# Evaluation of Dynamic Properties of Trees Subjected to Induced Vibrations
## 3 Materials and Methods
### Field Measurements and Signal Elaboration  
Field measurements were carried out by considering the excitations induced in the tree by pulling tests and ground vibrations due to agricultural machinery. Both tests were performed at the end of January 2023 under wind-free conditions and dry soil. The accelerations exhibited by the tree were recorded by means of a single accelerometer (acceleration range: \(\pm 16\) g; acceleration resolution: 0.005 g; sample frequency: 100 Hz) once placed at section H1 and once at section H2 of the trunk (Figures 1 and 2), along NS direction and EO direction. Sections H1 and H2 were selected in order to place the accelerometer near the removed branch (section H1) and near the two main branches. Regarding the pulling tests, they were carried out by imposing a lateral displacement to the trunk of about 5 cm (either along NS direction or along EO direction) throughout a cord fixed at section H2, and then suddenly releasing it by inducing free vibrations to the whole tree. In addition, pulling tests were performed before and after removing the branch located near the base of the tree (Figure 3). In detail, as reported in Table 1, four pulling tests were performed on the tree before removing the branch, and the same tests were performed after the removal of the branch located next to the base. Regarding the field measurements in the case of ground vibrations, they concerned the accelerations induced to the tree (in this case the accelerometer was placed at section H2 only) by agricultural machinery (rotary filler) working in proximity to the tree. In particular, the measurements were performed once along EO and once along NS directions (Table 1) when, in both cases, the agricultural machinery followed a squared path around the tree with a distance of about 2 m from the trunk. As shown in Table 1, in order to identify the different tests, labels containing the indication of the removal of the branch (symbol 'r1'), the direction of the applied displacement (symbol 'T' followed by 'NS' or 'EO', respectively, for the directions north-south and east-west), and the direction of the recorded signal (symbol 'S' followed by 'NS' or 'EO',  
Figure 3: Schematization of the tree assumed as case study (trunk and main branches). respectively, for the directions north-south and east-west) were employed for pulling tests. In the case of ground vibration, the symbol 'T' was replaced by 'GV'. The results deduced from field measurements consisted of acceleration time histories recorded with a sample frequency of 100 Hz. In order to perform dynamic identification in the frequency domain of the main frequencies characterizing the dynamic response of the tree, the power spectrum density (PSD) function was evaluated for each measurement. In particular, PSD was computed by using [PERSON]'s method [22]. It consists of the following main steps:  
* Partition of the data sequence X into a number K of segments (or batches), each composed of M number of points. The M-point sequence represents the window function. Among the commonly used window functions, they are the Rectangular, Harm or Hanning, Hamming, Blackman, Blackman-Harris, and Kaiser-Bessel. On the other hand, the parameter K represents the number of periodograms that are averaged together to form the PSD estimate;
* Computation for each segment of a windowed discrete Fourier transform (DFT) at some frequency values;
* Evaluation of the modified periodogram value from the discrete Fourier transform;
* Average of the periodogram values to obtain [PERSON]'s estimate of the PSD. This method is also called the Weighted Overlapped Segment Averaging (WOSA) method. Indeed, it requires the introduction of a parameter S referring to the number of points to shift between segments (it corresponds to the number of new points in each segment or batch). Consequently, two adjacent segments have M-S points in common or, in other words, they are overlapped by M-S points. This means that, in the case of M = S, the adjacent segments do not overlap.