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CSDL-R-2088 

CONTROL OF FLEXIBLE STRUCTURES -II (COFS-II) 
FLIGHT CONTROL, STRUCTURE, AND GIMBAL SYSTEM 
INTERACTION STUDY 

by 

Stanley Fay, Stephen Gates, Timothy Henderson, 
Lester Sackett, Kim Kirchwey, Isaac Stoddard, 
Joel Storch 

September 1988 



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The Charles Stark Draper Laboratory, Inc. 



555 Technology Square 
Cambridge, Massachusetts 02139 



R-2088 



CONTROL OF FLEXIBLE STRUCTURES-H (COFS-II) 

FLIGHT CONTROL, STRUCTURE, AND GIMBAL SYSTEM 

INTERACTION STUDY 



By 



Stanley Fay, Stephen Gates, Tim Henderson, Christopher Kirchwey, 
Lester Sackett, Isaac Stoddard, Joel Storch 



September 1988 




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Stanley Fay v \j 

Program Manajer 



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Approved:/^- 1 ^— ^-\JSL£l£4r--- Approved: — — '— — --^ 

Norm Sears Eli Gai 

' Space Programs Control & Decision 

Director Systems, Director 



The Charles Stark Draper Laboratory, Inc. 
Cambridge, Massachusetts 02139 



ACKNOWLEDGEMENT 



The report was prepared by the Charles Stark Draper Laboratory, Inc., for the Langley 
Research Center of the National Aeronautics and Space Administration under contract No. 
NAS9- 17560. The contract monitor at LaRC was Mr. Claude Keckler and the project manager 
at CSDL was Mr. Stanley Fay. 

The authors wish to thank Claude Keckler, Jon I'yle, Victor Cooley and others at LaRC 
for their help and direction in meeting the contract objectives. 



The principal authors of the report were as follows: 



Section 1: 
Section 2: 
Section 3: 
Section 4: 
Section 5: 
Section 6: 
Section 7: 
Section 8: 
Section 9: 
Section 10: 
Section 11: 



All 

Christopher Kirchwey, Lester Sackett 

Lester Sackett 

Tim Henderson 

Christopher Kirchwey 

Steve Gates, Joel Storch 

Isaac Stoddard 

Christopher Kirchwey 

Lester Sackett 

Stanley Fay 

All 



The publication of this report does not imply approval by the sponsor of the content and 
conclusions contained herein. 



Table of Contents 

Section Eag£ 

1 INTRODUCTION 1 

2 THE FLIGHT CONTROL SYSTEM 15 

3 ORBITER POINTING REQUIREMENTS 1 

3.1 The Problem 1 

3.2 Thermal Constraints 1] 

3.3 Communication Constraints 1 

3.4 IMU Alignment 1 

3.5 Orbiter Attitude 12 

3.6 Ventings 12 

3.7 Crew Motion 12 

3.8 Free Drift Time 12 

4 FIXED CONFIGURATION COFS-II MODELS 17 

4.1 Introduction 17 

4.2 Model Description 17 

5 SIMULATOR FOR THE FIXED CONFIGURATIONS 55 

5.1 Introduction 55 

5.2 Simulation Overview 55 

5.2.1 Flight Control System 55 

5.2.2 Dynamics Model 55 

5.2.3 Inputs and Initialization 56 

5.2.4 Output Plotting and Printing 56 

6 VARIABLE CONFIGURATION MODEL 59 

6.1 General System Description 59 

6.2 Mechanical idealization 59 

6.2.1 Antenaa 59 

6.2.2 Gimbal System and Offset Structure 65 

6.2.3 Mast 65 

6.2.4 Orbiter 66 

6.3 DISCOS Model 66 

7 SIMULATOR FOR THE VARIABLE CONFIGURATION MODEL 79 

7.1 Introduction 79 

7.2 Simulation System Description 79 



11 



7.3 System Dynamics and Control Functions 80 

7.3.1 SDAP Inputs: Configuration from Simulated Cockpit 80 

7.3.2 SDAP Inputs: Sensed Attitude 81 

7.3.3 Simulation Inputs: Execution Control 81 

7.3.4 Payload Inputs: Gimbal Control Torques 81 

7.4 Sensor Model: IMU 82 

7.5 Actuator Model: Jets 82 

7.6 Simulation System Checkout 84 

8 RESULTS FOR THE FIXED CONFIGURATION 87 

8.1 Introduction 87 

8.2 Interaction Overview 87 

8.3 Analytic Techniques 87 

8.4 Analytic Results 89 

8.5 Simulation Results 89 

8.5.1 Excitation/Stability Results 92 

8.5.2 Maneuver Results 101 

8.5.3 Attitude Hold Results 108 

9 RESULTS FOR THE VARIABLE CONFIGURATION SYSTEM 117 

10 GIMBAL SERVO/STRUCTURAL DYNAMICS INTERACTION 131 

11 CONCLUSIONS I41 

APPENDICES 

Appendix A: Selection of Spring Constants for the Three Rigid Body Model of the 
COFS-II Hoop Column 

Appendix B: Analysis of Free Vibration Characteristics of the COFS-II Mast 
Appendix C: Orbiter/COFS-II Pitch Plane Dynamics 



in 



Figures page 

1.1 Orbiter/COFS/II flight configuration 2 

2.1 FCS functional block diagram 6 

2.2 Phase plane controller g 

3.1 Example of two-sided limit cycle phase plane trajectory 13 

3.2 Example of one-sided limit cycle phase plane trajectory, small disturbance 14 

4.1 Offset structure and gimbal elements 19 

4.2 Harris hoop/column finite element model 21 

4.3 Harris hoop/column finite element model 22 

4.4 COFS-II finite element model: Configuration #1 23 

4.5 COFS-II finite element model: Configuration #2 24 

4.6 COFS-II finite element model: Configuration #3 25 

4.7 Orbiter mass properties 26 

4.8 COFS-II mass properties: Configuration # I zero gimbal angles 27 

4.9 Orbiter and COFS-II combined mass properties: Configuration #1 28 

4.10 COFS-II mass properties: Configuraiton #2 29 

4.11 Orbiter and COFS-II combined mass properties: Configuration #2 30 

4.12 COFS-II mass properties: Configuration #3 31 

4.13 Orbiter and COFS-II combined mass properties: Configuration #3 32 

4.14 Internal force sign conventions 3g 

4.15 COFS-II Configuration #1: Mode 7. (Part 1 of 5) 39 

4.15 COFS-II Configuration #1: Mode 8. (Part 2 of 5) 40 

4.15 COFS-II Configuration #1: Mode 9. (Part 3 of 5) 41 

4.15 COFS-II Configuration #1: Mode 10. (Part 4 of 5) 42 

4.15 COFS-II Configuration #1: Mode 11. (Part 5 of 5) 43 

4.16 COFS-II Configuration #2: Mode 7. (Part 1 of 5) 44 

4.16 COFS-II Configuration #2: Mode 8. (Part 2 of 5) 45 

4.16 COFS-II Configuration #2: Mode 9. (Part 3 of 5) 46 

4.16 COFS-II Configuration #2: Mode 10. (Part 4 of 5) 47 

4.16 COFS-II Configuration #2: Mode 11. (Part 5 of 5) 48 

4.17 COFS-II Configuration #3: Mode 7. (Part 1 of 5) 49 

4.17 COFS-II Configuration #3: Mode 8. (Part 2 of 5) 50 



XV 



4.17 COFS-II Configuration #3: Mode 9. (Part 3 of 5}.. 51 

4.17 COFS-II Configuration #3: Mode 10. (Part 4 of 5) 52 

4.17 COFS-II Configuration #3: Mode 11. (Part 5 of 5) 53 

6.1 Flight configuration of shuttle/COFS-II system planar view 60 

6.2 Diametrical cross section view of hoop-column artenna 61 

6.3 Finite element model cantilevered mode shapes... 62 

6.4 Three rigid-body antenna idealization, side and top views 63 

6.5 Three rigid-body antenna model cantilevered mode shapes 64 

6.6 Offset structure and gimbal system 65 

6 7 System topology and DISCOS model reference frames for nominal configuration 

67 

6.8 Orbiter geometric and mass properties 68 

6.9 Mast geometric, mass, and material properties 69 

6.10 Offset structure and gimbal base composite body geometric and mass 

71 

properties ' ' 

6.11 Gimbal payload platform geometric and mass properties 72 

6.12 Antenna column geometric and mass properties 73 

6.13 Feed mast and horm geometric and mass properties 74 

6.14 Antenna hoop and mass properties 75 

6.15 Antenna spring and dashpot coefficients 76 

7.1 Reference frames for simulation 83 

7.2 Impulse profile 83 

8.1a Initial rate case, roll phase plane 96 

8.1b Initial rate case, roll firing command 97 

8.1c Initial rate case, generalized coordinate, flexible mode 1 98 

8. Id Initial rate case, generalized coordinate, flexible node 4 99 

8.1e Initial rate case, roll moment at mast base 100 

8.2a Maneuver case, with flexure, roll phase plane 103 

8.2b Maneuver case, with flexure, roll firing commani 104 

8.3a Maneuver case, rigid body, roll phase plane 106 

8.3b Maneuver case, roll firing command 107 

8.4a Attitude hold, configuration 1 three-axis torque ;ase, roll phase plane 110 

8.4b Attitude hold, configuration 1 three-axis torque case, roll firing command Ill 



8.4b Attitude hold, configuration 1 three-axis torque case, roll firing command 1 1 1 

8.5a Attitude hold, configuration 3 roll-axis torque case, roll phase plane 112 

8.5b Attitude hold, configuration 3 roll-axis torque case, roll firing command 113 

9.1 Run 5 - pitch phase plane 121 

9.2 Run 5- rate and disturbance acceleration estimates 122 

9.3 Run 5 - pitch jet torque and mast base load 123 

9.4 Run 13 - roll phase plane 125 

9.5 Run 13 - actual and estimated roll rates 126 

9.6 Run 13 - jet torques 127 

9.7 Run 13 - mast base loads 128 

9.8 Run 17 - pitch phase plane 130 

10.1 Mechanical admittance function at gimbal servo 132 

10.2 High frequency servo loop closures 133 

10.3 Basic quadratic gimbal servo 135 

10.4 Quadratic gimbal servo in canonical form 136 



VI 



Tables EaS£ 

4-1 COFS-II mast instrumentation package mass properties 18 

4-2 Offset structure and gimbal mass properties 20 

4-3 Natural frequencies: orbiter attached COFS-II: Configuration #1 33 

4-4 Natural frequencies: orbiter attached COFS-II: Configuration #2 34 

4-5 Natural frequencies: orbiter frequencies: orbiter attached COFS-II: Configura- 
tion #3 35 

4-6 Node point descriptions 36 

6-1 Nodal masses and torsional moments of inertia 70 

6-2 Vibration characteristics of cantilevered mast 70 

6-3 Antenna spring and dashpot coefficients 76 

7-1 Impulse profile epochs and events 84 

8-1 RHC excitation analysis summary 90 

8-2 RHC excitation analysis summary 91 

8-3 Excitation/stability simulation results summary 93 

8-4 Maneuver simulation results summary 102 

8-5 Attitude hold simulation results summary 109 

9-1 Antenna slew simulation results summary 118 



VI 1 



SECTION 1 
INTRODUCTIO N 

This report documents the work done at CSDL 01 NASA contract NASA9- 17560 for the 
Langley Research Center for the task of COFS-II Flight Experiment Definition Support and 
specifically for Task 2, Dynamic Interaction of COFS- 31 Experiment and Shuttle Orbiter. Task 
1, Computer Requirements Definition, was reported elsewhere. 

The COFS-II flight experiment was expected to be the second Control of Flexible 
Structures (COFS) flight experiment. The first was to have included only the COFS mast plus a 
compact tip mass. The second experiment and the subject of this study includes the COFS-I 
mast and the Langley 15-meter hoop/column antenna attached to the tip of the mast by means 
of an adapter structure and a two degree-of-freedom gimbal. The gimbal to be used is based 
on the Sperry Advanced Gimbal System with 110 degrees deflection plus and minus in elevation 
and lateral angles. The maximum gimbal slew rate is 4 deg/s with 33.9 N-m (25 ft-lb) 
maximum torque. The mast will be mounted in the Space Shuttle Orbiter payload bay. A set of 
proof-mass dampers will be placed on the mast for experimental damping of flexure. The 
dampers may be inactive or may provide approximately the equivalent of 5% structural damping 
in the first several flexure modes. An illustration of the Shuttle/COFS-II configuration is 
shown in Figure 1-1. 

The Shuttle Orbiter Flight Control System (FCS) controls the firings of reaction control 
system (RCS) jets for attitude control and also translational maneuvering. The attitude control 
system may be active with the COFS-II mast and antenna deployed. It was assumed that only 
the low thrust vernier RCS (VRCS) would be used. The VRCS can be used for automatic 
attitude hold and for manual or automatic attitude maneuvering. Because the COFS-II system is 
flexible, there exists a concern about possible dynamic interaction between the flexible structure 
and the flight control system. Probably the Orbiter would be in free drift during experimental 
periods and during antenna slewing. There is also a concern about the loads on the COFS-II 
caused by RCS firings. 

The goals of the dynamic interaction study included the following. 

To determine the Orbiter pointing requirements. This task involves Shuttle operational 
procedures and affects the free drift time that would be allowed for COFS-II experiments. 

To determine the interaction between the FCS and the flexible COFS-II with and without 
active mast dampers. The interaction includes the stability of the FCS given the flexible 
structure, other interactions during attitude holds or maneuvers, loads produced by RCS 
firings on the base of the mast, tip of the mast, and base of the antenna, and the effect 
of adding the mast dampers on stability and dan-ping. 

To study the interaction of the gimbal servos an 1 the flexible structure assuming the FCS 
is inactive. 



HOOP-COLUMN ANTENNA 



MAST 



OFFSET 
STRUCTURE 




Figure 1-1. Orbiter/COFS-II flight configuration. 



The study was limited to 100% mast deployment length. Two model sets were created, 
one with a high fidelity structure in fixed configurations and one with a lower fidelity structure 
with a steerable antenna. Three fixed configurations were assessed. In the nominal configura- 
tion the antenna is facing aft with its column perpendicular to the COFS mast (as depicted in 
Figure 1-1). In the second configuration, the antenna is pointing up from the payload bay. In 
the third configuration, the antenna is pointed 45 degrees from the nominal configuration in 
both elevation and lateral gimbals. The effect of antenna slewing in which the initial orienta- 
tion was one of the three, but the pointing of the antenna changed, was also investigated. 
Slewing in only one axis at a time was considered. Because of the limited capability of the 
gimbal motors, the servo was assumed to be saturated with the maximum 33.9 N-m torque dur- 
ing gimbal slew. The maximum 4 deg/s slew rate can almost never be reached given the gimbal 
angular range and the mass properties of the antenna. 

Locations in the Space Shuttle Orbiter and Orbiter mass properties are commonly given 
with respect to the Fabrication coordinate frame. The origin of the Fabrication coordinate sys- 
tem is in the Orbiter plane of symmetry, 10.16 m (400 in.) below the centerline. Positive sense 
is from the nose toward the tail of the Orbiter. The Z-axis is in the Orbiter plane of symmetry 
perpendicular to the X-axis. Positive sense is upward in the Orbiter landing configuration. 
The Y-axis is out the right wing, completing a rotating, right-handed Cartesian coordinate sys- 
tem. 

FCS quantities are usually given in the Vehicle coordinate frame. The origin of the 
Vehicle body coordinate system, like that of the Fabrication frame, is fixed relative to the 
vehicle. It is located near the tail of the Orbiter with coordinates relative to the Fabrication 
from of (38.1, 0, 10.16) m or (1500, 0, 400) inches. In the Vehicle frame, the X-axis points 
toward the nose, Y is out the right wing, and Z is down. It is a right-handed, rotating Carte- 
sian coordinate frame. 

Other coordinate frames are introduced in this report as needed. 

The following topics are addressed in the remaining sections of this report. For better 
understanding of the sections which follow, the FCS is described in Section 2. Section 3 dis- 
cusses Shuttle Orbiter pointing requirements. Much of the information is taken from NASA 
documentation on Shuttle requirements and capabilities from flight experience, and from 
limited analysis. In order to assess the dynamic interaction of the COFS-II and the FCS, struc- 
tural models of the Orbiter/COFS-II were necessary. The fixed configuration models using 
finite element methods are described in Section 4. The dynamic interaction was investigated 
primarily by simulation. The simulator and FCS software used for investigation of the fixed 
configurations are discussed in Section 5. For antenna slewing studies, a model of the articu- 
lated system was necessary. The development of the articulated system model is described in 
the Section 6. That model was input to the industry-known flexible body dynamics program, 
DISCOS, and combined with an FCS software library, as described in Section 7. Results of the 
extensive simulation studies are then given, first for the fixed configurations in Section 8 and 



then for the articulated system in Section 9. In Section 10, the modeling, analysis, and 
simulation of the gimbal servo with the flexible structure and the results are discussed. Finally 
there are a concluding section and appendices. 



SECTION 2 
THE FLIGHT CONTROL SYSTEM 

The Orbiter Flight Control System controls the firing of RCS jets for attitude and transi- 
tional control. There are thirty-eight 3871.5 N (870 pound) thrust primary jets and six 111.25 
N (25-pound) vernier jets. For this study only the VRCS jets were assumed to be used. 

Figure 2-1 shows the FCS functional block diagram and its relationship to the vehicle 
control loop. The FCS elements included for this study are (from sensor to effectors) an inertial 
measurement unit (IMU), an attitude state estimator, selectable closed-loop manual and auto- 
matic maneuver logic, a phase plane switching controller, vernier jet selection logic, and the 
VRCS jets. 

The IMU is an attitude sensor with gimbal kinematics followed by an analog-to-digital 
converter. There is a hardware plus software transport time lag between an attitude reading and 
the resulting application of force by the VRCS. The state estimator generates body-axis vector 
estimates of Orbiter attitude, angular rate and disturbance angular acceleration from the IMU 
gimbal and angle data, and from jet firing information supplied by the jet selection logic which 
helps compensate for the transport lag. 

The closed loop manual mode generates an angular rate command for each body rotation 
axis (roll, pitch, and yaw) in response to corresponding deflections of the rotational hand con- 
troller (RHC). The command has the value -MR, +MR, or for negative, positive, or zero 
RHC deflection, where MR is the crew-selected maneuver rate. In each axis, the desired 
attitude is obtained by integration of the desired rate, and is reset equal to the current attitude 
whenever the RHC is moved out of or into the zero (center detent) position (which initiates or 
halts a maneuver about that axis). Because of the attitude integration, this logic implements a 
"rate hold" (accurate long-term average rate maintenance) during maneuvers and an attitude hold 
at other times. 

The closed loop automatic maneuver logic issues rate and attitude commands to perform a 
rotation to any target attitude about a single rotation axis (SRA), which ideally is fixed in both 
the inertial reference axes and the Orbiter body axes. The SRA is cyclically recomputed to 
allow for non-ideal response to the commands. The vector magnitude of the rate command is 
equal to the crew-specified maneuver rate (MR). When the vector magnitude (AM) of the dif- 
ference between the current and target attitudes becomes less than the size of the per-axis atti- 
tude deadband being used in the phase plane (see below), the logic switches to the attitude hold 
mode, commanding zero rate and the target attitude. During attitude hold, if disturbances cause 
AM to exceed twice the phase plane deadband, the logic returns to the maneuver mode. 

In the remaining sections of the FCS, attitude and angular rate errors are formed by com- 
paring the desired values with the corresponding estimated values, and the phase plane switcher 
in turn compares the errors with permissible error limits, referred to as a deadband and a rate 
limit. Depending on the outcome of these comparison-; and on the value of the estimated dis- 
turbance acceleration, the phase plane may command a jet firing to reduce errors in one or 
more of the body control axes. If the errors in a particular axis do not warrant a firing, the 
phase plane indicates a "preferred" value of residual acceleration for that axis in case a firing is 



r 






CONTROL LAWS 












" 1 




STEERING 
PROCESSOR 


OIRECT ROTATION 


RCS PROCESSOR 




1 

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VERNIER 

JET 

SELECT 










1 






< 
m 

n 
r— 
m 

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> 

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Vi 




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JET ON/OFF 


VERNIER 

RCS 

JETS 




MANUAL 1 


COMMAND 
ATTITUDE ERROR 










COMMANDS 








PHASE 
PLANE 






1 


ROTATIONi 


ANGULAR RATE 














command! 










AUTO ' 
ROTATIONi 


ERROR 








COMMANDI 


















DISTU 
ANGUI 
ACCEL 
ESTIMi 


R8ANCE 
.AR 

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ES 


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TIMATE 


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STATE 
ESTIMATOR 


EXPECTED A ANGULAR RATE 












IMU 






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ATTITUDE DATA 1 


























>— 
























J 







FCS 



Figure 2-1. FCS functional block diagram. 



commanded in any other axis. See Figure 2-2 for a diagram of the phase plane and the switch- 
ing lines. The jet selection logic then chooses up to thr.« VRCS jets whose acceleration vec- 
tors) provide a reasonable match with the command vector from the phase plane. The VRCS 
jets generate constant steady state forces with uniform buildup and tailoff profiles caused by 
electrical valve open/close delays and jet ignition characteristics. They can be modeled with 
little loss of fidelity as constant forces and torques applied for integer multiples of the 80-ms 
FCS computational cycle, with either time or thrust magnitude adjustments made to the first 
and last 80-ms portions. References 2-1, 2-2, and 2-3 provide a more detailed description of 
the on-orbit FCS. 



HATE ERROR 
AXIS 



FIRE 
♦RATE LIMIT ^ 



COAST* 



IF ALREAOY FIRING 1 
CONTINUE; ELSE COAST 




FIRE 



' OEAOBANO 





IF ALREADY FIRING t 
CONTINUE; ELSE COAST 



COAST* 



-RATE LIMIT 




FIRE 



— - TYPICAL LOCATIONS FOR MOVABLE CUTOFF LINE SI1 



Figure 2-2. Phase plane controller. 



REFERENCES 

2-1 L. L. Sackett and C. B. Kirchwey, "Dynamic Interaction of the Shuttle On-Orbit Flight 
Control System with Deployed Flexible Payloads". AIAA Paper 82-1535, August 1982. 

2-2 Shuttle On-Orbit Flight Control Characterization (Simplified Distal Autopilot), prepared 
by The Charles Stark Draper Laboratory, Inc., NASA JSC-18511, August 1982. 

2-3 Space Shuttle Orbiter Operational Level C_ Functio nal Subsystem Software Requirements, 
Guidance. Navigation, and Control. Part £, Fligh t Control Qrfrit DAP, Rockwell Interna- 
tional, STS 83-0009B, June 30, 1987. 



10 



SECTION 3 
ORBITER POINTING REQUIREMENTS 

3.1 The Problem 

Part of the task was to determine the Shuttle Orbiter pointing requirements when the 
COFS-II experiment is deployed and to estimate the maKimum testing period during which the 
Orbiter may be maintained in a free drift mode. The experimenters would like to have a long 
period in which there are no firings so that the ability of the structure, the gimbal servos, and 
the mass dampers to dampen vibrations can be investigated. 

3.2 Thermal Constraints 

According to Ref. 3-1, there exist limitations on ihe duration of an arbitrary attitude due 
to thermal constraints. Portions of the Shuttle may get either too cold or too hot. In general, 
thermal constraints may limit the Orbiter to 6 hours in in arbitrary attitude. Under many con- 
ditions, the length of free drift time could be longer. 

3.3 Communication Constraints 

The Orbiter uses several S-band antennas for general communication with the earth. 
There are no attitude constraints or pointing requirements for use of the S-band antennas (Ref. 
3-1). The Ku-band antenna is located in the payload bay. It may be used for communication 
with the TDRS. If so, the Orbiter or the deployed payload must not shadow the antenna or 
electrically interfere with it. Normally the Orbiter will not have a requirement for communica- 
tion with the TDRS, although the payload may. Thus there are no general Orbiter communica- 
tion constraints on the length of free drift time. 

3.4 IMU Alignment 

According to Ref. 3-2, IMU alignments occur ev<ry 10-14 hours. In general, an attitude 
maneuver precedes the alignment. 

3.5 Orbiter Attitude 

If it is desired to maintain an inertial attitude, then the FCS must be active. If the 
Orbiter is in free drift it will tend to align itself in a "stable" gravity gradient/aero torque bal- 
ance state (tail generally toward the earth). If the Orbiter is initialized in such a state, it will 
tend to oscillate about an equilibrium due to the varying forcing of the aero forces. According 
to flight experiments reported in Refs. 3-2 and 3-3, the roll attitude will diverge but remain 
within 2 deg of its initial state for about 15 minutes. After 6 hours the oscillations will typi- 
cally have an amplitude of 10 deg but could be as much as 20 deg or more. The pitch and yaw 
attitudes will remain within about 2 deg of nominal. 

If the FCS is used to maintain an attitude when the disturbance torques are negligible, a 
two-sided limit cycle will ensue. Typically, minimum impulse firings will occur. A simple 
analysis which assumes that there is a two sided limit cycle with minimum impulses at either 
attitude error deadband limit in three axes predicts a firing approximately every 10 minutes for 
a deadband of 1 deg. The period of the firings is proportional to the deadband. Thus for a 2 
deg deadband the firings would be about every 20 minutes. However, the firings would be 



PREC EDDSfG PAGE BLANK NOT FILMED 



more frequent if occasionally a firing was longer than one minimum impulse. Also, generally 
there will be some disturbance acceleration due to small gravity gradient torques, the varying 
aero torques, ventings, crew motion, and payload flexure. These may cause more frequent fir- 
ings than would occur for the minimum impulse two-sided limit cycle. See Figure 3-1 for an 
illustration of the attitude error/rate error phase plane during the 2-sided limit cycle. (Ref. 
3-4) 

If it is desired to maintain the Orbiter in an arbitrary attitude, then the gravity gradient 
torques could be large and cause more frequent jet firings to maintain the attitude and rate 
errors within a given rate limit and deadband. For the Orbiter with deployed COFS-II, maxi- 
mum gravity gradient accelerations are about 0.00008, 0.00005, 0.00010 deg/s/s in the roll, 
pitch, and yaw axes, respectively. If the altitude of the orbit is over 150 miles, the aero torques 
will be small compared to the gravity gradient torques. If there is a disturbance acceleration in 
a control axis, then typically a one-sided limit cycle results (see Figure 3-2 for an illustration). 
For example, if the gravity gradient torque is at a maximum in the yaw axis, and if the dead- 
band is 1 deg and the rate error deadband (rate limit) is 0.02 deg/s or greater, then a simple 
analysis predicts that there will be jet firings (equal to several minimum impulses) about every 6 
minutes. The frequency will vary depending on the deadband, rate limit, the exact values of 
the disturbances in each axis, etc. Payload flexibility will tend to cause more frequent firings 
also. Generally, firings every 2-5 minutes may be expected for typical deadbands or rate limits. 

