complex_vibration_modes_DE.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='top'></a>\n",
    "\n",
    "# Complex vibration modes\n",
    "\n",
    "Complex vibration modes arise in experimental research and numerical simulations with non proportional damping. In such cases the eigenproblem is [nonlinear](http://en.wikipedia.org/wiki/Nonlinear_eigenproblem), thus requiring additional work in order to obtain the vibration frequencies and mode shapes of the dynamic system.\n",
    "\n",
    "The solution to the free vibration response of the dynamic equilibrium equation involves some form of linearization of the [quadratic eigenvalue problem](http://en.wikipedia.org/wiki/Quadratic_eigenvalue_problem) it yields. For more information on this subject see for example [this report](http://prod.sandia.gov/techlib/access-control.cgi/2007/072072.pdf).\n",
    "\n",
    "## Table of contents\n",
    "\n",
    "[Preamble](#Preamble)\n",
    "\n",
    "[Dynamic equilibrium equation](#Dynamic-equilibrium-equation)\n",
    "\n",
    "[State space formulation](#State-space-formulation)\n",
    "\n",
    "[Dynamic system setup](#Dynamic-system-setup)\n",
    "\n",
    "[Undamped system](#Undamped-system)\n",
    "\n",
    "[Proportionally damped system](#Proportionally-damped-system)\n",
    "\n",
    "[Non proportionally damped system](#Non-proportionally-damped-system)\n",
    "\n",
    "[Conclusions](#Conclusions)\n",
    "\n",
    "[Odds and ends](#Odds-and-ends)\n",
    "\n",
    "## Preamble\n",
    "\n",
    "We will start by setting up the computational environment for this notebook. Furthermore, we will need numpy and scipy for the numerical simulations and matplotlib for the plots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "System: 3.6.1 |Anaconda 4.4.0 (64-bit)| (default, May 11 2017, 13:25:24) [MSC v.1900 64 bit (AMD64)]\n",
      "numpy version: 1.12.1\n",
      "scipy version: 0.19.0\n",
      "matplotlib version: 2.0.2\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib as mpl\n",
    "\n",
    "print('System: {}'.format(sys.version))\n",
    "print('numpy version: {}'.format(np.__version__))\n",
    "print('scipy version: {}'.format(sp.__version__))\n",
    "print('matplotlib version: {}'.format(mpl.__version__))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will also need a couple of specific modules and a litle \"IPython magic\" to show the plots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy import linalg as LA\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to top](#top)\n",
    "\n",
    "## Dynamic equilibrium equation\n",
    "\n",
    "In structural dynamics the second order differential dynamic equilibrium equation can be written in terms of generalized coordinates (d[isplacement]) and their first (v[elocity]) and second (a[cceleration]) time derivatives:\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{M} \\times \\mathbf{a(t)} + \\mathbf{C} \\times \\mathbf{v(t)} + \\mathbf{K} \\times \\mathbf{d(t)} = \\mathbf{F(t)}\n",
    "\\end{equation}\n",
    "\n",
    "where:\n",
    "\n",
    "> $\\mathbf{M}$ is the mass matrix  \n",
    "> $\\mathbf{C}$ is the damping matrix  \n",
    "> $\\mathbf{K}$ is the stiffness matrix  \n",
    "> $\\mathbf{a(t)}$ is the acceleration vector  \n",
    "> $\\mathbf{v(t)}$ is the velocity vector  \n",
    "> $\\mathbf{d(t)}$ is the displacement vector  \n",
    "> $\\mathbf{F(t)}$ is the force input vector  \n",
    "\n",
    "Considering a dynamic system with $NDOF$ is the number of generalized degrees of freedom, the vectors will have dimensions of $NDOF \\times 1$ and the matrices $NDOF \\times NDOF$.\n",
    "\n",
    "When the system is undamped, the damping matrix will be null. In this case the eigenproblem is linear:\n",
    "\n",
    "\\begin{equation}\n",
