Problem $1$:

Consider the following model for a mechanical dynamic system consisting of a linear spring with stiffness

$1500\frac{N}{m}$ and a linear visocus damper with damping $450N-\frac{s}{m}$ that are connected between rigid wall and a

$30$ kg moving mass $($frictionless rollers for part $($a$)$ and of course there is no friction as well for part$($b$)).$

Note: $x=0$ is considered as static equilibrium position. Note: With $x=0$ the as static equilibrium, your

equation of motion will be exactly the same for both part$($a$)$ and part$($b$)$ as mg will also not appear in the

equation of motion of part$($b$).$

a$)$ Draw a free body diagram and derive the equation of motion by Newton's method.

b$)$ Write the kinetic energy and potential energy of the system and derive the equation of motion by

Euler$-$Lagrange.

c$)$ Use the simulink of MATLAB to simulate the response of this system for $30$ s $.$

$($a$)$

$($b$)$