en forma matricial $$ \begin{equation*} v = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} = \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \end{equation*} $$
la deformación angular es $$ \begin{equation*} \theta = \frac{d v}{ d x} = \alpha_{1} + 2 \alpha_{2} x + 3 \alpha_{3} x^{2} \end{equation*} $$
reemplazando \( x_{1} = 0 \) y \( x_{2} = L \) $$ \begin{align*} \alpha_{0} + \alpha_{1} (0) + \alpha_{2} (0)^{2} + \alpha_{3} (0)^{3} &= v_{1} \\ \alpha_{1} + 2 \alpha_{2} (0) + 3 \alpha_{3} (0)^{2} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} (L) + \alpha_{2} (L)^{2} + \alpha_{3} (L)^{3} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} (L) + 3 \alpha_{3} (L)^{2} &= \theta_{2} \end{align*} $$
simplificando $$ \begin{align*} \alpha_{0} &= v_{1} \\ \alpha_{1} &= \theta_{1} \\ \alpha_{0} + L \alpha_{1} + L^{2} \alpha_{2} + L^{3} \alpha_{3} &= v_{2} \\ \alpha_{1} + 2 L \alpha_{2} + 3 L^{2} \alpha_{3} &= \theta_{2} \end{align*} $$
en forma matricial $$ \begin{equation*} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & L & L^{2} & L^{3} \\ 0 & 1 & 2 L & 3 L^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*} $$
resolviendo el sistema $$ \begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{3}{L^{2}} & -\frac{2}{L} & \frac{3}{L^{2}} & -\frac{1}{L} \\ \frac{2}{L^{3}} & \frac{1}{L^{2}} & -\frac{2}{L^{3}} & \frac{1}{L^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*} $$
reemplazando las incógnitas $$ \begin{align*} v &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{3}{L^{2}} & -\frac{2}{L} & \frac{3}{L^{2}} & -\frac{1}{L} \\ \frac{2}{L^{3}} & \frac{1}{L^{2}} & -\frac{2}{L^{3}} & \frac{1}{L^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &=\begin{bmatrix} 1 - \frac{3}{L^{2}} x^{2} + \frac{2}{L^{3}} x^{3} & x - \frac{2}{L} x^{2} + \frac{1}{L^{2}} x^{3} & \frac{3}{L^{2}} x^{2} - \frac{2}{L^{3}} x^{3} & -\frac{1}{L} x^{2} + \frac{1}{L^{2}} x^{3} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &=\begin{bmatrix} N_{1} & N_{2} & N_{3} & N_{4} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{align*} $$