HW 1 - 755180-MOD-HW1

# This code block installs ffmpeg to your Deepnote Machine # Allows Deepnote to write MP4 !apt update -y !apt install ffmpeg -y

#HW1 Problem 1 import numpy as np from numpy.linalg import solve import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle

#Define Constants g = 9.8 m = 1 #define time dt = 0.001 # Time Step sim_time = 10 # Length of Simulation t_vec = np.arange(0, sim_time, dt) #Initialize State Vectors vec_size = len(t_vec) x_vec = np.zeros(vec_size) dx_vec = np.zeros(vec_size) y_vec = np.zeros(vec_size) dy_vec = np.zeros(vec_size) lam_vec = np.zeros(vec_size) #Initial Conditions x_vec[0] = -1 dx_vec[0] = 0 y_vec[0] = x_vec[0]**2 dy_vec[0] = 2*x_vec[0]*dx_vec[0] lam_vec[0] = 0 #Lets start simulating some stuff 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)]) for i in range(1, vec_size): #Only Update Top right and bottom left A[0, 2] = 2*x_vec[i-1] A[2, 0] = 2*x_vec[i-1] #Last element of b must be updated B[2] = -2*(dx_vec[i-1] ** 2) [ddx, ddy, lam_vec[i]] = np.linalg.solve(A, B) # Use ddx and ddtheta to solve: x_vec[i] = x_vec[i - 1] + dx_vec[i - 1] * dt y_vec[i] = y_vec[i - 1] + dy_vec[i - 1] * dt dx_vec[i] = dx_vec[i - 1] + ddx * dt dy_vec[i] = dy_vec[i - 1] + ddy * dt

#Generate the X and Y position plots plt.plot(t_vec, x_vec) plt.xlabel('Time (s)') plt.ylabel('X Position') plt.title("X Position over Time") plt.show() plt.plot(t_vec, y_vec) plt.xlabel('Time (s)') plt.ylabel('Y Position') plt.title("Y Position over Time") plt.show() plt.plot(t_vec, lam_vec) plt.xlabel('Time (s)') plt.ylabel('Constraint Force (N)') plt.title("Constraint Force over Time") plt.show()

#Set Up Figure for animation fig, ax = plt.subplots() # Create a plot on those axis fig_size = 2 p, = ax.plot([], [], color = 'cornflowerblue') #Initializes empty plot ax.set_xlim([-fig_size, fig_size]) ax.set_ylim([-1, 3]) ax.axis('equal') ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Bead Simulation') video_title = 'my_sim_p1' #Define information for the video FPS = 20 dpi = 300 #Create the bead c = Circle((0, 0), radius = 0.1, color = 'cornflowerblue') ax.add_patch(c) #Add the circle to the patch #Define the sample rate and create a writer object sample_rate = int(1/(FPS*dt)) #Samples to skip writerObj = FFMpegWriter(fps = FPS) #Create and plot the wire x_wire_vec = np.arange(-fig_size, fig_size, 0.1) y_wire_vec = [number ** 2 for number in x_wire_vec] p.set_data(x_wire_vec, y_wire_vec) # Create vectors for the simulation matrices simulation_size = len(t_vec) #number of time points # Now make a video with writerObj.saving(fig, video_title+".mp4", dpi): # Loop through the frames we want to animate for i in range(0, simulation_size, sample_rate): #Update the bob circle patch_center = x_vec[i], y_vec[i] #No brackets, just comma, creates a list c.center = patch_center; #Update drawing fig.canvas.draw() writerObj.grab_frame()

#HW1 Problem 2 import numpy as np from numpy.linalg import solve import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle, Rectangle from matplotlib.lines import Line2D

