Problem Statement 2
[35, CO3]
The motion of a damped vertical spring system with a sudden inclusion of the mass is described
by the following ordinary differential equation:
$m\frac{d^2y(t)}{dt^2} + c\frac{dy(t)}{dt} + ky(t) = mg; \quad t \ge 0$
where $y(t)$ = displacement of the mass from the equilibrium position, $t$ = time (s), $m$ = 5kg
mass, and $c$ = the damping coefficient (Ns/m). Assume, the damping coefficient $c$ takes on
three values of 2.5 (under-damped), 20 (critically damped), and 100 (over-damped). The spring
constant $k$ = 20 N/m and the acceleration due to gravity, $g$ = 9.81 ms$^2$. The initial displacement
and the initial velocity both are zero (the spring was at rest before the inclusion of the 'mg'
weight on it).
Solve this equation using RK-4 method over the time period $0 \le t \le 40s$. Plot the displacement
versus time curves for each of the three values of the damping coefficient on the same curve.
Also plot the velocity versus time curves in the similar manner. Estimate the peak value, peak
time, rise time, overshoot and the final value of the displacement curves for each values of $c$.
[Hints: If there is no oscillation found in the curve, overshoot will be zero, and the peak value
and peak time will be undefined. However, by definition, the overshoot is,
Overshoot $= \frac{Peak \quad Value - Final \quad Value}{Final \quad Value} \times 100%$
However, rise time is defined as the time required for the curve to reach the 90% of its final value
from the 10% of its final value.]
