# Pendulum Animation
# Install ffmpeg (allows us to create videos)
!apt update -y
!apt install ffmpeg -y
# Normal Python Mathy Code
# Import Libraries:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FFMpegWriter
from matplotlib.patches import Circle
# Pendulum Parameters
g = 9.8
L = 1
# We need an array of time points from 0 to 10 in increments of 0.01 seconds
dt = 0.001
t_vec = np.arange(0,10,dt)
# Initialize a vector of zeros
theta_vec = np.zeros(len(t_vec))
dtheta_vec = np.zeros(len(t_vec))
# Set our initial condition
theta_vec[0] = np.pi/4 # initial angle
dtheta_vec[0] = 0 # initial angular velocity
# Loop through time
# Euler's Method (approximately integrates the differential equation)
for i in range(1, len(t_vec)):
theta_vec[i] = theta_vec[i-1] + dtheta_vec[i-1]*dt
dtheta_vec[i] = dtheta_vec[i-1] + (-g/L*np.sin(theta_vec[i-1]))*dt
plt.plot(t_vec,theta_vec)
plt.show()
# Set up our Figure for drawing our Pendulum
fig, ax = plt.subplots()
# Create a plot on those axes, which is currently empty
p, = ax.plot([],[], color='cornflowerblue') #initializes an empyt plot
ax.axis('equal')
ax.set_xlim([-3, 3]) # x limits
ax.set_ylim([-3, 3]) # y limits
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_title('Pendulum Simulation')
video_title = "simulation"
# Now we want to plot stuff on those axes
c = Circle((0, 0), radius=0.1, color='cornflowerblue')
ax.add_patch(c)
# Define information for our Animation
FPS = 20
sample_rate = int(1/(FPS*dt))
dpi = 300 # Quality of the video
writerObj = FFMpegWriter(fps=FPS)
# Now we're putting the rod and bob in the right place
# Initialize an array containing the
# positions of the pendululm over time
simulation_size = len(t_vec) # number of sim time points
x_pendulum_arm = np.zeros(simulation_size)
y_pendulum_arm = np.zeros(simulation_size)
for i in range(0, simulation_size):
x_pendulum_arm[i] = L*np.sin(theta_vec[i])
y_pendulum_arm[i] = -L*np.cos(theta_vec[i])
# We've computed all the pendulum positions
# Now we need to update the plot in a loop and store each frame in a video
# Plot and Create Animation:
with writerObj.saving(fig, video_title+".mp4", dpi):
# We want to create video that represents the simulation
# So we need to sample only a few of the frames:
for i in range(0, simulation_size, sample_rate):
# Update Pendulum Arm:
x_data_points = [0, x_pendulum_arm[i]]
y_data_points = [0, y_pendulum_arm[i]]
# We want to avoid creating new plots to make an animation (Very Slow)
# Instead lets take the plot we made earlier and just update it with new data.
p.set_data(x_data_points, y_data_points) # Update plot with set_data
# Update Pendulum Patch:
patch_center = x_pendulum_arm[i], y_pendulum_arm[i]
# ^ note: the commas without brackets create a LIST
c.center = patch_center # Same idea here, instead of drawing a new patch update its location
# ^ updates the circle
# Update Drawing:
fig.canvas.draw() # Update the figure with the new changes
# Grab and Save Frame:
writerObj.grab_frame()
# Iport the video you just made and run it on the notebook
from IPython.display import Video
Video("/work/simulation.mp4", embed=True, width=640, height=480)
# You can find the video's path by right clicking on it in Notebooks & Files
# and selecting "Copy path to clipboard"