# Evaluation of Dynamic Properties of Trees Subjected to Induced Vibrations
## 3 Materials and Methods
### Numerical Models  
In order to analyze the results deduced from the dynamic identifications, two simplified models were employed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Pulling Tests** & & \\ \hline \multirow{2}{*}{**Label**} & **With/Without** & **Pulling Section** & **Pulling Direction** & **Accelerometer Direction** \\  & **Branch (r1)** & **(H1/H2)** & **(TNS/TEO)** & **(SNS/SEO)** \\ \hline H1-TNS-SNS & with branch & section H1 & north–south & north–south \\ H1-TEO-SED & with branch & section H1 & east–west & east–west \\ H2-TNS-SNS & with branch & section H2 & north–south & north–south \\ H2-TEO-SEO & with branch & section H2 & east–west & east–west \\ \hline r1-H1-TNS-SNS & without branch & section H1 & north–south & north–south \\ r1-H1-TEO-SEO & without branch & section H1 & east–west & east–west \\ r1-H2-TNS-SNS & without branch & section H2 & north–south & north–south \\ r1-H2-TEO-SEO & without branch & section H2 & east–west & east–west \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performed field measurements. The first one consisted of approximating the dynamic response of the tree due to free vibrations induced by pulling tests throughout the dynamics of a single-degree-of-freedom (SDOF) viscously damped system subjected to free vibrations. Then, starting from the equation of motion governing the free vibration of SDOF,  
\[m\ddot{u}+c\dot{u}+ku=0 \tag{1}\]  
where \(m\) is the lumped mass of SDOF; \(c\) is the damping constant, representing a measure of the energy dissipated in a cycle of free vibration; \(k\) is the stiffness of SDOF; \(u,\ \dot{u},\ \ddot{u}\) are the displacement, velocity, and acceleration functions of the time t, respectively, introducing the initial conditions in terms of displacement and velocity characterizing the pulling test:  
\[u(0)=u_{0}\] \[\dot{u}(0)=0 \tag{2}\]  
and considering the case of underdamped systems (\(c<2\sqrt{k\cdot m}\)), the solution of Equation (1) in terms of displacement can be obtained:  
\[u(t)=\rho\cdot cos(\omega_{D}t+\theta)exp(-\xi\omega_{n}t) \tag{3}\]  
where:  
\(\xi=\frac{c}{2\sqrt{km}}\): equivalent damping ratio;  
\(\omega_{n}=\sqrt{\frac{k}{m}}\): natural circular frequency;  
\(\omega_{D}=\omega_{n}\sqrt{1-\xi^{2}}\): damped circular frequency. \(\rho\) and \(\theta\) are, respectively, the amplitude of motion and the phase angle that, considering the initial conditions of Equation (2), in particular, the initial velocity equal to zero, result in the following:  
\[\rho=\sqrt{u(0)^{2}+\left(\frac{\dot{u}(0)+u(0)\xi\omega_{n}}{\omega_{D}} \right)^{2}}=\sqrt{u(0)^{2}+\left(\frac{u(0)\xi\omega_{n}}{\omega_{D}}\right) ^{2}} \tag{4}\]  
\[\theta=-tan^{-1}\left(\frac{\dot{u}(0)+u(0)\xi\omega_{n}}{u(0)\ \omega_{D}} \right)=-tan^{-1}\left(\frac{\xi\omega_{n}}{\omega_{D}}\right) \tag{5}\]  
Consequently, the corresponding solution in terms of acceleration can also be obtained:  
\[\ddot{u}(t)=\rho\cdot\left[cos(\omega_{D}t+\theta)\left(\xi^{2}\omega_{n}^{2} -\omega_{D}^{2}\right)+2 sin(\omega_{D}t+\theta)\xi\omega_{D}\omega_{n}\right] exp(-\xi\omega_{n}t) \tag{6}\]  
The obtained solution describes the free-vibration response of SDOF in terms of accelerations. Then, in case the response in terms of acceleration is experimentally available and, moreover, frequency is deduced from a dynamic identification process, the above equation could be directly used for estimating the level of damping. This approach is presented in this paper where the measured response of the tree in terms of acceleration (time history provided by the accelerometer) has been fitted by the numerical response of an SDOF system considering the values of the identified frequencies and opportunely setting the level of damping ratio. This approach resulted in being particularly useful in providing qualitative information about the influence of the architecture of the tree on both the dynamic response and, moreover, the level of modal damping.