Vibrational System Simulation

A Python simulation of a damped harmonic oscillator with forced vibration analysis using FFT.

Table of Contents

• Description
• Mathematical Model
• Dependencies
• Installation
• Usage
• Parameters
• Output
• Customization
• Theory
• License

Description

This program simulates the behavior of a damped harmonic oscillator under forced vibration, then analyzes the displacement using Fast Fourier Transform (FFT). The simulation includes:

• Time-domain analysis of displacement and force
• Frequency-domain analysis via FFT
• Visualization of both time and frequency responses

Mathematical Model

The system is modeled using the second-order differential equation: mẍ + cẋ + kx = F₀cos(ωt)

text

Converted to state-space representation: Aẋ + Bx = F

text where:

• A = [[m, 0], [0, 1]]
• B = [[c, k], [-1, 0]]
• F = [F₀cos(ωt), 0]

Dependencies

• Python 3.x
• NumPy (>=1.20)
• SciPy (>=1.7)
• Matplotlib (>=3.4)

Installation

1. Clone the repository:

git clone https://github.com/yourusername/vibration-simulation.git
cd vibration-simulation

Install required packages:

pip install numpy scipy matplotlib

Usage

Run the simulation:

python vibration_simulation.py

Parameters

Parameter	Description	Default Value
m	Mass (kg)	2.0
k	Spring constant (N/m)	2.0
c	Damping coefficient (Ns/m)	0.2
omega	Forcing frequency (rad/s)	1.0
F0	Force amplitude (N)	1.0
delta_t	Time step (s)	0.001
time	Simulation duration (s)	100

Output

The program generates two plots:

Time Domain:

• Displacement (x) vs. Time
• Force (F) vs. Time

Frequency Domain:

• FFT magnitude of the displacement
• Shows dominant frequency components

Customization

To modify the simulation:

Change system parameters at the top of the script:

m = 2.0  # Mass
k = 2.0  # Spring constant
c = 0.2  # Damping

Adjust forcing function:

if t <= 15:  # Force duration
    F[0] = F0 * np.cos(omega*t)
else:
    F[0] = 0.0  # Force removal

Change simulation duration and time step:

delta_t = 0.001  # Time step
time = np.arange(0.0, 100.0, delta_t)  # Simulation time

Theory

Numerical Integration

The state-space equations are solved using forward Euler integration:

x = x + delta_t * inv(A).dot(F - B.dot(x))

Fast Fourier Transform

The displacement data is analyzed using SciPy's FFT:

Y = fft(X)

License

Free for academic, personal, and research use.