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Damage Detection in Structural Beams Using Model Strain Energy Method and Wavelet Transform Approach
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 5 (2018) 19565–19575
www.materialstoday.com/proceedings
ICMPC_2018
Damage Detection In Structural Beams Using Model Strain Energy Method And Wavelet Transform Approach Lanka Ramesh*a, Putti Srinivasa Raob a
Mechanical Engineering Department, Gudlavalleru Engineering College, Sheshadri Rao Knowledge Village, Gudlavalleru- 521356 Krishna District, Andhra Pradesh ,INDIA b Mechanical Engineering Department, Andhra University, Visakha Patnam-530003, Andhra Pradesh, INDIA
Abstract Damage identification method of surface cracks in beams using strain energy method and its spatial wavelet transform is investigated in this paper. Experimental modal analysis (EMA) is conducted on a aluminium cantilever beam to obtain the mode shapes before and after damage under a fixed-free boundary condition by Laser Doppler Vibrometer (LDV). A small perturbation is created by using Electro Discharge Machine (EDM) and simulated crack by separation of nodes at the crack location in beams. The purpose of the study is to present damage index method and spatial wavelet transform approaches to identify the location and quantification of surface crack in beam structure. A finite element analysis (FEA) is also performed to access this approaches and illustrate a attainable process for the experimental work. First, the displacement mode shapes obtained from the intact and damaged beams which are used to obtain strain energy ratio of the beam before and after damage and then these spatially distributed signals are analyzed by wavelet transformation. Good correlation between FEA and EMA results is obtained. Both methods damage index (DI) and wavelet transform (WT) successfully identify the location of surface crack in the aluminium beam. it is also demonstrated that the location and quantification of single and multi cracks can be detected by this methods. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization. Keywords: damage detection, Experimental modal analysis,Strain energy, wavelet transform,.
1. Introduction Vibration based Structural Health Monitoring (SHM) became one of the fundamental approaches in structural damage detection from past 30 years. It is based totally on the fact that methods allowing to existence of damage detection and its location at earlier stage become an intensive investigation thought the mechanical. civil and aerospace engineering domain from the last two decades. The existence of damage in structure introduces local flexibility that would have an effect on dynamic characteristics of whole structure. * Corresponding author. E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.
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Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
It causes in reduction of local flexibility that would have an effect on dynamic characteristics of whole structure. It causes in reduction of natural frequencies and change in displacement mode shapes of structure member. This property leads to notice existence of crack and its location and depth in the structural members. An extensive literature review [1] of the state of the artwork of damage detection and health monitoring from model parameters has recently been published. From this review it is clear that there are a lot work has been done in the area of damage detecting from modal parameters but, unfortunately, many of these methods require a modal parameters before and after damage. The method proposed in this paper avoids modal parameter requires before the damage . Pandey et al. [2] suggested that absolute changes in curvature of the deflected shape of beam can be a desirable indicator of damage compared to change in the frequencies. Cawley and Adams [3] simply extended same technique to detect the damage in two dimensional composite structures based the frequency shifts for different modes. Scott W. Doebling [3] developed vibration-based damage detection technique for damage identification and , even to predict location and level of damage in structures based the information of frequency shifts, changes in mode shape, changes in curvature mode shape, and modal strain energy change. Shi et al. [4-5] have considered the change of modal strain energy ratio to define a damage index which effective indicator to identification of damage location and quantify the damage level. The quality of this indicator is improved by summing the contributions of multiple modes in a normalized fashion. Cornwell et al. [7,8] derived damage detection algorithm for an index of damages in plate-like structure characterized by two dimensional curvatures by strain Energy approach. In their approach, the fractional strain energy of the plate before and after damaged was used to define a damage index which can successfully locate the area with stiffness reduction as low as 10% using relatively few modes. Liew and Wang [9] proposed an application of spatial wavelet theory to displacement mode shape for damage identification of beamtype structures has been extensively investigated. They calculated the wavelet coefficients along the length of the beam primarily based on the numerical solution for the deflection of the beam. the damage location and level was then indicated with aid of a height wavelet transform modulus in the coefficients of the wavelets along the length of the beam. Q. Douka et al. , Loutridis et al. [10] performed essessement of damage identification based wavelet transform approaches on mode shapes obtained by standard experimental modal analysis using accelerometers and a modal hammer.This work is provide new technique to apply strain energy method and spatial wavelet transform to highlight the sensitivity for detection and quantification and localization of damage in a beam structure, for that damage index method and wavelet-based damage identification approach was investigated experimentally by using non contact LDV and numerically on aluminium cantilever beam with damage in the form of surface crack of depth 30%, 20%,and 10% of the beam height .Also presented multi crack analysis using Strain Energy method on the few mode shapes as shown in below Fig 1& 2. 2. Theory of Strain Energy of a beam and damage index For intact beam : The strain energy of whole beam is given by =
(1)
For a particular mode shape, ψ(x) displacement at each node for a particular mode shape = 1
(2) j
2
Nd
....
