decaying exponentials with no oscillation. This case occurs for , and is referred to as overdamped. Situations where overdamping is practical tend to have tragic outcomes if overshooting occurs, usually electrical rather than mechanical. For example, landing a plane in autopilot: if the system overshoots and releases landing gear too late, the outcome would be a disaster. Critically damped The case where is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). Q factor and decay rate The Q factor, damping ratio ζ, and exponential decay rate α are related such that When a second-order system has (that is, when the system is underdamped), it has two complex conjugate poles that each have a real part of ; that is, the decay rate parameter represents the rate of exponential decay of the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. For example, a high quality tuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer. Logarithmic decrement For underdamped vibrations, the damping ratio is also related to the logarithmic decrement . The damping ratio can be found for any two peaks, even if they are not adjacent. For adjacent peaks: where where x0 and x1 are amplitudes of any two successive peaks. As shown in the right figure: where , are amplitudes of two successive positive peaks and , are amplitudes of two successive negative peaks. Percentage overshoot In control theory, overshoot refers to an output exceeding its final, steady-state value. For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. The percentage overshoot (PO) is related to damping ratio (ζ) by: Conversely, the damping ratio (ζ) that yields a given percentage overshoot is given by: Examples and Applications Viscous Drag When an object is falling through the air, the only force opposing its freefall is air resistance. An object falling through water or oil would slow down at a greater