STUDY OF STABILITY AND CONTROL MOMENT GYRO WOBBLE DAMPING OF FLEXIBLE, SPINNING SPACE STATIONS GRUMMA OFFICE OF PRIME RESPONSIBILITY https://ntrs.nasa.gov/search.jsp?R=19720014242 2020-03-11T18:54:50+00:00Z STUDY OF STABILITY AND CONTROL MOMENT GYRO WOBBLE DAMPING OF FLEXIBLE, SPINNING SPACE STATIONS. Prepared for National Aeronautics and Space Administration Manned Spacecraft Center Houston, Texas Contract No. - NAS 9-11991 Prepared by GRUMMAN AEROSPACE CORPORATION Bethpage, New York February, 1972 Authors: Howard Berman Dr. Joel Markowitz Will Holmer Approved: C. (S. Witt - Section Head Grumman Guidance and Control Section 11 FOREWORD This Final Report is submitted to the NASA Manned Spacecraft Center in partial fulfillment of Contract NAS 9-11991. Presented herein are the results of the technical investigations along with conclusions and recommendations for further action in the area of stabilization and control of flexible, spinning space stations. This report has been segmented into three parts. PART I, which can be considered an Executive Summary, presents an introduction, a concise summary of significant results, and the principal conclusions and recommendations. PART II presents a description of the analyses along with.an in-depth discussion of the results. PART III is a User's Guide for the digital computer program that simulates the flexible, spinning space station. The development and submittal of this computer program is a contractual requirement. The study program began on June 28, 1971. Mr. K. Lindsay of the Guidance and Control Division at NASA/MSC was Contract Technical Monitor. The overall study was directed and coordinated by Mr. H. Berman, Project Engineer. Mr. Berman also led the control analysis activities with aid from Mr. W. Holmer. Dr. J. Markowitz was responsible for the deriva- tion of the dynamic equations of motion and the modal analysis. Dr. F. Austin provided valuable advice in various aspects of the program. Other contributors to the successful completion of this task were: George Zetkov of the Guidance and Control Section, Arthur Schloth of Loads and Dynamics, and Howard Flicker of Grumman Data Systems. iii TABLE OF CONTENTS PART I - Introduction, Summary, and Conclusions page 1.0 Introduction 1 2.0 Summary of Results 3 3.0 Conclusions and Recommendations 6 PART II - Analysis U.O Physical Vehicle Properties 8 k.l Vehicle Configurations 8 U.2 Mass Properties 11 U.3 Structural Properties 15 5.0 Wobble Damping Requirements and Disturbances 18 5.1 Damping Requirements 18 5.2 Disturbances 21 6.0 Vehicle Math Model 25 6.1 Equations of Motion 25 6.2 Formulation of Structural Characteristics k8 6.3 Rotor-Hub Bearing Math Model 1*9 7.0 Computer Programs 5^ 8.0 Uncontrolled (Open-Loop) Dynamics 55 8.1 Preliminaries 55 8.2 Stability Sensitivity to Bearing Parameters 58 8.3 Significant Mode Shapes 70 8.U General Flexible Body Dynamics 78 9.0 CMG Wobble Damping 111 9.1 Preliminaries 111 9.2 Wobble Damping of the Flexible Space Station 116 9.3 CMG Requirements ll|0 iv Page PART III - User Manual for - Space Station General Flexible Body Simulation — Stage H 10.0 Introduction 10.1 Time History Program 1^ 4 10.1.1 Variable Definitions ll*5 10.1.2 Program Description (with listings) 153 10.1.3 Operating Instructions 207 10.1.U Sample Run 211 10.2 Modal Service Program 222 10.2.1 Program Description (with listings) 222 10.2.2 Operating Instructions 227 10.2.3 Sample Run 29^ 11.0 References Appendix A Modes, Frequencies, Stiffness and Damping Matrices 315 Appendix B Frequency Shift Technique 337 List of Tables Page U.I Mass Property Data 12 k.2 Shuttle Orbiter - Delta Wing -25K Payload, Mass Properties 12 5.1 Navigation and Control Requirements 19 5.2 Wobble Rate Specification 20 5.3 Wobble Induced by Mass Motion 22 6.1 Notation 29 9.1 CMG System Requirements 10.1 Definitions of Program Variables 10.2 Basic Data Identification 209 10.3 Structural Data Identification 211 10.h Sample Run Input Data 212 10.5 THP Sample Output 215 10.6 Rotor Input Data Guide 28U 10.7 Sample Rotor Input Data Setup 28? 10.8 Hub Input Data Guide 290 10.9 Sample Hub Input Data Setup 293 10.10 Sample Putput; "T" Baseline Rotor 295 10.11 Sample Putput; "T" Baseline Hub 309 VI Lict of Illustrations 4 4 4 4 5 6 6 6 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 .1 .2 ; .3 .4 .1 .1 .2 .3 ' .1 .2 .3 "T" Space Station "Y" Space Station "Y" Configuration Idealization/Mass Properties "T" Configuration Idealization/Mass Properties Shuttle Docking Vehicle Idealization Elastic Characteristics of Bearing Idealization Used to Asses the Effect of Rotation Bearing Damping Wobble Vector Projected on Rotor Coordinates 9 10 13 14 23 26 50 on 52 57 Inertial Motion of a Symmetric Spinning Space Station 57 Simplified Flexible Body Equations 60 .4 Decay Time Constant for "Y" Outer Race Bearing Damping 6l .5 .6 .7 .8 .9 .10 .9-8.14 .15 .16 .17 .18 .19 .20 .21 .22 Decay Time Constant for "Y" Inner Race Bearing Damping 6l Time Constant for Divergence for "T" Outer Race Bearing 62 Damping Stability Boundary for "T" - Inner Race Bearing Damping 63 Time Constant for "T" - Inner Race Bearing Damping 64 "Y" With Bearing Flexibility j ^ Histories "T" With Bearing Flexibility \ Rotor Mode Shapes Hub Mode Shapes "Y" - Baseline Structure, Uncontrolled "Y" - Nominal Structure, Uncontrolled "Y"/Shuttle, Nominal Uncontrolled -"T" - Baseline, Uncontrolled "T" - Nominal, Uncontrolled "T" With Rigid Hub - Nominal Rotor, Uncontrolled "T "/Shuttle - Nominal, Uncontrolled 66 68 71 77 80 84 89 Time 94 Histories QQ 102 107 vii List of Illustrations (Con't) Page 9.1 CMG 90-Degree H-Lag Law 112 9.2 CMG Wobble Damping Logic and Simulated Dynamics 117 9.3 "Y" - Nominal, CMG Control 9.U "Y"/Shuttle - Nominal, CMG Control 9.5 "T" - Nominal, CMG Control (H = 10,000). 9.6 "T" - Nominal, CMG Control (H = 12,000) 9-7 "T"/Shuttle - Nominal CMG Control 118 123 Time 128 Histories 132 136 10.1 Time History Program Flow Diagram 152 10.2-10.25 Listings of Time History Program Routines 158 10.26 "Y"/Shuttle - CMG Control (Time History) 218 10.27 Model Service Program Flow Chart 223 10.28-10.5-2 Listings of Modal Service Program Routines 228 10.53 Rotor Arm Geometrical Data Input Properties 283 10.5*1 Hub Geometrical Data Input Properties 289 PART I Introduction, Summary, and Conclusions 1.0 Introduction The open-loop (uncontrolled) stability of the space station and the control moment gyro (CMG) requirements for damping the wobble motion of the space station can be influenced significantly by structural flexibility and vehicle mass distribution. The purpose of this study is to provide an early evaluation of open-loop stability which will allow for sufficient lead time to modify, if necessary, the basic structural design of the space station. If the uncontrolled vehicle is unstable and if changes cannot be made to render the uncontrolled vehicle stable, then CMG reliability would assume vastly greater importance. In addition, early awareness of the CMG system require- ments will provide sufficient lead time for hardware development. As a result, the following study objectives are formulated: o Determine the effects of structural flexibility on the uncontrolled stability of two dual-spin space station configurations. o Investigate CMG wobble damping with an emphasis qn control interaction with the structure. o Develop and deliver to MSC a digital simulation of the flexible, spinning space station. The newly developed simulation will be used as an analysis tool to satisfy the first two study objectives; .also, it will be available to MSC personnel for future studies. NASA/M9C has selected for investigation two typical modular space station configurations with extremely different mass distributions. Both configurations consist of a spinning member (a rotor) rotating about a near stationary member (the hub). In order to accomplish the above outlined objectives for the two general vehicles chosen for investigation, the following study approach is adopted: o Establish wobble damping requirements, and the level of wobble induced disturbances. o Formulate the general flexible-body equations of motion and develop the corresponding digital simulation. o Perform preliminary open-loop stability analyses and CMC wobble damping analyses using digital computer programs developed dur- ing previous Grumman studies. o Use the newly developed simulation to evaluate open-loop stability and analyze sensitivity to variations in the space station structural properties, o Use the newly developed simulation to evaluate the performance of the CMS wobble damping concept developed during a previous Grumman study. 2.0 Summary of Results The two space station configurations selected for investigation are referred to as the "T" and "Y." The "T" and "Y" configurations were chosen so that each general class of rotating satellites with extremely different mass distributions is represented; specifically, "T" is a "Min I" vehicle (the inertia about the rotor spin axis is smaller than either transverse inertia), while "Y" is "Max l" (the inertia about the rotor spin axis is greater than either trans- verse inertia). The majority of the analyses were performed using two simulations: the Simplified Flexible Body Simulation (developed during a previous study for NASA/MSC) and the General Flexible Body - StageII simula- tion (developed during this study). The Simplified Flexible Body simulation idealizes the vehicle as a rigid rotor connected to a rigid hub by a flexible bearing. In the new simulation, the struc- ture is idealized by segmenting the vehicle into several flexible and rigid sections. The sources of structural flexibility considered are: o The bearing, connecting the rotor to the hub o The long, thin walled sections that comprise the rotor arms and hub o The space station-shuttle attachment point. Much of the previous stability analyses of flexible rotating satel- lites are based upon many simplifying assumptions, and non-rigorous energy sink methods. Hence, results of the present analyses, using high fidelity structural simulations, enhances the previous knowledge learned from the earlier studies. Using the Simplified Flexible Body Simulation, which physically represents a vehicle with a highly flexible bearing compared to relatively stiff rotor arms and hub, the following uncontrolled stability information is uncovered: o "Y" is always stable o "T" is unstable with damping in the bearing outer race o A stability boundary exists for the "T" configuration with bearing inner race damping. Using a range of stiffness values for rotor arm and hub flexibility in conjunction with the newly developed simulation, the following occurred: o The uncontrolled "Y" configuration is stable even with a flexibly attached shuttle. It should be noted that a "Y" station with a shuttle is a "Min I" vehicle, where stability is always questionable. o The uncontrolled "T" configuration is marginally stable. It appears that there is sufficient energy dissipation in the non-rotating portion of the vehicle to over-power the unstable influence of energy dissipation in the rotor. The latter is reaffirmed by demonstrating an instability for an uncontrolled "T" with a rigid hub. The wobble damping requirements are based upon achieving a 10 g environment throughout the hub within a specified time from the initiation of the wobble induced disturbance. Although CMG wobble damping is investigated in response to shuttle docking disturbances, CMG requirements will not be based upon the relatively infrequent dockings; rather, successful damping of the wobble induced by typical internal mass motion will determine the CMG requirements. CMG wobble damping is investigated using a unique CMG configuration that was developed during Grumman's previous study on space-base wobble damping. The CMG control technique employs a single-gimbal CMG with an associated control strategy that forces the wheel spin axis to lag the wobble vector "by 9° degrees in inertial space. The interaction of the CMG wobble damping scheme with the flexible space station is evaluated using the General Flexible Body Simulation - Stage II. The results of the investigations are summarized below. Little modification to the CMG system is required when operating with the "Y" station. A low pass filter with a time constant of 2 seconds is added to decrease the actuator duty cycle. The CMG system required the following modifications in order to perform successfully with the "T" station: o A negative limit on gimbal rate to prevent a control instability. o A low pass filter with a time constant of 10 seconds is needed to effectively decrease the actuator duty cycle. o A bias on the gimbal angle command is required in order to minimize the CMG tracking error introduced by the filter. o The criterion for stopping active CMG damping is raised in order to avoid interacting with the structrual modes at low wobble amplitudes. In addition to the above noted CMG system modifications required for successful wobble damping of the "T" station, the "T" configuration, as compared with "Y", will in general require a significantly larger CMG and greater torquer motor output power. Specifically, in re- sponse to a typical profile of internal mass motion the following requirements are established: Configuration Wheel Size Angular Momentum (ft-lb-sec) Maximum, Average Power Output (watts) Uoo 12,000 0.009 5.15 The development of the General Flexible Body - Stage II simulation was a contractual requirement. A user's guide for this simulation has been prepared and is presented in Part III of this report. 3.0 Conclusions and Recommendations There should be no concern over the uncontrolled vehicle stability of a "Y" type space station. The results of all investigations indicate that the vehicle is stable regardless of the amount or loca- tion of energy dissipative structural material. In fact, the time duration for wobble convergence will decrease as the amount of structural flexibility is increased; whether the increase in flexibility is in the rotating portion of the vehicle or non-rotating portion, is unimportant. In contrast, the uncontrolled vehicle stability of a "T" type space station is extremely sensitive to the amount and location of energy dissipative material. For example, it is shown that an increase in structural flexibility in the spinning portion has a de-stabilizing influence while an increase in the flexibility of the non-rotating section has a stabilizing effect. Therefore, to enhance the chances for uncontrolled vehicle stability, additional stiffening members should be added to the "T" rotor. It should be noted that the nominal "T" configuration ;is marginally stable. How- ever, the flexible parameters assumed as nominal represent preliminary estimates. As more structural data is made available, the uncontrolled stability of the "T" configuration should be reaffirmed. The CMG requirements associated with the "T" configuration are extremely unattractive. Furthermore, to satisfy the wheel size requirement with one CMG would be beyond the "state-of-the-art." In contrast, the CMG requirements associated with the "Y" station can be satisfied without developing additional CMG technology. Due to the stabilization and control problems associated with the "T" space station, a configuration of this type should be avoided, assuming the option is available. If the option is unrealistic, then early development of a highly reliable CMG system with a very large angular momentum capacity is recommended. Future Studies The work accomplished as part of this study solves just a small part of the complex space station control problem. As a result, the following directly related areas of study are recommended for future investigation. It has previously been established that along with a CMC wobble damping system, the space station will require the following additional spin vector control systems: o Mass Balancing System o RCS Attitude Control for Pointing o Spin Rate Control Aside from it being important to study the interaction of each control system with the flexible structure and uncover any possible problems, it is equally important to study the integrated spin vector control system and uncover any problems associated with the interaction between control systems. It is noted that a complex spin vector control system of this type would be advancing present technology and should be studied as early as possible. In addition to studies in the space station technology area, the simula- tion that has just been developed as part of this study can be used to investigate the flexible body dynamics of other space vehicles. Speci- fically, by manipulating the input data, the program can be used to simu- late a varity of vehicles with rotating and non-rotating sections, whether it be one rigid body with six degrees of freedom to fifteen flexibly connected bodies with eighty-four degrees of freedom. With minor modifica- tions to the program, the simulation can be made to represent dual spin satellites with an arbitrary number of rotor arms, as well as a high fidelity simulation of a flexible Shuttle Orbiter. PART II 0o ANALYSIS k.O Physical Vehicle Properties U.I Vehicle Configurations NASA/MSC has selected for investigation two typical modular space station con- figurations. Both configurations consist of a spinning member (the rotor) rotating about a near stationary member (the hub). The end module of each rotor arm serves as living quarters within which an artificial gravity environ- ment is experienced for the life of the station. Inside the hub, where most of the scientific experiments are performed, a near zero-g environment is provided. The two configurations chosen for study are the "T" and "Y" con- figurations, and are illustrated in Figures U.I and U.2, respectively. For reference purposes, these two configurations can be found on official MSC drawings SAYUU101266, and SAYUU10126U. The "T" and "Y" configurations were chosen so that each general class of rotating satellites is represented; specifically, "T" is a "Min l" vehicle (the inertia about the rotor spin axis is smaller than either transverse inertia), while "Y" is "Max I" (the inertia about the rotor spin axis is greater than either transverse inertia). The "Y" configuration may be considered an outgrowth from "T", in that the "T" rotor is comprised of two arms, l80° apart, while the "Y" has a three armed rotor, 120° between arms. In order to study the dynamics of a flexible space station, the structure is idealized by segmenting the vehicle into several flexible and rigid sections. The sources of structural flexibility considered are: 1. The bearing, connecting the rotor to the hub 2. The long, thin-walled sections that comprise the rotor arms and hub. 3. The shuttle-orbiter attachment point. The structural properties of each flexible member are detailed in Section U.3. Section U.2 presents their mass properties. <=>r\ HUB Figure 4-1. T Space Station 10 Figure 4-2. Y Space Station 11 4.2 Mass Properties Mass property data for the main bodies (rotor and hub) of each configuration was supplied by MSC in Reference 1. A duplicate table, obtained from Reference 1, is shown as Table 4.1 for completeness. The vehicle idealization used for the dynamic analyses of the flexible space station is a vehicle segmented into several lumped mass sections as shown in Figures 4.3 and 4.4. Specifically, the idealized vehicle is comprised of: 0 3-lumped masses per rotor arm 0 1-rotor ring 0 3-lumped masses for the flexible hub section 0 1-rigid hub section Mass property data for each lumped mass was developed based upon modular, or building block, data provided in Reference 1. Figures 4.3 and 4.4 contain tabulated mass property data for each lumped mass section. The mass 'property data shown is in the proper form for direct use in the simulation that was developed during this study. The analyses that included a shuttle orbiter attached to the station used orbiter mass properties corresponding to a delta wing-25K payload orbiter as obtained from Reference 2, and listed in Table 4.2. TABLE l*.l MASS PROPERTY DATA 12 CONFIGURATION T Y I Rotor Hub Total Rotor Hub Total WEIGHT (Ibsx 10~3) 285 181* 1*69 U31 156 587 C.M.-(FTr' X 0 0 0 0 0 0 Y 0 0 0 0 0 0 Z 0 140.7 55.1 0 - 58.3 - 15.5 INERTIAS (SLUG-FT2 X 10 f2' Ixx .302 .637 .939 .7^5 .220 .965 iyy 1.127 .661 1.788 .7^5 .256 1.001 Izz .889 .025 .9lU 1.330 .037 1.33 NOTES: 1 The center of the xyz coordinate system is in the rotor plane. 2 Inertias are measured relative to the total CM. Z axis is spin TABLE U.2 SHUTTLE ORBITER - DELTA WING - 25 K PAYLOAD MASS PROPERTIES Weight (lb) 3169UO. x sta * (inches) 1521.8 Moment of Inertia slug- ft2 x 10" 6 Ixx 2.725 iyy 19.883 Izz 21.1*95 •^Measured from nose of orbiter -106.25'• I I I I I FLEXIBLE ROTOR ARM FLEXIBLEJ HUB \ ROTOR Q) ROTOR 120°, 120°' 120° 13 •23.33 -6.25' RIGID HUB ^ ROTOR "RING 1 s --(:---(::) s 9 8 >ROTOR (D Element Number 1 2 3 if 5 6 7 8 Element Type rotor ring rotor rotor rotor rigid hub section hub hub hub Ixx (Slug-Ft2) 12,692.0 22,731.1 22,731.3 469,481.85 2,007,596.1 I,4l6.43 1,005.43 1,644,085.22 lyy (Slug-Ft2) 32,781.4 22,731.3 22,731.3 2,018,651.07 2,686,897.86 5,757.926 3,120.47 2,034,929.53 Izz (Slug-Ft2) 32,781.4 2,083. 2,083. 1,642,578.22 682,636.59 5,757.926 3,120.47 485,759.91 Weight (Lbs) 19,000 7,250 7,250 123,000 23,000 4,500 3,500 125,000 FIGURE 4.3 Y CONFIGURATION IDEALIZATION/MASS PROPERTIES 23.3 6.25 FLEXIBLE ROTOR ARM ' , i c i1 Ei >_ 1?-*- > CM ' ' ro - 1CO 0 s. 1 I 00 RIGID HUB, SECTION -105.23'- -*• Z ROTOR RING Y AXIS OUT > OF PAPER FLEXIBLE HUB Element Number 1 2 3 4 5 6 7 8 Element Type rotor ring rotor rotor rotor hub hub hub rigid hub section Ixx (Slug-Ft2) 11,296 22,731.3 22,731.3 2,034,929.53 1,735,329.45 1,733,430.57 1,644,085.2 9,007.36 lyy (Slug-Ft2) 15,586 22,731.3 22,731.3 485,759.91 2,012,994.74 1,829,283.28 2,034,929.53 146,270.93 Izz ( Slug-Ft2) 15,586 2,083 2,083 1,644,685*22 292,228.82 98,517.36 485,759.91 146,270.93 Weight (Lbs) 15,000 7,250 7,250 125,000 23,886.03 17,275.74 125,000. 16,067.25 FIGURE 4.4 T CONFIGURATION IDEALIZATION/MASS PROPERTIES 15 4.3 Structural Properties U.3.1 Rotor and Hub Extension Arms The extension arms for both the rotor and hub are constructed from 7 foot outer diameter cylindrical sections of Aluminum alloy (2219-T8?) . Moments of inertia (l and j) and elastic moduli (E and G) for these sections were provided by NASA and are listed below: 0 J = 26,892 in** 0 G = 4 x 10 psi 0 E = 107 psi Based upon the above polar moment of inertia (j) the computed cross-sectional area is: 0 A = 15.8 in2i l The computed stiffnesses are as follows: 8 20 Bending El = 9.3^ x 10 Ib-ft 0 Torsion GJ = 7-^7 x 10 Ib-ft2 The above values are considered baseline, lower, stiffness limits. For ah "upper ""bound which will also be considered nominal stiffness, the above values were doubled.' The" rationale for doubling the calculated stiffnesses and using the result for nominal is that stiffening members were excluded from the original calculations. The above stiffnesses in combination with the geometry of the extension arms are used as input data for the development of a stiffness matrix for both the rotor and hub (refer to Appendix A for the definition of the stiffness matrices). The re- sulting stiffness matrix is combined with the mode shapes, frequencies, and the per- centage of critical damping (as shown in Appendix A ), to obtain the damping matrix. One percent structural damping in the extension arms is considered nominal and used for the majority of the analyses. 16 U.3.2 Rotor-Hub Bearing The rotor is connected to the hub through a massive bearing that no doubt will be highly flexible. The assumed bearing model is the same as in Reference 3, and is described in section 6.3 of this report. A reasonable extimate for the elastic characteristics of the bearing can be obtained by assuming a structural natural frequency (f,™) of 0.25 cps. By applying, KB 2(D ' where K,, = Bearing stiffness JR = Maximum transverse inertia of the rotor, the following estimate for nominal bearing stiffness is obtained, 1 "B - "«6 » »8 !£* Most likely, as part of the bearing fabrication process the rotor arms will either be riveted, bolted, or welded to the bearing outer race. The material at these attachment points will be highly dissapative. Therefore, it is assumed that 10$ of critical damping is considered nominal damping for the bearing. By applying, (1) EB = 2V*B~^ • (^ 2) where E_. = Damping coefficient13 • y = Percent critical damping . The following estimate for nominal bearing damping is obtained, The derivation of the elastic parameters for the bearing is based upon a simple model where the hub is fixed and the relative angle between the rotor and hub (e) is given by, R 17 Although the above values are presently thought to be nominal they are subject to change as more structural details become available. It is for this reason that a parametric study was performed to determine the variation in vehicle performance as a function of the bearing parameters (refer to section 8.2). i It is noted that nominal bearing damping, as described above, locates dissipative material in the rotating section of the vehicle. As part of this study there were investigations performed to determine any performance differences associated with locating damping material in the non-rotating section of the bearing (refer to sec- tion 8.2). 4.3.3 Shuttle Docking Mechanism During the analyses that included a Shuttle orbiter attached to the station, a i rigid orbiter was!assumed to be flexibly attached to the rigid hub section. Nominal values for the elastic coefficients associated with the docking mechanism are obtained in the same manner as the bearing coefficients were obtained. However, the natural frequency for the shuttle connection is assumed to be 0.5 cps, and the structural damping is assumed to be nominally at 1%. The resulting values for the stiffness and damping coefficients are: KJJ = 2.119 x 10 ft-lb/rad £„ = 1.35 x 10 ft-lb/rad/sec n h.k Vehicle Spin A nominal rotor spin rate equal to 2.5 RPM (.262 rad/sec) is used throughout this study, as per an MSC recommendation. 18 5.0 Wobble Damping Requirements and Disturbances 5.1 Damping Requirements The desired motion of the space station is pure spin. Any torque free oscillatory motion of the vehicle in inertial space other than pure spin is commonly referred to as WOBBLE. The function of the CMG will be to remove the wobble and in turn restore the vehicle to a pure spin state. Wobble damping requirements consists of I an acceptable wobble level along with the time required to damp within the acceptable range. The term wobble level is used synonymously with wobble rate, or the oscillatory component of angular rate viewed on axes perpendicular to the nominal spin axis (i.e. transverse axes'). After reviewing the available control system requirements (as stated in References k and 5) and the physiological requirements (established in Reference 6) it is determined that the experimental requirement for _5 a 10 "ig environment imposes the most stringent restriction on the wobble level. Table 5.1 contains a summary of the Navigation and Control requirements for an .isrtificial gravity space station. In order to obtain a specification for the maximum acceptable — s wobble level (u^ ) based upon the 10~^ g requirement, the following inequality is solved: * lo"5g (5-1) Where, r = maximum distance from CM of total space station to the farthest point in the hub (feet) uij, = transverse angular acceleration (rad/sec) 2 It should be noted that the ^ r contribution to the total linear . 2 acceleration is negligible (i.e., \'WA » uw,) during the periods of interest characterized by low amplitude wobble. Making the appro- priate substitutions in (5-1) for considers only a nominal (or 1 sigma) docking disturbance. Internal Mass Motion The wobble amplitude induced by internal mass motion will be a function of the size of the moving mass and its motion profile. MSC has performed a rigid body analysis for CMG sizing purposes using a particular moving mass profile (Reference 7) that induced wobble amplitudes as listed in Table 5-3- TABLE 5-3 WOBBLE INDUCE BY MASS MOTION Configuration. JTransvers Rate (rad/sec) O.U869 x 10"2 "Y" 0.262 x 10"3 23 CM (STATION PLUS SHUTTLE) MAXIMUM DOCKING PARAMETERS (REFERENCE 4): ID = 5 INCHES 0M = 4 DEC U)s = 0.5 DEG/SEC VA = 1.0 FPS VL = 0.25 FPS VEHICLE PARAMETERS (SECTION 4) SHUTTLE INERTIA = l = 21.5 X 106 SLUG - FT2 CONFIGURATION 'CM (FEET) 'TOTAL SHUTTLE/STATION (SLUG - FT2) 126.5 4.39 X 108 133.99 3.49 X 108 1 SIGMA WOBBLE AMPLITUDE CONFIGURATION 1a TOTAL 0.43 X 10-3 0.59 X 10-3 Figure 5-1. Shuttle Docking These sane wobble amplitudes are used as initial conditions for the flexible body analyses performed as part of the present study. The rationale for using identical initial conditions is that the angular momentum transferred to a rigid vehicle by internal mass motion will be approximately equal to the momentum transferred to a flexible vehicle. 25 6.0 Vehicle Math Model 6.1 Equations of Motion This section contains the equations of motion of the flexible space station. The analytical formulation of these equations was based upon a model consisting of the following components (See Figure 6-1): • A rigid section of hub with 6 degrees of freedom (6 d.o.f.) • A flexible section of hub containing p masses interconnected by massless structure (6p d.o.f.) • A rigid rotor ring pinned to the rigid section of hub by means of a flexible bearing (3 d.o.f.) • q flexible rotor arms, each with s masses interconnected by massless structure (6qs d.o.f.) • A rigid shuttle pinned to the rigid section of hub by means of a flexible docking mechanism (3 d.o.f.) The equations of motion are presented in Equations 6-1 through 6-8; associated relationships required for the formation of these equations are presented in Equations 6-9 through 6-31. Table 6-1 contains the definitions for all nota- tions. These equations were linearized only with respect to those coordinates which describe the flexible motion of the vehicle; all other nonlinear effects were retained. Mass properties of all bodies were completely general; that is, no assumptions were made regarding symmetries, position of mass centers, etc. Control torques were included on all masses. Equation 6-^ 3 contains the matrix equations of motion for the two vehicle con- 2 figurations used in the simulation study ; the characteristics of these vehicles are as follows: 1 - Linearization of the flexible motion was not total; that is, nonlinear terms were eliminated only if significant simplification resulted. 