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Damped 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 damped spring harmonic oscillator experiences a frictional force (damping) proportional to the velocity.

The angular frequency for a damped spring harmonic oscillator equals 2\[Pi] times the frequency. It also equals the square root of 1 minus the damping ratio squared times the natural angular 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.

Formula

{QuantityVariable["ω", "AngularFrequency"] == Sqrt[1 - QuantityVariable["ζ", "Unitless"]^2]*QuantityVariable[Subscript["ω", "0"], "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)}

symbol description physical quantity
ω angular frequency "AngularFrequency"
ζ damping ratio "Unitless"
ω0 natural angular frequency "AngularFrequency"
k spring constant "SpringConstant"
m mass "Mass"
f frequency "Frequency"
T period "Period"

Forms

Examples

Get the resource:

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

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

In[3]:=
FormulaData[ ResourceObject[ "Damped Spring Harmonic Oscillator"], {QuantityVariable[ "k","SpringConstant"] -> Quantity[1, ("Newtons")/("Meters")], QuantityVariable["\[Zeta]","Unitless"] -> 0.2`, QuantityVariable["f","Frequency"] -> Quantity[1, "Hertz"]}]
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