Analytical Investigation of the Dynamics 

of Tethered Chmtellations in Earth Orbit (Phase II) 

Contract NAS8-36606 

Quarterly Report # 9 

For the period 1 April 1987 through 30 June 1987 

Principal Investigator 

Dr. Enrico C. Lomini 

July 1987 

Prepared for 

National Aeronautics and Space Administration 

Marshall Space Flight Center, Alabama 35812 

Smithsonian Institution 

Astrophysical Observatory 

Cambridge, Massachusetts 02138 

The Smithsonian Astrophysical Observatory 

is a member of the 

Harvard-Smithsonian Center for Astrophysics 
Analytical Investigation of the Dynamics 

of Tethered Constellations in Earth Orbit (Phase 11) 

The Smithsonian Astrophysical Observatory 

is a member of the 

- Harvard-Smithsonian Center for Astrophysics 

Contract NAS8-36606 

Quarterly Report # 9 

For the period 1 April 1987 through 30 June 1987 

Principal Investigator 

Dr. Enrico C. Lorenzini 

Co-Investigators 

Mr. David A. Arnold 

Dr. Mario D. Grossi 

Dr. Gordon E. Gullahorn 

July 1987 

Prepared for 

National Aeronautics and Space Administration 

Marshall Space Flight Center, Alabama 35812 

Smithsonian Institution 

Astrophysical Observatory 

Cambridge, Massachusetts 02138 
CONTENTS 

Page 

Summary ....................... 1 

Figure Captions ..................... 2 

SECTION 1.0 INTRODUCTION ........... 3 

2.0 TECHNICAL ACTIVITY DURING REPORTING 

PERIOD AND PROGRAM STATUS ... 3 

2.1 Short Length Crawling Maneuvers Of Space 

Elevator ............. 3 

2.1.1 Introductory Remarks ........ 3 

2.1.2 An Alternative Control Law For Elevator’s 

Crawling Maneuvers ......... 4 

2.1.3 Parametric Analysis Of Short Length Crawling 

Maneuvers ............ 12 

2.1.4 Aerodynamic And Thermal Contributions To 

The Acceleration Levels ....... 27 

2.1.5 Concluding Remarks ......... 36 

i 
CONTENTS (Cont.) 

2.2 

Page 

Tether Applications Simulation Working Group 

Support By G.E. Gullahorn ..... 37 

SECTION 2.2.1 Introduction ............ 37 

2.2.2 Questionnaire Subcommittee ...... 38 

2.2.3 Analytic Solutions Subcommittee .... 39 

2.2.4 Simulator Test Cases ......... 41 

2.2.5 Concluding Remarks ......... 42 

Appendix A: Documents in Support of Tether Applications Simulation 

Working Group .................... 43 

SECTION 3.0 PROBLEMS ENCOUNTERED DURING 

REPORTING PERIOD ......... 51 

4.0 ACTIVITY PLANNED FOR NEXT REPORTING 

PERIOD ............. 51 

ii 
Summary 

In summary the content of this Quarterly Report is as follows: 

A new control law has been developed to control the Elevator during shortdistance-maneuvers. 

This control law (called retared exponential or RE) has been analyzed 

parametrically in order to assess which control parameters provide a good dynamic 

response and a smooth time history of the acceleration on board the Elevator. 

The short-distance-maneuver under investigation consists of a slow crawling of the 

Elevator over a distance of 10 m that represents a typical maneuver for fine 

tuning the acceleration level on board the Elevator. 

The contribution of aerodynamic and thermal perturbations upon acceleration levels has also been evaluated and acceleration levels obtained when such 

perturbations are taken into account have been compared to those obtained by 

neglecting the thermal and aerodynamic forces. 

With regard to the tasks supervised by G.E. Gullahorn, the preparation 

of a tether simulation questionnaire is illustrated. Analytic solutions to be 

compared to numerical cases and simulator test cases are also discussed. 
Page 2 

Figure Captions 

Figures la-lg. The dynamic response obtained by adopting an MHT 

(modified-hyperbolic-tangent) control law is compared to that 

obtained by adopting an RE (retarded exponential) control 

law. Control parameters are as follows: total distance 

traveled by the Elevator A&T = 10 m, time constant l/a = 

400 sec, shape parameter 7 = 4 and, for the RE control law, 

time-delay-parameter At = 1000 sec. The aerodynamic and 

thermal effects are neglected. 

Figures 2a-22. Parametric analysis of the RE control law for the same 

control parameters as in the previous set of figures except for 

the time constant and the time-delay-parameter which are 

varied as follows: a) l/a = 100 sec and AT = 500 sec. 2) 

l/a = 200 sec and At = 500 sec. 3) l/a = 400 sec and 

At = 1000 sec. In all these three simulation runs the 

aerodynamic and thermal effects are neglected. 

