Solving second order ODEs

AIM

To write a code and execute the simulation of simple pendulum usind second order equation and create a animation using python.

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.

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

Simulate the motion between 0-20 sec, for angular displacement=0,angular velocity=3 rad/sec at time t=0.

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

#Defining Function that return 'dtheta_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


#Inputs 
b = 0.05
g = 9.81
l = 1.0
m = 1.0

#Initial condition
theta_0 = [0,3]

t = np.linspace(0,20,250)

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

#Plot results
plt.plot(t,theta[:,0], 'g', label=r'displacement')
plt.plot(t,theta[:,1], 'b', label=r'velocity')
plt.xlim([0,20])
plt.ylim([-3,3])
plt.legend(loc='best')
plt.xlabel('time')
plt.ylabel('Plot')
plt.savefig('output.png')
plt.show()

#Animation of the pendulum

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([-0.25,0.25],[0,0], "blue", linewidth=5)
	plt.plot([x0,x1],[y0,y1],"red")
	plt.plot(x1,y1,'o', markersize=15)
	plt.xlim([-1,1])
	plt.ylim([-1.5,1])
	filename = str(ct) + '.png'
	plt.savefig(filename)
	

EXPLANATION OF CODE

• create a new file and import math, mathplotlib,scipy and numpy
• define a function module which have the inputs b,g,l,m , theta, t and return dtheta_dt
• defina initial parameters
• solve the differntial equation using the odeint which is imported from scipy.integrate
• theta is solution ODE containing two column vector
• first is angular displacement and second is angular velocity.
• plot the angular displacement and velocity against the time.
• to create animation of simple pendulum set the plots for different angular displacement
• save the figure and create gif file.

RESULTS AND ANIMATION