Chapter 25, Problem 16P

The motion of a damped spring-mass system (Fig. P25.16) is described by the following ordinary differential equation:

$m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$ $x=$

where $x=$ displacement from equilibrium position (m), $t=$ time (s), $m=$20-kg mass, and $c=$ the damping coefficient $\left(\text{N}\cdot \text{s/m}\right)$. The damping coefficient c takes on three values of 5 (under- damped), 40 (critically damped), and 200 (overdamped). The spring constant $k=20\text{ N/m}$. The initial velocity is zero, and the initial displacement $x=1$m. Solve this equation using a numerical method over the time period $0\le t\ge 15\text{ s}$. Plot the displacement versus time for each of the three values of the damping coefficient on the same curve.