## 3 Numerical Modelling

Table 1: Division into nodes and finite elements of the measurement bases. In the case of modal analysis conducted with the use of engineering software based on the Finite Element Method algorithm, it is possible to determine modal parameters of structure without defining any boundary conditions. However, the mode shapes obtained in this way (so-called free modes of structure) will be far from reality because they will in no way consider the interaction between the ground and the measurement bases embedded in the ground. Another possible solution is to model the structure on an elastic foundation or blocking selected degrees of freedom of construction (for example, in the case of piles, a common method is a pile fixed at the head [39]). In the case of modal analyses of small-scale bases for measuring vibration in the ground, this solution, unfortunately, cannot be used. For this reason, in the planned numerical analyses, it was initially decided on a simplified solution in the form of simulating interaction at the interface of the ground and the element embedded in it by applying boundary conditions in the form of \"flexible support\", wherever there is contact between the measuring base and the ground. Given the very small cross-section of the spikes of the measurement bases embedded in the ground (significantly smaller not only than piles, but also smaller than micro piles), the elastic support values from a much wider range (from 0.005 N/mm\({}^{3}\) to 0.1 N/mm\({}^{3}\)) were adopted for numerical analyses than it would result from design practice in the case of modelling flexible foundation on the basis of various types of buildings. The analysis of preliminary results showed that in the case of the lowest eigenmodes frequencies (with frequencies in the range from 0 Hz to several dozen Hz) rigid body motion occurs. Further mode shapes, however, could not be considered reliable. Unfortunately, it is not possible to perform a reliable modal analysis for three-dimensional measurement bases anchored in a soil medium, considering the non-linear interaction that occurs at the interface between the measurement base and the ground. In modal analysis it is assumed that the structure being the subject of analysis or experimental research is linear. The soil medium is not linear. Even if the soil medium is modelled as homogeneous and isotropic, and the contact of the ground with the measurement base is assumed to be of the \"bounded\" or \"no separation\" type, the mass of the soil itself will cause the identified eigenmodes to be largely vibrations of the ground body. Therefore, in the next stage of numerical analyses, a compromise solution was decided, in which the measurement bases were modelled in a cylindrical soil block with real rigidity and mass selected so that it did not exceed 1% of the mass of the measuring bases with the heaviest seismic transducers mounted. Figure 2: The measuring bases after meshing: (**a**) No. 1, (**b**) No. 2, (**c**) No. 3. Considering the computational capabilities of modern efficient workstations and the real small range of interaction occurring between the measurement bases, a soil body with a diameter of 200 cm and a height of 100 cm was adopted for numerical simulations. The adopted dimensions of the soil body correspond to the conditions that we can provide in laboratory conditions for the needs of any possible experimental or operational modal analyses. Soil parameters for numerical simulations were adopted from geotechnical tests carried out at the Logistics Center Lososna in North-Eastern Poland [40] (homogeneous loose and medium dense coarse sands (cSa) with a density 1844 kg/m\({}^{3}\) (\(\rho\)), a Young's modulus E\({}_{0}\) in the range of 90.1-116.2 MPa and a Poisson's ratio v = 0.264-0.309). Eventually, the following, averaged mechanical soil parameters were adopted for numerical analyses: E\({}_{0}\) = 100 MPa and v = 0.3.