diff --git a/Finalproject_EnriqueLopez/finalproject_enriquelopez_reviewed.ipynb b/Finalproject_EnriqueLopez/finalproject_enriquelopez_reviewed.ipynb
new file mode 100644
index 0000000..6fb4798
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+++ b/Finalproject_EnriqueLopez/finalproject_enriquelopez_reviewed.ipynb
@@ -0,0 +1,1353 @@
+{
+ "metadata": {
+  "name": "finalproject_enriquelopez_reviewed.ipynb"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "heading",
+     "level": 6,
+     "metadata": {},
+     "source": [
+      "Content under Creative Commons Attribution license CC-BY 4.0 version, code under MIT license (c) 2014 by Enrique A. L\u00f3pez-Guerra."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 1,
+     "metadata": {},
+     "source": [
+      "Dynamics of a tapping probe that aims at seeing atoms"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Purpose of the notebook: show an application of numerical methods to simulate the dynamics of a probe in atomic force microscopy.\n",
+      "\n",
+      "Requirements to take the best advantage of this notebook: knowing the fundamentals of  [Harmonic Oscillators](http://en.wikipedia.org/wiki/Harmonic_oscillsator) in clasical mechanics and [Fundamentals of Vibrations](http://en.wikipedia.org/wiki/Vibration). "
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Introduction"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Since the atomic force microscope (AFM) was invented in 1986 it has become one of the main tools to study matter at the micro and nanoscale. This powerful tool is so versatile that it can be used to study a wide variety of materials, ranging from stiff inorganic surfaces to soft biological samples. \n",
+      "\n",
+      "In its early stages the AFM was used in permanent contact with the sample (the probe is dragged over the sample during the whole operation), which brought about important drawbacks, such as rapid probe wear and often sample damage, but these obstacles have been overcome with the development of dynamic techniques.\n",
+      "\n",
+      "In this Ipython notebook, we will focus on the operation of the probe in dynamic mode which is the most frequently used nowadays due to its advantages over the permanent contact techniques."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image;\n",
+      "Image(filename='C:/Users/enrique/Github Repositories/FinalProjectMAE6286_2/FinalProjectMAE6286/Fig1.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
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2pbwLKgdnPZQl3IfoGrD13XfFX/BX34pyaNos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\nJX5taP8A8Fs/iL+zffx6Z+0N8D/Enh1UQD+2NJiLQykBdxKvhM5ZSQr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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 38,
+       "text": [
+        "<IPython.core.display.Image at 0x246a28d0>"
+       ]
+      }
+     ],
+     "prompt_number": 38
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Figure 1.   Schematics of the setup of a atomic force microscope (Adapted from reference 6)"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In AFM the interacting probe is in general a rectangular cantilever (please check the image above that shows the AFM setup where you will be able to see the probe!). \n",
+      "Probably the most used dynamic technique in AFM is the Tapping Mode. In this method the probe taps a surface in intermittent contact fashion. The purpose of tapping the probe over the surface instead of dragging it is to reduce frictional forces that may cause damage of soft samples and wear of the tip. Besides with the tapping mode we can get more information about the sample!  HOW???"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In Tapping Mode AFM the cantilever is shaken to oscillate up and down at a specific frequency (most of the time shaken at its natural frequency). Then the deflection of the tip is measured at that frequency to get information about the sample. Besides acquiring the topography of the sample, the phase lag between the excitation and the response of the cantilever can be related to compositional material properties! \n",
+      "In other words one can simultaneously get information about how the surface looks and also get compositional mapping of the surface!  THAT SOUNDS POWERFUL!!!"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image;\n",
+      "Image(filename=\"C:/Users/enrique/Github Repositories/FinalProjectMAE6286_2/FinalProjectMAE6286/Fig2DHO.jpg\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 35,
+       "text": [
+        "<IPython.core.display.Image at 0x246a2ef0>"
+       ]
+      }
+     ],
+     "prompt_number": 35
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Figure 2. Schematics of a damped harmonic oscillator without tip-sample interactions"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Analytical Solution"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The motion of the probe can be derived using [Euler-Bernoulli's](http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory) equation. However that equation has partial derivatives (it depends on time and space) because it deals with finding the position of each point of the beam in a certain time, which cant make the problem too expensive computationally for our purposes. In our case, we have the advantage that we are only concerned about the position of the tip (which is the only part of the probe that will interact with the sample). As a consequence many researchers in AFM have successfully made approximations using a simple mass point model approximation [see ref. 2] like the one in figure 2 (with of course the addition of tip sample forces!  We will see more about this later).\n",
+      "\n",
+      "First we will study the system of figure 2 AS IS (without addition of tip-sample force term), WHY?  Because we want to get an analytical solution to get a reference of how our integration schemes are working, and the addition of tip sample forces to our equation will prevent the acquisition of straightforward analytical solutions :(\n",
+      "\n",
+      "Then, the equation of motion of the damped harmonic oscillator of figure 2, which is DRIVEN COSINUSOIDALLY (remember that we are exciting our probe during the scanning process) is:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\begin{equation}\n",
+      "m \\frac{d^2z}{dt^2} = - k z - \\frac{m\\omega_0}{Q}\\frac{dz}{dt} + F_0\\cos(\\omega t)\n",
+      "\\end{equation}$$\n",
+      "where k is the stiffness of the cantilever, z is the vertical position of the tip with respect to the cantilever base position, Q is the quality factor (which is related to the damping of the system), $F_0$ is the driving force amplitude, $\\omega_0$ is the resonance frequency of the oscillator, and $\\omega$ is the frequency of the oscillating force.\n",
+      "\n",
+      "The analytical solution of the above ODE is composed by a transient term and a steady state term. We are only interested in the steady state part because during the scanning process it is assumed that the probe has achieved that state.\n",
+      "\n",
+      "The steady state solution is given by:\n",
+      "$$\\begin{equation}\n",
+      "A\\cos (\\omega t - \\phi)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "where A is the steady state amplitude of the oscillation response, which depends on the cantilever parameters and the driving parameters, as can be seen in the following relation:\n",
+      "$$\\begin{equation}\n",
+      "A = \\frac{F_0/m}{\\sqrt{(\\omega_0^2-\\omega^2)^2+(\\frac{\\omega\\omega_0}{Q})^2}}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "and $\\phi$ is given by:\n",
+      "$$\\begin{equation}\n",
+      "\\phi = \\arctan \\big( \\frac{\\omega\\omega_0/Q}{\\omega_0^2 - \\omega^2} \\big)\n",
+      "\\end{equation}$$\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Let's first name the variables that we are going to use. Because we are dealing with a damped harmonic oscillator model we have to include variables suche as: spring stiffness, resonance frequency, quality factor (related to damping coefficient), target oscillation amplitude, etc."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy\n",
+      "import matplotlib.pyplot as plt;\n",
+      "\n",
+      "k = 10.\n",
+      "fo = 45000\n",
+      "wo = 2.0*numpy.pi*fo\n",
+      "Q = 25.\n",
+      "\n",
+      "period = 1./fo\n",
+      "m = k/(wo**2)\n",
+      "Ao = 60.e-9\n",
+      "Fd = k*Ao/Q\n",
+      "\n",
+      "spp = 28. # time steps per period \n",
+      "dt = period/spp #Intentionally chosen to be quite big\n",
+      "#you can decrease dt by increasing the number of steps per period\n",
+      "\n",
+      "simultime = 100.*period\n",
+      "N = int(simultime/dt)\n",
+      "\n",
+      "#Analytical solution\n",
+      "\n",
+      "time_an = numpy.linspace(0,simultime,N)  #time array for the analytical solution\n",
+      "z_an = numpy.zeros(N)  #position array for the analytical solution\n",
+      "\n",
+      "#Driving force amplitude this gives us 60nm of amp response (A_target*k/Q)\n",
+      "Fo_an = 24.0e-9  \n",
+      "\n",
+      "A_an = Fo_an*Q/k  #when driven at resonance A is simply Fo*Q/k\n",
+      "phi = numpy.pi/2  #when driven at resonance the phase is pi/2\n",
+      "\n",
+      "z_an[:] = A_an*numpy.cos(wo*time_an[:] - phi) #this gets the analytical solution\n",
+      "\n",
+      "#slicing the array to include only steady state (only the last 10 periods)\n",
+      "z_an_steady = z_an[(90.*period/dt):]\n",
+      "time_an_steady = time_an[(90.*period/dt):]\n",
+      "\n",
+      "plt.title('Plot 1    Analytical Steady State Solution of Eq 1', fontsize=20);\n",
+      "plt.xlabel('time, ms', fontsize=18);\n",
+      "plt.ylabel('z_Analytical, nm', fontsize=18);\n",
+      "plt.plot(time_an_steady*1e3, z_an_steady*1e9, 'b--');\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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bPHtG8M8/8j1rYSHB5csETZsS7Nsnfv6m2mLmTIJx4+R7TjV8tMUOf156oQ0A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\nkJAgXLuZq0Cx5jYGgKNH2d+bn09dmQ4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ZatfmV7YSREZSBUc+QluJCWheHpCYCAQE8CtX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\nBEFYCaS0CYIgCMJK+H8LH6zXOrv8OwAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x897a1d0>"
+       ]
+      }
+     ],
+     "prompt_number": 26
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Approximating through Euler's method"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This method was already discussed in the second notebook called \"Pughoid Oscillation\" located on the second IPython Notebook of the series _\"The phugoid model of glider flight\"_, the first learning module of the course [**\"Practical Numerical Methods with Python.\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about).\n",
+      "We will review it briefly since the intention is to make this notebook self-contained. Let's begin...\n",
+      "\n",
+      "If we perform a Taylor series expansion of $z_{n+1}$ around $z_{n}$ we get:\n",
+      "\n",
+      "$$z_{n+1} = z_{n} + \\Delta t\\frac{dz}{dt}\\big|_n + {\\mathcal O}(\\Delta t^2)$$\n",
+      "\n",
+      "The Euler formula neglects terms in the order of two or higher, ending up as:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n+1} = z_{n} + \\Delta t\\frac{dz}{dt}\\big|_n\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "It can be easily seen that the truncation error of the Euler algorithm is in the order of ${\\mathcal O}(\\Delta t^2)$.\n",
+      "\n",
+      "\n",
+      "\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This is a second order ODE, but we can convert it to a system of two coupled 1st order differential equations. To do it we will define $\\frac{dz}{dt} = v$. Then equation (1) will be decomposed as:\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dz}{dt} = v\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dv}{dt} = -kz-\\frac{m\\omega_0}{Q}+F_o\\cos(\\omega t)\n",
+      "\\end{equation}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "These coupled equations will be used during Euler's aproximation and also during our integration using Runge Kutta 4 method."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "t= numpy.linspace(0,simultime,N) #time grid for Euler method\n",
+      "    \n",
+      "#Initializing variables for Euler\n",
+      "vdot_E = numpy.zeros(N)\n",
+      "v_E = numpy.zeros(N)\n",
+      "z_E = numpy.zeros(N)\n",
+      "\n",
+      "#Initial conditions\n",
+      "z_E[0]= 0.0\n",
+      "v_E[0]=0.0\n",
+      "\n",
+      "for i in range (N-1):\n",
+      "    vdot_E[i] =( ( -k*z_E[i] - (m*wo/Q)*(v_E[i])   +\\\n",
+      "                    Fd*numpy.cos(wo*t[i]) ) / m)  #Equation 7\n",
+      "    v_E[i+1] = v_E[i] + dt*vdot_E[i]  #Based on equation 5\n",
+      "    z_E[i+1] = z_E[i] + v_E[i]*dt  #Equation 5\n",
+      "\n",
+      "plt.title('Plot 2  Eulers approximation of Equation1', fontsize=20);        \n",
+      "plt.plot(t*1e3,z_E*1e9);\n",
+      "plt.xlabel('time, s', fontsize=18);\n",
+      "plt.ylabel('z_Euler, nm', fontsize=18);\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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mJa3Ro0cjNTUVY8eORYcOHeDp6alUXERENkVbXbEjhrLMSlpJSUkYM2YMZs2a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\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x8994908>"
+       ]
+      }
+     ],
+     "prompt_number": 27
