F⃗_{air} = −bv⃗

F_{1} = −kx_{1} + k(x_{2} − x_{1}) − bv_{1} = −2kx_{1} + kx_{2} − bv_{1}

F_{2} = −k(x_{2} − x_{1}) − kx_{2} − bv_{2} = kx_{1} − 2kx_{2} − bv_{2}

m\frac{d^{2}x_{1}}{dt^{2}} = −2kx_{1} + kx_{2} − b \frac{dx_{1}}{dt}

m\frac{d^{2}x_{2}}{dt^{2}} = kx_{1} − 2kx_{2} − b \frac{dx_{2}}{dt}

m\frac{d^{2}(x_{1} + x_{2})}{dt^{2}} = −k(x_{1} + x_{2}) − b \frac{d(x_{1} + x_{2})}{dt}

s_{1} ≡ x_{1} + x_{2}

m\frac{d^{2}s_{1}}{dt^{2}} = −ks_{1} − b\frac{ds_{1}}{dt}

s_{1}(t) = A_{1} cos(ω_{1}t + ϕ_{1})e^{−t/τ}

s_{1}(t) = A_{1}e^{i(ω_{1}t+ϕ_{1})−(t/τ)}

\frac{ds_{1}}{dt} = (iω_{1} − \frac{1}{τ} ) A_{1}e^{i(ω_{1}t+ϕ_{1})−(t/τ)}

\frac{d^{2}s_{1}}{dt^{2}} = (−ω_{1}^{2} − i\frac{2ω_{1}}{τ} + \frac{1}{τ^{2}} ) A_{1}e^{i(ω_{1}t+ϕ_{1})−(t/τ)}

m\frac{d^{2}s_{1}}{dt^{2}} + b\frac{ds_{1}}{dt} + ks_{1} = 0

m (−ω_{1}^{2} − i\frac{2ω_{1}}{τ} + \frac{1}{τ^{2}} ) + b (iω_{1} − \frac{1}{τ} ) + k = 0

−\frac{2ω_{1}}{τ} + bω_{1} = 0

b = \frac{2m}{τ}  or  τ = \frac{2m}{b}

−ω_{1}^{2} + \frac{1}{τ^{2}} − \frac{b}{mτ} + \frac{k}{m} = 0

ω_{1}^{2} = \frac{k}{m} − \frac{b^{2}}{4m^{2}}

m\frac{d^{2}(x_{2} − x_{1})}{dt^{2}} = −3k(x_{2} − x_{1}) − b \frac{d(x_{2} − x_{1})}{dt}

m\frac{d^{2}s_{2}}{dt^{2}} + b\frac{ds_{2}}{dt} + 3ks_{2} = 0

s_{2}(t) = A_{2}e^{i(ω_{2}t+ϕ_{2})−(t/τ)}

ω_{2}^{2} = \frac{3k}{m} − \frac{b^{2}}{4m^{2}}

\begin{aligned}s_{1} & = x_{1} + x_{2} \\ s_{2} & = x_{2} − x_{1} \end{aligned}

\begin{aligned}x_{1} & = \frac{1}{2}(s_{1} − s_{2}) \\ x_{2} & = \frac{1}{2}(s_{1} + s_{2}) \end{aligned}

x_{1} = \frac{1}{2} (A_{1}e^{i(ω_{1}t+ϕ_{1})−(t/τ)} − A_{2}e^{i(ω_{2}t+ϕ_{2})−(t/τ)})

x_{2} = \frac{1}{2} (A_{1}e^{i(ω_{1}t+ϕ_{1})−(t/τ)} + A_{2}e^{i(ω_{2}t+ϕ_{2})−(t/τ)})

x_{1}(t) = A_{1}e^{i(ω_{1}t+ϕ_{1})−(t/τ)}

\frac{dx_{1}}{dt} = (iω − \frac{1}{τ} ) x_{1}

\begin{aligned}\frac{d^{2}x_{1}}{dt^{2}} & = (iω − \frac{1}{τ} )^{2} x_{1} \\ & = (−ω^{2} − i\frac{2ω}{τ} + \frac{1}{τ^{2}} ) x_{1} \end{aligned}

( \begin{matrix}−2\frac{k}{m} + \frac{b}{mτ} − i\frac{bω}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} + \frac{b}{mτ} − i\frac{bω}{m} \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) = (−ω^{2} − i\frac{2ω}{τ} + \frac{1}{τ^{2}} ) ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} )

( \begin{matrix}−2\frac{k}{m} + \frac{b}{mτ} − i\frac{bω}{m} & \frac{k}{m} \\ \frac{k}{m} & −2\frac{k}{m} + \frac{b}{mτ} − i\frac{bω}{m} \end{matrix} ) ( \begin{matrix}p \\ q \end{matrix} ) = (−ω^{2} − i\frac{2ω}{τ} + \frac{1}{τ^{2}} ) ( \begin{matrix}p \\ q \end{matrix} )

s_{1} = px_{1} + qx_{2}

s_{1} = 1x_{1} + 1x_{2}  ⇒  s_{1} = ( \begin{matrix}1 \\ 1 \end{matrix} )

s_{2} = −1x_{1} + 1x_{2}  ⇒  s_{2} = ( \begin{matrix}−1 \\ 1 \end{matrix} )