Damage Detection in Beams Using Frequency Response Function Curvatures Near Resonating Frequencies
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The document presents a novel algorithm for structural damage detection using frequency response function (FRF) curvature at frequencies near resonant peaks, enhancing detection sensitivity compared to traditional methods. The authors investigate the efficacy of this approach on a simply supported beam to identify damage, demonstrating that using FRF curvatures away from natural frequencies can improve noise resistance and localization of damage. Numerical simulations illustrate the impact of random noise on detection accuracy, affirming the robustness of the proposed methodology under various conditions.
![can be carried out. Recently, due to the rapid expansion of infrastructural facilities
as well as deterioration of the already existing infrastructures, the magnitude of the
problem has become enormous to the civil engineering community. Detection of
damages using various local and global approaches has been explored in current
literature. The measured dynamical properties have been used effectively for
detecting damages. The dynamical responses of structures can be very precisely
measured using modern hardware and a large amount of data can be stored for
further post processing to subsequently detect damages. The damage detection
problem can be classified as identification or detection of damage, location of
damage, severity of damage and at the last-estimation of the remaining service life
of a structure and its possible ultimate failure modes. During the last three decades
significant research has been conducted on damage detection using modal prop-
erties (frequencies, mode shapes and damping etc.). The mostly referred paper on
damage detection using dynamical responses is due to Deobling et al. [1] which
give a vivid account of all the methodologies of structural damage detection using
vibration signature until 90 s. Damage detection using changes in frequency has
been surveyed by Salawu and Williums [2]. The main drawback of detecting
damages using only frequency information is the lack of sensitivity for the small
damage cases. The main advantage of this method is that, frequency being a global
quantity it can be measured by placing the response sensor such as an accelerometer
at any position. Mode shapes can also be effectively used along with frequency
information to locate damage [3, 4], but the major drawback is that mode shape is
susceptible to the environment noise much more than the frequency. Moreover,
mode shapes being a normalized quantity is less sensitive to the localized changes
in stiffness. The random noise can be averaged out to some extent but systematic
noise cannot be fully eliminated. Furthermore, in vibration based damage detection
methodologies, depending upon the location the damage may or may not be
detected if it falls on the node point of that particular mode. Lower modes some-
times remain less sensitive to localized damages and measurement of higher modes
are almost always is necessary which is more difficult in practice.
In contrast with frequency and mode shape based damage detection, method-
ologies using mode shape curvature, arising from the second order differentiation of
the measured displacement mode shape is considered more effective for detecting
cracks in beams [5]. Wahab and Roeck [6] showed that damage detection using
modal curvature is more accurate in lower mode than the higher mode. Whalen [7]
also used higher order mode shape derivatives for damage detection and showed
that damage produce global changes in the mode shapes, rendering them less
effective at locating local damages. Curvature mode shapes also have a noticeable
drawback of susceptibility to noise, caused by these second order differentiation of
mode shapes. This differentiation process may amplify lower level of noise to such
an extent to produce noise-dominated curvature mode shapes [8] with obscured
damage signature. Most recently, Cao et al. [9] identified multiple damages of
beams using a robust curvature mode shape based methodology.
In recent years, many methods of damage detection based on changes in
dynamical properties have been developed and implemented for various
1564 S. Mondal et al.](https://image.slidesharecdn.com/c2-sec-2014-chp-3a10-160318142910/85/Damage-Detection-in-Beams-Using-Frequency-Response-Function-Curvatures-Near-Resonating-Frequencies-2-320.jpg)
![complicated structural forms. Wavelet transformation is one of the recent popular
techniques for damage detection in local level, although its performance to detect
small cracks is questioned [10, 11].
