Time response of the <a href="https://en.wikipedia.org/wiki/Canonical_form" class="extiw" title="en:Canonical form">normalised</a> <a href="https://en.wikipedia.org/wiki/Vibration" class="extiw" title="en:Vibration">damped harmonic</a> equation 
<dl><dd><i><b>ẍ + 2ζωẋ + ω<sup>2</sup>x = 0</b></i>
<dl><dd>with initial conditions
<dl><dd><i><b>x = 1</b></i></dd>
<dd><i><b>ẋ = 0</b></i> (where <i><b>ẋ</b></i> is shorthand for <i><b>dx/dt</b></i>)</dd></dl></dd>
<dd>for various values of <i><b>ζ</b></i> (<i><b>ζ ≥ 0</b></i>) showing how the damped <a href="https://en.wikipedia.org/wiki/Natural_frequency" class="extiw" title="en:Natural frequency">natural frequency</a> increases as damping increases (<i><b>ζ</b></i> becomes larger). If <i><b>ζ &lt; 1</b></i>, then the system is said to be <a href="https://en.wikipedia.org/wiki/Damping" class="extiw" title="en:Damping">under-damped</a>; if <span style="white-space:nowrap"><i><b>ζ = 1</b></i></span>, the system is said to be critically damped and if <span style="white-space:nowrap"><i><b>ζ &gt; 1</b></i></span>, the system is said to be over-damped.</dd></dl></dd></dl><p><br><b>Derivation</b>
</p>
<dl><dd>The standard equation for damped harmonic motion is:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><dl><dd><i><b>mẍ + cẋ + kx = 0</b></i>.</dd>
<dd>where <i><b>m</b></i>, <i><b>c</b></i> and <i><b>k</b></i> have standard meanings - for example, in a damped spring system <i><b>m</b></i> is mass, <i><b>c</b></i> is the velocity damping constant and <i><b>k</b></i> is the spring constant.</dd></dl></dd></dl><dl><dd>This equation is transformed to the normalised form by setting:
<dl><dd><i><b>ω<sup>2</sup> = k/m</b></i> and <i><b>ζ<sup>2</sup> = c<sup>2</sup>/4mk</b></i></dd></dl></dd></dl><dl><dd>The <a href="https://en.wikipedia.org/wiki/Closed-form_expression" class="extiw" title="en:Closed-form expression">analytic solutions</a> to the standard damped harmonic equation are:
<dl><dd><ul><li>if <i><b>ζ &gt; 1</b></i> then <span style="white-space:nowrap"><i><b>x = c<sub>1</sub> e<sup>α<sub>1</sub>t</sup> + c<sub>2</sub> e<sup>α<sub>2</sub>t</sup></b></i></span> where <i><b>α<sub>1</sub></b></i> and <i><b>α<sub>2</sub></b></i> are the solutions to the <a href="https://en.wikipedia.org/wiki/Parabola" class="extiw" title="en:Parabola">equation</a> <span style="white-space:nowrap"><i><b>α<sup>2</sup> + 2ωζα + ω<sup>2</sup> = 0</b></i></span></li>
<li>if <i><b>ζ = 1;</b></i> then <span style="white-space:nowrap"><i><b>x = c<sub>1</sub> e<sup>-ωt</sup> + c<sub>2</sub>t e<sup>-ωt</sup></b></i></span></li>
<li>if <i><b>ζ &lt; 1;</b></i> then <span style="white-space:nowrap"><i><b>x = e<sup>-ωζt</sup> (c<sub>1</sub> cos(γt) + c<sub>2</sub>sin(γt))</b></i></span> where <i><b>γ<sup>2</sup> = ω<sup>2</sup>(1 - ζ<sup>2</sup>)</b></i></li></ul></dd>
<dd>where <i><b>c<sub>1</sub></b></i> and <i><b>c<sub>2</sub></b></i> are <a href="https://en.wikipedia.org/wiki/Constant_of_integration" class="extiw" title="en:Constant of integration">constants of integration</a>.</dd></dl></dd></dl><p><br><b>Reference</b>
</p>
<div class="reflist" style="list-style-type: decimal;">
<ol class="references"><li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Neil, Ethan. <a rel="nofollow" class="external text" href="https://physicscourses.colorado.edu/phys2210/phys2210_fa20/lecture/lec36-damped-oscillator/">Damped Harmonic Oscillator</a>. University of Colorado Boulder. Retrieved on <span class="mw-formatted-date" title="2023-06-05">5 June 2023</span>.</span>
</li>
</ol></div>
<a href="//commons.wikimedia.org/wiki/User:Martinvl" title="User:Martinvl">Martinvl</a>
<span class="int-own-work" lang="en">Own work</span>