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Problem No$.2-$ Coulomb damping using the Runge Kutta Method $(20$ pts$)$

A spring mass system shown below is pulled to the right to a position of $0\mathrm{.}5$ m with a velocity of

$2\mathrm{.}5\frac{m}{s}$ and then suddenly released. It undergoes free vibration with energy dissipated due to

both friction and viscous damping.

A$)$ Write the equation of motion for the system as a second order differential equation

using equilibrium principles. The friction force opposes the motion, it is equal to the

friction coefficient time the normal force. Note that the sign of the frictional

resistance is always opposite to the direction of the velocity. The sign$($ v $)$ function

does this $($try it$).$

B$)$ Determine the D matrix of the system that can be used to calculate the system

response in state space form.

C$)$ Given $k=4,000\frac{N}{m},m=200kg,c=200N\frac{s}{m}$ and a friction coefficient between base

and the mass of $0\mathrm{.}1.$ Present graphs for $4$ seconds of the displacement, $x(t)$ and

velocity, $v(t)$ on the same graph

D$)$ Plot the phase plane versus $x(t).$

E$)$ Estimate the time for the system to stop moving, assume the static and kinetic

coefficients are equal.