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Harmonograph
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<h2 id="history">History</h2> <p>In the 1870s, the term <i>harmonograph</i> is attested in connection with A. E. Donkin and devices built by Samuel Charles Tisley.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="blackburn-pendulum">Blackburn pendulum</h2> <p>A Blackburn pendulum is a device for illustrating <a href="/facts/Simple_harmonic_motion/u2cGAd91">simple harmonic motion</a>, it was named after <a href="/facts/Hugh_Blackburn/CUMy137g">Hugh Blackburn</a>, who described it in 1844. This was first discussed by James Dean in 1815 and analyzed mathematically by Nathaniel Bowditch in the same year.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> A bob is suspended from a string that in turn hangs from a V-shaped pair of strings, so that the pendulum oscillates simultaneously in two perpendicular directions with different periods. The bob consequently follows a path resembling a <a href="/facts/Lissajous_curve/vbsovdjN">Lissajous curve</a>; it belongs to the family of mechanical devices known as harmonographs.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Mid-20th century physics textbooks sometimes refer to this type of pendulum as a <a href="/facts/Double_pendulum/UHHUHpT8">double pendulum</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h2 id="computer-generated-harmonograph-figure">Computer-generated harmonograph figure</h2> <p>A harmonograph creates its figures using the movements of damped pendulums. The movement of a damped pendulum is described by the equation </p> x ( t ) = A sin ( t f + p ) e − d t , {\displaystyle x(t)=A\sin(tf+p)e^{-dt},} <p>in which f {\displaystyle f} represents frequency, p {\displaystyle p} represents phase, A {\displaystyle A} represents amplitude, d {\displaystyle d} represents damping and t {\displaystyle t} represents time. If that pendulum can move about two axes (in a circular or elliptical shape), due to the principle of superposition, the motion of a rod connected to the bottom of the pendulum along one axes will be described by the equation </p> x ( t ) = A 1 sin ( t f 1 + p 1 ) e − d 1 t + A 2 sin ( t f 2 + p 2 ) e − d 2 t . {\displaystyle x(t)=A_{1}\sin(tf_{1}+p_{1})e^{-d_{1}t}+A_{2}\sin(tf_{2}+p_{2})e^{-d_{2}t}.} <p>A typical harmonograph has two pendulums that move in such a fashion, and a pen that is moved by two perpendicular rods connected to these pendulums. Therefore, the path of the harmonograph figure is described by the parametric equations </p> x ( t ) = A 1 sin ( t f 1 + p 1 ) e − d 1 t + A 2 sin ( t f 2 + p 2 ) e − d 2 t , y ( t ) = A 3 sin ( t f 3 + p 3 ) e − d 3 t + A 4 sin ( t f 4 + p 4 ) e − d 4 t . {\displaystyle {\begin{aligned}x(t)&=A_{1}\sin(tf_{1}+p_{1})e^{-d_{1}t}+A_{2}\sin(tf_{2}+p_{2})e^{-d_{2}t},\\y(t)&=A_{3}\sin(tf_{3}+p_{3})e^{-d_{3}t}+A_{4}\sin(tf_{4}+p_{4})e^{-d_{4}t}.\end{aligned}}} <p>An appropriate computer program can translate these equations into a graph that emulates a harmonograph. Applying the first equation a second time to each equation can emulate a moving piece of paper (see the figure below). </p> <h2 id="gallery">Gallery</h2> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Spirograph/c6p3lTSo">Spirograph</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="external-links">External links</h2> Wikimedia Commons has media related to Harmonograph. <ul><li><a href="http://www.jonathan.lansey.net/pastimes/pendulum/index.html">A complex harmonograph with a unique single pendulum design</a></li> <li><a href="https://web.archive.org/web/20091004061155/http://local.wasp.uwa.edu.au/~pbourke/geometry/harmonograph/">Harmonograph background, equations, and illustrations</a></li> <li><a href="http://www.karlsims.com/harmonograph/">How to build a 3-pendulum rotary harmonograph</a></li> <li><a href="http://andygiger.com/science/harmonograph/index.html">Interactive JavaScript simulation of a 3-pendulum rotary harmonograph</a></li> <li><a href="https://web.archive.org/web/20100904003022/http://hernan.amiune.com/labs/harmonograph/animated-harmonograph.html">HTML5 Animated Harmonograph</a></li> <li><a href="http://www.worldtreesoftware.com/apps/harmonograph/app.html">Virtual Harmonograph web application</a></li> <li><a href="http://www.excelunusual.com/harmonograph-video-demo/">An Animated Harmonograph Model in MS Excel</a></li> <li><a href="https://itunes.apple.com/us/app/pintograph-mesmerizing-line/id548056488?mt=8&uo=4">An interactive Pintograph for iOS</a>[<i>dead link</i>]</li> <li><a href="http://www.fxmtech.com/harmonog.html">Harmonographs, pintographs, and Excel models</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Turner, Steven (February 1997). "Demonstrating Harmony: Some of the Many Devices Used To Produce Lissajous Curves Before the Oscilloscope". Rittenhouse. 11 (42): 41. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Whitaker, Robert J. (February 2001). "Harmonographs. II. Circular design". American Journal of Physics. 69 (2): 174–183. Bibcode:2001AmJPh..69..174W. doi:10.1119/1.1309522. <a href="https://bearworks.missouristate.edu/cgi/viewcontent.cgi?article=4356&context=articles-cnas" target="_blank">https://bearworks.missouristate.edu/cgi/viewcontent.cgi?article=4356&context=articles-cnas</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Pook, Leslie Philip (2011). Understanding Pendulums: A Brief Introduction. Springer. ISBN 978-9-40-073634-4. <a href="978-9-40-073634-4" target="_blank">978-9-40-073634-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Baker, Gregory L.; Blackburn, James A. (2005). The Pendulum: a case study in physics. Oxford. ISBN 978-0-19-156530-4. <a href="978-0-19-156530-4" target="_blank">978-0-19-156530-4</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Francis Sears and Mark W. Zemansky (1964). University Physics (3rd ed.). Addison-Wesley Publishing Company. <a href="/wiki/Francis_Sears" target="_blank">/wiki/Francis_Sears</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>