Note: Descriptions are shown in the official language in which they were submitted.
~547~
BACKGROUND OF THE INVENTION
1. Field of the Invention: This invention relates in
general to earth boring drill bits, and in particular to
the arrangement of the cutting elements.
~. Description of the Prior Art: The most common type
of earth boring drill bits for oil and gas wells are
cutters that rotate about an axis and roll around the
bottom in a path or kerf as the bit rotates. The cut-
ters have rows of tee~h that disintegrate the earth for-
mation ~hrough force applied on the cutter. The teeth
are spaced in rows and spaced to disintegrate as much
of the bottom as possible in a single rotation. The
prior art earth drilling bits include various features
designed to avoid a problem known as "tracking". This
problem arises when the spacing of the teeth on a ro-
tatable cutter enables the teeth to fall repetitivelywithin previous tooth impreSSiGnS in the earth. Even-
tually, ridges and peaks are formed in the earth, and as a
result, the cutter experiences accelerated abrasive wear.
The teeth are thus worn prematurely and unevenly. In
bits ~ith teeth of hard metal inserts retained by in-
terference fit in drilled holes, the supporting metal may
wear prematurely and the inserts may be lost.
Solutions to tracking are shown in U.S. Patent No.
3,726,350, R.C.O. Pessier, April 10, 1973, and in U. S.
patent No. 4,316,515, P~.C.O. Pessier, issued
February 23, 1972. Another solution is suggested in
U.S. Patent No. 4,187,922, F.E. Phelps, February 12, 1980.
In each of the above inventions, the inserts are ar-
ranged in circumferential rows, with varying spacing among
inserts to prevent tracking. These prior art inserts are
arranged in groups, with similar spacing in a group, but
differing spacing in other groups; or the spacing in each
row progresses from a minimum ~o a maximum and back to the
minimum; or the insert spacing is varied in each row so
_3_
that each pair of inserts is separated by a space differ-
ent from the space between all other pairs of inserts in
the row.
In each of the prior art solutions discussed above,
the inserts are arranged in circumferential rows. The
rows are separated by a minimum spacing to provide ade-
quate supporting metal for the inserts. To prevent the
generation of a ridge between rows, another cutter posi-
tioned in the same kerf or path may have staggered rows
arranged to remove the earth where such ridges would
otherwise form. Another method is to stagger the cutter
itself from the other cutter in the kerf~ ,such as shown
in U. S. patent No. 4~316~515~ RoC~O~ Pessier.
Occasionally, bits some-
times rotate "off-center", meaning that the rotational
axis of the bit becomes displaced during drilling from
the central axis of the borehole. One result of this
phenomenon is the generation of ridges, even between
staggered rows of the various cutters.
There are regions of prior art cutters which have
annular rows that overlap without intervening spaces.
In U. S. Patent No. 3,726,350, the cutter has half rows
offset from each other. E. A. Morlan disclosed in U. S.
Patent 2,774,571, December 18, 19~6, the use of an in-
ner end or "nose" of each cutter which has such an ar-
rangement. J. H. Howard et al disclosed in U. S. Patent
2,230,569, February 4, 1941, a large number of arrange-
ments for cutters with milled teeth, including helical
rows of teeth. Also, shaft cutters with helical rows
have been used in the prior art.
In all art known to applicant, the teeth or inserts
are arranged in rows. The rows may be circumferen~ial and
perpendicular to the cutter axis, or the inserts in the
~5~7~
row may only extend partially around ~he cutter. The
rows may be parallel with the cutter axls, or the rows
may be helical as menti.oned. All of the various arrange-
ments, however, cannot completely eliminate tracking and
provide full coverage in a single kerf with a single
cutter.
5~L7~'~
SUMMAR~ OF THE INVENTION
The object of this invention is to provide a drill
bit for earth boring with cutters having inserts dis-
persed over the cutter surface such that only one cutter
may be used in a selected ker~, and providing more effi-
cient rock fragmentation and balanced wear on the cut-
ting elements.
Another object is to avoid tracking and eliminate ~-
the generation of annular ridges, even during off-center
running.
These objects are achie-ved in the preferred embodi-
~ment ~y spacing the inserts in a dispersed pattern that
eliminates rows and achieves widely varied spacing. To
provide adequate strength of the metal supporting the
inserts, a minimum distance is established around each
15 insert as one constraint on the insert spacing. To
achieve an interaction between adJacent impressions on
the borehole bottom, a maxlmum distance is established
around each insert. The maximum distance is a function
of the rock properties and the size of the inserts. Thus,
20 a boundary zone is established around each insert and in
these zones the inserts are dispersed.
In choosing the ]ocation of the inserts in the pre-
erred method, first an insert is arbitrarily located at
any point within the selected region of the cutter shell.
~5 Then the location of the second insert is selected within
the boundary zone surrounding the first insert by using
in the preferred method a random number generator. The
third insert is located in the same manner within the L
boundary æone surrounding the second insert. However,
30 the third insert may not be located closer to the first
insert than the desired minimum distance between inserts.
The location of each succeeding insert is chosen in the
same manner.
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BRIEF DESCRIPTION OF THE DRAWINGS
Fig. 1 is a frontal view partially in section of a
raise drill reamer, having cutter assemblies constructed
in accordance with this invention and shown in phantom as
being rotated into the plane of the section to illustrate
relative radial positions.
Fig. 2 is a schematic illustrating the insert posi-
tions`of one of the intermediate cutters of Fig. 1.
Fig. 3 is a graph indicating the insert density of
one of the intermediate cutters of Fig. 1.
Fig. 4 is a sectional view of a cutter shell for
one of the intermediate cutters of Fig. 1.
Fig. 5 is a schematic illustration of a method of
locating inserts in accordance with this invention.
Fig. 6 is a sectional view of a cutter shell for
one of the inner cutters or gage cutters.
Fig. 7 is a schematic layout of-one of the rows of
inserts in one of the gage cutters or inner cutters
of Fig. 1. L.
Fig. 8 is a schematic layout of two of the rows of
20 inserts in one of the gage cutters or inner cutters of ~-
Fig.`l.
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DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring to Fig. 1, a raise drill bit or reamer 11
is shown boring a shaft 13, being drawn upward through a
previously drilled pilot hole 15. Raise drill reamer 11
includes a cutter support member or plate 17 secured to
be normal to a cylindrical stem 19. Stem 19 is secured
to drill pipe (not shown) and has a longitudinal or ro-
tational axis concentric with that of plate 17.
A plurality of cutter assemblies 21 are mounted to
the plate 17 by cutter mounts 23. Each cutter mount 23
has two arms 25 spaced apart from each other and facing
away from the cutter support plate 17. Arms 25 define
` a saddle or cradle for receiving a cutter assembly 21.
Cutter assemblies 21 include an inner cutter 27,
several intermediate cutters 29, and several outer or
gage cutters 31. Inner cutters 27 and the gage cutters
31 are preferably identical. Also, the cutting struc-
ture of the inner cutters 27 and of the gage cutters 31
in the preferred embodiment is less than the width of the
cutting structure of the intermediate cutters 29.
Each cutter assembly 21 comprises a cutter shell
mounted on a bearing, $uch as shown in U.S. Patent
No. 4,316,515, R.C.O. Pessier.
The cutter shell 33 for ~he intermediate
cutters 29 is shown in section in Fig. 4. Each cutter
shell 33 is generally conical and truncated perpendicular
to rotational axis 35 to form a frusto-conical outer
surface in roIling contact with the earth. Tlle inner
side 37 of the cutter shell 33 is closer to stem 19 (Fig.
1~ and is smaller in outer diameter than the outer side
39.
Each cutter shell 33 has a nose region, an in~er-
mediate region, and a gage region. Nose region 41 is an
annular frusto-conical surface formed at the edge of in-
ner side 37. ~he surface of nose region 41 is formed at
an angle of fifty-four degrees with respect to axis 35.
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Gage region 43 is a frusto-conical surface formed a~ the
edge of outer side 39. The surface oE gage region 43 is
formed at an angle of sixty degrees with respect to axis
35. The intermediate region 45 includes an annular sec-
tion 45a next to gage region 43 that is cylindrical and
parallel with axis 35. A frust~-conical surface 45b joins
sur~ace 45a, it being formed at seven and one-half ~egrees
with respect to axis 35 in the preferred embodiment.
Another frusto-conical surface 45c, between surface 45b
and nose region 41, is formed at a twelve and one-half
degree angle with respect to axis 35. Mose and ga~e re-
gions are defined herein to refer to surfaces immediately
joining the inner side and outer sides, respectively,
separated by the intermediate region and formed at sub-
stantially greater angles with respect to the axis of
rotation than the intermediate region.
Intermediate region 45 contains a plurality of holes
47 (only one shown) drilled normal to its surface for
containing hard metal inserts 49 (Fig. 1), preferably
constructed from sintered tungsten carbide. In the pre-
ferred embodiment for intermediate cutters 29, there areno inserts located in the nose region 41 or heel region
43. The bottom hole pattern of the insert: holes 47 is
shown schematically in Fig. 2, which represents the ap-
pearance of the bottom of the boreholes ii- one cutter is
rolled for one revolution. The left side of tlle drawing
of Fig. 2 represents the inner side of the intermediate
region 45, at ~.he intersection of surface 45c with the
` nose region 41. The right side of the drawing oE Fig. 2
represents the outer side of the intermediate region 45,
at the intersection of surface 45a with gage region 43.
