New tutorials in documentation

docs/user_guide/example_25.ipynb

@@ -5,7 +5,7 @@
    "id": "73c0b6bc-eca1-4cb1-a36b-a90cd4c29520",
    "metadata": {},
    "source": [
-    "# Example 23 - Overhung Rotor\n",
+    "# Example 25 - Coaxial rotor\n",
     "\n",
     "This example is based on Example 6.8.1 from {cite}`friswell2010dynamics`.\n",
     "\n",

docs/user_guide/example_26.ipynb

@@ -14,7 +14,7 @@
    "id": "eb6566cf-32ab-4537-9a64-8877fa3ea2c0",
    "metadata": {},
    "source": [
-    "This example is based on Example 6.3.1.a page 253 from [Friswell, 2010]."
+    "This example is based on Example 6.3.1.a page 253 from {cite}`friswell2010dynamics`."
    ]
   },
   {
@@ -151344,9 +151344,9 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python [conda env:py312] *",
+   "display_name": "rs",
    "language": "python",
-   "name": "conda-env-py312-py"
+   "name": "python3"
   },
   "language_info": {
    "codemirror_mode": {
@@ -151358,7 +151358,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.4"
+   "version": "3.9.16"
   }
  },
  "nbformat": 4,

docs/user_guide/example_27.ipynb

@@ -14,7 +14,7 @@
    "id": "eb6566cf-32ab-4537-9a64-8877fa3ea2c0",
    "metadata": {},
    "source": [
-    "This example is based on Example 6.3.1.b page 253 from [Friswell, 2010]."
+    "This example is based on Example 6.3.1.b page 253 from {cite}`friswell2010dynamics`."
    ]
   },
   {

docs/user_guide/fluid_flow_elliptical_bearing.ipynb

@@ -15,24 +15,33 @@
     "\n",
     "The elliptical bearing or \"lemon bearing\", as it is also known, is a variation of the cylindrical bearing with axial groove and reduced clearance in one direction.\n",
     "\n",
-    "<img src=\"https://docs.google.com/uc?id=1EMhfdEnZkAHdd-yfL8VqrNTprkuxHRus\" width=\"350\"/>\n",
+    "![alt text](../_static/img/img_examplo_ff_eliptical.png)\n",
+    "<style type=\"text/css\">\n",
+    "    img {\n",
+    "        width: 350px;\n",
+    "    }\n",
+    "</style>\n",
+    "\n",
+    "<!-- <img src=\"https://docs.google.com/uc?id=1EMhfdEnZkAHdd-yfL8VqrNTprkuxHRus\" width=\"350\"/> -->\n",
     "\n",
     "For the inclusion of this new geometry, adaptations to the stator radius are necessary, as it will no longer be constant in $\\theta$. As seen in the figure above, the new stator is composed of the arc $C_{1}$, with center in $O_{1}$, joined to the arc $C_{2}$, centered in $O_{2}$, both with radius $R_{o}$. In this new configuration, the centers are at a distance $\\epsilon$ from the origin, called ellipticity.\n",
     "\n",
     "It is necessary to describe the stator from the origin. This new distance will be called $R_{o}^{*} $ and it varies along the angular position:\n",
     "\n",
     "$$R_o^* = \\sqrt{R_o ^2 - \\epsilon^2 \\sin^2{\\alpha}} + \\epsilon \\cos{\\alpha}$$\n",
     "\n",
-    "where $\\alpha =\\begin{cases} \n",
+    "where \n",
+    "$$\\alpha =\\begin{cases} \n",
     "\\pi/2 + \\theta \\text{,} \n",
-    "&\\mbox{if} \\quad \\theta \\in 1^{\\circ} \\text{quadrant} \\\\ \n",
+    "&\\text{if} \\quad \\theta \\in 1^{\\circ} \\text{quadrant} \\\\\n",
     "3\\pi/2 + \\theta \\text{,} \n",
-    "&\\mbox{if} \\quad \\theta \\in 2^{\\circ} \\text{quadrant} \\\\\n",
+    "&\\text{if} \\quad \\theta \\in 2^{\\circ} \\text{quadrant} \\\\\n",
     "\\theta - \\pi/2 \\text{,} \n",
-    "&\\mbox{if} \\quad \\theta \\in 3^{\\circ} \\text{quadrant} \\\\\n",
+    "&\\text{if} \\quad \\theta \\in 3^{\\circ} \\text{quadrant} \\\\\n",
     "5\\pi/2 -\\theta \\text{,} \n",
-    "&\\mbox{if} \\quad \\theta \\in 4^{\\circ} \\text{quadrant}\n",
-    "\\end{cases}$.\n",
+    "&\\text{if} \\quad \\theta \\in 4^{\\circ} \\text{quadrant}\n",
+    "\\end{cases}$$\n",
+    ".\n",
     "\n",
     "Another important parameter to be defined is the $ m $ preload which, in this text, will be established as:\n",
     "\n",
@@ -247,7 +256,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.16"
+   "version": "3.11.9"
   }
  },
  "nbformat": 4,