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This looks totally unphysical! We were expecting to have a steady state oscillation of 60 nm and we got a huge oscillation that keeps growing. Can it be due to the scheme? The timestep that we have chosen is quite big with respect to the oscillation period. We have intentionally set it to ONLY 28 time steps per period (That could be the reason why the scheme can't capture the physics of the problem). That's quite discouraging. However the timestep is quite big and it really gets better as you decrease the time step. Try it! Reduce the time step and see how the numerical solution acquires an amplitude of 60 nm as the analytical one. At this point we can't state anything about accuracy before doing an analysis of error (we will make this soon). But first, let's try to analyze if another more efficient scheme can capture the physics of our damped harmonic oscillator even with this large time step."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Let's try to get more accurate... Verlet Algorithm"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This is a very popular algorithm widely used in molecular dynamics simulations. Its popularity has been related to high stability when compared to the simple Euler method, it is also very simple to implement and accurate as we will see soon!  Verlet integration can be seen as using the central difference approximation to the second derivative. Consider the Taylor expansion of $z_{n+1}$ and $z_{n-1}$ around $z_n$:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n+1} = z_n + \\Delta t \\frac{dz}{dt}\\big|_n + \\frac{\\Delta t^2}{2} \\frac{d^2 z}{d t^2}\\big|_n + \\frac{\\Delta t^3}{6} \\frac{d^3 z}{d t^3}\\big|_n + {\\mathcal O}(\\Delta t^4)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n-1} = z_n - \\Delta t \\frac{dz}{dt}\\big|_n + \\frac{\\Delta t^2}{2} \\frac{d^2 z}{dt^2}\\big|_n - \\frac{\\Delta t^3}{6} \n",
+      "\\frac{d^3 z}{d t^3}\\big|_n + {\\mathcal O}(\\Delta t^4)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "Adding up these two expansions and solving for $z_{n+1}$ we get:\n",
+      "\n",
+      "$$z_{n+1}= 2z_{n} - z_{n-1} + \\frac{d^2 z}{d t^2} \\Delta t^2\\big|_n + {\\mathcal O}(\\Delta t^4) $$\n",
+      "\n",
+      "Verlet algorithm neglects terms on the order of 4 or higher, ending up with:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n+1}= 2z_{n} - z_{n-1} + \\frac{d^2 z}{d t^2} \\Delta t^2\\big|_n\n",
+      "\\end{equation}$$\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This looks nice; it seems that the straightforward calculation of the second derivative will give us good results. BUT have you seen that we also need the value of the first derivative (velocity) to put it into the equation of motion that we are integrating (see equation 1). YES, that's a main drawback of this scheme and therefore it's mainly used in applications where the equation to be integrated doesn't have first derivative. But don't panic we will see what can we do...\n",
+      "\n",
+      "What about subtracting equations 8 and 9 and then solving for $\\frac{dz}{dt}\\big|_n$:\n",
+      "$$\n",
+      "\\frac{dz}{dt}\\big|_n =  \\frac{z_{n+1} - z_{n-1}}{2\\Delta t} + {\\mathcal O}(\\Delta t^2)\n",
+      "$$\n",
+      "If we neglect terms on the order of 2 or higher we can calculate velocity:\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dz}{dt}\\big|_n =  \\frac{z_{n+1} - z_{n-1}}{2\\Delta t}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "This way of calculating velocity is pretty common in Verlet integration in applications where velocity is not explicit in the equation of motion. However for our purposes of solving equation 1 (where first derivative is explicitly present) it seems that we will lose accuracy because of the velocity, we will discuss more about this soon after..."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Have you noticed that we need a value $z_{n-1}$? Does it sound familiar?  YES!  This is not a self-starting method. As a result we will have to overcome the issue by setting the initial conditions of the first step using Euler approximation. This is a bit annoying, but a couple of extra lines of code won't kill you :)"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "time_V = numpy.linspace(0,simultime,N)\n",
+      "\n",
+      "#Initializing variables for Verlet\n",
+      "zdoubledot_V = numpy.zeros(N)\n",
+      "zdot_V = numpy.zeros(N)\n",
+      "z_V = numpy.zeros(N)\n",
+      "\n",
+      "#Initial conditions Verlet. Look how we use Euler for the first step approximation!\n",
+      "z_V[0] = 0.0\n",
+      "zdot_V[0] = 0.0\n",
+      "zdoubledot_V[0] = (   ( -k*z_V[0] - (m*wo/Q)*zdot_V[0] +\\\n",
+      "                            Fd*numpy.cos(wo*t[0]) )  ) / m\n",
+      "zdot_V[1] = zdot_V[0] + zdoubledot_V[0]*dt\n",
+      "z_V[1] = z_V[0] + zdot_V[0]*dt\n",
+      "zdoubledot_V[1] = (   ( -k*z_V[1] - (m*wo/Q)*zdot_V[1] +\\\n",
+      "                            Fd*numpy.cos(wo*t[1]) )  ) / m\n",
+      "\n",
+      "#VERLET ALGORITHM\n",
+      "\n",
+      "for i in range(2,N):\n",
+      "    z_V[i] = 2*z_V[i-1] - z_V[i-2] + zdoubledot_V[i-1]*dt**2 #Eq 10\n",
+      "    zdot_V[i] = (z_V[i]-z_V[i-2])/(2.0*dt) #Eq 11\n",
+      "    zdoubledot_V[i] = (   ( -k*z_V[i] - (m*wo/Q)*zdot_V[i] +\\\n",
+      "                               Fd*numpy.cos(wo*t[i]) )  ) / m #from eq 1\n",
+      "    \n",
+      "plt.title('Plot 3  Verlet approximation of Equation1', fontsize=20); \n",
+      "plt.xlabel('time, ms', fontsize=18);\n",
+      "plt.ylabel('z_Verlet, nm', fontsize=18);\n",
+      "plt.plot(time_V*1e3, z_V*1e9, 'g-');\n",
+      "plt.ylim(-65,65);\n",
+      " \n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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7vfLwt4Z75WtfvdYrX/LiJV75/OfO98pnTzrbK5/+l9O9cqcJnbzysb8/1isf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\nOPHEE9GhQwc88MADbJ0t5Vuwp3IPWjRo4aXfNyxs6H1rkU2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WqMn9kacCkkNLYp2WthaS\nROQkHX8XDjYUJV+LqhN3LUeqT6ktk3XOSAjRJHW/kGQSR9rZvoDnpupvas5Q/S3OTzkCIn9XDhtb\nJINYRDZixAg4jhMYIAo9Oe7ree1p9gBCRFaSKiGTPfx1MS6Dkf1xTCZhQ/VjmuL7X06KIBQuJEit\nr1GTzzC0KHuj8qT0jV0khJRnaBEAa/hkIygfp8Je/jk5Y04ZNZnMI8ROZHeKZfI8FIlIISf/hXXq\nWtx12TqgQ12R60qqQDxnSG0zafk6JUURIqe28k0mCZFRQs+NcjRNQpE6EpTPmXWyoXFhkQxiEVmx\nb/XvKzIAyLiZ9t3sW0+gLFNGJnu4jht9dyxmmZscfvKG+/nGwtlUNHmD+v0yjig5Ax35rkdMSk1o\nUbxuZL3ENDzIKDWgsE2DA6Pp8S8+UyQihoSUaoVQc5yikRVf5Dy5tmB8kU4E9G0zqmMQYpMdHznx\nQ3YiSBVuqqQUddhQpEEySSH3re1Lcc1LF4qU6rR64SWBlJsKEahIfBbJoFcFaR04EUV2ou3zrEUp\n2UMktbw2BwaxzqVQamLISSQ70ssT4/3y2ow0KUMqjAgtcgaaW3OQ1ZZMKKrwoOpdMJXh0xnE0MI/\nYcgi98oRnEKtRLxwou9lhyI4DwjjGEdJmdQh2hxZmyGuG3JYhDqc2mLXrYhEERO1RalqZVo+pEzF\nPO878kyYsLuyDtF/8nXJ8LDQHotkkFdPbt68GUuWLMEtt9yC//7v/wYAHDt2DK+88gqOHDmSaAOT\nRDaVRdptF6EpNxX82q1MXpH3yJiySbKHMvHDISaN7P0JXiEZDiFCguTk49SWcFw1QeXJx4UW2fAg\n6CQQE9Xmn4cziCEDpzBqsnrivHYjL5xYF5PrUM8z5CzIDohOUZjUYdovOymcohBVhMpx4JRUSFXl\no7YUySR+2FgmBSq0yN23rIxkwgqNEV0okltrJdatI6T2eXsskkEsImtra8PcuXMxdepUrFixAmvW\nrMG+ffsAAKlUCrNmzcLKlSs7paFJQRw8Iqn5isx13A5S81PxhTBjaN9FwySQiPIiyEKuwymHCAlS\nxpogR061hUjQ4Q2iaJSBjsSPQkOLXMJGyNhBIi/CIAL0Lz6ryIgjakr16J4bZ7xIT12so3AcQgbX\nwPuP037ynIJR1qXi56OkdHXk5y86RMHrFJJzxjmOMrnIJBIhLMJZVI0j8bsuouekCJRqm0UyiEVk\nDz30EH72s5/hkUceCX5U00efPn3wla98Bc8//3zijUwKn7Z+GpRzXi4gMjFr0U/wCK2XKV6IDoUB\nmbI8yeQy+ztl0pqailzIiagJLWont6q+uF4WJzwokZTK2Pll+Zyh406YyHSGLEI0cb12j1YCJKlB\nEa7SKCmWKAtVHcwaGaU6gGimavB8TJRUHqn7fpSEcohk1SsSMufYmYYWI4RPhBZ1ToFRaJGIblgU\njlg9+eSTT+LGG2/EokWLMHjw4MjnY8aMwXvvvZdY45LEX47/S3xjwjdCxwJFRoUWc3zWol9H3q4q\n5bQv6srH/UnGpeuTHp88UaiYPTEpZQOtCy3Kk1vn/cteZ5z0e9OEDSq0qErpVxkUFSlQhkkmdk4l\nUY6Jto4cDlMoqVCIiiBKTkmZtp9rJ6U6ROfCJxojJaVwZII6jFPDOUTkuFckclBOoSpMG+mzfJ0C\nwmmyocXOQ6ysxd27d+Puu+9mPx8wYECPXSP78cwfh/72PC8SWvQnkm69rCxTFpCRSEwRgjAgBcrj\nc8EYXCJm35JrCU0y0tNUhBZ1qoMlWS/8S8Oi8RKNlEhSokEUM8+MU/flkKMccuLaK5cVIdgIscsG\nUXq26XQ6SoIEsUaIQ2oD+WzlnxWRVAenpOT2y6pA5chQ49HoJWhVWr5JHYVTA0TVNvk8mfvgyEVL\n8oxTIG/erHQ0xJCz6FzmbGgxScRSZP369cPhw4fZz//3f/8Xp556asGN6gr43iWAYK9FoH3CiBmM\nPllwZYrIdOFE0bDKdVRhxsh55DAZ5UVKhjsS6pCJkqlPGQDXie7TZxpa9I2gkWpj1lfEBA/OKaDI\nhVIlrEKR+4DwwkkjG6MOF+Kl2sDWIcYUGYaTDCvXNrFspKSotPyY2alUuc1rf/E55aTozQUkxUSV\nZdIWlR07HxinhnIKqPnGOS/ycYtkEIvILrnkEjz11FPI5XKRz44cOYI1a9Zg6tSpiTWus+Gn4vsv\nQQOfqzP5nbI2fouqfFP02eQNTlFw4UTCGJFeoUKNcGEPcfKFvEvZgH7utecbWtS+m0Ssr/gGTlYr\nVBhISS4aJRXpA8lxCPWfpo7WcaDqyG3T1JGdDt29yCqPu18gnKnKhZNDZSKpw/QlaPFaEVVFjdek\nSF6heo3HlEc8B4YELZJBrJ5csmQJduzYgWnTpuEXv/gFAGDbtm1YtWoVxo8fj2PHjuHb3/52pzQ0\naWTcTIi8xPUykXT8rEWZ1MQ6Doj1LyecpEGVZVXFGS928knERK0hRM6vWcuJnIdJApHDNvnu7CEa\nLC51n1JzlIGT+4ZUpRyZM4bJ2CDmYhhN4ZxUHVJVx13jMbwXuT5Xxy/nm5bP1Q/VocLGqnXOmI6A\nCclzjl1ofhIKS+nggH5WDqwiSwqx1sgmTpyIn/3sZ1iwYAEaGhoAAPfccw8AYOjQofj5z3+Oc845\nJ/lWdgJ8YgLaX5T2Q4v+uhjQQWpy1iKlwlg1RAxwmdTavPC6GFmHm3yEMSUJizL6cqIDFe9nCFE2\nsgEBgc485LxwIL+dPUwVjYosZGMkE1+w24rcT1JfRkhQZTQN6nDjiHoOoXU0jWFljbtJHd9hUSgp\nyjHRpfFrw8bcmq7m2XJKSiZtWeWJeyH620lRoUXOGYmENAnHUSxbJAMtkT344IO46aabUFFRAQC4\n+uqrsXv3brz44otBCn5NTQ1mzJgR2kC4p8MnLgCBh+gfl0lNJpd8y5z3TKXuyxMlQnwOrZ5UE51V\nI4QHqjMAspH1+4wMP0lJIKIX7oBI9jDY2YMzItx6kLzWx97H506Kqg9IhZDT95nOsEb6WCQvA0Mc\n+XVxjXEnDa6KAHJtrJIKOSxSqJhSW2L9UCak9F4gq7AUzzYy3xgS8ev4Tmqb14aMk1E7i9TcI6IC\nJo6j69jQYlLQEtnixYvxt3/7t5gxYwYaGhowc+ZMlJaW4tprr8W1117bFW3sFIiDyPcW/eOyIgM6\nFJwuwcNBNMxIEZZu3YUzjkrVwUx01nvNaSYo6PBcZFJ60cQPXWiR9MINdvbwnxFFTFS/cspLJiP/\nPtJIR/uMU22SWlDVD5Eg1EowriGmnonOuHPPnDyndI8Aos9Qenldq7YUoeXQ2CWcP67PIopJt974\ned+ImacRZ0ETilTNYdlxjDgOsIosKWhdgp/85Ce48MILsXHjRsyZMweVlZW455578O6773ZF+zoN\nn7V+FvpbTvbwy/7EFclIDDlShCUOZF0WYmAQKSPCxea5cIvK8MkGDpKR0IQWOa89Yuxy9D59unfB\nfC/fL1M7e0Q8dZmE4WjLpAEnjBGbWMCt2ShUniv8jhznqcuOBmU0SfLN8c8kYpQF4845RHIdqp3s\nc6aeIZG6zz1/MQtV6YwYhBYjfWmwFiqToNx/kb6U57BEUtRcleenRTLQEtn8+fPxyiuvYMeOHbj/\n/vuRyWTwyCOP4JxzzsFFF12E1atX49ixY13R1kQh7vLhTyBAUmFimFHIbKRILZ8f2QzWxYi1M8dx\nOsKJcCOkwxpTzpDJxlrlhTMGgCRB2dgxCQGcUosQFreDh5viSYpRiqQRJsJGrDEi1KqsFijlEFk7\ni3FOLkRFlSmjKd+LUsERDlGce+eepxwqVKXuU6qduydd+I5ST/Lc4NQcS4Kq+owCjoRvhTEot9Mi\nGRj3ZHV1NZYvX473338fzz//PObOnYutW7fi1ltvRUVFBebPn49XX301kUZ961vfwtixY1FfX4/Z\ns2fjo48+Cj5bsWIFRo8ejTFjxmDTpk15nX/B+AVYNHlR8HfKSUVejvbLAakR6sxk7UwOXYW8YUda\nF2NIKhI+owwcHN7oKBSiOEHl+lQdahLLRlw0WKK37RNThKQMQouBp04QE7vuQ/R9xOAziQ26cJL8\nDPMJOXHnZAna5JzSvXAkTj5nw3v3y9QLzqowoy4t35+L7D0RioZ02hBVWJxTE1oPJJw88rhOATOh\nRUq1WSSD2C6B67qYMWMGnn76aXzwwQd47LHHcPbZZ+OJJ57AZZddhpqamoIb9eUvfxnvvPMO3nzz\nTdTU1GDFihUAgO3bt+OZZ57B9u3b8cILL+D2228n32nTYfXM1fjmhd/suCcnvC5GqTN/kAJ8yJEr\nyxOI8oxVXiFlfOVJwxkv2WvnQotU/F6sE/JS5UlJGFkAZiRlotqkBA8lMTGqhAydESEnqs9U/REy\nrHK/ItpOVZu558/dF1vfIJRGOUSm9+6X/Tkjrn8GjoziHcFQEojwbP35xN4TR+Ya9Rm7P+Q5ZqqA\nhX4K1Xc65p5MxBbJoCBtO3DgQCxcuBA///nP8dWvfhUAEtlrcfr06XDd9qZNnjwZe/fuBQCsX78e\n8+bNQyaTwciRI1FdXY0tW7YUfD1xXSxEakQqvl9HLosJBHGyGf0QIkdMIUJkDJzsIUaMHRFKixAf\n4e1SnmnEqyUMnGzsVCQVITUijdvvZ5VyUSkRzshTJGykjBDuG2W/EgqI6kv5nFxYSq6vC6Wp7otb\n59SF0sQyIP2kD+GwyMeptHyuD0i1JUUcVKFFlWJW9ocmtEiOO0apieRI9Z9F4Yj1HpmIEydO4Lnn\nnsPatWvxq1/9CrlcDsOGDcNNN92UZPuwZs0azJs3DwCwb98+XHjhhcFnVVVVaGp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4o4kopJcHJITZslaJKAoVJPButZ1NoWS3wFrtP5z1mZWl/IO2KmCR4xScokLV+EfD/UGOCUFBdO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\nUeP/ATvUclXfeqm4AAAAAElFTkSuQmCC\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x89362b0>"
+       ]
+      }
+     ],
+     "prompt_number": 28
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It WAS ABLE to capture the physics! Even with the big time step that we use with Euler scheme!\n",
+      "\n",
+      "As you can see, and as we previously discussed the harmonic response is composed of a transient and a steady part. We are only concerned about the steady-state, since it is assumed that the probe achieves steady state motion during the imaging process. Therefore, we are going to slice our array in order to show only the last 10 oscillations, and we will see if it resembles the analytical solution."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#Slicing the full response vector to get the steady state response\n",
+      "z_steady_V = z_V[(90*period/dt):]\n",
+      "time_steady_V = time_V[(90*period/dt):]\n",
+      "\n",
+      "plt.title('Plot 3  Verlet approx. of steady state sol. of Eq 1', fontsize=20); \n",
+      "plt.xlabel('time, ms', fontsize=18);\n",
+      "plt.ylabel('z_Verlet, nm', fontsize=18);\n",
+      "plt.plot(time_steady_V*1e3, z_steady_V*1e9, 'g-');\n",
+      "plt.ylim(-65,65);\n",
+      "plt.show();\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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75ptvXL02pIwhLS3NtTWy7j8nd90OgPUhSoFzcNBAXQUuFnUVOAeHGDj3J8+d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\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x8758668>"
+       ]
+      }
+     ],
+     "prompt_number": 29
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Let's use now one of the most popular schemes...     The Runge Kutta 4!"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The Runge Kutta 4 (RK4) method is very popular for the solution of ODEs. This method is designed to solve 1st order differential equations. We have converted our 2nd order ODE to a system of two coupled 1st order ODEs when we implemented the Euler scheme (equations 5 and 6). And we will  have to use these equations for the RK4 algorithm.\n",
+      "\n",
+      "In order to clearly see the RK4 implementation we are going to put equations 5 and 6 in the following form:\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dz}{dt}=v \\Rightarrow f1(t,z,v)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dv}{dt} = -kz-\\frac{m\\omega_0}{Q}+F_ocos(\\omega t) \\Rightarrow f2(t,z,v)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "It can be clearly seen that we have two coupled equations f1 and f2 and both depend in t, z, and v.\n",
+      "\n",
+      "The RK4 equations for our special case where we have two coupled equations, are the following:\n",
+      "$$\\begin{equation}\n",