2 - Actually, Equation 6-!4-3 can be used for any vehicle, with or without shuttle, for which p = 3 and q = 2 or 3- 26 N CO N CMN 27 • The flexible section of hub consists of three masses (p = 3)> the inner two of which are mass points having rotary inertia. The five reference points in the hub, namely the four mass centers and the bearing connec- tion point, are aligned in the undeformed vehicle.t • The two types of symmetric rotors studied are designated as T and Y. The T rotor has two identical arms (q=2) l80° apart, and the Y rotor has three identical arms (q=3) 120° apart. In both cases, a rotor arm con- sists of three masses (s=3), the inner two of which are mass points having rotary inertia. All mass centers, including that of the rotor ring, lie in a plane; also, the rotor ring mass center coincides with the bearing connection point. • A docked shuttle may or may not be present; the docking mechanism exhibits elastic and viscous effects (see Equation 6-37) • The flexible bearing between the hub and the rotor ring exhibits elastic and viscous effects; associated stiffness and damping expressions are given by Equations 6-32 to 6-36. This bearing is discussed in detail in Section 6.3 • The massless structures of the flexible hub and the rotor arms consist of straight beam segments of circular cross-section; the associated stiffness and damping matrices for these structures, discussed in Section 6-2,are integrated into the equations of motion by means of Equations 6-38 to 6-1*2. Equations 6-^ 3 contains 8U equations in 8U variables, all of which are needed to describe the motion of the largest simulation vehicle, viz., the vehicle with the Y rotor and the docked shuttle. For computer programming purposes, however, it was decided to preserve the size of the problem when a smaller vehicle was to be studied. This was accomplished by incorporating scalar parameters nq and IL into those equations of motion associated with the shuttle and with masses in the third arm of the rotor. For example, when the shuttle is not present (DL = 0, Cl ]=0) n is set to zero and Euler's equations for the shuttle reduce to I (YJ=o ; then starting from initial conditions {Y} = (YJ=0, the components of {Y} become meaningless variables with no effect in the analysis. When the shuttle is present, ng is set to unity and Euler's equations for the shuttle are retained in their usual form. Parameter ^ plays the same role for the third arm of the rotor. 1 - Arbitrary scalar I is included in order to avoid introducing ill-conditioning effects into the mass matrix. 28 Although the equations of 6-43 do constitute a legitimate set of equations of motion, they have certain shortcomings for the simulation study. First, they are difficult to use if the motion is constrained in some way, as in the case of a rigid rotor, etc. Second, when used in conjunction with a modal reduction scheme they do not lead to a particularly good reduced set of equations. Both of these difficulties stem from the fact that the forces on the right-hand side of 6-43 do not constitute a set of generalized forces for the problem. Accord- ingly, it was decided to premultiply both sides of Equation 6-1*3 by a trans- formation matrix such that the resulting right-hand side would contain a set of generalized forces . This matrix, denoted as COn™/)* is presented in Equation (6-44); the associated expressions needed to form [(L, 1 are given by Equation (6-45). The large size of the problem (84 degrees of freedom) plus the fact that the "X" and "Q" matrices vary with time made it impractical to use the entire set of equations for the time history study. Accordingly, the modal reduction technique was used to obtain a reduced set of approximate equations in terms of modal coordinates; the degree of approximation depends, of course, on the number of modes used. The essential relationships are presented in Equations (6-46) through (6-49). Equation (6-1*9) contains two alternate expressions for determining initial values of the modal coordinates. The simpler of the two, (6-49a), was adequate for the needs of the simulation study and was therefore incorporated into the computer program. However, if more complicated initial conditions are to be studied in the future, it would be advisable to use (6-49b). 1 - Specifically, the resulting set of generalized forces are associated with the following coordinates: (zj, {v}, {¥}, {u^}, f^ l), .. .(1^ 3), (0C3), {9}, (uDll3i (0Dll), ....[uD33], {0D33}. 29 TABLE 6-1 NOTATION Points | A Mass center of rigid section of hub B Fixed point in hub about which rotor rotates C Mass center of j mass module in flexible section of hub D Mass center of k module in i rotor arm E Connection between flexible section of hub and end module of hub G Connection between flexible section of hub and right section of hub H Connection between hub and shuttle L Mass center of rotor ring IT Intersection of rotor ring and i rotor arm 0 Origin of inertial frame S Mass center of shuttle V Connection between i rotor arm and associated end module Coordinate Frames £ Inertial frame fixed at 0 Y Frame fixed in rigid section of hub at A; the Y axis is ~ parallel to the nominal spin axis of hub % Frame fixed in rotor ring at B; the 3C axis is noninally aligned with Y-. x^ 1 Frame fixed in rotor ring at B; xi, coincides with 3C and zk is nominally aligned with i rotor arm ^ Frame fixed in j mass module of flexible hub at CJ; this frame is nominally aligned with £ £ Frame fixed in k mass module of i rotor arm at D , this frame is nominally aligned with x £ Frame fixed in shuttle at S. 30 Displacements {X} Ordered rotation which comprise transformation from !Z to Y; i) Yl about Z > ii) Y about "carried" Z > iii) Y^ about "twice carried" Z_ (9} Ordered rotations comprising transformation from Y to X a Constant rotation about X^ transforming from X to £ £0 j.} small elastic rotations comprising transformation from Jf to yj (0 ik3 small elastic rotations comprising transformation from x to £ £p} Constant ordered rotations which comprise transformation from Y to nominal position of r /•w *^ Small elastic rotations comprising transformation from nominal position of r to instantaneous position of £ <-w Small elastic displacements of CJ measured in the Y frame Small elastic displacements of D1 measured in the x1 frame Angular Velocities •y £ou } Angular velocity of Y frame projected onto itself •y{ou } Angular velocity of X frame projected onto itself {oux J Angular velocity of x1 frame projected onto itself j {(«y } Angular velocity of x frame projected onto itself ik „ Cioz J Angular velocity of £ frame projected onto itself {u> } Angular velocity of £ frame projected onto itself Transformation Matrices [a(Y)], [a(9)],... transformation matrices associated with ^ • 9 ,... [b(Y)], [b(9)],... transformation matrices associated with Y , 62,... Cc(Y)], [c(9)],... transformation matrices associated with Y 0 ... [rr(Y)] transformation matrix going from Z^ to Yj 31 transformation matrix going from X/ to X [TT(AJ)] transformation matrix going from Y^to yj5 C • i ik[n(0Jk)] transformation matrix going from x to z [11(3)] transformation matrix going from Y to nominal position of r [TT(Y)] transformation matrix going from nominal position of :r to final faCy1)"! tranformation matrix going from x to x1L ><•< ' J w ~ ~ ns Physical Properties m. mass of rigid hub section nu mass of jth module in flexible part of hub m, mass of rotor ring OLik mass of kth module in ith rotor arm nu mass of shuttle [I.] inertia matrix of m. with respect to A (in Y frame)A A f** [I-j] inertia matrix of n^j with respect to C (in y frame) [I,,"! inertia matrix of DL. with respect to B (in X frame) B J Li ~ . ' [Xik] inertia matrix of ni ikwith respect to Dik (in z frame)u D u ~ [!„] inertia matrix of HL, with respect to H (in r frame) nJ o ~ Forces and Couples (a) Structural -{FpJ} structural force acting on fflpj (in Y frame) -{F ik} structural force acting onm ik (in x frame) -{F.,} structural force exerted by flexible hub on m at point G (in Y frame) Gr A /v* structural moment acting on nuj about CJ (in Y frame) structural moment acting on ^ik about D (in x frame) structural moment (exerted by all rotor arms) acting, on m^ about point B (in X frame) r^i structural moment (exerted by flexible hub) acting on m. about point G (in Y frame) 32 (b) Controls ft i] control moment applied to m j (in y3 frame) c c ~ {T ik} control moment applied to nuik (in z frame) frq} control moment applied torn- (in r frame) {T .} control moment applied to m.A (in Y frame) {T.} control moment applied to n^ (in X frame) (c) Bearing and Docking total moment exerted by bearing on rotor ring (in X frame) elastic part of viscous part of moment exerted by docking mechanism on mo about point H (in r frame) ;• ' ^ rw Stiffnesses, Viscosities, etc. Kp,,,, K-aao linear stiffness coefficients of bearing bearing damping matrix bearing damping matrix when all damping material is fixed to hub bearing damping matrix when all damping material is fixed to rotor stiffness matrix of shuttle docking mechanism damping matrix of shuttle docking mechanism [KTA], [KYA] stiffness matrices of "T" and "Y" rotors, respectively [CTA], [CYA] damping matrices of "T" and "Y" rotors, respectively [K6HG], [K?H] hub stiffness matrices [CHG], [CH] hub damping matrices Miscellaneous , {Y }, etc. coordinates of any point (here taken to be B) in the X, Y, etc. coordinate frames •* r*s 33 [run cross-product matrix, formed from the elements of one vector, say \, involved in the cross-product, e.g., if X = then [ T ( X ) ] = 0 -I 0 [S(YJ], [S(9)] matrices used in conversion between angular velocities •y y « •{uu }, {CD } and rates of Euler angles fy], {9} 0 • 6 fv} B nominal spin speed of rotor instantaneous spin speed of rotor minus nominal spin speed of rotor similar to fe] but with 6 replacing 9, angular misalignment be teen Y.. and X_ axes "R ns m I value of 6B below which 1C applies and above which applie s scalar constant between zero and one; T| = 1 if all bearing damping material is attached to hub T) = 0 if all bearing damping material is attached to rotor ring n- = 1 for a three arm rotor; n_ = 0 for a two arm rotor nc = 1 when a docked shuttle is present; n_ = 0 when no O b shuttle is present dummy mass value used when rv, = 0 dummy inertia value used when ru = 0 or n<, = 0 [X], [M], [N(e)] Co}, [X], (a) miscellaneous combinations of terms appearing in the analysis matrices by means of which the equations of motion are presented matrix which premultiplies the equations of motion so as to eliminate forces of constraint * * * [A ], [B ], ...[J ] matrices needed to construct CM 00 CO s -p CJ -p o gg PP. ^} 8 o H EH I W CO 8CO ><<• o£o •o ro je ct e ft CO 0 -p CJ ^£j a CJ a co _i ^T -* ^ ^M ^ *• _l •^ > J •1 J ^ J4 J r j o rs pr oj ec te d o n to Y a xe s /•*> •p d ^ "3 v^~M 1 ^ a) *o rH Q . ^rf^ ^"^ 'x .'-o ' H ./< . cu 4 i •> |^ \^ ™» O X CO .~*~~, i .x?. 5 •*• '° j-j-i— JS =MJ* <§ ^ *^ ™ -^^ CO C O •H -P 1 0) CO C 040 D g to rs pr oj ec te d o n to x a xe s CJ £ 3 CO • «\ jj O -p 0 -p •rt a -p «H 0 CO CO a j^**\ M -p c2 COC O . -H 0) CO •r C oj_3 g ^ "tr V-r- it r * •57 -f r— "3rc? v-r- ^•^—^^ J<_, •*• Q "^ •^ ^ • 'Jj . . /C o •H O •p -pl§ CO T) O CO -P CJ (-4 CO CO ''^ d ° 3 ^ W ft CO •rl Q •p C •H oft O •8 c -p CO s 0* s43 CO f-l op o FH +3 •H 1 O CO 10 cO S f* 4^ ^ JS43 oCM to 'go •H +> 1 Q) CO —£*4 CO •du CO 9 y ^t^ C o43 o TJ 43 CJ CO «n O ft ^t '& II r^*i ^& .*Q^ I * .-^ fXM XT'.L H- 1>" ^^ "^^ ^^ •"•~ i}^ ,, v* L/ •o 0fcPi coh 0 -p 0£ 3 CO • «\pq -pfl _ «H f^ — O Li* ^ i -P u 0 Sr~ J^ 1* f5 • __. 0) \^ »M _ CO1 s* ^ ^^ Wa ^ 0) •^•^ ••^ •J rj i O^ •^r ° 3 **>bO *-«"• .g 1 Q,' fn XC* ^ . . O -p •+• o r*>~~ ^ j£ | br ~~*F & .<< . g o •H -P CU CO *• £H CUp-j 3 v-' CO s •p o CU > 3 05 • «\ PQ . -PC •H O Pi -P3 O « c1 -p CO r— S r*- ^ 1^7 O Ur— -P •I .n r . c • ^ ^^ ,_J •l^y *"*f"™ o5 CC0 «H ^S4 rQ a/^ a a i a *T* r ^ d 0) -p o a> •o 0 ^w W H M H Essential Equations and Definitions The following set of relationships are required for the calculation of the "x" and "a" matrices which constitute the fundamental matrix equations of motion, 6-1 through 6-8. t i Coordinate Transformation Matrices 36 \ 0 COS -SIM o SIM * COS COS X2 0 SIM)^ "COS *3 -SIM K3 0 o ( 0 S.MX3 COS ^3 O -SINX, 0 cos^2 0 o i (6r9a) (6-9b) (6-9c) (6-10) 37 Relationships between angular velocities and time rates of Euler angles K? ' * = [TT( fci (6-11) (6-12) (6-13) (6-14) (6-15) (6-16) 5(3) COS-63 COS 62 cose5 o o o l (6-17) -SIM cose2 o o cosoa (6-18) Miscellaneous Expressions 38 e, (6-19) (6-20) f 0 0 0 f 0 0 0 1 (6-21) (6-22) (6-23) (6-2*0 o e5 o -e3 o o Qo 0 O (6-25) (6-26) 39 -[M,*][r (i6 N [M.] (6-2- (6-28; Tabulated "a" Vectors (6 - 30) 1 . 1*1 Tabulated "\" Matrices = ms M - (rryjj |m0ik ) [r (YB)"] - 1 Oiy (rCYci + = C>K»]N(«)](>retf mc» [r Ci6 i-;in a L to 1*3 Expressions for Flexible Bearing and Flexible Docking Mechanism The following equations for the bearing and docking mechanism were used in the vehicle simulation study: Bearing Damping BV J - • [§ 1 0wl* 0 Bearing Stiffness COS CO& f ft o * -6Note: 6_ should not be less than 10 rad. D Total Bearing Moment Docking Moment - C «"««]{ V} - [C £ n Poc j(6-32a) (6-32b) X6-33) , (6-3*0 (6-35a) (6-35b) (6-36) (6-37) Structural Forces The following expressions are for the hub and rotor structures used in the simulation study: Hub where Rotor fcftere J( noe f T T Me Mf + [CVA] fa •JftTCcr(v , J Mo" ^o" tt»« (6-38) (6-39) (6-UO) (6-1+2) rO vi 1 1 1 1 1 , * 1 t 1 ft 1 1* 1 1 -* 1 •* 1 1 \i 1 if \ \ \ 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 l 1 1 l I .. I ,. I ,. I ,. I . i l 1 l 1 1 I.I 1 1 1 l l l l l II 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 e 1 .t 1 .j> 1 .i 1 d 1 if 1 J> 1 .i 1 .3> 1 .> 1 ii 1 .1 t .i 1 «S 1 .1 1 J> 1 ti v 1 >, J11 •aT* J-L ."-'. . 3 -'- TT' J-L i. •^r TT •7- '-7- •==• J£. 2 . " >: 1 T , — ^, "?.' I 1 i JL- ~r" .1, 'i' i 77- -r f \ ^L , ( ™ i ^ '-;' f Ji . . ^ i J-L , — , *£-. 1 *~ . , , ^L, I ™ , — . ^L- "T iii .i •n" , — , -^ T?* [ 3L T? ^_ - 1 - •rr f ^L TT , , •2- JL i> i JL 7T , — . •±~ tw S, TT . , •i- I TT TT . — . -=- I ^L IT , , . •* . "sj* i •5T 7T , , -a- 'aj' | *st* IT "sT | i 7T __ .^ K s. TT , , •^ 1 2 TT _ •Ji- 1- Ji j "0" 1 IT i s "-S i .}. ? 2 V '*;' jf. .*. j iL "f , , ^=- 1 "_Uf '"> «• i ^5i, •of •i- 1 1 1 3 # i ,6 8? E o 0i M s A' B1 C' ^ \ U1 \ U3 S \s^ ) J" t hi \ h2 °12 s h3 G13 s fc ' H1 \. fcl H1 s bl Hi s "•21 ^21 \ ••22 U22 % 1 "•23 ^•23 \ "=2 H2 s fc2 H2 \ 4 H2 s ••31 U31 s •-32 b32 \ K33 ^33 \ b3 H3 s "=3 H3 "=3 H3 S (8-44! I°GEN' MATni)< f°R SIMULATION VEHICLES Stibnatrices of CffO (6-U5, Modal Reduction Original equations of motion in terms of 8k physical coordinates fx}: (6-1*6) 1*8 Modal approximation; h modes [0]; n mod 1 coordinates {q}: Reduced set of approximate equations in model coordinates: Initial conditions in modal coordinates: or [X] f 6.2 Formulation of Structural Characteristics The details of the procedure by means of which structural modes, frequencies, stiffness matrices and damping matrices were computed are lengthy and are pre sented in Appendix A. Major features of the formulation are as follows: • The rigid section of hub was considered to be fixed in space; consequently, cantilever modes were computed for the flexible section of hub. • Nonrotating modes for the rotors were computed assuming that the rotor ring was pinned to ground (rigid section of hub); rotation of the rotor ring out of the spin plane was resisted by rotory springs having the elastic characteristics of the bearing. • The stiffening effects due to centrifugal forces in a spinning rotor were accounted for in the analysis by incorporating an incremental beam stiffness matrix; for the modal analysis, the centrifugal forces were represented by a properly oriented set of external loads. • Due to symmetry of the rotor, some repeated eigenvalues and coupled modes were anticipated. Although mathematically acceptable, this condition could make the mode shapes difficult to interpret. To avoid this problem, rotor motion was divided into its symmetric and antisymmetric parts and two separate eigenvalue problems were solved for each rotor. • The rotor had one rigid body mode, namely.simple rotation in the spin plane. Since the eigenvalue algorithm used could not deal with a zero frequency, it was necessary to eliminate . this frequency from the equations. This was accomplished by using a simple frequency shift technique (See Appendix B) rather than the conventional reduction technique. • With some minor modification, structural damping matrices were based on modal damping. 6.3 Rotor-Hub Bearing Math Model Calculations for the bearing stiffness and damping parameters are based on an assumed model illustrated in Figure 6-2. In order to assure a tight fit and to compensate for wear, it was assumed that the bearing will be assembled by heating the outer race and permitting it to shrink over the inner bearing structure as it cools. In Section U.3.2 an approximate analysis was conducted to determine reasonable elastic and damping parameters, K^ g, and EB, for the bearing. It may be assumed that stiffness Kggi is suitable for deflections •x- 6fi below some critical value, 6fi . For larger 6B a section of the elastic dissipative material is assumed to be completely compressed causing bottoming of the outer race against the rotor; hence the structure then has the higher stiffness Ko* Tnese characteristics may be summarized by the following equations: -for 1-14 ;BEARING-ROTOR ^CONNECTION IS > IDEALIZED AS AN |JELASTICDISSIPATIVEi IMATERIAL RACE! DEFLECTION DIAGRAM 1 = MBS + MBV WHERE MBS = ELASTIC RESTORING TORQUE j MBV = DAMPING TORQUE Figure 6-2. Elastic Characteristics of Bearing 51 By setting KW.I = 0, the above equations can also be used to investigate the behavior of a structure with a bearing which is so badly worn that slop occurs. The damping torque exerted on the rotor by the hub is a function of the spin rates of the viscous materials located in the bearing; also, the damping properties of a bearing with dissipative material which is fixed relative to the hub differ from the damping properties when the dissipative material is fixed relative to the rotor. This point will be explained by using the idealizations illustrated in Figure 6-3. In Figure 6-3a all of the damping material in the bearing is fixed to the hub. A slip ring is assumed to separate the rotor from the non-rotating structure. -.* The vector u> denotes the angular velocity of the rotor relative to the hub. If -. i0) PQ § § 1 o .0) x e. o Ul Ul H CO CQ (U 03 CQ TJ applies for "Max I" vehicles (l > I , I > I ), and the lower sign applies for "Min I" vehicles (I ) is defined as the projection of the station's angular velocity on the transverse plane; i.e., It is noted that the phase angle (p) can be omitted without a significant loss of generality. Examining the above relationship for fu)T} it is seen that the vector traces an elliptical path as illustrated in Figure 8-1. The frequency of the ellipse, referred to as the wobble frequency (Xp), will increase as the separation between the transverse inertias and the spin inertia is increased (see equation 8-2). Furthermore, it is noted that the ellipse degenerates into a circle for symmetric vehicles (i.e., when or = l). Finally, the rotation of the vector is in the same direction as the vehicle spin for a Max I configuration and opposite for Min 1. It is easier to visualize the vehicle motion by observing the wobble rate as seen in despun hub coordinates. The transverse components of the wobble rate as viewed in the hub, is given by _ _ 10 €M< ( f W r H J f ) an £pt)+(fU>rH*}) «*((*) Substituting the expression in (8-3) for {(!>_] and rearranging terms results in, (8-5) The upper sign in the above expression applies for Max I vehicles and the lower 57 Figure 8-1. Wobble Vector Projected on Rotor Coordinates P(1+X) RAD/SEC, MAX I P(1-\) RAD/SEC, WIN I Figure 8-2. Ihertial Motion Of A Symmetric Spinning Space Station 58 sign applies for Min I vehicles. The general expression (8-5) consists of the sum of two sinusoids: one with a frequency equal to the spin plus the wobble frequency, and the other frequency equal to the spin minus the wobble frequency. For most spinning satellites a is close to unity and, in fact, the deviation from unity first becomes appreciable for satellites whose inertia characteristics are close to an "Intermediate Inertia" vehicle. Therefore, the predominate inertial wobble mode for most Max I vehicles is at a frequency equal to spin plus wobble frequency (i.e., p + Xp). In order to visualize wobble motion let us examine the motion of a symmetric vehicle (a. = l) which is characterized by a pure coning motion in inertial space plus the spin motion of the rotor (as shown in Figure 8-2). For an unsymmetric vehicle (a ^ l) small cyclic motions are superimposed on the coning motion shown in Figure 8-2. When the structure is made flexible and energy dissipation exists then the coning motion, or wobble, will either converge or diverge depending upon the location of the energy absorbing material. The stability of each basic space station was investigated with two sources of structural flexibility: • Bearing • Extension Arms (both hub and rotor) The influence of a flexibly attached shuttle on vehicle stability was also studied. Section 8.2 presents the results of analyses performed on a vehicle whose only source of flexibility is its bearing. These results allow for the separation of the stability trends due to just a flexible bearing, as well as being representative of total system stability for space stations with extremely stiff extension arms. The results of total system stability investigations is presented in Section 8.U. A description of what are considered the signfi- cant mode shapes is included in Section 8.3. This information will allow the reader to visualize the predominate structural motion of the flexible rotor and flexible hub. 8.2 Stability Sensitivity To Bearing Parameters The bearing assembly, that connects the rotor to the hub, is idealized as in Reference 3, and is described in Section 6.0 of this report. As previously 59 noted (in Section 6.0), the bearing model permits the inclusion of energy- dissipative material in either the bearing outer race connection to the rotor or the bearing inner race connection to the hub. The nominal values for bearing stiffness and damping have been derived in Section .^3.2; the effects of variations about these nominal values are investigated. A characteristic root analysis was performed using the set of unforced Simplified Flexible Body Equations as presented in Reference 3 (Section U.3.1), and repeated in Figure 8-3 for completeness. The mass properties for each space station, were held constant at their nominal values as presented in Section U.2. The results of the analysis are as follows: "Y" Space Station - The "Y" configuration is stable for any stiffness and damping combination which includes damping material located on either the bearing inner race or outer race. The time constant associated with the decay of the wobble mode as a function of bearing stiffness and damping is shown in Figures 8-h and 8-5 corresponding to outer race damping and inner race damping, respectively. The time constant resulting from using nominal bearing coefficients is 18.6 hr. As seen from Figures Q-h and 8-5 dissipative material on the inner race has more of an influence on wobble damping then dissipative material on the outer race; that is for the same bearing coefficients the time constant resulting from inner race damping is appreciably less than that of outer race damping. "T" Space Station - The "T" configuration is unstable for outer race damping for any value of stiffness or damping. The time constant associated with the exponential divergence of the wobble mode is shown in Figure 8-6 as a function of bearing stiffness and damping. The time constant for divergence corre- sponding to nominal bearing parameters is 5.8 hrs. The "T" station with inner race damping is either stable or unstable depending upon the amount of damping and corresponding stiffness. A stability boundary is constructed and is shown in Figure 8-7. In general, a low bearing stiffness in combination with a high damping coefficient will improve the possibility of vehicle stability. Figure 8-8 presents the time constant associated with either an exponential decay or divergence as a function of the bearing parameters. 6o where: I. , I?, I = moement of inertia of rotor (spin is 1 axis) I. = transverse moment inertia of hub (euqations assume hub symmetry) = mass of rotor, hub a,,, aH = Distance for combined system CM to CM of rotor, hub p = spin rate {h} = angular moment urn of entire system (0} = ordered rotations from hub to rotor lOp, (I)-, = transverse angular velocity components of rotor Kp = Bearing Stiffness = Bearing Damping nw = o - Damping material on bearing outer race ^ D f l - Damping material on bearing inner race FIGURE 8-3 SIMPLIFIED FLEXIBLE BODY EQUATIONS 61 cc X to O CJ LU 320 240 160 o LU Q 80 ffi m m fp i±n:r BEARING) _ _ ,_7 DAMPING f EB - 10 •hj-h- •* *±tT FT-LB RAD/SEC 10 BEARING SPRING CONSTANT, Kg X 10~7 f ^^-J Figure 8-4. Decay Time Constant for "Y" Outer Race Bearing Damping 100 ccI 00 I < o 20 BEARING SPRING CONSTANT, KB X 10'7 Figure 8-5. Decay Time Constant For "Y" Inner Race Bearing Damping ® TIME CONSTANT (T) AS IN, e~t/T 62 e/j oc o C/) o(J fcJD t Q a* • I-*P ri V PQ CDI bn fH CD o 0) a •M •H CO 00 0) § § § 8 (OVd/aTld) c_0l X a» '1NV1SNOO DNIddS 9NldV38 CM 00 63 10' 2X 107 3X 107 4X 107 BEARING SPRING CONSTANT, Kg (FT-LB/RAD) Figure 8-7. Stability Boundary For "T" - Inner Race Bearing Damping o HI V) OO Aa at Q bO .a Si <0 ffl <0 I CO§ O 0) a •r-l H oo oo I •E (C1VU/BT.LJ) g_0l X 8>J '1NV1SNOO ONIUdS ONIUV38 65 Until this point in the report the rotational dynamics have been described by words, equations, illustrations and characteristic roots. A useful tool for examining vehicle dynamics is the Time History Simulation and the resulting time history plots. However, before proceeding with analyzing the time history plot resulting from the General Flexible Body simulation (which includes extension arm flexibility), let us first examine the time history plots from a simulation whose only flexible member is the bearing. Figures 8-9 and 8-10 are time histories of the transverse angular rates for the nominal "Y" and "T" configurations, respectively. The "Y" simulation was initialized with a transverse component of angular momentum in the hub while the "T" run was initialized with an angular momentum component in the rotor. The high frequency flexible mode, representing relative motion between the rotor and hub, is seen to damp out completely within the first 20 seconds. The remaining oscillitory motion is wobble. Although un- observable on these plots, the wobble motion of the "Y" station is con- verging while the wobble of the "T" is diverging. Due to the size of the time constant, convergence or divergence can only be observed by examining the printed data of the wobble magnitude | [CD }| . It is interesting to note that since a for "Y" is closer to one, a predominant frequency equal to spin plus wobble frequency (p + Xp) can be observed when viewing hub transverse rates. In contrast, the "T" station is close to being an "intermediate Inertia" configuration (i.e., a = 5.85) and as a result the hub rates exhibit two frequency components (p + \p) of relatively equal weights. These time history plots will be used as a basis of comparison when examining the time histories of vehicles with additional flexible members (refer to Section &.h). It should be noted that when a shuttle is rigidly attached to the station, the "Y" configuration as well as "T" are "Min I". As a result, both stations when combined with the shuttle are unstable for the condition of nominal bearing flexibility provided there is no other source of energy dissipation. (oas/ova) 3SJ3ASUBJJ, 70 8.3 Significant Mode Shapes Figures 8-9 through 8-15 show the rotor and hub modes used for the majority of simulation studies. The listed modal frequencies cor- respond to the "Y" space station with baseline properties. Six rotor modes and two hub modes were used for each vehicle. For the rotor, the predominant motions are i) in plane bending, ii) out-of-plane bending, and iii) torsion. Torsion in the T-rotor occurs indepen- dently of bending and is easy to visualize (Figures 8-l^ a and b); for the Y-rotor, however, some torsion accompanies out-of-plane bending and is difficult to show; e.g. there is some torsion in arms two and three in Figure 8-9(b). Predominant hub motions are bending in each of the two transverse planes. As was expected, axial extension of the beams occurs at very high frequency and is not present in any of the modes shown.. 71 a) MODE1: RIGID BODY MODE, IN PLANE (f = 0) SPIN PLANE I b) MODE 2: OUT OF PLANE BENDING (f = 0.12881 CPS) SPIN PLANE Figure 8-9. Y Rotor Modes 72 Figure 8-lo v r> *. ' Y R ° tor (Cont.) 73 SPIN PLANE a) MODE 5: IN PLANE BENDING (f = 0.14155 CPS) SPIN PLANE b) MODE 6: OUT OF PLANE BENDING (f = 0.14196 CPS) Figure 8-11. Y Rotor Modes (Cont.) SPIN PLANE a) MODE 1: RIGID BODY MODE, IN PLANE (f = 0) SPIN PLANE b) MODE 2: OUT OF PLANE BENDING (f = 0.13007 CPS) Figure 8-12. T Rotor Modes 75 | j SPIN PLANE *> MODERN PLANE BENlD,NG,f = 0, •13719CPS) SPIN PLANE = 0.14816) "««• 8-13. T Roto, Modes (Cort.) 76 SPIN PLANE MODE 5: TORSION (f = 0.39320 CPS) SPIN PLANE MODE 6: TORSION (f = 0.41336 CPS) Figure 8-14. T Rotor Modes (Cont.) 77 MODE 2: T-HUB,f = 0.07574 CPS Y-HUB,f = 0.22635 CPS MODE 1: T-HUB,f = 0.07289 CPS Y-HUB,f = 0.21063 CPS Figure 8-15. Hub Modes (T And Y) 78 8.U General Flexible Body Dynamics The flexible body dynamics of both the "T" and "l" configurations were investigated using the General Flexible Body Simulation - Stage II. This simulation, specifically developed for this study program, is described briefly in Section 7.0, and is presented in more detail in Part III of this report in the form of a User's Guide. The simulated equations of motion are presented in Section 6.0. These equations have been formulated to include the effects of the following flexible structural members: rotor arms, hub, bearing, and shuttle docking mechanism. An exact solution to the equations of motions is obtained when all the flexible modes are numerically integrated. Unfortunately, such a task would require an unacceptable amount of digital computer time. A good compromise is to find the minimum number of low frequency flexible modes that can represent the vehicle motion with a negligible error. It should be noted that the low frequency modes are considered most important since they generally exhibit larger flexible motions, they don't damp as fast, and they are most troublesome from a control point of view. The approach used to determine the minimum number of significant modes was to systematically perform simulation runs with fewer and fewer modes until significant changes in the dynamics were observed. It was determined that at least 6 rotor modes and 2 hub modes are required for an accurate representation of both "T" and "Y" dynamics. Section 8.3, contains a detailed description of the significant mode shape s. Stability is determined by observing either the convergence or divergence of the magnitude of the transverse angular rate as viewed in the rotor ring | « i—! -H . vr:ViA :l ::rn- ll;Sl-iiiil i ! M | : ; ! - j i . IjiiiijiiiiilL; ; i :- rf- t^*-:,,,...!-];.!:] itfTlTT- -"j • i iTfT*: • -'^ ^\ :\ :r.;_;:j.;"?'r.n .••:,:.rL'i4"M" '^.':f.'-':.'3xi;'p.rr+"l-"n—4 i H.-._i_:_i ^ _,Ti.J.4_i ,|,-. ,..i.|'.ij- d^i(4n3rt.j-j-j-i-i -i41 ':J±!l4Jl-i.'i:- tit."