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DYNAMICS OF THE ARECIBO RADIO TELESCOPE

Ramy Rashad 110030106
Department of Mechanical Engineering McGill University Montreal, Quebec, Canada February 2005

Under the supervision of Professor Meyer Nahon


Abstract
The fo llowing thesis presents a co mputer and mathemat ical model o f the dynamics of the tethered subsystem o f the Arecibo Radio Telescope. The computer and mathematical model for this part of the Arecibo Radio Telescope invo lves the study o f the dynamic equations governing the motion of the system. It is developed in its various co mponents; the cables, towers, and platform are each modeled in successio n. The cable, wind, and numerical integrat ion models stem fro m an earlier versio n of a dynamics model created for a different radio telescope; the Large Adapt ive Reflector (LAR) system. The study begins by convert ing the cable model o f the LAR system to the configuration required for the Arecibo Radio Telescope. The cable model uses a lumped mass approach in which the cables are discret ized into a number of cable elements. The tower motion is modeled by evaluat ing the co mbined effect ive st iffness o f the towers and their supporting backstay cables. A drag model of the triangular truss platform is then introduced and the rotational equat ions of mot ion o f the plat form as a rigid body are considered. The translat ional and rotational governing equat ions of mot ion, once developed, present a set of coupled non-linear different ial equat ions of motion which are integrated numerically using a fourth-order Runge-Kutta integration scheme. In this manner, the motion of the system is observed over time.

A set of performance metrics o f the Arecibo Radio Telescope is defined and these metrics are evaluated under a variet y o f wind speeds, direct ions, and turbulent condit ions. The general configurat ions o f the Arecibo Radio Telescope, before and after its two major upgrades, are also compared.

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Finally, a sensit ivit y analysis is carried out in order to ident ify which phys ical parameters of the system, if changed or redesigned, could improve the system's performance. The fo llowing six parameters are investigated: number of mainstay cables, tower radius (cable length), effect ive tower stiffness (number o f backstay cables), plat form mass, cable-platform attachment points, and mainstay cable properties.

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Acknowledgements
I would like to sincerely thank Professor Meyer Nahon, my supervisor and mentor throughout this thesis. I would like to thank him for his excellent guidance and advice, as well as for his pat ience, mot ivat ion, and reso lve. For always giving me the time of day, and for accepting to supervise my project, I am in his debt. I would also like to extend my thanks to Dr. Steve Torchinsky, the Head o f Astronomy at the Arecibo Observatory, for providing invaluable informat ion needed throughout the course of this project.

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Contents

Abstract Acknowledgments List of Figures List of Tables

i iii vii x

Chapter 1

Introduction

1

1.1 Radio Telescopes ............................................................. 1 1.2 Radio Telescope Simulations................................................. 2 1.3 Arecibo Construction Overview............................................. 3 1.4 Arecibo Upgrades.............................................................. 5 1.5 LAR Model and Previous Work............................................. 7 1.6 Scope of Thesis................................................................ 9 Chapter 2 The Arecibo Model 11

2.1 Overview........................................................................ 11 2.2 Lumped Mass Approach...................................................... 12 2.3 Simulation Basics............................................................. 14 2.3.1 2.3.2. Chapter 3 State Vector.......................................................... 14 Numerical Approach............................................... 16 19

Cable Model

3.1 Cable Properties................................................................ 19 3.2 Coordinate Systems........................................................... 20 3.3 Cable Kinemat ics.............................................................. 21 3.4 Cable Dynamics............................................................... 22

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3.4.1 3.4.2 3.4.3 Chapter 4

Internal Forces...................................................... 22 External Forces..................................................... 26 Translat ional Equat ions of Motion.............................. 27 29

Tower Model

4.1 Construction Details...........................................................29 4.2 Tower Properties............................................................... 31 4.3 Effect ive Stiffness............................................................. 32 4.3.1 4.3.2 4.3.3 4.3.4 Chapter 5 Tower Contribut ion................................................ 33 Backstay Cable Contribution..................................... 34 Combined Effect ive Stiffness.................................... 36 Tower-Top Motion.................................................36 40

Platform Model

5.1 Construction Details...........................................................40 5.2 Coordinate Systems........................................................... 41 5.3 Platform Properties............................................................ 42 5.4 Cable Attachment Points..................................................... 45 5.5 Platform Drag..................................................................46 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5 Factors Affect ing Drag............................................ 47 Side View of Truss..................................................48 Top View of Truss................................................. 51 Platform Angle of Attack.......................................... 54 Drag of Gregorian System......................................... 57

5.6 Translat ional Motion.......................................................... 59 5.7 Rotational Motion............................................................. 60 5.7.1 5.7.2 5.7.3 The Z-Y-X Euler Angles.......................................... 61 Transformat ions.................................................... 62 Rotational Equations of Motion.................................. 65

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Chapter 6 Performance Evaluation

68

6.1 Introduction..................................................................... 68 6.2 Performance Metrics.......................................................... 68 6.3 Addit io nal Parameters of Interest........................................... 72 6.4 Indicators....................................................................... 72 6.5 Arecibo Model Configurat ion................................................ 73 6.6 Equilibrium Condit io n........................................................ 74 6.7 Dynamic Runs................................................................. 76 6.7.1 6.7.2 6.7.3 Effect of Wind Speed.............................................. 77 Effect of Wind Direct ion.......................................... 81 Effect of Turbulence............................................... 83

6.8 Upgraded Arecibo Configurat ion............................................ 86 6.9 Selected Cases................................................................. 89 Chapter 7 Sensitivity Analysis 95

7.1 Test Matrix..................................................................... 96 7.2 Number of Mainstay Cables................................................. 97 7.3 Tower Radius.................................................................. 99 7.4 Effect ive Tower Stiffness.................................................... 100 7.5 Platform Mass................................................................. 102 7.6 Cable-Plat form Attachment Points.......................................... 105 7.7 Mainstay Cable Properties: Plasma Rope.................................. 107 7.8 Summary........................................................................ 111 Chapter 8 Conclusion 112

8.1 Final Remarks.................................................................. 112 8.2 Future Recommendat ions.................................................... 113

References

115

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List of Figures
1.1 1.2 1.3 1.4 2.1 2.2 2.3 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Single-Dish Radio Telescopes Located Around the World....................... 2 The Arecibo Radio Telescope........................................................ 4 Tower Base and the Welco me Center ...............................................4 The LAR and the Arecibo Radio Telescope........................................ 8 The Lumped Mass Approach in the LAR and the Arecibo....................... 13 Node and Element Numbering Scheme............................................. 13 Flowchart of Simulat ion Overview................................................... 18 Body-Fixed Coordinate System for a Cable Element.............................. 20 Unstretched Length Configurat ion...................................................23 Schemat ic Representation of the Internal Forces................................... 25 Nodal Force Body Diagram........................................................... 28 Tower and Backstay Cable Configurat ions......................................... 30 Tower Cross Section................................................................... 31 View of Tower-Top, Mainstay and Backstay Cables.............................. 33 Effect ive Tower Stiffness Model..................................................... 33 Geometry of the Perturbation Approach............................................. 35 Tower Orientation in the Inertial Frame of Reference............................. 37 A Picture of the Arecibo Platform................................................... 41 The Plat form Model's Body-Fixed Coordinate System........................... 42 Cable-Plat form Attachment Points................................................... 45 Flat Plate Normal to the Flow......................................................... 46 Drag Component Breakdown.........................................................48 Truss Pair Correction Factors......................................................... 50 Finite to Infinite Aspect Ratio Correction Factors................................. 51 Plan View Photograph of the Arecibo Plat form.................................... 52 vii


5.9 5.10 5.11 5.12 5.13 5.14 5.15 6.1 6.2 6.3 6.4 6.5 6.6 6.6 6.6 6.7 6.7 6.7 6.7 6.7 6.8 6.8 6.8 6.9 6.10 6.11 6.12 6.13 6.14

Correction Factors for Adjacent Truss Frames Used for the Top View of the Platform.................................................................................. 53 Angle of Attack......................................................................... 55 Platform Cd*A versus Angle of Attack............................................. 57 Flow Around Sphere for Low Reyno lds Number & High Reyno lds Number..58 Using the Z-Y-X Euler Angle Set.................................................... 61 Angular Velocit ies in the Plat form's Body-Fixed Frame......................... 62 Dynamics Model Flowchart........................................................... 67 The Focal Plane and Hemisphere..................................................... 69 Tilt Angle Performance Metric....................................................... 70 Performance Metrics to Equilibrium.................................................75 Tower-Top Posit ions to Equilibrium................................................ 76 Wind Direction and Tower Numbering............................................. 77 (a) Error in the Focal Plane vs Wind Speed......................................... 79 (b) Error Out of the Focal Plane vs Wind Speed................................... 79 (c) Platform Tilt Angle vs Wind Speed............................................. 80 (a) Error In the Focal Plane vs Wind Direct ion.................................... 81 (b) Error Out of the Focal Plane vs Wind Direction............................... 81 (c) Platform Tilt Angle vs Wind Direct ion.......................................... 81 (d) Average Cable Tensio n vs Wind Direct ion..................................... 82 (e) Average Tower Deflect ion vs Wind Direct ion................................. 83 (a) Error in the Focal Plane vs Turbulent Mean Wind Speed..................... 84 (b) Error out of the Focal Plane vs Turbulent Mean Wind Speed............... 84 (c) Platform Tilt Angle vs Turbulent Mean Wind Speed......................... 85 Average Error in the Focal Plane vs Wind Speed.................................. 88 Average Platform Tilt Angle vs Wind Speed....................................... 88 Case A Performance Metrics......................................................... 90 Case A Plat form Rotation............................................................. 91 Case A Plat form Translation.......................................................... 91 Case B Performance Metrics..........................................................92

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6.15 6.16 6.17 7.1 7.2 7.3 7.4 7.4 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.10 7.10

Case B Plat form Rotation............................................................. 93 Case B Plat form Translat ion.......................................................... 93 Case B: Tower-Top Motion........................................................... 94 Equilibrium Plat form Height vs No. of Mainstay Cables......................... 97 Equilibrium Tensio ns per Cable vs Number of Cables............................98 Error in the Focal Plane vs Tower Radius.......................................... 99 (a) Error in the Focal Plane vs Number of Backstay Cables...................... 101 (b) Platform Tilt Angle vs Number of Backstay Cables........................... 101 (c) Average Tower Deflect ion vs Number of Backstay Cables.................. 102 Equilibrium Plat form Height vs Platform Mass....................................103 Platform Tilt Angle vs Platform Mass............................................... 104 Cable-Plat form Attachment Points................................................... 105 Error in the Focal Plane vs Platform-Cable Attachment Radius................. 106 Platform Tilt Angle vs Platform-Cable Attachment Radius...................... 107 (a) Average Error in the Focal Plane vs. Wind Speed for Plasma Rope.........109 (b) Average Error out the Focal Plane vs. Wind Speed for Plasma Rope...... 110 (c) Tilt Angle vs. Wind Speed for Plasma Rope.................................... 110

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List of Tables
1.1 3.1 4.1 4.2 5.1 5.2 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Summary o f Construction Details....................................................6 Mainstay Cable Properties............................................................ 19 Tower and Backstay Cable Configurat ions......................................... 30 Tower Properties....................................................................... 32 Physical Properties of Plat form.......................................................43 Summary o f Drag Parameters......................................................... 54 The Original Arecibo Configuration................................................. 73 Wind Speed Test Matrix............................................................... 78 The Upgraded Arecibo Configurat ion............................................... 86 Upgraded Plat form Mass Moment of Inertias.......................................87 Test Matrix for the Sensit ivit y Analysis............................................. 96 Mainstay Cable Effect ive Areas...................................................... 97 Average Tower Effect ive Stiffness................................................... 100 Platform Mass-Mo ment of Inertia....................................................103 Minimum Breaking Strength of Cables............................................. 104 Plasma Rope Properties............................................................... 108 Qualitat ive Summary of Sensit ivit y Analys is...................................... 111

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Chapter 1 ­ Introduction

1.1

Radio Telescopes
Radio telescopes are used to detect and image electromagnet ic radiat ion in the

radio wave range. They generally consist of some components that collect the radiatio n and a receiver to detect the radiat ion [1]. Radio telescopes are used by radio astronomers to study our planet's at mosphere as well as asteroids and even distant galaxies [2]. A radio telescope has the difficult task of co llect ing and detecting very weak radio wave signals and hence, bigger is better when it comes to the collecting area o f the telescope [1]. Some examples o f the various shapes and sizes of radio telescopes fro m around the world are shown in Figure 1.1. In order to be o f pract ical use in detecting the electromagnet ic radiat ion fro m celest ial bodies, a radio telescope's receiver must be ver y accurately held in posit ion and orientation. Located on the island o f Puerto Rico, the Arecibo Radio Telescope (commonly called the "Arecibo Observatory" and often refereed to herein as simply "Arecibo") is the largest single-dish radio telescope in the world [3].

