                        The Single Helix

    Some toys are destined to be broken in the child's attempt to
discover the source of their magic sway.  No doubt many engineers
have sprung from such troublesome children, surrounded by the
empty wreckage and clinging dust of a trashed Etch-a-Sketch.
This gives us proposition one: a good toy is one with high
mortality rates.
    Furthermore, a plaything's power rises in proportion to the
simplicity with which it suggests elementary social or physical
phenomena.  Social toys date since the first time a stick was
raised in mock attack (war) and a doll with the vaguest of human
outlines was cradled (love).  But in a second, more Apollonian
category are objects--are they toys?--intended basically for
admiring.  Here's a list of current contenders and their
provenance:  Suspended metal clacking balls--Newtonian mechanics;
the lava lamp--heat transfer and cavitation with non-Newtonian,
low-Reynolds-number fluid dynamics; and the Slinky--wave
propagation, somersaults, and the mysterious, entrancing
animation of nothing but a dinky piece of metal tape.
    One Slinky expert's epiphany reveals the toy's pull.  In 1946
W.J. Cunningham was midway in his first year as a professor in
Yale University's electrical engineering department when someone
gave him a Magi-Koil.  It was a helical steel spring with about
80 turns.  The helix was a ribbon of thin, rectangular steel, and
if left undisturbed each coil stacked almost entirely on the
next.  The diameter was about 2-1/2 inches and the whole thing
about the size of the box a baseball comes in.
     Oscillated with two hands it looks like pouring water, as
energy is resupplied as the captivated user pretends he is
weighing fruit. Or, you put it on a top step, pull one end to the
step below, and let it do the rest.  According to the ads, it
could "walk as though it were alive."  Today it would be
difficult to find a child who disputes that.  The Magi-Koil was
renamed the Slinky, a mechanical engineer made a mint on it, and
today it's essentially unchanged.  Prof. Cunningham still hasn't
figured out how it works.
    Forty years ago Cunningham had printed a paper quantifying
the actions of his intriguing new toy.  Today he still teaches at
Yale, and also is chairman of the board of editors of American
Scientist magazine.  Writing in the May-June 1987 issue of the
magazine, he denigrates his earlier self-assurance, now that he
is "less sure he really understands all that goes on."
     In an interview he elaborated on the problems.  "The general
idea is pretty obvious, but even with all the physical
measurements I wanted I couldn't tell you the minimum height of
step to kick it off on a walk down the stairs."  It's tied up
with the damping in the spring, which is hard to measure to begin
with.  "What's clear is the larger the damping in the spring, the
larger the step height, and I can tell you how long it will take
to go down the step.  But working from first principles I have
not been abe to set up a predictive mathematical model." 
    Before looking at the analytical theory of Slinkyonics (as
we'll see, there is no one scientific domain within which its
actions can be described), we can get a sense of its odd-ball
usefuleness.  In Vietnam, American soldiers used it as antennae
for radios; its jiggles have been been used to predict the onset
of an earthquake; while being observed intently from Earth, Space
Shuttle astronauts have used it to while away the orbiting hours
and prove--in case you were worried--conservation laws.
    Who needs the $4 billion supercollider when you have the
Slinky?  One paper in the American Journal of Physics suggested
that a moving Slinky, when hung in the air by thread at various
points, models the  types of waves in gas plasma.  Cunningham
recently received a physics paper on "dispersion waves," from a
woman who attached one end to the side of a door, which acts as
a soundboard.  She stretched the thing out, and snapped the wire
at the far end.  Try this at home.
    The first sound that comes out, says Cunningham, is called a
"whistler"--a high-frequency whoop-whoop, like a descending bird
cry with a sharp ascent, or, to my ears, an eerily synthesizer
sound-alike suitable for a Star Trek episode.  A short time
later, said Cunningham, warming to the imitation, comes a deep
voiced phew-phew, a sharply descending cry from the bird's older
brother.  There is far too much ambient noise in my apartment for
me to ever actually hear these frequencies.  But the point is
that the high frequencies travel along the spring faster than the
low ones, each at a particular dispersion.