+      "k_1 = f1(t_i, z_i, v_i)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "m_1 = f2(t_i, z_i, v_i)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_2 = f1(t_i +1/2\\Delta t, z_i + 1/2k_1\\Delta t, v_i + 1/2m_1\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "m_2 = f2(t_i +1/2\\Delta t, z_i + 1/2k_1\\Delta t, v_i + 1/2m_1\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_3 = f1(t_i +1/2\\Delta t, z_i + k_2\\Delta t, v_i + 1/2m_2\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "m_3 = f2(t_i +1/2\\Delta t, z_i + 1/2k_2\\Delta t, v_i + 1/2m_2\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_4 = f1(t_i + \\Delta t, z_i + k_3\\Delta t, v_i + m_3\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_4 = f2(t_i + \\Delta t, z_i + k_3\\Delta t, v_i + m_3\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "f1_{n+1} = f1_n + \\Delta t/6(k_1+2k_2+2k_3+k_4)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "f2_{n+1} = f2_n + \\Delta t/6(m_1+2m_2+2m_3+m_4)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "Please notice how k values and m values are used sequentially, since it is crucial in the implementation of the method!\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#Definition of v, z, vectors\n",
+      "vdot_RK4 = numpy.zeros(N)\n",
+      "v_RK4 = numpy.zeros(N)\n",
+      "z_RK4 = numpy.zeros(N)\n",
+      "k1v_RK4 = numpy.zeros(N)\n",
+      "k2v_RK4 = numpy.zeros(N)\n",
+      "k3v_RK4 = numpy.zeros(N)\n",
+      "k4v_RK4 = numpy.zeros(N)\n",
+      "\n",
+      "k1z_RK4 = numpy.zeros(N)\n",
+      "k2z_RK4 = numpy.zeros(N)\n",
+      "k3z_RK4 = numpy.zeros(N)\n",
+      "k4z_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "#calculation of velocities RK4\n",
+      "\n",
+      "#INITIAL CONDITIONS\n",
+      "v_RK4[0] = 0\n",
+      "z_RK4[0] = 0\n",
+      "\n",
+      "   \n",
+      "for i in range (1,N):\n",
+      "       #RK4\n",
+      "        k1z_RK4[i] = v_RK4[i-1]   #k1 Equation 14 \n",
+      "        k1v_RK4[i] = ((   ( -k*z_RK4[i-1] - (m*wo/Q)*v_RK4[i-1] + \\\n",
+      "                           Fd*numpy.cos(wo*t[i-1]) )  ) / m )   #m1 Equation 15\n",
+      "        \n",
+      "        k2z_RK4[i] = ((v_RK4[i-1])+k1v_RK4[i]/2.*dt) #k2 Equation 16\n",
+      "        k2v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k1z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] +k1v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m ) #m2 Eq 17\n",
+      "        \n",
+      "        k3z_RK4[i] = ((v_RK4[i-1])+k2v_RK4[i]/2.*dt) #k3, Equation 18\n",
+      "        k3v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k2z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                            (v_RK4[i-1] +k2v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                            numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m ) #m3, Eq 19\n",
+      "                     \n",
+      "        k4z_RK4[i] = ((v_RK4[i-1])+k3v_RK4[i]*dt) #k4, Equation 20\n",
+      "        k4v_RK4[i] = ((   ( -k*(z_RK4[i-1] + k3z_RK4[i]*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] + k3v_RK4[i]*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt)) )  ) / m )#m4, Eq 21\n",
+      "        \n",
+      "        #Calculation of velocity, Equation 23\n",
+      "        v_RK4[i] = v_RK4[i-1] + 1./6*dt*(k1v_RK4[i] + 2.*k2v_RK4[i] +\\\n",
+      "                        2.*k3v_RK4[i] + k4v_RK4[i] )   \n",
+      "        #calculation of position, Equation 22\n",
+      "        z_RK4 [i] = z_RK4[i-1] + 1./6*dt*(k1z_RK4[i] + 2.*k2z_RK4[i] +\\\n",
+      "                        2.*k3z_RK4[i] + k4z_RK4[i] ) \n",
+      "\n",
+      "#slicing array to get steady state\n",
+      "z_steady_RK4 = z_RK4[(90.*period/dt):]\n",
+      "time_steady_RK4 = t[(90.*period/dt):]\n",
+      "    \n",
+      "plt.title('Plot 3  RK4 approx. of steady state sol. of Eq 1', fontsize=20); \n",
+      "plt.xlabel('time, ms', fontsize=18);\n",
+      "plt.ylabel('z_RK4, nm', fontsize=18);\n",
+      "plt.plot(time_steady_RK4 *1e3, z_steady_RK4*1e9, 'r-');\n",
+      "plt.ylim(-65,65);\n",
+      "plt.show();\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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OmDED1dXV+OCDD1S556+//hrvvfcejh49iry8PFRWVhq/EwQBubm5FuG9jh07Ws2y7tOn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\nEELoSBqvEEIIoSNpvEIIIYSO/g/oQttYxhXYvAAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x8994ac8>"
+       ]
+      }
+     ],
+     "prompt_number": 30
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Error Analysis"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Let's plot together our solutions using the different schemes along with our analytical reference."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plt.title('Plot 4  Schemes comparison with analytical sol.', fontsize=20);\n",
+      "plt.plot(time_an_steady*1e3, z_an_steady*1e9, 'b--' );\n",
+      "plt.plot(time_steady_V*1e3, z_steady_V*1e9, 'g-' );\n",
+      "plt.plot(time_steady_RK4*1e3, z_steady_RK4*1e9, 'r-');\n",
+      "plt.xlim(2.0, 2.06);\n",
+      "plt.legend(['Analytical solution', 'Verlet method', 'Runge Kutta 4']);\n",
+      "plt.xlabel('time, ms', fontsize=18);\n",
+      "plt.ylabel('z_position, nm', fontsize=18);\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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I7/LLcCgsCieU/PGab4jS9GcylWPhwoUNkr/m8msu+SsrL0P8k3j8EPsDPLd4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\n+g0UJaKYknvDEQAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x8849eb8>"
+       ]
+      }
+     ],
+     "prompt_number": 31
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It was pointless to include Euler in the last plot because it was not following the physics at all for this given time step. REMEMBER that Euler can give fair approximations, but you MUST decrease the time step in this particular case if you want to see the sinusoidal trajectory!\n",
+      "It seems our different schemes are giving different quality in approximating the solution. However it's hard to conclude something strong based on this qualitative observations. In order to state something stronger we have to perform an error analysis. We will choose L1 norm for this purpose (You can find more information about this [L1](http://en.wikipedia.org/wiki/Taxicab_geometry) ) and it was also discussed on the second IPython Notebook of the series _\"The phugoid model of glider flight\"_, the first learning module of the course [**\"Practical Numerical Methods with Python.\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about)."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "\"\"\"ERROR ANALYSIS EULER, VERLET AND RK4\"\"\"\n",
+      "\n",
+      "# time-increment array\n",
+      "dt_values = numpy.array([8.0e-7, 2.0e-7, 0.5e-7, 1e-8, 0.1e-8])\n",
+      "\n",
+      "# array that will contain solution of each grid\n",
+      "z_values_E = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "z_values_V = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "z_values_RK4 = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "z_values_an = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "\n",
+      "for n, dt in enumerate(dt_values):\n",
+      "    simultime = 100*period\n",
+      "    timestep = dt\n",
+      "    N = int(simultime/dt)\n",
+      "    t = numpy.linspace(0.0, simultime, N)\n",
+      "   \n",
+      "    #Initializing variables for Verlet\n",
+      "    zdoubledot_V = numpy.zeros(N)\n",
+      "    zdot_V = numpy.zeros(N)\n",
+      "    z_V = numpy.zeros(N)\n",
+      "    \n",
+      "    #Initializing variables for RK4\n",
+      "    vdot_RK4 = numpy.zeros(N)\n",
+      "    v_RK4 = numpy.zeros(N)\n",
+      "    z_RK4 = numpy.zeros(N)\n",
+      "    k1v_RK4 = numpy.zeros(N) \n",
+      "    k2v_RK4 = numpy.zeros(N)\n",
+      "    k3v_RK4 = numpy.zeros(N)\n",
+      "    k4v_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "    k1z_RK4 = numpy.zeros(N)\n",
+      "    k2z_RK4 = numpy.zeros(N)\n",
+      "    k3z_RK4 = numpy.zeros(N)\n",
+      "    k4z_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "          \n",
+      "    #Initial conditions Verlet (started with Euler approximation)\n",
+      "    z_V[0] = 0.0\n",
+      "    zdot_V[0] = 0.0\n",
+      "    zdoubledot_V[0] = (   ( -k*z_V[0] - (m*wo/Q)*zdot_V[0] + \\\n",
+      "                           Fd*numpy.cos(wo*t[0]) )  ) / m\n",
+      "    zdot_V[1] = zdot_V[0] + zdoubledot_V[0]*timestep**2\n",
+      "    z_V[1] = z_V[0] + zdot_V[0]*dt\n",
+      "    zdoubledot_V[1] = (   ( -k*z_V[1] - (m*wo/Q)*zdot_V[1] + \\\n",
+      "                           Fd*numpy.cos(wo*t[1]) )  ) / m\n",
+      "    \n",
+      "    \n",
+      "    #Initial conditions Runge Kutta\n",
+      "    v_RK4[1] = 0\n",
+      "    z_RK4[1] = 0 \n",
+      "    \n",
+      "    #Initialization variables for Analytical solution\n",
+      "    z_an = numpy.zeros(N)\n",
+      "       \n",
+      "    # time loop \n",
+      "    for i in range(2,N):\n",
+      "             \n",
+      "        #Verlet\n",
+      "        z_V[i] = 2*z_V[i-1] - z_V[i-2] + zdoubledot_V[i-1]*dt**2 #Eq 10\n",
+      "        zdot_V[i] = (z_V[i]-z_V[i-2])/(2.0*dt) #Eq 11\n",
+      "        zdoubledot_V[i] = (   ( -k*z_V[i] - (m*wo/Q)*zdot_V[i] +\\\n",
+      "                                   Fd*numpy.cos(wo*t[i]) )  ) / m #from eq 1\n",
+      "        \n",
+      "        #RK4\n",
+      "        k1z_RK4[i] = v_RK4[i-1]   #k1 Equation 14 \n",
+      "        k1v_RK4[i] = ((   ( -k*z_RK4[i-1] - (m*wo/Q)*v_RK4[i-1] + \\\n",
+      "                           Fd*numpy.cos(wo*t[i-1]) )  ) / m )   #m1 Equation 15\n",
+      "        \n",
+      "        k2z_RK4[i] = ((v_RK4[i-1])+k1v_RK4[i]/2.*dt) #k2 Equation 16\n",
+      "        k2v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k1z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] +k1v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m )  #m2 Eq 17\n",
+      "        \n",
+      "        k3z_RK4[i] = ((v_RK4[i-1])+k2v_RK4[i]/2.*dt) #k3, Equation 18\n",
+      "        k3v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k2z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                            (v_RK4[i-1] +k2v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                            numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m ) #m3, Eq 19\n",
+      "                     \n",
+      "        k4z_RK4[i] = ((v_RK4[i-1])+k3v_RK4[i]*dt) #k4, Equation 20\n",
+      "        k4v_RK4[i] = ((   ( -k*(z_RK4[i-1] + k3z_RK4[i]*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] + k3v_RK4[i]*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt)) )  ) / m )#m4, Equation 21\n",
+      "        \n",
+      "        #Calculation of velocity, Equation 23\n",
+      "        v_RK4[i] = v_RK4[i-1] + 1./6*dt*(k1v_RK4[i] + 2.*k2v_RK4[i] +\\\n",
+      "                        2.*k3v_RK4[i] + k4v_RK4[i] )   \n",
+      "        #calculation of position, Equation 22\n",
+      "        z_RK4 [i] = z_RK4[i-1] + 1./6*dt*(k1z_RK4[i] + 2.*k2z_RK4[i] +\\\n",
+      "                         2.*k3z_RK4[i] + k4z_RK4[i] ) \n",
+      "\n",
+      "             \n",
+      "        #Analytical solution\n",
+      "        A_an = Fo_an*Q/k  #when driven at resonance A is simply Fo*Q/k\n",
+      "        phi = numpy.pi/2  #when driven at resonance the phase is pi/2\n",
+      "        z_an[i] = A_an*numpy.cos(wo*t[i] - phi)  #Analytical solution eq. 1\n",
+      "    \n",
+      "          \n",
+      "    #Slicing the full response vector to get the steady state response\n",
+      "    z_steady_V = z_V[(80*period/timestep):]\n",
+      "    z_an_steady = z_an[(80*period/timestep):]\n",
+      "    z_steady_RK4 = z_RK4[(80*period/timestep):]\n",
+      "    time_steady = t[(80*period/timestep):]\n",
+      "    \n",
+      "    z_values_V[n] = z_steady_V.copy()  # error for certain value of timestep\n",
+      "    z_values_RK4[n] = z_steady_RK4.copy() #error for certain value of timestep\n",
+      "    z_values_an[n] = z_an_steady.copy()  #error for certain value of timestep\n",
+      "\n",
+      "\n",
+      "def get_error(z, z_exact, dt):\n",
+      "    #Returns the error with respect to the analytical solution using L1 norm\n",
+      "       \n",
+      "    return dt * numpy.sum(numpy.abs(z-z_exact))\n",
+      "    \n",
+      "#NOW CALCULATE THE ERROR FOR EACH RESPECTIVE DELTA T\n",
+      "error_values_V = numpy.zeros_like(dt_values)\n",
+      "error_values_RK4 = numpy.zeros_like(dt_values)\n",
+      "\n",
+      "for i, dt in enumerate(dt_values):\n",
+      "    ### call the function get_error() ###\n",
+      "    error_values_V[i] = get_error(z_values_V[i], z_values_an[i], dt)\n",
+      "    error_values_RK4[i] = get_error(z_values_RK4[i], z_values_an[i], dt)\n",
+      "\n",
+      "\n",
+      "plt.figure(1);\n",
+      "plt.title('Plot 5  Error analysis Verlet based on L1 norm', fontsize=20);\n",
+      "plt.tick_params(axis='both', labelsize=14); \n",
+      "plt.grid(True);                         #turn on grid lines\n",
+      "plt.xlabel('$\\Delta t$ Verlet', fontsize=16);  #x label\n",
+      "plt.ylabel('Error Verlet', fontsize=16);       #y label\n",
+      "plt.loglog(dt_values, error_values_V, 'go-');  #log-log plot\n",
+      "plt.axis('equal');                      #make axes scale equally;\n",
+      "\n",
+      "plt.figure(2);\n",
+      "plt.title('Plot 6  Error analysis RK4 based on L1 norm', fontsize=20); \n",
+      "plt.tick_params(axis='both', labelsize=14); \n",
+      "plt.grid(True);     #turn on grid lines\n",
+      "plt.xlabel('$\\Delta t$ RK4', fontsize=16);  #x label\n",
+      "plt.ylabel('Error RK4', fontsize=16);       #y label\n",
+      "plt.loglog(dt_values, error_values_RK4,  'co-');  #log-log plot\n",
+      "plt.axis('equal');                      #make axes scale equally;\n",
+      "\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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vKvmVl2tF85hyfrpuO2Pmp2tuppKfobadPvIzp/emOeR38eJFg/XL5shki6up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\nbN0AAAAASUVORK5CYII=\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x8932f28>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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qPXyh/EybkPMTcm6A4fJLTk4W3REmbzdfl0qlGm9KLjdnzhy93iT6SWLb+IQQ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\nXSHo1asX5s+fj6FDh8Le3h79+/eHRCKM/bdLly5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+       "text": [