#Define Constants g = 9.8 m1 = 1 m2 = 2 L = 1 k = 10 H = 1.5 L_s = 1 #define time dt = 0.001 # Time Step sim_time = 10 # Length of Simulation t_vec = np.arange(0, sim_time, dt) #Initialize State Vectors vec_size = len(t_vec) x_vec = np.zeros(vec_size) dx_vec = np.zeros(vec_size) theta_vec = np.zeros(vec_size) dtheta_vec = np.zeros(vec_size) #Initial Conditions x_vec[0] = 0 dx_vec[0] = 0 theta_vec[0] = 1 dtheta_vec[0] = 0 #Define information for the video FPS = 20 dpi = 300 #Lets start simulating some stuff A = np.array([[m2, 0], [0, m1*L**2]]) B = np.array([0, 0]) for i in range(1, vec_size): #B must be updated every iteration den = (L**2 - 2*L*x_vec[i-1]*np.sin(theta_vec[i-1]) + x_vec[i-1]**2 + H**2 - 2*H*L*np.cos(theta_vec[i-1]))**(0.5) B[0] = k*L*np.sin(theta_vec[i-1]) - k*x_vec[i-1] + k*L_s*(x_vec[i-1] - L*np.sin(theta_vec[i-1]))/den B[1] = (-m1*g*L*np.sin(theta_vec[i-1]) + k*L*x_vec[i-1]*np.cos(theta_vec[i-1]) - k*H*L*np.sin(theta_vec[i-1])) + k*L_s*(H*L*np.sin(theta_vec[i-1]) - L*x_vec[i-1]*np.cos(theta_vec[i-1]))/den [ddx, ddtheta] = np.linalg.solve(A, B) x_vec[i] = x_vec[i - 1] + dx_vec[i - 1] * dt theta_vec[i] = theta_vec[i - 1] + dtheta_vec[i - 1] * dt dx_vec[i] = dx_vec[i - 1] + ddx * dt dtheta_vec[i] = dtheta_vec[i - 1] + ddtheta * dt

#Generate the X and Y position plots plt.plot(t_vec, x_vec) plt.xlabel('Time (s)') plt.ylabel('X Position') plt.title("X Position over Time") plt.show() plt.plot(t_vec, theta_vec) plt.xlabel('Time (s)') plt.ylabel('Theta') plt.title("Theta over Time") plt.show()

#Set Up Figure for animation fig, ax = plt.subplots() # Create a plot on those axis p, = ax.plot([], [], color = 'cornflowerblue') #Initializes empty plot ax.set_xlim([-1, 1]) ax.set_ylim([-1, 3]) ax.axis('equal') ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Pendulum Simulation') video_title = 'my_sim_p2' #Create the Pendulum mass c = Circle((0, 0), radius = 0.1, color = 'cornflowerblue') ax.add_patch(c) #Add the circle to the patch #Create Block Mass width = 0.2 # Width of Block height = width / 2 # Height of Block x_cart = (x_vec[0] - (width / 2), 0) # Bottom Left Corner of Cart r = Rectangle(x_cart, width, height, color='cornflowerblue') # Rectangle Patch ax.add_patch(r) # Add Patch to Plot #Create Swivel x_cart = (-0.1 / 2, H-0.1/2) # Bottom Left Corner of Cart r2 = Rectangle(x_cart, 0.1, 0.1, color='black') # Rectangle Patch ax.add_patch(r2) # Add Patch to Plot #Create Lines pend_arm = Line2D([0, 0], [0, 0]) ax.add_line(pend_arm) spring = Line2D([0, 0], [0, 0]) ax.add_line(spring) #Define the sample rate and create a writer object sample_rate = int(1/(FPS*dt)) #Samples to skip writerObj = FFMpegWriter(fps = FPS) # Create vectors for the simulation matrices simulation_size = len(t_vec) #number of time points # Now make a video with writerObj.saving(fig, video_title+".mp4", dpi): # Loop through the frames we want to animate for i in range(0, simulation_size, sample_rate): # Update Pendulum Arm: x_pole_arm = [0, L*np.sin(theta_vec[i])] y_pole_arm = [H, H-L*np.cos(theta_vec[i])] pend_arm.set_data([x_pole_arm, y_pole_arm]) # Update Spring x_spring = [L*np.sin(theta_vec[i]), x_vec[i]] y_spring = [H-L*np.cos(theta_vec[i]), height/2] spring.set_data([x_spring, y_spring]) r.set(xy=(x_vec[i] - width / 2, 0)) #Update the bob circle patch_center = L*np.sin(theta_vec[i]), 1.5-L*np.cos(theta_vec[i]) #No brackets, just comma, creates a list c.center = patch_center; #Update drawing fig.canvas.draw() writerObj.grab_frame()

