Week 3 - Solving second order ODEs

Aim :- To Write a program in Python that will simulate the pendulum motion and solving the following ode which represents the equation of motion of a simple pendulum with damping.

Theory :- In Engineering, ODE is used to describe the transient behavior of a system. A simple example is a pendulum.

The way the pendulum moves depends on the Newtons second law. When this law is written down, we get a second order Ordinary Differential Equation that describes the position of the "ball" w.r.t time. Similarly, there are hundreds of ODEs that describe key phenomena.

From 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.

Using the given values

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

The Main

#importing required modules
import math
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import numpy as np


#function
def model(theta,t,b,g,m,l):
    theta1=theta[0]
    theta2=theta[1]

    dtheta1_dt=theta2
    dtheta2_dt=-(b/m)*theta2-(g/1)*math.sin(theta1)
    dtheta_dt=[dtheta1_dt,dtheta2_dt]
    return dtheta_dt


#inputs required
b=0.05
g=9.81
l=1
m=1


theta_0=[0,3]
t=np.linspace(0,20,200)


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


#plotting result
plt.figure()
plt.plot(t,theta[:,0],'g',label=r'Angular displacement')
plt.plot(t,theta[:,1],'b',label=r'Angular velocity')
plt.xlabel('plot')
plt.ylabel('Time')
plt.legend(loc='best')
plt.show()


#Animation step
ct=1
x0=5
y0=5
for i in range(len(t)):
    x1=x0+1*math.sin(theta[i,0])
    y1=y0-1*math.cos(theta[i,0])
    filename='pendulum'+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

we had imported the plot using the 'import'

for creating the animation we use for loop.theta_0 rages from 0-3m/s2 and images were saved.

Errors Faced

There were error faced while running

which we resolved later

and then we run simulation.

output

Run