aj
a
j 1
Figure 2. Sketch showing a beam's N d sub-divisions. . If N d is the number of divisions of the beam, then energy associated with each sub-region j due to the kth mode is given by
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
(
=
19567
)
(3)
The fractional energy is therefore = For damaged beam : Similarly, same quantities can be defined , where ()* indicates quantity estimated for damaged beam. Energy associated with particular mode shape kth . ∗
=
1 2
(4)
Ψ∗
∗
Enrgy associated with jth sub-regional strain energy of the kth mode shape for damaged beam. ∗
=
(
)∗
∗
(5)
The fractional energies of the beam are defined as ∗
=
∗
(6)
∗
Considering all the measured modes, m, in the calculation, the damage index in sub-region j is defined to be ∗
= ∑
/∑
(10)
A normalized damage index is obtained using =( where
−
)/
(11)
k and k represent the mean and standard deviation of the damage indices, respectively. Equation (6) is
used to predict the damage location in beam structures. 3. Continuous Wavelet Transform in Damage Detection This section shortly recapitulates the theoretical background of the continuous wavelet transform and its application to detection of singularities. For a given one-dimensional signal the continuous wavelet transform can be defined as (e.g.Mallat, 1998) ( , )=
1
( )
∗
−
√ Where ( ) is continuous wavelet function, x is the time or spatial variable, ∗ ( ) is the complex conjugate of wavelet function. Wavelets transform is the derivatives of a continuous function ∅( )usually called the scaling function i.e. ( ) = The wavelet transform can be defined as: ( , )=
1
( )∅∗
−
=
1
( )∅∗
−
√ √ ψ u,s(x) and the real numbers s and u denote the scale and the translation parameter, respectively. Therefore the wavelet with n vanishing moments can be rewritten as the nth order derivative of the smoothing function f(x), and the resulting wavelet transform can be expressed as a multiscale differential operator (Mallat,1998)
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Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
( , )=
( ( )∗
( )) ( )
( )=
1
√
−
(8)
Where f(x)* s(x) denotes convolution of functions. Equation (7) reveals that the wavelet transform is the nth derivative of the signal f(x) smoothed by the function s(x) at the scale s. Singularities in a signal f(x) can be detected by finding the abscissa where the maxima of the wavelet transform modulus (WTM) |Wf(u, s)| converge at fine scales (Mallat, 1998). 4. Numerical analysis of a cantilever beam In this steel cantilever beam with dimensions t = 0.06m, h = 0.7m, w = 0.06m is considered as the testing model with Fixed-Free support conditions as shown in Fig.3. Material parameters determined experimentally are: the modulus of elasticity E = 69GPa and the mass density = 2735 kg/m3 and Poisson ratio = 0.33 respectively. The element type selected for modelling beam is solid182, 4-node element having three degrees of freedom at each node. A surface crack has been introduced in beam with 40mm long, 0.5 wide and 1mm by separating the nodes at the distance x d measured from the crack centre to the fixed end as shown in Fig.03. The beam was divided into 400 elements of length mm. H. W.Hu [15] found that the resolution damage indices at surface crack has been significantly increased when damage indices are obtained from FEA by using the finer meshes.The normal mode shapes of the beam were computed by the finite element method using the classical Euler-Bernoulli beam theory.
