Damage Detection in Beams Using Frequency Response Function Curvatures Near Resonating Frequencies

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 Frequency response function curvature 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 B. Mondal A. Bhutia S. Chakraborty (&)

Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India

e-mail: sushanta@civil.iitkgp.ernet.in

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 Williums². 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 beams⁵. Wahab and Roeck⁶ showed that damage detection using modal curvature is more accurate in lower mode than the higher mode. Whalen⁷ 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 shapes⁸ 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 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. Ratcliffe¹² 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 Young¹⁴ 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. Bhutia¹⁵ 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 sensitivity, 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.

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-

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where ½M ; ½K and ½C are the mass, stiffness and viscous damping matrices with constant coefficients and fx :ðtÞg; fxðtÞg 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 equation¹⁸.

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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 mode¹⁸. 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.

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Fig. 1 Dimension and damage location of simply supported beam

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Fig. 2 Location of FRF measurement along the center line of beam

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Fig. 3 Mode shape and frequency of undamaged and damaged beam

The FRFs of the undamaged and the damaged beam has been overlaid in 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

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Fig. 4.Comparisons of point FRFs (point 11) of undamaged and damaged beam

frequencies) for undamaged and damage cases are determined. The procedure to compute the FRF curvature is a central difference scheme and is given below

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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.

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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.

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Fig. 6 FRF curvature beam without noise

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

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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 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.

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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

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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

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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

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.

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