The inserts in the intermediate region 45 are dis-
persed or irregularly located within the limits of bound-
ary zones so as to eliminate circumferential rows. Each
insert hole 47 in the intermediate region 45 has a bound-
ary zone that surrounds the insert. The boundary æonefor a first selected hole 47' is shown schematically with
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dashed lines in Fig. 5 and consists of a first loop 53
corresponding to the minimum desired distance between
centerlines of inser~s, and a second loop 55 correspond-
ing to the maximum desired distance between the center-
lines of inserts. In the preferred method and apparatus,the boundary zone loops 53, 55 are concentric circles and
identical for each insert hole 47 located in the inter-
media~te region 45.
The minimum distance is empirically determined by
the necessary cutter shell metal needed to retain an
insert. The maximum distance is determined by the ex-
tent a typlcal earth formation is disturbed by a single
insert. These minimum and maximum distances between ~-
centerlines will also depend upon the cutter circumfer-
ence, the insert shape and siæe, and the amount the in-
sert protrudes from the cutter shell. In the preferred
embodiment, for a cutter diameter of 13.496 inch at the
inner side of intermediate region 45c, a diameter of
15.540 inch at the in~ermediate surface 45a, a hole 47
2a diameter of 0.6250 inch, and hole 47 depth of 0.500 inch,
the minimum spacing between centerlines of inserts is
0.80~ inch. Thus the radius of loop 53 is 0.800 inch.
The maximum spacing between centerlines of inserts is
l.350 inch for this cutter. Thus the radius of loop
25 55 is l.350 inch.
In the preferred method of selecting the location
of the inserts, the location of the fi.rst hole 47' is
arbitrarily selected at any point in the intermediate
region ~5. Then, referring to the example of Fig. 5,
the location of the centerline of a second hole 47" israndomly selected within the boundary zone loops 53 and
55 of the first hole ~7' as determined by a typical com-
puter resident random number generator. The word "ran-
dom" refers generally to an irregular selection that has
no specific pattern within the specified boundary zones.
.
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Boundary zone inner loops 53' ancl 55' are then ap-
plied around the centerline of the second insert 47", as
indicated by the dotted lines in Fig. 5. The centerline
of third hole 47"` is randomly located within the bound-
ary zone of the second hole 47". However, the third hole
47"' may not be located closer to the first insert hole
47' than the desired minimum distance between inserts.
The portion of the boundary zone of the second hole 47"
that is too close to the first hole 47' is indicated by
10 the cross-hatched lines. This procedure is carried out
witEI each succeeding insert location being randomly cho-
sen within the boundary zone of the preceding insert, but c
not closer to any previously selected insert than the de-
sired minimum spacing between inserts. The procedure is
15 repeated until the intermediate region is completely cov-
ered. Because of the space limits of the interme~iate
region, there will be a few spaces that are greater than
the desired maximum distance from inserts, but yet pro- _
vide insufficient space to place an additional inser-t
20 without being too close to an existing insert. The min- r~
imum distance must always be observed.
~he selection process can be performed manually or
by a computer~ In the computer methocl, a random number
generator is used to select the locations within bound-
25 ary zone limits. In a pure mathematical sense, the pro-
gram is not random since in a true random selection, re-
peats will occur. The random number generator used with
the program will generate approximately 50,000 numbers be-
~ore repeating a number. This is sometimes called pseudo-
30 random selection. The computer program in Fortran lan- L
guage is set forth at the end of the specification. In
the program, the intermediate region 45 was assumed to
be a single angle conical surface, rather than having
multiple angles in the sections 45a, 45b and 45c.
35In selecting locations, certain of the insert holes
47 will fall close to the edge of the intermediate region
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45. This is permissible so long as the cyli.ndrical sur-
face of the hole 47 is no closer than about 1/64 inch
from an edge of intermediate region 45. If the boundary
zone of a preceding insert falls across an edge of the
intermediate region 45, only the por~ion of the boundary
zone inside the intermediate region may be used ~o locate
an insert.
~he result is a cutter with an intermediate region
45 wherein rows are deliberately avoided. Preferably ~~
10 the spacing is dispersed such that there are no groups
of three adjacent inserts wherein a single plane can be
passed through the points where their centerlines inter-
~sect the cutter surface. While i~ is possible for one or
more groups to occur in the preferred method, such occur-
15 rence is expected to be rare. Fig. 3 is a graph indicat-
ing the approximate uniformity of coverage of the cutting
structure. This graph has been prepared by starting at
the nose region 41 and making a plot of the relative in-
sert density as one proceeds outward to the gage region
20 43. The relative density represents the approximate total
linear distance of inserts through which a selected plane
passes, divided by the associated circumference of the
cutter shell at the selected plane. The selected plane
must be perpendicular to the axis 35 of the cutter shell
25 33. For example, a plane passing through the intermediate
region 45c about one-half inch from nose region 41 and
perpendicular to axis 35 would pass through a number of
inserts 49. The plane might pass through and bisect some
inserts while passing through only a segment of other
30 inserts. The distance that the plane cuts through each
insert at a point flush with the cutter shell 33 is added.
When summed, these distances divided by the associated
circumference yields about 0.2~ at a point one-half inch F
from nose region 41. If the inserts were spaced in a
35 circumferential row at this point, and had no cutter metal
between them, then the relative density would be 1.0 or
100%.,
pp.
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Note that the coverage is fairly uniform, in that
once past the first one quarter inch or so at both edges
of the intermediate region 45, the clensity varies between
about 0.15 and 0.28, and preferably does not drop below
0.10. This indicates that all possible planes passing
perpendicular through the a~is 35 will pass through a
portion o~ at least one insert. If there were circum-
ferential rows, then the graph of Flg. 3 would register
zeros between the rows, since the planes at these points
10 would fail to pass through any inserts.
Table No. 1, attached, lists the precise location
of each insert 49 in the insert holes 47 in the inter-
mediate region 45 for a cutter havîng dimensions de-
scribed above. The column marked "A" represents the
15 distance along the axis 35 from the outer side 39 to ev
the point where the insert is located. The angle ~ is
a radial measurement of the cutter shell 33 about its
axis 35, beginning with an arbitrary first point. The
difference between any of the angles a is proportional
20 to the circumferential distance along the cutter's in-
termediate region 45 in a plane perpendicular to the F
axis'35. Although not necessary to the invention, note
that, to three decimal points, each insert: hole 47 is
located at a different distance from the outer side 39
25 than all others. Also, each insert hole 47, to three
decimal points, is located on a different radial plane
than all other insert holes.
The insert locations were not selected by the com-
puter in ~he numerical order shown in the table. That
30 is, the second i.nsert location chosen by the computer is
not necessarily the insert number 2 in the table. Insert
number 3 in the table is not within the boundary zone of
insert number 2 in the table. Rather the table conven-
iently lists the inserts by increasing angle ~ . The
inserts numbered 292 through 294 are indicated in Fig. 3
to correlate Fig. 3 with the table. All of the insert
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holes 47 are drilled normal to the surface that they are
located on, except for holes that fall across the inter- :'
section of intermediate region 45a with the intermediate
region 45b, and the intersection of intermediate region
45b with intermediate region 45c. With these holes, the
hole is drilled normal to the surface that contains more
than half of the diameter. of the hole. ~~
~ igure 6 discloses a sectional view of an inner cut-
ter 27 or a gage cutter 31 (Fig. '1), these cutters being
identical to each other but considerably different from
the'intermediate cutters 29. One reason is that the gage
cutter 23 needs an extra high density of inserts on its
~outer edge for cutting the sidewall of the shaft 13. Also,
the inner cutter 23 needs a row of inserts on its nose re- f'i'
gion for cutting the edge of the pilot hole 15. For inter-
changeability, the inner cutter 27 and gage cutter 31 are
- made identical to each other, with rows of inserts being
located both on the nose region and near the heel region.
The inner cutter 27 or gage cutter 31 comprises a
cutter shell 53 that is generally conical and truncated
perpendicular to its rotational axis 54. The bearings
for the cutter shell 53 are of the same structure as
used with intermediate cutters 29, Cutter shell 53 has
an inner side 55 that is closer to stem 19 (Fig. 1~ than
its outer side 57. Each cutter shell 53 has a nose re-
gion, an intermediate region, and a gage region, as pre-
viously defined,in connection with intermediate cutters
29. Nose region 59 is an annular frusto-conical surface
formed at the edge of inner side 55 at an angle of thirty-
five degrees with respect to the axis 54. Gage region 61is an annular frusto-conical surface formed at the edge of
outer side 57 at an angle of sixty degrees with respect to
axis 54. The intermediate region 63 includes an annular
section 63a next to gage region 61 that is formed at an
angle of five degrees with respect to a~is 54. A frusto-
conical surface 63b joins surface 63a and is fcrmed at an
angle of seven and one-half degrees with respect to axis
f l~
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~i9L7S2
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54. Another ~rusto-conical surface 63c, between nose re-
gion 59 and surface 63~, is ~ormed at an an~le of twenty
degrees with respect to axis 54.
Nose region 59 contains a row 65 of holes drilled
and reamed for inserts 49 (Fig. 1). Row 65 contains
thirty-seven holes, all spaced ~he same distance from
the outer side 57. The pitch is defined herein to be
the distance between centerlines of the inserts at the
shell 53 surface. The pitch is varied in row 65 to a-
void tracking in accordance witk the teachings in U.S.patent No. 4,316,515, R.C.O. Pessier.
