Wave Propagation Analysis

Changes in the propagation of elastic waves have been successfully used in various systems for assessing the technical conditions of mechanical structures. For that purpose, numerical simulations may be an alternative method with which to avoid unexpected costs of maintenance. Modelling of elastic wave propagation requires the use of numerical models, which ensure the correct representation of the structure dynamics, especially for higher frequencies, in order to avoid any signal losses or distortions. This is why scientific research on the development of numerical methods that enable the analysis of this phenomenon has been very popular among various groups in the world.

For this purpose, simulations have been carried out to verify the developed numerical models in regard to their application to the analysis of the elastic wave propagation phenomenon. Similar to the described analysis of dynamic parameters, rod models have been developed by the use of elementary theory and the same approximation polynomials have been studied. An example of analysed excitation signals is shown in Figure 7, which illustrates the time signals and their normalised power spectral densities (*psd*). In both cases (the carrier frequency *fc* = 75 kHz and *fc* = 150 kHz) it was a sinusoidal

signal modulated by a Hanning window. The signal carrier frequencies *fc* have been selected to visualise the effect of the periodicity of various numerical models on the correctness of the results obtained. The value of the carrier frequency *fc* = 75 kHz is outside any spectrum discontinuities, whereas the carrier frequency *fc* = 150 kHz is located near such a discontinuity. It should be also noticed that in each case of the excitation signals, a certain range of frequencies <sup>&</sup>lt; *fc* <sup>−</sup> <sup>2</sup> *fm*, *fc* <sup>+</sup> <sup>2</sup> *fm* <sup>&</sup>gt; (*fm*—modulation frequency equal to *fc*/*m*, *m*—modulation equal to 10) has been simulated. For the carrier frequency of 75 kHz signal excited frequencies fall into the frequency range starting from 60 kHz and ending at 90 kHz, whereas in the case of the carrier frequency of 150 kHz, signal excited frequencies fall into the frequency range starting from 120 kHz and ending at 180 kHz. In both cases, the excitation amplitude was 1 N. Figure 7 shows the time and power spectra of the excitation signals.

**Figure 7.** Excitation signals for two different carrier frequencies *fc* in the time domain (**left**) and their power density spectra in the frequency domain (**right**).

In the following figures the examples of patters of propagating elastic waves have been shown. Thus, Figure 8 illustrates the changes registered for the rod under investigation in the case in which the rod is excited by a signal of the carrier frequency of 75 kHz. Illustrated signals have been registered at the excited end of the rod. The time of analysis has been set to 0.8 *ms* + 2/ *fm* and it was divided into 2<sup>13</sup> equal time steps. For each signal, an appropriate time window is marked, within which the propagating wave packet should be located at the end of the analysis. As a solution method for the equations of motion, the Newmark method has been chosen (*α* = 0.5, *β* = 0.25). Material damping has been neglected. It is worth mentioning that the selected carrier frequency of 75 kHz is beyond the boundaries of frequency spectrum discontinuity areas. As it can be noticed in the presented figure, the obtained patterns of propagating elastic waves, except the one obtained for Chebyshev polynomials of the first order *p* = 1, seem to be correct; i.e., there is no visible influence of the periodicity of numerical models used on the representation of the calculated patterns of elastic waves.

The next example (Figure 9) demonstrates similar changes, but registered for the excitation signal of the carrier frequency equal to 150 kHz. This carrier frequency is located close to the frequency band only slightly visible in the diagrams in Figures 5 and 6. A narrower time window is directly related to the shorter excitation time. In the discussed case it can be clearly noticed that a higher frequency of the excitation signal definitely requires reliable numerical models. In all of the numerical models based on the use of Chebyshev approximation polynomials, the registered signals have fallen outside the time window. One can see at least two examples of the registered signals, which may indicate the discursive nature of the wave propagation phenomena. However, this is very much misleading because the applied rod theory is non-dispersive [63]. The signals obtained by the use of numerical models based on the other approximation polynomials are represented in a correct manner.

**Figure 8.** Wave propagation patterns of the rod calculated based on different approximation polynomials for the carrier frequency *fc* = 75 kHz, in the case of the rod of free ends.

**Figure 9.** Wave propagation patterns of the rod calculated based on different approximation polynomials for the carrier frequency *fc* = 150 kHz, in the case of the rod of free ends.