Stepping out.  The Slinky in its preeminent role, as sinuous
stair descender, exhibits different characteristics, most
prominently what is called an extensional disturbance wave.  A
spring is a medium with distributed mass and stiffness.  In a
spring with a linear medium, an impulse--disturbance--will travel
at a speed of the square root of the ratio of stiffness to
density.  But is the Slinky doing its stuff linearly?
    Midway through its jaunt down the steps, the Slinky has two
essentially straight axes connected by a wide-linked, U-shaped
arch through which the metal snake seems to draw itself.  The
mid-air portion of the spring is stretched out, compared to the
compact back foot (the empyting pile) on the upper stair
unseating itself, and the collecting pile of turns on the lower
stair.
    Despite the event's apparent nonlinearity, Cunningham
considers it analyzable as a linear case of small deflections
along the length of the helical wire itself.  In his study, he
considers an idealized Slinky resting horizontally on a
frictionless tabletop.  (Surely Plato himself must play in the
frictionless land stocked with idealized toys.)  Say the right
side of the spring is jerked to the right.  An extensional
disturbance travels along the wire with a constant speed.  Each
turn moves briefly to the right with a certain "particle
velocity."  The last turn of the slinky finishes up at twice the
speed of the first one.  (We'll see why the speed doubles in a
moment.)
    Let's move from the heady world of ideals to the hall
staircase.  The researcher, or child, has piled the Slinky on one
step and obligingly placed the free end on the step below. 
Inertial and elastic effects cause a wave to travel through the
arched coil upwards--although it looks like the spring is pouring
downwards--and the last turn is lifted off the step with velocity
twice that of its brothers resting on the step above.  (It's hard
to keep it visualized that the spring is a continuum through
which the wave passes, even though the visible parts of the coils
seem to be knocked against each other one by one.)
    If the force is high enough at the critical take-off instant,
two events occur: the arch stays arched due to centrifugal force,
and the last rung vaults over to the next lower step.  The arch
inverts, and by the time the last rung lands, a few adjacent
turns are thrown in contact.  The turns pour onto the once high
flying turn, now the bottom of the pile.  A new disturbance has
begun in the opposite direction.
    It may seem that having the free end of the coil moving at
twice the speed of the initial coil, while receiving no
additional energy, violates conservation laws.  But the
explanation is related to the theory of transmission lines.  As
the pulse comes down the line, the quantity of material that is
moving becomes smaller.  With a decrease in mass, the velocity
has to increase to conserve the energy.  It's  similar to when
you snap a towel--considering here the snap made in mid-air, not
on someone's wet skin--or, even better, to popping a whip.  
Massive energy is needed to whip the massive stock of a bullwhip. 
A uniform amount of energy travels down the steeply tapered whip. 
The crack of the whip is the shockwave when the featherweight
flick-end breaks the speed of sound.
    A physical case closer to that of the Slinky--propagation
made visible by discrete points along the line, and uniform
material dimensions--is apparent in curtains made of hanging
beads.  (A locale of an exotic boudouir comes to mind.)  Large
bead curtains also have a nicer planar aspect and lovely
billowing effects, a phenomenom I once noticed as I stood from
the second-floor balcony of Avery Fischer concert hall in New
York, trying to flick, in exhilarating slow motion, strands of
the enormous bead rope against the ankle of a dowager in the
lobby.
    But a Slinky in action is different than the flick of a
beadstring.  The Slinky's two ends are constantly changing their
relative situation: one end initiates the pulse, and one end
awaits it, and then back again.  Energy is certainly lost inside
the spring itself, although with steel it's probably fairly
small; energy is also lost when the free end makes an inelastic
impact with the step.
    The takeoff speed and the material design of the Slinky are
critical.  The speed must be high enough to propel it down and
over two steps, but the entire rippling effect must be slow
enough to be visible.  To slow it down, you need relatively more
mass per unit length.  How do you keep down stiffness per length
(not the lateral stiffness)?  Edge-wind it: flatten the wire
(give it rectangular cross-section) which simply reduces the
ratio of stiffness to mass.
    The edge-winding of the Slinky reduces the axial length for a
fixed mumber of turns, helps the windings stack, and gives a
larger lateral stiffness to resist shearing forces--which keeps
the slinky from slip-sliding around as it pours into its
invisible glass.  The scaling factors for the Slinky are linear,
for those of you familiar with the Slinky Jr., a half-size, half
speed offspring.  The plastic, brightly-colored Slinky now in the
stores moves twice as slowly as the steel one, and is better for
engineering demonstrations (and children who gnaw on everything)
but is a loser as a toy compared to Old Reliable.
    Cunningham's first Slinky is still the only one in his eyes. 
Today's brass model, for example, doesn't work nearly as well as
the steel, he claims.  The brass doesn't have the the right
relationship between Young's modulus and the density--it's not
stiff enough against lateral deflection.  Perhaps we need
something like the original-instrument movement among the music
buffs; otherwise we will never know the true stuff of the device.

Birth and transfiguration.   In November 1945, Gimbel's (remember
Gimbel's?) sold out 400 of the brand-new items in 90 minutes. 
Two years later a patent for the Slinky was given to Richard
James, a Penn-State mechanical engineering graduate working for
Newport News Shipbuilding.  The original design was first
licensed out to one Leroy Shane, who marketed the Magi-Koil.  (As
we will see, there are some grey areas in the genesis story of
the Slinky.)  Eventually, the enterprising engineer founded James
Industries Inc., Holidaysburg, Pa., now run by James's widow
Betty.  The company's flagship model is made of "cold-rolled
spring steel," as divulged by a tight-lipped Mrs. James,
fabricated from round wire rolled out at zero tension, flattened,
and twisted into the helix.  The company turns out 6000 Slinkys a
day.
    What engendered the idea for the Slinky?  At one time voice
coils in loudspeakers were made edge-wound, like a Slinky, in
order that as much metal as possible could be within the magnetic
field.  So goes one theory for the initial design concept, its
adaptor to toydom unknown.  But according to Mrs. James, Richard
James got the idea when he dropped a "torsion spring."  The
quotes you see around the words are because I could not discover
exactly what this object is, and Mrs. James could not, or would
not qualify her information any further.  Interestingly, Prof.
Cunningham received a letter containing a third story, one with
darker overtones, but plausible nonetheless.
    According to this Deep-throat-delivered story, someone ran a
machine shop in Philadelphia making helical piston rings for
small gas engines. (Within each engine cylinder, placed over the
piston, are two or three springy helical coils of tape that bear
against the cylinder wall.  Their expansion keeps the cylinder
gas-tight, and the flat coil keeps lubricating oil away from the
burning fuel.)  The story goes that, after slicing off the tops
of the steel helixes to make the rings, whoever ran the machine
shop noticed the properties that now we all know.  The identity
of this "whoever" remains shrouded.  Some time later Richard
James got wind of the doctored piston rings, and marketed
the concept.  The relationship of humanity and staircase was
irrevocably altered. 



Copyright 1989
Copyleft 1989
Robert Braham
Scitech Publishing Services

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(An abbreviated, edited, and unsigned version of this text,
Slinky.txt, appeared in 1988 in Mechanical Engineering magazine.)
 

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