# Ecuaciones Lámina 42:A
#1
\det[\mathbb{K} - \omega ^2\mathbb{M}] = 0
#2
\det
\begin{bmatrix}
3k-\omega^2(2m) &k\\
-k & k - \omega^2(m)
\end{bmatrix} = 0
#3
(3k-2\omega^2m)(k - \omega^2m)-k^2 = 0
#4 Polinomio
3k^2-3km\omega^2-2km\omega^2+2m^2\omega^4-k^2 = 0
#5 Polinomio
2m^2(\omega^2)^2-5mk\omega^2+2k^2 = 0
#6 Resolvente
\omega^2_{1,2} = \frac{5mk \pm \sqrt{25m^2k^2 - 4(2m^2)(2k^2)}}{4m^2}
#7
\omega^2_{1,2} = \frac{5k}{4m} \pm \frac{3k}{4m}
#8
\frac{2k}{m}
#9
\frac{1}{2}\frac{k}{m}
#10
\omega_1 = \pm \sqrt{\frac{2k}{m}}
#11
\omega_2 = \pm \sqrt{\frac{k}{2m}}

# Ecuaciones Lámina 42:B
#1
\begin{bmatrix}
3k-\omega^2(2m) &k\\
-k & k - \omega^2(m)
\end{bmatrix}
\begin{bmatrix}
\Phi_{J1}\\
\Phi_{J2}\\
\end{bmatrix} =
\begin{bmatrix}
0\\
0\\
\end{bmatrix}
#2
-k\Phi_{J1}+(k-\omega_J^2m)\Phi_{J2} = 0
#3 Para omega_1
\begin{bmatrix}
2k &-k\\
-k & \frac{1}{2}k
\end{bmatrix}
\begin{bmatrix}
\Phi_{J1}\\
\Phi_{J2}\\
\end{bmatrix} =
\begin{bmatrix}
0\\
0\\
\end{bmatrix}

#4 Phi_1 = Phi_2

\Phi_{J,1} =    \left[   \frac{k-\omega_J^2m}{k}     \right] \Phi_{J,2}

# 5 J = 1
\Phi_{1,1} =    \left[   \frac{k -\frac{k}{2}}{k}     \right] 1  = \frac{1}{2}
#6
\Phi_{1,1} =
\begin{bmatrix}
\frac{1}{2}\\
1\\
\end{bmatrix}

# 7 J = 2
\Phi_{2,1} =    \left[   \frac{k -2k}{k}     \right] 1  = -1
#8
\Phi_{2,1} =
\begin{bmatrix}
-1\\
1\\
\end{bmatrix} =



##################################################################
## 42
$$
\frac{\partial D}{\partial \dot{x_1}} = C_1 \dot{x_1} + C_2 (\dot{x_2}- \dot{x_1})(-1) = (C_1 + C_2)\dot{x_1} - (C_2)\dot{x_2}\\
\frac{\partial D}{\partial \dot{x_2}} = - (C_2)\dot{x_1} +  (C_2 + C_3)\dot{x_2} - (C_3)\dot{x_3}\\
\frac{\partial D}{\partial \dot{x_3}} = - C_3(\dot{x_2}) -  (C_3 + C_4)\dot{x_3}
$$


# Sistema de EDOS de 2do orden Linela (No Homogeneo)
$$
\begin{bmatrix}
F_1\\
F_2\\
F_3\\
\end{bmatrix} =
\begin{bmatrix}
m_1\ddot{x_1}\\
m_2\ddot{x_2}\\
m_3\ddot{x_3}\\
\end{bmatrix} +
\begin{bmatrix}
C_1+C_2&-C_2&0\\
-C_2&C_2+C_3&-C_3\\
0&-C_3&C_3+C_4\\
\end{bmatrix}
\begin{bmatrix}
\dot{x_1}\\
\dot{x_2}\\
\dot{x_3}\\
\end{bmatrix}+
\begin{bmatrix}
k_1+K_2&-K_2&0\\
-K_2&K_2+K_3&-K_3\\
0&-K_3&K_3+K_4\\
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
x_3\\
\end{bmatrix}
$$

Mass Matrix
$$
\begin{bmatrix}
m_1&0&0\\
0&m_2&0\\
0&0&m_3\\
\end{bmatrix}
\begin{bmatrix}
\ddot{x_1}\\
\ddot{x_2}\\
\ddot{x_3}\\
\end{bmatrix}
$$

$$Letras
\mathbb{M}; \mathbb{C}; \mathbb{K}
$$

$$ Vector de Posición
\vec{x}(t) =
\begin{bmatrix}
x_1(t)\\
x_2(t)\\
x_3(t)\\
\end{bmatrix}
$$

$$ Vector de Velocidad
\vec{v}(t) =
\begin{bmatrix}
\dot{x_1}(t)\\
\dot{x_2}(t)\\
\dot{x_3}(t)\\
\end{bmatrix}
$$

$$
\mathbb{M} \ddot{\vec{x}} + \mathbb{C} \dot{\vec{x}} + \mathbb{K} \vec{x} = \vec{F}
$$

$$
\mathbb{M} \ddot{\vec{x}}  + \mathbb{K} \vec{x} = \vec{0}
$$

$$Condiciones iniciales
\vec{x}_o =  \vec{x}_o\\
\dot{\vec{x}}_o =  \vec{V}_o
$$

$$Solución Propuesta
\vec{x}(t) = \vec{\Phi} e^{i \omega t}
$$

$$Velocidad
\dot{\vec{x}}(t) = i \omega \vec{\Phi} e^{i \omega t}
$$

$$Aceleración
\ddot{\vec{x}}(t) = - \omega^2 \vec{\Phi} e^{i \omega t}
$$

##########################################################################
$$
-\omega^2 \mathbb{M} \vec{\Phi} e^{i \omega t} + \mathbb{K} \vec{\Phi} e^{i \omega t} = \vec{0}\\
(\mathbb{K} -\omega^2 \mathbb{M})\vec{\Phi} e^{i \omega t} = \vec{0}
$$
$$ Problema de Autovaloes y Autovectores
[\mathbb{K} -\omega^2 \mathbb{M}]\vec{\Phi}  = \vec{0}
$$
$$ Problema de Autovaloes y Autovectores (Forma Estándar)
[\mathbb{A} -\lambda \mathbb{I}]\vec{\Phi}  = \vec{0}
$$

$$Si solo si
[\mathbb{K} -\omega^2 \mathbb{M}]\vec{\Phi}  = \vec{0} \iff \mathbb{K}\vec{\Phi} =   \omega^2 \mathbb{M}\vec{\Phi}
$$

$$Solucion Trivial
\vec{\Phi}  = \vec{0}
$$

$$Solucion Trivial Pt.2
\vec{\Phi}  = [\mathbb{K} -\omega^2 \mathbb{M}]^{-1}\vec{0} = \vec{0}
$$


$$No Inversa
\det[\mathbb{K} -\omega^2 \mathbb{M}] = 0
$$


$$Respuesta libre no amortiguada
\vec{F} = \vec{0}
\mathbb{C} = [0]
$$

$$Aceleración
\ddot{\vec{x}}(t) = - \omega^2 \vec{x}
$$