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Waves
Waves

Damped Oscillation

x=Ae−bt2mcos⁡(ω′t)x = Ae^{-\frac{bt}{2m}}\cos(\omega' t)

Displacement of a damped harmonic oscillator (underdamped case).

Variables

x
Displacement
Position from equilibrium
m
A
Initial amplitude
Amplitude at t = 0
m
b
Damping coefficient
Damping constant
kg/s
t
Time
Time elapsed
s
m
Mass
Oscillating mass
kg
ω'
Damped angular frequency
ω' = √(ω₀² - (b/2m)²)
rad/s

Derivation

Solution to m(d²x/dt²) + b(dx/dt) + kx = 0. The damped frequency is ω' = √(k/m - (b/2m)²).

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Common questions

What is the Damped Oscillation formula?

Damped Oscillation is written x = Ae⁻bt/²mcos(ω' t). Displacement of a damped harmonic oscillator (underdamped case). It belongs to Waves, under Simple Harmonic Motion.

What are the units in x = Ae⁻bt/²mcos(ω' t)?

In SI units, x is displacement, measured in m; A is initial amplitude, measured in m; b is damping coefficient, measured in kg/s; t is time, measured in s; m is mass, measured in kg; ω' is damped angular frequency, measured in rad/s.

How is the Damped Oscillation formula derived?

Solution to m(d²x/dt²) + b(dx/dt) + kx = 0. The damped frequency is ω' = √(k/m - (b/2m)²). The result is x = Ae⁻bt/²mcos(ω' t).