+        "<matplotlib.figure.Figure at 0x88497f0>"
+       ]
+      }
+     ],
+     "prompt_number": 18
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "As we can see Runge Kutta 4 converges faster than Verlet for the range of time steps studied. And the difference between both is near one order of magnitude. One additional advantage with Runge Kutta 4 is that the method is very stable, even with big time steps (eg. 10 time steps per period) the method is able to catch up the physics of the oscillation, something where Verlet is not so good at."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Let's add a sample and oscillate our probe over it!"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It is very common in the field of probe microscopy to model the tip sample interactions through DMT contact mechanics. \n",
+      "DMT stands for Derjaguin, Muller and Toporov who were the scientists that developed the model (see ref 1). This model uses Hertz contact mechanics (see ref 2) with the addition of long range tip-sample interactions. These long range tip-sample interactions are ascribed to intermolecular interactions between the atoms of the tip and the upper atoms of the surface, and include mainly the contribution of van de Waals forces and Pauli repulsion from electronic clouds when the atoms of the tip meet closely the atoms of the surface. Figure 2 displays a force vs distance curve (FD curve) where it is shown how the forces between the tip and the sample behave with respect to the separation. It can be seen that at positive distances the tip starts \"feeling\" attraction from the tip (from the contribution of van der Waals forces) where the slope of the curve is positive and at some minimum distance ($a_0$) the tip starts experiencing repulsive interactions arising from electronic cloud repulsion (area where the slope of the curve is negative and the forces are negative). At lower distances, an area known as \"contact area\" arises and it is characterized by a negative slope and an emerging positive force."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image;\n",
+      "Image(filename=\"C:/Users/enrique/Github Repositories/FinalProjectMAE6286_2/FinalProjectMAE6286/Fig3FDcurve.jpg\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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Ieh/+m+CvBf2iP27vjLpP/BQG4+BHww8L+BNVvr7w/b6rYahrxuooNMbcWnmu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\nLnX4tf8AYWtv/bivuWvhr/g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rMWiXd0ddcWy3MMzw2UEnlyGMurHYqExEjBbFfQvxH/AOC4vw1/aP8A2Fvj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\nv6YUUUVBYUUUUAFFFFABRRRQAUUUUAFFFFABRRRQB//Z\n",
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 40,
+       "text": [
+        "<IPython.core.display.Image at 0x87555f8>"
+       ]
+      }
+     ],
+     "prompt_number": 40
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Figure 3.   Force vs Distance profile depicting tip-sample interactions in AFM (Adapted from reference 6)"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In Hertz contact mechanics, one central aspect is to consider that the contact area increases as the sphere is pressed against an elastic surface, and this increase of the contact area \"modulates\" the effective stiffness of the sample. This concept is represented in figure 4 where the sample is depicted as comprised by a series of springs that are activated as the tip goes deeper into the sample. In other words, the deeper the sample goes, the larger the contact area and therefore more springs are activated (see more about this on reference 5). \n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image;\n",
+      "Image(filename=\"C:/Users/enrique/Github Repositories/FinalProjectMAE6286_2/FinalProjectMAE6286/Fig4Hertzspring.jpg\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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/8co/4bH+EX/RVPhx/wCFLZf/AByvnc54fy7MpKtUfJVj8NSEuWcfmt15O68j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\nVrCy1TTroBZrW7gWeGYAggMjAqeQDyOoFXaKKAP/2Q==\n",
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 39,
+       "text": [
+        "<IPython.core.display.Image at 0x86bcb70>"
+       ]
+      }
+     ],
+     "prompt_number": 39
+    },
+    {
+     "cell_type": "raw",
+     "metadata": {},
+     "source": [
+      "####Figure 4.  Conceptual representation of Hertz contact mechanics"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This concept is represented mathematically by a non-linear spring whose elastic coefficient is a function of the contact area which at the same time depends on the sample indentation ( k(d)  ).\n",
+      "$$F_{ts} = k(d)d$$\n",
+      "where\n",
+      "$$k(d) = 4/3E*\\sqrt{Rd}$$\n",
+      "being $\\sqrt{Rd}$ the contact area when a sphere of radius R indents a half-space to depth d.\n",
+      "$E*$ is the effective Young's modulus of the tip-sample interaction. "
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The long range attractive forces are derived using Hamaker's equation (see reference 4): $if$ $d > a_0$\n",
+      "$$F_{ts} = \\frac{-HR}{6d^2}$$\n",
+      "\n",
+      "where H is the Hamaker constant, R the tip radius and d the tip sample distance. $a_0$ is defined as the intermolecular distance and normally is chosen to be 0.2 nm."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In summary the equations that we will include in our code to take care of the tip sample interactions are the following:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "Fts_{DMT} = \\begin{cases} \\frac{-HR}{6d^2} \\quad \\quad d \\leq{a_0}\\\\ \\\\\n",
+      "\\frac{-HR}{6d^2} + 4/3E*R^{1/2}d^{3/2}   \\quad \\quad d> a_0 \\end{cases}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "where the effective Young's modulus E* is defined by:\n",
+      "$$\\begin{equation}\n",
+      "1/E* = \\frac{1-\\nu^2}{E_t}+\\frac{1-\\nu^2}{E_s}\n",
+      "\\end{equation}$$\n",
+      "where $E_t$ and $E_s$ are the tip and sample Young's modulus respectively. $\\nu_t$ and $\\nu_s$ are tip and sample Poisson ratios, respectively.\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Enough theory, Let's make our code!"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now we will have to solve equation (1) but with the addition of tip-sample interactions which are described by equation (5). So we have a second order non-linear ODE which is no longer analytically straightforward:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "m \\frac{d^2z}{dt^2} = - k z - \\frac{m\\omega_0}{Q}\\frac{dz}{dt} + F_0 cos(\\omega t) + Fts_{DMT}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "Therefore we have to use numerical methods to solve it. RK4 has shown to be more accurate to solve equation (1) among the methods reviewed in the previous section of the notebook, and therefore it is going to be the chosen method to solve equation (6).\n",
+      "\n",
+      "Now we have to declare all the variables related to the tip-sample forces. Since we are modeling our tip-sample forces using Hertz contact mechanics with addition of long range Van der Waals forces we have to define the Young's modulus of the tip and sample, the diameter of the tip of our probe, Poisson ratio, etc."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#DMT parameters  (Hertz contact mechanics with long range Van der Waals forces added\n",
+      "a=0.2e-9 #intermolecular parameter\n",
+      "H=6.4e-20 #hamaker constant of sample\n",
+      "R=20e-9 #tip radius of the cantilever\n",
+      "Es=70e6 #elastic modulus of sample\n",
+      "Et=130e9 #elastic modulus of the tip\n",
+      "vt=0.3 #Poisson coefficient for tip\n",
+      "vs=0.3 #Poisson coefficient for sample\n",
+      "E_star= 1/((1-pow(vt,2))/Et+(1-pow(vs,2))/Es)  #Effective Young Modulus"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 1
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now let's declare the timestep, the simulation time and let's oscillate our probe!"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#IMPORTANT distance where you place the probe above the sample\n",
+      "z_base = 40.e-9 \n",
+      "\n",
+      "spp = 280. # time steps per period \n",
+      "dt = period/spp \n",
+      "\n",
+      "simultime = 100.*period\n",
+      "N = int(simultime/dt)\n",
+      "t = numpy.linspace(0,simultime,N)\n",
+      "\n",
+      "#Initializing variables for RK4\n",
+      "v_RK4 = numpy.zeros(N)\n",
+      "z_RK4 = numpy.zeros(N)\n",
+      "k1v_RK4 = numpy.zeros(N) \n",
+      "k2v_RK4 = numpy.zeros(N)\n",
+      "k3v_RK4 = numpy.zeros(N)\n",
+      "k4v_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "k1z_RK4 = numpy.zeros(N)\n",
+      "k2z_RK4 = numpy.zeros(N)\n",
+      "k3z_RK4 = numpy.zeros(N)\n",
+      "k4z_RK4 = numpy.zeros(N)\n",
+      "\n",
+      "TipPos = numpy.zeros(N)\n",
+      "Fts = numpy.zeros(N)\n",
+      "Fcos = numpy.zeros(N)\n",
+      "\n",
+      "for i in range(1,N):\n",
+      "   #RK4\n",
+      "    k1z_RK4[i] = v_RK4[i-1]   #k1 Equation 14 \n",
+      "    k1v_RK4[i] = ((   ( -k*z_RK4[i-1] - (m*wo/Q)*v_RK4[i-1] + \\\n",
+      "                       Fd*numpy.cos(wo*t[i-1]) +Fts[i-1])  ) / m )   #m1 Equation 15\n",
+      "    \n",
+      "    k2z_RK4[i] = ((v_RK4[i-1])+k1v_RK4[i]/2.*dt) #k2 Equation 16\n",
+      "    k2v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k1z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                       (v_RK4[i-1] +k1v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                       numpy.cos(wo*(t[i-1] + dt/2.))  +Fts[i-1])  ) / m )  #m2 Eq 17\n",
+      "    \n",
+      "    k3z_RK4[i] = ((v_RK4[i-1])+k2v_RK4[i]/2.*dt) #k3, Equation 18\n",
+      "    k3v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k2z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                        (v_RK4[i-1] +k2v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                        numpy.cos(wo*(t[i-1] + dt/2.))  +Fts[i-1])  ) / m ) #m3, Eq19\n",
+      "                 \n",
+      "    k4z_RK4[i] = ((v_RK4[i-1])+k3v_RK4[i]*dt) #k4, Equation 20\n",
+      "    k4v_RK4[i] = ((   ( -k*(z_RK4[i-1] + k3z_RK4[i]*dt) - (m*wo/Q)*\\\n",
+      "                       (v_RK4[i-1] + k3v_RK4[i]*dt) + Fd*\\\n",
+      "                       numpy.cos(wo*(t[i-1] + dt))  +Fts[i-1])  ) / m )#m4, Eq 21\n",
+      "    \n",
+      "    #Calculation of velocity, Equation 23\n",
+      "    v_RK4[i] = v_RK4[i-1] + 1./6*dt*(k1v_RK4[i] + 2.*k2v_RK4[i] +\\\n",
+      "                    2.*k3v_RK4[i] + k4v_RK4[i] )   \n",
+      "    #calculation of position, Equation 22\n",
+      "    z_RK4 [i] = z_RK4[i-1] + 1./6*dt*(k1z_RK4[i] + 2.*k2z_RK4[i] +\\\n",
+      "                     2.*k3z_RK4[i] + k4z_RK4[i] ) \n",
+      "           \n",
+      "    TipPos[i] = z_base + z_RK4[i] #Adding base position to z position\n",
+      "     \n",
+      "    #calculation of DMT force\n",
+      "\n",
+      "    if  TipPos[i] > a: #this defines the attractive regime\n",
+      "        Fts[i] = -H*R/(6*(TipPos[i])**2)\n",
+      "    else:  #this defines the repulsive regime\n",
+      "        Fts[i] = -H*R/(6*a**2)+4./3*E_star*numpy.sqrt(R)*(a-TipPos[i])**1.5\n",
+      "    \n",
+      "          \n",
+      "    Fcos[i] = Fd*numpy.cos(wo*t[i])  #Driving force (this will be helpful to plot the driving force)\n",
+      "\n",
+      "#Slicing arrays to get steady state\n",
+      "TipPos_steady = TipPos[(95*period/dt):] \n",
+      "t_steady = t[(95*period/dt):]   \n",
+      "Fcos_steady = Fcos[(95*period/dt):]   \n",
+      "Fts_steady = Fts[(95*period/dt):] \n",
+      "\n",
+      "plt.figure(1);\n",
+      "fig, ax1 = plt.subplots();\n",
+      "ax2 = ax1.twinx();\n",
+      "ax1.plot(t_steady*1e3,TipPos_steady*1e9, 'g-');\n",
+      "ax2.plot(t_steady*1e3, Fcos_steady*1e9, 'b-');\n",
+      "ax1.set_xlabel('Time,s');\n",
+      "ax1.set_ylabel('Tip position (nm)', color='g');\n",
+      "ax2.set_ylabel('Drive Force (nm)', color='b');\n",
+      "plt.title('Plot 7   Tip response and driving force', fontsize = 20);\n",
+      "plt.show();\n",
+      "\n",
+      "plt.figure(2);\n",
+      "plt.title('Plot 8  Force-Distance curve', fontsize=20);\n",
+      "plt.plot(TipPos*1e9, Fts*1e9, 'b--' );\n",
+      "plt.xlabel('Tip Position, nm', fontsize=18);\n",
+      "plt.ylabel('Force, nN', fontsize=18);\n",
+      "plt.xlim(-20, 30);\n",
+      "\n",
+      "\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "text": [
+        "<matplotlib.figure.Figure at 0xcd29c88>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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H9sfhosMy9sx+TpacRIhXCKJ01xaGqxgVMnplYG/uXhtnui6GZgPOVZzDLdHm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+       "text": [
+        "<matplotlib.figure.Figure at 0x853c128>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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TJRAhBCIjI+Ht7a1KIAAwbtw4xMXFAQDi4uJM0n1mKkwcRNTQVNsSKSgowIcf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\nEZHBmESIiMhgTCJERGQwJhEiIjLY/we1Rb7tjo3fbgAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x2469f400>"
+       ]
+      }
+     ],
+     "prompt_number": 22
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Check that we have two sinusoidals. The one in green (the output) is the response signal of the tip (the tip trajectory in time) while the blue one (the input) is the cosinusoidal driving force that we are using to excite the tip. When the tip is excited in free air (without tip sample interactions) the phase lag between the output and the input is 90 degrees. You can test that with the previous code by only changing the position of the base to a high-enough position that it does not interact with the sample. However in the above plot the phase lag is less than 90 degrees. Interestingly the phase can give relative information about the material properties of the sample. There is a well-developed theory of this in tapping mode AFM and it's called phase spectroscopy. If you are interested in this topic you can read reference 1.\n",
