# Numerical and Experimental Investigation of Guided Wave Propagation in a Multi-Wire Cable
## 3 Transient Finite Element Simulations  
The effective performance of the conventional transient FE technique for modeling UGW propagation in various structures has previously been demonstrated. Here, the transient solver ABAQUS(r) (Dassault System Inc., Paris, France) was employed to visually simulate UGW propagation in two different structures: an individual wire and two contact wires. The accuracy of the results is affected by the temporal and spatial resolution of signal acquisition in the FE simulation model. The time step for dynamic analysis (i.e., temporal sampling) and finite element size (i.e., spatial sampling) were determined by the maximum frequency (i.e., minimum wavelength) of the wave in the structure, which should meet the requirements of the Nyquist-Shannon sampling theorem [31]. An optimal spatial resolution is at least eight mesh nodes per wavelength [32; 33]. The temporal sampling rate should be 10 times higher than the highest frequency component of the signal. The parameters of the FE simulation model are summarized as Table 2.  
The dispersion of the dispersive guided wave propagating in a waveguide is relevant to the bandwidth of the excitation signal. The wider the bandwidth of the excitation signal, the greater the dispersion of the wave. In this paper, two types of excitation signals were used in the transient FE simulation to investigate the dispersion performance of the guided waves in the wire. First, the time-dependent triangular pulse signal was employed to excite a broadband signal with a frequency bandwidth exceeding 500 kHz, as shown in Figure 5a,b. Second, the Hann-windowed, five-cycle, 60 kHz sinusoidal tone burst was designed to excite the narrow bandwidth signal with a specific center frequency, as shown in Figure 5c,d.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Diameter** & **Length** & **Density** & **Young Modulus** & **Poisson Ratio** \\ \hline
7 mm & 500 mm & 7800 kg/m\({}^{3}\) & 209 GPa & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The parameters of the transient FE simulation model.  
Figure 4: A model of the wave energy transfer between two adjacent wires. The results of the active wire are plotted by a red line, and the results of the passive wire are plotted by a blue line. The transient finite element (FE) simulation data of the two wires obtained in Section 3.2. are marked by asterisks.  
The category of the UGW mode generation in a waveguide is related to the direction of the exciting loads. In order to achieve optimal excitation performance, the direction of vibration displacement of the loading was set to be consistent with the wave structure of the wave mode to be excited. According to the analysis of the wave structures of longitudinal and flexural guided waves in Section 2.1, the oblique imposed force with uniform components in the three directions was chosen to excite longitudinal and flexural mode waves. The perpendicular imposed force with a single component along the \(z\)-direction was chosen to excite pure longitudinal mode waves, as shown in Figure 6.  
Figure 5: The excitation signal and fast Fourier transform (FFT) spectrum: the triangular pulse with a broadband of 500 kHz in the (**a**) time domain and (**b**) frequency domain; the Hann-windowed, five-cycle, 60 kHz sinusoidal tone burst in the (**c**) time domain signal and (**d**) frequency domain.  
Figure 6: Guided waves in a single wire were excited using three loading methods: (**a**) an oblique triangular pulse with three equal components along the \(x\), \(y\), and \(z\) directions, (**b**) a perpendicular triangular pulse along the \(z\)-direction, and (**c**) a perpendicular sinusoidal tone burst along the \(z\)-direction.