Week 3 - Solving second order ODEs

1. AIM:- Write a programme for solving second order differential equation for a pendulum and also transient behaviour of a pendulum .

2.Governing Equation :

3.OBJECTIVE OF PROJECT: To To study how the time period of a simple changes when its amplitude (θ ) is changed.

4.BODY:-

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

#function that return dz/dt

def model (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

b= 0.05
g = 9.81
l=1
m=1

#initial conddition
theta_0 = [math.pi/2,0]
#time points

t = np.linspace(0,10,150)

#solve ODE
theta = odeint(model,theta_0,t,args = (b,g,l,m))

#plot results
plt.plot(t,theta[:,0],'b-',label=r'$\frac{d\theta_1}{dt}=\theta2$')
plt.plot(t,theta[:,1],'r--',label=r'$\frac{d\theta2}{dt}=-\frac{b}{m}\theta_2-\frac{g}{L}sin\theta_1$')
plt.ylabel('Plot')
plt.xlabel('time')
#plt.legend(Loc='best')
plt.show()

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)
	plt.figure()
	plt.plot([-0.5,0.5],[0,0],linewidth=8,color='y')
	plt.plot([x_start,x_end],[y_start,y_end],linewidth=3)
	plt.plot(x_end,y_end,'o',markersize=20,color='r')
	plt.plot([0,0],[0,-1.5],'--',color='k')
	plt.xlim([-1.25,1.25])
	plt.ylim([-2,1])
	plt.title('MOTION STUDY OF SIMPLE PENDULUM')
	filename = 'test%005d.png' % ct
	plt.savefig(filename)
	ct = ct + 1

STEPS TAKEN TO WRITE A PROGRAMME

1. Import math command, matplotlib command, and also Numpy command.
2. Function Model is created for required arguments, inside which the term is broken to solve secondary order.
3. Give The value of b,g,l,m.
4. give the range of THETa
5. USE Odient command to solve ode function.
6. PLOt the result using plotting command
7. save the file in .png format
8. use image magic to convert .png to gif.

PLOT

SCREENSHOT:-

YOUTUBE LINK:-

https://youtu.be/A9KgtwVuMmg

5.CONCLUSION

1. From getting a simulation result clear the concept of Simple Pendulum.
2. ALso gave how the above equation is important to get a result
3. its show hoe time and amplitude depend