Crack Localization Using Transmissibility of Operational Deflection Shape and Its Application in Cantilever Beam

Due to the nonlinearity of breathing crack, cracked structure under excitation of a single frequency always generates higher harmonic components. In this paper, operational deflection shape (ODS) at excitation frequency and its higher harmonic components are used to map the deflection pattern of cracked structure. While ODS is sensitive to local variation of structure in nature, a new concept named transmissibility of operational deflection shape (TODS) has been defined for crack localization using beam-like structure. The transmissibility indicates the energy transfer from basic frequency to higher frequency. Then, Teager energy operator (TEO) is employed as a singularity detector to reveal and characterize the features of TODS. Numerical and experimental analysis in cantilever beam show that TODS has strong sensitivity to crack and can locate the crack correctly.

maged beam structures are not needed if they are geometrically smooth and made of materials that have no stiffness and mass discontinuities. The damage index was shown as a contour plot of frequency versus position. Yoon et al. [3] expanded the GSM and developed a Global Fitting Method by introducing a globally optimized smooth shape using ODS to improve the performance of damage detection for various damage. Zhang et al. [4] proposed the Global Filtering Method based on ODS curvature extracted from dynamic response of a passing vehicle. Only ODS at few frequencies near the first natural frequency is needed and pre-filtering process is applied on the ODS to reduce the numerical error. A vibration testing method using superposed waveform method (SWM) is proposed by Feng and Li [5] for damage detection in structures. Propagating wave data were acquired by a scanning laser Doppler vibrometer to calculate the ODSs at each frequency using the SWM.
Since it is not always possible to locate cracks in structures only using the ODS [6], a method combining the ODS with transmissibility is proposed in this study. Zhang et al. [7] and Mottershead [8] demonstrated that transmissibility is determined solely by system zeros, which makes the transmissibility sensitive to the local variation in nature. ODSs provide the geometric feature of structure, while conventional transmissibility shows the energy transfer from point to point. Under the excitation of a single frequency, structure with nonlinear breathing crack will generate higher harmonic components, which contain the information of damage location. A new concept named transmissibility of operational deflection shape (TODS) has been defined in present paper. The new transmissibility is estimated using the ratio between the higher harmonic ODS and basic harmonic ODS, which shows the energy flow. Energy transformation in the crack position will be discontinuous. Teager energy operator (TEO) is employed as a singularity detector to reveal this feature of TODS.
The paper is organized as follows. Section 2 presents the definition of TODS based on the conventional transmissibility, and the procedure for crack location using the dynamic response. Section 3 shows several numerical simulations to verify the effectiveness of indicator. Section 4 describes the experiment used to validate the new method in cantilever beam. Finally, conclusions are presented in Section 5. ( ) k F ω is the input spectrum. It can be observed that the common denominator, whose roots are the system's poles, vanishes by taking the ratio of the two response spectra. Consequently, the poles of the transmissibility equal the zeroes of transfer function. In general, the peaks in the amplitude of transmissibility do not coincide with the resonances of the system. From the Equation (1), one important property of transmissibility can be derived. Transmissibility is independent to the input and system poles, but is dependent on the location of input and system zeros. While system poles is function of all the system dynamic parameters, system zeros is influenced by the local subset. Hence, transmissibility is sensitive to the local variation in nature, which gives the advantage for damage detection.

Transmissibility of Operational Deflection Shape
A breathing crack opens and closes alternatively during every cycle of loading and consequently produces the nonlinear phenomenon of the cracked structure.
When the cracked structure is excited at a single frequency, higher harmonics of the exciting frequency are generated due to the nonlinear dynamic. Since the nonlinearity is generated by the crack, higher harmonics can be used to locate the crack. The ODS of the cracked beam can be generated at the exciting frequency and the higher harmonic frequencies to map the deflection of the cracked structure. However, it has been observed from [6] that the ODSs at the higher harmonics cannot always distinguish the crack location due to the influence by the ODS at the frequency of excitation. As a result, in order to reduce the effect of the basic harmonic and the nonlinearity due to the breathing crack, transmissibility of the ODSs (TODSs) between the higher harmonics ODSs and the basic harmonic ODS are proposed in this study.
is the m-th harmonic ODS at mode k.

Teager Energy Operator of TODS
The energy transfer will generate the singular phenomenon. To detect and high-

Numerical Simulation
In this section, the proposed method is applied in a beam structure with different kinds of breathing crack. Numerical simulation is conducted to verify the effectiveness of the proposed damage indicator.  Table 1. Five different crack scenarios are considered in this simulation. Intact scenario I is utilized to compare the natural frequencies with the other damage scenarios. It can be seen that natural frequencies change slightly with the increase of crack number and depth ratio, which means that it is difficult to detect the crack only from the natural frequencies.
The cantilever beam is excited by a sinusoidal input at its first natural frequency according to Table 1. The breathing crack was assumed to be closed when the displacement of the nodes of the cracked element in y-direction is greater than zero for the applied excitation and to be open when that displacement

Experimental Verification
The approach proposed in the preceding section is validated experimentally only using the response of beam structures under sine excitation in this section.
Cracked beam clamped at its right end is considered in this experiment. Figure 8 shows the test setup. Seven acceleration sensors are installed at equal interval of 6 cm along the beam from the fixed end. Three crack scenarios are studied in this experiment as shown in Table 2. One single crack is created by saw cut between the 3 th and 4 th sensors with different depth. The corresponding natural frequencies are listed in Table 2.
After the experimental system is assembled, test is implemented follow on.
Step 1. Random excitation is generated to drive the shaker and vibration responses from all the seven acceleration sensors are recorded. The sampling frequency is 1024 Hz.
Step 2. OMA method is used to calculate the natural frequencies from the time domain responses.
Step 3. Sinusoidal excitation with first natural frequency is generated to drive  Step 4. The proposed method in Section 2 is used to identify the crack position with the responses.
The experimental data are processed using the proposed method for all the three crack scenarios. For crack scenario I, the crack location can be identified exactly from the TTODS in Figure 9(a). For crack scenario II, the negative peak of TTODS appears near the 3 th sensor in Figure 9 Further research is carried out using the comparison with the modal curvature [10]. Modal curvature is a popular conventional damage detection method based on modal shape, and can be computed by the second-order central difference of the mode shape. Since the excitation frequency is the first natural frequency, ODS at 1x is the first modal shape. So, modal curvature can be applied here. All three crack scenarios are calculated using modal curvature as shown in Figure 10. It can be seen that the modal curvature can't locate the crack position at all. One reason might be there are not enough sensors. Another reason is the modal curvature is susceptible to noise. Thus, the proposed method using transmissibility of ODS has the capacity to locate the damage position.

Conclusion
Structure with breathing crack will generate higher harmonics of the frequency