Week 3 - Solving second order ODEs

Code - To solve second order differential equation using python.

#program to solve 2nd order differential equation

import math

import matplotlib.pyplot as plt

import numpy as np

from scipy.integrate import odeint

import os

import imageio.v2 as imageio

def model1(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

#damping coefficient (b) = 0.05, length of string (l) = 1 m, mass of ball (m) = 1 kg, gravity = 9.81 m/s2

b = 0.05

l = 1

m = 1

g = 9.81

#initial conditions

theta_0 = [0,3]

#time points

t = np.linspace(0,20,150)

#solve ODE

theta = odeint(model1,theta_0,t,args=(b,g,l,m))

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

plt.plot(t,theta[:,1],'r--',label=r'$\frac{d\theta2}{dt}=-\frac{b}{m}\theta_2-\frac{g}{l}sin\theta_1$')

plt.xlabel('Time')

plt.ylabel('Plot')

plt.legend(loc='best')

plt.show()

#Pendulam animation

count = 1

for Theta in theta[:,0]:

x0 = 0

y0 = 0

x1 = l*math.sin(Theta)

y1 = -l*math.cos(Theta)

filename = '%03d.png'%count

plt.figure()

plt.plot([x0,x1],[y0,y1],"blue",linewidth = 3)

plt.plot(x0,y0,'*', color = "black", markersize = 10)

plt.plot(x1,y1,'o', color = "red", markersize = 18)

plt.xlim([-1.5,1.5])

plt.ylim([-1.5,1.5])

plt.savefig(filename)

count = count + 1

# Create a list of filenames in sorted order

filenames = ['%03d.png' % i for i in range(1, count)]

# Create a GIF animation

with imageio.get_writer('pendulum_animation.gif', mode='I', duration=0.05) as writer:

for filename in filenames:

image = imageio.imread(filename)

writer.append_data(image)

print("Animation saved as pendulum_animation.gif")

Image of oscillation in waves:

Explanation:

Importing necessary libraries:

math: Provides mathematical functions like sin() and cos().

matplotlib.pyplot: For plotting graphs and visualizing the pendulum.

numpy: Used for array handling and numerical operations.

scipy.integrate.odeint: Solves systems of ordinary differential equations (ODEs).

os: For file operations (used later for cleanup).

imageio.v2: Reads and writes image files; used to create GIF animations.

Defining function to solve 2nd order differential equation:

This function represents the system of 2 first-order differential equations derived from the second-order equation for a damped pendulum:

• θ'' = (b/m) * θ' - (g/l) * sin(θ)

• It's rewritten as two first-order ODEs to be solved using odeint.

Later defining damping coefficient, length of string, mass of ball, gravity values, angular velocity from 0 - 3 rad/sec for 20 sec with 150 instance.

Further function is called and 2ODE is solved for given inputs as mentioned above.

All plot statements plots angular velocity over a period of time in which

To create animation of pendulum variable "count" with initial value 1 is defined, then for each value of angle pendulum second end (x1, y1) position is computed as first end (x0, y0) remains fixed.

Declaring variable "filename" to give name with max 3 digit with ".png" extension to each image created for different position of pendulum.

Further

plt.figure() - to create fresh image for each position

plt.plot([x0,x1],[y0,y1],"blue",linewidth = 3) - plots string position with line color as blue and its width.

plt.plot(x0,y0,'*', color = "black", markersize = 10) - "*" shape defined for fixed end with black color and size 10.

plt.plot(x1,y1,'o', color = "red", markersize = 18) - "o" shape defined for rotating end with red color and size 18.

plt.xlim([-1.5,1.5]) - limits of graph

plt.ylim([-1.5,1.5])

plt.savefig(filename) - saving each image

count = count + 1 - increasing count after each image is created.

Now sorting/arranging images from 1 to 150 and with

imageio.get.writer - creates a writer object for saving the animation to a file (pendulum_animation.gif).

mode = I - means you are creating a multi-image (animated) files

duration = 0.05 - specifies the time between each frame in seconds (i.e., 20 frames per second).

imageio.imread(filename) - reads each saved .png image.

writer.append_data(image) - adds each image as a frame in the final GIF.

In the end just printing statement to notify animation has been saved.

Animation: attached gif file.

Errors occurred:

Mostly it were typing error which were corrected after reading error prompts.