Problem #$1(25$ pts$)$: Mode Superposition Method: Decoupling Equations of Motion

The mode superposition method allows us to decouple our equations of motion for MDOF systems by using generalized force, mass, stiffness, and damping matrices in modal coordinates. Consider a $3$DOF undamped spring mass system where the equations of motion in physical coordinates are defined by:

$\left[\begin{array}{ccc}{m}_1& 0& 0\\ 0& m_2& 0\\ 0& 0& m_3\end{array}\right]\{[u_1^{}],[u_2^{}],[u_3^{}]\}+\left[\begin{array}{ccc}{k}_1+k_2+k_3& -k_2& 0\\ -k_2& k_2+k_3& -k_3\\ 0& -k_3& k_3\end{array}\right]\{[u_1],[u_2],[u_3]\}=\{[F_1],[F_2],[F_3]\}**f(t)$

or

$[M]\{[u_1^{}],[u_2^{}],[u_3^{}]\}+[K]\{[u_1],[u_2],[u_3]\}=\{F\}**f(t)$

a$)$ Are the equations of motion in $(1)$ coupled to one another? Explain why they are or aren't. If they are, are they statically of dynamically coupled? Why?

b$)$ Make use of the eigenvector expansion theorem and the orthogonality properties of eigenvectors to de$-$couple the equations of motion. Put the equations in modal coordinates $q_r$ and generalized mass, stiffness, and force matrices. Write your answer in matrix form.

c$)$ Derive the initial conditions $q_r(0)$ and $q_r^{}(0)$ starting from the eigenvector expansion theorem at $t=0.$ That is: $u(0)={\displaystyle \underset{r=1}{\overset{N}{}}}\{{}_r\}{q}_r(0).$