Week 3 - Solving second order ODEs

AIM: To write a program for the motion of simpe pendulum using ODE.

THEORY: A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. 

 This ODE represents the equation of motion of a simple pendulum with damping

In the above equation,

g = gravity in m/s2,

L = length of the pendulum in m,

m = mass of the ball in kg,

b=damping coefficient.

PYTHON PROGRAMME:

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

#function program
def model(theta,t,b,g,m,l):
	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

#input parameters
b= 0.05
g= 9.81
l= 1
m= 1
theta_0= [math.pi/2,0]
t= np.linspace(0,20,200)

#Solving the ordinary diff. equation
theta= odeint(model,theta_0,t,args=(b,g,l,m))

#Plotting the values
plt.figure()
plt.plot(t,theta[:,0],'g',label=r'Angular Displacement')
plt.plot(t,theta[:,1],'r',label=r'Angular velocity')
plt.xlabel('plot')
plt.ylabel('Time')
plt.legend(loc='best')
#plt.show()

#Animating the outputs
ct=1
x0= 5
y0= 5
for i in range(len(t)):
	x1= x0+l*math.sin(theta[i,0])
	y1= y0-l*math.cos(theta[i,0])
	filename= 'pendulum motion'+str(ct)+'.png'

	plt.figure()
	plt.plot([4,6],[5,5],'k-')
	plt.plot([x0,x1],[y0,y1])
	plt.plot(x1,y1,marker='o',markersize='10')
	plt.axis([3.5,6.5,3,6])
	plt.savefig(filename)
	ct= ct+1

EXPLANATION OF CODES:

1. Import the required math libraries.
2. Function module is created in which required arguments have been inserted to calculate the ODE.
3. Inputs like g,mass,length,etc have been defined.
4. Theta_0 calculated from range 0-90.
5. Differential equation has been calculated using odeint command.
6. After that plots have been generated for angular displacemant and angular velocity.
7. After that animation is created using the for loop using the length of t viz .20.
8. Finally, images are being saved using savefif command.

OUTPUT PLOTS:

ERRORS FACED: No error encountered.

ANIMATION FILE:

                                    

REFERENCES:

www.wikipedia.com