Chapter 28, Problem 47P

A forced damped spring-mass system (Fig. P28.47) has the following ordinary differential equation of motion:

$m\frac{d^{2}x}{d{t}^2}+a\left|\frac{dx}{dt}\right|\frac{dx}{dt}+kx=F_o\sin \left(\omega t\right)$

where $x=$ displacement from the equilibrium position, $t=$ time, $m=2\text{ kg}$ mass, $a=5\text{ N/}{\left(\text{m/s}\right)}^2,\text{ and }k=6\text{ N/m}$. The damping term is nonlinear and represents air damping. The forcing function $F_o\sin \left(\omega t\right)$ has values of $F_o=2.5\text{ N}$ and $\omega =0.5$ rad/sec. The initial conditions are

Initial velocity $\frac{dx}{dt}=0\text{ m/s}$

Initial displacement $x=1\text{ m}$

Solve this equation using a numerical method over the time period $0\le t\le 15\text{ s}$. Plot the displacement and velocity versus time, and plot the forcing function on the same curve. Also, develop a separate plot of velocity versus displacement.

FIGURE P28.47