Fig. 3. Surface crack in FE In this study, spatial mode shapes of the beam are investigated at three different depth of the crack are 10%, 20% and 30% of the beam height. Numerical simulations were performed on the first eight mode shapes for un-damaged and damaged beam, first eight modes crack free beam present in Fig.6.Then, the wavelet analysis is observed by fed the spatial signals to the wavelet transform here damaged modes are treated as input signals for wavelet analysis (spatial domain). 5. Experimental Investigation In this project an experimental analysis is employed on aluminium beam has been performed by using the Laser Doppler Vibrometer to validate the numerical model as well as damage detection methods. Here cantilever beam 800 mm length and 25 x 10 mm2 cross-section was clamped at a vibrating table. The beams surface has been cleaned and prepared for straightness before the experimental study. Subsequently, single transverse cracks is deemed on the beam at 160 mm from fixed end of the beam by using Electro discharge Machine as shown in Fig. 5., the depth of cracks are varied 10%, 20% and 30% of height of the beam. The beam is uniformly divided into 26x3 elements by the grid lines as shown in Fig. 4. The beam is subjected to a dynamic pulse load applied at each grid point using modally tuned hammer (PCB 652B10). The entire beam was then scanned using 3-D Polytec PDV-100 scanning system with the maximum measurable frequency bandwidth of 1 MHz. The response measurements are made using pulse generator and amplified to the maximum 100 V to record the response of the structure. Modal analysis was performed to obtain the Frequency Response Function (FRF) and displacement modes. VIBE, software is used to extract the mode shapes and natural frequencies of beam from the FRFs.
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
Damage
(75cm
0 05
Fig. 4.Test plate with 26 × 3 grid
Fig. 5. Crack embedded in beam by using EDM
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Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig.6. Numerical Mode shapes
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
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5.2 Damage detection procedure Localized changes in stiffness result reduce natural frequencies and change in mode shapes, regarding the natural frequencies; a change in their value is a significant indicator of presence of damage, however this information was not sufficient to indicate the location and quantification of the damage because and that change in mode shapes leads to localized change in slope, therefore, this feature will be studied as a significant parameter for crack detection purposes. In this work, first modal analysis is carried out and the model is solved for the first eight natural frequencies. The mass normalized mode shapes corresponding to the first eight natural frequencies are obtained for different damaged conditions. A one-dimensional strain energy based method damage detection algorithm has been generalized for beam like structures this method requires only model parameter before and after damage of the structure. Then the normalized damage index is obtained from all the sub-regions by using the average and standard deviation of all the damage indices The sketch presented in Fig.7 illustrates the scheme of operation of the wavelet-based damage detection technique. Wavelet function defect
( ) Mode Shape
L Fig.7. (a) Geometry of the analysed cantilever beam;
(b) sketch showing the wavelet-based damage detection technique. Then, the wavelet analysis is conducted on beam displacement mode shapes which are treated as a spatially distributed signal by the Gaussian wavelet family. To detect the damage location, the mode shape is wavelet transformed using Gaussian with four vanishing moments. M. Rucka, K. Wilde [13] suggested that selection of an appropriate type of the wavelet function and the number of its vanishing moments is very essential for the effective use of the wavelet analysis in damage detection.The applications of wavelets which create strong non-zero values in places where damage occurs that are facilitate damage identification. The reason is that there is the geometric discontinuity at that position so the changes of deflection are larger at that region. In this case, the wavelet transform damage index calculated from the first eight numerical mode shapes are shown in Fig.8. It has been observed that the best candidates to damage identification with the one-dimensional continuous wavelet transform. 6. Results and discussions case study 1: Damage Identification In the first simulation, the notch has been introduced at the distance xd = 0.16m. 1D-Continuous wavelet transform was carried out on the first eight displacement mode shapes by the Gaussian wavelet family with four vanishing points. The chosen wavelet functions create the maximum number of wavelet coefficients that are close to zero, and non-zero values (peaks) dominate at the position of damage. Figure.8. shows the computed WDI(wavelet damage index) for the 1st and 4th mode shape for 10% of the crack depth. The notch position can be very well located at xd = 0.16m for the 1st and 4th mode shapes. It should also be noticed that for the chosen wavelet (e.g. gaus4), the higher mode (see mode 1st and 4th in Fig. 8), the higher value of the wavelet damage index.
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Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
Fig.8. Wavelet damage index of the first eight numerically mode shapes for damage at the distance 0.16 m from fixed end. case study 2: Damage quantification And Fig.9. presents the Wavelet damag index of the 1st mode shapes computed using gaus4 wavelets with varying crack depth of 10% to 30% of height of the beam, it is evident from the plots wavelet transform modulus increases with increase crack depth. it has also been observed that the wavelet function with a smaller number of vanishing moments causes that some non-zero values are observed beyond the defect position.However, the maximum value of the WDI in the defect place has a larger value for the wavelet with the smaller number of vanishing moments than for the wavelet with the larger number of vanishing moments. For the 1st mode, the maximum value of WDI is 8x10-7 when crack depth 10% of the beam height ,increases with crack depth 20% and 30% as WDI are 3x10-5 and 2x10-4 .
