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Driven Torsion Harmonic Oscillator

A torsion harmonic oscillator is a twisting system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. A driven torsion 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 natural frequency equals the square root of the torsional constant divided by moment of inertia. 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["κ", "TorsionalConstant"]/QuantityVariable["I", "MomentOfInertia"]], 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"
I moment of inertia "MomentOfInertia"
κ torsional constant "TorsionalConstant"
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 Torsion Harmonic Oscillator"]
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Get the formula:

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

In[3]:=
FormulaData[ ResourceObject[ "Driven Torsion Harmonic Oscillator"], {QuantityVariable[ \!\(\*SubscriptBox[\("\[Omega]"\), \("d"\)]\),"AngularFrequency"] -> Quantity[10, ("Radians")/("Seconds")], QuantityVariable["\[Omega]","AngularFrequency"] -> Quantity[6, ("Radians")/("Seconds")], QuantityVariable["\[Phi]","Angle"] -> Quantity[0, "Radians"]}]
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