Week 3 - Solving second order ODEs

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

'''
damping: effects that reduce the amplitude of the vibrations
'''

def ode_func(theta,t,b,g,l,m):

theta1=theta[0]
theta2=theta[1]
#first ode
dtheta1_dt=theta2
#second ode
dtheta2_dt=-((b/m)*(theta2))-((g/l)*math.sin(theta1));
dtheta_dt=[dtheta1_dt,dtheta2_dt]
return dtheta_dt

# material with high damping coefficient used as the shock absorbers
b=0.05

# acceleration due to the gravity
g=9.81

#length
l=1

#mass
m=1

#intial conditions
theta_0=[0,3]

# time plot
t=np.linspace(0,20,200)

#T = len(t)

# solvimg the ode
theta = odeint(ode_func,theta_0,t,args=(b,g,l,m))
plt.figure(1)
plt.plot(t,theta[:,0],'b--',label=r'$\frac{d\theta_1}{dt}=\theta2$')
plt.plot(t,theta[:,1],'r--',label=r'$\frac{d\theta_2}{dt}=-\frac{b}{m}\theta_2-\frac{g}{L}sin\theta_1$')
plt.xlabel('time')
plt.ylabel('plot')
plt.legend(loc='best')
plt.show()
ite = 1

# Create a loop for the animation
for t in theta[:,0]:
x0=0
y0=0
x1=(l*math.sin(t))
y1 = -(l*math.cos(t))
filename = str(ite)+'.png'
ite=ite+1

plt.figure()
plt.plot([-0.5,0.5],[0,0],'black',linewidth=5)
plt.plot([x0,x1],[y0,y1],linewidth=3,color='r')
plt.plot(x0,y0,'o',markersize=5)
plt.plot(x1,y1,'o',markersize=30,color='blue')
plt.xlim([-1.5,1.5])
plt.ylim([-1.5,1.5])
plt.title('Animation of of Simple Pendulum')
plt.savefig(filename)

youtube link

https://youtu.be/8qyJ_HYfJPk