IFigures 3a-3n. The Elevator is controlled according to an RE control law 

with control parameters as in the first set of figures. The 

dynamic response when the aerodynamic and thermal effects 

are taken into account are compared to the dynamic response 

when such effects are neglected. 
Page 3 

1.0 INTRODUCTION 

This is the ninth Quarterly Report submitted by SA0 under contract NAS8- 

36606, "Analytical Investigation of the Dynamics of Tethered Constellations in 

Earth Orbit (Phase 11)," Dr. Enrico C. Lorenzini, PI. This report covers the 

period from 1 April 1987 through 30 June 1987. 

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND 

PROGRAM STATUS 

2.1 Short Length Crawling Maneuvers Of Space Elevator 

2.1.1 Introductory Remarks - 

The previous report (Quarterly Report #8) dealt with the dynamics of a 4- 

mass tethered system when the elevator moves along the tether over distances of a 

few kilometers. This quarterly report investigates the dynamics of the same 

tethered system when the elevator moves over distances as short as a few meters. 

These short maneuvers are required for fine tuning of the acceleration level on 

board the elevator. In the case of the short, as opposed to long, maneuvers the 

dynamics of the system (and the accelerations on board the elevator) are 

dominated by the elasticity of the tether along which the elevator is crawling. The 

lengths traveled by the elevator along the tether in order to reach the desired 
Page 4 

distance from the Space Station, for example, are shorter than the elastic 

deformations of the. tether itself. A parametric analysis of suitable control laws to 

perform such short maneuvers has been carried out and the results are presented 

in this quarterly report. 

2.1.2 An Alternative Control Law For Elevator’s Crawling Maneuvers - 

The modified-hyperbolic-tangent control law that was illustrated in the 

previous quarterly report (Quarterly Report #8) proved to be very effective in 

performing crawling maneuvers over long distances. The modified-hyperbolictangent law, however, has one minor drawback: the modulus of the maximum 

acceleration is higher than the modulus of the maximum deceleration. The 

maximum acceleration has a tolerable effect on the acceleration level on board the 

elevator when the traveled distance is long because the motion-induced-acceleration 

on board the elevator is small compared to the overall variation of the gravitygradient-acceleration from the start to the end of the maneuver. In the case of 

short maneuvers the overall variation of the gravity-gradient-acceleration is small 

and the motion-induced-acceleration becomes much more important. In order 

to maintain the same ratio between the above mentioned accelerations, time 

constants equal to those adopted for long length maneuvers should also be adopted 

for the short length maneuvers. This strategy would result in very slow 

maneuvers to cover a distance of only a few meters. We decided, therefore, 
Page 5 

to modify further the control law with the goal of reducing the maximum 

acceleration induced, by the elevator’s motion without having to increase the value 

of the time constant. 

The variation of the traveled tether length At, according to the new control 

law is 

where A&T is the total traveled tether length, a is the rate parameter, 7 is the 

shape factor and At is a delay time. The delay time has the purpose of reducing 

the value of the motion-induced-acceleration at the start of the maneuver to a 

value close to zero. From now on we will call this control law retarded 

exponential (RE), while we will use the acronym MHP for the modified-hyperbolictangent control law. By taking the time derivatives of equation (1) we obtain the 

expression of the crawling velocity 

-(7+1) a(At- t) Aic = dt,~ l+ea(At-t)] e } 

and of the crawling acceleration or motion-induced-acceleration 
Page 6 

It is clear from equation (3) that for At = 0 and t = 0 the acceleration is 

different from zero and is given by 

By increasing At the acceleration at t = 0 becomes smaller and smaller. 

We have run a simulation to compare the dynamic response of a 4-mass 

tethered system, when the elevator crawls according to a RE control law, to the 

case when the elevator crawls according to a comparable MHP law. The elevator 

starts at a distance of 1 km above the Space Station and moves upwards covering a 

distance of 10 m. We have adopted a time constant l/a = 400 sec (the reason 

for this choice will become clear later on in this report) and a shape factor 7 = 4. 

The time delay At for the RE law is equal to 1000 sec in order to provide a 

smooth start of the maneuver. In the MHP control law we have adopted a 

constant-velocity-phase that covers 80% of the maneuver (1 -X = 0.8). The 

crawling maneuver of the elevator starts, in both cases, after a time lag of 1000 

sec that allows for the settling down of the initial transient oscillations. 
Page 7 

Figure la depicts the controlled and the actual tether length of tether 

segment 2 (that connects the Space Station to the Elevator) for the RE and MHP 

control law respectively. The difference between the controlled and the actual 

length is the sum of the elastic stretch and the deformation of the longitudinal 

damper. Since the longitudinal dampers are tuned to the frequencies of the 

associated tether segments at the start of the maneuver, the elastic stretch of 

tether segment 2 is equal to the deformation of longitudinal damper 2 at the start 

of the maneuver. Figure la also points out that the sum of the elastic stretch and 

the damper’s deformation is comparable to the variation of the controlled length 

during the maneuver. Figure lb shows the velocity of tether 2 for the two control 

laws. The maximum speed for the RE control law is greater than for the MHT 

control law, and no constant velocity phase is used for the RE control law. 

