Question $1$:

Consider a spring$-$mass system with damper. $f(t)$ is the force exerted on the mass $M$ that faces a

displacement of $y(t)$ due to the force. $K$ and $B$ are the spring constant and damping co$-$efficient.

Equating the forces along $x$ axis we get $f(t)=B\frac{dy(t)}{dt}+Ky(t)+M{d}^{2}y(t)$

What is$/$are state variables for this system?

a$.$ Displacement of the mass

b$.$ Displacement, Acceleration, Speed of the mass

c$.$ Speed, Acceleration of the mass

d$.$ Displacement, Speed of the mass

Question $2$:

See the image and choose the correct option.

Question $3$:

Find out the steady state response value of the system using Matlab for a unit step input to the open loop system for M$=1$kg$,$B$=1($N$)/($m$)/$s K$=1($N$)/($m$).$Give the numeric answer with solution.

Question $4$:

Find out H$($A$)$ for the open loop spring$-$mass$-$damper system given in Question $1$ when we are using Ackerman's method to place system poles at $[-0\mathrm{.}9,-0\mathrm{.}9]$ Consider M$=1$kg$,$B$=1($N$)/($m$)/$s K$=1($N$)/($m$).($Hint: Refer to Ackerman's method$).$Give numeric answer with solution.