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Driven Harmonic Oscillator

A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. A driven harmonic oscillator experiences an external time-dependent force driving the system.

The angular frequency for a damped driven harmonic oscillator equals the driving angular frequency, as well as 2\[Pi] times the frequency. The frequency equals the reciprocal of the period. The amplitude is directly proportional to the driving amplitude, and maximizes when the natural angular frequency equals the driving frequency. The phase depends on the difference between the natural angular frequency and driving frequency.

Formula

{QuantityVariable["ω", "AngularFrequency"] == QuantityVariable[Subscript["ω", "d"], "AngularFrequency"], QuantityVariable["ω", "AngularFrequency"] == 2*Pi*QuantityVariable["f", "Frequency"], QuantityVariable["f", "Frequency"] == QuantityVariable["T", "Period"]^(-1), QuantityVariable["A", "Unitless"] == QuantityVariable[Subscript["A", "d"], "Unitless"]/Abs[1 - QuantityVariable[Subscript["ω", "d"], "AngularFrequency"]^2/QuantityVariable[Subscript["ω", "0"], "AngularFrequency"]^2], QuantityVariable["ϕ", "Angle"] == Pi*HeavisideTheta[Quantity[1, "Seconds"^2/"Radians"^2]*(-QuantityVariable[Subscript["ω", "0"], "AngularFrequency"]^2 + QuantityVariable[Subscript["ω", "d"], "AngularFrequency"]^2)]}

symbol description physical quantity
ω angular frequency "AngularFrequency"
ωd driving angular frequency "AngularFrequency"
f frequency "Frequency"
T period "Period"
A amplitude "Unitless"
ω0 natural angular frequency "AngularFrequency"
Ad driving amplitude "Unitless"
ϕ phase "Angle"

Forms

Examples

Get the resource:

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

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

In[3]:=
FormulaData[ ResourceObject[ "Driven Harmonic Oscillator"], {QuantityVariable["f","Frequency"] -> Quantity[1, "Hertz"], QuantityVariable["\[Omega]","AngularFrequency"] -> Quantity[6, ("Radians")/("Seconds")]}]
Out[3]=

Publisher Information