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Figure 1.1 ­ Single-Dish Radio Telescopes Located Around the World. A) 100 m Effelsberg in Germany B) 64 m Parkes in Australia C)100 m Green Bank in U.S.A. D) 305 m Arecibo in Puerto Rico

1.2

Radio Telescope Simulations
When a new radio telescope is to be designed and constructed in this day in age,

the structural and mechanical design process often calls on co mputer models and dynamic simulat ions; a cost-efficient and convenient tool. Wit h the advance o f co mputer models, it is o f particular interest to model a design already in use, namely the Arecibo Radio Telescope. The Arecibo Radio Telescope was designed and constructed in the early 1960's [4] and with no use of dynamic simulat ion whatsoever. In developing a computer simulat ion, the system's behavior and performance may be observed over t ime, under various wind and turbulence condit ions, without the expense and complications o f experimental work. A sensit ivit y analys is may also be carried out in order to identify the parameters that may improve or deteriorate the system's performance, and all o f this done on a personal co mputer workstation. 2


1.3

Arecibo Construction Overview
The Arecibo Radio Telescope was conceived by William E. Gordon who at the

time was a professor at Cornell Universit y [5]. The original Arecibo configurat ion became operational in November of 1963 [5]. Since that time the telescope has seen two major upgrades (one in 1974 and the other in 1997), which will be described in the next sect ion. Figure 1.2 shows a photograph of the Arecibo Radio Telescope next to a close up of one the supporting towers. To observe the enormit y o f the structure notice, in Figure 1.3, the doorway entrance to the Welco me Center that is found at the base o f one of the towers. Arecibo 's structural configuration consists of a 305 m diameter reflector dish; o f spherical shape, with a radius of curvature of 265 m [2]. The spherical reflector always points straight up and unlike many other radio telescopes; it cannot be steered to a different direct ion. A natural bowl in the landscape of the regio n was found to aid in the construction effort of the enormous mesh surface dish. The telescope's triangular feed support structure (referred to herein as the "plat form") is suspended approximately 150 m above the surface o f the reflector [5]. In the original configurat ion, the triangular platform had a mass of 550 tons, and was suspended by 12 main cables (each of 3 inch diameter - braided steel) fro m 3 towers. The 3 towers, whose tops are all o f equal elevat ion, are posit ioned at a radial distance o f 213 m fro m the center of the receiver. Due to the region's landscape, two of the three towers are of an equal length, 76.2 m, with the third having a length of 111.2 m. Supporting each tower, 5 backstay cables (each of 3.5 inch diameter - braided steel) run fro m the tower tops to concrete anchorages in the ground. Running fro m each corner o f the platform are two 1.5 inch cables called tie downs (functioning as catenaries) which are anchored to the ground just along the rim o f the reflector [5].

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Figure 1.2 ­ The Arecibo Radio Telescope

Figure 1.3 ­ Tower Base and the Welcome Center

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1.4

Arecibo Upgrades
The Arecibo Radio Telescope has undergone two major upgrades since it was first

constructed. The first upgrade, completed in 1974, was carried out to improve the smoothness o f the reflect ing surface to an accuracy o f 2.5 mm r. m.s [2]. Upgrades at that time also included "the addit ion of a powerful transmitter at 2380 MHz designed for radar studies of the so lar system." [2]. The second Arecibo upgrade, completed in 1997, was the implementation of the new Gregorian system (receiver/transmitter), whose weight and addit io nal structure increased the total mass of the plat form form 550,000 kg to 815,000 kg. [2]. To support this much heavier plat form, using the same three towers, additions to the number o f cables were made: 6 auxiliary mainstay cables (each of 3 1 inch diameter) were added 4 fro m the tower tops (2 per tower) to the plat form and 6 auxiliary backstay cables (each o f 3 5 8 inch diameter) were added from the tower tops to the ground. A new Gregorian system was installed on the triangular plat form which brings rays fro m the spherica l primary reflector to a point focus through a series of reflect ions [2]. The ent ire Gregorian enclo sure is spherical in shape (as can be seen in Figure 1.2). Finally, to meet the new po inting requirements o f the telescope beam, three new pairs o f vert ically-oriented tiedown cables [2] were added at each corner of the triangular plat form, anchored to the ground below the reflector. These new t iedown cables replaced the exist ing catenaries. Each tiedown is anchored to controllable jacks which can provide active control (using co mputer models) to the tiedown by exerting up to a "60 tons of vert ical force" cables [2]. Table 1.1 summarizes and co mpares so me o f the important construction details o f the Arecibo Radio Telescope. Throughout this work the "original" Arecibo will refer the state of the Arecibo upon init ial construction and the "upgraded" Arecibo will refer the state of the Arecibo as it is found today (after both major upgrades).

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Construction of the Original Arecibo
platf orm mass receiver drag Mainstay Cables Backstay Cables Main Auxiliary Cables Backstay Auxiliary Cables 550 000 kg Line Feeds 4 cables per tower d = 3" 5 cables per tower d = 3.25" N/A

Construction of the Upgraded Arecibo
815 000 kg Gregorian System 4 cables per tower d = 3" 5 cables per tower d = 3.25" 2 cables per tower d = 3.25" Attached 2/3 of the way along the sides of the triangular truss 2 cables per tower d = 3.625" 6 cables total d = 1.5" Two cables run v ertically from each corner of the platf orm and are anchored directly below the dish. Activ e Control Jacks which can exert up to 60 tons of vertical f orce.

N/A 6 cables total d=1.5" Two off-vertical cables (f unctioning as catenaries) run from each corner of the platf orm and anchored to the ground near the rim of the reflector

Tie Down Cables

Table 1.1 - Summary of Construction Details

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1.5

LAR Model and Previous Work
The Arecibo dynamics model stems fro m an earlier versio n o f a model created for

the proposed "Large Adaptive Reflector (LAR)" [6]. The LAR dynamics model, developed by Dr. Meyer Nahon, includes the ent ire cable and wind models as well as the numerical integration scheme (all developed in C++). Telescope. The part icular versio n o f the LAR model used in the conversio n includes the fo llo wing physical components: 3 plasma rope tethers, a payload (the telescope's receiver), a leash, and a lift ing aerostat (balloon). The aerostat's buo yancy provides enough lift ing force (approximately 40 kN) [6] to keep the telescope's receiver in it s elevated position and hence the three tethers (which are anchored to the ground) in tension. Figure 1.4 shows the LAR system (shown wit h 6 tethers) and its various components along side the Arecibo Radio Telescope. The fo llowing is a brie f descript ion of the LAR system and it s various differences wit h the Arecibo Radio

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Figure 1.4 ­ The LAR (above) and the Arecibo Radio Telescope (below). The basic differences between the two systems, amo ng factors affect ing the dynamics, are summarized as fo llows: (i) The LAR tether base po ints are at fixed locations (anchored to the ground), whereas the high tower tops of the Arecibo cannot be assumed stationary, and in fact may sway as the tensio ns in the cables change. (ii) The LAR design calls for lightweight materials in order to minimize the buo yancy requirements of the aerostat. (iii) On the other hand, the Arecibo plat form was originally constructed having a mass of 550 tons. The large plat form - a triangular truss structure to which Arecibo 's cables are attached - is to be considered a rigid body subject to the rotational equat ions

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of motion. On the contrary, in the version o f the LAR model used here, the LAR paylo ad and aerostat are both considered subject to translat ional mot ion only. (iv) In the versio n of the LAR model used here, the LAR payload and aerostat are both considered spherical for the purposes of their drag models. A new drag model for Arecibo 's large triangular truss plat form is required.

1.6

Scope of Thesis
In the context of this undergraduate honours thesis, a basic model for the

dynamics of the tethered subsystem of the Arecibo Radio Telescope is first developed. The co mputer model o f this subsystem is then subjected to a performance evaluat ion and a sensit ivit y analysis. In Chapters 2 to 5 the dynamics model is described in detail. We begin the model descript ion with an overview and introduction to the modeling techniques and the simulat ion basics. In Chapter 3, the cable properties, kinematics and dynamics model are discussed. Chapter 4 deals with the tower model, construction details, properties, effect ive tower stiffness, and tower-top motion. Finally, in Chapter 5 the plat form model is described which includes the platform drag model and the rotational equat ions of motion. In Chapter 6, the various performance metrics of the Arecibo system are defined, and the motion and performance o f the "original" Arecibo system are observed and evaluated under a variet y o f wind condit io ns (specifically wind speed, wind direct ion, and turbulence). Also in Chapter 6 is a general comparison between the "original" and "upgraded" configurat ions o f the Arecibo Radio Telescope. In Chapter 7, a sensit ivit y analys is is carried out on the "original" Arecibo configuration. The sensit ivit y analysis invo lves changing the system's configuration one parameter at a time. Six different physical parameters are discussed, including: the number of mainstay cables, the tower radius, the effect ive tower stiffness, the plat form mass, the cable-plat form attachment points, and the mainstay cable properties. In clo sing, Chapter 8 contains the final remarks

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and conclusio ns, as well as the reco mmendat ions for future work and development of the Arecibo model. The modeled configurations o f the Arecibo radio telescope are simplified in that they do not include the tiedown cables (neit her the off-vert ical catenaries in the origina l configuration nor the vert ically oriented cables in the upgraded configurat ion). A large effort was indeed made to incorporate the tiedown cables into the model, and alt hough we were close, it was decided that the tiedowns were beyo nd the scope of this thesis and consequent ly they have been left to future work.

