Q$\mathrm{.}1$ An undamped system has,m $=10$ kg$,$ k $=1000$ N$/$m$.$ If the system is subjected to initial displacement and velocity of $0\mathrm{.}1$ m and $10$ m$/$s respectively, and it is excited by a sinusoidal harmonic force having an amplitude F$0=50$ N$.$ a$)$Plot the system's total and steady$-$state response $($separately$)$ for excitation frequencies of $=5$ rad$/$s and $15$ rad$/$s$.$b$)$For both the frequency values, determine $($i$)$ the frequency ratio $,($ii$)$ the amplitude of the forced response Xp$,($iii$)$ the magnification factor Rd and $($iv$)$ phase angle $.$

Q$.2$ If the same system given in Question $1$ has a damping of c $=80$ Ns$/$m$,$a$)$Plot the total and steady$-$state damped response for excitation frequencies of $=5$ rad$/$s and $15$ rad$/$s$.$b$)$Determine $($i$)$ the amplitude of the forced response Xp$,($ii$)$ the magnification factor Rd and $($iii$)$ phase angle the damped system for both the forcing frequencies.

Q$.3$ Find the response of a damped SDOF system using the Interpolation of excitation and Central difference method with $500$ kg mass, $200$ Ns$/$m damping and $750$ N$/$m spring stiffness for a rectangular pulse force of Fo $=2000$ N and td $=9$ sec$.$ Compare the response with the theoretical response for up to t $=12$ sec$.$ Use t $=1$sec$.$