NATIONAL RADIO ASTRONOMY OBSERVATORY

Charlottesville, Virginia

Engineering Division Internal Report No. 110

The Design of the Deformable Subreflector

for the 140 ft. Radio Telescope

W-Y. Wong

January, 1979

Number of copies: 100

The Design of a Deformable Subreflector for

the 140-ft. Radio Telescope

W-Y. Wong

TABLE OF CONTENTS

Page

I. SUMMARY.. . . . • • • • ............................ ...........1

II. THE SUBREFLECTOR . . . .. • • •

•

2

General Description . . . • • • • • •

•

2

Geometry. . . . . . . . . . . . . . . . . 3

Material .................................... • • • • • . • • • • . ............ 5

Stress Analyses ........................ . . . ............................. . 6

Error Budget ................ ...................................................................7

Deforming the Subreflector Surface . . 0 • 9

The Motion of the Ideal Actuators. • • . 0 • • . . . 10

Evaluations and Measurements of the SubreflectorSurface. . • • • • . 11

Evaluationd and Measurements of the ExperimentalActuators. . . . 12

The Motion of the Experimental Actuators .................... 13

III. DEFORMING SYSTEM . • 0 • . . . • • . 15

General Description. . . 15

Electronic Control . . . . 15

Force Requirements on Actuators ........................................ 17

Type of Experimental Actuators . . 18

The Linkage ................................................................................ 19

The Switching Off of the Actuator .................................... 19

TABLE OF CONTENTS

(Continued)

Page

IV. BACK UP FRAME. . • • • • • • . . 20

General Description. . . • • • • • • • . 20

The Design . . . 20

V. SOFTWARE PROGRAMMING . . • • • • . . 22

General Description. • • • • • 0 • • • •

The Input and the Output . . • • • •

VI. MEASURING JIG. . O O O O O O O O O O O •

ACKNOWLEDGEMENT. . • • • • •

REFERENCE. . . ......... . • • •

LIST OF TABLE. . • • • • • • •

• 0 0 • •

• • • .

• • .

• 6 • a

22

. 22

. 24

. 25

. 26

. 28

FIGURE CAPTIONS .. • • . • • . 33

APPENDIX

A. Engineering Drawings of the SubreflectorSurface . • 0 • . . 40

B. Engineering Drawings of the Back up Frame. . . 48

C. Engineering Drawings of the Deforming System . . 58

D. Engineering Drawings of the Measuring Jig. . . . 63

E. Best Fit of a Hyperboloid ..................................................... 70

1

SUMMARY

It was suggested that the strong gravitational astigmatism of large

radio telescopes can be corrected for by a shaped subreflector [1], if

the error profile of the main reflector is reproduced at the subreflector

and the reflecting rays do not cross each other [2]. Experiments by Cowles

and Parker, using a 0.3m shaped subreflector on a 2.8m reflector show that

the aperture efficiency was improved by a factor of two at 34 GHz [3].

The gravitational astigmatism of the 140 ft telescope was first indicated

by structural analyses, and subsequently verified by S. Von Hoerner,

using an elongated feed horn [4]. A 3.2m subreflector was built, with the

profile mechanically deformed and electronically controlled. The amount

of deformation varies with the telescope pointing.

This second subreflector includes all the features of the first sub-

reflector: it is asymmetrical and tiltable. This report describes the de-

tailed mechanical design, and ground level evaluation, of the subreflector.

The electronic aspect of this subreflector: design of controller and the

interface to the computer, is described in a separate report by R. Lacasse

[5]. The test on the telescope with seven unresolved radio sources at

22.3 GHz gives considerable amount of improvements, especially far south

and east. The best result shows the aperture efficiency was increased by

a factor between two and three [6].

2

11. THE SUBREFLECTOR

General Description

The deformable subreflector has to follow the criteria that 1) the

subreflector is asymmetrical due to the off-set feed arrangement; 2) the

subreflector is tiltable for beam switching technique in observation; and

3) the subreflector is deformable to correct the gravitational astigmatism.

The first consideration was entirely geometric, readily defined

when the first subreflector was implemented to the telescope system. The

second consideration means that the weight and the moment of inertia of

the new system has to be kept equal to, if not less than, the previous sys-

tem. For the load capacity of the sterling mount, nominally set to 2500

lbs., was already exceeded. The third consideration was new to the system.

Four servo controlled motor-actuators are installed at four points behind

the subreflector. The amount of actuation is determined by the elevation

position of the telescope.

The fiberglass, aluminum honeycomb core sandwich construction method

is proven acceptable, in terms of cost and surface accuracy, by the first

subreflector. The force required for the deformation of the surface,

solved analytically, was found reasonable. Hence the same material and

fabrication approach for the new surface was adopted. The long term be-

havior of this composite material is not entirely known, however. Aging,

temperature, environment and cyclic loading effects might degrade the sur-

face accuracy and reduce its usable life time. The surface of the first sub-

reflector has degraded from the original error of 0.25 mm rms to 0.43 mm rms

after four years.

The stiffening rib arrangement of these two gubreflectors are also

different. The first subreflector has a stiffening ring, whereas the second

3

one has two stiff diagonals. For both cases, however, the main structural

stiffness is derived from the back-up frame which serves as an interface

between the beam-tilt system and the subreflector surface.

Due to the off-set receiver arrangement, the deforming feature is

applicable only to the feed position of the K-band maser receiver (22.3 GHz,

X = 1.23 cm) plus three others, rotated by 90°. For other receiver positions,

the subreflector's profile must be returned to its undeformed position.

The subreflector system can be dismounted from the telescope so that

the optic of the system can be modified within a reasonably short period

of time.

The front face of the subreflector is coated with a thin layer of

aluminum film for reflection. The subreflector is painted white. The

back-up frame is unpainted. All moving parts of the deforming system are

protected by water-tight covers.

Geometry

A schematic arrangement of the Cassegrain system for the 140-f

telescope is shown in Figure 1. Some important parameters are given in

the following list, describing the Cassegrain optics:

Average subreflector diameter = 3.213m (126.4")

Distance from the prime focus to secondary focus: a = 13.48m

(530.89")

Distance from the prime focus to subreflector vertex:

b = 1.30 (51.262")

Magnification: in = 9.348

Eccentricity: e = 1.240

Angle between paraboloic axis and hyperbolic axis: 4.148°

= (1)

4

The hyperboloid is given in eq. (1), describing the profile of the

subreflector in an untilted system, with Z-axis as the hyperbolic axis.

where r2 = x

2 + y

2 and c = a-b.

The hyperbolic axis and the parabolic axis do not coincide. They

subtend an angle of 4.148°. A cluster of feeds are arranged in a ring

with a radius of 0.98m (38.58") and they are 4.57m (180") above the ver-

tex [7]. The subreflectors' axis is aligned with the phase center of the

feed in use by rotating the subreflector about the axis of the main re-

flector at the Sterling mount. The rim of the subreflector, which is

the intersection of the hyperboloid and the cone defined by the line of

sight of the prime system, is not a circle. The coordinates of the rim

were computed by P. J. Napier and are listed in Table I for reference.