Referring to Fig. 7, row 65 is
divided into groups of increasing pitch, marked "I" and
decreasing pitch, marked "D", in a counterclockwise di-
rection. The pitch gradually increases in the increasing
groups and gradually decreases in the decreasing groups.
The inserts marked with an asterisk fill in the space
between the last insert in the last group in row 65 and
the first insert in the first group.
The amount of increase in pitch, decrease in pitch
and the number in each group are selected according to
sever~l criteria. Firs~, there is a minimum pitch de-
termined by the necessary cutter shell metal needed to
hold the insert in place. The maximum amount of pitch
is determined by the extent a typical earth forma~ion
is disturbed by a single insert. This will be greater
than the diameter of the insert 49 and depends also on
the cutter she~l 53 circumference, and the size, shape
and amount the insert protrudes from the cutter shell
~0 exterior.
The number of inserts within the group depends upon
the desired change from insert to insert. To have an ap-
preciable difference between the pitch from one insert to
its adjacent inserts, generally groups from about three to
seven inse~ts are used. To calculate the precise position,
the number of spaces between inserts in tlle group, less
one, is divided into ~he total increase in pitch. This
i
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constant number is allotted to each space between inserts
;n the group. Consequently, in an increa~sing group, any
space between insert centerlines will be the same as the
preceding space in the group plus the constant number.
5 In a decreasing group, any space between insert center-
lines will be the same as the preceding space less the
constant number. Preferably the same maximum and mini-
mum a~re used for each group within a single row.
Referring still to Fig. 7, row 65 has nine insert
lO groups, five increasing and four decreasing. Two in-
creasing groups are Eollowed by two clecreasing groups
respectively. Each group contains five inserts, yield-
ing four spaces between inserts in each group for vary- t
ing pitch. Also, when an increasing group is followed
15 by a decreasing group, the groups overlap with the last ~,
space of the increasing group being also the first space
of the decreasing group.
Fig. 7 discloses the relative angular positions of
the inserts in row 65, as indicated i.n the Table No. 2,
2Q set forth subsequently. Cutter shell 53 (Fig. 6) uses
the same size of inserts 49 (Fig. 4) as cutter shell 33 F~
(Fig..4). However, it has different dimensions, it being
5.500 inches from inner side 55 to outer side 57, 15.601
inches in diameter at the inner edge of the gage region 61
and 14.262 inches in diameter at the outer eclge of the nose
region 59. The angle ~ in Fig. 7 begins at zero with the
~ertical axis 67. The insert hole 65' located on the axis
67 is indicated in this table as insert tlO. 2, all of the
inserts in row 65 ~or this particular cutter size being
5.219 inches from the outer side 57 as show~ in the "A"
column. The next insert hole 65" in row 65 is insert No. ~`
7 in Table No. 2, located 8.560 degrees rotationally from
the centerline of the first insert hole 65' and from axis
67. The thircl insert hole 65"' is insert no. 13 in Table
No. 2, located 17.940 degrees from axis 67 or 9.430 degrees
from the centerline of insert hole 65".
The gradual increase and decrease in pitch and the
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insert locations can be determined through Table No. 2 in
this manner. The other numbers listed in Table No. 2
disclose locations for other inserts on cutter shell 53,
discussed subsequently.
Referring again to Fig. 6, a staggered row 69 of
inserts is located in the intermediate region sec-tion 63b
near the edge with intermediate section 63a. Fig. 8 is a
layout similar to Fig. 7, disclosing the relative posi-
tions of rows 69 and 71. ~11 of the insert centerlines of
row 69 are located 1.874 inches from the outer side 57
while all of the insert centerlines of row 71 are located
1.581 inches from outer side 57. The centerlines are thus
0.~93 inches apart when measured along the axis 54. Since
the diameter of the holes for these inserts is 0.625
inches, there will be overlapping coverage of approxi-
mately one-half the insert's diameter. To assure some
overlapping the axial distance between row 69 and 71
insert centerlines should not exceed the insert diameter.
The eighteen inserts of row 69 are divided in-to three
groups of six inserts each. Each group of row 69 is a de-
creasing pitch group, when considered counterclockwise.
The positioning of these inserts is selected as set forth
in the discussion of row 65 and is set forth in Table No.
2. Each group of row 69 alternates and is circumferen-
tially separated by a group of inserts frc,m row 71. Thefirst insert hole 69' of row 69 is listed as insert number
38 in Table No. 2, and is located 54.290 degrees from axis
73, which is the same axis as axis 67. The second insert
hole 69" is listed as insert no. 46 as is located 63.430
degrees from axis 73.
The twenty-one insert holes of row 71 are divided
into four groups, three of which have five inserts and one
has six inserts. The groups of row 71 have uniform
pitch between inserts. The first insert hole 71' of row
71 is listed in Table No. 2 as insert no. 5, located
4.940 degrees from axis 73. The second insert hole 71"
~;4~75;~
-17-
of row 71 is listed in Table No. 2 as inser~ no. 12,
located 14.810 degrees from axis 73.
Referring again to Fig. 6, a fourth row 75 of in- _
serts is located in the intermediate section 63a. The
centerlines of all of insert holes of row 75 are spaced
1.015 inches from the outerside 57. There are forty
insert holes in row 75 and they are divided into three
increasing groups of seven inserts each or six spaces
between inserts. Tle pitch of these groups is calculated
as set forth in the discussion of row 65. Inserts are
eqùally spaced between ~hese three goups. The precise
positions are shown in Table No. 2, with all row 75
insert holes being found in the "A" column under the
distance 1.015 inches.
Note, that for an insert of 0.625 inches diameter,
the coverage of heel row 75 overlaps with the inserts of
the staggered row 71 since they are only 0.566 axial
inches apart. To allow this overlap, each insert of
staggered row 71 is spaced between two inserts of heel
row 75. The overlap prevents buildup between the heel
ro~ 75 and staggered row 71. r-
~ Referring to Fig. 6, a gage row 77 of gage inserts
is located in the gage region 61. The gage inserts (not
shown), differ from inserts 49 (Fig. 1) in that they have
flat top surfaces. The gage inserts are mounted with
their top surfaces flush with the gage region 61. Pref-
erably there are thirty-nine equally spaced inserts in row
77, and these lnserts are not listed in Table No.2.
Referring to Fig. 6, a plurality of holes 79 (only
one shown) are dispersed in the intermediate region sec-
tions 63b and 63c. The locations for holes 79 are select-
ed in the region between the nose region 59 and boundary
zones of rows 69 and 71. Holes 79 are selected within the
same maximum and minimum limits for the boundary zone as
discussed in connection with the intermediate cutters 29.
The same computer program as previ.ously set forth is used
for selecting the locations o.f holes 79, with different
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numbers used for the dimensions of the intermediate re-
gion. The locations of all o~ the randomly selec~ed in-
serts in the cutter shell 53 are set forth in Table No. 2.
T~e numeral differences between the computer program
for the ;ntermediate cutter shell 33 and t:he gage or in-
ner cutter shell 53 are only in lines 00620, 00630, 006~0,
00650. The changes are as follows:
Line Shell 33 Shell 53
00620 7.074 7 ~73
00620 8.0?0 8.024
00620 395 274
00630 169 96
00630 0 21
00640 0 21
00640 9.893 11.0
` 00650 5.506 2.~94
00~50 6.561 4.498
00650 0 18
Also, lines 00660C through 00720C do not have the "C"
included in their designation.
Because of the irregular boundary provided by rows 69
and 71, there will be no circumferential space between
rows 69 and 71 and the dispersed holes 79. That is, any
plane passing perpendicular to the axis 54 in the inter-
mediate region 63 will necessarily cut through a portion
of at least one insert. Since the staggered rows 69 and
71 prevent any circumferential spaces to exist between
these rows and heel row 75, there will be no spaces in
the intermediate region 63 through which a perpendicular
plane could pass without striking a portion of at least
one insert. A circumferential space does exist in the
nose region 59, inward from the nose row 65. The rela-
tive density of inserts across the cutter shell 53 is
fairly uniform, and preferably does not drop below 0.10,
as previously defined in connection with cutter shell 33.
In operation, stem 19 (Fig. l) is rotated clockwise
and urged upward. This causes cutter assemblies 21 to
~.
~' . . .
L'752
rotate, creating an annular path about t:he borehole face
51. The inserts 49 disintegrate the earth, creating shaft
13.
The invention has significant advantages. In the in-
termediate portion of the borehole, between the gage and
inner cutters, only one cutter is required to cover an an-
nular section of the borehole face, since the insert po-
sitioning does not allow ridge buildup that might otherwise
occur in the prior art between rows. Without the need for
overlapping or staggering cutters, greater pressure can be
exerted through the inserts, since there will be fewer cut-
ters for transmitting the force imposed on the bit. Fewer
cutters reduce maintenance required in shaft drilling. The
shaft face is evenly covered, providing efficient fragmen-
tation and avoiding uncut bottom due to off-center running
conditions~ Since overlapping cutters are not required in
the intermediate portion, tracking between cutters is avoid-
ed.
The combination of the dispersed pattern with rows of
inserts with varying pitch for the gage and inner cutters
evenly covers the borehole face. The rows provide higher
carbi`de density for the pilot hole and sidewall areas of
the borehole. The varying pitch in these rows avoids
tracking.
