time$/$s

Problem $4.$ Consider the nonlinear mass$-$spring$-$dashpot system

$m{q}^{}(t)+c|q^{}(t)|q^{}(t)+k_{1}q(t)+k_2{q}^3(t)=u(t)$

where $q(t)$ is the position of the mass, $u(t)=5sin3tN$ is an external force, $m=2kg$ is the mass, $c=0\mathrm{.}5k\frac{g}{m}$ is the drag damping coefficient, $k_1=3k\frac{g}{s^2}$ is the linear stiffness, $k_2=5k\frac{g}{m^2}\frac{?}{s^2}$ is the cubic stiffness, and $|*|$ denotes absolute value. The differential equation $(1)$ is nonlinear and difficult to solve analytically.

Use Matlab's ODE$45$ function or Simulink to solve the differential equation $(1),$ where the initial conditions are $q(0)=0\mathrm{.}2m$ and $q^{}(0)=0\mathrm{.}1\frac{m}{s}.$ Plot $q(t)$ and $q^{}(t)$ for $10$ s $.$ Please turn in a hardcopy of: $($i$)$ the Matlab m$-$file that runs the simulation, $($ii$)$ the Matlab function or Simulink model that codes the differential equations, and $($iii$)$ the plot of $q(t)$ and $q^{}(t)$ with all axes correctly labeled and with units specified.

Hint: The Matlab function for absolute value is 'abs$()\text{'}$ and the absolute value block in Simulink is under 'Math Operations'.