+      "Also look at the above plot and see that the response amplitude is no longer 60 nm as we initially set (in this case is near 45 nm!). It means that we have experienced a significant amplitude reduction due to the tip sample interactions.\n",
+      "Besides with the data acquired we are able to plot a Force-curve as the one shown in Figure 3! It shows the attractive and repulsive interactions of our probe with the surface.\n",
+      "\n",
+      "We have arrived to the end of the notebook. I hope you have found it interesting!"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.core.display import HTML\n",
+      "css_file = 'C:/Users/enrique/Github Repositories/FinalProjectMAE6286_2/FinalProjectMAE6286/numericalmoocstyle.css'\n",
+      "HTML(open(css_file, \"r\").read())\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
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+      }
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+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "REFERENCES\n",
+      "\n",
+      "1. Garc\u0131\u0301a, Ricardo, and Ruben Perez. \"Dynamic atomic force microscopy methods.\" Surface science reports 47.6 (2002): 197-301.\n",
+      "2. B. V. Derjaguin, V. M. Muller, and Y. P. Toporov, J. Colloid\n",
+      "Interface Sci. 53, 314  (1975)\n",
+      "3. Hertz, H. R., 1882, Ueber die Beruehrung elastischer Koerper (On Contact Between Elastic Bodies), in Gesammelte Werke (Collected Works), Vol. 1, Leipzig, Germany, 1895.\n",
+      "4. Van Oss, Carel J., Manoj K. Chaudhury, and Robert J. Good. \"Interfacial Lifshitz-van der Waals and polar interactions in macroscopic systems.\" Chemical Reviews 88.6 (1988): 927-941.\n",
+      "5. Enrique A. L\u00f3pez-Guerra, and Santiago D. Solares. \"Modeling viscoelasticity through spring\u2013dashpot models in intermittent-contact atomic force microscopy.\" Beilstein journal of nanotechnology 5, no. 1 (2014): 2149-2163.\n",
+      "6. Enrique A. L\u00f3pez-Guerra, and Santiago D. Solares, \"El microscopio de Fuerza At\u00f3mica: Metodos y Aplicaciones.\" Revista UVG (2013) No. 28, 14-23.."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}
\ No newline at end of file
diff --git a/Finalproject_EnriqueLopez/readme.txt.txt b/Finalproject_EnriqueLopez/readme.txt.txt
new file mode 100644
index 0000000..2043e29
--- /dev/null
+++ b/Finalproject_EnriqueLopez/readme.txt.txt
@@ -0,0 +1 @@
+This bonus notebook has the purpose of introducing to Atomic Force Microscopy (AFM) simulations in Tapping mode. The notebook compares three different integration schemes (Euler, Verlet and Runge Kutta 4) and compares them in terms of accuracy to solve an ordinary differential equation that describes the motion of the probe in dynamic AFM.
\ No newline at end of file
diff --git a/scanningprobe_finalproject_enriquelopez.ipynb b/scanningprobe_finalproject_enriquelopez.ipynb
new file mode 100644
index 0000000..ea70c07
--- /dev/null
+++ b/scanningprobe_finalproject_enriquelopez.ipynb
@@ -0,0 +1,1384 @@
+{
+ "metadata": {
+  "name": "scanningprobe_finalproject_enriquelopez.ipynb"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "heading",
+     "level": 6,
+     "metadata": {},
+     "source": [
+      "Content under Creative Commons Attribution license CC-BY 4.0 version, by Enrique A. L\u00f3pez-Guerra."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 1,
+     "metadata": {},
+     "source": [
+      "Dynamics of a tapping probe that aims at seeing atoms"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Purpose of the notebook: show an application of numerical methods to simulate the dynamics of a probe in atomic force microscopy.\n",
+      "\n",
+      "Requirements to take the best advantage of this notebook: knowing the fundamentals of  [Harmonic Oscillators](http://en.wikipedia.org/wiki/Harmonic_oscillator) in clasical mechanics and [Fundamentals of Vibrations](http://en.wikipedia.org/wiki/Vibration). "
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Since the atomic force microscope (AFM) was invented in 1986 it has become one of the main tools to study matter at the micro and nanoscale. This powerful tool is so versatile that it can be used to study a wide variety of materials, ranging from stiff inorganic surfaces to soft biological samples. \n",
+      "\n",
+      "In its early stages the AFM was used in permanent contact with the sample (the probe is dragged over the sample during the whole operation), which brought about important drawbacks, such as rapid probe wear and often sample damage, but these obstacles have been overcome with the development of dynamic techniques.\n",
+      "\n",
+      "Inn this Ipython notebook we will focus on the operation of the probe in dynamic mode which is the mainly used nowadays due to the significant advantages over the permanent contact techniques."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image\n",
+      "Image(filename='C:/Users/Enrique Alejandro/Documents/GitHub/FinalProjectMAE6286/finalproject/Fig1.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
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2pbwLKgdnPZQl3IfoGrD13XfFX/BX34pyaNos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\nJX5taP8A8Fs/iL+zffx6Z+0N8D/Enh1UQD+2NJiLQykBdxKvhM5ZSQr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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 28,
+       "text": [
+        "<IPython.core.display.Image at 0x7d3e358>"
+       ]
+      }
+     ],
+     "prompt_number": 28
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Figure 1.   Schematics of the setup of a atomic force microscope"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In AFM the interacting probe is in general a rectangular cantilever (please check the image above that shows the AFM setup where you will be able to see the probe!). \n",
+      "Probably the most used dynamic technique in AFM is the Tapping Mode. In this method the probe taps a surface in intermittent contact fashion. The purpose of tapping the probe over the surface instead of dragging it is to reduce frictional forces that may cause damage of soft samples and wear of the tip. Besides with the tapping mode we can get more information about the sample!  HOW???"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In Tapping Mode AFM the cantilever is shaken to oscillate up and down at a specific frequency (most of the time shaken at its natural frequency). Then the deflection of the tip is measured at that frequency to get information about the sample. Besides acquiring the topography of the sample, the phase lag between the excitation and the response of the cantilever can be related to compositional material properties! \n",
+      "In other words one can simultaneously get information about how the surface looks and also get compositional mapping of the surface!  THAT SOUNDS POWERFUL!!!"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image\n",
+      "Image(filename=\"C:/Users/Enrique Alejandro/Documents/GitHub/FinalProjectMAE6286\\FinalProject/Fig2DHO.jpg\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 9,
+       "text": [
+        "<IPython.core.display.Image at 0x81d5a20>"
+       ]
+      }
+     ],
+     "prompt_number": 9
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Figure 2. Schematics of a damped harmonic oscillator without tip-sample interactions"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "3.1 Analytical Solution"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The motion of the probe can be derived using [Euler-Bernoulli's](http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory) equation. However that equation has partial derivatives (it depends on time and space) because it deals with finding the position of each point of the beam in a certain time, which can make the problem too expensive computanionally for our purposes. In our case, we have the advantage that we are only concerned about the position of the tip (which is the only part of the probe that will interact with the sample). As a consequence many reserachers in AFM have succesfully made approximations using a simple mass point model approximation [see ref. 2] like the one in figure 2 (with of course the addition of tip sample forces!  we will see more about thi on section 4).\n",
+      "\n",
+      "First we will study the system of figure 2 AS IS (without addition of tip-sample force term), WHY?  Because we want to get an anlytical solution to get a reference of how our integration schemes are working, and the addition of tip sample forces to our equation will prevent the acquisition of straightforward analytical solutions :(\n",
+      "\n",
+      "So, the equation of motion of the damped harmonic oscillator of figure 2, that is besides DRIVEN COSINUSOIDALLY (remember that we are exciting our probe during the scanning process) is:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\begin{equation}\n",
+      "m \\frac{d^2z}{dt^2} = - k z - \\frac{m\\omega_0}{Q}\\frac{dz}{dt} + F_0 cos(\\omega t)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "The analytical solution of the above ODE is composed by a transient term and a steady state term. We are only interested in the steady state part because during the scanning process it is assumed that the probe has achieved that state.\n",
+      "\n",
+      "The steady state solution is given by:\n",
+      "$$\\begin{equation}\n",
+      "A cos (\\omega t - \\phi)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "where A is the steady state amplitude of the oscillation response which depends on the cantilever parameters and the driving parameters as can be seen in the following relation:\n",
+      "$$\\begin{equation}\n",
+      "A = \\frac{F_0/m}{\\sqrt{(\\omega_0^2-\\omega^2)^2+(\\frac{\\omega\\omega_0}{Q})^2}}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "and $\\phi$ is given by:\n",
+      "$$\\begin{equation}\n",
+      "\\phi = arctan \\big( \\frac{\\omega\\omega_0/Q}{\\omega_0^2 - \\omega^2} \\big)\n",
+      "\\end{equation}$$\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Let's first name the variables that we are going to use. Because we are dealing with a damped harmonic oscillator model we have to include variables suche as: spring stiffness, resonance frequency, quality factor (related to damping coefficient), target oscillation amplitude, etc."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy\n",
+      "import matplotlib.pyplot as plt\n",
+      "\n",
+      "k = 10.\n",
+      "fo = 45000\n",
+      "wo = 2.0*numpy.pi*fo\n",
+      "Q = 25.\n",
+      "\n",
+      "period = 1./fo\n",
+      "m = k/(wo**2)\n",
+      "Ao = 60.e-9\n",
+      "Fd = k*Ao/Q\n",
+      "\n",
+      "spp = 28. # time steps per period \n",
+      "dt = period/spp #Intentionally chosen to be quite big\n",
+      "#you can decrease dt by increasing the number of steps per period\n",
+      "\n",
+      "simultime = 100.*period\n",
+      "N = int(simultime/dt)\n",
+      "\n",
+      "#Analytical solution\n",
+      "\n",
+      "time_an = numpy.linspace(0,simultime,N)  #time array for the analytical solution\n",
+      "z_an = numpy.zeros(N)  #position array for the analytical solution\n",
+      "\n",
+      "#Driving force amplitude this gives us 60nm of amp response (A_target*k/Q)\n",
+      "Fo_an = 24.0e-9  \n",
+      "\n",
+      "A_an = Fo_an*Q/k  #when driven at resonance A is simply Fo*Q/k\n",
+      "phi = numpy.pi/2  #when driven at resonance the phase is pi/2\n",
+      "\n",
+      "z_an[:] = A_an*numpy.cos(wo*time_an[:] - phi) #this gets the analytical solution\n",
+      "\n",
+      "#slicing the array to include only steady state (only the last 10 periods)\n",
+      "z_an_steady = z_an[(90.*period/dt):]\n",
+      "time_an_steady = time_an[(90.*period/dt):]\n",
+      "\n",
+      "plt.title('Plot 1    Analytical Steady State Solution of Eq 1', fontsize=20)\n",
+      "plt.xlabel('time, ms', fontsize=18)\n",
+      "plt.ylabel('z_Analytical, nm', fontsize=18)\n",
+      "plt.plot(time_an_steady*1e3, z_an_steady*1e9, 'b--')\n",
+      "#plt.show()\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 19,
+       "text": [
+        "[<matplotlib.lines.Line2D at 0x7e29550>]"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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bPHtG8M8/8j1rYSHB5csETZsS7Nsnfv6m2mLmTIJx4+R7TjV8tMUOf156oQ0A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\nkJAgXLuZq0Cx5jYGgKNH2d+bn09dmQ4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ZatfmV7YSREZSBUc+QluJCWheHpCYCAQE8CtX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\nBEFYCaS0CYIgCMJK+H8LH6zXOrv8OwAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x85632b0>"
+       ]
+      }
+     ],
+     "prompt_number": 19
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "3.2 Approximating through Euler's method"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This method was already discussed in the second notebook called \"Pughoid Oscillation\" located on the second IPython Notebook of the series _\"The phugoid model of glider flight\"_, the first learning module of the course [**\"Practical Numerical Methods with Python.\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about).\n",
+      "We will review it briefly since the intention is to make this notebook as most self-contained as possible. Let's begin...\n",
+      "\n",
+      "If we perform a Taylor series expansion of $z_{n+1}$ around $z_{n}$ we get:\n",
+      "\n",
+      "$$z_{n+1} = z_{n} + \\Delta t\\frac{dz}{dt}\\big|_n + {\\mathcal O}(\\Delta t^2)$$\n",