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Chapter 2 ­ The Arecibo Model
2.1 Overview
A dynamics model of the Arecibo Radio Telescope has been developed in Visual Studio C++. The model is described in several sections, which co me together to form the 3-dimensio nal non-linear dynamics simulat ion. The Arecibo model will be presented in the fo llowing sect ions: (i) (ii) (iii) (iv) (v) Lumped Mass Approach Simulation Basics Cable Model (Chapter 3) Tower Model (Chapter 4) Platform Model (Chapter 5) To make the development of the Arecibo dynamics model tractable requires the use o f certain assumptions and approximat ions. The kinematics and dynamics o f Arecibo 's physical co mponents (which include the cables, the towers, and the plat form) will be described in detail wit h attention drawn to the various assumpt ions and approximat ions. In terms o f the overall dynamics of the system; the development of the r r translat ional equat ions of mot ion is governed by Newton's second Law, F = ma ; while the development of the rotational equat ions of mot ion fo llow Euler's equat ions, r rr r rr & M = I + â I .

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2.2

Lumped Mass Approach
The kinemat ics and dynamics of the Arecibo Radio Telescope are modeled using

the same approach as that of the LAR system; the so-called "lumped-mass approach". This model has been validated for the LAR tethered aerostat system, as well as a number of underwater applicat ions [6]. The basic principle is to divide the total unstretched length o f the cables into a number o f discrete cable elements, forming a set of nodes bounding each element. The mass of each cable element is then lumped into its end nodes, which are in turn treated as po int masses. Further details o f how we use the lumped mass approach in developing the governing equat ions o f motion are discussed throughout this sect ion. For now it suffices to become co mfortable wit h the geometry and methodology used in the numbering o f the nodes and cable elements. Figure 2.1 shows the geometry of the lumped mass approach as used in both the LAR and Arecibo models. The numbering system for the nodes and the cable elements is very important to the organizat ion within the code of the dynamics model. Figure 2.2 shows the numbering system for the Arecibo model as used in the performance evaluat ion, that is, wit h 5 nodes used for each o f the 3 cables (Section 3.1 will rationalize the use o f only 3 cables). Notice that the tower-top posit ions are in fact not considered to be nodes, in that they are not included in the node numbering system. The equations governing the motion o f the tower-tops are determined separately fro m the other nodes and are discussed in detail in the tower model (Chapter 4). Also notice the very important shared confluence po int, which is actually the center of mass o f the Arecibo plat form. In terms of the numbering system, the confluence po int is considered to be the last node (#15). However, also sharing this locat ion are the nodes #5 and 10. Wit h the number of nodes per cable chosen as 5 here, the number of discrete cable elements is 5 per cable with numbering as shown in Figure 2.2.

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Figure 2.1 ­ The Lumped Mass Approach in the LAR (left) and the Arecibo (right)

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Figure 2.2 ­ Node and Element Numbering Scheme

2.3

Simulation Basics
Our goal is to develop a model capable o f simulating the motion o f the Arecibo

system over a given period of t ime while under particular wind and turbulence condit ions. The dynamics model o f the Arecibo Radio Telescope has been reduced ­ through the lumped mass approach ­ to the dynamics of a finite number of po int masses, subject to translat ional mot ion, and a rigid body plat form, subject also to rotationa l mot ion. We begin by defining the state vector for the system and then present an overview o f the numerical approach to solving the result ing non-linear, second order, different ial equat ions o f mot ion. The basics o f the simulat ion are being presented early on, so that we may have a clear understanding of the overall modeling process invo lved.

2.3.1 The State Vector
The state of the Arecibo system may be co mpletely defined at any t ime by the posit ion and the velocit y of its "n" nodes, in addit ion to the orientation and angular velocit y if its plat form. We define the first 6n elements of the state vector as fo llows [7]:

X(t ) =

& x1 (t ) x1 (t ) & y1 (t ) y1 (t ) & z1 (t ) z1 (t ) M
dx(t ) gives the velocit y o f the dt

(2.1)

& Where x(t ) gives the posit ion of the nodes and x(t ) =

nodes. The very last 6 elements of the state vector define the platform's orientation and angular velocit y in the inertial frame:

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X(t ) =

(t ) (t ) (t ) (t ) & (t ) (t )

& &

(2.2)

Where , , form the Z-Y-X Euler angle set which will be described in more detail in the platform model descript ion (Chapter 5). The length o f the state vector depends on the number o f individual nodes that must be described. We must be careful to subtract the nodes which are shared at the plat form's centre of mass (recall Figure 2.2). We must also add the 6 rotational state variables required to describe the rotational motion of the platform. The total length o f the state vector array is thus given by: nodes =# of nodes per cable = 5
cables = 3

length(X) = 6 [nodes cables - (cables - 2) + 1] = 6 [5 3 - (3 - 2) + 1] = 84

(2.3)

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2.3.2 Numerical Approach
The non-linear, second order different ial equat ions of motion that result fro m the analys is in Chapter 3, are integrated numerically in order to observe the state of the Arecibo system over time. Any second order differential equation may be reduced to a system o f first order differential equat ions by introducing a new variable [8]. The new variable in our case is the velocit y. We then have a set of coupled first order different ial equations that may be written in their most general form as fo llows [8]:

v i (t ) =

dx i (t ) dt

(2.4) (2.5)

dv i ( t ) = f i (t , x, v ) dt

In order to implement a numerical integrat ion scheme, we must know the init ia l values o f the state vector at some time to, and also the funct ions on the right hand side o f Equation 2.5. The Arecibo model uses a fourth order Runge-Kutta integration scheme, which was already implemented in the LAR dynamics model by Nahon [6]. The RungeKutta scheme requires the fo llowing as input: the start and stop times (to and tf), the init ia l

& state vector X and derivat ive o f the state vector X at time to, and the t ime step t . The
integrat ion scheme will then return the state vector values for each time step unt il tf. In a step-wise manner, we can observe the state of the Arecibo system over t ime. Figure 2.3 shows a flow chart of the overall process by which the simulat ion runs.

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The remaining sect ions describing the Arecibo model (Chapters 3 to 5) will focus primarily on how to determine the functions on the right hand side of the genera l Equation 2.5. In other words, we must evaluate the kinemat ics and dyna mics o f the system to develop the governing equat ions of motion that will determine the accelerat ions of the system (which will be used as input to the Runge-Kutta integration scheme).

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Figure 2.3 ­ Flowchart of Simulation Overview

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Chapter 3 ­ Cable Model
As discussed in Sect ion 2.2, the lumped mass approach is used to model the Arecibo cable dynamics. The cable properties, coordinate systems, kinemat ics and dynamics are all discussed in this sect ion. Much of what fo llows ma y be found in more detail in Nahon's work on the LAR system [6]. The final results and equat ions fro m [6] are presented here in order to demonstrate the basics of the cable dynamics model and its use for the Arecibo Radio Telescope.

3.1

Cable Properties
The "original" Arecibo cable configuration consists of 12 braided steel cables,

each of 3-inch diameters. Each tower-top (considered the cable base points) has 4 of these closely spaced cables running toward the near corners of the triangular platform. These 4 mainstay cables are modeled together as one cable having an equivalent effect ive area given by:

n (d c ) Aeff = c 4

2

(3.1)

Where Aeff is the effect ive area of the cable, nc is number of actual mainstay cables per tower and dc is the diameter of each actual mainstay cable. Hence the system is

immediately simplified to one in which only 3 main cables are supported by the 3 towers. Other cable properties include the modulus of elasticit y for braided steel cables. This is taken to be half o f the modulus o f elast icit y for steel, as is co mmo nly used for braided cable construction [9]. The cable properties used in the Arecibo model are given in the Table 3.1.
Mainstay Cable Properties Parameter Symbol Value Individual cable Diameters Density Elastic Modulus dc 0.0762 7850 1.00E+02 Units m kg/m3 GPa


E

Table 3.1 - Mainstay Cable Properties

19


3.2

Coordinate Systems

There are two different frames of reference used when describing the posit ion o f the cable nodes and the orientation of the cable elements. The first is the inertial frame of reference, in which the orthogonal coordinate axes are considered to be fixed wit h respect to the Earth. The origin is located at the bottom-centre of the collector dish (i.e. at an elevat ion equal to the lowermost point of the collector dish). The Zi-axis po ints vertically toward the sky, while the Xi-axis po ints toward one o f the three towers and the Yi-axis completes the coordinate system by the right-hand-rule. The second frame o f reference will be called the "body-fixed" frame which is fixed relat ive to each cable element. The q-axis is tangent ial and in the direct ion o f the cable element itself, while the p1 and p2axes co mplete the orthogonal coordinate system by forming a plane whose normal is a vector in the direct ion o f the q-axis. The body-fixed coordinate system is shown for a given cable element in Figure 3.1.

Figure 3.1 ­ Body-Fixed Coordinate System for a Cable Element

20


3.3

Cable Kinematics

The posit ions o f the nodes are described using the inertial frame and the corresponding (x,y,z) coordinates. The orientations o f the elements are described using an Euler angle set. Specifically, the Z-Y-X Euler angle set is used with the angles denoted by (, , ) . For a more thorough explanat ion o f the Z-Y-X Euler angle set, and how it is to be used, the reader is referred to the plat form model descript ion in Sectio n 5.7.1. The rotation matrix that brings any vector from the body-fixed frame to the
cos i Ri = 0 - sin sin i sin cos
i i i i

inert ial frame is given by [6].
sin i cos i - sin i cos i cos i

(3.2)

cos i sin

The fo llowing equat ions [6] are used to calculate the Euler angles. superscript i and i-1 refer to the two nodes bounding the ith element:

The

i = a tan 2( x i - x i -1 , z i - z i -1 )
( z i - z i -1 ) if cos sin i = a tan 2- ( y i - y i -1 ), cos i

(3.3)

i

(3.4)

( x i - x i -1 ) if cos < sin i = a tan 2- ( y i - y i-1 ), sin i

i

(3.5)

The angle (a rotation about the Z-axis) is taken as zero since the torsion in the elements is neglected (as discussed in the next sectio n).

