The trunk of the tree shown in Fig. 2.91 can be assumed to be a uniform cylinder of diameter \(d=0.25 \mathrm{~m}\) with a density of \(ho=800 \mathrm{~kg} / \mathrm{m}^{3}\) and Young's modulus of \(E=1.2 \mathrm{GPa}\), and the crown of the tree has a mass of \(m_{c}=100 \mathrm{~kg}\).

a. If the maximum deflection of the trunk, \(\delta\), due to a transverse force \(\left(F_{0}\right)\) applied at the top of the trunk is equal to \(40 \%\) of the buckling length needed for bucking to occur when the weight of the crown of the tree acts along the axial direction at the top of the trunk.

b. Find the natural frequency of vibration of the tree in sway motion.

State the assumptions made in the solution.

Figure 2.91:-

Transcribed Image Text:

Fo d- Crown of tree mc Top of trunk Bottom of trunk (a) FIGURE 2.91 Trunk of a tree subjected to force. Fb d (b)