A structure vibrates on its own during resonance at high amplitude and therefore
the FRFs become very sensitive to noise. Ratcliffe [12] explored the frequency
response function sensitivities at all frequencies rather than at just the resonant
frequencies to define a suitable damage index which can be used in a robust manner
in presence of inevitable experimental noise. Sampaio et al. [13] have given an
account of the frequency response function curvature methods for damage detec-
tion. Pai and Young [14] detected small damages in beams employing the opera-
tional defected shapes (ODS) using a boundary effect detection method. Scanning
laser vibrometer was used for measuring the mode shapes. Bhutia [15] and Mondal
[16] have also investigated damage detection using operational deflected shapes and
using FRFs at frequencies other than natural frequencies respectively.
Therefore, it appears that at frequencies slightly away from the natural frequency
(either above or below), it may be somewhat less affected by measurement noise.
But, it is to be also remembered that the sensitivity of FRFs to damage will also fall
down at the frequencies other than the natural frequencies. Hence, the FRFs at
frequencies other than natural frequencies, although less noise prone is less sen-
sitive to damage as well. With all probability, there might exist an optimum location
in FRF curve nearer to the resonant peaks where the measured FRFs still have
enough sensitivity towards damage yet have substantial less sensitivity to noise. It
must also be noted that most of the existing damage detection algorithm works well
when the damage is severe, because the level of stiffness changes will be substantial
for such damages and will be easily detectable. Such damages can be detected
easily by other means such as direct visual observations. The real challenge in the
research field of structural damage detection is to test the damage indicator’s
sensitivity for small damages in presence of inevitable measurement noise. Most
algorithms are observed to give spurious indications of damage when the noise
level becomes somewhat higher.
In this paper FRF curvature is used at frequencies different than the natural
frequencies to detect damage. Thus the fundamental principle behind this damage
detection methodology is to exploit the relative gain in terms of lower noise sen-
sitivity, sacrificing a bit in terms of resonant response magnitude. Although the
concept appears to be attractive, the current literature does not provide enough
guidance in this regard. The present paper tries to explore the same through an
example beam in simply supported condition.
The present study concentrates on a forward problem of simulating damage
scenarios, considering the FRF curvatures as the damage indicators to see if it
performs better than the methods employing FRFs at natural frequencies. The key
question is the robustness of the algorithm, i.e. whether the results obtained will
remain unique in the presence of real experimental noise, especially under the
condition of modal and coordinate sparsity. Finite element analysis using ABAQUS
[17] has been used to generate the required vibration responses for this simulated
study. Simulated noises into the data are added as a percentage of FRF magnitude.
Damage Detection in Beams Using Frequency … 1565](https://image.slidesharecdn.com/c2-sec-2014-chp-3a10-160318142910/85/Damage-Detection-in-Beams-Using-Frequency-Response-Function-Curvatures-Near-Resonating-Frequencies-3-320.jpg)
![2 Theoretical Background of the Present Methodology
The mass, stiffness and damping properties of a linear vibrating structure are related
to the time varying applied force by the second order differential force equilibrium
equations involving the displacement, velocity and acceleration of a structure. The
corresponding homogenized equation can be written in discretized form-
M½ Šfx
::
ðtÞg þ ½CŠf_xðtÞg þ ½KŠfxðtÞg ¼ f0g ð1Þ
where ½MŠ; ½KŠ and ½CŠ are the mass, stiffness and viscous damping matrices with
constant coefficients and fx
::
ðtÞg; f_xðtÞg and fxðtÞg are the acceleration, velocity and
displacement vectors respectively as functions of time. The eigensolution of the
undamped homogenized equation gives the natural frequencies and mode shapes.
If the damping is small, the form of FRF can be expressed by the following
equation [18].
HjkðxÞ ¼
Xj
Fk
¼
XN
r¼1
rAjk
k2
r À x2
ð2Þ
Here, Hjk(ω) is the frequency response functions, rAjk is the modal constant, λr is
the natural frequency at mode r and ω is the frequency.