3.6 Ventings 

Ventings are another source of disturbances which can cause jet firings if control is 
active, or can disturb the payload even if the FCS is in free drift. However, according to 
information in Ref. 3-1, at least 6 hours can pass between scheduled ventings, often much 
longer. In any case, the torques are quite small. There are some contingency or failure vent- 
ings which produce large torques, but these need not be considered for nominal operations. 

3.7 Crew Motion 

Crew motion can cause significant disturbances according to Ref. 3-2 and 3-3. These 
disturbances can influence the frequency of jet firings if the FCS is active, or can disturb the 
payload even if the Orbiter is in free drift. During critical experiment periods, it may be 
desirable to minimize crew motion. 

3.8 Free Drift Time 

Based on the factors discussed above, from an Orbiter requirements point of view, the 
Orbiter may be in free drift for at least 6 hours. 



12 




Figure 3-1. Example of two-sided limit cycle 
phase plane trajectory. 



13 




Figure 3-2. Example of one-sided limit cycle 
phase plane trajectory, small disturbance. 



14 



REFERENCES 

3-1 Shuttle Operational Data Book. VoL L NASA JSC. JSC-08934(Vol. 1) Rev. D, October 

1984, with revisions to April 1987. 
3-2 R. Schlundt, et. al., §PJ Space Shuttle Based Experiments for Acquisition, Tracking and 

Pointing: Definition of Space Shuttle Operational Environment. C. S. Draper Lab., CSDL- 

R-1868, 15 April 1986. 
3-3 J. Miller, "Shuttle Pointing Error Reduction", C.S. Draper Lab., Memo no. CSDL-ATP-16, 

18 December 1985. 
3-4 L. L. Sackett and C. B. Kirchwey, "Dynamic Interaction of the Shuttle On-Orbit Flight 

System with Deployed Flexible Payloads", AIAA Paper 82-155, August 1982 (also CSDL- 

P-1581). 



15 



16 



SECTION 4 
FIXED CONFIGURATION C OFS-II MODELS 

4.1 Introduction 

This chapter contains a description of the finite element models used in the fixed config- 
uration COFS-II simulations. In these models, the orbiter is represented as a rigid body with 
the appropriate mass and inertia properties. The mast and antenna are represented with flexible 
finite element models. These three structures were combined to form three different finite 
element models corresponding to the three COFS-II configurations. The three configurations 
are: 

Configuration #1: zero gimbal angles - antenna pointing aft 

Configuration #2: elevation gimbal angle = 90 - antenna pointing up 

Configuration #3 elevation gimbal and lateral gimbal angles = 45 

In all cases the gimbal are assumed to be locked. 

Each model was analyzed using the MSC/NASTRAN finite element program to compute 
the undamped natural frequencies and mode shapes of the system. The modal data was 
expanded to include the internal forces in the mast structure at selected locations. The model 
has been modified to allow recovery of the. total internal load at the top and bottom of the mast 
and at the base of the antenna. Since the mast and antenna base were modeled by equivalent 
beam elements, these loads are the total forces on the itructural sections, not the forces in 
individual members. Included in this section are descriptions of the COFS-II finite element 
models, mass properties of the COFS-II, orbiter, and the combined system and the natural fre- 
quencies and mode shapes of all three configurations. 

4.2 Model Description 

The finite element model of the COFS-II system was assembled using the data in Refer- 
ences 4-1, 4-2, & 4-3 and the configuration described in Section 6. The COFS-II system con- 
sists of five major components: the mast, offset structure, two axis gimbal, the 15 meter 
antenna, and the orbiter. The model of each component will be described in the following 
paragraphs. 

The mast was modeled by 27 equivalent beam elements with additional lumped masses 
added at the sensor/actuator instrumentation package locations. Each beam element represents 
two bays of the deployed truss. The mass and stiffness characteristics of the beams, as defined 
in Reference 4-3, are: 

Mass/Length = 4.641 kg/m 

EA = 124.5 x 10 6 n 

GA =2.11 x 10 6 n 

EI X = 28.63 x 10 6 n-m2 

EI y = 32.29 x 10 6 n-m 2 

GK = 0.40 x 10 6 n-m 2 

fKEUiSDlNU i'AGE BLANK NOT FILMED 

17 



where EI X and EI y refere to bending about the fabrication frame x and y axes, EA is the axial 
stiffness, GA is the transverse shear stiffness, and GK is the portional stiffness. The mass 
properties of the instrumentation packages are given in Table 4-1 by reference to the bay num- 
ber in the mast and the node number in the finite element model. In this table I„ is the mass 
moment of inertia about the axis of the mast. 

Table 4-1. COFS-II Mast Instrumentation package mass properties. 



Bay # 


Node # 


Mass (kg) 


I„ (kg-m2) 


12 


5006 


50.1 


2.8 


24 


5012 


14.4 


2.8 


30 


5015 


50.1 


2.8 


38 


5019 


14.4 


2.8 


44 


5022 


50.1 


2.8 


54 


5027 


147.1 


21.6 



The offset from the mast to the gimbal system, the gimbal system, and the payload plat- 
form were modeled as a series of rigid bodies connected y rigid elements. These elements are 
known in Figure 4-1 and the mass properties of these components are given in Table 4-2. 

The finite element model of the 15 meter Harris-Hoop-Column antenna was provided by 
NASA/Langley. The mesh antenna surface was not included in this model since it does not 
contribute significantly to the response of the antenna in the frequency range of interest and 
would greatly increase the size and complexity of the finite element model. The finite element 
model of the antenna is shown in Figure 4-2, and Figure 4-3. The model of the antenna 
includes the stiffening effects of the pretensioned cables attached to the rim. The orbiter was 
modeled as a rigid body with its mass and inertia properties lumped at node point 4900 located 
at the orbiter center of mass. The location of the base of the COFS-II mast, node 5000, in the 
fabrication frame is: 

x = 22.634 meters 
y = 0.00 meters 
z = 9.007 meters 

Node 4900, the orbiter center of mass, is rigidly attached to node 5000. The mass properties of 
the empty orbiter are given in Figure 4-7. 



18 



5027 =028 



Offset Truss 



50:; 



03 5030 5040 5041 10000 



7 TT~T/ 1 



Gimbal Base 



Glutei S Antenna Base 



Gimbal Hinge 

Payload Platform 



Figure 4-1. Offset structure and gimbal elements. 



19 



Table 4-2. Offset structure and gimbal mass properties. 



Offset Truss: 



Node 5028 



Gimbal Base: 



Mass = 11.788 kg 
1^ = 4.826 kg-m 2 
Iyy = 25.35 kg-m 2 
I„ = 25.35 kg-m 2 

Node 5029 



Gimbal: 



Mass = 90.621 kg 
1^ = 1.436 kg-m 2 
I yy = 5.395 kg-m 2 
I„ = 5.395 kg-m 2 

Node 5040 



Payload Platform: 



Mass = 58.900 kg 
I** = 0.475 kg-m 2 
I yy = 3.087 kg-m 2 
I„ = 3.087 kg-m 2 

Node 5041 



Mass = 113.28 kg 
Ixx = 11.84 kg-m 2 
I yy = 6.072 kg-m 2 
I„ = 6.072 kg-m 2 

MSC/NASTRAN finite elements models of the three COFS-II/orbiter configurations were con- 
structed using the data provided and are shown in Figure 4-4, Figure 4-5, and Figure 4-6. 

The mass properties of the three COFS-II payload configurations are given in Figure 4-8, 
Figure 4-10, and Figure 4-12, and the mass properties of the combined COFS-II/orbiter system 
are given in Figure 4-9, Figure 4-11, and Figure 4-13. The inertias are given with respect to 
the center of mass and the products of inertia are given as positive integrals. The principal 
mass moments of inertia and the transformation matrix from the fabrication frame to the prin- 
cipal axes are also given. 



20 



2o? 



lea 




Figure 4-2. Harris hoop/column finite element model. 



21 



'" .- ,--;-" STY 

OF r^X'* ^"^* 



207 
205 

5BS 

Z03 
2D2 
201 




Figure 4-3. Harris hoop/column finite element model. 



22 



5027 



5022 




5019 



5012 



5006 „ 



5000 



4900 



Figure 4-4. COFS-II finite element model: Configuration #1. 



<: 



23 




<> 



Figure 4-5. COFS-II finite element model: Configuration #2. 



24 




1 



Figure 4-6. COFS-II finite element model: Configuration #3. 



25 



Or biter 



Mass: 

Center of Mass: 

Inertia: 



84831.40 kg 

(28.020, 0.019, 9.211) m 



Principal Inertia: 



Principal Axes: 



1^= 1.2533 x 106 kg-m2 
I yy = 8.9134 x 106 kg-m2 
I„ = 9.4325 x 106 kg-m2 
Pxy = 1.2307 x 10< kg-m2 
P„ = 3.2695 x 105 kg-m2 
P yB - 3.9910 x 103 kg-m2 



I p xx = 8.9134 x 106 kg-m2 
lP yy = 9.4449 x 106 kg-m2 
Ip m = 1.2410 x 106kg-m2 



Rotation Matrix From Fabrication Frame to Principal 
Axes 



R = 



0.0019 -.9999 -.0065 

0.0386 0.0066 -.9992 

J0.9993 0.0017 0.0386 



Figure 4-7. Orbiter mass properties. 



26 



COFS-II 



Mass: 

Center of Mass: 

Inertia: 



Principal Inertia: 



Principal Axes: 



1246.90 kg 

(26.63, 0.00, 58.81) m 

Ixx = 3.8026 x 105 kg-m2 
Iyy = 4.1276 x 10 5 kg-m2 
I„ = 3.9353 x 10* kg-m2 
Pxy = 5.934 x 10-* kg-m2 
Px, = 5.4290 x 10* kg-m2 
P yi = 1.108 x 10- 5 kg-m2 

Ip*^ 3.8870 x 105 kg-m2 
iPyy = 4.1276 x 105 kg-m2 
Ip„ = 3.0915 x 10* kg-m2 



Rotation Matrix From Fabrication Frame to Principal 
Axes 



R = 



0.98814 0.0 0.15356 

0.0 1.0000 0.0 

L- 1.5356 0.0 1.0000 



Figure 4-8. COFS-II mass properties: Configuration #1 
zero gimbal angles. 



27 



ORBITER & COFS-II 



Mass: 

Center of Mass: 

Inertia: 



86078.21 kg 
(28.00, 0.02 9.93) m 



Principal Inertia: 



Principal Axes: 



Ixx = 4.6568 x 106 kg-m2 
I yy = 1.2352 x 107 kg-m2 
I„ = 9.4743 x 106 kg-m2 
Pxy = 1.2339 x 104 kg-m2 
P„ = 2.8679 x 105 kg-m2 
P y , = 2.8143 x 103 kg-m2 



I Pxx = 9.4912 x 106 kg- m 2 
lP yy = 1.2352 x 10? kg-m2 
Ip m = 4.6397 x 10 6 kg-m2 



Rotation Matrix From Fabrication Frame to Principal 
Axes 



R = 



"0.0592 


-.0016 


0.9982 


-.0007 


0.9999 


0.0017 


. -.9982 


-.0008 


0.0592 



Figure 4-9. Orbiter and COFS-II combined mass properties: 
Configuration #1. 



28 



COFS-II 



Mass: 

Center of Mass: 

Inertia: 



1246.90 kg 

(24.29, 0.00, 61.16) m 



Principal Inertia: 



Principal Axes: 



Ixx = 4.6328 x 105 kg-m2 
I yy = 4.6659 x 105 kg-m2 
I„« 1.0173 x 10* kg-m2 
Pxy = 1.025 x 10-3 kg-m2 
P„ = 2.7368 x 104 kg-m2 
P yB = 1,666 x 10-3 kg-m2 



I p xx = 4.6493 x 105 kg-m2 
lP yy = 4.6659 x 105 kg-m2 
Ip„ = 8.5256 x 103 kg-m2 



Rotation Matrix From Fabrication Frame to Principal 
Axes 



R = 



"0.99819 


0.0 


0.06007 • 


0.0 


1.0000 


0.0 


--.06007 


0.0 


0.99819- 



Figure 4-10. COFS-II mass properties: 
Configuration #2. 



29 



Orbiter & COFS-II 



Mass: 

Center of Mass: 

Inertia: 



86078.21 kg 

(27.97, 0.02, 9.96) m 



Principal Inertia: 



Principal Axes: 



Ixx 


= 5.0321 x 106 


kg-m 2 


Iyy 


= 1.2713 x 107 


kg-m2 


I.. 


= 9.4598 x 106 


kg-m 2 


Pxy 


= 1.2396 x 10< 


kg-m2 


P« 


= 1.0630 x 105 


kg-m 2 


Py. 


= 2.7588 x 103 


kg-m 2 



I p xx - 9.4624 x 106 kg-m2 
lP yy = 1.2713 x 107 kg-m 2 
Ip„ = 5.0295 x 10 6 kg-m 2 



Rotation Matrix From Fabrication Frame to Principal 
Axes 



R = 



0.0240 -.0017 0.9997 

-.0007 0.99990 0.0012 

.-.9997 -.0008 0.0240 _ 



Figure 4-11. Orbiter and COFS-II combined mass properties: 
Configuration #2. 



30 



COFS-II 



Mass: 

Center of Mass: 

Inertia: 



Principal Inertia: 



Principal Axes: 



1246.90 kg 

(27.97, 0.02, 9.96) m 

1^ = 4.2663 x 105 kg-m2 
Iyy = 4.2996 x 105 kg-m2 
I„ = 2.9613 x 104 kg-m2 
Pxy = 1.0315 x 104 Rg-m2 
Px, = 4.5676 x 104 kg-m2 
P yt = 2.9357 x 104 kg-m2 



Ip^ = 4.3895 x 105 kg-m2 
lp yy = 4.2511 x 105 kg-m2 
Ip„ = 2.2170 x 104 kg-m2 



Rotation Matrix From Fabrication Frame to Principal 
Axes 



R = 



"0.69828 


0.70673 


0.11373" 


-.71534 


0.69482 


0.00743 


.-.02649 


-.01333 


0.99073. 



Figure 4-12. COFS-II mass properties: 
Configuration #3. 



31 



Orbiter & COFS-II 



Mass: 

Center of Mass: 

Inertia: 



86078.21 kg 
(27.98, 0.04, 9.95) m 



Principal Inertia: 



Principal Axes: 



Ixx = 4.8509 x 106 kg-m2 
Iyy = 1.2519 x 107 kg-m2 
I„ = 9.4735 x 10« kg-m2 
Pxy = 1.7473 x 10* kg-m2 
Px, = 2.0306 x 105 kg-m2 
P y£ = 1.3554 x 105 kg-m2 



iPxx = 4.8420 x 106 kg-m2 
lP yy = 1.2526 x 107 kg-m2 
Ip„ = 9.4764 x 106 kg-m2 



Rotation Matrix From Fabrication Frame to Principal 
Axes 



R = 



-.99903 -.00105 0.04398 
-.00300 0.99902 -.04420 
-.04389 -.04429 -.99805 



Figure 4-13. Orbiter and COFS-II combined mass properties: 
Configuration #3. 



32 



4.3 Analysis Results 

The natural frequencies and mode shapes of the combined COFS-II/orbiter system were 
computed using the MSC/NASTRAN finite element program. The natural frequencies for the 
three configurations are listed in Table 4-3, Table 4-4, and Table 4-5. 

Table 4-3. Natural frequencies: orbiter attached COFS-II: 
Configuration #1. 



Mode # 


Frequency (hz) 


1-6 


0.0 


7 


0.069 


8 


0.080 


9 


0.127 


10 


0.226 


11 


0.240 


12 


0.941 


13 


1.237 


14 


1.394 


15 


1.749 


16 


2.037 


17 


3.382 


18 


3.793 


19 


4.059 


20 


5.048 


21 


5.304 


22 


5.572 


23 


5.921 


24 


5.996 


25 


6.420 


26 


6.832 


27 


6.832 


28 


7.061 


29 


7.061 


30 


7.174 


31 


7.808 



33 



Table 4-4. Natural frequencies: orbiter attached COFS-II: 
Configuration #2. 



Mode # 


Frequency (hz) 


1-6 


0.0 


7 


0.072 


8 


0.117 


9 


0.117 


10 


0.162 


11 


0.314 


12 


0.427 


13 


1.225 


14 


1.380 


15 


1.787 


16 


1.789 


17 


3.137 


18 


3.792 


19 


4.071 


20 


4.872 


21 


5.304 


22 


5.572 


23 


5.749 


24 


5.752 


25 


6.438 


26 


6.832 


27 


6.832 


28 


7.017 


29 


7.061 


30 


7.061 


31 


7.808 



34 



Table 4-5. Natural frequencies: orbiter attached COFS-II: 
Configuration #3. 



Mode # 


Frequency (hz) 


1-6 


0.0 


7 


0.070 


8 


0.086 


9 


0.136 


10 


0.215 


11 


0.268 


12 


0.862 


13 


1.247 


14 


1.391 


15 


1.760 


16 


1.932 


17 


3.359 


18 


3.793 


19 


4.062 


20 


5.021 


21 


5.302 


'22 


5.571 


23 


5.839 


24 


5.895 


25 


6.430 


26 


6.832 


27 


6.832 


28 


7.061 


29 


7.061 


30 


7.099 


31 


7.808 



35 



Plots of the first five flexible modes for each configuration are given in Figure 4-15, 
Figure 4-16 and Figure 4-17. Only the lowest twenty modes are valid because of possible 
interaction of the higher modes with the flexible modes of the orbiter. The node points are 
described in Table 4-6. 

Table 4-6. Node point descriptions. 



Node # 


Description 


4900 


Orbiter Center of Mass 


1-24 


Hoop 


25-48 


Top Stay Attachment Ring 


49-72 


Bottom Stay Attachment Ring 


73-132 


Column 


133 


Bottom of Column 


134 


Top of Column 


200 


Bottom of Feed Mast 


201-208 


Feed Mast 


209 


Top of Feed Mast 


210-212 


Feed Horn 


401-425 


Not Used 


673-675 


Extra Mast Points 


1000-1002 


Not used 


5000 


Bottom of Mast 


5001-5026 


Mast 


5027 


Top of Mast 


5028 


Rigid Offset Center of Mass 


5029 


Gimbal Base Center of Mass 


5030 


Gimbal 


5040 


Upper Gimbal Center of Mass 


5041 


Gimbal Platform Center of Mass 


10000 


Platform/Antenna Interface 


25000 


Forces on Base of mast (node 5000) 


25027 


Forces o Top of Mast (node 5027) 


30000 


Forces on Base of Antenna (node 10000) 



The first 213 points represent structural nodes and the last 3 points are additional non- 
structural points which are used to include the nodal forces at the lower and upper ends of the 
mast and at the base of the antenna. All grid points have six degrees of freedom. For the 
structural node points they correspond to three translations (Ax, Ay,Az), and three rotations 
( Gx, 6y, 0z) in the orbiter fabrication frame. In the case of the extra force node points, the 
six degrees of freedom correspond to the internal forces in the following manner: 



36 



Degree of Freedom #1 === F x Force in Local > -Direction 
Degree of Freedom #2 === F y Force in Local Y Direction 
Degree of Freedom #3 === F, Force in Local Z- Direction 
Degree of Freedom #4 === M x Moment About Local X-Axis 
Degree of Freedom #5 === 
Degree of Freedom #6 === 



M y Moment About Local Y-Axis 
M E Moment About Local Z-Axis 



The local axes are parallel to the fabrication frame axes and are centered at the three 
node points as is shown in Figure 4-14. 



37 




1 

Sooo 










\ocoo 



Figure 4-14. Internal force sign conventions. 



<: 



38 



MODE 7 

FREQ = 0.053 HZ 




Figure 4-15. COFS-II Configuration #1: Mode 7. (Part 1 of 5) 



39 



OQ 

B 

n 
i 

VI 



o 



O 

O 

?! 
(A 
i 



o 

o 

B 



O 
B 



3 
o 
a 
<» 

00 




K> 

O 

VI 



mo 
urn 



n 



m 



CD 



CD 
CO 

o 






TO 

e 



i 



<ji 



o 



o 

o 



on 

c 



o 

B 



2 

o 










o 



"n 3 

»o 
mo 
om 

ii 



Irt 



M 






M 



•*1 



4k 

I 



en 






n 
o 

CO 
I 



o 

o 
a 

e 
■I 
» 

o 

D 




2 

o 
o. 
« 






At 

o 



"nil 
» o 
mo 
om 



(A 



II 



M 



O 
cn 






MODE 11 

FREQ = 0.240 HZ 




Figure 4-15. COFS-II Configuration #1: Mode 11. (Part 5 of 5) 



43 



MODE 7 

FRE3 = 0-072 HZ 




Figure 4-16. COFS-II Configuration #2: Mode 7. (Part 1 of 5) 



44 



MODE 8 
FRE3 = C 



1 17 HZ 




CvW 



Figure 4-16. CGFS-II Configuration #2: Mode 8. (Part 2 of 5) 



45 



MOOE S 

FREQ = 0.162 HZ 




Figure 4-16. COFS-II Configuration #2: Mode 9. (Part 3 of 5) 



46 



MODE 10 

FREQ = 0.305 Hi 




Figure 4-16. COFS-II Configuration # I: Mode 10. (Part 4 of 5) 



47 



MODE 11 

FREQ = 0-314 HZ 




Figure 4-16. COFS-II Configuration #2. Mode 11. (Part 5 of 5) 



48 



rODE 
FREQ = 



7 



0-070 HZ 




Figure 4-17. COFS-II Configuration #3: Mode 7. (Part 1 of 5) 



49 



MODE 8 

FRE3 = C..085 Hi 




Figure 4-17. COFS-II Configuration #3: Mode 8. (Part 2 of 5) 



50 






to 
en 



en 
ii 

uio 

OUJ 



rvj 



o 










o 

s 



B 
O 



3 

.2? 

B 
O 

U 



i 

O 



I 






ere 

E 



I 



K3 



n 
o 

I 



o 

o 

B 
Si 
TO* 

B 



O 

a 
* 




2 

o 
a. 






o 



"H3 

mo 
turn 

ii 



fvl 



o 
en 

M 



MODE 11 

FRE3 = 0.253 HZ 




Figure 4-17. COFS-II Configuration #3: Mode 11. (Part 5 of 5) 



53 



REFERENCES 

4-1 Pyle, J.S., Montgomery, R., "COFS-II 3-D Dynamics and Controls Technology", NASA 
CP-2447, pp. 327-345, Presented at First NASA/DOD CSI Technology Conference, Nov. 
18-21, 1986, Norfolk, Virginia. 

4-2 Belvin, WK., Edighoffer, H.H., "15-Meter Hoop-Column Antenna Dynamics: Test and 

Analysis", NASA CP-2447, pp. 167-185, Presented at First NASA/DOD CSDL Technology 
Conference, Nov. 18-21, 1986, Norfolk, Virginia. 

4-3 Lenzi, D.C., Shipley, J.W., "Mast Flight System Beam Structure and Beam Structure Per- 
formance", NASA CO-2447, pp. 265-279, Presented at First NASA/DOD CSI Technology 
Conference, Nov. 18-21, 1986, Norfolk, Virginia. 



54 



SECTION 5 
SIMULATOR FOR THE FIXED CONFIGURATIONS 

5.1 Introduction 

Part of the study of flight control system (FCS) interaction with the Orbiter/COFS-II sys- 
tem was carried out using the three fixed configuration finite element models described m Sec- 
tion 4, in conjunction with the orbital control functional simulator (OCFS), a high fidelity 
engineering simulation which includes attitude and structural dynamics and a model of the FCS. 
The OCFS accepts structural models with up to 50 flexible modes included. There is extensive, 
easily modified output capability. The OCFS or its precursors have been used in dynamic inter- 
action studies for such systems as the Waves in Space Plasma experiment (WISP Ref. 5-1. a 
long, Orbiter-attached dipole antenna), and the Stabilized Payload Deployment System (SPDS, 
Ref. 5-2). The remainder of this section describes the OCFS in more detail. 

5.2 Simulation Overview 

The simulation consists of essential elements of the Shuttle on-orbit FCS used for attitude 
control coupled to a dynamics model. An input interface enables specification of the test con- 
ditions, and outputs consist of time plots of key variables and printouts of initial and terminal 
conditions. 
5.2.1 Flight Control System 

The FCS elements simulated are described in Sect.on 2. The IMU is modeled as an error- 
free attitude sensor. A higher fidelity modeling option, which was used in the maneuver and 
attitude hold simulations reported in Section 8, includes IMU gimbal kinematics, 
analog-to-digital converter quantization, and FCS software to convert the quantized gimbal 
angles to Orbiter body axis attitude data. The digital autopilot is constructed per the Section 2 
description The VRCS jets are modeled as constant forces and torques applied for integer 
multiples of the 80 ms FCS computational cycle, with added time lags representing the thrust 
buildup/tailoff profiles. The overall lag between reading of the IMU and application of jet 
forces due to the resulting commands can be adjusted to equal the actual total (hardware plus 
software) lag. 
5.2.2 Dynamics Model 

The dynamics model receives jet forces and torques from the FCS, and separately com- 
putes the rigid-body and bending responses of the Orbiter/COFS-II system. The Orbiter atti- 
tude change due to bending is added to the rigid-body attitude to obtain total Orbiter attitude 
which is fed back to the IMU model in the FCS for attitude estimation and closed-loop control 
if desired. 

The rigid-body attitude equations include effect: of jet torques, nonlinear Euler rotation 
coupling and a user-specified constant external torque. Total system moments and products of 
inertia are necessary inputs to these computations. 



