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Pendulum with Damping and Forcing

Overview

The forced damped pendulum is a classic example of a nonlinear dynamic system that exhibits a wide range of behaviors, including periodic motion, quasiperiodicity, and chaos. It is governed by the following equation:

\[ \ddot{\theta} + b \dot{\theta} + c \sin(\theta) = A \cos(\omega t) \]

Where: - \(\theta\) is the angular displacement. - \(b\) is the damping coefficient. - \(c\) is the gravitational restoring torque coefficient. - \(A\) is the amplitude of the periodic driving force. - \(\omega\) is the angular frequency of the driving force.

Python Code Example: Simulating and Visualizing the Forced Damped Pendulum

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

# Define the forced damped pendulum equations
def forced_damped_pendulum(t, state, b, c, A, omega):
    theta, omega_theta = state
    dtheta_dt = omega_theta
    domega_dt = -b * omega_theta - c * np.sin(theta) + A * np.cos(omega * t)
    return [dtheta_dt, domega_dt]

# Parameters
b = 0.5  # Damping coefficient
c = 9.8  # Restoring torque coefficient
A = 1.2  # Amplitude of the driving force
omega = 2.0  # Angular frequency of the driving force

# Initial conditions
initial_state = [0.1, 0.0]  # Initial angle and angular velocity

# Time span for the simulation
time_span = (0, 50)
time_eval = np.linspace(time_span[0], time_span[1], 10000)

# Solve the equations
solution = solve_ivp(forced_damped_pendulum, time_span, initial_state, args=(b, c, A, omega), t_eval=time_eval, method='RK45')

# Extract the results
theta, omega_theta = solution.y

# Unwrap the angle for better visualization
theta = np.unwrap(theta)

# Plot the angular displacement over time
plt.figure(figsize=(12, 6))
plt.plot(solution.t, theta, label="Angular Displacement", color="blue")
plt.title("Forced Damped Pendulum: Angular Displacement Over Time")
plt.xlabel("Time")
plt.ylabel("Angular Displacement (radians)")
plt.grid()
plt.legend()
plt.savefig("docs/Mechanics/pic/forced_damped_pendulum_time.png")
plt.show()

# Plot the phase plane
plt.figure(figsize=(8, 8))
plt.plot(theta, omega_theta, color="purple")
plt.title("Phase Plane of the Forced Damped Pendulum")
plt.xlabel("Angular Displacement (radians)")
plt.ylabel("Angular Velocity")
plt.grid()
plt.savefig("docs/Mechanics/pic/forced_damped_pendulum_phase.png")
plt.show()

Key Insights

  1. Nonlinear Dynamics: The interaction between damping, restoring force, and driving force creates complex dynamics.

  2. Phase Space Behavior: The phase plane reveals different regimes: limit cycles, quasiperiodic orbits, and chaotic attractors.

  3. Sensitivity to Parameters: Small changes in \(b\), \(c\), \(A\), or \(\omega\) can lead to drastically different behaviors.

Suggested Projects

  1. Parameter Exploration: Investigate the effects of varying \(A\) and \(\omega\) to explore transitions between periodic, quasiperiodic, and chaotic motion.

  2. Poincaré Section: Visualize the system's dynamics using a Poincaré section to identify periodic orbits and chaos.

  3. Energy Analysis: Study the energy evolution in the system and identify energy dissipation and driving force contributions.

  4. Comparison with Real Pendulums: Build a physical forced pendulum and compare its motion to the simulation.

The forced damped pendulum serves as a rich system for studying nonlinear dynamics. Experiment with the parameters and initial conditions to uncover its fascinating behaviors!