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Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782. Detection of damage in beams and composite plates by harmonic excitation and time-frequency analysis 1 Ouahabi A. , 1Thomas M and 2Lakis A.A.. 1 Department of Mechanical Engineering, École de technologie supérieure, 1100, Notre-Dame Street West, Montreal, Quebec, H3C 1K3. Canada. mailto: [email protected] 2 Department of Mechanical Engineering, École Polytechnique de Montréal, Montreal, Quebec, H3C 3A7, Canada ABSTRACT In structural health monitoring, multiple non destructive techniques such as thermography, eddy current and/or vibration monitoring may be used for early detection of structural changes caused by corrosion, rivet damage or structural cracks. The aim of this study is to investigate an effective monitoring technique for detecting structural damage when it occurs. To address this problem, many studies have been based on the principle that natural frequencies decrease when a weakness occurs in the structure. The modal approach may work when the crack remains open however natural frequency monitoring has generally been proven ineffective at the early stage of crack occurrence. In this study, we are especially interested in fatigue breathing cracks. A breathing crack opens and closes periodically under an external excitation. It results in periodic structural stiffness variations which can be described as non-linear vibration. Consequently, the natural frequencies tend to shift periodically. These stiffness changes induce variations in the critical damping and damping rates. The result is an amplitude response variation during one cycle that produces modulation frequencies and harmonics in the frequency domain. Consequently, we have based our method on an external input to the structure that produces a harmonic excitation at a specific frequency which is equal to half of its first natural frequency. When there is no crack, the structure responds with a pure harmonic signal at this excitation frequency. When a crack appears, a second harmonic of the excitation frequency occurs with its amplitude amplified by coincidence with the natural frequency. Because the natural frequency moves at each half cycle, the amplitude at the second harmonic is modulated and a timefrequency analysis is required. The theoretical and experimental approach of this new method has been developed for cracked beams and theoretically applied to composite plates with delamination. 1. INTRODUCTION The presence of a crack in a structure creates a local flexibility which affects its dynamic response. Fatigue cracks often exist in structural members that are subjected to repeated loading, which eventually degrades their structural integrity. Moreover, the crack opens and closes over time according to the conditions of load and the amplitude of vibration. The corresponding changes of the dynamic characteristics can be measured and identified with structural changes, which can thereafter be useful for the detection of a presence of a crack [1]. Several analytical, numerical 775

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Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782. and experimental investigations have been carried out. The majority of the researchers assume in their work that the crack in a structural element remains open during the vibrations. This assumption avoids confronting complexities which result from the nonlinear characteristics of breathing cracks [2]. In fact, during the period of vibration of a cracked structure, the crack does not always remain open and the system necessarily becomes nonlinear. Actis et al [3] used the finite element method to study a beam simply supported with a breathing crack. They made the assumption that when the bending moment changes sign, the crack changes state: open to closed or closed to open. Thomas et al [4] presented a theoretical and numerical study on the vibratory behaviour of a cantilever beam with variable rigidity under a harmonic excitation. Qian et al. [5] studied effects of the opening and closing of a crack on the dynamic behaviour of structures using a finite element model. Chondros et al [6] developed a theory of vibration of continuous cracked beams for the lateral vibration of Euler-Bernoulli beams with one crack or two cracks. Li et al [7] studied the presence of cracks and zones corroded into homogeneous plates. Leonard et al [8] have shown that time-frequency analysis was useful for identifying cracks. In this paper, a breathing crack model is developed. For the detection of breathing cracks, a harmonic excitation at half of the natural frequency is applied with an analysis in the frequency domain at the natural frequency. Analyses are carried out in both time and frequency domains. The method has been developed for beam cracks with different depths and then theoretically applied for identifying delamination in composite plates. 2. STIFFNESS VARIATION MEASUREMENT In order to understand and develop a model for the stiffness variation of breathing damaged structures, compression and tensile tests were carried out on four cantilever beams. The beam sections were 25 x 25 mm with a length of 180 mm. Three of them were cracked at various crack depths (10.4 %, 13.6% and 14.4 % of the beam height) while one was kept intact. Figure 1 shows the results obtained from these experiments. Beam 1 was healthy and stiffness was constant when applying the load. As expected, the stiffness of cracked beams decreases when the load is applied in tension and retrieves its initial value when the load is removed. The decreasing stiffness is more pronounced for large cracks (Beam 4: 14.4%). It can be noticed that Beam 2, which has a low value of crack depth, (10.4%) retrieves its initial stiffness more rapidly than the others when load is removed. The stiffness variation has been empirically modelled from stiffness measurements.