21


3.4

Cable Dynamics

The forces act ing on the cable elements must be applied to the point masses located at the cable nodes. It is important to distinguish that although the forces are applied throughout the continuous cable, they are treated as act ing only at the nodes. The various forces act ing on the cable elements (but acting at the nodes) are: (i) (ii) (iii) (iv) Cable Tensio n Cable Damping Aerodynamic Drag Gravit y

The above forces are divided into the internal (tensio n and damping) and external (aerodynamic drag and gravit y) forces acting on the cables. The effect of the bending stiffness in the cables is much smaller in magnitude than the tensile stresses and axia l stiffness [9] and is therefore neglected. zero [6]. An important result of this, to the cable kinemat ics, is that the angle of rotation about the inertial Z-axis, , is always equal to

3.4.1 Internal Forces
The internal forces are those forces that are generated within the braided cables. The first force to consider is the tension. Each individual cable element is treated as an elast ic element [6] subject only to axial deformat ion. To calculate the unstretched lengths of the elements, we consider the platform's centre of mass to be at equal elevat ion as the tower tops, and take the distance from the `cable-platform' attachment points to the `cable-tower' attachment points, as shown in Figure 3.2

22


Figure 3.2 ­ Unstretched Length Configuration The unstretched length of the entire cable is found to be: lu = rad t - rad
p

= 213.36 m - 38.01 m = 175.35 m

(3.6)

Wit h 5 elements per cable the unstretched length of each element is taken to be: lu =
i

lu = 35.07 m No. of elements

(3.7)

At equilibrium the elevation o f the plat form's centre of mass is approximately 35 m below the tower tops. The actual lengths of the cable elements, at a given instant, must now be determined in order to calculate the strain. The length of the ith cable element is given by:

l i = ( x i - x i-1 ) 2 + ( y i - y i-1 ) 2 + ( z i - z i -1 )

2

(3.8)

r Where x, y, z are the components of the posit ion vector r which spans from the

inert ial origin to the given node. The except ions to the above length calculat ion are the

23


first or uppermost cable elements (#1, 6, 11) and the last or lowermost cable elements (#5, 10, 15). The length calculat ions for the last elements are based on the second-last nodes (#4, 9, 14) and the cable-plat form attachment points (see platform model Sect ion 5.4 for details). The cable-platform attachment points are not explicit ly considered nodes; however the inert ial coordinates of these "fict itious nodes" are used to determine the lengths and Euler angles of the lowermost elements. The length calculat ions for the uppermost elements are based on the tower-top locations and the first node locations. Once the lengths have been calculated, the axial tensio n forces in the cable elements may be found as [6]:
Tq = Aeff E
i i i u

li - l where = i lu
i

(3.9)

The except ion to using Equation 3.9 direct ly is only in the first cable elements of each cable; the tower-top positions and the tension in elements (#1, 6, 11) depend implicit ly on each other. The equations for the tower-top motion and the first element tensions are explained and so lved simultaneously in the Tower Model sect ion. The second internal force to be considered is the damping generated in the cables. "The energy dissipat ion in the cables is due primarily to the frict ion generated in their braided construction" [6]. The viscous damping is considered proportional to the tangent ial velocit y o f one bounding node relat ive to the element's other bounding node. The damping force is calculated as [6]:
i i i i-1

Pq = Cv (vq - vq )

(3.10)

Where Pq is the damping force in the ith element, Cv is the damping coefficient, and vq is
i i

the tangent ial velo cit y o f the ith node. The damping coefficient, as compared to the mass and the st iffness of a system, is generally a more difficult parameter to measure direct ly [10]. Taken from Nahon's work, the damping coefficients used in the Arecibo cable elements are not considered constant, but rather depend on the tensio n force in each cable element:

24


Cv = 2

i

Tq Aeff E g

i

i

where = 0.015

(3.11)

Where T denotes the tensio n, g is the gravitat ional constant (taken as 9.81 m / s 2 ), and

is the damping rat io taken as 0.015 [11]. Figure 3.3 shows a schemat ic o f the springdamper system in which the spring represents the cable st iffness and the dashpot is used to indicate the presence of the damping force [10].

Figure 3.3 ­ Schematic Representation of the Internal Forces [6]. Finally, a condit io n is imposed in the dynamics model that will always set the tension and damping forces equal to zero if or when the tensio n is calculated as negative. That is, if t he cable element is physically slack then there will be no tensio n or damping forces in that cable element. In realit y the cable elements should never be slack, however in the simulat ion, as the system co mes to a static equilibrium fro m its unstretched lengt h configuration, slack cable elements may be found.

25


3.4.2 External Forces
The external forces act ing on the cable system are the aerodynamic drag and the Earth's gravit y [6]. First considering the gravitational forces, we calculate the weight acting in the negative inertial Z direct ion. The lumped mass approach requires the calculat ion of the mass o f each cable element:
mc =
i i i i u

steel c

V where: Vc = Aeff l

(3.12)

Where Vc is the vo lume o f the cable element calculated based on the unstretched length of the element, lu . The gravitat ional force act ing on the ith cable element is calculated
i

i

using the gravitat ional constant as:
W i = mc g
i i

(3.13)

Where W i and mc refers to the weight and mass of the ith element respectively. However, it is important to note (as will be discussed later) that the gravitat ional force acts at the nodal po ints bounding each cable element. The aerodynamic drag act ing on the cables is found as a funct ion of the relat ive velocit ies of the cable element midpo ints to that of the surrounding air. The coefficient of drag for the cylindrical cable is taken as, C dc = 1.2 [6]. The drag force acting on the ith cable element is calculated as fo llows [6]:

Di = D

i2 p1

+D

i2 p2

i + Dq

2

(3.14)

Where the components are calculated as:
i p1 2

1 i D = - air C d d c lu f p v 2
i p1

i2

v

(v ip1 ) 2 + (v ip 2 )

(3.15)

26


D

i p2

1 i = - air C d d c lu f p v 2

i2

v

i p2 2

(v ip1 ) 2 + (v ip 2 )

(3.16)

1 i i Dq = - air Cd d c lu f q v 2

i2

(3.17)

i Where v i (and its co mponents v ip1 , v ip 2 , vq ) refers to the velocity of the cable element's

midpo int relative to the surrounding air.

The f

p

and f

q

are the required loading

funct ions that "account for the nonlinear breakup of drag between the normal and tangent ial direct ions"[6]. For a more rigorous descript ion o f the loading funct ions and the relat ive velocit y calculat ions the reader is referred to [6].

3.4.3 Translational Equations of Motion
We are now ready to set-up the translat ional equations o f mot ion for the noda l point masses, which fo llow Newton's second law; F = ma . The approach is to apply the external and internal forces at the nodes. In order to apply the drag and weight to the nodes, we must take only half of the drag and weight of the cable elements to the right, and similarly fro m the cable element to the left of the given node. For example, the mass of the ith node is equal to half o f the mass of the element to its right plus half of the mass of the element to its left. The approach is more easily understood by observing the force body diagram of a given node as shown in Figure 3.4.

27


Figure 3.4 ­ Nodal Force Body Diagram The equat ions governing the mot ion of the cable nodes fo llow Newton's second law, such that:
1 + P i +1 ) - (T i + P i ) + (D 2 1 i g) + (Di + mc g) 2

M i&&i = (T r

i +1

i +1

+m

i +1 c

(3.18)

r Where M i is the mass o f the ith node and &&i is the acceleration vector of the ith node. The
fo llo wing po ints should be noted regarding the governing equations of motion: (i) (ii) (iii) The tensio n, damping, and aerodynamic drag forces must be transformed into the inertial frame of reference. The governing equat ion is a vector equation which represents 3 equations in the inertial frame of reference The governing equat ion is a non-linear second order different ial equat ion that must be so lved using numerical techniques (as discussed in Sect ion 2.3.2)

28


Chapter 4 ­ Tower Model

The Arecibo Radio Telescope, being the largest single-dish radio telescope in the world, emplo ys the dist inct tower support structure. In this section, the constructio n details of the towers are discussed and their properties determined. In modeling the towers we first calculate their effect ive stiffness while keeping in mind that our final goa l is to solve the equat ions of motion for the tower-tops (cable base po ints) to be implemented in the dynamics model.

4.1

Construction Details
The three tall towers are each made o f reinforced concrete with a cross-shaped

cross-sectional area. To keep the platform level, their tower-tops were designed to all be at the same elevat ion. Due to the uneven landscape, two of the towers are 250 ft in height and the third is 365 ft in height [12]. The construction o f the towers themselves was a tedious task, rising at a slow rate of less than 10 inches per hour [13]. It took about 16 days of cement pouring to construct one of the two shorter towers [13]. To facilitate their construction, a concrete production plant was installed at the Arecibo construction site. The original Arecibo construction includes 5 backstay cables running fro m the tower-tops to the ground where they are anchored to cement blocks. The main purpose of the backstay cables is to support the towers (in bending) while carrying the heavy load o f the plat form. The 5 backstay cables are each of diameter 3.25 inches. The anchorage locat ions (both in radius and elevation) are unique to each tower; again due to the uneven landscape. Figure 4.1 shows the general construction o f the three towers with their backstay cables. "The towers are labeled as T4, T8, and T12 following the numbers on the face o f a watch, T12 being the one due north." [13]. Table 4.1 gives the dimensio ns and angles o f each tower-backstay configuration.

29


Figure 4.1 ­ Tower and Backstay Cable Configurations

Tower

T8

T4

T12

Property Tower Height Anchorage Radius Backstay Cable Length Tower-Anchorage Elevation Backstay Cable Angle Tower Height Anchorage Radius Backstay Cable Length Tower-Anchorage Elevation Backstay Cable Angle Tower Height Anchorage Radius Backstay Cable Length Tower-Anchorage Elevation Backstay Cable Angle

Symbol L a b c L a b c L a b c

Value 111.252 118.872 58.3692 132.4293 26.1522 76.2 138.684 99.5172 170.6954 35.6626 76.2 115.824 83.058 142.5266 35.6445

Units [m] [m] [m] [m] [deg] [m] [m] [m] [m] [deg] [m] [m] [m] [m] [deg]

Table 4.1 ­ Tower and Backstay Cable Configurations 30


4.2

Tower Properties
The tower cross-sectional dimensio ns are not constant throughout their lengths.

In fact, the cross sectional area decreases fro m base to top both continuously and in clearly vis ible increments at particular heights. The dimensions and geo metry o f the cross-section is required to determine the mo ment of inert ia o f the towers to be used in determining their st iffness in bending. As an approximation, the tower cross-sect ion is assumed to be constant based on dimensio ns that are available from the AutoCAD drawings o f the Towers [14]. Figure 4.2 shows the geo metry o f the tower cross sect ion. Table 4.2 gives the constant tower properties as used in the Arecibo model

Figure 4.2 ­ Tower Cross Section

For the given cross-sectional geo metry (which is symmetric) the area mo ment of inertia is given by: [15] BH 3 + ( H - B) B I= 12
3

(4.1)

31


Property Cross Sectional Dimension Cross Sectional Dimension Area Moment of Inertia Concrete Elastic Modulus Tower Height (2) Tower Height (1)

Symbol H B I E L L

Value 2.74 1.83 2.64 25 76.20 111.25

Units [m] [m] [m4] [GPa] [m] [m]

Table 4.2 ­ Tower Properties

4.3

Effective Stiffness

Important to the cable and tower models is the fact that the 4 mainstay cables are completely different than the 5 backstay cables. "The reason for the different number of cables and diameter is related to the different angles at which the cables carry the loads" [13]. Figure 4.3 shows a photograph of one of the three tower-tops as seen from ground level. Notice the 5 backstay cables running to the right, the 4 mainstay cables to the left, as well as the main and backstay auxiliary cables installed during the second Arecibo upgrade. Since the mainstay cables are terminated at the tower-tops we may model the combined effect of both the tower itself and the backstay cables as one. For the purpose of modeling the system, the tower and backstay cables are replaced by a single-degree-offreedo m spring wit h an effect ive st iffness, keff , and no damping, as shown in Figure 4.4. In order to calculate the effect ive st iffness, we first separate the physical problem into two parts: the contribut ion o f the tower's st iffness in bending and the contribut ion o f the backstay cables.

32


Figure 4.3 ­ View of Tower-Top, Mainstay and Backstay Cables.

