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Патент USA US2404294

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July 16, 1946.
2,404,294
w. B. JORDAN
INTEGRATOR
Filed Sept. 18, 1944
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*Y /l- Inventor*
William B. JoT' dan ,
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His Attorneg.
Patented July 16, 1946
2,404,294
UNITED STATES PATENT OFFICE
2,404,294
IN TEGRATOR
William B. Jordan, Scotia, N. Y., assigner to Gen
eral Electric Company, a corporation of New
York
Application September 18, 1944, Serial No. 554,661
4 Claims. (Cl. 23S-61)
1
2
This invention relates to integrators, and it
has for its object the provision of an improved
integrator which is positive and accurate in its
grator comprising a variable o1' integration input
shaft I0, an integrand input shaft II, and a
final integral output shaft I2. The specific ein
bodiment of the invention illustrated integrates
operation.
More specifically, this invention contemplates
with respect to time (t) the magnitude of a
variable rate of change (œ), which may be the
rate of change of range to a target. Thus, the
the provision of a mechanical integrator which
computes its integral by the trapezoídal rule;
variable of integration (t) is introduced by the
shaft I0, the integrand (œ) is introduced by the
series of trapezoids; computing the area of each 10 shaft Il, while the final output of the integral
shaft I2 is (y), (y) being related to a: and t-by
trapezoid by multiplying one-half the sum of the
the equation: y=lfxdt.
lengths of two sides thereof by the base length;
The shaft I0 operates a first reversing mecha
and finally by adding the calculated areas of the
that is, it mechanically computes the desired area
under a curve by sub-dividing the area into a
trapezoids together.
nism I3; as shown, the shaft I0 is geared to
In accordance with this invention, there is pro 15 drive an input gear I4 of the reverser I3 by means
of a shaft I5 to which the shaft I0 is geared by
vided an integrand input shaft and a variable
bevel gears I6, and by a shaft I'I to which the
of integration input shaft. Means are provided
gear I4 is secured and which shaft is geared to
for sub-dividing the total variable of integration
shaft I5 by bevel gears I8. The gear I4 is in
input to determine the interval of integration.
And also means are provided for determining the 20 mesh with gears I9 and 2U which it rotates in
opposite directions and which are mounted on
product of this interval and the values of the
a shaft 2I to rotate freely with reference to it.
integrand introduced by the integrand input
A shuttle 22, which is slidably mounted on the
shaft so as to determine the integral of the
shaft 2I but which is arranged to rotate it, as
integrand with reference to the variable of in
tegration,
25 by means of a splined connection (not shown),
is arranged to be operated selectively by the two
More specifically, a first reverser is provided
reversely rotating gears I9 and 20; for this pur
which is operated by the variable of integration
pose, the shuttle is provided on its opposite ends
input shaft; it periodically reverses the direction
with clutch teeth 23 and 24 which are adapted
of rotation of the input thereto, and thereby sub
divides the total of the variable of integration 30 to mesh with sets of teeth 23a and 24a. `formed
on the gears I9 and 20, respectively. The shuttle
input in order to determine the interval of in
is shown in its neutral position in Fig. 1, and
tegration. This interval is multiplied by the
when moved upwardly to cause its tooth 23 to
integrand values fed in by the integrand input
engage gear tooth 23a it causes the shaft 2| to
shaft by a suitable multiplying mechanism in
order to determine the areas of the various sub
35 rotate in one direction, and when moved down
divisions. The output of the multiplier is fed to
an output shaft which, because of the reversals of
the reversing mechanism ,is driven first in one di
rection and then in the other, and the total motion
of the shaft is imparted to a final and integral 40
output shaft by means of a second reverser. This
second reverser always operates the final shaft
in one direction, until the integrand changes
sign, so that the total movement of the shaft is
the combined forward and backward movements
of the multiplier output shaft.
For a more complete understanding of this
invention, reference should be had to the accom
wardly to cause its tooth 24 to mesh with the
gear teeth 24a it causes the shaft 2| to rotate in
the opposite direction.
The shuttle is moved to its upper and lower
driving means periodically by means of a cam 25
which operates between- a pair of spaced collars
25a and 25D on the shuttle. This cam 25 is peri
odically operated by the input shaft I0.
This
is accomplished through a Geneva gear train 26
having three stages 26a, 2Gb, and 26e. This gear
train is, in effect, a revolution counter, and every
time the shaft l0 turns over a predetermined
number of revolutions the cam 25 is operated to
move the shuttle to reverse the output of the re
panying drawing in which Fig. 1 is a perspective
view illustrating an integrator arranged in ac 50 verser mechanism; as shown, the output of the
gear 21 of the Geneva train meshes with a gear
cordance with this invention; and Fig. 2 is a
diagrammatic view illustrating the principle of
operation of the mechanism.
