Problem #$1.(40$ points$).$ For the $2^{nd}-$order dynamic system represented by the block diagram as shown in Figure #$1($a$),$ the under$-$damped response for a step$-$input is as shown in Figure #$1($b$)$ below. Find a$,$ b$,$ and c$.$

Figure #$1.$ Second$-$Order Dynamic System Response for Problem #$1.$

Problem #$2(30$ points$)$: For the system given in Problem #$1,$ answer the following questions:

$(5$ points$).$ The overshoot, in percentage, $\%Mp.$

$(5$ points$).$ The damping ratio, $,$ given the overshoot observed in item $(1).$

$(5$ points$).$ The peak time, in seconds.

$(5$ points$).$ The rise time, in seconds.

$(5$ points$).$ The settling time, in seconds, for $2\%$ offset error tolerance.

$(5$ points$).$ Verify your answers, $1$ through $5,$ using MATLAB$.$

Problem #$3.(30$ points$).$ Figure #$2$ below shows a translational mechanical system where external force $u(t)$ is the input to the system, and the displacement $y(t)$ of the mass $(m)$ is the output. The damper constant is $b$ and spring constant is $k.$

$(15$ points$).$ Draw the free$-$body diagram and write the ordinary differential equation $($ODE$)$ of the dynamic system assuming that the initial conditions are zero.

$(15$ points$).$ Find the transfer function $\frac{Y(s)}{U(s)}$ of the translational mechanical system where $y(t)$ is the displacement of the mass and $u(t)$ is the external force applied to the system.

Figure #L$.$ I rans$$at$$onal mecnan$$cal system.

Problem #$1.(40$ points$).$ For the $2^{nd}-$order dynamic system represented by the block diagram as shown in Figure #$1($a$),$ the under$-$damped response for a step$-$input is as shown in Figure #$1($b$)$ below. Find $a,b,$ and $c.$

$($a$)$

Figure #$1.$ Second$-$Order Dynamic System Response for Problem #$1.$

Problem #$2(30$ points$)$: For the system given in Problem #$1,$ answer the following questions:

$(5$ points$).$ The overshoot, in percentage, $\%Mp.$

$(5$ points$).$ The damping ratio, $,$ given the overshoot observed in item $(1).$

$(5$ points$).$ The peak time, in seconds.

$(5$ points$).$ The rise time, in seconds.

$(5$ points$).$ The settling time, in seconds, for $2\%$ offset error tolerance.

$(5$ points$).$ Verify your answers, $1$ through $5,$ using MATLAB$.$

Problem #$3.(30$ points$).$ Figure #$2$ below shows a translational mechanical system where external force $u(t)$ is the input to the system, and the displacement $y(t)$ of the mass $(m)$ is the output. The damper constant is $b$ and spring constant is $k.$

$(15$ points$).$ Draw the free$-$body diagram and write the ordinary differential equation $($ODE$)$ of the dynamic system assuming that the initial conditions are zero.

$(15$ points$).$ Find the transfer function $\frac{Y(s)}{U(s)}$ of the translational mechanical system where $y(t)$ is the displacement of the mass and $u(t)$ is the external force applied to the system.

Figure #$2.$ Translational Mechanical System.