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
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Fig. 9 : Wavelet damage index of the first eight numerically mode shapes for damage quantification at crack depth 10% , 20%, 30% of the height of the beam at the distance 0.16 m from fixed end Case study 3: Different crack locations Figure 10, 11, 12 Shows the damage indices of experiment of same aluminium beam having surface crack location changes from 20% to 80% (0.163 to 0.64m) with the step of 20% (0.16m) of length of beam of first mode1and second mode. It clear that in all three cases the height of damage indices well identify the damage location. This proposed method also applied to multiple damages in beam , it has been observed that which clearly locate the damage location on beam structure as shown in Fig.13.
Crack location at 20% of length of beam 0.000025 0.00002
ΔK
0.000015 0.00001 0.000005 0 -0.000005 0
50
100
150
-0.00001 -0.000015 -0.00002
LENGTH OF BEAM
Fig. 10.Length of beam vs. ΔK values for crack at 20% of length for mode-1
200
19574
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
Crack location at 50% of length of beam 0.000015 0.00001 ΔK
0.000005 0 -0.000005 0
50
100
150
200
-0.00001 -0.000015
LENGTH OF BEAM
Figure: 11. Length of beam vs.ΔK values1st mode Crack location at 60% of length of beam 0.000015 0.00001
ΔK
0.000005 0 -0.000005 0
50
100
150
200
-0.00001 -0.000015 LENGTH OF BEAM
Fig.12.Length of beam vs. ΔK values for crack at 60% of length for mode-2
Crack location at 2 different places(20% &40%of length) 0.000002
ΔK
0.000001 0 -0.000001 -0.000002
0
50
100
150
200
ΔKM01
LENGTH OF BEAM
Fig. 13. Length of beam vs.ΔK values for crack at 20% and 40% of lengths for 1st mode
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
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7. Conclusion The objective of this paper is to present damage detection methods of strain energy and spatial wavelet transform on identification of a multi-damage location, quantification and to highlight the sensitivity of higher modes toward the damage in a beam structure, for that strain energy and wavelet-based damage detection technique were investigated on a cantilever beam with damage in the form of surface crack of depth 30%, 20%, and 10% of the beam height. Also damage index compute the surface crack at different and multi crack. This outcomes of research provides evident that both methods are efficient approaches to identify the damage in structure members and also the effectiveness of the wavelet and strain energy based damage identification methods leads to the following conclusion:
For the established wavelet function, wavelet damage index higher with increase the mode number, this indicates that higher modes are more sensitive towards the damage. The quantification of damage for the same mode shape analyzed it has been observed that crack depth increases wavelet damage index increases, WDI contains a small number of Non-zero values are observed only in places where the damage occurs. The strain energy method effectively used to locate the different and multi-damage locations in the global structures. It conclude that the proposed method can clearly locate the different and multiple damage locations on the beam structures.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Doebling, S. W., Farrar, C. R. and Prime, M. B., “A Summary Review of Vibration-Based Damage Identification Methods,” The Shock and Vibration Digest, 30, pp. 91-105 (1998). Pandey AK, Biswas M, Samman MM. Damage detection form changes in curvature mode shapes. J Sound Vib 1991;145:321–32. Cawley P, Adams RD. The location of defects in structure from measurements of natural frequencies. J Strain Anal 1979;14:49–57. Doebling, S. W., Farrar, C. R., Prime, M. B., and Shevitz, D. W. (1996).“Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review.” Los Alamos National Laboratory Rep. No. LA13070-MS, Los Alamos, N.M. Shi, Z. Y., Law, S. S. and Zhang, L. M., “Structural Damage Localization from Modal Strain Energy Change,” J. Sound Vib., 218, pp. 825844 (1998). Shi, Z. Y., Law, S. S. and Zhang, L. M., “Structural Damage Detection from Modal Strain Energy Change,” J. Eng. Mech., 126, pp. 12161223 (2000). Cornwell, P. J., Dobeling, S. W. and Farrar, C. R., “Application of the Strain Energy Damage Detection Method to Plate-Like Structures,” Proc. 15th Intel. Modal Anal. Conf., Orlando, FL, pp. 1312 1318 (1997). Cornwell, P. J., Dobeling, S. W. and Farrar, C. R., “Application of the Strain Energy Damage Detection Method to Plate-Like Structures,” J. Sound Vib., 224, pp. 359-374 (1999). Liew, K. W. and Wang, Q. 1998. Application of wavelet theory for crack Identification in structures. Journal of Engineering Mechanics, 124: 152-157. Dimarogonas, A.D., 1976. Vibration Engineering. West Publishers, St Paul, Minesota. Adams, A.D., Cawley, P., 1979. The location of defects in structures from measurements of natural frequencies. Journal of Strain Analysis 14, 49–57. Pandey A.K.,M.Biswas, and M.M.Samman."Damage detection from changes in curvature mode shapes." Journal of sound and vibration 145.2 (1991): 321-332. Q. Wang, X. Deng, Damage detection with spatial wavelets, International Journal of Solids and Structures 36 (1999) 3443–3468. M. Rucka_, K. Wilde, Application of continuous wavelet transform in vibration based damage detection method for beams and plates, Journal of Sound and Vibration 297 (2006) 536–550. Chih-Chieh Chang, Lien-Wen Chen, Damage detection of a rectangular plate by spatial wavelet based approach, Applied Acoustics 65 (2004) 819–832 W.L. Bayissa, N. Haritosa, S. Thelandersson, Vibration-based structural damage identification using wavelet transform , Mechanical Systems and Signal Processing 22 (2008) 1194–1215.
ScienceDirect Materials Today: Proceedings 5 (2018) 19565–19575
www.materialstoday.com/proceedings
ICMPC_2018
Damage Detection In Structural Beams Using Model Strain Energy Method And Wavelet Transform Approach Lanka Ramesh*a, Putti Srinivasa Raob a
Mechanical Engineering Department, Gudlavalleru Engineering College, Sheshadri Rao Knowledge Village, Gudlavalleru- 521356 Krishna District, Andhra Pradesh ,INDIA b Mechanical Engineering Department, Andhra University, Visakha Patnam-530003, Andhra Pradesh, INDIA
Abstract Damage identification method of surface cracks in beams using strain energy method and its spatial wavelet transform is investigated in this paper. Experimental modal analysis (EMA) is conducted on a aluminium cantilever beam to obtain the mode shapes before and after damage under a fixed-free boundary condition by Laser Doppler Vibrometer (LDV). A small perturbation is created by using Electro Discharge Machine (EDM) and simulated crack by separation of nodes at the crack location in beams. The purpose of the study is to present damage index method and spatial wavelet transform approaches to identify the location and quantification of surface crack in beam structure. A finite element analysis (FEA) is also performed to access this approaches and illustrate a attainable process for the experimental work. First, the displacement mode shapes obtained from the intact and damaged beams which are used to obtain strain energy ratio of the beam before and after damage and then these spatially distributed signals are analyzed by wavelet transformation. Good correlation between FEA and EMA results is obtained. Both methods damage index (DI) and wavelet transform (WT) successfully identify the location of surface crack in the aluminium beam. it is also demonstrated that the location and quantification of single and multi cracks can be detected by this methods. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization. Keywords: damage detection, Experimental modal analysis,Strain energy, wavelet transform,.
1. Introduction Vibration based Structural Health Monitoring (SHM) became one of the fundamental approaches in structural damage detection from past 30 years. It is based totally on the fact that methods allowing to existence of damage detection and its location at earlier stage become an intensive investigation thought the mechanical. civil and aerospace engineering domain from the last two decades. The existence of damage in structure introduces local flexibility that would have an effect on dynamic characteristics of whole structure. * Corresponding author. E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.
