APERIODIC MOTION
OF A SUSPENDED MAGNET BY G.
ARMOUR
E.
MARSH
INSTITUTE OF TECHNOLOGY
1912 mim^^-
538
M 35
mo3f
tostatuee
of Techi'miogy
UNIVERc
.3MRIES
AT 263
The
Aperiodio of
a
Suspended A
motion
Ma£:;net.
Thesis
Presented George
E.
by
l''arRh
the
to
President
and
Faoultj'-
of Anaonr
Institute
For
the
Electrical
of
Degree
Technology of
Enrcineer.
u ^^p^^ M.yi
.^e a^
ciK^.
.
7
^ ^
-
m be deBicnater! by
rn
and
then the particular Bolutions aro
S a.nd
_
--
^ '^'^r e^'''^
'"^'
-^'' :>
the ger.ep'^l solution Is given by
S
=
A
6
^
Ae
e'^'-^i(a +
Ce^'"^'
^.^-T-
K+r3e
(.^-v.,.^. 3^p^>)
There are three separate oases in the detailed
consideration of
J'n.
(5/
)
and they are deterrrdneri
by the value of the radical,
t>;at
is, by the relative
magnitude of n and f. In general, then, vhen
m^ are real quantities and unequal, of the form
and of the form
y.'hen
S
=
/^ &
S
solution in
^^
+
--(.^
t:ie
r:i,aj;d'
t
l^tji^
^"^-^ ^
the roots are enual. If the roots be ir-aginar:,^
It
shall be our problern to consider in
sorae
19
detail the o'aar^^GteriRtics of thf:se equations unrter varioxiP appurnptionR «.nd to that end. re shall divide
the invest ic'ttion into three ijartn.
nf
.
Fo.( ^^).
Partjii^
n=f.
Eo.(r>.'^).
Part
I,
Part I.
'^he
xr.^
and
vhevp
radical is imaginary and if the tro roots
rcL^he
i
vrritten
iLj^-n -ig =-n -r; xri^-=.-n-tlr- -n+r;
= /=r, 2 ^ V f - n"
,
and r =i£: ^•V'n^-f^
,
then
our equation may oe v/ritten
If the initial oonditions
s=a
and s'
=
are trae
for the time t^O, then the constantB have the values
B
—-ha(n+r)
/Pt j
and accordingly
tanecus
lis
]
v-e
h^vp the emjation for the ins+rsn-
acenent in t>p form
.
Also
the.
the
find
vrlooitv
.
)
.
^
j.
)
)
^^
acielemtion
It muBt nor ue
Rhofnt^rjt
the la'tior tn nneptlon
in opriodic irA for th'^t piirponp let ur
t.'^Kp
usf-
of
the pair of exi^reBRionr'. e
=cop-~t + inin£;t,
e'^*
- ooRfft
The PAilstitution of
ab o
•>'
e,
e qu'^.t io
-
i-sin-gt.
trigonometric
thf:
terr-in
in the
coBgt
+ i-Ringt
g iv e r
)
-Ht
R - e
(
A(
-e
(
(
coBgt - i-Riiigt
A+n)cofvgt -
i(
)
+
T^(
)
A-B)singt
Incorporating the valuer, of the oonstantR A and p gives R
=r
ae
(
GORgt
-f
n /gf?ingt
If n/g be reolaced by tan(-°^), _
beoonep
r
=•
ae
{*/o).
)
t'e inrt eXi^rer-sKion
/(
f /g.Ring(
t
-
^
(¥o)
) )
The angu.lar ^'elocity nov uecor.en
in
g( t -^
)
)
O^-^^
.
.
21 In the capt^ of Ep.(40) the '^^flpction ip zero vhen
t-l/gt°n
(-3/h)anrl
If the
+.= o
thiat
dai.'iping
k and hence if
ef^^eot
s
ivresent
in a single
be the undar;.ped and s the
I-.
'
,
?;5
or
then
vfilMP of an oRoillation.
^maijR'-I
'^J'^
^
= e
,
-^R
log Ro=
lo2>:=> •^/'"
e
,
-hlo£:
^
sjr
Aocorclinjly
ir.
R
---
.Mi-^i'>;.i- >,c!+
-
/TT
-
)
f?'^-
v'hich is the ordinal' o^orn\ila.
Rh" 11 noY!
"'e
PXi.reBsion for the
S
=
s =
A
'
=
•",
.
t
=
.
uo the
da.'.ijing
/!UAA(^5t-c/
valv.e e
^
It
e
'
-
I'alv.e
of the true
beoomeR equal to the ao^roxirnate
for the "alue of A-oonly.
if?
interesting to note in passing that
a relation that enables the constant n to be determined.
And fron the second eouation
the other enu^itional constant
inay
be found.
'
1
/f PftCCor
~
3H NIov:
for thr
'i
re
th^-t
-lerivei t'\e tr-ne nv ,reRrion
h-^-A^e
factor, it vill be -irRirable
'^•^mping
ooi'^rison of fie
tlty ip encoT-mtered
foms
vario-if?
ar.ri.
in vhioh thin nu^.n-
uRed. T'ith the
y^ttention rill first be onid to
correot one,
Its relation to
Hence
feux
of Prof.
^"
.
\-hiGh has £;one before
t"'.a+
=~
'(^^ -T—
chit
1't.t
th'^t
vhore
orif^-in
Stroud.
rr.ay
be
ih .the follov'i]~c ray. Nov, clearly, ve
eaRili'' sho^'n
have
dr»
the'
form
tvie
ras i.ore recently/ ijrosed by N.K.Sriith is to be olaced to
to nnke
%/ A
— *^ -
/a
(^0-
~
Y'here h in a const aiit.
and as the tnie :"elition
-==-
betreen t^e. damped and the undaiuued deflections is given by
S^-S^G.
