#Parabolic Constrained Harmonic Motion #install ffmpeg !apt update -y !apt install ffmpeg -y # Import Libraries: import numpy as np import matplotlib.pyplot as plt import sympy as sp from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle # Pendulum Parameters g = -9.81 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)) dx_vec = np.zeros(len(t_vec)) ddx_vec = np.zeros(len(t_vec)) y_vec = np.zeros(len(t_vec)) dy_vec = np.zeros(len(t_vec)) ddy_vec = np.zeros(len(t_vec)) cons_vec = np.zeros(len(t_vec)) # Set our initial condition x_vec[0] = -1 # initial X location dx_vec[0] = 0 # initial horizontal velocity y_vec[0] = 1 dy_vec[0] = 0 # Loop through time # Euler's Method (approximately integrates the differential equation) for i in range(1, len(t_vec)): x_vec[i] = x_vec[i-1] + dx_vec[i-1]*dt ddx_vec[i-1] = -(x_vec[i-1]*(200*(dx_vec[i-1]**2) + 981))/(50*(4*(x_vec[i-1]**2) + 1)) dx_vec[i] = dx_vec[i-1] + ddx_vec[i-1]*dt y_vec[i] = x_vec[i]**2 cons_vec[i-1] = -(200*(dx_vec[i-1]**2) - 981)/(100*(4*(x_vec[i-1]**2) + 1)) fig1= plt.plot(t_vec, x_vec) plt.show() fig2 = plt.plot(t_vec, y_vec) plt.show() fig3 = plt.plot(t_vec, cons_vec) plt.show() # Setup Figure: Initialize Figure / Axe Handles fig, ax = plt.subplots() # Initialize Figure and Axes p, = ax.plot([], [], color='royalblue') # Initialize Empty Plot 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 Bead Simulation:') # Plot Title video_title = "parabolic" # name of the mp4 # Initialize Patch: c = Circle((-1, 1), radius=0.1, color='red') # Draws a circle of radius 0.1 at (X, Y) location (0, 0) 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 #### x_func = np.linspace(-2, 2, 2000) y_func = x_func**2 fig4 = plt.plot(x_func, y_func) plt.xlim([-2, 2]) plt.ylim([-0.5, 2.5]) plt.show() simulation_size = len(t_vec) with writerObj.saving(fig, video_title+".mp4", dpi): for i in range(0, simulation_size, sample_rate): p.set_data(x_vec[i], y_vec[i]) patch_center = x_vec[i], y_vec[i] c.center = patch_center fig.canvas.draw() writerObj.grab_frame() from IPython.display import Video Video("/work/parabolic.mp4",embed=True, width=640,height=480)