Week 3 - Solving second order ODEs

Gdrive: https://drive.google.com/drive/folders/1mwlGUaGJyMrs0zaj7CcsVDp8T3jkt2CQ

AIM

• Your objective is to write a program that solves the following ODE. This ODE represents the equation of motion of a simple pendulum with damping.

Program

• import numpy as np
  from scipy.integrate import odeint # for intagration purpose
  import matplotlib.pyplot as plt
  import math
  
  def model (theta,t,b,g,l,m):
  	theta1= theta[0] # coverting the second order ODE into simpleform
  	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
  
  b=0.02 #damping coefficient.
  g=9.81 #gravity m/sec2
  l=1	   # length of the pendulum in m,	
  m=0.1  #mass of the ball in kg,
  
  #initial conditions
  theta_0=[0,5] #the angle between string and plane is 5
  
  #time points
  t=np.linspace(0,20,150)
  
  #solve ode
  theta= odeint(model,theta_0,t,args=(b,g,l,m))
  #plot results
  plt.plot(t,theta[:,0])
  plt.plot(t,theta[:,1])
  plt.ylabel('plot')
  plt.xlabel('time')
  plt.show()
  
  
  ct=1
  for theta in theta[:,0]:
  	x0=0
  	y0=0
  	x1=l*math.sin(theta)
  	y1=-l*math.cos(theta)
  	plt.figure()
  	plt.plot([-1,1],[0,0])
  	plt.plot([0,x1],[0,y1])
  	plt.plot(x1,y1,marker='o', markersize =30)
  	plt.xlim([-1.5,1.5])
  	plt.ylim([-1.5,1.5])
  	#plt.show()
  	
  	filename='test%05d.png' % ct
  	plt.savefig(filename)
  	ct=ct+1
  
  
  
  

Procedure

• Import the required module. as in case we need the ODE scipy module
• convert the second order ODE into simple single order ODE
• Applying or define the inputs in python such mass,gravity,daming factor 
• Intial displacement is 0 and velcoity is 5m/s
• the total time is 20ms and the input 20 is spillited into 150 to plot curve
• 
• the Orange line indicate the velcoity and the bule is displacement

OUTPUT

Result

from this assignment we learn the how to slove the second order ode by using the python