Problem $7(30\text{'})$

The single$-$degree$-$of$-$freedom system shown below is made up of a particle of mass, $m,$ three springs $($having stiffness of $2k,3k$ and $3$ k $)$ and a dashpot with a damping coefficient c $.$ Let x represent the motion of the particle measure from a position at which the springs are unstretched. If $m=20kg,k=1000\frac{N}{m}$ and $c=60k\frac{g}{s}.$

Determine:

the undamped natural frequency ${}_n({10}^{\text{'}})$

the damping ratio $(5^{\text{'}})$

the damped natural frequency ${}_d(5)$

Assume there is an external harmonic force applied on the mass with $F(t)=10sin(2t),$ and the initial condition is $x_0=1mm$ and $v_0=50m\frac{m}{s}.$ Compute the total response of the system, $x(t)*(10)$