F_{1} = −kx_{1} + (x_{2} − x_{1})k_{c}

F_{2} = −k_{c}(x_{2} − x_{1}) − kx_{2}

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

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

( \begin{matrix}− (\frac{k}{m} + \frac{k_{c}}{m} ) & \frac{k_{c}}{m} \\ \frac{k_{c}}{m} & − (\frac{k}{m} + \frac{k_{c}}{m} ) \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) = ( \frac{d^{2}x_{1}}{dt^{2}} ; \frac{d^{2}x_{2}}{dt^{2}} )

s = ax_{1} + bx_{2}

( \begin{matrix}− (\frac{k}{m} + \frac{k_{c}}{m} ) & \frac{k_{c}}{m} \\ \frac{k_{c}}{m} & − (\frac{k}{m} + \frac{k_{c}}{m} ) \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) = −ω^{2} ( \begin{matrix}a \\ b \end{matrix} )

γ ≡ (\frac{k}{m} + \frac{k_{c}}{m} )

Δ ≡ \frac{k_{c}}{m}

( \begin{matrix}−γ & Δ \\ Δ & −γ \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) = −ω^{2} ( \begin{matrix}a \\ b \end{matrix} )

−γa + Δb + ω^{2}a = 0

Δa − γb + ω^{2}b = 0

b = \frac{(γ − ω^{2})}{Δ}a

Δ a − γ (\frac{γ − ω^{2}}{Δ} ) a + ω^{2} (\frac{γ − ω^{2}}{Δ} ) a = 0

(Δ^{2} − γ^{2}) + 2γω^{2} − ω^{4} = 0

λ ≡ ω^{2}

(Δ^{2} − γ^{2}) + 2γλ − λ^{2} = 0

λ = γ ± Δ = (\frac{k}{m} + \frac{k_{c}}{m} ) ± \frac{k_{c}}{m}

λ = \frac{k}{m} + \frac{2k_{c}}{m}  or  λ = \frac{k}{m}

ω_{1} = \sqrt{\frac{k}{m}}

ω_{2} = \sqrt{\frac{k}{m} + \frac{2k_{c}}{m}}

−γa + Δb + (\frac{k}{m} + \frac{2k_{c}}{m} ) a = 0

− (\frac{k}{m} + \frac{k_{c}}{m} ) a + (\frac{k_{c}}{m} ) b + (\frac{k}{m} + \frac{2k_{c}}{m} ) a = 0

b = −a

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

b = a

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

s_{1}(t) = A_{1} cos(ω_{1}t + ϕ_{1}) s_{2}(t) = A_{2} cos(ω_{2}t + ϕ_{2})

s_{1}(t) = A_{1} cos(ω_{1}t + ϕ_{1}) = x_{1}(t) + x_{2}(t) s_{2}(t) = A_{2} cos(ω_{2}t + ϕ_{2}) = x_{2}(t) − x_{1}(t)

a = a

x_{1}(t) = \frac{1}{2}(A_{1} cos(ω_{1}t + ϕ_{1}) − A_{2} cos(ω_{2}t + ϕ_{2}))

x_{2}(t) = \frac{1}{2}(A_{1} cos(ω_{1}t + ϕ_{1}) + A_{2} cos(ω_{2}t + ϕ_{2}))

x_{1}(t) = \frac{L}{2}[cos(ω_{1}t) − cos(ω_{2}t)]

x_{2}(t) = \frac{L}{2}[cos(ω_{1}t) + cos(ω_{2}t)]