# AERO3220 HW4 Program Flow Block Diagram

```mermaid
flowchart TD
    A([Start Script]) --> B[Clear Workspace: clear, close all, clc]
    B --> C[Set Parameters: k1, k2, m1, m2]
    C --> D[Set Time Grid: t0, tf, dt, tspan]
    D --> E[Set Initial State Vector x0 with 4 states]
    E --> F[Define Input Force Functions: F1, F2, F3]

    F --> G[Case 1: b=100, F=F1]
    G --> H[Run ode45 with eom_two_mass]

    H --> I[Case 2: b=100, F=F2]
    I --> J[Run ode45 with eom_two_mass]

    J --> K[Case 3: b=0, F=F2]
    K --> L[Run ode45 with eom_two_mass]

    L --> M[Case 4: b=100, F=F3]
    M --> N[Run ode45 with eom_two_mass]

    N --> O[Case 5: b=0, F=F3]
    O --> P[Run ode45 with eom_two_mass]

    P --> Q[Build Legend Strings]
    Q --> R[Plot Figure 1: z1 vs time for 5 cases]
    R --> S[Plot Figure 2: z2 vs time for 5 cases]
    S --> T[Plot Figure 3: z1dot vs time for 5 cases]
    T --> U[Plot Figure 4: z2dot vs time for 5 cases]
    U --> V[Plot Figure 5: z1 and z2 together by case]

    V --> W[Local Function eom_two_mass]
    W --> X[Unpack States: z1, z1dot, z2, z2dot]
    X --> Y[Evaluate Input u = Ffun of t]
    Y --> Z[Compute Accelerations: z1ddot and z2ddot]
    Z --> AA[Return State Derivative Vector dx with 4 elements]
    AA --> AB([End])
```

## ODE Model Used in Each Case

\[
\ddot{z}_1 = -\frac{k_1}{m_1}z_1 + \frac{k_1}{m_1}z_2
\]

\[
\ddot{z}_2 = \frac{k_1}{m_2}z_1 - \frac{k_1+k_2}{m_2}z_2 - \frac{b}{m_2}\dot{z}_2 + \frac{1}{m_2}u(t)
\]

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
```

## 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/m1]
    G11 --> S1[Sum z1ddot]
    Z2[State z2] --> G12[Gain +k1/m1]
    G12 --> S1
    S1 --> I11[Transfer block 1/s]
    I11 --> Z1D[State z1dot]
    Z1D --> I12[Transfer block 1/s]
    I12 --> Z1

    U[Input force u] --> G2U[Gain 1/m2]
    G2U --> S2[Sum z2ddot]
    Z1 --> G21[Gain k1/m2]
    G21 --> S2
    Z2 --> G22[Gain -(k1+k2)/m2]
    G22 --> S2
    Z2D[State z2dot] --> G23[Gain -b/m2]
    G23 --> S2
    S2 --> I21[Transfer block 1/s]
    I21 --> Z2D
    Z2D --> I22[Transfer block 1/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

\[
x = \begin{bmatrix} z_1 \\ \dot{z}_1 \\ z_2 \\ \dot{z}_2 \end{bmatrix}
\]

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
flowchart TD
    A([Start]) --> B[Define constants and masses: k1, k2, m1, m2]
    B --> C[Define simulation time: t0, tf, dt, tspan]
    C --> D[Set initial state vector x0 with 4 states]
    D --> E[Define force inputs F1 constant, F2 sine slow, F3 sine fast]

    E --> F[Create case table with case b and force]
    F --> G{i <= Number of Cases?}

    G -- Yes --> H[Load case i values: b_i, F_i]
    H --> I[Call ode45 with eom_two_mass and current case]
    I --> J[Store outputs: t_i and x_i]
    J --> K[i = i + 1]
    K --> G

    G -- No --> L[Generate legend labels]
    L --> M[Plot z1 vs time for all cases]
    M --> N[Plot z2 vs time for all cases]
    N --> O[Plot z1dot vs time for all cases]
    O --> P[Plot z2dot vs time for all cases]
    P --> Q[Plot combined z1 and z2 by case]
    Q --> R([End])
```

### Solver Subsystem (Inside `eom_two_mass`)

```mermaid
flowchart LR
    A1[Inputs: t, x, m1, m2, k1, k2, b, Ffun] --> A2[Unpack states: z1, z1dot, z2, z2dot]
    A2 --> A3[Compute input force u from Ffun and t]
    A3 --> A4[Compute z1ddot from mass 1 equation]
    A4 --> A5[Compute z2ddot from mass 2 equation]
    A5 --> A6[Assemble derivative vector dx with 4 elements]
    A6 --> A7[Return dx to ode45]
```