# Numerical and Experimental Investigation of Guided Wave Propagation in a Multi-Wire Cable
## 3 Transient Finite Element Simulations
### Two Contact Wires  
When the two ends of the multi-wire cable are subjected to an axial tension, the wire-to-wire contact stress will appear between the adjacent wires. The magnitudes of the contact stresses are proportional to the axial tension. A simplified wire-to-wire contact model was employed to study the effect of contact stresses on UGW propagation in a multi-wire cable. The FE model was made up of two 7-mm-diameter parallel straight steel wires with the same length of 500 mm. One of the wires was active and the other was passive. This pair of wires was mechanically coupled in contact using uniform stresses. The contact stresses were provided by a static radial pressure equaling 70% of the ultimate tensile strength (UTS) of the wire. These contact stresses are called Hertzian contact stresses [35]. The contact condition between the two wires was modeled using a penalty friction formulation with the friction coefficient of 0.6. The loading conditions in the FE simulations involved two stages: static load and dynamic load. The linear ramp signals were uniformly preloaded into two wires along the wire axes with opposite radial directions of the two wires. The static wire-to-wire contact and friction stresses were generated in this load step. In the second step, guided waves were excited by the dynamic load. An investigation of the wave propagation was conducted in two cases. In Case 1, the active wire was excited with an obliquely imposed triangular pulse force on one end face with equal components (\(F_{x}=F_{y}=F_{z}\)) along three directions. In Case 2, the active wire was excited with a Hann-windowed, five-cycle, 60 kHz sinusoidal tone burst perpendicular to the end face along the \(z\)-direction. Acceleration data were recorded at the surface nodes of the FE elements of the two wires along the \(z\)-axis. The interval of the recorded points was set as 1 mm. The 2D FFT algorithm proposed in Section 3.1 was used again to compare the two cases to observe the change in the dispersion properties of UGWs in a pair of mechanically coupled wires. The 2D FFT transform spectra for the two cases are shown in Figures 9 and 10. In Figure 9, the dispersion nephograms in the 2D FFT spectra coincide with the theoretical phase velocity dispersion curves (see Figure 1a), which is consistent with the results obtained in the previously presented single wire case. The longitudinal mode (i.e., L(0,1)) and all flexural modes (i.e., F(1,1), F(1,2), and F(1,3)) below 500 kHz were generated in the active and passive wires. Guided waves in the passive wire are transferred by contact and friction stresses between two wires. The acoustic energy can be transferred into the adjacent wire when sufficient contact stresses occur on the interface of two wires. As a result of the leakage of energy, the guided wave modes and energy eventually trend to be the same, with the wave propagation in the two-wire system. Figure 8: The phase velocity dispersion spectrum from nodal accelerations with a Hann-windowed, five-cycle, 60 kHz sinusoidal tone burst along the \(z\)-direction. The theoretical phase velocity dispersion curve of the L(0,1) mode obtained in Figure 1a is represented by a black dotted line. The CAN model described in Section 2.2. and the energy transfer model described in Section 2.3. provide two different perspectives and methods to analyze the dispersion properties and propagation behaviors of the guided wave in mechanically coupled structures. Acceleration data recorded on two  
Figure 10: The phase velocity dispersion spectra of wires from nodal accelerations with a Hann-windowed, five-cycle, 60 kHz sinusoidal tone burst along the \(z\)-axial direction: (**a**) the active wire and (**b**) the passive wire. The theoretical dispersion curve of L(0,1) obtained in Figure 1a is represented by white dotted lines. (**c**) The FFT spectra of a representative received signal of the active wire and (**d**) part of the representative acceleration time-domain-received signals in the active wire.