################## Homework 1 Problem 1 ########################### # 992759-MOD-HW1 !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 g = 9.8 # m/s^2 L = 1 # m m = 1 # kg # define our time vector for simulation dt = .001 t_vec = np.arange(0,10,dt) # define vectors that will hold our state variables over time (initializing vectors) 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)) lambda_vec = np.zeros(len(t_vec)) # constraint # inital conditions x_vec[0] = -1 # position dx_vec[0] = 0 # velocity y_vec[0] = 1 dy_vec[0] = 0 # loop for x 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 # loop for y 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 # loop for constraint 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) ############ Plotting ############# # x plot plt.plot(t_vec,x_vec) plt.title('X positon vs time') plt.xlabel('time (s)') plt.ylabel('x (m)') plt.show() # y plot plt.plot(t_vec,y_vec) plt.title('Y postion vs time') plt.xlabel('time (s)') plt.ylabel('y (m)') plt.show() # constraint plot plt.plot(t_vec,lambda_vec) plt.title('Constraint force vs time') plt.xlabel('time (s)') plt.show() ######### Animation ############# fig, ax = plt.subplots() p, = ax.plot([], [], color='navy') ax.axis('equal') ax.set_xlim([-3, 3]) ax.set_ylim([-3, 3]) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_title('Problem 1 Simulation') video_title = "Problem 1 Simulation" c = Circle((0, 0), radius=0.2, color='cornflowerblue') ax.add_patch(c) # Setup Animation Writer: FPS = 20 sample_rate = int(1 / (dt * FPS)) dpi = 300 writerObj = FFMpegWriter(fps=FPS) # calculating pendulum location in cartesian coords simulation_size = len(t_vec) x_var = np.linspace(-2,2,2000) y_var = x_var**2 plt.plot(x_var, y_var, color='black') plt.show() # Plot and Create Animation: with writerObj.saving(fig, video_title+".mp4", dpi): for i in range(0, simulation_size, sample_rate): x_data = x_vec[i] y_data = y_vec[i] p.set_data(x_data, y_data) # updating plot # patching patch_center = x_vec[i], y_vec[i] c.center = patch_center fig.canvas.draw() # update drawing writerObj.grab_frame() from IPython.display import Video Video("/work/Problem 1 Simulation.mp4", embed=True, width=640, height=480)