Bead on Paraboloid: HW 1.3

# Install ffmpeg (allows us to create videos) !apt update -y !apt install ffmpeg -y import numpy as np import matplotlib.pyplot as plt import sympy from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle, Rectangle from mpl_toolkits import mplot3d

# Parameters m = 1.0 # mass of bead (kg) g = 9.8 # gravity # Simulate System: x0 = 1, 0, 0, 1 # x, dx, y, dy dt = 0.001 sim_time = 10 time = np.arange(0, sim_time, dt) sim_length = len(time) # Initialize Arrays: y_vec = np.zeros(sim_length) dy_vec = np.zeros(sim_length) x_vec = np.zeros(sim_length) dx_vec = np.zeros(sim_length) z_vec = np.zeros(sim_length) # Evaluate Initial Conditions: x_vec[0] = x0[0] dx_vec[0] = x0[1] y_vec[0] = x0[2] dy_vec[0] = x0[3] z_vec[0] = x_vec[0]**2 + y_vec[0]**2 # Euler Integration: for i in range(1, sim_length): # Evaluate Dynamics: ddx = -(g*x_vec[i-1] - 2*x_vec[i-1]*dx_vec[i-1]**2 - 2*x_vec[i-1]*dy_vec[i-1]**2)/(2*x_vec[i-1]**2+2*y_vec[i-1]**2) ddy = -(g*y_vec[i-1] - 2*y_vec[i-1]*dx_vec[i-1]**2 - 2*y_vec[i-1]*dy_vec[i-1]**2)/(2*x_vec[i-1]**2+2*y_vec[i-1]**2) # Euler Step Integration: x_vec[i] = x_vec[i-1] + dx_vec[i-1] * dt dx_vec[i] = dx_vec[i-1] + ddx * dt y_vec[i] = y_vec[i-1] + dy_vec[i-1] * dt dy_vec[i] = dy_vec[i-1] + ddy * dt z_vec[i] = x_vec[i]**2 + y_vec[i]**2 plt.plot(time,x_vec) plt.title("X versus time") plt.show() plt.plot(time,y_vec) plt.title("Y versus time") plt.show() # fig = plt.figure() # ax = plt.axes(projection='3d') # ax.plot3D(x_vec, y_vec, z_vec)

# Create Animation: # Setup Figure: Initialize Figure / Axe Handles fig, ax = plt.subplots() lb, ub = -1, 1 ax.axis('equal') ax.set_xlim([lb, ub]) ax.set_xlabel('X') # X Label ax.set_ylabel('Y') # Y Label ax.set_title('Bead on Paraboloid Simulation:') video_title = "simulation" # Setup Animation Writer: FPS = 20 sample_rate = int(1 / (dt * FPS)) dpi = 300 writerObj = FFMpegWriter(fps=FPS) # Initialize Patch: Pendulum 1 and 2 bead = Circle((0, 0), radius=0.1, color='cornflowerblue', zorder=10) ax.add_patch(bead) # Plot and Create Animation: with writerObj.saving(fig, video_title+".mp4", dpi): for i in range(0, sim_length, sample_rate): # Draw Pendulum Arm: x_pendulum_arm = x_vec[i] y_pendulum_arm = y_vec[i] # Update Pendulum Patches: pendulum_center = x_vec[i], y_vec[i] bead.center = pendulum_center # Update Drawing: fig.canvas.draw() # Grab and Save Frame: writerObj.grab_frame()