Simple Pendulum Modeling

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Learning Objectives

By the end of this lecture, you should be able to:

1. Set up the mathematical model for a simple pendulum
2. Understand the small angle approximation and when to use it
3. Implement the pendulum equation using numerical methods
4. Create an animated visualization of pendulum motion
5. Explore how different parameters affect pendulum behavior

The Physical System

A simple pendulum consists of a mass (bob) attached to a string of length L, swinging under the influence of gravity.

Key Variables:

• θ (theta): Angular displacement from vertical
• L: Length of the pendulum
• m: Mass of the bob
• g: Gravitational acceleration (9.81 m/s²)

Mathematical Model

The motion of a pendulum is governed by the equation:

For small angles (θ < 15°), we can use the approximation sin(θ) ≈ θ:

This gives us simple harmonic motion with period:

Building the Simulation

Let’s implement the pendulum model step by step:

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

# Physical parameters
L = 1.0      # Length (m)
g = 9.81     # Gravity (m/s²)
theta0 = 0.3 # Initial angle (radians)
omega0 = 0.0 # Initial angular velocity

# Time parameters
dt = 0.01    # Time step
t_max = 10   # Total time
time = np.arange(0, t_max, dt)

def pendulum_ode(theta, omega, t):
    """Pendulum equations of motion"""
    dtheta_dt = omega
    domega_dt = -(g/L) * np.sin(theta)
    return dtheta_dt, domega_dt

# Solve using Euler method
theta = np.zeros(len(time))
omega = np.zeros(len(time))

theta[0] = theta0
omega[0] = omega0

for i in range(len(time)-1):
    dtheta, domega = pendulum_ode(theta[i], omega[i], time[i])
    theta[i+1] = theta[i] + dtheta * dt
    omega[i+1] = omega[i] + domega * dt

# Create animation
fig, ax = plt.subplots()
ax.set_xlim(-1.5, 1.5)
ax.set_ylim(-1.5, 1.5)
ax.set_aspect('equal')

line, = ax.plot([], [], 'o-', lw=2, markersize=8)
time_text = ax.text(0.02, 0.95, '', transform=ax.transAxes)

def animate(frame):
    # Calculate position
    x = L * np.sin(theta[frame])
    y = -L * np.cos(theta[frame])

    # Update pendulum
    line.set_data([0, x], [0, y])
    time_text.set_text(f'Time = {time[frame]:.1f}s')
    return line, time_text

ani = FuncAnimation(fig, animate, frames=len(time),
                   interval=50, blit=True)
plt.show()

Parameter Studies

Explore how different parameters affect pendulum behavior:

Example Problem

Assignment

Key Takeaways

• Mathematical modeling translates physical principles into equations
• Numerical methods solve equations that lack analytical solutions
• Visualization helps understand system behavior
• Parameter studies reveal how design choices affect performance

Next Lecture

In the next lecture, we’ll explore spring-mass systems and learn about resonance, natural frequencies, and how these concepts apply to vibration control in engineering systems.