Problem $1$: Impulse Response of a Single$-$Degree$-$of$-$Freedom $($SDOF$)$ System

Consider a single$-$degree$-$of$-$freedom $($SDOF$)$ mass$-$spring$-$damper system with:

Mass $m=500kg$

Stiffness $k=1500\frac{N}{m}$

Damping coefficient $c=50N\frac{s}{m}$

The system is subjected to a sinusoidal input force of the form $f(t)=F_{0}sin(t)$ with $F_0=100N$ and $=$

$2ra\frac{d}{s}$

Derive the analytical solution for the system's response to the sinusoidal input, assuming zero initial

conditions $($i$.$e$.,$ no initial displacement or velocity$).$

In MATLAB, simulate the response to the same sinusoidal input using Duhamel's integral $($manual

convolution of the input with the system's impulse response function$),$ again assuming zero initial

conditions.

Note: Use the convolution integral to compute the system's response numerically. Implement the

integral manually in MATLAB $($without using built$-$in functions for convolution$).$

Using the convolution integral, determine the analytical solution for the system's response to the

sinusoidal input.

Plot both the analytical solution and the numerical solution obtained through convolution for the

sinusoidal input on the same plot.