The rim is marked at every 15° with respect to the hyperbolic

axis. A collimating mirror with cross-hair defines the hyperbolic axis.

Another cross is marked 94.3 mm (1.714 in) from the hyperbolic axis along

the plane of symmetry which defines the parabolic axis of the subreflector.

The arrangement of the actuators (Al - A4) and the stationary con-

nections (51-54) are also illustrated in Figure 1, showing their positions

relative to the main reflector. The direction of the beam tilting is perpen-

dicular to the parabolic and hyperbolic axes. The relative position of

the subreflector to the main reflector is defined when a particular receiver

is used. For example, the actuator Al is aligned to the northern rim of

the main reflector only when the K-band maser receiver is in use.

5

Material

The subreflector shell structure is composed of fiberglass cloths,

and an aluminum honeycomb core. This sandwich construction is bonded

together by epoxy rasin cured by hardener, or curing agent. There are

three plies of fiberglass cloth laminates on the front and the back of

the shell. The thickness of each ply is 0.28mm (0.011 'n). The honeycomb

core is 25.4mm (1.0 in) thick.

The shell structure is laid out in steps in a mold and bonded

together with the epoxy. The curing takes place inside the mold.

Suction was constantly applied between the mold and the front surface to

insure the reproduction of the profile in the mold. The curing agent was

selected so that room temperature curing is adequate to achieve the neces-

sary material strength. Shrinkage of the curing agent is claimed to be

less than 27.

A thin layer (about 0.15mm) of aluminum film was flame-sprayed

onto the mold before the epoxy-fiberglass was applied to make the surface

reflective.

The quality of this composite construction depends on the material

properties of the components as well as on the skill of the fabrication,

the type of curing agent and the duration of the curing. Temperature,

pressure and types of fiberglass cloth also affect the strength of the

product. Moreover, there is a wide selection of size, foil thickness and

alloy of the aluminum honeycomb core. Because of the many possible combina-

tions, the material properties of the final product is best determined by

direct measurements. On the other hand, tests on specimens are usually

6

made by the manufacturer only when a large quantity of the product is

involved. There were no such measurements made for this subreflector.

The material properties used during the analyses are based on the best

available data provided by the manufacturer. The results of the cal-

culations should be considered good within 20%. The components of the

subreflector are listed in Table II. The Antenna System Incorporated of

San Jose, California, was responsible for the construction of the shell

structure.

Stress Analyses

The in-house feasibility studies started in 1975, determining the

actuator arrangement and basic stiffening geometry. The studies also

provided a prediction on the improvement of the telescope surface error

due to the astigmatic correction. All detail computations are filed in

reference [8].

The final analysis was done by an outside firm - Control Data Cor-

poration. The CDC computations [9] used an accurate mathmatical model

of the shell structure, with the geometry, material properties and loading

conditions specified by N.R.A.O. The model has 180 finite elements and

108 nodes (Figure 2), made in one quadrant by assuming the subreflector

is symmetrical. The model is constrained at node 57, and actuated at

nodes 52 and 62. The four loading conditions considered In the analyses

are:

1) Astigmatic deformation: node 52 and 62 were actuated in opposite

directions by an amount of 1.57mm. (0.062 in.);

7

2) Survival load: mode 52 and 62 were actuated in the same direc-

tion by 3.8mm (0.150 in.);

3) Acceleration due to tilting: the angular acceleration is 34.4

rad/sec2, equivalent to the tilting of the subreflector at

= 2cm observation, and

4) Thermal load of 10°F difference between the front and the back

faces of the subreflector.

The analytical results are summarized in Table III. The large safety

margin in the survival condition shows that the subreflector will survive

an erroneous actuation, when all actuators actuate in the same direction.

This mode of actuation is in fact a higher order astigmatic correction

suggested in ref. [9], but presently not possible to produce.

The stiffness at the point of actuation are:

1st mode astigmatic correction = 201 N/mm(2)

2nd mode astigmatic correction = 417 N/mm 1

The accuracy of the analyses was proven to be good. Figure 3 shows

a comparison of the measured and the computed profiles. When the subref-

lector is actuated by 1.57mm, the difference of these two profiles is 0.14

mm rms.

Error Budget

The surface error distribution of the 140-ft for wavelength A = 1.3

cm was discussed in detail by S. von Hoerner[11]. The error allocated

specifically for the deformable subreflector is 0.42mm rms.

8

During the design of the subreflector, sources of error taken into

account are:

1) the residual path length error after the astigmatic correction;

2) the deflection of the back-up frame and linkage causing an

actuation differ and from the commanded magnitude;

3) the manufacturing inaccuracy of the subreflector surface and

4) the inaccuracy of the electronic control.

The first item was derived by analytical means. The difference

of an astigmatic correction and the surface error of the main reflector

is computed at various telescope positions. The average of these differ-

ences is found to be 0.26mm rms [1][11].

The second item was found by direct measurements after the subreflec-

tor was assembled. These measurements were obtained by placing dial in-

dicators at various points of the frame, and recorded the deflection when

the subreflector is deformed in a given amount. Due to the insufficient

design of these parts, the average efficiency of the deforming system is

only 60%. The averaged peak deflection of the linkage is 1.02mm, and the

standard deviation is considered to be 0.34mm rms.

The surface error measured at the plant and in Green Bank produced

an average value of 0.25mm rms. Refer to section "surface evaluation and

measurement" for details.

The error derived from the electronics is small: 0.04mm rms. The

error contributed mainly by the non-linearity of the position transducer,

the temperature effect and the noise of the system. It was discussed in

detail in a separate internal report by R. Lacasse [5].

9

All errors mentioned above are summarized in Table IV for quick

reference. The combined error, a root sum square of the above mentioned

contribution, is:

Total error from the subreflector = 0.50mm (3)

Deforming the Subreflector Surface

The first order astigmatic deformation is defined by the rim deflec-

tion 'a' at four points 900 apart. The deflection of the circumference

forms a two-cycle sine wave.

Figure 4 summarized the analytical and measured rim deflections in

terms of the elevation positioning the telescope. The computations were

based on the surface-panel adjustment at zero hour angle, +4.4° declina-

tion, with the dish illuminated with 9 and 15 db taper. There is good

agreement between the analytical curves and the measurement with narrow

feed horn, in which the edge taper of 8.7 db was used. The Cassegrain feed

presently used has a 10 db taper. The rim deflection of the subreflector

approximates the measured curve, with 'a' defined as the rim deflection, and

(1) the elevation angle of the telescope:

a(mm) = -1.72 + 8.70 (1-sin 0 (4)

The first term gives the absolute value of rim deflection at zenith

position, and the constant of the second term denotes the total rim deflec-

tion required as the telescope moves from zenith position to horizontal

position. The sign convention of the rim deflection is given in Table V.