-l'!:: rt' 3 :nH.Jir^ i^firjii::ri oo 3 a P7*F Wui MEM .. .. • IU-J :: -' - - - j J 4 4 _ 4 i _ L .; . - . - I . ' : , * : . . 1,. '". - ; ;.,/../'d-r Hfe-M jw 8 I ...I ___U iJ. iLi^^-'-^Ji^i^ lliJik-"Jiiili...Jii:-::idj LiiIi •_•_: i '.' J".3jlB1 S^jlilSij LiMti-fS^®'1 ;, . , ( . 1 . , ^ ^ . . ; ^ . . .. . , . . : : . ! : : , . . : ; : ; : . : ; :. :. i : -.|-p.:i , 3 7 4 . ; :i .- ':. / '!.: J.:::r; JJJ -i::±r: ni imi^ .j:]^ :^ CO CO3 1 s •H n} by 90 degrees in inertial space. This strategy is called the 90 degree H-Lag Law. The above control technique was chosen because it accomplishes the following desriable control characteristics: o The transverse component of the control torque is always in opposition to the wobble vector. I o The total torque impulse (or angular momentum) applied in 1 opposition to the wobble, per wobble period, is near the maximum deliverable as limited by the wheel size, h. The latter characteristics can be demonstrated as follows: The control torque {T } is given by, (9-1) where i • fh} = the inertial time derivative of 'the CMG angular momentum vector. fuu, } = total inertial angular rate of the CMG = feu} + fu)G} = total vehicle angular rate = P fi} + f(wr} = angular rate of the CMG gimbal relative to the vehicle SPACE-BASE SPIN AXIS TORQUE REQUIRED TO PRECESS GYRO ANGULAR-VELOCITY VECTOR WOBBLE VECTOR ELLIPSE TRACED BY WOBBLE VECTOR GYRO ANGULAR- MOMENTUM VECTOR —. SATELLITE 112 1-7 Figure 9-1. CMG 90-deg H-Lag Law 113 and the transverse torque component fTn } is, "T fTr } = - (P + u>J fi} x fh] (9-2) If fh} lags fuK,} by 90 degrees then it is easily seen that the above torque component will always oppose fow,}> provided & > - P. Furthermore, in order to track the wobble vector and maintain the proper 90 degree phase relationship, the gimbal rate, uv,5 is con-(j strained to equal the rotation rate of the wobble vector, which will always be greater than minus spin. The wobble vector rotation rate * f can be derived by defining $ as, (% tan f = — (9-3) «fe • Using equation (8-1) for the wobble components ouy. and u)75 differen- tiating the result and making the appropriate trigonometric sub- stitutions results in, $ = ± (a +.1) + (1 - a ) cos 2\Pt Where _ are both equal to the wobble frequency, \p. The total torque impulse, !L, applied in opposition to the wobble, per inert ial wobble period, is given by, 1 "0 Forj a symmetric vehicle uj_ = ± \P, so that(3 IT = h, (9-5) which is the maximum deliverable angular momentum. A previous analysis performed at Grumman (Reference 10) proved, using an energy sink method, that the 90 degree-H-Lag Law is time optimum for a symmetric vehicle and is an averaging approach to the time optimal law for non-symmetric vehicles. In fact, for most vehicles, symmetric or unsymmetric, the damping time penality associated with 90 degree-H-Lag Law is negligible. Furthermore, to mechanize the true optimal law would not be practical (as noted in Reference 10). In order to stop the application of control torques, the CMG gimbal is driven so that the CMG momentum vector is stationary in inertial space. That is, when the CMG is located in the rotating portion of the vehicle it is driven at a rate equal, to minus the spin rate. The stop criterion is based upon whether the wobble amplitude is within the Target Ellipse, or not. The semi-major and semi-minor axes of the Target Ellipse are defined as a O and o , respectively; for a <1, the definitions are reversed. Furthermore, -, for a s 1 (9-6) for Where the values for the maximum allowable wobble levels (m ), corresponding to both "T" and "Y" configurations, are given in Table 5-2. . . The basic CMG wobble damping concept, as presented in the above paragraphs evolved from rigid body analyses. Applying this basic concept to the flexible space station resulted in interactions with the structure which necessitated a few modifications. The results of the flexible body analyses are presented in Section 9-2. Before proceeding to the flexible body analysis, an estimate for the required CMG wheel size (h) is obtained. The wobble damping task is specified by stipulating that the amplitude of a component of wobble rate (Q) is to be reduced within a specified time (t_) to a value below a specified threshold, n(tf). Based upon a rigid body analysis, Reference 10 shows that the 90 degree-H-Lag-Law control 115 concept will require a wheel size that is given by, h = n(o) - n(tf) KtT (9-7) where 21 F(k); 1 / 2 - l) 0 <;a <; 1, k1 = (1 - F(k) = the complete elliptic integral of the first kind. All other variables are defined as in Section 8.1. Applying the above relationships to the nominal "T" and "Y" configura- tions (as defined in Section ^ .0) in combination with the wobble damping requirements as specified in Section 5-1 results in the following specifications for the CMG wheel size. Configuration "yll "y" "Y "/Shuttle "T "/Shuttle in response to mass motion^- in response to mass motion - in response to docking^ in response to docking^ internal internal one sigma one sigma Required Wheel Size h (ft-lb-sec) 385 9,670 11,170 17,870 1 Mass motion is defined in Reference 7• Initial disturbance is presented in Section 5.2. 2 Docking disturbance is derived in. Section 5-2. 116 9.2 Wobble Damping of the Flexible Space Station The CMG wobble damping concept, previously described, was applied to the General Flexible Body Simulation - Stage II. A functional block diagram of the math model for the torquer motor and gimbal dynamics along with the associated control logic is shown in Figure 9-2. The natural frequency and damping for the torquer-gimbal com- bination used for all studies were 3. rad/sec and 0.8, respectively. The components of the applied control torque are derived in the simulation as follows: T = h(u^ sin 0 + UK cos 0) 1 T = - i ? (%,» (Wo ~ are vehicle angular rates of the body that the CMG is within 0, 0 - CMG gimbal angle, rate. Where 0 is measured positively from axis 2, and 0 ^ 0 ^ 2rr. Nominal values for all the physical vehicle properties (refer to Section 4.0) were used during the flexible body analyses with CMG control. The initial disturbance, for CMG sizing purposes, is the result of a particular mass motion profile corresponding to a 2,000 lb mass moving within rotor arm #1. The initial transverse angular rate resulting from this disturbance was derived in Reference 7 and presented in Section 5.2 for completeness. Applying the basic CMG wobble damping concept to the "Y" space station resulted in successful wobble damping without modification to the control system. A low pass filter, with a two second time constant, was added to the system so as to avoid unnecessary inter- action with the structure and in turn decrease the actuator duty cycle. Figure 9-3 are the resulting time history traces with a 400 ft-lb-sec CMG placed in the "Y" rotor. It is noted that all motion 117 G IM BA L AN G LE oa. II - A Jfilttfi ' "' I M *~ If te-ift «f i «irr: iif i : 1 . | j^ J-' J-l-1 n- i.:. _:[ . i ; i ! £.| III :! i If: !"il I IT JJTIT j l l ' - :ii: |!^ -r1-; -!r -HJ Hii it • ; i ! j ! :;;! iii •r-T~"'^' ; ; :• ; .. r - K . •r; j[ ii if; ,_ SI I K i |T in ill§:iiITlr-iiji sxxv OT^DS a sffBi X 8Uf "iir rhr i ! ii! -wi l.)-L H :fs H ' -i '-. 1 I TilJ-l-1 ' ; ; -'• illJ I' ! 1 f 3E; tffi I I.-.1+] ;| - ] . • : ; T ; j-j. . 3 : > : ft •y •M- i iif 122 is damped including the relative flexible motion as exemplified by the traces of relative linear velocities (u and u ). Referring to the time history for gimbal rate (p) it is seen that the wobble damping task is satisfied within the required time (l80 sec). For example, when the desired wobble level is reached, the CMG is driven at minus spin rate. Although the gimbal is 90 degrees out of phase at the start of the simulation, the CMG gimbal is at its desired position within 10 seconds and maintains track until the wobble is damped within the desired level. Applying the same CMG system to a "Y" station with the Shuttle attached, and disturbed by a 1 sigma docking, results in the time histories shown in Figure- 9-^ - Although the gimbal does lock-on and track the wobble vector, the amount of damping during the run duration is slight due to the small CMG wheel size (^ 00 ft-lb-sec). It is estimated that it would take 5,000 seconds to completely damp the wobble induced by the docking using a 1*00 ft-lb-sec CMG. It should be noted that it is considered unreasonable to size the CMG to accommodate the relatively infrequent docking disturbances. If faster damping is required in response to a docking, then the reaction jet system can be used. Unlike the immediate successful results experienced with the "Y" station, the application of the basic wobble damping concept to the "T" station rotor required modifications for proper operation. The first difficulty encountered was a high frequency limit cycle re- sulting from the gimbal rate being less than minus spin rate. It is noted that a rigid "T" station requires a gimbal rate close to nega- tive spin and with the addition of small flexible motions there are periods of a commanded gimbal rate below negative spin. The result of this phenomenon is a reversal of the direction of rotation of the wheel angular momentum vector in inertia! space; in turn,control torques are aiding the wobble rather than opposing it. It should be noted that for any rigid body vehicle this anomaly would not occur. In order to fix this problem a negative rate limit equal to minus the spin rate is incorporated. ; ptTB 3 31ft. DO p3q.OSpOJn | • The maximum average power is defined as the average output power required at the beginning of the wobble damping task. Using actual values for a and wobble frequency (\P), as experienced during the flexible body simulations, estimates for required output power are obtained and are listed in Table 9-1. (9-10) It is interesting to note that if the CMG system were placed in the hub, rather than in the rotor, the average output power is given by, Where the plus sign is used with "Max l" vehicles, and the minus sign with "Min I" vehicles. Comparing the above equation with equation (9-10) it is seen that: o The output power requirement for "Max I" vehicles will always be greater for hub mounted CMG's as compared with rotor mounted CMG's. o The output power requirement for "Min l" vehicles will be greater for hub mounted CMG's only if the average gimbal rate is less than 1/2 the spin rate. Corresponding to the nominal "Y" and "T" configurations, the required output power for a hub mounted CMG would be 0.0^ 7 watts and 15.25 watts, respectively. Referring to Table 9-1, it is seen that a rotor mounted CMG requires less output power for both the "T" and "Y" configurations. PART III USER MANUAL FOR - SPACE STATION GENERAL FLEXIBLE BODY SIMULATION STAGE II 10.0 Introduction The digital computer simulation described herein was developed as partial fulfillment of the requirements of Contract NAS 9-11991. The digital program is a special purpose simulation of the flexible body dynamics of the "T" and "Y" space station configurations. The version of the program submitted to MSC, includes the simulation of a Control Moment Gyroscope along with the corresponding control logic used for wobble damping. Although the digital program is presently assembled to simulate two particular space station configurations, by manipulating the input data, the program can be used to simulate a varity of vehicles with rotating and non-rotating sections, whether it be one rigid body with six degrees of freedom to fifteen flexibly connected bodies with eighty four degrees of freedom. With minor modifications to the pro- gram, the simulation can be made to represent dual spin satellites with an arbitrary number of rotor arms (presently limited to three), as well as a high fidelity simulation of a flexible Shuttle Orbiter. The present version of the program is set up to accept external torques to be applied to any of the fifteen bodies. Thus, torque applying actuators can be placed within any of the bodies. A minor modification to the program is required to allow external forces to be applied to any of the individual bodies. The digital program is divided into two main sub-programs: 1. The TIME HISTORY PROGRAM (THP) is a numerical integration of the flexible body equations of motion presented in Section 6.0. Wobble damping control is added in the form of a subroutine called CMC (refer to Section 9.2 for the model description). A modal reduction technique is used to decrease the number of coordinates (or integrations) required to des- cribe the vehicle motion. 2. The MODAL SERVICE PROGRAM (MSP) is used to determine the mode shapes, frequencies, stiffness matrices, and damping matrices corresponding to the flexible rotor and the section of flexible hub. This data is required for use by the THP. Output ' from the MSP is in the form of punched cards, in the proper format, for direct input to the THP. The programming language used was FORTRAN IV. Although the program was developed using the IBM 360/75, particular care was taken to insure compatibility with the UNIVAC 1108 compiler. A description of the THP and MSP are found in Sections 10.1 and 10.2, respectively. 10.1 TIME HISTORY PROGRAM (THP) The THP accomplishes the numerical integration of the flexible body equations presented in Section 6.0. The program is structured as follows: INPUT DATA AND INITIALIZATION _L CALCULATE COEFFICIENTS FOR EQUATIONS OF MOTION NUMERICAL INTEGRATION YES The basic program functions, as pictured above, is accomplished by several subroutines which are described in Section 10.1.2. The program user can observe at a time increment of his choice, the displacements, rates, and accelerations of each of the 8k coordinates that describe the motion of the vehicle. The number of variables that are actually numerically integrated is equal to the number of flexible modes, chosen by the user, plus eight rigid body modes. The numerical integration technique, known as "Modified Adams," is a corrector-predictor technique that uses a fixed integration-ififjer-ral. A good rule of thumb for chosing a value for DT is that it should be less than j(l/20th) the period corresponding to the highest frequency. The operating procedures for the THP are presented in Section 10.1.3. 10.1.1 Variable Definitions Table 10.1 contains the definitions of all the program variables used in the Time History Program. The notation used in the equa- tions of motion are used as definitions for program variables that are directly correlated with the equations of motion (refer to Table for definitions). TABLE 10.1 Definitions of Program Variables NOTE; The superscripts included in the table below are defined as: j - lumped hub mass number ~~k - lumpeB~rb'tor' mass "number — — .»_... i - rotor arm number Symbol Definition AALPHA (I, J, i) ALPHA (i) ANTTWO B BDE BDED, BDEDD BDEF BDR BETA (I) BIAS DD(I) DEL x (I), DEL x (I, -i)., DEL x (I, k, i) DEL (I), DELD (i) DELTB A (a1) i a Logical variable to initialize problem to second antisymmetric mode of Y-rotor CMG gimbal angle Desired gimbal angle Gimbal rate, gimbal acceleration Filtered desired gimbal angle Limited gimbal rate 0 CMG commanded angle bias, filter compensation. Input data array Ik V , V1 6*' Symbol Definition DT DTPLOT DTPRNT CH (I, 3) CHG (I, 4) CONTR CR (I, 40 ETA FC (I, j) FD (I, k, i) FG (I) GAM (I), GAMD (l) GPSID (I, J) GYB (I, J) GXL (I, J) GYBM J) • Inert ial angular rates are calculated based upon the vehicle state. The vector variable is determined. FLX11 (Figure 10.U) .- Calculate the M matrices and start the formu- lation of the Xic matrices. FLX12 (Figure 10.5) - Complete the formulation of all the xk matrices. FLX13 (Figure 10.6) - Start setting up the right side of the equations of motion by computing Afc- FLXl^ (Figure 10.7) - Continue formulating the right side of the equa- tions of motion by computing £k. FLX15 (Figure 10.8) - Complete the formulation of the right side of the equations of motion. All elastic forces and moments are computed by multiplying the stiffness and damping matrices by the appropriate relative displacements and rates. Finally, u^ is Computed. PREINT (Figure 10.9) - The purpose of this subroutine is to solve for. the second derivates of the modal coordinates which are sub- sequently integrated in subroutine INTIG. The process is outlined below: Set up total mass matrix \ , and right side cr. = fa] Where Y is the physical coordinates, or state vector. 155 CALL GEKIM CALL GENR (entry to GENM) . Transform the equations, via Q, , to eliminate all constraint forces .from the equations of motion. CQ] C X] fY} = CQ] fa) Reduce the nuniber of equations via mode shapes C$T] [Q ][ X ][*] f X ) - [$T] [Q ]fa} Where X is the reduced set of coordinates, and is related to the physical coordinates through the mode shapes, fYj = [*] 1 SOLVE FOR X, XVECDD IHTIG (Figure 10.10) - Perform a double integration, via CALL RIEMAN, to obtain the displacements and rates in modal coordinates. Transform modal displacements and rates to physical displacements and rates by, - [*] {*}, {Y} = •[$] {X} The modal displacements and rates are initialized within INTIG by applying, . f X(0) (A)~V[Y(0)} Finally, Euler rates that aren't included in the state vector (eJg. yj and 6-,) are integrated to obtain the corresponding Euler angles. GEHM (Figure 10.11) - Eliminates all constraint forces from the equations of motion. CMG (Figure 10.12) - Develops control torques, TQ. for wobble damping. The simulated CMG configuration and associated control logic is presented in Section 9.2. The user has the option to place the CMG in any of the bodies that form .the flexible vehicle. This is accomplished by the firs.t and last three executable statements in the subroutine. The first three statements equate the angular rates of the body to the rates (10-1) (10-2) 156 used within the control equations. The last three statements equate the external torques applied to the particular body to the derived control torques. Control variables that require numerical integration is accomplished via a direct call to STLJES from CMG. It is noted that an additional common block, called CONT, is used to communicate between MAIN and CMG. 10.1.2.3 Utility Subroutines RIEMAN (Figure lO.lS) - A Modified-Adams numerical integration, for a second order differential equation, using a fixed integra- tion interval. Basically, the integrated variable at t + &t is approximated by the first three terms of a Taylor series expansion about At. That is, f(t + At) = f ( t ) + f ( t ) At + f (t) &- . . (10-3) The integration scheme is a corrector-predictor technique where the predicted value (i.e., calculated at t for t + £t) is corrected during the next iteration by applying current rates and accelera- tions to the above expansion. STLJES (Figure 10.11+) - A special version of RIEMAN for the purpose of integrating a first order differential equation. GAMMA (Figure 10.15) - The GAMMA matrix, r(a), is formed to facilitate the vector cross product operation. Specifically, {a} x {b} = r(a){b} .. (10-U) STACK (Figure 10.l6) - Places small matrices at specified locations within a large matrix. ABCPI (Figure 10.17) - Formulates the coordinate transformation (n(9)) for three ordered rotations (0]_, Gg, 93), and/or the coor- dinate transformation for a single rotation A(8i). SMPY (Figure 10.18) - Multiplies a scalar by a matrix. (Part of IBM Sientific Subroutine Package). GMADD (Figure 10.19) - Matrix Addition (IBM Scientific Subroutine Package). SCLA (Figure 10.20) - Set each element of a matrix equal to a given scalar. (IBM Scientific Subroutine Package). GMTRA (Figure 10.21) - Matrix transpose. (Part of IBM Scientific Subroutine Package') GMPRD (Figure 10.22) - Multiplies two" matrices (Part of IBM Scientific Subroutine Package) 157 (MSUB (Figure 10.23) - Matrix subtraction. (Part of IM Scientific Subroutine Package). SREVN1 (Figure 10.2*0 - Inverts a real, single precision matrix. The method used is Gauss-Jordan elimination with partial pivoting. WILLY (Figure 10.25) - Prints a matrix in an easily readable format. FIGURE 10.2 MAIN 158 >()• i ( Aur, 71 ) OS/363 F O R T R A N H COMP-T.LEP OPTIONS - N.AME = M A I N « CFT=0 2 , LI NECNT = 60 ,S IZ E=Q3 OOtC . _, SOURCE.BCD.NOLIST .DECK ,LOAO.MAP, NOEOIT, ID.NDXREF I M P L t C T T RFAL ( A - Z ) COMMON/TI VF /T .DT . N C Y V . A N T T W C COMMON/CONT/ ,TAUF, «TAPT , STPP.H , VDNG , ZETG , 809, B, F F, BOE, BDE F, BOEDD, BO ED.*TC( *B TA *E *4 A L P H A ( 7,3, 7) ,ALPHAf ) , BETA (3 ) .DELI (3 ) . DFL2( 3 ,7 ) , DE_ 3( 3, 3, 3), *) ,nFl 5(1 ,3,3) .OFLM3) ,OEL7(7) , OEL3( 3) ,DE_TB, ) ,CH( i8.1 P> ,CHG(6,1 S) .DEL (3 ) ,rt!ELO(3 ), IP( 7) ,FH ,ER ,ETA ,FC(3,7) , F 0( 3, 3 , 3 ) . ^G( 3 ) , G AM( 3), GAMCM 3).GH, JP( 3) ,r, P « T D (3 ,3) ,C,P ,r,YnB(3,3> ,C,XL( 3,3 ), O0( 500 >, ,r,WP(3 ,3 ) ,<~, V * X ( 3 ,3) ,GW X»A ( 3 , 3 , 3 ) , G W V ( 3 , 3 ) , C,WYC( 3, 3, 3), ) ,R VB (,? ,1) ,GYRW (? ,3 ) ,GYH( 3,3) ,HCMA5S( 3), H It HJt HMASS LX/ *C, COMMON' < BS2. *KBVH(3 .3 ) ,KBVP(3 *LM?(3 *LM ,L*4 (3 .3 ) .LM5(3 .3»3, 3 ),LM6( 3. 3 ).LM7( 3, 3,_3) » :,3) ,LM/ *M .M! ( 7 .7 ) ,M?(3 ,7 ,?) ,V7(3 .3 ,3 .3) , M4 ( 3 , 3 ) . MM 3, 3) , M AST , M ASS( 8 4. 8~4 )", *MF< ( 7) , M B F A P ( 7 ) , M C ( " ' . 7 ) « M O ( 3 . 3 .3)«M'">OC<( 3 ) » M G ( 3 ) ,^ROT,NHUB, *N ,MS,o ,OA SS ,PIBF TA (7 ,3 ) .P IGA»»(3 ,3 ) . PIPH I C( 3, 3, 3), PIPHIO( 3, 3, 3, 3). *P IP!?I (.T..3) «PT THF T (3 ,7) ,PHIC(3 .3) ,PHICD(3,3) . PHI DC 3, 3. 3 ) . *PHIDO (7,3 .3) ,PSI (7) .PSI 0(3) , PHI MR (5 7 . 57 ) ,PH IM H( 13, 18)tNTHET( 3.3) COMMON/FL>/ * P D M A « ; c ( T , 3 ) ,DH(7 ) ,PI ,PJ .PNASS , RPM AS S . PT S (84 ) , S 11 ( 3. 3 ) ,S I 2( 3). *S I7 ( T ,7 ,3 ) ,c iA (7 ) ,STG1 (3) .S I G2 (3 . 3) .5 IG 3( 3. 3, 3 ). 3 IG4{ 3. 3). * SIG?( 7 ,3 .3) .STG6 (3) ,SIG7(3) ,S I G8 ( 3 ) .5 G AM ( 3, 3) , SM ASS , STHETJ 3. 3). * SUM 1 ( "3 ,3.3) ,TAUA (3) ,TAUC(3 .3) ,T AU D( 3, 3 , 3 ) ,T AJL( 3 ) ,TAJS( 3). *THFT( 7) , THE TO (?) ,PHO(3) ,STHE(3,3) .SPIN COMMON/FLX/ __ *UC( 7, 7) ,UCO (.3,3) ,LO (3 ,3 ,3 ) . UOD (3 , 3 , 3 ) , W OKI ( 3, 3 ) , * OK 2( 84•, 84) , * W O K 7 ( P&) , WOK4 (3 ,7) , *O K5 (3 ,3 ) ,WCK5 (3 . 3 ), W R( 3 ) , rf< Rf 3), rfXRA( 3, 3). ^ *XL( 7) ,XLL(04) , X V E C ( R 4 ) , X V E C n ( 8 4 ) , XV ECOD ( 84 ) , V C( 3, 3 ) . Y B( 3 ) VfE3( 3)". * Y G ( 3 ) ,YH(3 ) . 2A (3 ) , ZAO (3) ,ZV3 (3 ,3) NAMFLI ?7 /T I V/T N A M F U ! C T / ? T A T E / 7 A O , / r A , W Y H , G A M .UCD.UC.PH ICO, PHIC, OELO, DEL. W X R , *Unn ,(in .DHTOO ,PH IP ,W7 ,XVEC , X VE CC,X VECDD, RTS,*r MM NAMFLT ?=T/SH LT/PSI n ,PSI ,*R , JT , NDOCK, PIPS I ,T A'J5 NAMFLI ST/TF STC/AI PH A, RET A, ETA .HMASS .HCMASS ..I A. IB, 1C. ID. IH, *KRS1 , K B S 2 , K B V H , K B V R , K H S , K H V , K .MAST.NROT . NHJ B, N. N5 , P, PASS. *GXL ,GPH ,G VB ,G YBM ,GYH ,PI BE TA. f i DM ASS, PH, R M ASS , RRMA5 Sj» SM ASS . S3 IHj, *XN, VI, ,YB , YF1* ,YG »YH , Z V 3 , C P ,CH , CHG , KHG, KP , KH, PHIMR, PHIMH CT / T = cT1 /OF LI ,OEL2 .DEL3 ,OEL4,OEL5 . DEL6, DEL 7 , DE_ 8, FC, FD, FG, YGB ,G HXPA ,G»> ,GWVS.L*1 .LW? .LM3.LM4.LM5, *LM6 ,LM-r ,L(«y .LVP.LN1 0 ,LM1 1 ,L»»1 2 ,LM13 ,LM1 4 ,LM 15 , LM1 6, LM 1 7,'LM 1 8, L*fl 9, *LM?O,LM?I .LNSP ,L^23 ,Lw?4 ,LM25 ,LM26 .LMAB.LMI 3B.LM1 68 NAMFLI ST/1E ST2/M1 ,^2 .V3 ,V4 , K5 . WB, MBEAP, MC.MD, MG. a IGAM. *PIPHTC .PIPHIO ,oi THFT .NTHET , SI 1 ,S I 2 . S I 3 . S I4.SIG1, 5 IG2, S VG3, *SIG&, FIGS .SIC6.Sir,^ .SIGfi .SG AN .STHET ,TAJ A.T A'JC.T AJ D.T AUL.THET, ,PHO , FIGURE 10.2 (Cont'd) X59. NAMFL I ST/TF S T3/P T «= , XVFC , X VtTCO , XVF.COO NAMFL I «=T/CMGPR/VOP ,FE ,PDF ,ROEF , P , I1DEDD, RDEO, P DP, T C IMTFT.FP I T , JJ .KK ,Ll. .NN INTfGFP I HUT, I PI N ,IDN ,NRUN T N T F G F P . N P O T , N H U B ^ « V t N F C l N T , N S Y M INTEGER N»O*(5) ,CTL COMMON/BUF/BUFFEP (5! 2 ) DIMENSION V A L U ( ^ ) D IMFNSTON O N A M ( 1 1 ) sO. 0 DO \ 2 \ 1 I t=l ,500 nn( T T ) = 0. C P F A D < T H I N ,32) NPUN , O N A V F DP MA T ( I 1 ,! 1 A4) WP r T? r rn tT ,??) FORMA T( 1H1 , «Pi;N T F ( M P t;N.C,T. G ) G O ^o. o. , O N A N MJMREP=« . 13 .5 X ,1 1 A4 / 1 TC ?5 r, r TC ??4 26 CALL FVI T R E A D ( I O I N , 2 6 ) I O N , V A L , F OP MA T ( I 5 »E 2 0. 5 , 1 J A4 I IF( IDN.LE.0.)GO TT ?7 D O C T D N ) = V A L .20) IDN .VAL 27 F OP MA T ( "5 X , « DO ( « , I ? , • GO TO 25 D TPi,.OT=r>r> (i ) JF(^DTPLOT,LE. 0. Q. CR» PI "" !=• ,F?0 .5 ,5X QAC PLOT ROUTINES REPLACE BY EQU: MSC PLOT ROUTINES NE. 0 .3 )GO TO 75 C C PAU«?FD = 1. 0 7«: C DNTI NIJF INPLT OA TA APRA > ITERATION TIME O 1=DO (PI RUN TIMF C C ?FPC NO SHUTTLE PP I NT I N^FP VAL D 1PPNT=OD (4) MASS PROPERTY DATA PI GTP HUB MA <_« HMA c<= =DO ( 7) /?2« 1 74 SHljTTLF M A S S - SFT SMA ««=OD (0) /3^. I 74 SHUTTLF ATTACHED NS = I. ,NCT A T T A C H E D NS=O. NS=DD (5) NUMBER O^ ROTOR A R M S P =nn ( f-. } RHTHP PING M A S S C C DO 10 I T = I , .? M A S S OF LUMPED HUB W A S S HCM/iSMI I >=OD (o* T! )/32. 174 M A S S OP LUMPED FflTOP MASS-EACH P D M A S S f II ,1 )=DD (12 + T I ) / 3 2 , 1 7 4 . P D M A S ^ d l ,? )=Rr>MA.cs( I I ,1) ARM IS S A M E FIGURE 10*2 (Cont'd) 160 IF(P.F0.2.)GO TO » o D V A S S d l ,3 )=PDMASS( I 1,2) GO TO J C B P Q M A S 5 - ( T I » 3 ) = 0«0 DO 15 II =! ,3 O o 1.5 J J = 1 ,3 C PIGIP HUB TNFFTIA NN = ! 5+ J J+ 3* (T T-l ) I A ( JJ ,1 I ) ^ fMNNl C SHUTTLE MASS - INPUT ZERC FOR NO SHJTfLE IH< JJ ,1 I ) =DO(NN4 c) C pn TTP PING INERTIA IP( JJ , T I )=DO(NM* ! P) DO 1 ^ YF3( I I ) =DD( 1 0«+T T ) C LOCATE M A S S CENTER CF POTC1 XL( IT )=DD (t 1 1-H T ) C L O C A T E FLEX ARM - ALL A R * S XN( IT ,1 ) =PP> (1 1 A+ T I ) X N ( I T , 2 ) = X N < T 1 , 1 ) X N ( I T , ? ) = X N ( I I , 2 ) 7 V 3 < I T , 1 ) = O D ( J 1 7 + I T ) 7 V 3 C I I . 2 ) = 7 V 3 ( I T ,1 » Z V 3 ( I I , • » » = ? V3 f I I ,2) DO 30 J J-1 ,3 N N= 1 2 C+ J J+ 3 * (I I - 1 ) C LOCATF HUP LUMPED M A S S YC ( JJ , t I I =DO{NN+. 5C) C L O C A T ^ POTOP LUMPFO VASS XD( JJ , T T ,1 ) =OD (h 'N) •=0 XD( JJ ,TI ,3) =00 (NN) C ARM POTA TI ON AL°HA (1 ) = C? ALPHA (2)=OD (! 30) II REPRESENTS EACH LJMPEO MASS ARMS I DENT ICAL RING IDENTICAL . CHUTTLF P O T A T T T N FIGURE 10.2 (Cont'd) pO 40 I.T -=1 ,3 AC. R F T A { I I )=DO (1 11 + 1 I ) C STIFFNESS AND V ISCOSIT IES C P F A R T N G ' KBS2=DD<1 36) C O ISrONTT NLT TY DEFLFCTIOK r»FL in =00 ( 1^7) C AMOUNT OF DAMPING CN I NNF E PACE (ZERO TO ONE) 5 TA =n>n ( i 3») OO "50 11=1,3 nn ^o jj=i.i C nut~P PACF DAMPING MNsJ.J*''* ( TT -! )-«-1 7fl K rlVP ( J,l . I T ) =OO ( \'N ) r TNMFn' PACF D A M p f K C , K P V H C JJ , T I) = c «HUTTLF P T V O T KHMJ J9! 1 ;.)=DD (NN+lfl) c «HU^TL.E PIVOT DAMPING fO KHV( J J,I I )=DD (NN+?7) 124 1125 ' • ?6 ._ 127 1 2P 129 130 13! 1 32 133 V4 115 1 36 1 37 138 139 140 C ROTOR A R M ? — ALL I DENT C F EP=DD ( 175) C I R 1=00 (176) r (^ GP=CD ( 177) C J P J=DD (1 76) C HtW S E C T I O N S c r F. Hr=DD ( 7 7 0 ) C I H I =DD ( 190 ) C G GH=DD (I 81 ) C J H J=DO ( 1 82 > C EQUATING F AC H S E C T I O N F IR (1 )=ER*RT G JP ( 1 )=GP*P J E IH(1 )iFM*Hl GJH( 1 )=GH*H J DC 60 11=2,3,1 E If (I I )=EIP (1 ) G J" ( I I ) =G JR ( 1 ) E IH(I I )=EIH (1 ) 60 G JH( I I )=G JH (1 ) NOMINAL SPIN RATF . 1 )GO T T . - M 5 IFfMRCT.F.O.DD (?! 0) )GO TC 315 ?FPO ONI MP FOR NFW WCDAL D A T A DO 71 C II =1 t57 DO 71 C JJ = 1 i57 310 OHIMO (J J.II )=0.0 •* 1. 5 CONTINUE FIGURE 10.2 (Cont;'d) i6e C NUMRFP OF R O T O R MODF5? 0150 N P O T = n n ( ? } Q } C NIJMRCTF QF- H LR MP.DF « 01=1 N.HUR =00(211 ) C NUMBFP OF <=YM. POTCP VCDES 015.2 N«YM=00 (212 ) C ANTTWO=1. TO T M TI A LIA7E THE PCTOP 2ND A N f l - S Y M MODE 0153 ANTTWO=DP (2 1 3) C NUMRFP OF PLOT R L O C K C 0 1 * 4 PRLOC K=OD {?. 14 ) C O A T A FOP CMG CCNTPCL C FOR CMC CONTPCiL SF T CONT ,GT. 0.0 CONTP =DD (220) C. «TOP P A W P T N G FT.P hCPRLE .LT. STCP POP *rF>riLE «GT. TIMF C O N S T A N T OF G I NRAL ANGLE COMMAND LOrf PASS FILTERtNO FILTER IF / TAOF = O O (223) . . ' . . . . _ . ANGLt_AR MOVFNTL'V PF ViHEFL H=OO ( ??4) N A T L P A L FPFO. OF TCROUEP N C T C P 5 T A P T DAMPING 7FTr, = OF TOROUEP F F R F C VT MJS FCP MIM NT G I W R 4 L ANGLE BIAS W A I T OPTION FOR A C T I V F C C N T R T L S F T V 8 A T T = M I N . FPPOP( IK RAO)PEFCRF W A T T=00(2?«) I M T T I A L I 7 F f iTAT^ V F C T T F nP TO T T =1 « ? 7A( I T )=OO ( 3 C O - M I ) D «I (I I ) = D D ( 306+1 I ) OO 70 JJ= l t3 IJC ( J J , I IT ) =D 0 ( NN+ 3 C 0 ) OO 70 KK = 1 . 3 LL = 30 + KK+3* (JJ-1 )+l 8* (I T-l ) UD(KK,JJ,I I )=OD(LL+300) 70 PHIO(KK ,JJ,II )=DO(LL+309) INITIAI I7F O F P I V A T I V F CF S T A T E DP PO T T =1 , 3 ? A D ( I T ) = O O ( 4 0 0 + I I ) WYH( ! I ) =DD( 403+T ! ) P S I O ( II )=00 (&06-H I ) W X Y U I )=OD( A?7-H I ) DO PO -JJ = 1 ,3 NN = o+ JJ+ 3*( IT-! ) l/ro ( J J,f f >=00 (NN4-400I PHICO ( JJ , I I ) =OO (K'N + 409) OO f lO KK=1 .3 LL = 30+»n PHIDO (i^K , JJ ,T I ) =DO (LL+4C9) TN IT T A L I 7F F(JLFP ACTIVE CONTR IS ALLOWED FIGURE 10.2 (Cont'd) 163 r c. ANO D A M P I N G 205 MATRICES DO CO r, A M ( I I ) =f CO THFT( II )=OO (4F7-H T 7FPO MODAL. , STIFFNESS IF (NP L'M.G T. 1 )r,p -r< 50 20.1 JJ = 1 »*7 OP ?0 '. I T =1 t*^ 7 O HIMp (II , JJ ) r 0. 0 KP f T T , -JJ) =0. C PC! CP< I I t JJ) =0.0 DO 204 JJ=1 .1 8 DO 207 I f =1 «1 3 PHIMH (IT , JJ)=0. 0 K H ( I T ,JJ)=0.0 ' 2C"1 CMC I I , J J) =0. C DO 204 I I =1 ,6 KHG (I T , JJ) = C. 0 204 CMGd I . JJ) = C. 0 9 r= COM TI \'(.F INPUT MPD»l. .STI FF KF S 5 AND DAMPING VATRICE5 ?C^ P F A D ( TOT N ,?C7)rr( . , ( N P C W ( JJ) ,VALU( JJ) . JJ= 1 ,5 ), I DN 2C7 FOPMA T ( T 3 ,5 (13 .F12.5) ,12) TF( TDN.F.O.O)GO TC 100 t 1=1. TF( T r»N) .GT. l )GO TC ?08 L I. = 1 Q • C C c c. r C. r c c r. T P ( C n L « G T . N S V M ) ^ C TC 206 GO TT ?0fl . CCL=C CL-LL+ N? >V I F(CH L.G T. NROT) r C TO 206 ?. C ° T F ( NR C t * ( ! T ) . F o. 0 ) G t T C 2 06 GO TD (21 O.?20 t?30 ,?40.?50 t260 t270 .230) , ION TON I NFUT D A T A TO- 1 PHIMP 2 PH I VH • » K T A C P K Y A 4 CT CR CV «S . KM 6 CM 7 KHG =VALU( I I ) 0 2 4 2 2?C 11=11+1 0244 IF( I I ,LT. *) GO TO 208 0246 GO TO ?06 0247 -?00 CONTIN'L'F C I N I T I A L I S A T I O N PLCC K C T T M F WTMM= C. 0 AD") PLOT T N T F P V A L S T. 0. ) NT = 0. 0 TPLOT=0. 0 N PH I N T x 0 GIMpAU ANGLE AND G I V P A L PATF * =0. 0 R OP = 0 . 0 F X "^ P N A L T OR C UP. S DO A6C I 1=1 .1 TAOL ( T I )=0. 0 TAUA ( i T ) =0. c T A U S ( I I )=0. C DO 46C JJ=t t"' TAUC ( JJ ,1 I ) =0.0 DO 46 C KK=1 ,3 460 TAljn ( KK t JJ«II )=0» 0 c c M A I N I T F P A T I C N !SO CALL FL*1 CALt CALL CALL CALL LOOP FLX1 1 FLX12 FLX1 3 FLX14 I F ( C O N T R . F . O . 0 . )GC CALL. CMG 151 C O N T I N U E CALL FLX15 CALCLLATE WOBBLE MAGMTUDE TG 151 MAX WOBRLF CVFO O P I K T OCLE 152 IF( T .LT WPI TF ( l O U T t TI W) CALL PPEINT IF( T.LT. T D P I N T ) G O TO T P P T N T = TPPT N T - f O T P P N T WP ITF (I OUT , S T A T F ) TC 152 155 FIGURE 10.2 (Cont'd) 165 1 22 I F t C O N T p . E Q . C . ) G O TO 156 CCNTI NUF TC 1 53 CALL W T L L X ' LAy • .LAV ,94 , 34 , 1 ,34 , 1 , 84> WP f TF ( T O U T , TF <=T! ) WPI TC ( I OUT, t^^T? ) WPT T- UOt'T . ir ^T? ) f F(PA "SS.GT. C. )«0 TP 154 WPTTF. ( IOUT.TFSTC ) TF(NS,FO. 0. )r,C TH 1 «55 wo T T^ ( T O U T , «HUT) C PNTJ NIJC TF (C> T"LOT.LF. C. ) GC TC 125 T F ( P A ««:.BLCC K , TFNO , OT PLOT CCINTI NL^ . T F( T. i. T. TPt_PT)r, n TO 12=1 i^( Nprr NT.f;p, e'roi oc Tt 125 MPO IM T = NPOT NT-H WP I TE (1 )T , W X P ( ? ) ,WXP(3 ) ,WVH(2 .» WPt TF (1 ) »VH C?) ,V»>C (3 ,3) ,UCO(2 ,3) w»i TF n >P find ) ,unn(i ,3,2 » ,uooO, 3,2 C ON TI IF (T*OT .LT. T E K D J G C TO 120 TFCDTPLOT . LF.O. )C,C! TC 121 E ND F T LF 1 . CAM r, n T r niMT CALL IMTIG T = T4r> T PAS 5=1. en m i. 5 o E NT> (AUG 71) FIGURE 10.3 FLX1 OS/360 FORTRAN H 166 COMDII..EP T O O ? 