SOLVING SECOND ORDER ODE'S

AIM: TO SIMULATE THE TRANSIENT BEHAVIOUR OF A SIMPLE PENDULUM AND TO CREATE AN ANIMATION OF IT’S MOTION USING PYTHON

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 Newton's second law. When this law is written down, we get a second order Ordinary Differential Equation that describes the position of the "bob" w.r.t time.

GOVERNING EQUATIONS:

• Ordinary Differential Equation (ODE) representing equation of motion of a simple pendulum with damping –

`(d^2 θ)/(dt^2 )+ b/m×(dθ)/dt+ g/L sinθ=0`

where,

g = gravity (m/s²)

L = length of the pendulum (m)

m = mass of the ball (kg)

b = damping coefficient

OBJECTIVES:

1. To study the transient behavior of the pendulum system.
2. Animate the motion of the simple pendulum.

INPUT VALUES:

L = 1 m

m = 1 kg

b = 0.05

g = 9.81 m/s²

FUNCTION PROGRAM:

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

# Function that returns dz/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

b = 0.05
g = 9.81
l = 1
m = 1

# Initial condition
theta_0 = [0,3]

# Time points
t = np.linspace(0,20,300)

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

# Plot results
plt.plot(t,theta[:,0],'b-',label = r'$\frac{d\theta_1}{dt}=\theta2$')
plt.plot(t,theta[:,1],'r--',label = r'$\frac{d\theta_2}{dt}=-\frac{b}{m}\theta_2-\frac{g}{L}sin\theta1$')
plt.ylabel('Plot')
plt.xlabel('Time')
plt.legend(loc='best')
plt.show()

# Animation using image stitching
# Defining counter variable
ct = 1
for th in theta[:,0]:
	x_start = 0
	y_start = 0

	x_end = l*math.sin(th)
	y_end = -l*math.cos(th)

	# Naming the figures using the counter variable and saving them in '.png' format
	filename = 'test%05d.png' %ct
	ct = ct + 1

	# Plotting figures and saving them with particular filename
	plt.figure()

	#Plotting the support
	plt.plot([-1,1],[0,0], linewidth=4, color='red')

	#Plotting the string through which the bob is attached
	plt.plot([x_start,x_end],[y_start,y_end], linewidth=2, color='blue')

	# Plotting the bob
	plt.plot(x_end,y_end, marker='o', markersize=25, color='green')
	
	plt.xlim([-2,2])
	plt.ylim([-1.5,0.5])
	plt.savefig(filename)

EXPLANATION OF PROGRAM:

1. Initially, writing a function program where a specified function is mentioned covering all the parameters and the variables.
2. The given values are specified and the initial condition is imposed.
3. Timespan over which the pendulum’s transient behavior has to be observed is also defined.
4. Now, the 'odeint' command acts as a solver for the first-order differential equations extracted out of a single second-order differential equations
5. Animation of the pendulum motion is done using ImageMagick by stitching the images or plots obtained.

TRANSIENT BEHAVIOUR:

PLOTS:

PLOTS SAVED WITH A DEFINITE FILENAME FORMAT INTO ANIMATION FOLDER

ANIMATION: The link for the animation has been mentioned below.

https://youtu.be/NDkpOkGge2Y

CONCLUSION: Establishing a clear view of the transient behavior of the simple pendulum by solving the second-order ODE defining its motion. Also, the animation of the pendulum motion is observed by stitching the successive plots into a gif and thereby converting into a '.avi' file format.