Damped Oscillation

x = Ae^{-\frac{bt}{2m}}\cos(\omega' t)

Displacement of a damped harmonic oscillator (underdamped case).

Variables

Displacement

Position from equilibrium

Initial amplitude

Amplitude at t = 0

Damping coefficient

Damping constant

kg/s

Time

Time elapsed

Mass

Oscillating mass

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).

Your notes

0/5000

Other waves formulas with the same quantities

SHM Velocity (Simple Harmonic Motion) shares Time
SHM Acceleration (Simple Harmonic Motion) shares Displacement
Spring Oscillation Period (Simple Harmonic Motion) shares Mass
SHM Total Energy (Simple Harmonic Motion) shares Mass
Physical Pendulum Period (Simple Harmonic Motion) shares Mass

Where this formula is listed

Waves formulas, under Simple Harmonic Motion, alongside SHM Maximum Velocity, Pendulum Period, Torsional Oscillation Period and Resonance Amplitude.
In the waves list it comes after Torsional Oscillation Period and before Quality Factor (Q Factor).
Waves formula sheet, one of 37 formulas printed with symbol keys and SI units.
The full formula index, grouped by category and topic.

Common questions

What is the Damped Oscillation formula?

Damped Oscillation is written x = Ae^(−bt/(2m)) cos(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/(2m)) cos(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/(2m)) cos(ω′t).