Week 3 - Solving second order ODEs

AIM:

To simulate the transient behaviour of a simple pendulum using Python.

OBJECTIVES:

• To write a program that solves the second order ODE which represents the equation of motion of a simple pendulum with damping.
• To simulate the motion of the simple pendulum with damping & creating an animation.

Program:

# Program for solving second order ODE

import matplotlib.pyplot as plt 
import math
import numpy as np 
from scipy.integrate import odeint 
 
 
def ode_function(theta,t,b,g,l,m):  
	theta1 = theta[0]  
	theta2 = theta[1]  
	dtheta1_dt = theta2  
	dtheta2_dt = -(b/m)*theta2-(g/l)*math.sin(theta1)  
	dtheta_dt = [dtheta1_dt,dtheta2_dt]  
	return dtheta_dt 

# Inputs

# damping coefficient 
b = 0.05 

# acceleration due to gravity(m/s^2) 
g = 9.81

# length of string (m)                                                                
l = 1

# mass of bob (kg)                                                                   
m = 1       
 
# Initial Conditions 
theta_0 = [0,3]  

# time points 
t_span = np.linspace(0,20,100) 
 
# Solving ODE 
 
theta = odeint(ode_function,theta_0,t_span,args = (b,g,l,m)) 
 
# Plotting the graph 
plt.plot(t_span,theta[:,0]) 
plt.plot(t_span,theta[:,1]) 
plt.xlabel('Time (second)') 
plt.ylabel('Plot') 
plt.title('Damped Harmonic motion of a Simple Pendulum')  
plt.ylim([-4,4]) 
plt.xlim([0,20]) 
plt.grid() 
plt.show() 
 
# for loop to create animation of motion of pendulum 
 
ct =1 
range = theta[:,0] 

for i in range:  
	x_start = 0                                                             
	y_start = 0  
	x_end =  l*math.sin(i)                                            
	y_end = -l*math.cos(i)  

	# Plotting for animation  
	plt.figure()    

	# Plotting the rigid support through which string of pendulum is attached  
	plt.plot([-0.5,0.5],[0,0],linewidth=3,color='b') 

	# Plotting String of pendulum  
	plt.plot([x_start,x_end],[y_start,y_end],linewidth=3)

	# Plotting Bob of pendulum  
	plt.plot(x_end,y_end,'o',markersize=15,color='r')  
	plt.xlim([-1.5,1.5])  
	plt.ylim([-1.5,0.5])  
	plt.title('Transient behaviour of a Simple Pendulum')  
	filename = 'test%05d.png' % ct  
	plt.savefig(filename)  
	ct = ct + 1

Explanation:

• Firstly, import all the modules required.
• Define the function as it will be used to solve repeatedly & provide the inputs.
• Run the program & check whether it has error or not.
• If errors come read the error & try to debug it.
• Check whether graph & images obtained are proper or not.
• Convert the images into animation.

Errors encountered:

Module error.

Name error.

syntax error.

Result:

https://drive.google.com/file/d/1BYKKEHlm6AJMOznASKbDUxXrJDgkNp8S/view?usp=share_link

Conclusion:

Python Program for ODE is done & output is shown.