Figure 4.4 ­ Effective Tower Stiffness Model

4.3.1 Tower Contribution
To calculate the contribution o f the tower's stiffness in bending, we treat the tower as a simple beam subject to the transverse load of the mainstay cable tensio n. The deflect ion of the cant ilever beam is given by the fo llowing equation [16]:

33


xtop

TL3 = 3EI

(4.2)

Where L is the length or total height of the towers, E is the modulus o f elast icit y o f the concrete material, and I is the area mo ment of inertia o f the towers. In treating the towertops as 1-dimensio nal springs, whose mot ion is constrained to the horizontal plane, the contribution o f the stiffness may be calculated using Hook's Law:
T =k xtop

tower

(4.3)

k

tower

=

T 3EI =3 xtop L

(4.4)

4.3.2 Backstay Cable Contribution
In order to determine the contribution o f the backstay cables to the overall effect ive stiffness, we use a perturbation approach. The problem is that of so lving the angles and lengths of the backstay cables geo metrically before and after a horizontal perturbation o f the tower-top. The perturbation acts to stretch the backstay cables fro m their init ial length, thus creating a reaction force (a tensio n) from which the effect ive stiffness may be calculated. Figure 4.5 is a schemat ic of the geo metry. The init ia l length, L1 , and angle to the horizontal, 1 , of the backstay cables are found using the design details o f the AutoCAD drawings [14]. The cosine law is used to calculate the length o f the backstay cables after the tower-top is displaced horizontally by a distance
xtop . . The actual magnitude of the perturbation has no effect on the final backstay cable

contribution (so long as the perturbation is small, i.e. less than 1 m)

2 L2 = L1 + xtop - 2 L1 cos

2

where = 180 - 1

(4.5)

34


Figure 4.5 ­ Geometry of the Perturbation Approach The 5 backstay cables are treated as one cable whose area is calculated by: nbc (d bc ) 4
2

Aeff =

(4.6)

The change in tension as a result of the tower-top moving a distance xtop . may then be found using: L2 - L1 L1

T = Aeff E

where =

(4.7)

In the horizontal direct ion: Tx = T cos 1 (4.8)

Finally the contribut ion of the backstay cables to the overall effect ive st iffness may be found using:

35


kbc =

Tx x

(4.9)

4.3.3 Combined Effective Stiffness
We may now calculate the combined effect ive stiffness of the towers and the backstay cables. The springs are taken as act ing in paralle l such that the total effect ive stiffness is the sum o f the two parts:
k
eff

= kbc + k

tower

(4.10)

As an example, the values found for tower T4 are
k
tower

kbc = 1.03 â 10 7 N / m ,

= 4.57 â 105 N / m and k

eff

= 1.08 â 10 7 N / m . It should be noted that the contribut ion

of the backstay cables is 2 orders of magnitude greater than the tower itself. As shown in Figure 4.1 and Table 4.1 the geometry o f each tower and its backstay cables are each unique. This implies that the amount of tower deflect ion for each tower, under the static loads would each be different. In realit y the tensio ns in the backstay cables are pretensioned such that, upon init ial erection, the plat form's posit ion and orientation is centered and level above the dish. In the Arecibo model, for all three tower-top locations, an average of the three cases is used as the overall effect ive st iffness. The final co mbined and averaged effective st iffness is presented in table 7.1 o f the sensit ivit y analysis (Section 7.4).

4.3.4 Tower-Top Motion
As ment ioned in Section 3.4.1, the tension in the first (uppermost) elements o f each cable and the length o f the first cable elements depend implicit ly on each other. That is, we need to know the length, relative to the unstretched length, in order to calculate the tensio n force, however the tensio n force must be used to determine the tower-top location and hence the length, as per Equation 3.8. Simply put, at a give n instant, they both must be calculated simultaneously. A system o f equat ions is set-up and

36


solved analyt ically, in closed form, to arrive at a new expression for the tensio n in the first elements and to govern the tower-top motion. The fo llowing equat ions describe the tension and the tower-top locations. T = EAeff ( l - lu ) lu
2

(4.11)

where l = ( xt - xn ) 2 + ( yt - yn ) 2 + ( zt - z n )

(4.12) (4.13) (4.14)

xt = rad t cos - yt = rad t sin -

T cos k eff T sin k eff

In the above system of equations, the unknown quant it ies are the tensio n, T, and the posit ions o f the tower-tops in the horizontal plane of motion, xt and yt (in the inert ial frame o f reference). All other quant ities are considered known. An increase in the mainstay cable tensio n acts to move the tower-tops closer to the centre of the co llector dish, or in other words, it acts to decrease the tower-top radial distance in the horizontal plane. The angle is calculated in the plan view of the Arecibo, based on the tower orientation, as shown in Figure 4.6.

Figure 4.6 ­ Tower Orientation in the Inertial Frame of Reference

37


After so me lengthy algebra, we arrive at a quadratic equation of T in terms of the known quant ities. The final so lut ion is of the form:

AT 2 + BT + C = 0 Where:

quadratic formula

T=

- B - B 2 - 2 AC 2A

(4.15)

A= B=-
2

1 k
2 eff

(4.16)

2rad t 2 x n 2y cos + n sin + k eff k eff k eff
2 n

(4.17)
2 n

C = rad t - 2 x n rad t cos - 2 y n rad t sin + x

+y

+ ( z1 - z 2 )

2

(4.18)

It turns out that the negat ive square-root of the quadratic equat ion yields tensio ns that make sense phys ically with the actual Arecibo system, whereas the posit ive squareroot yields unreasonable magnitudes. We may now back-subst itute, using Equations 4.13 and 4.14, to calculate the tower-top inert ial coordinates in the horizontal plane. As an approximat ion, the angle that the mainstay cables make wit h the horizontal plane o f motion o f the tower-tops is not taken into account. In realit y, the angles at equilibrium are in fact small (i.e. less than 10 degrees), the cosine of the angle is always greater than 0.98, and we make the assumptio n it is approximately 1.0. in the final equilibrium posit ion o f the tower-tops. This simplificat ion in the model, allows for a closed form so lut ion, but introduces a small error In fact, the difference in the equilibrium posit ion of the tower-tops (when making this simplificat ion) is approximately 1.8 cm of a total 94 cm deflect ion fro m the tower-tops init ial start position being that of zero cable tensio n. Note that when the Arecibo was init ially constructed, the cables were all pre-tensioned and the towers did not deflect more than 2 inches throughout the process [13]. The percent error in the equilibrium posit ion may be calculated as: % Error = 1.8 cm â 100% = 1.9 % 94 cm 38 (4.19)


To eliminate this simplificat ion, the angle that the first cable elements make wit h the horizontal plane o f the tower-top motion would also have to be so lved simultaneously with the tensions and tower-top posit ions, making the problem (specifically the mathemat ics) even more difficult. Current ly, the angles o f all the elements must be calculated using the known posit ions of each end of the elements (as is done for the LAR system). With the effect of the towers and backstay cables co mbined as an effect ive stiffness and the tower-top equations of motion derived, we move onto the final physica l component of the Arecibo Telescope; the platform.

39


Chapter 5 ­ Platform Model

The receiver o f the Arecibo Radio Telescope is held alo ft by a very large triangular truss section, which will be referred to herein as the "platform". The mot ion o f the receiver, which is important to the astronomer (see performance metrics Section 6.2), will in fact be modeled by evaluating the dynamics of the plat form's centre of mass. To begin, the construction details and the platform properties will be discussed. Next a model for the aerodynamic drag experienced by t he platform will be presented. Finally, the plat form will be treated as a rigid body, and the rotational equat ions o f motion governing its orientation in space will be discussed.

5.1

Construction Details

The plat form is a very heavy structure which upon init ial erection had a total suspended mass o f approximately 550 000 kg (or 550 tons). After the second upgrade the plat form's mass was increased to 815 tons [2]. The platform is constructed as a truss structure on which the receiver may be posit io ned with millimeter precisio n [2]. The receiver may mo ve alo ng an azimuth arm which may rotate about a circular track in order to take on various zenith and azimuth angles. Since Arecibo 's co llector dish is spherical, the incident radiat ion does not reflect to a single fo cal po int, but rather reflects onto a line [5]. For the Arecibo Radio Telescope, the receivers/transmitters are either in the form o f a line feed or a set of enclosed secondary and tertiary reflectors, known as the Gregoria n (as was implemented in the second Arecibo Upgrade co mpleted in 1997) [2]. Important to the model is the fact that Arecibo's plat form does not itself take on different azimuth and zenit h angles; meaning that there is no need to consider these angles in the cable model (unlike the LAR in which the cable lengths are changed using winches to allow the receiver to take on different azimuth and zenit h posit ions). Figure 5.1 shows a picture of 40


the rather complex truss structure. The azimuth arm and circular track can be seen, alo ng with the two types of receivers/transmitters (the line feed on the left and the spherica l Gregorian on the right).

Figure 5.1 ­ A Picture of the Arecibo Platform

5.2

Coordinate Systems
In order to model the plat form, we first define the coordinate systems used to

describe its posit ion and orientation in space. Similar to the cable model we use the inert ial frame o f reference wit h axes Xi-Yi-Zi (as defined in Sect ion 3.2). We also use the "body-fixed" frame oriented such that the Zb-axis points upward (toward the sky) in a 41


direct ion normal to the plane of the triangular shape truss. The Xb-axis po ints towards one of the corners of the isosceles triangle and the Yb-axis co mpletes the orthogonal coordinate system using the right-hand-rule. The origin o f the body-fixed coordinate system is located at the plat form's centre of mass. The centre of mass will be defined in the next section. Figure 5.2 shows the body-fixed coordinate system.

Figure 5.2 ­ The Platform Model's Body-Fixed Coordinate System

5.3

Platform Properties
The fo llowing simplificat ions are imposed on the complex truss structure in order

to model the plat form and evaluate its properties. (i) The plat form's structure (which includes the azimuth arm is reduced to a slice of an isosceles triangular shaped sect ion. The section is o f equal dimensio ns to the outer triangular truss frame of the actual Arecibo platform (may be see n in Figure 5.1).

42


(ii)

The centre of mass is considered to be at the geometric centre of the triangular sect ion. Important to the dynamics model is the fact that the lower-most node (i.e. the confluence po int #15) is located at this centre of mass of the modeled plat form.

(iii)

The platform is o f uniform densit y such that its mass (defined as the total suspended mass ­ including the receivers) is evenly distributed over its vo lume.

Table 5.1 gives the plat form's phys ical dimensions and properties as used in the Arecibo Model:
Property Mass Base Height Vertical width Gross Volume Unif orm Density Symbol mp b h w V Value 550 000 65.84 57.02 9.14 34 313 16.03 Units [kg] [m] [m] [m[ [m3] [kg/m3]

Table 5.1 ­ Physical Properties of Platform In the Sect ion 5.7 we will treat the plat form as a rigid body subject to both translat ional and rotational motion. The rotational equat ions of mot ion require that we know the plat form's mass mo ment of inertia. The mass mo ment of inertia of any 3dimensio nal body is a measure of the body's resistance to angular accelerat ions [17] and is given by [7].
I xx = ( y 2 + z 2 )dm
M

I xy = I yx = xydm
M

I

yy

= ( z 2 + x 2 )dm
M

I xz = I zx = xzdm
M

(5.1)

I zz = ( x 2 + y 2 )dm
M

I yz = I zy = yzdm
M

43


The mass mo ment of inertia o f the plat form is convenient ly expressed in a 3x3 matrix, known as the inertia matrix (or tensor), for use in the rotational equat ions o f mot ion [18]: I xx I = - I yx - I zx -I
xy

I yy - I zy

- I xz - I yz I zz

(5.2)

Rather than trying to evaluate the above integrals analyt ically, and wit h no success in finding expressio ns for the mass mo ment of inertia of a co mmon triangular shaped sect ion such as the plat form, our attention was turned to available CAD so ftware. The plat form was created as a 3-dimensio nal part in ProE® software, into which the uniform densit y o f the platform could be entered as a material property. The result ing inert ia matrix, for the properties in Table 5.1, is found to be: 1.032 â 10 I= 0 0 0 1.987 â 10 8 0

8

0 1.032 â 10 0
8

(5.3)

The cross terms in the above inert ia matrix are all zero, implying that there are two planes o f symmetry in the modeled platform. This is indeed the case with one plane of symmetry being the x-z plane and the second being the x-y plane. It should also be noted that when the sensit ivit y analysis is performed (Chapter 7), and the plat form mass is changed, the mass mo ment of inertia must be recalculated for each case.