55 



The bending equations are driven by jet forces and torques only. Thus static centrifugal 
deflection, differential Euler coupling and differential gravity gradient torques are among the 
inertml effects that are neglected. In the rigid Orbiter/flexible appendage formulation, equiva- 
lent forces and torques at the Orbiter center of mass are first computed. The resulting 6-D 
vector is shaped by multiplication by a constant influence coefficient matrix to yield a vector of 
forcing functions, one per flexible mode. For each mode, application of the forcing function 
and integration of the bending equations yields a modal displacement, or generalized coordinate. 
Various constant matrices multiply the vector of generalized coordinates to provide physi- 
cal displacements of the Orbiter and of various nodes of the COFS-II structure and loads at 
selected points. The constant matrices are the output of the finite element modeling process 
described in Section 4. These matrices, together with the modal frequencies and damping ratios 
comprise the flexibility data input to the simulation. 
5.2.3 Inputs and Initialization 

In setting up the simulation for a particular run, the user specifies the initial body-axis 
angular rate of the Orbiter/COFS-II system and the external disturbance torque. The state esti 
mator can optionally be initialized such that its outputs and internal variables "agree" with these 
disturbances. For special studies requiring initial modal excitation, the first derivative of 
selected generalized coordinates can be specified. 

Other inputs specify run conditions and FCS mode and control parameters The 
simulated-time duration, system configuration, node point indexes, and inclusion or exclusion of 
flex modeling comprise a typical run-condition specification. The FCS mode is selected from 
auto, manual or open-loop. FCS parameters include the maneuver rate, phase plane deadband 
and rate limit, expected available per-axis control acceleration, expected VRCS jet accelerations 
and maneuver commands. The user may input the mass property-dependent FCS parameters ' 
(i.e., expected per-axis and jet accelerations) directly, or may request that "ideal" values be 
computed from total system mass properties and actual jet forces, locations and autopilot- 
generated jet commands. Angular rate commands (in manual mode) or a new target attitude (in 
auto mode) can be input at any time in the run. 

5.2.4 Output plotting and Printing 

The plotted outputs indicate behavior of the Orbiter/COFS-II system and of the FCS 

nThTn T f nT Ce indiCat ° rS " thC Pl ° tted d3ta 3re thC angU,ar acce, ^tion, rate and attitude 
of both the Orbiter and the composite system, and deflections and loads at selected points of the 
COFS-II structure. It should be noted that the deflections given in Section 8 are always with 
respect to the composite body, not the Orbiter. 

Indications of FCS behavior are provided by plots of attitude error, rate error, estimated 
disturbance acceleration, phase plane output commands, individual VRCS jet activity and 
cumulative fuel consumption. The first six generalized coordinates are also plotted to provide 
insight into which modes are contributing significantly to loads and deflections 



56 



The printed output consists of input echoes (for verification of successful read- in of 
desired conditions and flex data), derived initial conditions, and terminal conditions. The 
derived initial conditions are mass property-dependent FCS parameters and disturbance depen- 
dent state estimator outputs. Useful terminal condition data are inertial attitude (all simulation 
runs start with the composite body axes aligned with the inertial reference axes) and VRCS 
usage statistics (per-jet and total firings and fuel consumption). 



57 



REFERENCES 

5-1 Kirchwey, C, Sackett, L., and Satter, C, "Wisp Antenna Dynamics and Orbiter Control 
System Interaction", Charles Stark Draper Report CSDL-R-1763, March, 1985. 

5-2 Sackett, L., and Kirchwey, K., "SPDS Dynamic Interaction Study", Charles Stark Draper 
Laboratory Memorandum DI 87-5, April 9, 1987. 



58 



SECTION 6 
VARIABLE CONFIGURATION MODEL 

This section describes the dynamics model employed to simulate the motion of the 
COFS-II system during large angle maneuvers of the antenna relative to the Shuttle Orbiter. 
This model when combined with the algorithmic descriptions of the gimbal servo-motors and 
the Shuttle attitude control system formed the integrated simulation used for studies involving 
Shuttle-antenna reconfiguration. 

To simulate the COFS-II system undergoing these maneuvers, an articulated multibody 
dynamics model was used. The system was modelled as an assembly of rigid and flexible bodies 
with carefully defined interconnections. The general purpose multibody dynamics and control 
simulation program, DlSCOSt 6 " 1 ! was used to numerically synthesize and integrate the equations 
of motion and provide the interbody forces and torques. 

A detailed description of the idealized mechanical model and the values of the parameters 
implemented in DISCOS follow. Further information on the integrated simulation is provided in 
Section 7. Results from simulation case studies are presented in Sections 9 and 10. 

6.1 General System Description 

Figure 6-1 illustrates a planar view of the COFS-II system in the reference configuration. 
Cantilevered to the Orbiter cargo bay is a large deployable truss structure, considered to be 
identical to the COFS-I mast described in Reference 6-2. An offset structure, having the same 
truss design as the mast, is fixed to the mast tip. A two-axis gimbal system controls antenna 
pointing in elevation and lateral degrees of freedom. Mounted to the offset structure, this 
system is based on the Sperry Advanced Gimbal System, described in Reference 6-3. The 
NASA Langley/Harris 15m Hoop Column antenna is attached to the gimbal system payload 
platform. This large lightweight axisymmetric structure is described in References 6-4 and 6-5. 
A finite element model of the antenna was provided to CSDL by NASA/LaRC. 

6.2 Mechanical Idealization 

6.2.1 Antenna 

Figure 6-2 shows a cross-sectional view of the Hoop Column antenna, indicating various 
elements of the structure. The column is considered to be cantilevered to the base which 
represents its mounting to the gimbal system payload platform. 

According to Reference 6-4, 97% of the antenna mass is contained in its three major 
components: the hoop (33%), the column (34%), and the feed mast and horn (30%). These 
components are each complex structures with intricate interconnections. After examination of 
the free vibration characteristics of the antenna obtained from the LaRC finite element model, 
the first five mode shapes of which are shown in Figure 6-3, we chose to idealize the antenna 
as the simple rigid body assembly shown in Figure 6-4. The three primary components are 
portrayed as separate rigid bodies interconnected by discrete massless torsional springs and 



59 



HOOP-COLUMN ANTENNA 



MAST 



OFFSET 
STRUCTURE 




Figure 6-1. Flight configuration of shuttle/COFS II system planar view. 



60 



FEED MAST & HORN 



UPPER HOOP 
SUPPORT CABLES 



SURFACE 
TENSIONING CORDS 




HOOP EDGE 



BASE 



Figure 6-2. Diametrical cross section view of hoop — column antenna. 

dashpots. While this representation ignores the inertial effects of the surface mesh, tensioning 
cords, and the hoop support cables, it does capture their essential stiffening influence. The 
dashpots, arranged in parallel with the torsional springs, model the intrinsic damping of the 
structure. 

For the idealized antenna of Figure 6-4, the column is connected to the base through a 
hinge which permits rotation about two mutually orthogonal axes oriented perpendicular to the 
column's nominal longitudinal axis. These two rotational degrees of freedom are resisted by 
identical pairs of springs and dashpots. The hoop is constrained to lie in a plane perpendicular 
to the column's longitudinal axis at a fixed distance above the base. During deformation the 
hoop follows the column such that the only relative displacement between them is a simple 
rotation of the hoop in its plane about the column axis. This relative angular motion is resisted 
by spring and dashpot pair. The feed mast and horn combination is connected to the column 
top through a hinge. The hinge permits rotation of the feed body relative to the column about 
two mutually orthogonal axes perpendicular to the column's longitudinal axis. These two 
degrees of freedom are resisted by identical pairs of springs and dashpots. 

The idealized antenna model has five degrees of freedom. These degrees of freedom are 
discrete representations of selected structural deflections, and as such are meaningful only when 
they are small in an engineering sense. 



61 




M0DE1 

HOOP TORSION 

o> 1 = 0.08 Hz 




MODES 2 & 3 

1st PLANAR BENDING 

u>2 = "3 = 0.24 Hz 




MODES 4 & 5 

2nd PLANAR BENDING 

o> 4 = w 5 = 1.74 Hz 



Figure 6-3. Finite element model cantilevered mode shapes. 



62 



COLUMN 



TORSIONAL 
SPRING & 
DASHPOT 



,/FEED MAST & HORN 




TORSIONAL SPRING 

& DASHPOT HOOP 



/" \ 



BASE 




Figure 6-4. Three rigid-body antenna idealization, side and top views. 



The method used to select numerical values for the antenna s spring constants is presented in Appendix 
A. Those values appear together with all other parameter v; dues later in this section. 

The undamped mode shapes and natural frequencies for the three-rigid-body model of the 
antenna are portrayed in Figure 6-5. Those characteristics show good agreement with the corresponding 
modes and frequencies of the LaRC finite element model (shown in Figure 6-3). 

For each of the parallel spring-dashpot pairs, the dashpot coefficients were chosen to be directly 
proportional to the corresponding spring constants. This simple approach permitted the introduction of 
damping into each of the vibration modes. For the dashpot coefficients selected, (see Table 6-3) the 
modal damping was: Ci = C 2 = C 3 = 0.005, < 4 - C 5 - 0.036 (where the 1 th modal coordinate, t, v 
for damped free vibration satisfies: *• + 2^ a> { n { + co? ?i = 0, with ^ being the modal fre- 
quency). 



63 



HOOP TORSION 
u 1 = 0.08 Hz 




1st PLANAR BENDING 
u>~ = cj- = 0.24 Hz 




2nd PLANAR BENDING 
w 4 = w 5 = 1i74 ^ z 



Figure 6-5. Three rigid-body antenna model cantilevered mode shapes. 



64 



6.2.2 Gimbal System and Offset Structure 

The gimbal system is considered to consist of three primary sections: the base, joint 
assembly, and payload platform (Figure 6-6). The payload platform is represented as a distinct 
rigid body. The two axis gimbal assembly is idealized as; a two degree of freedom pivot point 
joining the payload platform to the gimbal base. The respective gimbals are capable of large 
angular displacements in response to motor torques, or, their motions can be specified by 
rheonomic constraints. The gimbal base and offset structure are considered to be a single 
composite rigid body, rigidly attached to the mast tip. 



TRUSS OFFSET STRUCTURE 



ELEVATION AND 
AZIMUTH GIMBALS 



GIMBAL 

SYSTEM 

BASE 





PAYLOAD 
PLATFORM 



Figure 6-6. Offset structure and gimbal system. 



6.2.3 Mast 



The mast structure with the sensor and actuator instrumentation packages is idealized as a 
uniform cantilevered beam carrying a set of compact rigid bodies fixed along its length. The 
beam is considered to be inextensible and is permitted small transverse bending deflections in 
two orthogonal planes as well as torsional deflections about its longitudinal axis. Appendix B 
presents a free vibration analysis of this structure. 

In the DISCOS program the beam with point bodies was modelled as a single flexible 
body described by its first five mode shapes and natural frequencies. 1 These mode shapes, 
which were generated from a lumped mass finite element model, included the first and second 
bending modes for each of the two orthogonal planes and one torsion mode. 



1 While it is recognized that there are inaccuracies associated with such portrayals under certain 
circumstances I 6 " 6 ), the conditions for the studies reported here reduce their impact. 



65 



To portray both the intrinsic damping of the structure, as well as the enhanced damping 
provided by the action of the mast damping control system, simple modal damping was 
employed. The natural damping was assumed to provide a uniform modal damping factor for 
each mode of £ = 0.005. The modal damping factor was assumed to increase for each mode to 
£*> 0.05 when the vibration control system was active. 

6.2.4 Obiter 

The Shuttle Orbiter is treated as a rigid body and is provided six unrestricted degrees of 
freedom. External forces and torques act on the Orbiter as a result of the action of the 
vehicle's attitude control thrusters. A more detailed description of the flight control system 
appears in Sections 2 and 7. 

6.3 DISCOS Model 

An exploded-view of the idealized COFS-II system appears in Figure 6-7, with the 
DISCOS model hinges and references frames identified. 

Many other possible choices exist for idealizing the system, each of which entails 
trade-offs between different aspects of the simulation. As an example, the Orbiter, mast, and 
offset structure-gimbal base could have been treated as a single composite flexible body. While 
this model would produce a faster simulation, the DISCOS program would no longer compute 
the forces and torques acting between the Orbiter and mast and between the mast and offset, 
which were desired quantities. Alternate idealizations such as this, do however, provide a 
means for corroborating implementations of complicated models. 

The DISCOS model was subjected to a hierarchy of validation tests, beginning with simple 
situations for which the correct response was known a priori, and progressing through 
comparisons between distinct simulations for increasingly complicated conditions. The 
simulation comparisons were made between the seven-body DISCOS model and: 

1) Finite element based models, for fixed configuration cases. These models and associated 
simulations are described elsewhere in this report. Good agreement was obtained for 
attitude motions and structural loads and deflections. 

2) A two-body DISCOS model, for variable configuration cases. This model treated the 
Orbiter + mast + offset structure as one flexible body and the payload platform + antenna 
as another. Excellent agreement was obtained for attitude motions and antenna pointing 
motions. 

This chapter concludes with the specification of the geometric, mass, and stiffness 
parameters implemented in the DISCOS model. The data is given in Figures 6-8 through 6-15, 
and Tables 6-1 through 6-3. The information was derived primarily from References 6-2 
through 6-5, and the antenna finite element model provided by NASA LaRC. Note that in the 
following, the respective body fixed frames are parallel, in the reference configuration, to the 
Orbiter fabrication frame. 



66 



Hinge 4 



Hinge 5 



Hinge 3 



Body 3 Rel. Frame Hinge 4/ 

Hinge 3/Body 3 Frame Body 3 Frame 




Body 5 Hinge 6 

Ret. Frame 

I Hinge 7/ 

J Body 5 Frame " ,n ? e c 6/ 
f ', Body 5 Frame 

4r£ =t 



Hinge 7 




Body 7 Rel. Frame 
Hinge 7/Body 7 Frame 



Hinge 6/ 
z Body 6 Frame 



Body 6 Ref. Frame 



oo 

m » 

o 2: 

o > 

> o 

I- in 

H _ 
-< U% 



Hinge 2 



Figure 6-7. System topology and DISCOS model reference frames for nominal configuration. 



ORBITER 



Mast Attachment 
Point 




Body Fixed 
Frame — Body 1 




Mass = 84831.4 kg 



Inertia matrix with respect to the orbiter mass center and the body 
fixed frame axes: 



Hi] = 



1253282.0 -12307.0 -326951.0 
8913427.0 -3991.0 
sym 9432545.0 



kg — m' 



Position of mast attachment point relative to the orbiter body fixed 
reference frame: 

(-5.386, 0.0, -0.205) m 

Figure 6-8. Orbiter geometric and mass properties. 



68 



MAST 



28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 



60.693 m 



BODY FIXED 
FRAME - BODY 2 



28 node, lumped mass, finite element model. 

27 beam elements of equal length. 

Cantilevered @ node 1, free @ node 28. 

Transverse bending deflections permitted in 
x— z, and y— z planes. 

Torsional deflection about z— axis permitted. 

Rotary inertia in bending neglected. 

Uniform geometric and material properties: 

Bending stiffness y—z plane 
E« x = 28.63* 10* N-m 2 

Bending stiffness x— z plane 
Ei y = 32.39 •1(TN-m 2 

Torsional stiffness 

GK = 0.5»10 6 N-m 2 

Mass/length = 4.641 kg/m 



Moment of inertia about z-axis/length 
= 1 .9 kg — m 2 /m 

Instrumentation packages located at nodes: 
#7, 13, 16, 20, 23, and 28 



Figure 6-9. Mast geometric, mass, and material properties. 



69 



NODAL MASSES AND TORSIONAL 
MOMENTS OF INERTIA 



Node Number 


Mass (kg) 


Moment of Inertia 
about z— axis (kg— m ) 


1 


5.2162 


2.1355 


2-6, 8-12, 14, 15 
17-19,21,22,24-27 


10.4325 


4.2710 


7,16,23 


60.5325 


7.071 


13,20 


24.8325 


5.271 


28 


152.3162 


23.7355 



Table 6-1. 



VIBRATION CHARACTERISTICS OF CANTILEVERED MAST 



Mode 


Description 


Natural Frequency 
(Hz) 


1 


1 st bending mode y— z plane 


0.194 


2 


1 st bending mode x— z plane 


0.206 


3 


2 nd bending mode y— z plane 


1.359 


4 


2 nd bending mode x— z plane 


1.436 


5 


1 st torsion mode about z 


1.727 



Table 6-2. 



70 



OFFSET STRUCTURE & GIMBAL BASE 



Offset Truss Gimbal 

Structure Composite Base 

Mass Center - 



Body Fixed 
Frame — Body 3 



Mast 

Attachment 

Point 




« li 0.3556 m 



2.540 m 



2.742 m 



3.3274 m- 



Gimbal Pivot Point 




Total mass = 102.409 kg 



Inertia matrix with respect to the body fixed frame origin and axes: 



[« 3 ] = 



6.262 


0.0 


0.0 




810.683 


0.0 


sym 




810.683 



kg — m" 



Figure 6-10. Offset structure and gimbal base composite body geometric and mass properties. 



71 



GIMBAL PAYLOAD PLATFORM 



Body Fixed Frame 
— Body 4 



Gimbal Pivot Point 



0.358 m-H 




T 
0.254 m 

1 0.914 m 



h ^ 

0.508 m u Antenna Attachment Point 



Total mass = 172.176 kg 



Inertia matrix with respect to body fixed frame origin and axes: 



[l 4 ] = 



12.315 



sym 



0.0 
31.54 



0.0 

0.0 

31.54 



kg — m" 



Figure 6-11. Gimbal payload platform geometric and mass properties. 



72 



ANTENNA COLUMN 



Body Fixed Frame 
— Body 5 



Feed Attachment Point 




Gimbal Payload 

Platform 

Attachment 

Point 




i.i23 mF r— Hoop Plane 
5.296 m — 



Mass = 126.951 kg 



Inertia matrix with respect to the mass center and the body fixed frame 
axes: 



["si = 



11.264 



sym 



0.0 
1743.736 



0.0 

0.0 

1743.736 



kg — m' 



Figure 6-12. Antenna column geometric and mass properties. 



73 



FEED MAST & HORN 



Body Fixed Frame 
—Body 6 




£>, 



3.388 m 



Antenna Column 
Attachment Point 



Mass = 117.234 kg 



Inertia matrix with respect to mass center and body fixed frame axes: 



[l 6 ] = 



0.853 



sym 



0.0 
34.861 



0.0 

0.0 

34.861 



kg — m" 



Figure 6-13. Feed mast and horn geometric and mass properties. 



74 



ANTENNA HOOP 




Antenna Column 
Attachment Point 



Body Fixed Frame 
-Body 7 



Mass = 118.337 kg 



Inertia matrix with respect to mass center and body fixed frame axes: 



[l 7 l = 



6631.537 0.0 0.0 

3315.772 0.0 

sym 3315.772 



kg — m' 



Figure 6-14 Antenna hoop and mass properties. 



75 



o 



COLUMN 



TORSIONAL 
SPRING & 
DASHPOT 




^TEED MAST & HORN 



s, 



TORSIONAL SPRING 
& DASHPOT 



HOOP 



f \ 



BASE 




Figure 6-15 Antenna Spring and Dashpot Coefficients. 



Table 6-3. Antenna spring and dashpot coefficients. 



Spring-Dashpot 
Connection 


Spring Stiffness 
(N-m/rad) 


Dashpot Coefficient 
(N-m-s/rad) 


Column-Base 


71422.17 


471.518 


Feed-Column 


57829.16 


381.738 


Hoop-Column 


1698.141 


33.556 



76 



REFERENCES 

6-1 Bodley, C.S., Dever, A.D., Park, A.C., and Frisch, H.P., "A Digital Computer Program 

for the Dynamic Interaction Simulation of Controls and Structure (DISCOS)," Vols. I 
and II, NASA Technical Paper 1219, May 1978. 

6-2 Lenzi, D.C., and Shipley, J.W., "Mast Flight System Beam Structure and Beam Struc- 

ture Performance," NASA CP-2447, pp. 265-279, Presented at the First NASA/DoD 
CSI Technology Conference, November 18-21, 1986, Norfolk, Viginia. 

6-3 "AGS Control System Design and Pointing Performance Report," Books 1 and 2, 

Sperry Flight Systems, Phoenix, Arizona, December 1982. 

6-4 Belvin, W.K., and Edighoffer, H.H., "15-Meter Hoop-Column Antenna Dynamics: 

Test and Analysis," NASA CP-2447, pp. 167-185, Presented at the First NASA/DoD 
CSI Technology Conference, November 18-21, 1986, Norfolk, Virginia. 

6-5 "Development of the 15-Meter Diameter Hoop-Column Antenna," NASA Contractor 

Report 4038, December 1986. 

6-6 Ryan, R.R., "Flexibility Modelling Methods in Multibody Dynamics," AAS Paper 

87-431, Presented at the AAS/AIAA Astrodyn amies Specialist Conference, Kalispell, 
Montana, August 1987. 



77 



78 



SECTION 7 
SIMULATOR FOR THE VARIABLE C ONFIGURATION SYSTEM 

7.1 Introduction 

This section describes the integrated Space Shuttle Orbiter and COFS-II payload dynamics 
and control simulation. It was installed on the CSDL IBM 3090 MVS computer system, and was 
written in IBM FORTRAN 77. 

This simulation was built to study the mutual interactions between the Orbiter's attitude 
control functions and the COFS-II payload. Results obtained using the simulation are presented 
in Sections 9 and 10. 

The simulation was based on version 2 of the DISCOS multibody dynamics and control 
analysis program (Ref. 7-1). As described in Section 6 of this report, the COFS-II system is 
modelled as a collection of interconnected rigid and flexible bodies. At their interconnections, 
they are excited by internal forces and torques, and they are disturbed by external forces and 
torques. This implementation represents Orbiter jet firings as external disturbance forces and 
torques, and antenna control torques as internal torques. The system is controlled by a combi- 
nation of simplified DAP (SDAP) and antenna gimbal control laws. The algorithms defining 
these controllers are interfaced to DISCOS through user -supplied subroutines. 

The SDAP emulates the portion of the Space Shuttle flight control system which is active 
during on-orbit operations, when the payload is unlatched from its ascent position, but may still 
be connected to the Orbiter. SDAP, because of its simplification, permits only rotational con- 
trol. SDAP receives its input from the IMU model as an attitude matrix, and constructs rotation 
rates from its history. Given the switch settings and gains configuring the SDAP, its output is 
then a series of jet firing commands to the Reaction Control System (RCS) model. 

In preparing an experiment, data inputs are classed as for SDAP configuration, Orbiter 
motion commands, gimbal control torque parameters, payload motion commands, and simulation 
execution control. 

In all experiments, a comprehensive set of plots is produced for each simulation run, so 
that the behavior of either the Orbiter or the payload may be reviewed more easily. Printout 
simulation time interval and amount listed can be varied to suit the experiment. 

7.2 Simulation System Description 

The integrated simulation system consists of the program DISCOS, its associated user- 
supplied subroutines, and the model of the Space Shuttle on-orbit digital autopilot SDAP. 

DISCOS is a multibody dynamics and controls analysis package, developed for NASA, and 
distributed by COSMIC. It permits a user to model the dynamics of a system of articulated 
rigid and/or flexible bodies, subject to user-defined constraints, controls, and external actions. 
A dynamics problem is formulated as a topological tree of flexible bodies, then routines are 
added to represent the action of model actuators and sensors. The problem is constructed by 
first creating a data file, which describes the topology of body interconnection and orientation 



PKBUEDING PAGE BLANK NOT FILMED 
79 



and specifies the interbody hinge degrees of freedom, body mass and geometric parameters, and 
then writing FORTRAN subroutines to define disturbance and control forces and torques based 
on user-selected conditions. 

The program DISCOS, as used here, numerically synthesizes and integrates the equations 
of motion governing the mechanical system which the user has defined. 

The user-supplied subroutines specify forces and torques acting between adjacent bodies, 
and those disturbances exerted by the environment on the bodies. For this simulation, the 
external forces and torques acting on the Orbiter include those due to the firing of the attitude 
control thrusters. The choice of which thrusters to fire and how long to fire them is deter- 
mined by the model of SDAP. 

The subroutine for internal force and torque models is used to define the effects of tor- 
sional springs and dashpots acting between the antenna components, as well as the control 
torques from the gimbal motors. It also computes the potential energy. Nominally, any device 
which develops forces or torques between adjacent bodies must be defined in this subroutine. 

The autopilot subroutine performs as a nonlinear state-space controller, and may be set to 
a variety of different configurations, such as holding attitude, performing a maneuver automati- 
cally, or performing a manual maneuver, as described in Section 2 of this report. The autopilot 
receives attitude dynamics, samples the control panel settings for the input data if necessary, 
determines the allowable motion from the controller phase plane limits and availability of jets, 
and returns a set of appropriate jet firing commands. 

7.3 System Dynamics and Control Functions 

As mentioned in earlier sections, the SDAP may be viewed as a feedback controller in a 
typical plant-sensor-collector-actuator feedback control system. The SDAP is driven by inputs 
from both sensors and users (i.e., simulated crew). While the attitude sensor inputs vary contin- 
uously, the user inputs generally do not. The SDAP outputs are binary commands to turn jets 
either on or off. Each jet acts as a force actuator, with accompanying torque due to the jet's 
position on the vehicle. The vehicle model reacts, changing its attitude, the sensors detect the 
attitude change, and the cycle continues. 

7.3.1 SDAP Inputs: Configuration from Simulated Cockpit 

Performance of the SDAP may be altered by simulated crew inputs, depending on the 
requirements of the task. There are several categories of inputs which may be changed either at 
the keyboard or by switches: configuration constants, maneuver variables, and mode switches. 

Most constants are used in the state space controller section. They may be varied from 
mission to mission. For further information, see Reference 7-2, Table 1. 

Maneuver variables can be specific to a particular maneuver. They provide commanded 
attitude and rate, controller rate limit and attitude deadband, and allowable coupling from com- 
manded motion to other axes. For further information, see Reference 7-2, Table 3. 



80 



Mode switches provide automatic or manual performance selection, jet group exclusion, 
mass property set choice, and position of the Rotational Hand Controller (RHC). For further 
information, see Reference 7-2, Table 4. 