19566
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
It causes in reduction of local flexibility that would have an effect on dynamic characteristics of whole structure. It causes in reduction of natural frequencies and change in displacement mode shapes of structure member. This property leads to notice existence of crack and its location and depth in the structural members. An extensive literature review [1] of the state of the artwork of damage detection and health monitoring from model parameters has recently been published. From this review it is clear that there are a lot work has been done in the area of damage detecting from modal parameters but, unfortunately, many of these methods require a modal parameters before and after damage. The method proposed in this paper avoids modal parameter requires before the damage . Pandey et al. [2] suggested that absolute changes in curvature of the deflected shape of beam can be a desirable indicator of damage compared to change in the frequencies. Cawley and Adams [3] simply extended same technique to detect the damage in two dimensional composite structures based the frequency shifts for different modes. Scott W. Doebling [3] developed vibration-based damage detection technique for damage identification and , even to predict location and level of damage in structures based the information of frequency shifts, changes in mode shape, changes in curvature mode shape, and modal strain energy change. Shi et al. [4-5] have considered the change of modal strain energy ratio to define a damage index which effective indicator to identification of damage location and quantify the damage level. The quality of this indicator is improved by summing the contributions of multiple modes in a normalized fashion. Cornwell et al. [7,8] derived damage detection algorithm for an index of damages in plate-like structure characterized by two dimensional curvatures by strain Energy approach. In their approach, the fractional strain energy of the plate before and after damaged was used to define a damage index which can successfully locate the area with stiffness reduction as low as 10% using relatively few modes. Liew and Wang [9] proposed an application of spatial wavelet theory to displacement mode shape for damage identification of beamtype structures has been extensively investigated. They calculated the wavelet coefficients along the length of the beam primarily based on the numerical solution for the deflection of the beam. the damage location and level was then indicated with aid of a height wavelet transform modulus in the coefficients of the wavelets along the length of the beam. Q. Douka et al. , Loutridis et al. [10] performed essessement of damage identification based wavelet transform approaches on mode shapes obtained by standard experimental modal analysis using accelerometers and a modal hammer.This work is provide new technique to apply strain energy method and spatial wavelet transform to highlight the sensitivity for detection and quantification and localization of damage in a beam structure, for that damage index method and wavelet-based damage identification approach was investigated experimentally by using non contact LDV and numerically on aluminium cantilever beam with damage in the form of surface crack of depth 30%, 20%,and 10% of the beam height .Also presented multi crack analysis using Strain Energy method on the few mode shapes as shown in below Fig 1& 2. 2. Theory of Strain Energy of a beam and damage index For intact beam : The strain energy of whole beam is given by =
(1)
For a particular mode shape, ψ(x) displacement at each node for a particular mode shape = 1
(2) j
2
Nd
....
aj
a
j 1
Figure 2. Sketch showing a beam's N d sub-divisions. . If N d is the number of divisions of the beam, then energy associated with each sub-region j due to the kth mode is given by
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
(
=
19567
)
(3)
The fractional energy is therefore = For damaged beam : Similarly, same quantities can be defined , where ()* indicates quantity estimated for damaged beam. Energy associated with particular mode shape kth . ∗
=
1 2
(4)
Ψ∗
∗
Enrgy associated with jth sub-regional strain energy of the kth mode shape for damaged beam. ∗
=
(
)∗
∗
(5)
The fractional energies of the beam are defined as ∗
=
∗
(6)
∗
Considering all the measured modes, m, in the calculation, the damage index in sub-region j is defined to be ∗
= ∑
/∑
(10)
A normalized damage index is obtained using =( where
−
)/
(11)
k and k represent the mean and standard deviation of the damage indices, respectively. Equation (6) is
used to predict the damage location in beam structures. 3. Continuous Wavelet Transform in Damage Detection This section shortly recapitulates the theoretical background of the continuous wavelet transform and its application to detection of singularities. For a given one-dimensional signal the continuous wavelet transform can be defined as (e.g.Mallat, 1998) ( , )=
1
( )
∗
−
√ Where ( ) is continuous wavelet function, x is the time or spatial variable, ∗ ( ) is the complex conjugate of wavelet function. Wavelets transform is the derivatives of a continuous function ∅( )usually called the scaling function i.e. ( ) = The wavelet transform can be defined as: ( , )=
1
( )∅∗
−
=
1
( )∅∗
−
√ √ ψ u,s(x) and the real numbers s and u denote the scale and the translation parameter, respectively. Therefore the wavelet with n vanishing moments can be rewritten as the nth order derivative of the smoothing function f(x), and the resulting wavelet transform can be expressed as a multiscale differential operator (Mallat,1998)
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Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
( , )=
( ( )∗
( )) ( )
( )=
1
√
−
(8)
Where f(x)* s(x) denotes convolution of functions. Equation (7) reveals that the wavelet transform is the nth derivative of the signal f(x) smoothed by the function s(x) at the scale s. Singularities in a signal f(x) can be detected by finding the abscissa where the maxima of the wavelet transform modulus (WTM) |Wf(u, s)| converge at fine scales (Mallat, 1998). 4. Numerical analysis of a cantilever beam In this steel cantilever beam with dimensions t = 0.06m, h = 0.7m, w = 0.06m is considered as the testing model with Fixed-Free support conditions as shown in Fig.3. Material parameters determined experimentally are: the modulus of elasticity E = 69GPa and the mass density = 2735 kg/m3 and Poisson ratio = 0.33 respectively. The element type selected for modelling beam is solid182, 4-node element having three degrees of freedom at each node. A surface crack has been introduced in beam with 40mm long, 0.5 wide and 1mm by separating the nodes at the distance x d measured from the crack centre to the fixed end as shown in Fig.03. The beam was divided into 400 elements of length mm. H. W.Hu [15] found that the resolution damage indices at surface crack has been significantly increased when damage indices are obtained from FEA by using the finer meshes.The normal mode shapes of the beam were computed by the finite element method using the classical Euler-Bernoulli beam theory.