10:04 - NAME = M A T N ,OPT=0 2 , L I NECNT = 60 , 3 IZE= 0 D SOURCE ,f?CO .NOLI ST ,D = CK .LOAD, MAP, NOEOI «U*FOt.TlNE FLX1 T M f n i r t T PT At. (A- 7) CO-IMHN /FL »/ * A ALPHA (3 ,3 , 1) .ALPHA (3) .BETA <3 ) .OEL1 (3 > , OEL?.( 3 *DFL«( 3,!_) .DELSC* .3.3) ,DEL6(3) ,OEL7( 3) ,OEL8( 3) * C ^ ( 57 ,*7) ,CH(! 8 ,1 fi) ,CHf-, (6 ,18) ,OEL(3 ), DEL DC .3), *F IH( 7 ) ,P ID( 3) ,FH ,FP .ETA ,FC(3 ,3) ,FO( 3,3 , 3 ) , FG( *C,JH( 3) ,GJP( ?) . G P S T D (3 ,3) ,GP,r ,YGR(3.3) ,GXL( 3,3 *GPH{3 ,3) , G W P < 3 ,3 ) ,r, V*X (3 ,3) ,G<* XRA (3 . 3 , 3 ) , G W Y ( 3 *G WYM 3.3) ,C, VP (3 ,3 ) ,GYBW (3 ,3 ) ,GYH(3, 3) ,HCMASS( COMMON/^LX/ * I A ( 3 , 3 ) . I B ( .3 , 3 ) , I C ( 3 ,3 , 3 ) , I C ( 3 ,3 , 3 , 3 ) , I H ( 3 , 3 ) *KPVH( ^ ,3 ) ,KPVP(3 ,3) ,KHS<3 ,3) ,KHV(3,3) , L AM ( 34 , *LM.?( 3 .3) ,LM3(3.3 ,3) ,J.M4 (3 ,3 ) , L V5 ( 3 , 3, 3 . 3 ) , LMS *LMflC 3,3.3) ,LM , ? ) , *t»Hir-r» (.3,3 .3 COMMON/TL. x/ *PDMAS5(3 ,3) *SI3( 3,3.3) , *Sir,5( 3,3,3) *SUM1 ( ->,3 ,3) *THFT( ->) .THF ?) (3 ,3,3 .3) , M*( 3,3) (3. 3 ) , M AST , M ASS( 8 4, 84 ) , (3) ,MC (3 ,3) ,VD(3 ,3 ,3) . VDOCK( 3), MG( 3 ) , * ROT , NHU8 , ,DTBFT/ * (3 ,3) .PIT, A V ( 3 , 3 >, PIPHIC( 3. 3. 3), PIPHIDC 3, 3. 3.3). PI THFT (? ,3) ,PHtC(3 ,3) ,PHICD( 3, 3) ,PHIO( 3, 3, 3), ) ,PST (?) ,p<=in(3) .PHI MR (5 7, 57 > ,PHIMH( IB , is ) ,NTHET( 3, 3) C C SFT ,PH(3) ,R I , P J ,P> ASS , PRM AS S , RT S (84 ) , S 1 1 ( 3, 3 ) . S I 2( 3 ), 514 ( -») ,STG1 (3) .STG2 (3 .3) .S I G 3 ( 3, 3, 3 ) , 5 IG4( 3. 3 ) , .STGM3) ,SIG7(3) ,SIG3 (3) .5 GA M( 3, 3 ) . SM ASS, ST_HET£ 3, 3 ). , T A U A ( 3 ) , T AUC ( 3 , 3 ) ,T AU D( 3 , 3 , 3 ) , T AJ L( 3 ) , T AUS( 3 ) , TD (3) ,RHC(3) .5 THE (3 ,3) .SPIN *UC(3. 3) ,UCO (3 ,3) ,tn (3 ,3 ,3) ,UOD (3 , 3 . 3) , W OKI ( 3, 3 ) , H OK 2( 84. 84) , " * W O K ^ ( P<1) , v»OK4 (3 ,3) ,V»OK5(3 ,3 ) ,WHK5 (3 .3 ) , WR( 3 ) . WX R( 3 ) . « X R A ( 3, 3). * W X Y ( 3 ) , W Y C ( 3 . 3 ) ,V»YH (3) , W 7 ( 3 ,3 ,3) , X C ( 3 4 ) ,XCO( =t 4 ) , < D( 3, J3, 3 ) , XN(_ 3L3^ * X L ( 3 ) .XLL(64) ,.XVEC (84 ) , X V E C O ( 8 4 ) , XV ECOO ( 84 ) , Y C( 3, 3 ) « Y B ( 3), YE3( 3), *Yf, ( 3) ,YH(3 ) ,?A (-! ) , 7 A D ( 3 ) . Z V 3 ( 3 ,? ) INTFGFf- I T , JJ ,KK .LL.NN DIMENSION OtMMY (3 ,3) UP TR ANSFPP. VA TI ON V A T P T C E S TPANSFntJMATION A OF ALPHA-STEPl IF (PA 5S.GT. C. )GT TC 10 " .. . . . . . on ? IT =1 ,3 nn i jj=i ,3 A ALPHA( Jj',1 I ,! ) -0 .0 ........ A «LPHA(I I ,1 I ,! ) =1 .0 KK=P 5 CALL Anroi ( AALHHA (1 ,1 ,1 I ) .DUMMY, ALPHA( I I ),0 ) ADJL? !T A A L P H A F CP NUVF^ICAL I N A C C U R A C I E S IF(P,FO. 3. ) GO TO 6 A ALPHA( 2, 2.3) =-i , C AALPHA (3 ,2 ,?) =0. 0 A ALPHA (?, 1 ,2) =0. 0 AALPHA ( 3 , 3 , ? > =-i . c GO TO 7 FIGURE 10.3 (Cont'd) 16? c r AND f. A A L P H A ( 2 , 2 , 2 ) - - • 50 A ALPHA M, ? , 2) =-. A ALPHA ( 2 » 7 » ?) =+• A ALPHA (i, •«, ?) =-. «c A AI.PHA( ? , 2 . 7 ) =-. "SC A A L P H A (3 ,2 ,7)=4. f i f i60?54 A ALPHA ( 2 • " * » ? ) •••=—• Pft *S 0.? 54 A ALPHA (? . , •> , ?l =-. so 7 C P N T T NUF T R A N S F O R M A T I O N PI CF RFTA - STEPS CALI A P C P T (DUMMY , P I ^ E T A , R P T A , 1 ) NTI N»;E TRANSFORMATIONS PI OF r, AN WA ,THET A , PS I , PH I C ALL ARC PI ( DUMMY , P T C i A * , G A W , l ) CALL A«CPI (DUMMY ,FITHET ,THFT ,1 ) CALI. A R C P I (DUMMY ,PI PSI ,CSI ,1 ) DO" 20 11=1 ,3 CALL ABCPI (DUMMY ,PIPHIC (1 ,1 .1 I ) , PHI C( 1 , I I ) , 1 ) no 20 J.J = i_«^ ?0 CALL A R C P T (DUMMY ,FT^HTD (1 ,1 , J J,I I ) , PH I D ( 1, JJ, II), 1 ) T R 4 N S F O P V A T I ON J -FRCV HUP TC SHUTTLE - STEP 7 C. AH. r,MPpr>(PT PSI ,PTPF TA , JT ,7 ,3 »1) T O C A»4D D.-STEPS 5.6 "=FT L<"> F(j|_rp O A 7 P C A L C U L A T I CNS - r ?=r n « (GA M( ?) ) r "=rn « (C-A v( ~) ) S«AM( 1 , 1 ) =C 3/C? S G A M ( 2,1) =«3 S<;AMC •», i ) =-c?*S2 /c? C.A M( ^ ,?) ( ? , 2 '( i ,?) =o. S G A M ( 2,?) =0. ,SGAM( ?,1) =1 . S?=PT N( THET(2 ) c^=c i N. (THFT (->) C 2=COC STHFT( ! ,1 )=C STHrT(2, l )=S STHFT( •*,! ')=- T ( i ,2)=- S THF T ( 3 , .? )= 5? *S? /C? (! ,3) = C. (2 ,^) =C. (^, ?) =1. S THP T S1HF ( 1 ,1 ) =C 2*C2 STHr ( 2,1 )=-S3 STHFf 2, !)-=«? STHF ( 2,2) =C.^ STHF ( 3 .2 ) =0. ( 1,?) =0. FIGURE 10.3 (Cont'd) 168 5.079 9.0 pa 00*5 308* OOP7 OOfifl r r ( ,?) =o. ,?) =1 . NTHF Tr ALT ULA N THF T(i ,J ) = I.-CTHF (1,1) NTHET(2 ,1 )=-STHF (2,1) MTHET( .7,1 )=-STHE: o.i ) N TMF T ( 1 , 2_) = C. 0 N THFT(2 ,2) =1. C MTHFT( 3,2) =C. 0 NTHFT (1 ,3) = C. 0 N T H C T C ? ,3)=C. 0 MTHET(3,3)=1. 0 T N ' F R T I A L ANG1..LAC RA1FS-STEP 9 CALL. ^MPPD( PI THT T ,V*YH ,WCK1 ,3,1,1) j r - ( P A CC.PO. c« »<"-r ir TO CALL C,VPRD( STHF , THFTD , V C X Y «.l «3 ,1 ) CALL. GMAOD< V » X V , W C K 1 ,WXR ,^ ,1 ) . W X R . W X Y - ^ O T Q R RING 30 C r KK =P WR - SHt'TTLF PA TF S ~? rF(NS.fTO. C. J f ^ G Tf AC CALL GMPPDC JT ,WYH ,WCK1 ,1 ,3 ,1 ) CALL GMAOO( WOK1 ,P«ID ,WP ,? ,1 > . A C O f V « O T T = I . 3 WYC - P A T F 5 CF HUP LL^OED V ASSES C ALl GMPRDCPIPHIC (! ,1 ,1 I) ,W YH , WOK1 , 3, 3 , 1 ) CALL GMAOD ( *CK] , PHI CO (1 , 1 I » , W YC( 1 , I I ) . 3 , 1 ) WXRA. - R A T E S OF POTCP A P V S IF ( IT . G T . K K )GO TC 50 C A L L GMORn( AALPHA (1 ,1 ,1 I) t W X R , W X R A ( 1 , T I ) ,3t 3, 1 ) 5C CON T TN1JP W7 - f A T E S OF RHTOP LUMPFO MASSES 00 CS JJ = 1 , KK on «?s n =1 «3 CALL GMPRO(PTPHIO (1 ,1 ,1 I . JJ) , W X R A ( 1 ,JJ) ,«O<1. 3, 3, CALt CJMAOOC »OK1 ,PHFDD (I ,1 I t JJ) tWZU ,1 I , JJ), 3, I ) 5C C HN TI NLIE P A T r c F I R S T D A ? ? ONL > CALL CMPRD (PI THE T ,VIYH ,WCK1 ,3 ,3 ,1 I T F ( P A ^S.GT. 0, )GO TO TO GAMMA-DOT CALL GMPPO( ?GAW , V» \H ,GAMO ,3 .3,1 ) THE TA -DPT C A L L GVPRO ( 5THET , V» X V , THETC . 3 , 3 ,1 I CALCULATE INITIAL V A L U F S FCP DEL AND DEL( 1 )=0 OELO DEL ( 3) =THFT (? ) DELD ( 1 )=THETO (1 ) -SPIN+WOK1 (1,1) 3ELO ( ?) =THF. TD (2) O ELD ( 2 ) =THE TD ( 3 ) 70 CONTINUE CALCLLATF PHO C&LL GA VMA ( HOKA , \» XY) CALL GMDRD ( VIOK4 ,V>OK1 ,WOK5 ,3 ,3,1) CALL GWPRD ( f^THF T ,V(Ct<5 ,WCK1 ,3 ,3 ,1 ) FIGURE 10.3 (Cont'd) 0.1.27. PHD( ! )..= THETD..(3) *.THE.TO (2)- \»CKl (1,1) 012R PHO(? )=-THETD (3) * THE TO ( l ) - W C K l (2,1) 01 29 PHO(3)=THETD (2)*7HETD (1 ) -*CKl (1,1) _ - 0130 PFTURN' 01 "3 1 F MD FIGURE 10.4 FLX11 170 1 ( ft UG 71 I cs/360 F O R T R A N H COMPILER OPTION6; - NAME = M A I K , OPT = C ? , L I MECNT = 60 . 5 I ZE=0 D OOK , enuRCr ,RCO ,NOLI ST .DECK .LOAD.MAP, NOEDIT, ID, NJ3 E FLX11 T T PF AL ( A - ? ) ____________ _ /FL »/ (3, 3, 3) .ALPHA (3) , PET A (3) .DELI (3), DEL 2(3, 3), DEL 3( 3. 3. 31. j.3J . .DELS (3 .3 ,3 ) .OFL6 ( 3 > , CELT ( 3 ) . D EL8 ( 3 > , DELT B,, ..... ____________ ', ______ *?) ,CH'U8,! 8) ,CHG (6 ,1*) ,C£L(3 ) ,OELO(3) , ,FIR(3) ,EH ,ER .f.TA ,FC(3 .3) ,FD( 3.3 ,3 ) , FG( 3). GAM( 3), GAMD( 3),GH, »GJR( 3) ,GPSIO(3 ,3) , GR ,G YG8 ( 3 , 3 ) ,GXL( 3.3 ) .DD(500), _ _ __ 3) ,GWR(3 '.3 ) .GvTx(3 ,3) ,G V» XRA ( 3 , 3 . 3 ) . GWY ( 3 . 3 ) . Grf Y C0 (3 ,3) .GYBM3 ,3) .GYHC3.3) ,HCMAS5( 3)* H I.HJ, HMASS /ft »/ ' .................... _ ________ _ __ •__ ____ ) .IP ( 3,1) , 1C (3 .3 ,3) .1 0(3 .3 .3.3) . IH(3.3) . Jf ( 3. 3). KBS 1. KBS2. ,3) ,KBVP f ,3) ,KHS (3 ,3 ) ,KHV(3 ,3 ) , L /> V( 8*. 84 ) . LM 1 ( 3, 3), ?) ,LM3(3 ,3 ,?) ,LW4 C? ,3 ) , L V5 ( 3 , 3 , 3 , 3 ),L^5( 3. 3 >.LM7( 3, 3, 3J>_,__ 3 ,3) ,LW?(3 ,3 .3) .LW1 0 (3 .3 .3 • 3 > .LM1 1(3. 3, 3,3) ,LM12( 3. 3, 3. 3). .3,3,3) ,LM14 (3 .3,3) . L VI 5 (3 , 3 , 3 ) , L M 16 ( 3, 3, 3, 3),LM 17f 3. 3, 3. 3). .7V.LM «!(3 ,3) ,LM20(3 ,3) ,LM21 (3 ,3) ,LM22( 3, 3 ) , LM23( .3. _3) ». ____ ' __ ___ _ ,3) ,LV?5( ? ,3) , LW26(3 ,3) ,LM4B(3,3) ,LM13B( 3 . 3 , 3, 3 ) . K HG( 6, 18), .1 ,3) ,KC(S7 .c-') ,KH (1 8,13) C c I _ MO L. I C COMMON *A ALPHA *P FL A <.l *CP( ST. *F JH( 3 ) *G JH( 3) *GPH(3, * G W Y S ( 3 COM WON * I A( 7, !? *KF»yH( *LMJ>(3 , *LM8( 3. *L Ml. 3 ( 3 *LV! (3 _ _ _ *M ,Mt. (YVl) ,M2(3.3 ,2) , V 3 ( 3 , 3.3 ,3) ,M4(3.3) , V5( 3, 3 ) . M AST . M "AS sTa 4",""84 » . *MB( 3) .MBEAR (3) . MC (3 ,3 ) ,WD(3 .3 .3) ,MDOCK( 3 ), MG( 3), M ROT , NHU8, *N ,KS».P.«.?.A.S5 •l=rBF ,T.A(3 ,3) , P I G A W (3,3 ) . PI;.PHIC( 3,3, 3). PIPHID( 3. 3. 3.3), *P I^ST (3 ,3) ,PT THF1 (3,3) ,PHIC (3 .3) ,PH1CD( 3 ,3) ,PHID( 3. 3,3), *PHIDO (3 .3 .3 ) .P!?I (3) ,PS. IO<3) .PHIMR(57,57 ) ,PHIMH( 13 » 18), NT MET ( 3, 3) * R n M A S S ( 3 , 3 ) ,PH(3 ) ,RT ,RJ ,P»»A5S , FRM ASS. RT S (84 ) . S I 1( 3, 3 ) . s2( 3, * S T 3 ( 3,3,3) . S T 4 ( 3 ) , SIG1 (3) ,SIG2(3 ,3) .5 IG3(3,3, 3), 5 IG4( 3..31. *SIG5( 7t3.»;3J_«,SI.G ) ,YH(3) , 7 A ( 3 ) , Z A P ( 3 ) INTEGFP I I , JJ ,KK ,LL,Nt D TMFN5?!ON W 7 ( 3 .3) COMPUTE THE M VA TRICES THAT F O R M U L A T I O N - STEP 11 IF(PA SS.GT. C. )GC TC 10 CALL G A M M A ( G X L .VL ) ( PI THET ,WCK1 .3,3 ) ( ViOKl ,G>L .Ml ,3 ,3 ,3) Y ( M 1 ,-1 . ,V1 ,3 ,3) OK 2 (" 3 ). 84 ", T84 ) A( 3, , 31 . Y B( 3 _ E3( 3)" WILL VI BE USED AS PARf OF THE MASS MATRIX 1 C CALL CALL CALL M4.M X K=P FIGURE 10.I* .(Cont'd) 171 0037 00 ?P 00 20 0030 ! '003! -Q P 3 ? 007? 0034 0035 D O 3 0 T I = l i K K ' . . ' • . . . CALl r,MPRD( A A L P H A (t .1 ,1 I > , P I THPT , WOKl ,1 ,3, 3) C A|, I CM7PA < VOK1 , V? (1 ,1 , M ) .1 ,1) OH 10 -IJ=! «•* CALL G M A O O C >0 <1 , J „ ,1 I ) ,UO(1 , J J , I I ), 5U Ml ( I, J.J , I I), 3« 1 ) CALL < S A M M A < »OK1 .SUM1 (1 t JJt l I)') C ALL 6MPRO(.1»OK1 , A / S L P H A ( 1 ,1 , I I ) ,WOK4 ,3«3 , 3> CALL GMPPD( (V? (I ,1 ,1 f » ,WCK4 ,WCK3 t3 .3 ,3 ) CALL « M P Y ( V « C K 3 ,-!. ,W3 U ,1 ,JJ f I !) ,.3,3» CONTINLE 001*? 00.33 00 4 4 004 5 0046 0047 004* 004=IC (KK , JJ ,1 I ) OO ^0 NN=1,3 LH1 2(NN,KK, JJ,I T )=0.0 LM1 7( KK, JJ» II ,NN)=ID (KK .JJ ,1 I . NN) OO 51 KK = 1 ,3 .V ?!. 2 * **.»-!< K»_4J».1 T )= fOMASS( JJ .1 I) . . _ „ „ . LM?0( JJ ,1 T ) = T H ( J J ,1 T ) L M 3 ( J J . J J . I I ) =HC.» A 5 S ( 11 > L.MO(JJ, J J ,11) =HCMASS( II ) C O N T I N U E PETWN F ^ ^o 20,1 (AUG 71 ) FIGURE 10.5 FLX12 cs/360 F O R T R A N H 1T2 CpMPILFP OPTION 000' 0003 0004 OOO? - NAMF= V A I N , OPT = 0 2 i L I NFCNT=*0 i S I ZE=OD OOK , <=OIJPCF .PCD .NOLI ST ,OFCK .LOAD. MA", NOEOir , ID.NOXREF MflPOLTT NF FLX l? I M P I I C I T PFAl (A-7 . ) CnMMON/FLX/ ' ) ,OEU (3 I , OEL2< 3 » 3) • CO. 3< 3, 3. 31, ,nEL7(3) ,DELS(3) .DELT6, ,DFL(3 ) ,DELO(3) , * A ALPHA ('3,3, 3) .ALPHA (3) .RET A *PEL4( 3 r?) ,DEL5(3 ,3,3) *C:-P( 57,1=7) ,CH(1S,1 8) ,CHG *F IH(3) .TIP (3) .EH ,EP ,FTA ,FC(3 ,3) ,FD( 3,3, 3 > , FG{ 3). GAM( 3), GAMD( 3> .G^ *r,JH( 1) ,r,jP( 3) ,GP = T D ( 3 ,3) ,GP ,GVGB(3,3) ,GXL( 3 , 3 ) , DD ( 500 ) . *GRH(3 1 ,?Y,GWP<3,3> . G W X ( 3 ,3) , G W X R A ( 3 , 3 .3 ) ,GV. PT THf? 1 (3.3) .FHIC(3 ,3) ,PH1CD( 3.3),PHID( AST,MASS( &l ROT, NHU8. PIPHICX 3 .3»3»3)» '3. 3."3"V» ) ,PHIMH( 13 , 18 ),NTHET( 3. 3) 3 ,3,3) ,1 D( 3,3,3, 3) . IH< 1, 3), JT( 3,3),KBSi. s2« ,KPVP (? ,3) ,KHS(3 ,3 ) ,KHV(3 ,3 ) . L fi V( 84, 34 > . LM 1 ( 3, 3). L M 3 ( 3 ,7 ,7) ,L>4 (3 ,3) ,LW5 (3. 3,3, 3 ) ,LM6(3,3 ),LM7( 3, 3, 3) , > .LM9(3 .3 ,3) ,LW10 (3 ,3,3 ,3) ,LM1 1( 3, 3, 3, 3) ._VI12( 3. 3, 3* 3). 3.3) .LMJ A (3 .3 .3) ,L Wl 5 (3 , 3 , 3) ,L M16( 3. 3, 3, 3 ), LM 17( 3, 3, 3t 3) ,LM1 C(3 ,.?) 5 L f2q(3 ,3 ) ,LM2 1 ( 3.3) ,LM22( 3,3) ,LM23{ 3, 3), 18), *LM! ^q (•» .3 .3 COMMON/PL. >/ *M .Ml ( 7,3) ,M *MB( 3) .N'BP AP *M ,N?,P,r»ACi *PI"?I (7 ,1) *PHir>D (3 ,3 ,3) ,PSI (3) ,P5in(3) CPMMON/FLX/ *RDMA«5?(3 .3 ) ,PH(3) ,"I ,RJ .PNASS .KRMASS. RTS(84 ) , 5 I1C 3. 3).S I2( 3). * S T 3 ( 3 , 3 . 3 ) , STA (3 ) .SIG1 (3) .SIG2 (3 .3) ,5 IG3(3,3, 3), 5 IG4( 3, 3). *STG* ( .? .3 .3 ) ,/=IG6(3) ,S!G7(3) , S I G<3 ( 3 ) tS GA M ( 3, 3 ) , SMASS , STHETJ 3,J )± *SUM1 ( 3,?,11) ,TAUA (') ,TAUC(3 ,3 ) ,TAJD( 3,3. .3 ),T AJ L( 3 ) .TAJ5( 3), *TH" rT( 3) .THETD (3) ,PHC (3) »STHE(3 ,3) .SOIN C OMMOK'/FL XX *UC( 3. 7) ,UCO (7 ,3) ,LO (3 ,3 ,3) ,UDD (3 ,3 , 3 ) ,V* CK1 ( 3, 3), « OK 2( 84, 84). *WOK3( P4) ,WOK& (3 .3-) , WO K5 (3 ,3 ) .VKGK6 (3 .3 ), WR( 3), W X R C 3), WXRA( 3, 3», » W X Y ( 3 ) ,UiVC(3 ,3) ,¥YH (3) ,»Z(3 ,3 ,3) » X C ( 8 4 ) . X C O ( 8 4 ) V X D( 3. 3. 3 ), XN< 3, 3) *XL( ?) ,XLL "(**~4) , X V F C ( 8 4 ) , X V E C O ( 3 4 ) , XV ECDO ( 34 ) , / C( 3, 3 ). Y B( 3 ), Y> E3( 3 ). *YG(3 ) ,YH(3) , 7 A ( ? ) ,7AD (3 ) , Z V 3 ( 3 .3) IMTFGFP I I . JJ ,KK ,LL ,Kh _ _ _ _ _ DIMFNSTON W7(3 ,3 ) ........ ," KK=P CALL «MPV(PIGAM , f cAST ,LV1 ,3 ,3) L M f l . P A P T T A L LM2 CALL SCLA (LV? ,0. .3 ,3) DD 70 IT =1,3 CALL <5M.AOr>_P ,PMA5-<; ,3 ,3) FIGURE 10.5 (Cont'd) 173 .00? 7 oo?e OO?Q '0030 00*1 00 3 ? • 0 0 3 3 . 0034 003 = 0036 0037 ;003e 00^ 7 0040 0041 004 .? 004^ 0044 ,0045 004* 0047 0040 004-3 00*0 00 !: I 005? 00 "! 3 0054 0055 005f 0057 0 0 5 P 0050 00*0 00*1 0062 006? 00*4 00** 0067 006fl ,0069 0070 0071 007-9 0073 00.74 0075 007* 0077 9C CALL GMSUR < LM? ,G Yf iV ,LN2 ,3 ,3 ) C LM4,l.MS,Cr>MPLFt 7F L*2 CALL SMPY(M1 ,PPNASS,LN4 ,3 ,3 » nn i or r r =1 .I^K on i oc JJ=i ,3 , " C A L L ?MPY(M.3O ,1 , J J.I I) ,PDVASS ( JJ ,i i) ,« OKI, 3, 3) CALL GMADIMI.M4 ,*CK1 «L»»4 ,3 ,3) 1 00 C A L V C V P Y ( M 2 ( 1 ,1 ,1 I ) , R O V A « ' 3 < JJ.l M ,LM=>< 1 ,t , JJ, II ) , 3, 3) CALL GMPPO ( LVA ,NTHFT ,wcK4 ,3 ,3 ,3) C A L L GVPPHf XCK4 .PITHFT , W C » < t ,3 ,3, 3 1 CALL GMADOC \»PK1 ,L^2 «LV2 ,3 ,3 ) CALL S M P Y t L W A ,1. ,LM4B,3,3) CALL GMPPDCLM4R, *THE,LW4, 3 , 3 , 3 ) CALL =MP Y ( W ? , SMA 55 ,LV6 ,3 ,3) C LM7 DO 11 C II =1 ,3 tlC CALL S W P Y ( P I G A M , H C M A S S ( T I ) ,LV7 (1 , 1, II ), 3, 3) C LM10 ,LM1 1 ,LW12 ,LM13 . CALL GMPPn( PT THE f ,PIG/1M,»CK1 ,3 ,3 ,3) C A L L GMPPO PT THE T ,G YR ,WCK4 .3 ,3 ,3 ) OO 12C II- ,KK CALL GVPPD AALPHA (! ,1 ,1 I ) ,WCK1 ,WOK5 ,3,3 . 3) """ C A L L GMPPD AALPHA (1 .1 ,1 I ) ,WCK4 ,tf C K6 ,3,3, 3> C A L L GVPRO A A L P H A {! ,1 ,1 I ) ,PITHET ,W7 ,3,3 , 1) no i ?o J J= ,7 CALl C M O Y ( W C f ^ , P O N A ? S ( J J , II ) , L VI 3 (1 ,1 , J J, I I ), 3, 3) C A|. 1 5MP'»<*'7 ,QDWA?:< : : (JJ, I I ) , W C K 3 , 3 , 3 ) CALL GMPPDC 1«OK3 ,W3 (1 ,1 , JJ,I I ) ,LMl 3^ ( 1 , 1 , J J , I I ) , 3, 3, 3 ) CALL SMPY(V»CK* , -BDWASSC JJ ,1 I ) ,LM1 1{ 1 ,1 , JJ, II) ,3. 3) C«LL GMPRp(LM13R (1 ,1 » JJ ,1 I) ,KTHET ,WOX3, 3.3, 3) CALL GMPPD( *TK3 ,FITHET,*CK2 ,3 ,3 ,3) CALL GWAnn( \»OK2 ,LV! 1 (1 ,1 , JJ.I I) ,LM1 1(1 , 1 ,JJ, I I), 3, 3) 1 ?C CALL GMPPDM.M1 3n (! ,1 , j J ,1 I) ,STHE,LM13( 1 , 1 , J J, I I ), 3, 3, 3 ) 00 13C I I =1 ,3 1 3C CALL GMPRD( 1C (l ,1 ,11) ,PIFHI C ( 1 ,1 , 1 1 ) , LM 1 4( 1 , 1 , 1 1 ) , 3, 3, 3) " ___ C LMl fi " D O ! 4 C 1 I = 1 , K K nn 1 4 c J J=1 ,^ CALL GWPPDC PIPHIDd ,1 .JJ.II) , A ALPHA (1 ,1 , II ),*QK1, 3, 3, 3) CALL GMPPO( ID (1 ,1 ,JJ,l I ) ,WCK1 ,LM16B(1 ,1 , J J . I I ), 3, 3.3) 140 CALL GMPRD(LM1*8 (1 ,! , J J ,1 I ) ,STHE , LMl 5 ( 1 , 1 , J J, I I ) , 3, 3, 3_) C LMl 8 ,LM1 c I F (MS«TO. 0. )C 0 Tf l^O C«LL C,MDPO(GPH , JT ,\»CK1 ,3 ,3 ,3) CALL GMPPO(*OK1 , P I G A M ,ta(CK4 ,3 .3 ,3) C*l.L SMPV(WTK4 ,- e v A S S ,LM fl ,3 ,3 ) CALL GMPRO<*OK1 ,GYH ,«j C X l . , L M l < ? ,3 .3) C LM21 ,LM?? ,LM;>3 1 ?0 CALL GMPRD(GXL, PI THET ,WCK1 ,3,3,3) CALL GMPPD(»nt 0094 0095 009* 0007 009^ 009P 0100 0101 0102 010"7 0! 04 O l O f n 010* OlO^ 010* 010° 160 C ALL C/»LL CALL CALL GWP.RDC IB ,NTHET ,WCK4 ,3 ,3 t 3 ) ,3 .3,3 ,3) ,3) C V»OK1 ,L*»2?tLM2? ,1 ( IP ,5THE ,LM2-=J ,3 CALL SM»V(G VR ,-HVASS,LM24 ,3 CALL £MPY(LW24 ,1 . ,V«OK1 ,3 .3) CALL «WP>( I A ,1. ,LN?5 ,3 t3) CAL1 fCLA r,o TC 160 CALL C,WSLB( 1H , VB ,*0(<1 .3 ,1 ) CALL GAMVA ( \«nK4 ,V»TK1 ) CALL 5MPY ( W C K 4 , e » i A S S , V ( O K l ,3 C A L L c,MAnn( v»rxi CALL C.wnpn ( CALL CALL CALL 3) ,3 .3 .3) ,3 ) t3 » .3 ) ,3)C,MAOD( LM^S ,*»CK4 ,LV25 t3 G V T R A ( JT ,WOK1 ,3 .3) GMOPO( VsPXl ,IH «VSOK4 .3 «3 t3 » r,MAnn( v»CK«i • L K ^ A t LV26 .3 .3 > C,VPRO ( V»TK4 , JT .VUCKS t3 f3 ,3 ) r,VADO( LVSS.mrKS .LW25 .3 ,3 > C,wopr>( KOK1 ,GPH , WC K4 ,3 i3 , 3 ) GMPPO( »OK4 , JT .WCK1 .3 t3 ,31 S M P Y f W C K J ,«N/»S5 t W C K A ,3 ,3) C,M<:tR( LM24 ,V»rK4 ,WOK1 ,3 ,3) CIMPPD ( I«CKA ,G YH ,I«CK5 ,3,3,1) C,MAOO( LM?5 , W C K 5 ,LV2f? ,3 ,3 ) C O N T I N U E CALL C,MPRO(»nKl , P I G A W ,L V24 , 3 , 3 ,3 ) CALL C ALL CALL CALL CALL CALL CALL CALL CALL FMO 20.1 CD .NCLIST .DECK .LOAD.MAP, NOEDIT, ID.NOXPEF SLflFOL'TINF FLX13 IMPL IC IT cp AL ( A - Z ) 000* 0006 000*7 O O O Q 00!. 0 001 1 00 T 2 0017 0014 -O.P * * OO16 0017 C C C * A ALPHA (7 ,3 ,^) ,ALFHA(3 ) , « ? E T A ( 3 ) tOELl (3 ) , DEL2( 3 , 3 ) , DEL 3( 3t 3, 3), *DFL M 7 ,3) .OEL5C3 ,3.3) ,OEL6 (3) ,OEL7(3) ,DEL8( 3), OE-TB, ___ *CP(?7, * -M ,CH<18 ,1P) ,CHG (fi ,1 8) ,f!EL(3 ), DELC( 3), *F IH( 3) , P T P ( 7 ) ,F.H , F R , E T A , F C ( 3 .7) ,FD( 3,3, 3 ),FG( 3 ) , G AM( 3 ) , G AMD ( 3 ) • Gl *G JH(3 ) ,G JP( 7) ,r, P 5IPX3 ,3) »GR .GYC.B (3 , 3) , GXL( 3, 3 ) , DD( 500 ), *GPH(7 .7) ,C,WPf» ,3 ) ,GWX{.3 .3) , G W X P A ( 3 , 3 . 3 ) . G W Y C 3 . 3 ) , GWYCC 3, 3, 3), * r, u v S ( 7 . 3 ) ,C, \R ( 3 , 7 ) ,C, V R N ( 3 ,3 ) , G VH ( 3 , 3 ) , H C MAS 5 < 3 ) , H I , H J , HM ASS C O M M O N / F L X / *T A ( ? ,3) , I" ( -» ,7.) , If ( 7 ,3 ,3 ) . I O ( 3 . 3 , 3 , 7 ) , I H C 3 , 3 ) . J T C 3, 3 ) . K BS I , KB S2. **r'nvH (•?,?) ,KPVR<3 ,7) ,KHS (3 ,3 ) ,KHV(3,3) « L A M ( 84, 84 ) , LM 1 ( 3, 3), *L«?<3 .3) ,t. M .•?•(? ,3 .^J ,LVA (3 ,3 ) ,LW5 (3,3,3, 3 ),LM6( 3,3 ).LM7{ 3, 3, 3)» *LM8< 3,3,3) ,LMQ(? ,3 .3) ,L M 0 ( 3 ,3 ,3 . 3 ) ,LM1 1 ( 3, 3, 3. 3),LM12( 3, 3, 3» 3), *LMi ?( 3,3,3 ,3) ,LW14 (3 ,3 ,3) ,LN15 (3 ,3 ,3) ,LM16( 3, 3» 3, 3I,LM17( 3, 3t 3» 3 *L«l fi< ? ,3 ) ,LM. < ? ( ? .3) ,L»*20(3 *3) ,LM31 (3 ,3 ) ,LM22< 3,3) ,LM23( 3, 3), *L M?M 3 ,3 ) ,LV?C (7 i?) ,LV26 (3 ,3) , LM* R( 3 , 3 ) , LM1 3R( 3 , 3 , 3, 3), KHG( 6, *L«1 ^P (3 ,3 ,7 ,-M ,KP (57 ,5^> ,KH (18,18 ) COMMON' /PL >/ *M ,M1 ( 7, 7) ,M?(3 ,3 ,3) ,K3 (3 ,3 ,3 ,3) ,M4(3,3) , W5( 3. 3) , M AST , M ASS( 8"A, *MP( T) ,MREAP (3) ,VC (3 ,3) ,VD(3 ,3 ,3) ,MDOCK( 3),MG( 3 ) , N ROT , NHU B, *N ,NS.P,PA Sf5 ,PIBFTA(3.3) ,F IGAK(3 .3 ) , PI PH I C( 3 , 3 , 3 ) , P IPM IDC 3,_3, 3,_3 )j *p T O d j (-*,"M ,PI THFT(3 ,3) ,PHIC(3 ,3) . PHICO(3 ,3) ,PHID( 3, 3, 3), *PHTDD (7,3 ,7 ) .n^I (7) ,P«IC (3) ,PH I MR (5 7 ,5^ ) , PH IMH( 18, 18 ),NTHET( 3, 3) 18 3,7) ,PH(3 ) ,PT ,PJ , P W A S S .RRMASS.PfS (84 ), SI 1C 3, 3 ) ,S I 2( 3). ) , «ini (3) ,SIG2 (3 ,3) o I G3 C 3, 3, 3 ) , 5 !G4( 3, 3), 7,3,7) ,CTC.6 (') , SIC, 7 ( 3 ) ,SIG8(3) ,SGAM(3 , 3) , SM ASS , STHET( 3, 3), 7 .7 »? ) ,TAIJA (3 ) , T A U C ( 3 , 7 ) , T A J D ( 3 , 3 , 3 ) , T A J L ( 3 ) , T A J S ( 3 ) , T( 7) ,THP ID (3) ,PHT (3) (3 ,3) ,5 *UC( 3,7) ,UCH (7 ,7) ,UD(3 ,3 ,3) ,UDD (3 , 3 . 3 ) . * OKI ( 3, 3) , « OK2C 84 , *WOK3( Pfl) , VKOK4 (3 ,3) , WCK«5 (3'.3) , WOKS (3 ,3 ) , W F.( 3 ) , rfX RC 3), W X R A C * W X Y < 3 ) , W Y C (3 ,3) , ¥ Y H ( 3 ) , V » 2 ( 3 ,3 .3) ,XC(34) , XCD( 3 4 ) ,J( DC 3» 3,3) *XL( 7) ,XLL(84) ,XVFC(P4) .XVECOC1}*) , XV ECDD ( 84 ) , Y C( 3 , ~3)VYBT31) *YG( 3) ,YH(3) ,7A (7) ,7AO (3 ) ,ZV3 (3 ,3) IMTFGFC i T , jj ,KK ,LL ,NN SET UP THF RIGHT SIDE OF THE ECN COMPUTE LOWER C A S E P.SI -STEP13 S T 1 T A L L G f i M M A ( G W Y . W Y H ) CALL C-MPpn(GV*Y ,Gfc > ,GW>rS ,3,7 ,3) P o I 0 I I = 1 , 7 CALL GMAPDC >C (1 ,1 I) ,UC (1 ,1 I ) ,WOK1 ,3 ,1 ) CALL GMPRP (G*YS , VtCKl , V»0*4 ,3,3,1) CALL G M P R D C G W Y . U C O d ,11 ) , \ « C K 1 ,3,3 ,1 ) _ C A L L < = M PY(WOK1 i?8 ,SI 1 (1 ,1 T ) ,3 ,1 ) 1 r CALL GMAOO ( « I 1 ( 1 , I I ) , W C K4 ,511(1,11), 3,1) 0018 0019 0020 002 1 002? 0023 CALL G A M M A ( G W X , V » X P ) CALL GMPRO(GV*X,G * > ,V(OK1 ,3,3,3) CALL GMPPO ( *HK1 , XL .WCKA ,3 ,3 ,1 ) CALL GMTRA ( PI THFT ,WOK1 ,3 ,3) CALL GMpeo(¥oi 0044 0045 0046 0047 004P 0049 0050 0051.' 0052 .0053. 0051 CALL G M A O D ( « I? ,WtK l ,SI2',3 «T K K =P nn 20 TI= 1 ,KK CALL G A M V A ( G V t X R A ( 1 ,1 ,11 ) , W X R A (1 , I I) ) CALL GMPRO(GWXPA <1 ,1 , I I ) , G W X R A ( 1 , 1 , I I ) , WOK4, 3, 3, 3 ) DO 20 JJM «3 C A L L GM4DO ( >D (1 , J „> ,1 I ) ,tn (1 , J J ,1 I ) ,WOK5 ,3,1 ) C A L L GVPRD ( G V k X R A (1 ,1 ,T t ) ,UDC (1 ,JJ ,1 I) ,*OK6, 3, 3. I ) CALL GMADDt VCK3 ,V»CK5 ,V«CK6 ,3 ,1 ) C ALL GMPPO ( *?. (1 ,1 ,! I ) ,V»CK6 ,V«OK5 ,3 ,3 ,1 ) ?0 CALL. GMADO ( U C K l , V ) C K 5 , S I 3 ( i , J J , T I ) t 3 , l ) SI4 CALL 5CI.A (,SI4 ,0. ,3 ,! ) I F C N S . P O . 0. )GO TC 30 CALL GMPRD(GWYS, >H , ,G >B (3 , COMMON/FL>/ *I a( -> , -") ,1^(3 ,?) , *KRVM( 3 ,3 ) , K R V P ( 3 *LM?(3 ,3) ,LM7(3 .3 *LMP( 3,3,3) ,LMEL6(3) ,OEL7( 1 P) ,CHG (6 ,13) ,OEL(3 ,FP ,ETA ,FC (3 .3) ,FO< STr>n ,3) ,GR ,GYGB(3. ) , G W X ( 3 ,3) , GW XPA ( 3 , 3) ,GVBM(3 ,3) ,GYH<3 , 1C (7 ,3 ,3) ,1 D(3 ,3 .3, ,:•»> .KHS (3 ,3) ,KHV(3, ,.3) ,LV4 (3 ,3 ) ,LW5 O, ,3 ,3) ,L M 0 ( 3 ,3 ,3 ,3 ) ! a (3 ,3 ,3) ,LW15 (3 ,3, 3 .3) ,LM21 ( 3 ,3) , ,57) ,KH (13 (3 ), OEL2(3» 31, DEL 3( 3, 3,3), 3) ,DEL8( 3),OFLT8, ) ,DELC(3>, 3.3,3 ),FG( 3), GAM( 3), GAMD( 3) ,G 3> ,GXL( 3. 3 ) , DD( 500 ). ___ 3 , 3 ) , G W Y ( 3 , 3 ) , GWY~C( "3, 3. 3 >»"" 3) ,HCMASS( 3>,HI,HJ. HMASS ___ 3) , IH(75, 3), JT( 3,3 ) ,KBSl ,KBS2, 3 1 ,L A M ( S 4 , 94 ) , LM 1 ( 3, 3 ) , 3,3, .3 ),LM5( 3,3 ),LM7( 3, 3,_3),_ ,LM1 J ( 3, 3, 3, 3) . LM 1 2( 3. 373, T), 3) ,LMlft(3. 3, 3, 3),LM17< 3, 3/3, 3 3,3) ,LM22( 3,3) »LM23( 3, 3), ___ 3,3 ) ,LMI3B( 3, 3, 3, 3T,~KHGT6, 18) *SIG5( 3 C( 3 , V B ( 3 E3( *M ,M1( '3\3) ,M2(3,3 ,3) ,M3(3,3 ,3 ,3) , M4 ( 3 , 3 ) , M5(3, 3 ), MAST . MAISSC 84^841", *MR( ?) ,MBEAR (3) ,WC (3 ,3) , *0 (3 .3 ,3) ,MDCCK(3 ) , MG( 3 ) . NP.OT , NHU B, *N ,N55,P,PA«S ,PIBFTA(3,3) ,P IGAV(3 ,3 ), PI PH I C( 3 , 3 . 3 I _, P IPH1 _OC 3,_3»_3, 3)i *P TP'SI (3 ,3 ) ,P! THFT ^ <3 :,3) , PH I C (3 ,3) ,PHICD{ 3.3) , PHIoTs"," 3,"3 ) , ......... *PHTPD (3,3 ,3) ,PSI (3) ,P«IO(3) , PHI MR (57 ,57 ) ,PHIMH( 13 , 18),NTHET( 3, 3) COMMnN/^t. >/ . . . * R D M A S S ( 3 , 3 ) ,RH(3) ,PI ,RJ .RVASS .RRMASS.RTS (64) ,S IK 3, 3),Sf2( 3)7 ..... ~T «I 4(3 ) ,«5IG1 (3) ,SIG2 (3 ,3) ,S IG3(3,3, 3), 5 IG4( 3, 3), . .SIGfr (3) ,SIG7(3) ,SIG8(3) ,SGAy(3 ,3 ) t SMASS ,_STHETC_3_,^_),____ ,TAUA (,3) ,TAUC(3 ,3) ,TAUO( 3, 3, 3 ) ,TAJL( 3) ,T A~US("3), *THF T( 3) .TH^TO (?) .CHT (3) ,«5THF(3 ,3) .SPIN COMMON' X^L X/ __ *UC( 3. 3) ,UCD (3 ,3) ,UO (3 ,3 ,3 ) ,UDD (3 ,3 , 3 ) . rf OKI ( 3, 3) , * OK 2( 84784). *WOK?( P4) , VIOK4 (3 ,3) ,%OK5 (3 ,3 ) , WOKS (3 ,3 ) » W R( 3 ) , WX R( 3 ), «fXRA< 3, 3), * W X Y ( 3 ) , W V C ( 3 . 3 ) ,»YH(3) ,»Z(3 .3 ,3) , X C ( 8 4 ) , X C D ( 9 4 ) , X D ( 3, 3, 3) . < N( 3, 3 1 *XL( 3)",Vl".L (94) ,XVFC(«4 ) ,XVECO( f l4 ) , XV ECDO(B4), *VG (?) ,VH (7) , 7 A ( 3 ) ,ZAD (3 ) , Z V 3 ( 3 ,3) IMTFGFP I I , JJ ,KK ,LL,N^ C CONTINUE ' TO" SET UP PIGHT SIDE OF EOM C COMPUT" DELTA-STEP 14 C, DELI ,OFL2,DEL3 KK=P CALL «CLA (DEL! ,0. ,3 ,1 ) _ CALL. «CLA ( W C K t ,o, ,3,3) on 40 JJ=I.KK . CALL GMPPO( AALPH A (1 ,1 ,J J) .PITHFT ,WQK4 ,3 , 2,3) nn A ° ! l =1 *- CALl FMPV(SI? f 1 ,1 I , JJ) . R D W A S S ( Tl . J J ) , rf OK5 , 3. 1 ) CflLl GM^RD( ViCKA .WPK5 ,DFL3 (1 ,1 I ,JJ).3,3, 1 ) 40 CALL GMAnO(CFL1 , V C K 5 ,PFL1 ,3 ,1 ) OO 41 JJ=1,3 CALL «MOY(S I1 (1 ,J») , H C V A S S ( JJ) ,DEL2 (1 , J J ).3, 1 T 41 CALL G M A O O ( *OK1 ,DEL2(1 ,JJ) , V * C K 1 ,3 ,1 ) CALL GMA.r>D(OFLl ,V»TK! ,OEL1 .3 »1 ) ' . C«LL < M P V ( S T 9 ,RP^A«S ,^C>t ,3 ,1 ) C A L L G M A O O f O F t J ,*CK! ,DFL1 ,3 ,1 ) FIGURE 10.7 (Cont'd) 178 0025. 00?7 oo ?e 00? P , W C K 1 ,3 ,! , JJ, II ) CALL C A L L CALL CALL C ALL CALL on e. c ALI C A L L CALL CALL CALL C A L L C Al.l 60 CALL HEL6 0 r,Mppn ( (1 ,1 ,11) ,WOK4 ,3.3 . 1 3*1 > ,WOK4 ,3*3 . 1 ) C A L L CALL c ALL CALL CALL CALL CALL. CALL CALL CALl CALL r,McUR GMPPO( TH.GPSID.wnK l ,3,3,3) ,PI"FTA ,V«CK4 .3 ,3,3) ,V»H,WCK1,3 ,3 ,1 ) GMSU"(OFL6 , fcCKl ,DEL6 ,3 ,1 ) C,MOPO(GPH , JT ,V»OK1 ,3,3,3) ,3 ,1 ,1) GMPPOCVvOKd ,YH ,WOK1 ,3 ,3 ,1 ) GMSIJFMDFL6 ,DEL6 ,3,1) C DEL7 CALL CAL l C A L L CALL CALL CALL CALL GMPPO ( G t e X . I F ,VtCKl ,3 ,3 .3 ) c;vpPr>(unKl ,V»XP,DEL7 ,3 ,3 .1 ) r,MPPO(GVL .PTTHFT ,WCK1 ,3 ,3 ,3) ,G» YS ,V»OK4 ,3 ,3 ,3) ,YP ,V»CK1 ,3 ,3 .1 ) V»rK4 ,3 ,1 ) ,DFL7 ,3 ,1 ) C DELfl C A L l - G W P P O ( G W V , I A ,WCM ,3,3,3) CALL GMPPn(»PKl ,V» YH ,DFLQ ,3 ,3 ,1 ) IFCNS.FO, 0. )GH TC RO CALL Gf*PPD(G*P, IH ,WOK1 ,3 ,3 , 3) CALL GMPPD ( »C*l , ViP ,ViCK4 ,3 ,3 ,1 ) C AH. C M P P r > ( T H , G P f I O , V 8 C K l ,3 ,3,3) CALL GMPPrMvtCK! , P I Rr T A , WCK5 ,3 , 3 , 3 ) C A L L GMPPHC teCK5 ,V»YH , W C K 1 ,3 .3 .1 ) CALL GMSUP( V>CKa , *CK1 ,V»OK5 ,3 ,1 ) CALL G M T R A ( JT.WOK1 ,3 ,3) CALL GMPPncmc*! * » C K B .tiCKA ,3 *3 .1) CALL G M / v o n c o ^ L P . w C K A ,nEL9 ,3 ,1 ) CALL GMSUO ( YH , VH ,WPK5 ,3 ,1 ) C A L L G A M M A { t»CK4 ,V« CK^) FIGURE 10.7 (Cont'd) 179 ..ope?. 001? ~~OOBt 0087 ...QP.88. 00*9 0090 00 0 1 C ALL f?MPPD ( ¥OK4 .SI A ,*ICK5 ,3,3.1 ) CALl CMPRO ( V»CK4 , JT , W C K J ,3 .3 ,3 ) CALL GVPOfM 1»CK1. ,C » VS »«O»<4 ,3 ,3 ,3) CALL GMPRn ( »OK4 ,>H ,WOtEI . 5 ( 3 .3 ,3) .DEL6C3) ,QEL"M *CC( f ,c?) ,CH( V9 ,t P) ,CHG (6 ,18) .CFL (3 * r i H ( 7 ) , r i t ' < - > ) , E H , E f : , F T A , F C ( 3 , 3 ) , F O ( *G JH( •*) ,GJR( 3) ,r.F > M P ( ^ ,3) ,GR.GYr,B(3. *GPH(1 ,3) , G V » P < 3 .3 ) .GVIX(3 ,3) ,G* XFA (3 , * G W V ? ( ^ .T ) »n YP (3 ,3 ) ,GVPV (3 ,3> ,GYH(3, rOMMPN/FLX/ * t A ( •", 3) .in ( 3 ,•*) ,IC (.3 ,3 ,3) ,1 0 (3 , .3 ,3 , *KPVH( ? ,1) ,KBVP {? ,?) ,KHS (3 ,?) ,KHV<3, *L M ? < 3 » r l) ,LM3(3 .? ,-M ,L*4 (3 ,^) ,LM5<3, *LMfl( 3 ,3 ,3 ) , LV9(3 .3 ,3) ,LV1 0(3 .3 ,3 ,3) *LM13( 3 ,3 ,3 ,3 ) ,LV14(3,3 ,3) ,LW15(3,3, *LM1 ?( 3, 3> ,L HI ,HCMASS( 3) ,H I. HJ, MM ASS 3) ,IH(3. 3> , JT ( 3, 3 ), K BS 1, K BS2, 3) . L A V (8 4 , 84 K L M 1 ( 3, 3), 3,3, 3 l,LM5< 3,3 ).LM7( 3»J»_3)_»_ ,LM1 1 (3, 3. 3. 3),LM12( 3. 3. 3t 3), 3) ,LM16( 3, 3.3, 3 ) ,LMJ7C 3, 3, 3, 3 3,3 ) , LM22( 3 , 3 ) , L M 2 3(_3, 3J .^_J ____ 3,3) ,LM139( 3, 3, i 3, ^ 3),~KHG( 6, 18), *M ,M *MP( *N .N *p!To< 0007 : TO *TH' rT( ,3) ,M2(3,3 .3) .V3(3 ,3 ,3 ,3) ,M4(3,3) , M5(3, 3), M VREAP (?) ,MC (3 ,3) , WO (3 ,3 ,3) , Vt>OCK( 3 ) , MG( 3) ,,MROT,NHU8, ^ASS ,PTBF TA (.3 ,3) .PIGAMO.3 >. Pl»HIC( 3,3, 3 ) , P IPH IO< 3, _3s ^ ^ ,3 ) ,pf THP T (3 ,3 ) , PH I T (3 ,3) ,PHICD( .3,3),PHI.O( 3, 3, 3), -*,-* ,^) ,051 (T) , r>sin(3) ,PHTMR(57,S7 ) ,PHIMH( 13. 18) ,NTHET( /^t. > / (3 ,7) ,RH(-> ) ,RT ,RJ , R ^ A S S ,RPMA5S,nrs (84 ),SIl( 3, 3 ) ,S I 2( 3 ) 3,3) , = 1 4 ( 3 ) ,S IG1 (3) , < ^ I G 2 (3 .3) , S I G 3 ( 3 , 3, 3 ) , 5 IG4( 3, 3), , •>• , •«> ,c;tr, M3) .SIG7C*) ,S IGB(3) , ; G A M ( 3 , 3 ) , SM ASS, ST HET( 3 , , 3 , 3 » , T A « J A (3) , T A U C < ? ,3 I ,TAUO( 3,3, 3 ).T AJL( 3) ,TAUS( 3», ) , THE 7H (3) ,PHC( 1) , S T H F ( 3 ,3) ,SPIN OOOfi 000? 0010 0011 0013 001 « 0015 O O l f i 0017 001 9 00 ?0 00?2 '0023 3* 31 1, 3,3) , 3), *UC( 3,3) ^ ,UCD (3 ,3 > , ID (3 ,3 ,3) ,UOO(3 .3,3) . r fOKH 3, 3) ,«OK2( 8*,~a*J~,~~ ....... * W O K ? ( 1=4) , WOK4 (3 ,3) , VkOKB (3 ,3 ) ,WOKS (3 , 3 ) . W R( 3 ) , WX R( 3 ) . tfXR A( 3, 3>, * W X V ( 3 > , W V C ( 3 , 3 ) , V»YH(3 ) , W Z ( 3 ,3 ,3) , X C ( 8 4 ) ,XCO( 3 4 ) , < 0( 3. 3. 3|O( WJ3, 3 J * X L ( 7) ,XLL («4» , X V F C ( R 4 ) , X V E C C ( B 4 ) , XV ECDD ( 34 ) , / C( 3. ~3),YBr3), / E3( 31." *YT, ( 3) ,YH (3) ,ZA (3) ,7 AD (3 ) , 2 V 3 ( 3 ,3) DIMENSION M B S ( ? ) , W R V ( 3 ) ,KBV(3 ,3) _ __ INTFGEP I T , JJ ,KK ,LL,^^ r COMPLETE THE RIGHT *10 CF EO C CALCULATE E L A S T I C MnVENT AT DOCKING MECHANISM .__ C MDOC K IF(MS.r-o. 0. )GO Tf ! 0 CALl GMPRO( KH.c; ,rcj ,WOK1 ,3 ,3 ,1 ) CALL. GWPPO( KHV ,P«=Tn , W C K 4 ,3 ,3,1 ) «V»CK4 ,i»OKl ,3 .1 ) . ,MnCCK,3 ,1 ) CALL GMAOr>( CALL ? V P Y ( W C K 1 ,-1 1.0 C(?NTINLF C A L C U L A T E HEAPING MRS- SPRING DL TB=APC MRS( 1 )=0« 0 SPRING DI «CCNTINUI t Y IF(r>L TP.G T. On T P ) G C KP = =-VPCI GO TO 70 * C C S ( T I - E T ( 3) TC 2C FIGURE 10.8 (Cont'd) 181 0024 0025 0026 C C 00 ?7 00 2 P. 0030 0031 007? 0033 0034 0035 00^6 C "6037 C c r 0039 003«5 0040 0041 004? O O 4 3 O044 0045 0046 0047 004«=> 004<3 0050 005! 0053 0054 0055 0056 0057 005P 0050 0060 0061 006? 0063 0064 c C 0065 0066 0067 006R 0069 007Q 0071 007? 0073 0074 ?r K.n S = ' I.L=P DO fC KK=1,LL on co J J = i , 3 DO ^0 T T = 1 . . 3 XC( T I+NN) =UD ( I I , J J .KK) XC( II+NN4- 3) =PHID (II t J J . K K ) : XCD( T T+NN)=LDD (I T ,J J ,KK) 50 XCD (I T + NN+ 3)=PHIPD (I I ,J J , K K ) IF(P.FO« 3. ) NN'=57 DO 55 T I =1 ,NN W O K 3 ( II ) = C, C DO 55 KK = 1 , NN c = WOK 3( I 1 ) = W O K 3 ( T T ) + K R ( T T ,KK) *XC ( KK ) •»• CR ( I I ,KK) *XCD( K< ) 60 P.n 62 11=1,3 6? MP( T I > = W O « 3 (I T ) nn 65 KK=1 ,LL DO 65 JJ=1,3 D'O 65 T T = 1 , 3 FD(II , J J .KK) =V*OK-« (1 I-f NN) 65 MD{ 1 I , J J ,KK )=V* rSK3 (I I + NN+3) CALCULATE FORCES AND vrMFNTS DUE TO HUB BENDING ~ SET UP TEMP HLB S T A T E IN XC AND ITS DERIV IN XCD DO 70 JJ = 1 ,3 DO 7Q II =1 ,3 NN=6* ( J J-l ) X.r( I I +NN)=UC (I I , J J) XCD ( I I+NN ) = UCD(T T , J J) " "' XC( IT- fNNH-3) =PHIC (I I . J J) 7C XTD( I I-»NN+3)=PHICO(T I ,,J J) CALL GMPRD( KH , XC ,WOK2 ,18,18,1) CALL GMPPD (CM ,XCD ,WCK3 ,18 ,1« ,1 ) CALL GMADD(»OK3 .**CK2 .HOK3 fl 3 ,1 ) FIGURE 10.8 (Cont'd) 182 0.0.7 5 007f t 007? 007fl O C 7 < 5 O O f l O ,0081 oon? 00*3 O O P - 6 OOP 7 ooaV DO 7.5. JJ = l ,3 no 7S TT =1 ,3 NN = 6* (JJ-1 ) r r ( T I , j j) = W C K " < ( T T wr ( H , j j ) = v « r K 3 u i* M^+3) r ALL OMORD ( KHO , vc ,«CK? ,6 c c c r ALL ,xcn ,v« V»CK? r> n FT, 90 1 1 T I = 1 (T T ) (I T-f 1 CALCULATE THF «! C AH. GMPRO ( LM4R , PHC t W C K l ,3 t3 tl ) q ALL SMPY (DEI. }, ,-1 . . «I Gl ,3 tl I CALL GMSUP ( 5IC,1 , \ »CK1 .51 Gl ,1 ,1 ) KK=P 009 1 0093 0095 0096 00 n c c T T = 1 , TA UC PI PH TF( JJ.GT. K K ) G O 0 C c c1 C S I G 5 C c c ALL ALL AH ALL ALL ALL ALL ( I T SMPY (DEl.3(! OMSC'B( r,vppn ( GVc U R ( GMSUB { GMPRD( GMStB( *OK1 , L M 1 3 R WOKJ TAUD PIPH ¥OK1 . ( T *CALL GMPPnfLv l fcR c1 IP ,RHC , W T K 1 ,3,3,1 ) T F ( P A ?«.GT. C. )on in 200 CALL GMSUP ( \<~- ,VR , VlCKl ,3 C A L L Ga VVA (G VGB , » T K 1 ) 1 ) FIGURE 10.8 (Coni'd) 183 ,.0127 0!2<5 P 1 3 0 01 31 .01 32 . .PIT! 01 34 0 1 3 * 01 37 o i 3 e 0139 0140 0141. 2CC CALL CALl CALL CALL CALL CALl CALL 1 F ( N S CAL l C A L L CALl. ? V O C O N T T P F T LP F NO GMPPD (G>Gn , GMSL'O ( TA UA , GMCUR ( « T G a , GMTRA ( GMPRD ( G M 5 UP ( »ro. o. r, M T R A ( r,Mppn ( GMADDC NIIF. N FT THE ViOKl , J - T G 8 , )C D T J"' ,V«C v>nwi , = roe. FG .fcOKl .3 ,3 .1 ) \»Ckl ,SIG8 .3 .1 ) *T, .SIGfl .3 ,1 ) T ,WDK1 ,3,3) WPEAP ,WCK4 ,3,3,1 ) V iCK4 ,S!<;P .3,1) Kl ,3 ,3) T A (j s § to C ^ A t ."^ 1 3 • 1 ) HCKA ,e iG8«3 .1 ) ?0 . 1 ( A UG 71) FIGURE 10.9 PREINT OS/160 F T P T R A N H CnMPILFP 000' 000"? 0005 0007 000R COMMHN/FLX/ * I A ( 3 , 3 ) t T P ( T , *KP\/H ( •> ,3) ,KBV *LiM?( 3 , ~ ) ,LM3( *LMP( 3,7, 3) ,L *LM1?( ?,3 ,3.3) *l Ml P( ?,3) ,LV1 *L w?M 7,-«) ,L *LM1^R ( 7 , 3 , 3 .ALPHA (3) .GET Am ,DELI m . OELSO, 3), DELIC 3, 3, 3), M •> .1.1) ,DEL*(3) ,OEL7(3) ,OEl_S( 3) ,DELT8, 1 8 . ! R) .CHG (6 ,1 3) ,OFL(3 ).DELO( 3), «FH , E P . F 1 A , F C ( 3 . 3 ) , F D ( 3 , 3 , 3 ),FG( 3), GAM( 3 ) , G AM D( 3 ) t Gh . G P c f O (3 .3) ,GK ,GYGR(3 , 3) . GXL( 3 , 3 ) . DD ( 500 ) , 7 ,3 ) ,GWX (3 ,3) , GW.XRA ( 3 . 3 , 3 ) . GWV (3, 3). GWYCC 3« 3. 3). (3 .7 ) ,GYRV (3 .3 ) ,GYH (3 ,3) .H CMASS( 3) , H I, HJ , HMASS 3) , IC (3 ,7 ,3 ) , I D ( 3 , 3 , 3 , 3 ) . I H ( 3 , 3 ) , J T ( 3, 3 ) , K BS 1 . KB 52, ^ {3 ,3) .KHS (3 ,3 ) , KHV( 3 .3) , L A v (94, 84 ) , LM 1 ( 3, 3), 3,.3 ,?) ,.L»*a (3 ,3) ,L ^ 5 ( 3 , 3 . 3 . 3 ) ,LM6 ( 3 . 3 ) , LM7( 3, 3, 3). = (7 »3 ,3) ,L VI 0 ( 3 , 3 ,3 . 3 ) ,l_Ml 1 ( 3 , 3. 3 , 3 ) . LM 1 2( 3, 3. 3« 3). ,LVl 6(3 ,3 ,3) .LV15 (3 .3.3) ,LM16( 3, 3, 3. 3).LM17( 3. 3. 3, 3) .3) ,L^20(3 ,3) ,LM21 ( 3 , 3 ) ,LM?,?( 3,3) ,LM23( 3. 3)9 ,3) , L W ? 6 ( 3 .3) ,LM4B(3,3) ,LM13B( 3. 3. 3. 3 ) , K HG( 6, 1 8 ) , ST) ,KH( lR , i a» 9( * M , M 1 ( - > , 3 ) , M 2 ( 3 , 3 , 3 ) , M - » ( 3 . 3 . 3 , 3 | , M 4 < 3 , 3 ) *MR( 3) .MBEAP (3) «NC (3 .3 ) . *O ( 3 ,3 ,3) . MDOCK( *N ,N<=,D,PACS,PIBF TA (3.3) . P I G A V (3,3 ) .PI PH ( 3,3) , PT THFT (3 ,3) ,PHIC (3 .3) ,PHICD( ( 7 . 