Damped harmonic motionEdit
with a characteristic equation
Depending on the value of the discriminant (c2 - 4mk), the characteristic equation can have two real, one real, or two complex solutions. These states are known as overdamping, critical damping, and underdamping respectively.
If the discriminant is negative, the solution will take the form
Underdamped systems will oscillate but the amplitude of the oscillations approaches zero with time.
Critically damped systems approach zero in the fastest possible time without oscillating. They are important in many engineering applications, as most shock absorbers are designed to be critically damped. They will follow the equation
C1 and C2 are determined by the initial conditions of the system.
Overdamped systems do not oscillate, but take more time to approach zero due to excessive damping. They follow the equation