44


5.4

Cable Attachment Points
The mainstay cable attachment po ints are considered to be in line wit h the

plat form's centre of mass, as previously define d. That is, they are considered to be located at Z b = 0 in the body-fixed coordinate system and at a radius o f rad p = 38.01 m in the direct ion of the three towers. As per the Arecibo AutoCAD drawings, the mainsta y cable attachment points are in fact in line wit h the plat form's arm, the receiver, and in fact all other extruding truss sect ions into the simpler triangular sect ion. Figure 5.3 shows the cable attachment points in the actual platform vs. the model.

Figure 5.3 ­ Cable-Platform Attachment Points 45


5.5

Platform Drag
The model for evaluat ing the aerodynamic drag forces act ing on the triangular

truss plat form is here described.

In general, despite the wide applications o f truss

frameworks subjected to wind forces, calculat ing the drag coefficients o f such structures is in so me cases "inconsistent and in disagreement with experimental results" [19]. In other words, without experimental results, obtaining accurate drag coefficients o f a specific or a particular truss structure can be a very difficult task. That being said, we must still attempt to at least approximate the drag forces act ing on the plat form and to do this we make various simplifying assumpt ions to the structure itself. Arecibo 's plat form alt hough a huge structure is composed of largely spaced truss members. The plat form is treated as being very poor at generating a lift force. So muc h so, that the induced drag (the component of drag due to lift generat ion) and the lift force itself may be neglected. Hence, the drag acting on the plat form is assumed to be caused ent irely by form t ype drag. In general, form drag results fro m the pressure distributio n normal to the body's surface [20]. The case o f pure form drag act ing on a flat plate normal to the flow is shown in Figure 5.4. In the case o f bluff bodies with sharp edges, such as this, the drag coefficients tends to be approximately constant over a large range o f Reyno lds numbers [20].

Figure 5.4 ­ Flat Plate Normal to the Flow [20].

46


5.5.1 Factors Affecting Drag
The main factors to be considered when determining the drag force acting on a trussed framework in general, at a given wind speed, are listed in Simiu and Scanlan [19]. i) The aspect ratio, : the ratio of the length o f the overall framework to its width. Which is used to consider either the two-dimensional ( = ) or threedimensio nal ( = finite) nature of the truss section ii) The so lidit y rat io of the trussed framework, : the ratio of the effective area to the gross area (bounded by the outer truss members). The effect ive area is here defined as the area that the shadow of the truss members would project onto a plane which is behind the framework and perpendicular to the airflow.
Aeff = A
gross

(5.4)

iii) iv)

The angle o f attack, , of the plat form wit h respect to the oncoming wind Truss frame shielding: the shielding of portions of the framework by other portions o f the framework located upwind. We will consider the effects o f having two adjacent truss frames normal to the wind. The first step in approximat ing the drag coefficients of the plat form is to separate

the horizontal and vert ical sides o f the plat form and consider them separately with respect to the oncoming wind direct ion. The spherical Gregorian receiver (introduced after the second upgrade) is also considered separately (in a later section) and added to the overall drag force only when considering the "upgraded" Arecibo configurat ion. Figure 5.5 shows the deco mposit ion o f the various co mponents of the platform. For the truss frames we use Equation 5.4 to calculate the effect ive area of the given truss face. Considering the wind direct ion normal to the given truss face, we consider its drag coefficient equal to that of a 3-dimensio nal flat plate (also oriented normal to the wind). The area of the 3dimensio nal flat plate is considered equal to the effect ive area of the given truss face. When we take the angle o f attack of the two truss faces into considerat ion (Section 5.5.4),

47


we will see how the drag forces of the two platform faces are co mbined to find the overall plat form drag.

Figure 5.5 ­ Drag Component Breakdown The coefficient of drag for a 3-dimensional flat plat, with an aspect ratio of 1, normal to the wind is taken from [20] as:
C
(1) D

= 1.18

(5.5)

Where the superscript (1) indicates that this is the drag coefficient for a single truss frame normal to the wind (i.e. no truss frame shielding is taken into account).

5.5.2 Side View of Truss
To obtain the drag coefficient of the vert ical face, #1 of Figure 5.5, we consider the wind force to act normal to the vert ical truss face o f the platform. The so lidit y rat io was approximated using the available AutoCAD drawings [14] and various photographs of the truss structure. Figure 5.3 shows an AutoCAD drawing of the platform's vertica l face wit h the azimuth arm perpendicular to the wind direct ion (which is always assumed).

48


For the vertical truss face, #1, the solidit y rat io is taken as = 0.3 . The effect ive area is then calculated as:

A1eff = A1

gross

= (b w) (5.6)
2

= 0.3 (65.84 9.14) = 180.53 m

Where, the dimensio ns of the platform (b, h, and w) are defined by Figure 5.2 and Table 5.1. The aspect ratio of the vertical truss face is given by: b w 65.84 = 9.14 = 7.2

1 =

(5.7)

As a first approximat ion to the effects of truss frame shielding (factor iv as listed above) we model the vertical face as consist ing o f two parallel truss frameworks adjacent to one another and separated by a distance "e". The separation o f the two vertical truss frames is taken as h/2 of the plat form; e =
h 57.02 = = 28.51 m . Both truss frames are 2 2

considered normal to the wind. The rat io of the spacing to the width of the truss frames, will prove to be useful, and is denoted by e / d [19]. The width of the truss frames is taken as the dimensio n w of the plat form, d = w = 9.14 m . Taking the width as `w'

rather than `b' (as per Figure 5.2), we more closely represent an infinite aspect ratio truss; a correction factor for the infinite aspect ratio assumpt ion will later be considered. For the vertical face the ratio e / d is thus given by:
e 28.51 = = 3.12 d 9.14

(5.8)

In Simiu and Scanlan [19] this problem was looked at by considering that the overall drag coeffic ient of the two vertical trusses (denoted C
( 2)

D

) may be obtained fro m

the drag coeffic ient of a sing truss frame, when mult iplied by so me correction factor, k:
C
( 2)

D

= k C

(1)

D

(5.9)

49


The approximate correction factors are presented in Figure 5.6, which plots the correction factor k =
C C
( 2)

D
(1)

versus the so lidit y rat ios for various ratios o f

D

e > 1 . For d

the vertical face o f the plat form, we find that the correction factor is given approximately by: k = 1.7. Therefore, we find the drag coefficient of the side view o f the plat form whe n normal to the wind is approximated by:
(2) D1 (1) D

C

= k C = 2. 0

= 1.7 â 1.18

Figure 5.6 ­ Truss Pair Correction Factors [19].

50


The values o f C

( 2) D

obtained by Figure 5.6 are given for infinite aspect ratio truss frames. C D ( ) is very C D ( = )

It has been found in [19] that for small so lidit y ratios, the ratio of

nearly constant (and close to one) for a wide range of aspect ratios. Figure 5.7 is used to find a second correction factor that will give us the final co mbined drag coefficient for the side view of the plat form:

Figure 5.7 ­ Finite to Infinite Aspect Ratio Correction Factors [19]. Therefore the final drag coefficient for the vert ical face of the platform is given by:
C
(2) D

= 0.96 * 2.0 = 1.9

(5.10)

5.5.3 Top View of Truss
The top side of the plat form is considered to be far less "so lid" than the vert ical side. It has a much larger gross area and smaller aspect ratio than the vert ical surface o f the platform. To approximate the drag coefficient of the top face of the truss, we assume that it is placed normal to the wind (it should be noted that this condit ion is never realized by the Arecibo plat form). For the top truss face, the solidit y rat io is taken as = 0.1 . A photograph of the plan view of the platform is shown in Figure 5.8. 51


Figure 5.8 ­ Plan View Photograph of the Arecibo Platform The effect ive area of the top face is calculated as: A2
eff

bh ) 2 65.84 57.02 = 0.1 ( ) 2 = 187.71 m 2 = A2
gross

= (

(5.11)

We model the vert ical face as consist ing of two parallel truss frameworks adjacent to one another and separated by a distance e = w = 9.14 m , equal to the vertical width o f the plat form, w. The length and width o f the truss frame are approximat ly equal, as if it
2 where a square. Hence the aspect ratio is taken as 1, with d = b = 43.89 m . The ratio of 3

the spacing between truss fames to the width of the truss frames is given by:
e 9.14 = = 0. 2 d 43.89

For ratios of

e < 1 we find the coefficient of drag of the pair of trusses using the same d
( 2) D (1) D

principle as the vert ical side of truss, that is:
C = k C

Where k is given here by the sum o f two correction factors plotted in Figure 5.9 k = (I + II ) (5.12)

52


Figure 5.9 ­ Correction Factors for Adjacent Truss Frames Used for the Top View of the Platform [19]. As can be seen for the vert ical face having a so lidit y ratio less than 0.2 and e / d = 0.2 , the correction factors are taken as: 1 1 II 0
k =1+ 0 =1

(5.13) (5.14)

Therefore the combined drag coefficient of the top truss face pair is given by:
C
( 2) D

C

(1) D

1.18

(5.15)

Correcting for the infinite aspect ratio, for which the correction factors are based, and using Figure 5.7, we obtain:
( 2) D2

C

= 0.97 â 1.18 = 1.14

(5.16)

53


In summary, Table 5.2 gives the final drag properties of both the side and top faces of the plat form:
Property Coefficient of Drag, CD
2 (2)

Side View 1.9 180.53 0.3 7.2 3.12

Top View 1.14 187.71 0.1 1 0.2

Eff ective Area, Aeff [m ] Solidity Ratio, Aspect Ratio, e/d

Table 5.2 ­ Summary of Drag Parameters

5.5.4 Platform Angle of Attack
The wind and turbulence model used in the Arecibo model was previously developed and used by Nahon in his model o f the LAR system [6]. It considers the mean wind to move only in the horizontal plane, such that the Zi - co mponent of the mean wind velocit y is always zero. The magnitude o f the wind velocit y may be calculated as:

Vw = VWx + VWy

2

2

(5.17)

So far we have found approximate values for the plat form truss sect ion given two particular cases: the wind is normal to either the top of the platform or the sides o f the plat form. In order to calculate the angle of attack at any instant in t ime, we need to know the orientation of the plat form and the co mponents of the wind velocit y vector. The components of the wind velocit y vector are known fro m the wind model (developed for the LAR system). The unit vector in the direction o f the wind velocit y (which does not consider the Z-component of the wind that would be introduced by the turbulence model) is given by:
V V ^ ^ Vw = wx ^ + wy ^ + 0 k i j Vw Vw

(5.18)

54


Wit h regards to the orientation of the platform, we need to know the components of the unit vector in the same direct ion as the platform's body-fixed Zb ­ axis. Here we introduce the rotation matrix that brings a vector's components in the body-fixed frame to components in the inert ial frame as [24]:
cos sin cos cos cos sin sin cos - cos sin cos sin sin sin sin - cos cos cos sin sin - sin cos (5.19) T= - sin cos sin cos cos

I B

The 3rd co lumn of the above rotation matrix gives the direction o f the Zb axis in the inert ial frame [7], which is exact ly what we need. The unit vector, in the direct ion of the Zb axis, is given in the inertial frame of reference as:
^ ^ Z B = (cos sin cos )^ + (cos sin sin - sin cos )^ + (cos cos )k i j

(5.20)

The angle o f attack of the platform is defined relative to the vertical sides of the plat form (i.e. the side view) as shown in Figure 5.10.