Fig. 3. Surface crack in FE In this study, spatial mode shapes of the beam are investigated at three different depth of the crack are 10%, 20% and 30% of the beam height. Numerical simulations were performed on the first eight mode shapes for un-damaged and damaged beam, first eight modes crack free beam present in Fig.6.Then, the wavelet analysis is observed by fed the spatial signals to the wavelet transform here damaged modes are treated as input signals for wavelet analysis (spatial domain). 5. Experimental Investigation In this project an experimental analysis is employed on aluminium beam has been performed by using the Laser Doppler Vibrometer to validate the numerical model as well as damage detection methods. Here cantilever beam 800 mm length and 25 x 10 mm2 cross-section was clamped at a vibrating table. The beams surface has been cleaned and prepared for straightness before the experimental study. Subsequently, single transverse cracks is deemed on the beam at 160 mm from fixed end of the beam by using Electro discharge Machine as shown in Fig. 5., the depth of cracks are varied 10%, 20% and 30% of height of the beam. The beam is uniformly divided into 26x3 elements by the grid lines as shown in Fig. 4. The beam is subjected to a dynamic pulse load applied at each grid point using modally tuned hammer (PCB 652B10). The entire beam was then scanned using 3-D Polytec PDV-100 scanning system with the maximum measurable frequency bandwidth of 1 MHz. The response measurements are made using pulse generator and amplified to the maximum 100 V to record the response of the structure. Modal analysis was performed to obtain the Frequency Response Function (FRF) and displacement modes. VIBE, software is used to extract the mode shapes and natural frequencies of beam from the FRFs.
Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
Damage
(75cm
0 05
Fig. 4.Test plate with 26 × 3 grid
Fig. 5. Crack embedded in beam by using EDM
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Fig.6. Numerical Mode shapes
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5.2 Damage detection procedure Localized changes in stiffness result reduce natural frequencies and change in mode shapes, regarding the natural frequencies; a change in their value is a significant indicator of presence of damage, however this information was not sufficient to indicate the location and quantification of the damage because and that change in mode shapes leads to localized change in slope, therefore, this feature will be studied as a significant parameter for crack detection purposes. In this work, first modal analysis is carried out and the model is solved for the first eight natural frequencies. The mass normalized mode shapes corresponding to the first eight natural frequencies are obtained for different damaged conditions. A one-dimensional strain energy based method damage detection algorithm has been generalized for beam like structures this method requires only model parameter before and after damage of the structure. Then the normalized damage index is obtained from all the sub-regions by using the average and standard deviation of all the damage indices The sketch presented in Fig.7 illustrates the scheme of operation of the wavelet-based damage detection technique. Wavelet function defect
( ) Mode Shape
L Fig.7. (a) Geometry of the analysed cantilever beam;
(b) sketch showing the wavelet-based damage detection technique. Then, the wavelet analysis is conducted on beam displacement mode shapes which are treated as a spatially distributed signal by the Gaussian wavelet family. To detect the damage location, the mode shape is wavelet transformed using Gaussian with four vanishing moments. M. Rucka, K. Wilde [13] suggested that selection of an appropriate type of the wavelet function and the number of its vanishing moments is very essential for the effective use of the wavelet analysis in damage detection.The applications of wavelets which create strong non-zero values in places where damage occurs that are facilitate damage identification. The reason is that there is the geometric discontinuity at that position so the changes of deflection are larger at that region. In this case, the wavelet transform damage index calculated from the first eight numerical mode shapes are shown in Fig.8. It has been observed that the best candidates to damage identification with the one-dimensional continuous wavelet transform. 6. Results and discussions case study 1: Damage Identification In the first simulation, the notch has been introduced at the distance xd = 0.16m. 1D-Continuous wavelet transform was carried out on the first eight displacement mode shapes by the Gaussian wavelet family with four vanishing points. The chosen wavelet functions create the maximum number of wavelet coefficients that are close to zero, and non-zero values (peaks) dominate at the position of damage. Figure.8. shows the computed WDI(wavelet damage index) for the 1st and 4th mode shape for 10% of the crack depth. The notch position can be very well located at xd = 0.16m for the 1st and 4th mode shapes. It should also be noticed that for the chosen wavelet (e.g. gaus4), the higher mode (see mode 1st and 4th in Fig. 8), the higher value of the wavelet damage index.