10

Positive denotes a deflection towards the main reflector; whereas negative

denotes a deflection away from the main reflector. Plus sign means a re-

duction of path length, and negative an addition to the path length.

The Motion of the Ideal Actuators

Actuators Al thru ALF are arranged in a circle with a radius equal to

0.99m (39 in.) from the parabolic axis. The value 'a' from the analyses,

also given in Figure 4, are based on a circular aperture, in which the

radius of the rim equals 1.58m (62.28 in The amount of actuation

required for each actuator are reduced by a geometric constant g where

g = 39/62.28 = 0.626 (5)

so that the amount of actuation 'd' in mm at Al thru A4 are

d=axg= -1.078 + 5.446 (1-sin 0 (6)

Since the most useful range of the telescope are positions above

20° elevation, the total amount of rim deflection A, taken from Figure

4, is

Ao

= 6.8mm(7)

The corresponding total amount of actuation D o required at Al thru

A4 is

Do =gxA

o=4 .3mm (8)

11

Evaluation and Measurements of the Subreflector Surface

The surface manufacturing tolerance specified was 0.17mm rms. The

measurement at the factory during the acceptance tests was 0.24mm rms

[12]. In spite of this discrepancy, the subreflector was accepted and we

proceeded with the distorsion tests, which was, as agreed beforehand, no

longer the responsibility of the manufacturer. These additional tests

indicated no noticeable hysteresis. Surface contours repeated well after

three cycles of deformation.

The surface was measured again in Green Bank with the in-home designed

jig. The measuring accuracy is considered higher than that of the factory,

and the data was processed with a best-fit program, included in Appendix E.

The results of the nine measurements was summarized in Table VI,

with detail computations in ref. [13]. The first three measurements

consistantly yielded a value of 0.21mm rms. The surface became slightly

degraded after some cyclic distorsions. The nominal rms deviation from

the best fit hyperbolid AZ is considered

rms (AZ) = 0.25mm (9)

It is advisable to re-adjust the subreflector at times so that the value

0.25mm rms or less can be maintained. Figure 5 shows the surface con-

tours of three measurements.

There are six dials on the template which give six readings on each

radius. The readings were taken on every 15°, providing 24 sets of radial

readings. Each surface measurements hence produced 144 single dial readings.

These readings were record ed manually, and data are punched on cards

as input to the computer.

12

The zero-readings on the template profile were done prior to each

measurement. These six additional data are required by the program.

The first subreflector of the 140-ft was also measured. The purpose

of this measurement was to provide the proper position of the bearings on

the beam-tilting shaft, so that the alignments of these two subreflectors

are "identical" when installed on the telescope. It is interesting to note

that the profile of this first subreflector had deteriorated from the ori-

ginal error of 0.25mm rms to 0.43mm rms after four years of service [14].

Evaluation and Measurement of the Ex erimental Actuators

Equation (6) represents the amount of actuation required for each

actuator at a given telescope elevation (P. It is assumed that the actua-

tors are supported by a perfectly rigid frame. In reality, the bending of

the frame and the linkage cause a reduction of the efficiency. In order

to match the input and the output, calibration constants are incorporated

to each individual actuator.

The most direct way to measure the efficiency is to place a dial

indicator at the surface underneath each actuator, and to compare the command-

ed (input through the controller) and the dial reading. The ratio of the

output and the input of the four actuators are listed in the third column

of Table VII.

The deflection of the frame was measured in the similar fashion.

The ratio of output and input are listed in column no. 1 of Table VII.

The amount unaccounted for is considered the deflection of the link-

age. These values are listed in column no. 2 of Table VII, so that the

sum of the column 1 thru 3 is 100% for each actuator.

13

Figure 6 illustrates a comparison of commanded actuation and the

measured deformation under the dial no. 4 and 6. The solid line is the

measurements, whereas the dashed lines are the ideal readings, based on

the locations of the dial, assuming the supporting frame and linkage are

perfectly rigid. The ratio of the two different slopes at dial no. 4

Indicates a reduction of efficiency. The results, called indirect measure-

ments, are listed in the forth column of Table VII.

Figure 6 also illustrates the linearity of the subreflector surface

when deformed, showing the structure is behaving in its elactic range.

The average values of the direct and indirect measurements are listed

in column 5, with the invert listed in column 6.

It is clear that the linkage arms and hinges are not adequately

designed. Improvement is possible and desirable..

The Motion of the Experimental Actuators

The actuators used for the experiment have a maximum travel D o of

6.35mro.. The range was reduced by half in order to increase the rated

force from 445 N (100 lbs) to 890 N (200 lbs). Hence in theory the

actuation at the output is D2' where

D2 = 6.35 4 2 = 3.18mm•

The amount given in (10) was not totally realized. The actual travel

capabled through the controller is limited to a value D3 where

D3 = 2.06mm 4 . 0.50mm = 2.56mm

The zero crossing of D3

was set tp 0.50mm in the controller as well

as at the actuators.

(10)

14

The efficiencies of these actuators were measured as tabulated in

Table VII. The averaged efficiency is found to be 60%. Hence the

maximum travel D

4' capable by the present actuators at the output is

D4 = (2.06+0.50) x 60% = 1.24+0.30=1.54mm (12)

The corresponding rim deflection A

3' due to the ranges of the

actuators described in (12) is

A3 = (1.24 + 0.30)/0.626

= 1.98 + 0.48 (13)

= 2.45mm

The amount of (13) is marked in Figure 4. Compared with demanded

range of 6.8mm eq. (7), the present actuators are short of the design

goal by a factor of 2.8.

15

III. DEFORMING SYSTEM

General Description

The deformation of the subreflector is controlled by four indepen-

dent and identical systems. Each system controls the contour of one quad-

rant of the surface. Each system has its own position control. The in-

fluence of deformation in one quadrant to the others is reduced because

of the two stiff diagonals on the back of the subreflector (Figure 1).

The deforming system has four major parts: an electronic control,

a position transducer, a motor actuator and a lever arm linkage. The first

three items are covered in the internal report 139 in detail by R. Lacasse

[5]. The lever arm linkage is illustrated in a detail drawing, no. 30D00017

in Appendix C.

This deforming system is an experimental version. Once this deform-

able subreflector is proven workable, the deforming system should be modi-

fied to incorporate actuators with a larger amount of actuations. A reverse

in the direction of actuation will make the subreflector more versitile:

it will be usable for receivers placed + 90 0 and 1800 apart from the present

K,band maser receiver location.

Electronic Control

For the completeness of this report, a block diagram of the control

system is included for reference, illustrated in Figure 7. Each actuator's

position is monitored by a LVDT (Linear Voltage Displacement Transducer).

A position given by the computer is compared with the measured position.

16

The difference of these quantities is reduced to an amount of + 0.01m.