While the invention has been shown in only one of its
forms, it should be apparent that it is not so limited, but
is susceptible to various modifications and changes without
departing from the spirit thereof.
!
~ .
i~'
~ ,,,,',,.
A ~ r
~-- _ __ _ ~! ~7~ ~
~1 n s e r c~ A _~, U 4 ( ~ f, 6 4,7 ~~
~ __w~ _~
41, I . I 11 S 2. 3G 7
. 1.727 6 274
J ~51.7 U 7 1.037
2 1 987 S ? 4 t,
4 3~ 3. 379 (),7
3 2.5 t, R : . (), S !~ '~ _ 1 . fl 57
4 6, 5 b 2 3 . 2 ' 1 4 Sj 4.7 S 9 1 ~ 2 L
8.099 3.99 J ,6~ t .2. ( ) ( () ~ r
6 8.254 I 4 1 (, 47~, 1. ( ) 7 ', l, . f) U I
7 8 7hl . 33'
', ~3 ~ 3. )f~3 _ _ ] ~ 337
8 1~).850 5 665 ~ 496().7()1 ?.856
9 11.505 _'I . ~ 38 _ 5~) (1. ~f~9 S . i')6
0 17.788 ~ ~ t).',22 ~ 5~tj2.()26 _ ',.4',7
11 13.931 2.797 ¦ 5262.9?f 2 ()i3l
12 14. -31~) 3.623 ~3tj 3.630 3. i304
_ ~
3 14.507 1.931 5',67.f,07 1.288
14 ~6 832 S.324 55~;9.053 3.36
_
17.41~1 1.166 S~ 69.584 5.9f~
~ P~
J6 19.18b 6.289 57 70.J15 4.702
17 19.422 4.576 ~5fl 71.737 2.096
18 2 ~ .17(~ L .864 59 75.719 1. ',3',
__ _
19 22. g 6 ~, ¦ 7 () 8 6 ()76. () L 4 '" 16 L L
_ _.
21~ S .827 3.95 () h 176. n 7 '. 5.800
21 25. f37 U ~ . f3 f37 6277. L ~1 1 2. ~,35
, 29.~2~ 1 217 6 3 77.',30 tj.()28
23 30.19~ ~.()()f. ~4f~0.~;3() 6.461
24 3U.~5; 2.4] ? Cr) 8(1.5f9 2.]?1
._ .
2~ 3 .1f34 1. '. (. fl6I . ~)(,, 1. ()63
26 32. ', ~ ~, 42 c~ ',731. ? 53 3.44
27 32.505 ' .111 r7f~~32.() ,1~ 4.ll2
~ _,
2~3 35.~() 2.7~35 69 8~.7(-1 , .7l()
. _ . ......
29 35.9~3;. , .6 ~ ~ _ 70f~7.51 ~) I .3~()
37.475 3.613 7J f37.G37 4.888
31 37.591 1.849 72 87.891 2.5 ?7
_ __ _~___ _
32 38.839 ~.072 _ 73 ~9 0~() ,4-1
33 39.066 4.410 74 90.031 3.?.74
. _ _ ,.
34 41.853 2.~103 75 9L?.552 6.29',
.._ ._
3542.694 fj.l97 7~, Y .9'5 1.041
J~,43.273 5.232 ~~ 77 93.(~1lJ 5. '~S
346.! 43 I . h 31 7~3 95.5S6 1.994
:~t3 ~ 7. 35U -3 ~( l, ~ 7g 95.9~6 4. ',26
~ ~ r
-~c~ t ~ 8 ~ j :, . f S I 80 ~ 07 2 922
- :- -- ~ __ _ ~ _
- --
~_ ,~ 1
_ _ 1 2 I 1 ll 8. ',9 /~ 3.1
'34
8] ~ 9.53~ ( . ~ ______ ~
. .
I ~'~ I '~ .7'~) ~" 7'J~
8 ' 9 (~ .7 ~ ~' 'j .3 '31 ___
I ~ :~ 1 j ] 59f. l . '~ 7 ~3
83 101 9()4 ].~91 _ _ . __
1-24 I')1.~355 2.4~S
8S 102.276 2.039
12'i 154.56(~ 3.769
103.43 ' 3.5 'j 7 ______
12~ ~ 54.59'2 1 'J53
86 104.270 ll. i8n ______
127 157 4n8 6.105
87 108.129 3.02'~ __
______ 128 1 S7.8/ln 3. n84
88 108.255 l .701
______ _ 129 15~3.~66 4. ~6'3
89 108.354 5.835 _ ~
130 160.213 1.85G
111.146 1 3.968 _
91 112.865 5.0~5 I3] _ 160.267 _ 1.056
92 113.4~` (~.4~4 13;' 16().727 '3.96]
93 ~14.497 2.999 1'33 164.258 5.831
94 116.327 2.~05_ 134 164 671 _ 4.7/~3 ~ t
116.353 1.403 ~ 35 1(~6.019 1.48~
l '36 166 U`J 2 ~ R l 4
96 ] 17.798 4.25l _ _ . . r
97 11B.02' ` S .668 1:37 163 348 'I .842
9~3 12'1.284 3.455 138 1 `~U 730 5.196
99 121.914 '4. R96 1 '39 173.285 2.211
lC0 122 057 1.286- I~.~ 17~.5~l~ '3.2~9
01 123.157 6-.427 ' 14l J7'3.921 6.l;4 _
10~ 125 587 - ' 2.28-~ 1/.2 17/.762___ 4.289 r
_ ] ~ 1 175.55~ ] 29
103 126.'509 4.073 __ .
104 128.166 5.~22 1~' 177.134 ').~ 30
105 128.441 1.463 '~ 145 179.328 2.~ 0/~
106 131.087 3.182 146 119.995 6.442
. _ I1~7 180.189 3 ~6
107 131.15() ~ 01 __ . ~
_ 148 181.2 '36 1.0~ 8
1~)~ 133 733 6 412 _____~ ~ ______________
. 149 ] 8~ .58~3 4. ~80
109 134 106 ' 1.129 ______ ~ _____________
_ _ 15() 184.897 ~ .667
110 134.464 5.578 - _ ~__ __ __, ___ - - - _ _ 151 185.716 5.566
111 135.683 2.215 ______ ~
______ _ _ _ 152 186.530 6.389
112 136.234 3.7~0 ~ __ _ _ _____________
~ 5 '3 187.729 2.801
113 138.749 4.50~ ______ ~ _____________
154 1~9.803 3.8'36 ~r
114 139.945 3.112 _ . ~ _~
1~5 1(~2.064 S.113
115 141.179 1.6~(~ ______ ~ ___ __ _
_ l S" 192. ~S 1.9~6
116 142 368 ' 5.'352 _ . __ _ _
_ ] ' 7 1 '~2.17S 3.112
117 142.952 6.ll2~ _ _____________
_ _ _ 1 S.~ ~ 95.765 5.992
118 144.681 3.9~ ~ ~_____
LS~ I'J5~83l 3 307
119 145.3~2 ~ .207 _ _ . F'
_1~,0 196 134 '~ ~55
~120 348.l~62 1.'3~.3 - __
r~ . ......... _ .. 2~ 75Z
`_ _ =__ _ 'I 1,~',_O. 1
I t)I l'~f~.~f~ `~ ' J7 1 _ _ ,,~("~ ("421 .
,~() 2 ~)'ll)
t~ 2 ¦ 9 ~) ~ ) 5 , _ _ _
1 t 3 ; 99.71 :3 ,` t~ ~ (J ___ _ _ _ f~ )
. __
I 64 ~ ( ) 2. b 2 I /, . :~.1 . ( ) ~ ~ ~ `4 Y . l fG ') 3.72 f 3
_ .~
1 b 5 2 O 4.354 I .818___ 2 / '),47 ') 5.257
1 ~.6 20! .759 (~ 37 S.` (I fj ? 51.746 I .128
. .
