Wave Propagation Analysis

In order to illustrate the changes in the patterns of propagating flexural waves in the aluminium beam element, certain results have already been presented in Figure 13. The excitation signals used have been presented in Figure 7. Similarly to the examples discussed above related to the wave propagation analysis in the rod element, the signal carrier frequencies *fc* have been selected to take into account the effect of the periodicity of the numerical model on the correctness of the results obtained. Since the group and the phase velocities are different in the case of the elementary theory of the beam employed by the authors, some signal dispersion should be observed. As a consequence, the group speed of propagating waves also depends on the frequency, as it is equal to *cg* = 4854.1 m/s for *fc* = 150 kHz and *cg* = 3432.3 m/s for *fc* = 75 kHz. Therefore the time of the analysis have been adjusted accordingly in each particular case.

**Figure 13.** Wave propagation patterns of the beam calculated based on different approximation polynomials for the carrier frequencies *fc* = 75 kHz and *fc* = 150 kHz, in the case of a beam with free ends.

The last of the presented diagrams illustrates the changes in the patters of propagating waves in relation to the type of assumed approximation polynomials employed by numerical modes of the beam. The results presented in Figure 13 concern both excitation frequencies *fc* = 75 and *fc* = 150 kHz. Based on the results obtained, it can be concluded that in the case of the carrier frequency of 75 kHz the result fits quite well to the selected time window. On the other hand, in the case of the analysis carried out for the signal of the carrier frequency equal to 150 kHz, the influence of the periodic nature of the numerical model on the correctness of the obtained results is clearly visible. Signal frequency components near the frequency band gap are proportionally misrepresented in both aspects: their propagation speed related to increased values of natural frequencies and distorted shapes of modes of natural vibrations.
