A harmonograph is a mechanical apparatus that employs

pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...

s to create a geometric image. The drawings created typically are

Lissajous curve A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dire ...

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 Bailie Hugh Blackburn (; 2 July 1823, Craigflower, Torryburn, Fife – 9 October 1909, Roshven, Inverness-shire) was a Scottish mathematician. A lifelong friend of William Thomson (later Lord Kelvin), and the husband of illustrator Jemima Bla ...

, a professor of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

at the

University of Glasgow , image = UofG Coat of Arms.png , image_size = 150px , caption = Coat of arms Flag , latin_name = Universitas Glasguensis , motto = la, Via, Veritas, Vita , ...

, is commonly believed to be the official inventor. A simple, so-called "lateral" harmonograph uses two pendulums to control the movement of a

pen A pen is a common writing instrument that applies ink to a surface, usually paper, for writing or drawing. Early pens such as reed pens, quill pens, dip pens and ruling pens held a small amount of ink on a nib or in a small void or cavity whic ...

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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

axis. By varying the

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

and phase of the pendulums relative to one another, different patterns are created. Even a simple harmonograph as described can create

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are: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 A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...

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 (as opposed to a

pantograph A pantograph (, from their original use for copying writing) is a mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line dr ...

, 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.

Blackburn pendulum

A Blackburn pendulum is a device for illustrating

simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...

, it was named after

, 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

; it belongs to the family of mechanical devices known as harmonographs. Mid-20th century physics textbooks sometimes refer to this type of pendulum as a

double pendulum In physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaos pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a ...

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 :

x(t) = A \sin (tf + p) e^,

in which

f

represents frequency,

p

represents phase,

A

represents amplitude,

d

represents damping and

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 :

x(t) = A_1 \sin (tf_1 + p_1) e^ + A_2 \sin (tf_2 + p_2) e^.

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 :

\begin
 x(t) &= A_1 \sin (tf_1 + p_1) e^ + A_2 \sin (tf_2 + p_2) e^,\\
 y(t) &= A_3 \sin (tf_3 + p_3) e^ + A_4 \sin (tf_4 + p_4) e^.
\end

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

File:Harmonograph.jpg, A harmonograph at

Questacon Questacon – The National Science and Technology Centre is an interactive science communication facility in Canberra, Australia. It is a museum with more than 200 interactive exhibits relating to Science, technology and society, science and ...

Canberra Canberra ( ) is the capital city of Australia. Founded following the federation of the colonies of Australia as the seat of government for the new nation, it is Australia's largest inland city and the eighth-largest city overall. The ci ...

Australia Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasmania, and numerous List of islands of Australia, sma ...

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

Notes

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External links