Referring to the drawing, this invention has
been shown in one form as applied to an inte
28 which is fixed to and drives a gear 29. The
gear unit 28 and 29 rotates freely on the shaft
I0. The gear 29 drives a gear 30 which in turn
55 operates the shuttle cam 25 through a shaft 3l,
2,404,294
a
¿i
bevel gears 32, shaft 3f, bevel gears 34 and cam
differential 63 by ring gear 6l. Differential 69
shaft
In the specific embodiment of the invention
illustrated where the input of the shaft is is time
(t), the mechanism
operates the shuttle everyT
(it) seconds, and therefore, in effect, mechani
cally sub-divides the total variable of integration
also has an input gear 'il which is driven by shaft
El 'to introduce elim; as shown, it is driven from
shaft li by bevel gears l2, shaft "i3, spur gear 'M
fixed to shaft ‘f3 and an idler gear lâ meshing
with the gears 'El and lli. The gear ratio between
shaft l i and gear ‘H is such that the rotation of
the gear li measures the product ‘li-Kaz. This
mi’
(t) in order to determine the interval of inte
gration (ît). '
>
The output of the reverser i3 is added to the
input (r) of the integrand input shaft l l in a dif
ferential 36. This differential, as shown, has an
input gear 3l, which is driven by gear 38 attached
to the shaft 2l; an input gear 39 driven by a
gear ¿iii attached to the shaft il ; and a set of 15
planetary gears lli which drive the output shaft
£2 of the differential, which output is the sum-`
mation:
t<of shaft 10) +r<of shaft 11)
Another similar differential 43 subtracts the
input (x) from input (t). This differential has
an input gear fill- driven by the output gear-3B
product is subtracted from the product 4MM-K)
in the differential 6% so that the differential out
put gear lâ measures 43st.
It will be observed, in View of the foregoing,
that the differentials ¿if and (ai, the two cam
mechanisms
and ff, and the two differentials
and ¿it constitute a multiplier wherein the
inputs thereto from the reverser i3 and the in
put shaft il are multiplied so that the output
becomes :the product 40st.
The output ¿fait of the multiplier is fed to a
shaft 'Il from gear it, which gear meshes with
gear lla, fixed to the shaft 'Ví so that the motion
of the shaft ‘il measures the product 496i.
The shaft l? is utilized to drive the ñnal in
of reverser i3 through the gear 3l; an input gear
tegral output shaft l2.
¿l5 driven by the gear il@ of input shaft Il; and
the reverser E3 in periodically reversing, the mo
tion of the shaft ll is ñrst positive and then
a planetary gear set e6 which drives the differ
ential output shaft ¿il to measure the difference
t-œ.
The output shafts @2 and 4l of the two differ
entials 36 and ¿i3 are fed into two cam mech
anisms fill and ¿i9 respectively, which mechanisms
generate quadratic functions of the inputs. The
two mechanisms ¿i8 and i9 comprise cams 5!! and
El respectively having formed therein a series of
gear slots 52 and- 53 arranged in spiral paths,
and with which toothed gear wheels 54 and 55
mesh respectively. The cam plate 50 has a pe
ripheral gear 5t which meshes with gear El driven
by shaft ¿i2 while cam 5l has a peripheral gear
driven by gear 59 of shaft lll.
Therefore, `the summation of t-|~5c is fed into
gear cam 5&3, while the difference t-:r is fed into
gear cam 5l.
Because of the action of
negative. It is desirable that the final output
shaft if always be driven in the same direction,
and that it add up the total motions of the shaft
'il in its two directions; therefore, the shaft 'Il
drives `the final output shaft l2 through a second
reverser TES. This reverser comprises input gears
le and 88 which are driven in opposite directions
by the shaft 'if-the gear l@ by means of gear
8l ñxed to the shaft il, and the gear 86 from
the gear 8l through gear 32 fixed to a shaft 83,
and a gear Sit also fixed to the shaft 83 and
meshing with the gear 8f?, This reverser fur
ther comprises a shuttle 85 with two sets of teeth
8l' and e3 which are arranged to mesh with sets
of teeth 3S and 89 formed on the two input gears
‘F9 and e@ respectively~ The shuttle is splined to
the shaft i2 and, therefore, when it rotates it ro
In order to always keep positive the radii of
tates this shaftl The position of the shuttle 85 is
the two spiral paths of gear slots 52 and 53, a 45 controlled by means of a cam 9E! that operates be
constant (K) is added to each input so that the
tween the shuttle collars âìl and 92. The cam 90
input to the cam 50 becomes t-i-K-l-œ, while that
is driven by the shaft 3i so that the reverser 'I8
to cam 5l becomes t-i-K-au It will be under
operates to reverse periodically and simultane
stood that this constant (K) is generated by off
ously with the reversals of the reverser I3, where
setting the Zero position between the outputs of
by even though the output shaft l? of the mul
the differentials 36 and ¿i3 and two cams 50 and
tiplier moves first in one direction and then in
fil respectively.