The individual terms of the Frequency Response Functions (FRF) are summed
taking contribution from each mode [18]. At a particular natural frequency, one of
the terms containing that particular frequency predominates and sum total of the
others form a small residue. But for a FRF at frequency just slightly away from the
natural frequency, the other terms also starts contributing somewhat significantly,
thereby remaining sensitive to stiffness changes of the structure. For localizing the
damage, FRF curvature method can more effectively be used than the FRFs
themselves.
3 Numerical Investigation
In this current investigation a simply supported aluminum beam has been modeled
using the C3D20R element (20 noded solid brick element). Eigensolutions have
been found out using the Block Lanchoz algorithm with appropriately converged
mess sizes for the modes under consideration. The material properties of beam are
assumed to be E = 70 GPa and NEU = 0.33. Then, damage has been inflicted with a
deep narrow cut of width 2.5 mm thick and 5 mm deep as shown in Fig. 1. The
FRFs are computed at 21 evenly spaced locations as shown in Fig. 2.
Figure 3 shows the natural frequencies and the corresponding mode shapes of
the ‘undamaged’ beam and damaged beam.
1566 S. Mondal et al.](https://image.slidesharecdn.com/c2-sec-2014-chp-3a10-160318142910/85/Damage-Detection-in-Beams-Using-Frequency-Response-Function-Curvatures-Near-Resonating-Frequencies-4-320.jpg)
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Damage Detection in Beams Using Frequency Response Function Curvatures Near Resonating Frequencies
- 1. Damage Detection in Beams Using Frequency Response Function Curvatures Near Resonating Frequencies Subhajit Mondal, Bidyut Mondal, Anila Bhutia and Sushanta Chakraborty Abstract Structural damage detection from measured vibration responses has gain popularity among the research community for a long time. Damage is identified in structures as reduction of stiffness and is determined from its sensitivity towards the changes in modal properties such as frequency, mode shape or damping values with respect to the corresponding undamaged state. Damage can also be detected directly from observed changes in frequency response function (FRF) or its derivatives and has become popular in recent time. A damage detection algorithm based on FRF curvature is presented here which can identify both the existence of damage as well as the location of damage very easily. The novelty of the present method is that the curvatures of FRF at frequencies other than natural frequencies are used for detecting damage. This paper tries to identify the most effective zone of frequency ranges to determine the FRF curvature for identifying damages. A numerical example has been presented involving a beam in simply supported boundary condition to prove the concept. The effect of random noise on the damage detection using the present algorithm has been verified. Keywords Structural damage detection Á Frequency response function curvature Á Finite element analysis 1 Introduction Damage detection, condition assessment and health monitoring of structures and machines are always a concern to the engineers. For a long time, engineers have tried to devise methodology through which damage or deterioration of structures can be detected at an earliest possible stage so that necessary repair and retrofitting S. Mondal Á B. Mondal Á A. Bhutia Á S. Chakraborty (&) Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India e-mail: [email protected] © Springer India 2015 V. Matsagar (ed.), Advances in Structural Engineering, DOI 10.1007/978-81-322-2193-7_119 1563
- 2. can be carried out. Recently, due to the rapid expansion of infrastructural facilities as well as deterioration of the already existing infrastructures, the magnitude of the problem has become enormous to the civil engineering community. Detection of damages using various local and global approaches has been explored in current literature. The measured dynamical properties have been used effectively for detecting damages. The dynamical responses of structures can be very precisely measured using modern hardware and a large amount of data can be stored for further post processing to subsequently detect damages. The damage detection problem can be classified as identification or detection of damage, location of damage, severity of damage and at the last-estimation of the remaining service life of a structure and its possible ultimate failure modes. During the last three decades significant research has been conducted on damage detection using modal prop- erties (frequencies, mode shapes and damping etc.). The mostly referred paper on damage detection using dynamical responses is due to Deobling et al. [1] which give a vivid account of all the methodologies of structural damage detection using vibration signature until 90 s. Damage detection using changes in frequency has been surveyed by Salawu and Williums [2]. The main drawback of detecting damages using only frequency information is the lack of sensitivity for the small damage cases. The