Part II $(15$ points$)$: S$=$ la to thw

Considering the following equation of motion:

$x^{}+2{}_n{x}^{}+{}_n^{2}x=F_{0}cos(t)$

The steady$-$state solution of this equation of motion can be written as:

$x(t)=xsin(t+)$

where $\frac{x}{{}_{st}}=\frac{1}{\sqrt[\phantom{2}]{(1-r^2{)}^2+(2r{)}^2}}$;$r=\frac{}{{}_n}$; and ${}_n=62\mathrm{.}8ra\frac{d}{s}$

Plotting $\frac{x}{{}_{st}}$ as a function of $$ for two distinct damping ratios $($configuration $1$ shown in Figure $7$ and configuration $2$ shown in Figure $8)$; one obtains the following graphs:

$2$

Questions: $($keep $3$ digits after decimal point in your calculations$)$

$1-$ If ${}_{st}=3mm,$ determine the value of the excitation acceleration $F_0$ for each configuration. $(3$ points for each configuration$)$

$2-$ Determine the quality factor of the dynamical systems in Figures $7$ and $8$ using three methods for each configuration $($one of the methods should be the original definition of the quality factor$).(2$ points for each method$)$

$3-$ Why is there a difference between the obtained quality factors for the configuration shown in Figure $7($configuration $1)?(3$ points$)$