docs/user_guide/fluid_flow_theory.ipynb

@@ -41,20 +41,27 @@
     "\n",
     "Fluid flow occurs in the annular space between the shaft and the bearing, both of $ L $ length. These structures are called rotor and stator, respectively. The stator is fixed with radius $R_{o}$ and the rotor, with radius $R_{i} $, is a rigid body with rotation speed $\\omega$, as shown in the figure below.\n",
     "\n",
-    "<img src=\"https://docs.google.com/uc?id=1ZVqsNZEBQ8PhKZ5v4IlXHpwUTkgO4sM8\" width=\"350\"/>\n",
+    "![alt text](../_static/img/img_examplo_ff_theory.png)\n",
+    "<style type=\"text/css\">\n",
+    "    img {\n",
+    "        width: 350px;\n",
+    "    }\n",
+    "</style>\n",
     "\n",
     "Due to the rotation of the rotor, a pressure field is set in the lubricating oil film, developing fluid forces that act on the rotor surface. For a constant speed of rotation, these forces displace the rotor to a location inside the stator called the _equilibrium position_. In this position, the stator and rotor are eccentric, with a distance between centers $e$ and an attitude angle $\\beta$, formed between the axis connecting both centers and the vertical axis.\n",
     "\n",
     "Based on the eccentricity and attitude angle, the cosine law can be used to describe the position of the rotor surface $R_{\\theta}$ with respect to the center of the stator:\n",
     "\n",
     "$$ R_{\\theta} = \\sqrt{R_i ^2 - e^2 \\sin^2{\\alpha}} + e \\cos{\\alpha},$$\n",
     "\n",
-    "where $\\alpha =\\begin{cases} \n",
+    "where \n",
+    "$$\\alpha =\\begin{cases} \n",
     "\\dfrac{3\\pi}{2} - \\theta + \\beta \\text{,} \n",
-    "&\\mbox{se } \\dfrac{\\pi}{2} + \\beta \\leq \\theta < \\dfrac{3\\pi}{2} + \\beta \\\\ \\\\\n",
+    "&\\text{se } \\dfrac{\\pi}{2} + \\beta \\leq \\theta < \\dfrac{3\\pi}{2} + \\beta \\\\ \\\\\n",
     "- \\left(\\dfrac{3\\pi}{2} - \\theta + \\beta\\right) \\text{,} \n",
-    "& \\mbox{se } \\dfrac{3\\pi}{2} + \\beta \\leq \\theta < \\dfrac{5\\pi}{2} + \\beta \n",
-    "\\end{cases}$."
+    "& \\text{se } \\dfrac{3\\pi}{2} + \\beta \\leq \\theta < \\dfrac{5\\pi}{2} + \\beta \n",
+    "\\end{cases}$$\n",
+    "."
    ]
   },
   {
@@ -370,8 +377,12 @@
     "\n",
     "where $e_{r}$ and $e_{\\theta}$ are unit vectors of the cylindrical coordinate system. This is shown in the figure below.\n",
     "\n",
-    "<img src=\"https://docs.google.com/uc?id=1M9hvJYa6Hg_-PD8HpP9AriD59vR64e-P\" width=\"350\"/>\n",
-    "\n",
+    "![alt text](../_static/img/img_example_ff_theory2.png)\n",
+    "<style type=\"text/css\">\n",
+    "    img {\n",
+    "        width: 350px;\n",
+    "    }\n",
+    "</style>\n",
     "   \n",
     "Now, consider the position vector $a=R_{\\theta} e_r$, from the stator center to any point in the rotor surface. Its time derivative, relative to an inertial frame, is the total speed $v_{tot}$:\n",
     "\n",
@@ -407,50 +418,45 @@
     "    \n",
     "where \n",
     "\n",
-    "$$\\mathbf{C_0} = \n",
+    "$$\\mathbf{C_0} =\n",
     "    - \\omega R_{i} R_{\\theta}\n",
-    "    \\left[\n",
+    "    [\n",
     "        \\ln{\\left(\\frac{R_{o}}{R_{\\theta}}\\right)}\n",
-    "        \\left(\n",
-    "            1 +\n",
-    "            \\frac{R_{\\theta}^2}{(R_{o}^2-R_{\\theta}^2)}\n",
-    "        \\right)\n",
+    "        (1 + \\frac{R_{\\theta}^2}{(R_{o}^2-R_{\\theta}^2)})\n",
     "        -\\dfrac{1}{2}\n",
-    "    \\right]\\label{eq:C_0}\\text{,}$$\n",
-    "    \n",
+    "    ] $$ ,\n",
+    "\n",
+    "  \n",
     "$$\\mathbf{C_1} =\n",
     "    \\dfrac{1}{4\\mu}\n",
-    "    \\left\\{\n",
-    "        \\left[\n",
-    "            R_{o}^2 \\ln{R_{o}}\n",
-    "            - R_{\\theta}^2 \\ln{R_{\\theta}}\n",
+    "    {[R_{o}^2 \\ln{R_{o}} \n",
+    "            - R_{\\theta}^2 \\ln{R_{\\theta}} \n",
     "            + (R_{o}^2-R_{\\theta}^2)(k-1)\n",
-    "        \\right]\n",
+    "        ]\n",
     "        - 2R_{o}^2\n",
-    "        \\left[\n",
-    "            \\left(\n",
-    "                \\ln{R_{o}}+k-\\dfrac{1}{2}\n",
-    "            \\right)\n",
-    "            \\ln{\\left(\\frac{R_{o}}{R_{\\theta}}\\right)}\n",
-    "        \\right]\n",
-    "    \\right\\}\\label{eq:C_1}\\text{,}$$\n",
-    "\n",
+    "        [\n",
