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{{Merge from|Blackburn pendulum|discuss=Talk:Harmonograph#Merge_proposal|date=October 2011}}
{{oneref|date=January 2010}}
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[[Image:Harmonograph 2.jpg|thumb|200px|A harmonograph output]]
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.
 
==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> represent phase, <math>A</math> represent 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 axes 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>x(t) = A_1 \sin (tf_1 + p_1) e^{-d_1t} + A_2 \sin (tf_2 + p_2) e^{-d_2t}, \,\!</math>
:<math>y(t) = A_3 \sin (tf_3 + p_3) e^{-d_3t} + A_4 \sin (tf_4 + p_4) e^{-d_4t}. \,\!</math>
 
An appropriate computer program can translate those 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==
*[http://www.jonathan.lansey.net/pastimes/pendulum/index.html A complex harmonograph with a unique single pendulum design]
*[http://www.sequences.org.uk/harmonograph/ Online interactive Harmonograph]
*[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://hernan.amiune.com/labs/harmonograph/animated-harmonograph.html HTML5 Animated Harmonograph]
*[http://www.worldtreesoftware.com/apps/harmonograph/harmonograph.html Virtual Harmonograph web application]
*[http://excelunusual.com/archive/2011/07/harmonograph-video-demo/ An Animated Harmonogharph Model in MS Excel]
*[https://itunes.apple.com/us/app/pintograph-mesmerizing-line/id548056488?mt=8&uo=4 An interactive Pintograph for iOS]
 
[[Category:Curves]]
[[Category:Pendulums]]

Revision as of 10:30, 13 February 2014

For your offense, you might attain Gunboats which can easily shoot at enemy resistance coming from a very long range and Landing Work which you must refill when you train equipment for example Rifleman, Heavy, Zooka, Warrior and Fish tanks. To your village defenses, you might have structures like Mortar, Washer Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Drive Launcher to assist your family eradicate enemies.

In the instance as a parent you're concerned with movie game content, control what downloadable mods are put on the inside sport. These down loadable mods are usually created by players, perhaps not your current gaming businesses, therefore there's no ranking system. What you thought was a realistically un-risky game can switch off all electronics a lot worse offering any of these mods.

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