!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 # Defining our Parameters g = 9.8 # Gravity (m/s^2) L = 1 # Length (m) m = 1 # In Kg # Creating the time array for 10 second run time dt = .001 # Seconds (s) t_vec = np.arange(0,10,dt) # Time Vector # Define vectors that will hold our state variables over time # x Vectors of Zero x_vec = np.zeros(len(t_vec)) dx_vec = np.zeros(len(t_vec)) ddx_vec = np.zeros(len(t_vec)) # y Vectors of Zero y_vec = np.zeros(len(t_vec)) dy_vec = np.zeros(len(t_vec)) ddy_vec = np.zeros(len(t_vec)) # Lambda Vector (Constraint Force) lambda_vec = np.zeros(len(t_vec)) # Establishing the inital conditions for x & y x_vec[0] = -1 # x Initial Position dx_vec[0] = 0 # dx Initial Velocity y_vec[0] = 1 # y Initial Position dy_vec[0] = 0 # dy Initial Velocity # Loop Through 10 second time interval of equations solved from Matrix # Creating Loop For the x Direction for i in range(1, len(t_vec)): x_vec[i] = x_vec[i-1] + dx_vec[i-1]*dt ddx_vec[i-1] = -(2*x_vec[i-1]*(2*(dx_vec[i-1]**2) + g))/(4*(x_vec[i-1]**2) + 1) dx_vec[i] = dx_vec[i-1] + ddx_vec[i-1]*dt # Creating Loop For the y Direction for i in range(1, len(t_vec)): y_vec[i] = y_vec[i-1] + dy_vec[i-1]*dt ddy_vec[i-1] = -(2*(2*g*x_vec[i-1]**2 - dx_vec[i-1]**2))/(4*x_vec[i-1]**2 + 1) dy_vec[i] = dy_vec[i-1] + ddy_vec[i-1]*dt # Creating Loop For the Lambda Component for i in range(1, len(t_vec)): lambda_vec[i] = (2*m*dx_vec[i-1]**2 + g*m)/(4*dx_vec[i-1]**2 + 1) # Setting Up the x-axis plt.plot(t_vec,x_vec) plt.title('x-Positon') plt.xlabel('time') plt.ylabel('x') plt.show() # Setting Up the y-axis plt.plot(t_vec,y_vec) plt.title('y-Postion') plt.xlabel('time') plt.ylabel('y') plt.show() # Setting Up the Lambda plt.plot(t_vec,lambda_vec) plt.title('Constraint Force') plt.xlabel('time') plt.show() # SETTING UP THE SIMULATION PORTION # Setup Figure: Initialize Figure / Axle Handles fig, ax = plt.subplots() # Initialize Figure and Axes p, = ax.plot([], [], color='navy') ax.axis('equal') ax.set_xlim([-3, 3]) # X Lim ax.set_ylim([-3, 3]) # Y Lim ax.set_xlabel('x') # X Label ax.set_ylabel('y') # Y Label ax.set_title('Parabolic Wire Simulation:') # Plot Title video_title = "Simulation Q1 Part C" # Naming of Animation # Initialize Patch: (Circle On Wire) # Creating Circle On Plot of Desired Location c = Circle((0, 0), radius=0.2, color='palegreen') ax.add_patch(c) # Add the patch to the axes # Setup Animation Writer: FPS = 20 # Frames per second for the video sample_rate = int(1 / (dt * FPS)) # Rate at which we sample the simulation to visualize dpi = 300 # Quality of the Video writerObj = FFMpegWriter(fps=FPS) # Video Object # We need to calculate the cartesian location of the pendulum: # Initialize Array: simulation_size = len(t_vec) # We use this alot, so lets store it. x_var = np.linspace(-2,2,2000) # Creating Array for x placements y_var = x_var**2 # What y would equal plt.plot(x_var, y_var, color='black') plt.show() # 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 = x_vec[i] y_data_points = y_vec[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_vec[i], y_vec[i] c.center = patch_center # Same idea here, instead of drawing a new patch update its location # Update Drawing: fig.canvas.draw() # Update the figure with the new changes # Grab and Save Frame: writerObj.grab_frame()