From 38d89c1182a61ca4e024e0834ae9187db76c67f8 Mon Sep 17 00:00:00 2001
From: mattkae <mattkae@protonmail.com>
Date: Sat, 26 Feb 2022 20:08:53 -0500
Subject: Fixing spring simulations and some write ups

---
 2d/softbody/softbody_1.html.content | 87 ++++++++++++++++++++++++++++++++-----
 1 file changed, 76 insertions(+), 11 deletions(-)

(limited to '2d/softbody/softbody_1.html.content')

diff --git a/2d/softbody/softbody_1.html.content b/2d/softbody/softbody_1.html.content
index 602d0a8..8314bbf 100644
--- a/2d/softbody/softbody_1.html.content
+++ b/2d/softbody/softbody_1.html.content
@@ -131,17 +131,54 @@
   <section>
 	<p>
     It is time to investigate what it means to have a deformation in our physics system. By deformation, I mean that the
-    vertices of our shapes are no longer fixed to one point within the model. We will begin with what I believe
-    by investigating the most common of all deformations: a weight attached to a spring in two dimensions.
+    vertices of our shapes are no longer fixed to one point within the model. Towards this end, we will begin by investigating
+    springs.
 	</p>
   </section>
   <section>
-	<h2></h2>
+	  <h2>Undamped Spring Explanation</h2>
+    <p>
+      An <b>undamped spring</b> is one that never stops oscillating. Once an undamped spring starts moving back and forth,
+      it moves back and forth forever.
+    </p>
+    <p>
+      From Newton's second law, we know that the sum of all the forces acting on an object will equal zero. We know that one
+      of these forces is mass times acceleration, but for a spring we will add another force which uses the <i>spring constant</i>.
+      This force grows in proportion to the displacment that spring is currently set at. So, if the spring is currently displaced
+      10m, the effect of this force will be greater than if the spring was at <i>equlibrium</i>, which means that it is displaced by
+      0m. This should make sense intuitively.
+    </p>
+    <p>
+      One thing to note is that a damped spring has three cases:
+      <ul>
+        <li><b>Overdamped</b>: the spring will decompress immediately to the equlibrium position</li>
+        <li><b>Underdamped</b>: the spring will do one oscillation before returning to the equlibrium position</li>
+        <li><b>Critically damped: the spring will oscilate, but each oscillation will bring it closer and closer to the equlibrium position</b></li>
+      </ul>
+      I will not go into the mathematical details as to why this happens, as the links at the end of this video can very easily tell you. Just know
+      that they are dependent on your values of <i>c</i> and <i>k</i>, so choose wisely for your use case.
+    </p>
+    <p>
+      The equation is given as such:
+      <div class="formula">
+        <math class="formula">
+          ma + kx = 0
+        </math>
+      </div>
+      where <i>m</i> is the mass, <i>a</i> is acceleration, <i>k</i> is the spring constant, and <i>x</i> is the current displacement
+      from the rest position.
+    </p>
+    <p>
+      We can define an undamped 1D spring in code like so:
+      #SNIPPET softbody_1/snippet1.cpp
+
+      Note that we are just using Euler Integration here.
+    </p>
   </section>
   <section>
-	<h2>
-	  Undamped Springs
-	</h2>
+    <h2>
+      Undamped Springs Example
+    </h2>
     <p>
       <span class='widget_container'>
         <label for='undamped_spring_length'>Spring Length (m)</label>
@@ -179,9 +216,40 @@
     </div>
   </section>
 
+  <section>
+	  <h2>Damped Spring Explanation</h2>
+    <p>
+      Undamped springs are loads of fun, but if we want to make a softbody physics system, we're going to want
+      our simulation to stop oscillating at some point. That is where the <b>viscous damping constant</b> (<i>c</i>) force
+      comes in. The damping force grows in proportion to the velocity. The equation is given as such:
+
+      <div class="formula">
+        <math class="formula">
+          ma + cv + kx = 0
+        </math>
+      </div>
+
+      where <i>m</i> is the mass, <i>a</i> is acceleration, <i>c</i> is the damping constant, <i>v</i> is the velocity
+      <i>k</i> is the spring constant, and <i>x</i> is the current displacement from the rest position.
+    </p>
+    <p>
+      This force produces the most effect when the velocity is largest. Let's try to intuit exactly what this means. When
+      our spring is extended downward, it will want to move upward to return to its equlibrium position. At this point, 
+      the displacement (<i>x</i>) will be negative and the velocity (<i>v</i>) will be positive. Since the signs are different,
+      we can see that our spring force will be mitigated by the damping force. Hence, we damp!
+    </p>
+
+    <p>
+      We can define a damped 1D spring in code like so:
+      #SNIPPET softbody_1/snippet2.cpp
+
+      Note that we are just using Euler Integration here.
+    </p>
+  </section>
+
   <section>
 	<h2>
-	  Damped Springs
+	  Damped Springs Example
 	</h2>
     <p>
       <span class='widget_container'>
@@ -204,7 +272,7 @@
 
 	  <span class='widget_container'>
         <label for='damped_viscous_constant'>Viscous Damping Constant (N / m)</label>
-        <input type='range' id='damped_viscous_constant' min='0' max='1000.0' value='1.0' step='0.1'/>
+        <input type='range' id='damped_viscous_constant' min='0' max='100.0' value='1.0' step='0.1'/>
         <span></span>
       </span>
 
@@ -238,9 +306,6 @@
       <li>
         <a href='http://ambrsoft.com/CalcPhysics/Spring/SpringData.htm'>List of Equations for Spring Motion</a>
       </li>
-      <li>
-        <a href='https://www.ryanjuckett.com/damped-springs/'>Ryan Juckett's Explanation of Damped Springs</a>
-      </li>
     </ul>
   </footer>
 </article>
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