Add transfer-function form of system block diagram

AERO3220_HW4_block_diagram.md

@@ -76,6 +76,45 @@ flowchart LR
     Z2D --> E2
 ```
 
+## Transfer Function Block Diagram Form
+
+This version shows the same coupled dynamics using gain blocks and integrator transfer functions.
+
+```mermaid
+flowchart LR
+    Z1[State z1] --> G11[Gain -k1 over m1]
+    G11 --> S1[Sum z1ddot]
+    Z2[State z2] --> G12[Gain plus k1 over m1]
+    G12 --> S1
+    S1 --> I11[Transfer block 1 over s]
+    I11 --> Z1D[State z1dot]
+    Z1D --> I12[Transfer block 1 over s]
+    I12 --> Z1
+
+    U[Input force u] --> G2U[Gain 1 over m2]
+    G2U --> S2[Sum z2ddot]
+    Z1 --> G21[Gain k1 over m2]
+    G21 --> S2
+    Z2 --> G22[Gain minus k1 plus k2 over m2]
+    G22 --> S2
+    Z2D[State z2dot] --> G23[Gain minus b over m2]
+    G23 --> S2
+    S2 --> I21[Transfer block 1 over s]
+    I21 --> Z2D
+    Z2D --> I22[Transfer block 1 over s]
+    I22 --> Z2
+```
+
+Equivalent transfer relation notes:
+
+\[
+Z_1 = \frac{1}{s^2}\left(-\frac{k_1}{m_1} Z_1 + \frac{k_1}{m_1} Z_2\right)
+\]
+
+\[
+Z_2 = \frac{1}{s^2}\left(\frac{k_1}{m_2} Z_1 - \frac{k_1+k_2}{m_2} Z_2 - \frac{b}{m_2} s Z_2 + \frac{1}{m_2} U\right)
+\]
+
 ### Coupled State Space Form
 
 Let