Explain where the formulas come from and why we use those formulas fornthis problem, it is confusing. Thsnk you.

Can you please explain the steps i am confused and do not understand the problem. Thank you.

A fan of $45$ kg has an unbalance that creates a harmonic force. A spring$-$damper system

is designed to minimize the force transmitted to the base of the fan. A damper is used

having a damping ratio of $=0\mathrm{.}2.$ Calculate the required spring stiffness so that only

$10\%$ of the force is transmitted to the ground when the fan is running at $10,000rpm.$

Solution: The equation of motion of the fan is

$m{x}^{}+c{x}^{}+kx=m_{0}e{}^{2}sin(t+)$

The steady state solution as given by equation $(2\mathrm{.}84)$ is

$x(t)=\frac{m_{0}e}{m}\frac{r^2}{\sqrt[\phantom{2}]{(1-r^2{)}^2+(2r{)}^2}}sint$

where $r$ is the standard frequency ratio. The force transmitted to the ground is

$F(t)=kx+c{x}^{}=\frac{m_{0}e}{m}\frac{k{r}^2}{\sqrt[\phantom{2}]{(1-r^2{)}^2+(2r{)}^2}}sint+\frac{m_{0}e}{m}\frac{c{r}^2}{\sqrt[\phantom{2}]{(1-r^2{)}^2+(2r{)}^2}}cost$

Taking the magnitude of this quantity, the magnitude of the force transmitted becomes

$F_0=\frac{m_{0}e}{m}\frac{r^2\sqrt[\phantom{2}]{k^2+c^2{}^2}}{\sqrt[\phantom{2}]{(1-r^2{)}^2+(2r{)}^2}}=m_{0}e\frac{\sqrt[\phantom{2}]{1+(2r{)}^2}}{\sqrt[\phantom{2}]{(1-r^2{)}^2+(2r{)}^2}}$

From equation $(2\mathrm{.}81)$ the magnitude of the force generated by the rotating mass $F_r$ is

$F_r=m_{0}e{}^2$

The limitation stated in the problem is that $F_0=0\mathrm{.}1F_r,$ or

$m_{0}e{}^2\frac{\sqrt[\phantom{2}]{1+(2r{)}^2}}{\sqrt[\phantom{2}]{(1-r^2{)}^2+(2r{)}^2}}=0\mathrm{.}1m_{0}e{}^2$

Setting $=0\mathrm{.}2$ and solving for $r$ yields:

$r^4-17\mathrm{.}84r^2-99=0$

which yields only one positive solution for $r^2,$ which is

$r^2=22\mathrm{.}28=\frac{{}^2}{\frac{k}{m}}=>\frac{k}{m}=(\frac{100002}{60}{)}^2\frac{1}{22\mathrm{.}28}$

$=>k=45(\frac{100002}{60}{)}^2\frac{1}{22\mathrm{.}28}=2\mathrm{.}21{10}^6\frac{N}{m}$