Translational control is not included in the SDAP model. Only rotational motion is 
sensed, commanded, and controlled. Jet-ON failures miy be simulated, but this function was 
not exercised outside the benchmark runs. 

Motion is commanded either automatically or manually. Manual control is triggered by 
operation of the RHC. Automatic control is engaged by setting control panel switches and 
entering keyboard inputs to the guidance computer. 

Additional details on inputs are also available in Reference 7-3. 

7.3.2 SDAP Inputs: Sensed Attitude 

The SDAP requires attitude inputs to be in the Body reference frame, but the attitude 
rates available from DISCOS were in the inertial frame at the body e.g., parallel to the Fabrica- 
tion frame axes, so there were some rotations required at the interface. 

All frames are shown in Figure 7-1. 

The Fabrication frame is a prime reference for much Orbiter-reJated kinematics and 
dynamics. It is centered a distance ahead of the Orbiter nose, with the X-axis along the longi- 
tudinal direction from nose to tail, the Y-axis pointing from the X-axis out the starboard wing, 
and the Z-axis pointing from belly pan to cargo bay. 

The Vehicle frame is defined at station (38.1,0,10.16) meters or (1500,0,400) inches in the 
Fabrication frame, shown as r F F / V in Figure 7-1, near the mass center of the empty Orbiter. 
Its X-axis points from tail to nose, Y-axis out the starboard wing as that of the Fabrication 
frame, and Z-axis from inside the vehicle out through the belly pan. 

The Body reference frame has its origin at the Orbiter mass center. Its axes are parallel 
to those of the Vehicle frame. 

7.3.3 Simulation Inputs: Execution Control 

Controlling the simulation is a matter of choosing the start and finish times, initial body 
attitude, position, and rates, and integration interval. 

The integration interval, or integrator step time, must be an integer divisor of the 80 ms 
SDAP clock period in this simulation. This has an impact on the jet model construction, 
described later in this section. The choice for experimental runs was 20 ms, or a quarter 
period. 

7.3.4 Payload Inputs: Gimbal Control Torques 

The only payload actuators are elevation and lateral gimbals. They can be controlled one 
at a time in this simulation. Varying the antenna gimbai control law parameters can change the 
characteristic response of the antenna to a steering command. 



81 



Simple servo control laws, discussed more fully in Section 10 of this document, were 
implemented to model the antenna gimbal torque motors. Identical and independent control 
loops were assumed for both gimbal axes. 

The control law outputs torque as a function of commanded angle, gimbal angle error, 
and error rate. The maximum torque available is 33.894 N-m (25 foot-pounds), so the output 
saturates easily. The assessment in Section 10 assumed operation in the linear range of output. 
For large angle slews, the torque would saturate almost immediately. 

Slews were simulated by applying a constant torque for a given period and then reversing 
the polarity of the torque for an equal period. Due to the limit on the torque available from 
the motor as modelled, the speed of "fast" slewing was relatively slow. A test slew of several 
tens of seconds was usually required to sweep a 45 to 90 degree angle. 

7.4 Sensor Model: IMU 

The IMU was modelled as a simple noiseless process. It serves to transform attitude data 
from the DISCOS inertial reference frame, at the Orbiter body mass center but parallel to the 
Fabrication frame, to the SDAP Body frame. 

7.5 Actuator Model: Jets 

One major design problem for the simulation was to resolve how best to model jet per- 
formance to fit the coarseness of the dynamic model, yet retain compatibility with respect to 
the more complicated model in the CSDL Statement-Level Simulator (SLS; Ref. 7-4) model used 
as performance benchmark. The jet model design goals required conservation of impulse as 
well as frequency content of the jet's output. The problem may be outlined as follows. 

The Orbiter flight control system operates with an 80 ms cycle time. The SLS models 
actual start-up and tail-off delays. The PRCS on-delay is 34 ms, and off-delay is 22 ms; the 
VRCS has 15 ms and 10 ms times. The simulation time step was constrained to be an integral 
divisor of 80 ms for practical purposes. 

The pulse output shape and phasing for jet firings was affected by the integration time 
step size. Alternative situations could be handled in the simulation. 

The simulation allows the inclusion of turn-on and turn-off delays which are integer mul- 
tiples of the integrator period. For the 20 ms integrator interval, the delays were 40 ms and 20 
ms, respectively for PRCS jets, and both 20 ms for VRCS jets. 

A 2 ms clock was tried, but had an unusually large CPU/simulation clock time ratio, even 
on the CSDL IBM 3090 model 200, hence was deemed impractical for the serious experiments in 
this study. 

The SLS and DISCOS simulated impulses are compared in Figure 7-2. The SLS impulse is 
shown as a solid line, that of DISCOS as dashes. The epochs A through F are described in 
Table 7-1. 



82 



Y 
Fabrication 




DISCOS Inertial (at C.G.) 



dy (at C.G.) 



Figure 7-1. Reference frames for simulation. 



Impulse amplitude 



H x - 
H 2 - 

H + 



__i_ ... k 



B B' 



-i 1 — -j r 



F C D' D 



—time 



Figure 7-2. Impulse profile- 



83 



Epoch 


Event 


! A 


Ignition command issued 


B 


Jet ignition (SLS) 


B' 


Jet ignition (DISCOS) 


C 


Turn-off command issued 


D 
D' 

E,"F 

H 


Turn-off response (SLS) 

Turn-off response (DISCOS) 

SDAP cycle clock event 

on non-minimum impulse burns 

level of nominal thrust 


H x 


level of first simulated DISCOS impulse 


H 2 


level of last simulated DISCOS impulse 



Table 7-1. Impulse profile epochs and events. 



In designing the jet emulation, it was considered most important to match epochs B with 
B' and D with D'. The reason for attempting to match epochs to the same millisecond, is that 
the closer the SLS and DISCOS simulations' event sequences are to each other, the closer the 
results will be (all other things being equal), and the more confidence will support the results. 
This is where the simulation clock pulse duration came into play. Ideally, all simulations and 
the real thing would fire the jets and turn them off at exactly the same time. Since that was 
impractical in these circumstances, the impulse off-nominal amplitude was chosen to provide 
equivalent total impulse in the case of a minimum impulse firing. The details may be noted 
with reference to the figure. 

7.6 Simulation System Checkout 

To provide benchmarks of performance and accuracy for the COFS-II simulation, the 
software system was tested first with a rigid-body Orbiter without payload. Resultant system 
performance of a given maneuver was compared against the same maneuver on the SLS. 

For all test runs, the Orbiter was configured at simulation start in a nominal attitude of 
payload bay open to earth, nose along orbital path, and rotating once per orbit with respect to 
earth reference. Nominal orbital parameters were given in any run where appropriate. 

Gravity gradient torques were neglected in all but one test run. In order to match one 
SLS benchmark, they were emulated by an external torque, which was applied as a constant 
independent of attitude and altitude. 

Aerodynamic torques and solar pressure torques were neglected. 



84 



REFERENCES 

7-1 Carl S. Bodley, A. Darrell Devers, A. Colton Park and Harold P. Frisch. A digital com- 
puter program for the Dynamic Interaction Simulation of Controls and Structure (DIS- 
COS). Technical Paper 1219, NASA, May 1978. Volumes I, II, III, IV. 

7-2 P. Hattis, C. Kirchwey, H. Malchow, D. Sargent, and S. Tavan. Simplified Model of the 
Space Shuttle On-Orbit Flight Control System. Report CSDL-R-1562, The Charles Stark 
Draper Laboratory, Inc., July 1982. 

7-3 Isaac A. Stoddard. Manual for COFS-II Payload Interaction Simulation. Report unpub- 
lished, The Charles Stark Draper Laboratory, Inc., January 1988. 

7-4 Leonard W. Silver (ed.) ESIM Model for the C.S. Draper Laboratory Statement Level Sim- 
ulator. Report CSDL-R-776, revision 5, The Charles Stark Draper Laboratory, Inc., April 
1981. 



85 



86 



SECTION 8 
RESULTS FOR THE FIXED C ONFIGURATIONS 

8.1 Introduction 

The three rigid Orbiter/flexible COFS-II fixed configuration models described in Section 
4 were subjected to analysis and simulation to determine the effects of interaction with the FCS. 
Of concern were the effects both on the COFS-II structure and on FCS performance. The 
remainder of this section provides an overview of the interaction problem (both its mechanism 
and its potential ill effects) followed by results of the analytical and simulation studies. 

8.2 Interaction Overview 

The FCS constitutes a source of flexural excitation to the combined Orbiter/COFS-II sys- 
tem, raising the possibility of undesired structural deflections and loads. When the FCS is oper- 
ating as a closed loop controller, there is the additional possibility of flexural rotation of the 
rigid Orbiter being fed back through the IMU into the autopilot, and the effect on FCS 
performance can range from negligible to catastrophic. 

A nonnegligible but noncatastrophic effect would be inefficient attitude maneuvering, as 
evidenced by a few excess jet firings and greater fuel consumption compared to rigid body 
performance, perhaps accompanied by degraded maneuver path control accuracy. A more 
severe effect would be a high-energy phase plane limit cycle, augmented from the expected 
rigid body cycle by flexure, which could result in a large excess of jet firings and fuel use, 
especially during a long-term attitude hold. Many excess firings can shorten the lifespan of the 
jets, and unexpectedly high fuel consumption could force early mission termination. 

If the FCS/structure closed loop gain and phase characteristics at some structural fre- 
quency permit, the high-energy cycle could "run away," with jet firings driven by phase plane 
commands of alternating polarity becoming locked to or near that frequency, and structural 
deflections and loads increasing either to failure or to limits determined by damping. 

CSDL has developed a set of analytical tools (Ref:s. 8-1, 8-1, 8-3) for predicting the pos- 
sibility of such runaway behavior and recommending autopilot parameters and mission timelines 
that will prevent it. Accordingly, before beginning the simulation effort, analysis of the 
FCS/Orbiter/COFS-II system was performed to suggest initial values of autopilot parameters and 
the degree of jet-induced excitation needed to demonstrate instability, and to propose alternate 
parameter sets and stratagems for its avoidance. 

8.3 Analytic Techniques 

The analytical tools enable prediction of the possibility of unstable feedback interaction 
for a given set of phase plane control parameters DB (deadband) and RL (rate limit) and for a 
given RCS jet option (in this case limited to the VRCS jets). The "possibility" is output in the 
form of two indicators. 



PREOKD 



■jV.-.jj ' 



87 



The first indicator is the location of each system flexible mode on a parameter plane con- 
taining a boundary defining "stable" and "unstable" areas. The axes of the plane are the natural 
frequency of the mode and a parameter /3 that characterizes the flexural rotation response of 
the Orbiter, at that mode, to excitation by the selected jet option. The stable/unstable boundary 
is specific to a particular combination of DB, RL, and a parameter y defining the rigid body 
acceleration of the chosen jet option. 

For each flexible mode, attitude control axis and jet option, /3 and y are assessed using the 
rigid body mass properties, the Orbiter flexural response data that is an output of the finite 
element modelling process, and the jet forces and torques. The corresponding point on the 
j3-frequency plane is then marked, and the stable/unstable boundary for y and the desired DB 
and RL is overlaid. Many of these boundaries have been generated, assuming a degree of 
structural excitation just sufficient to exceed either the selected DB or RL by a factor of two, 
and a damping coefficient i of 0.005. The result of this process is a go/no-go indication of 
whether bipolar flexural feedback-driven jet firings, once started, will drive the structure to 
larger-amplitude vibrations (unstable), or allow the vibrations to decrease (stable). 

The second indicator gives insight into the likelihood of achieving sufficient excitation, on 
the premise that the only credible source of excitation is the jets themselves. The inputs to this 
process are /3 , y, natural frequency, f , and either DB or RL. The result is the number N of 
successive worst case jet firings needed to generate sufficient flexural Orbiter rotation to just 
exceed either DB or RL in a bipolar fashion. We define a worst case firing sequence as a train 
of contiguous, alternating polarity jet force pulses, each pulse having a duration of one half the 
modal period. Note that, neglecting losses due to damping, any odd-integer multiple of this 
duration is also worst case. Usually, N- values less than 10 are considered to deserve special 
attention, since some manual maneuvering scenarios can require repeated application of pulses, 
and closed loop attitude maintenance in the presence of a steady state torque disturbance can 
generate cyclic firings. Very low values of N indicate that sufficient excitation may be 
achieved in the course of normal maneuvers, if the commanded maneuver rate MR is large 
enough to require start and stop firings of sufficient duration to approximate the worst case 
pulse definition. 

It should be pointed out that instability can only be considered improbable when both DB 
and RL are selected appropriately. Unstable interaction is quite possible with a combination of 
wide DB and too-narrow RL or vice versa. Another caveat is that the analysis considers one 
mode at a time, and does not allow for the effect of additive mode responses. Also, as will be 
shown in subsection 8.5, there exists a region of the phase plane control logic in which the 
effective rate limit is much smaller than the value of RL and thus sensitivity to flexure is much 
greater than elsewhere in the phase plane. Operation in this region can be avoided through FCS 
parameter selection. 



88 



8.4 Analytic Results 

The results of the preceding assessments predicted instability and relatively great sensitiv- 
ity to excitation for a number of modes in each configuration when the deadband was relatively 
narrow (0.1 deg) or when the rate limit was the smallest allowed for VRCS (0.01 deg/sec), and £ 
was assumed equal to 0.005, as shown in Table 8-1. The situations (i.e., combinations of mode, 
control axis and DB or RL) in this table for which N is less than 30 are unstable. (This is not a 
general rule, but happens to be true for the FCS and structural configurations under study). 
Increasing the deadband to 1 deg made it very difficult to excite deadband oscillations, and 
stabilized the subsequent closed loop response. Increasing the rate limit to 0.02 deg/sec reduced 
but did not eliminate the unstable rate limit modes, and made excitation more difficult. The 
0.01 and 0.02 deg/sec RL values are generally preferred to larger values which can cause sloppy 
maneuver and attitude hold performance by allowing large unwanted rates, and by generating 
large rate-change commands when operation is temporarily outside the deadband. 

The damping coefficient £ of the modes associated with mast bending was then set to 0.05 
to emulate the expected damping present when the experiment proof mass actuators are active. 
As Table 8-2 shows, the total number of easily excited modes for DB = 0.1 deg or RL = 0.01 
deg/sec declined to three, so the assessment was repeated for DB = 0.05 deg, which brought the 
excitable mode count to seven. Stable/unstable boundaries have not been generated for 
£ = 0.05, but the three or four smallest numbers in Table 8-2 strongly suggest possible instabil- 
ity. 

Both roll and pitch unstable modes exist. Roll modes are generally more easily excited, 
which can be attributed to the Orbiter's much smaller roll moment of inertia. The analysis 
showed negligible yaw rotation of the Orbiter due to flexure. 

8.5 Simulation Results 

Using the findings of the foregoing analysis, simulations were run using the OCFS (Sec- 
tion 5) to investigate three main areas of FCS/payload interaction, which are reported in the 
following three subsections. First, excitability and stability were studied using relatively 
stressed conditions — either deliberate excitation, or the normal closed loop response to an ini- 
tial condition of high angular rate. Next, various attitude maneuvers were studied to evaluate 
the effect of flexibility on performance, and to assess typical loads associated with maneuvering. 
Finally, several long term attitude holds were simulated to determine the likelihood of achieving 
sufficient excitation for instability under unstressed conditions. A goal common to the three 
areas of study was to obtain a set of FCS parameters which provided acceptable performance 
and adequate immunity to unstable behavior. 



89 



o 



Table 8-1. RHC excitation analysis summary ( £ = 0.005). 



Number of half -period pulses 
required to excite given DB or RL 



Config 


Mode 


Axis 


DB=0.1 


DB=1.0 


RL=0.01 


RL=0.02 


1 


1 


r 


2.4 


30. 


1 .7 


3.5 


1 


3 


P 


9.7 


- 


12. 


27. 


1 


4 


r 


7.1 


- 


16. 


38. 


2 


1* 


r 


74. 


- 


46. 


- 


2 


2 


P 


7.3 


- 


8.3 


18. 


2 


3 


r 


3.7 


52. 


5.7 


12. 


3 


1 


r 


7.1 


- 


5.1 


11. 


3 


2* 


r 


14. 


- 


12. 


26. 


3 


3 


r 


14. 


- 


19. 


47. 


3 


4 


r 


9.8 


- 


22. 


56. 


3 


5* 


r 


150. 


- 


- 


- 



* = mode is not considered a "mast" (i.e., 5% dampable) mode. 



Table 8-2. RHC excitation analysis summary (£ = 0.05). 



Number of half-period pulses 
required to excite given DB or RL 



Conf ig Mode 

1 1 

1 3 

1 4 

2 2 

2 3 
^ 1 

3 4 



Axis 


DB=0.05 


DB=0.1 


RL=0.01 


RL=0.02 


r 


1 .3 


3.0 


2.0 


4.9 


P 


7.9 


- 


- 


- 


r 


4.8 


- 


- 


- 


P 


4.9 


- 


- 


- 


r 


2.1 


5.2 


13. 


- 


r 


4.« 


- 


9, "* 


- 


r 


8.0 


_ 


_ 


_ 



8.5.1 Excitation/Stability Results 

The immediate aim of the excitation/stability simulations was to corroborate the findings 
of analysis; i.e., to demonstrate that unstable interaction in any of the three configurations was 
not only possible but also easily excited if the FCS attitude control parameters DB and RL were 
sufficiently tight. Another desire was to determine the degree of stabilization provided by the 
active proof mass actuators with their assumed £ of 0.05. 

Table 8-3 summarizes the results of the excitation/stability simulations for all three con- 
figurations. Case numbers in the first column are for reference in this subsection. The second 
column in the table gives the configuration number (1 = antenna facing aft; 2 = antenna facing 
up; 3 = antenna rotated up 45 deg and to the right side of the Orbiter 45 deg from the aft 
facing position). Configuration 1 was selected for the majority of the test cases, because it 
became available for use earliest and because the analysis indicated that one of its modes was 
the most susceptible to excitation. 

The next two columns in Table 8-3 describe the excitation applied to the system; i.e., 
which mode was excited, and the method used to obtain the excitation. "Modal" excitation 
makes use of the capability of the OCFS to initialize the first derivative of any selected mode(s) 
to a specified level. While use of this feature provides no information on ease of excitation, it 
does give a quick indication of stable or unstable response once sufficient excitation is achieved. 
The modal excitation cases used the minimum value for the first derivative needed to produce 
Orbiter flexural rotation about the roll control axis sufficient to exceed either DB or RL by a 
factor of two. 

"RHC" (rotational hand controller) excitation simulates the insertion of manual rotation 
commands by the crew. In these simulations, we assumed worst case crew inputs producing 
rotation commands that resonate the desired mode, each command being one half modal period 
in duration. The control axis the commands were issued in (r = roll, p = pitch) and the number 
of manual commands input are also cited in the method column. 

Several simulations investigated a region of the phase plane (Figure 2-2) known to be 
potentially sensitive to rate error oscillations. The out-of-deadband coast "corridor" is intended 
to maintain the rate error at a value that will drive the phase point back inside the deadband 
without exceeding RL. When the phase point is above or below the corridor, the logic produces 
firing commands to drive it back inside the corridor. Within the corridor, the firings are cut 
off. The corridor has only 0.2 times as large a rate deadzone as the in-deadband region (i.e., 
0.4 RL vs. 2 RL), hence the greater sensitivity. 

When attitude hold is commanded, the attitude current at the time of the command is 
"snapshot" as the desired attitude, and the attitude error seen by the phase plane is the deviation 
from this reference. If the angular rate at the time of the attitude hold command is large and 
DB is small, the attitude error can go beyond the deadband by the time the rate is nulled, and 
the phase point will be driven into the corridor. The time spent in the corridor depends on the 



92 



Case Cfg, 



-Excitation- 
mode method 1 



DB, 
deg 



RL, 
d/s 



Closed- 
locp axes 



Mast-mode 
damping 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 



1 
1 
1 
1 
1 
1 
1 
1 
i 
i 
l 
l 
l 
l 
l 
l 
l 
l 
l 
l 
l 
l 
l 
l 
2 
2 
2 
2 
.2 
2 
2 
2 
2 
3 
3 
3 



1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
4 
4 
4 
4 
4 
4 



2 
2 
3 

3 
3 
3 
3 
3 
3 
1 
1 
4 



modal/r 

modal/r 

RHC/rl 

RHC/r2 

RHC/r2 

RHC/r4 

RHC/r4 

RHC/r4 

RHC/r4 

RHC/r4 

modal/r 

modal/r 

RHC/rll 

RHC/rll 

RHC/rl6 

RHC/r23 

wr o =0.03 

cjr o =0.04 

a)r o =0.05 

ojr o =0.06 

ajr o =0.07 

cur o =0.08 

ojr =0.09 

cjr o = 0.l0 

RHC/p4 

RHC/p6 ■ 

modal/r 

RHC/r2 

RHC/r2 

RHC/r6 

RHC/r6 

RHC/r6 

RHC/r6 

RHC/r4 

RHC/r6 

modal/r 



0. 

0. 

0. 

1. 

1. 

1. 

1. 

1. 

0. 

1. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0, 

0, 

0, 

0, 

0, 



0. 







1 
1 







1 

2 

1 







1 



1 
1 
1 
1 

2 

2 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

.1 

,1 

,0 

,0 

,1 
,1 
,1 



0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0, 

0, 

0, 

0, 

0, 

0, 



0. 





















01 

02 

01 

01 

01 

01 

02 

02 

01 

01 

01 

01 

01 

01 

02 

02 

01 

01 

01 

01 

01 

01 

01 

01 

01 

01 

01 

01 

01 

,01 

,01 

,01 

,01 

,01 

,01 

.01 



-p-y 
,p»y 

r,P,Y 
only 

,PrY 
,PrY 

only 

rP,y 

,P,Y 
,PrY 

,p,y 

,PrY 
,PrY 
,P,Y 
,P,Y 
,P,Y 
,P,Y 

■»p»y 
■#p#y 

'/PrY 

■ ,p,y 

■ ,PrY 

■ ,P,Y 

•»p»y 

" • p » y 

:■ . P f Y 

■■ ,P,Y 
r only 
r only 

r,p,y 

r , P , y 

r only 

~ ,P>Y 
r only 

• »p»y 



005 
005 
005 
005. 
005 
005 
005 
,005 
,05 
,05 
,005 
,05 
,005 
,05 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.005 
.05 
.005 
0.005 
0.005 
0.005 
0.005 
0.005 



































































Results 2 

divergent 

divergent 

divergent 

insuf. exc. 

divergent 

divergent 

insuf. exc. 

divergent 

sustained 

sustained 

divergent 

damped 

divergent 

insuf. exc. 

sustained 

divergent 

damped 

damped 

damped 

damped 

divergent 

divergent 

damped 

divergent 

sustained 

divergent 

divergent 

insuf. exc, 

divergent 

divergent 

divergent 

insuf. exc 

divergent 

insuf. exc 

sustained 

divergent 



Notes: 1. (Excitation method) "modal" = mode first derivative set such that Orbiter flex 
rotation about the indicated axis is the minimum needed to exceed either DB or 
RL by a factor of 2; "RHC" = manual rotation +/- command sequence at modal 
frequency, with axis and number of commands as indicated; "u>r " = initial rigid 
body roll rate as indicated (in deg/sec). 
2. (Results) "divergent" = closed loop firings appreciably increase flex amplitude; 

"sustained" = firings maintain amplitude approx. same as caused by excitation; 
"damped" = firings occur but allow appreciable decrease of amplitude (and fir- 
ings may stop); "insuf. exc." = excitation does not provoke significant closed loop 
firings. 



Table 8-3. Excitation/stability simulation results summary. 



93 



attitude error excess beyond the deadband and the average rate inside the corridor. Some excit- 
ation/stability cases used initial roll rates with DB = 0.1 deg in the hope of provoking unstable 
behavior in the corridor. These are identified in the excitation method column with 
"cor " and the initial composite body angular rate in deg/sec. 

Remaining columns in Table 8-3 give values for DB and RL, mention which FCS control 
axes were active (r = roll, p = pitch, y = yaw), specify the damping coefficient J for the mast 
modes, and summarize the results of the closed loop interaction. Although three-axis attitude 
control (the normal configuration in an actual mission) was generally used, it was occasionally 
necessary in RHC roll excitation cases to close only the roll axis in order to achieve sufficient 
excitation for feedback interaction when using the predicted number of pulses. The reason is 
that flexure and coupling from the roll jets into other axes create uncommanded rates causing 
the three-axis control system to issue additional commands in pitch or yaw. These commands 
are realized as either added or removed jets that reduce the effectiveness of the roll excitation. 
Analytical prediction of the number N of firings needed assumes maximum effectiveness; i.e., 
that only a single-axis command is present. The FCS allows such axis-by-axis mode selection 
when in manual control. 

The results column shows that many unstable or divergent cases exist when either DB = 
0.1 deg or RL = 0.01 deg/sec. Mode 1 of configuration 1 is particularly sensitive. Case 3 in 
Table 8-3 shows this clearly (only one RHC command needed for divergence) and also illus- 
trates two effects not accounted for in the analysis (which predicted 1.7 or 2.4 pulses were 
needed). Although only one rate command was input via the RHC, the closed loop manual 
mode behavior described in Section 2 caused a second, opposed firing to partially null the rate 
as soon as the RHC returned to center position. This increased the structural excitation to the 
equivalent of about 1.6 worst case pulses, which the analysis predicted was nearly, but not 
quite, enough to start a closed loop feedback cycle. However, another FCS response to the 
removal of the RHC command is the resetting of the desired attitude to the current attitude, 
thus setting the attitude error to zero. The timing of the worst case command pulse is such that 
absolute flexural displacement is a maximum when the pulse terminates and the desired attitude 
is reset. Thus if the vibrational cycle continues freely, the attitude error oscillates between the 
extremes of zero and twice the flexural amplitude (neglecting damping). 

