Problem $3$:

In lecture, is was shown that equation of motion for a damped forced mechanical oscillator with

harmonic forcing is

$m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=F_{o}cos(t)$

a$)$ Use the Method of Undetermined Coefficients to show that the particular solution $x_p(t)$ is

$x_p(t)=\frac{(\frac{F_o}{k})cos(t+{}_p)}{\sqrt[\phantom{2}]{4{}^2(\frac{}{{}_o}{)}^2+[1-(\frac{}{{}_o}{)}^2{]}^2}},$ such that $,{}_o=\sqrt[\phantom{2}]{\frac{k}{m}},$ and $,=\frac{c}{\sqrt[\phantom{2}]{4mk}}$

by assuming that the solutions is of the form

$x_p(t)=$acos$(t)+$bsin$(t)$

Note: show all steps

b$)$ The ratio $\frac{A}{\frac{F_o}{k}}$ is often called the Dynamic Magnification Factor $($DMF$)$

$\frac{A}{\frac{F_o}{k}}=\frac{1}{\sqrt[\phantom{2}]{4{}^2(\frac{}{{}_o}{)}^2+[1-(\frac{}{{}_o}{)}^2{]}^2}}$

On same $log-$log plot, plot the variation of DMF with respect to $\frac{}{{}_o}$ within the range

$0\mathrm{.}1\frac{}{{}_o}10,$ for the following damping ratios, $$:$0,0\mathrm{.}05,0,1,0\mathrm{.}2,0\mathrm{.}5,$ and $1\mathrm{.}0.$

What the effect of damping ratio on DMF$?$