Derive the relationship between residue matrix $)$ and modal vector $(\{{\}}_r)$ for a two$-$degree$-$of$-$freedom system.

Hints:

$($a$)$ Transfer function matrix $H(s)$ or impedance matrix $B(s)$ can be represented as a modal form:

$[H(s)]=[B(s){]}^{-1}={\displaystyle \underset{r=1}{\overset{2}{}}}(\frac{[A{]}_r}{(s-{}_r)}+\frac{[A^{**}{]}_r}{(s-{}_r^{**})})$

${}_r$ is the $r^{th}$ mode modal frequency. $[A{]}_r$ is the $r^{th}$ mode residue matrix which can be represented as a column vector form:

$[A{]}_r=[\{[a_1],[a_2]{\}}_r,\{[a_2],[a_3]{\}}_r]$

$($b$)[B(s)][B(s){]}^{-1}=[I]=>[B(s)][H(s)]=[I]$

$($c$)$ Eigenvalue problem:

$[B({}_r)]\{{\}}_r=[B({}_r)]\{[{}_1],[{}_2]{\}}_r=\{0\}$

${}_r$ : the $r^{th}$ mode modal frequency $($eigenvalue$)$

$\{{\}}_r=\{[{}_1],[{}_2]{\}}_r$ : the $r^{th}$ mode modal vector $($eigenvector$)$