Week 3 - Solving second order ODEs
ASSIGNMENT 4
Solving second order ODEs
OBJECTIVE:
The objective of Assignment 4, is to write a program that solves the following ODE.
This ODE represents the equation of motion of a simple pendulum with damping and to simulate the pendulum motion, by using Python program,subline as text editor and imagemagick.
PROBLEM STATEMENT:
In this assignment we have to simulate the pendulum motion.. And the files should be saved in a separte folder otherwise, if not the animation will not takes place.
The png files are in a zigzag form in the command promt, we have to arrange them in a right format based on the time.
Assigning of theta angle should be noted carefully.
SOFTWARE USED:
Python3.6 (64bit), subline text3 as text editor.
THEORY:
The below example shows, how to simulate the motion of a simple pendulum using python programme. Solving the second order differential equation and Derivind the equation of motion of the pendulum and finally creating the animation of a simple pendulum by using image magick.
second order differential equation:
pendulum equation of motion:
The pendulum is a simple mechanical system that follows a differential equation. The pendulum is initially at rest in a vertical position. When the pendulum is displaced by an angle θ and released, the force of gravity pulls it back towards its resting position. Its momentum causes it to overshoot and come to an angle −θ (if there are no frictional forces), and so on. The restoring force along the motion of the pendulum due to gravity is −mgsinθ. Thus, according to Newton's second law, the mass times the acceleration must equal −mgsinθ. Let X0,Y0 will be the one end of the pendulum which is fixed and another end will be the X1,Y1 which is connected to the bob, and X1Y1 position have to be calculated.
X0 = 0
Y0 = 0
X1 = ?
Y1 = ?
Sinθ = X1/L
X1 = L*Sinθ.......1
cosθ = Y1/L
Y1 = L*COSθ.........2
PROCEDURE: to simulate the pendulum equation of motion,
1. Open subline text editor.
2. Open a new file and save it as ----.py
3. Write the comment that what is the program is about, which the line is never executed. And write the comment first followed by #(pound) or double quotation codes in starting and as well as at the end of comments. then
import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import math
import matplotlib.markers as mk
4. Define the model which is a function, that represents the ODE system, in which the two ordinary differential equation is defined. [displacement, velocity]
5. Define the damping coefficient, gravity, length of pendulum, mass as follows.
b = 0.05
g = 9.81
l = 1
m = 1
6. Define the initial condition [displacement , velocity] displacement=0, velocity=3
initial condition
theta_0 = [0,3]
7. Define the time points and the no. of values, here we are defining the time intervals between 0-20 and producing 150 values for the animation.
time points
t = np.linspace(0,20,150)
8. Solve the second order differential equation by using odeint command, which is a command in the scipy module.
without importing scipy module this command will not work.
9. Then plot the results by using limit of x and y as well as use the legends.
plt.plot(t,theta[:,0],'-',label=r'$\frac{d\theta_1}{dt}=\theta2$')
plt.plot(t,theta[:,1],'--',label=r'$\frac{d\theta2}{dt} = -\frac{b}{m}\theta_2-\frac{g}{L}sin\theta_1$')
plt.ylabel('plot')
plt.xlabel('time')
plt.legend(loc='best')
plt.show()
10. Define the positions of the pendulum by defining the string starting pint as x0,y0 and x1,y1 values which is calculated in the theory part. Also define the line width as well.
x1=0;y1=0
x2=l*math.sin(i)
y2=-l*math.cos(i)
file='test%05d.png'%ct
plt.figure()
plt.plot([-1.5,1.5],[0,0],linewidth=8)
plt.plot([x1,x2],[y1,y2],linewidth=3)
plt.plot(x2,y2,'o',markersize=15)
plt.xlim([-1.5,1.5])
plt.ylim([-1.5,1.5])
plt.savefig(file)
ct=ct+1
PROGRAM SCREEN SHOT
FIG PROGRAM
# second order differential equaion and pendulum animation renukumar
import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import math
import matplotlib.markers as mk
# 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/1)*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(0,20,150)
#solve ODE
theta=odeint(model,theta_0,t,args=(b,g,l,m))
#plot results
plt.plot(t,theta[:,0],'-',label=r'$\frac{d\theta_1}{dt}=\theta2$')
plt.plot(t,theta[:,1],'--',label=r'$\frac{d\theta2}{dt} = -\frac{b}{m}\theta_2-\frac{g}{L}sin\theta_1$')
plt.ylabel('plot')
plt.xlabel('time')
plt.legend(loc='best')
plt.show()
ct=1
for i in theta[:,0]:
x1=0;y1=0
x2=l*math.sin(i)
y2=-l*math.cos(i)
file='test%05d.png'%ct
plt.figure()
plt.plot([-1.5,1.5],[0,0],linewidth=8)
plt.plot([x1,x2],[y1,y2],linewidth=3)
plt.plot(x2,y2,'o',markersize=15)
plt.xlim([-1.5,1.5])
plt.ylim([-1.5,1.5])
plt.savefig(file)
ct=ct+1GRAPH:
PROCEDURE TO CREATE THE AIMATION:
Microsoft Windows [Version 10.0.18362.1139]
(c) 2019 Microsoft Corporation. All rights reserved.
C:\Users\Meenasri Kalisetty>cd C:\NEW RENU ANIMATION PENDULUM
C:\NEW RENU ANIMATION PENDULUM>dir
Volume in drive C is OS
Volume Serial Number is 4AC9-1ADB
Directory of C:\NEW RENU ANIMATION PENDULUM
28-12-2020 13:12 <DIR> .
28-12-2020 13:12 <DIR> ..
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150 File(s) 1,710,421 bytes
2 Dir(s) 52,108,357,632 bytes free
C:\NEW RENU ANIMATION PENDULUM>magick *.png output.gif
Result:
FIG: ANIMATION
DISCUSSION:
The above animation shows the motion of a simple pendulum with respect to time. The second order differential equation is converted to the first order differential equation and the respected values are defined and the position of the pendulum are also calculated and the motion of the pendulum is animated.
Difficulties faced:
Difficulties faced in syntax error that a bracket is not closed in the odeint command and an error is shown in the another line.
Difficulties in creating an animation that the png which are generated in the system are in a zig zag form when it has invited in the command promt. dir /od can helps the png's to form an asending order in the coomand promt but it is also not worked. Finally the way of saving the png is modified and saved. Where the errror get solved.
LINK: https://drive.google.com/drive/folders/1q5nbx24cfwQPVT1rcV8nxkWRhFJJtxa5?usp=sharing
UTUBE LINK: https://youtu.be/yxAvSkWoDSg