!b7 207.()2~ 3. ' ;'~ _ ')~ ;'54.~)69 ).()Sf~
1~ 8 207.179 'i . I h 5 / ) q 2 5 I .6 / 4 4.2 G 3
I f.9 2 O 8. l f.8 2.6 ~ 9 ) J 255. (!/ ? 6. I 4
170 ? () i~ .272 ~ . ( ) 8 :3 ~ ~ __ _ 1. g iS~ (J
71 21 I .294 6 . I)9 3 ~ I i ~ ~. . sn,~ , . 33 I
72 213.234 3.1~4 ,1~ _'-'1,21~ 3.~Ui
73 213.906 S . l 69 ~ I I ~5~ . 2()/
~74 213,9~f~) 3.9i~1f~-- _ 14 2f~().if)f) ~ 95
17~ 213.951 I .663 1) 1 ', )l) I . '.()`~ 'i, 'J4(,
176 2 I f~) .771 ' . b 7 4 ~ 2 h l . i~33 'J / . 351
77 !]9.103 I ,0')1 .`I 7 2f-5.(~] I 1. /21 yr
~7i~1 `1~).3f~ 4.-527 ~ ~t)l~.4'1) 2.f~
! 7 q ? 1 ~ .4 '~ 1 5, 6 0 b _ ~ ! (J .I () f` !~ 6 ~ -
'2(). b l~, I . '-66 ,';'l~ Lf ~.0' U (~. ()qfj
I fJ 1 2 21.733 _ ' ,' I ' 6 f1; 'j ,~ ~ _---
I ~- '' ! 2 ' . 3 ? 2 S . 1 f 4 I ~ ,~ ? 6 '3 . 1 ~' 3 3.597
If31 ;'2').4~3 4,305 ?7 3 '. l56
I ~:`, q, ~ ? .7 :~ 7 .' `, ,) ~ ~ o ~ /, ~ . S ~) 6
I f35 2) / .34] (' .1 ()5 .' ' 5 27 '~,1 (17 ___ _
~t) ' ' f~,9 f~() 1, b 94 . ~ ~ 75 ~ 6f~ 5 3 n I s
i~37 qq() .899 7 __ 2 ' 7 ;~ 77 . ~6 ) . 052
~ !df~ ~ 3() ~ 958 4 . 7 9 b ... 177. fl/. I 6 . I~4 1
1 ~i 3 L 3.` ,054 . , f 3 `16 ) 27 J .129 ;~ ,2
.` ~ (! 2 i 't .49 ;2 3.831
33.387 ` .()75 _~ ~ ~ ___o_
, _ _ _ _ _ 7 11 ~f; 1.7 ~ I I ()6
1~1 2 ~3.723 5.942 ~ ~ _ _
~ :~,.' ~f~ )2/~ ~ l)
1'32 2~3b.159 4.103 _ ~ ___
.'~ A~f~ y) () 3fjl
193 23b.676 1.073 _ ~ ,_ _
~ _ ___ ~ ~ !, , ~16, f~ ~ 1 S .; '~
194 q 37.128 1.296_ _ __ _
_ ~ `flfj,9('~1 4.
13 S L 37.445 S .153 ~ ~ f, ~liii~- 2
1 S~ 6 2 !1 0 . 8 L 5 ) . S ? 1 .
i97 `41. '10---- f,),4 38 ~ ~7 ;'1)l . 31)U I .U')~
.__ ~ ~ 38() 2 '~
~1 f3 L 4 1 . f~ 7 5 I ~ 5 L 6 ___.__ r~
__ 7~q 1`(12 5~2 S ~'30
~9~ 243.0l~3 5.579 ,_ _ _ -~ 9F
` O-' 2 4 3 . ~ ~ 7 3, 2 13 ? .~1/ ) ? '`~ '., 0 / 7 _
- _. _. _
. . . , _, . . .
~ -- - - -23~ 75~
_ _ T~L~ M(). 1
24~ 294.542 - 2.227 __ 730 ;~ 2.l 48 - -
2C 2 295.964 3.76(. 28~ 3~ 4.643 4.023
_ _ j. . . . .___
243 298.867 1.180 282 3/ 6.415 5.099
. _ . _ . .. ..
244 300 366 4.822 283 ~46.658 I.559
~45 300.779 S .707 284 3~ i.-3-~4 -- 6.076
. ` _
246 30l 7ao 2.72~ 285 3 7. ~ 93 2 ~ 01
24~ 303.590 1.855 2~35 34~.~313 3.~57_ . ._ ~- .
248 3n~.609 1 7l9 287 35l.194 4.21
_ . ~ _ ~ . . ~
243 305.979 1.102 2~8 :35~.5/6 2.543
250 307.224 6.4~4 289 3 ~ 3.593 5.100
._ . ___ 11
251 3~8.434 5. ~22 2~0 3~4.036 5.361
__ _ ..... _ . 11
252 308.552 3.129 291 354.275 I.fi53
~ .. ... . ___
253 308.800 4.414 292 357.772 3.395
._ . . . . .-- .
254 309.961 2.272 ~93 358.809 4.5~ 2
255 312.821 1.581 ~29~. 359.549 - 1.0~3l ----- -
. _
256 315.905 3.903
257 316.163 3.078
.
258 /~ 5.172
259 317.126 6.132
260 318.283 2.074
261 318.994 1.198
_
262 3~2.3~4 2. fi85
2~3 323.195 4.104
264 324.405 1.578
265 325.035 6.305
_
266 327.417 5.273
267 327.625 3.5~4
2~8 329.556 1.060
329.768 2. ~fiO
_ ~
27n 33l .292 4.655
. ~,
271 331.871 6. ~57
27~ 333.889 1.639
273 335.23~ 3.53~3
~74 335.820 2.670
_ _ _
2 ~S 336.249 5.696
. . ..... .
276 337.643 4 580
. __
277 338.635 6.437
_
78 339.228 1.310
~79 341.522 3.314
..