The commanded position is up-dated by the output of the computer while

the telescope tracks a source. The up-date rate is set to be 1 new posi-

tion per second.

Figure 8 shows the front panel of the controller. The controller is

an interface between the H-316 on-line computer and the servo mechanism.

It's functions are receiver and transmitter data. Presently this interface

is connected to the power supply at the focus by cable and no. 45 and 34

in the control room.

The interface provides the option of manual and computer control.

A switch for this selection is located in the upper center of the panel.

Upper travel limit of 2.56mm is displayed under this switch. A dis-

play of travel in mm is located below this 'limit' window, with the select-

ing switch located on the right.

It is possible to monitor the commanded and measured positions of

actuators Al thru A4. It is also possible to monitor the temperature at

the LVDT in Celsius. Heating elements are provided to the position trans-

ducer to maintain continued level specified by the producer of the trans-

ducer.

The display of 'LOCK' is for debug purposes. It shows the differ-

ences between the sum of the commanded positions and the sum of the actual

positions.

The manual control enables one to input values into the thumb wheel

knobs, either individually or as a group (switch to 'ALL'). The step is

executed by pressing the 'UPDATE ENABLE' button.

17

All actuators are travel from an undeformed position to a prede-

termined direction (see Table V). The deforming option is applicable only

for the K-band maser receiver.

The size of the actuator depends on 1) the force required to produce

the given amount of deformation on the subreflector; 2) the weight of the

subreflector surface and 3) the dynamical force due to the periodic tilt of

the subreflector in the beam-tilting mode.

As shown in eq. (8), the range of actuation from 200 elevation angle

to 900, based on the measured value of astigmatism, is 4.3mm. Refer to

eq. (2), that the stiffness of the shell structure when actuated in a

astigmatic deflection, is 201 Wm. The force required for the full actua-

tion is

F (astigmatic) = 4.3mm x 201 N/mm = 864 N(194 lb) (14)

The weight of the shell is 668 N (150 lbs). Assuming that the weight

is equally distributed to the eight connections, the force on the actua-

tor due to dead weight is

F (dead weight) = 668 N/8 = 83 N (19 lb) (15)

The force due to the beam switching at A = 2.0 cm is derived as

follows:

B (HPBU) = 1.2 X/ D = 57.92 A (sec)

0 (beam tilt) = 3B

18

x M a (Rotation of the subreflector) - 1.15

- 8.13 e

T (transition time) = 40 ms

T (available torque) = 7000 ft-lb

4a 2A (angular acceleration) = --

2-= 34.3 rad/sec

lb-sec2

m (mass of the shell per connection) = 1501b/8 = 19 lb = .049

d (distance to the axis of tilt) = 39 in

F (beam tilt) = m-A•d = 294N (66 lb) (16)

The total force required for each actuator is the sum of the force

shown in eq. (14), (15) and (16), or

F (total) = 864+83+294 = 1241N(279 lb) (17)

The experimental version has actuators rated at 100 lbs. This

force was amplified by a factor of 2 by lever arm linkage, having rated

output of only 200 lbs.

Type of the Experimental Actuator

The size of the actuator is dependent on the required force as desc-

ribed in eq. (17) and the amount of actuation as in eq. (8). The avail-

able actuator for the experimental version of the deformable subreflector

has the following properties:

Manufacturer: AIRSEACH

Part No.: 34596

Weight (approx.): 2.5 lbs.

Max. Actuation: 0.25 in (6.35 mm)

Max. Load: 100 lb (445 N)

Motor Type: Series wound

Electrical: 20 VDC

in

19

The Linkage

The necessity of this linkage is due to the motor actuator available,

in conjunction with the requirement in force and displacement of the sub-

reflector. The final version of the deforming system could by-pass this

linkage to reduce the additional deflections.

The extension of the lever arm also provides an amplification in the

travel of the LVDT.

The Switchin Off of the Actuators

The subreflector deformation can be terminated either at the commence-

ment of an observation; or if erratic behavior is detected, such as the

monitored amount of actuation being different from the commanded quantities;

or the temperature readings at the actuator motors show a level higher

than a preset value.

The proper sequence is 1) to set all actuator command values to

0.50 mm at the four windows on the left hand side of the panel; 2) switch

the "manual/computer" switch to the manual position; 3) put the update

selection button to "all" position and 4) press the "update enable" button.

This will activate all actuators to drive the surface back to the undeformed

position. These positions will hold even if the power supply is turned off.

20

IV. BACKUP FRAME

General Description

The backup frame of the deformable subreflector provides the follow-

ing functions: to add structural stiffness to the subreflector; to inter-

face the subreflector to the existing beam tilting device; and to support

the actuators-linkage combinations.

The subreflector is connected to the frame at eight points. Four of

which are actuated by electro-mechanical devices, labeled as Al thru A4. Another

four conhections are mechanically fixed with rods and brackets, and labeled

as S1 through S4. Figure 1 shows the schematic arrangements of these con-

nections, and appendix B shows the detail. The stationary and the actuated

connections are alternately arranged for the astigmatic deformation of the

subreflector.

The subreflector-frame combination is attached to the beam tilting

device through two bearings and two connections to the push rods. These

connections are designed for quick installation so that the telescope can

be operated in either prime focus or Cassegrain system. All connections

are positioned beforehand by pins so that calibration procedures are kept

to a minimum after the subreflector is installed.

The Design

The material of the frame is aluminum. The limit of design is not

governed by the allowable stress but by the total weight and moment of inertia.

The detail of the design is included in Appendix B. The analyses of the struc-

ture is summarized from the detailed computation of reference [8] and listed

as follows:

21

Properties of the Subreflector Frame

Weight 6 in sch.10 weldment 134.4 lbBrackets 63.0 lbShaft & brgs 83.0 lb

Total 286.4 lb

Moment of inertia About the Shaft 3x104 lb-in2

_

Spring const. in thedirection parallel tothe telescope axis

Al, S2, A3, S4:Si, A2, S3, A4:

2.05x104 Clin

6.29x104 ii/in

Max. bending stress 612 psi

Max. shear stress 84 psi

Resonance frequency Frame aloneWith Subreflector

105 Hz33 Hz

The frame was fabricated in the G.B. shop. It was shipped to ASI

of California for the framing of the subreflector. The subreflectoriframe

combination was shipped back to G.B. as one integral part.

22

V. SOFTWARE PROGRAMMING

General Description

A subroutine was added to the 140-ft telescope control program in

the Honeywell-316 computer by Bob Vance. This subroutine gives the

user the option of using this deformable subreflector by switching on the

power supply of the servo control electronics and setting the manual/

computer switch (see fig. 8) to the computer control mode.

This subroutine was written based on the algorithm illustrated in

figure 9.

The Input and The Output

The amount of actuations on the subreflector surface is dependent

on the elevation angle 0 of the telescope. The values 4) are readily cal-

culated by the main control program.

