Week 3 - Solving second order ODEs

objective :

To write a program that solves the following ODE. 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.

Input Values

L=1 metre,

m=0.1 kg,

b=0.05.

g=9.81 m/s2.

Step 1 : Define the function

def model(theta,t,b,g,L,m)

theta1 = theta[0]

theta2 = theta[1]

dtheta1_dt = theta2

dtheta2_dt = -(b/m)*theta2 - (g/1)*math.sin(theta1)

dtheta_dt = [dtheta_dt,dtheta2_df]

return dtheta_dt

Program :

import math

import matplotlib.pyplot as plt

from scipy.integrate import odeint

import numpy as np

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

theta_0=[math.pi/2

t=np.linspace(0,50,100)

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

print(theta)

plt.plot(t,theta[:,0])

plt.plot(t,theta[:,1])

plt.ylabel('plot')

plt.xlabel('time')

plt.savefig('plt')

plt.show()