In response to a couple of further inquiries that I've received at home,
in addition to the one on-line from Karl Cunningham:

   Experimenting with flexures---  Some flat, good quality
spring-tempered flexure material which is readily available in a range
of thicknesses is feeler guages.  Although typically available only in a
single width (0.500 inch), feeler guage stock should at least allow you
to make some "mock-ups" for experimentation to determine what you need
for your final design.  Go to your local auto supply store.  All should
have the complete feeler guage sets, which have several blades typically
ranging in thickness from .0015 inch to .040 inch that pivot into the
handle for storage.  If you're lucky, the store may also have single
"replacement" blades for sale--- possibly in longer lengths;  ask, since
the single blades may be a "back-room" item.  (The replacement blades
allow you to obtain multiple flexures of the same thickness without
buying additional sets.) In any case, try to get a set which has
relatively long blades and has a screw-type pivot for easy disassembly. 
Most sets have flat blades, but there is also the "tappet-style" which
has an approximate 30 degree bend about 2/3 down it's length; this might
be useful for certain mounting situations.  In any case, a set should
run about $4-$8.  Although obviously a spring steel, and somewhat
"stainless" (although magnetic), I don't know what the material
designation is (and it may vary, depending on origin).  However, the
important material property in determining the spring constant (k) is
the elastic modulus (E), and this is essentially the same for all
steels.  Thus, your experimental results will be "accurate" to within a
few percent for any spring steel.  BeCu, phosphor bronze, and the
"exotics" (such as low-expansion, constant modulus materials which some
"purists" may want for their final design) can obviously be "ratioed"
from your test results--- see paragraph on "Calculations" below.

   Forming flexures--- If your flexure is going to be a simple,
constant-width, constant-thickness strip, and you can obtain the
material in that width and thickness, the only thing you'll need is a
pair of household scissors!  Not tin snips, not a sheet metal shear, not
a guillotine paper cutter, and not your kid's super-dull school
scissors.  You'll find that any relatively thin (say less that .012
inch) metal can be cut well with with a good (sharp) pair of scissors,
and that it actually cuts "cleaner" in the hardened (spring) state than
when soft. You can probably also "get by" with scissor cuts in the
"active" portion of the flexure (i.e., cutting to an exact width) if you
carefully "stone" the cut edges to remove any burrs or distortions.  If
you want mounting holes, obviously the normal drill bit won't do the job
on hardened metals; the three best answers are--- 1. drill first, harden
(heat treat) second, 2. use a hardened punch and die (total diametral
clearance no more than material thickness), or 3. chemical milling.  For
complex shapes, such as the "spider" suspensions in geophones or bands
with internal cutouts (such as in the "Cardan hinge"?), photochemical
milling (etching) is really the only way to go.  Basically, this
consists of coating the material with a photosensitive material,
exposing it to light with a mask providing the desired pattern, and then
spraying with etchant chemicals.  As mentioned in the earlier message,
BeCu is easily etched; stainless steels are difficult.  If you do use
photochemical processing, additional features--- such as holes, slots,
mounting "ticks", etc. are easily incorporated.

   Mounting--- Usually, thin flexures are attached at both ends (one end
on the stationary base, the other to the moving assembly) using clamp
bars and machine screws.  The clamp bars need to be sturdy (e.g., at
least 1/4 inch thick steel for 1/2 inch wide flexures) to maintain good
contact across the entire width, and the screws well tightened.  Having
at least one of the screws through a hole or slot in the flexure can be
helpful (particularly on wider flexures), as are somewhat roughened
surfaces on the mating parts.  Karl asked about adhesives; my feeling is
"only if proven necessary in addition to the clamping".

   Distortion and/or "oil-canning"--- My experience with this would
indicate that there is a some other problem present--- lack of clamp bar
(or one which isn't sturdy enough), flexure material which wasn't
initially flat, unequal weight distribution during assembly, too thin
flexures for the weight being supported, etc.  Having said all of this,
I do kinda' like Karl's thought of AVOIDING ZERO stress/strain on these
seismic devices.  In other words, since the motion is very small, why
not provide a mounting such that the flexure never returns to being
completely flat.  One way of doing this with crossed flexures would be
to have 90 degrees between flexures on one (stationary?) end, and
somewhat more (or less)-- perhaps 100 degrees-- on the other.  Thus,
both flexures would have a small initial curvature--- say 5 degrees if
both have the same thickness, width, and free length--- which is larger
than the maximum instrument motion.  I can't think of any reason why the
crossed flexures wouldn't work equally well.

   Calculations--- To a first approximation, the (torsional) spring
force, and thus the (torsional) spring constant, can be calculated for
flat flexures (or ones with small curvatures) using the "cantilever beam
with end loading" equations of Mechanics of Materials. (Note that you
will probably want to use the "slope at end", since the normal action of
a flexure is a pivot.)  For those who would prefer to do it entirely
experimentally, your rules of thumb are: Force is directly proportional
to the cube of thickness (i.e., doubling the thickness multiplies the
force by eight); force is directly proportional to flexure width;  force
is directly proportional to elastic modulus of the material (i.e., BeCu
will produce approximately 62% of the force of spring steels), and force
is inversely proportional to the square of the free length (i.e., a free
length of 1/4 inch is four times as "stiff" as a free length of 1/2
inch). 

   "Inverted Pendulum"--- As indicated in the earlier message, this is a
(potential?) method of making a long period horizontal seismic
instrument in a compact package.  The principle can best be understood
by clamping a flat flexure, such as a .004 inch thick feeler guage
blade, in a vise such that it is vertical.  At the upper (free) end,
start adding weight (a small bolt, washers, and nuts in the pivot hole
of the blade is a convenient method), keeping approximately equal weight
on each side of the flexure).  You'll note that as the weoght is
increased, the natural frequency changes.  If too much weight is added,
it becomes "bistable", leaning one way or the other when at rest.
However, with just the correct weight, it becomes a surprisingly long
period device (several seconds per cycle).  In my earlier message, I was
somewhat discouraging about the practicality of replacing the single
flexure with crossed flexures; further reflection on my part says that
the final adjustment could be just as indicated here--- adjustment of
the mass instead of the free length of the flexure(s)--- which would
then allow crossed flexures of a predetermined spring constant to be
used.

	---Dean E. Gladow---

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