main advantage of this method is that, frequency being a global quantity it can be measured by placing the response sensor such as an accelerometer at any position. Mode shapes can also be effectively used along with frequency information to locate damage [3, 4], but the major drawback is that mode shape is susceptible to the environment noise much more than the frequency. Moreover, mode shapes being a normalized quantity is less sensitive to the localized changes in stiffness. The random noise can be averaged out to some extent but systematic noise cannot be fully eliminated. Furthermore, in vibration based damage detection methodologies, depending upon the location the damage may or may not be detected if it falls on the node point of that particular mode. Lower modes some- times remain less sensitive to localized damages and measurement of higher modes are almost always is necessary which is more difficult in practice. In contrast with frequency and mode shape based damage detection, method- ologies using mode shape curvature, arising from the second order differentiation of the measured displacement mode shape is considered more effective for detecting cracks in beams [5]. Wahab and Roeck [6] showed that damage detection using modal curvature is more accurate in lower mode than the higher mode. Whalen [7] also used higher order mode shape derivatives for damage detection and showed that damage produce global changes in the mode shapes, rendering them less effective at locating local damages. Curvature mode shapes also have a noticeable drawback of susceptibility to noise, caused by these second order differentiation of mode shapes. This differentiation process may amplify lower level of noise to such an extent to produce noise-dominated curvature mode shapes [8] with obscured damage signature. Most recently, Cao et al. [9] identified multiple damages of beams using a robust curvature mode shape based methodology. In recent years, many methods of damage detection based on changes in dynamical properties have been developed and implemented for various 1564 S. Mondal et al.
- 3. complicated structural forms. Wavelet transformation is one of the recent popular techniques for damage detection in local level, although its performance to detect small cracks is questioned [10, 11]. A structure vibrates on its own during resonance at high amplitude and therefore the FRFs become very sensitive to noise. Ratcliffe [12] explored the frequency response function sensitivities at all frequencies rather than at just the resonant frequencies to define a suitable damage index which can be used in a robust manner in presence of inevitable experimental noise. Sampaio et al. [13] have given an account of the frequency response function curvature methods for damage detec- tion. Pai and Young [14] detected small damages in beams employing the opera- tional defected shapes (ODS) using a boundary effect detection method. Scanning laser vibrometer was used for measuring the mode shapes. Bhutia [15] and Mondal [16] have also investigated damage detection using operational deflected shapes and using FRFs at frequencies other than natural frequencies respectively. Therefore, it appears that at frequencies slightly away from the natural frequency (either above or below), it may be somewhat less affected by measurement noise. But, it is to be also remembered that the sensitivity of FRFs to damage will also fall down at the frequencies other than the natural frequencies. Hence, the FRFs at frequencies other than natural frequencies, although less noise prone is less sen- sitive to damage as well. With all probability, there might exist an optimum location in FRF curve nearer to the resonant peaks where the measured FRFs still have enough sensitivity towards damage yet have substantial less sensitivity to noise. It must also be noted that most of the existing damage detection algorithm works well when the damage is severe, because the level of stiffness changes will be substantial for such damages and will be easily detectable. Such damages can be detected easily by other means such as direct visual observations. The real challenge in the research field of structural damage detection is to test the damage indicator’s sensitivity for small damages in presence of inevitable measurement noise. Most algorithms are observed to give spurious indications of damage when the noise level becomes somewhat higher. In this paper FRF curvature is used at frequencies different than the natural frequencies to detect damage. Thus the fundamental principle behind this damage detection methodology is to exploit the relative gain in terms of lower noise sen- sitivity, sacrificing a bit in terms of resonant response magnitude. Although the concept appears to be attractive, the current literature does not provide enough guidance in this regard. The present paper tries to explore the same through an example beam in simply supported condition. The present study concentrates on a forward problem of simulating damage scenarios, considering the FRF curvatures as the damage indicators to see if it performs better than the methods employing FRFs at natural frequencies. The key question is the robustness of the algorithm, i.e. whether the results obtained will remain unique in the presence of real experimental noise, especially under the condition of modal and coordinate sparsity. Finite element analysis using ABAQUS [17] has been used to generate the required vibration responses for this simulated study. Simulated noises into the data are added as a percentage of FRF magnitude. Damage Detection in Beams Using Frequency … 1565