The acceleration on board the elevator for the two control laws is shown 

as follows: the front component (perpendicular to the tether in the orbital plane) 

in Figure IC, the side component in Figure Id and the longitudinal component 

(along the tether) in Figure le. The minimum (negative) value of the front 

component for the RE law is slightly greater than for the MHT law. This peak 

acceleration can be reduced by introducing a constant-velocity-phase in the RE 

control law. This modification, however, was not deemed necessary because the 

front and side components are orders of magnitude smaller than the longitudinal 

component of the acceleration and the behavior of the !atter is satisfactory. 

Figure IC also shows that the ringing of the system is much greater for the MHT 
Page 8 

control law than for the RE law. The greater ringing is most probably caused by 

the faster acceleration phase of the MHT law with respect to the RE law. Figure 

Id shows the side component of the acceleration. Comments stated with regard to 

the front acceleration component apply to the side component. 

Figure le depicts the longitudinal component of the acceleration. The 

superiority of the RE control law with respect to the MHT law is evident. By 

adopting the latter control law the acceleration measured on board the elevator 

overshoots soon after the beginning of the maneuver because of the relatively fast 

acceleration phase. There is also a ringing of the system that takes a long time to 

be damped out. These two phenomena disappear when the RE law is adopted. 

The steady-state sinusoidal oscillation of the longitudinal acceleration in Figure le 

(when the Elevator is standing still) is caused by the J2 component of the gravity 

field. 

~ 

Finally, Figures lf and lg show the three components of the acceleration on 

board the Space Station as a consequence of the Elevator’s motion for the RE and 

MHT control laws respectively. The comparatively large oscillation of the 

longitudinal component is caused by unbalanced initial conditions: the Space 

Station is initially at the system CMand the initial conditions have been computed 

by linearizing the gravity potential and therefore by assuming that the initial 

acceleration on board the Space Station is (incorrectly) equal to zero. 
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Page 11 

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Page 12 

2.1.3 Parametric Analysis Of Short Length Crawling Maneuvers - 

As a consequence of the conclusions stated in the previous subsection the RE 

control law has been adopted to perform crawling maneuvers over short distances. 

This subsection analyzes the effect of varying the control parameters upon 

the dynamics of the system and the acceleration levels on board the Elevator and 

the Space Station. In the following simulation runs the distance covered by the 

Elevator is 10 m long. Because of the tethers’ elasticity and the longitudinal 

dampers the controlled tether length (or length crawled by the Elevator along the 

tether) is shorter than the traveled distance. The controlled length that provides 

an actual traveled distance of 10 m is computed beforehand by the computer code. 

In the following set of simulation runs the time constant l/a is respectively 

equal to 100 sec, 200 sec and 400 sec while the shape parameter 7 = 4 for the 

three cases. The delay-time-parameter At = 500 sec for the first and the second 

case (l/a = 100 sec and 200 sec respectively) while At = 1000 sec in the third 

case (l/a = 400 sec) in order to have a small acceleration at the start of the 

maneuver in the three cases. The delay-time-parameter At should not be confused 

with the time lag between the beginning of the simulation and the start of the 

crawling maneuver. Such time lag is equal to 1000 sec for the three cases and it 

allows for the initial transient oscillations to be damped out. 
Page 13 

In the next three simulation runs the aerodynamic and the thermal effects 

have been set equal, to zero in order to speed up the CPU time to perform the 

simulation. The effect of aerodynamic and thermal perturbations is negligible 

when compared to the variations of the acceleration level on board the Elevator 

in the case of long-distance-maneuvers. In the present case of short-distancemaneuvers the aerodynamic and thermal induced accelerations are more important 

than in the previous case. The next subsection addresses this issue. 

Figure 2a depicts the controlled and actual tether lengths of tether 2 for the 

three case mentioned before. The Elevator starts to crawl at the initial time tl = 

1000 sec and it reaches the final distance on the tether at the final time t~ = 2330 

sec, 3160 sec and 5320 sec respectively. The simulation runs, however, are 

continued until t = 10,000 sec for the first two cases and until t = 12,000 sec for 

the third case in order to show the steady-state oscillations of the system. The 

duration of the crawling maneuver for the RE control law tc = tF - tz is given 

bY 

CY 

(5) 

In equation (5) T = Q/A&T where Q is the cut off distance and A& the total 

variation of the controlled tether length. and At = 500, 

500 and 1000 sec equation (5) gives tc = 1330, 2160 and 4320 sec for the three 

If we select T = 
Page 14 

cases respectively. The actual length of tether 2 differs from the controlled length 

because of tether elasticity and longitudinal damper’s stretch. In particular (see 

Figure 2a): a) The initial difference between the controlled and the actual tether 

length is greater than the variation of the controlled tether length during the 

maneuver. b) The fastest maneuver (l/a = 100 sec) excites (longitudinal/lateral) 

oscillations which are not excited in the other two cases. c) The comparatively 

large steady-state oscillation experienced by the actual tether length is caused by 

the 52 gravity term which forces the system to librate and eventually stretch the 

tethers according to the libration frequency. 

