Add system-equations block diagram and state-space form

AERO3220_HW4_block_diagram.md

@@ -50,6 +50,64 @@ flowchart TD
 
 where \(u(t)\) is one of the three forcing functions depending on the case.
 
+## System Equations Block Diagram
+
+This is a dynamics block diagram of the coupled equations, not a program flowchart.
+
+```mermaid
+flowchart LR
+    U[Input force u] --> E2[Compute z2ddot from z1 z2 z2dot u and parameters]
+
+    E1[Compute z1ddot from z1 z2 and parameters] --> I1v[Integrate to z1dot]
+    I1v --> Z1D[State z1dot]
+    Z1D --> I1x[Integrate to z1]
+    I1x --> Z1[State z1]
+
+    E2 --> I2v[Integrate to z2dot]
+    I2v --> Z2D[State z2dot]
+    Z2D --> I2x[Integrate to z2]
+    I2x --> Z2[State z2]
+
+    Z1 --> E1
+    Z2 --> E1
+
+    Z1 --> E2
+    Z2 --> E2
+    Z2D --> E2
+```
+
+### Coupled State Space Form
+
+Let
+
+\[
+x = \begin{bmatrix} z_1 \\ \dot{z}_1 \\ z_2 \\ \dot{z}_2 \end{bmatrix}
+\]
+
+then
+
+\[
+\dot{x} = A x + B u
+\]
+
+with
+
+\[
+A = \begin{bmatrix}
+0 & 1 & 0 & 0 \\
+-\frac{k_1}{m_1} & 0 & \frac{k_1}{m_1} & 0 \\
+0 & 0 & 0 & 1 \\
+\frac{k_1}{m_2} & 0 & -\frac{k_1+k_2}{m_2} & -\frac{b}{m_2}
+\end{bmatrix},
+\qquad
+B = \begin{bmatrix}
+0 \\
+0 \\
+0 \\
+\frac{1}{m_2}
+\end{bmatrix}
+\]
+
 ## Assignment-Style Block Diagram (Recommended For Submission)
 
 ```mermaid