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147:, 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 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
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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
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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
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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).
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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
897:{\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}}}
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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
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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
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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
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Turner, Steven (February 1997). "Demonstrating
Harmony: Some of the Many Devices Used To Produce Lissajous Curves Before the Oscilloscope".
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A harmonograph creates its figures using the movements of damped pendulums. The movement of a damped pendulum is described by the equation
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and phase of the pendulums relative to one another, different patterns are created. Even a simple harmonograph as described can create
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527:{\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}.}
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A Lissajous figure, made by releasing sand from a container at the end of a double pendulum
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A simple, so-called "lateral" harmonograph uses two pendulums to control the movement of a
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is attested in connection with A. E. Donkin and devices built by Samuel
Charles Tisley.
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Mid-20th century physics textbooks sometimes refer to this type of pendulum as a
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151:; it belongs to the family of mechanical devices known as harmonographs.
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Interactive JavaScript simulation of a 3-pendulum rotary harmonograph
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to create a geometric image. The drawings created typically are
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A complex harmonograph with a unique single pendulum design
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Harmonograph background, equations, and illustrations
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69:, is commonly believed to be the official inventor.
1130:(3rd ed.). Addison-Wesley Publishing Company.
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948:A figure produced by a simple lateral harmonograph
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139:A Blackburn pendulum is a device for illustrating
1098:Baker, Gregory L.; Blackburn, James A. (2005).
1155:How to build a 3-pendulum rotary harmonograph
1075:Understanding Pendulums: A Brief Introduction
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1185:Harmonographs, pintographs, and Excel models
1175:An Animated Harmonograph Model in MS Excel
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240:{\displaystyle x(t)=A\sin(tf+p)e^{-dt},}
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984:Computer-generated harmonograph figure
162:Computer-generated harmonograph figure
1100:The Pendulum: a case study in physics
1034:Whitaker, Robert J. (February 2001).
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1170:Virtual Harmonograph web application
1036:"Harmonographs. II. Circular design"
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1126:and Mark W. Zemansky (1964).
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1165:HTML5 Animated Harmonograph
1040:American Journal of Physics
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119:In the 1870s, the term
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343:{\displaystyle t}
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46:harmonograph
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23:Harmonograph
1017:Rittenhouse
63:mathematics
1194:Categories
1102:. Oxford.
996:Spirograph
109:pantograph
1205:Pendulums
1023:(42): 41.
935:Australia
927:Questacon
870:−
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250:in which
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82:frequency
50:pendulums
990:See also
931:Canberra
86:ellipses
1048:Bibcode
912:Gallery
115:History
102:gimbals
90:spirals
65:at the
1200:Curves
1106:
1081:
1002:Notes
1104:ISBN
1079:ISBN
1056:doi
929:in
827:sin
753:sin
656:sin
582:sin
461:sin
387:sin
195:sin
74:pen
1196::
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