+      "\n",
+      "The Euler formula neglects terms in the order of two or higher, ending up as:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n+1} = z_{n} + \\Delta t\\frac{dz}{dt}\\big|_n\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "It can be easily seen that the truncation error of the Euler algorithm is in the order of ${\\mathcal O}(\\Delta t^2)$.\n",
+      "\n",
+      "\n",
+      "\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This is a second order ODE, but we can convert it to a system of two coupled 1st order differential equations. To do it we will define $\\frac{dz}{dt} = v$. Then equation (1) will be decomposed as:\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dz}{dt} = v\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dv}{dt} = -kz-\\frac{m\\omega_0}{Q}+F_ocos(\\omega t)\n",
+      "\\end{equation}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This previous coupled equations will be used during Euler's aproximation and also during our integration using Runge Kutta methods in the section 3.4."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      " t= numpy.linspace(0,simultime,N) #time grid for Euler method\n",
+      "    \n",
+      "#Initializing variables for Euler\n",
+      "vdot_E = numpy.zeros(N)\n",
+      "v_E = numpy.zeros(N)\n",
+      "z_E = numpy.zeros(N)\n",
+      "\n",
+      "#Initial conditions\n",
+      "z_E[0]= 0.0\n",
+      "v_E[0]=0.0\n",
+      "\n",
+      "for i in range (N-1):\n",
+      "    vdot_E[i] =( ( -k*z_E[i] - (m*wo/Q)*(v_E[i])   +\\\n",
+      "                    Fd*numpy.cos(wo*t[i]) ) / m)  #Equation 7\n",
+      "    v_E[i+1] = v_E[i] + dt*vdot_E[i]  #Based on equation 5\n",
+      "    z_E[i+1] = z_E[i] + v_E[i]*dt  #Equation 5\n",
+      "\n",
+      "plt.title('Plot 2  Eulers approximation of Equation1', fontsize=20)        \n",
+      "plt.plot(t*1e3,z_E*1e9)\n",
+      "plt.xlabel('time, s', fontsize=18)\n",
+      "plt.ylabel('z_Euler, nm', fontsize=18)\n",
+      "#plt.ylim(-190,190)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 20,
+       "text": [
+        "<matplotlib.text.Text at 0x830f1d0>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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mJa3Ro0cjNTUVY8eORYcOHeDp6alUXERENkVbXbEjhrLMSlpJSUkYM2YMZs2a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\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x7e542b0>"
+       ]
+      }
+     ],
+     "prompt_number": 20
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This looks totally unphysical! We were expecting to have a steady state oscillation of 60 nm and we got a huge oscillation that keeps growing. Can it be due to the scheme? The timestep that we have chosen is quite big with relation to the oscillation period. we have set intentionally ONLY 28 time steps per period (That could be the reason why the scheme can't cath up the physics of the problem). That's quite discouraging. However the timestep is quite big and it really gets better as you decrease the time step. Try it! Reduce the time step and see how the numerical solution acquires an amplitude of 60 nm as the analytical one. At this point we can't state anything about accuracy before doing an analysis of error (we will make this soon). But first, let's try to analyze if another more efficient scheme can catch the physics of our damped harmonic oscillator even with this large time step."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "3.3 Let's try to get more accurate... Verlet Algorithm"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This is a very popular algorithm widely used in molecular dynamics simulations. Its popularity has been related to high stability when compared to the simple Euler method, it is also very simple to implement and accurate as we will see soon!  Verlet integration can be seen as using the central difference approximation to the second derivative. Consider the Taylor expansion of $z_{n+1}$ and $z_{n-1}$ around $z_i$:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n+1} = z_n + \\Delta t \\frac{dz}{dt}\\big|_n + \\frac{\\Delta t^2}{2} \\frac{d^2 z}{d t^2}\\big|_n + \\frac{\\Delta t^3}{3} \\frac{d^3 z}{d t^3}\\big|_n + {\\mathcal O}(\\Delta t^4)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n-1} = z_n - \\Delta t \\frac{dz}{dt}\\big|_n + \\frac{\\Delta t^2}{2} \\frac{d^2 z}{dt^2}\\big|_n - \\frac{\\Delta t^3}{3} \n",
+      "\\frac{d^3 z}{d t^3}\\big|_n + {\\mathcal O}(\\Delta t^4)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "Adding up these two expansions and solving for $z_{i+1}$ we get:\n",
+      "\n",
+      "$$z_{n+1}= 2z_{n} - z_{n-1} + \\frac{d^2 z}{d t^2} \\Delta t^2\\big|_n + {\\mathcal O}(\\Delta t^4) $$\n",
+      "\n",
+      "Verlet algorithm neglects terms on the order of 4 or higher, ending up being:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "z_{n+1}= 2z_{n} - z_{n-1} + \\frac{d^2 z}{d t^2} \\Delta t^2\\big|_n\n",
+      "\\end{equation}$$\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This looks nice, it seems that the straightforward calculation of the second derivative will give us good results. BUT have you seen that we also need the value of the first derivative (velocity) to put into the equation of motion that we are integrating (see equation 1). YES, that's a main drawback of this scheme and therefore is mainly used in applications where the equation to integrated doesn't have first derivative. But don't panic we will see what can we do...\n",
+      "\n",
+      "What about substracting equations 8 and 9 and then solving for $\\frac{dz}{dt}\\big|_n$:\n",
+      "$$\n",
+      "\\frac{dz}{dt}\\big|_n =  \\frac{z_{n+1} - z_{n-1}}{2\\Delta t} + {\\mathcal O}(\\Delta t^3)\n",
+      "$$\n",
+      "If we neglest terms on the order of 3 or higher we can calculate velocity:\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dz}{dt}\\big|_n =  \\frac{z_{n+1} - z_{n-1}}{2\\Delta t}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "This way of calculating velocity is pretty common in Verlet integration in applications where velocity is not explicit in the equation of motion. However for our purposes of solving equation 1 (where first derivative is explicitly present) it seems that we will loose accuracy because of the velocity, we will discuss more about this soon after..."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Have you noticed that we need a value $z_{n-1}$? Does it sound familiar?  YES!  This is not a self starting method. As a result we will have to overcome the issue by setting the initial conditions of the first step using Euler approximation. This is a bit annoying, but a couple of extra lines of code won't kill you :)"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "time_V = numpy.linspace(0,simultime,N)\n",
+      "\n",
+      "#Initializing variables for Verlet\n",
+      "zdoubledot_V = numpy.zeros(N)\n",
+      "zdot_V = numpy.zeros(N)\n",
+      "z_V = numpy.zeros(N)\n",
+      "\n",
+      "#Initial conditions Verlet. Look how we use Euler for the first step approximation!\n",
+      "z_V[0] = 0.0\n",
+      "zdot_V[0] = 0.0\n",
+      "zdoubledot_V[0] = (   ( -k*z_V[0] - (m*wo/Q)*zdot_V[0] +\\\n",
+      "                            Fd*numpy.cos(wo*t[0]) )  ) / m\n",
+      "zdot_V[1] = zdot_V[0] + zdoubledot_V[0]*dt\n",
+      "z_V[1] = z_V[0] + zdot_V[0]*dt\n",
+      "zdoubledot_V[1] = (   ( -k*z_V[1] - (m*wo/Q)*zdot_V[1] +\\\n",
+      "                            Fd*numpy.cos(wo*t[1]) )  ) / m\n",
+      "\n",
+      "#VERLET ALGORITHM\n",
+      "\n",
+      "for i in range(2,N):\n",
+      "    z_V[i] = 2*z_V[i-1] - z_V[i-2] + zdoubledot_V[i-1]*dt**2 #Eq 10\n",
+      "    zdot_V[i] = (z_V[i]-z_V[i-2])/(2.0*dt) #Eq 11\n",
+      "    zdoubledot_V[i] = (   ( -k*z_V[i] - (m*wo/Q)*zdot_V[i] +\\\n",
+      "                               Fd*numpy.cos(wo*t[i]) )  ) / m #from eq 1\n",
+      "    \n",
+      "plt.title('Plot 3  Verlet approximation of Equation1', fontsize=20) \n",
+      "plt.xlabel('time, ms', fontsize=18)\n",
+      "plt.ylabel('z_Verlet, nm', fontsize=18)\n",
+      "plt.plot(time_V*1e3, z_V*1e9, 'g-')\n",
+      "plt.ylim(-65,65)\n",
+      " \n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 21,
+       "text": [
+        "(-65, 65)"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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7vfLwt4Z75WtfvdYrX/LiJV75/OfO98pnTzrbK5/+l9O9cqcJnbzysb8/1isf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\nOPHEE9GhQwc88MADbJ0t5Vuwp3IPWjRo4aXfNyxs6H1rkU2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WqMn9kacCkkNLYp2WthaS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4o4kopJcHJITZslaJKAoVJPButZ1NoWS3wFrtP5z1mZWl/IO2KmCR4xScokLV+EfD/UGOCUFBdO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\nmWcwc+bMU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+       "text": [
+        "<matplotlib.figure.Figure at 0x8563b70>"
+       ]
+      }
+     ],
+     "prompt_number": 21
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It WAS ABLE cath up the physics! even with the big time step that we use with Euler scheme!\n",
+      "\n",
+      "As you can see, and as we previously discussed the harmonic response is composed by a transient and a steady part. We are only concerned about the steady state, since it is assumed that the probe achieves steady state motion during the imaging process. Therefore, we are going to slice our array in order to show only the last 10 oscillations, and we will see if it resembles the analytical solution."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#Slicing the full response vector to get the steady state response\n",
+      "z_steady_V = z_V[(90*period/dt):]\n",
+      "time_steady_V = time_V[(90*period/dt):]\n",
+      "\n",
+      "plt.title('Plot 3  Verlet approx. of steady state sol. of Eq 1', fontsize=20) \n",
+      "plt.xlabel('time, ms', fontsize=18)\n",
+      "plt.ylabel('z_Verlet, nm', fontsize=18)\n",
+      "plt.plot(time_steady_V*1e3, z_steady_V*1e9, 'g-')\n",
+      "plt.ylim(-65,65)\n",
+      "plt.show()\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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75ptvXL02pIwhLS3NtTWy7j8nd90OgPUhSoFzcNBAXQUuFnUVOAeHGDj3J8+d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\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x7e29630>"
+       ]
+      }
+     ],
+     "prompt_number": 22
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "3.4 Let's use now one of the most popular schemes...     The Runge Kutta 4!"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The Runge Kutta 4 (RK4) method is very popular for the solution of ODEs. This method is designed to solve 1st order differential equations. However we have converted our 2nd order ODE to a system of two coupled 1st order ODEs when we implemented the Euler shceme (equations 5 and 6). And we will  have to use this equations for the RK4 algorithm.\n",
+      "\n",
+      "In order to clearly see the RK4 implementation we are going to put equations 5 and 6 in the following form:\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dz}{dt}=v \\Rightarrow f1(t,z,v)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "\\frac{dv}{dt} = -kz-\\frac{m\\omega_0}{Q}+F_ocos(\\omega t) \\Rightarrow f2(t,z,v)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "It can be clearly seen that we have two coupled equations f1 and f2 and both depend in t, z, and v.\n",
+      "\n",
+      "The RK4 equations for our special case where we have two coupled equations, are the following:\n",
+      "$$\\begin{equation}\n",
+      "k_1 = f1(t_i, z_i, v_i)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "m_1 = f2(t_i, z_i, v_i)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_2 = f1(t_i +1/2\\Delta t, z_i + 1/2k_1\\Delta t, v_i + 1/2m_1\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "m_2 = f2(t_i +1/2\\Delta t, z_i + 1/2k_1\\Delta t, v_i + 1/2m_1\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_3 = f1(t_i +1/2\\Delta t, z_i + k_2\\Delta t, v_i + 1/2m_2\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "m_3 = f2(t_i +1/2\\Delta t, z_i + 1/2k_2\\Delta t, v_i + 1/2m_2\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_4 = f1(t_i + \\Delta t, z_i + k_3\\Delta t, v_i + m_3\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "k_4 = f2(t_i + \\Delta t, z_i + k_3\\Delta t, v_i + m_3\\Delta t)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "f1_{n+1} = f1_n + \\Delta t/6(k_1+2k_2+2k_3+k_4)\n",
+      "\\end{equation}$$\n",
+      "$$\\begin{equation}\n",
+      "f2_{n+1} = f2_n + \\Delta t/6(m_1+2m_2+2m_3+m_4)\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "Please notice how k values and m values are intercalated, since it is crucial in the implementation of the method!\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#Definition of v, z, vectors\n",
+      "vdot_RK4 = numpy.zeros(N)\n",
+      "v_RK4 = numpy.zeros(N)\n",
+      "z_RK4 = numpy.zeros(N)\n",
+      "k1v_RK4 = numpy.zeros(N)\n",
+      "k2v_RK4 = numpy.zeros(N)\n",
+      "k3v_RK4 = numpy.zeros(N)\n",
+      "k4v_RK4 = numpy.zeros(N)\n",
+      "\n",
+      "k1z_RK4 = numpy.zeros(N)\n",
+      "k2z_RK4 = numpy.zeros(N)\n",
+      "k3z_RK4 = numpy.zeros(N)\n",
+      "k4z_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "#calculation of velocities RK4\n",
+      "\n",
+      "#INITIAL CONDITIONS\n",
+      "v_RK4[0] = 0\n",
+      "z_RK4[0] = 0\n",
+      "\n",
+      "   \n",
+      "for i in range (1,N):\n",
+      "       #RK4\n",
+      "        k1z_RK4[i] = v_RK4[i-1]   #k1 Equation 14 \n",
+      "        k1v_RK4[i] = ((   ( -k*z_RK4[i-1] - (m*wo/Q)*v_RK4[i-1] + \\\n",
+      "                           Fd*numpy.cos(wo*t[i-1]) )  ) / m )   #m1 Equation 15\n",
+      "        \n",
+      "        k2z_RK4[i] = ((v_RK4[i-1])+k1v_RK4[i]/2.*dt) #k2 Equation 16\n",
