!apt update -y !apt install ffmpeg -y

# Cart Pole Code: import numpy as np from numpy.linalg import solve import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Rectangle from matplotlib.patches import Circle # Pendulum Parameters g = 9.8 m = 1 # Define Time Vector for Simulation dt = 0.001 # Time Step sim_time = 10 # Length of Simulation t_vec = np.arange(0, sim_time, dt) # Initialize State Vectors: vec_size = len(t_vec) # We use this value alot so lets store it x_vec = np.zeros(vec_size) dx_vec = np.zeros(vec_size) y_vec = np.zeros(vec_size) dy_vec = np.zeros(vec_size) l_vec = np.zeros(vec_size) x_vec[0] = -1.0 dx_vec[0] = 0.0 y_vec[0] = 1.0 dy_vec[0] = 0.0 l_vec[0] = 0.0 # Euler Simulation: Using Matrix Form (A * x = B) # Initialize A and B: for i in range(1, vec_size): A = np.array([[m, 0.0, -2*x_vec[i-1]], [0.0, m, 1.0], [-x_vec[i-1], 0.5, 0.0]]) B = np.array([0.0, -m*g, dx_vec[i-1]**2]) [ddx, ddy, l] = np.linalg.solve(A, B) # Use ddx and ddtheta to solve: 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 l_vec[i] = l plt.xlabel('t(sec)') plt.ylabel('x(m)') plt.title('x(t)') plt.plot(t_vec,x_vec) plt.show() plt.xlabel('t(sec)') plt.ylabel('y(m)') plt.title('y(t)') plt.plot(t_vec,y_vec) plt.show() plt.xlabel('t(sec)') plt.ylabel(r'$\lambda$(N)') plt.title(r'$\lambda$(t)') plt.plot(t_vec,l_vec) plt.show()

# set up our figure for the drawing of the pendulum fig, ax = plt.subplots() # create a plot on those axes which is currently empty p, = ax.plot([], [], color='cornflowerblue') # Initialize Empty Plot ax.set_xlim([-3,3]) # x lim ax.set_ylim([-10,10]) # y lim ax.axis('equal') ax.set_xlabel('X') # x label ax.set_ylabel('Y') # y label ax.set_title('problem 1 - bead on a parabola') # Plot Title video_title = "simulation" # name of the mp4 # now we want to add stuff on the plot c = Circle((0,0), radius=0.1, color = 'cornflowerblue') ax.add_patch(c) # define information of our animation FPS = 20 sample_rate = int(1/(FPS*dt)) dpi = 300 # quality of the video writerObj = FFMpegWriter(fps = FPS) # now we are putting the rod and bob at the right place # initialize an array containing the position of pendulum over time simulation_size = len(t_vec) 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] = x_vec[i] y_pendulum_arm[i] = y_vec[i] # we have computed all the pendulum positions xx = np.linspace(-1.5, 1.5, 100) yy = xx**2 ax.plot(xx, yy) # now we want to update the plot and save frames and create the video # Plot and Create Animation: with writerObj.saving(fig, video_title+".mp4", dpi): # Loop thru the frames we want to animate for i in range(0, simulation_size, sample_rate): # Update Pendulum Arm(rod) #x_data_points = [0, x_pendulum_arm[i]] #y_data_points = [0, y_pendulum_arm[i]] # update the data in plotting object #p.set_data(x_data_points, y_data_points) # Update Pendulum bob: patch_center = x_pendulum_arm[i], y_pendulum_arm[i] # note: commas without brackets create list c.center = patch_center # Update Drawing: fig.canvas.draw() # Update the figure with the new changes # Grab and Save Frame: writerObj.grab_frame() # We can also run it inline with the following block of code: 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"