Figure 5.10 ­ Angle of Attack

55


^ The angle of attack is calculated as the angle between the unit vectors Z B and ^ Vw . This angle is found by taking the dot product of the two unit vectors such that:

Z ·V ^ ^ w = cos B Z V ^ B ^w
-1



(5.21)

We are now ready to evaluate the overall drag force act ing on the plat form at any given angle of attack. In theory, the angle of attack of the plat form covers the range of 90 to 0 degrees. In realit y the angle of attack relative to the vertical face of the truss is always very close to 90o. In other words, the drag of the plat form will be due mainly to the vertical face of the plat form normal to the wind. However we have st ill created a model capable o f taking into account the ent ire range o f angles o f attack. This is done by varying the product of the coefficients o f drag times their respect ive effect ive areas fro m one extreme value o f to the next. The reason for doing this is because the drag coefficients, C
( 2)
D1

(for the side view) and C

( 2)
D2

(for the top view), are based o n

different effect ive areas. The way in which we vary this product of C D A is such that a smooth curve is generated by the fo llowing relat ion [21]. Figure 5.11 shows the variat ion of the product of C D A with angle of attack.
( 2) D1

C D A = (C

A1eff ) sin 2 + (C

( 2)
D2

A2eff ) cos 2

(5.22)

The drag force acting on the platform at all angles of attack is treated as act ing through the platform's centre of mass. Finally, for the "original" Arecibo configuration, the drag acting on the platform is given by:
1 airVw2 (C D A) 2

Dc

of M

= Dp =

(5.23)

56


Platform CDA vs Angle of Attack
360

340

320

Cd*Area (effective) [m2]

300

280

260

240

220

200 0 10 20 30 40 50 60 70 80 90 Angle of Attack relative to Vertical Face [deg]

Figure 5.11 ­ Platform Cd*A versus Angle of Attack

5.5.5 Drag of Gregorian System
The drag coefficient of the Gregorian receiver is calculated based on the drag around a sphere. The model is used fro m Nahon's previous model o f the LAR paylo ad treated as a sphere. The Reyno lds number (based on the diameter of the sphere) is important to the drag coefficients in this case, since the separation po int of the flow ma y change with Reyno lds number as shown in Figure 5.12.

57


Figure 5.12 ­ Flow Around a Sphere for (a) Low Reynolds Number and (b) High Reynolds Number [20]. For the various ranges o f Reyno lds number (corresponding to laminar, transit ion, and turbulent flow) the fo llowing drag coefficients are used to calculate the drag force acting on the Gregorian receiver: For Re > 631000 : C = 0.15
= 0.4 - 0.25 â log(Re) - 5.4 0.4

DG

(5.24a) (5.24b) (5.24c)

For 251000 Re 631000 : C For Re < 251000 : C

DG

DG

= 0.4

The drag force on the Gregorian is then calculated based on:
1 DG = C 2
DG

AG Vw

2

(5.25)

58


Where AG is the cross sectional area of the sphere with diameter dG = 19.81m :
AG =

d

2 G 2

4 (19.81 m) = 4 = 308.22 m 2

(5.26)

We only consider the drag o f the Gregorian receiver when we are modeling the "upgraded" Arecibo configurat ion. For the "original" Arecibo configurat ion the drag caused by the narrow line feed is neglected as compared to the drag caused by the rest of the plat form structure. A final note regarding the Gregorian drag is that it is treated as acting through the plat form's centre of mass, thereby causing no mo ments on the plat form. When modeling the "upgraded" Arecibo, the total drag force acting on the node located at the platform's centre of mass is given by:
DC = DP + DG

of M

(5.27)

5.6

Translational Motion

The translat ional mot ion o f the platform is defined by t he accelerat ion of its centre of mass, denoted here with the subscript G, in the inertial frame. The centre of mass of the plat form is node #15 according to Figure 2.2. The equat ions of motion are governed by Newton's second law such that [22]:
F = ma G =





Fx = m(aG ) Fy = m(aG ) Fz = m(aG )

x y z

(5.28)

The sum o f the forces act ing on the centre of mass is the vector summat ion o f the components of the forces act ing on the platform. The forces act ing on the platform include the cable forces (as considered in the cable dynamics section), the platform's

59


weight, and the aerodynamic drag (which is assumed to act through the plat form's centre of mass). The cable forces are taken as those forces that would be act ing at the nodes #5, 10, and 15 if t hey were each considered the last node of their respect ive cables. The last (and shared) node being the centre of mass of the platform is taken as having a mass equal to that of the ent ire platform; 550 000 kg. The accelerat ion o f the plat form's centre of mass, at a given instant in t ime, is then calculated by dividing the sum o f the forces at that instant in time by the total platform mass. These accelerat ions are entered into the
& appropriate elements of X(t ) ; the derivat ive of the state vector.

5.7

Rotational Motion
The LAR model, fro m which the Arecibo model is developed, considers its

"payload" (equivalent to Arecibo's plat form) to be spherical and subject only to translat ional motion. We will now consider the effect of the mo ments caused by the forces acting at a particular locat ion on the plat form. In order to do so, the plat form is first converted fro m a po int mass to a rigid body, and the cable-platform attachment points are defined (Section 5.4). A rigid body is such that its changes in shape may be neglected as co mpared to the overall dimensio ns of the body [17]. The plat form is defined in space by not only the posit ion o f its centre of mass, but also by its orientation. The plat form must satisfy Euler's rotational equat ions of motion given by [25]: r rrrrr & M CM = I CM + â I CM

(5.29)

Before discussing the strategy for setting up and numerically solving Euler's rotational equations of motion, we will first consider the definit io n of the Euler angle set as well as the transformat ions required between the body-fixed and inertial coordinate systems.

60


5.7.1 The Z-Y-X Euler Angles
A new set of state variables, in the inert ial frame, are defined to account for the
& & & plat form's orientation and rotational mot ion in space; (, , , , , , ) . That is, the

plat form's state (in general) is co mpletely defined by its posit ion and velocit y, and also by its orientation, and angular velocit y. As is commo nly used in Aeronautics and Flight Mechanics [20], the Z-Y-X Euler angle set is used in the Arecibo model. The main difference from the Z-Y-X angle set as used in flight mechanics is the orientation of the Z-axis (which is pointed toward the earth in flight mechanics [23] but is po inted toward the sky for the Arecibo Radio Telescope). The orientation o f the platform is given by the Z-Y-X Euler angles of , , using the fo llowing procedure (taken direct ly fro m [24]) and is also shown schemat ically in Figure 5.13 (i) (ii) (iii) Rotation by an angle about the inert ial Z-axis Rotation by an angle about the result ing Y-axis (from the previous rotation) Rotation by an angle about the result ing X-axis (from the previous rotation)

Figure 5.13 ­ Using the Z-Y-X Euler Angle Set [24]

61


Note that using the right hand rule gives the directions for posit ive angles o f rotation. Care should be taken here since the final orientation of the platform does in fact depend on which order these rotations take place. Wit h no aerodynamic forces or init ia l perturbations act ing on the plat form the inert ial and body-fixed frames will remain aligned and the Euler angles will all be zero [20].

5.7.2 Transformations
In the body-fixed frame the angular velocit ies are defined as p, q, and r as shown in Figure 5.14. We must define the transformat ion matrices for both vector and rotationa l parameters in order to allow us to move back and forth fro m the body-fixed to the inert ia l frame as required. For example, this is necessary when the cable-platform attachment points are defined in the body-fixed frame and their coordinates are required in the inert ial frame of reference. Since the platform is treated as a rigid-body, these inert ia l coordinates are determined fro m the Euler angle set. The transformation matrices have been used in the simulat ion of a mult i-tethered aerostat system by Zhao [24]. The fo llo wing equat ions, transformat ions, and convent ions are taken direct ly fro m this work.

Figure 5.14 ­ Angular Velocities in the Platform's Body-Fixed Frame

62


The rotation matrix that brings a vector's components in the body-fixed frame to components in the inert ial frame is given by [24]:


xi xb = IT y yi B b zb zi

(5.30)

Where:
cos sin cos cos cos sin sin cos - cos sin cos sin sin sin sin - cos cos cos sin sin - sin cos T= - sin cos sin cos cos

I B

(5.31)

The lower and upper subscripts of B and I indicate that the given rotation matrix is used when go ing fro m the body-fixed to the inert ial coordinate systems. Notice that the rotation matrix used in the cable model Equat ion 3.2 is obtained fro m the above rotation matrix, Equation 5.31, by setting = 0 [24]. To move fro m vector components in the inertial frame to the body-fixed frame the inverse o f the rotation matrix is required and obtained by the transpose of IBT , since the rotation matrix is orthogonal:
B I

T = IBT

-1

= IBT

T

(5.32)

B I

cos cos cos sin - sin sin sin cos - cos sin sin sin sin - cos cos cos sin T= cos sin cos cos sin sin - sin cos cos cos

(5.33)

63


Finally, we must also form the transformat ion matrix that will allo w us to move back and forth between the t ime derivatives o f the Euler angles in t he inert ial frame and the angular velocit ies p, q, and r of the platform in the body-fixed frame.
p = Tw q & r & &



(5.34)

Where:
1 sin tan Tw = 0 cos 0 sin / cos

cos tan - sin cos / cos

(5.35)

The inverse transformat ion, which allows us to obtain the angular velocit ies o f the plat form fro m the time derivat ive of the Euler angles, is given by:
0 1 = 0 cos 0 - sin - sin cos sin cos cos

Tw

-1

(5.36)

A different set of Euler angles using the Z-Y-Z convent ion was init ially developed for the Arecibo platform. However, since the transformat ion matrix, Tw , in ZY-Z convent ion beco mes degenerate whenever sin = = 0 , problems arose in trying to avo id that particular condit io n. If we instead use the Z-Y-X Euler angle set, we see that the transformat ion matrix, Tw , in this case beco mes degenerate whenever cos = 0 or

= 90 o , however this case is never physically realized by the Arecibo platform. Hence,
the Z-Y-X convent ion has been chosen to model the platform's rotation.