Since the analysis assumes attitude error oscillations that are symmetrical about zero, the 
pulse number N it predicts as necessary to exceed a given DB is an overestimate for this situa- 
tion. The result is that the deadband can be exceeded at least once with fewer than N pulses, 
causing at least one additional pulse that increases the excitation. If these additional pulses can 
increase the excitation to the equivalent of N pulses by the time they have driven the rigid body 
attitude error to zero, a bipolar deadband firing cycle will occur. This additional-pulse mecha- 
nism has a greater relative effect when N is small. Thus in case 3, a single command pulse 
caused one additional pulse due to normal manual mode behavior, and another pulse due to a 
combination of normal behavior and flexure, and the result was sufficient excitation to start a 
feedback cycle. 



94 



The initial rate/coast corridor simulations (cases 17 through 24) provide an illustration 
both of the sensitivity of this region as previously described, and of the apparently chaot.c 
nature of the excitation obtained from the closed-loop, nonlinear FCS in the pre-d.vergence 
phase Cases 21, 22 and 24 ( cor = 0.07, 0.08 and 0.10 deg/sec) were unstable, while the 
remaining five cases (including one "bracketed" by the unstable cases) were stable. 

All of these cases started with a firing to damp the roll rate. These firings varied from 
14 to 32 seconds in duration and produced excitations of modes 1 and 4 that ranged from negli- 
gible to nearly the equivalent of a worst case single pulse. The unstable cases resulted from 
near-worst case equivalent mode 1 excitations which initiated a low duty factor bipolar firing 
cycle. 

Figure 8-1 illustrates the evolution of the divergent bipolar firing cycle for a typical case, 
that with wr = 0.10 deg/sec. Shown are plots of the phase plane trajectory, roll firing com- 
mands generalized coordinates for flexible modes 1 and 4, and the resulting roll moment at the 
base of the mast. (The numerical values of the generalized coordinates do not represent 
physical quantities). The firing command in any axis is primarily a function of the rate error, 
and shows the effect of both rigid body and flexural rates. Its value can range from -1.0 to 
1.0, but it only generates a jet firing when it or the command from another axis has an absolute 
value of 1.0. 

The initial rate damping command lasted 32 seconds, and its termination provided some 
reinforcement to the oscillations started by its initiation, for both modes 1 and 4. During the 
next 50 seconds of travel along the coast corridor, uniformly spaced command pulses of alter- 
nating polarity occurred at an average rate of one pulse per 6.45 seconds. (The closely spaced 
positive pulse pair at about 45 seconds is considered a single pulse.) This rate corresponds to a 
frequency of 0775 Hz, which is 12% higher than mode 1, but whose third multiple is only 3% 
higher than mode 4. Thus while initially driven primarily by the mode 1 component of the rate 
error, the firings nearly resonated mode 4, which can be seen to increase in amplitude as mode 
1 decreases in the 30-80 second range. 

Although analysis predicted that mode 4 would require the equivalent of three worst case 
excitation pulses to start a bipolar cycle in the corridor, while mode 1 would require less than 
one worst case equivalent pulse, the emerging dominance of mode 4 can be explained by the 
durations of the VRCS pulses during this time in addition to their repetition rate. The average 
pulse duration was about 1 second, as compared to the worst case pulse lengths of 7.2 seconds 
for mode 1 and 2.2 seconds for mode 4. Thus each pulse was a substantial fraction of the worst 
case duration for mode 4, in addition to being in a tram of pulses having a large mode 4 
repetition- rate content. 

The following two pulse pairs, at about 85 and 100 seconds, were closely spaced, nearly 
worst case doublets for mode 4, and their effect on the mode 4 generalized coordinate ampli- 
tude is visible, as is their damping effect on mode 1. Finally, at about 110 seconds, a third 
mode 4 doublet precipitated a full-scale instability, at which point the jet command duty factor 
became nearly 100%. The roll moment at the base of the mast (which is almost entirely due to 
mode 4) then started to increase rapidly, exceeding 9096 n-m at about 180 seconds. The phase 



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270 



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plane trajectory for this case shows counterclockwise rotation due to phase lag in the rate esti- 
mate from the filter in the state estimator. The usual rigid body trajectory is effectively clock- 
wise; i.e., attitude error increasing when rate error is positive and vice versa. 

The results of the simulations summarized in Table 8-3, and previously discussed consid- 
erations favoring small rate limits, led to a tentative adoption of DB = 1.0 deg, RL = 0.02 
deg/sec as the "baseline" phase plane parameter set for the maneuver and attitude hold 
simulations to follow. Although instability did occur under conditions of deliberate excitation 
with DB = 1.0 deg, RL = 0.02 deg/sec, the likelihood of exciting unstable behavior with this set 
is acceptably low, as long as manual maneuvers are avoided and automatic maneuvers are 
assessed in advance of being performed. The 1.0 deg DB gives a comfortable excitation margin, 
as Table 8-1 showed. The 0.02 deg/sec RL, with a 3.5 worst pulse excitation level in the worst 
case of the three configurations tested, provides a margin that should be adequate for most 
automatic maneuvers and attitude holds. When this panimeter set was used in case 7 in Table 
8-3, four worst pulses failed to achieve sufficient excitation for instability to occur when in the 
three-axis (e.g., automatic) control mode. However, the approximately 3 to 45 second firing 
durations typical of the start and stop phases of automatic maneuvers easily span the worst case 
durations, and maneuvers at the high end of the tested MR range of 0.05 to 0.2 deg/sec (see 
subsection 8.5.2) could possibly cause operation in the out-of-deadband coast channel of the 
phase plane, which should be avoided. Thus a "baseline" value of MR = 0.05 deg/sec was also 
adopted to complete the initial FCS parameter selection. 

8.5.2 Maneuver Results 

Five basic maneuver types were selected as representative of the range of stresses likely to 
occur during normal operation. The simulations performed and results obtained are summarized 
in Table 8-4. The selected maneuvers consisted of a 5 deg total maneuver angle command, with 
the axis of rotation varied to explore the effects of jet off-axis torques, the composite body 
moment of inertia, and flexure. The maneuver column in Table 8-4 indicates the axis of rota- 
tion for each case. The "+r, +p, +y" cases commanded clockwise rotation about an axis that was 
equally displaced from the forward-pointing roll axis, the rightward-pointing pitch axis, and the 
downward-pointing yaw axis. Similarly, the "+r, +p, -y" cases commanded clockwise rotation 
about a forward-rightward-upward axis. The "+roll," "+pitch" and "+yaw" cases commanded 
clockwise rotation about single vehicle basis axes. If the effects of off-axis rotations are "fa- 
vorable" (e.g., if +roll, +pitch and +yaw are commanded, and a combination of jets can be found 
to provide the desired accelerations in approximately the desired proportions), the rate changes 
needed to start and stop the maneuver should be accomplished with a few long firings, whereas 
"unfavorable" coupling should cause more, shorter firings and reversals of angular acceleration. 
Thus using both relative polarities of roll and yaw commands should produce results that cover 
a range of performance and firing signatures. 

Flexure caused some of the 0.05 deg/sec multiaxis maneuvers to exhibit inefficient behav- 
ior, as evidenced by excessive roll firings and fuel consumption when compared to their rigid 
body equivalents. Figure 8-2 shows the roll phase plane trajectory and the roll firing command 



101 



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Figure 8-2b. Maneuver case, with flexure, roll firing command. 



104 



for a typical case, the +r, +p, -y maneuver using configuration 3. The maneuver started reason- 
ably cleanly, but the combination of rigid body Euler coupling and flexure produced a rate 
limit firing at about 72 seconds, which was reversed aboat 7 seconds later by the maneuver 
termination command. In the rigid body equivalent shown in Figure 8-3, the pre-termination 
rate limit firing did not occur, due to the absence of flexure and the slightly different rates 
established at the start of the maneuver. 

In general the single-axis maneuvers were better behaved than the three-axis maneuvers, 
although the effects of flexure could be seen. However, there is probably no advantage to 
performing maneuvers one axis at a time (i.e., in an Euler sequence) rather than as single equiv- 
alent (eigenaxis) rotations, because of the additional time and fuel expenditure required for 
three sequential maneuvers. Furthermore, flexure does not always degrade all aspects of 
multiaxis maneuvers, as shown, for instance, by the slightly reduced fuel consumption of the 
configuration-2 +r, +p, +y maneuver with flexure present. Other maneuvers (with different 
total maneuver angles, maneuver rates and eigenaxes) might be better or worse behaved than 
those simulated. 

The effect of maneuver rate on performance and stability (especially involving the phase 
plane coast corridor as discussed in subsection 8.5.1) was assessed in six simulations using the 
+F) +p? _ y maneuver at 0.1 and 0.2 deg/sec for each configuration. The larger rate commands 
produced longer firings at the start of the maneuvers, but load values differed insignificantly. 
In all maneuvers, bending moments measured at the base of the mast never exceeded 2000 N-m. 
The maximum torsion at that location was 130 N-m. At the tip of the mast and the base of the 
antenna, the maximum moments were 140 N-m. 

None of the maneuvers exhibited instability (i.e., & bipolar jet firing cycle). However, for 
the 0.2 deg/sec maneuvers, this was fortuitous. The mass properties of the composite system, 
the VRCS jet forces, and the nature of the VRCS jet selection logic are such that roll accelera- 
tion, when simultaneous roll, pitch and yaw commands exist, is much smaller than the pitch and 
yaw accelerations. Thus during these relatively small maneuvers, the roll rate never reached 
more than about one half the commanded rate of 0.115 deg/sec (the single-axis component of 
the vector of magnitude 0.2 deg/sec). With so large a rate error seen by the phase plane during 
most of the maneuver, the roll phase point was never inside the coast corridor. However, the 
attitude error (essentially the integral of the rate error) grew quite large, going well beyond the 
deadband. Had the maneuver been larger, the roll rate error would eventually have been driven 
down, and the phase point would have entered the coast corridor, probably remaining there for 
the remainder of the maneuver, and providing an opportunity for instability similar to that seen 
in Figure 8-1. 

From the results described in this subsection we can conclude that highly efficient per- 
formance of maneuvers with the COFS-II erected is a goal that is unlikely to be achieved. The 
fuel budget must allow for greater than normal expenditures due to flexure. The mission 
timeline should provide adequate time for maneuvering at low rates, preferably 0.05 deg/sec. 
Extensive simulations of any planned maneuvers are necessary to assure stability. 



105 



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Figure 8-3a. Maneuver case, rigid body, roll phase plane. 



106 



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Figure 8-3b. Maneuver case, rigid body, roll firing command. 



107 



8.5.3 Attitude Hold Results 

Attitude hold simulation cases were run for the tentative "baseline" phase plane parameter 
set of DB = 1.0 deg, RL = 0.02 deg/sec, and for a set offering tighter control, DB = 0.1 deg, 
RL = 0.01 deg/sec. For each parameter set, the three configurations were tested with roll-only 
and three-axis gravity gradient torques. These were simulated as constant torques whose values 
were obtained by averaging the maximum single-axis values for the three configurations. Devi- 
ations from average did not exceed ±9% for any configuration. The three-axis torques were 
"worse than worst case," since no earth- relative attitude can produce simultaneous maxima in 
two or three axes. (The roll axis values were also applied in the maneuver simulations). 

Ideal single-axis phase plane attitude hold performance in the presence of a constant dis- 
turbance torque produces a "one-sided" limit cycle; i.e., generation of regularly spaced unipolar 
commands that oppose the disturbance. If the RL/DB ratio exceeds a minimum value (which it 
does in both parameter sets used in these simulations), a stable cycle is established in which the 
disturbance drives the phase point beyond the deadband (never the rate limit) and the resulting 
command drives the rate error from its current value approximately to its negative. The distur- 
bance then acts alone to move the phase point in a parabolic trajectory until the deadband is 
exceeded again, completing the cycle. Multiaxis disturbance torques, off-axis torque coupling 
and contamination of the disturbance acceleration estimate supplied by the state estimator will 
cause less than ideal performance. In these simulations, flexure was expected to be an apprecia- 
ble source of contamination. 

The results of the attitude hold simulations are listed in Table 8-5. Performance with the 
tentative baseline phase plane parameter set was stable, although mode 1 of configuration 1 
caused one or two reversals of the roll firing commands in both the roll-only and the three-axis 
torque cases. These reversals did not cause significant fuel waste or excessive firings. The roll 
phase plane trajectory and firing command history for the configuration 1 three-axis torque 
case are shown in Figure 8-4. (The jagged appearance of the phase plane plot is due to the 
combined effects of flexure and the sampling rate of the plotting routine, which is limited to a 
total of 1000 data points.) The nonideal behavior is readily apparent as gross deviations from a 
single trajectory, and the flexure amplitude is large. Flexure caused the phase point to reach 
the negative rate limit once, as shown by the largest negative spike in the phase plane plot, 
causing a wrong-polarity firing command. The firing commands are irregularly spaced. In 
contrast, Figure 8-5 shows the roll phase plane trajectory and firing command history for the 
configuration 2 roll-axis torque case. The phase plane trajectory comes close to the ideal 
model, traversing a series of cycles that closely resemble each other. The firing commands 
occur with regularity, and flexure amplitude is small. In spite of such variations in perform- 
ance, mast base bending loads in all the baseline parameter set attitude hold cases were less than 
1300 N-m. 

The tight control parameter set produced divergence for configurations 1 and 2 in both 
the roll-only and three-axis disturbance torque cases. It is interesting to note that the configu- 
ration 1 cases resonated mode 1 but did not progress to dominance by mode 4, in contrast with 
the coast-corridor instability case (subsection 8.5.1) which used the same configuration and 



108 





DB, 


RL, 


Tgg 1 


RCS act 


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Mast base moment, 


Conf iq 


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1 


1.0 


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roll 


226 


10.1 


0.84 


2 


1.0 


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roll 


190 


9.1 


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3 


1.0 


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roll 


215 


9.7 


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1 


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14.5 


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15.8 


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roll 


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roll 


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22.2 3 


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roll 


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Notes: 1. Tgg = gravity gradient torque 

= 6.4 N-m (57 in-lbf) roll 

= 9.8 N-m (87 in-lbf) pitch (if used) 

= 16.3 N-m (144 in-lbf) yaw (if used) 

2. Divergent case; divergence largely due to mode 1; 
load largely due to modes 1 and 4 (mast modes) 

3. Divergent case; divergence and load largely due to 
mode 3 (a mast mode) 



Table 8-5. Attitude hold simulation results summary. (AH simulations used run time = 4000 

sec, mast-mode damping = 0.005 x critical). 



109 



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110 



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400 goo 1200 ' 1600 2000 2400 2800 3200 3600 4QCG 

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112 



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400 800 1200 1600 2000 2400 2800 3200 3600 400Q 



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Figure 8-Sb. Attitude hold, configuration 3 roll-axis torque case, roll firing command. 



113 



parameter set. As a result, the bending loads at the mast base approached about 3500 N-m as 
an asymptotic value, well below the assumed 9096 N-m limit. (The asymptotic load value due 
to mode 4 in the coast corridor instability case was about 14,000 N-m.) 

The reason mode 1 retained dominance lies in the different nature of the firing commands 
preceding the divergence. The attitude hold firings occurred at an average rate of about once 
per 60 seconds, compared to once per 6.5 seconds in the coast-corridor case, whereas the dura- 
tions of the firing commands were about the same (1 second) in both cases. The greatly 
reduced rate and similar duration of the pulses would tend to give mode 1 a relative advantage 
in the attitude hold case because of the greater proportional damping losses between pulses for 
the higher frequency mode. 

The configuration 2 cases resonated mode 3 and produced asymptotic load values of about 
22,000 N-m. Differences in fuel use and jet firings among the four divergent cases reflect only 
the different times of onset of divergence. The jet firing patterns before onset were similar in 
all of these cases, as were the firing patterns after onset. 

The baseline parameter set provides greater immunity to excitation than the tight control 
set, in direct proportion to the sizes of DB and RL. The simulations showed that it also pro- 
vides a much longer time span between firings (200 to 400 seconds vs. 60 to 80 seconds) allow- 
ing greater damping losses. The 0.02 deg/sec rate limit is the only source of concern in the 
baseline set. Although some cases showed rate limit firings caused by flexure, none of these 
firings drove the phase point near the opposite rate limit to create an unstable cycle. The base- 
line set should provide stable if not ideal performance in attitude hold situations. 



114 



REFERENCES 

8-1 "Shuttle On-Orbit FCS Screening Criterion for 'Pivot' Type Employable Payloads," 

NASA/JSC Avionics Systems Division Internal Note JSC-20104, prepared by Charles Stark 

Draper Laboratory, September 1984. 
8-2 Kirchwey, C, and Sackett, L., "Stability of the Shuttle On-Orbit Flight Control System for 

a Class of Flexible Payloads," Charles Stark Draper Laboratory Report CSDL-P-1708, June 

1983. 
8-3 Kirchwey, C, "Computer Programs for 3-D Stability Gain Evaluation," Charles Stark 

Draper Laboratory Internal Memorandum KBK 85-1, August 6, 1985. 



115 



SECTION 9 
RESULTS FOR THE VARIABLE CONFIGURATION SYSTEM 

The DISCOS/SDAP simulation was used to investigate the dynamic interaction of the 
COFS-II system during antenna slewing with the FCS active. Although it is unlikely that the 
FCS will be active during antenna slewing, the results of this section consider that possibility. 
The simulator was not configured to include an active FCS and an active gimbal angle hold 
simultaneously; therefore, only slewing was considered. Because of the very low torque level at 
which the gimbals saturate, it was assumed that during a slew the servo would be saturated at 
33.9 N-m (25 ft-lb). Slewing was limited to one axis at i time, either the elevation (EL) or 
lateral (LAT) gimbal. Active mast damping was included in some cases. 

Slewing always began with the antenna pointing in one of the three "standard" configura- 
tions: antenna pointing aft, antenna pointing upward away from the payload bay, or antenna 
gimballed 45 deg in EL and LAT from the aft direction. Slewing was performed by 
commanding an open loop torque at the gimbal location. A maximum accelerating torque was 
commanded followed by the reversed polarity decelerating torque for an equal length of time. 
For example, a 40 s acceleration followed by a 40 s deceleration results in a slew of about 90 
deg. The maximum slew rate in that example is about 2 deg/s. Given the limited gimbal range 
it is unlikely that the given maximum gimbal rate of 4 deg/s would ever be reached. There- 
fore, there would be no "coast" period, and none was incfuded in any of the simulations. 

The input to the FCS includes mission dependent parameters called I-loads. These include 
expected available per-axis control accelerations and expected individual VRCS jet accelerations 
used by the phase plane and jet select. Actual accelerations depend on the mass properties of 
the composite vehicle and change as the configuration changes. Some of the simulations used a 
set of jet acceleration values which were based on the first configuration, antenna pointing aft. 
Other simulations used a set of jet accelerations which were chosen by assessing actual accelera- 
tions for the three standard configurations and by picking a set expected to give reasonable 
performance for all three configurations. 

The attitude error deadband and the attitude rate cieadband (rate limit) may be changed 
by the crew for different mission requirements. For most of our simulations, we used typical 
values for VRCS operations of 1.0 deg and 0.02 deg/s, respectively. For slewing, it was 
assumed that the FCS would be in an automatic attitude hold mode. 

Fourteen cases were simulated illustrating typical slews with FCS attitude hold. Some 
characteristics of the cases and selected results are summarized in Table 9-1. The FCS attitude 
error deadband ranged from 0.2 deg for cases 1-4, to 2.0 deg in cases 1 1 and 14. The rate limit 
was 0.02 deg/s except in cases 11 and 14. The slew acceleration was 10 s followed by a slew 
deceleration of 10 s in cases 1-4; acceleration and deceleration were 40 s each for other cases. 
A deceleration was followed by a 10 s period with free gimballing, since the software had no 
capability to either lock the gimbal or to perform an active servo hold following a slew. The 
gimbal axis about which the slew occurred is given (EL, parallel to the Orbiter pitch axis, and 
LAT, often parallel to the roll/yaw axes). The I-load choice is shown: 1 refers to jet accelera- 
tions based on configuration 1 mass properties, 4 refers to a selected set based on all three 



117 



Case 



Deadband 
(deg) 



1 


0.2 


2 


0.2 


3 


0.2 


4 


0.2 


5 


1.0 


6 


1.0 


8 


1.0 


11 


2.0 


12 


1.0 


13 


1.0 


14 


2.0 


15 


1.0 


16 


1.0 


17 


1.0 



Rate 

Limit 

(deg/s) 



0.02 


20 


0.02 


20 


0.02 


20 


0.02 


20 


0.02 


80 


0.02 


80 


0.02 


80 


0.05 


80 


0.02 


80 


0.02 


80 


0.10 


80 


0.02 


80 


0.02 


80 


0.02 


80 



Slew 

Period 

(s) 



Gimbal 

Axis 

(EL, LAT) 



ALT 

AZ 

ALT 

AZ 

ALT 

AZ 

AZ 

AZ 

AZ 

AZ 

AZ 

AZ 

AZ 

ALT 



I -Loads 



Initial 
Configura- 
tion 



Mast 

Damping 

(%) 



0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
5.0 
5.0 
5.0 



Maximum Torque 
Magnitudes (N-m) 



Mast 
Base 



195 
999 
757 
752 

2209 
781 

1817 

1308 
945 
992 
978 

1278 
709 

1402 



Mast 
Tip 



328 

83 

67 

63 

297 

307 

144 

114 

230 

230 

230 

101 

230 

235 



Antenna 
Base 



77 

55 

75 

68 

77 

75 

89 

87 

100 

114 

109 

82 

76 

77 



Maximum 
Antenna 

Base 
Deflection 
(millideg) 



58 
21 
59 
17 
61 
45 
71 
71 
80 
91 
87 
61 
61 
61 



Antenna 
Slew 
Angle 
(deg) 



6.0 

6.3 

6.8 

6.8 

93.0 

92.1 

84.8 

95.1 

78.2 

78. 3 

87.6 

84.5 

78.5 

93.7 



No. 
of 
jet 
firings 





27 



33 

8 

1 36 

70 

32 

1 14 

1 16 

24 

76 

113 

8 



RCS 
Fuel 
Use 
(Kg) 




1.2 


1. 3 
0. 7 
5.2 
4.1 
4.4 
5.6 
5.6 
4.0 
4.2 
5.6 
1.0 



• -Slew period includes slew acceleration followed 
by equal time of slew deceleration 

• Gimbal axes 

EL: elevation 
LAT: lateral 



Moads used by FCS phase plane, slate estimator, 
and |et select, based on composite mass 
properties 

1: configuration 1 mass properties (antenna 
facing aft) 

4: selected lor better performance in all 3 
initial configurations 



Initial configurations 

(1) antenna facing aft 

(2) antenna facing upwards 

(3) antenna gimbailed 45 deg in EL and LAT 



Table 9-1. Antenna slew simulation results summary. 



initial configurations. The initial configuration is indicated: 1 refers to the antenna facing aft, 
2 to the antenna facing upwards, and 3 to the antenna girnballed +45 deg in EL and LAT 
(upward and toward the right wing). The mast damping is listed. Most cases assumed only the 
standard structural damping of 0.5% of critical. Cases 15-17 included 5% damping in the mast 
flex modes to approximate the effect of the active proof mass mast dampers. 

The remaining columns show simulation output quantities. The maximum torque magni- 
tudes are given for three locations of interest: the mast tase, the mast tip, and the antenna 
base. The mast base maximum torque ranged from 195 N-m to 2209 N-m for the 14 cases. 
Based on data obtained on COFS-I, the maximum allowed mast base load is 9096 N-m. The 
mast base loads for the 14 simulations were well below that limit. The mast tip maximum 
torque ranged from 63 to 628 N-m, and the antenna base maximum torque ranged from 55 to 
114 N-m. No load limits were provided for these locations. The maximum antenna base 
deflection with respect to the platform ranged from 17 to 91 mdeg. Because of the nature of 
the antenna model used in DISCOS, the antenna deflection is actually the model's column base 
deflection. The total antenna slew angle excursion is shown. The 20 s slews resulted in 6 to 6.8 
deg gimballing. The 80 s slews resulted in 78.2 to 95.1 deg gimballing. It varied because the 
torques caused by RCS firings were added to the 33.9 N-m saturated servo torque over the 
period of the slew. The number of VRCS jet firings ranged from to 136 and fuel use ranged 
from to 5.6 kg. 

In the short runs of cases 1 and 3, there were no RCS jet firings. For the other cases, 
VRCS firings occurred when the rate error estimate reached the rate deadband. There was no 
external torque disturbance such as gravity gradient. At the initial time, the servo torque was 
applied as a step function. As the antenna moved, so did the Orbiter. The Orbiter IMU 
detected the motion and its measurements drove the state estimator. Both rate and disturbance 
acceleration are estimated and input to the attitude controller. Typically, the rate limit was 
reached before the attitude deadband. A jet firing then followed to maintain the phase point 
within the rate limit. The estimate of disturbance acceleration (caused by the estimator- 
unmodelled antenna motion) affected the firing termination time. Both the rate estimate and 
the disturbance acceleration estimate lag the actual value: of each due to the dynamics of the 
estimator. This characteristic can cause poor FCS performance when there are unmodelled dis- 
turbance accelerations such as flexibility and antenna gimballing. During the antenna slew, the 
FCS is trying to maintain Orbiter pointing within the attitude and rate deadbands. 

The system pitch moment of inertia is much larger than the roll moment of inertia. The 
moments of inertia of the antenna about the EL and LAT axes are about the same. Thus slew- 
ing of the antenna in an axis parallel to the pitch axis tends to have less effect on the Orbiter. 
The EL axis is parallel to the pitch axis for configurations 1 and 2. If the disturbance were 
confined to the pitch axis, it would take a while for the phase point to reach a firing line. 
There would be a correcting firing and then a non-firing period again. Slewing in LAT typi- 
cally caused Orbiter response in the roll and yaw axes. Because of the smaller roll moment of 
inertia, and the firing lines were reached more quickly. Typically this contributed to longer 
firings, more firings, and more fuel use. 



119 



A few cases were run with deadbands and rate limits increased over the nominal values. 
Because of the long firing times for any rate limit associated with LAT slews, the effect was 
small. Larger rate limits did tend to reduce the number of firings for pitch plane slews or small 
angle slews. 

