15.13:
Types of Damping
Consider a wood log floating on the surface of the water. When it is pushed downward and allowed to bob up and down, it oscillates about its mean position. Over time, the dissipative force reduces the amplitude of oscillation, and it can be described by the equation of damped harmonic motion.
On solving, if the imaginary part is non-zero, the general solution for the differential equation relative to the position and the angular frequency within the time-dependent exponential decay envelope is obtained.
Depending on the angular frequency, the system exhibits different types of damping conditions. When the damping force is low, the angular frequency resembles the natural frequency. Therefore, in underdamped conditions, the system oscillates with decaying amplitude after making some possible cycles.
In critical damping conditions, the damping and restoring forces are equal, and the angular frequency becomes zero, making the system oscillations fade out exponentially.
On the other hand, in overdamped conditions, the damping force is relatively large, which makes the system slowly reach equilibrium.
15.13:
Types of Damping
If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about the equilibrium point as it does so. In contrast, a critically damped system moves as quickly as possible towards equilibrium without oscillating about the equilibrium point.
Generally, critical damping is often desired because such a system not only returns to equilibrium rapidly, but remains at equilibrium too. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible, without overshooting or oscillating about the new position. For example, when a person stands on a bathroom scale that has a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling. Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity. However, many damping forces depend on velocity—sometimes in complex ways and sometimes simply being proportional to velocity.
This text is adapted from Openstax, College Physics, Section 16.7: Damped Harmonic Motion and Openstax, University Physics Volume 1, Section 15.4: Damped Oscillations.