Firstly, the wave propagation response and the velocity dispersion in classical rod structure and in Rayleigh-love rod are shown in Fig. Figure 10 a, b , because the waves of each frequency component propagate at the same rod speed. However, for the Rayleigh-Love rod theory, it is can be seen from Fig.

The group velocities of waves in the vicinity of Hz change slowly, while those of in the vicinity of Hz have lower speeds and change more quickly. When the ratio of the cross-section size to the wavelength is less than 0. Otherwise, it is necessary to develop and apply the complex multi-dimension theory. And it is an important preparation for SHM to select the proper frequency and mode of wave according to the dispersion property of waves.

Figure 11 shows the ultrasonic guided wave propagation in the rod with crack in middle of rod, in which the excitation signal is shown in Eq. Also as the time goes, the waveform reflected by the crack is more and more obvious, while the excitation wave would go across the crack and propagate along the rod structure until it is reflected by the right end of the rod again, at the same time, the wave reflected by the crack would propagate along the rod to the left end and is reflected by the left end of the rod. The time frequency analysis results of the wave propagation signal along the rod structure shows the central frequency is moving along the rod, and the central frequency is separated into two parts as the wave signal is passing the damage.

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Furthermore, we would like to study the influence of the crack depth on the wave propagation in the rod, so several different ratios of crack depth are compared in our manuscript. Here we would fix the crack in the middle of the rod while the crack depths are set with different ratios, the following plots show the received wave plots at the right end of the rod. The percentage of damage is evaluated as the ratio of crack depth with respect to the height of rod.

As we could see from the left plot in Fig.

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The amplitude of the flaw signal is proportional to the crack depth, the amplitude of the flaw signal would go up as the crack depth increases. Also the amplitude of direct wave signal would decrease when the crack depth increases if we take a look at the right plot of Fig. Another important factor that we studied in this manuscript is the crack location, which is shown in Fig.

In this case, we applied the same excitation signal on the left end and receive the signal on the right end of the rod, the depth of the cracks are set as 0. Also the direct waves received by the right end of rod are the same. As shown in the plot, different amplitudes of different waves are received, which shows that the crack wave amplitude is much smaller than direct received wave, and the wave signal shows much more complex than those signals received for only one crack.

In this manuscript, a novel numerical Laplace based wavelet finite element method is proposed for ultrasonic wave propagation and nondestructive testing in rod structures. Laplace transform is a more advanced method than fast Fourier transform that Laplace transform does not depend on the periodic assumption while Fourier transform does. Also BSWI is a wavelet based finite element method that has been used in ultrasonic wave propagation and has a lot of advantages.

By combining the advantages of the two methods, the following conclusions could be achieved:. Laplace transform is a symbol-based transform method, but still could be achieved via numerical method, but Laplace transform could abandon the periodic assumption of FFT. By comparing the group velocity and wave propagation in rod, we could see that LWFEM is a very reliable numerical method that could be used in ultrasonic wave propagation and nondestructive testing of rod structures. By studying the sensitivity of mesh size and time interval with different numerical methods, we could conclude that LWFEM has much lower element size and time interval requirement than traditional FEM but could still provide the necessary accuracy of results.

Although it shows similar results with FFT based FEM, this is because both methods are solved in frequency domain and the two methods have similarities.

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The velocity dispersion could not be clearly recognized and the waveforms almost have no change in the process of propagation in the classical rod element theory, while the Rayleigh-Love rod theory is able to give a good approximation for the dispersion when the ratio of the cross-section size to the wavelength is less than 0.

LWFEM is a reliable numerical method for nondestructive testing in rod structure and could recognize both small and large damages in the rod structure. Also the crack location also has a great influence on the received signals on the rod structure.

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Also the amplitudes of the flaw signal received by the rod are highly influenced by crack depth, which was proved by FFT based BSWI simulation results. Chen, B. Chen, Y. Secure in-cache execution. Springer, Cham. Chen, H. Chen, X. Journal of Vibration and Acoustics, 5 , A condensation method for the dynamic analysis of vertical vehicle-track interaction considering vehicle flexibility.

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## Guided Waves in Structures for SHM: The Time - domain Spectral Element Method

Energy and Buildings, , pp. Ma, J. Marfurt, K. Mitra, M. Gopalakrishnan Moser, F. Park, J. Numerical analysis of helical dielectric elastomer actuator. International Society for Optics and Photonics. Przemieniecki, J. Theory of matrix structural analysis, Courier Corporation. Rose, J. Royston, T. Estimating material viscoelastic properties based on surface wave measurements: A comparison of techniques and modeling assumptions.

The Journal of the Acoustical Society of America, 6 , pp. Saenger, E. Seemann, W. Shen, W. Tada, H. Tang, Q. Finite element simulation of ultrasonic waves in corroded reinforced concrete for early-stage corrosion detection. Finite element simulation of photoacoustic fiber optic sensors for surface corrosion detection on a steel rod. Xiang, J. Yam, L. Zhang, Z. Entropy, 20 1 , Applied Sciences, 7 2 , A thermography-based method for fatigue behavior evaluation of coupling beam damper. Fracture and Structural Integrity, 40 , This is an open-access article distributed under the terms of the Creative Commons Attribution License.

Services on Demand Journal. Original article Nondestructive ultrasonic testing in rod structure with a novel numerical Laplace based wavelet finite element method. Abstract Rod structure has been widely used in aerospace engineering and civil engineering. Numerical Models 3. Numerical Examples Several numerical examples of wave propagation simulation in rod structures are proposed to validate the Laplace based wavelet finite element method, a uniform rod is used in the numerical simulation, the geometry parameter and material properties are shown in Table 1.

Lamb wave interaction with a notch is investigated by using this method, and the results obtained are with respect to transmission, reflection and mode conversion. Because of the symmetric mode shape, S0 is more sensitive to the shallow notch than A0. By making use of the fact that the reflection increases with increase in notch depth and mode conversion are maximized when the notch is around half through the thickness of the plate, the reflection and conversion coefficients can be used to characterize the depth of the notch.

Gresil and V. Request Permissions. New York Plenum Press, I: Wiley Inter-sciences publication, Cambridge University: Cambridge University Press, All Rights Reserved. Log In. Paper Titles. Guided Waves for Aircraft Panel Monitoring p. Article Preview. Key Engineering Materials Volume Main Theme:. Structural Health Monitoring: Research and Applications. Edited by:.

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Online since:. June Matthieu Gresil , Victor Giurgiutiu. Add to Cart. Cited by. Related Articles. Paper Title Pages. By using S-FEM technique, only local mesh should be re-meshed and it becomes easy to simulate crack growth. By combining with auto-meshing technique, local mesh is re-meshed automatically, and curved crack path is modeled easily. Plural fatigue crack problem is solved by this technique. For two parallel crack problem, criteria of crack coalescence are proposed. By simulating this problem by S-FEM, it is verified these criteria are conservative ones.

Authors: Hong Shuang Zhang.