There are four outputs from this subroutine: a, b, c and d. Each

is a eight-bit word, containing a max. value of 255. These outputs are

channeled to the controller, and ultimately translated into displacements

of the subreflector shell. The expressions in eq. (18) are modifications

of eq. (6), with zero crossing set at 0.50 mm:

P1 = V

1 + 2 (1-sin = -0.578+5.446 (1-sin (I))

P2 = V

3 + V

4 (1-sin (1)) = 0.

(18)P

3 = V

5 + V

6 (1-sin = 0.

P4 = V

7 + V

8 (1-sin (1)) = 0.

and subsequently, the four output words are expressed as:

23

a = (P +P +P)xC at A1 2 3 1 1b = (P

1 + P

2 - P

3) x C

3 at A

3

C = (-P +P +P)xC at A1 2 4 2 2d= (-P +P -P)xC at A1 2 4 4 4

PI represents the first order astigmatic deformation. P2 thru P4 are

values for higher order deformations, which are not being con-

sidered during the design phase of the subreflector. Presently P2 thru P4

are dummys and the values of P 2 thru P4 (or V3 thru V8 in eq. (18)) are

set to zero. These values can be redefined once it is proven that the sub-

reflector is capable of these types of deformations.

thru C4 are calibration constants, correspondent to actuatorsCl

A1 thru A

4. The numerical values of C 1 thru C

4 are listed in table VII,

column 6.

(19)

24

VI. MEASURING JTG

Fig. 10 shows the general setup of the measuring jig, with the detailed de-

sign included in Appendix D. It consists of a supporting frame for the sub-

reflector and a template. The measuring jig serves the following purposes.

1) to verify the subreflector's surface accuracy,

2) to measure the contour of the deformed subreflector,

3) to evaluate the efficiency of the actuators, and

4) to provide a reference system for future adjustments.

The subreflector was held in place and was so adjusted that its

hyperbolic axis is aligned with gravity. The central axis of the jig is

defined by an optical plumb, with the three adjustment screws at the sup-

porting legs to correct the tilt, and mild forces required to shift the

jig in lateral direction so that the axes of the subreflector and the jig

are coinciding. Few repeating cycles are required until these two axes

will actually meet. The optical plumb defines the direction of gravity,

and with an eyepiece to collimate the central mirror of the subreflector.

The horizontal reference of the template was monitored by an op-

tical level (Wild N3) to insure the line remains horizontal as the jig

rotates. There is no fine adjustment for the counter weight, but good

balance can be achieved by using heavy bolts or nuts clamped on the jig.

The 1/8 inch steel plate template has a cut-out hyperbolic profile.

This profile was also measured by optical means so that the reference is

well established. Fig. 11 shows the arrangements of the dial indicators,

the reference profile and the proper corrections for the zero readings of

each dial.

25

The dials are placed on the template vertically. The dials are

product of MITUTOYO, part no. 3052. The graduation is 0.010 mm. The

total travel is 30 mm.

There was no effort to evaluate the accuracy of the system. Rough

estimates of the measure accuracy is about + 0.050 mm.

ACKNOWLEDGEMENT

The author wishes to thank the Green Bank workshop, under the

supervision of Albert Steinemann and Martin Barkley, for the assembly

of the subreflector stiffening frame; the maintenance group with Ron

Gordon, Don Gordon and Herb Hanes for the mechanical work; Sidney Smith

for the help during the measurement of the surface. The Controller was

designed and assembled by R. Lacasse of the Electronic Division. R. Vance

was responsible for the programming. Also thanks for advice from R.

Fisher. The moving force of the project shall be accredited entirely to

Dr. S. von Hoerner.

26

REFERENCE

[1] S. von Hoerner and W-Y. Wong, "Gravitational Deformation and

Astigmation of Tiltable Radio Telescopes", IEEE Trans. Antennas

Propagat., Vol. AP-23, pp. 689-695, 1975.

[2] S. von Hoerner, "The Design of Correcting Secondary Reflectors,"

IEEE Trans. Antennas Propagat., Vol. AP-24, pp. 336-340, 1976.

[3] P. R. Cowles and E. A. Parker, "Reflector Surface Error Compensa-

tion in Cassegrain Antennas," IEEE Trans. Antennas Propagat., Vol.

AR-23, p. 323, 1975.

[4] S. von Hoerner, "Measuring the Gravitational Astigmatism of a

Radio Telescope," IEEE Trans. Antennas Propagat., Vol. AP-26, pp.

315-318, 1978.

[5] R. J. Lacasse, "Correctable Subreflector Controller," N.R.A.O.

Electronic Division Internal Report No. 193, 1978.

[6] S. von Hoerner, R. Fisher, W-Y. Wong, "140-ft Observations with

Deformable Subreflector," N.R.A.O. Engineering Division Internal

Report No. 109, 1978.

[7] Peter J. Napier, "Cassegrain System for 140-ft Telescope," N.R.A.O.

Memo, 1972.

[8] Computer output file no. DS-001, "N.R.A.O. Stren Analyses of the

Subreflector".

[9] Computer output file no. DS-002, "C.D.C. Stress Analyses of the

Subreflector, Dec. 28, 1976.

[10] C.D.C. Report, "Stress Analyses of A Deformable Subreflector",

Control Data Corp., Dec. 28, 1976.

27

[11] S. von Hoerner, "140-ft Deformable Subreflector", N.R.A.O. Memo,

October 28, 1975.

[12] Computer output file no. DS-004, "Antenna System Inc. Surface Mea-

surement",

[13] Computer output file no. DS-005, "Green Bank Measurements of the

Deformable Subreflector".

[14] Computer output file no. DS-003, "Green Bank Measurements of the

First Subreflector".

2 8

LIST OF TABLES

TABLE I SUBREFLECTOR EDGE COORDINATES. DEFINITION OF ANGLE 0 IS

SHOWN IN FIG. 7, LINEAR DIMENSIONS ARE IN INCH

TABLE II MATERIAL LIST OF THE SUBREFLECTOR

TABLE III ANALYTICAL RESULTS BY CDC

TABLE IV ERROR CONTRIBUTION OF THE SUBREFLECTOR

TABLE V LOCATION AND SIGN CONVENTION OF THE ACTUATIONS

TABLE VI G. B. MEASUREMENTS OF THE SUBREFLECTOR SURFACE

TABLE VII EFFICIENCY OF THE ACTUATORS

29

TABLE I. SUBREFLECTOR EDGE COORDINATES. DEFINITION OFANGLE cp IS SHOWN IN FIG. 1, LINEAR DIMENSIONS ARE IN INCH

Z(R) c1 R Z(R)

