## 3 Results
### Prototype Characteristics
#### 3.1.3 Multimodal Dynamic Characteristics  
The FEM of the prototype with correct leaf mass was used to identify the modes of the branches and the trunk. A total of 208 vibrational modes were calculated, where a branch mode is defined as the motion when a single or multiple branches sway independently without the trunk swaying (or with the trunk swaying slightly), while a trunk mode is defined as when the trunk sways significantly. Figure 6a illustrates the frequencies of the first twenty-four vibrational modes in orthogonal directions. The first twenty-two vibrational modes are associated with branches, while the 23 rd and 24 th vibrational modes are associated with the trunk. This indicates that the branch modes are characterized by a high modal density. To further identify these dense branch modes, Figure 6b presents the effective mass fraction of each mode. It can be seen that the vibrational modes with large effective mass fraction are the 1 st, 2 nd, 3 rd, 4 th, 23 rd and 24 th modes, which account for 13%, 14%, 9%, 10%, 17% and 18%, respectively. The cumulative effective mass fraction of the remaining branch modes accounts for less than 1%. This indicates that the contribution to dynamic response of the first four branch modes is far larger than that of other branch modes. Through checking the mode shape of each branch mode, it is found that branch modes with a small effective mass fraction have localized vibration of a few branches, while branch modes with a larger effective mass fraction are collaborative vibration of multiple branches. This indicates that the number of branches swaying in phase determines the magnitude of effective mass fraction of the branch mode.  
The mode shape of branch mode one and two at 1.48 Hz are for all when all branches sway in phase (Figure 6c), while the mode shape of branch modes three and four at 1.83 Hz are when multiple branches sway in phase with a few branches swaying out of phase (Figure 6d). The mode shape of trunk modes 23 and 24 at 5.18 Hz are for when the trunk sways out of phase with the crown (Figure 6e). The cumulative effective mass fraction of the aforementioned six modes is 81%, which indicates that it is sufficient to reflect the dynamic response of the prototype for the aeroelastic model of the decurrent tree to match crown and trunk frequencies (i.e., 1.49-1.83 Hz, 5.18 Hz) of the prototype. The parameters of dynamic characteristics of the prototype are summarized in Table 3. Because the damping of the prototype at the crown frequencies is not identified, the damping of broad-leaved trees at fundamental frequencies is provided as a reference [34].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Prototype** & **Aeroelastic Model (Full Scale)** \\ \hline Total leaf mass & 15.41 kg & 0–23.09 kg \\ Crown frequency & 1.49–1.83 Hz & 1.14–1.98 Hz \\ Trunk frequency & 5.18 Hz & 4.81–6.49 Hz \\ Crown damping & 0.06–0.08 [34] & 0.085 \\ Trunk damping & 0.054 & 0.033 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of dynamic characteristics of the prototype and the aeroelastic model.  
Figure 5: Relationship between fundamental trunk frequency (\(f_{trunk1 st}\)) and total leaf mass (\(m_{leaf}\) ).