- 4. 2 Theoretical Background of the Present Methodology The mass, stiffness and damping properties of a linear vibrating structure are related to the time varying applied force by the second order differential force equilibrium equations involving the displacement, velocity and acceleration of a structure. The corresponding homogenized equation can be written in discretized form- M½ Šfx :: ðtÞg þ ½CŠf_xðtÞg þ ½KŠfxðtÞg ¼ f0g ð1Þ where ½MŠ; ½KŠ and ½CŠ are the mass, stiffness and viscous damping matrices with constant coefficients and fx :: ðtÞg; f_xðtÞg and fxðtÞg are the acceleration, velocity and displacement vectors respectively as functions of time. The eigensolution of the undamped homogenized equation gives the natural frequencies and mode shapes. If the damping is small, the form of FRF can be expressed by the following equation [18]. HjkðxÞ ¼ Xj Fk ¼ XN r¼1 rAjk k2 r À x2 ð2Þ Here, Hjk(ω) is the frequency response functions, rAjk is the modal constant, λr is the natural frequency at mode r and ω is the frequency. The individual terms of the Frequency Response Functions (FRF) are summed taking contribution from each mode [18]. At a particular natural frequency, one of the terms containing that particular frequency predominates and sum total of the others form a small residue. But for a FRF at frequency just slightly away from the natural frequency, the other terms also starts contributing somewhat significantly, thereby remaining sensitive to stiffness changes of the structure. For localizing the damage, FRF curvature method can more effectively be used than the FRFs themselves. 3 Numerical Investigation In this current investigation a simply supported aluminum beam has been modeled using the C3D20R element (20 noded solid brick element). Eigensolutions have been found out using the Block Lanchoz algorithm with appropriately converged mess sizes for the modes under consideration. The material properties of beam are assumed to be E = 70 GPa and NEU = 0.33. Then, damage has been inflicted with a deep narrow cut of width 2.5 mm thick and 5 mm deep as shown in Fig. 1. The FRFs are computed at 21 evenly spaced locations as shown in Fig. 2. Figure 3 shows the natural frequencies and the corresponding mode shapes of the ‘undamaged’ beam and damaged beam. 1566 S. Mondal et al.
- 5. The FRFs of the undamaged and the damaged beam has been overlaid in Fig. 4. The difference is noticeable in some modes, indicating more damage sensitivity. Curvature of FRFs, i.e. the rate of change of FRFs measured at twenty one locations and at different frequency (lies in between 90 and 110 % of the natural Fig. 1 Dimension and damage location of simply supported beam Fig. 2 Location of FRF measurement along the center line of beam Fig. 3 Mode shape and frequency of undamaged and damaged beam Damage Detection in Beams Using Frequency … 1567
- 6. frequencies) for undamaged and damage cases are determined. The procedure to compute the FRF curvature is a central difference scheme and is given below H00 ijðxÞ ¼ Hðiþ1ÞjðxÞ À 2HijðxÞ þ HðiÀ1ÞjðxÞ ðDhÞ2 ð3Þ For example, as shown in the Fig. 5 FRF curvature was taken at 90, 95, and 98 % of 1st natural frequency for both the undamaged and damaged cases and this process was continued for twenty one different location of the beam. The difference was taken as absolute difference of the curvature. Since there is a frequency ‘shift’ due to damage, a mapping scheme has been adopted as shown. Many references just directly compare FRFs without accounting for such frequency shifts and may not truly represent the effect of FRF changes due to damage. Figure 5 shows the absolute change in FRFs at different frequencies around the first natural frequency without considering the noise. 3.1 Damage Detection Using FRF Curvature Near the First Fundamental Mode Figure 6 shows the FRF curvatures at various locations along the beam length for different values of frequencies away from the natural frequencies as a percentage of the resonant frequency. Thereafter, random noise is added to the FRFs and the same methodology is applied to determine the sensitivities. Figure 7a–f shows that as the noise level increases the FRF curvatures show pseudo peaks of much higher magnitudes to obscure the actual damages, however the effect is minimum around FRF curvature computed at 95 % of natural frequency. 0 1 2 3 4 5 6 7 8 0 500 1000 1500 2000 2500 3000 Magnitude Frequency (Hz) FRF for undamaged beam FRF for damaged beam Fig. 4 Comparisons of point FRFs (point 11) of undamaged and damaged beam 1568 S. Mondal et al.