-90.000 67.842 19.187- 88.000 67.839 19.185-86.000 67.827 19.179-84.000 67.809 19.169- 82.000 67.783 19.155- 80.000 67.749 19.137-78.000 67.708 19.114- 76.000 67.660 19.088-74.000 67.605 19.058- 72.000 67.543 19.025-70.000 67.473 18.987-68.000 67.397 19.946- 66.000 67.314 18.901-64.000 67.224 18.853- 62.000 67.128 18.801- 60.000 67.025 18.746- 58.000 66.916 18.688- 56.000 66.801 18.626-54.000 66.681 18.561-52.000 66.554 18.494- 50.000 66.422 18.424- 48.000 66.285 18.350-46.000 66.143 18.275- 44.000 65.995 18.197-42.000 65.843 18.116-40.000 65.687 18.034- 38.000 65.526 17.949-36.000 65.362 17.862-34.000 65.194 17.774-32.000 65.002 17.684-30.000 64.847 17.593-28.000 64.668 17.500-26.000 64.487 17.406-24.000 64.30 17.310-22.000 64.117 17.214- 20.000 63.930 17.117-18.000 63.741 17.020-16.000 63.550 16.921- 14.000 63.357 16.823- 12.000 63.164 16.724-10.000 62.970 16.825- 8,000 62.775 16.526- 6.000 62.580 16.428- 4.000 62.386 16.329- 2.000 62.191 16.231

0.0 61.996 16.1332.000 61.803 16.0364.000 61.610 15.9406.000 61.419 15.8448.000 61.229 15.749

10.000 61.040 15.65612.000 60.853 15.56314.000 60.669 15.47216.000 60.486 15.38318.000 60.306 15.29420.000 60.129 15.20722.000 59.954 15.12224.000 59.783 15.03826.000 59.614 14.95628.000 59.449 14.87730.000 59.288 14.79932.000 59.130 14.72234.000 58.976 14.64836.000 58.827 14.57638.000 58.681 14.50740.000 58.540 14.43942.000 58.403 14.37444.000 58.271 14.31146.000 58.143 14.25048.000 58.020 14.19250.000 57.903 14.13652.000 57.790 14.08354.000 57.683 14.03256.000 57.580 13.98458.000 57.484 13.93960.000 57.392 13.89662.000 57.306 13,85564.000 57.226 13.81766.000 57.151 13.78368.000 57.082 13.75070.000 57.019 13.72172.000 56.962 13.69474.000 56.911 13.67076.000 56.865 13.64978.000 56.826 13.63180.000 56.792 13.61582.000 56.765 13.60284.000 56.744 13.59386.000 56.728 13.58588.000 56.719 13.58190.000 56.716 13.580

Loading Condition Reaction at actuatorN(1b)

Astigmatic correction of 1.57 mm

Survival load of 3.8 mm

Beam tilt at 34.4 rad/sec2

AT = 10°F

16 (71)

80 (356) *

4 (16)

3 (12)

30

TABLE II. MATERIAL LIST OF THE SUBREFLECTOR

Item

Epoxy rasin and curing agent

Fiberglass cloth

Honeycomb core

Collimating mirror

Product

HEXCEL

J. P. STEVENS

HEXCEL

BRUNSON

Part N

Hexcelite HP-302

No. 141

5052/F40-0013-2.1

185-1

TABLE III. ANALYTICAL RESULTS BY CDC

* Safety margin is 63% for the laminates, 54% for the core.

Actuator

Al

A2

A3

A4

Location

North

West

South

East

Edge Movement

31

TABLE IV. ERROR CONTRIBUTION OF THE SUBREFLECTOR

Source ] Error (inm rms)_

Residue path length error after astigmatic correction

Deflection of the frame, linkage

Manufacturing error of the surface

Electronic

TOTAL

TABLE V. LOCATION & SIGN CONVENTION OF THE ACTUATORS

+: deflect towards the main reflector to shorten the path length

-: deflect away from the main reflector to lengthen the path length

0.26

0.34

0.25

0.04

0.50

16% 26% 58%

17% 25% 58%

18% 15% 67%

15% 35% 50%

63% 60%

62% 60%

69% 68%

54% 52%

Avg 60%

C

1

= 167

C2 = 1.67

C3 = 1.47

C4 = 1.92

32

TABLE VI. G. B. MEASUREMENTS OF THE SUBREFLECTOR SURFACE

Date

12- 5-77

12- 7-77

12-13-77

12-14-77

12-14-77

3-15-78

3-15-78

3-15-78

3-15-78

Measurement

GB2-2

-3

-4

- 5

-6

-12

-13

- 14

- 15

Error (min rms)

0.21

0.21

0.21

0.26

0.22

0.26

0.24

Actuation (mm)

•••••••••

•■■■••

2.54

■11.1."11.11

1.94

•••••••••

Actuator

A-1

A-2

A-3

A-4

TABLE VII. EFFICIENCY OF THE ACTUATORS

Frame Linkage Direct IndirectDeflection Deflection Measurement Measurement

1 2 3 4

CalibrationAvg Constant

5 6

FIGURE CAPTIONS

Figure 1 Schematic drawings of the arrangements of the 140-ft telescope

Cassegrain system. The general dimension of the deformable

subreflector is given, and its relative position to the main

reflector when the K-band maser receiver is in use.

Figure 2 Finite element model used by C.D.C., contains 180 elements,

108 nodes. The subreflector is considered symmetrical, ac-

tuated at nodes no. 52 and 62, and held at no. 57.

Figure 3 Comparison of the deformed subreflector profile to the best-

fit deviation of the main reflector at elevation angle of 28°

with the radius of the main reflector normalized to 62.3 inches.

Figure 4 Comparison of rim deflection of the deformable subreflector

with the telescope positions. Dashed line denote the analytical

results of 0 db, 9 db and 15 db edge taper. Solid line shows the

measured result with an elongated feed. Ao denotes the de-

sirable range of rim deflection; A3 shows the range presently

covered by the experimental actuators.

Figure 5 Profile contour of measurements GB2-4, GB2-14 and GB2-15, with

contour level of 0.005 inch.

Figure 6 Comparison of the commanded actuations (vertical scale) and

measured deflections (horizontal scale). Solid line is

the measurement, dashed line is the ideal.

34

Figure 7 Deformable subreflector's electronic block diagram.

Figure 8 Panel arrangement of the controller.

Figure 9 Subroutine flow chart implemented to the telescope control

program.

Figure 10 Green Bank measuring jig for the deformable subreflector.

It includes a supporting frame (1), a template (2), 6

dial indicators (3), optical plumb (4) and a rotating arm

with counter wt (5).

Figure 11 The error on the profile and the reference system of the

template used in the measurement.

Fi'et

ILO

elece's

474-/:r.07its,

V iew from .the kopof The 14cY--f-1. -tele6c,ope

No-fa fico(1) -444 :

Acq uaieci cuppo)450611-/g4:

S ikalionory cuppori5

0Coll "mai p.s tn;rror

Sec4fon thrt.1 ,Eobrefleckorplone Leimmegin

y4 NRAO SUBREFLECTOR

107

106

104

103

9 it. ga 1 / I 62 4'7,.. 82 i 92 \\,,k7.......-----7•

%

---_*.......... I / 73 149 // ‘, "le:1 1

/I 54 \ / 48 , , ; , \I , 44 / .... ,.;iee 72 , 61 ....),c4.5 °

1 // 1' 63 /31(..'• / 74 1

■it I 4.6' I '''' , /35........ / N.

