AcademyOscillations
Academy
Damping
Level 1 - Physics topic page in Oscillations.
Principle
Damping removes oscillator energy through a velocity-dependent resistive force.
The undamped SHM model conserves mechanical energy. Damping adds a force that opposes motion.
Notation
\(x\)
displacement from equilibrium
\(\mathrm{m}\)
\(b\)
linear damping coefficient
\(\mathrm{kg\,s^{-1}}\)
\(\beta\)
damping rate
\(\mathrm{s^{-1}}\)
\(\omega_0\)
undamped angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(\omega_d\)
damped angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(A_0\)
initial envelope amplitude
\(\mathrm{m}\)
Method
Derivation 1: Add a resistive force
For linear damping, the resistive force is proportional to velocity and points opposite the velocity.
Damping force
\[F_d=-b\frac{dx}{dt}\]
Spring force
\[F_s=-kx\]
Newton model
\[m\frac{d^2x}{dt^2}=-b\frac{dx}{dt}-kx\]
Standard form
\[m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0\]
Derivation 2: Identify the damped oscillation
The exponential envelope comes from energy loss. The cosine remains only in the underdamped case.
Damping rate
\[\beta=\frac{b}{2m}\]
Natural frequency
\[\omega_0=\sqrt{\frac{k}{m}}\]
Underdamped condition
\[\beta<\omega_0\]
Damped frequency
\[\omega_d=\sqrt{\omega_0^2-\beta^2}\]
Displacement
\[x=A_0e^{-\beta t}\cos(\omega_dt+\phi)\]
The graph is a time history. The oscillation remains periodic-like, but the amplitude envelope shrinks.
Rules
These are the compact results from the derivations above.
Damped oscillator
\[m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0\]
Damping rate
\[\beta=\frac{b}{2m}\]
Underdamped condition
\[\beta<\omega_0\]
Damped frequency
\[\omega_d=\sqrt{\omega_0^2-\beta^2}\]
Underdamped motion
\[x=A_0e^{-\beta t}\cos(\omega_dt+\phi)\]
Checks
- Damping force opposes velocity.
- Use the cosine form only for underdamping.
- Damping lowers frequency slightly.
- Mechanical energy decreases with time.