+    "            (\\ln{R_{o}}+k-\\dfrac{1}{2})\n",
+    "            \\ln{(\\frac{R_{o}}{R_{\\theta}})}\n",
+    "        ]\n",
+    "    }\n",
+    "$$ ,\n",
+    "  \n",
     "$$\\mathbf{C_2} =\n",
     "    - \\dfrac{R_{\\theta}^2}{8\\mu}\n",
-    "    \\left\\{\n",
-    "        \\left[\n",
+    "    {\n",
+    "        [\n",
     "            R_{o}^2-R_{\\theta}^2\n",
     "            -\\dfrac{(R_{o}^4-R_{\\theta}^4)}{2R_{\\theta}^2}\n",
-    "        \\right]\n",
+    "        ]\n",
     "        +\n",
-    "        \\left(\n",
+    "        (\n",
     "            \\dfrac{R_{o}^2-R_{\\theta}^2}{R_{\\theta}^2 \\ln{\\left(\\dfrac{R_{o}}{R_{\\theta}}\\right)}}\n",
-    "        \\right)\n",
-    "        \\left[\n",
-    "            R_{o}^2\\ln{\\left(\\dfrac{R_{o}}{R_{\\theta}}\\right)}\n",
+    "        )\n",
+    "        [\n",
+    "            R_{o}^2\\ln{(\\dfrac{R_{o}}{R_{\\theta}})}\n",
     "            -\\dfrac{(R_{o}^2-R_{\\theta}^2)}{2}\n",
-    "        \\right]\n",
-    "    \\right\\}\\label{eq:C_2}\\text{.}$$\n",
+    "        ]\n",
+    "    }$$\n",
     "\n",
     "This is a elliptic partial differential equation and its solution detemines the pressure field $p$."
    ]
@@ -463,7 +469,12 @@
     "\n",
     "The partial differential equation is solved using finite centered differences method. It is applied to a regular rectangular mesh with $𝑁_{z}$ nodes in the axial direction and $𝑁_{\\theta}$ nodes in the tangential direction, as shown in the figure below.\n",
     "\n",
-    "<img src=\"https://docs.google.com/uc?id=1PTvlpNx6QlZq7wPWfwjKLsOa9M_qR9pp\" width=\"350\"/>\n",
+    "![alt text](../_static/img/img_example_ff_theory3.png)\n",
+    "<style type=\"text/css\">\n",
+    "    img {\n",
+    "        width: 350px;\n",
+    "    }\n",
+    "</style>\n",
     "\n",
     "The discretized equation is given by\n",
     "\n",
@@ -481,7 +492,12 @@
     "\n",
     "It is important to note that this change in pressure behavior due to cavitation does not necessarily start at the point of least thickness in the annular space. Several studies seek to establish the appropriate boundary conditions to describe the beginning of cavitation in the fluid. ISHIDA and YAMAMOTO (2012) [2] indicate that the condition of Gumbel is widely used because of its simplicity. Using the argument that lubricant evaporation and axial air flow from both ends can occur, the pressure in the region $\\pi < \\theta < 2\\pi$ is considered to be almost zero (that is, the atmospheric pressure ). $p = 0 $ is then defined across the divergent region.\n",
     "\n",
-    "<img src=\"https://docs.google.com/uc?id=1pmKtHS0sW4t6kW0qq3sxwNIlBMJRQUaz\" width=\"250\"/>\n",
+    "![alt text](../_static/img/img_example_ff_theory4.png)\n",
+    "<style type=\"text/css\">\n",
+    "    img {\n",
+    "        width: 350px;\n",
+    "    }\n",
+    "</style>\n",
     "\n",
     "In addition, according to SANTOS (1995) [3], although it violates mass conservation, this condition presents acceptable errors in the global parameters of the bearing. For these reasons, the present study adopts the Gumbel boundary condition to describe the phenomenon of cavitation."
    ]
@@ -722,7 +738,12 @@
     "    \n",
     "Knowing the external load $ W $, it is possible to reach the equilibrium position using an iterative method. Starting at an initial position, the residues between the forces at the current position and external forces are calculated. If the residue is greater than a defined tolerance, the position of the rotor is varied systematically,  inside the fourth quadrant, until the desired tolerance is reached. In other words, a local minimum of the forces function is reached. The Python tool *optimize.least\\_squares* was used for this purpose. The method is shown in the image below.\n",
     "\n",
-    "<img src=\"https://docs.google.com/uc?id=1KQImAbLR8uyFHpl1pGL-sJ4BCiUT0Ok3\" height=\"300\" />"
+    "![alt text](../_static/img/img_example_ff_theory5.png)\n",
+    "<style type=\"text/css\">\n",
+    "    img {\n",
+    "        width: 350px;\n",
+    "    }\n",
+    "</style>"
    ]
   },
   {
@@ -939,7 +960,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.16"
+   "version": "3.11.9"
   }
  },
  "nbformat": 4,

