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A load of variable frequency $\text{'}\text{'}$ acts on the $2$ DOF spring mass system above. Construct a spreadsheet to determine the maximum displacement amplitude of the $5kg[u_1(t)]$ and $]$ masses as functions of $\text{'}\text{'}.$ Complete the table below and plot the displacement frequency response function $($maximum amplitude vs$.$ frequency$)$ for each mass. The damping values for modes $1$ and $2$ are ${}_1=0\mathrm{.}05$ and ${}_2=0\mathrm{.}03$ respectively. What do you notice about the frequency v $.$ amplitude response plot?

$\backslash$table$[[$Frequency$,$Max. Amplitude$],[(\text{'}\text{'}),,]]$

Hints:

calculate natural frequencies and modes of vibration

damping is low so take ${}_{ni}={}_{di}$

normalise the mode shapes with respect to the mass matrix

use mode superposition

write out equations for $Y_1(t)$ and $Y_2(t)Y_i^{}(t)+2{}_i{}_i{Y}_i^{}(t)+{}_i^2{Y}_i(t)=\{{}_i{\}}^{T}P(t)$

set up your spreadsheet solution so that you can recalculate for varying $\text{'}\text{'}$

use time stepping procedure to solve for $Y_i(t)$; choose $t$ carefully & use same value for $Y_1(t)$ and $Y_2(t)$

then $U(t)=\{[u_1(t)],[u_2(t)]\}={\displaystyle \underset{i=1}{\overset{2}{}}}\{{}_i\}{Y}_i(t)$

plot $u_1(t)$ and $u_2(t)$ and collect maximum absolute displacement amplitude for each value of $\text{'}\text{'}$

complete adjacent table and plot max. amplitude $($for $u_1$&$u_2)$ versus $\text{'}\text{'}$

to avoid mistakes build your solution incrementally in Excel!