ω_{0} = \sqrt{\frac{k}{m}}

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}

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}} )