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Lanka Ramesh et al./ Materials Today: Proceedings 5 (2018) 19565–19575
Fig.8. Wavelet damage index of the first eight numerically mode shapes for damage at the distance 0.16 m from fixed end. case study 2: Damage quantification And Fig.9. presents the Wavelet damag index of the 1st mode shapes computed using gaus4 wavelets with varying crack depth of 10% to 30% of height of the beam, it is evident from the plots wavelet transform modulus increases with increase crack depth. it has also been observed that the wavelet function with a smaller number of vanishing moments causes that some non-zero values are observed beyond the defect position.However, the maximum value of the WDI in the defect place has a larger value for the wavelet with the smaller number of vanishing moments than for the wavelet with the larger number of vanishing moments. For the 1st mode, the maximum value of WDI is 8x10-7 when crack depth 10% of the beam height ,increases with crack depth 20% and 30% as WDI are 3x10-5 and 2x10-4 .
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Fig. 9 : Wavelet damage index of the first eight numerically mode shapes for damage quantification at crack depth 10% , 20%, 30% of the height of the beam at the distance 0.16 m from fixed end Case study 3: Different crack locations Figure 10, 11, 12 Shows the damage indices of experiment of same aluminium beam having surface crack location changes from 20% to 80% (0.163 to 0.64m) with the step of 20% (0.16m) of length of beam of first mode1and second mode. It clear that in all three cases the height of damage indices well identify the damage location. This proposed method also applied to multiple damages in beam , it has been observed that which clearly locate the damage location on beam structure as shown in Fig.13.
Crack location at 20% of length of beam 0.000025 0.00002
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Fig. 10.Length of beam vs. ΔK values for crack at 20% of length for mode-1
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Crack location at 50% of length of beam 0.000015 0.00001 ΔK
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Figure: 11. Length of beam vs.ΔK values1st mode Crack location at 60% of length of beam 0.000015 0.00001
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Fig.12.Length of beam vs. ΔK values for crack at 60% of length for mode-2
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Fig. 13. Length of beam vs.ΔK values for crack at 20% and 40% of lengths for 1st mode
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7. Conclusion The objective of this paper is to present damage detection methods of strain energy and spatial wavelet transform on identification of a multi-damage location, quantification and to highlight the sensitivity of higher modes toward the damage in a beam structure, for that strain energy and wavelet-based damage detection technique were investigated on a cantilever beam with damage in the form of surface crack of depth 30%, 20%, and 10% of the beam height. Also damage index compute the surface crack at different and multi crack. This outcomes of research provides evident that both methods are efficient approaches to identify the damage in structure members and also the effectiveness of the wavelet and strain energy based damage identification methods leads to the following conclusion:
For the established wavelet function, wavelet damage index higher with increase the mode number, this indicates that higher modes are more sensitive towards the damage. The quantification of damage for the same mode shape analyzed it has been observed that crack depth increases wavelet damage index increases, WDI contains a small number of Non-zero values are observed only in places where the damage occurs. The strain energy method effectively used to locate the different and multi-damage locations in the global structures. It conclude that the proposed method can clearly locate the different and multiple damage locations on the beam structures.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
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