~ _ _
7.5;~
-24
CCMPUl'ER Pi~O&RAM
00120 C INPUT TO l~E PRO~RAM ARE (IN CONSISI~NT UNIl'S~
00130 C Rl=INNER CONE RADIUS AT TIP OF CCMPACT
00140 C R2=OUTER CONE RADIUS AT TIP OF CCMPACT
00150 C NC~hY=~Y. Nl~ER OF CCMPACI'S INCLUDING PREFIXED CO~PACrS ~-
00160 C NC~DN=MIN. NUMBF.R OF CCMPACTS INCLUDING PREFIXFD CCMPACTS
00170 C DM~Y=MAXIML~I COMPACT SPACIN& TIP TO TIP
00180 C rMlN=MINDMMM COMPACT SPACING TIP TO TIP
OGl90 C N=NUMBER OF PREFLYED C~MPACTS ON INNER AND OUTER RoW ONLY
00200 C D=COMPACT DIA~I~R
00210 C H=COMPACT HEIGHT
00220 C FSEED=STARTIN& "SEED" FOR RANDoM NUMBER &ENERATION
00230 C NCl=NUMBER OF PREFLYED COMP~CI~. ON INNF,R R~W
0024~ C NC2=NUMBER OF PREFLXED CCMPACTS ON OUTER P~W
00250 C THETA=CONE AN&LE IN DEGREES (NOT INCLUDED)
00260 C PI,TOPI=3.1415927,6.2831854
00270 C WIDTH-CONE LENGTH ALONG GENERATOR LINE FIRST TO LAST CCMPACT RoW
00280 C TOI~D=CONE LENGTH ~ ONG GFNFJRATOR LINF. FIRST RCW TO RACKFACF,
00290 C ALPHA(I)--ANGLF,S OF PREFLYED COMPACTS ON INNER AND OUTER
00300 C RoW IN DEGRE,F.S STARTING WITH INNFR R~W
00310 C BETA(I)--ANGLES OF ADDITIONAL PREFIXED COMPACTS IN DEGREES
00320 C RBETA(I)-`RAD~L POSITION OF COMPACTS AT T~E TIP
00330 C NMORE=NUMBER OF ADDITIONAL PREFIXED C~MPACTS ~-
00340 C $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
00350 C IF NO RCWS OF CCiMPACTS ARE PREFIXED, THE PRC~E~M MAKF,S A RANDOM
00360 C S~ FCTION OF THE FIRST CCiMPACT. THE RANDOM NUMBERS ARE GENERATED
00370 C BY THE USE OF lHE M~,TIPLICATIVE LINEAR CONGRUENl`IAL MElHOD:
00380 C U(N~lj=FRACTIONAL PART OF 997-~''U(N). U(l) IS CALLED SEED. THE
00390 C PRC~RAM ASSUMF.S .5284163 AS SEED (ALTHOUGII ~THER APPROPRIATE
00400 C SEEDS MAY BE USED). T~E PERIOD OF l~E GENFRATOR USING THIS SEED
00410 C HAS A LENGTH OF 500,000 NUMBERS. THIS IS USING 10 M GITS
00420 C CALCULATIONS. T~5E M~ST SIGNIFICANr DIGITS (THE LEFT HAND M GITS) ~-
00430 C ~RE THE ~ST RANDOM DIGITS.
00440 C $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
00450 C OUTPUr FROM T~E PROGRAM ARE:
00460 C N=COMPACT COUNTER
00470 C RHO(N)=RADIAL COMPACT POSITION AT ~ E TIP
00480 C ALPHA(N)=ANGULAR COMPACT POSITION IN DEGRE~S
00490 C M ST(N)=DISTAN OE CENTER OF CARBIDE HOLF. ~ro BACKFAOE
00500 C Y(N),X(N)=COMPACT COORDI~l`ES FOR PLOITER
00510 C` $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$~$$$$$$$
00520 C THIS PRDGRMM ALSO C~LCUTATES THE CARBIDE DENSIlY ACROSS THE
00530 C SURFA OE OF T~E CONE SHELL AND STORF.S THE DATA IN THE SAME ~ATA
00540 C SET (PACCA.DATA) FOR USE BY T~IE PLOTTING PROG~AM "UNIVPLTC" T~
00550 C PLOT BOrH COMPACT LOCATIONS AND CARBIDE DENSITY.
00560 C --00--00--00--00--oo--oo--oo oo oo oo oo--oo--oo--oo--oo-
00570 C --00--00--00--00--00--00--00--00--00--00--00--00--00--00--00-
00580 C --00--00--00--00--00--00--00--
00590 IMPLICIT REAL*8(A-II,O-Z)
00600 DIMENSION Y(500),ALPHA(500),~HO(500),DX(500),XS(500),YC(500) P-
00610 DIMENSION XD(500),YD(500),1HI(13),BETA(300),RBE~A(300)
00620 DATA Rl,R2,NCMAX,NCMlN,DMAX,DMIN,N,D,H,FSF~D/7.074,8.020,395,
00630 169,1 .~50,0 800,O, 625, 750 52~41~3/
00640 DATA NCl,NC2,THETA;PI,TOPI ~b,0,9.893,3.1415927,6.2831854/
00650 DATA WIDIH,TChlD,N~lORE/2.84,4.98,1/ 5.506,6.561,0/
00660 C DATA BETA/54.29,63.43,71.84,79.52,86.47,92.69,151 91,161.05,
00670 C C169.46,177.14,18'~.n9,190.~l,249~3,258.67,267.08,274.76,281.71,
~475~
-25-
00680 C C287.93,282`'`-0.DO/
00690 C n~TA RBF.TA/18*7.967DO,282`''0.DO/ ,
00700 C DATA AlE~l4.94,14.81,24.68,34:55,44.42,l02.56,112.43,122.30,
00710 C C132.17,142.04,200.18,210.05,219.92,229.79,239.66,297.80,307.67,
00720 C C317.54,327.41,337.28,347.15,479`''0.~0/
00730 C DATA ALP~0.0,7.14,15.12,23.93,33.59,44.08,51.22,
00740 C C59.2,68.01,77.67,88.16,97.82,106.63,114.61,121.75,132.2~ 1.9,
00750 C C150,71,158.69,165.83,172,97,180.95,189.76,].99.42,209.91,217.05,
00760 C C255.03,233.84,243.5,253.99,263.65,272.~6,280.4~,287.58,298.07,
00770 C C307.73,316.54,324.52,331~66,340.~7,350.13,
00780 C C.200.0,206.22,213.17,
00790 C C220.85~229.26~238.4~244.62~251.57~259.25~267.66~276.8~285.21~
00800 C C292.89,299.84,306.06,315.2,323.61,331.29,33~3.24,344.46,350.68, r-
00810 C C357.63,5.31,13.72,22.86,29.08,36.03,43.71,52.12,61.26,69.67,
00820 C C77.35,84.3,90.52,99.66,108.07,115.75,122.7,:l28.92,135.1~,142.09,
00830 C C149.77,~58.18,167.32,175.0,183.41,192.55,412~0.DO/
00840 DATA IHI/'R.M.','I. ','6/05','/79 ',' 3"',' CUT','TER ','343',
00850 C'COMP','ACTS','--NO',' RCW','S '/
00860 ~RITE(6,200)
00870 200 FORMAT(////,1~X,'N',16X,'RHO(N)',lOX,'AIE~(N)'jlOX,'DIST(N)',
00880 C9X,'Y(N)',14X,'X(N)
00890 CON=PI/180.
00900 I~ETA=IHETA*OON
00910 ZROT=PI~';DSIN(T~EIA)
00920 RM=Rl-.25`~`L`cos(THETA)-~TowI~l;DsIN(T~ETA)
00930 REWIND 2
00940 WRITE(2,307)ZROT,R1,R2,PI,IHETA,D
00950 307 FORMAT(6F12.7)
00960 ~RITE(2,318)IHI
00970 318 FOR~T~13A4) ~_
00980 IF(N.EQØAND.NMORE.EQ.0) GO TO 15
00990 .IF(N.EQ.0) GO TO 12
01000 DO 1 I=1,N
01010 ALFHA(I)=AIPHA(I)~'CON
01020 1 CONTINUE
01030 IF(NC1.EQ.0) GO TO 6
01040 RH=R1-.25'''DCOS(T~ETA)
01050 DO 5 I=1,NC1
01060 DX(I)=(RM-RH)/DI~N(TI~ETA)
01070 R~IO(I)=Rl
01080 CALL POS(Rl,THETA,CON,AIPHA(I),ZROT,ZETA,Y(I),XS(I),YC(I),ANGLE)
01090 WRITE(6,250)I,Rl,ANGLE,DX(I),YC(I),XS(I)
01100 5 CONTINUE
01110 IF(NC2.EQ.0) GO TO 11
01120 6 NCS~NC1+1
01130 KH=R2-.25`;L~OS(THETA)
01140 DO 10 I=NCSUM,N
01150 RHO(I)=R2
01160 DX(I)=(RM-RH)/DTAN(T~.TA)
01170 CALL POS(R2,T~ETA,CON,ALPHA(I),ZROT, ÆTA,Y(I),XS(I),YC(I),ANGLE)
01180 WRITE(6,250)I,R2,A~GLE,DX(I),YC(I),XS(I)
01190 10 CONTINUE
01200 11 IF(NMDRE.EQ.0) GO TO 16
01210 12 DO 99 I=].,N~RE
01220 BETA(I)=BF.TA(I)~''CON
01230 99 CONTINUE
01240 IIN=N~1 ` Fr
~5~'7~
-26-
01250 IFIN=N~NM~RE
01260 DO 13 I=IIN,IFIN
01270 ALPHA(I)=BETA(I-N)
01280 RHO(I)=RBETA(I-N)
01290 RH=RHO(I)-.25~DCOS(TÆ TA)
01300 DX(I)=(RM-RH)/~T~N(THETA)
01310 R=RHO(I)
01320 CALL POS(R,TÆ TA,CON,ALPHA(I),ZROT,ZETA,Y(I),XS(I),YC(I),ANGLE)
01330 WRITE(6,250)I,RHO(I),ANGLE,DX(I),YC(I),XS(I)
01340 13 CONTINUE
01350 ~N+~MORE
01360 GO TO 16 ~_
01370 15 N=l
01380 RAN~ANY(FS~D)
01390 FSEEDCRAND
01400 RHO(N)=RA~nr`;(R2-Rl)+Rl
01410 RH=RIIO(N)-.25`';DCOS(THETA)
01420 DX(N)=(RM-RH)/DrAN(n ETA)
01430 RAND=ANY(FSFED)
01440 FSEEDCRAND
01450 ALPHA(N)-RA~n~ TOPI ~,
01460 R=RHO(N)
01470 CALL POS(R,THETA,CON,A ~ ~A(N),ZROT,ZErA,Y(N),XS(N),YC(N),ANGLE)
01480 WRITE(6,250)N,RHO(N),ANGLE,DX(N),YC(N),X~S(N)
01490 16 DAM~=~MAX/Rl
01500 KN=O
01510 18 DO 60 Kl=l,N
01520 RAND=ANY(FSEED)
01530 ADIST-RAN~(DMAX-DMIN)~ IN
01540 FSEEL=RAND
01550 KTIME=50
01560 ~O 40 J=1,100
01570 KTIME=KTI~E+l
01580 IF(KTIME.LT.100) GO TO 20
01590 CALL TI~ER(KII~E)
01600 KTI~E=0
01610 20 r~l.0
01620 ICOUNr=0
01630 KKK=0
01640 RANDtANY(FS~Fn)
01650 AKT=RA~'~OP.I
01660 FSEED=RAND
01670 IF(A~r.LT.PI) GO TO 21
01680 KKK=l
01690 AKT=TOPI-AKT
01700 21 SINE=DSQRT(Y(Kl)'W'2+ADIST~'~2-2.~Y(Kl)~ DlST`~'`lX~OS(AK~'))
01710 IF(A~r~E~ pI~oR AKT~EQ~o~o~oR~AKT~EQKropI) DAN=0.0
01720 IF(DAN.EQØ0) C~ TO 27 ~_
01730 IFCY(Kl).NE.SINE) GO TO 24
01740 23 ETA=PI-2.`'AKT
01750 GO TO 26
01760 24 ETA=DARSIN(ADIST'~DSIN(AKT)/SINE)
01770 26 DAN=ErrA/DSIN(THETA)
01780 IF(DAN.~r.DAMA) CO TO 40
01790 27 RAr~SL~ 'DSIN(I~ErA)
01800 IF(NCl.EQØAND.NC2.EQ.0) GO ro 30 F
7S;i~
-27
01810 IF(NCl.EQ.0) GO TO 28
01820 IF(NC2.EQ.0) GO TO 29
01830 28 IF(RAD.LT.Rl.OR.RAD.OE .R2) GO TO 40
01840 GO TO 31
01850 29 IF(RAD.LE.Rl.OR.RAD.GT.R2) GO TO 40
01860 GO TO 31
01870 30 IF(RAD.LT.Rl.OR.RAD.GT.R2) GO TO 40
01880 31 IF (KKK)32,32,33
01890 32 ANGLE=ALPHA(Kl)-DAN
01900 IF(ANGLE.LEØ0)ANGLE=TOPI+ANGLE
01910 GO TO 34
01920 33 ANGLE--ALPHA(Kl)+DAN
01930 IF(ANGLE.GT.TOPI)ANGLE=ANGLE-TOPI
01940 34 DO 35 K2=l,N
01950 DIST=R Wr(ALPHA(1~2),ANGLE,Y(K2),SINF.,PI,TOPI,THElA)
01960 IF (DIST.LT.DMIN) GO TO 40
01970 IF(DIST.GE.DMIN.AND.M ST.LE.DMAX)ICOUNT--ICOU~r+l
01980 35 CONTINUE t-
01990 IF(N.EQ.l) G0 TO 37
02000 36 IF(ICOUNT.LT.2) GO TO 40
02010 37 N-N+l
02020 Y(N)=SINE
02030 RHO(N)=RAD
02040 RH=RHO(N)-.25`~'DCOS(THETA)
02050 DX(N)=(RM-RH)/D~AN(THETA)
02060 AIE~A(N)=ANGLE
02070 Z~TA=ANGLE~`DSIN(I~ETA)
02080 XS(N)=SINE`~`DSIN(ZETA-ZRDT)
02090 YC(N)=SINE''`DCOS(ZETA-ZROT)
02100 ANGLE--ANGLE'`~180./PI
02110 WRITE(6,250)N,RAD,ANGLE,DX(N),YC(N),XS(N) ~-
02120 250 ~OR~AT(lOX,I5,15X,F6.3,10X,F8.3,10X,F6.3,10X,F6.3,10X,F8.3)
02130 GO TO 60
02140 40 CONTINUE
02150 60 CONTINUE
02160 IF(N.GE.NCM~uY) G0 TO 70
02170 IF(N.LE.NCMIN) GO TO 18
02180 IDIFF=N-KN
02190 IF(IDIFF.EQ.0) GO TO 70
02200 KN=N
02210 GO TO 18
02220 70 WRITE(2,327)N
02230 327 FOR~T(I5)
022~0 L=N-l
02250 75 DO 76 IS=l,L
02260 J=IS-H
02270 IF(A ~ ~A~IS).LE.ALE~IA(J)) GO TO 76
02280 A=AIEHA(IS) ,~
02290 B=DX(IS)
02300 ALPHA(IS)--A~HA(J)
02310 DX(IS)=DX(J)
02320 AIPHA(J)=A
02330 DY(J)=B
02340 GO TO 75
02350 76 CONTINUE
02360 ~RITE(6,330) r
02370 330 FoRMAT(35(l)~2ox~N~llx~ALpHA(N)~7x~DIsT(N)~)
~ , . .