If in Ea.(^/
)
iR nerlifible
,
re
Tna3'-
rrite
^2,)
iP.
vO^J-
Xis negligible con^^ared toir, then h ii^.
corr.p -orison
tolT/p and the-^eforr
iR negligible rit'i resj^ect to unity.
Eq.(
S"o--S,e >e.
PenRibl:r
en-'iPl
to
^o
=
I^lA
Accordingly, >S,
e
'-^
(^3).
.
."^9
3iit
by defijiition,
Strong
r,
and hence vp have as
- o
/f'>
form for the damping factor
'n
an pri.rp?^Fi.on
th?=it
±nz the oase v'hen
can on'.y be \ ip
T>e~ar'i.e'l
as represent-
rrnall in con^jTriPon T'ithTT.
An eTanination of En.(
^i'^f
)
and F,n.{fR/)
fihovrp
that the value of the nndaiiiijed swing as ^iven oy the forLier is alv^'^ys larger thaii that oaloulated by the
aid of the latter,
"^he
error int-^oduaed by
th(^
use
of the Stroud- FiL'iith forrr.ula increases v-ith the logarithniic decreraent and the ratio of the tr^o values
r
Bearing in of y
,
^
riind that h is
re see
only Y'hen
th-'^t
a
directly varying fvuiotion
the tro s^'inrs become the same "
y is ?ero
The ppr cent error
b'c^'*^^^'^
in'
J
any cast; is gi^pr. by
-
f
y
»,.
il.'«*^«1l'fS
40 The wraph of thip. last eniiation Pi'ablep t^e
error oresent in the une of
obtair.P'-''.
tween
y,
thlr.
rtii.i^lo
fori",
to
l-e
by insi^eotit)n. This ennation. plo+teri beof^r
an*!
ofnt error,
if:
given on
ith.irvf-Rht^jet
No.XT Lsstly,
!+•
na"
Been that
'ue
p
'''"
np shovn in
the oorreotior-f actor to the oori^e^tion-
EQ.(f.5~).
i-^
f actor.
In critical
'-iaaping;,
r?aenX-^ the error
becomes infinite. We shall nov pass on to the consideration
the stp?n9.ar4 forrr.nl^
^o
"
^rv
''"
v4 3
-
'
of
vhich consists
of the first tvo terras of the expanion of
e
,
PS vas shov:i. in t^e earlier ^art of the chaj..ter.
The error present in this case is clt arly given by
This equation He:^e.
if?
as before,
^.lottei^.
on Ou.rve-sheet Mo. XLL
the error lof^cones infinite vith
^imft
«
^^^^^' »it
K=v«.
•
'' .
•?
1
41
Gritical
rir-n.jinc.
But ve knar that
?^=
c ~p..718r^^ in
!^^
instance,
thir,
Fron the graph of it ip seen that
V'.e
for Ruch
of K
valrxp.^
t::iF i^ercerit
r^.asnit\ide
^"e
of V:p error
Ip.
Pmall
are ordinarily nnconnterrd,
^^.
and acoordingly this forj; of
used in fne a^^erage
error ennation
daiaxjing
fnotor
ca::
le
oaf^e.
to consider the enaijirical forraula '?«^ SJ. s, . i(
have
nov:
^-^)
Since
s, /s
-
e
Sj^ v:e
sr-iall
,
"oer
«
--
X+
-
-
)
if higher ijovrers te neglected, rS. s„'^. + C '^4 •^' fron En.(^7
yy
,
J>
then
there results The
(i-+
= l-h
have(,V.^ *^^j.d
If X be
njid
^r
)
^j/^.i-" Gouj^arable
S^ -
S,
cent error in
'^'Tfl^i'
gi'^'en
^z)
b""
Q.E.P.
•
^
unity
\at'-.
.
,_,
and an inP:j->e
arrani^c"! in
tabular forra
the niimerioal value«^ of dorrection factors for dauping as calculate
1
by
Vv
'^^ever'^l
jreceedin^
I'orriulap.
Correction /
^ 1.
X
riA t
'"''rictorR,
4? (olMWft'*^
t;kLlYir»(
iitVCt^'i
/.-te-'T^
K
i-fY^
1^-
\
of gen-
eral interest it supplieRthe oonclurfing chautt-rB that
rcake the riiacup^sion
Bearing in ';n'^erl3''ing
complete
iriin^_
that the fun-iarnental equa-jrions
aperiodic motion are S-e* CAe of danplng
Thev. the coefficipr.t
1?^
coefficient of re^titM+lon, and
-t-'Se
.
,1
J-
greater than the
s_-
e
*"
(^t
t2>7^}^
L^?))-
when the tro ooof "icle]-t8 are equal, ve have tvo con-
ditions to deal rith and rhich the first and
Let
rhich
nr.
Re-^or.d
aRRui.ie
Trill give it
i-ill he ^lesignated as
capes teBpectively,
that the initial velocity is
a
direction the
sa-ae
— c,
as that of
4o the restoring force, at
t—
and s'-_c, re obtain as
Then^as t—
0.
t>ie
0,
valtic of the constants
of integratioj;,A,B, in eouationp
(
j"3
)
,(
,
The notion is still
s = a,
'
a^^eriodlc as may be seen
vhen t-«D;tho, as
v;e
shall see,
greater than a certain critical nagnitude
causes the body to pass
thjr.x
position on the other side
zero, attain a Maximum
and.
then creep bncK. to the
zero position, unrler no circumstances can oscillat-
ions
ta?.e
pla
)e.
The velocities nre easily found to be
and S
-
e
.^C^{h^~c)t -cj
•
iP)