    "\\left[ -\\mathbf{M} \\times \\mathbf{\\omega^2} + \\mathbf{K} \\right] \\times \\mathbf{v} = \\mathbf{0}\n",
    "\\end{equation}\n",
    "\n",
    "In a proportionally damped system, the damping matrix is proportional to the mass and stiffness matrices:\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{C} = \\alpha \\times \\mathbf{M} + \\beta \\times \\mathbf{K}\n",
    "\\end{equation}\n",
    "\n",
    "where $\\alpha$ and $\\beta$ are mass and stiffness proportionality coefficients. These are typically very small positive numbers. The resulting eigenproblem is still linear because the damping matrix can be decomposed by the modal vectors.\n",
    "\n",
    "In a system with non proportional damping, the damping matrix will not be proportional neither to the mass nor to the stiffness matrices."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to top](#top)\n",
    "\n",
    "## Dynamic system setup\n",
    "\n",
    "In this example we will use the folowing mass and stiffness matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.]\n",
      " [ 0.  2.]]\n"
     ]
    }
   ],
   "source": [
    "MM = np.matrix(np.diag([1., 2.]))\n",
    "print(MM)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 20. -10.]\n",
      " [-10.  10.]]\n"
     ]
    }
   ],
   "source": [
    "KK = np.matrix([[20., -10.], [-10., 10.]])\n",
    "print(KK)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will also setup two damping matrices, one proportional to the mass and stiffness matrices (C1) and the other non proportional (C2):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.9 -0.4]\n",
      " [-0.4  0.6]]\n"
     ]
    }
   ],
   "source": [
    "C1 = 0.1*MM+0.04*KK\n",
    "print(C1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.1  0.2]\n",
      " [ 0.2  0.2]]\n"
     ]
    }
   ],
   "source": [
    "C2 = np.matrix([[0.1, 0.2], [0.2, 0.2]])\n",
    "print(C2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to top](#top)\n",
    "\n",
    "## Undamped system\n",
    "\n",
    "In the undamped system the damping matrix is all zeros and therefore the eigenproblem is a linear one as it involves only the mass and stiffness matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.1922  22.8078]\n",
      "[[ 0.4896  0.9628]\n",
      " [ 0.8719 -0.2703]]\n"
     ]
    }
   ],
   "source": [
    "W2, F1 = LA.eig(LA.solve(MM,KK)) # eigenanalysis\n",
    "ix = np.argsort(np.absolute(W2)) # sort eigenvalues in ascending order\n",
    "W2 = W2[ix] # sorted eigenvalues\n",
    "F1 = F1[:,ix] # sorted eigenvectors\n",
    "print(np.round_(W2, 4))\n",
    "print(np.round_(F1, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The angular frequencies are computed as the square root of the eigenvalues:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.48062012  4.77574749]\n"
     ]
    }
   ],
   "source": [
    "print(np.sqrt(W2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The modal vectors, the columns of the modal matrix, have unit norm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  1.]\n"
     ]
    }
   ],
   "source": [
    "print(LA.norm(F1, axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Contrary to what is normaly done, we will visualize the modal vectors in a polar plot of the corresponding amplitudes and angles of equivalent complex values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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1Ql9CROjr60NLSwukUilSUlIQHR3tk1G3L0bLRAS9Xo+WlhaYTCakpKQgNjZ2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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x204d55ceb00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 2, subplot_kw=dict(polar=True))\n",
    "for mode in range(2):\n",
    "    ax[mode].set_title('Mode #{}'.format(mode+1))\n",
    "    for dof in range(2):\n",