docs/user_guide/fluid_flow_wear_bearing.ipynb

@@ -15,21 +15,30 @@
     "\n",
     "Although lubrication reduces the friction between the metal surfaces of the bearing, these structures usually suffer wear after a long operating period or else due to a certain number of repetitions of the starting cycles.\n",
     "\n",
-    "<img src=\"https://docs.google.com/uc?id=1ZGYl4aCO3WTx-hp_Bkh9PpQMEWCKOwsL\" width=\"350\"/>\n",
+    "![alt text](../_static/img/img_examplo_ff_wear.png)\n",
+    "<style type=\"text/css\">\n",
+    "    img {\n",
+    "        width: 350px;\n",
+    "    }\n",
+    "</style>\n",
+    "\n",
+    "<!-- <img src=\"https://docs.google.com/uc?id=1ZGYl4aCO3WTx-hp_Bkh9PpQMEWCKOwsL\" width=\"350\"/> -->\n",
     "\n",
     "The wear geometry that will be used in the FluidFlow has been adapted from the version presented by MACHADO; CAVALCA (2015) [1]. To include wear in the geometry, it is necessary to make some adaptations to the stator radius. Considering that the fault starts at the angular position $\\theta = \\theta_{s}$ and ends at $\\theta = \\theta_{f}$, the stator description from the origin is defined as:\n",
     "\n",
     "$$R_o^* = R_o + d_{\\theta}$$\n",
     "\n",
-    "where $d_{\\theta} =\\begin{cases} \n",
+    "where \n",
+    "$$d_{\\theta} =\\begin{cases} \n",
     "    0 \\text{,} \n",
-    "    &\\mbox{if} \n",
+    "    &\\text{if} \n",
     "    \\quad 0 \\leq \\theta \\leq \\theta_s\\text{,} \\quad\n",
     "    \\theta_f \\leq \\theta \\leq 2\\pi \\\\ \n",
     "    d_0 - F \\left(1 + \\cos{\\left(\\theta - \\pi/2\\right)} \\right) \\text{,} \n",
-    "    &\\mbox{if} \n",
+    "    &\\text{if} \n",
     "    \\quad \\theta_s < \\theta < \\theta_f\n",
-    "\\end{cases}$.\n",
+    "\\end{cases}$$\n",
+    ".\n",
     "\n",
     "In $\\theta_{s}$ and $\\theta_{f}$, the wear depth is zero, so the location of the edges can be defined as follows:\n",
     "\n",

docs/user_guide/user_guide.md

@@ -40,6 +40,8 @@ example_24
 example_25
 example_26
 example_27
+example_28
+example_29
 fluid_flow_elliptical_bearing
 fluid_flow_short_bearing
 fluid_flow_theory