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Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782. 5.00E+06

Stiffness [N/m]

4.50E+06 4.00E+06 3.50E+06 3.00E+06 2.50E+06 2.00E+06 1.50E+06 Intact Beam

Beam 3

Beam 4

Beam 2

Figure 1. Variation of stiffness of the beams. The stiffness variation has been interpolated by a sine wave that can also be modelled using the polynomial form: k  k0  100 Pf  a0  a1t  a2t 2   k0  100 Pf sin t  (1) where    

k0 represents the stiffness for an undamaged beam (4.5 106 N/m for beam 1) Pf is the depth of crack (mm); ai are constants that depend on crack depth  is the excitation pulsation (rad/s).

3. MODELING OF THE BREATHING CRACK The changes generated by the presence of the cracks have an influence on the intrinsic properties of the beam, and their effect on the response in displacements of the beam has been investigated. The application of a harmonic force at the free end of a cantilever beam produces oscillations with compressive and traction stresses. In the case of compression, the crack is closed and the equation of motion for our structure is the same as an intact structure. It is only in a state of traction that the crack opens. Consequently, stiffness decreases as a function of time. According to equation (1), the variation of the stiffness can be expressed as: k0 si t  T 2   k t     2t  k  k0  100 Pf a0  a1t  a2t 2   k0  100 Pf sin  si t  T 2  T  

(2)

where T is the period of the excitation (s). For non-linear stiffness variation the equation of motion can be expressed by assuming a one degree of freedom behaviour, as follows: (3) mut   cu t   k t  u t   mut   cu t   k0 1  100Pf sin t u t   F t 

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Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782. As the stiffness varies, natural frequency and critical damping vary. This variation of the critical damping affects the damping rate. The variation in time of the stiffness, natural frequency and damping rate is described in Figure 2.

Figure 2. Variations of the intrinsic properties of the cracked beam

4. NUMERICAL SIMULATION OF CRACKED BEAMS Fatigue tests were carried out on three steel beams to create fatigue cracks. The fourth one was kept intact. The modal parameters were measured and the crack for each beam was numerically modelled using the finite element method and by using contact elements. Figure 3 shows the effect of a crack (14.4% of the height) on natural frequencies compared with those from the intact beam.

Figure 3. Harmonic response of a cracked versus intact beam. It can be noticed that the decrease in numerical frequency is more sensitive at the third frequency than at the first one. Comparison of these numerical natural frequencies with experimental measurements indicates that they are quite close. 778


Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782.

5. HARMONIC EXCITATION AT HALF THE NATURAL FREQUENCY The proposed strategy consists in exciting the structure with a harmonic excitation at half the natural frequency of the cracked beam. The intact beam shows a pure harmonic vibration while the cracked beam exhibits a more complex periodic vibration in the time domain. The Fast Fourier Transform of time signals shows that in the frequency domain, the intact beam only provides a response at the excitation frequency while the cracked beams exhibit the excitation frequency and its harmonics due to the non-linear behaviour of the stiffness (Figure 4). Furthermore, the second harmonic of the excitation frequency aligns with the natural frequency and consequently is amplified. It can be noticed that monitoring the vibration amplitude at the second harmonic of the excitation frequency allows identification of a damaged structure and this amplitude increases with damage severity.

a) Intact beam

c) crack depth; 13.6% of beam height

b) crack depth; 10.4 % of beam height

d) crack depth; 14.4 % of beam height

Figure 4. Frequency response of beams: a) intact, b) 2, c) 3 and d) 4. An analysis by time-frequency (STFT) is shown in Figure 5. The spectrogram is effective in providing information on amplitude modulation at the natural frequency when a crack is present. For a healthy beam, the signal is purely harmonic at the excitation frequency. 779


Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782.

a) healthy beam b) cracked beam Figure 5: spectrogram of healthy and cracked beams for an excitation at half the natural frequency

6. APPLICATION TO IDENTIFICATION OF COMPOSITE PLATE DELAMINATION The developed method has been applied to a composite plate in order to identify a delamination (Figure 6). The investigated sample (Figure10) was 457 mm  395 mm  3.4 mm and the defect was located at a = 190 mm and b = 170 mm. The default type is a delamination with thin layers. The damaged plate has been numerically modelled using finite elements. An 8-layered plate has been considered with a symmetric ply angle of [0°/45°/-45°/90°]. The following material properties have been considered: E1  120 GPa, E2  E3  6.9 GPa, G12  G23  G13  5.2 GPa, 12  0.3, 23   13  0.01 a b

