Unique Identification of Damage Mode and Damage Parameters in Structures using Daubechies Wavelets...

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Unique Identification of Damage Mode and Damage Parameters in Structures usingDaubechies Wavelets with Spectral Element Method

Summary: The current study aims at developing an analytical tool for damage detection toreduce maintenance costs. For this purpose unique identification of mode of failure and its

quantification in isotropic and layered media, including composite materials, is done usingDaubechies' compactly supported wavelets and spectral methods.

1. Introduction

Structural health monitoring can be defined as the acquisition, validation and analysis of technical data to facilitate life cycle management decisions [1]. The analysis techniques for dynamic response integrate: (a) Either of frequency analysis, modal analysis, finite elementanalysis (FEA), spectral analysis among other methods [2] and (b) Neural networks or optimization algorithms such as evolutionary algorithms(EA) or genetic algorithms(GA).

Extensive literature is available and several methods have been developed over time for damagedetection. Ryter, 1992 [3] defined four levels of damage identification: (a) Verification of

presence of damage, (b) Locating the geometric position of damage, (c) Quantification of severity of damage, (d) Estimation of remaining service life of structure.

The aim of this study is to computationally evaluate the differences in response of a specimen tospecific forcing conditions. The difference in response and the variation of response in time arethen studied to obtain geometric location of the damage along with its type, size and intensity.Although, such studies have been performed for different modes of damage/failure individually,we look to present a comprehensive work that uniquely determines the properties of mode of failure with efficiency.

2. Methodology

Daubechies compactly supported wavelets coupled with the spectral element method are chosenfor our formulation.

2.1 Formulation for an undamaged rod

Taking the elementary example of a one-dimensional rod with density and cross-sectional area

having a Young's modulus and a damping ratio . The response of the rod to an end

displacement , is given by the wave equation,

UE, A ,

Figure 1: Illustration of division of rod into sections. Shaded region marks the damage zone



(2.1)

| | (2.2)

where (2.1) gives the boundary conditions for this particular case considered. Following theDaubechies scaling function approximation for time dependent axial displacement(2.3)-(2.7),decoupling of governing equations using eigenvalue analysis(2.8)-(2.11), waveletimplementation and spectral element formulation as in [4], the required transient solution can beobtained. Non-periodic time boundary conditions [5] are used to resolve the non-periodic forcedend-displacement.

Following standard derivations [4], the equation can be simplified to,

{ } (, - , -)*+ (2.7)

Using eigenvalue analysis for the matrix , we can write the matrix as, , where is the eigenvector matrix of and is a diagonal matrix containing corresponding eigenvalues

of . This gives . Hence,

( ) (2.9)

Boundary conditions transform to,

or, (2.10)

The spatial part is approximated as the exact solution of (2.9)

(2.12)

where, ( ) (2.13)

2.2 Analysis for a Damaged rod

We use a simple model for introduction of a damage in the rod with a damage of intensity .We assume its effect on the local stiffness as . Assuming the damage zone

begins at and ends at , the size of the damage is said to be . We shall vary theintensity and location of this imperfection along the length of the rod and observe its effect onresponse of the rod. Using these responses, we will analyze on how to accomplish the goal of detection of damage location, type, intensity, etc.



The damaged part is considered as a separate material with a new stiffness modulus upto thedamage zone. We will be obtaining responses for two damage zones. Thus, as seen in fig. 2, therod is divided into three sections. In our case, we have .

Now, our from (2.12) is written in the p th domain as:

(2.14a)

Using boundary conditions and additional continuity constraints at the 'nodes', i.e. thedisplacement continuity and the displacement derivative continuity, we solve the system of equations.

| | | | (2.15a)

Similarly, this study will encompass the analysis of response behavior of an ideal beam, layered beam. In-house codes have been developed and results have been presented using MATLAB.

3. Preliminary results

3.1 Validation case

The response of an Aluminum rod (E=70 GPa, =2700 kg/m 3, cross sectional area=0.01*0.01m2, length=1 m) to a time-varying forced displacement at the tip is given in fig. 3 and theresponse in fig. 4. All results are produced for the undamped case, i.e. .

