# Bead on a Wire # Install ffmpeg (allows us to create videos) # deepnote.com/@jeh15 !apt update -y !apt install ffmpeg -y # Import Libraries import numpy as np import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle # Parameters g = 9.8 m = 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 x_vec = np.zeros(len(t_vec)) y_vec = np.zeros(len(t_vec)) dx_vec = np.zeros(len(t_vec)) dy_vec = np.zeros(len(t_vec)) lam_vec = np.zeros(len(t_vec)) # Set our initial condition x_vec[0] = 1 # initial x position dx_vec[0] = 0 # initial angular velocity # Set the initial y values y_vec[0] = x_vec[0] ** 2 # initial y position dy_vec[0] = 0 # initial angular velocity # Initialize Matrix and Constants Ax = B A = np.array([[m, 0, 0],[0, m, -1],[0, -1, 0]], dtype = float) B = np.array([[0],[-m*g],[0]], dtype = float) # Loop through time # Euler's Method (approximately integrates the differential equation) for i in range(1, len(t_vec)): # for i in range(1, 10): # Update the Matrix A[0,2] = 2 * x_vec[i-1] A[2,0] = 2 * x_vec[i-1] B[2,0] = -2 * dx_vec[i-1] ** 2 # Solve the equation Ax = B [ddx, ddy, lam] = np.linalg.solve(A, B) # Integrate Using Eulers Method x_vec[i] = x_vec[i-1] + dx_vec[i-1]*dt y_vec[i] = y_vec[i-1] + dy_vec[i-1]*dt dx_vec[i] = dx_vec[i-1] + ddx*dt dy_vec[i] = dy_vec[i-1] + ddy*dt lam_vec[i] = lam # Setup our figure for drawing # fig, axs = plt.subplots(3) # fig.suptitle('Positions and Forces') # axs[0].plot(t_vec, x_vec) # axs[0].set_ylabel('X Position') # axs[1].plot(t_vec, y_vec) # axs[1].set_ylabel('Y Position') # axs[2].plot(t_vec, lam_vec) # axs[2].set_ylabel('Lambda Value') # Create a plot on those axes which is currently empty fig, ax = plt.subplots() p, = ax.plot([],[],color='black') # Initializes an empty plot ax.set_xlim([-2, 2]) # x limits ax.set_ylim([-1, 3]) # y limits ax.axis('equal') ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Bead on Wire Simulation') video_title = "simulation" # Now we want add the circle to the plot c = Circle((0,0), radius=0.04, color='cornflowerblue') ax.add_patch(c) # puts the circle on that set of axes # Now we want add the wire x = np.arange(-2,2,0.1) wire = x, x**2 p.set_data(x, x**2) # Define Information for our animation FPS = 20 sample_rate = int(1/(FPS*dt)) vec_size = len(t_vec) dpi = 300 # Quality of the video writerObj = FFMpegWriter(fps = FPS) # Plot and Create Animation with writerObj.saving(fig, video_title+".mp4", dpi): # Loop through the frames we want to animate for i in range(0,vec_size,sample_rate): # Update the bob circle patch_center = x_vec[i], y_vec[i] # NOTE: commas w/o bracket create a list # update the circle c.center = patch_center # Update the Drawing fig.canvas.draw() # Grab and save the frame writerObj.grab_frame() # Import 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)