Fixed equations format on notebook

2 files changed, 42 insertions, 36 deletions

diff --git a/tutorials/module_3/system_of_equations.md b/tutorials/module_3/system_of_equations.md
index 2320d08..e3ae314 100644
--- a/tutorials/module_3/system_of_equations.md
+++ b/tutorials/module_3/system_of_equations.md
@@ -1 +1,2 @@
-# System of Equations
\ No newline at end of file
+# System of Equations
+
diff --git a/tutorials/module_3/vibrations.ipynb b/tutorials/module_3/vibrations.ipynb
index 5198d06..4b7eee4 100644
--- a/tutorials/module_3/vibrations.ipynb
+++ b/tutorials/module_3/vibrations.ipynb
@@ -13,59 +13,58 @@
     "\n",
     "**Given:**\n",
     "\n",
-    "    Mass of the structure (roof and upper level):\n",
-    "    m=20,000 kgm=20,000 kg\n",
+    "Mass of the structure (roof and upper level):\n",
+    "    $m=20,000 kgm=20,000 kg$\n",
     "\n",
-    "    Stiffness of the supporting structure (walls/columns):\n",
-    "    k=8×106 N/mk=8×106 N/m\n",
+    "Stiffness of the supporting structure (walls/columns):\n",
+    "    $k=8×106 N/mk=8×106 N/m$\n",
     "\n",
-    "    Damping ratio:\n",
-    "    ζ=0.05ζ=0.05 (5%)\n",
+    "Damping ratio:\n",
+    "    $ζ=0.05ζ=0.05 (5%)$\n",
     "\n",
-    "    Ground acceleration during an earthquake is modeled as:\n",
-    "    y¨(t)=Asin⁡(ωgt)y¨​(t)=Asin(ωg​t), where:\n",
-    "    A=3 m/s2A=3 m/s2, ωg=5 rad/sωg​=5 rad/s\n",
+    "Ground acceleration during an earthquake is modeled as:\n",
+    "    $y¨(t)=Asin⁡(ωgt)y¨​(t)=Asin(ωg​t)$ ,  where: $A=3 m/s2$, $ωg=5 rad/s$\n",
     "\n",
     "**Tasks:**\n",
     "\n",
-    "    Natural Frequency:\n",
+    "Natural Frequency:\n",
+    "Compute the natural frequency ωn and natural period Tn of the structure.\n",
     "\n",
-    "        Compute the natural frequency ωnωn​ and natural period TnTn​ of the structure.\n",
+    "Equation of Motion (EOM):\n",
+    "Write the equation of motion of the system in terms of relative displacement $x(t)$ between the mass and the moving base.\n",
     "\n",
-    "    Equation of Motion (EOM):\n",
+    "Steady-State Response:\n",
+    "Determine the steady-state amplitude of the relative displacement using the frequency response function (assume sinusoidal base excitation).\n",
     "\n",
-    "        Write the equation of motion of the system in terms of relative displacement x(t)x(t) between the mass and the moving base.\n",
+    "Resonance Check:\n",
+    "Discuss whether resonance is likely to occur with the given excitation. What could engineers do to reduce the risk?\n",
     "\n",
-    "    Steady-State Response:\n",
+    "Plotting (Optional for coding extension):\n",
+    "Plot the amplitude of the relative displacement vs. excitation frequency $ωg$ in the range $0–10 rad/s$.\n",
     "\n",
-    "        Determine the steady-state amplitude of the relative displacement using the frequency response function (assume sinusoidal base excitation).\n",
-    "\n",
-    "    Resonance Check:\n",
-    "\n",
-    "        Discuss whether resonance is likely to occur with the given excitation. What could engineers do to reduce the risk?\n",
-    "\n",
-    "    Plotting (Optional for coding extension):\n",
+    "Hints:\n",
+    "The relative EOM under base excitation is:\n",
     "\n",
-    "        Plot the amplitude of the relative displacement vs. excitation frequency ωgωg​ in the range 0–10 rad/s.\n",
+    "```math\n",
+    ":label: my_label\n",
+    "mx¨+cx˙+kx=−my¨(t)\n",
+    "```\n",
+    "$mx¨+cx˙+kx=−my¨(t)$\n",
     "\n",
-    "Hints:\n",
+    "Use the frequency response function (FRF) for an SDOF system to solve for steady-state amplitude:\n",
     "\n",
-    "    The relative EOM under base excitation is:\n",
-    "    mx¨+cx˙+kx=−my¨(t)\n",
-    "    mx¨+cx˙+kx=−my¨​(t)\n",
+    "$X(ω)=mA(k−mω2)2+(cω)2$\n",
     "\n",
-    "    Use the frequency response function (FRF) for an SDOF system to solve for steady-state amplitude:\n",
-    "    X(ω)=mA(k−mω2)2+(cω)2\n",
-    "    X(ω)=(k−mω2)2+(cω)2\n",
+    "$X(ω)=(k−mω2)2+(cω)2$\n",
     "\n",
     "​mA​\n",
     "\n",
-    "Damping coefficient c=2ζmωnc=2ζmωn​"
+    "Damping coefficient $c=2ζmωn$"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 2,
    "id": "5c55c97d-7a2c-42aa-921c-f9de5838dc46",
    "metadata": {},
    "outputs": [
@@ -146,11 +145,9 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "ea11273e-edb3-456f-b180-fa28af671be0",
+   "cell_type": "markdown",
+   "id": "92f52a0e-5c70-4762-9514-4dab65a12f01",
    "metadata": {},
-   "outputs": [],
    "source": [
     "\n",
     "1. Given\n",
@@ -163,6 +160,14 @@
     "\n",
     "5. Comment"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "eb76cf13-6fb1-41ef-9869-f65ad43fca32",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {