Problem $3)($Bonus $-20$ pts $)$ The result of an experiment carried on a one$-$port mechanical element is shown in the following plot. $($Note that $x_{21}={\displaystyle \underset{0}{\overset{t}{}}}{v}_{21}.dt)$

a$)$ As far as the basic characteristics are concerned, what type of element is this?

b$)$ Determine the amount of energy stored in this element when the applied force is increased from zero to $k_1{x}_0$ value.

c$)$ Determine the net energy flow into this element when the applied force is changed such that first increased from zero to $k_1{x}_0,$ then decreased from $k_1{x}_0$ to $-k_1{x}_0,$ and finally made zero.

d$)$ Is this a pure element? Explain the reason.

e$)$ Comment on the energetic behavior of this one$-$port element.