Simulation of Transient behaviour of a Simple Pendulum

OBJECTIVE – To write a program to simulate the transient behaviour of a simple pendulum and to create an animation of its motion.

 

PROCEDURE –

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 the Newtons second law. When this law is written down, we get a second order Ordinary Differential Equation that describes the position of the "ball" w.r.t time. 

This ODE represents the equation of motion of a simple pendulum with damping

ASSUPTIONS-

In the above equation,

g = gravity in m/s2,

L = length of the pendulum in m,

m = mass of the ball in kg,

b=damping coefficient.

Assumptions made for calculation purposes,

L=1 metre,

m=1 kg,

b=0.05.

g=9.81 m/s2.

 

PROGRAM AND EXPLANATION –

  • The required modules like numpy, matplotlib.pyplot and scipy modules are imported.
  • The governing equation of motion for the simple pendulum are utilized to create the state variable and defined under a function.
  • Input variables like length of string, damping coefficient ,acceleration due to gravity, mass of ball are initiated.
  • The initial conditioned and time span are defined.
  • Odeint command is called and inputs are specified.
  • A plot of angular displacement and angular velocity are created as functions of time.
  • For loop which iterates through the values of angular displacement and angular velocity is defined.
  • Vector decomposition is used to determine the location of the moving pendulum.
  • Counter is setup in the loop,a filename is devised to take on the name of the count for ever subsequent o=plot of the pendulum position that is generated.
  • The plt.savefig() command is used to save the plots in series under the filename and corresponding count.
  • The plot images are stitched together using ImageMagick to generate an animation of the simple pendulum.
#code for simulation of a simple pendulum

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

def pendulum(theta, t, b, g, l, m):
			theta2 = theta[0]
			omega =theta[1]
			theta_dot = [omega, -(b/m)*omega -(g/l)*math.sin(theta2)]
			return(theta_dot)


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

IC = [0,3]
 
t = np.linspace(0,20,250)

theta = odeint(pendulum,IC,t,args =(b,g,l,m))
theta1 = theta[:,0]
omega1 = theta[:,1]

plt.plot(t,theta1,'b-',Label='Angular Displacement')
plt.plot(t,omega1,'r--',Label='Angular Velocity')
plt.ylabel('Anugular Variables')
plt.xlabel('Time')
plt.legend(loc='best')
plt.show()

ct = 1
for angle in theta1:
 			x0 = 0
 			y0 = 0
 			x1 = l*math.sin(angle)
 			y1 = -l*math.cos(angle)
 			filename = 'swing%05d.png'%ct
 			ct = ct + 1
 			#plots
 			plt.figure()
 			plt.plot([-1.5,-1.5],[0,0],linewidth = 7)
 			plt.plot([x0,x1],[y0,y1],linewidth = 1)
 			plt.plot(x1,y1,'o',markersize = 20)
 			plt.xlim([-2,2])
 			plt.ylim([-1.5,1.5])
 			plt.show()
 			plt.savefig(filename)

OUTPUT GRAPH-

 

The youtube link to the animation file is given below,

https://www.youtube.com/watch?v=mh7sjryn_vU&feature=youtu.be

 

 

CONCLUSION – Thus the program is written to simulate the transient behaviour of a simple pendulum and animation file is created.


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