3 , 3 ) , P c : T (3) ,P«I C (3) ,FHIMR(57,57 C OMMOM/F L X^ *ppMAS?( 3 ,3) ,^H(3) ,PT ,PJ , P N A S S , P n M A S S , R T * « J T 3 < 3.3,3) .514 (3 ) ,STG1 (3) ,5IG2 (3 .3) ,5 IG * S T G 5 ( 3 , 3 ,3 ) ,S IG6(3) , S T G 7 ( 3 ) .SIGfl (3 ) ,SGA *Pl)Ml ( ? , .3 ,3> ,TAUA (3) , T A U C ( 3 ,3) ,TALIO( 3, 3. *THPT( ?) , THF TO (3) ,PHT (3) , = 1HE(3 ,3) .SPIN r HMMON /=•!_ */ *UC( 7 ,7) .UCn (7 ,3) ,UD (3 .3 .3 ) ,UDC ( 3 . 3 , 3 ) , ti *WOK-» ( 84) . V t O K A (3 ,3) ,WOK5 (3 ,3) ,WCK5 (3 ,3), * W X Y ( 7) , W Y C ( 3 ,3) , *YH (3) , « » Z ( 3 ,3 .3) , XC (34 ) *Xt ( 3) ,XLL(R4) , X V F C ( R 4 ) , X V F C D ( 3 4 ) ,X * Y G ( 3) ,YH (3) ,7A <3) ,?AD (3 ) , 2 V 3 (3 ,3) I I , JJ ,KK ,LL .NN ,M5(3, 3 ) . M A S T , M A S S ( 8 4 , 8 4 ) . 3 ) .MG( 3). N ROT, N HUB. I C( 3, 3,3) , PIPHID( 3.3. 3, 3), 3. 3). PHID( 3, 3, 3), ) , P H f M H ( l B , 1 8 ) . N T H E T ( 3 , 3 ) S ( 84 ) , S I 1 ( 3. 3 1 , S I 2( 3 ), 3 C 3 , 3, 3), 5 1G4( 3. 3), W ( 3 , 3) , SM ASS, STHET( 3 . 3), 3 ) , T AJL ( 3) ,TAJS(" 3), HK 1 ( 3, 3). * OK2( 84, 84), W R ( 3 ) , *XR( 3 ) , W X R A ( 3, 3) , , X CD( 8 4 ) , X D( 3. 3. 3 ) , X N( 3, 3) (84) , VC( 3, 3), YB( 3),VE3( 3), INTFGEP TNTEGCP NPOT.NHUB FQPM THF IF ,7 ,3) M A T R I X LAM84, ! 34) r r. c 0013 opts 001 f 001 7 001 p • 0019 0020 00?1 00?A MOO IF Y MA £<=; TFPNS Fnr- M5 = 0. ?=HLTTLF NOT IF(NS.fO. 1. )GO TC DP 511 =1 .3 nn ti j.j=i .^ I. Ml F( I T . J J) =0. 0 L vi o(i i , j j) -n. o 4 LM?C( II , J J) =0. = L M * C ( T T ,1 I ) = 2 5 0 0 C C . 10 rCNT OFLETICNS ATT ACHEC , NS=1 , 10 SHUrfLE A T T A C H E D A -rvrn I F ( P . E 0 . 3 . ) G O TO DO 1 ? I T =1 ,3 P C T C R FIGURE 10.9 (Conf d) 185 .002.? 00?* 00?.7 oo?« 00?9 0070 ...Q.P 11 0032 0033 003* 0035 00 3€ 0037 0030 0061 00*? 00*3 00*4 .00*5 006* 0067 po*e 0069 0070 007! 0072 007.3 0074 0073 007* 0077 007* 0079 ooao C SrT DO 12 JJ = 1 ,3 I Ml Of I T , JJ, KK. ,T) =C« LMi i ( I I , J J, KK ,?) =0. l _ M t ? ( i r , J J . K K , 3 ) = C . LM13( I T , JJ .KK ,3)=0. LM1 *(I I ,JJ,KK ,3) =C. L J M ? 7 < I T , JJ. 'KK ,3)=C« 0 Lwi 7(T T , JJ ,K« ,3 )=0 .0 C nNTI NLF UP M A S ? M A T P I X DO 25 I 1 = 1 . P4 on ?B jj = t ,g4 L A M ( T T , J J ) = C . O M = ** 4* CALC LLAT CALL. GMPPO( NTHFT .PTTHET , W C K 1 .3 ,3,3) DO 35 JJ = 1 , KK T T =! ,3 R { i ,1 , i i , J j) , W C K 1 , I_M27< 1 , 1 , I I, JJ ), 3, 3, 3) ? T A C K ( LV1 ,l AN .3 ,7 ,84 ,1 ,1 ) « T A C K ( L M ? ,LA»» ,3 , .3 ,P4 ,1 ,4 I STACK (LM6 ,LAV ,3,3,«4,! ,7) « T A C K ( L K ' a ,LA»» ,3 .9,94 .1 ,10) C T A C K ( LV4 ,1 A^ ,3 ,3 ,f?4 ,1 ,23) T ALl C A L l <-|ALl. CALL f^ALL r'ALI. C|ALL r-ALl CJALl C'ALL CALL C A L l «TAC*<( L V i n . l . A W , ? ,T ,«4 ,7 ,1 ) CALL 5 TAC K( LW1 Q ,t A* ,3 ,1 ,B4 ,7 ,4 ) CALL « = T A C K ( L M 2 0 . L A W , 3 ,3 ,84 ,7 ,7) on so jj=i , 3 r » n 5 0 K K = i ,3 on 50 T T = I , 3 NJNJ^T T * ">* ( KK-J )-f « A^ ,3 ,0,^4 ,\ ,^ i ) .1 ,1 ,?) , L ^ W ,3 ,Q ,1 ,t ,3) , L A V ,3 ,f» ,L AW, 3 ,3 ,e4,4 , i ) 5 T A C M L W 2 5 . L A V ,3 ,3 ,84 ,4 ,4 ) 34 ,4? ,fi7 LAM(NN,.IJ-f 3) =LM« (II , JJ .KK) LAM (MN' ,MM)=LM9( I T , J J . K K ) LAM (NNJ+ O , VM+0) =1. VIS (II ,» J ,KK) CAL l C ALL CALL on f-o nn f-.Q on «o ! r> n f o NN=II « TAT K ( L V?\ , L A V .7 ,? S T A C K ( LW?? . L A W , 3 ,7 S T A C K ( LW?.? , L A V ,1 ,3 I J = l ,3 JJ=1 . 3 KK=1 . 3 11=1.3 + 1* (KK-1 )+! e* (T J-l .84 ,?B .R4,?9 ,R4 ,28 ) + 30 ,4 ) LAM(MN, JJ )=LV10 (T I , J J , K K ,1 J) LAM (NK',.JJ+3 )=LV1 ! (II , J J , K k , ! J ) I. A M ( N M , J J+ ? 7 ) =L M 1 3 ( T T . J J . K K ,IJ) FIGURE 10.9 (Cont'd) 186 0,0.9.1 003 ? .00 "i 3 00*4 0085 0096 00 9P OOP 9 0090 009! 009? 0094 00/3F 00 9 .- * ( K K - 1 ) + i R * ( T J - l ) + 3 0 L4M (NN'*MM)=L"1 2 (T I . J J , K K , I J ) f-. 0 L * M ( NN+ vyLT, IS C A R R I E D OUT BELOW TO A V O I D U N N E C E S S A R Y 2 E C C MULTIPLICATIONS N=q.+NPOT+NHUP ^O 70 T T = 1 ,N - DO 70 'JJ=1 , 66 70, WOK;2< I I , J J) =0. 0 :.;"£OW...-'l "^HPlJ.e . ' . . - . . - • . ; ,D O 7 S I T = 1 , "?. ' 0 P 71 J J - \ , \ P • • . 7 1 W O K ? ( T I , J.I) = L A V ( I T , JJ) _0100 05 0 1 0102 6 ic i 0104 0107 010P 01 09 01 1 0 0 J 1 1 01 12 72 WOK? (I I . J J). =l.AV (T I , JJ) . piO 73 JJ = 49,57,1 7? 1VO 7 W O K J > ( I I , JJ) =LAM(I I , JJ) 'B,O_W n 11= i Tj^ti_i >n \ \ r . J jA 00 10": K *:|$f:/L L : jf*> i 05 w1*"1* 2 (11 + N >®i> j) =wr.it®#i i ..110 . nn O,o =1 P:-';<^^ v*.ff • • • '&;••: ~J-'-? •A!-''" -..;V"-' • •r«-. s:-" . ) *LAM< K< +27, JJ ) 1 ! ! = V l C K ? ( M + ^ ^ , J J ) ^ • f t H I M R ( K K , I I ) * L A M ( « 427, JJ ) FIGURE 10.9 (Cont'd) 18? .^jO:i34. 0135 0! 36 _01.37 0136 0139 dQJLAft-.. 0141 0142 0143 b"l44 0145 JJ-31 ,84 ,1 KK=1 ,3 DO .112 on j . t 2 W O K ? ( T on H15 JJ=?e ,30 00 11 5 KK=1 ,LL WOK?( I I4NN,JJ )= (I I4-NN ,J J) +PHI MP (KK. I I ) *LAM(K< 427, JJ ) ?c T J=l T T =1 0146 0147 rj>o on on 13C Jj=i ,3 oo T§'C"KK-=I .3' 130 W C K 2 ( II4-NN, JJ4-LL430) •= * * 30 4L L . J J4/L t* 3 0) (I I 4- hK , J J+LL + 30 ) + PHI MRC KtC-M-L +3, I I ) *L AM ( 140 C "c. c .MUL TIPLY MODE BY WCK2 TC COMPLEfE THE PEDJCEO 01 4«? 0150 01 s r 0152 0153 0155 01517 oise 015«J 01*0 01*51 M A S S MA TPI X - M A « S ( N ,N) .P_Q_..?_PJL_Tjr..= i .,N On ?05 JJ=1 ,N C5 MA e«( j j , Jj) =0. o COL. 1 THPU ? D,O 207 ,JJ=1 ,8 ' DO 20? I I =1 ,N , JJ) =WOK2 (I I , J J) NHUR PLUS fl NH CB-*- »? TF^MHUP.EO. C ) C - O 1C 215 DO 21 G I T =1 ,N D|O 21 C JJ=J. ,NHUB . ..-0.n_.21C.jKK=l ,lrfi 0 M|A<;<;( T I , JJ4- P) =MAS« (IT , J J48 )+WCK2 ( II COL.i O THPU NN^ ) *»HI MH( < K , J J 64 0167 01 vo ..... 1 021 1 0212 0214 021 5 0217 .0219 0220 0221 , 022.2 • ._.P2J*3._. 0224 • 022S P TS( I T-f 27) =SIG7( I I) OO ige jj=i ,3 NN=30+JJ+3* (I 1-1 ) P T S ( N N l = S I G 3( JJ tl I ,1) R TS(NN-f 9) =STC,5 ( J J ,11 ,1) |F GO TO 195 195s COMTTNUF. 200 CONTINUE CALL GFNP C FORM PFOUCED RIGHT SIDE A NO STORE IN WOK.3 DO 30CO I I =1 ,N 3000 WOK3( ! I )=C» C rlo 30C 1 1 =1 ,e 300 WOK •»( I I ) = P T S (I I ) T,F(NHtP.FO. C ) G O TC 320 00 31 C II =1 ,NHU" DO 31 C KK=1 ,1 6 IF( NPOT.FO, 0) GO TC 350 LJ L = 3 0 IiFCP.EO. 3. ll.U=57 rvO 330 I I =1 ,NPOT 00 33 C KK=1 ,LL 3.30 WnK3< I I-f NN) =WOK3 ( T T + N N) + PHI MP ( KX , I I ) *RT S 0227 PF. TLH7N EiND _ j FIGURE 10.10 189 20. 1 ( AUG 71 ) OS/360 FORTRAN H , 0 0 0 2 _Q-003_ .0004 ''0:00s . PP.T..mN5-T..NAJfE-= WA IIV. C.FT-.CL2 ,LI NECNT=60.,S IZ E= 00 0.0K.,..._ „_ SOURCE,nCD ,NOLIST .DECK .LOAD,MAP, NOEDIT, ID,NOXREF SUBROUTINE INTIG . I.MP.LIC.LI...REAL ( A - 2 ) COMMON/TI ME/T.DT , f ^ S Y V , A ^ T W C " - - - - - - COMMON/FLXX • _ . ._ *.AAkPHA.(_3.,,3.r2) .ALPHA (3) .EETA (3) .DELI (3) , DEL2( 3 . 3 ).._.DE1_3J J3._3,_3J_._ *C|EL4(3,3) ,DFL5<.3,2,3) ,DEL6<3> ,DEL7(3) ,OEL8( 3) , OELTB, *CR(57,57) ,CH(18'. 1R) ,CHG(* ,18) ,OEL(3 ) ,DELO( 3), *F IH( 3 )..E_I_R( 3) .EH ,FR ,E TA ,PC ( 3 , 3 ) , FO( 3 , 3 , 3 ) , FG( 3 ) , G AM( 3), GAM^p( 3 *GJH(3) ,GJR( 3) ,GP?ID (3 ,3) ,GR ,G YGB (3 , 3) ,GXL( 3 . 3 ) , DO ( 500 ), *GRH(3 ,3 ) ,GWR(3 ,3) ,G WX (3 ,3 ) , G W X R A ( 3 , 3, 3 ) , G W Y C 3 . 3 ) , GWYC( 3,3,3), . . . *RW.YS(3,3) ,GYP (3 .3) ,GYBW(3 ,3) ,GYH(3,3) ,HCMASS( 3) , H I, HJ , HM ASS^ c, i " ' *I ' 0006 0007 oooe 0009 0010 0011 J)0 \2 0013 0014 ~OQ I 5 0017 ._op_ie.. 0019 00 ?. 1 _ 0022 "ooTi 002(5 *L A ( 3 , 3 ) , I B ( 3 , 3 ) , 1C (3 ,3,3) , I 0 ( 3 ,3 .3 , 3 ) , I H( 3, 3) , JT( 3, 3 ) , KBS1, KB S2, *KPVH( 3,3) .KB VP (3 ,3) ,KHS(3 ,3 ) , K H V ( 3 , 3 ) ,L A M(84 , 84 ) , LM 1 ( 3, 3 ) , M2(3 .3 ) ,LM3(3,3 ,3) , ___ (3 ,3 ) ,L V5 ( 3 , 3, 3 , 3 ) ,LM6 ( 3, 3 ) ,LM7{ 3,3i"3)» 3,3,3) ,LMS<3 ,3 ,3) ,LMO(3,3.3 ,3) ,LM1 1 (3, 3. 3, 3) ,LM12( 3, 3, 3, 3), *LjM1313,3,3,3) ,LM1 A(3 ,3 ,3 > ,L VI 5 (3 ,3 , 3) ,L M16 ( 3, 3, 3, 3>,LM17C 3, 3, 3. 31 *L,M1 8 (3 ,3 ) ,L VI Q(3 ,3) ,LV?0(3 ,3) *LM2U3,3I ,LM22( 3". 3j ' t LTM~23( 3,3), *LM2A{ 3,3) ,LW25(3 ,3) ,LV26 (3 ,3 ) ,LM48(3,3) ,LM138( 3, 3 , 3, 3 ) , < HG( 6, 18 ) , *LM1 ftR (3,3 ,3 ,3) ,KC (57,57) ,KH(18 ,18 ) CbMMON/FLX/ ....... *M|fM K 3,3) .M2C3.3 ,3) ,V3 (3 ,3,3 ,3) , M4 ( 3 , 3 ) , MS( 3. 3 ) , M AST , M ASS( 8 4, 84) , .* M?<..^>.»_MBEAP.m.tMC 13 .3) ,*D(3 ,3 ,3) .MDOCKC 3) ,MG( 3 ) , N ROT , NHUB, *N ,MS,P,PA«S,PIBFTA (3,3) , P I G A W ( 3 , 3 ) , PI PHIC(3,3,3),PIPHID( 3,3, 3,3), (3,3) ,PT THET (3 , 3 ) , PH I C (3 , 3 ) ,PHTCD( 3 , 3 ) , PH I O( 3, 3, 3 ) , (3 ,3 ,3) ,PSI (3) ,P«I C < 3 ) ,P HI MR (5 7, 57 ) ,PHIMH( 1R, 18),NTHET( 3,3) ....... r #P ,3) ,PH(3) .PI ,PJ ,R*ASS,PF!MASS.RTS (84 ),S IK 3, 3) ,S I 2( 3), *.S.T3J 3,3,3) ,? I4(3) .SIG1 (3) ,SIG2(3 .3) ,S I G3 ( 3 . 3, 3 ) , 5 IG4( 3, 3), __ *slir,5( 3.3.3) ,«IG6(3) ,SIG7(3) ,SIGB(3) ,SGA W ( 3, 3) ,SM ASS, STHET("3,~3 *SUM1( - " ,3*3 ) ,TAUA (3) . T A U C C 3 .3) ,T AJ D( 3, 3 , 3 ) , T AJL( 3) ,TAUS( 3), *TI^F T( 3) , THF TD (3 ) ,PH C (3 ) ,? THE (3 ,3 ) ,S PI N COMMON/FLx/ - - 31 ,UCD (3,3) ,UD(3 ,3 ,3) ,UDD(3 , 3, 3) , * OKI ( 3, 3) , * OK2( 84, 84) , -§*jL»_Vl.n.KA<.3 .»3 > »WOK5 (3 ,3 ) ,WCK6 (3 ,3 ) , WR( 3 ) , «XR(_3 > • » X RA f _3,_3_)_ * W X V ( 3 ) ,V (YC< .? ,3 ) , * Y H < 3 ) , *Z(3 ,3 ,3) , X C ( B 4 ) , XCD{ 84) , X 6T sTs", 3l"i7NT3, 3 ) *XL (3 ) .XLLT84) , X VF C ( 84 ) , X VEC 0 ( 34 ) , XV ECDD ( 84 ), / C( 3, 3), YB( 3 ) , C , C TO 1 OR DO 1 05 I T =1 .57 109 X C O d t + 2 7 ) = P H I M P (II :._1.C.P C O N T I N U E C I N I J T I A L C O N D I T I O N S C F W C D A L C C C K O I N A T E S *OQ (A 30 ) /<2HI MP( 3, N S Y M + 2 ) r 0037 _oo.M_ 0039 Q040 004? COMPUTE THE INVERSE OF DO 1 1 C _JJ.-1 »N o"6 u c fi =1 VN 110 W O K 2 ( I I . JJ) =0. 0 (FHI ( T R A N S ) P H I ) A N C STQRF I N K O K 2 p_ 1.1 s = » * 0044 _QPJL?._ 0046 0048 0040 0050 0052 -flOJL?™ 0054 0055 t 0057 005? 0059 Ojp«>0 oo e t 0062 OO"6? 0066 .0067 006? 0070 .00^71 007?. 0074 0075 JP-Q7JL. 0077 0078 0079 J008CL ooei OOS2 O O P * 115 W O K 2 ( I I ,1 I) =1 . 0 I F ( N H L ' O . F O . OCO TC 125 D n 1 2 C II =1 »NHUR 5JO 12C JJ=1 .NHUP DO 12C K K = 1 ,1 8 J_?O.JW|QJ<,2J JJL±JL»jJJ±8) = WC K?_(.I I.*8_. J J>8 ).+ PHI MH ( < K , I I ) *;PHJ M H< KK. JJ ) 125 I F ( N R O T . F O » OGO TC 135 I F C P . F Q . 3. 1 It. =5 7 DO 11C 11=1 .NROT OO 1.3Q,.J_J^=1_.NROT D|O 13C K K = 1 ,l.L K ? < T I + N N , J J + N N ) = 3 = C O N T T N I J F C C 4-PHIMf i (KK» I I ) *PHIMR(KK,JJ ) A L L S P E V N 1 ( W O K 2 . N » V ( O K 3 , M t N I X ) CpAPUTE __(PH IJTPA N S ) «;XC > .. A_MP ST CRE I N _W.OK 3 DjO ! ACC I I -1 tN 1 4 C C W O K 3 ( I I ) = C « C DO 14C I I =1 ,« 140 W J O K 3 ( I I ) = > C (T I ) T F ( N H U P . P O . 0)^0 TC 150 DO 145 K K = 1 ,1 S I T + g ) = V i n k ' 3 ( I I - » - 8 ) + P H I N H ( K K , I l ) * X C ( K K + 9 ) I F ( N P O T . E O . C) CO TO 160 DO 15.5 I I =1 , N P O T OP 1/55 KK = l ,LL ••C O N T I NUE ,ULA TF XVEC (0 ) = V i C K 2 . DO 17CO I T = 1 , N 17CC X (VEC (II ) = 0. C DO 17C I I =1 ,N KK+27 17C X V E r ( I T ) = >VEC ( I I C C COMPUTE ( I I KK ( T ) . X C O ) A N D STCPF I N W O K 3 D O ?4CO 1 1 = 1 , N 2400 WOK.K II ) = 0. C .I_I,-1 .B I I ) = >C D ( T I ) T F ( N H U R » E O « C ) r , O TC ?50 DO 245 I I =1 , N H U R FIGURE 10.10 (Cont'd) 191 DO 2.4« KK*1 tl 345 WPKM IT + fl) ( I VH (KK ,1 I ) *XCO( K.K4-9) C DO ?=>«= I I =1 .MBOT DO 25= KK=1 ,l.L 25? W O K 3 ( I T + KN) *WCK3 < IT* KM + FHI VB 2.60. CONTINUE CALCULATE XVECO ( 0) = \»CK2« V»OK3 DO 77CC I 1=1 ,N > 7 C C XVECD (I T )=0.0 .DO ?7C II =1 ,N OH 27C KK=1 «N ? 7 C XVFCD < T I V = X V E C D ( I t ) + W C K 2 M I » K K ) * W O K 3 < K K ) II » *XCO(KK+27) c C 1:0 DO ?o T i = 5.»i*- ?P CALL P IEMAN(XVr r (II ) ,>VECn(I I ) .XVECDDt I I ),DT, NPiXCDI(J_I *I ),XLl. (T I )) TRANSFORM MODAL C O C P O I N A T E S INTO PHYSI CAL CCORDI NATES F in>?T DFP I V A T T VE «= >. TE ( r»HI.>VECn) AND STCPE IN WOK3 D . n 3 0 C O I T = ! . f l 4 . W O K 3 i l I )=0. 0 ;,-. - D O . 30 C II =1 .8 3CC W O K 3 C II >=>VECDU I ) I{ I IFCNHCR.EQ. It 320 - DO ?j. C It =1 ,1 8 DO I 31 C KK=1 ,NHUB 310 WOK3< 11*9) =WOK3 (I 1+9) 4-PHI VH ( I I ,KK ) *XV EC 0 ( KK+3 ) :..-;j : ;>GO.iTp .330 _; , ' . ;. ;. • ' ' . . . : . , , . ' . _ . . . " '32'0'. DO 335- T-I =1 .1 8 • 3? = ' .370 WOK TNPOT . 335 WOK?( I I-»-27) =WOK^ (I G O ' T) •» 4 c 3.40 DP 34? T I =2 7,»4.'il ;>4? WOK3( II )=0. C ' 3 4 5 ' D O 50 T I ='1 i'3 ,,.. 7AD O T >. = V«OK3.(I I ) WVH(.i I )=wnK3( I ! + 3) ECD(K<*NN> '40 0 O ' fl 0 J J = 1 . 3 UCD ( JJ.t I )=VO«3 ( f>M PHICD ( JJ • II )=WOK3 (NN-t-9) DO ^5 JJ = 1 ,3 N N = 7 0+ J J* "** ( T I - 1 » UODCJJ .T I ,1 ) = W C K 3 ( N K ) PHTDD ( JJ.II ,1 ) =*CK3 yPD C J J_» II .2 ) -WC K3 <>• N* 1 8 ) IF(P.FO. 2. )OO TP. 45 UDO ( J J .1 I .3 ) =WCK3 (NK+36 ) FIGlHffi 10.10 (Cont'd) 192 fiiAP 0141 01 4? A? A3 0144 0145 0_14fi 0147" pi** 01J5P 01~5 ! 015? 01 -5 i. .'; PHIDD (jj ,11 »3) =v»r K.3 c 0 C ON TI N I.F r C S T A T f c V6C TOR C CALCULATr (PHr .XVrC) AND STCRF IN WOK3 ....° n 35CO I f =1 «P4 3 S C O 'wn<-»( II ) =C. C no 35C I 1=1 t8 ^ 5 0 W O K T ( T I ) = 1 V F C ( I T ) rF(NHCO.EO. OGO TC 370 D" 36C I T =1 tl C . 1 ft C K K = ! t NH UH .KK) *XVEC( 0155 5155- 0156 01<39 OJlftp 0161 0162 no TO 7 3 c . 37C OO 37= I I =1 t! 8 1C 00 ?P«? I T =1 tLL + PHI , K K ) * X V E C < K K + NN 0164 0166 0167 0168 £!«-?.- 0170 0171 01JJ? 0~173" 0174 0175 0176 0177 °JLZ?~. 9 tab" 0191 OL?_2_ 0163 O"l84 oiee o i i a GO TO -»<55 7ia-> wnK ^ ( I T ) =0. C 195 OO TO T I '= 1 t 3 7A( IT ) = y i q K 3 _ ( T _ T ) " P SI (ft ) s »OK 2(11 + 6) UC( JJ t !T >= f > H T C ( JJ.I _ U0( J T,I I iT) V*CK3 (^N+21 > PHIOC JJtl I ,1 )=WOK3 (NN+301 r c . ..... - . - . - P H in ( j J , I T . 2 ) =WO K 3 ( t^ N+4 q) TF(P,E0.2. ) GO TO fO Un( JJ ,1 I , 2 ) -=WGK3 (NN+57) ....... ..... ______________ ....^ _ L. PHTO(JJ, II ,3)=wr«3(NN+66) • *C CONTT NUE '_ 70' CCNTT NUP . . ....... . -- — ......... ______________ ^ - TNTEGPATF THETD( l ) TALL STLJF^(THET(1 ) ,THFTn(l ) .OT.NPASSd ) ,XS ( 1 ) . X 5L( 1 ) ) TNTEGPATP GAMD . ._. . . . . . . _____ DO 1 12 1 1=1 t3 112 CALL eTLJES(r ,AV(I I ) .GAMOdI ) ,DT.NPAS5( I I +1 ) , K S( I I +1 ) t XS_ ( II>1» . .? TP lIMR_7(2) t!HFT(3) .. ... ______ __ ____ ^_ CALL GMPRD(PT THPT ,*\H ,VKCK1 ,3.3,1) THFTD( 1 )=OELO(1 )+SPI N-WCK1 (1 .1 ) THETD (3)=OELD (3) THET( 2)=DEL (2) 0192 019? FOPM GAMD CALL GMPPO( SGAM RFTUPN , G A V D » 3 •3 t l ) FIGURE 10.11 GENM 193 1 ( A UG 71) CS/360 FORTRAN H V A T N . qPT=02 , LI NECNT=60 .5 I ZE=00 OOK, ------------- , --- : ---- _ PCn.NaLIST , DECK, LOAD, MAP, NOEDIT, ID,^3XR£F clfl"OLTT NE GFNy TM?n. T C T T PEAL ( A - 7 ) . . . . . . . . . . . roMMON/Ft x/ * A A L P H A < 3 . 3 , 3 ) .ALPHAH) , BE T A (1 ) ,DEL1 (3), D6L?>(3,3>, DEL 3( 3. 3.3). *r>FL4( 7.1) ,OEL5(^,3,3) ,DEL6(3) ,OEL7(3) ,OEL8(3) . OELTB. ------------------------ * rP (?7 , r7 ) ,CHUP .1 P> .CMC, (6 .IB) .DEL (3 ),DELC( 3). *r THf) ,F IR<3> .FH ,FP,FTA,FC(3.3) ,FO( 3. 3 . 3 ) . F« 3» . G AM< 3 ) , GAM D( 3 ) , GH, *r,JH( 1 ) .GJPO) .GPMOC3 ,3) .GR.GYf iBO,^) ,G XL ( 3 , 3 ) , DD ( 500 ) , ...... ...... *GPH<3 .?) .GMPC3 .3) ,C,WX(3 .3 ) . GW X PA ( 3, 3. 3 ) . GW V ( 3 . 3 ) i G* YC( 3, 3. 3 ). * G W V S ( 3 i 3 ) .G>B(1 .3) .GYPM3 .3 > ,GVH( 3 , 3 > , H CM ASS( 3 ) . H I. MJ . HMASS ~*IM3".-«) '."IB (3, 3) .1C (3 .3 .3) .1 C C3 ,3 .3. 31 . I H( 3. 3) . JT( 3. 3 ). K BS 1. KB S2. * K B W « < 3 , 3 ) ,KBVP(3 ,3) ,KHS (3 .3 ) . KHV ( 3 . 3 ) , L AM( 84, 84 ) . LM 1( 3. 3 ) . * L M 2 ( 3 , - > ) .LM3C3.3 .3) . L W A f S .31 .LM5C3.3 .3 .3) .LM6f 3. 3 ) .LM7( 3, 3, ,_3J _, __________ *LMB(3.3.3)',LM9{3.3,3) ,L^1 0 ( 3 .3 ,3 .3 ) .LM1 1 ( 3. 3. 3. 3 ) . LM 1 2( 3. 3, 3, 3 Ir *LM13(3,3,3,3) ,LN!&(3.3.3) .L Nl 5 (3 . 3 . 3 ) ,LM16( 3, 3,3, 3),LM17< 3, 3,3.3). .3) , L W 2 0 < 3 , 3 > ,LM2 1 ( 3, 3 ) . LM22( 3, 3) .LM23( S.J) , _____ _:_ .3) .L '^fi (3 ,3) ,LM4 8( 3, 3 ) , L*l 3fl( 3.3.3. 3) ,KHG( 6. 18). COMP.TLEP QPTY...ONS - NAME = *LM1 ^ (^ ,3 ,3 ,?) ,KC(S7 ,S7) ,KH(18 ,18) r n^MDN /FLX/ ..... . ........ -------------- ..— *M.M 1(^ ,3) .M2C3.3 .3 ) .V3(3.3.3.3I .M4(3,3) , N5 ( 3, 3 ), M AST , M ASS( 84, 84 ). *«R(3 ) .M^FAP (?> ,MC (3 .3) .HO (3, 3 ,3) , MD OCK ( 3 ) . MG( 3 ) . M ROT , N HUB, *N ,N* ,P ,PACS,PTBFTA (3.3) . FI G A V (3 . 3 ) . PI PH I C( 3, 3 . 3 ) . P IPH IO( 3, 3. 3, 31 *o r^<;T ( 7 , 3 ) .01 T H F T C 3 .?) .PHICC3 .31 ,PH T C O C 3 , 3 I . PH IO( 3, 3, 3), .-* ,^> .°^T (^1 ,P«IO(3) ,FHIMR(57,=57 ) ,PHIMH( IB, 18),N4THET( 3,3) ^L x/ v ---------- * o r » M A « ! 5 ( 3 . 3 ) .PH<3) .PI . P J . P f r A S S . P R M A S S . P T S (84) .S III 3.3).SI« 3).. * SI 3 (3 , 3. 3) . S I 4 C 3 I ,SIG1 (3) .SIG2C3 ,3) ,SIG3(3,3, 3). 3 IG4( 3, 3), *SIG5(3.3.3) ,SIG6(3) .SIG713I ,SIG9(3) ,S GA V< 3, 3 ) , SM A^S . ST HETJ 3, 3 ). *MJM1 ( •»,3,"3')",TAUA{<»') , T A U C ( 3 .3) ,TAUO( 3. 3 , 3 ) .T AJL( 3) ,T AUS { 3 > . *THCT( ->) .THE TO (3) ,PHO(3) ,STHE(3 .3) .SPIN COMMON/FLX/ .......... ------ . - -. *UC(3 . - "» ,UCD C3 ,3 ) ,1.0(3,3 ,3) ,UOO(3 ,3.3) . W OKI ( 3 , 3 ) , * OK2( 84, 84 ) . *wnK3( R4) ,*OKA(3 .3) ,WO»«5(3.3) . W CKS (3 .3 ) , W R( 3 ) . *X R( 3 ) .WXRA( 3, 3 ) . * W X V ( 3 ) ,W>C(3.3) .»YH(3) .»2(3 .3 ,3) . XC (84 ) , XCD( 8 A ) , X C( 3. ._3._2 M Lr^LL *XL( 3) ,XLL(84> ".XVEC (P4) , » V E C n ( 8 A ) , XV ECOO ( 84 ) , Y C( 3. 3),YB( 3 ) .YE3(3) , *YG( 7) ,YH (3) ,7A(3 ) ,7AO (3 ) ,2V3(3 ,3 ) INTEGFP I I •. JJ.KK .LL.NN . ______ ........... OIMFN«inN A (3,3) ,B (3,3) .0(3,3.3) ,C( 3.3) ,E( 3,3,3 ), F( 3.3,3, 3), *G(3.7,3.3) .H(3.3 CALL G M T P A ( F I G A V S 2 = ? I N ( G A M ( 2) ) ,GX(3 .84) ,GY(3 ,57) ,3 ,3) C. S,'S=CT N(GAM( 3) ) c 2=r n « (G A * ( 2 ) ) r ?=rosC (1 ,1 I ) ,UC(1 .11 ) .WHKl ,3 ,1 ) CALl GMSUfU *OK1 ,YB ,WCK4 ,3 ,1 ) CALL GA M M A < *C«1 , V C K A ) 1 CALL GMP«D(C,WCK1 .0(1 ,1 ,! I) .3 ,3,3 ) - FOPM E : . •--.:. „•.•/:,;.; KK=P ' . " ". "• • ' •; •'• on « n =t ,KK CALL GMTRA ( PI THET ,*OKI ,3 .3) CALL G M T R A f AALPHA (1 .1 ,1 I) « W C K 4 .3.3) CALL GVPRP( *OK1 ,V»CK4 ,*OK5 ,3 .3 ,3) CALL r,MPPD(C ,WCK5 ,E (.1 ,1 ,1 I) .3,3,3) ^OPM H CALL G M T P A C STHF » V » C K 6 ,3 ,3) C A L L GUPPD ( V»OK6 .VCK4 ,H { 1 ,1 , I I ) ,3.3, 3) ^OFM r- CALl r,WORD( A A L P H A C 1 ,1 ,1 I) , P I THET , WOK* , 3 , 3, 3) CALL GMf>Pt> ( taO<4 , ve , W C K 6 .3 ,3 ,1 ) CALL CMr>PD(R ,V»CK5 ,V«OK1 ,3 .3 .3) nn r jj = i ,3 C A L L GMADD ( >P (1 ,.J J ,1 I ) ,UD (1 ,J J.I I ) . W O K 4 , 3, 1) c ALL GWADD ( \»CK . CALL GMPRO(C .taDKl ,WCK4 ,3 ,3 ,3) :'':'". : on co jj = t .? r»n FO i T =1 .3 G V ( T I , J J ) = R ( I I , J J ) r,xu i ,jj+3) =c L A M ( I I , J . J ) = L A M ( T I t J J ) + « ( M . K K ) *MASS (KK, JJ) 00 ?OC JJ=t ,LL l . A M ( T 1 + 1 , JJ) =0. OO ?00 .KK=1 ,|.L LAM ( I 1 + 3 , J J )=LAV (I 1 + 3 . J J )+GX( I I, KK) *MA3S «K, J J ) no :*oc jj=i ,\L DO ^0 C I I = I ».3 LAV ( I 1 + 27 , J J) =0. 0 DO 30C KK=1 ,NN L A M ( I I + ? 7 , JJ)=LAV( I I+27 ,JJ )+GY( I I ,KK)*MASS(KK +27,JJ ) TRAN«-FPP'M RIGHT SI OF DO 50 C II =1 ?LL SCO XC( I I )=R1«(II ) 00 60C JJ = 1 «3 DO 60 C KK=1 ,3 6CO P T ^ ( J J ) = P T 5 , ( J J ) + A ( J J t K K ) * X C ( K K ) 00 70 C JJ=1 »? P 1( J J t K K ) * X C ( K K ) DO QOC JJ=1 .3 P T S ( J J+2^) = C. 0 OO 80 C K K = T «NN PCC R 1«C J J + 2 7 ) = R T S < JJ + 27)+G V ( J J , KK) *X C( KK+2 7 RETURN! END FIGURE 10.12 CMG IS 20. 1 < AUG 7i ) OS/360 FORTRAN H 000? 0001 0004 .cnM.ej-L5.Ff: JHP.T.ION$...-. .NAM.F= MA.i.N_fjoPT=oa..LiNECfsrr_=.6.Q..*s IZE=.Q.DP_O_K,_ SOURCE .BCD .NOLI«T ,DECK .LOAD,MAP,MQEDIT. ID.NOXREF SUBPni-'TINE CMG IMPLICIT REAL (A -7 ) 0006 O007 oooo 0010 001 I 001'?•; : - c C C CflMMON/TI lyE/T ,OT COMMON/FLX/ * A ALPHA (3 .3 , 2) jALPHA (3) , B F T A ( 3 ) ,DEL1 (3) , OEL2C3, 3), DEL 3( 3...3. J; ).,____ *r>FLAC»,- l) .DF.LM3 ,3 .3) ,OEL6<3) ,OEL7{ 3) .OEL8C 3) .DELT8, *CR< ?7,57) ,CHU8 ,lfi) ,CHG (6,19) .DEL (3 ) »O£LC( 3). *i? IH< 1) ,EIP( ?) ,EH .EP.ETA ,FC(3,3) ,FO< 3,3, 3).FG< 3), G AM( 3), GAMMDC 3) *«5JH(3) ,G JRC?) ,GP«ID(3 .3) .GR . G Y G B C 3 . 3 ) .GXLC 3, 3 ), DD< 500). *GPH(3,3) , G W R ( 3 . 3 ) ,GV(X<3 .3) , G W X R A ( 3 . 3 , 3 ) , GWY ( 3 , 3 I *GWYCC 3. 3» 3)t * G W Y S ( 3.3) .G VB (3 ,3) ,GYRM(3 .3) .GYH(3.3) .HCMASS( 3) . H I. HJ..JHMASS COMMON/PL X/ * T A ( 3 . 3 ) . T B ( 3 . 3 ) . I C ( 3 . 3 . 3 ) . I D ( 3 . 3 . . 3 . 3 ) , ! H ( 3 . 3 ) . J T ( 3 . 3 ) . K B S 1 . K B S *KRVH( 3,3) ,KPVR(3 ,.3) ,KHS(3 ,3) , KHV ( 3.3) . L AM(84. 84), L*«l( 3._3K_ _ *LM2(3 .?T.LM3C3 .3 ,3) ,L^4 (3 ,3) .L*5 < 3 , 3 . 3 , 3 ) . LM6C 3. 3) ,LM7< 3, 3.73). *L M«( ? ,? ,3 ) ,LM9(3 .3 .3) *L»10(3 .3 ,3 ,3) ,LMl 1 (3, 3, 3, 3) ,LM12( 3, 3. 3.4 3 *LMl ?( 3.1,?..3) ,LM14(3.3.3) ,LM15 (3... 3 . 3 ) ,L M16( 3. 3,3, 3 ), LM 17( 3, J. 3i *LM1 P(7 ,3) ,LV? PC3 ,3) ,LV20(3 .3) ,LM21 (3.1) ,LM22( 3, 3) , LM 23( 3, 3"), ? ,3) , L W 2 6 ( 3 , 3 ) ,LM4B(3,3) ,LM13B(3, 3, 3, 3) ,KHG( 6. H COMMON/FL* / *M ,M1( ?,3) ,M2(3.3 ,3) ,M3(3,3 ,3 ,3) ,M4(3,3) , «5 ( 3 . 3 ) , M A S T , M A S S C 84. ' 8.' *MP( 3) .MBFAR (3) ,MC (3 ,3 ) . VD (3 .3 ,3) ,MDOCK( 3),MG( 3 ) , NROT , NHU8, . •• '_ *N ,>g = ,P,r»A ec; .piBFf M3 ,3) .PIGAM3 .3 ). PIPHIC( 3,3, 3>. P IP»-ID( 3^3, 37 * P > i n ^ T ( ? , T ) , P T THFT (3 .3) .PHI-C (3 ,3) »PHICO( 3 ,3 ) .PHIDC 3, 3, 3), *»HTDn (? ,3 ,7) , PSI (3) ,PS IC<3 ) ,PHI MR (57,^7 ) ,»HIMH( 18, 1B)»NTHETC_ 3_t_. COMMON/FL>/ v '"•""" ~ * f ? D M A S S ( 3 , 3 ) ,PH(3) ,RI ,RJ ,R fASS ,PRM ASS,RT S (84 ) , S II ( 3.3),SI2( 3), *SI3( :3,3,3) ,«I4(3) ,STG1 (3) ,SIG2(3,3) , S T G 3 ( 3 , 3, 3) .5 IG4C 3«_3)_._ *<=IGB(i73 ,3) ,?IGft CM-.siG7(3) .S IG8C3) ,S GAM( 3, 3) , SM ASS. ST HET( 3, 3~) *SUM1 ( -?,3,3) .TAUA (3) . TAIJCC3 ,3) .TAUO(3.1, 3 ) . T AJL( 3) ,TAUS( 3), *THET( 7) , THETD (3) ,PHC(3) ,STHE(3,3) .SPIN COMMOW/FLX/ " ' *UC(3, 3) .UCD (3 ,3) . L'O (3 .3 ,3) ,UOO(3 .3 . 3) ,W CK1 { 3, 3 ) . H OK2( 84,84), *WqK3( 64) ,WOK4 (3 ,3) ,V»OK5 (3 .3 ) . WCK5 (3 , 3 ) , W R( 3 ) . *XR( 3) .«XRA( 3. 3)» * W X Y ( 3 ) ,*>C ( 3 .3) , V» YH (3 )• ,»2 ( 3 ,3 ,3) , XC (84 ) , X C D C 8 4'T. x"D("3, 3, 2 ), XN( 2\ *XL(^) .XLLC84) ,XVFC(84 ) , > V E C C ( 3 4 ) , XV ECOO ( 34 ) , YC( 3, 3), YB( 3 ) ,YE3C *Vr,(3) ,YH(3) ,7A(3 ) , Z A O ( 3 ) ,ZV3(3 ,3 ) '". COMMON/CONT/ - - — - .. *TC(3) ,TAUF.START,STCP,H,*NG,2ETG,BOR,B,FE,BOE.BOEF,BOEOO, BDED, *B_IAS. W A I T ;\ ' . __J . INTFGEP IT ,NP THF S SURPOL7INF OFVFLCPS THE CCNTROL T ORQU ES , T C. FOR HOBBLE DAMP I* THg WOBBLE DAMPING TFCHNICUE CONSISTS FO A S GCM G T H AT IS DR~I V EM 1 THAT TT:«S. ANGULAR I»CNFNTVJW VECTOR LAGS THE WOBBLE VECTOR BY 90 01 C ANGULAR R A T E S OF BOD> THAT .C*G IS WITHIN .0015 O'Q i e 'O0vl8, 0020 002'.! 0022 IF(PA SS.GT. C. )GO TO 400 15 ONE =1, =0. 0 FIGURE 10.12 (Cont'd) 197 00 ?4 002"! 00 37 0028 oo TO .00 •> 1 00?2 00? 3 00? 4 0015 00 IP 0040 0041 0042 00 6 '? 0 0 4 4 004S 0046 0047 0.0 4 fl 0040 00 «5 0 .OO1?? 00 B? 00 'lA 005^ 00^7 60 s e '.0 OJ59 OOM 00 £2 006.? 0064 0065 0066. 0067 006=0» 0 r T F C T crop < : T A P T ANO S T C P 400 T F(r>ONr.FJQ. C. >GO TO 431 t430 GO TO 41 • . A - > I T F ( i A IT.FQ. c. )GO 1C 30 I F ( A B « ( E R > . G T . W A I T ) G O TC 430 70 BDF,DD=F.B* *NG**2- (2» *7ETG*V8NG 332 IF (AR S(BOEO )-!• ) 34,34,33 33 BD^zSIGNd. .BDED ) GO TO 4Q *8DED C OC 40 CONTINUE LIMIT N E G A T I V E PA TF TC MINUS SPIN ' f ,?C IF(RDFO.L T. -*«1 ) POR=-Wc ' l DEG H-LAG-LA* P = AKGLE P T W N IN'NER GMBL A X I S AND THE 2 AX IS TCf 21 =(-WP! -BOP) *H#COS(f?) TC ( 3)=(-ViPl -BOP) *H*SIMP) CALL 5TLJFS (ROEO .BOFnO.DT , NP ( 1 ) , XCQR( 1 ) , XL AS ( 1 ) > C A L L , XL A3 ( 2) ) CALL STLJES (BDEF ,BDEFD,OT ,NP(3) ,XCOR(3) ,XLAS( 3) ) TOPOUES TO BODY THAT CNG IS LOCATED WITHIN TA UL ( 1. ) =_TC f 1 > _ TAUL( 2)=TC (2) TAUL ( ^«)=TC (31 RETtRN END ' . " "" FIGUBE 10.33 RIEMAN 199 SUBROUTINE RIEMAN(Y.VI ,Y2.OT,NPASS,VIC.YC.Y2L) REI0001 C REI0002 C THIS SUBROUTINE IS A MOD IF I ED-ADAMS INTEGRATION FOR A SECOND REI0003 C OFDER DIFFERENTIAL EQUATION. THE VARIABLES USED ARE AS FOLLOWS — REI0004 C REI0005 C * V- DEPENDENT VARIABLE AS PREDICTED BY THIS INTEGRATION REI0006 C * VI- FIRST DERIVATIVE OF Y O X = U FOR 1 = 1,10 REI0028 C DT2 I I DT I I I REI0029 C REI0030 C REI0031 IF (NPASS) 5, 3.5 REI0032 3 Y2L=Y2 REI0033 Y1C=Y1 REI0034 YC=Y REI0035 NP*SS=1 REI0036 GO TO 6 REI0037 5 YC=YC*Y1C*DT + (Y2+2.*Y2L)/6.#OT**2 RE1O038 Y1C=Y1C*OT*( Y24Y2D/2. REI0039 6 Y=YC+YlC*DT+OT**2*(4.*Y2-Y2L)/6. REI0040 YI=YIC+DT*( 3.*Y2-Y2L)/2. - - REnoa*i Y2L=Y2 REI0042 PETUPN FIGURE 10.1^ STLJES REI0043 END -.———————- .- RET0044 St'EROUTINE STLJES(Y.Y1,DT,NPASS,YC.Y1L) STLOOOI IF ( NPASS .NE. 0 ) GO TO 5 STL0002 •'~ Y1L=YI STL0003 YC=Y STL0004 NPASS=I STL0005 -" GO TO 6 STL 0006 5 YC=YC*DT*(Y 1+Y1D/2. STL0007 6 Y=YC+DT*( 3.*YI-YlL)/<. STL0008 Y1L==Y1 STL0009 RETURN STL0010 EhO RTi not I FIGURE 10.15 GAMMA 200 X'SUBROUTING GAMMA SUBROUTINE GAMMA (GAU.ANG) THIS ROUTINE SETS UP THE GAMMAC c C c c c c ARRAY FROM THREE ANGLES PARAMETERS ARE AS FOLLOWS, -GAM 3X3 3X1 GAMMA ANGLE ARRAY ARRAY DIMENSION GAM(9) ,ANG(3) DO 5 1 = 1,9,4 GAM I )=0.0 6AM(2 ) = ANG( 3) CAM( 3)=-ANG(2 ) GAM 4)=-ANG( 3) G A M ( 6 ) = A N G ( 1 ) C A M ( 7 ) = A N G < 2) 6AM8)=-ANG< 1 ) RETURN E N D FLX 00880 FLX00890 FLXOO9OO FLX00910 FLX 00920 FLX00940 FLX00950 FLX 00960 FLX00970 FLX00980 FLTT00990" FLX01000 FLX 01010 FLX 01 020 FLX 01 030 FLX01040 FLX01060 FIGURE 10.16 STACK c s c c c c c c c c ""C~ ' ' c c. "C~' r • ='; - (U6ROUTINE STACK "-SUBROUTINE STACKCMATIN, MATOUT, MINR. MINC.MCUTOR. MCI .MC2) THIS ROUTINE STACKS SMALL MATRICIES IN A LARGE MATRIX 'PARAMETERS ARE AS FOLLOWS... - ' • . MTIN INPUT SMALL MATRIX MAtOUT OUTPUT MATRIX MINR NUMBER OF ROWS IN INPUT M A T R I X MINC NUMBER OF COLUMNS IN INPUT MATRIX MOUTOR ORDER OF OUTPUT MATRIX MCI ROW COORDINATE OF STARTING LOCATION MC2 COLUMN COORDINATE OF STARTING LOCATION DIMENSION MAT IM MINR. MINCJ. MATOtT(MOUTOR , MOUTOR) DO 10 J=l .MINC DO 10 1 = 1. MINR . 10 MATOUT (MCI- 1*1. MC2-1+J)=MAT INU.J) RETURN •—' r E N D • ' ' ' ' • ' ' . . . . . . . . . . . - . FLX 01 310 FLXOr720 FLX01330 FLX01340 FLX 01 350 FLX 01 360 FLX01370 FLX01380 FLX01390 FLX01400 _. FLXOT4Tflr FLXO1420 FLX01430 FLX 01 440" FLX01450 FLX 01 460 FLX 01 470 FLX 01 480 FLX 01 490 FLX 01 500 FIGURE 10.17 ABCP1 201 •c- c c c c c c c c c c SUBROUTINE ABCPI SUBROUTINE ABCP I < A. P I • ANG tK ) ; TtfIS ROUT INE GENERATES THE 3X3 '' ' ' A ' ". • 8. ' "-' ' "' C PI K - CONTROL LOGIC - 0 CALCULATE A M A T R I X - 1 CALCULATE A,PI M A T R I X ANG MATRICIES A AND PI 3X3 REAL MATRIX A 3X3 REAL MATRIX B 3X3 REAL MATRIX C 3X3 REAL MATRIX PI 3 ELEMENT ARRAY OF ANGLES C I MEN 5 ION T E M P { 9 ) , A C 9 ) , B < 9 ) ,C < 9) • ANGC 3 I .PI <9) CO 1 1=2,4 Cr-CCOS(DBLE( ANG( 1) M S1 = CSIN(OBLE(ANG( 1) ) ) / > < 7 ) = 0. A ( t ) = t . A ( 5 ) a C l IF(K.EQ.O) GO TO 100 CO 2 1=2,8,2 B( I) = 0. C2=OCCS ) ) S3=DSIN(D8LE(ANGC3) )I C(3)=0. CC 100 C(9)=1.0 CALL GMPRC CALL GMPRC 'RETURN END (C,B.TEMP,3,3,3) (TEMP,A,PI,3,3,3) FLX0001 FLXOOIO FLX0015 FLXOO16 FLX0017 FLXOO18 FLXOOT9 FLX0020 FLXOO14 FLX0053 FLX0056 FLX0057 FLX0058 FLXOO59 FLX0060 FLX0061 FLX0064 FLXOO 67 FLX0068 FLX0069 FLX0070 FLX0071 FLX0074 FLX0077 FLX0078 FLXOO79 FLX0080 FLX0081 FLX 0082 FLX0085 FLXT)051 FIGURE 10.18 SMPY 202 C SUBROUTINE SMPY • ----- C PUPPOSE C MULTIPLY EACH ELEMENT OF A MATRIX BY A SCALAR TO FORM A C RESULTANT MATRIX C C USAGE C CALL SMPY( AtC.R.N.MJ C C DESCRIPTION OF PARAMETERS C A - NAME OF INPUT MATRIX C C - SCALAR C R - NAME OF OUTPUT M A T R I X C N - NUMBER OF ROWS IN MATRIX A AND R C M - NUMBER OF COLUMNS IN MATRIX A AND R C SUEPCUTINE SMPY{ A,C.R,N,M1 --, DIMENSION A( 1 ).R( 1) DO 1 1=1, IT R{ I ) = A ( I)*C RETURN ENC FIGURE 10.19 GMADD C C C C C C C C C ' C C C C C SUBROUTINE GMADD PURPOSE ACD TtoO GENERAL MATRICES TO FORM USAGE CALL GMADDC A.B.R.N.M) DESCRIPTION OF PARAMETERS A - NAME OF FIRST INPUT MATRIX B - NAME OF SECOND INPUT MATRI R - NAME OF OUTPUT MATRIX N - NUMBER OF R O W S IN A,B,R M - NUMBER OF COLUMNS IN A .B*R X RESU.TANT GENERAL MATRIX SUEROUTINE GMADD( A *B ,R,N, M) DIMENSION A( 1 ) .BC1 J *R(1 ) DO 10 1=1, NM 10 R{ I ) = A ( I )*6< I ) RETURN EKfO FIGURE 10.20 SCIA 203 SCEROUTINE SCLA PURPOSE SET EACH ELEMENT OF A MATRIX EQUAL TO A GIVEN SCALAR i i USAGE CALL SCLAC A.C.N.M) DESCRIPTION OF PARAMETERS A - NAME OF INPUT MATRIX C - SCALAR N - NUMBER OF ROWS IN MATRIX A M - NUMBER OF COLUMNS IN MATRIX A SUBROUTINE SCL A{ A, C , N. M ) C IMENSION A( 1 ) IT=N*M 00 1 1= 1, IT 1 A . . - ,-fHF FORTRAN A P P A Y - TMC F IRST OIMF.NSICN •_. j - ' ' " THF , SECOND DIMENSI ' - TMr F f p c t P f f c QF D - THC LA5!7 rr» CF P -THF r iPT, T CCI.UMN OF ""-'.' > .THE L A f T r HLUWN OF •Ai- 1NAME(? ) ,P(NP ,NC) fTFGEP ,Sn,EP»SC»EC < = sc - ! 0 CF F CN CF T0 BE TT PE P TO 'P TO EACH ABOVE T H E M A T R I X , AN INTEGER P . AN INTEGER PPINTED , AN IMTEGER PR I N'T FO ,AN INTEGER BF PRINTED , AN INfEGER RE PRINTED . AN INTEGER" I C NL=MI N.'C . on AO i=SP,ER WRITE ( ^ , 3 0 ) (T «
