the system settles down to within

$5\mathrm{.}2$ Consider the two independent single degree$-$of$-$freedom systems in Figure $E{S}_{S_2}$ that are each being forced to vibrate harmonically at the same frequency $0_0,$ The excitation on system $1$ starts at $t=0,$ and the excitation on system $2$ starts at ${?}_{at}t=t_o,$ that is$,$

$f_1(t)=F_{1}sin(t)u(t)$

$f_2(t)=F_{2}sin([t-t_o])u(t-t_o)$

Use Eqs. $(5\mathrm{.}1)$ to $(5\mathrm{.}9)$ to show that the steady$-$state responses of the two systems are

$x_{1ss}()=\frac{F_1}{k_1}H(,{}_1)sin(-(,{}_1))$

$x_{2ss}()=\frac{F_2}{k_2}H(\frac{}{{}_r},{}_1)sin(\frac{}{{}_r}(-{}_o)-(\frac{}{{}_r},{}_1))$

where $=\frac{}{{}_{n1}},{}_r=\frac{{}_{n2}}{{}_{n1}},=\frac{{}_2}{{}_1},$ and

$H(a,b)=\frac{1}{\sqrt[\phantom{2}]{(1-a^2{)}^2+(2ba{)}^2}}$

$(a,b)=ta{n}^{-1}(\frac{2ba}{1-a^2})$

If both systems are operating in their respective mass$-$dominated regions, then what is the ratio of the magnitudes of the amplitudes of system $2$ to that of system $1$ and their relative phase?

ure E$5\mathrm{.}2.$