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Damped Harmonic Oscillator

A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. A damped harmonic oscillator experiences a frictional force (damping) proportional to the velocity.

The angular frequency for a damped driven harmonic oscillator equals the natural angular frequency times the square root of 1 minus the damping ratio squared, as well as 2\[Pi] times the frequency. The frequency equals the reciprocal of the period.

Formula

{QuantityVariable["ω", "AngularFrequency"] == Sqrt[1 - QuantityVariable["ζ", "Unitless"]^2]*QuantityVariable[Subscript["ω", "0"], "AngularFrequency"], QuantityVariable["ω", "AngularFrequency"] == 2*Pi*QuantityVariable["f", "Frequency"], QuantityVariable["f", "Frequency"] == QuantityVariable["T", "Period"]^(-1)}

symbol description physical quantity
ω angular frequency "AngularFrequency"
ζ damping ratio "Unitless"
ω0 natural angular frequency "AngularFrequency"
f frequency "Frequency"
T period "Period"

Forms

Examples

Get the resource:

In[1]:=
ResourceObject["Damped Harmonic Oscillator"]
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Get the formula:

In[2]:=
FormulaData[ResourceObject["Damped Harmonic Oscillator"]]
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Use some values:

In[3]:=
FormulaData[ ResourceObject["Damped Harmonic Oscillator"], {QuantityVariable[ \!\(\*SubscriptBox[\("\[Omega]"\), \("0"\)]\),"AngularFrequency"] -> Quantity[6, ("Radians")/("Seconds")], QuantityVariable["\[Omega]","AngularFrequency"] -> Quantity[6, ("Radians")/("Seconds")]}]
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