    "        r = np.array([0, np.absolute(F1[dof,mode])])\n",
    "        t = np.array([0, np.angle(F1[dof,mode])])\n",
    "        ax[mode].plot(t, r, 'o-', label='DOF #{}'.format(dof+1))\n",
    "plt.legend(loc='lower left', bbox_to_anchor=(1., 0.))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to top](#top)\n",
    "\n",
    "## Proportionally damped system\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This damping matrix is orthogonal because the mass and stiffness matrices are also orthogonal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.3304  0.    ]\n",
      " [ 0.      1.0863]]\n"
     ]
    }
   ],
   "source": [
    "print(np.round_(F1.T*C1*F1, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The system and input matrices are the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.   0.   1.   0. ]\n",
      " [ 0.   0.   0.   2. ]\n",
      " [ 1.   0.   0.9 -0.4]\n",
      " [ 0.   2.  -0.4  0.6]]\n"
     ]
    }
   ],
   "source": [
    "A = np.bmat([[np.zeros_like(MM), MM], [MM, C1]])\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  1.   0.   0.   0.]\n",
      " [  0.   2.   0.   0.]\n",
      " [  0.   0. -20.  10.]\n",
      " [  0.   0.  10. -10.]]\n"
     ]
    }
   ],
   "source": [
    "B = np.bmat([[MM, np.zeros_like(MM)], [np.zeros_like(MM), -KK]])\n",
    "print(B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The eigenanalysis yields the eigenvalues and eigenvectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.0938+1.4776j -0.0938-1.4776j -0.5062+4.7488j -0.5062-4.7488j]\n",
      "[[ 0.4058-0.j      0.4058+0.j      0.9423+0.j      0.9423-0.j    ]\n",
      " [ 0.7226+0.j      0.7226-0.j     -0.2646-0.j     -0.2646+0.j    ]\n",
      " [-0.0174-0.2735j -0.0174+0.2735j -0.0209-0.1962j -0.0209+0.1962j]\n",
      " [-0.0309-0.487j  -0.0309+0.487j   0.0059+0.0551j  0.0059-0.0551j]]\n"
     ]
    }
   ],
   "source": [
    "w1, v1 = LA.eig(LA.solve(A,B))\n",
    "ix = np.argsort(np.absolute(w1))\n",
    "w1 = w1[ix]\n",
    "v1 = v1[:,ix]\n",
    "print(np.round_(w1, 4))\n",
    "print(np.round_(v1, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the eigenvalues come in complex conjugate pairs. Let us take only the ones in the [upper half-plane](http://en.wikipedia.org/wiki/Upper_half-plane):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.0938+1.4776j -0.5062+4.7488j]\n"
     ]
    }
   ],
   "source": [
    "print(np.round_(w1[::2], 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These complex eigenvalues can be decomposed into angular frequency and damping coefficient:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular frequency: [ 1.48062012  4.77574749]\n",
      "Damping coefficient: [ 0.06338204  0.10598452]\n"
     ]
    }
   ],
   "source": [
    "zw = -w1.real # damping coefficient time angular frequency\n",
    "wD = w1.imag # damped angular frequency\n",
    "zn = 1./np.sqrt(1.+(wD/-zw)**2) # the minus sign is formally correct!\n",
    "wn = zw/zn # undamped angular frequency\n",
    "print('Angular frequency: {}'.format(wn[[0,2]]))\n",
    "print('Damping coefficient: {}'.format(zn[[0,2]]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The columns of the modal matrix, the modal vectors, also come in conjugate pairs, each vector having unit norm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  1.]\n"
     ]
    }
   ],
   "source": [
    "print(LA.norm(v1[:,::2], axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Moreover, the modal matrix is composed of four blocks, each with $NDOF \\times NDOF$ dimension. Some column reordering is necessary in order to match both modal matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.4058-0.j      0.4058+0.j      0.9423+0.j      0.9423-0.j    ]\n",
      " [ 0.7226+0.j      0.7226-0.j     -0.2646-0.j     -0.2646+0.j    ]\n",