The first four cases were short runs with total slew periods of 20 seconds. Two initial 
configurations and two slew axes were included. For the EL slews of cases 1 and 3, there were 
no RCS firings. Orbiter motion was confined to the pitch plane, and because of the large 
Orbiter pitch moment of inertia, the phase point did not reach a firing line for the small slew 
angle. The relatively large mast tip torque in case 1 was due to the transient start-up torque of 
the servo step function, which may not be realistic. For the LAT slews of cases 2 and 4 the 
phase point did reach a firing line in the roll control axis and so there were jet firings. There 
was not much difference in the results due to the different initial configurations. 

The initial conditions of case 5 were the same as in case 3, but the slew was 80 seconds 
long. There were jet firings. Figure 9-1 shows the pitch axis phase plane. The phase point 
remains well within the attitude error deadband of 1.0 deg. As the Orbiter rate decreases in 
response to the antenna slew, the phase point approaches the lower rate limit of 0.02 deg/s. 
When it reaches the rate limit, there is a firing which moves the phase point to the SI 1 cut-off 
line, near the positive rate limit. The location of the Sll line is affected by the disturbance 
acceleration. Following that firing, the antenna slew rate begins to decrease and so the Orbiter 
rate error estimate begins to increase. When the positive rate limit is reached there is a firing 
which cuts off when the Sll line is reached. Due to estimator lag, the Sll line is slowly mov- 
ing downward. The phase point again drifts up to the rate limit. There is another firing, 
longer this time, since SI 1 has moved further. The undulations in the phase point are due to 
flexibility. 

Figure 9-2 shows time histories of the estimated pitch rate and the estimated pitch distur- 
bance acceleration for case 5. The sharp vertical slopes on the rate plot indicate jet firing 
times. The disturbance estimate lags the actual disturbance. The estimate is shown to change 
sign at about the time of the termination of the third pitch firing. Figure 9-3 shows the 
Orbiter pitch torque and the mast base y-axis (pitch) torque. Note that the three pulses each 
reinforced the base load. 

The initial conditions of case 6 were the same as in case 5; however, the LAT gimbal was 
slewed. The Orbiter roll axis experienced quite a bit more jet firing activity than the pitch axis 
had in case 5. The number of jet firings and the fuel use was considerably larger. There was 
not the flex reinforcement that occurred in case 5, and thus the mast base load was smaller. 

Cases 8 and 1 1 were similar except that the deadband and rate limit were increased over 
the nominal values for case 1 1 . The principal effect was a change in the timing and number of 
the firings, causing some difference in loads and fuel use. 

Cases 1-12 and 17 used I-loads based on configuration 1 mass properties. Cases 13-16 
used I-loads based on inspection of the mass property effect of the three initial configurations. 



120 



00 

e 



i 



» 

B 
B 

m 

i 

P °. 
n 

BT* 






ta 

a 





i 


2.50 




O ~ 

-fc. 






en 






i 
CD- 






co 






o 






i 






_ 






en 




"0 






<-+ 


o_ 




o 


o 




or 


o 




> 


o 




«-♦■ 


• 




r-h 


en 




r-¥ 






c 






CL 


o 




CD 


o 




m 







Pitch Estimated Rate Error (deg/s)/100 

-2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 

-" h 



o 
en 




i • ■ 

o 


en 




H 


o 




CD 
CQ 


CD 

o 






o 
en 

o 

CO 

o 


1 1 1 i , 1 _ 



a 



-C o 

°- In 
■a o)°. 

CD a 

| . 

to "&° 



10 



20 



H ' r- 



30 



40 50 

TIME 



60 




70 



80 



90 



100 




a 



40 50 

TIME 



Figure 9-2. Run 5 - rate and disturbance acceleration estimates. 



122 




10 



20 



30 



40 50 

TIME 



60 



70 



80 



90 



100 



03 


Q 

a 

o 


• ^Hi 


en 


X 




CD 




1 


*»—-. 


> 


E 
■ 


03 


1 

z 


CO 




CD 







13 


03 


or 

1_ a 


CO 


o g 




10 



20 



30 



40 50 

TIME 



60 



70 



100 



Figure 9-3. Run 5 - pitch jet torque and mast base load. 



123 



The difference between these two sets of I-loads was not great, although I-load differences 
based on the other configurations would be greater. Cases 12 and 13 were similar except for 
the use of the two sets of I-loads. There were minor differences in the results. 

Case 13 illustrates one cause of poor FCS performance during an antenna slew. Figure 
9-4 shows the roll phase plane for case 13. The phase point remains within the attitude error 
deadband, but as the Orbiter attitude changes as the antenna slews, the phase point moves 
toward the lower rate limit. When it reaches the rate limit, jets begin to fire. Jets continue to 
fire until nearly the end of the run. With the large roll moment of inertia of the COFS-II, the 
roll jet torque authority is relatively small. Because of the initial antenna location, antenna 
slewing causes attitude error build-up in all three control axes. Although a roll axis command 
continues, the actual jet combinations selected change as conditions and commands from the 
pitch and yaw axes influence the jet selection. In the roll phase plane, the phase point is driven 
upward by the jet firings. After crossing the zero rate line, the phase point slope is affected by 
the antenna slew deceleration which begins at 40 s. The firing continues until Sll, which is 
close to the upper rate limit, is reached. By this time, however, the actual disturbance accelera- 
tion has changed sign, although the disturbance estimate lags. The Sll line starts to move 
downwards, but the phase point is pushed upward now, and the roll jet command changes sign. 
After some chattering between the rate limit and the nearby Sll line, the phase point moves 
above the rate limit and the roll command remains on. Axis cross-coupling causes the actual jet 
combinations to vary over time. Gradually, the disturbance decreases (as the slew decelerates) 
and the phase point can be controlled to within the rate limit. The SI 1 line has moved below 
the zero rate axis and the jet firings carry the phase point downward. 

Figure 9-5 shows the estimated roll rate and the actual roll rate for case 13. The flexure 
is more noticeable in the roll axis because of the smaller Orbiter roll moment of inertia. The 
estimator's smoothing and lag characteristics cause distinct differences in the two quantities. 
Figure 9-6 shows the jet torques in the orbiter roll, pitch, and yaw axes as various jet combina- 
tions are selected. Figure 9-7 shows the mast base load torques in the x-axis and the y-axis. 
The effect of the varying jet combination firings and the flexure is indicated. The peak load 
occurs at about 57 seconds. Because the jets have been on almost continuously, there was high 
fuel use of 5.6 kg. 

The deadband and rate limit were increased above nominal in case 14; otherwise case 14 is 
similar to case 13. There were fewer firings and less fuel used in case 14, but the loads were 
similar. 

Cases 1-14 included nominal structural damping of 0.5% of critical. Cases 15-17 
increased the damping to 5% in the mast to approximate the effect of the active proof-mass 
dampers. Case 15 is similar to case 8 except for the increased damping and the I-load change. 
There is a lower mast base load, although that is primarily due to random differences in the 
firing patterns. Cases 16 and 13 differ only in the damping included. The differences in the 
results are small. 



124 




H h 



-Q.60 -0-45 



Roll Attitude Error (deg) 



Figure 9-4. Run 13 - roH phase plane. 



125 





a 
on 


CD 


a 


4-" 




CT3 




cc 


o 




r"~ 


^^^ 


^ a 


o 


aT ° 


cc 


^s a 




D5 


T3 





CD 


T3 


4-> 


^m^ 


CD 


a 


E 


a. 


■ ^^ 




+-> 





CO 




LU 






40 50 

TINE 



00 



CD 




a 
ao 


4-> 




■ 


03 


o 


a 


CC 


\ 






«*^^ 


n 


^^ 


CO 


a 


o 


\ 


n 


cc 


O) 






CD 

"a 


a 

03 


(U 




* 


D 




a 

i,. 


+-» 




1 


O 






< 








40 50 

TIME 



100 



Figure 9-5. Run 13 - actual and estimated roll rates. 



126 



_+ i ) 1- 



orjg 




_< 1 *- 




r 



-H 1 1 1- 



T 



o 

-. ! l-O 



o 

CO 



-n 1 1 ^- 



O 
03 



O 



O 



o 
in 



LU 



--OI 



o 



o 
eg 



_.o 



4 1 1 1- 




I I 1 »- 





-I 1 1 1- 



o 
o 



o 
en 



o 

03 



o 



o 



o 
in 



LU 



o 



o 

Cvl 



009- 



OOOfr 



OOOtr- 0003 



0002- 



(lu-n) onbjoi ||oy (ui-n) anbjoi ipiid (uj-N) snbjoi mba 



o 



I 
f> 

a 
s 

»\ 

<u 

■_ 

3 
OS 



CM 



Y-axis Torque (N-m) x-axis Torque (N-m) 



oo 



3 
«5' 



I 



W 

B 
3 

>— ' 
U> 

I 

s 

to 



S" 

a. 

(A 




Case 17 is similar to case 5 except for the mast damping. The mast base load is consider- 
ably smaller for case 17 than for case 5. This is not caused directly by the difference in damp- 
ing. Basically, it is a "chaotic" effect. In both cases 5 and 17, the disturbance acceleration 
estimates (Figure 9-2) are close to zero at the time of the third major roll firing. But they were 
of opposite signs in the two cases. The sign difference caused a large change in the SI 1 switch 
line location and thus a large change in the duration of the firing. (Compare the phase planes 
of Figure 9-8 and 9-2.) In case 5, there was considerable reinforcement of the bending, which 
did not occur in case 17. These cases illustrate the observation that the resulting firing histories 
and loads can differ greatly for small differences in initial conditions or other inputs. 

These runs did not investigate FCS instability directly. However, instability is unlikely. 
For instability to occur the flex oscillations must build up to a fairly large magnitude. Gener- 
ally, many firings timed to reinforce the oscillations are needed. That is unlikely to occur given 
the chaotic firings observed in these runs. Although the active mast dampers had little effect 
on the runs performed, increased damping would reduce the probability of instability further. 

Generally the effects seen in these runs were due :o slew attitude disturbances and the 
resulting RCS jet firings and not due to flexibility. The VRCS jets did not cause much flexure 
and there was minimal flexure feedback to the FCS. 

There were undesirable effects due to having an active FCS during the gimbal slewing. 
The FCS wastes fuel as it tries to correct for the unmodelled disturbance caused by the antenna 
motion. The RCS firings cause unnecessary stress on the systems, including the loads on the 
structure and on the gimbals and servo motors. The antenna slew angle is affected by the jet 
torques. Since the servo will probably be operated in an open loop saturated mode during a 
slew, the jet firings make it difficult to obtain a desired slew angle. Therefore, it is recom- 
mended that the FCS be in free drift mode during antenna slewing. At the end of a slew, the 
Orbiter will have changed attitude slightly. If desired, the attitude could be adjusted with RCS 
firings with the gimbals locked. 



129 




-3-20 -2-40 -1.60 -0-80 0-00 0-80 1-60 2-40 3-20 

Pitch Attitude Error (deg/10) 



4.0Q 



4-80 



Figure 9-8. Run 17 - pitch phase plane. 



130 



SECTION 10 
GIMBAL CTPyr ySTmiCTURAL DYNAMICS INTERACTION 



One of the original goals of the COFS-II servo study was to determine the extent and 
impact of dynamic interaction between the structural dynamics and the servo and then the 
interaction between the servo and the orbiter flight control system. A second goal was to 
quantify the role played by the servos in damping structural vibrations. 

It quickly became obvious, however, that the hardware parameters, specifically the ±33.9 
Newton meters (±25 ft lbs) torque saturation and ±4 degree per second slew saturation of the 
Sperry Advanced Gimbal System (AGS), were not compatible with the high moment of inertia 
of the COFS-II antenna. Any attempt to achieve a servo closed loop bandwidth of the , desired 
10 rad/sec resulted in a system which saturated at the extremely small error signal of 0.0005 
decrees Thus, a conventional servo design would yield a highly oscillatory system. If the 
constraint to use the Sperry AGS is maintained, the proper design would entail the incorporation 
of a nonlinear, minimum transfer-time technique which involves dividing a desired angular 
repositioning distance in half, and accelerating (not slewing) at saturated torque for that ha 
angle then decelerating at negative saturated torque for the remaining half angle. Essential y, 
this is an open loop operation. Following this phase, the positional servo loop is automatically 
closed and any residual, small error signal can be stably eliminated since the load angular 
velocity should be close to zero. 

Thus investigating the dynamic interaction of this small angle, small angular velocity 
servo system with the COFS-II structural dynamics is a necessary precursor to quantifying the 
impact of the servos on structural vibrational damping. 

An excellent method to describe the contribution io servo loop dynamics by the structural 
dynamics is to define a mechanical admittance function of gimbal servo motor angular velocity 
per unit motor torque, £ «•>. as portrayed in Fig. 10-1 (The solid lines are asymptotes of the 
function). With such a system, trying to close the servo loop with stability in the high 
frequency" region of structural poles and zeros requires careful network compensation as 
portrayed in Fig. 10-2. Specifically, note the need for network compensation to achieve 
adequate phase margin,* m, every time the amplitude spectrum has a downward zero db 
crossing. 

For the COFS-II system, this mechanical admittance function was derived analytically 
from a multiple body idealization of the mechanical system and then verified by obtaining the 
mechanical admittance function amplitude and phase spectra via NASTRAN using a finite 
element model of the mechanical system truncated beycnd 25 modes. 



131 



3 







.a 
E 



u 

B 
3 






E 

08 



B 
JS 

u 



0* 



132 



cei 




Analytically, the derived (see Appendix C) pitch gimbal angle per unit torque was: 



— (s) = (s) = 

T M sT M 



1.617 x 10 



•2 <J2 



(s 2 + co\ ) (s 2 + o>§ ) (s 2 + <o% ) 
tl _f2 2_ 

(s 2 + <yf>) (s 2 + <yf>) (s 2 + <y 2 ,) 



cw z = 0.80, <o z = 1.55, (O'l = 11 
where 1 2 ^3 rad / s 

cop = 0.82, <y p = 10.6, <o p = 36 

■i- a o 



Note the near cancellation of two pole-zero pairs, co^ with cop and a>p with cov 



The asymptotes of the amplitude spectrum of this function may be depicted as follows: 




^/o 



y 



QJ 



U4 



Any servo loop closure above the 1.55 rad/sec "zero," even with network compensation, would 
be futile with the existing hardware since the 25 ft lbs torque saturation coupled with the large 
moment of inertia of the COFS-II antenna produces a servo which saturates at minuscule frac- 
tions of a degree of command angle. A 10 rad/sec bandwidth servo would saturate and become 
bang-bang at any command over 0.0005 degrees. Thus, for the present purposes of examining 
servo contribution to structural damping, loop closure was achieved on the low frequency 
asymptote of the mechanical admittance function at 1 rad/sec. 

Figure 10-3 assumes a basic quadratic portrayal for the two gimbal servo, where the units 
of the constants are: 



K-i 
A 

K t 

Tm 

Tl 

Jm 

Ji 

k t 

©c 



e 



'M 



Transducer constant, volts per radian 
Amplifier gain, amps per volt 
Torque motor constant, ft lbs per amp 
Torque output of torque motor, ft lbs 
Load torque of structure on torque motor 
Motor moment of inertia, ft lb sec 2 
Payload platform moment of inertia 
Tachometer constant, radians per rad/sec 
Command angle, radians 
Motor angle, radians 




siwciural dtjmwfcs 



/ 



(%+?,)* 



SI 



M. 



9. 



M 



Figure 10-3. Basic quadratic gimbal servo. 



135 



Figure 10-4 is the same servo loop of Figure 10-3 expressed in the canonical form of the 
closed loop parameters urn (servo bandwidth), and f (dimensionless damping ratio). Note that 
J L represents the entire rigidized-structure moment of inertia, whereas Jx is only the payload 
platform moment of inertia. 




T 



jk- 



a- 



^( j ^m^j, 




% =±2Sft./is 




n 



s 



a 



A7 



— ~T/,°/ 



Figure 10-4. Quadratic gimbal servo in canonical form. 

The format of Figure 10-4 assumes servo loop gains consistent with stability at loop 
closure low-band on the structural spectrum. Thus, from Figure 10-4: 



T M = <4 < J M + Jl> <*c " % " |f *M> 



where T M = +. 25 ft lbs and 5 M = ± 4 deg/s 
max max 



Note that gimbal motor torque is impressed outward upon the antenna structure, and 
equal and oppositely downward upon the mast structure. (That is, the servo ba§£ is compliant, 
not rigid.) 

A series of computer runs were submitted to investigate system stability and the ability of 
the servo to dampen structural vibrations. Of course, the 25 ft lbs torque saturation, when 
faced with the massiveness of the COFS-II antenna, did not allow much in the way of fast servo 
response. For a given error signal, torque saturation varies with the square of the servo 
bandwidth and proportional to the structural moment of inertia. Thus, for the given ±25 ft lb 
torque saturation, a 10 rad/sec servo coupled with the COFS-II antenna reaches saturation at 



136 



only 0.0005 degrees of error signal! A bandwidth of 1 rad/sec is 100 times more forgiving and 
saturates at 0.05 degrees. Notwithstanding these low saturation levels, the existing hardware still 
allows examination of the ability of the gimbal servo to dampen structural vibrations. 

The following four computer simulation runs quantitatively demonstrate the effectiveness 
of the servo to improve structural damping. 

Run 1 

With the mast initially undeflected, the servo commanded angle (with respect to the top of 
the mast) was set to zero, the elevation gimbal of the antenna was given an initial deflection of 
0.03 degrees and released at t=o+. The system constants were: 



jfev \ 003 ' 

.— J^gL_ ' Servo B.W. = 1 rad/sec 

1^ J „ , Servo £ = 0.5 

; deflected „ , .. n nnc 

I platform Structural^ = 0.005 

Servo damping via tachometer 




Results: The antenna platform vibration was attenuated consistent with a dimensionless 
damping ratio, £ of 0.026 



Run 2 



To verify that the servo actually contributed to the- antenna vibrational attenuation, the 
above run was repeated, but with the servo locked . The initial condition was deflection of the 
antenna base spring by 0.03 degrees and then release at t=o+. 

Results: The measured effective £ was 0.005, simply the structural intrinsic damping, thus 
demonstrating the damping effect of the servo when operating in Run 1. 

Run 3 

Next, to investigate the servo contribution to damping of mast vibration, the mast was 
now deflected and released. However, under these conditions the antenna elevation gimbal sees 
no initial deflection with respect to the top of the mast and hence there is no initial servo error 
signal as a result of the deflection. To remedy this situation, we must realize that we are 
concerned with stabilizing the antenna line of sight with respect to inertial space and not with 
respect to the top of the mast. Thus, we shall assume the employment of gyroscopic 
measurements to define the gimbal angles and commands with respect to inertial space. 
Consequently, the elevation gimbal command at t=o+ with respect to inertial space was set to 



137 



zero. Also, since the mast was deflected in its first mode shape with a tip deflection of 0.03 
degrees, the elevation gimbal deflection (or more exactly, the elevation platform angle) with 
respect to inertial space at t=o+ was 0.03 degrees. 

As seen in the accompanying configuration sketch, the top of the mast is initially 
deflected 0.03 degrees in elevation with the antenna gimbal elevation angle initially maintaining 
the antenna column orthogonal to the top of the mast. The deflected mast is then released at 
t=o+. The system constants were once again: 




\ deflected SeTV0 BW - " l rad / sec 

\ mast Servo f = 0.5 

\ ygr Structural £ = 0.005 

_j f£i Servo damping via tachometer 



Results: Since we are interested in how well we are maintaining the antenna line of sight 
with respect to inertial space, we monitored the antenna platform angle with 
respect to inertial space and measured the vibrational atennuation. The effective £ 
was only 0.008. Similarly, observations of the vibration of the tip of the mast 
showed attenuation consistent with a £ of 0.008. Thus, the active servo was only 
somewhat damping the entire structural system which has intrinsic damping of 
f = 0.005. By making a proper change in servo design, much better results may 
be obtained as seen in the next run. 

Run 4 

This run is identical to Run 3 above except that the servo damping signal was obtained 
from a rate gyro instead of a tachometer. That is, servo viscosity was referenced to inertial 
space instead of the top of the mast. Thus: 



\ 



0<°(o+) = 

i§^f \ 003 ' 0$ (o+) = 0.03° via deflection of the mast tip 

by 0.03° 



rv c 



deflected ^M is ^"^ res Pect to inertial space 

i mast Servo BW = 1 rad/sec 

| /$ Servo C = 0.5 

g ^_ ~= H Structural f = 0.005 



Servo damping via rate gyro 



138 



Results: Monitoring the vibrating antenna platform angle with respect to inertial space 
exhibited an attenuation of vibration consistent with an effective £ of 0.012! 

Of significance here is the greatly increa- ed structural damping achieved by the 
servo when its own damping signal was obtained from a rate gyro rather than a 
tachometer. This is understandable when one realizes that the tachometer signal 
can be zero if the servo gimbal angle remains fixed with respect to the top of 
the mast even with the mast vibrating. 



139 



140 



SECTION U 
CONCLUSIONS 

The COFS-II system has been assessed from the standpoint of dynamic interaction with 
the Shuttle Orbiter flight control system. Issues of FCS stability in the presence of the flexible 
mast/antenna system as well as internal loads and general FCS performance have been 
investigated with analysis and simulation. Two simulations were used. One assumes a fixed 
configuration (locked gimbals) of the COFS-II system; the other allows gimbal motion. Because 
of the small torque capability, the linear range of the servo is quite small. A study was made of 
the servo characteristics over this linear range. 

A review was made of Shuttle requirements that wc uld influence the amount of time the 
Orbiter could be in a free drift mode. Experimenters desire periods of free drift to minimize 
disturbances on the COFS-II system. NASA STS documentation, flight experience, other 
payload simulation experience, and simple analysis indicate that with no payload constraints 
given, the Orbiter may be in free drift for at least six hours under nominal conditions. 
Thermal, communications, and IMU alignment requirements were considered. During free 
drift, the attitude will vary depending on the effects of gravity gradient and aerodynamic 
torques and other disturbances. 

Three fixed configurations were assessed for FCS stability, FCS performance, loads due to 
RCS firings and flexure, and the effect of adding the act ve mast damping system. It was 
found that FCS instability is possible. Several structural modes are potentially unstable. 
Typically, instability requires a large periodic disturbance for initiation, unless tight deadbands 
and rate limits are used. Experiment dampers somewhat reduce that probability of instability. 
FCS performance is affected by payload flexibility. However, for stable conditions, the effect 
is not large. Maneuvers should be simulated in advance of flight to minimize poor performance 
and reduce the probability of initiating an instability. Maneuver rates should be kept as small 
as practicable. Mast base loads appear to be acceptable for typical attitude maneuvers and 
attitude holds. Simulations also output mast tip loads and antenna base loads. Since load limits 
for those locations were not available they were not evaluated. Unstable oscillations can lead to 
large loads. 

Simulations of antenna slew with the FCS active were performed. Because of the limited 
torque capability of the gimbal servo, the servo will probably be saturated during a slew. For 
most slew angles, the peak slew rate of 4 deg/s cannot be reached. Therefore, for slew 
simulations, the servo behavior was approximated by an open loop torque command at the 
gimbal location. It is recommended that slewing occur with the FCS inactive. An attitude hold 
during a slew wastes fuel, and the jet torques overcome the servo motor. Thus an open loop 
commanded slew can result in various slew angles, depending on the jet firing influence. The 
mast base loads were acceptable with active FCS for all the slew simulations performed. The 
simulations were limited to single axis, open loop, servo torque-saturated slews. 



PRECEDING PAGE BLANK NOT FILMED 



141 



The servo interaction study was limited because of the small linear operating region of the 
servo. A closed loop bandwidth of 10 rad/s results in a system which saturates at only 0.0005 
deg. Consequently, a conventional servo design is highly oscillatory. The impact of the servos 
on structural damping was investigated and it was found that reasonable damping could be 
obtained if the servos themselves were stabilized by rate gyros and if the servo was operating in 
its linear range. 

Our study indicates possible undesirable interaction between the Orbiter FCS and the 
flexible, articulated COFS-II mast/antenna system, even when restricted to VRCS jets. 
Undesirable conditions can probably be avoided with careful planning, pre-flight analysis and 
simulations, and flight operational constraints. 



142 



Appendix A 



Selection of Spring Constants for the Three Rigid Body Model of the COFS-II Hoop Column 

Antenna 



143 




w 



TUfiL/m wmmmmm 



■Ml 



DI-87-09 

TO: Stan Fay 
FROM: Steve Gates 
DATE: 20 July 1987 

SUBJECT: Selection of Spring Constants for the Three Rigid Body Model of 
the COFS-II Hoop-Column Antenna 

This memorandum presents the analysis performed to determine the 
torsional spring constants for the articulated three rigid body model of 
the COFS-II Hoop-Column Antenna . 

The mechanical idealization of the COFS-II Hoop-Column Antenna, 
described in Reference 1, is shown in Figure 1. The model consists of 
three distinct axisymmetric rigid bodies, labeled the column, feed, and 
hoop. These bodies are interconnected at pivot points by discrete massless 
torsional springs. 




Figure 1. Idealized Hoop-Column Antenna 



144 



FOR CHARLES ST ARK DRAPER LABORATORY USE ONLY 



Refering to Figure 1, the column is connected to the base through a 
frictionless pivot which permits rotation aboi t two mutually orthogonal 
axes oriented perpendicular to the column's nominal longitudinal axis. 
These two rotational degrees of freedom are resisted by identical torsional 
springs. The hoop is constrained to move in such a way that it has a 
single degree of freedom relative to the column. Specifically, the hoop 
lies in a plane perpendicular to the column's longitudinal axis at a fixed 
distance above the base. When the column deflects it is assumed that the 
hoop follows along such that there is no relative displacement between the 
two bodies except for a simple rotation of the hoop in its plane about the 
column axis. This angular displacement is resisted by a torsional spring 
acting between the hoop and column. The rigid, feed is appended to the 
column top through another frictionless pivot. This joint allows rotations 
of the feed relative to the column about two mutually orthogonal axes 
perpendicular to the column's longitudinal axis. These two relative 
angular deflections are counteracted by identical torsional stiffnesses. 

Equations of Motion for Small Vibrations 

Nomenclature 

We establish an inertial reference frame, with axes X Y Z and unit 
vectors ? 5 £, fixed to the base. For each of the bodies, we define a body 
fixed reference frame with origin at the respective mass center. The 
subscripts "c", "f", and "h" will be used to 3istinguish the coordinate 
frame axes, (x, y, z), and unit vectors, (i, f, £) . associated with the 
column, feed, and hoop respectively. Figure 2 illustrates the respective 
frames of reference. Note that for each body the respective x-axis 
corresponds to the axis of inertial symmetry. 

