Python -Transient behavior of a simple pendulum

Transient behaviour of a simple pendulum

Objective

To write a program in python to simulate the transient behaviour of a simple pendulum and to create an animation of it's motion using the ODE for the motion of a pendulum.

Procedure

Formulae:

g = gravity in m/s^2

L = length of the pendulum in m

m = mass of the ball in kg

b=damping coefficient

Values used in the program

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s^2.

By importing the odeint module from scipy.integrate library we are able to calculate the complex differential equations using python. However the ODE for a simple pendulum is of the second order and second order ODE must be simplified into the first order form before computing it in python

`theta = theta1`

`d theta/ dt = d theta1/dt = theta2` Eq.1

`d^2 theta/dt^2 = d theta2/dt`

`d theta2/dt = -(b/m) theta 2 - (g/L)sintheta1`` Eq.2

`theta 1` = angular displacement , `theta2` = angular velocity

We then use the main code along with conditions,

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

theta_0 = [0,3]

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

where ,

model: is the function containing Eq.1 and Eq.2 and will define the ODE with theta1 in the 0th column and theta2 in the 1st column

theta_0': this array is used to denote the initial conditions at t=0 to pass on to the function model. in this case we have given zero angular displacement and 3m/s angulat velocity

t: is used to pass the time conditions , in this particular program we are using t =0 to t=20 secs with 150 intervals between them

Python program to simulate the motion of a pendulum:

"""
Simple Pendulum ODE
""" 


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


b = 0.05
g = 9.81
l = 1 #metre
m = 1 #kg

#initial condition
theta_0 = [0,3]


#timepoints

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

# function returning 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



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


x0 = 0
y0 = 0
ct = 1 

for i in range(100):
	x1 = l*math.sin(theta[i,0])
	y1 = -l*math.cos(theta[i,0])

	plt.figure()
	plt.plot([-1,1],[0,0], color = 'red', linewidth = 1 )
	plt.scatter(x1,y1,color = 'blue', s = 300)
	plt.plot([x0,x1],[y0,y1], color = 'black')

	plt.xlim([-2,2])
	plt.ylim([-2,2])
	filename = 'test%05d.png'%ct
	plt.savefig(filename)
	ct = ct + 1 

Simulating motion of the pendulum:

Initiate conditions for the pivot point at co-ordinates (0,0)

x0 = 0

y0 = 0

we use a for loop and increment for a certain value 'i' to get the position of the pendulum for each ofthe values stored in the first column which contains all the angular displacement values

The co-ordinates of the bob are ,

`x1 = l*sintheta`

`y1 = l*costheta`

Fig: Image from the simulation

Conclusion

The ODE for the motion of pendulum was successfully made into a python program and the animation images for the motion were obtained.