5~L75~
2~ _.
02380 DO 77 IST=1,N
02390 ALPHA(IST)=AI.PIIA(IST)/CON
02400 1~RITE(6,335)IST,ALPHA(IS'r),DX(IST)
02410 335 FORMAT(16X,I5,9X,F8.3,10X,F6.3)
02420 77 CONTIN~E
02430 `DO 80 I=1,N
02440 ~RITE(2,396)XS(I),YC(I)
02450 396 EORMAT(2F15.7)
02460 80 C0NTINUE
02470 ~WqDTH+~+.1
02480 ND=1~/.1
02490 WRITE(2,440)W,ND ~_
02500 440 FORMAT(F15.7,I5)
02510 B1=(Rl-.25~';DCOS(THETA))/DSIN(THErA)-D/2.
02520 B=B1
02530 DO 95 JD=1,ND
02540 C~LL TDMER(100)
02550 SUM~-0.0
02560 DO 90 l~L=1,N
02570 YUP-Y(KL)-.25/DTAN(THETA)+D/2.
02580 YLCW'-Y(KL)-.25/DIAN(T~TA)-D/2.
02590 IF(B.LE.YLOW.OR.B.GE.YUP) GO TO 90
02600 T=B-Y(KL)+.25/DTAN(THETA)
02610 C=D-2.`''DABS(T)
02620 SUM=SU~C
02630 90 CONTINUE
02640 XD(JD)=B-B1
02650 YD(JD)=(SUMV(TOPI'''B'''DSIN(THETA)))`l'10.0
02660 B=B+.1
02670 1~RITE(2,396)XD(JD),YD(JD) r
02680 95 CONTINUE
02690 ~TOP
02700 END
02710 FUNClION ANY(FSEED)
02720 I~LICIT RE~L`~8(A-H,O-Z)
02730 RAN=997.`~`FSEED
02740 M~RAN
02750 ANY=RAN-FLOAT~
02760 RETll~N
02770 END
02780 SUBROUTINE POS(R,THETA,CON,ALPHA,ZROT,ZETA,YjXS,YC,ANGLE)
02790 IMPLICIT RE~L*8(A-H,0-Z)
02800 Y--R/DSIN(THETA)
02810 ZETA=ALPI~A'~`DSIN(THETA) L
02820 XS--Y~"DSIN(ZETA-ZROT)
02830 YC--Y~`DCOS(ZETA-ZROT) ~-
02840 ANGLE=ALEHA~CON
02850 RE~RN
02860 END
02870 FUNCTION ROOT(ALE~lA,ANGLE,Y,SlNE,PI,'1~PI,1~-D~r~)
02880 IMPLICIT REAL*8(A-H,O-Z)
02890 PHALA=ALPHA
0290Q GLANE=ANGLE
02910 IF(DABS(PI~ALA-GLANE).LE.PI) GO TO 10
02920 IF(PHAlA.GT.PI) PHAIA=PHALA-TOPI
02930 IF(GLANE.GT.PI) GLANE.-GLANE-TOPI
02940 10 UNA=DBS(GLANE-PHAIA)
.
5~7~
-29-
02950 UNA--UNA`~'`DSIN (THErA~
02960 R~I)SQRI (Y`~2+SI~;'`~2- 2 . '''Y~`-SINE;'''DCOS (UNA) )
02970 R~'TURN
02980 END
READY
,
2 -`
_ -30-
___ TABLE N0. 2
_ ~ .. ~ __ _ _
o
N0 oC + .02 A + 015
_ _ __
1 0,000 1,0] 5 23 3~.62~) 3./196
I __
2 0.000 5.219 2'l 3~s-55() I.5~31
_ I _
3~ ()7 ~.6~2
3 1.641 40413
__ . __ ___. L.
4 ~3.5233.651 _26 _ _ 39.13() __ _ 5 - Z19_ _
. . . _. ____
14 9401 581 27 39.303 2.()23 r
~ . . . ._ __ ~ _ _ _ _
6 6.505 2.814 28 39.480 1.01S
~ __. _.__
7 8.560 5.219 29 39.520 4.414 ' r
: . .. __ ____. _ __ __ _.. ._.__
8 9.870 1.015 30 ~0.567 2 967
.__ ____ ___..... _._. _ _._ _ .
9 11.714 4.131 __ ___ _ 4l~.~20 1.58
_
11.727 2.399 32 46.61i 7 3.4~ ~
_ _ ___ __ ___ _ _____.__ _ _
1-1 13.983_ _3.31_ 33 l7.690 502l9
12 14.8]01.581 34 L 4~.-338 4.271
_ ___.______ __ . ___ _ .__ __
13 `17.94~5.219 ~5 ~.(34~ 2.65~
, . __ .. ._._ _ ~____ _. ___._ _ ___ __ _ __ .
14 19.740 1.015 1 3~ 49.350 1. Ul j
_ ._ . _. . _. _._ ___ . . .. . _ _. ___ ___
~ 15 19.827 4.030 _ _ _ _ _ _
; __ _ _
i 3n 5l~ .29n ] . ~7~l
16 120.261 2.377 _ __ _ __ _____
__ __ ___ ___
17 123.781 3.239 ___ _ 54. gOS 3.0~7
_ I
18 24.680 1.581 40 55.570 1.~15
. _ ~ ._ _ . . _ . _ .. _ .. . . . _ . .. .. . _
19 128.031 4,210 41 _57,07Q__ _ 5~
__ __ ._~ ___. _ . _ _
28.106 2.486 42 6~.251 2.720
. _ _ _ _ _._._ .. _ .. __ .____ __ ~
21 28. l 30 5.219 l~3 61.524 l~ .408
~ _ ~_ _ ._.___~_
22 29.610 1.015 44 62.52() 1.015
_ _____ _._.__ ______ ___ _ _,_ __________
` - - - - . .. _
~LS~'75i~
-3:L-
TARl~E MO. 21 _
¦ C_~ ~ .02 ¦ A + .015
,, I . i - ..
45` 63.112 3.632 6~3 1 97.8:30 5.219 __
_ __. - .
46 63.430 1.874 69 1 99.95() 2.(171
__ _ .. ___ _ __ ___ .~ . _ . ._ ~
47 66.885 3.011 70 1~0.497 4.2~3
__ __ , ._ ___ ____.___ . __ _.____ e~
48 67.260 S~ 21g ~ I 10~ . s(,n ~ ~ 5~.
_ _. _ __. _ _
49 7U .200 1 ~ OlS 7~ 104.091 3.609
_ . _ . _ _ __.__ ___
71.692 3.819 73 ~06.'390 5,21')
. ... ___ . . .. - I , . _
' Sl 71.8401.874 74 107.490 1.015
. _ _ ,, .. .
52 73.3132.809I 75 108.J5] 2.461 ,
53 78.2605.219 76 112.123 4.115
_ _ . . ___ __ ____ ______.__
54 78,375 3.230 77 112.~30 ~ .5~1~
. _ _. ._ .. _ _ ___ _______._ . __.__ __.___.___ ______.. :____.. _._
-78.610 1.015 78 113,99l~ 3.154
_ ___ ___ ___ ___ _ _ ___ . .
56 79.520 1.874 F-
_7" 117. '3~0L .~l5 _
57 ~ ~0.372 4.125
` _ _80 _ l l 7.390 __ _5. .1 '3 __,
58 82,533 2.6~3 ~1 ll8. S6'3 2.3~3
. ._~. _ .~___ ._. _ ___ . . __. . __. . ___ . _ _ _
5986.470 1.874 ~32 11'3.55'3 4.0~ ~
. ...... _ _ _ ... _ ___ _ .. _ ._ _ ._ __ . ._ .