Figure 2b shows the rate of change of the controlled length for tether 2. It 

is readily clear from the figure that the maximum speed is directly proportional to 

the rate parameter a. Figure 2b also indicates that the RE control law can be 

easily modified in order to include a constant velocity phase in between the 

acceleration and the deceleration phase. For short maneuvers, however, this 

additional modification has been deemed unnecessary because the maximum tether 

speed is low enough. The controlled speed of tether 3 (not shown here) is like 

that of tether 2, with an opposite sign. The controlled speed of tether 1 (not 

shown here) can be obtained by scaling down the speed of tether 2 by a factor of 

two. It should be reminded that the length of tether 1 (the downward tether) is 

controlled, during the Elevator maneuver, in such a way as to maintain the system 

C! at its initial position. 
Page 15 

The three components of the acceleration on board the Elevator for the 

three cases are depicted as follows: the front component in Figure 2c, the side 

component in Figure 2d and the longitudinal component in Figure 2e. The front 

component is generated primarily by Coriolis forces. It is interesting to notice that 

for l/a = 100 and 200 sec (longitudinal/lateral) oscillations are excited by the 

Elevator’s motion. The same oscillations are not excited in the case l/a = 400 

sec, and the front component of the acceleration has a smooth time history. 

The side component is much smaller than the front component. Since, as pointed 

out in Section 2.3 of Quarterly Report #7, it is not convenient to damp the out-ofplane oscillations, the side component of the acceleration shows an undamped 

behavior. The value of such component, however, is so small that it can be 

neglected. The longitudinal component (Figure 2e) shows a large overshoot for 

l/a! = 100 and 200 sec. Higher frequency oscillations are also excited in those 

two cases. For l/a! = 400 sec the behavior is very smooth and the initial 

overshoot is quite small. The acceleration reaches steady-state conditions as soon 

as the Elevator stops and the higher frequency oscillations are not excited (the 

settling down time is zero). 

The components of the acceleration on board the Space Station are shown in 

Figures 2f, 2g and 2h for l/a = 100, 200 and 400 sec respectively. As previously 

stated the (damped) oscillation of the longitudinal component before the Elevator 

starts moving is caused by the initial conditions at the start of the simulation run. 

The fluctuations of the acceleration once the Elevator starts crawling are more 
Page 16 

relevant to our analysis. In all the three cases such fluctuations are much smaller 

than g (that is.the required limit value for microgravity experiment on board 

the Space Station). Such fluctuations decrease as the time constant of the control 

law increases. For short maneuvers of the Elevator the acceleration levels at the 

Space Station are not a critical issue. 

In order to complete the presentation of data about the dynamic response 

Figure 2i shows the in-plane (8) and out-of-plane (p) libration angles vs. time. 

The libration dynamics of the system is the same for the three cases under 

investigation (within the resolution of the plot) and only one plot is, therefore, 

presented. 

Figures 2j, 2k and 24 depict the in-plane components of the lateral 

deflections Eli (Space Station) and ~2i (Elevator) with respect to the line through 

the end masses for the cases l/a = 100, 200 and 400 sec respectively. The 

increase of the time constant has a dramatic effect on these degrees of freedom 

and such effect is eventually reflected in the acceleration level on board the 

Elevator. Figure 2m shows the out-of-plane components of the deflections €10 and 

€20. Such components are not (appreciably) affected by the variation of the time 

constant. The moduli of the 

lateral deflections €1 and €2 are shown in Figures 2n, 2p and 2q for l/a = 100, 200 

and 400 sec respectively. The moduli reflect the variations, already pointed out, of 

the in-plane components. 

This plot is therefore valid for the three cases. 
Page 17 

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Page 24 

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Page 25 

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Page 26 

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Page 27 

Finally the tether tensions are shown as follows: tether 1 tension in 

Figures 2r, 2s and 2t; tether 3 

tension in Figures 2x, 2y and 22 for the three values of the time constant 

respectively. 

tether 2 tension in Figures 2u, 2v and 2w; 

2.1.4 Aerodynamic And Thermal Contributions To The Acceleration Levels - 

The last case (l/a = 400 sec), examined in the previous subsection, has 

been rerun by adding aerodynamic and thermal effects. The results are compared 

to those obtained without taking into account such effects in the following figures. 

