*3.2. Beam Structure*

It should be emphasised that the results of numerical investigations presented so far have been directly related to the propagation of longitudinal elastic waves in rod structural elements. However, the conclusions drawn can be easily generalised and extended onto other types of structural elements, such as beams, plates, shells, or solid elements. As the next numerical example, a beam of a uniform cross-section has been selected, as presented in Figure 3. All geometrical and material properties, as well as the form and type of boundary conditions and excitation signals were the same as in the case of the analysed rod element. The modelled beam has been divided into the same number of spectral finite elements. The elementary theory for beams has been employed in this analysis.

Figure 10 presents the results of natural frequency calculations of the beam modelled by means of several numerical models using different approximation polynomials: Hermite polynomials of the third order *p* = 3, Hermite polynomials of the fifth order *p* = 5, and B-spline approximation polynomials of the third order *p* = 3. Again, the letter *N* stands for the number of finite elements used in each analysed case. On the basis of the results presented, it may be concluded that numerical models based on Hermite approximation polynomials show certain discontinuities in their frequency spectra. This phenomenon is particularly visible near the boundaries of frequency bands related to the total number of finite elements used in the analysis. As before, these are the so-called frequency band gaps and their appearance is a direct proof of the periodical nature of the numerical models under investigation [63].

**Figure 10.** Natural frequencies of the beam calculated based on different approximation polynomials, in the case of a beam with simply-supported ends.

In order to determine the extent to which such frequency gaps influence the correctness of the results obtained, the following diagrams (Figure 11) have been presented to illustrate changes in the relative error of the beam's natural frequency spectrum. It is clearly seen that Hermite polynomials of the third order *p* = 3 result in significant increases in the relative errors in the parts of the frequency spectra corresponding to higher natural frequencies. On the other hand, Hermite polynomials of the fifth degree *p* = 5 result in an increase in the relative error that reaches only 10%. Moreover, the minimal changes in the relative frequency error are noticed for B-spline approximation, for the third degree of approximation polynomials *p* = 3. It can be summarised that the nature of the chances in the relative frequency error appears as independent of the order of approximation polynomials or node distribution—non-equidistant or equidistant—but it is dependent on the order of the smoothness of the displacement field in relation to the order of approximation polynomials.

**Figure 11.** Relative errors of natural frequencies of the beam calculated based on different approximation polynomials, in the case of a beam with simply-supported ends.

Additionally, in the case of beam elements, in order to verify the correctness of the results achieved by the proposed numerical models, errors in the representation of the mode shapes have been determined (Figure 12). Again, the fitness value equal to 1 indicates the maximum degree of fitness (i.e., the coefficient of determination) of the determined modes of natural vibrations, while values smaller than 1 indicate inaccurate fitness.

It can be seen from the diagrams presented in Figure 12 that in each case considered the fitness decreases for higher frequencies. Moreover, in both Hermite approximation polynomials there are are strict boundaries of total incompatibility of the determined modes. These discontinuities appear at natural frequency numbers that are multiples of the number of finite elements of numerical models. As before, in the case of the beam under investigation it should be emphasised that the correctness of numerical calculations is significantly determined by both the number of finite elements and the degree of the polynomial approximation function, as these parameters determine the extent of the usable part of the frequency spectrum that is up as the first frequency band gap. Only in the case of B-spline approximation polynomials of the third degree *p* = 3 are there no visible frequency band gaps in the frequency spectra, and the line representing fitness of the calculated modes remains smooth. Additionally, in this case the usable part of the frequency spectrum is the greatest.

**Figure 12.** Fitness of natural vibration modes of the beam calculated based on different approximation polynomials, in the case of a beam with simply-supported ends.