87.629 3.684 83 12? . -30() l .581
61 87.750 __ 1.015 __84 __ 1~'3 7-/9 '3.'337
62 88.450 5.219
_ 85 ~ 26 137 ~ 2Sf~
, _____ __ ___:_ __ __ __
63 90.068 2.805
_ _ ___~6 ~ ~) .6~)9 2. '352 __
64 92.690 1.874 __ 137 l 27.23~ - 1. '.~IS
93.839 4.370 ~38 127,5~3~ 5. ) I ')
______._ .... _ . __
66 194.454 3.510 ___ __ l3~ 6 '3.769
67 197.620 1.015 9() I 3~.170 ] .5~1 r
. .. . . .. .. .. .. . .
V
75~
TABLE NO 2
NO o~ + .02 A + 015
.. , _ . .
~4 1 ~66.4~9 ~.7]~
___ 133 752 2.846 __ _ . _ __: _ _
115 ~ 7.82~ 1. ()15
92 136.696 2.075 _ ______ _ _ _____ _
~__ __ _ _ _ _
93 136.960 5.219 __ ~ ) I . ~7~ _
~7 l6').~' ~,.223
94 ~ 137.100 1.015 ____ __
- - _ llX1 170.25j 3 354
95 137.424 4.387 _ __ _ _
. . . _ . .
_96 140.~348 --779 ____ l19173 6;() 5.219
~ '01 174. ~ 2.769
'~7 141.096 3.760 __ __ ~ ______ _ __ ~-
121176.2)() ~ 015
98 142.040 1 581 _ _____ ~r
99 145.436 4.405 1~2177. l4r) ] .X-/~
....... ___ __. ___
lQ0 145.520 5.219 123t- l78.358__ 3.737
. .__ ~ . ._.
101 146.970 1.015 12'I-32.~37~. 4.377
1251 ~3. 5~ 2.74
102 147.941 2.604
, , , ~_ _ _ .
103 , 149.654 3.648 126 1~34.090 ~.~74
.. .. __ _. _ _ ,__ ___ _. __ _ _ . _
104 151.839 4 399 I27 18~....... 650 50219
_ __. _ ___ . ._. .__. .. _ ___ _ .... _ . .
105 151.910 1.874 ; 128 1~5.370 I.()l 5
_ ____... .__ .. _ _ .__. _ .__.. _ .. _. _ .. .......
106 153.190 1.015 l ~9 18~ 493 3 ~53
___;......... _.. _ .. __. _. .. _.. '. __ . _.. ._.. _.. -............ ...
07 154.080 5.219 130 l~9.~37~ 2.81 f)
_. ___ .____ ____ .___ _._ ._._ .. _ __.
108 ~53.089 4.054 13~ 1'30~ l.~74
__ . _ __ _ . _ . . __ _ . __ _ _. _ __ _ . . _ ___. ._.. _ _ _ . . _
1091 158.699 3.026 132 193 2l0 5.'1~
___ _ r
110 160.140 _ 1.015 : I'~ '~ .32~ '~ .332
111 161.050 1.874 13~1 l95.240 1.01,
_._._ .___ _ _ . _ ___.____ ____ _
____
112 ~63.460 5.219 135 1'~5.86'3 3.329
_ ___.__. .______._~ ._.. _ ._ ________. _ __ _. _____ .__._____
113 164.354 3.560 136 196.470 2.543
_ .__ __ __ . . _ _ _ . _ _ __ . __
~4'7~
-33--
TABLE N0. 2
I I o
N0 c~ + .02 ~ + .015
. _ ~= __ I _ _
137 200.180 1.581 160 L23~.... 720 1.0~5
.___ _ ._ _ . __ _ --~ -- 1- ---------------
161 2,6 r) I ~ 3.715
138 2~1; J'l5 3. ~17 __ _ _ _ __ _ _ _ ___ __
.. . ____ _ _.
162 239.660 1.5~1
139 202.590 ___ __ _ _ _ __ ___ _ _ ___ _ ____ _
163 1 _:39,~1# 2,5~5
1401 203.419 2.736 - t- ~_ _
i64 1 2~3.279 3.860
141. 205 110 __ 1.015 __ _ _ _ _ __ _ _ _ _ __ ___
~65 1 243.350 5.219
142207.630 3.636 _ - ___ _
.. __ ______
143210.050 1.58116() ' 2~l~i.590 1.01;
__ ___ __ . .. .. _ .. _ .. __ _ _ _ _ .. . _ _ . h
] f~7 ~ 2~1f~ .4f~5 3.013
144210.6~0 _4 386 _ _ ._______
145212.7805.219 16~3 ~3. ~'J3 4.408
_ _ __
146213.4092.565 _ I169 1~19.530 1. ~37~
17~ 250. U l~ l .O~ S
14721409801.015 _ _ _ ____ __ _ _ ____ __ __ _
I_
171 '~51.9~() 5.219
148216.6323,419 _ _~ r
149. 219.5914 121 172 1 252.289 3.68~
` ......... _ .... I .. __.. __ ....... _ ___
150219.9201.58~ l ] 7 ~ 25~ 357 2, f~3~
~_____ ______ _ _ __-----f
, 174 255`2'~5 4.~06
151222.127 _ 2.31 L __ ' ____ ___ __ ______________
152223.780 5.219 175 257.760 1.01~
. _ I _ . . .
153224.850 1.015 176 258.377 3.700
_ _ __ ._ _______
177 258.670 1. ~37
154226.714 3.432
__ _._____ _ _. ._ __. _.____. .
155~27.463 4.334 178 26] .421 2.940
.~ ___ __ ___ _.___ ._._. ._
156 227.869 2.47fi 179 262~6~l 4.393 ~_
____ _ __ __ . _ . _ _ _ _ . .. . _ _ . _ . _ _._ .. ._ . .. _ .. . . .. .
157 229.79~ ] .581 18U 262.~10 5.21'J
_ __ _ _____ _______ ___.__
233.891 2.523 ______ 265.440 ___ 1.()15 _
159 1 233,970 1 5.219 __1s~ )7.()~() 1.8/~
11_ --- - __ .
.
~15~752
. ` -34--
TABLE N0. 2 _ _
NOCYC t 2 A -~ 015
. _ . ............... . I i
2l)() 1~()2 7~0 1 ~L5
183267.480 2.973 __ __ ' . r-
. . . ___ ~
207 303 5~l3 2 273
184268.016 3.855 _ __ _ __ __ _ __
.__ .. _
208 303.757 3.214
185273.100 5,219 _ __ _
2()9 307.670 1,581
186273. B50 I,015 ' _ _
. _ ____ _
187274,244 3,717 'lO ¦ 30R,371 3,97Y p_
_ _ . _
211 30~3. ')80 S .219
188274.277 2.675 _ _ _ _ _ ___ ______
._ __ . .
j 18g~7~ " ~ 0 1,874 2 L2 3 L ',392 2. L73
.__ _. ...........
213 312.600 1,015
190 281,37 l ~,179 _ __ _____
_ _._ .. ... ..
L91 281.7L0 1.87~ ~I4-_ 312,~) _~ 6
192 282,~80 5,219 215L316.4t)C _ _3. -~97
_
193 282.990 1,0]5 '1~ 31~,5~ 1,'~1
. _ . .... .... __ .. . .... . . .. . .... _
217319.0hl 2,857
194 283,177 3,063 ___I__ ____ _ r
_ __ _ l
195 286.821 3.747 21~ 319,170 5.219
196 287,930 1,874 219 1 322,471~ 1.015
..... ~ _ .. __. _ ___. __
' 197 290,555 2,745 , 22() 322.952 4,062
. 221 324,~l49 2~ 7
198 291 040 5.219 ______________
. . _ _
199 291.477 ' 4.276 222 326~ 3~t3 3,367
200 292,860 1,015 223 32i,410 1,581
___. --- ~ L,
.
201 296.741 2.788 22~ 330.17() j 5.2~ 9
225 330.372 4.017 ~-
202 297.799 3.720
l__ . ._. .___
203 297,800 1.581 226 331, l9'~ 2.40/3
I . _ ___ . ._.____ . ___ _ . .
204 299.600 5.219 22 ~ 332, ~ ) 1, ~15
__ ______ _ __ .. _ _ _ __ . __ _ __._.. __ .
205 301,114 4,381 2 '~, 332,482 3. ~
__ . . .___ , _ _ . ___ __ _._ _ ,___ . .. ... . .. .
. .
`:
.
~i54'7~
TA]3L~ NO. 2
_ . _ ., , ~ . . .. .. . ....................... _ .
N0 Lo~ -~ .02 ~_ _ _ ____
L~9 33~,16~ ll,3')~
~ . . _._ _._.___ . _ ~. . . _
'30 137.2~() 1 5)3l
~31 _~_.333 _ ~ 7~__
232 338.730 ~ .219
_ . __ _ _ _ ~ _ V--
233 339.433 _ - 9
234 342,210 1,015
. ' ~. __ ____,_
235 342 237 2,()15
236 342,684 4, L~i3 ~-
_ _ ~
237 346,008 2, fi')0
_. _~ _ __ . : _ __ .__. _.. .. .. .. ._ ..
238 346,570 3,54~
___ _ __. _ _
~39 347,150 ~,5~1 _
240 348,110 5, ? 19
_ _ __ . _ __ _ ___ .. ..
2~ 3~9, C~0 _ _ _,307
242 352,08(~ i .015
__ __ __ _~
243_ 353.573 1.~3~7
.4435~,114 ___ 3 010 _
245357,263 3, ~35C !
_ _ ~ L
24fi - 358,741- ~
5~