Figure 3a shows the controlled and the actual tether length of tether 

segment 2. The controlled length is the same for both cases whereas the actual 

length experiences an additional stretch that is caused primarily by the thermal 

expansion and secondarily by the aerodynamic forces. The initial temperature of 

the tethers is arbitrarily assumed equal to 290 OK. As the tethers cool to achieve 

the thermal equilibrium the tether lengths increase because of the negative 

expansion coefficient of the tether material. This small variation of tether lengths 

result in a slight increase of acceleration level on board the Elevator. This latter 

effect may be compensated by modifying slightly the distance traveled by the 

Elevator along the upper tether (this distance is also called controlled tether length 

of tether segment 2). The components of the acceleration on board the Elevator 
Page 28 

are compared, for the two cases with/without aerodynamic and thermal effects, in 

the following figures: ' the front component in Figure 3b, the side component in 

Figure 3c and the longitudinal component in Figure 3d. The aerodynamic 

deceleration (since we are plotting the acceleration on board the vehicle, a 

deceleration is plotted as positive) is primarily responsible for the differences in the 

front component of the acceleration between the two cases. The side component 

is almost unaffected by the thermal and aerodynamic forces. The effect of 

thermal perturbations upon the longitudinal coznponent is evident in Figure 3d: at 

each crossing of the terminator a sudden thermal contraction or expansion of the 

tethers takes place and a longitudinal oscillation is initiated. This oscillation 

causes an appreciable variation of the longitudinal component of the acceleration 

measured on board the Elevator. In the case of short-length-maneuvers, therefore, 

the effect of thermal and aerodynamic perturbations is important and can not be 

neglected when evaluating the acceleration levels on board the Elevator. 

Figures 3e and 3f show the components of the acceleration on board the 

Space Station for the two cases without and with thermal and aerodynamic 

perturbations respectively. The effect of the aerodynamic drag upon the front 

component and the effect of thermal perturbations upon the longitudinal 

component of the acceleration is immediately seen by comparing the two figures. 

The following figures show the remaining dynamic quantities for the 

simulation run with thermal and aerodynamic forces: The in-plane (e) and out-of- 
Page 29 

plane (cp) libration angles in Figure 3g. The in-plane components of the lateral 

displacements Eli, ~2j 'in Figure 3h. The out-of-plane components of the lateral 

displacements €10, €20 in Figure 3i and the moduli €1, €2 in Figure 3j. The tether 

tensions in Figures 3k, 3l and 3m for the three tether segments respectively. The 

temperature of tether segment 2 (the temperature of the three tether segments is 

almost the same) in Figure 3n. By comparing Figure 3n with Figures 3d and 3f it 

is readily seen that the thermal shock (at the crossing of the terminator) is the 

cause of the ripples in the longihdinal components of the accelerations on board 

both the Elevator and the Space Station. 

2.1.5 Concluding Remarks - 

The new control law (retarded exponential or RE), developed during this 

reporting period, has proved to be more effective than the MHT (modified 

hyperbolic tangent) control law for the short-distance-maneuvers of the Elevator. 

When the BE control law is adopted the dynamic response of the system and the 

acceleration level on board the Elevator are smoother than for a comparable MHT 

control law. 

The results of a parametric analysis of the RE control law, over distances 

covered by the Elevator of 10 m? indicate that a time constant l/a = 400 sec, 

a time-delay-parameter At = 1000 sec and a shape factor 7 = 4 provide a 
Page 30 

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Page 31 

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Page 32 

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Page 33 

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Page 35 
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Page 37 

smooth dynamic response. 

In the case of short-distance-maneuvers, thermal and aerodynamic perturbations can not be neglected since their effect upon the acceleration levels on board 

the Elevator is comparable to the variation of the gravity gradient acceleration 

during the Elevator’s crawling maneuver. 

2.2 Tether Applications Simulation Working Group Support By G.E. Gullahorn 

2.2.1 Introduction - 

One of the co-investigators (Gullahorn) is working in support of the Tether 

Applications Simulation Working Group. He has attended the two meetings of the 

Group at General Research Corporation in McLean, VA, and is working on a 

variety of tasks. In particular, he has been made chair of two subcommittees, 

one to circulate a questionnaire to ascertain the variety and status of simulation 

programs, and the other to provide a number of analytically precise solutions to 

TSS dynamics problems to which numerical solutions can be compared. He is also 

working in support of a committee to provide suggested test cases for 

intercomparison among the various simulators, and sample results in these cases. 

A further task shall be to define suggested environmental models for inclusion in 

current and future simulators, both simple ones (such as spherical or J2 gravity 

models) for rapid calculation and more complex realistic models. 
Page 38 

2.2.2 Questionnaire Subcommittee - 

On the basis of reaction to a preliminary questionnaire and cover letter 

presented at the 29 May committee meeting, a new questionnaire and letter have 

been designed. The questionnaire is designed (1) to be as specific as possible so 

that answers will be truly comparable, and (2) to ease the burden of filling it out 

by using as many simple "check off" responses as possible. The questionnaire and 

cover letter are shown in Appendix A. 