40 CONTINUE I^CNL .LT, FC) GC TC 10 PFTUPN ' F-ND 20? 10.1.3 Operating Instructions Program operation is governed by three groups of input data cards: RUN CONTROL CARD, BASIC DATA CARDS, and STRUCTURAL DATA CARDS. The required order for these groups is shown below along with the required format and corresponding information provided by each data card. STRUCT. DATA MA7 — *•• JMN — "- _j (13) 3 -< 0 cc z 113) 6 VALU IE12.5) 18 =: 0 cc z (13) 21 VALU (E 12.51 33 :? O cc z (13) 36 VALU (E12.5) 18 =: o CC Z (13) 51 VALU (El 2.5) 63 f O 1 then the STRUCTURAL DATA used for the previous run is acceptable. QNAM is an alphanumeric variable used to identify the run to be executed. BASIC DATA - Defines: vehicle geometry, mass properties, initial state, CMG control logic and hardware characteristics, and general simulation timing logic for printing, plotting and integration. Each data card defines a single input variable whose value is VAL, and is identified by it's identification number IDN. All BASIC DATA are initialized to zero internal to the program, so that only values other than zero need be read-in. Subsequent runs require data cards only to define changes to BASIC DATA. A blank card is required at the end of the BASIC DATA group so as to terminate data read-in. Subsequent runs that do not require changes to BASIC DATA still require a blank card in the. BASIC DATA position of the data set. Table 10.2 provides a correlation between identification numbers and the corresponding Basic Data variables. These variables have been defined in Table 10.1, however, a remarks column is added for additional clarification. For the case of a multiple dimensioned input array, the range of IDN is given. It should be noted that a multiple dimensioned array is internally stored in a stacked column arrangement; for example, a triple array A (3, 3, 3) will have an IDN range from (n + l) to (n + 2?) where the A(3,1>1) element will have a corresponding (n + 3) identification number, and the A(l,l,3) element will have a corresponding (tt + 19) IDN, etc. ftNAM is an alphanumeric variable used to describe -the BASIC DATA variable . STRUCTURAL DATA - Defines: Modal Matrix, Stiffness Matrix, and Damping Matrix for the rotor and flexible portion of hub. This data is provided by the Modal Service Program described in Section 10.2. All the Structrual Data matrices are initialized to zero; therefore, only those elements with non-zero values require input data. Each card can contain data for as much as five elements of the matrix that belong t,o a common column (COL) . VALU represents the data value contained in element (NROW, COL) ; where NROW is the row number that immediately precedes it's corresponding VALU on the data card. 209 BASIC DATA IDENTIFICATION NOTE: The range of all IDN I Program Variable 1 2 3 k 5 6 7 8 9 10-12 13-15- 16-2U 25-33 , 3^-42 U3-69 70-96 97-99 100-102 103-105 106-108 109-111 112-11^ 115-117 118-120 . 201-209 121-129 130 131 132-13^ DTPLOT DT TEND DTPRNT NS P HMASS SMASS EEMASS HCMASS (I) RDMASS (I, 1) IA (I, J) IH (I, J) IB (I, J) 1C (I, J, K) ID .(I, J, K, 1) YH (I) RH (I) YB (I) YG (I) YE3 (I) XL (I) XN (I. 1) ZV3 (I, 1) YC (I, J) XD (I, J, 1) 2 )a, \ «* ) 0 indices is 3« Remarks IF DTPLOT = 0, no plots will be generated IF DTPLOT > 0, calcomp plots are scaled at DTPLOT seconds per inch. Iteration Time (sec). Should not be greater than 1/20 the period of the largest freq.. Run Duration (seconds) Print every DTPRKT sec. 0. Shuttle not attached 1. Shuttle attached Number of rotor arms. 2 for T, 3 for Weight (ibs) of each body. Note ; Only the weight of the lumped in the first arm are read- in RDMASS(l, l) . Masses of other arms set eq.ua! to arm #1. o Inertia (slug- ft ) of each body. Y. masses are Note: Inertia properties of each arm is assumed identical. Vehicle Geometry, (feet) Locate second arm (rad} locate third arm (rad) . Note: first arm is the zero reference. Locate shuttle (radians) 210 IDN 135 136 137 138 139- 1W 11+8-156 157-165 166-17^ 200 210 211 212 213 211+ 220 221 222 223 221+ 225 226 . 227 228 301-302 370-309 310-318 319-327 331-339 3110-31+8 3*+9-357 358-366 . 367-375 1 376-381+ 1+01-1+02 lj.0l+-l+06 1+07-1+09 1+10-1+18 1+19-^ 7 1+28-1+30 1+31-1+39 1+1+0-1+1+8 1(49- 14.57 1+58-1+66 1+67-1+75 1+76-1+81+ 1+85-1+87 1+88-1+90 Program Variable Remarks \ KBS1 KBS2 DELTB ETA KBVR (I, J) ' KBVH (I, J) KHS (I, J) KHV (I, J) SPIN NROT NHUB NSYM ANTTWO PBLOCK CONTR STOP START TAUF H WNG ZETG BIAS WAIT ZA (I) V PSI (I) UC (I, J) PHIC (I, J) UD (I, J, 1) PHID (I, J. 1) UD (I, J, 2) PHID (I, J, .2) UD (I, J, 3) PHID (I, J, 3) MD (I) WYH (I) PSID (I) / LCD (I, J) I PHICD (I, J) I WXY (I) 1 UDD (I, J, 1) 1 PHIDD (I, J, 1) 1 Stiffness and Damping at beaming and Shuttle connection, (feet, Ib, rad, sec) (RAD/SEC) -minimum is one for spin. ) Number of flexible modes . -normally set to zero. -3 variables plotted per block (rad/sec) (rad/sec) (sec) (ft-lb-sec) CMG (rad/sec) Variables = wobble freq. (rad/sec). + for Max I, -for Mini. For no bias set =0. (radians) ' Initial vehicle state (feet, RAD, SEC) UDD (I, J,2) I PHIDD (I, J, 2) | UDD '(I, J, 3) H PHIDD (I, J, 3) II GAM (I) I THET (I) / 211 There attgul"d~tre~mrl)^ank~fi"e±ds~lretween one MOW/VAfcU-contbima tion and another. In addition, although all the NROW/VALU fields on one card need not be filled, data must be placed on the card from left to right. The data found on a particular card belongs to the matrix identified by IDN. The correlation between IDN and the cor- responding structural data matrix, is shown in Table 10.3. It is noted that IDN will always appear in column 80. A zero in column 80 will terminate STRUCTURAL DATA read-in; hence, a blank card is required at the end of the STRUCTURAL DATA group. For subsequent runs that do not require new structural data, a blank card is needed in place of the STRUCTURAL DATA group. If new data is desired, then an entire STRUCTURAL DATA group is required along with NRUN set to 1. There is one exception to the above outlined multiple run procedure and that is when the number of rotor modes (either total or symmetric) is changed. For this case, only new rotor modal data .(PHIMR) need be read-in, provided NRUN is not set to 1. TABLE 10.3 STRUCTURAL DATA IDENTIFICATION IDN 1 2 3 h 5 6 7 8 Structural Data Matrix FHIMR PHIMH KTA or KYA CT or CY KH CH KHG CHG Description Rotor Modal Matrix Hub Modal Matrix Rotor Stiffness Matrix Rotor Damping Matrix Hub Stiffness Matrix Hub Damping Matrix Hub Stiffness About Point G Hub Damping About 'Point G. SAMPLE RUN . A CMG controlled Y space station with a shuttle docked to its hub is the set of conditions chosen to demonstrate THP operation. This condition was chosen since it is one that yields the maximum number of print variables per print cycle. The RUN CONTROL CARD and the BASIC DATA group is printed at the beginning of the run as shown in Table 10.k. Referri-ng to Table 10.k, the argument of the DD array is IDN of the corresponding Basic Data variable. Any Basic Data omitted from the print indicates a zero value. For futher clarification of the variable definitions refer to Tables 10.1 and 10.2. TABLE 10.»» SAMPLE RUN; INPUT DATA RUN NUNS 00 ( OC( OC( OC( D0( OD( 00< DC ( O C ( 00< 00 < DD( OC< OC( DC( OC( DOC OO( D0( DIM OC( DC( DC( OD( 00 ( " ' 00( OC( CD( O C ( D0( D0( "DD( OO( O0( OD( DD( DD< bb ( OD( DD( 00 (" OC( DC( DD( D0( OO( bo( DC( OC( ob( D0( DD( 00 < DC( DC( DC( DD( DD( ~ D0( 3 ER = 1 1 > = 2) = 3) = 4) = 5) = 6) = 7) = 8 »= 9) = 10) = 12) = 13) = 14) = . 15) = 16) = 20) = 24) = 25) = 29 33 34 38 42 43 47 51 52 56 ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = 60)- 61 66 69 70 74 ) = ) = ) = > = > = 78) = 79) = 83 87 88 92 96 97 102 103 106 109 117 120 1 = ) = ) = i=)= > =) =) = > = \ -m j ) = ) = 1231 = 126 129 1 33 157 161 166 ) = ) = ) = ) = ) = ) = 170) = 201 204 ) = ) = Y CMC CONTROL 0.200COH 0. 1 O O C O E 0.200COE 0.50000E 0.10000F 0.30000F 0. 2 3 0 C 0 E 0. 3 l6 c -4£ 0.190COE 0.450COE 0.35000E 0.125COE 0, 72500E 0, 725COE 0. 123COE 0.20076E b«26"8C9E 0.68264E 0.18038E 0.17877E 0.27250E 0* 12CS2E 0.32 76 IE 0.32781E 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. o. 0. 0. 0. 0. 0. 0. -0. -o. -0. 0. 0. -0. 0. 0. 0. 0. 0. 0. 0. 0. -0. -0. 14 160E 575 COE 575 P. OF iocs OE: 312COE 31200E 16441E 20349E 46576E 22731E 22 73 IE 20830E 22731E 22731E 20830E 46S48E 201 67E 16426E 590 OOE 127COE 581ECE 68150E 186COE 625COE 320COE 232ZOE 57330E 10625F 1S7C6E 21 1 93 E 211 S3E 13'SOOE 13SCOE 809ZOE 10015E 02 00 03 01 01 01 05 06 05 04 04 06 04 04 06 07 07 06 09 09 07 05 05 05 04 04 04 C4 04 04 07 07 06 05 05 04 05 05 04 06 07 07 01 03 02 02 02 01 02 02 02 03 01 09 C9 07 07 02 03 DTPLOT TINE PER INCH OF PLT DT TEND OTPRNT NS SHUTTLE A T T A C I - E O P - NC, OF A R M S HMASS SMASS SHUTTLE *ASS RRMASS - ROTOR RING MASS (LBS) HCMASS HCMASS HCMASS R O M A S S ( l ) - LUMPED ROTOR MASSES(LBS) R O M A S S ( 2 ) - LUMPED RCTCfi MASSES(L8S)" R D M A S S ( 3 ) - LLNPED RCTCP MASSES(LBS) IA I A IA IH( 1 IH(2, IH(3, IB(2, 18(3, IC( 1 IC(2 , . .1 2 1 2 3 1. 2 . IC(3. 3, IC( 1 IC(2 IC(J3 IC( 1 IC(|2 IC(3 10(1 ID(2 10(3 I0( 1 10(2 10(3 IO( 1 10(2 10(3 YH(1 , . t * * , . . . * . , , * . ) 1. 2. 3. 1. 2. 3 * 1 . 2. 3. 1. 2 . 3 * 1 . 2, 3. - RH(3) - YB Y G ( 1 YE3( XN(3 ( ) 1 ) Z V 3 ( 3 X D ( 3 X D ( 3 . « XD(3 . BETA KHS( ( 1 KHS(2 KHV KHV YC (1 YC (1 , . 1 ) - ) SHUTTLE INERTIA SHUTTLE INERTIA SHLTTLE INEPT IA - ROTOR RING INERTIA - POTOR RING INERTIA - R C T C P KING INERTIA 1) - I N E R T I A OF 1) - I N E R T I A OF 1) - I N E R T I A CF 2) - INERTIA OF 2) - INERTIA CF 2) - INERTIA OF 3) - INERTIA OF 3) - I NERTI A OF 3) - INERTIA OF 1): - INERTIA CF 1 ) - INERTIA CF 1) - INERTIA OF 2) - INERTIA OF 2) - INERTIA OF 2) - I NERTI A OF 3) - I NERTI A OF 3) - INERTIA CF 3) - INERTIA CF LOCATE ShLTTLE L O C A T E ShLTTLE LOCATE G LUMPED LUMPED LUMPED LUMPED LUMPED LUMPED LUMPED L'uMPlb LUMPED LUMPED L UMPED LUMPED LUMPED LUMPED LUMPED LUMPED LUMPED LUMPED HUB MASS HUB MASS HUB MASS HUB MASS HUB MASS HUB MASS HUB MASS HUB "MASS HUB MASS ROTOR MASS ROTOR MASS ROTOR MASS ROTOR MASS ROTOR MASS ROTOR MASS ROTOR MASS ROTOlR'M'ASS "" ROTOR MASS ~ ' - • - ) - LCCATE E - LOCATE RCTCR AF M - PT.N ) - LOCATE AP7 - FT.V 1 2 •a 2 t .2 1 ) 2) - L O C A T E POTOR - L C C A T E POTOR - L C C A T E POTOR ) ) MASSES MASSES MASSES - • •"*•- •"• • - ., ., _.„ , - . - L C C A T E HLE MASSES - L C C A T E HUE M A S S E S TABLE 10.4 (Cont'd) 213 DC( D C ( DD( DOC OO( D0( DD< DD( DC( OC( DD( DD( O0( OC( DC( DC( 00< OD( 00 ( OD( DD( 207) = 130) = 131 ) = 135) = 137) = 143) = 147) = 200) = 210) = 21 1 ) = 212) = 214) = 221) = 222)= 222 ) = 224)= 225) = 226) = 406) = 426) = 220) = -0.13955E 03 0.20S44E 01 0.418676 01 0.21156E 09 0.10000E 04 0.26950E 06 0.26950E 08 0.262COE 00 0.60CCOE 01 0.200COE 01 0.30000E 01 Os400COE 01 0»11000E-04 0« 15000E-04 C.200COE 01 0 .100COE C5 0.300COE 01 0.80000E 00 O.S9000E-03 0.262COE 00 0.10000E 01 Y C C 1 . 3 ) - LCCATE HUB MASSES" ALPHA (2) - LOC/kTE ARM 2 ALPHA(l) - L O C A T E ARM 3 K8S1 - BEARING SPRING CONSTANT • OELTB - D ISCCNTINL ITY CEFLECTION OF BEARIN K B V R ( 2 t 2 ) - CC1EK PACE CAMPING K8VRC2.3) - CUTER RACE CAMPING SPIN - NOMINAL SPIN RATE CRAD/SEC) NROT NHUB NUMBER OF HUB KCDES " NSV.M PBLCCK STOP ' - •- START TAUF H- CMC WHEEL SIZE" ' ' " «NG- CMC NATURAL FREQ ZETG- CMC DAMPING WYH(3) DCCKtNG DISTURBANCE WXY(l) - INITIAL SPIN RATE (RAO/SEC) COHTR The Structural Data matrices \ised~dwi~rig~the~~rim~are-part— of the namelist TESTC, which is printed once at the beginning of each run. The printed output of TESTC for this aaraple run is not shown here because it is felt to be too lejigthy for the purpose that it would serve. However, the stiffness data used was twice baseline stiffness (i.e. nominal stiff- ness as described in Section U.3) with 1$ structrual damping for both rotor and hub. At the end of each print cycle the variables in the following four namelists are printed: o TIM o STATE o CMGPR o SHUT TIM - Contains only T, the problem time (in seconds). STATE - Contains physical.displacements and rates (in feet, rad, and seconds), modal displacements, rates and. accelera- tions (i.e. XVEC, XVECD, XVECDD), and the maximum rotor. , ring transverse angular rate over one print cycle (i.e. WTMM in rad/sec). The latter variable is used as an indicator for stability. CMGPR - Contains the simulation variables associated with CMG control. This namelist is printed only when there is active CMG wobble danping (i.e. CONTR ?K>). The units are: feet, Ibs, rad, sec. SHUT - Contains the simulation variables associated with a flexibly attached shuttle. This namelist is printed only when NS ^ 0. The units are: feet, Ibs, rad, sec. As indicated, by the basic data variable DTPRWT, the above namelists are printed every 5 sec. The printed output at the 100 second time point is shown in Table 10.5. As seen in table 10.5, the name of each data array is printed followed by the values presently stored in the array. Each output variable is printed in row column order; for example,, the elements of a N by M.array .will be printed as follows: (l,l), (2,1)...(N,1) ,(l,2)...(N,2)...(l,M)...(N,M). For arrays that are not filled (e.g. XVEC, XVECD, and XVECDD) "garbage" is printed in place of the unused areas, and should be ignored. It should be noted that although some arrays are not filled for a particular run, these arrays may be filled during a future run. Selected time history plots, for the above described sample run, are shown in Figure 10.26. Four plot blocks are shown (PBLOCK « U) with three variables per block. The particular variables chosen for plotting are determined by the following statements, that appear in MAIN: WRITE (1) T, WXR (2), WXR (3), WYH (2) WRITE (1) WYH (3). WTC (3,3) UCD (2,3) WRITE (1) PSID (1), UDD (1,3,2), UDD (2,3,2) WRITE (1) WZ (2,3,1) BDR, B OU T O H IX " I OJ . o o p- •0 n1 o ui i •O Ul0 o - 4. 01 00 •O o» 4 N O 0>(0 4 . Ol o . o u I O . O 3 O O O O 1 I o o o i O O N O O O | o o o o o o o o o > o o o o o o o o o o r > o U . I I I U J • 111 1 1 X| | I P - I1 O 01 0 o o 1 0 o 1 o 0 in o UJ 4 O J 4 P I O J O i a> 4 IOD — j >4 o! uj ui JUJ uj i :- p- S o N in 1 O r~ 'O O o'O «l 1 O oo «|0> »l? O OJO O O OIO Oi j O ;Ol o O l O O.1 oj o I Ul Ul oj p> o ioo — 03 - iP) PI . to — U) IO1 p. to 4 O O 1*4 0 IM P- KJ P) iVO tO III P- — O ItO 01 4 o> P- in !p- CJ> O 4 — — I . . P) I . • . ll*l o I ;o o o ! 91 o «j> in 10 o oo . 'tO PI P- • O IM o a oo oo • » n PI . • K) PI PI V «' P- OU O* ^ *•• 4* *J* O 0> p. UI 4 * u> 0> O 4 P- (VI m N P) to ol> PI i*' PI .00 fM 41 • HI OU 41 PI o> M in • • OB • • • if) 41 UI OJ *O O O | ' • • ' . ' . 1 -. . • ' «-t . • O 'illp. ro (j> N o> vo in : UI 4 v u> 03 4 4) o * in O >0 P- PI O< 03 . 00 4 •• 4 4 a) V « iO * P- 4 N (VI ivl — •< >O . (\J (M — • • • u I o o o 1 1 ^ 05 * in in N PJ , — o> P. :N 4 4 P> ' \0 -0 N 'O 03 (VI S PI rn (vi o o PI oc . 4 0> N \0 P) 4 PI oo PI • * in 03 >O >0 0* *U O UI 4 fV) . .03* . *<*P) P I P ) ( V | C U • — • O 1 1 O O 1 1 1 UJ o o O III P- " • IU 4 U* 4 41 • P- . . n ja ip -ux l i 1 1 o o o o r- III » — (VI O 0 1 1 'JJ UJ • •03 . in tn 4 " cu UI 1C (VI .... PI | 0 O O O 1 1 1 ' ' , • :? UJ ( X I C | t * ^ ( V I > O P ) P ) O | 00 ** O P* in 4 'ID (^ ( O . . P | . . ( V I ( V I I 0 O 00 | 1 (VI •« O 0 I I UJ UJ n -• ••« au -* ui p- 4 • 0> 03 p- W .0 )4 o « w»< n n • • • •4 . . . * O> N •• O 0 O O 1 1 1 1 • w* 03 00 <0 p- N II 03 = 2 .6 6 6 6 3 1 7 . - J 5 5 .7 2 C 4 6 3 2 2 9 9 0 6 . 8 D E F . 7 7 8 .8 2 4 2 2 3 2 . B O E = 2 . r c = 3 .2 6 4 0 2 6 6 :- O ;.F E= c. e o e e c - 0 .1 7 8 6 9 2 1 6 . 1 E -0 3 . 0 .2 6 1 6 5 7 8 7 E 7 7 9 9 6 4 3 , S C R = - , - C . 2 4 9 7 6 8 4 0 ! L 3 E -0 2 .B D E D = - C . 1 C E N O E C M G P R * X R = 0 .2 6 1 9 9 9 9 1 e O E O O = 0 .7 0 2 8 3 4 4 21b o •1 • UI. 4 • (VI > PI * 03 OP- 00 ui n o> 4 (VI 4 O1 Cj • (71 O* O O O 0-4 a o> o " . . 4 . O 4 0 0 it ;o> M u j • o> In m ; • o> a »vi • o> a N . . 4 O PI (VI 0 0 - 1i •• £ 6 tl 7 F -C 5 . 0 .0 , 0 .8 8 8 7 7 8 3 6 E 7 S . O . C . O . J 7 8 5 8 5 9 9 E -O S r 6 2 7 0 E -0 4 .- O .3 7 e - 0 4 . 0 .9 9 9 9 9 9 8 2 » 7 , 3 0 4 .9 9 8 , 0 .2 3 4 7 6 2 4 1 E -0 4 4 UJ III PI •* til (VI 4 . • (VI * O * P- 4 I P- O> O> II 4 4 O> M PI (I* in (vi o< U. . II W i u y « u • 41 O O 0 X 2 E -0 5 . 0 .0 . J T = 0 .2 8 5 0 7 9 3 0 E - . 0 .2 3 5 0 8 0 2 0 E -0 6 . o .e 8e .7 r8 36 E -io . 0 . 0 1 E - 0 6 . - 0 . 1 9 6 5 6 6 8 . - 0 .3 1 7 2 5 2 2 2 E -0 6 . 0 .8 2 1 8 5 2 1 7 E -T 0 , - 0 .3 7 e 5 8 6 0 8 E -0 5 . 0 . 0 tE N O & S H U T P S ID = -0 . £ 7 9 5 3 0 2 - 6 . 3 S 1 S 7 2 8 8 E -0 3 - 0 .9 9 9 9 5 5 8 8 0 .0 T A O S = 0 .0 * • . ' ' O UJ u 21? These plot-variables are defined as follows: T - Time (must always appear as first variable in first plot block) WXR (2), WXR (3) - Inertial angular rate of rotor ring pro- , jected on the two and three axes in the rotor ring. WYH (2), WYH (3) - Inertial angular rate of rigid hub section projected on the two and three axes in the rigid hub section. WYC (3,3) - Inertial angular rate of the third flexible hub mass, projected on the three axis in the third lumped mass in the hub. UCD (2,3) ' - The linear velocity of the third flexible hub mass relative to the rigid hub section, projected on the two axis in the third hub mass. PSID (l) - Shuttle angular rate relative to the rigid hub section, projected on the one axis in the Shuttle. UDD (1,3,2), UDD(2,3,2) - Linear velocity of the third lumped mass in the second rotor arm, projected on the mass1 one and two axes. WZ (2,3,1) - Inertial angular rate of the third lumped I mass in the first rotor arm, projected • - on the mass' two axis. BDR, B - Gimbal rate, gimbal angle. It should be noted that the CALCOMP subroutines that are included as part of the program package have been designed specifically for operation at the Grumman facility. Modification to these routines would most likely be required for use at another facility. pus c :q .^ uo PPH PT8TH JO sqq. uo p9^oafoja ' PT8TH qi>H paTHI aui- Jo « (OHS/OVH) -Sfxv 3 .ss'BW am tto 'ni.iv ^SJTJ st« HI SSBW aqq. jo 222 10.2 Modal Service Program The Modal Service Program (MSP) of the General Flexible-Body Stage II digital simulation establishes the following Structural Data for use "by the Time History Program: o Mode Shapes o Stiffness Matrices o Damping Matrices The MSP is special purpose in that it provides the above data for only the "T" and "Y" space station configurations. The method used for formulating this data is outlined in Section 6.2. Structural data for the hub and rotor may be obtained, during one program execution. Printed output of the above structural data is provided as well as punched data cards with the proper format for direct input to the Time History Program. In addition, modal frequencies are provided so as to aid the user of the Time History Program in choosing the signfleant mode shapes. Operating insutructions for the MSP are found in Section 10.2.2. 10.2.1 Program Description (with Listings) i As in the case of the Time History Program, program functions are accomplished by several subroutines. Figure 10.27 presents the logical flow between operational subroutines. The logical tests along with the calling of these subroutines is accomplished by the MAIN (or Executive) routine. A brief description of each routine is presented below. 223 DATA READ MAIN ROUTINE EXECUTE CONTROL CALCULATE STRUCTURAL DATA NO YES DATA READ IN YES SUBROUTINE BEMST ESTABLISH BASIC AND INCREMENTAL ROTOR ARM STIFFNESS SUBROUTINE SMATRX ESTABLISH STIFFNESS DATA PRINT-PUNCH STRUCTURAL DATA PROGRAM EXIT G ESTABLISH STIFFNESS DATA FOR MODAL MATRIX CAL'S Figure 10-27. Model Service Program, Flow Chart 221* 0 SUBROUTINE RMMODC ESTABLISH SYMMETRIC AND ANTI- SYMMETRIC STIFFNESS AND MASS MATRIX I SUBROUTINE RSMODI ESTABLISH MODAL MATRIX DATA PRINT-PUNCH MODAL CONFIGURATION DATA CALL RDAMPI - ESTABLISH DAMPING MATRIX DATA PRINT-PUNCH CONFIGURATION DATA NO SUBROUTINE BEMST ESTABLISH BASIC HUB STIFFNESS SUBROUTINE HMMOD ESTABLISH STIFFNESS DATA MODAL MATRIX DATA CALL HDAMD ESTABLISH DAMPING DATA PRINT-PUNCH-HUB CONFIGURATION DATA Figure 10-27. Model Service Program, Flow Chart (Cont.) 225 Main Routine ( listing is found in Figure 10.28) The main program accepts input data and identifies the data with the program variables. This routine establishes the calling sequence for the subroutines in which the Rotor and/or Hub stiffness, damping and modal matrices are formed. Subroutine BEMST (figure 10.29) Establishes an intermediate basic and incremental stiffness matrix for the rotor and an intermediate stiffness matrix for the Hub. Subroutine SMATRX (figure 10.30) Establishes the final rotor stiffness matrix for use by the THP, and an intermediate stiffness matrix that is used to formulate mode shapes. Subroutine RMMODC (figure 10.31) Establishes the rotor symmetric and antisymmetric stiffness and mass data arrays which are utilized in the .computation of system modes and mode shapes. Subroutine RSMODI (figure 10.32) Computes the rotor modal matrix to be used by the THP. Subroutine RDAMP (figure 10.33) Establishes the rotor damping matrix to be used by the THP. Subroutine HMMOD (figure 10. 3*0 Computes the Hub stiffness matrix and modal matrix to be used by the THP. Subroutine HMMP (figure 10.35) Computes the Hub damping matrix to be used by the THP. This subroutine is called from HMMDD. ; Subroutine LATE (figure 10. 36) Generates Rotor and Hub modal frequencies and mode shapes in the following manner : The problem to be solves is, where [M][§] = \[K][«] (10-5) (I) 226 and [K-] - Strifness Matrix - (Positive Definite Symmetric) [M] - Mass Matrix (Symmetric) - Matrix of eigen vectors (mode shapes) - Frequency (Rad/sec) - Eigen value. The first program operation is to factor the stiffness matrix [K] into the form of a lower triangular matrix [L] such that [K] = [L] [L] T (10 -6) This above factorization is performed in subroutine "FUTILE" (figure 10.37). Subroutine "DAGGER" (figure 10.38) is then called repeatedly to form, [L] "1[M][ L] "T = [ C] (10 -7) where array [C] is defined by the following standard eigen value problem, [C] Y = [D] [Y] (10 -8) and [Y] = [ L J T [ 0 ] [D] = Diagonal array composed of eigen values \ ^ A. 2, X - . . . Matrix [C] is then converted in subroutine "SWITCH" (figure 10.39) to a form acceptable to subroutine "SYMEIG" (figure 10.1*0). SYMEIG solves the standard eigen value problem. Subroutine TRIEQ (figure 10. hi) is used to transform the eigenvectors to obtain [ 0] . Additional supporting routines required for subroutine (LATE) are: . TFORM (figure 10. 1 . STURM (figure 10.1*3) . PREP (figure 10. 1*1*) . QSVEC (figure 10.^ 5) . AND (figure 10.1*6) . DOTPRO (figure 10.1*7) . RDM (figure 10.1*8) It should be noted that mode shapes will not be calculated if the stiffness matrix is not positive definite. Under these circumstances an error message "Stiffness Array not positive definite - Eigenvalues-modal vectors not computed- check input data" is printed. At this point the program run is terminated and the next data set is read. An error message indicating a negative eigenvalue may occur. If the printed magnitude of the error indicator "CHEC" is close to zero, this error can be ignored and attributed to computer round-off 227 Subroutine COMER (Figure 10.1*9) Converts double dimensioned array into a column vector (or a single dimensioned array). Subroutine ABCP (Figure 10.50) Generates the transformation matrix for a single angle rotation. Subroutine "Anorm" (Figure 10.51) , . ! . - . Normalizes each column of a matrix. Subroutine "PWRIT" (Figure 10.52) Generates punched data in the proper format for direct input to the THP. Refer to section 10.1.2 for a description of the following utility subroutines: o WILLY o STACK ' o GAMMA o SREVW1 10.2.2 Operating Instructions Program operation is controlled by input data cards exactly as described in Section 10.1.3 for the THP. The only difference is that the STRUCTURAL DATA block is not used. Specifically, a RUN CONTROL card is required followed by the BASIC BATA group. The required format for these data groups is identical to the THP. ROTOR BASIC DATA All arms are assumed to have identical geometric and mass properties. The rotor arm geometric properties that have to be considered are illustrated in .Figure 10.53. Input data card idntification (-ID) numbers are included in Figure 10.53. Table 10.6 defines all rotor input data variables and associated ID numbers. A sample input card set-up to obtain rotor configuration data is illustrated in Table 10.7. HUB BASIC DATA The hub geometrical properties that must be considered along with their corresponding ID numbers are shown in Figure 10.5U. Table 10.8 identifies all hub input data parameters and associated ID numbers. A sample input data set- up to obtain Hub structural data is illustrated in Table 10.9. FIGURE 10.28 MAIN 228 M.- . . IV G LEVEL 1. MOD 1 MAIN. . . 0AT.E.= 7.13.6.3 .....L3-.-1A. INTEGER P.ER.EC COMMON /PUNCH/ IR I T . TREUG..TPHHF... TS MAT , T.CT YA ,T K 7H ,.T CH. TK6WG, T-CHG_-... COMMON /BEAM1/ *HL< 3 ) * A L ( 3 ) .PSSC3) .HUB .El *AE,GJ COMMQN /RHMST/ , .. . _ _ .. * W O R K 6 ( 7 2 , 7 2 ) . W O R K 8 ( 7 2 , 6 0 ) ,WCHK9(72.60) , W O R K 1 1 C 57.57) COMMON /RRMID/ *RRIB( 3,3) ,DI I (3 ,3) ,Dia(3.t3) »DI3(3.,3) iD.Kl ,DM2,DW3 COMMON /RMMST1/ *XRN(3 ) .XRD1 (3) ,XRD2(3) .ZV(3) .ALPH1 (3 .3 ) . GXN( 3. 3 ) . GZP( 3 , 3 ) , ANG{ 2) * *PT1 .PT2.PT3.PVBKS1 . TE ST , OARRA (54 j3) ,XRD3(3) . „ COMMON /EIGVEC/ *COLK(435) ,COLM(435) ,E 1C, (29,29) .QMEGA1 (281 .QMEG/>2( 29),TOL I, FREQ „ _ COMMON. ./HUB ID/ . *CI 1(3.3) .C I 2(3,3) ,CI 3(3.3) ,^HE(3) ,CM1 . CM2 ,CM3, VHAC 3) ,Y HGt 3 >, *VHC1( 3 ) ,YHC2(3) ,YHC3(3» .. ' ._. . . COMMON /HO A.MPC/ _ __ __„„_.___._.__......„„ *DAMPH COMMON /ROAMPC/ *DAMPR C O M M O N / P R I N T / . * I O U T . I O I N COMMON /TERUP/BRAK DIMENSION O D ( ? 0 0 ) - V DIMENSION O N A M ( 7 ) DIMENSION X H P ( 3 ) W R I T E ( 6 . 9 0 I ) 901 F O R M A T ( ' TYPE FILE NUMBER TO BE USED1) R E A D ( 5 . 9 0 0 ) I 01 N 900 F O P M A T ( I 1) IOUT=R ^ • DO 1 1 = 1,200 . . . „ _!.,.,.« 1 DOC I ) = 0. 21 REAO(IOIN.22) NRUN.QNAM 22 FORMAT( I3..7A4) . _ ..._ _. _„„ WRITE( IOUT .23 ) NRUN.QNAM 23 FORMAT( 1H1 , 'RUN NUMBER = • .1 3 ,5 X .7 A4/) IF(NRUN.GT.O) GO TO 25 . _. _ CALL EXIT 25 READ(IOIN.26) ID .VAL .QNAM 26 FORMAT( I5,E20.5,7A^) . ... IF(ID.LE.O) GO TO 27 DOC ID)=VAL WR ITE( IOUT.28) ID ,DD(Ip) ,QNA^ _ __. 28 F O P M A T ( 5 X , » D D ( • ,15, • )= • .G20 . 7 ,5 X .7 A4 ) GO TO 25 27 CONTINUE . . C C INPUT DATA A R R A Y C. . . ^ . . .. .'.... DO 40 1=1.3 XRDKI )=DD(I+1) _, . . XRD2J I )=S>0( I* ..€ _ i .. ..- _ •. • :._.. C HUB INPUT D A T A C HUBCAL=D.D( 1.0.0) _.. _. . DO 35 1=1.3 Y H A ( I ) = D D < 1 0 0 + 1 ) VHG( I )=DD( 1 03*1 ) '__ YHCK I )=DD( 106+1 ) YHC2( I )=DD( 109+1 ) YHC3( IJ=DD< 1 1,2 + 1 ) ... .. : 35 YHE( I)=DD(1 1 5+1 I 1 = 119 DO 36 J = l .3 .. . . _ ; _. _„, DO 36 L=i.3 CI 1(L.J)=DD ( I ) _ .36. 1=1+1: . . . . _- . ._ 1=128 DO 37 J=1.3 DO 37 L = l,,3 . : CI 2 ) . ._ , .___ HUB=o. TEST=1 . FIGURE 10.28 (Cont'd) 231 IV G LEVEL 1. MOD 1 MAIN DATE = 71363 13.16. C C C C C C STIFFNESS CALCULATIONS 45 50 499 PT1=DMI*XRD1 <3)*PRS P T2=DM2* XRO 2(3) *PR S P T 3 =DM 3* XRO 3 ( 3 ) *PR S P.SSC 1)=PT1«-PT2+PT3 PSS( 2)=PT2+PT3 PSS(3 )=PT3 IFlHUBCALtEQ.l .) GO TO 45 BRAK=0. CALL BEMST CALL ....... SMA..TRX. ...... CALL RMMOOC CALL RSMOD1 IF(HU8CAL.EQ.O.) GO TO 50 HUD CALCULATIONS CONTINUE HUfl=l. BRAK=0. CALL HEMST CALL HMMOD CONTINUF: IF( IP IT.LE.6) GO TO 21 IV = 0 WR ITE{ IRIT.,499). I V . ' . . ' . . FORMAT( 78X, 12) GO TO 21 END. ..... _. .„. ........ . ......... FIGURE 10.29 BEM3T IV G , CEVE.L. 1» MOO 1 .. B.EMS.T .DATE = 7.1.3.6.3 ..... _. _________ 1.3... 19. .2.7.. T I NE BEMST COMMON XHEAMI / *HL(3 ) .AL<3) ,PSS<3) ,HUB,EI .AE.GJ COMMON /RHMST/ *WORK6( 72.72) t W O R K 8 ( 7 2 .60) • tt.CPK9(72 .60K.WOBKI K.57. .57..)... ....... ..... _- ....... DIMENSION BK1 (3) ,BK2(3J ,BK3(3> ,BK4(3) , BK6(3) ,POL( 3 I , SK ( 12, 12),SSK( * 12, 12) ,TSK< 12,12) ,BK5(3) ......... DIMENSION. ARYX.3 ,3) .............. ... . ....... „ ...... _______ . ______________ .......... IFCHUB .EG. I. ) GO TO 40 DO 1 10 J=l ,3 BK l< J)=AE/AL< J) . , . . . . . - ........ ..... _____________ j._._ ...... BK2( J)=12.*EI /AL( J)**3 BK 3( J)=6.*E I /AL< J) **2 BK4< J)=6J/AL2; 3SK(LtJ>=0. .' . • . •, - . • ' . C • INCREMENTAL * BASIC C C A R R A Y K* : '• c .DO 1 1.6 J SK( l . l ) SK ( 5 , 1 ) ... :,„ SKJ.7, 1 ) SK( 11,1) SKC2 .2 ) .,:.- _•: .i "-... '. SK't 4, .2) . SK ( 8 , 2 ) SK ( 1 0 , 2 ) _.. i. . SM.3,3) s«g.3) ' ' . SK ( 4t ^ ) / "'. .:., ;, SK.'C8.4.) - S K ( 1 0 « 4 ) C A L S = li3 =8K2( J) =BK3< J) •=-BK2( J) =BK3( J) =8K2< J) =-BK3( J) =-BK2( J) =-BK3( J) =BK1 ( J) =-BKl ( J) =8K5l J) =BK3(J) =BK6( J) STIFFNESS A R R A Y CALS FIGURE 10.29 (Cont'd) 233 , • - • • I . . • • • : • -IV .G_ LEVEL. 1. MOD 1 BEMS.T. . . J3AIE_s 7.1.3L&3. SMS.5) =8K5(J) _,.,..T.:.. .„.._:.., SK.C7-..5.1.~=_-BK3ii'j) ..,. ..'..-'.- 1.,. ..-.. ! • SK< li ,5) =BK6( J) SKC6.6) =BK4( J) _, SKC.12,6) =-BK4.(J) . . ... ._ I S K ( 7 . 7 ) =6K2(J) | S K ( 1 1 « 7 ) =-BK3O) .. . .SK(8 , 8) =BK2.(J) .... SKf 10* 8) =BK3< J) SK<9,<3) =BK1(J) SM10.1C) =BK5(J> SK( 1 I , 11 ) =BK5( J) SKf 12. 12) =BKM J) 1 IF (HUB .EQ. L. > GO TO 80 . C . • • - . C A R R A Y K** CALS ; c ..„ . • .. • POL( J) =PSS( J) /( 10,*AL< Ji ) SSK(l. l ) =12.*POL(J) ...,.,. _ _.. ^ §§K(.5Lf.l) =AL( J|.*POLC J) ._ SSKC7.1 ) =-SSK(l,U SSK(11 ,1 ) =SSK(5 .1) SSK(2,2) =SSK(1,1) . . S S K ( A , 2 ) = - S S K ( 5 « l ) S S K ( 8 , 2 ) = -SSK(2 ,2) S S K ( 1 0 , 2 > =SSK(^. ,2) _ _ SSK(4 .A ) =4.*AL< J»**2/3.*POL(J) SSK(8,4) =SSK(5.1) S S K < 1 0 . 4 > SSK(5 ,5 ) SSK(7 ,5 ) =SSK( * , 2 ) SS.K11 I t5) . ._. =.S5K * S S K ( L , K ) GO TO (101,102,103) ,J K=l GO TO 102 L=13 104 FIGURE 10.29 (Cont'd) .G-.. LEVEL 1. MOO I ,-. B.E.M.&T..: -DATE =71363 -13 .1.9.. £7. K = l .GO TO 104 . ..._ • _ „... 103 L = 25 104- CALL STACKCTSK,l*ORK6.12j l2.72.L.K) 116 CONTINUE ' IF(HUB .EQ.O. ) GO TO 200 _ ..... _ . . _ ..... DO ...1.4,9. L=1.12 ......... _ . . ' . . DO 149 J=20,31 149 >ORK6=0. DO. 153 K = l ,12 153 WQRK6.(LN, J39) =WORK6(LN,J39) . tV(CRK6(LN.K>*WGRK6(K, J19) N=N*12 IFCN.LE.24) GO TO 152 c . . . - . . • • . . . • . . . . . . . . . . . . . . . . . . . C W O R K 6 ( 1-36.40-51 ) =ARR A Y ( KS'T/»F*d» ) C ..... DO 1.54 L = 1.12. ................. . ........ L19-L-H9 00 154 J=20.31 . , j.i3=J-7 ' . . .• . . : . . . . ..... . _. . ........ _ 154 WORK6C J13,L19)=V(ORK6{L ,J) C jC __________ WO.RK6.( 13-24.20-31) =ARR A_Y._C_« ( TRANSFCSEJ! _______ ......... . ............ 'C N = 0 , 155. CONTINUE . .. .. _________ DO 1.56 L = l .12 LN=L+N ... ________ ,L.1.2=L-H2. . . . . . . . ....... __________ .,. DO 156 J=l . 12 WQRK6(LN.J)=0. DO 156 K-l ,12 FIGURE 10.29 (Cont'd) 235 IV G..LEVEL. .1*. MOD 1 BENST DAT E = 71363- _ 13.13.. I KN=K«-N .„,.. . . . . . . . . . . , 156 WORK6 N=N+12 -i . IF(N.LE.24) GO TO 155 | . . . . 200 CONTINUE i DO I 17 I-= 1 .24 . DO. 11 7 J=25«48 .. . . , 117 W O R K 6 ( L t J ) = 0 . DO 1 1 8 L=l » 12 DO 118 J = l .12 118 WORK6(L.24+ J) =WORK6(L , J) DO 1 19 L = 7. 1 8 DO 119 J = 7 t l 8 I 19 W O R K 6 < L » 2 4 + J ) = W O R K 6 ( L » 2 4 * J ) +*CRK6 » *PTI ,PT2.PT3.P,BKS1,TEST,QARRA(54»3)»XR03<3) COMMON /PRINT/ *IOUT, IOIN DIMENSION X R N M < 3 ) , Z V M < 3 ) DO 122 L=l,24 _ . - . DO 121 J=1.24 121 WORK6CL,J )=0 . 122 WORK6CL.L)=1. DO 108 L = l, 3 XRNM(L )=-XRN+UORK6(L«K+24)**CfiK6(K.JU . C C WORK6C1-24 ,49-72)=ARRAY(K4*Ql ) C ] . _. . DO 124 L = l,24 DO 124 J = l, 24 « * 124 W O R K 6 C L , J + 24)=WORK6( J»L) . ... '."._ C - C WORK6C l -24,25-48)=ARRAY Ql ( TRANSPOSE) C ' " , ' . ' . . . . . . . . . . . DO 125 L = l, 24 ; DO 125 J = i, 24 *» . WORK6(L , J )=0 . DO 125 K = l, 24 125 W O R K 6 C L , J)=WORK6(L. J )+WORK6 (L, K*24) *WCRK6 C K , J+48) C . .._ C WORK6( 1-24, 1-24|=ARRAY K5 { C WORK6( 4 ,4)=WOPK6( 4,4)+PSS( 1)*XRN(3) WORK6( 5, 5)=WORK6( 5,5)+PSS( 1)*XRN<3) WORK6( 22*22) =WORK 6(22, 22 )-PT3*ZV(3) WORK6( 23,23) = WORK6(23,23)-PT3*ZV<3) .....•: C C WORK6( 1-24, 1-24)=ARRAY K5* i FIGURE 10.30 (Cont'd) 237 FILEO SMATPX FORTRAN PI CALLOATA TIME - SH C 00 220 L = l. 24 00 220 J = l , 2 4 220 WORK1 KL tJ )=WORK6(L. J) 221 CONTINUE ' j . I CALL ABCPCALPH1,ANG( 1) 1 DO 126 L*l. 24 DP 126 J = l. 24 *ORK6(L+24,J)=0. 126 CONTINUE CA.L STACK CALL STACM ALPH1.WORK6.3.3. 72.28.4) ' DO 127 L=7, 24 127 WORK6(L+24,L)=1 . I C C WORK6< 25-48. 1-24) =ARRAY Q2 C . : . . . . . . . . . . . . . . . DO 128 L = l ,24 DO 128 J si « 24 ; WORK6(L*48. Jl=0. j DO 128 K = l. 24 123 W O R K 6 < L+48, J )=WORK6(L + 48. J )+WORK6(L.K) *t»CPK6(K+24,J) ' C ' . ' . . . . . . . . C WORK6C 49-72. 1-24 ) =ARR A Y< K5*Q2) ' C ' . . ,;' DO 129 L = l. 24 DO 129 J = l. 24 129 WORK6C J+24.L*48)=WORK6(L*24.J) • :c .:.:"..- ' : . . . ; ' • . • . . . . • ' • . . . . . . . . . C W3RK6( 25-48, 49-72)=ARRAY 02( TRANSPOSE ) c ' • ' . • • • . . DO 130 L=1.24 .. DO 130 Jalt 24 WORK6(L*24. J*24)=0. DO 130 K = l. 24 130 WORK6(L*24, J +24 )=WORK6(L*24 , J+24 )+ WOR K6 (L+24 ,K+48 ) *WOFK6 ( K* A8 . J ) C . ..C... 1 .. WORK6( 25-48. 25-48)=ARRAY K6 •<:•- . . . .• • I F C P . E 0 . 2 ) GO TO 145 -c'.. • • ' . ' . ' . - ' . . . . - • . • ' . . : . . C. , THREE ARM ROTOR CALCULATIONS c ' . ' • • ' • • ' • • .CALL ABCPC ALPH1,ANG( 21 > . , . DO 131 L = l. 24 ;: '..•"•,-• DO 131 J = l. 24 131 CONTINUE CAu-L STACK( ALPH1.WORK6.3.3,72.49.1 ) CALL STACK+|\»ORK6=ARRAY K6* C DO 1 3 6 L = 1. 24 DO 136 J = l, 24 WORK 61 L+24, J >=0. WORK6=l. DO 140 L = 1,6 WORK8(L+24 ,L ) = 1 , 140 WORK8(L+48,L1 = 1 . ... .... _ C C WORK8=ARRAY Q3Y C DO 141 L = l, 72 DO 141 J = 1, 60 W O P K 9 C L . J )=0, .... . . DO 14 1 K = I, 72 141 WORK9CL,J)=WORK9(L,J)+WORK6(L,K)**ORK8(K,J) DO 142 L=l, 72 DO 142 J=l,60 : 142 WORK6( J iL)=WORK 8(L, J) FIGURE 10.30 (Cont'd) FILED SMATRX FORTRAN PI CALLOATA TIME -,-S.HX • i C C WORK6( l-60t 1-72»=ARRAY Q3Y=0. j C WORK6C 1-48. l -48)=APRAY K7T C DO 147 L = l,48 DO 147 J = 1.42 147 WORK8(L ,J )=0 . DO 148 L = l. 24 148 WORK8(L,L1=1. _ DO 149 L = l, 18 ! 149 WORK8=ARRAY Q3T( TRANSPOSE) ' C DO 153 L=1.42 DO 153 J= 1*42 WORK8CL,J )=0. DO 153 K=1.48 _ 153 W O R K 8 C L , J )=WORK8 - • \ • ' ' . . . - • DO Y58 L=l,57 OD 157 J= 1. 57 157 WORK 1 1(L.J »=0. 158 WORK 1 HLtL 1 = 1 . . • . CALL GAMMAt GXN.XRD1 ) CALL STACM GXN, WORK1 1,3,3. 57.4,1) DO 259 L = l. 3 WORK 1 KL+6.L )=- 1 • W O R K 1 HL + 12.L ).