the other in accordance with the product 43st,
As pointed out above, the two mechanisms 43
the ñnal output shaft i2 always moves in the
and itâ generate quadratic functions of their in
same direction and measures the total motions
puts; in the speciñc embodiment of the invention 55 of the shaft l1.
illustrated, they generate the squares of their in
And this ñnal output of the shaft l2 is
puts; thus, the mechanism ¿i8 generates and its
g/:fœdt
output shaft ‘dfi measures (t-l-K-i-œ)2, and the
The theory of operation of this integrator may
mechanism £9 generates and its output shaft 5l
be better understood by reference to Fig. 2. The
measures (t+K-;c)2. These two outputs are 60 total area under the curve, which represents the
subtracted in a differential Gla.
equation x=î<t), is equal to the sum of 'the sub
This differential lila, as shown, has an input
divided areas of the trapezoids S1, S2, etc. The
gear 62 driven by gear h3 fixed to shaft 60; an
area of
input gear Se driven by gear 65 fixed to shaft
el, and a planetary gear system 66 having a
that of
ring ge‘ar Gl, which measures the aforemen
S2=1/2(x1-}-$2)h
tioned difference (t-i-K-Hw 2-(t-{-K-«x)2; this
difference equals: ext-PMK or 4MM-K) .
The constant K is deleted in a differential 68.
and so on.
Now at the instant of time tu the instantane
by subtracting ¿1Km from ¿HUH-K) so that the 70 ous position of the output shaft 2l of the reverser
output of the differential 68 is the product ext.
i3 is -l/gh and the reverser has just thrown
As shown, the differential 63 has a planetary
gear system e9 which includes a ring gear 70 which
meshes with output ring gear 6l of differential
over to its forward position; and at the instant
of time t1, the instantaneous position
the out
put shaft 2l is -l-l/ah and the reverser throws to
te; in other’words, 4r(t-|-.K)_is introduced into 75 its reverse position and reverses its output; the
.2,404,294
5
6
be calculated by this integrator using the trape
zero position of the shaft 2| corresponds to the
zoidal rule. It is also possible, however, to use
time point exactly mid-way between to and t1 i11
it for computing integrals by Simpson’s one
dicated by the vertical dotted line in Fig. 2.
third rule or by other well-known mechanical
When the shaft 2| is at position -1/2h, the :n
input shaft I I is at its position œo, and the output 'ci quadrature formulas by an adjustment of the
Zero position of the multiplier. For example, if
shaft 'I'I of the multiplying mechanism is at its
the shaft 2I is at position -l?gh when the .7: in
position èl/zhxo.
put shaft II is at its position ro, and the re
Then at the end of one-half of an operating
versers I3 and 'I8 have just thrown to their for
_cycle of the mechanism, that is, at the end of the
ward driving positions at this instant, then the
time interval h=t1-îo, the position of shaft 2i
integrator will calculate the integral fœdt based
is +1/„>h, that of shaft II is an, and that of shaft
on Simpson’s one-third rule.
»
'I1 is -i-l/ghsci.
While I have shown a particular embodiment
The net rotation of the shaft ‘I7 at the end of
of my invention, it will be understood, of course,
the first half cycle, that is, at the end of time
that I do not wish to be limited thereto since
t1-to, is the difference betweenits position at
many modifications may be made, and I, there
the end of the first half cycle and its position at
fore, contemplate by the appended claims to
the beginning of the ñrst half cycle, or
cover any such modifications as fall within the
true spirit and scope of my invention.
20
Thus, during the first half cycle, the shaft 2I
moves from its initial forward position corre
sponding to »Mah to its extremefposition in its
forward motion corresponding to -i-ï/gh, and as a
result the movement of shaft 'I'I is 1/2(œo-|-sci)h.
