A sprung table has been designed to support a measurement device to isolate the table from the surroundings. Before the measurement device is added, tests are made on the table to determine is vibration and damping characteristics. It can be assumed that the table has a mass of $24$ kg $,$ and is supported by springs that deflect $15$ mm under the static loading. A damper is fitted to the table, and in a free vibration test, it is found that the amplitude of the oscillation decays to $\backslash (10\backslash \%\backslash )$ of the original value after $3$ complete oscillations.

Calculate the spring stiffness $($assume the total spring stiffness$)$ and the natural frequency of the table, $\backslash (\backslash$mathrm$\{$k$\}=\backslash )\backslash (\backslash$mathrm$\{$N$\}/\backslash$mathrm$\{$m$\},\backslash$omega$\_\{$n$\}=\backslash )\backslash (\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash )$

Determine also:

The logarithmic decrement, $\backslash (\backslash$delta$=\backslash )$

The damping ratio, $\backslash (\backslash$zeta$=\backslash )$

and the damping coefficient, $\backslash (\backslash$mathrm$\{$c$\}=\backslash )\backslash (\backslash$mathrm$\{$Ns$\}/\backslash$mathrm$\{$m$\}\backslash )$