      " [-0.0381+0.5996j -0.0381-0.5996j -0.4770+4.475j  -0.4770-4.475j ]\n",
      " [-0.0678+1.0677j -0.0678-1.0677j  0.1339-1.2565j  0.1339+1.2565j]]\n"
     ]
    }
   ],
   "source": [
    "AA = v1[:2,[0,2]]\n",
    "AB = AA.conjugate()\n",
    "BA = np.multiply(AA,w1[[0,2]])\n",
    "BB = BA.conjugate()\n",
    "v1_new = np.bmat([[AA, AB], [BA, BB]])\n",
    "print(np.round_(v1_new[:,[0,2,1,3]], 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will visualize again the complex valued modal vectors with a polar plot of the corresponding amplitudes and angles:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Vw4yxpQJ1UUQAIlYJMsZyVCrVw08++eTUas2csNlsaGhowOrVq9HQ0DAvKdYu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\ngICAgMAVy/8HZSdu2K461lcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x204d5752ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 2, subplot_kw=dict(polar=True))\n",
    "for mode in range(2):\n",
    "    ax[mode].set_title('Mode #{}'.format(mode+1))\n",
    "    for dof in range(2):\n",
    "        r = np.array([0, np.absolute(v1[dof,2*mode])])\n",
    "        t = np.array([0, np.angle(v1[dof,2*mode])])\n",
    "        ax[mode].plot(t, r, 'o-', label='DOF #{}'.format(dof+1))\n",
    "plt.legend(loc='lower left', bbox_to_anchor=(1., 0.))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to top](#top)\n",
    "\n",
    "## Non proportionally damped system\n",
    "\n",
    "In non proportionally damped systems the damping matrix is not proportional neither to the mass matrix nor the stiffness matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Non proportinal damping carries the fact that the damping matrix is not orthogonal anymore:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.3468  0.1414]\n",
      " [ 0.1414  0.0032]]\n"
     ]
    }
   ],
   "source": [
    "print(np.round_(F1.T*C2*F1, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The system and input matrices are the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.   0.   1.   0. ]\n",
      " [ 0.   0.   0.   2. ]\n",
      " [ 1.   0.   0.1  0.2]\n",
      " [ 0.   2.   0.2  0.2]]\n"
     ]
    }
   ],
   "source": [
    "A = np.bmat([[np.zeros_like(MM), MM], [MM, C2]])\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  1.   0.   0.   0.]\n",
      " [  0.   2.   0.   0.]\n",
      " [  0.   0. -20.  10.]\n",
      " [  0.   0.  10. -10.]]\n"
     ]
    }
   ],
   "source": [
    "B = np.bmat([[MM, np.zeros_like(MM)], [np.zeros_like(MM), -KK]])\n",
    "print(B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The eigenanalysis yields the eigenvalues and eigenvectors of the system matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.0986+1.4777j -0.0986-1.4777j -0.0014+4.7745j -0.0014-4.7745j]\n",
      "[[  4.06300000e-01-0.0087j   4.06300000e-01+0.0087j  -9.42000000e-01+0.j\n",
      "   -9.42000000e-01-0.j    ]\n",
      " [  7.22300000e-01+0.j       7.22300000e-01-0.j       2.65100000e-01-0.0184j\n",
      "    2.65100000e-01+0.0184j]\n",
      " [ -2.41000000e-02-0.2733j  -2.41000000e-02+0.2733j\n",
      "    1.00000000e-04+0.1973j   1.00000000e-04-0.1973j]\n",
      " [ -3.25000000e-02-0.4866j  -3.25000000e-02+0.4866j\n",
      "   -3.90000000e-03-0.0555j  -3.90000000e-03+0.0555j]]\n"
     ]
    }
   ],
   "source": [
    "w2, v2 = LA.eig(LA.solve(A,B))\n",
    "ix = np.argsort(np.absolute(w2))\n",
    "w2 = w2[ix]\n",
    "v2 = v2[:,ix]\n",
    "print(np.round_(w2, 4))\n",
    "print(np.round_(v2, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the eigenvalues come in complex conjugate pairs. Again, let us take only the ones in the [upper half-plane](http://en.wikipedia.org/wiki/Upper_half-plane):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.0986+1.4777j -0.0014+4.7745j]\n"
     ]
    }
   ],
   "source": [
