Solving Second Order Differential Equation using Python - Animation of Simple Pendulum

AIM:- To solve Second Order Differential Equation write a code. Plot the Graph and show the animation of Simple Pendulum using python.

Follow this condition in order to solve ODE.

initial condition at time(t=0) where initial angular displacement is 0 , initial angular velocity is 3 rad/s.

In Engineering, ODE is used to describe the Transient Behaviour of system.

SOLUTION

Coding:-

#Solving Second Order Differential Equation- Animation of Simple Pendulum  with python
#Program by Dakshkumar Jadav

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

#function that return dz/dt
def model(theta,d,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 = 0.02
g = 9.18
l = 1
m = 1

# initial Conditions
theta_0 = [0,3]

#Time Point
t = np.linspace(0,20,200)

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

#Plot Result

#plotting Angular Displacement
plt.plot(t,theta[:,0],'b-',label=r'$\frac{d\theta_1}{dt}=\theta2$')

#plotting Angular Velocity
plt.plot(t,theta[:,1],'r--',label=r'$\frac{d\theta_1}{dt}=-\frac{b}{m}\theta_2-\frac{g}{L}sin\theta_1$')

#limiting X-axis
plt.xlim([0,20])

#Limiting Y-axis
plt.ylim([-3,3])

plt.ylabel('plot')
plt.xlabel('time')
plt.legend(loc="best")
plt.show()

#for Counter Variable

ct=1
for Theta in theta[:,0]:

	#co-ordinates of Fixed point of string
	x0=0
	y0=0
	#co-ordinate of End point of string
	x1= l*math.sin(Theta)
	y1= -l*math.cos(Theta)

	filename = str(ct) + '.png'

	ct = ct + 1
	plt.figure()

	#for Horizontal line Plotting
	plt.plot([-0.5,0.5],[0,0],'g',linewidth=5)

	#for Plotting String of Pendulum
	plt.plot([x0,x1],[y0,y1],'r')

	#Plotting Ball of Pendulum
	plt.plot(x1,y1,'o',markersize=30)

	#limiting X-axis
	plt.xlim([-1.5,1.5])

	#limiting y-axis
	plt.ylim([-1.5,1.5])

	#for Title
	plt.title('Animation of Simple Pendulum Using Python')

	#Saving each Figure with its file name
	plt.savefig(filename)

Graph:-

#Animation of Simple Pendulum

Comments:-

1, first of all Import numpy module, scipy module, matplotlib.pyplot module & math module.

2,Define a function which is the system of ordinary differential equations.

3,split the Second order ODE into two First order ODEs.

4, Assigning a given values to inputs.(b,g,l,m)

5, Assigning an initial condition at time(t=0) where initial angular displacement is 0 and initial angular velocity is 3 as shown in the question.

6, Time span is 0 to 20 sec. So, Assign 200 equally spaced values from 0 to 20 in variable " t ".

7, "odeint" command is in the scipy module. "odeint" is basically a function that integrate system of ordinary differential equations. For that defined a system('model' function) above which is the system of two first order ODEs.

8, Assign a function , initial conditions , time and input arguments to the "odeint".

9, Plot Angular Displacement & Angular Velocity.

10, Used legend to describe the graph.

11, Use xlim & ylim to limit the X-axis & Y-axis and Use xlabel & ylabel to indicate both axes.

12, Assigning value '1' to Counter Variable 'ct'.

13, theta[:,0] is the set of values of Angular Displacement.

14, Assigning a co-ordinates to fixed point & end point of string of pendulum by using simple trigonometry.

15, ct is the counter variable which is used to assign a number to the filename(with one increment) with every iteration.

16, Plotting Fixed Horizontal Line , String & Ball of Pendulum.

17, Used 'xlim' & 'ylim' to limit the X-axis & Y-axis.

18, Used 'title' to indicate this animation.

19, savefig(filename) is used to save each position of pendulum in seperate figure with its filename.