Week 3 - Solving second order ODEs

AIM:

to write a program to simulate the transient behaviour of a simple pendulam and creat an animation of its motion

PROCEDURE:

1. importing math ,matplot, numpy

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

#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

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

# initial condition
theta_0 = [0,3]

# time points
t = np.linspace(0,20,200)

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

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

# making animation
THETA3 = []
for theta3 in theta[:,0]:
	THETA3.append(theta3)

ct = 1

x0 = 0
y0 = 0

for THETA4 in THETA3:
	x1 = l*math.sin(THETA4)
	y1 = -l*math.cos(THETA4)
	filename= str(ct) +'.png' 
	ct = ct+1

	plt.figure()
	plt.plot([-1.5,1.5],[0,0],linewidth=8)
	plt.plot([x0,x1],[y0,y1],linewidth=3)
	plt.plot(x1,y1,'o',markersize = 15) 
	plt.xlim([-1.5,1.5])
	plt.ylim([-1.5,1])
	plt.savefig(filename)

2. input values are,

damping coefficient,

b = 0.05

gravity,

g = 9.81

length of pendulam ,

l = 1

mass of ball,

m = 1

3. defining the function ,

def model(theta,t,b,g,l,m)

when the second oder equation can split into 2 parts ,

dtheta1_dt = theta2
dtheta2_dt = -(b/m)*theta2 - (g/l)*math.sin(theta1)

4. at the initial condition theta_0 can change from 0 to 3

- considering the time is 0 to 20 sec ,an d time can split into equal 200 parts using np.linspace

5. when solving the ode to calculate the values for theta by using odeint function,

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

6.ploting the plots by using the values in displacement and time

the output plot is,

7. creating th animation when considering theta3 and theta4

-for creating the loop for theta3 at theta[:,0]

- making another loop in theta3 by theta4

-at initialy x0,y0 is zero

- at perticular point in

x1 = l*math.sin(THETA4)
y1 = -l*math.cos(THETA4)

- making each plots seperatly by using plt.plot command

8. each plots can covert into a single folder

- command prompt is used to make a animation of all plot

the animation link is;

https://drive.google.com/file/d/1X1PcZmNj4O19sSIYnppSpAHlirGPVOsS/view?usp=sharing

CONCLUSION:

can be write a program to simulate the simple pendulam and create an animation of pendulam