$3$Q: Using the above calculated matrices and the following mode shapes:

${}_1=\{[0\mathrm{.}3961,0\mathrm{.}6083,0\mathrm{.}6878]\}$

${}_2=\{[-0\mathrm{.}4948,-0\mathrm{.}1058,0\mathrm{.}8626]\}$

${}_3=\{[-0\mathrm{.}3476,0\mathrm{.}8501,-0\mathrm{.}3955]\}$

a$.$ Compute the unity normalized mode shapes $(3)$

b$.$ Compute the mass normalized mode shapes $(3)$

c$.$ Calculate the modal mass $(3)$

d$.$ Calculate the modal stiffnesses $(3)$

e$.$ Compute the mass participation ratios $(4)$

f$.$ Based on a minimum total mass participation of $90\%,$ how many modes would be needed for

the analysis? $(1)$

Mass participation using normalized modes, for the $j^{th}$ mode:

$M{M}_j$:$=\frac{[(x^T*M*r{)}_j{]}^2}{(x^T*M*x{)}_{j,j}},$ Note: $to$ create ratio, divide $by$ total mass

Assume $r$ is a vector of influence coefficient, each term equal to $1$

Mass normalization of modes:

${}_n^{TT}*M*{}_n=1$