created in the steel beam in four stages and the mode shapes were determined after damage for each stage. The damage was also located and severity was predicted with the help of the extracted modes shape. It was found that location predicted by proposed method was almost correct. This technique is useful for sensitive structures.

Based on the Bernoulli–Euler theory, the vibration characteristics of a clamped-elastic pinned beam under the elastic constraint and the compressive axial loads are derived and verified by finite element method. The results of examples show that the natural frequency of the beam decreases with loads increase and the frequency increases with the increase of the elastic constraint stiffness. When the constraint stiffness increases from 104 to 108 N/m, the first-order natural frequency becomes 4.24 times, the second-order natural frequency becomes 2.19 times, and the third-order natural frequency becomes 1.57 times; when the constraint is weak, the loads change is mainly reflected in the first mode shape. When the constraint stiffness is 104 N/m, the first-order natural frequency decreases by 20%, the second-order modal natural frequency changes by 3%, and the third-order natural frequency changes by less than 1%. The range of the elastic constraints with significant changes in the natural frequencies of the higher-order modes is larger. When the first-order frequency is taken to 100EI/l3, the change tends to be flat, the second-order frequency is about five times that of the first-order, and the range of the third-order frequency is 15 times or more.

Structural damage identification using static test data and changes in frequencies

A dynamic mesh-refinement method (DMRM) of multiple cracks identification is presented, which is based on the relationship of natural frequency change ratios with crack parameters in a beam. The beam is first divided into a coare mesh and the crack is represented by a rotational spring, and expressions relating changes in natural frequencies and the damage index matrix are formulated via an influence matrix. The damage index matrix can be solved by the resulting system of linear algebraic equations. The elements corresponding to the damage indexes with positive values are predicted to contain the crack. Then, these damage elements are refined with the successive subdivision method. The influence matrix is reconstructed through integrating a new subdomain of the damage elements. The new damage index matrix for the refinement meh can focus the crack on the refinement element. In this way, elements are gradually refined until the desired element size is found. The final refinement result indicates the crack location. In each tiny damage element, the spring crack model can be used to quantify the size of the descrete crack. Numerical examples are proposed to demonstrate the effectiveness of the presented method.

A generalized flexibility matrix based approach for structural damage detection

Stiffness-based structural health monitoring methods are widely used for detecting the damage in a structure. These stiffness-based structural health monitoring methods uses change in natural frequencies and modeshapes for damage detection. These methods are based on identifying the change in stiffness of the healthy and damage structure to predict the damage in the structure. These stiffness-based methods are not efficient for detecting a small damage in a structure as there is a negligible change in natural frequencies and modeshapes due to a small damage in a structure, however the damping characteristics of the structure are highly sensitive to the damage in a structure. In this paper, new damping-based damage detection procedure has been proposed. In the proposed procedure, the changes in damping matrix of the structure has been used to detect the damage in the structure. The proposed procedure is able (or can) to detect both the location of the damage and the extend of the damage in the structure. The proposed procedure of damping-based damage detection is a 2-step procedure. In the first step, damping matrices of both the healthy and damage structure are identified and in the second step, the identified damping matrices are used for damage detection. Numerical and experimental case studies are presented to demonstrate the effectiveness of the proposed procedure. The results have shown that the proposed damping-based damage detection procedure can be used for detecting damage in a structure with confidence.

Detection of multiple cracks using frequency measurements

Securing the structural safety of blades has become crucial, owing to the increasing size and weight of blades resulting from the recent development of large wind turbines. Composites are primarily used for blade manufacturing because of their high specific strength and specific stiffness. However, in composite blades, joints may experience fractures from the loads generated during wind turbine operation, leading to deformation caused by changes in structural stiffness. In this study, 7132 debonding damage data, classified by damage type, position, and size, were selected to predict debonding damage based on natural frequency. The change in the natural frequency caused by debonding damage was acquired through finite element (FE) modeling and modal analysis. Synchronization between the FE analysis model and manufactured blades was achieved through modal testing and data analysis. Finally, the relationship between debonding damage and the change in natural frequency was examined using artificial neural network techniques.

An efficient method utilizing the multi-stage improved differential evolution algorithm (MSIDEA) as an optimization solver is presented here to detect the multiple-damage of structural systems. Natural frequency changes of a structure are considered as a criterion for damage occurrence. The structural damage detection problem is first transmuted into a standard optimization problem dealing with continuous variables, and then the MSIDEA is utilized to solve the optimization problem for finding the site and severity of structural damage. In order to assess the performance of the proposed method for damage identification, an experimental study and two numerical examples with considering measurement noise are considered. All the results demonstrate the effectiveness of the proposed method for accurately determining the site and severity of multiple-damage. Also, the performance of the MSIDEA for damage detection compared to the standard differential evolution algorithm (DEA) is confirmed by test examples.

It is very important to monitor the occurrence of damage of various structures. By early detection of damage and by taking the proper corrective measures the failure of damage could be avoided. When damage occurs in structure it results in change in stiffness properties of the structure which in turn result in change in modal properties like natural frequency, mode shape change, mode shape curvature and strain energy of a structure. These parameters are used as an input to train the artificial neural network(ANN). The efficiency of the ANN to predict the damage at single and multiple damage cases using the change in natural frequency and strain energy was evaluated in this study. The efficiency of individual modes and combination of frequency and strain energy to detect damage is also considered in this study. The modal parameters were obtained by carrying out modal analysis using ANSYS 2021 R1 software and the ANN was developed using MATLAB software. The results indicated that the strain energy accurately predict the damage location and varies linearly with severity of damage. It is better to use combination of frequency and strain energy as indicator to detect damages.

Two major challenges associated with a vibration-based damage detection method using changes in natural frequencies are addressed: accurate modeling of structures and the development of a robust inverse algorithm to detect damage, which are defined as the forward and inverse problems, respectively. To resolve the forward problem, new physics-based finite element modeling techniques are developed for fillets in thin-walled beams and for bolted joints, so that complex structures can be accurately modeled with a reasonable model size. To resolve the inverse problem, a logistic function transformation is introduced to convert the constrained optimization problem to an unconstrained one, and a robust iterative algorithm using the Levenberg-Marquardt method is developed to accurately detect the locations and extent of damage. The new methodology can ensure global convergence of the iterative algorithm in solving under-determined system equations and deal with damage detection problems with relatively large modeling error and measurement noise. It is applied to various engineering structures including lightning masts, a space frame structure and one of its components, and a pipeline. The exact locations and extent of damage can be detected in the numerical simulation, and the locations and extent of damage can be successfully detected in experimental damage detection.

In many elasto-mechanical systems the natural frequencies and shapes of the lower-range modes are not changed very much by the onset of damage. This insensitivity is the root of many problems with methods that seek to assess the operational integrity of such systems by using low-frequency vibration measurements. In this paper it is demonstrated that the beat frequencies, which are directly measurable, can be a significantly better indicator of damage than natural frequencies. The determination of small changes in natural frequencies requires very high resolution in digitally produced frequency response functions. Methods which can assign vibration nodes may be helpful in determining the location of damage sites.

Modeling and analysis of a cracked composite cantilever beam vibrating in coupled bending and torsion