QUESTION $1$

$($a$)$ A three$-$degree$-$of$-$freedom system is shown in Figure Q$1.$

Finure $1$

where

$k=48\frac{N}{m},m=3kg,F_1=F_2=5cos2tN,F_3=0N$

Knowing that the natural frequencies are:

${}_1=8\mathrm{.}6636ra\frac{d}{s},{}_2=2\mathrm{.}3172ra\frac{d}{s},{}_3=5\mathrm{.}9642ra\frac{d}{s}$

and the mode shapes are:

$\{x{\}}_1=\{[1\mathrm{.}0000],[-0\mathrm{.}2304],[0\mathrm{.}0856]\}$ and $\{x{\}}_2=\{[1\mathrm{.}0000],[1\mathrm{.}2215],[0\mathrm{.}7339]\},$ and $\{x{\}}_3=\{[1\mathrm{.}0000],[0\mathrm{.}5922],[-2\mathrm{.}6528]\}$

it can be shown that the modal masses are:

$M_1=3\mathrm{.}6809,M_2=24\mathrm{.}1354,M_3=49\mathrm{.}4337$

Using modal analysis, find the steady$-$state response of the system. The damping

ratio for each mode is ${}_i=0\mathrm{.}05,i=1,2,3.$

The mathematical solution to the equation:

$q_i^{}+2{}_i{}_i{q}_i^{}+{}_j^2{q}_i=F_{i}cost,i=1,2,3$

can be written as:

$q_i(t)=A_{i}cos(t-{}_i),i=1,2,3$

where

$A_i=\frac{F_i}{{}_i^2}\frac{1}{\sqrt[\phantom{2}]{[1-(\frac{}{{}_i}{)}^2{]}^2+(2\frac{}{{}_i}{)}^2}}$ and ${}_i=ta{n}^{-1}(\frac{2{}_j\frac{}{{}_i}}{1-(\frac{}{{}_i}{)}^2})$

$($b$)$ What are the advantages of Theoretical Modal Analysis when it is used to find the

response, to an excitation, of a MDOF system? Giving a practical example, explain

the main disadvantage of the method.