Free vibrations can be defined as oscillations about a system’s equilibrium position that occur in the absence of an external excitation. In physics, perpetual motion is impossible. Dry friction, viscous friction, internal friction of the system, aerodynamic drag, etc eventually dissipate the energy (damping). For example, in one degree of freedom system, a mass is suspended by a spring, which is attached to a ceiling. If we pull the mass down and release it. the mass will oscillate and eventually return to its equilibrium position.
Damping is related to any effect that reduces the amplitude of oscillation. In mathematical term, damping is presented by symbol c, unit is N s/m. Damping ratio is a parameter that measures how fast the vibration magnitude decays over time. Damping ratio is given by:
ζ = c/(2mω)
m = mass.
ω = natural frequency , √(k/m). k is stiffness of the system.
Free vibrations are divided into three categories:
- Underdamped free vibrations, ζ<1 happen when the mass oscillates (overshooting) about its equilibrium position. Some energy in the system is dissipated in each cycle, and finally oscillations die towards zero at its equilibrium position. An example of underdamped system is oscillations of mass suspended by a spring.
- Overdamped free vibrations, ζ>1 happen when there is no oscillation or overshooting, and the mass could slowly return to its original position.
- Critically damped vibrations, ζ=1 are similar to overdamped free vibrations, except the system returns to its equilibrium in the minimum amount of time. An example of critically damped vibrations is the closing door mechanism in public buildings.
Viscous damping has been widely used in many critically damped systems. It also leads to positive effects when added to systems undergoing forced excitation. I’ll discuss forced vibrations in the next XYO blog since they are common systems in the field of vibration.