Cas Lab #3

Joshua Cook

This lab was prepared using Pythons Interactive Terminal.

Joshuas-MacBook-Air:Desktop joshuacook$ pythonPython 2.7.5 (default, Aug 25 2013, 00:04:04)[GCC 4.2.1 Compatible Apple LLVM 5.0 (clang-500.0.68)] on darwinType "help", "copyright", "credits" or "license" for more information.

Python Basics

In order to prepare these plots, several critical libraries must be imported. Ihave also added a custom save script to the initial import.

import osimport numpy as npimport matplotlib.pyplot as pltimport scipyimport scipy.special as spimport pylabfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib import cm

Problem 1

define Bessel Function Approximations

def BesselJ(v,x):""" Defines a Bessel Function of the first kind J_v"""return np.sqrt(2/(np.pi*x))*np.cos(x-np.pi/4-v*np.pi/2)

def BesselY(v,x):""" Defines a Bessel Function of the second kind Y_v"""return np.sqrt(2/(np.pi*x))*np.sin(x-np.pi/4-v*np.pi/2)

Create an evenly sampled time vector at 200ms intervals

t = np.arange(0., 100., 0.2)

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Prepare Plots

Plot 1 - First Order

p1,= plt.plot(t, BesselJ(4,t), r--)p2,= plt.plot(t, sp.jn(4,t), gs)plt.title(User generated Bessel functions v Scipy Bessel \n of first order with v = 4)plt.axis([0, 100, -2, 2])plt.legend([p2, p1], ["scipy", "user"])plt.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_1a)

Figure 1: Generated Plot - First Order

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Plot 2 - Second Order

p1,= plt.plot(t, BesselY(4,t), r--)p2,= plt.plot(t, sp.yn(4,t), gs)plt.title(User generated Bessel functions v Scipy Bessel \n of second order with v = 4)plt.axis([0, 100, -2, 2])plt.legend([p2, p1], ["scipy", "user"])plt.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_1b)

Figure 2: Generated Plot - Second Order

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Problem 2

Python was rebooted for this problem.

Joshuas-MacBook-Air:Desktop joshuacook$ pythonPython 2.7.5 (default, Aug 25 2013, 00:04:04)[GCC 4.2.1 Compatible Apple LLVM 5.0 (clang-500.0.68)] on darwinType "help", "copyright", "credits" or "license" for more information.>>>

Import Libraries

import numpy as npfrom scipy import *from scipy.special import jn, jn_zerosimport pylabfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib import cm

Define function for modes of oscillation

def u(m,n,distance,angle,t):""" Defines a user function for a graph of a Bessel function in time, as in say, for a drumhead"""nth_zero = jn_zeros(m,n)return jn(m,distance*nth_zero[n-1])*np.cos(m*angle)*np.cos(t)

Set up spatial increments

theta = r_[0:2*pi:50j]radius = r_[0:1:50j]x = array([r*cos(theta) for r in radius])y = array([r*sin(theta) for r in radius])

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Prepare Plots

Plot 1 - m = 0, n = 1

z = np.array([u(0,1,r,theta,0) for r in radius])fig = pylab.figure()ax = Axes3D(fig)ax.plot_surface(x,y,z,rstride=1,cstride=1,cmap=cm.jet)ax.set_xlabel(X)ax.set_ylabel(Y)ax.set_zlabel(Z)pylab.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_2a)

Figure 3: Generated Plot - m = 0, n = 1

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Plot 2 - m = 0, n = 2

z = np.array([u(0,2,r,theta,0) for r in radius])fig = pylab.figure()ax = Axes3D(fig)ax.plot_surface(x,y,z,rstride=1,cstride=1,cmap=cm.jet)ax.set_xlabel(X)ax.set_ylabel(Y)ax.set_zlabel(Z)fig.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_2b)

Figure 4: Generated Plot - m = 0, n = 2

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Plot 3 - m = 0, n = 3

z = np.array([u(0,3,r,theta,0) for r in radius])fig = pylab.figure()ax = Axes3D(fig)ax.plot_surface(x,y,z,rstride=1,cstride=1,cmap=cm.jet)ax.set_xlabel(X)ax.set_ylabel(Y)ax.set_zlabel(Z)fig.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_2c)

Figure 5: Generated Plot - m = 0, n = 3

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Plot 4 - m = 3, n = 1

z = np.array([u(3,1,r,theta,0) for r in radius])fig = pylab.figure()ax = Axes3D(fig)ax.plot_surface(x,y,z,rstride=1,cstride=1,cmap=cm.jet)ax.set_xlabel(X)ax.set_ylabel(Y)ax.set_zlabel(Z)fig.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_2d)

Figure 6: Generated Plot - m = 3, n = 1

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Plot 5 - m = 3, n = 2

z = np.array([u(3,2,r,theta,0) for r in radius])fig = pylab.figure()ax = Axes3D(fig)ax.plot_surface(x,y,z,rstride=1,cstride=1,cmap=cm.jet)ax.set_xlabel(X)ax.set_ylabel(Y)ax.set_zlabel(Z)fig.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_2e)

Figure 7: Generated Plot - m = 3, n = 2

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Problem 3

Here is a plot of a drumhead with m = 5 and n = 4

z = np.array([u(5,4,r,theta,0) for r in radius])fig = pylab.figure()ax = Axes3D(fig)ax.plot_surface(x,y,z,rstride=1,cstride=1,cmap=cm.jet)ax.set_xlabel(X)ax.set_ylabel(Y)ax.set_zlabel(Z)fig.savefig(/Users/joshuacook/Desktop/cas_lab_3_plot_3)

Figure 8: Generated Plot - m = 5, n = 4

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Number of Nodal Curves

Four circular nodal curves can be seen.

Figure 9: Generated Plot - m = 5, n = 4, alternate angle

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Cas Lab #3Joshua CookPython BasicsProblem 1define Bessel Function ApproximationsCreate an evenly sampled time vector at 200ms intervalsPrepare Plots

Problem 2Import LibrariesDefine function for modes of oscillationSet up spatial incrementsPrepare Plots

Problem 3Number of Nodal Curves