Figure 3: Transient forced rodtip-displacement

(a) (b)Figure 4: Axial displacement of undamaged rod at (a) 5 0 μs (b) 100 μs

Figure 2: Illustration of division of rod into sections. Shaded region marks the damage zone

Ue E, A , ρ E', ρ



(a) (b) (c)Figure 5: Axial displacement of the rod for damage at 0.6m from root, d=0.5,

L 2-L 1=4mm, at (a) 50 μs (b) 90.625 μs (c) 100 μs

3.2 Verification of presence of damage

The first level of damage identification as described by Ryter [3] is the verification of presenceof damage. The knowledge that there is a difference is the response of two different rods with thesame material and geometric properties to the same impulse under the exact same conditions can

be treated as conclusive proof that damage is present in one of the rods.

Fig.5 shows response of damaged rod with time and fig. 6 shows the difference in response of anundamaged and damaged rod with time thereby corroborating presence of damage. The damageis of intensity 50% and its size is 4mm in length, located at 0.6m from the root.

Figure 6: Difference in response of undamaged and damaged rod,for damage at 0.6m from root, d=0.5, L 2-L 1=4mm at (a) 50 μs , (b) 100 μs

3.3 Locating the geometric position of damage

When the impulse travels along the rod, it produces a change in response when it interacts withthe damaged zone. The interaction produces a pulse which travels in the opposite direction as can

be seen by looking at figs. 5a, 5b, 5c, in that order. Since this reflected pulse is observed as soonas the travelling impulse reaches the damage zone, we can easily locate the geometric position of



damage and are automatically registered analytically and successfully reach the second level of damage identification

3.4 Quantification of severity of damage

The intensity of the damage in the case of a rod can be determined by amplitude of the reflected pulse. As expected, the higher the difference in elastic modulus of the undamaged and damagedzones, the higher the amplitude. This is also verified as shown in fig. 7.

(a) (b)

Figure 7: Axial displacement of the rod at 100 μs for damage at 0.6m from root, L 2-L 1=4mm,(a) d=0.5, (b) d=0.7

The size of the damage is quantified by the size of the pulse in the axial direction. It is observedthat as damage size increases, the pulse begins to separate. Depending on the duration of theimpulse, we have a critical size above which both reflections from the two edges of the damagezone are clearly visible as shown in fig. 8.

(a) (b)

Figure 8: Axial displacement of the rod at 100 μs for damage at 0.6m from root, d=0.5,(a) L 2-L 1=4mm, (b) L 2-L 1=20mm



4. Scope of our study

This study entails the detection of damage parameters of a rod with multiple damages as well.The parameters for each damage are captured in a very similar manner as the case of multipledamages. Results for response of composites and layered media such as a bi- and tri-material

strips with and without damage have not been included in this abstract. The work is beingextended to encompass anisotropic.

5. Conclusions

The Spectral Element Method and Daubechies wavelets work well in tandem to capture theresponse of undamaged as well as the damaged rod. Differences in response can be used todetermine the location of damage as well as damage parameters such as type, intensity andextent.

References

[1] Kessler S.S., Spearing S. M., Damage Detection in built-up composite structures using Lambwave methods, submitted to Journal of Intelligent Material Systems and Structures, Dec 2001

[2] Doebling, S.W., Farrar, C.R., Prime, M.B., Shevitz, D.W., 1995, Report LA-13070-MS, LosAlamos, NM, Damage Identification and health monitoring of structural and mechanicalsystems from changes in their vibrational characteristics: A literature review

[3] Rytter, A., Vibration based inspection of civil engineering structures, Ph.D. Dissertation,Department of Building Technology and Structural Engineering, Aalborg University,Denmark

[4] Gopalakrishnan, S., Mitra, M., Wavelet methods for dynamical problems: With applicationto metallic, composite, and nano-composite structures, CRC Press, 2010

[5] Williams, J.R., Amaratunga, K., A discrete wavelet transform without edge effects usingwavelet extrapolation, Journal of Fourier Analysis and Applications Vol. 3, No. 4, 1997 435-449

Unique Identification of Damage Mode and Damage Parameters in Structures using Daubechies Wavelets...

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