$(3)$ A motor is supported by a set of rubber pads with overall stiffness $k($Case $1)$ while it has an additional identical set in parallel $($Case $2),$ as shown in the figure. The damping constants, masses and the sinusoidal forcing functions are the same for both cases. The forcing frequency is equal to the natural frequency of the motor in Case $1,$ i$.$e$.,{}_1={}_2={}_{n1},$ where the subscripts $1$ and $2$ refer to the two cases and $n$ denotes the natural frequency $($i$.$e$.,F_{0}sin{}_{n1}t$ is applied in both cases, where $F_0$ denotes the forcing amplitude and $t$ denotes the time$).$

$($a$)$ Determine the overall stiffness in Case $2,$ the frequency ratio $\frac{?}{\text{b}\text{a}\text{r}}({)}_2=\frac{{}_2}{{}_{n2}},$ and the ratio $\frac{{}_1}{{}_2},$ where ${}_1$ and ${}_2$ refer to the damping ratios of the two cases.

$c$

$(15$ marks$)$

$($b$)$ Determine the actual values of ${}_1$ and ${}_2,$ if the ratio of the amplitudes in steady motion for the two cases is $\frac{x_1}{x_2}=3.$

$(20$ marks$)$

Hint: You may use the magnification ratio formula:

$\frac{x}{\frac{F_0}{k}}=\frac{1}{\sqrt[\phantom{2}]{(1\frac{-}{b}ar({)}^2{)}^2+(2\frac{}{b}ar(){)}^2}}$

where the symbols have the usual meanings.