It is intended that a matrix in which the features of the various simulators 

are displayed will be compiled. A preliminary version has been prepared on the 

basis of response to a previous questionnaire, and is being sent with the 

questionnaire package; the matrix and an abbreviation key are also in the 

Appendix. 

Experience in filling out this matrix indicates that the features displayed 

(rows) and options (abbreviations) will require substantial modification. It will also 

be desirable to devise a scheme to categorize simulators: e.g., general purpose, 

control, etc. 

The questionnaire package is in the process of being mailed to some 350 

organizations on a mailing list provided by General Research. For distribution to 

European groups, a sample package and mailing labels are being sent to Dr. 

Albert0 Loria of PSN/CNR, who has volunteered to handle the European mailing. 
Page 39 

2.2.3 Analytic Solutions Subcommittee - 

The purpose of the "analytic solutions" is to provide precise basemarks 

against which numerical simulations (suitably restricted) can be compared. (We 

put "analytic" in quotes because numerical work must still sometimes be done, 

though the computations are more precise and simpler than in a full simulation.) 

Often it will be the case that when the degree of computational effort 

is increased, as when the number of masses in a bead model is increased, the 

theoretical solution should be approached. E.g., the modal frequencies of a <mass, 

linear tether, fixed or infinite mass> system can be precisely calculated; by 

making the Shuttle mass very large and allowing the simulation to ring in response 

to an initial perturbation, these modal frequencies should be discernable by Fourier 

analysis. In such cases, the rapidity with which the true solution is approached 

with increased effort would make for a measure of program efficiency. 

In other cases, the analytic solution should be approached as the simulation 

parameters become in some sense small: e.g., librations become analytically 

known in the limits of short, rigid tethers and small angular displacements from 

vertical equilibrium. 

A third group of analytic solutions should be precisely simulated unless 

there are program bugs (or other known effects, which should have been 

consciously and judiciously chosen). E.g., a system started in the equatorial plane 
Page 40 

of a planet with cylindrically symmetric gravity field and with no out-of -plane 

velocities should remain in this plane. 

It should be noted that these analytic solutions primarily provide consistency checks in various limits, and cannot by their nature provide validation in 

the cases where simulations are most necessary, with complex behavior and fully 

non-equilibrium and non-linear configurations. These latter situations will require 

inter-comparison of results from several simulators. 

A preliminary outline of analytic solution categories was prepared and 

presented at the 29 May meeting. This outline is included with the documents in 

Appendix A. A list of references to analytic solutions is being prepared, as well as 

simply writing down explicitly the solutions in simple cases (e.g. librations). 

These references will be annotated placed in context in the outline. There was 

discussion at the 29 May meeting as to whether simply providing references was 

sufficient, or whether providing summary detail would also be desirable. It 

appeared to be the sense of the Group that such detail would be desirable, 

complete and explicit enough to allow someone writing a simulator to perform 

validation activities without actually referring to the original reference. Time and 

resources permitting, such summaries will be prepared for selected solutions 

(references); we envision one or two pages each, explicitly setting forth the 

physical problem and system, and restrictions involved (e.g. short tether), and the 

solution. 
Page 41 

2.2.4 Simulator Test Cases - 

In a telecon on 16 June the test case subcommittee made several 

recommendations for the SA0 team (Arnold, chair, and Gullahorn) to follow up. 

(1) The cases should be specified as physical situations, with the simulation 

performer free to simulate them as s/he sees fit. (2) A “data base” of such items 

as c.m. orbit height and inclination and system characteristics should be prepared, 

to be used in all simulations. (3) A uniform output format should be specified to 

allow for as nearly direct comparison as possible; the implementation (whether 

directly in the program or through post processors) is to be left to the simulator. 

(4) The simulations should be chosen so that as many groups as possible can 

perform them. (5) The results should concentrate on the TSS itself, that is on the 

end masses and tether shape rather than, say, the ground track. (6) Test cases 

should be chosen so that each simulation will demonstrate several phenomena. 

Suggested test cases were: (1) Large out-of-plane initial condition, at rest 

with zero tension. This would demonstrate out-of-plane librations, coupling to inplane librations, and tether bobbing. (2) Masses vertical, but tether bowed in a 

combination of in- and out-of-plane initial configuration. (3) An impulse to the 

subsatellite from an initial hanging equilibrium configuration. 

Further discussion and recommendations are in a series of memoranda 

among members of the subcommittee. 
Page 42 

Effort to define such test cases is beginning. The test cases will be 

distributed following an initial telephone canvassing of possible respondents. The 

cover letter to the questionnaire asks respondents if they will be willing to perform 

such test runs. SA0 will perform the test case simulations with SKYHOOK and 

other available programs. 