    "print(np.round_(w2[[0,2]], 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These complex eigenvalues can be decomposed into angular frequency and damping coefficient much like in the propotional damping case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular frequency: [ 1.48099973  4.77452335]\n",
      "Damping coefficient: [ 0.06655163  0.00030098]\n"
     ]
    }
   ],
   "source": [
    "zw = -w2.real # damping coefficient times angular frequency\n",
    "wD = w2.imag # damped angular frequency\n",
    "zn = 1./np.sqrt(1.+(wD/-zw)**2) # the minus sign is formally correct!\n",
    "wn = zw/zn # undamped angular frequency\n",
    "print('Angular frequency: {}'.format(wn[[0,2]]))\n",
    "print('Damping coefficient: {}'.format(zn[[0,2]]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, the columns of the modal matrix, the modal vectors, come in conjugate pairs, and each vector has unit norm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  1.]\n"
     ]
    }
   ],
   "source": [
    "print(LA.norm(v2[:,[0,2]], axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Moreover, the modal matrix is composed of four blocks, each with $NDOF \\times NDOF$ dimension. Some column reordering is necessary in order to match both modal matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.4063-0.0087j  0.4063+0.0087j -0.9420+0.j     -0.9420-0.j    ]\n",
      " [ 0.7223+0.j      0.7223-0.j      0.2651-0.0184j  0.2651+0.0184j]\n",
      " [-0.0272+0.6012j -0.0272-0.6012j  0.0014-4.4975j  0.0014+4.4975j]\n",
      " [-0.0712+1.0673j -0.0712-1.0673j  0.0873+1.266j   0.0873-1.266j ]]\n"
     ]
    }
   ],
   "source": [
    "AA = v2[:2,[0,2]]\n",
    "AB = AA.conjugate()\n",
    "BA = np.multiply(AA,w2[[0,2]])\n",
    "BB = BA.conjugate()\n",
    "v2_new = np.bmat([[AA, AB], [BA, BB]])\n",
    "print(np.round_(v2_new[:,[0,2,1,3]], 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once more we will visualize the complex valued modal vectors through a polar plot of the corresponding amplitudes and angles:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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79u1feuONN2JCzfTpcrlgMplgs9m8xW63w+Vyr4JktVrR0NAAkUjk7SSkUinkcjkiIiIg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\nNDY2wm6fn3elrq4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purp60v41NTV05ZVXkt1uJ6PRSFlZWXTmzBlau3atHkA6hUDapakl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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x204d5b04c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 2, subplot_kw=dict(polar=True))\n",
    "for mode in range(2):\n",
    "    ax[mode].set_title('Mode #{}'.format(mode+1))\n",
    "    for dof in range(2):\n",
    "        r = np.array([0, np.absolute(v2[dof,2*mode])])\n",
    "        t = np.array([0, np.angle(v2[dof,2*mode])])\n",
    "        ax[mode].plot(t, r, 'o-', label='DOF #{}'.format(dof+1))\n",
    "plt.legend(loc='lower left', bbox_to_anchor=(1., 0.))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Back to top](#top)\n",
    "\n",
    "## Conclusions\n",
    "\n",
    "Several conclusion can be drawn from this very simple example. First of all, damping changes the vibration frequencies and mode shapes. Furthermore, the polar plots show clearly that:\n",
    "\n",
    "1. In the undamped system the vibration modes are fully \"orthogonal\";\n",
    "2. In the proportionally damped system the vibration modes are no longer \"orthogonal\";\n",
    "3. In the non proportionally damped system each DOF has a different phase angle.\n",
    "\n",
    "[Back to top](#top)\n",
    "\n",
    "## Odds and ends\n",
    "\n",
    "This notebook was created by Paulo Xavier Candeias.\n",
    "\n",
    "[Back to top](#top)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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