7.3 Applications  (Page 4/13)

Case 2: $b^2=4mk$

In this case, we say the system is critically damped . The general solution has the form

where ${\lambda }_1$ is less than zero. The motion of a critically damped system is very similar to that of an overdamped system. It does not oscillate. However, with a critically damped system, if the damping is reduced even a little, oscillatory behavior results. From a practical perspective, physical systems are almost always either overdamped or underdamped (case 3, which we consider next). It is impossible to fine-tune the characteristics of a physical system so that $b^2$ and $4mk$ are exactly equal. [link] shows what typical critically damped behavior looks like.

Case 3: $b^2<4mk$

In this case, we say the system is underdamped . The general solution has the form

where $\alpha$ is less than zero. Underdamped systems do oscillate because of the sine and cosine terms in the solution. However, the exponential term dominates eventually, so the amplitude of the oscillations decreases over time. [link] shows what typical underdamped behavior looks like.

Note that for all damped systems, $\underset{t\to \infty }{\text{lim}}x(t)=0.$ The system always approaches the equilibrium position over time.