# This code block installs ffmpeg to your Deepnote Machine # Allows Deepnote to write MP4 !apt update -y !apt install ffmpeg -y # Bead on a wire code: import numpy as np from numpy.linalg import solve import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle # Defining parameters g = 9.8 # gravity (m/s^2) m = 1 # mass of the bead (kg) # Define Time Vector for Simulation dt = 0.001 # Time Step (s) sim_time = 10 # Length of Simulation (s) t_vec = np.arange(0, sim_time, dt) # Initialize State Vectors: vec_size = len(t_vec) x_vec = np.zeros(vec_size) dx_vec = np.zeros(vec_size) y_vec = np.zeros(vec_size) dy_vec = np.zeros(vec_size) # Assign Intial Values: x_vec[0] = -1 dx_vec[0] = 0 y_vec[0] = x_vec[0]**2 # ** is to the power of dy_vec[0] = 0 # Euler Simulation: Using Matrix Form (A*x = B) # Initialize A and B: A = np.array([[m, 0, 2*x_vec[0]], [0, m, -1], [2*x_vec[0], -1, 0]]) B = np.array([0, -m*g, 0]) # Initialize constraint Force lagrangeMultiplier = np.zeros(vec_size) for i in range(1, vec_size): # Updating the arrays A[0, 2] = 2*x_vec[i-1] A[2, 0] = 2*x_vec[i-1] B[2] = -2*(dx_vec[i-1]**2) # Find the accelerations and the Lagrange Multiplier [ddx, ddy, lMultiplier] = np.linalg.solve(A, B) lagrangeMultiplier[i] = lMultiplier # Use accelerations to find next position and velocities 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 # Plotting x(t), y(t), and lambda plt.plot(t_vec, x_vec) plt.title("X Position versus Time") plt.xlabel("Time [s]") plt.ylabel("X Position [m]") plt.show() plt.plot(t_vec, y_vec) plt.title("Y Position versus Time") plt.xlabel("Time [s]") plt.ylabel("Y Position [m]") plt.show() plt.plot(t_vec, x_vec) plt.title("Normal Force versus Time") plt.xlabel("Time [s]") plt.ylabel("Normal Force [N]") plt.show() # Create animation # Setup figure: fig, ax = plt.subplots() p, = ax.plot([],[],color='dimgray') ax.axis('equal') ax.set_xlim([-3, 3]) ax.set_ylim([-3, 3]) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Bead on a Wire Simulation') title = "simulation" # Setup animation writer FPS = 20 sample_rate = int(1/(dt*FPS)) # Real time playback dpi = 300 writerObj = FFMpegWriter(fps=FPS) # Initialize Patch (bead) c = Circle((x_vec[0], y_vec[0]), radius=0.1, color='cornflowerblue') ax.add_patch(c) # Draw the wire x_wire = np.arange(-1.3, 1.3, dt) y_wire = x_wire**2 # Plot and Create Animation with writerObj.saving(fig, title+".mp4", dpi): p.set_data(x_wire, y_wire) # Sample only a few frames # Update position of bead on the wire for i in range(0, vec_size, sample_rate): patch_center = x_vec[i], y_vec[i] c.center = patch_center # Update drawing, grab and save frame fig.canvas.draw() writerObj.grab_frame()