!apt update -y !apt install ffmpeg -y import numpy as np import sympy import matplotlib.pyplot as plt from matplotlib.patches import Circle from matplotlib.animation import FFMpegWriter # Parameters m = 1 # kg g = 9.8 # m/s^2 # time vector dt = 0.001 # sec t_vec = np.arange(0,10,dt) # State variable vectors initialized to zero x_vec = np.zeros(len(t_vec)) dx_vec = np.zeros(len(t_vec)) y_vec = np.zeros(len(t_vec)) dy_vec = np.zeros(len(t_vec)) lam = np.zeros(len(t_vec)) # Initial Conditions x_vec[0] = -1 # initial x-position (m) dx_vec[0] = 0 # x-velocity (m/s) y_vec[0] = (x_vec[0])**2 # initial y-position (m) dy_vec[0] = 0 # intial y-velocity (m/s) # Initialize A and B: A = np.array([[m,0,-2*x_vec[0]],[0,m,-1],[2*x_vec[0],-1,0]]) B = np.array([[0], [-m*g], [-2*(dx_vec[0])**2]]) # Euler Simulation with A*x=b: for i in range(1,len(t_vec)): x_vec[i] = x_vec[i-1]+dx_vec[i-1]*dt y_vec[i] = y_vec[i-1]+dy_vec[i-1]*dt # update A and B where necessary A[0,2] = -2*x_vec[i-1] A[2,0] = -2*x_vec[i-1] B[2] = -2*(dx_vec[i-1])**2 # solve for ddx, ddy, and lam [ddx,ddy,lam[i-1]] = np.linalg.solve(A,B) dx_vec[i] = dx_vec[i-1] + (-2*lam[i-1]*x_vec[i-1])*dt dy_vec[i] = dy_vec[i-1] + ((-m*g+lam[i-1])/m)*dt plt.plot(t_vec,x_vec) plt.show() plt.plot(t_vec,y_vec) plt.show() plt.plot(t_vec,lam) plt.show()

# Animation # Initialize Figure / Axe Hanndles fig, ax = plt.subplots() # Initialize Figure and Axes p, = ax.plot([],[],color='cornflowerblue') # Initialize Empy Plot ax.axis('equal') ax.set_xlim([-2,2]) ax.set_ylim([0,2]) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Question 1 Simulation:') video_title = "MOD-HW1-Q1-Simulation" # Initialize Patch: c = Circle((-1,1), radius = 0.1, color='cornflowerblue') ax.add_patch(c) # Setup Animation Writer: FPS = 20 sample_rate = int(1/(dt*FPS)) dpi = 300 writerObj = FFMpegWriter(fps=FPS) simulation_size = len(t_vec) #plotting parabola line x = np.linspace(-10, 10, 500) y = x**2 parabola_line, = plt.plot(x,y) with writerObj.saving(fig,video_title+".mp4",dpi): for i in range(0,simulation_size,sample_rate): 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/MOD-HW1-Q1-Simulation.mp4",embed=True,width=640,height=480)