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https://github.com/inventwithdean/damped_harmonic_oscillator

A solution to Spring-Mass-Damper using Forward Euler Integration
https://github.com/inventwithdean/damped_harmonic_oscillator

differential-equations dynamical-systems euler-integration

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A solution to Spring-Mass-Damper using Forward Euler Integration

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README 

        
# 🌊 Damped Harmonic Oscillator Integration 

*A beautiful dance of physics and mathematics* ✨



Based on Steve Brunton's amazing lecture: [Second-Order ODE: Spring-Mass-Damper](https://youtu.be/r1eWerqrcqo) 🎓



## 🧮 The Beautiful Theory

The damped harmonic oscillator is like a poetry in physics, described by this elegant equation:



$$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$$



What's dancing in this equation? 🩰

- m = mass (our main character)

- c = damping coefficient (the resistance)

- k = spring constant (the restoring force)

- x = position (where we are)

- dx/dt = velocity (how fast we're moving)

- d²x/dt² = acceleration (how our speed changes)



### 🔄 State Space Magic

Here's where it gets interesting! We can transform our 2nd order ODE into two 1st order ODEs using the state space sorcery ✨



Let's say $x$ be the position and $v = \dot{x}$ be the velocity.



And voilà! Our system becomes this beautiful matrix:

$$\begin{bmatrix} \dot{x} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -\omega^2 & -\zeta \end{bmatrix} \begin{bmatrix} x \\ v \end{bmatrix}$$



The magical ingredients 🪄:

- $\omega = \sqrt{\frac{k}{m}}$ (natural frequency - the system's heartbeat 💓)



- $\zeta = {\frac{c}{m}}$ (damping ratio - how quickly the dance fades 🌙)



## 📊 Understanding Our Creation



![output](https://github.com/user-attachments/assets/8459782d-6dfb-4981-8774-8c9e751d3412)



### 📈 Time Domain Plot

Watch the beautiful dance unfold:

- 🔵 Blue line: Position gracefully oscillating and settling

- ❤️ Red dashed line: Velocity leading the dance

- Both gradually calm down, like waves on a peaceful shore 🌊



### 🌀 Phase Space Plot (The Most Beautiful Part!)

This is where mathematics meets art:

- A mesmerizing spiral in position-velocity space 🌌

- Each loop tells a story of one complete oscillation ⭕

- The journey to equilibrium is like a leaf gently falling 🍃

- The spiral's shape reveals the system's character 🎭



### 🎯 Key Insights

1. 🎯 Matrix multiplication → instant rates of change

2. ⏱️ Small time steps → future prediction

3. 📐 Smaller steps = better accuracy (but more computation!)

4. ⚖️ Newton's F = ma holds every single moment!



## 🌟 The Beauty of Physics

This simple system shows us how nature works in perfect harmony:

- Energy gradually dissipates 📉

- Oscillations find their natural rhythm 🎵

- Mathematics predicts reality with stunning accuracy ✨



## 📚 Want to Learn More?

- 🎥 Watch Steve Brunton's amazing Spring-Mass-Damper: [Video Link](https://youtu.be/r1eWerqrcqo)

- 📺 Full playlist for more Differential Equations: [Differential Equations and Dynamical Systems](https://youtube.com/playlist?list=PLMrJAkhIeNNTYaOnVI3QpH7jgULnAmvPA&feature=shared)



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*Remember: In every equation, there's a story. In every oscillation, there's a dance. And in every simulation, there's a bit of magic!* ✨🪄