■ I ,

/ 1 ,...i. I ,

' i • 1 \/ i ‘ '.

1 \ i.,

lae N. \

1

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, ,...511_ 112 / \ 122 // \ 153 / \

i 91 %/ \

I 102 \iu..4 a .... _ i \I _1173/_92 _ \_lx.,4

/42 • I

12 76.....3 .... I/ tat

i

i •/ • / / •

••

1 ‘ //75

/ \

• /

• 53 1 43 ;A‘‘ " // 6CI ‘‘ 445,,....---- /-.‘41t: - - - - - \.W/

-_ •• / 17369-----._,....‘e- ,, i . I

„ , ,cn ,\**15%---9

.1".-.--,• 39 / 1 •,.. I ,47 .....--)t / ■ ill i \ /21 II \ ---...

• / N.1 / ‘ &ig.,_ ....... / ,k, 1 \

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--1 70 1 \ I 90 \ / 100 ‘ \ `"". i\

.3/4. 100/ 59 \ / / 1/- 33 1 \ \ SS\ / 131 1

1. / ' 182\

\ i ____g. 21/21 ‘,9 *-'/ \ 52 1 \

ir- -- ..... 1 ,... / \ 142 ‘#)0...1 -- -- 4'1 - -----/7\-- ‘ g6

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23 ' % 1 9‘ 7 ----- ' Y7 /

1 5 1 # 27 I 21 '',_.'12•.`"........''./ \.."-".."*-'.."`" 4.52-"...'"'"..•• If. \54/ 99 \/28 1

1 /#i1 ,, i/ 1 22 "*.''g

\ 5' 46

i1 32 /I 37 \\ 1 •'# ‘ 69 /58

\ ,/ ‘ 110

-0\, / \ // 89\ / ‘ %

• (3\ /

/ \ 120 i 15,:. 1\ :: 172

/i.... .....

, 1.2 9c / • • / %, / 1 .... _ :fit• /I 1 1 ..Y4 •,Vt" .-.... / 1 ■is .0 ' , N. 1 .....',

.1C‘ 31 / • 51/

\ 68 / •/ / 78 •-. .--"-*`-, It/ / 12 \ ‘..,, i 2g \ i

i 36 ■ i/41

\ / 57 \ // 109 "*;? '''' ■ 130 \ /# I 99

. / \ / \ / ..........92,....... 42 I , 1, I .

' '' % / /40 1''' ,,,-,75/61/ as ,/t7 ; /

g...... 1k16 / 11 • 12_...-2*(2. 1z/

/ .x- t 67 "I' "'..*"`..I\

/,

a. i 119 ,/t-,, /6 rr

• 1 .....-~"/ ,, 25 / \ 30 s '-.. . 1

• :; u I / I ...It

a. 15 )r.I.5 I /,,,, ‘ ,/ 1 „. i •,, I . 1 77■ I /# i ...,, 1 s. • 19 \ „1' i •,1 56 I1,‘ 4, . .■

c , 1,/, ‘ .„.6 ........,;-,y...... „ • \ . 35 1 50 )1" -%1 •*/ I 171

I/ I '''''.....

II

A4'''''o- ; 108 '' ''''ITI4 129 1 t 5a ,"I .1)7,4f ." � ‘ ``' ‘ n2 Jt

e.'"f "^"

."3'

T -1 4 / 1 24----;*1:7 I s I 66 * '.7 1 #■ # I •

'i

i \., A ‘ ■

/9 li Ls, -ra 1 1 ,fr-, I ' 40 1 'Sei I -76-'-, 1 1/ I .-' 7 0 . 139. I 168.

I / %■.., I `...

2.a". 140 163 '■ 1.7e 85 ...

Figure 2 Finite element model used by C.D.C., contains 180 elements,

108 nodes. The subreflector is considered symmetrical, ac-

tuated at nodes no. 52 and 62, and held at no. 57.

• 18

80

.2.." .

• 9.Z......---# /....-IC., 7 \ / % --4e 1 ‘ 165 /e/ " 135 •., tu,...--° 1/ % 105 ,,n- .., I 1• . // •1 • i ..... ,...-"" I \\ 155 12 % 1 \

\p % , I

144 \\\ I

/ 175-_,...........----- ......ifro% / 115 1 104 „..""51‘ c

71,..! ".. 1 •■• t

• 96 1 1 -.4',... l .... I 124 ; .

; /34 \ I• / 95 1 •„_191 1021, / ••• 1''',.. 1 114...# I

. N.\ I 5 ,, /1

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%/ ' / ,8 I e 1 I 103 . i .........-- -/C \

% s I 94 , ■ 1 ..les_-- I ., 154 // \ 16451 VO 1 / 84 I 4, 1 . /; . I \---)6 , I „/'' . I ,,.--"" / ‘ 123 I e, \ ‘ / , ‘ /65 / • I 93 _x(P i /4., ‘ / 174/ 64 / I °"°'•......... 1

4,8 83 i ••••• -i' 113 i \ i 133 ‘

I

‘

• 138 / • ---"--/ 157 % 146 N'...._*9.3 1■126 //

\ 137 / I /66 1

/t .ea'- " ‘

Na■ .e'. . 176/ • / v 74 • 7'3 \ / 1361 .;*".)16,,. 156 .----X---__ ■ 72' 126 . It 145 ■ ; ..---,,,, / \ --314-, 1 .• .1 .. . .wil • 106 / 1 '''•-,,, 1 •• .,,/ \ /

'.\ /80 #

/ %

\ / •170; / • 179

/.

%• // 169 \• . %97 •)?8 • /. 1/ \

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/ • 147 1/ ./1-.....5 ‘ / 158 % / 1 ■..,,,----- 2 4 % # 1 ■/ .---.

gm. fingicifitcal1.111101111111101110111111411P

I <leg I1 =c(i)z•eas red

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/9 no itri- ka I (E/ev. z: 28 c)(RE14413)

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Figure 3 Comparison of the deformed subreflector profile to the best-

fit deviation of the main reflector at elevation angle of 28°

with the radius of the main reflector normalized to 62.3 inches.

Das

hed

line

den

ote

the

anal

ytic

alw

ith

the

tele

scop

e po

siti

ons.

.1

m

easu

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calc

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per

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.153 2-/4 Deformed surface

Ail R3

R2, 4.

Where,' — ' means kigiherof po4k) and 14'

ShOrten ir25 0-f P24h

/er19-70),

-/ 1 94 Thrn

Surface, error = 0, Z4 Thrn rMSZS X asOZThrn4y 44. 04- mri

gd3 rnrn

19)( = -40.2 � /6-6- rad.

19r 29,o /0--7 rad.