Figure 3 displays the essential geometric parameters: 

a - distance from column base to column tip 
b - distance from feed base to feed centroid 
c - distance from column base to column centroid 
h - distance from column base to hoop plane 



145 




Figure 2. System Reference Frames 




Figure 3. Geometric Parameters 



n 



146 



The respective inertial parameters for tine system are: 



[I C ] = 



m , m f , m. - mass of column, feed, and hoop 



centroidal inertia matrix of the 
columr. w.r.t. the column frame 



[i' 



- centroidal inertia matrix of the 
feed w.r.t. the feed frame 



[I h ] 



- centroidal inertia matrix of the 
hoop w.r.t. the hoop frame 



We define also the torsional spring constants: 

kj - stiffness of column/base spring 
*2 - stiffness of feed/column spring 
k 3 - stiffness of hoop/column spring 

Degrees of Freedom 

Let the angular deflection of the column frame relative to the 
inertial frame be measured by the angles $ lr and tpi, which correspond to a 
"2-3" Euler rotation sequence shown below: 



147 





*■ L 



Since we are ultimately interested in the small oscillations of the 
system about the nominal equilibrium configuration, we shall restrict our 
attention to situations for which (f^, and ip 1# are "small" angles. Under 
these conditions, the relations between the column frame and inertial frame 
can be expressed as 






it 



1 


*1 


-♦r 


it 


*1 


1 





r 


♦1 





1 


( * 



In a completely analogous fashion we define the orientation of the 
feed relative to the column by the angles <J>2' $2> and consider these angles 
to be small. The relations between the feed frame and column frame vector 
bases is then given by 






X. 



"+n 



*2 "*2 



*2 ° 




1 



1 



148 



The planar rotation of the hoop relative to the column will be 
measured by the angle Y, which will be treated as a small quantity. The 
hoop and column frames are related by 



1, 






k. 



1 







1 

-Y 





Y 

1 



+ 

:„ 



->■ 



The five degrees of freedom of the antenna model described above are 
capable of portraying "bending" deflections of the column and feed in two 
orthogonal planes, as well as simple torsional rotation of the hoop 
relative to the column. 

Lagrange 's Equations 

We will obtain the free vibration motion equations for the system from 
Lagrange's equations. To this end we record :.he centroidal velocities and 
angular velocities of the respective components: 



V - inertial velocity of the centroid o: body i. 
i 

to - angular velocity of body i relative to the inertial frame, 
i 



01 = 



V = 



0), = 



v. = 



♦l 3 c + *1 k c 



(^ + l 2 ) J f + fy + * 2 ) *f 

[(a+b)^ + b $ 2 ] f f ~ [(a+b)^ + b$ 2 ] )? f 



149 



\ = * \ + I, 3 + h + ^ k + h 



V h - h ^ 4 - h ^ k* h 



Note that the expressions for the angular velocities reflect the added 
approximation of ignoring terms involving products of: (angular 
deflections) x (angular rates). This approximation, which together with 
the small angle assumption is in keeping with the usual "small motions" 
analysis, will yield the desired linear motion equations. 

The respective component kinetic energy expressions, 



T f =| [m f (a + b) 2 + I*] [^ + ^) 
+ \ (» £ b 2 + I*) {\\ + %) 
+ [m f (a+b) b + l[] (^$2 + t^ij 



1 ( ,2 T h w «2 -2, 1 T h •: 
T h - 2 K h + Z t ) (*1 + *J* 2 X s ^ 



together, give the system kinetic energy 



T-l {q} T [M] {q} 

where {q } = {y, ^ , <j> 2 , ^ , i|» 2 } 

[M] - [Mf 

M„ = I h 
1 1 s 

M 12 =M 13 =M 14 =M 15 -° 

M 22 " M 44 - K° 2 + m f^ +h)2 + ^ + I l + lf t + ^ 



150 



M 23 = M45 = K (a+b)b + I*] 

M 24 = M 25 = ° 

M 33 = M 55 = K b +1 t } 
M 34 = M 35 " ° 
The potential energy for the system is given 

or 

V=ifq} T [K] {q} 

[k] = diagonal (kj, k.,, k 2# k^ k 2 J 

c «-v,«» f n) - a i <5vstem kinetic and potential 
Using the expressions for the total system kx 

ahnw Laaranqes equations in the independent generalized 
energies given above, Lagranges. evi"°>- 

coordinates, T. ♦,. H. ♦.. ♦* '« «' » to "- s^""' ^.^ ^ 
.otion equations for .-11 vibrations about the equilibrium conf.curat^n 

[M] (q) + [K] fq) • !°} 



where 



It is evic 



c 4-u, m ac= anri stiffness matricies 
ident from the structure of tin; »ass ana sturn. 

th ,t Ze -bendin,- deflections In th. XZ an,, X, planes, Measured by 
,♦„♦,) and (♦,.♦,). respectively, are uncoupled fro. each other as well as 
f!ol L torsional motion of the hoop. Also, since th. mass and st.ffness 
coefficients corresponding to each plane of bending are identical, 
necessary to consider only one of the plane, for the analysis of the free 
vibration characteristics. 

Hoop Torsion 

Considering the hoop torsion equation, (the first in the above set), 
it follows immediately that the natural frequency, » T . associated with 
this degree of freedom is given by 



151 



The spring constant required to produce a given natural f 



requency is simply 



k = to I 
3 t s 



:d 



Bending Vibration 



Considering the equations corresponding to the bending vibrations in 
the XZ plane; 



M 22 M 23 



M M 

n 23 M 33 



k 1 



k. 





' 1 




' » 




♦l 







< 




' a ■ 






*2 









t J 




i 



it follows that the characteristic equation for the natural frequencies, 
u i» (i=1» 2), is given by 



"i (M 22 M 33 " M 23 ) " ( k i M 33 + k 2 M 22 ) 4 + k ^ k 2 = ° 

We wish to determine spring constants k, and k 2 which produce given natural 
frequencies U1 , and u> 2 . The natural frequencies satisfy 



"l ^ M 22 M 33 " M 2 3 ) " ( k 1 M 33 + k 2 M 2 2 ) 4 + k 1 k 2 " ° 



(2a) 



U 2 ^ M 22 M 33 " M 23^ - ( k 1 M 33 + k 2 M 2 2 ) 4 + k 1 k 2 = ° (2b) 

Subtracting (2b) from (2a) we obtain 



152 



M 



23 



k 2 = (03 1 + ( o 2 )(M 33 -— ) 



- k 



33 



1 M 



(3) 



22 



Adding (2a) and (2b), and eliminating k 2 from the result, leads to a 
quadratic equation in k-| , which has roots 



-B ±VB 2 - 4AC 
2A 



(4) 



where 



A = M 



33 



B = -{** + ^)(M 22 M 33 - M^) 



2 
*23 



C = u] * 2 2 K 22 (M 22 M 33 - M 2 23 ) 



The possibility thus exists for there to be two real positive values of 
k 1# (and a corresponding pair of k 2 's) which yield the desired natural 
frequencies for a given system. This was indeed found to be the case for 
the parameters of the system treated below. The question of which pair of 
spring constants to use was resolved by considering the mode shapes 
associated with each pair. 

Let, {$i}, denote the eigenvector associated with the eigenvalue 
u> 2 , (i=1, 2). Then 



k, - a). M_. 
1 l 22 



2 
-u>. M_, 
1 23 



2 M 

l 23 



k 2 " \ M 33 



# 1i 



\ 2i 



from which we can 



determine the eigenvector to within a constant factor, 



as 



153 



{*} = 



k. - 0). M__ 
1 1 22 



O) 2 M_, 
l 23 



(5) 



Numerical Results 

Given below are mass properties and geometric parameters for the three 
body antenna model which were derived from a high fidelity NASTRAN finite 
element model of the canti levered Hoop-Column Antenna, provided by NASA 
LaRC. 



m = 126.951 kg 
c ^ 



[I C ] 



11 .264 



1743.736 



1743.736 



kg-m 



m = 117.234 kg 



I*] 



0.853 



34.861 



34.861 



kg-m 



itl = 118.337 kg 



[I h ] 



6631.537 



315.772 

3315.772 



kg-m 



a 


= 9.4715 


m 


b 


= 3.388 


m 


c 


= 4.1755 


m 


h 


= 5.2982 


m 



The first five natural frequencies and corresponding mode shapes of the 
finite element model are given in Table 1, and Figure 4 respectively. 



154 



Mode 1 

Hoop Torsion 




Modes 2 & 3 

1st Planar Bending 




Modes 4 & 5 

2nd Planar Bending 




(S^b 



ggj, 



Figure 4. Finite Element Model Mode Shapes 



155 



ON 



c 

H 

fD 





CO 




rt 


CO 




(D T) 




rt (-• 




» 




l-H 3 




M M 




H 




CO 




x> w 




ns 




3 D. 




OQ H- 









OOQ 




O 




3 3 




en o 




rt a. 




3 co 




rt 




CO 






NJ 




a 




CU 





CO 

fD T) 



CO R 
T) 

H- fD 

3 3 

00 Cu 
H- 

O 3 

O OQ 

3 _ 

co 3 

rt o 

& p* 

3 fD 

rt CO 
CO 



ISJ 

3 

a. 




Natural Frequency (H ) 



a I 



= 0.08 



Wo = W3 = 0» 24 
uii^ = 0)5 = 1.74 



Mode Shape 



Torsion of Hoop w.r.t. Column 



First Beiding Mode (in 2 planes) 



Second Bending Mode (in 2 planes) 



Table 1. 

Using the system parameters and natural frequencies given above, Eqs , 
(1), (4) and (3) yield the spring constants 



k 3 = 1698.141 



N-r, 



I) ki = 1257562.126 N-rt 



1 
k 2 



3285.846 



N-n 



II) k t = 71442.17 N-r, 
k 2 = 57839.166 N-n 

Figure 5 illustrates the mode shapes, Eq. (5), corresponding to each pair 
of "bending" spring constants. It is clear from those results that the 
second pair yields mode shapes which are more in accord with those provided 
by the finite element model. Thus our choice of the second pair of 
constants for the three body model's bending spring stiffnesses. 



Reference 

1. Gates, S. and J. Storch, "Mechanical Idealization of the Orbiter/COFS 
II Structural System," CSDL Intralab Memorandum No. DI-87-02, 
Feb. 5, 1987. 



157 



158 



Appendix B 



Analysis of Free Vibration Characteristics of tie COFS-II Mast Structure 



159 



PRECEDING PAGE BLANK NOT FILMED 




lEMA 



@^«U^ 



MEMO NO: DI-87-03 

TO: S. Fay 

FROM: J. Storch and S. Gates 

DATE: April 7, 1987 

SUBJECT: Analysis of Free Vibration Characteristics of the COFS-II Mast 
Structure 



This memo presents analyses of the torsional and transverse free 
vibration characteristics of a hybrid continuum/discrete model of the COFS 
mast structure. These results are central to our treatment of this major 
structural component in the formulation of the complete COFS-II system 
motion equations. 

The idealization of the COFS mast structure treated here is somewhat 
more general than that described originally in Reference 1 . Specifically, 
it is assumed to be a uniform long slender inextensible continuous beam 
carrying a set of compact rigid bodies fixed along its span. Torsion and 
transverse bending are permitted independently. The discrete bodies may 
possess both mass and rotary inertia, however, their spatial dimensions are 
assumed negligible. The free vibration characteristics obtained from this 
model, provide the means for achieving a high fidelity representation of 
the mast at a minimum cost in the number of degrees of freedom. 

The analyses which follow, respectively treat the torsional and 
transverse bending vibration problems. For each case, two independent 
analyses determining the natural frequencies and eigenfunctions are 
presented. One of the analyses is analytically exact, the other, 
analytically approximate. The former employs singularity functions, while 
the latter implements the assumed modes method. The exact results are 
intended for use in the ultimate system level dynamics model, where they 
will contribute to a simpler, more accurate and efficient set of final 
equations. The results of the approximate analyses serve to validate those 



160 



FOR CHARLES ST ARK DRAPER LABORATORY USE ONLY 



of the exact analyses. Each of the methods for each of the vibration 
problems have been implemented numerically in FORTRAN programs. Selected 
results from those programs are provided. 

Torsional Vibrations of a Bar with Concentrated Inertias 



Consider a uniform circular bar carrying a system of disks along its 
span as shown in Fig. 1. We wish to investigate the free torsional 
vibrations of the bar taking into account the noments of inertia of the 
disks about the bar's torsional axis. 

We take the x axis as the bar's neutral axis and assume that the disks 
are of negligible thickness, with centroids located at the points x=a^^ 
i=1, 2,..., N. The bar has length I, torsional stiffness GJ, and mass 
polar moment of inertia per unit length I. The disks have moments of 
inertia 1^ about the x axis. Let the sequence a-|, a2, • •• , *n be 
strictly increasing. For the purpose of the derivation we assume that 
CKa-j/fc; however the results remain valid for a disk at the bar tip 
(a N =£) by a simple continuity argument. The bar is clamped at x=0 and 
free at x=£. 



(0\ 



XJ 



a 



X7 



D- 



l 
A. 



Figure 1. Uniform Circular Shaft Carrying Discrete Rigid Bodies 



161 



The partial differential equation governing the free torsional 
vibration of the bar can be written as: 



GJ 



i!fi-[l + !l.6(x-..)]-^ ' m 



3x z i = 1 »t' 

where 9(x,t) represents the angle of twist of a cross section at the point 

x and at time t. Note the use of the Dirac delta function to represent the 

concentrated mass polar moment of inertia of the disks. The boundary 

conditions take the form 

jo (2 ) 

9(0, t) = 0, -g (£, t) = 

Seeking solutions to Eq. (1) of the form 9= 0(x) e 1 "* we obtain 



_d_2e_ + 1 _2 r N , _ ^ _ (3) 

dx 2 I 2 i-1 



-§- h + I*£ 6(x-a. )] 9(x) = 

7 . . 1 1 



where we have introduced the dimensionless parameters 

The eigenf unctions 0(x) are the nontrivial solutions to the differential 
equation, Eq. (3) satisfying the boundary conditions 

0(0) =0, -|£ U) = 
ax 

Taking the Laplace transform of Eq. (3) and observing the first boundary 
condition we have 

|2 (0) .2 N I* 0(a) e" a i S 

rfte(x), s} - -^-— - f I ^— ^ 



s 2 +6 2 /* 2 l i=1 s 2 +B 2 /£ 2 



Inverting, we obtain 

N 
6x 



0(x)= C sin P ^ - I D i sin(6(x-a i )/i) u(x-a i ) (4) 

i=1 



* d0 
where C, = -? — 

1 8 dx 



n and D.= 6 I.0(a.) i=1 , 2,..., N 
x=0 i ii 



u( ) represents the unit step function. 



162 



The free end condition at x=£ requires that 

J cosCSd-cU) • D i = ° '• { V a i /£) 
i=1 



cos 8 •C 1 



(5) 



in addition we have the N consistency relations 



lira B(x)»9(a.) 
x+ a . 



j-1, 2,..., N 



which can be written in the form 



sinSa 1 «C 1 



8 I. 






■i-1 



: 



sinBa. • C 
3 1 



Y sin(B(a.-o i )) ^ " 
i=1 



D.= 



(6) 



for j = 2, 3, . . . , N 

4.4 «nc f5)-<6) are homogenoeous linear equations in the 
The system of equations (5J-1&; are uumuy 

u»»n=™., C„ D„ D 2 D». I» order to have a nontrivial 

solution the determinant of the coefficient .atrix M t vanish. Th lS 

, „ a r^ from which the natural 
condition yields the eigenvalues Q u B 2 »"" 

frequencies can be obtained. 



GJ 



u). - 

i 



£ 2 I 



6. 



Per the special case of a single dl* locate,, at the tip of the bar ,-1. 
a 1= 1) we have the condition 



cos 6 -1 
-1 



sin 6 



BI 



= or tanB = 



which agrees 



with Meirovitch's result [2]. 



163 



If for each eigenvalue ^ we assume that the coefficient matrix has 
rank N then we can set C, =1 (assuming that the twisting moment at x=0 is 
nonzero) and solve for the remaining coefficients D., , D 2 , ..., D N . 
Equation (4) then gives the non-normalized eigenfunction. 



Orthogonality Condition 

Let Ojnfx) and O^x) be eigenfunctions corresponding to the 
distinct eigenvalues m and (^ respectively. With the aid of Eq. (3) 
we can write 

1 S n l N 

' G m G n dx + -^ / f 1+ I I* * <5(x-a. ) JO 9 dx = 

o l 2 o i = 1 i n m 

Integrating the first term by parts, and invoking the boundary conditions 
we can write the above as 

% r ' ' 6 n l 6 2 N 4 

" / 9 n 9 m dx + — / G n 9 m dx + ~T I h V a i ) ' 9 (a. ) - 
o £ o i=i i n i mi 

Writing a similar equation with m and n interchanged and subtracting, we 
obtain 

1 Z N 

I / V n dx + J i h VV VV = ° ("« * n) 



From Eq. (4) we see that the eigenfunction can be regarded as a function of 
the dimensionless variable £ = x/£. 



e n U> = sin6 n C - J D in sinS n (C-a.). u(C-a.) 



N 

y 

i=1 

With this understanding the orthogonality condition is 
1 N * 

/ e n (c> e n ( 5 ) dc + ^i.e (a. ) (a) = < B * „> 



o " ^=1 i m i n i 



(7) 



164 



Tr ansverse Vibrations of a Beam With Concentrated Masses and Rotary 
Inertias 



We wish to investigate the transverse vibration of a uniform beam 
carrying a system of heavy bodies along its span. The beam is clamped at 
x=0 and free at x=2, where the x-axis coincides, with the beams neutral axis 
in the undeformed state. The bodies are idealized as point masses with 
rotary inertia situated at the points x=a^ in the undeformed state. As 
before, we assume that the sequence a-| , &2> • • - / a N is strictly 
increasing and for the purposes of the derivation that 0<a i < I. Again, 
the results remain valid for the case of a tip body (a N =£). 



rt, , J, 




H. J 



*» i 



Figure 2. Uniform Beam Carrying Discrete Rigid Bodies 



For a beam with bending stiffness EI (constant) and linear density 
p(x) the partial differential equation governing free vibration is 
(neglecting shear deformation and rotary inertia) the well known 
Euler-Bernoulli equation 

EI i\ + p(x ,i5t . o 

ax 4 at 2 

where w(x,t) denotes the transverse deflection. This equation is not 
directly applicable to our problem since its derivation assumes 
differentiability of the bending moment and shear force. The presence of 
the concentrated masses and inertias gives rise to discontinuities in the 



165 



shear force and bending moment. The conventional solution to this problem 
is to apply the above equation in each subdomain of 0<x< I separated by the 
points a-), a2»..« sn* This gives rise to a fourth order equation in 
each subdomain; thus a large number of integration constants appear in the 
solution. The evaluation of these constants is arrived at by applying the 
boundary conditions at the ends of the beam, writing trans lational and 
rotational equations for each body, and demanding continuity of the beam 
deflection and slope at the points x=a^_ . It is seen that the order of 
the determinant in the frequency equation is extremely large, even for 
moderate values of N. A much more compact solution can be realized by 
employing delta functions to represent the concentrated masses and 
inertias. This idea was first suggested by Pan [3,4]. 

If we follow the same derivation as in the Euler-Bernoulli equation 
but include rotary inertia we arrive at the so called Rayleigh beam 
equation 

EI i^ . _<l [j{x) _i!»_] + p(x) j5i = o 

ax 4 * 3x3t 2 3t 2 

where J(x) is rotary inertia per unit length. Using delta functions it is 

an easy matter to include the mass M^ and rotary inertia J^ of the ith 

body in the p(x) and J(x) distributions 

N 

p(x) ► p + y M. 5(x-a. ) 

i = 1 

N 

J(x) ► T J. 5(x-a. ) 

i = 1 L 

Here p represents the uniform mass density of the beam alone. The rotary 

inertia of the beam (alone) is neglected. Inserting these expressions for 

p(x) and J(x) and seeking solutions of the form e 1 ^ y(x) we arrive at 

the equation 

u N N 

EI S-X + u _S_ [y»( x ) Jj6(x-a)] -co [p + I M. 6(x-a. ) ] y(x) = (8) 
dx i=1 i=1 

with boundary conditions 

y(0)=y'(0)=0 , y"{ £)=y« ••{ £)=0 



166 



Taking the Laplace transform of Eq. (8), and observing the boundary 
conditions at x=0 we obtain 



-a. s 

s — K * 



-a. s _ 

+ JL. VM.y(a.) where k - -gf 

Inverting, we obtain 

1 y(x) = Cl 6' 2 [cosh(6 x/U- cos(8 x/4>] + C 2 f" 3 [sinh(B x/£>- sin(Bx/t)] 
B 2 y D.[cosh BU/l- a.) -cos 8(x/£- a.)] • u(x/l- cl) (g) 

y E. [sinh 8(x/£ - ct.) - sin B(x/l - cl ) ] • u(x/4 - *. ) 
'1 1 

8 = k* , M* - M./pt , J* = V p£ ' < \ = a i /£ ' 

1 =|y"(0) , C 2 = \ 



i=1 

N 

+ 6 
i=1 



D , 1 j* y . (a .) , E.= 4? M ^ (a ^ i = 1 ' 2 ' 3 N# 

1 2 1 1 l * x, 



1 l 



At the free end of the beam we have y- { = y"'U) = 0. With the aid of 
Eq. (9) these conditions can be written as 



(cosh 8 + cos 



8) • C + -^ (sinh 8 + sin 8) • C. 



-B** I [coshd - a )8 + cosd - a )8] D. 

i=1 (10) 



+ B 3 I [sinhd - ol )B + sind - cl )B] E. - 
i = 1 



167 



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The system of equations (10 - 14) are homogeneous linear equations in 
the unknowns: C-|, C2, D-j , D2, • • • , D^, E-| , E2, • • • , E N . 
Setting the determinant of the coefficient matrix to zero yields the 
eigenvalues, 81, 62/ • • • / from which the natural frequencies of 
transverse vibration can be obtained. 

For the special case of a single tip body (N=1 , a-; =1 ) we have the 
condition 



(cosh 6 + cos 8) 
8(sinhB - sin8) 



■1 



(sinhB + sin8) 
(cosh 8 + cos 8) 



8~"(cosh8 - cos8) 8 (sinhB - sin8) 



-2 



-26 1 



8 (sinh8 + sin8) 6 (coshB - cos 8) -2/J 



28 l 



-2/M 



= 



Expanding this determinant we have the frequency equation 



M,J. B 4 (1 - cos 8 cosh8) + M 6 (cos8 si nh 6 - sin8 cosh8) 



11 



-J 8 (sinS cosh8 + sinh8 cos8) + 1 + cos 8 coshB = 
which agrees with Eq. (2-19) in [5] (with C* =- 0). 



In a manner completely analogous to that discussed for torsional 
vibration we set C-|=1 (assuming the root bend: ng moment is nonzero) and 
solve for the remaining coefficients. The eigenf unction then follows from 
Eq. (9). 

Orthogonality Condition 

Following a procedure similar to that described for the torsional 
vibration we can show with the aid of Eq. (8) that 



/py B (x) y n (x) dx + T M. y m (a.) y n (a.) + J J ± y^a,) y;(a.) = 
o 1=1 :i=1 



169 



where y m (x) and y n (x) are eigenfunctions corresponding to distinct 
eigenvalues. In terms of the nondimensional eigenfunction S n (£) we have 
y (x)=fc S (?) where S = x/t. The orthogonality condition takes on the 
form 



N * N _ * 



/S m U) S n (5) dE + .S M iW W + J.Vm'V S n (a i 
o i = 1 x 

where (') now indicates differentiation with respect to £. 



) = (15) 



Natural Frequencies and Eigenfunctions 
Assumed - Modes Method 



In this section we obtain approximate expressions for the natural 
frequencies and eigenfunctions of a uniform canti levered beam carrying a 
set of compact rigid bodies along its span. Both the torsional and 
transverse vibrations are considered, respectively. The formulations are 
implementations of the assumed modes method [2]. The results of this 
section serve to validate the previous exact analyses. 

Torsional Vibration 

Here we consider the pure torsional vibration of a uniform circular 
bar carrying a set of N rigid bodies along its length. While the bodies 
possess inertia about the torsion axis, their spatial dimensions are 
assumed negligible. The centroids of the bodies are restricted to lie 
along the neutral axis of the bar. 

Let the x-axis correspond to the neutral axis of the bar. The bar has 
length I, torsional stiffness GJ, and polar mass moment of inertia per unit 
length I. The i th rigid body is located at x=a i# and has polar mass 
moment of inertia I ± . Designating the angle of twist of the cross 



170 



section by, 9(x,t), the kinetic and strain energies for the system are, 

respectively 

I , a 2 , N - (16) 



T.i/.d?) *x^ i x i tf?^] 



i=1 



* .a«».« (17) 



V -2 GJ ' fe )dX 
o 

We choose to express the twist angle, 6<x,t>, as a series involving 

the eigenfunctions of a simple uniform clamped-f ree circular shaft. Those 

eigenf unctions, in terms of the nondimensional variable n = x/ 1, are given 

by 

* ( n) = s. r 2 sin a R Ti 



„ _ ( ?v-_i ) — (k = 1,2,...) 
where ^ - i^k u 2 

The eigenfunctions, ^(n), satisfy the eigenvalue problem 



d 2 $, 



~ + °k *k (n) = ° 



dn 2 

d#. 

and orthonormality condition 

1 

/*. (n) *.( n) dn = ^ . 

o 

where, 6^j is the Kronecker delta. 

