Damping force = $-c \frac{dx}{dt}$, it slows the motion, dissipating energy from the system

• Total force acting on the mass becomes restoring force + damping force = $-kc -d \frac{dx}{dt}$
• Newton’s second law of motion gives: $m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$, which gives damped harmonic motion
• The characteristic equation is $mr^2+cr+k = 0$

Underdamping

• When the discriminant of the characteristic equation is $<0$

  

Overdamping

• When the discriminant of the characteristic equation is $>0$

Critical Damping

• When the discriminant of the characteristic equation $=0$