Fief

Measuremen4 3454 - FrtNo, paraire4er5

6B2-4 Relaxed sur(ce

Surface error = 0,21 mm,6x = 0,o4 n--)mAy ---- 4/05 n m

777 fnt9x = 271 )(to' raci

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63

APPENDIX D

Engineering Drawings of the Measuring Jig and the Gentry DWG No.

37 D 00002 to 37 D 00007.

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70

APPENDIX E

The equation of the hyperboloid described in eq. (i) is repeat

here:

where r2= x2

+Y2 , and the coordinate system illustrated in figure 1.

The subreflector is small in size and stiff in a sense that

the defocusing type of deformation is considered neglegible. Hence, as

a rigid body motion, the five degree of freedom are:

AX: translation in X direction

AY: translation in Y direction

AZ: translation in Z direction (2)

A(I): rotation about the X axis, where Z=YA(1)

A*: rotation about the Y axis, where Z=XA1P

with small amount of motions, eq. (1 ) can be written as:

in which, without losing the accuracy, the mathematics is simplified by

letting

(7, X) 1 X,4 * -

-4 E)Z

2-(4)

(7)

2q4X 4

substituting (8) to (5)

1-46- +X, IP (9)

so the eq. ( 3 ) became

71

1..t. 76'4 ‘ 9c,sf."(d.t- i,e13- —

/6C"IP,Q• 'Y4, A 9S %4%14') (5)

Call

A = 1 .4c.—

with a general expression that

-2 7A#49c+(6)

combining (6) and (7)

(8)

Call

17= Cc-ta ) I V' cEq. (9) becomes

7 tA,c + 140,:4 (4) —17 (10)

EFI

6"-x/

Equation (10) can be rewritten as

72

1M(‘A *14-‘ 4 4' uRez v

Call

and the deviation from the design [H] is

So that eq. (11) is written as

11x

[F][G] = [H] (12)

Premultiplying the transpose of [F] to both sides of (12):

T

[F][G] = [F (13)

where

fet

and the unknown [G] is solved,

[G] = [ct [F] [11] (14)

The computer program listing, sample input and the result are also included.

t.-%1RAN IV G LEVEL 20

MAI1

DA

TE

= 7

73

46

15

/14

/52

PA

GE

000

1

C BEST FIT FOR HYPERBOLOID WITH 5 DOF

C DEC. 12,1977 A4=-.100, A-2=-.100,A-1=+.100,A-3=+.100

0001

DIMENSION A(144,5),B(144),AA(5,5),B8(5),S(5),C(144)

X,

X(144),Y(144),1(144),STM(72),R(144),R0(6)

X,

RA

DI(

6),

LL

(5),

MM

(5),

LB

F(1

44

),D

EV

(14

4)

X,

AD

J(6

)

CIN

ITIA

TIO

N

0002

DATA RADI / 266.7, 4

95

.3,

72

3.9

, 9

52

.5,

11

81

.1,

14

09

.7 /

0003

DATA AA, BB, S / 25*0., 5*0., 5*0. /

0004

DATA ADJ / -.09, -.04, .07, .01, -.09, -.031

0005

C1=214.185*25.4

0006

C2=24585.8338*25.4

0007

C3=C1/C2

CX(I), Y(I)

0008

K=0

0009

PH=0.

0010

ANG=15.*3.14161180.

0011

DO 60 1=1,24

0012

COSP=COS(PH)

0013

SINP=SIN(PH)

0014

00 50 J=1,6

0015

K=K+1

0016

X(K)=RADIIJI*COSP

0017

Y(K)=RADIIJ)*SINP

0018

50 CONTINUE

0019

PH=PH+ANG

0020

60 CONTINUE

C INPUT : STATEMENT C GAP READINGS

0021

READ(5,900) (STM(I),I=1,72)

0022

PRINT 914,

(STM(I)11=1,72)

0023

READ(5,901) (R0(I),I=1,6)

0024

DO 70 1=1,6

0025

R0(1)=R0(I)+ADJ(I)

0026

70 CONTINUE

C1(1)

0027

K=0

0028

AG=0.

0029

DO 200 1=1,24

0030

DO 100 J=1,6

0031

K=K+1

0032

READ(5,917) R(K)

0033

C(K)=SORTI1.+IXIKI*X(K)+Y(K)*Y(K))/C2)

0034

Z(K)=C1*(C(K)-1.)+R0(J)-R(K)

0035

AG=AG4RO(J)-R(K)

0036

100 CONTINUE

0037

200 CONTINUE

0038

AG=AG/144.

FORTRAN IV G LEVEL 20

MAIN

DATE = 77345

15114152

PAGE 0002

FORTRAN IV G LEVEL 20

MAIN

DATE = (7346

15/1

4/5

2P

AG

E 0

002

0039

PRINT 916,(R(1),I=1,144)

C PRIVT X,Y,Z C AVG. GAP

0040

PRINT 902,(I,X(I),Y(I)INI)11=1,144)

0041

PRINT 903,AG

C BEST FIT BY PARALLEL SHIFT

0042

SUM=0.

0043

DO 250 1=1,144

0044

ZBF(I)=C1*(C(I)-1.)+AG

0045

DEV(1)=11I)-ZBF(I)

0046

SUM=SUM+DEV(I)*DEV(I)

0047

250 CONTINUE

0048

RM

S=

S0R

T(S

UM

/144

.)0049

PRINT 915,(DEV(I),I=1,144)

0050

PRINT 904,RMS

C MAKING OF A(N,5) & B(N)

0051

DO 300 1=1,144

0052

4(I,1)=1.

0053

A(I,2)=-(C3*X(I)/C(1))

0054

A(1,3)=-(C3*Y(I)/C11))

0055

A(I,4)=YII)

0056

4(1,5)=X(I)

0057

B(1)=Z(I)-C11)*Cl+Cl

0058

300 CONTINUE

0059

PRINT 905,t(A(I,J),J=1,5),I=1,144)

0060

PRINT 906,(B(I),I=1,144)

C PRE-MULTIPLY A BY A-TRANSPOSE

0061

0062

0063

0064

AA(L,J)=AA(L,J)+A(I,L)*A(I,J)

0065

400 CONTINUE

0066

PRINT 9070(AA(I,J),J=1,5),I=1,5)

C PRE-MULTIPLY B BY A-TRANSPOSE

0067

DO 500 L=1,5

0058

DO 500 1=1,144

0069

BBIL)= BBIL)+A(I,L)*B(I)

0070

500 CONTINUE

0071

,PRINT 908,(BM1)11=1,5)

C INVERT Of AA

0072

CALL MINV(AA,5,D,LL,MM)

0073

PRINT 909,I(AA(I,J),J=1,5),I=1,5)

C SOLUTION

0074

DO 600 L=1,5

0075

DO

600 1

=1,5

DO40

0L

=1

,5DO

400

J=1,5

DO40

01=

1,1

44