- 7. Figure 7a shows the effect of noise on the damage localization using the FRF curvature at around 1st natural frequency. With 1 % noise, false peaks appear in addition to peak at 14th number point, so the localization sensitivity reduces. Addition of 2 % noise gives pseudo peaks at other points having much higher magnitudes which are however actually not damaged. Thus addition of noise has caused more and more false detection of damages as compared to the noise-free case. Addition of 3 % noise gives an even more unacceptable result with substantial increase in false detections apart from the actual damage at point 14. Figure 7b which is the plot of FRF curvature at 98 % of 1st natural frequency shows similar kind of result with very little improvement towards the noise resis- tance. However when curvature differences at 96 and 95 % of 1st natural frequency are explored, they show substantial increase in resistance towards the added random noise as is evident from Fig. 7c, d. It can be easily observed that the small peaks are Fig. 5 Mapping of FRF for undamaged and damaged beam at different frequency 0 0.00002 0.00004 0.00006 0.00008 0.0001 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 changeinFRFcurvature 90% 93% 95% 96% 97% 98% 99% Point number Fig. 6 FRF curvature beam without noise Damage Detection in Beams Using Frequency … 1569
- 8. relatively suppressed, thereby locating the damage much more uniquely at the designated point number 14. Further downward movement along frequency scale however could not fetch any benefit and in fact shows reduction in damage detection capacity. At 93 and 90 % of 1st natural frequency false peaks again started to predominate. Hence, Curvature difference away from 1st natural fre- quency shows very distinct damage localization capability, even with substantial level of added random noise. The peaks at the actual damage location are distinct enough to pin-point the actual damage location. Overall damage identification capability in presence of noise increases as we move away from resonant peak of FRF and damage detection is most robust within certain range of frequency, very close to the natural frequency. Similar phenomenon on other side of the FRF peak at natural frequency have been observed and are presented in Fig. 8. From Fig. 9a–f it is clear that damage detection can be done better between 104 and 105 % of natural frequency in noisy environment than the usual practice of using FRFs at resonant frequency. Fig. 7 Difference between curvatures of FRFs of undamaged and damaged (point 14) Simply supported beam for a 1st natural frequency b 98 %, c 96 %, d 95 %, e 93 % and f 90 % of 1st natural frequency for different percentage of noise. Input force at mid point 1570 S. Mondal et al.
