Meredith,
    My system also demonstrates two of the potential wells you mentioned; i=
..e.,
"the PG will "prefer" several locations along it's path.  With
a gentle slow finger push, the PG will speed up, slow down
as it crosses these (broad) area magnetic peaks and valleys."
    In the first figure of my article the PG plate is in the photograph pic=
tured at rest in equilibrium in one of the wells.
The other well, where a 2nd stable equilibrium was found to exist, is indic=
ated by the dashed rectangle.  I could, as you noted, move the plate with a=
 gently nudge of a finger, back and forth between the two stable positions.
    The pair represent the heart of a classic nonlinear oscillator, called =
the Duffing.  It has been studied extensively by chaos scientists during th=
e last twenty years or so, even though Mr. Duffing first encountered it man=
y years earlier.  I studied and wrote a paper about a mechanical form of th=
e Duffing oscillator, titled "Chaotic pendulum based on torsion and gravity=
 in opposition", published in the Amer. J. of Phys. Vol. 63 (12), 1995.
   If I had elected to operate with a single PG plate rather than the pair,=
 the nonlinearity of just one piece, rather than the two, would be responsi=
ble for considerable (undesirable) nonlinearity.  In other words, the asymm=
etry of the Duffing oscillator, when confined to a single one of its two we=
lls, gives rise to harmonic distortions that depart significantly from the =
ideal features of the parabolic (harmonic) well of the Hooke's law (idealiz=
ed) case with which we always seek to work.
    By connecting the two plates with a small copper wire I am able to 'cor=
rect' for the thickness-gradient effects of a single plate that you mention=
, while at the same time eliminate the asymmetry of the Duffing problems th=
at result from the use of a single well. Moreover, the sensitivity is incre=
ased because of the array form.  Who could ask for anything more!  I wasn't
expecting such a fortuitous consequence, and am again reminded of a saying =
by one of my Cajun buddies:  "even a blind hog sometimes finds an acorn".
  By the way, Dr. Martin Simon is trying to get in touch with you.  He does=
n't have a workable address for you and asked if I had a current email addr=
ess (for which my response was negative).  I'm sure he would be pleased to =
receive from you an email directed to him at
msimon@................
    I suspect that Martin, like myself, though functioning as a 'profession=
al' while working on related problems; must have been also the recipient of=
 rewarding contributions to his research from the 'amateur' community .  On=
e of my colleagues at Mercer (Matt Marone) admonished me years ago to get i=
nvolved with folks of the type who contribute to this list-serve.  His reco=
mmendation proved to be very beneficial to my transition into the world of =
seismology.  Some of my interactions that have happened during the (approx.=
) last five years have been the source of influence (even to the point of f=
riendships) that I will always treasure.
    So I encourage you folks to keep on offering your ideas amongst us, no =
matter how 'off the wall' they may seem from your own perspective.  Those f=
olks in the scientific world who continue to insist on maintaining the 'sta=
tus quo' of 'purist' refereed publications restricted to a relatively small=
 'ivory towered elitist world' have no idea what they've been missing.
     Randall



Meredith,=
    My system also demonstrates =
two of the potential wells you mentioned; i.e.,
“the PG will "prefer" several locations along it=
's path.  With
a gentle slow f=
inger push, the PG will speed up, slow down
as it crosses these (broad) area magnetic peaks and valleys.”=
;
    In the first fig=
ure of my article the PG plate is in the photograph pictured at rest in equ=
ilibrium in one of the wells.
The oth=
er well, where a 2nd stable equilibrium was found to exist, is i=
ndicated by the dashed rectangle.  I could, as you noted, move the pla=
te with a gently nudge of a finger, back and forth between the two stable p=
ositions.  
   &n=
bsp;The pair represent the heart of a classic nonlinear oscillator, called =
the Duffing.  It has been studied extensively by chaos scientists duri=
ng the last twenty years or so, even though Mr. Duffing first encountered i=
t many years earlier.  I studied and wrote a paper about a mechanical =
form of the Duffing oscillator, titled “Chaotic pendulum based on tor=
sion and gravity in opposition”, published in the Amer. J. of Phys. V=
ol. 63 (12), 1995. 
  &nb=
sp;If I had elected to operate with a single PG plate rather than the pair,=
 the nonlinearity of just one piece, rather than the two, would be responsi=
ble for considerable (undesirable) nonlinearity.  In other words, the =
asymmetry of the Duffing oscillator, when confined to a single one of its t=
wo wells, gives rise to harmonic distortions that depart significantly from=
 the ideal features of the parabolic (harmonic) well of the Hooke’s l=
aw (idealized) case with which we always seek to work.  
    By connecting the two plat=
es with a small copper wire I am able to ‘correct’ for the thic=
kness-gradient effects of a single plate that you mention, while at the sam=
e time eliminate the asymmetry of the Duffing problems that result from the=
 use of a single well. Moreover, the sensitivity is increased because of th=
e array form.  Who could ask for anything more!  I wasn’t
expecting such a fortuitous consequenc=
e, and am again reminded of a saying by one of my Cajun buddies:  R=
20;even a blind hog sometimes finds an acorn”.  
  By the way, Dr. Martin Simon is trying t=
o get in touch with you.  He doesn’t have a workable address for=
 you and asked if I had a current email address (for which my response was =
negative).  I’m sure he would be pleased to receive from you an =
email directed to him at
msimon@................
    I suspect that Martin, like myself, though functioning =
as a ‘professional’ while working on related problems; must hav=
e been also the recipient of rewarding contributions to his research from t=
he ‘amateur’ community .  One of my colleagues at Mercer (=
Matt Marone) admonished me years ago to get involved with folks of the type=
 who contribute to this list-serve.  His recommendation proved to be v=
ery beneficial to my transition into the world of seismology.  Some of=
 my interactions that have happened during the (approx.) last five years ha=
ve been the source of influence (even to the point of friendships) that I w=
ill always treasure. 
    So =
I encourage you folks to keep on offering your ideas amongst us, no matter =
how ‘off the wall’ they may seem from your own perspective.&nbs=
p; Those folks in the scientific world who continue to insist on maintainin=
g the ‘status quo’ of ‘purist’ refereed publication=
s restricted to a relatively small ‘ivory towered elitist world’=
; have no idea what they’ve been missing.
&=
nbsp;    Randall      
  =
      
=
 
 
=