A small single$-$story building, with stiffness $k,$ mass $m,$ and natural period of $0\mathrm{.}3sec,$ is found to be seismically vulnerable because it is estimated that the structure would displace too much during a medium$-$intensity earthquake and experience damage. The owner of the building hires a contractor to stirien the building. As his fix, the contractor installs an elastic link with stiffness $6k$ between the small building and an adjacent stiffer and bigger building, which has stiffness $4k$ and mass $4m($see sketch below$).$ If you need to$,$ assume identical damping ratio throughout, i$.$e$.$ for both individual buildings and when they are linked together.

About a year later an earthquake of medium$-$intensity occurs nearby. The displacement response spectrum for the ground motion recorded duing the earthquake near the buildings is given below. Assume that the buildings are very close to each other and therefore experience identical ground motion.

Unon inspection, the columns in the small sirigle$-$story building are found damaged. The owner sues the contractor claiming that the contractor's fix was useless. The contractor defends that his fix helped the small building. He claims that his fix reduced the peak displacement of the building compared to what it would have experienced without his fix. The judge on the case has invited you as an expert witness to tell the court whether the elastic link installed by the contractor was useful or not.

What will you say to the court? Was the contractor's fix useless or was it helpful? Do not forget to provide evidence proving your conclusion.

$[$small building linked to the bigger building by contractor's fix$]$

$[$displacen ont response spectrum$]$