New graphs.ipynb

Kernel: Python 3 (system-wide)

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

M=1
m=1
g=9.81

def Atwood(y,t,M,m,g):
    
    #theta'(t)=omega(t)
    #radius'(t)=Rho(t)
    
    #omega'(t)=(1/radius(t))(-g*sin(theta(t))-2(Rho(t)*omega(t))
    #Rho'(t)=(1/(M+m))(m*radius(t)*(omega(t)^2)-M*g+m*g*cos(theta(t)))
    
    theta, omega, radius, Rho = y
    
    #dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)),Rho, (1/(M+m))*(m*radius*(omega^2)-M*g+m*g*(np.cos(theta)))]
    dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)), Rho, (1/(M+m))*(m*radius*(omega**2)-M*g+m*g*(np.cos(theta)))]
    
    return dydt
                           
#y0=[theta, change in theta, radius, change in radius]
y0 = [np.pi/2,0 ,1 ,0]
#time points
t = np.linspace(0,100,1000000)
#Solving the ODE
sul = odeint(Atwood, y0, t, args=(M,m,g))


plt.plot(sul[:,0], sul[:,1], 'r')
plt.ylabel('Angular velocity')
plt.xlabel('Angular displacement')
plt.title('Fig 5.1 - Phase Plot')
plt.grid()
plt.show()

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

M=6
m=1
g=9.81

def Atwood(y,t,M,m,g):
    
    #theta'(t)=omega(t)
    #radius'(t)=Rho(t)
    
    #omega'(t)=(1/radius(t))(-g*sin(theta(t))-2(Rho(t)*omega(t))
    #Rho'(t)=(1/(M+m))(m*radius(t)*(omega(t)^2)-M*g+m*g*cos(theta(t)))
    
    theta, omega, radius, Rho = y
    
    #dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)),Rho, (1/(M+m))*(m*radius*(omega^2)-M*g+m*g*(np.cos(theta)))]
    dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)), Rho, (1/(M+m))*(m*radius*(omega**2)-M*g+m*g*(np.cos(theta)))]
    
    return dydt
                           
#y0=[theta, change in theta, radius, change in radius]
y0 = [np.pi/2,0 ,1 ,0]
#time points
t = np.linspace(0,100,1000000)
#Solving the ODE
sul = odeint(Atwood, y0, t, args=(M,m,g))


plt.plot(sul[:,0], sul[:,1], 'r')
plt.ylabel('Angular velocity')
plt.xlabel('Angular displacement')
plt.title('Fig 5.2 - Phase Plot')
plt.grid()
plt.show()

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

M=19
m=1
g=9.81

def Atwood(y,t,M,m,g):
    
    #theta'(t)=omega(t)
    #radius'(t)=Rho(t)
    
    #omega'(t)=(1/radius(t))(-g*sin(theta(t))-2(Rho(t)*omega(t))
    #Rho'(t)=(1/(M+m))(m*radius(t)*(omega(t)^2)-M*g+m*g*cos(theta(t)))
    
    theta, omega, radius, Rho = y
    
    #dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)),Rho, (1/(M+m))*(m*radius*(omega^2)-M*g+m*g*(np.cos(theta)))]
    dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)), Rho, (1/(M+m))*(m*radius*(omega**2)-M*g+m*g*(np.cos(theta)))]
    
    return dydt
                           
#y0=[theta, change in theta, radius, change in radius]
y0 = [np.pi/2,0 ,1 ,0]
#time points
t = np.linspace(0,10,10000)
#Solving the ODE
sul = odeint(Atwood, y0, t, args=(M,m,g))


plt.plot(sul[:,0], sul[:,1], 'r')
plt.ylabel('Angular velocity')
plt.xlabel('Angular displacement')
plt.title('Fig 5.3 - Phase Plot')
plt.grid()
plt.show()

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

M=9
m=1
g=9.81

def Atwood(y,t,M,m,g):
    theta, omega, radius, Rho = y
    dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)), Rho, (1/(M+m))*(m*radius*(omega**2)-M*g+m*g*(np.cos(theta)))]
    return dydt
    
A=12
a=1
g=9.81

def Chaotic(y,t,A,a,g):
    theta, omega, radius, Rho = y
    dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)), Rho, (1/(A+a))*(a*radius*(omega**2)-A*g+a*g*(np.cos(theta)))]
    return dydt

B=14
b=1
g=9.81

def Chaotic2(y,t,B,b,g):
    theta, omega, radius, Rho = y
    dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)), Rho, (1/(B+b))*(a*radius*(omega**2)-B*g+b*g*(np.cos(theta)))]
    return dydt
    
y0 = [np.pi/2 ,0 ,1 ,0]
t = np.linspace(0,100,1000000)
sul = odeint(Atwood, y0, t, args=(M,m,g))
chaos=odeint(Chaotic, y0, t, args=(A,a,g))
chaos2=odeint(Chaotic2, y0, t, args=(B,b,g))

plt.plot(sul[:,0], sul[:,1],'g')
plt.plot(chaos[:,0],chaos[:,1],'b')
plt.plot(chaos2[:,0],chaos2[:,1],'y')
plt.ylabel('Angular velocity')
plt.xlabel('Angular displacement')
plt.title('Fig 5.5 - Phase Plot of different systems showing Chaotic Motion')
plt.grid()
plt.show()

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

M=2.4
m=1
g=9.81

def Atwood(y,t,M,m,g):

    theta, omega, radius, Rho = y
    
    #dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)),Rho, (1/(M+m))*(m*radius*(omega^2)-M*g+m*g*(np.cos(theta)))]
    dydt = [omega, (1/radius)*(-g*(np.sin(theta))-2*(Rho*omega)), Rho, (1/(M+m))*(m*radius*(omega**2)-M*g+m*g*(np.cos(theta)))]
    
    return dydt
                           
#y0=[theta, change in theta, radius, change in radius]
y0 = [np.pi/2,0 ,1,0]
#time points
t = np.linspace(0,100,1000000)
#Solving the ODE
sul = odeint(Atwood, y0, t, args=(M,m,g))


plt.plot(sul[:,0], sul[:,1], 'r')
plt.ylabel('Angular velocity')
plt.xlabel('Angular displacement')
plt.title('Fig 5.1 - Phase Plot')
plt.grid()
plt.show()