+      "        k2v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k1z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] +k1v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m ) #m2 Eq 17\n",
+      "        \n",
+      "        k3z_RK4[i] = ((v_RK4[i-1])+k2v_RK4[i]/2.*dt) #k3, Equation 18\n",
+      "        k3v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k2z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                            (v_RK4[i-1] +k2v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                            numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m ) #m3, Eq 19\n",
+      "                     \n",
+      "        k4z_RK4[i] = ((v_RK4[i-1])+k3v_RK4[i]*dt) #k4, Equation 20\n",
+      "        k4v_RK4[i] = ((   ( -k*(z_RK4[i-1] + k3z_RK4[i]*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] + k3v_RK4[i]*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt)) )  ) / m )#m4, Eq 21\n",
+      "        \n",
+      "        #Calculation of velocity, Equation 23\n",
+      "        v_RK4[i] = v_RK4[i-1] + 1./6*dt*(k1v_RK4[i] + 2.*k2v_RK4[i] +\\\n",
+      "                        2.*k3v_RK4[i] + k4v_RK4[i] )   \n",
+      "        #calculation of position, Equation 22\n",
+      "        z_RK4 [i] = z_RK4[i-1] + 1./6*dt*(k1z_RK4[i] + 2.*k2z_RK4[i] +\\\n",
+      "                        2.*k3z_RK4[i] + k4z_RK4[i] ) \n",
+      "\n",
+      "#slicing array to get steady state\n",
+      "z_steady_RK4 = z_RK4[(90.*period/dt):]\n",
+      "time_steady_RK4 = t[(90.*period/dt):]\n",
+      "    \n",
+      "plt.title('Plot 3  RK4 approx. of steady state sol. of Eq 1', fontsize=20) \n",
+      "plt.xlabel('time, ms', fontsize=18)\n",
+      "plt.ylabel('z_RK4, nm', fontsize=18)\n",
+      "plt.plot(time_steady_RK4 *1e3, z_steady_RK4*1e9, 'r-')\n",
+      "plt.ylim(-65,65)\n",
+      "plt.show()\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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OmDED1dXV+OCDD1S556+//hrvvfcejh49iry8PFRWVhq/EwQBubm5FuG9jh07Ws2y7tOn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\nEELoSBqvEEIIoSNpvEIIIYSO/g/oQttYxhXYvAAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0xb2021d0>"
+       ]
+      }
+     ],
+     "prompt_number": 23
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "3.4 Error Analysis"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Let's plot together our solutions using the different schemes along with our analytical reference."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plt.title('Plot 4  Schemes comparison with analytical sol.', fontsize=20) \n",
+      "plt.plot(time_an_steady*1e3, z_an_steady*1e9, 'b--' )\n",
+      "plt.plot(time_steady_V*1e3, z_steady_V*1e9, 'g-' )\n",
+      "plt.plot(time_steady_RK4*1e3, z_steady_RK4*1e9, 'r-')\n",
+      "plt.xlim(2.0, 2.06)\n",
+      "plt.legend(['Analytical solution', 'Verlet method', 'Runge Kutta 4'])\n",
+      "plt.xlabel('time, ms', fontsize=18)\n",
+      "plt.ylabel('z_position, nm', fontsize=18)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 24,
+       "text": [
+        "<matplotlib.text.Text at 0xb1ffbe0>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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I7/LLcCgsCieU/PGab4jS9GcylWPhwoUNkr/m8msu+SsrL0P8k3j8EPsDPLd4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\n+g0UJaKYknvDEQAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x7d3e9b0>"
+       ]
+      }
+     ],
+     "prompt_number": 24
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It was pointless to include Euler in the last plot because it was not following the physics at all for this given time step. REMEMBER that Euler can give fair approximations, but you MUST decrease the time step in this particular case if you want to see the sinusoidal trajectory!\n",
+      "It seems our different schemes are making a different job quality in approximating the solution. However is hard to conclude something strong based on this qualitative obervations. In order to state something stronger we have to make an error analysis. We will choose L1 norm for this purpose (You can find more information about this [L1](http://en.wikipedia.org/wiki/Taxicab_geometry) ) and it was also discussed on the second IPython Notebook of the series _\"The phugoid model of glider flight\"_, the first learning module of the course [**\"Practical Numerical Methods with Python.\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about)."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "\"\"\"ERROR ANALYSIS EULER, VERLET AND RK4\"\"\"\n",
+      "\n",
+      "# time-increment array\n",
+      "dt_values = numpy.array([8.0e-7, 2.0e-7, 0.5e-7, 1e-8, 0.1e-8])\n",
+      "\n",
+      "# array that will contain solution of each grid\n",
+      "z_values_E = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "z_values_V = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "z_values_RK4 = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "z_values_an = numpy.zeros_like(dt_values, dtype=numpy.ndarray)\n",
+      "\n",
+      "for n, dt in enumerate(dt_values):\n",
+      "    simultime = 100*period\n",
+      "    timestep = dt\n",
+      "    N = int(simultime/dt)\n",
+      "    t = numpy.linspace(0.0, simultime, N)\n",
+      "   \n",
+      "    #Initializing variables for Verlet\n",
+      "    zdoubledot_V = numpy.zeros(N)\n",
+      "    zdot_V = numpy.zeros(N)\n",
+      "    z_V = numpy.zeros(N)\n",
+      "    \n",
+      "    #Initializing variables for RK4\n",
+      "    vdot_RK4 = numpy.zeros(N)\n",
+      "    v_RK4 = numpy.zeros(N)\n",
+      "    z_RK4 = numpy.zeros(N)\n",
+      "    k1v_RK4 = numpy.zeros(N) \n",
+      "    k2v_RK4 = numpy.zeros(N)\n",
+      "    k3v_RK4 = numpy.zeros(N)\n",
+      "    k4v_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "    k1z_RK4 = numpy.zeros(N)\n",
+      "    k2z_RK4 = numpy.zeros(N)\n",
+      "    k3z_RK4 = numpy.zeros(N)\n",
+      "    k4z_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "          \n",
+      "    #Initial conditions Verlet (started with Euler approximation)\n",
+      "    z_V[0] = 0.0\n",
+      "    zdot_V[0] = 0.0\n",
+      "    zdoubledot_V[0] = (   ( -k*z_V[0] - (m*wo/Q)*zdot_V[0] + \\\n",
+      "                           Fd*numpy.cos(wo*t[0]) )  ) / m\n",
+      "    zdot_V[1] = zdot_V[0] + zdoubledot_V[0]*timestep**2\n",
+      "    z_V[1] = z_V[0] + zdot_V[0]*dt\n",
+      "    zdoubledot_V[1] = (   ( -k*z_V[1] - (m*wo/Q)*zdot_V[1] + \\\n",
+      "                           Fd*numpy.cos(wo*t[1]) )  ) / m\n",
+      "    \n",
+      "    \n",
+      "    #Initial conditions Runge Kutta\n",
+      "    v_RK4[1] = 0\n",
+      "    z_RK4[1] = 0 \n",
+      "    \n",
+      "    #Initialization variables for Analytical solution\n",
+      "    z_an = numpy.zeros(N)\n",
+      "       \n",
+      "    # time loop \n",
+      "    for i in range(2,N):\n",
+      "             \n",
+      "        #Verlet\n",
+      "        z_V[i] = 2*z_V[i-1] - z_V[i-2] + zdoubledot_V[i-1]*dt**2 #Eq 10\n",
+      "        zdot_V[i] = (z_V[i]-z_V[i-2])/(2.0*dt) #Eq 11\n",
+      "        zdoubledot_V[i] = (   ( -k*z_V[i] - (m*wo/Q)*zdot_V[i] +\\\n",
+      "                                   Fd*numpy.cos(wo*t[i]) )  ) / m #from eq 1\n",
+      "        \n",
+      "        #RK4\n",
+      "        k1z_RK4[i] = v_RK4[i-1]   #k1 Equation 14 \n",
+      "        k1v_RK4[i] = ((   ( -k*z_RK4[i-1] - (m*wo/Q)*v_RK4[i-1] + \\\n",
+      "                           Fd*numpy.cos(wo*t[i-1]) )  ) / m )   #m1 Equation 15\n",
+      "        \n",
+      "        k2z_RK4[i] = ((v_RK4[i-1])+k1v_RK4[i]/2.*dt) #k2 Equation 16\n",
+      "        k2v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k1z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] +k1v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m )  #m2 Eq 17\n",
+      "        \n",
+      "        k3z_RK4[i] = ((v_RK4[i-1])+k2v_RK4[i]/2.*dt) #k3, Equation 18\n",
+      "        k3v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k2z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                            (v_RK4[i-1] +k2v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                            numpy.cos(wo*(t[i-1] + dt/2.)) )  ) / m ) #m3, Eq 19\n",
+      "                     \n",
+      "        k4z_RK4[i] = ((v_RK4[i-1])+k3v_RK4[i]*dt) #k4, Equation 20\n",
+      "        k4v_RK4[i] = ((   ( -k*(z_RK4[i-1] + k3z_RK4[i]*dt) - (m*wo/Q)*\\\n",
+      "                           (v_RK4[i-1] + k3v_RK4[i]*dt) + Fd*\\\n",
+      "                           numpy.cos(wo*(t[i-1] + dt)) )  ) / m )#m4, Equation 21\n",
+      "        \n",
+      "        #Calculation of velocity, Equation 23\n",
+      "        v_RK4[i] = v_RK4[i-1] + 1./6*dt*(k1v_RK4[i] + 2.*k2v_RK4[i] +\\\n",
+      "                        2.*k3v_RK4[i] + k4v_RK4[i] )   \n",
+      "        #calculation of position, Equation 22\n",
+      "        z_RK4 [i] = z_RK4[i-1] + 1./6*dt*(k1z_RK4[i] + 2.*k2z_RK4[i] +\\\n",
+      "                         2.*k3z_RK4[i] + k4z_RK4[i] ) \n",
+      "\n",
+      "             \n",
+      "        #Analytical solution\n",
+      "        A_an = Fo_an*Q/k  #when driven at resonance A is simply Fo*Q/k\n",
+      "        phi = numpy.pi/2  #when driven at resonance the phase is pi/2\n",
+      "        z_an[i] = A_an*numpy.cos(wo*t[i] - phi)  #Analytical solution eq. 1\n",
+      "    \n",
+      "          \n",
+      "    #Slicing the full response vector to get the steady state response\n",
+      "    z_steady_V = z_V[(80*period/timestep):]\n",
+      "    z_an_steady = z_an[(80*period/timestep):]\n",
+      "    z_steady_RK4 = z_RK4[(80*period/timestep):]\n",
+      "    time_steady = t[(80*period/timestep):]\n",
+      "    \n",
+      "    z_values_V[n] = z_steady_V.copy()  # error for certain value of timestep\n",
+      "    z_values_RK4[n] = z_steady_RK4.copy() #error for certain value of timestep\n",
+      "    z_values_an[n] = z_an_steady.copy()  #error for certain value of timestep\n",
+      "\n",
+      "\n",
+      "def get_error(z, z_exact, dt):\n",
+      "    #Returns the error with respect to the analytical solution using L1 norm\n",
+      "       \n",
+      "    return dt * numpy.sum(numpy.abs(z-z_exact))\n",
+      "    \n",
+      "#NOW CALCULATE THE ERROR FOR EACH RESPECTIVE DELTA T\n",
+      "error_values_V = numpy.zeros_like(dt_values)\n",
+      "error_values_RK4 = numpy.zeros_like(dt_values)\n",
+      "\n",
+      "for i, dt in enumerate(dt_values):\n",
+      "    ### call the function get_error() ###\n",
+      "    error_values_V[i] = get_error(z_values_V[i], z_values_an[i], dt)\n",
+      "    error_values_RK4[i] = get_error(z_values_RK4[i], z_values_an[i], dt)\n",
+      "\n",
+      "\n",
+      "plt.figure(1)\n",
+      "plt.title('Plot 5  Error analysis Verlet based on L1 norm', fontsize=20) \n",
+      "plt.tick_params(axis='both', labelsize=14) \n",
+      "plt.grid(True)                         #turn on grid lines\n",
+      "plt.xlabel('$\\Delta t$ Verlet', fontsize=16)  #x label\n",
+      "plt.ylabel('Error Verlet', fontsize=16)       #y label\n",
+      "plt.loglog(dt_values, error_values_V, 'go-')  #log-log plot\n",
+      "plt.axis('equal')                      #make axes scale equally;\n",
+      "\n",
+      "plt.figure(2)\n",
+      "plt.title('Plot 6  Error analysis RK4 based on L1 norm', fontsize=20) \n",
+      "plt.tick_params(axis='both', labelsize=14) \n",
+      "plt.grid(True)     #turn on grid lines\n",
+      "plt.xlabel('$\\Delta t$ RK4', fontsize=16)  #x label\n",
+      "plt.ylabel('Error RK4', fontsize=16)       #y label\n",
+      "plt.loglog(dt_values, error_values_RK4,  'co-')  #log-log plot\n",
+      "plt.axis('equal')                      #make axes scale equally;\n",
+      "\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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vKvmVl2tF85hyfrpuO2Pmp2tuppKfobadPvIzp/emOeR38eJFg/XL5shki6up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\nbN0AAAAASUVORK5CYII=\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0xb1f1b38>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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qPXyh/EybkPMTcm6A4fJLTk4W3REmbzdfl0qlGm9KLjdnzhy93iT6SWLb+IQQ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\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0xb21ee10>"
+       ]
+      }
+     ],
+     "prompt_number": 25
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "As we can see Runge Kutta 4 converges faster than Verlet for the range of time steps studied. And the difference between both is near one order of magnitude. One additional advantage with Runge Kutta 4 is that the method is very stable, even with big time steps (eg. 10 time steps per period) the method is able to catch up the physics of the oscillation, something where Verlet is not so good at."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Let's add a sample and oscillate our probe over it!"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "It is very common in the field of probe microscopy to model the tip sample interactions through DMT contact mechanics. \n",
+      "DMT stands for Derjaguin, Muller and Toporov who were the scientists that made the model (see ref 1). This model uses Hertz contact mechanics (see ref 2) with the addition of long range tip-sample interactions. This long range tip-sample interactions are ascribed to intermolecular interactions between the atoms of the tip and the upper atoms of the surface, and include mainly the contribution of van de Waals forces and Pauli repulsion from electronic clouds when the atoms of the tip meet closely the atoms of the surface. Figure 2 displays a force vs distance curve (FD curve) where it is shown how the forces between the tip and the sample behave with regards of the separation. It can be seen that at positive distances the tip starts \"feeling\" attraction from the tip (from the contribution of van der Waals forces) where the slope of the curve is positive and at some minimum distance ($a_0$) the tip starts experiencing repulsive interactions arisen from electronic cloud repulsion (area where the slope of the curve is negative and the forces are negative). At lower distances, an area known as \"contact area\" is arisen and it is characterized by a negative slope and a emerging positive force."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image\n",
