!apt update -y !apt install ffmpeg -y

# Import Libraries: import numpy as np import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle from matplotlib.patches import Rectangle # Parameters g = 9.8 k = 300 a = 0.5 L1 = 1 L2 = 0.5 m1 = 150 m2 = 50 b = a/0.8 # We need an array of time points from 0 to 10 in increments of 0.01 seconds # Simulation Parameters dt = 0.001 sim_time = 10 t_vec = np.arange(0,sim_time,dt) vec_size = len(t_vec) F = np.arange(0,sim_time,dt) #F = 0.10*np.sin(t_vec) for i in range(0,vec_size): F[i] = 0.10*np.sin(t_vec[i]) #print(F) # Initialize a vector of zeros th1_vec = np.zeros(len(t_vec)) dth1_vec = np.zeros(len(t_vec)) th2_vec = np.zeros(len(t_vec)) dth2_vec = np.zeros(len(t_vec)) # Set our initial condition th1_vec[0] = -np.pi/2 # Initial th1 dth1_vec[0] = 0 # Initial th1 velocity th2_vec[0] = 0 # Initial th2 dth2_vec[0] = 0 # Initial th2 velocity # Loop through time # Euler's Method (approximately integrates the differential equation) for i in range(1, len(t_vec)): th1_vec[i] = th1_vec[i-1] + dth1_vec[i-1]*dt dth1_vec[i] = dth1_vec[i-1] + (-(g/L1)*np.sin(th1_vec[i-1]) + ((k*a**2)/(m1*L1**2))*(th2_vec[i-1]-th1_vec[i-1]) + (b/m1)*np.sin(th1_vec[i-1])*np.cos(th1_vec[i-1]))*dt th2_vec[i] = th2_vec[i-1] + dth2_vec[i-1]*dt dth2_vec[i] = dth2_vec[i-1] + (-(g/L2)*np.sin(th2_vec[i-1]) - ((k*a**2)/(m2*L2**2))*(th2_vec[i-1]-th1_vec[i-1]) + (b/m2)*np.sin(th2_vec[i-1])*np.cos(th2_vec[i-1]))*dt th1_vec = th1_vec[::-1] dth1_vec = dth1_vec[::-1] th2_vec = th2_vec[::-1] dth2_vec = dth2_vec[::-1] plt.plot(t_vec,th1_vec) plt.title("th1 vs Time") plt.show() plt.plot(t_vec,dth1_vec) plt.title("dth1 vs Time") plt.show() plt.plot(t_vec,th2_vec) plt.title("th2 vs Time") plt.show() plt.plot(t_vec,dth2_vec) plt.title("dth vs Time") plt.show()

class Pendulum_Calculations: def __init__(self): # Constructor self.x_pendulum_arm = None self.y_pendulum_arm = None self.origin = None self.x_data_points = None self.y_data_points = None self.patch_center = None # Making each link in system an Object Pend1 = Pendulum_Calculations() Pend2 = Pendulum_Calculations() Spring = Pendulum_Calculations() # Create animation # Setup figure: fig, ax = plt.subplots() p1, = ax.plot([],[],color='dimgray') p2, = ax.plot([],[],color='dimgray') spring, = ax.plot([],[],color='red') min_lim = -3 max_lim = 3 ax.axis('equal') ax.set_xlim([min_lim, max_lim]) ax.set_ylim([min_lim, max_lim]) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Double Pendulum Spring System') title = "Double Pendulum Spring System" # Setup animation writer FPS = 20 sample_rate = int(1/(dt*FPS)) # Real time playback dpi = 300 writerObj = FFMpegWriter(fps=FPS) # 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. Pend1.origin = (-1/2, 0) # Pin Joint (X, Y) Location of pendulum 1 Pend2.origin = ( 1/2, 0) # Pin Joint (X, Y) Location of pendulum 2 Pend1.x_pendulum_arm = np.zeros(simulation_size) Pend1.y_pendulum_arm = np.zeros(simulation_size) Pend2.x_pendulum_arm = np.zeros(simulation_size) Pend2.y_pendulum_arm = np.zeros(simulation_size) Spring.x_pendulum_arm = np.zeros(simulation_size) Spring.y_pendulum_arm = np.zeros(simulation_size) # Find the X and Y trajectories of the pendulums: for i in range(0, simulation_size): Pend1.x_pendulum_arm[i] = L1 * np.sin(th1_vec[i]) + Pend1.origin[0] Pend1.y_pendulum_arm[i] = Pend1.origin[1] - L1 * np.cos(th1_vec[i]) Pend2.x_pendulum_arm[i] = L2 * np.sin(th2_vec[i]) + Pend2.origin[0] Pend2.y_pendulum_arm[i] = Pend2.origin[1] - L2 * np.cos(th2_vec[i]) Spring.x_pendulum_arm[i] = L1/2 * np.sin(th1_vec[i]) + Pend1.origin[0] Spring.y_pendulum_arm[i] = Pend1.y_pendulum_arm[i]/2 # Initialize Patch: (Pendulum 1) c1 = Circle(Pend1.origin, radius=0.1, color='cornflowerblue') # Draws a circle of radius 0.1 at (X, Y) location (0, 0) ax.add_patch(c1) # Add the patch to the axes hinge1 = Circle((Pend1.origin), radius=0.05, color='black') ax.add_patch(hinge1) # Initialize Patch: (Pendulum 2) c2 = Circle(Pend2.origin, radius=0.1, color='cornflowerblue') # Draws a circle of radius 0.1 at (X, Y) location (0, 0) ax.add_patch(c2) # Add the patch to the axes hinge2 = Circle((Pend2.origin), radius=0.05, color='black') ax.add_patch(hinge2) # Plot and Create Animation with writerObj.saving(fig, title+".mp4", dpi): for i in range(0, vec_size, sample_rate): # Update Pendulum Arm 1: Pend1.x_data_points = [Pend1.origin[0], Pend1.x_pendulum_arm[i]] Pend1.y_data_points = [Pend1.origin[1], Pend1.y_pendulum_arm[i]] p1.set_data(Pend1.x_data_points, Pend1.y_data_points) # Update Pendulum Patch: Pend1.patch_center = Pend1.x_pendulum_arm[i], Pend1.y_pendulum_arm[i] c1.center = Pend1.patch_center # Update Pendulum Arm 2: Pend2.x_data_points = [Pend2.origin[0], Pend2.x_pendulum_arm[i]] Pend2.y_data_points = [Pend2.origin[1], Pend2.y_pendulum_arm[i]] p2.set_data(Pend2.x_data_points, Pend2.y_data_points) # Update Pendulum Patch: Pend2.patch_center = Pend2.x_pendulum_arm[i], Pend2.y_pendulum_arm[i] c2.center = Pend2.patch_center # Update Spring: Spring.x_data_points = [Spring.x_pendulum_arm[i], Pend2.x_pendulum_arm[i]] Spring.y_data_points = [Spring.y_pendulum_arm[i], Pend2.y_pendulum_arm[i]] spring.set_data(Spring.x_data_points, Spring.y_data_points) # Update drawing, grab and save frame fig.canvas.draw() writerObj.grab_frame() from IPython.display import Video Video("/work/Double Pendulum Spring System.mp4", embed=True, width=640, height=480)