Damped oscillators and decay

Real oscillators lose energy

An ideal simple harmonic oscillator keeps moving forever because no energy is lost. Real oscillators are different. A swinging pendulum slows because of air resistance and pivot friction. A vibrating guitar string eventually becomes quiet. A car shock absorber reduces bouncing by dissipating energy.

Damping is any effect that removes mechanical energy from an oscillator. In many introductory models, the damping force is proportional to velocity and opposite motion:

$F_d=-b v$

where $b$ is the damping coefficient.

Equation of motion

For a mass-spring system with damping, the forces are spring force and damping force:

$F_s=-kx$

F_d=-brac{dx}{dt}

Newton's second law gives

mrac{d^2x}{dt^2}=-brac{dx}{dt}-kx

or

mrac{d^2x}{dt^2}+brac{dx}{dt}+kx=0

This is the damped harmonic oscillator equation.

It is often written in normalized form:

rac{d^2x}{dt^2}+2etarac{dx}{dt}+omega_0^2x=0

where

eta=rac{b}{2m}

and

omega_0=sqrt{rac{k}{m}}

Here $omega_0$ is the natural angular frequency of the undamped oscillator.

Underdamping

If damping is weak, the system still oscillates, but the amplitude decays over time. This is called underdamped motion. The solution has the form

x(t)=Ae^{-eta t}cos(omega_d t+phi)

where

omega_d=sqrt{omega_0^2-eta^2}

The factor e^{-eta t} is the decaying envelope. It gradually reduces the oscillation amplitude.

Critical damping

Critical damping is the boundary between oscillatory and non-oscillatory return to equilibrium. A critically damped system returns to equilibrium as quickly as possible without overshooting.

This is useful in instruments and engineering. A door closer should not oscillate back and forth. A measuring needle should settle quickly. Vehicle shock absorbers are designed to reduce oscillations efficiently.

Critical damping occurs when

eta=omega_0

in the normalized model.

Overdamping

If damping is stronger than critical, the system is overdamped. It returns to equilibrium without oscillating, but more slowly than in the critically damped case. Heavy damping resists motion so strongly that the system creeps back.

Overdamped behavior is useful when overshoot must be avoided, but it is inefficient if fast settling is desired.

Energy decay

In underdamped motion, amplitude decays like

A(t)=A_0e^{-eta t}

Since oscillator energy is proportional to amplitude squared,

$E(t)propto A(t)^2$

so energy decays approximately as

E(t)=E_0e^{-2eta t}

Energy disappears from the mechanical oscillator but is transformed into thermal energy, sound, or microscopic motion.

Damping in graphs

A damped oscillator graph shows oscillations inside a shrinking envelope. The time between peaks is related to the damped frequency $omega_d$, while the decreasing peak heights reveal the damping strength.

If damping is small, $omega_d$ is close to $omega_0$. Stronger damping reduces the oscillation frequency until oscillation disappears.

The big idea

Damping makes oscillations realistic by accounting for energy loss. The damped oscillator equation adds a velocity-dependent resistive term to the SHM equation. Depending on damping strength, motion can be underdamped, critically damped, or overdamped. These regimes are central in mechanical systems, electronics, acoustics, and engineering design.