What I claim as new and desire to secure by
Letters Patent of the United States, is:
'
1. An integrator comprising variable of integra
tion input means, integrand input means, revers
ing mechanism operable in accordance with the
operation of said variable of integration input
means for periodically reversing the direction of
the input, means for multiplying the output of
said reversing mechanism by the input of said in
tegrand input means, and means for periodically
position +1/2h, back to its position -1/271, and the
reversing the direction of the output of said mul
position of input shaft a: changes from :ci to x2.
tiplying means simultaneously with the reversals
The position of output shaft 'I'I at the start cf
of said ñrst reversing mechanism.
the second half cycle is -I-ï/ghœi and at the end
2. An integrator comprising a variable of inte
thereof is -l/ghœz, and the net change in the po
gration input shaft, a reversing mechanism op
sition of the shaft 11 during the second half cy
35 erated by said input shaft constructed and ar
cle is:
ranged to reverse periodically the direction of ro
tation of its output with reference to the input
thereto, an integrand input shaft, a multiplier
operated by the output of said reversing mecha
During all of this time, the final output shaft 40 nism and the input of said integrand input shaft
I2 is operated by the shaft 'I'I through the re
operating to multiply said output and input to
verser 18, this reverser always functioning to
gether, a final integral output shaft, and a sec
add the total forward and backward movements
ond reversing mechanism operating said final out
of the shaft 'I'I and imparting the sum te' the
put shaft and controlled to reverse periodically
shaft I2. Therefore, during the first half cycle 45 and simultaneously with said first reverser so as
when the reverser I3 is in its forward motion,
to periodically reverse the output of said multi
the change in position of shaft I2 is the same
plier to impart its motions to »said ñnal output
as that of shaft 11, that is, it is equal to
shaft.
1/2(:vo+œi)h; however, the change in position of
3. An integrator comprising a variable of inte
the final shaft I2 during the second half cycle
60 gration input shaft, an integrand input shaft, an
when the reverser 'I8 has reversed its operation
integral output shaft, a first reverser having an
is the negative of the change in position of the
output shaft, said reverser being operated by said
shaft 'I'I during the second half cycle, that is, it
variable of integration input shaft so that said
is the negative of -1/2h(:r1-I-:rz) , which, of course,
output shaft is periodically reversed with refer
55 ence to said variable of integration input shaft, the
is +1/2h(:r1-I-œ2).
Therefore, the total change in the position of
output shaft thereby measuring the interval of
the final output shaft I2 from time to to time t2 is
integration, first mechanism for adding the out
put of said output shaft with the input of said
integrand input shaft and for squaring said sum
60 mation, second mechanism for subtracting from
the output of said output shaft from the input of
Immediately upon the completion of the first
said integrand input shaft and for squaring the
cycle, both reversers I3 and 18 operate to reverse
difference, means for subtracting the output of
and the complete cycle is repeated. And in this
said second mechanism from that of said first
During the next half cycle the shaft 2I moves
in the reverse direction, and it moves from its
way the mechanism adds up the areas of all of 65 mechanism so as to obtain the product of the
the trapezoids under the curve. In other Words,
variable of integration input and the integrand
the total motion of the final output shaft I2
input, an output shaft driven first in one direc
measures the areas of all the trapezoids under the
tion and then in the other by the output of said
curve, and consequently measures the desired in
last-named means, and a second reverser for op
tegral.
70 erating said integral output shaft operated by said
variable of integration input shaft to reverse pe
It will be understood that this integrator will
riodically and simultaneously with said first re
operate to calculate the integral of x regardless
verser so that the total motion of said last-named
of whether :c is positive or negative.
output shaft in both directions is combined as a
The description of this integrator and of its
operation given above show how integrals may 75 motion in said final integral output shaft.
nu
2,404,294
4. An integrator comprising a variable of `inte
gration rinput shaft, an integrand input shaft,
an integral output shaft, a first reverser having an
output shaft, said reverser being operated by said
variable of integration input shaft so that said
output shaf t is periodically reversed with reference
to said variable of integration input shaft, the out
put shaft thereby measuring the interval of inte
8
operated by the output of said first differential
for generating the square of said summation, a
second cam mechanism operated by the output
of said second differential for generating the
square of said difference, a third differential op
erated by said two cam mechanisms having' an
output which generates the difference between
the outputs of said first and second cam mech
gration, a differential having an input driven by
anisrns to thereby generate the product of the in
said output shaft of said reverser, a second dif 10 tegrand input and the variable of integration in
ferential having an input driven by said output
put, a'shaft driven by the output of said third dif
shaft of said reverser, said two differentials also
ferential, a second reverser controlled by the op
eration of said variable of integration input shaft
having inputs operated by said integrand input
shaft, and the two differentials also having out
so as to reverse periodically and simultaneously
put shafts, the first diñerential however gener 15 with the reversals of said first reverser, and a driv
ating and delivering to its output shaft the sum
ing connection between said second reverser and
mation of the reverser output and the input of
said integral output shaft so that said shaft moves
said integrand shaft, while the second generates
through the total distance that said last-named
and delivers to its output shaft the difference be
shaft is driven by said third differential.
tween the output of said reverser and the input 20
of said integrand shaft, a first cam mechanism
WILLIAM B, JORDAN.
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