= -!. WORK! KL-H8.L 1=- 1 . 259 CONTINUE CALL GAMMA< GXN.XRD2) CALL STACK( GXN,WORK 11,3,3,57.1 0.1) CALL GAMMA(GXN,XRD3) CALL STACMGXN,WORK11,3,3,57,16.1) C C ..WOPKl 1(4-21, l -3)=ARRAY 04 C . CALL ABCP( ALPH1 ,ANG( 1) > DO 159 L=l, 18 DO 159 J = 1,3 W O R K 1 1(:L + 21,J )=0. ' x : DO 159 K = 1 , 3 159 W O R K 1 1=ARRAY QY? S C S DO 183 L = l. 57 S DO 183 J = 1,S7 S FIGURE 10.30.(Cont'd) FILED SMATRX FORTRAN PI CALLOATA TIME - Sh WORK8(L.J>=0. DO 183 K=l,57 133 WORK8(L,J )=WORK8(L,J)+WORK9*WCRK11 (K,J) C C WORK8( 1-57. 1-57)=ARRAY< (Kl 1 Y*)OY7) C DO 184 L = l, 57 DO 184 J=l. 57 184 WORK!1(L,J)=WORK8(L,J) C C WORK1 1( 1-57. 1-57)=ARRAY KYA C GO TO 191 C C TWO ARM ROTOR CALCULATIONS C 186 DO 187 L=4. 39 DO 187 J=l. 3 187 WORK11(L,J)=-WORKl1(L.J) C : ' C WORK 1 1( A-39. l -3 )=-ARRAY(Q4.Q5) C W O R K 1 1( 1-39. 1 -39>=ARRAY QT7 C DO 188 L = l, 39 I DO 188 J = l. 39 W O R K 8 ( L , J ) = 0. ' DO 188 K = 1.39 188 W O R K 8 ( L , J ) = W O R K 8 C L . J ) + WORK9=ARRAY K TA C c ' • . : . C STIFFNESS MATRIX PRINTOUT C 191 I F( P. ECU 2) GO TO 194 WRITEC IOUT. 192) 192 FOPMATC '0«. *TH?EE ARM ROTOR STIFFNESS M A T R I X ' ) ER=57 EC=57 GO TO 196 194 WRITE (IOUT,195) 195 FORMAT( "O1 . »TWO ARM ROTOR STIFFNESS M A T R I X ' ) ER= 39 EC=39 196 CONTINUE DO 279 L=1,ER 279 W O R K 1 1 ( L , 1 ) = 0 . K=53 IF(P.EQ.2) K=35 FIGURE 10.30 (Cont'd) FILED SMATPX FORTRAN PI CALLDATA TIME - SH L=5 M=17 280 CONTINUE WOPK1 I(L. 1 )=-PTl WORKl KL4-6, U=-PT2 , ' WORK I !(M, I )=-PT3 M=M-H8 L=L*18 IF(M.LE.K) GO TO 280 CA_L W ILLV( 'S -MATRIX* ,WORKII ,57.57.1 .EP . I .EC) CALL PWRIT(WORK11.57,57, l ,ER, l ,EC .3 . TSM AT , I RIT > GO TO 221 160 RETURN END FIGURE 10.31 RMMODC FILEO RMMODC FORTRAN PI CALLDATA TIME - SH, SUBROUTINE RMMOOC INTEGER P.ERtEC COMMON /RHMST/ *WORK6( 72 .72> .WORK8(72 .6C) .WORK9C 72,60) « VlCPK 1 1 (57 ,57 ) COMMON /RHMST I/ i , | . '. *XPM( 3)t XRDK 3 ) .XRD2< 3) *ZV( 3 ), .ALPH1 <3 , 3 > .GXM3 ,3 > ,GZP( 3 .3 ) , ANG( 2 ) , *PT1,PT2.PT3.P,BKS1 . TE ST.OARRA ( 54 .3) ,XRD3<3) COMMON /RRMID/ *RRIB< 3. 3),DI1( 3t 3 ) ,D I2C 3,3) ,013(3.3) ,DM ,DI*2,OW3 COMMON /EIGVEC/ *COLK( 435».COLM( 435 ) .E IG( 29 • 29) .OMEGA I (28) »C»«EGA2(29) .TCLltFPEC COMMON /PR INT/ ' *IOUT, IOIN C C ROTOR MASS M A T R I X CALCULATIONS C DO 161 L=l. 57 ! DO 161 J = l,57 161 WORK8(L , J )=0. DO 163 L=l,3 I DO 162 J = l,3 WORK8(L. J)=RRIB WORK6=ARRAY KYSYM C WORK6=0. DO 178 K = U57 173 »ORK6(L+39,J)=WORK6=ARRAY TANT2 DO 204 L=l, 39 DO 20 4 J = 1 , 19 WORKl 1CL,J )=0. DO 204 K = l,39 204 WORKl1 (L , J ) = W O R K l 1 ( L , J ) * W O R K 9 < L , K > * W O R K 6 < K , J) DO 205 L = l. 39 DO 205 J=20,39 W O R K l 1 C L , J >=0. DO 205 K = l, 39 205 WORKl 1(L,J ) = WORKl ML , J ) + WORK9( L * K ) * W Q R K 6 ( K , J ) WDRK1 1{ 1-39, 1 -19 )=ARRAY(K1 CT*TSYM21 WORK 1 K 1-39,20-39|=ARRAY(K10T*TANT2) DO 206 L = 1.39 DO 206 J = l, 19 WORK9(L ,J )=0 . DO 206 K = l,39 206 W O R K 9 ( L , J ) = W O R K 9 ( L , J ) * W O R K 8 ( L , K ) * W O R K 6 ( K . J ) DO 207 L = 1.39 DO 207 J = 20,39 WORK9(L ,J )=0 . DO 207 K = 1,39 207 W O R K 9 C L , J ) = W O R K 9 ( L , J ) + W O R K 8 ( L , K ) * W O R K 6 ( K , J) WORK9C 1-39. I- 19)=ARRA Y(M2T*TSYM2) W O R K 9 C 1-39,20-39»=ARRAY(M2T*TANT2) DO 208 L = l. 39 DO 208 J = 1.39 208 WORK8(J .L )=WORK6(L , J ) W O R K 8 C 1-19. 1-39)=ARRAY TSYM2 ( TRANSPOSE) WORK8( 20-39, 1-35)=ARRAY TANT2( TRANSPOSE) DO 209 L = l. 72 DO 209 J=1.72 209 W O R K 6 C L . J DO 210 L = DO 210 J = W O R K 6 ( L , J DO 210 K = 210 W O R K 6 ( L . J DO 21 1 L = DO 21 1 J = WORK6(L .J*39)=0. 00 21 1 K=l, 39 = 0. * 19 . 19 = 0. ,39 = W O R K 6 C L , J ) + W O R K 8 ( L . K ) * W O R K 1 1 ( K , J ) » 19 . 19 FIGURE 10.31 (Cont'd) PILED RMMODC FORTRAN PI CALLDATA TIME - SHX 211 W O R K 6 C L , J+39)=WORK6 DO 255 J=1,29 255 MORK6(L*39rJ)=MORK6(L+39,J) *QHEG2*W3RK6 (L+39, J*39) c ••"""" ~ ." '' ";•••• -;•; ~ ~ : ~~ : - C WOHK6 (40-68,1-29)=ARRAY KYANT-BAR c ! . • . '• GO TO 265 C FOR ANTI-SYHHETRIC PROBLEH ONLY c ' 260 DO 261 L=1,20 DO 261 J=1,20 261 WORK6(L+39,J)=WORK6(L+39,J)+QHEG2*HORK6(L*39,J+39) C _ _ _ • C «0¥K6T^ Or-59, 1-20) =ARRAY KTANT-BAH C GO TO 270 c ~ ~ " "• r C THREE ARM ROTOR EIGENVECTOR-EIGENVALUE CALCULATIONS ~C S Y H H E T R I C PROBLEH CALCULATIONS C 265 CALL C O H P R ( H O R K 6 , C O L K , 72, 1,1,28) _L CATL C O H " P R ( W O R K 6 / C O L H r 7 2 , 1 r a O , 2 8 ) CALL L A T E ( 2 8 r C O L K , 2 8 f T O L l , Q H E G A l , N I X 1 , C O L H , H O R K 6 , 7 2 , 1 , Q H E G 2 , E I G ) I F ( B R A K . E Q . 1 . ) R E T U R N C WORK6(1-28,1-28)=ARRAY BIG-YSYH1 C C " A N T I - S Y H H E T R I C PROBLEH CALCULATIONS FIGURE 10.32 (Cont'd) 250 • i • IV G LEVEL 1, HOD 1 BSHOD1 DATE = 71363 13.18.49 C CALL COMPB(WORK6,COLK,72,«0, 1,29) CALL COHPB(WOBK6,COLH,72yiK),UO,29) CALL LATE(29,COLK,29,TOL1,Q!!EGA2,NIX2,COLN,WORK6,72,<»0,QHEG2,EIG) IF(BRAK.EQ.1.) BETURN i | c '• ' ' ' ! i ."' " C WORK6(40-68,1-29)=ARBAY EIG-YANT1 C "" , DO 266 L=1,57 " " " ' " " ~~" "' " DO 266 J=1,28 »OBK8(L,J)=0. I | DO 266 K=1,28 ,"," 266 i3BK8(L, J) =WORK8 (L, J) »WOBK11 (L,K) *WOBK6(K, J) DO 267 L=1,57 DO 267 J=1,29 «OBK8(L,J4-28) =0. DO 267 K=29f57 267 WRW(L,J+28) =WORK8(L, J + 28) *WOBK11 (LrK)*MOBK6(K»11,J) ' C C WOBK8 (1-57, 1-28) =ARBAI EIG-^YSYB2 "C™ iORK8(1-57,29-57) =ABBAY EiG-rAHT2 ~ r~ C GO TO 275 . i - c "" ; "• .' ' . " ' '' " '': C TWO ASM ROTOR EIGENVECTOR-EIGENVALUE CALCULATIONS C 4..; . : C ~ S Y H H E T H I C PROBLEM CALCULATIONS C 270 CALL COMPR(WORK6,COLK,72,1,1,19) CALL COHPR(WORK6,COLM,72,1,aO, 19) CALL LATE(19rCOLK,19rTOL1,QMEGA1,NIXl,COLH,WORK6,72f1,QHBG2,EIG) IP(BRAK.EQ. 1.) RETURNc " . . . . . - ._. ,. C WORK6(1-19,1-19)=AHRAf EIG-TSYM1 C C ANTI-SYHHETRIC PROBLEM CALCULATIONS C CALL C O M P R ( W O R K 6 , C O L K , 72,40, 1,20) CALL COHPB(WOBk6,COLM,72/a6 ,<»0,20) CALL L A T E ( 2 0 , C O L K , 2 0 , T O L 1 , Q N E G A 2 , N I X 2 , C O L H , W O B K 6 , 7 2 f a O , Q H E G 2 . E I G ) IF(BRAK.EQ.1.) BETUBN _ '_ ,"" c " " ."" " ---.-— C WORK6(40-59,1-20)=ARBAY EIG-TANT1 C . _ _ _ _ _ • _ _ _ _ • _ _ DO 271 L= 1,39" " "' " DO 271 J=1,19 WORKS(L,J)=0. t DO 271 K=1,19 271 WORK8(L, J) =WOHK8(L,J)*WORK11 (L,K)*WORK6(K, J) DO 272 1=1,39 DO 272 J = 1,20 " WOBK8(L,J*19)=0. DO 272 K=20,39 "272 WORK8(L, jVi9)=WORK8(L,J*19) *WORK11 (L,K)*iOBK6(K*20,J)" " FIGURE 10.32 (Cont'd) 2ST' |y G LEVEL 1, HOD 1 RSHOD1 DATE = 71363 .._ 13.18A45! c C C W O R K S (1-39, 1-19) = A R H A Y EIG-TSYH2 W O R K 8 ( 1 - 3 9 , 2 0 - 3 9 ) = A R R A Y BIG-TAHT2 275 COHTIBOE DO 279 L=1,57 DO 276 J=1,57 276 WORK9(L, J) =0. 279 BpRK9(L,L)=1. "" ...... " ...... ~ ....... " .............. """' K=54 IPJP.EQ. 3) GO TO 400 , J = 2 0 i ....... , - — • - ; - - ........... - ........... • - • -—;••-. K=36 • 400_CONTIHUB ....... DO wr T=i,3 ..... ~"": ............. "• ..... .................................. 401 W O R K S (L,J)=»ORK9(L,1) DO 402 L=1 ,K 402 H O R K 8 { L + 3 / J ) = - Q A R H A { L , 1 ) C ? _elL*TIYE POORDINf tTES HODAL «*TRIX CALCOLAriOHSc .............. ~" ..... - . - - . , - . - . . - . ............... - ..... - IP(P.BQ.2) GO TO 280 C , ' ' C THREE ARM ROTOR CAL'S C I DO 277 L=1,54 ~" ......... bo~277 j=i/3 ........... " ........ " ........... . :• ..................... 277 «ORK9(L*3,J)=QARRA(L, J) C i C »ORK9{1-57V1-57)=ARRAY QY6 C ! DO 278 L=1,57 ~ ..... DO 278 J=Tr57 ..... " ....... ' ........ " ...... " •' ' " ....... '. .......... "'" IORK11(L,J)=0. DO 278 K=1,57 278 WORkl 1 (L, J) =BORK1 1 (L, J) VWORK9 (L,K) *VORK8 (X7J) C ' • . C WORK11 (1-57,1-57) =BODAL MATRIX EIG-YR ~c ^ ...... " ......... ."" ........... : ..... " ........... ------ CALL t A N O H M ( W O H K 6 , H O H K 1 1 , 57,72, 57) GO TO 285 c" ...... ............ """ ..... ' ' "~ C TWO ARH ROTOR CAL'S C T8i pOffSAf (* 0« , « ROTOR' lODAL~81f lIX*) CALL W I L L Y C B I G - T Y B P ' . H O B K l l , 57,57, 1,BB,1, EC) C • ' C IHSERT POHCH-TAPE BOOTIMET HERE CALL P«RIT(«ORK11,57,57,1 ,BR,1,EC,1,TREIG,IRII) CALL R D A H P l R E T U R N END FIGURE 10.33 RDAMP 255 IV G LEVEL 1* MOD 1 RDAMP DATE = 71363... .................. 1.3....1.9...1: SUBROUTINE RDAMP „_ . . . . COMMON /PUNCH/ IRI T « TREI G t T PHHF . TSMAT t.T CT YA • TK7H.J.CH.1 TK6HGt TCHG- _________ COMMON /PRINT/IOUT.IOIN COMMON /RDAMPC/ *DAMPR ! ....... COMMON /RHMST/ * WORK6( 72t72) .WORK 8 { 72 .60) .VHCRK9 ( 72 .60 ) ,WGRK1 1( 57.57) COMMON /RHMSTl/ _______ .. .......... _._ ..... _ .......... _. *XRN(3I , XRD1 (3) . XRD2O) ,2V (3) .ALPH1 (3 .3) ,GXN( 3 , 3 ) , GZ P { 3 . 3 ) . ANG( 2 ) , *PT1.PT2.PT3.P.8KS1 , TEST . OARK A (54 ,3) *XR03 (3 ) COMMON i/EIGVEC/ .................. _ _______ _. *COLK(43|5) ,COLM(435) .EIGC29.29) .OVEGA1 (28) ,OMEGA2( 29) ,TOL 1 , FREQ INTEGER P _ ......... ............ .DIMENSION A I N V ( 5 7 , 5 7 > .LCC(57) _ ................... . ....... ______________ M = 57 IF(P.EQ.2) M = 39 ..... DO 10 L = 1,M ...................... _.:_• ______________ DO 10 J^l.M 10 A I N V ( L , J ) = W O P K 8 ( L « J) ...C... . .. C C - . C C C . AINV( 1-3S/5 RETURN 1 ENTRY POINT i ENTRY RDAMP I QMEGA2C 1 )=0. LZ = 0 LL-28 ... ...... LK=57 . .. , . IF(P.EQ.3) GO TO 15 LL = 1 9 LM=LL+ I LK = 39 15 CONTINUE ..... DO 20 L=1..M DO 20 J=1,M 20 WORK9(L , J) = 0. DO 21 L = 1.LL 21 WORK9(L ,L )=QMEGA1 (L) DO 22 L=LM,LK LZ=LZ-H 22 WORK9(L.L)=QMEGA2 (LZ) C C WORK9( 1-39/57, 1-39/57) = A R R A Y OMEGA-T/Y C DO 3 5 L = 1 , M DO 35 J=1.M 35 WORK6(L. J )=WORK8(L, J) CALL ANORM( WORK6.WOWK6 ,72 ,72 .1") 00 23 L=l .M . DO 23 J=1.M FIGURE 10.33 (Cont'd) 256 I-Y G LEVEL 1* MOD 1 RDAMP - DATE = 71363... . - ia«JJ9-...1.3 23 W O R K 8 C L . J )=AI NV T / Y C ...... CALL SREVN1 (AJNV.M.LOC .57 ,KMs) .-- DO 25 L = l ,M DO 25 J=UM ..WORK.'l.l V (El GVECC NOFM)*A) = B c ' ! DAMPR2=2.'*DAMPP DO 27 L=l ,M . ..: DO 27 J-l,M WORKU C T / Y C DO 29 L=ltM , .DO 28 J=ltM 28 WORK9(L. J) =0. 29 WORK9(L.L)=1. MM=M-3 ... . „ „ _ DO 30 L=1,MM DO 30 J=1.3 30 WORK9(L4-3.J)=-QARRA(L, Jl ......„: C C WORK9( 1-39/57,1-39/57) =ARP A Y CT7/CY7 C ' DO 31 L=l,M DO 31 J=l.M AINV(L,J)=0. ;_ DO 31 K=l,M 31 AINV{L,J)=AINV(L.J)-H»CfiKll (L,K) *WCPK9 ( K, J) DO AO L-l.M 40 AINViL,1)=0. C C AINV( 1-39/57. l-39/57)=ARRAY CTA/CYA .... C FIGURE 10.33 (Cont'd) 257 I .IV G-LEV.BL 1. MOD 1 RDAMP D A T E = 713A3- - l.i._l.9. WR I TEC I OUT t 60) 60 FO«MAT< ' 0 « » 'ROTOR DAMP IN<3 V ATPI X • ) CALL WILLY( ' A P Y - C T Y A ' , A I K V « 5 7 . 5 7 .I.M.I ,M> C C INSERT TAPE-PUNCH ROUTINE HEPE , , CALL P W P I T C A I N V . 5 7 . 5 7 . 1 .»».l . V •-» , TCTYA , I HIT) .RETURN END FIGURE 10.31* HMMOD 258 •AN IV G LEVEL 1* MOD 1 HMMOO DATE = 7136.3 13..21..5.' SUBROUTINE HMMOO INTEGER P,ER,EC . __ _ COMMON /PUNCH/ IRIT.TREIG.TFHHF .TSMAT .TCTYA.TK7H.TCH.TK6HG.TCHG COMMON /BEAM1/ ii * H L < 3 ) f A L < 3 ) ,PSS<3) ,HUB ,EI ,AE ,GJ COMMON /RHMST/ *WORK6< 72.72) .WORK 8 ( 72 .CO) , VlCF K9 < 72 .60 ) .WCFK1 1( 57,57) COMMON /EIGVEC/ *COLK(435) .COLM<435) .E I G<29 .29) . 0*EGA1 < 28 ) ,QMEG/>2{ 29 ) .TCL 1, FREQ COMMON /HUB ID/ *C I 1( 3.3) .C I 2(3, 3) ,C I 3(3.3) ,YHE(3) .CM I , CV2 ,CM3 . YHA ( 3) . V >G( 3 > , _ . . * YHC 1(3) ,YHC2(3) »YHC3(3» COMMON /PRINT/ *IOUT»IOIN COMMON /TERUP/BRAK DIMENSION YHE3(3» ,GYHL(3 ,3) DO 10 L = l ,24 _. ....' DO ' 9 J = 1 , 2 A I 9 WOPK6(L , J)=0. 10 WORK6(L ,L )=1 . YHE3(3I = 0. ' YHE3<2)=0 . YHE3( 1)=-(YHE('1)-YHC3( 1) ) CALL. GAMMA (GYHL.YHE3) CALL S T A C K C G Y H L , W O R K 6 , 3 , 3 , 7 2 , 1 9 , 2 2 ) . . . . . . C . . . | • • . . . . - . . . . .....I . _ , C WORK6( 1-24, 1-24) =APRAY Q1H C ''• • , DO 12 L = l,24 DO ' 1 2 J = 1 , 2 4 ' WORK6(L. J*48)=0. DO 12 K=l,24 12 WORK6(L . J+48)=VKORK6=ARHAY K6H ......£. W O R K 8 < 1-6.1-18) = A R R A Y K6HG . .. . ............ . ....... _________ ..... _______ „ ...... C DO 16 L=l,72 00 16 J=l .72 16 W O R K 6 ( L , J)=0. ' DO I 7 L = 1 , 1 fl ......... DO 17 . J=l .18 ....... ..'.. ..... 17 WORK6CL, J )=WORK8(L*6« JJ C I C WOHK6< 1- 18. 1 - 1 8 ) = A R R A Y K7H C DO 1 0 L = 1 , 3 WOPK6(L.L*39)=CM1 , . . . . . . . . . . . . . . . . . ... . . . . . . . . . . _____________ . . . . . . . . 1. . WORK6C L+6,l_+45> =CM2 18 WORK6(L-H2,l_+5l )=CM3 CALL STACK1CII .WORK6.3 ,3 .72 ,A,^3) CALL S T A C K ( C I 2 . W O R K 6 « 3 « 3 .72 .10.49) CALL STACK(C I3 .WORK6.3 ,3 ,72 .16.55) C ......... I . . _. ....... . ........................ ________ l_. C V«ORK6( 1- 18.40-57) =ARRA> WH C CALL COMPR( WORK6.COLK.72 ,1 .1 .18) ...... ........... CALL COMPRC WORK6 ,COLM , 72 .1,40,18) CALL L A T E { Ifl.COLK.l S.TCLl .QNEGA1 , N I X3 , CCLM ,W OPK6 . 72 , 1.QMEG2.EIG) IF(BRAK.EQ. 1. ) RETURN .............. ............................ C ' i • . . " ' ' C WORK6( 1-18, 1-18)=ARRAY EIGVEC-HUB c ! ....... • ,. ..'. . . . . CALL ANORM( WORK1 1 .VKORK6 ,72 ,57 ,18 ) C C WORK 1 1 ( I- 18, 1-1 8) =NORMALI 2E A H R A Y EIGVEC-HUR .......... _ C HUB DAMPING M A T R I X CALCULATIONS C , , ' | CALL HDAMP ' C I c INSERT- TAPE-PUNCH ROUTINE HEKE CALL P W R I T ( WORKS, 72 ,60,1 ,6.1 ,18,7 .TK6HG, IRIT ) _ _________ CALL PWRIT( WORK8,72.60,7 ,24 ,1 ,18 »5 ,TK7H, IRIT ) C DO 19 L= 1.1 8 . ........ . . __ . . ........... DO 19 J=l ,1 8 19 WORK6(L, J)=0. _.. . Dn 20 «- = 1.3 . , . _ _________ ________ ___ WORK6(L ,L)=1 . WORK6(L+9,L*3)=1. WORK6(L+12,L*9) =1 . WORK6(L-»-6.L-»- 12) =1 . 2 0 W O P K 6 ( L + l 5 « L + l 5 ) = l . c . ' C WORK6( 1-18, 1-18)=ARRAY 08 C on ,21 L = i ,1 8 FIGURE 10.34 (Cont'd) 260 AN IV G LEVEL It MOD 1 HMMOO DATE = 7 1..3..63._..._ _. 13...2J..... 5 DO 2 I J=1 , 18 i WORK6(L ,J+18 )=0 . .. . . _.'.... DO 21 K=l,18 21 WOHK6(L,J-M 8) -=WORK6(L. J+1 8 > + * O R K 6 < L . K ) * W C R K 1 l(K. J) DO 22 L = 1,1.8 ' ,: ' . :.,_ DO 2 2 J = I « I 8 2?. W Q W K 11 (L t J) =WORK6(L t J* 18) C A L L A N O R M C W O R K 1 1 t W O R K l 1 ,57 ,57«18) C c W O R K i u i - 1 8 . 1 - 1 8 ) = A R R A > FH-HF c . . ....".. ER =18 EC=18 I - NL=0 NUMM=18 WR ITE( IOUT,24) 24 F O R M A T ( ' 0 ' . ' H U D - M O D A L V A T R I X E I G E N V A L U E S ' ) 26 NL=NL-HO IF(NL .GT.NUMM) NL = NVJMW N B = N B + 1 0 WP ITE( IOUT,28) U,I=NB,NL1 28 FORMAT< ' 0 ' . 7 X , ' O M E G A ' / / ' EIG. VAL-NO.• . 13 ,9 I 12) WR ITE( IOUT.30) (QMEGAl (K) ,K=N«,NL) 30 F O R M A T < 12X.10E12.5) IF(NL .LT.NUMM) GO TC 26 WR ITE( IOUT,40) 40 F O R M A T C « 0 ' , " H U B MODAL M A T R I X ' ) CALL WILLY! 'MODE-MAT' ,*ORK1 1 ,57.57 .1 »EF,1 ,EC> C C INSERT TAPE-PUNCH RCUTINE HEPE CALL PWRI T( WORK1 1 ,57,57.1 ,1 8 ,1 .1 8 ,2,TPHHF, IRIT ) C RETURN END FIGURE lOrl .... .. HOAMP FORTRAN PI CALLOATA TIME - SUBROUTINE HOAMP _ j COMMON! /PUNCH/ IR1T»TREIG,TPHHF, TSMAT , T C T Y A .TK7H,TCH,TK6HG,TCHG I COMMON /PRINT/IOUT.10 IN COMMON /HDAMPC/ . . . ' . ' *DAMPH COMMON /RHMST/ *WORK6< 7 2 , 7 2 ) . W Q R K 8 < 72.60) ,WORK9( 72,60) .WCPM 1 (57,57) COMMON /EIGVEC/ I ' *COi_K< 435)»COLM< 435) .EIG(29, 29) , OMEGA 1 (281 ,CPEGA2<29) «TGL1.FREO DIMENSION AINV{ 18,18), LOC ( 18) DO 51 L = 1. 18 DO 50 J = 1. 18 53... WORK'I KL + i9,j + i9).=o. 51 WORK1 1(L-H9.L + 19)=QMEGA1(L) c !' ' ' ' •• • • • C WORK 1 l( 20-37, 20-37)=ARRAY. OMEGA-H (WH) _ 1 .' C ' - DO 52 L = 1. 18 DO 52 J = 1,18 , _. . . . - . . . . 52 A INV(L» J ) = «ORKl 1CL. J) c ' . . ' • • ' • . C AINVC 1-18» 1-18)=ARRAY E IG.YE.C (.NOR MALI 2ED.1 .._ . ._ _.; C ' • . CA-L SREVNK A INV, 18,LOC, 18,NNX) DO 53 L = l. 18 .. „ . DO 53 .- J = 1 • 1 8 ' WORKJ1 1(L*19,J ) = 0. DO 53 K=l. 18 | 53 WORK1 1 (L + 19, J )=WORK1 1 ( L + 1 9 * J ) + WORK1 1(L+19,K-H9)*A1NV(K,J) C . I . ' ; ' C WORK 1 1( 20-37, 1 - 1 C )=ARR A Y( WH*OH* < I N VERSE ) ) = A c • • . • ' ; ' DO 54 L = 1, 18 i DO- 54 J-It 18 W O R K ! KL + 39, J ) = 0. DO 54 K = 1, 18 $l* W O R K 1 1 ( L * 3 9 » J )=WORK1 1 (L*39 , J ) + WORK 11 ( |_ » K ) * fcCRKl 1 ( K*J 9 , J ) .... .i C C W O R K ^ 1 ( 4 0 - 5 7 . 1 - 1 8 ) = A R R A Y < E I G V E C ( N O R M ) * A ) = B c . . . ' . . . . . . . . . . : . „ ' . DAMP2=2.*OAMPH DO 55 L=l. 18 DO 55 J = l, 18 . WORK1 1(L + 19.J+39)=0. DO 55 K = 1 , 18 55 WORK11(L*19. J * 3 9 ) = W O R K 1 I (L+ 1 9 , J«-39) +DAMP2 * WCRK6 (L ,K*39 ) *WORK11 { *9, J) C • C WORK 1 K 20-37, 40-57) =ARRAY CH c • . ' • . DO 60 L = l» 18 , DO 60 J = l» 18 53 ' A INV( L , J ) = W O R K 8 ( L - f 6 , J ) C C A INV{1 -18 , l - i a )=A«RAY K7H • . ' ' • • • • < • . / . _ FIGURE 10.35 (Cont'd) 2b2 FIUEO HDAMP FORTRAN PI CALLDATA TIME - SHAi C . . , . . . ...... . . . . . ...: \ CAuL SREVNKAINV. 18.LOC. 18.NX1 ) I DO 61 L = l. 18 . I .: .. .DO 61 J=1.18 ' -.-• ',. - ~-.- — —» HORK8(L,J+19)=0. I DO 61 K = l, 18 I 61 WORK8{L , J+19)=WORK8(L. J*19).*AI NV(L.K) 4.W.GRK 11 (K|19 , J.*391, I C ' ' . ' • . . I C WORK8< 1-18.20-37)=ARRAY(K7H(INV)*CH) . I C . . : . . • . . .. - - . . . , . , : ..-. - • _ — - ...» DO 62 L = 1, 6 » DO 62 J = l, 18 ' » WORK8CL,J+39J-0. » DO 62 K = l, 18 f 62 WOR^8( L ,J+39)=WORK8 CAUL W ILLY( •ARPY-K7H*.WORKS.72.60,7.2'4. I .18) » WRITE( IOUT,.8.1 ) ' , '. :....'...• , .- » 81 FORMAT( '0» . 'HUB DAMPING M A T R I X ' ) > CA-L W ILLY( 'ARRY-CH • , WORK 1 1 . 57, 5 7 .2 0 ,3 7 .40 ,57 ) f WRITE( IOUT, 82) ' ' ; . „ . . . . .': '..._ ..«: 82 F O R M A T ( ' 0 « » " H U B (ATTACHMENT POINT) STIFFNESS MATRIX* ) f CAUL W ILLYC 'ARY-K6HG1 .WORKS, 72, 60,1 .6*1 .18) f WR IT,E( IOUT, 83) . . • ^ _. _., •_._ ^.J 83 FO-RMAT( * 0 « , 'HUB (ATTACHMENT POINT) DAMPING MATRIX 1 ) * CA-L WILLY( 'ARRY-CHG' ,WORKS,72,60,1 .6.40,57) h _C._ •• , . . _...' ' - ' .-.._ *• C INSERT-TAPE-PUNCH ROUTINE HERE I- CAL.L PWR IT( WORK 1 1 , 57, 57,20. 37,40.57.6 .TCH.IRIT) t CA-L PWRIT( WORK 8. 72, 60,1 , 6, 40 ,57. 8 . TCHG . I R I T) ';..:. > c . ' ' ^ RETURN ^ E N D ' _ _ . . . . . _ »- FIGURE 10.36 LATE 263 PILE: LATE F O R T R A N P1 CALLDATA TIME - SHft! C C C C C C C C C C C C C C C C C C C •.C" C C C C C SUBROUTINE LATE(N,B,NUNEIG,TOL,QMEGA,NIX,A,ARRAY,NA,IR,COB,BIG) COHHON/PRINT/IOUT,IOIN COMMON /TERUP/BRAK SOLVE THE EIGENVALUE PROBLEM (A) (X) = (LAMBDA) (B) (X) WHERE- . ! A = THE NXN MASS MATRIX B = THE NXN STIFFNESS MATRIX Q = THE EIGENVALUES QMEGA = THE FREQUENCIES , ' BIG = THE EIGENVECTORS NUMEIG= NUMBER OF EIGENVALUES N = ORDER OF A,B TOL = TOLERANCE LIMIT 0.LE.TOL.LT.1E-9 NIX = B ARRAY POSITIVE-DEFINITE TEST =0 ARRAY OK =!+, OVERFLOW =- NOT POSITIVE DEFINITE ; ARRAY - EIGENVECTOR ARRAY OUTPUT NA - ORDER OF ARRAY IR - 'STARTING ROW OF EIGENVECTOR ARRAY COR - FREQ. SHIFT FACTOR ANTISYMMETRIC PROB. ONLY DIMENSION A{1) ,B(1) ,EIG(N,1) ,Q(210) DIMENSION ARRAY(NA,NA),QNEGA(NUNEIG) EQUIVALENCE (DOUBLE,Q (1).) DOUBLE PRECISION DOUBLE COMMON/WINTER/INDIC8 CONNON/BOND/M,L INDIC8=-1 L= 1 INDEX=(N*N*N)/2+1 INDEX1=INDEX-1 6 NIX=0 FORM CHOLESKY DECOMPOSITION OF B MATRIX CALL FUTILE(B,N,NIX) IF(NIX)2f7,7 2 WRITE(IOUT,U1) H FORMAT(»0',10X,"STIFFNESS ARRAY NOT POSITIVE DEFINITE',//,11X, *ENVALUES-HODAL VECTORS NOT COMPUTED1,//,11X,"CHECK INPUT DATA' BRAK=1. RETURN FOR,M A-MATRIX COLUMN INVERSES - STORE IN Q DO 10'I=1,N DO 11 11=1,1 IL=IKfI1 11 Q(I1)=B(IL) I I I I I I I I I I I I I I I I I I •BIG I I I L L I L L L L I L I L FIGURE 10.3.6? (Cbnt'd) PILE: LATE F O R T R A N P1 CALLDATA TINE - SHftE Q(I)=-1. I DO 20 J=1/I I 20 Q(J)=Q(J)/X , I 10 CALL DAGGER(A,N,Q,I ,BIG) I C ' '• l.j - > : , i ' • I C STORE (L) (A) (L-TRANSPOSB) IN A FORH SUITABLE FOR SYMBIG I C WHERE L IS THE INVERSE OP THE CHOLESKY HATRIX I C L CALL SHITCH(A,N) L CALL SYBEIG(A,N,1,NUHEIG,Q,TOL) , I C ! ' , ' ' .L C EIGENVALUES ARE IN Q L C : • ' ' ' •-• . . . - . / • L CALL1 SYHVEC(EIG,1,N,N) L DO 24 1=1,N I 2H CALL TRIEQ(B,EIG(1,I» I IF(IR.EQ.I) GO TO 26 I DO 25 I=1fNUMEIG L CHEC=1./Q(I)-COR L IP(CHEC.GE.O.) GO TO 25 L WRITE (IOUT,50) I,Q(I),CHEC ' L 50 FORMAT(«0»,7X,'NEGATIVE EIGENVALUE (ANTI-HODE)-SOBROOTINE L&TEV/1 L *X,'Q(«,I3,•)-',E1U.7,5X,«CHEC='fE1U.7) L CHEC=ABS(CHEC) L 25 QHEGA(I) =SQRT(CHEC) . L GO TO 29 , i L 26 DO 27 I=1fNUHBIG ' L 2.7 QHEGA (I)-SQRT (1./Q (I)) L 29 CONTINUE L DO 30 L = 1 , N L DO 30 J=1 ,N L JJ=J+IR-1 L 30 .ARRAY ( J J , L ) = E I G ( J , L ) L R E T U R N L END | ' . ' L FIGURE 10.37 FUTILE 265 N IV G L E 7 E L 1, HOD 1 POTILE DATE = 71363 13.17.28 SUBROUTINE PUTILE(A,N,NIX) DIMENSION A(1) DOUBLE PRECISION SUM EQUIVALENCE (SUN,SUM) K l i p 1 KK = 0 | DO 210 K = 1,N KK = KK * K IK = KK KK1 = KK - 1 IP (KK1) 60,50,60 50 ASSIGN 100 TO LEAP GO TO 70 60 ASSIGN 80 TO LEAP 70 Tl = K1 DO H*0 I = K,N SUM = -A (IK) GO TO LEAP, (80,100) 80 IJ = 11 . . _ SUM = SUM "V" A (IJ) *A (KJ) 90 IJ = IJ * 1 100 11 =11 +1 IP (I - K) 120,105,120 105 DENOH = -SUM IP (DENOH) 980,980,110 1 10 DE HO H = -S QRT ( DENO M ) A(IK) = -DENOH GO TO 130 120 A(IK) = SUM / DENOH 130 IK = IK * I CO N T I N UE K1 = Kf 210 CONTINUE NIX = 0 220 R E T U R N 980 NIX f -K GO_TO 220 "EHD ......... " FIGURE 10.38 DAGGER PILE: DAGGER FORTBAN P1 CALLDATA TIME - S ' ! ' I . . ' ' . ' SUBROUTINE DAGGER{A,H,P,KG,Q) DIMENSION A(1) ,P(1) ,QO) . . . . . . DOUBLE PRECISION SUM EQUIVALENCE (SUN,.SUM) K =j KG . -I. ' ' i i . ' - . K1 != K + 1 L=1 LL = 0 INDEX = 1 DO i130 I = 2,K1 LJ = INDEX SUM =0. j ' ', DO 90 J = 1 ,L SUM = SUM + A(LJ)*P(J) - ... - 90 LJ '= LJ + 1 IP (K - L) 100,120,100 100 LL != LL'.+ L LJ = LL '* L : DO 110 J = T,K SUM, = SPU t A(LJ)*P(J) i 110 LJ = LJ + J 120 Q fT-1) = SUN . INDEX = INDEX * L 130 L = T SUM = 0. DO '1UO T = 1,K SUM, = SUM * P(I) *Q(T) A (LL-H) = 0(T) : 140 LL j= LL + 1 A{LL) = SUV IF (1 - K) 150, 2:0, 150 150 DO 190 L = K1,M JL = LL+L-K ; . SUM, = 0. ' « . ' DO 180 J = 1,K > t SUM; = SUM * P(J)*A.(JL) » ,«. * • 180 JL=JL+1 LL;= LL + L - 1 . ... 190 A(LL) = SUM 200 CONTINUE R E T O H N . E N D FIGURE 10.39 SWITCH 267 IV G LEVEL 1, MOP 1 SWITCH SUBROUTINE SWITCH(A,H) DIMENSION A(1) If* IABS(M) IP (N - 2) 190,190,90 90 L - (N*(N+1)) / 2 h KEY * 1 LOCK = H/2 * 1 IP (H) 100,190,160 P1TJL?. 7 1 3 6 3 . _ 13.J7.36 IP (N - 3) 110,140,110 110 KKT ' 3 NKP = N - 1 IHAGB = L ". • INTO = L - 3 I_= 3 D0~130 K = 2,LOCK DO 120 IK = KKT,HKP ?_ r *(?*) A(IK) = AYlNTO) A(INTO) = X INTO = INTO - I T20 t = j TT- - — -— - -~= KKT = NKP * K I NKP = NKP * N - K IIHAGE = IMAGE -~K~ INTO = IMAGE 130 I_= K JW IP ' iKBYj 150,r90,150 150 KEY * 0 160 LOV2 * L / 2 ~" ~ ~~K~i"t - 2 ' : i' DO 170j I = 3,LOV2 X = A(H) ~ "" A (r)'"-""ii(K) ~: »(K) =, X 170 K = K. [• 1 "IP (KEY) 180 , 190, 180 180 KEY = 0 GO TO 100 END FIGURE 10/40 SYMEIG 268 IV G LEVJtL It MOD 1 SYWEIG DATE .71363 13.. 16*35 C C C C C C C C C C C C ..C'.. C C C C C C C C C C SUrjROUTJNE 5 Y M F I G < A , M , I N D E X ,NUMBP,R,TOL) ROUTINE .FOR EIGENVALUES AND ( CPT I O.N. ALLY.) £ IJGENV.E,CTQRS...OF... A REAL SYMMETRIC MATRIX STORED TRIANGULARLY IN CORE. THE HOUSEHOLDER REDUCTION TO TRIDIAGONAL FCR* (SUBRCUTINE TFORM) IS FOLLOWED BY A [BISECTION TECHNIQUE (SUBRCUTINE STURM) FOP THE ROOTS AN.G THEN .IN-... VERSF ITERATION (SUBROUTINE TRIVEC) FOR THE VECTORS. MEANING OF THE PARAMETERS. THE TRIANGULAR ARRAY CCNTAI N ING THE MATRIX. DURING EXEC CUTION THE ARRAY IS CHANGED. - M INDEX NUMBR R TOL ORDER OF A. - INDEX OF FIRST EIGENVALUE RECUIRED. -.NUMBER OF EIGENVALUES REQUIRED. - RFAL ARRAY FOR THE RCQTS AND WORK ING STORAGE. R MUST CON- TAIN AT LEAST Q*t> WORDS AND START ON A.'P..B.C. ..BJOM.NLCAR.Y...* ... - TOLERANCE. IF 8 IS THE KAG. OF LARGEST ROOT, THE ROOTS WILL USUALLY HAVE ERRCRS AS LARGE A3 G*TOV* - .-•-*,- t.. DIMENSION A(1),R(1) L D = 1 t M . . . , . _ LO = LD * M LS = LO + M LP. = LS + M .' . ' • . . . • • . ' . .' ' LO = LP * M LR = LQ + M 'CALL TFORM( A ,M,R(LD) » R ( L C ) » R ( L S ) .R(LP) I ]: ,:. J EPS = AMIN1 (TOL,1 .E-3) " * *•• CALL STURM( M« I NDE X , NUMBR ,R(LO) ,R(LCI ,R . R(LP ) , R, EPS ) RETURN ENTRY S Y M V F C ( X . L O W t K O U N T t V I D ) DIMENSION X (MID . l ) CALL QSVEC ( A ,R ( LO > , R ( LC ) ,R(LP) , R(LQ) ,R (LR» ,P( LS),.M) ROOT - R(LS-1 ) EPS = ROOT * 3.E-7 K = LOW ' • , . . DO 200 I = 1.KOUNT ROOT = AMIN1(ROOT-EPS.R(K)) CALL QWIEL(ROOT,X(1 .1 ) ) . „„. 200 ;K = K + 1 RETURN END ._ FIGURE 10.41 TRIEQ 269 V G LEVEL 1, MOD 1 TPIEQ DATE = 71363 13.17.45 100 105 110 120 125 130 140 145 15C 160 170 SUBROUTINE T R I E O ( A . Y ) REAL A ( 1 ) » Y ( 1 ) COMMON /WINTER/INDIC8 COMMON /BOND/ M.L DOUBLE PRECISION SUM EQUIVALENCE (SUN.SLM) LI = L LM1 = LI - 1 MM1 = M - 1 I F ( I N D I C a ) 1 3 0 , 1 0 0 . I O C Y ( L 1 > = Y (L1) / A(L1) IF ( M M 1 ) 105,125,105 I 1 = LI DO 120 I = L I , M M 1 £1 = I 1 + I SUM = - Y ( I * 1 ) IJ = II 'on 110 J = LI,i SUM = SUM 4- A ( I J ) * Y ( J ) , U = I J + 1 II = IJ ' Y( 1 + 1 ) = -SUM / A (1.1 ) ' IF ( INDIC8) 17C.140.17C | II = ( M*M + Ml / 2 <; - I = M Y( I) = Yd ) / A<1 I ) I I =11 - I 1 = 1-1 IF (I - LI) 170,1 5 C.I 5 C 'SUN .= . -'Y( !•»• i) IJ = I I '+ LI DO . 160 J = LI, I IY ( J ) = Y(J ) + SUN*A(IJ) IJ = IJ * 1 'GO TO 14 5 .RETURN END FIGURE 10. U2TFORM 27 ° I IV G LEVEL 1, MOD 1 TFORH PATB_= 71363 1.3.16.115 S U B H O U T I N E TFORH ( A, H,D, 0, S, P) DINBNSION A ( 1 ) ,D(1) ,0(1), S(1) BL = 0. ......... " ....... • BO = 0. OLD =0 . 1,H1 K1K1 = 1 H1 = N - 1 DO 230 K = KP1 = K + 1 KK * K1K1 KKP1 = KK * 1 HK = H - K KH = KK + HK ....... KfK1 = KH + 1 son =o. DO 100 KJ = KKP1,KH 100' SUM = SUN + A(KJ)*A(KJ) S(K) = SUM _ _ 8A"D = OLD T'HHO ...... BL = A M I N 1 ( B L , D ( K ) -RAD) BO = A H A X l ( B U , : p ( K ) + 8 A D ) l¥ (K - NIT 120,230,230 120 OLD = ;BHO 0(K) = RHO IP (SON) 150,230,150 150 A(KKP1) = A(KKP1) - RHO RHO = 1. / (RHO*A(KKP1)) AL?K)_" RHO IJ = KK DO 160 J = KP1,H IJ = IJ * 1 0(J) = A(IJ) 160 D(J) = 0. II = K1JS1 DO 190 I = KP1,H * II = II * HI NI = HI - 1 II 170 X = 0(1) DO 180 J = I,N1 I j = t j * 1 D(J*1) = D(J»1) * A(IJ)*X 180 D(I) = D(I) * A(IJ)*0(J*1) "1"g"CT D(I) = D (I) * RHO SON = 0. DO 200 I = KP1,H 200 SON » SOH * D(I)*0(I) FIGURE 10.1*2 (Cont'd) 271 I IV G tEVEL 1, HOD 1 TPOBH TAO = RHO * SDH * . DO 210 I = KP1 ,H 210 D(I) = D(I) * TAO*0 II = K 1 K 1 HI =, HK >" DO 220 I = K P 1 , N RHO = D( I ) TAO = 0(1) TJ * II II = II * HI HI = HI - 1 5 (I) DO 220 J = I,H A(I J ) = A(IJ) » RHO*0(J) 220 IJ = IJ * 1 230 b(K*1) = A ( K 1 K 1 ) DATE = 71363 13.16.45 * TAO*D(J) 0(N1) = A(KKP1) 0(H) * AMIN1 (BLrD(H)-RHO) S ( N) = A H A X 1 ( BO , D ( N) + R HO) RBTORN END FIGURE 10.1*3 STRUM 2?2 I.V G LEVEL It MOD 1 STURM DATE = 71363 13.16*5.6 SUBROUTINE STURM »S IGMA(1 ) ,.PFFO( I) DATA HALF X ,5/ BL = OFFO(N) BU = SECCN): | LIM2 = LIMI + NUMB - 1 j CALL PREPCN.D , SEC. ROOT ,LCPO) N1 = N - I _ ... IF (Nl) 16*200.200 200 TOL = A M A X 1 (-BL.BU) OFFD(N) = TOL j i. TOL = TOL * A M A X 1 ( 1 .E-15 .EPSI i DO 2 I = LI.M1 ,LI M2 SIGMA( I ) = BL 2 PFFD(I) - 3U LORD = 0 L = LIMI - 1 RUTE = 1.E20 GO TO 3 30.0 DO ^»00 I = K,L 400 S I G M A ( I ) = ROOT 3 K = L * I IF , (R04 , RF.4) Nl = N - 1 .'RETURN " •> 1 FNTMY OF T ' RDO = ROOT. LOW '= .0 . L A WD - 0 100 R02 = 0.00 RD4 = l .DO I = L O V K . N 1 1) - RDO - RD2 IF (RD4) 120,140,110 1 10 LAWO = l.AWD *• 1 120 RE2 = SEC(I + 1) / RFA 130 LORD •= LAWD RETURN 1AO LAWD = l.AWD * 1 IF ( R E P ) 150,160,150 150 I. = I * 1 160 LOW = 1 + 1 I F ( L O W - N l ) 1 0 0 . 1 0 0 , 1 3 0 END DO 120 R D 4 = D ( I PILE: QSVEC FIGURE 10.U5 QSVEC ' FORTRAN PI CALLDATA . TIME - SH SUBROUTINE QSVEC (A,D,OPPD,P, Q,R, S,N) C SYMMETRIC MATRIX EIGENVECTOR CALCULATION. C GIVEN THE ENTRIES (D AND OFPD) OF THE HOUSEHOLDER TRI-DIAGONAL PORH ; C B OF A REAL SYHMETRIC MATRIX A, AND GIVEN A GOOD APPROXIMATE ROOT OF C B (AND A) THIS FORTRAN «» SUBROUTINE COMPUTES A UNIT EIGENVECTOR X C OF B, THEN TRANSFORMS IT TO A UNIT VECTOR OF A, USING THE VECTORS ' c WSTORED IN THE A ARRAY. DIMENSION A(1) ,D(1) ,OFFD(1) , P ( 1) ,Q(1) ,R(1) ,S(1),X(1) DOUBLE PRECISION SUM ! j 1 I COMMON /INFO/ SUM,N,IX,IA C C PART 1. PRELIMINARIES. C I I IX = |1 IA = Jl N1 = N - 1 N2 = N - 2 RETURN ENTRY QWIEL(ROOT,X) ASSIGN 170 TO KOUNT TOL = 0. I . ' 00 100 I - I, N P(I) = D(I) '- ROOT Q(T) = OPFD(I) ' R(I) =0. TOL = AHAX1 (TOL,ABS (D(I») 100 X (I) = RDM (X) + . 1 TOL = (TOL * 1.E-15) * 1.E-15 C . ' i C PART 2. MATRIX DECOMPOSITION. C DO 150 I = 1,N1 T = ABS (P(I)) U = ABS (OFFD(I)) IF (T + 0 - TOL) 110,120,120 110 P(I) = TOL T = P ( I ) 120 IP (T - U) 130, 1UO, 1UO 130 S (I) = P ( I ) /OFFD(I ) I F ( A B S ( S ( I ) ) .LT.TOL) S ( I ) = T O L S{I) = O R ( S ( I ) , 1) TEMP = Q( I ) P(I) = OFFD(I) P(I*1) = TEMP - S (I)*Q(I) GO TO 150 1UO S (I) = OFFD(I)/P(I) S(I) = AND(S(I),-2) P (1*1) = P(I+1) - S 150 CONTINUE IF (ABS(P(N)) .LT. TOL) P (N) = TOL GO TO 210 C I FIGURE 10.1*5 (Cont'd) 275 FILE: QSVEC FORTRAN P1 CALLDATA -T-IHE-,* C PART 3. RIGHT SIDE MODIFICATION. C ' . ' 170 ASSIGN 330 TO KOUNT DO 200 I = 1,N1 TEHP = AND(S(I) ,1) IF (TEMP) 180,190,180 180 T * X(I)| X(I) = X{I+1) , i X (1*1) = T - S(I) *X(I) GO TO 200 190 X(I*1) = X(I+1) - S(I)*X(I) 200 CONTINUE C C , PART 4. TRIANGULAR SYSTEM SOLUTION. C ( • • 210 X(N) = X(N)/P(N) I X(N1) = (X(N1) - Q(N1)*X(H» / P(N1) _ ! DO 220 I = 2,N1 K = N - I 220 X(K) = (X(K) - Q(K) *X(K*1) - R(K) *X(K*2)) / P (K) C C PART 5. SCALING TO UNIT VECTOR. ...C . • ... . . . . . ' • - 230 SUM = O.DO fl = N SCALAR = SQRT(DOTPRO(X,X)) i DO 250 I = 1,N 250 X(I) = X(I) /SCALAR . GO TO KOUNT, (170,330,370) C PART 6. TRANSFORMATION BY ORTHOGONAL MATRICES. C.., 330 L = (N*(N*1))/2 - H DO 360 I = 1,N2 , ;'..HI = N - I SUM = O.DO M = I + 1 SCALAR = A(L-1) * DOTPHO (X(NI) , A (L)) IJ = L DO 350 J = NI,N _ _._.X(J) = X(J) •«• SCALAR*A(IJ) , 350 IJ = IJ * 1 360 L = L - I - 3 ASSIGN 370 TO KOUNT GO TO 230 370 RETURN ........ BflD FIGURE 10.46 AUD 2?6 FILED AND . FORTRAN Pi ; CALLDATA TIME - FUNCT ION ANO( X, Y) LOGICAL UNDtJA.NEIN REAL NO EOUIVALENCEOR=fc I RETURN END 100 110 120 FIGURE 10.U? DOTPRO 277 FUNCTION DOTPRO(X,Y) DIMENSION X(1),Y(1) DOUBLE PRECISION S,DOTPRO COMMON /INFO/ S,N,IX,IY IF (N) 120,120,100 JX JY 00 S = = 1 = 1 110 S JX = JX JY = JY DOTPRO = RETURN END J = 1,N X=C1 A 16)=-SI A ( 8 ) = S 1 ACi)=Cl GO TO 12 10 DO II 1=1,9 11 A ( I ) ~ 0 * A ( 11 = 1 . A { 5 ) =- 1 . A ( 9) =- 1 . 12 RETURN END 5*. FIGURE 10.,51 ANORM' 28l IV G LEVEL.. .1., HOD 1 A N O R M DATE = 7 1 3 6 3 13.21.1U S U B R O U T I N E A N O R M ( A O U T , A I M , N A , M B , N ) D I M E N S I O N A O U T ( N B , N B ) , A I N ( N A , N A ) 00 10 L = 1 , N A M X = 0 . DO 5. J»1 VN , . ' i 5 AHX = & f 1 A X 1 ( A B S ( A I N ( J , L ) ) , A M X ) I F ( A H X . E Q . O . ) A H X = 1 . DO 10 J = 1 , N 10 A O U T ( J , L ) = A I N ( J , L ) / A H X R E T U R N END , pflp i FIGURE 10.52 PWRIT i N IV G LEVEL 1, HOD 1 PWHIT DATE = 71363 1.3.21..46. S U B R O U T I N E P W R I T ( P , N R , N C , S R , B R , S C , E C , I V , T O L , I R I T ) R E A L P ( N R , N C ) D I M E N S I O N K C ( 5 ) , K A ( 5 ) INTEGER SR,ER,SC,EC IP(IRIT.EQ.O) R E T U R N i ' i J=SC noo=i 5 N=0 DO 30 L = S R , E R I F ( A B S ( P ( L , J ) ) .LT.TOL) GO TO 16 K C ( N ) = L I K A ( N ) = L - ( S R - 1 ) I P ( N . E Q . 5 ) GO TO 18 j_ ' " T(5i YF(L.EQ. EH. A N D . N . N E . O ) G O T O 18 " l~" GO TO 30 18 CONTINUE j W R I T E ( I R I T , 2 5 ) M O D , I V , (KA (I) , P (KC (I) , J) ,1= 1, N) ' 25 P O R H A T ( I 3 , T 7 9 , I 2 , T U , 5 ( I 3 , E 1 2 . 5 ) ) N=0 30 CONTINUE M O D = H O D + 1 IF ( J .LE .EC) GO TO 5 R E T U R N ( END 283 J. X . N Dl D2 £ I ,1 v , /6s z X-, Xg, X_ Fixed Frame in rotor ring. Xo COORDINATE is aligned with the #1 ARM OF (T or Y) ROTOR CONFIGURATION Fixed FRAME IN END module GEOMETRICAL DATA MEASURED WITH RESPECT TO X.