ω_{0} = \sqrt{\frac{k}{m}}

ω = \sqrt{\frac{k}{m} − \frac{b^{2}}{4m^{2}}}

F⃗ (air) = −bv⃗

F (t) = F_{max} e^{iωt}

m\frac{d^{2}x}{dt^{2}} + b\frac{dx}{dt} + kx = F_{max} e^{iωt}

\frac{d^{2}x}{dt^{2}} + \frac{b}{m} \frac{dx}{dt} + \frac{k}{m}x = 0

x(t) = Ae^{i(ωt+ϕ)}

\frac{dx}{dt} = iωAe^{i(ωt+ϕ)} = iω x(t)

\frac{d^{2}x}{dt^{2}} = −ω^{2}Ae^{i(ωt+ϕ)} = −ω^{2} x(t)

−mω^{2}Ae^{i(ωt+ϕ)} + ibωAe^{i(ωt+ϕ)} + kAe^{i(ωt+ϕ)} = F_{max}e^{iωt}

−mω^{2}Ae^{iϕ} + ibωAe^{iϕ} + kAe^{iϕ} = F_{max}

−ω^{2}Ae^{iϕ} + i\frac{bω}{m} Ae^{iϕ} + \frac{k}{m}Ae^{iϕ} = \frac{F_{max}}{m}

−ω^{2}Ae^{iϕ} + i\frac{bω}{m} Ae^{iϕ} + ω_{0}^{2}Ae^{iϕ} = \frac{F_{max}}{m}

−ω^{2}A + i\frac{bω}{m} A + ω_{0}^{2}A = \frac{F_{max}}{m} e^{−iϕ}

−ω^{2}A + i\frac{bω}{m} A + ω_{0}^{2}A = \frac{F_{max}}{m} [cos(ϕ) − i sin(ϕ)]

−ω^{2}A + ω_{0}^{2}A = (ω_{0}^{2} − ω^{2})A = \frac{F_{max}}{m} cos(ϕ)

\frac{bω}{m} A = −\frac{F_{max}}{m} sin(ϕ)

cos(ϕ) = \frac{Am(ω_{0}^{2} − ω^{2})}{F_{max}}

sin(ϕ) = −\frac{Ab ω}{F_{max}}

F_{max} = \sqrt{A^{2}m^{2}(ω_{0}^{2} − ω^{2})^{2} + A^{2}b^{2}ω^{2}}

F_{max} = A\sqrt{m^{2}(ω_{0}^{2} − ω^{2})^{2} + b^{2}ω^{2}}

A = \frac{F_{max}}{\sqrt{m^{2}(ω_{0}^{2} − ω^{2})^{2} + b^{2}ω^{2}}}

A = \frac{F_{max}/m}{\sqrt{(ω_{0}^{2} − ω^{2})^{2} + \frac{b^{2}}{m^{2}} ω^{2}}}

ϕ = tan^{−1} (\frac{(\frac{b}{m}) ω}{ω_{0}^{2} − ω^{2}} )

\frac{d^{2}x}{dt^{2}} + \frac{b}{m} \frac{dx}{dt} + \frac{k}{m}x = 0

x(t) = Ae^{i(ωt+ϕ)−(t/τ)}

τ = \frac{2m}{b}

SPE(x) = \frac{1}{2}k [x(t)]^{2} = \frac{1}{2}k A^{2}e^{2i[ωt+ϕ]−(2t/τ)}

amplitude time constant τ = \frac{2m}{b}

energy time constant \frac{τ}{2} = \frac{m}{b}

time to move one cycle P = \frac{2π}{ω_{0}}

time to move one radian \frac{P}{2π} = \frac{1}{ω_{0}}

Q ≡ \frac{τ/2}{1/ω_{0}} = \frac{ω_{0}m}{b}

Q = \frac{τω_{0}}{2}

amplitude ∝ e^{− \frac{t}{τ}}

decay in amplitude in one cycle = e^{−(π/Q)} = (\frac{1}{e})^{(π/Q)} decay in energy in one cycle = e^{−(2π/Q)} = (\frac{1}{e})^{(2π/Q)}

Δx_{static} = \frac{F_{max}}{k}

A = \frac{F_{max}/m}{(\frac{b}{m}) ω_{0}}

Q = \frac{ω_{0} m}{b}

A = \frac{F_{max}}{k/Q} = Q (\frac{F_{max}}{k} )

Q = \frac{oscillating displacement}{static displacement}

A = \frac{F_{max}/m}{\sqrt{(ω_{0}^{2} − ω^{2})^{2} + \frac{ω_{0}^{2}}{Q^{2}} ω^{2}}}

Power dissipated ∝ A^{2} ∝ \frac{F_{max}^{2}/m^{2}}{(ω_{0}^{2} − ω^{2})^{2} + \frac{ω_{0}^{2}}{Q^{2}} ω^{2}}

Peak power dissipated ∝ \frac{F_{max}^{2}/m^{2}}{\frac{ω_{0}^{2}}{Q^{2}} ω^{2}}

(ω_{0}^{2} − ω^{2})^{2} + \frac{ω_{0}^{2}}{Q^{2}} ω^{2} = 2 \frac{ω_{0}^{2}}{Q^{2}} ω^{2}

(ω_{0}^{2} − ω^{2}) = \frac{ω_{0}}{Q} ω

(ω_{0} + ω)(ω_{0} − ω) = \frac{ω_{0}}{Q} ω

2ω_{0}(ω_{0} − ω) ≃ \frac{ω_{0}}{Q} ω

(ω_{0} − ω) ≃ \frac{ω_{0}}{2Q}

\frac{Δω}{ω_{0}} ≃ \frac{1}{2Q}