Figure 6. Geometry of the composite plate with delamination The natural frequencies for the intact plate have been computed at: 116.3Hz, 196.6Hz, 273.9Hz, 332.4Hz and 342.4 Hz. An analysis of the time signal in the 780


Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782. frequency domain (Figure 7) reveals that the second harmonic of the excitation is amplified due to alignment with its natural frequency. This presence of harmonics with high amplitudes allows the identification of damage in the plate due to a stiffness non linearity. Other harmonics are present but their amplitudes are too low to be significant.

Figure 7. Fourier transform of the time response. The time-frequency analysis (SFTF) confirms the diagnostic by showing an amplitude modulation at twice the excitation frequency (Figure 8). On the spectrogram, a periodic phenomenon can be clearly noticed which represents the variation of the natural frequency in time while exciting it at a constant frequency. Figure 8 shows that, when investigating the vibration behaviour at twice the excitation frequency an amplitude decrease appears at each half cycle while it is kept constant during the other half cycle.

Figure 8. time-frequency analysis of the damaged composite plate.

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Proceedings of the 3rd European Workshop on Structural Health Monitoring 2006. Granada, Spain, pp 775-782. 7. CONCLUSION A new strategy for identifying cracks or delaminations in structures is proposed. It consists in exciting a structure at half its first natural frequency. A healthy structure exhibits a pure harmonic signal at the excitation frequency while damaged structures show harmonics of the excitation frequency due to their non-linear stiffness characteristics. In fact the structure stiffness decreases at each half cycle and is kept constant during the other half cycle. This results in variations of the natural frequency and damping rate. When the excitation frequency is tuned to half the natural frequency, the second harmonic aligns with it resulting in amplitude amplification at this frequency. Hence monitoring of the second harmonic can be used for identifying a fault in structure stiffness. Furthermore, due to the variation of the natural frequency, an excitation at a constant frequency close to the natural frequency results in amplitude modulation at this frequency and consequently the fault diagnostic can be confirmed by time-frequency analysis. This method has been developed for cracked cantilever beams and extended to the identification of delamination in composite plates with multiple degrees of freedom. These preliminary results seem promising for future works in on-line detection and location of damaged structures. REFERENCES 1.

2.

3.

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Thomas M., Nguyen H., Hamidi L., Massoud M. and Piaud J.B., , Detection of rotor cracks by experimental modal analysis, Transactions of the Canadian Society of Mechanical Engineering, 1995 19(2) : p 155-174. Thomas M., Lakis A.A., Hamidi L. and Massoud M., Rotor health monitoring by modal analysis, Proceeding of the 20th seminar on machinery vibration, Quebec, 2002, p 4.204.29. Actis, R.I. and A.D. Dimarogonas. Non-Linear Effects Due to Closing Cracks in Vibrating Beam. in 12th ASME Conference on Mechanical Engineering, Vibration and Noise. 1989. Montreal, Canada. Thomas, M. and A. A. Lakis. Breathing Crack Detection by Time-Frequency Analysis. in 30th International conference on Computers and Industrial Engineering. 2002. Tinos, Greece, 10 p. Qian, G.-L., S.-N. Gu, and J.-S. Jiang, The dynamic behaviour and crack detection of a beam with a crack. Journal of Sound and Vibration, 1990. 138(2): p. 233-243. Chondros, T.G., A.D. Dimarogonas, and J. Yao, A continuous cracked beam vibration theory. Journal of Sound and Vibration, 1998. 215(1): p. 17-34. Li, Y.Y., et al., Identification of damage locations for plate-like structures using damage sensitive indices: strain modal approach. Computers & Structures, 2002. 80(25): p. 18811894. Leonard F, Lanteigne J, Lalonde S and Turcotte Y., Vibration behavior of a cracked beam, 18th IMAC, San Antonio, Texas, 2000, 7 p.

ACKNOWLEDGMENTS The authors are pleased to acknowledge for their financial and technical support, Bombardier-aeronautic, the Consortium for Research and Innovation in Aerospace in Quebec (CRIAQ) and the research centre of Hydro-Quebec (IREQ). 782