+      "Image(filename=\"C:/Users/Enrique Alejandro/Documents/GitHub/FinalProjectMAE6286\\FinalProject/Fig3FDcurve.jpg\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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Ieh/+m+CvBf2iP27vjLpP/BQG4+BHww8L+BNVvr7w/b6rYahrxuooNMbcWnmu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\nLnX4tf8AYWtv/bivuWvhr/g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rMWiXd0ddcWy3MMzw2UEnlyGMurHYqExEjBbFfQvxH/AOC4vw1/aP8A2Fvj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\nv6YUUUVBYUUUUAFFFFABRRRQAUUUUAFFFFABRRRQB//Z\n",
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 11,
+       "text": [
+        "<IPython.core.display.Image at 0x81d5f98>"
+       ]
+      }
+     ],
+     "prompt_number": 11
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Figure 3.   Force vs Distance profile depicting tip-sample interactions in AFM"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In Hertz contact mechanics, one central aspect is to consider that the contact area increases as the sphere is pressed against an elastic surface, and this increase of the contact area \"modulates\" the effective stiffness of the sample. This concept is represented in figure 4 where the sample is depicted as comprised by a series of springs that are activated as the tip does deeper into the sample. In other words, the deeper the sample goes, the higher the contact area and therefore more springs are activated. \n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Image\n",
+      "Image(filename=\"C:/Users/Enrique Alejandro/Documents/GitHub/FinalProjectMAE6286\\FinalProject/Fig4Hertzspring.jpg\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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/8co/4bH+EX/RVPhx/wCFLZf/AByvnc54fy7MpKtUfJVj8NSEuWcfmt15O68j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\nVrCy1TTroBZrW7gWeGYAggMjAqeQDyOoFXaKKAP/2Q==\n",
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 12,
+       "text": [
+        "<IPython.core.display.Image at 0x81d5cf8>"
+       ]
+      }
+     ],
+     "prompt_number": 12
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "####Figure 4.  Conceptual representation of Hertz contact mechanics"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "This concept is represented mathematically by a non-linear spring whose elastic coefficient is a function of the contact area which at the same time depends on the sample indentation ( k(d)  ).\n",
+      "$$F_{ts} = k(d)d$$\n",
+      "where\n",
+      "$$k(d) = 4/3E*\\sqrt{Rd}$$\n",
+      "being $\\sqrt{Rd}$ the contact area when a sphere of radius R indents a half-sapace to depth d.\n",
+      "$E*$ is the effective Young's modulus of the tip-sample interaction. "
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The long range attractive forces are derived using Hamaker's equation (see ref. 4): $if$ $d > a_0$\n",
+      "$$F_{ts} = \\frac{-HR}{6d^2}$$\n",
+      "\n",
+      "where H is the Hamaker constant, R the tip radius and d the tip sample distance. $a_0$ is defined as the intermolecular distance and normally is chosen to be 0.2 nm."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "In summary the equations that we will include in our code to take care of the tip sample interactions are the following:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "Fts_{DMT} = \\begin{cases} \\frac{-HR}{6d^2} \\quad \\quad d \\leq{a_0}\\\\ \\\\\n",
+      "\\frac{-HR}{6d^2} + 4/3E*R^{1/2}d^{3/2}   \\quad \\quad d> a_0 \\end{cases}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "where the effective Young's modulus E* is defined by:\n",
+      "$$\\begin{equation}\n",
+      "1/E* = \\frac{1-\\nu^2}{E_t}+\\frac{1-\\nu^2}{E_s}\n",
+      "\\end{equation}$$\n",
+      "where $E_t$ and $E_s$ are the tip and sample Young's modulus respectively. $\\nu_t$ and $\\nu_s$ are tip and sample Poisson ratios, respectively.\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Enough theory, Let's make our code!"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now we will have to solve equation (1) but with the addition of tip-sample interactions which are described by equation (5). So we have a second order non-linear ODE which is no longer analitically straigthforward approachable:\n",
+      "\n",
+      "$$\\begin{equation}\n",
+      "m \\frac{d^2z}{dt^2} = - k z - \\frac{m\\omega_0}{Q}\\frac{dz}{dt} + F_0 cos(\\omega t) + Fts_{DMT}\n",
+      "\\end{equation}$$\n",
+      "\n",
+      "Therefore we have to use numerical methods to solve it. RK4 has shown to be more accurate to solve equation (1) among the methods reviewed in the previous section of the notebook, and therefore it is going to be the chosen method to solve equation (6).\n",
+      "\n",
+      "Now we have to declare all the variables related to the tip-sample forces. Since we are modelling our tip-sample forces using Hertz contact mechanics with addition of long range Van der Waals forces we have to define tha Young's modulus of the tip and sample, the diameter of the tip of our probe, Poisson ratio, etc."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#DMT parameters  (Hertz contact mechanics with long range Van der Waals forces added\n",
+      "a=0.2e-9; #intermolecular parameter\n",
+      "H=6.4e-20; #hamaker constant of sample\n",
+      "R=20e-9; #tip radius of the cantilever\n",
+      "Es=70e6; #elastic modulus of sample\n",
+      "Et=130e9; #elastic modulus of the tip\n",
+      "vt=0.3; #Poisson coefficient for tip\n",
+      "vs=0.3; #Poisson coefficient for sample\n",
+      "E_star= 1/((1-pow(vt,2))/Et+(1-pow(vs,2))/Es);   #Effective Young Modulus"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 17
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now let's declare the timestep, the simulation time and let's oscillate our probe!"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "#IMPORTANT distance where you place the probe above the sample\n",
+      "z_base = 40.e-9 \n",
+      "\n",
+      "spp = 280. # time steps per period \n",
+      "dt = period/spp \n",
+      "\n",
+      "simultime = 100.*period\n",
+      "N = int(simultime/dt)\n",
+      "t = numpy.linspace(0,simultime,N)\n",
+      "\n",
+      "#Initializing variables for RK4\n",
+      "v_RK4 = numpy.zeros(N)\n",
+      "z_RK4 = numpy.zeros(N)\n",
+      "k1v_RK4 = numpy.zeros(N) \n",
+      "k2v_RK4 = numpy.zeros(N)\n",
+      "k3v_RK4 = numpy.zeros(N)\n",
+      "k4v_RK4 = numpy.zeros(N)\n",
+      "    \n",
+      "k1z_RK4 = numpy.zeros(N)\n",
+      "k2z_RK4 = numpy.zeros(N)\n",
+      "k3z_RK4 = numpy.zeros(N)\n",
+      "k4z_RK4 = numpy.zeros(N)\n",
+      "\n",
+      "TipPos = numpy.zeros(N)\n",
+      "Fts = numpy.zeros(N)\n",
+      "Fcos = numpy.zeros(N)\n",
+      "\n",
+      "for i in range(1,N):\n",
+      "   #RK4\n",
+      "    k1z_RK4[i] = v_RK4[i-1]   #k1 Equation 14 \n",
+      "    k1v_RK4[i] = ((   ( -k*z_RK4[i-1] - (m*wo/Q)*v_RK4[i-1] + \\\n",
+      "                       Fd*numpy.cos(wo*t[i-1]) +Fts[i-1])  ) / m )   #m1 Equation 15\n",
+      "    \n",
+      "    k2z_RK4[i] = ((v_RK4[i-1])+k1v_RK4[i]/2.*dt) #k2 Equation 16\n",
+      "    k2v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k1z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                       (v_RK4[i-1] +k1v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                       numpy.cos(wo*(t[i-1] + dt/2.))  +Fts[i-1])  ) / m )  #m2 Eq 17\n",
+      "    \n",
+      "    k3z_RK4[i] = ((v_RK4[i-1])+k2v_RK4[i]/2.*dt) #k3, Equation 18\n",
+      "    k3v_RK4[i] = ((   ( -k*(z_RK4[i-1]+ k2z_RK4[i]/2.*dt) - (m*wo/Q)*\\\n",
+      "                        (v_RK4[i-1] +k2v_RK4[i]/2.*dt) + Fd*\\\n",
+      "                        numpy.cos(wo*(t[i-1] + dt/2.))  +Fts[i-1])  ) / m ) #m3, Eq19\n",
+      "                 \n",
+      "    k4z_RK4[i] = ((v_RK4[i-1])+k3v_RK4[i]*dt) #k4, Equation 20\n",
+      "    k4v_RK4[i] = ((   ( -k*(z_RK4[i-1] + k3z_RK4[i]*dt) - (m*wo/Q)*\\\n",
+      "                       (v_RK4[i-1] + k3v_RK4[i]*dt) + Fd*\\\n",
+      "                       numpy.cos(wo*(t[i-1] + dt))  +Fts[i-1])  ) / m )#m4, Eq 21\n",
+      "    \n",
+      "    #Calculation of velocity, Equation 23\n",
+      "    v_RK4[i] = v_RK4[i-1] + 1./6*dt*(k1v_RK4[i] + 2.*k2v_RK4[i] +\\\n",
+      "                    2.*k3v_RK4[i] + k4v_RK4[i] )   \n",
+      "    #calculation of position, Equation 22\n",
+      "    z_RK4 [i] = z_RK4[i-1] + 1./6*dt*(k1z_RK4[i] + 2.*k2z_RK4[i] +\\\n",
+      "                     2.*k3z_RK4[i] + k4z_RK4[i] ) \n",
+      "           \n",
+      "    TipPos[i] = z_base + z_RK4[i] #Adding base position to z position\n",
+      "     \n",
+      "    #calculation of DMT force\n",
+      "\n",
+      "    if  TipPos[i] > a: #this defines the attractive regime\n",
+      "        Fts[i] = -H*R/(6*(TipPos[i])**2)\n",
+      "    else:  #this defines the repulsive regime\n",
+      "        Fts[i] = -H*R/(6*a**2)+4./3*E_star*numpy.sqrt(R)*(a-TipPos[i])**1.5\n",
+      "    \n",
+      "          \n",
+      "    Fcos[i] = Fd*numpy.cos(wo*t[i])  #Driving force (this will be helpful to plot the driving force)\n",
+      "\n",
+      "#Slicing arrays to get steady state\n",
+      "TipPos_steady = TipPos[(95*period/dt):] \n",
+      "t_steady = t[(95*period/dt):]   \n",
+      "Fcos_steady = Fcos[(95*period/dt):]   \n",
+      "Fts_steady = Fts[(95*period/dt):] \n",
+      "\n",
+      "plt.figure(1)\n",
+      "fig, ax1 = plt.subplots()\n",
+      "ax2 = ax1.twinx()\n",
+      "ax1.plot(t_steady*1e3,TipPos_steady*1e9, 'g-')\n",
+      "ax2.plot(t_steady*1e3, Fcos_steady*1e9, 'b-')\n",
+      "ax1.set_xlabel('Time,s')\n",
+      "ax1.set_ylabel('Tip position (nm)', color='g')\n",
+      "ax2.set_ylabel('Drive Force (nm)', color='b')\n",
+      "plt.title('Plot 7   Tip response and driving force', fontsize = 20)\n",
+      "plt.show()\n",
+      "\n",
+      "plt.figure(2)\n",
+      "plt.title('Plot 8  Force-Distance curve', fontsize=20) \n",
+      "plt.plot(TipPos*1e9, Fts*1e9, 'b--' )\n",
+      "plt.xlabel('Tip Position, nm', fontsize=18)\n",
+      "plt.ylabel('Force, nN', fontsize=18)\n",
+      "plt.xlim(-20, 30)\n",
+      "\n",
+      "\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "text": [
+        "<matplotlib.figure.Figure at 0xa041f98>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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H9sfhosMy9sx+TpacRIhXCKJ01xaGqxgVMnplYG/uXhtnui6GZgPOVZzDLdHm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\nYjAYDIZi+X9R1nTjWMLqVgAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0xa041780>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 27,
+       "text": [
+        "(-20, 30)"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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TJRAhBCIjI+Ht7a1KIAAwbtw4xMXFAQDi4uJM0n1mKkwcRNTQVNsSKSgowIcf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\nEZHBmESIiMhgTCJERGQwJhEiIjLY/we1Rb7tjo3fbgAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x800c048>"
+       ]
+      }
+     ],
+     "prompt_number": 27
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Check that we have to sinusoidals. The one in green (the output) is the response signal of the tip (the tip trajectory in time) while the blue one (the input) is the cosinusoidal driving force that we are using to excite the tip. When the tip is excited in free air (without tip sample interactions) the phase lag between the output and the input is 90 degrees. You can test that with the previous code by only changing the position of the base to a high position enough that it does not interact with the sample. However in the above plot the phase lag is less than 90 degrees. Interestingly the phase can give relative about the material properties of the sample. There is a well developed theory if this in tapping mode AFM and is called phase spectroscopy. If you are interested on this topic you can read reference 1.\n",
+      "Also look at the above plot and see that the response amplitude is no longer 60 nm as we initially set (in this case is near 45!). It means that we have experienced a significant amplitude reduction due to the tip sample interactions.\n",
+      "Besides with the data acquired we are able to plot a Force-curve as the one shown in Figure 3! It shows the attractive and repulsive interactions of our probe with the surface.\n",
+      "\n",
+      "We have arrived to the end of the notebook. I hope you have found it interesting!"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.core.display import HTML\n",
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+      "HTML(open(css_file, \"r\").read())"
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+      "REFERENCES\n",
+      "\n",
+      "1. Garc\u0131\u0301a, Ricardo, and Ruben Perez. \"Dynamic atomic force microscopy methods.\" Surface science reports 47.6 (2002): 197-301.\n",
+      "2. B. V. Derjaguin, V. M. Muller, and Y. P. Toporov, J. Colloid\n",
+      "Interface Sci. 53, 314  (1975)\n",
+      "3. Hertz, H. R., 1882, Ueber die Beruehrung elastischer Koerper (On Contact Between Elastic Bodies), in Gesammelte Werke (Collected Works), Vol. 1, Leipzig, Germany, 1895.\n",
+      "4. Van Oss, Carel J., Manoj K. Chaudhury, and Robert J. Good. \"Interfacial Lifshitz-van der Waals and polar interactions in macroscopic systems.\" Chemical Reviews 88.6 (1988): 927-941."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}
\ No newline at end of file