Editing Harmonograph

{{Short description|Device using pendulums to create images}}
[[File:Harmonógrafo.jpg|thumb|Harmonograph]]
[[File:Harmonógrafo_02.webm|thumb|A harmonograph]]
A '''harmonograph''' is a mechanical apparatus that employs [[pendulum]]s to create a geometric image.  The drawings created typically are [[Lissajous curve]]s or related drawings of greater complexity. The devices, which began to appear in the mid-19th century and peaked in popularity in the 1890s, cannot be conclusively attributed to a single person, although [[Hugh Blackburn]], a professor of [[mathematics]] at the [[University of Glasgow]], is commonly believed to be the official inventor.<ref>{{cite journal | last = Turner |first = Steven | title = Demonstrating Harmony: Some of the Many Devices Used To Produce Lissajous Curves Before the Oscilloscope | journal = Rittenhouse | volume = 11 | issue = 42 | page = 41 |date=February 1997}}</ref>

A simple, so-called "lateral" harmonograph uses two pendulums to control the movement of a [[pen]] relative to a drawing surface.  One pendulum moves the pen back and forth along one axis, and the other pendulum moves the drawing surface back and forth along a [[perpendicular]] axis. By varying the [[frequency]] and phase of the pendulums relative to one another, different patterns are created. Even a simple harmonograph as described can create [[ellipse]]s, [[spiral]]s, [[wikt:figure eight|figure eights]] and other Lissajous figures.

More complex harmonographs incorporate three or more pendulums or linked pendulums together (for example, hanging one pendulum off another), or involve rotary motion, in which one or more pendulums is mounted on [[gimbal]]s to allow movement in any direction.

A particular type of harmonograph, a pintograph, is based on the relative motion of two rotating disks, as illustrated in the links below. (A pintograph is not to be confused with a [[pantograph]], which is a mechanical device used to enlarge figures.)

==History==

In the 1870s, the term ''harmonograph'' is attested in connection with A. E. Donkin and devices built by Samuel Charles Tisley.<ref>{{cite journal |last1=Whitaker |first1=Robert J. |title=Harmonographs. II. Circular design |journal=American Journal of Physics |date=February 2001 |volume=69 |issue=2 |pages=174–183 |doi=10.1119/1.1309522|bibcode=2001AmJPh..69..174W |url=https://bearworks.missouristate.edu/cgi/viewcontent.cgi?article=4356&context=articles-cnas |url-access=subscription }}</ref>

==Blackburn pendulum==
[[File:Lissajous figure - sand on paper.jpg|thumb|A Lissajous figure, made by releasing sand from a container at the end of a Blackburn pendulum]]
A Blackburn pendulum is a device for illustrating [[simple harmonic motion]], it was named after [[Hugh Blackburn]], who described it in 1844. This was first discussed by James Dean in 1815 and analyzed mathematically by Nathaniel Bowditch in the same year.<ref>{{cite book |title=Understanding Pendulums: A Brief Introduction |first=Leslie Philip |last=Pook |publisher=Springer |year=2011 |isbn=978-9-40-073634-4}}</ref> 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 [[Lissajous curve]]; it belongs to the family of mechanical devices known as harmonographs.<ref>{{cite book |title=The Pendulum: a case study in physics |first1=Gregory L. |last1=Baker |first2=James A. |last2=Blackburn |publisher=Oxford |year=2005 |isbn=978-0-19-156530-4}}</ref>

Mid-20th century physics textbooks sometimes refer to this type of pendulum as a [[double pendulum]].<ref>{{cite book |author=[[Francis Sears]] and Mark W. Zemansky |title=University Physics |edition=3rd |publisher=Addison-Wesley Publishing Company |year=1964}}</ref>

==Computer-generated harmonograph figure==
A harmonograph creates its figures using the movements of damped pendulums. The movement of a damped pendulum is described by the equation

: <math>x(t) = A \sin (tf + p) e^{-dt},</math>

in which <math>f</math> represents frequency, <math>p</math> represents phase, <math>A</math> represents amplitude, <math>d</math> represents damping and <math>t</math> 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 axis will be described by the equation

: <math>x(t) = A_1 \sin (tf_1 + p_1) e^{-d_1t} + A_2 \sin (tf_2 + p_2) e^{-d_2t}.</math>

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

: <math>\begin{align}
 x(t) &= A_1 \sin (tf_1 + p_1) e^{-d_1t} + A_2 \sin (tf_2 + p_2) e^{-d_2t},\\
 y(t) &= A_3 \sin (tf_3 + p_3) e^{-d_3t} + A_4 \sin (tf_4 + p_4) e^{-d_4t}.
\end{align}</math>

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).

==Gallery==
<gallery>
File:Harmonograph.jpg|A harmonograph at [[Questacon]] in [[Canberra]], [[Australia]]
Image:lateral2.jpg|A figure produced by a simple lateral harmonograph
Image:lateral3.jpg|A figure produced by a simple lateral harmonograph
Image:pgraph1.jpg|A figure produced by a pintograph
Image:Harmonograph 9998.png|Computer-generated harmonograph figure
</gallery>

==See also==
*[[Spirograph]]

==Notes==
{{reflist}}

==External links==
{{Commons}}
*[http://www.jonathan.lansey.net/pastimes/pendulum/index.html A complex harmonograph with a unique single pendulum design]
*[https://web.archive.org/web/20091004061155/http://local.wasp.uwa.edu.au/~pbourke/geometry/harmonograph/ Harmonograph background, equations, and illustrations]
*[http://www.karlsims.com/harmonograph/ How to build a 3-pendulum rotary harmonograph]
*[http://andygiger.com/science/harmonograph/index.html Interactive JavaScript simulation of a 3-pendulum rotary harmonograph]
*[https://web.archive.org/web/20100904003022/http://hernan.amiune.com/labs/harmonograph/animated-harmonograph.html HTML5 Animated Harmonograph]
*[http://www.worldtreesoftware.com/apps/harmonograph/app.html Virtual Harmonograph web application]
*[http://www.excelunusual.com/harmonograph-video-demo/ An Animated Harmonograph Model in MS Excel]
*[https://itunes.apple.com/us/app/pintograph-mesmerizing-line/id548056488?mt=8&uo=4 An interactive Pintograph for iOS]{{dead link|date=March 2025|bot=medic}}{{cbignore|bot=medic}}
*[http://www.fxmtech.com/harmonog.html Harmonographs, pintographs, and Excel models]

{{Authority control}}

[[Category:Curves]]
[[Category:Pendulums]]