2.2.5 Concluding Remarks - 

The preparation of the questionnaire and cover letter are complete. 

Distribution is in process. Relevant documents are in Appendix A. Work on the 

accumulation and elucidation of analytic solutions for program validation is 

underway, and results will be reported in the next quarterly. One document is 

included in the Appendix. 
Page 43 

Appendix A: Documents in Support of Tether Applications Simulation Working 

Group 

Center for Astrophysics 

60 Garden Street 

Cambridge, Massachusetts 02138 

Harvard College Observatory 

Smithaonian Astrophysical Observatory 

Mail Stop 59 

8 July 1987 

(617)495-7419 

Dear Colleague: 

Pursuant to the recommendations of the Tether Dynamics Simulation Workshop 

held 16 September 1986 in Arlington, VA, a Tether Applications Simulation Working Croup has been formed by NASA and PSN. 

The Working Croup has undertaken to assemble yet another catalog of tether 

simulation programs. The experience of previous efforts has resulted in an 

updated -- more explicit and detailed -- questionnaire, a copy of which is 

enclosed. 

previous surveys, and apologize for any duplicated effort this may require. 

Copies of previous responses we are aware of are enclosed; please feel free to 

"cut and paste" if appropriate. Note that many items can be completed by simply 

checking off the corresponding response. 

We would appreciate your response even if you have replied to 

Note that we are interested in programs under develovment as well as mature 

simulators; a response is provided to indicate stage of development. 

Please use a seDarate germ for each pronrap for which you respond. We 

suggest you keep a copy of the questionnaire in case you develop new programs or 

substantially update a current program. 

We also encourage you to distribute copies of the questionnaire (and this 

letter) to others you are aware of who have or are developing tether simulators. 

The responses to the questionnaire will be edited and issued as a report 

early next year. 

by filling out the form on the last page of the questionnaire. 

should help keep you abreast of developments in tether simulation. 

Respondees (and other interested parties) may requeste a copy 

This report 

A preliminary report will be presented at the Second International Conference on Tethers in Space in Venice, October 5-8. For inclusion in this Venice 

report please responde within two weeks. 

Also at the Venice meeting we shall be presenting results of various 

If you would be willing to perform such comparison runs (and possibly 

simulators for a set of standardized test cases; we also hope to continue this 

activity. 

present the results in Venice) please contact either myself at the number above 

or Mr. David Arnold at 495-7269. 
Page 2 

In parallel with this effort, the Working Group will be preparing "standard" or ffrecommended" environmental models (atmosphere, gravity, etc.) for use 

in simulation programs. We envision two categories: simple models, for comparison with theory and for economy; and "best possiblevf models. If you have been 

actively involved in selection or implementation of environment models in your 

programs, any input you may have as to criteria used, relative merits, etc., 

would be greatly appreciated. 

Part of the report will consist of a matrix of simulators vs. attributes. 

A sample portion of a preliminary matrix is included, along with a key for its 

interpretation. 

Sincerely yours, 

Cordon Giillahorn 
TETHER SIMULATOR QUESTIONNAIRE 

Tether Applications Simulation Working Group 

May 1987 

- Please fill out a separate questionnaire for each simulator. 

- Many questions have a number of likely responses listed; 

check off choice(s) given or expand as appropriate. 

c - Mail to the address at'the end of questionnaire; if you have any 

questions, write or telephone. 

- Distribution of this form is encouraged. 

PR'OGRAM/SIMULATION NAME: 

INSTITUTE: 

Address : 

Person responding: 

Contact person(s): 

phone 

phone 

SUMMARIZE, BRIEFLY, THE DISTINCTIVE FEATURES OF THIS SIMULATOR: 

(I.e., why did you write this program?) 

IS YOUR PROGRAM: 

BASED ON 

ADAPTED FROM 

AN IMPLEMENTATION OF 

ANOTHER PROGRAM? 

If so, 

Original program name and source: 

Differences in your version: 

Should your version be considered: 

a new program? or 

an added implementation of the original program? 

DEVELOPMENT STAGE : 

Mature. 

Operational, but still under development. 

Under development, programming. Under specification (writing equations, etc.). 

SYSTEM BEING MODELED: 

End Masses: Shuttle If 

point 

attitL 

fixed position 

simple drag aerodynamics 

other (specify) 

mass 

ide dynamicssatellite" 
Tk ther : 

Internal structure 

- none ( "massless" , "rigid" ) 

- finite element, physical - partial differential equ's - finite element 

- finite difference 

- modal synthesis 

Control : Does the program simulate: 

What algorithms are used? 

- reel motion 

- current control 

. 

ENVIRONMENT MODELS: 

atmosphere 

plasma 

magnetic field 

earth gravity 

lunar/solar grav- radiation ~ ~ ~- 

other (specify) 

Properties 

- axial elasticity - axial damping 

- bending stiffness 

- torsional stiffness 

- thermal 

- conducting 

Simple Complex * 

* if readily available, please provide (complex) model names, 

references, descriptions, etc. 

