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Simulation: Damped Harmonic Oscillator Ready

# Damped Harmonic Oscillator Simulation
# m*x'' + b*x' + k*x = 0
import numpy as np
import matplotlib.pyplot as plt

# Physical Parameters
m = 1.0     # Mass (kg)
k = 16.0    # Spring constant (N/m)
b = 0.5     # Damping coefficient (kg/s)
omega0 = np.sqrt(k / m)
gamma = b / (2 * m)

# Time array
t = np.linspace(0, 15, 600)
x0 = 1.0    # Initial displacement (m)
v0 = 0.0    # Initial velocity (m/s)

# Underdamped solution
omega = np.sqrt(omega0**2 - gamma**2)
x = np.exp(-gamma * t) * (x0 * np.cos(omega * t) + ((v0 + gamma * x0) / omega) * np.sin(omega * t))

print(f"Natural Frequency (omega0): {omega0:.3f} rad/s")
print(f"Damping Ratio (gamma): {gamma:.3f} s^-1")
print(f"Oscillation Frequency (omega): {omega:.3f} rad/s")

# Plot Waveform
plt.figure(figsize=(8, 4.5))
plt.plot(t, x, label='Displacement x(t)', color='#06b6d4', lw=2)
plt.plot(t, np.exp(-gamma * t), 'r--', label='Envelope', alpha=0.7)
plt.plot(t, -np.exp(-gamma * t), 'r--', alpha=0.7)
plt.title('Damped Harmonic Oscillator Trajectory', fontsize=12)
plt.xlabel('Time (s)')
plt.ylabel('Position (m)')
plt.grid(True, alpha=0.3)
plt.legend()
plt.tight_layout()
plt.show()

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Introduction to Python

Iterable Data Type

Basic Python Programs

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Ordinary Differential Equation-1

Introduction to Numpy

Solve Linear Equations

Introduction to Scipy

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Ordinary Differential Equation-2

Curve Fitting

Special Functions

Fourier Series

Partial Differential Equations

Shooting Method for Boundary Value Problem

TISE Solution

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