Q$.2$ The packaging arrangement for transporting a piece of valuable scientific instrument is represented by the mass and spring assembly shown in Figure Q$\mathrm{.}2.$ The artefact has a mass of $180$ kg and is enclosed within a $25$ kg inner packing case, which is$,$ in turn, packaged in an additional $55$ kg outer packing case. Each packing case is filled with a honeycomb material to protect the instrument from harmful shock effects, and can be modelled by means of springs having the same stiffness $\text{'}\backslash ($ k $\backslash )\text{'}(\backslash (\backslash$mathrm$\{$N$\}/\backslash$mathrm$\{$m$\}\backslash )).$ This packaging arrangement is transported by truck, with the interface between the truck and the packaging arrangement represented by three further springs each having a stiffness of $\backslash (\backslash$mathrm$\{$k$\}(\backslash$mathrm$\{$N$\}/\backslash$mathrm$\{$m$\})\backslash ).$ Consider vertical motion only.

i$)$ Draw a schematic of the mass$-$spring model being considered and determine the dynamic matrix describing the equations of motion for the complete assembly as described.

ii$)$ Due to attempted cost cutting, the original packing material between the instrument and the inner packing case is removed and replaced with a rigid support. Determine the stiffness $\backslash (\backslash$mathrm$\{$k$\}(\backslash$mathrm$\{$N$\}/\backslash$mathrm$\{$m$\})\backslash )$ of the packing material required to make the natural frequency of vibration not less than $0\mathrm{.}25$ Hz $.$

iii$)$ For the arrangement described in ii$)$ above, determine the displacements of the assembly if the outer packing case experiences a force of the form $\backslash (65\backslash$cdot $\backslash$sin $(12.$ t$)\backslash$mathrm$\{$N$\}\backslash ).$

Figure Q$.2$