- 9. 0 0.00002 0.00004 0.00006 0.00008 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Absolutechangeincurvature 101% 102% 103% 104% 105% 107% 110% Point number Fig. 8 Difference between curvatures of FRFs of undamaged and damaged (point 14) simply supported beam for different percentage (100–110 %) 1st natural frequency. Input force at mid point Fig. 9 Difference between curvatures of FRFs of undamaged and damaged (point 14) simply supported beam for a 1st natural frequency b 102 %, c 104 %, d 105 %, e 107 % and f 110 % of 1st natural frequency for different percentage of noise. Input force at mid point Damage Detection in Beams Using Frequency … 1571
- 10. 3.2 Damage Detection Using FRF Curvature Near the Second Mode The investigation is extended further to include the second mode also and similar results are obtained and are presented in Fig. 10. Most effective zone to detect damage is again found to be at 95–96 % of the resonant frequency (so also at 100–110 % of the second natural frequency) and is not presented here for brevity. 4 Conclusions An attempt has been made to detect the location of damage in a simply supported aluminum beam using FRF curvatures at frequencies other than natural frequencies and is found to be more robust as compared to method using FRFs at resonant frequencies when random noise are present in data. Upto 2–3 % of random noise in observed FRF data are tried. An optimum frequency zone at around 95–96 % (or 105–106 %) of the natural frequency has been identified as ideal to locate damage as they maintain the required sensitivity for damage detection yet being slightly offset from the peak value. Keeping all the above observations, we can conclude that damage detection using FRF curvature at other than natural frequency may be a better option if considerable measurement noise is present into the data. The method needs to be further explored with appropriate model of noise actually present in real modal testing of structures in various boundary conditions. Fig. 10 Difference between curvatures of FRFs of undamaged and damaged (point 14) Simply supported beam for a 2nd natural frequency, b 98 %, c 95 % and d 90 % of 2nd natural frequency for different percentage of noise. Input force at mid point 1572 S. Mondal et al.
- 11. References 1. Doebling SW, Farrar CR, Prime MB (1998) A summary review of vibration based damage identification methods. Shock Vib Dig 30(2):91–105 2. Salawu OS, Williams C (1994) Damage location using vibration mode shapes. In: Proceedings of the SPIE, Proceedings of the 12th international modal analysis conference, vol 2251, pp 933–941 3. Chen J, Garba JA (1988) On-orbit damage assessment for large space structures. AIAA J 26 (9):1119–1126 4. Pandey AK, Biswas M (1994) Damage detection in structures using changes in flexibility. J Sound Vib 169:3–17 5. Pandey AK, Biswas M, Samman MM (1991) Damage detection from changes in curvature mode shapes. J Sound Vib 145:321–332 6. Waheb MMA, Roeck GDE (1999) Damage detection in bridges using modal curvature: application to real world scenario. J Sound Vib 226(2):217–235 7. Whalen TM (2008) The behavior of higher order mode shape derivatives in damaged beam like structures. J Sound Vib 309(3–5):426–464 8. Cao MS, Cheng L, Su ZQ, Xu H (2012) A multi-scale pseudo-remodeling wavelet domain for identification of damage in structural components. Mech Syst Signal Process 28:638–659 9. Cao M, Radzieński M, Xu W, Ostachowicz W (2014) Identification of multiple damage in beams based on robust curvature mode shapes. Mech Syst Signal Process 46:268–280 10. Lee YY, Liew KM (2001) Detection of damage locations in a beam using the wavelet analysis. Int J Struct Stab Dyn 01(03):455–465 11. Liew KM, Wang Q (1998) Application of wavelets for crack identification in structures. J Eng Mech ASCE 124(2):152–157 12. Ratcliffe CP (2000) A frequency and curvature based experimental method for locating damage in structures. ASME J Vib Acoust 122(3):324–329 13. Sampaio RPC, Maia NMM, Silva JMM (1999) Damage detection using the frequency- response-function curvature method. J Sound Vib 226(5):1029–1042 14. Frank Pai P, Leyland G (2001) Young, damage detection of beams using operational deflection shapes. Int J Solids Struct 38:3161–3192 15. Bhutia A (2013) Damage detection using operational deflection shapes. M.Tech thesis, Department of Civil Engineering, IIT Kharagpur 16. Mondal B (2014) Damage detection of structures using frequency response functions at frequencies other than natural frequencies. M.Tech thesis, Department of Civil Engineering, IIT Kharagpur 17. ABAQUS/CAE 6.10-1, Dassault Systèmes Simulia Corp., Providence, RI, USA 18. Ewins DJ (2000) Modal testing: theory, practice and application. Research Studies Press Ltd, England Damage Detection in Beams Using Frequency … 1573