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diff --git a/2d/softbody/softbody_1.html b/2d/softbody/softbody_1.html index 578024e..576b9c4 100644 --- a/2d/softbody/softbody_1.html +++ b/2d/softbody/softbody_1.html @@ -185,17 +185,67 @@ <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: + <pre><code><span class="code_keyword">struct</span> Spring { + <span class="code_keyword">float</span> mass = 1.f; <span class="code_comment">// Mass of the weight on the end of the spring</span> + <span class="code_keyword">float</span> k = 4; <span class="code_comment">// Spring Constant, in N / m</span> + <span class="code_keyword">float</span> force = 0.f; + <span class="code_keyword">float</span> velocity = 0.f; + <span class="code_keyword">float</span> position = 0.f; +}; + +<span class="code_keyword">void</span> updateSpring(Spring* spring, <span class="code_keyword">float</span> dtSeconds) { + spring->force = spring->k * spring->position; + <span class="code_keyword">float</span> acceleration = spring->force / spring->mass; <span class="code_comment">// F = ma</span> + spring->velocity = spring->velocity + acceleration * dtSeconds; + spring->position = spring->position + spring->velocity * dtSeconds; +} +</code></pre> + 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> @@ -234,8 +284,53 @@ </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: + <pre><code><span class="code_keyword">struct</span> Spring { + <span class="code_keyword">float</span> mass = 1.f; <span class="code_comment">// Mass of the weight on the end of the spring</span> + <span class="code_keyword">float</span> k = 4; <span class="code_comment">// Spring Constant, in N / m</span> + <span class="code_keyword">float</span> c = 1; <span class="code_comment">// Viscous damping constant, in N / m/s</span> + <span class="code_keyword">float</span> force = 0.f; + <span class="code_keyword">float</span> velocity = 0.f; + <span class="code_keyword">float</span> position = 0.f; +}; + +<span class="code_keyword">void</span> updateSpring(Spring* spring, <span class="code_keyword">float</span> dtSeconds) { + spring->force = spring->c * spring->velocity + spring->k * spring->position; + <span class="code_keyword">float</span> acceleration = spring->force / spring->mass; <span class="code_comment">// F = ma</span> + spring->velocity = spring->velocity + acceleration * dtSeconds; + spring->position = spring->position + spring->velocity * dtSeconds; +} +</code></pre> + Note that we are just using Euler Integration here. + </p> + </section> + + <section> <h2> - Damped Springs + Damped Springs Example </h2> <p> <span class='widget_container'> @@ -258,7 +353,7 @@ <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> @@ -292,9 +387,6 @@ <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> |