PROGRAM VARIABLES WHOSE ENOLUTION IS COMPUTED: 

OTHER RESTRICTIONS, LIMITATIONS, DEFICIENCIES, PROBLEMS: 

Short tether? 

Small angles (from vertical)? 

Other? 

TYPICAL TETHER APPLICATIONS SIMULATED: 
PROGRAMMING LANGUAGE(S): 

Language : 

Does your code meet a portable standard, e.g. Fortrail 77? 

COMPUTER/OPERATING SYSTEMS ON WHICH IMPLEMENTED: 

At this institute: 

Other known implementations: 

I 

AVAILABILITY: 

Listing? 

Executable? tape, floppy disk (what format?), punch cards 

Source code? tape, floppy disk (what format?), punch cards 

Contract work? perform runs, studies; modify programs 

PRE/POST-PROCESSORS USED : 

(e.g. for generating initial conditions or plots) 

INPUT REQUIRED: (i.e., what is needed to run the program? or, how do you specify the input?) 

OUTPUT / PRESENTATION OF RESULTS: 

Program itself: 

- tabular file for inspection/plotting 

- direct screen display 

- other 

Postprocessor: 

IS THERE ANY INTERACTIVE CAPABILITY? 

For simulation setup and execution? 

To change course of simulation (e.g. control parameters) during 

run? 

NON-STANDARD HARDWARE, SOFTWARE: 

(Specify if used for main program or for post-processors.) Commercial or other software libraries, e.g. IMSL or NAG: 

Graphics terminals: 

Plotters : 
TYPICAL PROGRAM EXECUTION TIME(S): 

E.g., computer time per orbit. If tether is finite mass, use ten 

Other typical situations, ranges: 

masses/elements/modes: 

DOCUMENTATION AVAILABLE: 

Physical model : 

Numerical/computational techniques: 

Program structure: 

Program use: 

Source for documents: 

ANY OTHER COMMENTS: 

________________________________________--------------------------------------- 

Please return to: Gordon Gullahorn Mailstop 59 

Center for Astrophysics 

60 Garden St. 

Cambridge MA 02138 

Phone: (617)495-7419 or FTS 830-7419 

Bitnet: GORDON@CFAZ SPAN: CFA2::GORDON 

Telefax: Call above number to arrange. 

Fill in name and address below to receive a copy of summary report: 

(You need not fill out a questionnaire to receive the report.) 
MODEL: 

Shuttle 

Satellite 

Tether 

Tether Prop 

Control 

ENVIRONMENT : 

atmosphere 

plasma 

magnetic field 

earth gravity 

lun/solar grav 

radiation 

RESTRICTIONS: 

LANGUAGES : 

COMPUTER : 

AVAILABILITY: 

OUTPUT : 

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KEY FOR ABBREVIATIONS IN MATRIX 

MODEL : 

Shuttle Fixed; Point; Attitude; Drag; AErodynamics 

Satellite Fixed; Point; Attitude; Drag; AErodynamics 

Tether Rigid; Lumped mass; FiniteElement; FiniteDifference; Modal 

Tether Prop Elastic; Damped; Bending; Torsion; THermal; ELectrodynamic; Drag 

I 

Control Reel; Current; Thrusters 

atmosphere 

plasma 

magnetic field 

earth gravity 

lunar/solar gravity radiation 

ENVIRONMENT : Simple; Complex 

RESTRICTIONS: 

LANGUAGES : 

COMPUTER : 

AVAILABILITY: 

OUTPUT : 

INTERACTIVE: 

DOCUMENTATION: 

SHort tether; small Angles; nearly STraight tether 

Fortran C PAscal Assembler ( + = portable ) 

VAX; IBM mainframe; PC (ibm); Macintosh 

Listing; Source code; Executable image; Contract work 

Numerical file; Plots; Screen display 

Setup; Control 

physical Model; Numerical methods; Program; User's guide 
Page 51 

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD 

None. 

t 

4.0 ACTIVITY PLANNED FOR NEXT REPORTING PERIOD 

During the next reporting period, as requested by the technical monitor, a 

new control law will be implemented in our computer code. This control law, 

developed by F.R. Swenson of Tri-State University, is called mirror image motion 

control law or MIMCL. The MIMCL will be tested in the dynamics simulation 

computer code in a variety of cases involving long traveled distances (4 km) and 

short traveled distances (2 m and 9.75 m). Results will be compared to those 

obtained previously with the MHT and RE control laws. 

Regarding the tasks supervised by G.E. Gullahorn the activity for the 

next reporting period will be as follows. Effort on definition of the comparison 

test cases is beginning, and progress will be reported in the next quarterly. 

Preparation of a list of suggested environment models will be aided by the 

response to the questionnaire; effort on this task will begin in the next quarter.