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Driven Spring Harmonic Oscillator

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

The angular frequency for a driven spring harmonic oscillator equals the driving angular frequency as well as 2\[Pi] times the frequency. The frequency equals the reciprocal of the period. The natural angular frequency equals the square root of the ratio between the spring constant and the mass. 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[Subscript["ω", "0"], "AngularFrequency"] == Sqrt[QuantityVariable["k", "SpringConstant"]/QuantityVariable["m", "Mass"]], 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"
ω0 natural angular frequency "AngularFrequency"
k spring constant "SpringConstant"
m mass "Mass"
f frequency "Frequency"
T period "Period"
A amplitude "Unitless"
Ad driving amplitude "Unitless"
ϕ phase "Angle"

Forms

Examples

Get the resource:

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

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

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