Introducing the expansion 

m 
9(x,t) = I «L(n) P k (t) 
k = 1 

into the energy expressions Eqs. (16) and (17), and making the appropriate 



171 



coordinate transformation, one obtains 



. m N m m 

T = j II Ip 2 + - I I I I. $ (a)^(a)pp k ' 
3=1 1 = 1 3=1 k=1 J J 

1 GJ n 2 2 

where a . 

l 

i I 
Application of Lagrange's equations yields the free vibration motion 
equations 

GJ *.„2, r l r„i (18) 



where 



1^] {p*} + — 7 ^ {p} = fo} 
i a ] = hu + [p] i-i*j [p] T 



MJ = (m x m) identity matrix 

* * 

1' I 2' 



* * * * 

["~"I-J = diagonal (Ii, I 2 , • • • , I„) 



* 
I. 

l 



I. 



l 



[P] - (m x N), with (i,j ) element: P. . = $. ( a. ) 

s 2, .. n . 2 2 2 X 
ra^J = diagonal (a, a_, ..., a ) 

I z m 

Seeking harmonic solutions; {p } = {u } e^- ^ to Eqs. (18), leads to 

the algebraic eigenvalue problem 

Oct 2 J - \\& ]) {u} = {0} 

The eigenvalues, X r , which are roots of the characteristic equation 

det Oo?J - \{tf] ) = (19) 

are related to the natural frequencies of the bar with rigid bodies, fjj-, 



172 



These estimates of the torsional vibration frequencies will be upper bounds 

to the true values. 

Approximate expressions for the eigenfunct ions corresponding to the 
torsional vibrations of the bar with rigid bodies, can be expressed in 
terms of the nondimensional variable, n, as 

r (n) = I V n) u k 

k=1 k k 

where U (r) is the k th element of the eigenvector, {u (r) } corresponding to 

k 
eigenvalue X r . 

In order to directly compare (numerically'' the approximate 
eigenf unctions of this section with those of the exact analysis, we enforce 

the normalization 

1 N t 

f 6 2 (n) dn + y. I. [9 (ct )] 2 = 1 

o r i-1 
The approximate eigenfunctions that satisfy th-> above normality condition 

are given by 

1 r . , , „(c) 



(n) = — I \ (Tl) U 
r /V k = 1 



where 



Transverse Vibration 

Here we present an approximate analysis cf the planar transverse 
bending vibrations of a uniform cantilevered luler-Bernoulli beam carrying 
a set of N compact rigid bodies. The motion plane is taken to be a plane 
of inertial symmetry for the bodies. The bodies possess mass and inertia, 
however, their spatial dimensions are assumed negligible. 

The beam is taken to have length t, bending stiffness EI, and mass per 
unit length p. The rigid bodies are fixed to the beam with their centroids 
on the neutral axis. In the undeformed state, the beam's longitudinal axis 



173 



coincides with the coordinate x-axis, and the rigid body has position 
x = a^. Body i has mass M^, and centroidal mass moment of inertia 
about an axis normal to the motion plane, J^ . Letting, w(x,t), be the 
transverse displacement of the beam's neutral axis, the kinetic and strain 
energies for the system are 

o N 

T - 2 P /fe) dx + 2 A M ife (a i' fc) J 



o 



i=1 



1 n T r 3 2 w , „, i2 (20] 

i=1 



v = 1 EI / <r^T? dx (21) 

We will express the elastic displacement, w(x,t), in terms of a set of 
eigenfunctions corresponding to a simple uniform cantilevered-free beam. 
Vie record that eigenvalue problem, in terms of the dimensionless parameter, 

x 

r, = -, as 

l!i - B h s(n) = 
d^ 

S(0) = p- (0) = ^ (D = ^ (D = o 
dn dn 2 dr, 3 

The eigenfunction solutions can be expressed as 

sinS, - sinhB. 
S k (n) - cosh 8^ - cos 8^ + cos ^ + cosh ^ (sinh^n - sin^n) 

where 6^ corresponds to the k th root of the associated characteristic 
equation 

cosh 6 cos 6+1=0 

These eigenfunctions satisfy the orthonormality condition 

1 

/ s i (n) s (n) dn = 6 ij 



174 



Introducing the expansion 

n 



n 

w(x,t) = I J s k (n) c te (t) 
k = l 



into the energy expressions, Eqs . (20) and (21), and making the appropriate 
coordinate transformation, one obtains 

n o 

1 3 r * 2 

T = -r pi 5 1 q. 

j=1 3 

1 i=1 j=i k=1 J 

a . 
l , 
where a. = — r - . ana 

i £ n 

EI v o«+ „2 



= — y 6 H qf 



Application of Lagrange's equations yields the free vibration motion 
equations 

un {q ' } + — u M k } - fo 1 



(22) 



P* 4 



where 



[^i = r-i-j + is]rM*-jts] T + [s«]t^*-J[s'] T 



[--I-J - (n x n) identify matrix 

* * * -, 
r*M*-J = diagonal (M , h^,..., M N 

* * * , 
[~-J*-J = diagonal \J ^, J 2 » •• • » J N , 

* M i • J i 

Mi " pl i piT 

[S] - (n x N), with (i,j) element: S„ = SA a. ) 

dS. 
[S'] - (n x N), with (i,j) element: S!^ - — j^p °y 

444 
r~-6~J = diagonal ( 6 1 , 3 2 /««»» & n ) 



175 



Seeking harmonic solutions; {q } = {v } e to the Eqs. (22), leads 
to the algebraic eigenvalue problem 

(r-e^j - xt^/n ) {v} = {0} (23) 

The eigenvalues, X r , are related to the natural frequencies of the 
cantilevered beam with rigid bodies, Qj., by 



r W r 

These approximate natural frequencies will be upper bounds to the true 
values. 

Approximate expressions for the eigenfunctions of the cantilevered 
beam with rigid bodies can be written in terms of the nondimensional 
variable n, as 

n . . 

w (1> = I s v (Tl) V < r = 1 ' 2 "~ n) 

r k=1 K * 

where V is the k — element of the eigenvector, {v } , associated with 
eigenvalue X r . Clearly, these eigenfunctions have been scaled to the 
interval _< n _< 1. 

In order to directly compare (numerically) the approximate 
eigenfunctions of this section with those of the exact analysis, we enforce 
the normalization: 

1 N * N * 

/ [w(n)] 2 dn + T m. [w (a )1 2 + 7 J. [w'(a. ) "I 2 = 1 
L r J . L . i L r i J . L . i L ri J 

o 1=1 1 = 1 

The approximate eigenfunctions normalized to satisfy the above condition 
are given by 

w (n) = I I s. (n) v^ r) 

v^7 k=1 



where 



176 



I K r) i 2 + \ m* [ n i vvv^r 



k=1 * i»l k=1 

(D12 



i=1 k=1 



Selected Numerical Results 

Validation of Natu ral Frequencies 

Tables 1 and 2 provide the natural frequencies for the torsion and 
transverse bending vibrations, respectively, of a beam carrying six small 
rigid bodies. The data for the body distributions and their mass and 
inertia ratios are consistent with the parameters for the COFS-I 
B ast<6>. The exact results of Table ^ (for torsion), and Table 2 (for 
bending) are the solutions from the characteristic equations associated 
with the homogenous systems given by Eqs. (5) fi (6), and Eqs. (10) - (14), 
respectively. The exact results are clearly corroborated by the solutions 
obtained from the assumed modes method, which were yield from Eq. (19) in 
the case of torsion, and Eqs. (23) in the case of bending. 

Comparison of Eigenfunctions 

Figures 3 and 4 show plots which compare the corresponding 
eigenf unctions of a simple uniform clamped-free beam with those of a 
clamped-free beam carrying six small rigid bodies. The data for the rigid 
bodies, for both the torsion and bending cases, are the same as that gxven 
in Tables 1 and 2. Figure 3 presents the first four eigenfunctions for 
torsion, while Figure 4 gives the first four eigenfunctions for transverse 
bending. All the eigenfunctions have been normalized to have a maximum 
amplitude of unity. These figures are intended to simply illustrate the 
change in the eigenfunctions associated with the addition of several small 
rigid bodies. 



17 7 



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References 



1. S. Gates & J. Storch "Mechanical Idealization of the Orbiter/COFS II 
Structural System," CSDL Intralab Memo No. DI-87-02, February 5, 1987. 

2. L. Meirovitch, "Analytical Methods in Vibrations," Macmillan, 1971. 
pp. 156-60; pp. 233-235 

3. H.H. Pan, "Some Applications of Symbolic Functions on Beam Problems," 
Journal of the Franklin Institute , Vol. 303, 1963pp. 303-313 

4. H. H. Pan, "Transverse Vibration of an Euler Beam Carrying a System of 
Heavy Bodies," Journal of Applied Mechanics , June 1965, pp. 434-43 7. 



5. 



J. Storch & S. Gates "Planar Dynamics of Uniform Beam with Rigid 
Bodies Affixed to the Ends," Report CSDL-R-1 629, May, 1983. 



6. Lenzi, D.C. s Shipley, J.W. "Mast Flight System Beam Structure and 

Beam Structure Performance," NASA CP-2447, pp. 265-279, Presented at 
the First NASA/DOD CSI Technology Conference, Nov. 18-21, 1986, 
Norfolk, Virginia. 



182 



Appendix C 



Orbiter/COFS-II Pitch Plane Dynamics 



183 




The Charles Stark Draper Laboratory, Inc. 

555 Technology Square, Cambridge, Massachusetts 02139 Telephone (617) 258- 



DI-87-14 



TO: 


Stan Fay 


FROM: 


Joel S torch 


DATE: 


5 October 1987 



SUBJECT: Orbiter/COFS II Pitch Plane Dynamics 

In a previous memorandum (Draper memo # DI-87-02, Feb. 5, 1987) the 
mechanical idealization of the orbiter/COFS II structural system was 
described. Presently, we restrict our attention to the vehicle pitch plane 
and obtain linearized equations of motion, with the ultimate aim of 
obtaining the transfer function between the pitch servo torque and 
corresponding gimbal angle. For a representative set of vehicle parameters 
the zeros and poles of the transfer function are obtained numerically. 

Figure 1 depicts the situation in a general state of motion and 
deformation. All displacements (rigid and elastic) are treated as "small" 
and are restricted to a single plane. The term "small" will be understood 
to mean that the displacements are restricted in magnitude so as to result 
in a set of differential equations. of motion with constant coefficients. 

Vehicle Coordinate Frames 

Fi - Inertial frame. Rotation by angle 9 about z axis brings us to 

the orbiter body frame. 
Fj - Orbiter body frame with origin at orbiter mass centre, xj axis 

along orbiter roll axis, positive towards aft. Zj axis along 

orbiter pitch axis, positive toward port. The yj axis completes 

the right handed system. Note that this results in the mast's 

under formed axis to be along y j. 
F 2 - Frame rigidly attached to mast tip (P 2 ). Misalignment relative 

to F i due to transverse bending of mast in x,y, plane. 
F 3 - Frame with origin rigidly attached to mass centre of gimbal 

platform. Misalignment relative to F 2 due to rotation by gimbal 

angle a about Z 2 axis. 

184 



F 4 - Frame with origin rigidly attached to mass centre of antenna 
column. Misalignment relative to Fj due to rotation by 
restoration angle ifij about Z 3 axis. Note that the x^ axis is 
along the symmetry axis of the column. 

F 5 - Frame with origin rigidly attached to mass centre of feed 
mast/horn. Misalignment relative to F^ due to rotation by 
restoration angle ^2 (at P5) about Z ^ axis. Note that the X5 
axis is along the symmetry axis of the feed mast/horn. 

Vehicle Geometry and Mass Properties 

c_ - Vector from orbiter mass centre to mast attachment point Pj. 
(Resolved in Fj). 

L - Length of undeformed mast. 

2 
c_ - Vector from mast tip P2 to mass centre of offset structure 

(resolved in F2L 

2 

r_ - Vector from mast tip P2 to gimbal pivot P3 (resolved in F 2 ). 

c_ - Vector from gimbal pivot P3 to mass centre of pay load platform 

(resolved in F 3 ) . 

3 
r_ - Vector from gimbal pivot P 3 to base P^ of antenna column 

(resolved in F 3 ) . 
c - Distance from base P^ of antenna colamn to its mass centre 

(measured along X^). 
h - Distance from base Pt^ of antenna column to hoop's mid plane 

(measured along X^). 



a - Length of antenna column Pi+Ps* 

b - Distance from antenna column tip P 5 to mass centre of feed mast & 

horn. 

m^ - Mass of orbiter. 

( X ) 
I - Moment of inertia of orbiter about z, axis. 

zz 1 

p - Uniform mass per unit length of mast. 

ni2 - Mass of offset structure. 

( 2 ) 
I - Moment of inertia of offset structure about its mass center, axis 

ZZ 

parallel to Z2» 
m 3 - Mass of gimbal platform. 

185 



( 3 ) 



1^ - Moment of inertia of gimbal platform about z 3 axis. 
ra u - Mass of antenna column. 



("*) 
I - Moment of inertia of antenna column about z u axis . 
zz H 

m$ - Mass of feed mast £ horn. 

( 5 ) 
I - Moment of inertia of feed mast & horn about z c axis. 
zz 3 



mg - Mass of antenna hoop. 

< 6 ) 
I - Moment of inertia of hoop about diametrical axis . 
zz 



Elastic Moduli 

EI - Bending stiffness of mast in x-y plane (lb. ft ). 

Kj - Torsional spring modules at antenna column base P,^ (ft. lb./rad). 

K 2 - Torsional spring modules at antenna column tip P 5 (ft. lb./rad). 



186 



System Strain Energy 

The Strain energy V is stored in the two discrete torsional 
springs at P 4 , P5, and in the mast. 






where u(y,t) denotes the transverse deflection of the mast in the 
xy plane. Introduce the one term modal expansion. 



V-V^t) - L f It) S^ Ll ) 



with 5 =y/l. P(t) is an undetermined generalized coordinate, and 
S!(£) is the first eigenf unction corresponding to transverse 
vibration of* uniform cantilevered-free beam. 

At* 

The eigenvalues are the roots of the transcendental equation 
l+cos(> cosh£ = 0, with(J» = 1.8751. The eigenfunctions satisfy the 
orthonormality condition 



f S-Ct) SiU)^ * r -*' 



Inserting the modal expansion into the strain energy integral, 
integrating by parts twice, and observing the properties of Si(|) 
we obtain 



187 



Vtt) - \ ^» V It) V ^ 4i v ^(tj V ET-_ 



1L 



<a, M f'-i*) 



(i) 



Orbiter Kinetic Energy 

Let [X(t), Y(t)] t be the position vector of the mass centre 
of the orbiter relative to the inertial origin. The kinetic 
energy T^ of the orbiter is then given by 



T>- ^.Ci fc vt fc )v \l^l\ e 



(2) 



Kinetic Energy of Mast 

Under small angle assumptions, the transformation matrix 
from F-[ to Fi can be written as 



[ft 11 !- 



i e 



-© 



:) 



The inertial velocity of point P^ expressed in F^ is given by 



V 



<.f,) 



* t^M*) vi4.x c 



V.«1 



(.11 



\) 



tf«> - v x - c A & 



v v c w v " e 



where we have dropped the nonlinear terms ©X and «Y 



The masts transverse deflection 
u(y,t) is measured in F^ 



188 




The inertial velocity of a point along the mast's neutral axis is 
given by 



V*> V 



•I 

it d 



where we have neglected terms of the order structural deflection 
x angular rate. Using the above expression for vtP 1 ) this 
expression assumes the form 



* " c ? * v 75 - • 1 



i v c"' 6 



AST 



The kinetic energy of the mast due to transverse bending and base 
motion is given by 

T t - \f- O* VC '-*) V vti-c'ie) 1 ^ * 



Introducing the modal expansion and defining the modal parameters 






(3) 



we can write the kinetic energy in the form 



T,- ^/L^tY *C x O v ^^-^^) U 2 + 



189 






(4) 



Kinetic Energy of Offset Structure 



Defining the two additional modal parameters 



u. 1( = S",l\) = I. oo o 



\X M -= J/U) i 2.15 3 



(5) 



we can write the inertial velocity of point P2 as 



•\) "»* ^ [ * - ^ C \ VL ) • v LM -ai P 
Y v cV ft 



A*N d", 



and the angular velocity of F2 in the form 

The inertial velocity of the mass centre of the offset structure 
resolved in F2 can be written as 

X ~ ^6 vb, J, 
Where we have neglected non linear terms and define 



190 



The 



S v - c- VC (6) 

kinetic energy T 3 of the off- set structure is then given by 

(7) 



Kinetic Energy of Gimba l Platform 

The inertial velocity of point P 3 follows directly from that 
of the misl centre of the offset structure if we replace c2 with 
r2. 



\N\ tVx 









(8) 



\i> 



W. -- l^, v vv *» 



r l 



4 



The inertial velocity of the mass centre of the gimbal platform 
resolved in F 3 is given by 






CM 



191 



which can be written as 



* - 3* * vt if- 






AJf\ 



v * 3 t * ~ b M p V c L '> A 



7, 



again dropping nonlinear terms. 
Here 



(9) 



The kinetic energy T 4 of the gimbal platform can now be written 



as 






(10) 



Kinetic Energy of Antenna column s Hoop 

The inertial velocity of point p. resolved in F 3 is given by 



192 <- 



where ^ __ ^ v ,.v„ _ l,--t l VV^„ 

3,-- a, ^ i : , w- >*« t f"; * ^ ) . (11) 

-1 

Let C be the distance (measured along the X4 axis) from P 4 to the 
combined mass centre, and I the moment of inertia of the combined 
bodies about an axis parallel to the z direction passing through 
the composite centre of mass. 






<A> , ^ 1 ~ ? \ v . -r «.** - ^ 12 ) 



The inertial velocity of the mass centre of the combined system 
(antenna column & hoop) resolved in F4 is given by 

where 

(13) 



3-.--3«* c , J,^^' VC y k^b t 



WA.i> C 



The kinetic energy T5 of the antenna column* hoop . is then given 
by 

Ts" ^l*N** fc ) [>*-3-,» *fe,p-* lJ 3 ' i) v v 

J (14) 



193 



Kinetic Energy of Feed Mast t Horn 

The inertial velocity of point P5 resolved in F 4 is given by 



u » k 



The inertial velocity of the feed mast & horn mass centre 
resolved in F5 is given by 

where 



The kinetic energy Ts of the feed mast & horn is given by 
* V. " L »» <>~ U "P ** *t, * fj\ 



(16) 



System Stiffness and Mass Matrices 

The 7x1 vector of generalized coordinates q consists of 

(1) Orbiter x translation 

(2) Orbiter y translation 

(3) Orbiter pitch 

(4) Mast modal coordinate 



194 



(5) Pitch gimbal angle 

(6) Torsional angle at antenna column base 

(7) Torsional angle at antenna column tip 



\ '- O, Y, e, f, *, y,, v % y 



From eq.(l) the strain energy V(t) can be written as the 
quadratie form 



vfc)-* \ %J L^\_ 



where the stiffness matrix [K] is given by 



CK3* ^^0,0,0, «(3,\ *, ^,, * v ) < 17 > 



The total system kinetic energy T, can be obtained by summing 
equations (2), (4), (7), (10), (14) and (16). 



Writing T - \ % y £f\^ \ 

the elements of the mass matrix [M] follow. 

i 
»hfc - ° 

Mi-, ■=- O 195 



ORIGINAL Fa'S. !S 
Vyi v ^ ^ vvx v v OE POOR QUALITY 



x x. 






*.>> 









Mi 



-. - f O ^u, - PL- c™ ul >% - u„ <,*iU c\ v-i va, 






w> 



3? 



5t ^' ^^j^J K^^dj,^, * M fi»3u v 









™1«# 






W\..,- -^u 






196 



Wi 



v tf> 



4< 



-- -*„ (l\ % Vl)-^^«)^ <^ -^J„ t f 



"M 



m t 



-VLuX"' - «, U>, 



»l 



"w* 



I «t lf\)' V<J«i*M»^, v . VM. ; 5vX *"> s [tC , O v Vt.c' ;) )^ 



;-h 



tf") -. o) 



V X v X*' V X % 



it 






\S*1 IV 



_v 



V^U * X *l 



^ S) v x v^^H-v^t)^: ^va^Sj 



|3 



w\ 



6T 



- X^ * *V t ^u 



m 



m 



^ X ^ ' v *v ^ 



(V 



» U tWoJk £"3 iS ^w.^t^Wc 



197 



Generalized Forces 

At the gimbal pivot (P 3 ) a servo torque T (t) is present 
which when positive, tends to increase the gimbal angle <* (t) . No 
other noncoservative forces are acting upon the vehicle. During 
an arbitrary virtual displacement of the system the virtual work 
fw done by the servo is 

<fw" Ttt) £»- 



Equations of Motion & Transfer Function 

It follows from Lagranges • equations that 



1*2% V C*M - Si 



All elements of the generalized force vector Q are are zero 
except for Q s = T(t). Taking the Laplace transform of the 
equations of motion and assuming that <( (o\ - <£ ^) -s o we W*</ 1 



( *" C*3 v £vG) %} s ) - <k^) 




^ KS) 



where A (s) = det (S 2 [M] + [K] ) andA, (s) is the determinat of a 
matrix obtained by replacing the fifth column of (S C M 1 + t K ]) 
with (0,0,0,0,1,0,0)^ The transfer function will be of the form 






CO lr 0)2, and W3 are the non zero natural frequencies of the 
unforced vehicle. Note that in addition to these values use have 
four rigid body modes: orbiter translation (x and y) , orbiter 



198 



pitch, and rotation about the gimbal pivot, 



199 



a 
■p 

H 
3 
n 
a 
as 



10 

u 

•rl 
U 
9 



c 






9 
1 

rtt 

I 



Ul 









oo 

T 

Oo 

5 






o 






O 

■ _J 



r 



3- 

3" 

r 



i» 



oi 



O 



-> HI 



*- 

o 



rJ 
^ 



H 



3 



HI 

H 



•» « 



oi 



ll 
4 



3- 
O 






37 1» 



8- 
vi> 






** 






-v ~ 



r 
r 



oo 
»>• 
ft 

I' 



OO 

r- 



J' 



Hk 






r 



H 

or 

r 
i» 



3" 


3T 


r> 




•"" 


0« 


* 


1- 


O 


r» 


H 


— 


-* 


> 


-t- 




.» 


> 


n 


"> 


w> 


w/ 


•— • 




r 


T 


rJ 


vfl 


pJ 


. 


or 


rl 






r* 


*~ 


* 


> 


,i 


f> 




i/> 


^O 


_^» 


-w/ 




-^•^ 


— 


^ 


T 


o* 


X 


X 


r- 


s* 


u» 


• 


• 


4- 


Jr 


.» 


c 1 


*7 


v> 


_> 


^> 






II 



o 

o 

CM 



<S)'^ 



origin ;l p^qs fs 
OF PO )R QUALITY 

COFS-II Mechanical Admittancf Function 



2 

a. 



u. 



2 



T 
M 



^ 



S T. 



£4 



(s) 



1.617 x 1Q~ 2 (S 2 +■ 0-541) (S 2 + 2.42) (S 2 + 120) 



(S 2 +• 0.674) (S 2 + 112) (S 2 -I- 1293) 



2 
Y 



2 

Y 



2 

Y 



or 






0.30 



0.32 



«B 



1.55 



10.6 



m - 11 

3 rad/sec 

^-36 



Note the near cancellation of two pole-sera pairs, 
tii with ai_ and u^ with o^ . 




CJfcct 



^-/c 



'OP- CO 



y 



201 



Le> 



P 



o 

N3 



s 


O 

•xs 


•o 


e> 


8 


2* 


» 


t— 


•O 


Ti 


C 


> 


> 


C.) 


l- 


M 


H 


__ 


■< 


C/) 




TECHNICAL REPORT STANDARD TITLE PAGE 



1. Report No. 



2. Government Accession No. 



4. Title and Subtitle 

Control of Flexible Structures-II (C0FS-I1) 
Flight Control, Structure, and Gimbal System 
Interaction Study 



7. Aothor(s) s> Fayj s. Gates, T. Henderson, L. Sackett, 
C. Kirchwey, I. Stoddard, J. Storch 



9. Performing Organization Name and Address 

Charles Stark. Draper Laboratory, Inc. 
555 Technology Square 
Cambridge, MA 02139 



12. Sponsoring Agency Name and Address 

Langley Research Center 

National Aeronautics and Space Administration 

Hampton, VA 23665 



3. Recipient's Catalog No. 



5. Report Date 

September 1988 



6. Performing Organization Code 



8. Performing Organization Report No. 

R-2088 



10. Work Unit No. 



11. Contract or Grant No. 

NAS9-17560 



13. Type of Report and Period Covered 

Final Report 



14. Sponsoring Agency Code 



15. Supplementary Notes 



16. Abstract 

The second Control of Flexible Struct 
includes a long mast as in the first flight 
hoop-column antenna attached via a gimbal s} 
would be mounted in the Space Shuttle cargo 
could be used to point the antenna relative 
of the Shuttle Orbiter/COFS-II system with t 
System (FCS) and the gimbal pointing contro] 
and simulation. The Orbiter pointing requii 
impact on allowable free drift time for COF! 
configurations were investigated. Also simi 
behavior with active vernier jets during ant 
mast dampers was included. Control system 
various portions of the COFS-II structure w< 
possible undesirable interaction between th< 
articulated COFS-II mast/antenna system, evt 
jets. Undesirable conditions can probably 
flight analysis and simulations, and flight 
should be inactive during antenna slews. Ti 
for small gimbal angle excursions with no at 
torque authority of the gimbal motor is so : 



ures Flight Experiment (COFS-II) 
experiment, but with the Langley 15-m 
stem to the top of the mast. The mast 
bay. The servo-driven gimbal system 
to the mast. The dynamic interaction 
he Orbiter on-orbit Flight Control 

system has been studied using analysis 
ements have been assessed for their 

experiments. Three fixed antenna 
llated was Orbiter attitude control 
enna slewing. The effect of experiment 
stability and performance and loads on 
re investigated. The study indicates 

Orbiter FCS and the flexible, 
n when restricted to vernier reaction 
>e avoided with careful planning, pre- 
operational constraints. The FCS 
ie gimbal control system was analyzed 
:tive Orbiter attitude control. The 
imall that control is quickly saturated. 



17. Key Words Suggested by Author 

Orbiter Flight Control System Stability 
Dynamic Interaction, Large Space 
Structures 



19. Security Classif. (of this report) 



Unclassified 



18 Distribution Statement 



Unlimited Distribution 



20. Security Classif. (o+ this page) 

Unclassified 



21. No. of Pages 



22. Pr