*3.1. Rod Structure*

As a numerical example, a uniform structural element of a circular cross-section has been selected, as presented in Figure 3. It has been assumed that the length of the element is *L* = 2000 mm, while its

diameter is *d* = 50 mm. It has also been assumed that the element is made out of aluminium alloy (elastic modulus *E* = 67.5 GPa, Poisson ratio *ν* = 0.33, material density *ρ* = 2700 kg/m3). It has been assumed that the element is modelled by the use of elementary rod theory. For this type of element, the analytical solutions are commonly known. Therefore, it was easy to validate the correctness of the determined dynamic parameters; i.e., the natural frequencies and modes of natural vibrations. Several cases has been analysed for different approximation functions, in particular:


**Figure 3.** Geometry of a uniform aluminium rod.

It should be stressed that the application of Hermite and B-spline approximation polynomials enforces the continuity of the strain and stress fields. In the case of Hermite polynomials of the third degree, *p* = 3, this only comprises the first derivatives of the displacement fields (i.e., the strain and stress fields remain continuous but not necessarily smooth). In the case of Hermite polynomials of the fifth degree, *p* = 5, and for B-spline approximation polynomials of the third degree, *p* = 3; the first and second derivatives of the displacement fields are continuous (i.e., strain and stress fields remain continuous and smooth). It should be also added that in the the case of numerical analysis, three different types of boundary conditions have been employed. Firstly, for the analysis of natural frequencies and mode shapes, the clamped-clamped boundary conditions for rods and simply-supported boundary conditions for beams have been used. Secondly, for the analysis of wave propagation, the free–free type of boundary conditions have always been in use, despite the type of structural element.

Computational analysis of structural dynamics requires appropriate modelling of dynamic responses, which are directly linked with accurate modelling of structural natural frequencies and mode shapes. Figure 4 shows the results of numerical calculations of natural frequencies for the same rod (Figure 3) modelled by the listed numerical models. The letter *N* stands for the number of elements used in each analysed case. On the basis of the results presented, it may be concluded that some numerical models show certain discontinuities in their frequency spectra. This phenomenon is caused by the discontinuity of strain and stress fields expressed by spatial derivatives of the approximation functions [63]. That is particularly well visible in the case of Chebyshev approximation polynomials; however, the frequency spectra related to the use of Hermite or B-spline approximation polynomials seem free of such discontinuities.

**Figure 4.** Natural frequencies of the rod calculated based on different approximation polynomials, in the case of the rof with fixed ends.

In order to determine the extent to which the presence of such frequency gaps influence the correctness of the results obtained, the following diagrams (Figure 5) have been presented to illustrate the changes in the relative error of the rod natural frequency spectra. It is clearly seen that the use of a higher approximation Chebyshev polynomial results in a significant increase in the relative errors of frequency spectra corresponding to higher frequencies. On the other hand, for the same order of approximation polynomials, but in the case of Hermite polynomials, results in the increase of the relative error are on a level comparable to the error present in calculations obtained by the use of other approximation functions. Moreover, a small change in the relative frequency error may be noticed for all approximation functions analysed. The value of this error seems to be dependent not only on the smoothness of the approximation polynomials used, but the type of node distribution: non-equidistant (Chebyshev *p* = 3 or Chebyshev *p* = 5) or equidistant (Hermite *p* = 3, Hermite *p* = 5, or B-spline *p* = 3).

**Figure 5.** Relative errors of natural frequencies of the rod calculations based on different approximation polynomials, in the case of the rof with fixed ends.

In the analysis of the dynamics of the examined structure, properly represented modes of natural vibrations, obtained numerically, are of great importance. In order to verify the correctness of the results obtained by the proposed numerical models, the errors in the representations of calculated modes of natural vibrations have been determined in comparison to the modes known from analytical solutions for the elementary rod theory, as seen in Figure 6. 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 6 that in each case considered, the fitness decreases for higher frequencies. Moreover, there are such approximation polynomials (i.e., Chebyshev and Hermite), for which there are strict boundaries of total incompatibility of the determined modes. These discontinuities appear at natural frequency numbers that are multiples of the numbers of finite elements of numerical models [63]. Based on the results obtained, it can be concluded that the correctness of numerical calculations in the dynamics of the rod under investigation is significantly determined by the type of approximation polynomials. The appearance of the first frequency gap limits the usable part of the frequency spectrum in a much more significant manner than the observable decrease of the fitness for higher frequencies. Only in the cases of Chebyshev polynomials of the first degree (linear shape functions) *p* = 1, and B-spline approximation polynomials of the third degree *p* = 3, do the fitness lines remain smooth. In the first case out of these two, however, the values of fitness decrease for higher frequencies much faster; thus, the usable part of the frequency spectrum is smaller.

**Figure 6.** Fitness of natural vibration modes of the rod calculated based on different approximation polynomials, in the case of the rof with fixed ends.