^ Xg, X COORDINATES Location Point l N Dl D2 V D3 B Definition Attachment point of rotor arm to rotor ring (X, coord in feet) Mass center location of first mass module in rotor arm (X_ coord in feet) Mass center location of second mass module in rotor (X, coord in feet) i Connection point between rotor arm and associated end module (X,, coord in feet) Mass center location of end module (X- coord in feet) Fixed point in hub about which rotor rotates ID 60 h 7 13 10 Not utilized in program FIGURE 10.33 ROTOR ARM GEOMETRICAL DATA INPUT PROPERTIES 284 TABLE 10.6 POINT LOCATION Dl D2 D3 V Dl D2 COORDINATE REFERENCE X Y Z X Y Z X Y Z X Y Z XX YX ZX XY YY ZY XZ YZ ZZ XX YX ZX XY YY ZY ROTOR INPUT DATA GUIDE ID . NUMBER 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 DATA UNITS FT FT FT 1 FT- < Slug-Ft2 Slug-Ft2 i 1 DEFINITION First mass center location (see Figure 10.53) ' Second mass center location (see Figure 10.53) j End module location (see Figure 10.53) Rotor Arm - Associated end module Intersection - Location (see Figure 10.53) Number of arms on rotor 2. - T Configuration 3. - Y Configuration (l) Inertia property data - first mass center location (Coordinate frame nominally aligned to XI, X2, X3 (rotor ring) coordinates - Figure 10.53 • (l) Inertia property data - second mass center location (Coordinate frame nominally aligned to XI, X2, X3 (rotor ring) coordinates - Figure 10.53 285 POINT LOCATION D2 D3 B «•» — — — — -- — — COORDINATE REFERENCE XZ YZ , ' ZZ XX , XX zx XY YY ZY XZ YZ ZZ XX YX ZX XY YY ZY 1 XZ YZ ZZ «>« — — — — ID NUMBER 30 31 32 33 3^ 35 36 37 38 39 ho ki ^3 kk ^5 k6 ^9 50 51 52 51 52 53 5^ 55 56 57 • i DATA UNITS DEFINITION 2 Slug-Ft (i) Inertia property data - second mass center locationi ' ii i (Coordinate frame nominally alligned to XI, X2, X3 (rotor ring) coordinates - Figure 10.53 2Slug-Ft (i) Inertia property data - end module (Coordinate frame nominally aligned to XI, X2, X3 (rotor ring) coordinates - Figure 10.53 i : 2 Slug-Ft (i) Rotor ring inertia property data P In Cross-sectional area of beam (rotor arm and/or hub arm) In Area moment of inertia Lbs/In2 Modulus of rigidity (G) I Lbs/In Modulus of elasticity (E) In Polar moment of inertia CPS „ «..«!. Utilized in ant i- symmetricFreQ shift Elgen-value calculations - value set. to 5- Tolerance factor - Utilized in modal matrix calculations - value set to 0. Limits 0 to l.E-9 286 POINT LOCATION N Dl D2 D3 — -- — — I — 1 COORDINATE! IDREFERENCE NUMBERX 58 Y 59 z 60 61 62 ! 63 t ' 6k 67 68 100 ' 90 91 93 i ! 1 DATA UNITS FT Slugs Slugs Slugs Rad/Sec In-Lbs/Rad — — DEFINITION Rotor ring - rotor arm intersection location \ Point Dl mass Point D2 mass End module mass Rotor spin rate Bearing spring constant Structural damping ratio Program control - 0. - Calculate rotor structural i data only 1> - Calculate hub data onlyi 2. - Calculate rotor and Hub structural data - Program control - punch output data [ 0. - Do not punch data 7. - Punch data i Note: 7« is IBM 360 system output i device for punch output ] i Punch output data lower (magnitude) : limit ; control modal matrix output i Punch output data lower (magnitude) limit; control stiffness matrix i output Punch output data lower (magnitude) ; limit ; control damping matrix output ! I. 1 i i i . i 287 TABLE 10.7 SAMPLE ROTOR INPUT DATA SETUP CARD 1 2 3 1* 5 6 7 8 9 10 11 12 13 li* 15 16 17 18 19 20 21 22 23 2k 25 26 27 28 29 30 31 32 FORMATTED DATA INPUT 1 ROTOR T CONF. k 23.3 7 57.3 10 105.23 13 7^.25 1U 2. 15 : 22731. 3 19 22731-3 23 2083 . 2k 22731-3 28 22731.3 32 2083 . 33 2031+929.53 37 485759.91 1+1 16M*085.22 1*2 11296. 1+6 '15586. 50 15586. 51 15.823 52 131*1*6. 53 1+.0385 51+ 1.05 55 26892. 56 5. 60 6.25 6l 225.16 62 225.16 63 3881.99 61+ .2618 65 '3.11+159 67 2.1156 68 ' '.01 RUN NO. 1 XRD1(3) XRD2(3) XRD3(3) XRP(3) i P DI1(1,1) DI1(2,2) DI1(3,3) DI2(1,1) , DI2(2,2) DI2(3,3) ' DI3(1,1) ; DI3(2,2) DI3(3.3) RRIB(1,1) RRIB(2,2) HRIB(3,3) AREA AI E+6GBE E+7BE PMJ FREQ ' XRN(3) DML DM2 DM3 PR ANG(2)* E+9BSK1 ROTOR DAMPING FACTOR TYPE i Program Control Card R 0 T 0 R, D A ; T ' . A I N P. U T *Note: Program data card not required - angle data for rotor arms stored in program 288 CARD 33 3U 35 36 37 FORMATTED DATA INPUT 90 7. 91 1. 93, 10. 9^ 10. IE IT E-5TREIG TSMAT TCTYA BLANK CARD TYPE PUNCH OUTPUT CONTROL MODAL MATRIX TOL. LIMIT STIFFNESS MATRIX TOL. LIMIT , DAMPING MATRIX TOL. LIMIT PROGRAM CONTROL CARD 289 ' V Y3 Location Point Coordinate frame fixed in r.igid section of hub at A, the axis is parallel to the nominal spin axis of hub. Geometrical data is measured with respect to YI, Y_, Y_ coordinates. Definition Card Data Input Format - 15 I.D. Number B G Mass center of rigid section of hub Fixed Point in hub about which rotor rotates ! Location of connection between flexible and rigid hub sections Location - of first module mass center Location - of second module mass center i Location of connection between hub second module and end module ' i Location - of end module Data not required for program Data not required for program 107 110 116 113 FIGURE 10.5U HUB GEOMETRICAL DATA INPUT PROPERTIES 290 TABLE 10.8 POINT i COORDINATE LOCATION REFERENCE G Y, Y2 ; Y3, ci I Yi Y2 ! ' Y3 C Y2 j , 1 Y2 Y3 c3 YI /Y3 E i ' Y, ) , -1- Y2 Cl C2 Y3 XX YX ZX XY 1 YY 'ZY xz YZ ZZ XX YX ZX XY YY ZY XZ YZ ZZ HUB INPUT DATA GUIDE ID NUMBER lOU 105 106 107 108 109 110 111 112 113 115 116 117 118 119 120 121 122 123 121* 125 126 127 128 129 130 131 132 133 131* DATA UNITS 1 1 Ft Ft ! Ft Ft Ft Slug-Ft2 Slug-Ft2 , i DEFINITION i I Intersection of hub flexible and rigid sections First mass center location I Second mass center locationi i End mass center location 1 i i ' Hub second module - end module inter- 1 section 1 (l)i Inertia property data - 1 Coordinate frame nominally i Yx, Y2, Y (hub rigid body) Figure 10.5!* ' i (l)i Inertia property data - Coordinate frame nominally Yr . Yg, Y3 (hub rigid body) I 135 1 136 j first mass aligned to coordinates second mass aligned to coordinates POINT LOCATION C3 cl C2 C3 — — — — -- COORDINATE REFERENCE XX YX . zx XY YY ZY XZ S YZ ZZ — — _ _ ;; — — — -- — — 1 • • ID DATA NUMBER UNITS 137 Slug-Ft2 138 139 140 142 !' 144 145 ; 146 Slugs 147 Slugs 148 Slugs 149 51 In2 52 In* 53 Lbs/In2 54 Lbs/In2 55 In* 57 : 100 i90 -- ;i 291 DEFINITION (i) Inertia property data - end mass Coordinate frame nominally aligned to Y.J Yg, Y (hub rigid body) coordinates Point Cl mass Point C2 mass Point C3 mass Structural damping ratio Cross-sectional area of beam (hub arm and/ or Are Mod Mod Pol rotor arm) • a moment of inertia ulus of rigidity (G) ulus of elasticity (E) ar moment of inertia Tolerance factor-utilized in Bigen value matrix calculations - value set to 0. Limits 0 to 1. E-9 Program control 3. - Calculate only rotor structural data 1. - Calculate only hub data 2. - Calculate both rotor and hub Program control - punch output data 0. - Do not punch data 7. - Punch data Note: 7. .is IBM 3°0 system output device for punch output. t 292 POINT LOCATION COORDINATE REFERENCE ID NUMBER DATA UNITS DEFINITION 92 95 96 97 98 Punch output data lower magnitude limit; control modal matrix output Punch output data lower magnitude limit; control hub stiffness matrix Punch output data lower magnitude limit; control hub damping matrix Punch output data lower magnitude limit; control hub (attachment point) stiffness matrix | Punch output data lower magnitude limit; control hub (attachment point) damping matrix 293 TABLE 10.9 SAMPLE HUB INPUT DATA SETUP t CARD FORMATTED DATA 1 ! ' 1 1HUB T CONF. RUN 2 100 1. 3 10k -27.8 4 10? -63. 5 110 -122. 6 113 ; -175.1 7 116 -156.5 8 119 1735329.45 9 123 2012994.74 10 127 292228.82 11 128 1733430.57 12 132 1829283.28 13 136 98517.36 14 137 16^4085.2 15 141 2031+929.53 16 145 485759. 91 17 146 741.8022 18 147 536.5137 19 148 3881.9876 20 149 .01 21 51 15.823 22 52 13446. 23 53 4.0385 24 54 1.05 25 55 26892. 26 .'90 7. 27 92 l. 28 95 10. 29 96 10. 30 97 10. 31 98 10. i 32 BLANK CARD 33 BLANK CARD i INPUT NO. 1 HUB CALCULATIONS YHG(l) ' YHCl(l) YHC2(l) YHC3(D YHE(l) CI1(1,1) CI1(2,2) CI1(3,3) CI2(1,1) CI2(2,2) CI2(3,3) CI3(1,D CI3(2,2) CI3(3,3) CM1 CM2 CM3 DAMPH AREA AI E+6GBE E+7BE PMJ ZRIT E-5TPHHF TK7H TCH TK6HG TCHG TYPE PROGRAM CONTROL CARD H U B I N P U , T D A T A PUNCH OUTPUT CONTROL MODAL MATRIX TOL. LIMIT STIFFNESS MATRIX TOL. LIMIT DAMPING MATRIX TOL. LIMIT ATT. PT. STIF. MAT. TOL. LIMIT ATT. PT. DAMP MAT. TOL. LIMIT PROGRAM CONTROL CARD PROGRAM CONTROL CARD 10.2.3 Sample Run Printed output of Rotor and Hub "T" structural data is illustrated in i , ! • ' ' i ' 'Tables 10.10 and 10.11, respectively. At the start of each printed output the RUN CONTROL CARD along with the BASIC input data are. printed. Rotor Structural Data output will be printed in the following order: o Input Card Data o Stiffness Matrix : o Modes (rad/sec) i o Modal Matrix o Damping Matrix The, order of printing hub structural data output is as follows: • . I o Input Card Data o Hub Stiffness Matrix I o Hub Damping Matrix o Hub (attachment point) Stiffness Matrix o Hub (attachment point) damping Matrix o Hub Modes (rad/sec) o Hub Modal Matrix ' The modes are printed in ascending order of; frequency (in radians per second) with printed titles designating symmetric and antisymmetric modes. A mode shape, defined by the Modal Matrix columns, is linked-1 with a corresponding frequency whose Eigen value number is identical with the column number. i The modal, stiffness, and damping matricies will be punched on cards > i in the proper format for direct input to the Time History Program. The last card of each data set of pilnched output will only contain a zero in column 80.i Structural data that is less than the tolerance limit specified by the basic input data will be omitted from the punched output. ! I Ia o !' OS ! O ; «c c Z Q: —«- o E^'M »a xiUjtn t-. 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O:I , oi UJ lOin O I NT N f l i C M N t i C M . i . o o O!C I I U r- m CM i O I I U J I U l U* i «M f* •t'.-f CM in|— • coin o *>cr r-CM — ca rt . ,M . - eg 0 o 1 i IU UJ oo ro NO tO n o n to 0 o 1 II in uJ O|CD - • ,'tn on njm u) in O O O IO III I UJ Ul UJ IUJ sr c- eg — !» — in cT1 oo *o cr> — -a «!* ni CM P- oo vO ro in — —0:0 o o 1 , 1 I IUJ tu u u O ,CM 0s 'N(fl :r*". if) .ro NJ- « NT CM O fO r- •o • jroNT co o o o I o I UJ o i co o AJ cv • r— — >o ro .eg » '«T in NO in.in ^.— ej>]eg 10 in CM »o • • NO — « 1 I O Oi : o CM o o UJ UJ CJ* o* ov o f**T • o- -o o o o o o 1 o 0,0 O'O ! . o I I I CM 1 O 1 UJ ro CM ' 10 *o . * ro . \Q IM ^*'in NQ £ — 1* .INT r- o — • • o iCM in o o ' , i CM O eg — o • ejr< « r o c M C O .N- *t \ i 1 O O 1 1 O i ! i c o t M c g j . r ^ - c g e gi : ! ; oi . 1 ; it 00 \0 ItN- CM ' 0> I ' CM co — .in oi • 1 ; f) . JIlfN. NiO ] ^J- O* >Oi • «|o j NO in o o 1 1 o 10 o1 i . 1 I UJ LJ! . UJJUJ UJ 0* ro Ir— CMiO O'O — 0 P> C U1 . . O N O ' - N t ; o r^ 17s NT 00 ' 03 CO ^* — • W * ro — CJN NO *O * CO | CQ — 1*1 CM : O r*~ CM IO rO " CM CM 0 1 0 0 1 0 M to *C — NO o o c. o oI 1 f til UJ UJ UJ UJ NT ** tO ^ ^* NO 7> f- ro o co m "-QO o m oo in •T *o tn ^ ^ ro J- 00 — NT CM - o o o o o o 1 1 — — to O O *J 1 1 1j.i uj UJ ro oo . eg — eg . — — NO « \o •» — 1« CM o o- • in ro O l O O O I O I : Oi i i ; i . i ' — cv — — NCI— n i NT o o u o oio o o1 i I I i i I : i uj uj uJ uj UJ uj iu uJ t** * * C M ' T * * O N ^ O > I ' O C ^ ' N C ro r * - N O r * - r O N Q — NO— •— 'NO NO oo NO r- o CM — ^ NO N eg uVN i eg cu NT NO c? o n NT in ro n in — — — CM in •» N T - — r« v ~,*- 1 « — C i O I O O O . U O i l O ' O i O 1 1 1 1 : 1 ; . 1 C' 0 O 0.0 I i i i i ; i : U . Ul .UJ UJIUJ . ! 1 CM 01 UJ NO * o o s o> —in eg o o 1 eu CM O 01 1 • O;N NO'— ro rO sU :N NO ^ CM o> — 1— — leg ro i 0,0 cio o i , l i ' i — ; NT 'eg - NO !o o1 1 — in . 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IO'« ;ro ro mice o CM 00 to N^- ,— CU (^ — O O'lO — oo o r» « ; C M IT 00 *T in o, ro 10 NT oj in i I I 'UJ LJ OJJ--O CM .!'O in 0-! CM •!) in;"J CM — «' o a: i n ' ro 10 tn — eg O Ci O C-' O t > I i I I I I I uj ; uj uJ uj uj u'f- Mm CM NT NQiin e>i •'. -T co NT -o r-fo o N} eg NO in — NO r-' - - co - C' to co ro o- NO CM in —« r* in o !ci O i I I I } O O O I I • O I (.. o uI J 3 o u o a u i > u. • tt < I 'O o I •T CM CM o o . •j- in CM ro NO . — * INO °. I I , ro — 0.0 UJ 'UJ c.,tr. NO;coOlin NTiCO — . . o o Z I I -I G •u oio o : i i O I 'O I I CM O'[ I UVtO 4T • I^'NO DID IO NO O'm cj^'in ooiin o- • 11 • « — CM CM ' — • i— —.— «,CO o I i - «.-: i I ' fv''- • ; • o o 1 . 1 U UJ ^* ! fk * Q) fl 1 ^ j «I -T in'.cM oo -in eg en « ro CM NO K CM S l>- C71 ro I NO' NO . — • -T —'. . in cc • • ;o o o • i 0*0 .— o ;— o i n c r CM CM'OO Jeg IO'CM : r* C* ro \o in •NT'OO o NT'O ro —iCM 0 o o oJj v\ 1 ! UJi roioo — \ . <»i- NO O ml • «'o • In roico ol n o i . O OlO O'O I : iI UJ c/* .10 i. . O •» O r- N in in -m -ICMin! • n in — I— .: « CD . ooiro _ roic7< cr o< o «io oir- CM • .iU1 ro -T . —Ol ,1fO ' 1 O,O ~ - I N.,0 , aj u; o — .'in CM '. „ 4- _|m CM t •C oo eu!— f«- . — .to |o — CM n «r IT'NO r-:co ' i ' ! ' '• i . ii i j • i : i , i i i ' I i : ! ' i ! ! •'r ' 1 1 ! 1 ' -C: — PJI i i ' - 1 ! i oi o eli 1 : i ' I ' l l . [ ' i ; ' UJ UJ UJ i * ' *+ O -O • ") If i ' I i in oj N in N — 1 . PV oo oj >fr oj <*• i ! i C rj o O pl O «- O h- o o o o o o f-- o o o o ol o | h- in 04 c\j TO -^ •-' O O O O O O O O O O O | O | . O O O O.O: ' i - 1 i •r- ' GO c\j: I 1 O ( ' O ; i O 1 1UJ UJ [ t i l N. r- co c o <\i — j • r» o ' ; ** * in ^ \tj j ^ ••4" r ' T*" \ • \ ' ^ ™^ *O C*A P^ -i" . r*- ; \f\ ! | -^.00 o n r-icr* *Q O- >J" O O O1 U"^ * O O O O O O O IsO *O' ^ ** * * * ^ * ° O O O C O O O O O O O O O i l " 1 g : ' f*- re ; J oj1 Oj o o; o o o CJ i ' 1 . 1 • ! *— ^ ' UJ OJ • ' LLJ • UJ _-i r*-, N- i • ' \o in u^ c^ o o^ 3 r*-i c1 t , n in «-( \ j • >o • ^ h" ' j ••* . ^ ^ *^ 30 *^ O ' . r - t r \ i O ; f * » C T ' i o o > f " mo o -Jo ir> o T, o o o o o o 1 tr» «-" • in . r^ ' • 3 o o o o o o o o o o o o oo o I i gj CC I 1 sO OJ PI ^ " o • i o o o o 1 1 1 1 ; U i LU LU UJ UJ «c • o* c\j *4~ t o o o o j o o - j - o o o o o o ^ f\J o — -. -< • O O O O O O O O O O O O O O O O O ' I i i i i ! \O N- i "tf* ftj ' t> 0 U 0 : ! I l l UJ UJ I i ; UJ UJ ! *O ^*~' i CO Pj CD P5 r^ -^ ^~ ^ j >O W* CO ^- * O x c r < r ! ' ! P*- . — . ^- o PI 1-1 « O N - i (M CC O •« IP i CD a; n o T' oj o *r o ; P*. o o o o o i o v n oo r*- tn — cr^ - n < O O « ' O O O O C 1 O O O : O u O O | O | , ' O a l i . H i i o r~ ^ t*) pj u) o o • y i o o i " i i •1.1 L.' uJ c ui a; Z U ^ ' 2 O J « ^ P I O U » O t . 'y. i*- ^ • < i— t j in (*— u* o r*- • U. ^ ^J" Q. ' ^ • t - f l ^ * n . o « - > o r ^ y r ^ r ) O s T r — : o ^ H OJ C- <*J c^ C'. O >J" O.I C f (' O O O < • O.! > } • < > — ' - P) ! U"1 O O O O O O O O O O O O O ) O O O i — 1 ' ~ ; 1 i- r- >~ oj n -J- — 1 — Z ° Z O O O Olo "- •" 1 1 1 1 1 1 C UJ O uj UJ U) LJJUJ j • in o r*- m i -c I - U — O ' K O ^ ^ t n o ^ ' ^ zi |~- i 7 r -4- c> a in (M l " •j o — o o o o o o >o « o o c1 o o o m u '*- o « oj ^ oo!^ I i o o o o c - o i o o o o o ' o I >- i 1 0 0 0 0 : 0 u >- z i > i' i-' u. 2 i < u. ? a * < u. 7 : 1 - < D < T ) t - < " 5 H -J -I »- J < Q i < Q «. U O _ u c » - > ( v ; r n « i n M 3 » — M n « m vo CD » — oo n >i- in ^) 3 0 o , o oI o; tr x a ^ n r- o • r- !•» C^ OJ OO I>.,OJ OJ •^ • ' (\J • O I O 1 •*} 0 1 . UJ. tr •* oj f^ OJ • •o c- cc ^ n ^ ^ r^ ^ oj OJ "" o.o "" | t \£J C\J (\J 00 0 ill 1 UJ UJ Ui OJ r- IT -^ r- co 0s \ OJ 3s ! OJ •O — : O '. O 1 1 OJ : to V oin — in co r- jj 0 Oi o UJ O 30 O " o • r- n . vrt • • u t o o 1 n o (VIc> i i LLli UJ Ift ty oj f^ V** O O OJ o • o^j- oj m n c> t\i - 1 0 (VI OJ o o i i IUJ UJ n'oj in • 00 03 O n c* N <* CO n o — — t 0 0 oj! ?! uji ; ff\ \ u — * •j- r-; t — ! • • *- u. Z : 1 a y i « 0 .i t o i o » — OJ P5 0 a n r~-10 -• -• cc *4 • • CO O* (VI iC • rn cr* ~4 •« • h- in r- "* N VN o i .UJ • -a " 00 (t^ Cd - O OJ =0 1 0 1 — in 00 03 in n 0 0 1 u: o oj y *) CO V4 • • CO 1 J* • CC — o n oj • o n n c 0 1 n r+ *C O — o IX. • in o- CO ij" ^^0 1 UJ OJ 0> ,$• l>- p^ o 1 CP • ^> o r* n C* 0 _4 00 o 1 •a 1 o CO 00 o — • O j CM O ON —• i 01 o. HI' f" c CC lO • C.i- CM: P-in in1 m i IJ in, o; HJI O ' O ill! HI I PHI 'U o yi 03 HI, Si O LJ 7 O NO OJ r) 3? —f* . O O! O LJ CM p- N CO V4 . O CM ,O O in CM o HI NT CM O CM o -T O O NO CM 00 NT fM CM ^* . O CM O .J < > O UJ a D 9 1 0 3 6 7 0 C E -0 3 0 .2 2 7 6 6 E -0 6 a 7 5 3 8 5 K -0 9 C . o o o o o o li l i i i UJ HI HI HI HI UJ O NO n *T •- . G O O D — O p - o o p - i M i n i n c o • n'incM — p-n — o o o iO o o o l l i ,T — 00 NT r- 10 r> o o o o o i i i i 11 HI ID HI HI HI UJp N - c u O N n N j c o o i p . in— o -o o P- o> NO rN - ^p« -oo inp - i n in CM u" — « PI — w O 0 O O O O O 1 1in ao NT r- in — 0 00 000 1 i l I I I UJ LU ill 'j-1 'JJ HI o o o o o o o o i ' ' ' ' OLI ' o o o o o o o o o 1 1 1 1 1 1 1 1 1 111 'li! HI UJ U.i li" UJ HI HI — ion N • (\j nj o co p- — -Oi f " )O — — P - - T P M N T in 'co in n ON 71 CM ^ cu O1 .>O lO CM — ON O O^ NO O ' O O O O O O O O 1,1 1 1 1 1 CO lO CO 1C CO — *O o ; o o o o o o 1 , II 1 1 1 1 HI Hi LU HI Ul HI UJ vO — in in m « CM ON <» — 5 1 5 4 8 7 E -1 1 C . o i • HI N CM O 3 1 0 7 7 8 E -1 3 C 0 CM I Ul CM NO ooo o o o o 1 1 1 o o — o o o o o1 1 1 1 1 1 1 1 U J U J U J L U H J U J H I H I p - — i n o o o n n n c M N T i 7 - - - N O ' 0 O O O C' O O O O1 1 o — o o o o t I I I I I 111 UJ LU HI Hi HI a co a. in o n n in CM CM 13 — O O ^ m NO' — .-» cc o *o co .{y i t i -O u> n in IT — in* i i i i i 1 l 1 HI 'L' UJ HI UJ UJ UJ UJ M — NT -T n GO C7> i^ o o o o o o o o 1 1 1 ' •H IT4 — LJN — UJ O c; — o — O I I I I I 1 : HJ UJ HI Hi H! HI rji u) v o NT — o in.. CM ct; u; a* o r«> o *» O 'CN o f] r. C.N o o. I 1 1 ! C-.. rN <;. CC 0 CM Cj M NT n - c co re ro — NT, — NJ — 1 4 2 9 9 E -C 9 -0 . 4 2 8 R O E -0 2 -0 . 1 0 3 5 5 E - 0> i- 0 . 5 7 3 7 6 E - 1 1 -0 o o o o 1 1 o — rvj n 0 0 1 \v o o 1 1 HI UJ CO — in "I _ 2 1 0 7 5 E -0 5 C . . 5 9 0 7 8 E -0 1 -0 6 8 3 0 8 0 . 1 3 7 0 9 S -0 4 0 o in 0 i "* . 6 5 9 9 .3 E -0 6 0 0 o o o l I NT NO' — '0 00 : 1 1 1 : LU Hi 1 UJ C\! CM : — in co; CM in — ; n o n. m - -:CM J o o o HI, ,T CMi-- in in cM| in o! in — . 1 6 4 4 4 E -C 7 -0 . . 2 4 9 2 9 E - 0 1 - 0 . . 0 0 0 0 - 0 3 1 6 7 6 E -C 8 0 1 2 2 7 7 E -0 9 1 1 9 6 9 3 E -1 0 -0 , c o i i i 1 1 HJ HliCM -; O P~ CM O CM 0 IT. —1 o o' i ; NO P- o O ! 1 LU •st iO : CM i . 7 4 2 8 3 E -0 7 -0 . 1 0 8 4 3 E -C 6 -0 O : 1 O 1 '• HIin ; O 1 o 0 -6 0 -3 1 0 9 0 2 o IL 1 .£ C 00 a - . - - - . . . - - 51 09 9 o o o o o IN « 1 6 2 6 3 E -0 5 . 2 0 2 5 2 E -0 5 . 0 0 0 0 4 9 2 2 3 E -O C 6 4 1 4 7 E -0 6 - 0 2 4 4 9 2 E -0 5 0 6 3 9 4 2 - " 3 3 6 3 9 E -0 6 -0 6 2 0 6 3 9 E -0 6 1 12 91 8 ~ 0 . 2 1 8 0 4 E -0 3 -0 o |N- O1 H) NO in r— Ci 1 » . :O l LU O O! in OI-T n NT -o m oj NO 01 NT C' o o, pi rn in — , — o! PI — IN- n; in .j - - 0 0 1 NT in 0 lo Oi1 1 1 1 : NO p- co O"i i 3 3 2 1 3 E -1 C 6 9 1 9 2 E -0 8 . 1 3 9 2 0 E -0 7 1 1 4 1 8 E -1 C 2 3 4 0 2 E - 1 0 -0 5 0 S 9 0 E -0 8 -0 . 1 1 7 7 2 E -' C 7- C 8 0 4 4 0 E - 1 1 0 7 3 2 7 6 E -0 6 0 . 1 7 S 3 6 E -0 5 0 . 7 4 5 5 1 E -0 6 0 . 6 1 3 5 8 E -0 5 -0 . o o 1 CM NO O O 1 1 HI HIin — — Nil- CM p- Ul CM in NO •0 -9 0 -3 8 9 1 8 1 •0 IO -3 9 5 5 S E . 1 7 7 7 6 E -0 6 0 . . 6 9 4 7 7 E -0 3 -0 . . 2 4 3 9 0 E -C 3 -0 . . 1 4 68 1 E -0 6 -0 2 3 0 C 5 E -C 6 0 2 3 7 2 1 E -C 5 0 0 0 P- CM 0 0 1 1 Ul P- in — — o OO NT 7 1 9 9 4 E - 1 1 -0 . 4 2 B 0 8 E -0 8 0 . O 0' I 5 0 3 6 6 E -C 1 -C 4 9 8 2 1 E -0 6 0 31: P- NU o o 1 1 UJ '11 CM P- «T O CO P- 00 — 1 3 4 6 4 E -0 7 0 9 2 4 4 0 E -0 7 C 0 0 1 i IP o i HI P- ON U) — P- CM — NO . 4 2 2 2 7 F .- 0 4 0 . 6 5 7 3 5 E -0 1 0 . . 3 2 7 9 5 E -C 2 0 . 3 8 9 5 0 E -C 3 -0 . 0 0 l l NO —O Ct 1 1 HI HIin p~ O NT _ 0.' ON C\J 4 4 0 9 3 E -0 6 -0 . 1 7 5 3 1 E -0 _ 6 = 0 . 3 3 6 9 7 E -0 2 -0 . 2 8 3 1 4 E -0 4 C . o — l i HJ IUO NO N f- n NC ~ n| O O 1 co NO 0 0 1 1UJ U1 cc n. PI CO' CM -T| «T CM — UJ — 01 1 HI LU co n P— 171 CM NO CM in — .T 8 8 9 7 0 E -1 2 0 3 3 4 8 S E -C 8 -0 4 3 3 3 0 E -0 6 -0 . 2 2 0 9 S E -0 5 0 . 1 9 7 3 5 E -0 7 0 . 1 1 2 9 5 E -0 3 0 . . 3 9 9 2 8 E -0 6 C . . 6 3 3 4 8 E -0 2 0 0 O NO in o o - in in —in ^ 1 3 6 5 6 E -C 6 0 . 2 4 3 5 2 E -0 2 C . 10 67 IE - 1 1 - 0 . 2 6 4 0 5 E -0 8 -0 . o c,, o oio of I.I 1 , CM n NT 2 3 2 3 5 E -0 7 . 1 5 9 2 2 E -0 7 -0 o o LU CC o n -t •0 1 0 -3 9 2 6 5 2 . 1 5 4 5 9 E -0 3 0 . o HJ O 1 5 8 5 4 E -0 4 -0 . o : i o HI :to ; •-o ' e o • N 0 w*^— 11.0 REFERENCES 1. "Space Station Mass Properties," NASA/MSC Memo EG2-71-71*, March 9, 1971. 2. "Design Data for Space Shuttle Orbiter (NRl6l-C) ACPS and OMPS Control Equations," NASA/MSC Memo EG2-71-132, June Ik, 1971. 3. "Study of Control Moment Gyro Applications to Space Base Wobble Damping and Attitude Control," Grumman Report GCR70-U, September 1970, Under Contract to NASA/MSC. k. "Space Station Guidance, Navigation and Control Systems Requirements," MSC Internal Note No. EG-70-36, September 15, 1970. 5. Revision 1 to Reference k, July 15, 1971. 6. "Artificial Gravity Space Station Physiological Effects and Design Criteria," TRW Report 176l8-H125-RO-00, April 15, 1971. 7. "Control Momentum Gyro Wobble Damper Sizing for Candidate Modular Artificial -g Space Station Configurations," EG2-71-1U3, July 6, 1971. 8. "Stability Criteria for a Rotating Space Station with a Non- rotating Hub," Austin, F.; AIAA Journal, Vol. 6, No. 11, November 1968. 9. "Use of Floquet Theory to Determine the Stability of Linear Differential Equations with Periodic Coefficients," Austin, F.; Grumman Aerosapce Corp., Structural Mechanics IOM 70.38, May 1970. 10. "Optimum Wobble Damping of Rotating Satellites Using a Control- Moment Gyroscope," Austin, F.; Herman, H.; Grumman Report ADR06-05-71.1, June 1971. 315 APPENDIX A MODES, FREQUENCIES, STIFFNESS AND DAMPING MATRICES This appendix contains an outline of the procedure used to obtain the structural characteristics for both simulation vehicles. For each rotor a stiffness matrix will be formed for use in the time-history simulation. This matrix is applicable for a rotor in which tensile forces are present. In the time-history formulation the tensile forces naturally arise from centrifugal effects. However, to obtain more accurate modes, equivalent external forces were added to the nonrotating idealization used for the eigenvalue problem. As a result slightly different stiffness matrices are formed below for the time-history and eigenvalue problems. I. Rotor Basic beam element stiffness matrix (circular cross-section) We begin with the stiffness matrix for a beam segment located between two node points. This basic matrix does not include the effects of\tension due to cen- *A trifuged forces. -I* A = cross-sectional area I = moment of inertia of cross-section J = polar moment of inertia of cross-section L = length of beam E = modulus of elasticity G = modulus of rigidity displacement convention = [K1 ( >^ nr IB n *.] 59 _ . fft where 316 V V V A6 T V C3 •£ I* GJ t - 4S c Incremental Beam Element Stiffness Matrix .g. :The following incremental stiffness matrix will be added to [K ] to account for the stiffening effect due to the centrifugal forces = axial tensile load applied to the beam; the direction of this load does not change as beam bends P** n L -ft I n -i -cz -c ._. $ L 1* _.. * -C ^ j . ^1 (I -i /M n (, Ml n %l 7C 4t* )_ Stiffness Of A Rotor Arm Next the element stiffness matrices will be assembled to obtain the stiffness matrix for a rotor arm. _ ~ COLUMN < ANP MAIN IN BOVU4, COLUMN4 [92]' [d («(')] START* 'M ROW (, COLUMN I A / ANP A6AIN IN ROW 4, COLUMN 4 [02*3 - [Q2JT Assemble Rotor Next the stiffness matrix of the entire rotor is formed by noting that the displacements at point B are common to all arms. This is accomplished by carrying out the following steps: The previously obtained arm stiffness matrices are first stacked. T Rotor Y Rotor 46 CKTTA]OK = ([K7Tg7]) KSA OR 1A\ I I 1A KfaA OR (KfcB) n n [KTVA] OR = K5A or T 14 14 KfaA or \1A (K6B) 1 __ U_ 1 * 14 KfeA* Next the following matrices are formed to superimpose the displacements at point B. I 18 J 5TAKW IN BOUI I, CO.. ( 57 fO] STARTS IN BOW <,CO..I L-l [K9TB] Mass Matrix The mass matrix for one rotor arm is as follows: [MI] = \ \ 323 In order to obtain the mass matrix for the entire rotor the matrices for each arm and for the rotor ring are stacked as follows: Vt 'START* IN ROW (, COt. I MI 57 n «TART« IN ROW 1,COLOMWI Ml 16 Ml \t> Ml Rotor Eigenvalue Problem The eigenvalue problem for each simulation vehicle is formulated in this section. The equations of motion follow: T Rotor Y Rotor The displacements {q_} and (q ] are shown below: zo n "37 17 10 Ifr WWW "7<.4-^< *>*7< s' ir ir M T Rotor Symmetric motion (with respect to the X_-X plane) is characterized by: 325 Y Rotor Symmetric motion (with respect to the )C-X plane) is characterized by: torort • ' »* (9 ' ' "ft" OF ' ft8" - -fc* ' %»' «•»•-*« %*>• ft»» V 'V ft''*' 4»'° ft«- ft* »«• ° 6' %t T- 0 fl- i« V0 ,«• o I"'«" fffOUO AHP •tyltW AKl*S «lt ' «*> 8* • -? ««' ft« t»' ' "ft* ft»»' ft* 2» • -%» 7 " 8 " Antisymmetric motion (with respect to the X-X plane) is characterized by: Antisymmetric motion (with respect to the X-X_ plane) is characterized by: eoroe KMG • ' -gtfr »' f t " 10 ft" ' ft»f ft« •-%•»(Jf» • %« jw • -in %rt • 4» *"•' '«« «"' ft»» ft"' -g" ftf ft" fti'° «»•«» AKM I «4- O ft« ft»' %««• ft» ' 4tt' ««' ««•• ««•• 0 % I J ' $17 '«' ° «l • %» U- 0 ft* ' ft" at- -««< «• .OF 326 The matrices used to convert the basic equations into symmetric and anti- symmetric motion (viz [TSYM2], [YSYM2], l'TAWT2] and [YANT2]) are formed below: 16 I 0 f 0 f 0 1 0 f 0 [SNM] 327 [TS^MI] STAPTC luL Jeow(,coc.l 18 w x v«VM( y IU ROW I, COUIMW < I !». I 1 wf*" [TANT1] = 1 0 O I 00 [VAMT t]' ( 0 O O 0 1 (8 0 \ o ( 0 ( 0 ( 0 I 0 1 0( 328 *» •STASIS IN 10 [VANT2] [ 0VAWT <] The modal matrices are now partitioned into modes to be retained and modes to be ignored in the modal reduction procedure. (rttain) (ignore) (retain) (ignore) [0TANT2H0TAMT3; 0TANT4J (retain) (ignore) OSVMS; 0VSVM4 (retain) (ujnorej The symmetric and antisymmetric modal matrices of the n modes to be retainedt are: The symmetric and antisymmetric modal matrices of the n modes to bey returned are: The above mode shapes are expressed in local absolute coordinates, whereas the equations of motion in the time history program are in local relative coordinates. The mode shapes can be converted into relative coordinates by use of transformation matrices [QT6] and [QY6] defined as follows: 16 [0*1- [06] • [Q4][a(<*2)] IN SOW 4, COl. I IN ROW4, COU [«5] STARTS IN WW TZ.COU Qjfc] €1ABTS IN ROW 40, COU Next the modal matrices are transformed to relative coordinates: The stiffness matrices needed for the time history program must also be converted into relative coordinates. 57 ,COt.l ROW4,COL I nI 1 POVU17.COL. ( ?Cl] - eovu4o, Coi. ( 331 Rotor Damping It was decided to use modal damping to represent energy dissipation in the structure. Since inversion of the square modal matrix is used in manipulations below, all modes are now assembled. 57 The following frequency matrices will also be required. UftfMM to 10 IW6VM M 10 WV4MT The following physical damping matrix is based upon a fraction, Y> of critical modal damping: [CT] = 1 VCM^T][0TALl](>T][0TAa]-f - ^ -t Recalling that the bearing springs were incorporated into the stiffness matrices used for the modal analyses, it follows that the physical damping effects, as given by [CT] and [CY], will be transmitted to the hub. This unrealistic effect is deleted as follows: [CT*]-[QT7]T[CT][QT7] • The first 3 rows and columns of [CT. ] are deleted, and the result is called *•*-,[CT ]. The final damping matrix is: [CTA] - The first 3 rows and columns of [CY ] are #* deleted and the result is called [CY ]. The final damping matrix is: [CNA] 332 II. Hub The basic beam stiffness matrix, [K ], presented .^ in the rotor section is used to compute [K1H ], [K2H ] and [K3H ] for hub elements 1, 2 and 3, respectively. Note that the coordinate systems are not the same; that is, the Y.. axis is the torsional axis for the hub, whereas the X_ axis was the torsional axis in the rotor arm. Next the stiffness elements are stacked to form the stiffness matrix of the total hub. KIH« I I "1 I STABT* IN'{I.') [KIM] «TABT« IN (7.1) [K3M] Matrix [Kl*H] operates on displacements at points G, C , C and E. The trans- formation matrix [Q1H] is now formed to convert to displacements of G, C , C2 and C3; call result [K5H]. [QIM] IN [QIM]TC 333 To cantlever the hub at point G, the first six columns of [K5H] are deleted below. The result is called [K6H]. 14 16 [K6H] is partitioned into [K6HG] and[K7H] as shown below: / / , TO/ if // K«^y/ 18 [K?H] is the stiffness matrix for the flexible section of hub. [K5HG] can be used to obtain the elastic loads transmitted to the rigid section of hub at point G. The hub mass matrix is now stacked as follows: 16 16 MC1 1C' MC' [MH> rc1 MC' rc» 334 [MH] and [K?H] are used to formulate the eigenvalue problem, i.e., [MH] {„ = 0 The solution of the above problem provides the modal matrix [0H ] and frequencies, UJ. [0H ] is partitioned into n, modes to be retained for modal reduction and modes to be ignored. (retain) (ignore) [0H] »[ 0H i 0H**J Hub Damping As was the case for the rotor, modal damping is used in the hub. The complete * modal matrix [0H ] as well as the following frequency matrix are required to obtain the hub damping matrix: Finally the physical damping matrix is computed based upon a fraction, 6, of critical modal damping. The damping matrix for the flexible section of hub is: and the damping forces transmitted to the rigid section of hub can be computed using [CHG]. 16 335 Combined Modal Matrix - Total Vehicle The order of the displacements in the mode shapes must be made compatible with the order of the variables in the time history equations (Equations 6-^3). The following matrix is used to reorder these displacements: wweee ( 0 o 0 f 0 0 0 ( The hub modes, as required in the time history simulation, are «i. 18 similarly for rotor modes, For T Rotor Modes For Y Rotor Modes [CWT] = Q6 Q8 Q6 Q8 The rotor modes, as required in the time history simulation, are 336 Assemble Total Reduction Matrix For Total Vehicle In the time-history simulation, the hybrid coordinates used for numerical integration are the physical coordinates for the rigid, section of hub {Zf l} n Y and (W }, the two bending rotation of the shuttle Y, and Y?, the TL hub modal coordinates, and the nu or rJ^ rotor modal coordinates. In order to rigidize the shuttle torsional coordinate Y-,, the following constraint matrix is formed: i ^ 0 0 0 Then, the total matrix used to reduce the number of coordinates for numerical integration is: For T Configuration For Y Configuration Uf (6* V [0] = 0UF 0VRF 337 APPENDIX B FREQUENCY SHIFT TECHNIQUE The fundamental vibration equation may be written as where u>. is the frequency and {$.} is the associated mode shape. Since the eigenvalue algorithm used in this study cannot treat a zero frequency, this frequency was removed from the equations in the following manner. Let uJ,. be a new frequency which is related to u>. by the small constant shift frequency (B-2) The fundamental equation (B-l) may then be written as or where the new "stiffness" matrix [K] is given by Note that Equation (B-U) is the same form as (B-l) and also that the mode shapes f0.} have not been effected. Accordingly, the eigenvalue algorithm may now be applied to Equation (B-U), yielding the mode shapes {0.} and shifted frequencies til ; the original frequencies may then be recovered by using Equation (B-2). GRUMMAN m

Flexible rotor dynamic analysis Final report

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Flexible rotor dynamic analysis Final report

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