Engineers and scientists use mass$-$spring models to gain insight into structure dynamics under the

influence of disturbances such as earthquakes. A three$-$storey building was built with the system

detecting earthquakes as shown in Figure Q$2.$ Each floor mass is represented by $m_i$ and each floor

has stiffness by $k_i$ for $i=1$ to $2.$

Figure Q$2$: A two$-$storey building model.

a$)$ Using Figure Q$2,$ draw the free$-$body diagrams and form the equation of motion for the

dynamic force balances. Next, convert the equation to the matrix form by assuming the

nature of motions as $x_n=A_{n}cost.$ Then, reduce the matrix in the form of $[A]\{vec(x)\}-$

$\{vec(x)\}=0$

$[20$ marks$]$

b$)$ Given all $k\text{'}$s and $m\text{'}$s in Figure Q$2,$ form the characteristics equation and find the

eigenvalues. Repeat this problem using an iterative method of up to three iterations to

determine the dominant eigenvalue and the corresponding eigenvectors. Use an initial

value of the eigenvector of ${\left[\begin{array}{cc}2& 4\end{array}\right]}^T.$ Show all your calculation steps. Round the final

value to four significant figures.

c$)$ Compare your eigenvalues result in $($b$)$ by showing the relative error. How can you improve

the accuracy of the iterative method?