#HW1 Problem 3 import numpy as np import math from numpy.linalg import solve import matplotlib.pyplot as plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle, PathPatch from mpl_toolkits import mplot3d import mpl_toolkits.mplot3d.art3d as art3d

#Define Constants g = 9.8 m = 1 #define time dt = 0.0001 # Time Step sim_time = 10 # Length of Simulation t_vec = np.arange(0, sim_time, dt) #Initialize State Vectors vec_size = len(t_vec) x_vec = np.zeros(vec_size) dx_vec = np.zeros(vec_size) y_vec = np.zeros(vec_size) dy_vec = np.zeros(vec_size) z_vec = np.zeros(vec_size) dz_vec = np.zeros(vec_size) #Initial Conditions x_vec[0] = 1 dx_vec[0] = 0 y_vec[0] = 0 dy_vec[0] = 1 z_vec[0] = (x_vec[0] ** 2) + (y_vec[0] ** 2) dz_vec[0] = 2*(x_vec[0]*dx_vec[0] + y_vec[0]*dy_vec[0]) #Define information for the video FPS = 20 dpi = 300 #Lets start simulating some stuff #Fill in every element that wont change A = np.array([[m, 0, 0, 0], [0, m, 0, 0] , [ 0, 0, m, (-1 - 0.000001)], [ 0, 0, -1, 0]]) B = np.array([0, 0, -m*g, 0]) for i in range(1, vec_size): #Only Update Top right and bottom left A[0, 3] = 2*x_vec[i-1] A[1, 3] = 2*y_vec[i-1] A[3, 0] = 2*x_vec[i-1] A[3, 1] = 2*y_vec[i-1] #Last element of b must be updated B[3] = -2*((dx_vec[i-1] ** 2) + (dy_vec[i-1] ** 2)) [ddx, ddy, ddz, lam] = np.linalg.solve(A, B) # Use ddx and ddtheta to solve: x_vec[i] = x_vec[i - 1] + dx_vec[i - 1] * dt y_vec[i] = y_vec[i - 1] + dy_vec[i - 1] * dt z_vec[i] = z_vec[i - 1] + dz_vec[i - 1] * dt dx_vec[i] = dx_vec[i - 1] + ddx * dt dy_vec[i] = dy_vec[i - 1] + ddy * dt dz_vec[i] = dz_vec[i - 1] + ddz * dt

#Generate the X and Y position plots plt.plot(t_vec, x_vec) plt.xlabel('Time (s)') plt.ylabel('X Position') plt.title("X Position over Time") plt.show() plt.plot(t_vec, y_vec) plt.xlabel('Time (s)') plt.ylabel('Y Position') plt.title("Y Position over Time") plt.show() #plt.plot(t_vec, z_vec) #plt.xlabel('Time (s)') #plt.ylabel('Z Position') #plt.title("Z Position over Time") #plt.show()

def f(x, y): return x ** 2 + y ** 2 x = np.linspace(-1, 1, 30) y = np.linspace(-1, 1, 30) X, Y = np.meshgrid(x, y) Z = f(X, Y) #Set Up Figure for animation fig, ax = plt.subplots() plt.axis('off') # Create a plot on those axis ax = plt.axes(projection='3d') ax.contour3D(X, Y, Z, 30, cmap='binary') ax.set_xlim([-1, 1]) ax.set_ylim([-1, 1]) ax.set_zlim([0, 1]) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_zlabel('Z') ax.set_title('Paraboloid Simulation') video_title = 'my_sim_p3' ax.view_init(60, 35) #Define the sample rate and create a writer object sample_rate = int(1/(FPS*dt)) #Samples to skip writerObj = FFMpegWriter(fps = FPS) # Create vectors for the simulation matrices simulation_size = len(t_vec) #number of time points # Now make a video with writerObj.saving(fig, video_title+".mp4", dpi): # Loop through the frames we want to animate for i in range(0, simulation_size, sample_rate): scat = ax.scatter3D(x_vec[i], y_vec[i], z_vec[i], color = 'cornflowerblue' ) ax.view_init(math.atan2(z_vec[i], np.sqrt(y_vec[i]**2 + x_vec[i]**2) )*180/3.1415, math.atan2(y_vec[i], x_vec[i])*180/3.1415) #ax.view_init(35, 60) #Update drawing fig.canvas.draw() writerObj.grab_frame() scat.remove()