Week 3 - Solving second order ODEs

WEEK -3

PENDULUM -Solving second order ODEs

This is the following code for the given question :

import math
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import odeint

# Now we will first assign the inputs of the given damping eqaution

b = .2
g = 9.81 
l = 1
m = 1

# Now the initial condition has 2 values in which first one relates to displacment and second relates to velocity
# Also there are only 2values due to fact that equation has 2 degree 

initial_cond = [0,3]

# time array with 100 steps 

t = np.linspace(0,20,100)

# the differential equation has been splitted into 2 first order equation and after it the solution has been taken into matrix 

def  motion( theta ,t, b ,g ,l ,m  ) :
     
    dtheta1_dt = theta[1]  # this theta[1] shows that the output vector(theta) has 2 values out of which first value is assigned to the input variable
    dtheta2_dt = (-b/m)*theta[1]- (g/l)*math.sin(theta[0]) 
    dtheta_dt = [dtheta1_dt , dtheta2_dt] 

    return dtheta_dt

# Now this variable theta has 2 output values stored in it after passing the following parameters because the eqaution has also 2 degree

theta = odeint(motion, initial_cond , t , args = (b ,g ,l , m ))

# extracting the coloumn first from the array theta 

x = theta[:,0]

ct = 1

# the for loop has been created because to show that the pendulum has run for 20 sec for differnt values of theta . 

for i in x:
    x0 = 0
    y0 = 0

    x1 =  (x0 + l*math.sin(i))
    y1 = -(y0 + l*math.cos(i))

    # plot function

    plt.figure()
    plt.plot([-2 ,2] , [0,0] , color = "yellow" , linewidth = 3)
    plt.plot(x0,y0 , 'o' ,marker = 'o', markersize = 5  , color = "blue"  )
    plt.plot([ x0,x1] , [y0, y1] , linewidth = 3 , color = "red" )
    plt.plot( x1 ,y1 , 'o' ,marker = 'o', markersize = 30  , color = "purple")
    filename = 'test%05d.png'  % ct # name for saving the filename to create image one by one 
    plt.savefig(filename)
    ct =ct +1
   

screenshots :

This is a folder in which each image of pendulum is saved .

https://drive.google.com/file/d/1jtM6g4K5RQZmRmSVv2dcjvTHf0WevoU_/view?usp=sharing

This is the link for gif animation .

https://drive.google.com/drive/folders/1F_DX3Rh9S0TxxcE_hueyOScia_jkOTq2?usp=sharing

This is the folder in which all the images are saved .