4.6.1.5: Neutral frequency of Floating Bodies

This case is similar to pendulum (or mass attached to spring). The governing equation for the pendulum is \[ll\ddot{\beta} - g\beta = 0 \] Where here \(ll\) is length of the rode (or the line/wire) connecting the mass with the rotation point. Thus, the frequency of pendulum is \(\frac{1}{2\pi}\sqrt{\frac{g}{ll}}\) which measured in \(Hz\). The period of the cycle is \(2\pi \sqrt{ll/g}\). Similar situation exists in the case of floating bodies. The basic differential equation is used to balance and is \[I\ddot{\beta} - V \rho_{s} \overline{GM} \beta = 0 \] In the same fashion the frequency of the floating body is \[\frac{1}{2\pi}\sqrt{\frac{V\rho_{s}\overline{GM}}{I_{body}}}\] and the period time is \[2\pi\sqrt{\frac{I_{body}}{V\rho_{s}\overline{GM}}}\] In general, the larger \(\overline{GM}\) the more stable the floating body is. Increase in \(\overline{GM}\) increases the frequency of the floating body. If the floating body is used to transport humans and/or other creatures or sensitive cargo it requires to reduce the \(\overline{GM}\) so that the traveling will be smoother.