64


5.7.3 Rotational Equations of Motion
The equat ion that governs the plat form's rotation is known as Euler's equat ions o f mot ion, and is given by [22]:
r rr r rr & M = I + â I

(5.37) As previously mentioned, the

The mo ments acting on the Arecibo plat form must be evaluated in order to solve Euler's rotational equations of motion numerically. aerodynamic drag o f the plat form is assumed to act through the plat form's centre of mass, meaning that the wind force on the truss section does not apply any mo ment to the plat form. The cable-plat form attachment points however do indeed cause a mo ment, which is taken into account in the fo llowing manner: (i) (ii) Sum the cable forces acting alo ng the q-axis in the cable body-fixed frame. Transform these forces (act ing alo ng the cable q-axis) to the inertial frame o f reference and reso lve the forces of the three mainstay cables into their inert ia l components (iii) (iv) Transform the result ing sum o f inertial force co mponents into the platform's body-fixed frame o f reference. Find the components of the mo ments acting on the platform (in the plat formfixed frame) using the components of M =



ri â Fi .

r & We wish to calculate , the time derivat ive o f the angular velocit y vector, which will

be transformed into the second derivative of the Euler angles wit h respect to time. These accelerations alo ng with first derivat ive o f the Euler angles will form the last six elements
& of X(t ) ; the derivative o f the state vector. As discussed in Sect ion 2.3, the numerica l

integrat ion scheme may t hen be applied in a step-wise manner in t ime, to observe the r & rotational mot ion o f the platform. In order to calculate at a given instant, we first calculate the angular velocit ies o f the plat form in its body-fixed frame using the known first derivat ives of the Euler angles (from the state vector):

65


= Tw
Rearranging Equation 5.37 we have:

r

-1



&

& &

(5.38)

rr r r rr & I = M - â I

(5.39)

The terms on the right hand side of the equation are known. Hence, we obtain: r r r r rr & & = I -1 ( I ) = I -1 ( M - â I ) Equation 5.34:
r &r & = Tw + Tw && & & & &

(5.40)

Finally, the second derivat ives o f the Euler angles are found by applying the chain rule to

(5.41)

To summarize the ent ire dynamics model description of Chapters 3 to 5, Figure 5.15 shows a flo w diagram o f the overall process required to obtain the t ime derivat ive of the
& state vector X(t ) , which is then sent to the Runge-Kutta integration scheme.

66


Figure 5.15 ­ Dynamics Model Flowchart

67


Chapter 6 ­ Performance Evaluation

6.1

Introduction
The performance evaluation o f the Arecibo Radio Telescope is a most valuable

exercise that will help us assess the dynamic performance of the system as well as demonstrate the capabilit ies of our computer model. We begin by explaining how the "performance" of the Arecibo Radio Telescope is defined. We may then use our model to observe (over t ime) the mot ion, performance metrics, and other parameters of interest of the Arecibo Radio Telescope under turbulent and non-turbulent wind condit ions at a variet y o f wind speeds and direct ions. In fact, if we include the sensit ivit y analysis carried out in Chapter 7, the Arecibo model has been subjected to more than 80 different dynamic test cases which consider a variet y of system configurat ions and wind condit ions. The output and results will be presented in the form of tables and figures.

6.2

Performance Metrics
In order to evaluate the performance of the Arecibo system we must first

understand what "performance" actually means in terms o f the system's motion. A performance metric is defined as a parameter that allows us to quantitatively evaluate the system's performance. Important to the astronomer are the positional and rotational error of the system's receiver/transmitter (to and fro m which the electromagnet ic radio waves reflect from the co llector dish). The receiver is defined in our model as the centre of mass o f the plat form (node #15 in Figure 2.2). To the satisfact ion of Steve Torchinsk y [25] (Head of Astronomy at the Arecibo Observatory) the fo llowing parameters have been chosen to quantitatively define the dynamic performance of the Arecibo Radio Telescope:

68


(i) (ii) (iii)

Error of the receiver posit ion in the focal plane Error of the receiver posit ion out of the focal plane Tilt angle of the receiver relative to the inertial Z-axis. The so-called "focal plane" is defined as the plane locally tangent to a hemisphere

[6] of radius equal to the plat form's static equilibrium height above the bottom-centre of the co llector dish (i.e. the origin in t he inert ial frame). Since the triangular truss plat form does not itself take on any azimuth or zenith angles, the focal plane is in fact always horizontal. Figure 6.1 shows the hemisphere as defined.

Figure 6.1 ­ The Focal Plane and Hemisphere The error out of the focal plane at any instant is calculated based on the difference between the plat form's actual height and the height of the focal plane (which, to reemphasize, is the plat form's height when resting at static equilibrium):
Errorout = Z -Z

CM

focal plane

(6.1)

69


The error out of the focal plane is taken as posit ive if the platform is outside o f the hemisphere (above the focal plane) and negat ive when the plat form is located inside the hemisphere (below the focal plane). The error in the focal plane at any instant is calculated as the radius of the plat form's horizontal displacement in the focal plane. The error in the focal plane is always a posit ive quant it y and is best visualized if one where looking direct ly down on the platform from high above it. Errorin = X
2
CM

+ YCM

2

(6.2)

Finally, we define the tilt angle (denoted by ) as the angle between the plat form's Zebu-axis (in its own body-fixed frame) and the inert ial Zip-axis (which always points straight up), as shown in Figure 6.2.

Figure 6.2 ­ Tilt Angle Performance Metric

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In order to calculate , we use the same technique as that used to calculate the plat form's angle o f attack in Section 5.5.2. That is, we use the rotation matrix that brings any vector from the body-fixed coordinate system to the inertial coordinate system. The third co lumn o f this rotation matrix represents the co mponents of the unit vector in the direct ion of the platform's body-fixed Zebu-axis. This unit vector is given in the inertia l frame by equat ion (5.20) as:
^ ^ Z B = (cos sin cos )^ + (cos sin sin - sin cos )^ + (cos cos )k i j

(6.3)

The unit vector in the direct ion of the inertial Zip-axis is given by:
^ ZI = 0 ^ + 0 ^ + k i j^

(6.4)

The tilt angle is then found be evaluating the dot product of the above unit vectors:
Z ·Z ^ ^

I = cos -1 B Z Z ^^ BI

(6.5)

= cos -1 (cos cos )

(6.6)

The tilt angle is direct ly related to the so-called "point ing accuracy" (PA) of the receiver using the fo llowing approximat ion:
PA = Z tan

iCM

(6.7)

Where Z

iCM

is the elevat ion of the plat form's center of mass in the inert ial frame of

reference. However, the tilt angle itself has been chosen as the performance metric, while the point ing accuracy may be subsequent ly found using equation 6.7 if so desired.

71


6.3

Additional Parameters of Interest
Aside fro m the performance metrics, there are often addit io nal parameters that

may be of interest, perhaps to a structural engineer, rather than to the astronomer. The parameters observed in addit ion to the performance metrics include the tensio n force in the cables as well as the tower-top deflections. All tensio ns, unless stated otherwise, are presented for the ent ire effect ive area o f the mainstay cables. That is, the values in the plots that follow must be divided by the number of mainstay cables in order to find the tension per cable. Furthermore, the tensio n forces quoted are those acting specifically in the first cable elements (#1, 6, and 11 as per Figure 2.2). The tower deflect ions are given in the horizontal plane o f motion of the tower-tops relat ive to their equilibrium posit ion. The deflect ions are considered posit ive if the tower is brought closer to the centre of the system (i.e. due to an increase in tensio n) and negative if the opposite is true.

6.4

Indicators

There are three different indicators that are used in co mparing the results of the various test cases. These indicators are the average, the root-mean-square, and the peak values o f the performance metrics and/or addit ional parameters o f interest over the total sampling t ime. The average value o f a metric, X, is the sum o f that metric fro m t ime to to tf divided by the number of times it was sampled/stored.
n

X

avg

=


1

X (6.8)

n

For example, if the simulat ion run t ime is set for 10 seconds and the value of the metric is stored in an array every 0.1 seconds, then in this case the value of "n" would be equal to 100. Next we define the root-mean-square for our purposes to be:
X
RMS

=

1 n


1

n

(X - X

avg

)

2

(6.9)

72


Finally, the peak value of the metric is simply defined as the maximum error encountered over the given period of time. If the max error is in fact negative, than the peak error is also taken as negat ive:
X
peak

=X

max

(6.10)

6.5

Arecibo Model Configuration

Before present ing any results fro m the test cases, it will be made clear exactly which physical configurations of the telescope are being simulated and how they differ fro m the actual Arecibo Radio Telescope. Table 6.1 gives the details o f the "original" Arecibo model and how it differs fro m the actual Arecibo construction. The ent ire performance evaluat ion and sensit ivit y analysis has been performed on the origina l Arecibo configurat ion. The upgraded configurat ion is presented immediately fo llowing the performance evaluat ion (in Sect ion 6.8) in order to assess the impact of so me of the recent design changes made to the Arecibo Radio Telescope.
Model of the Original Arecibo
550 000 kg Line f eed neglected 4 cables per tower d = 3" 5 cables per tower d = 3.25" N/A N/A

Actual Construction of the Original Arecibo
platf orm mass receiver drag Mainstay Cables Backstay Cables Main Auxiliary Cables Backstay Auxiliary Cables 550 000 kg Line Feed 4 cables per tower d = 3" 5 cables per tower d = 3.25" N/A N/A 6 cables total d=1.5" Two off-vertical cables (f unctioning as catenaries) run f rom each corner of the platf orm and anchored near the rim of the reflector [5]

Tie Down Cables

N/A

Table 6.1 ­ The Original Arecibo Configuration

73


6.6

Equilibrium Condition

An important feature of the Arecibo model is the equilibrium condit io n of the system. This condit ion must be reached before subjecting the system to various wind configurations. Given the specific physical parameters of a given configurat ion o f interest, the system is init ially (at time = 0 seconds) released fro m the unstretched length configuration shown in Figure 3.2. The system's motion will eventually damp out over time and settle at its equilibrium height and orientation. The fo llowing Figures show how the performance metrics vary over the first 400 seconds o f the total 1500 seconds required for the original Arecibo configurat ion to come to equilibrium. The reason for letting the system run for 1500 seconds is to ensure that the vertical mot ion is of the order of millimeters or less (if possible) before subje cting the system to the various wind condit ions. Notice that the platform remains level and centered as it oscillates in a vert ical mot ion. In realit y, upon init ial construction, the supporting cables o f the Arecibo structure were all pre-tensio ned prior to lift ing the plat form into the desired posit ion [13]. The tower-top radial distances fro m the centre of the co llector dish are also plotted versus time as they reach their equilibrium values in the simulat ion. The tower numbering is explained in the next section through Figure 6.5

74


Figure 6.3 ­ Performance Metrics to Equilibrium

75


Figure 6.4 ­ Tower-Top Positions to Equilibrium

6.7

Dynamic Runs

In this sect ion, we discover the qualit y and variat ion in the performance of Arecibo under a variet y of wind speeds, wind directions, and turbulence. All test cases are carried out on the original Arecibo model. We define the wind direct ion as is shown in Figure 6.5. A wind direct ion of 0° implies the wind is directed toward posit ive infinit y on the inert ial Xi-axis.

76


Figure 6.5 ­ Wind Direction and Tower Numbering

6.7.1 Effect of Wind Speed
In varying the wind speed, we keep the wind direction constant at 0° and consider no turbulent wind effects. We wish to determine the magnitude of mot ion and the performance metrics wit h increasing wind speeds. The range o f wind speeds used was 030 m/s. The first real proof of the Arecibo Radio Telescope's stabilit y was on August 28, 1966, when the system remained stable while encountered by up to 70 mile per hour (30 m/s) winds caused by the passing of Hurricane Inez [13]. We also know that wind speed of up to 17 miles per hour (7.6 m/s) cause "no significant displacement of the structure" [5]. Significant displacement of the structure is considered to be of the order of millimeters [2], which is quite impressive for such a massive structure. Table 6.2 shows the test matrix used for the wind speed evaluat ion.

77


T est # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Wind Speed m/s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 25 30

Wind Direction deg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Turbulence No No No No No No No No No No No No No No No No No No No

Table 6.2 ­ Wind Speed Test Matrix

For each of the above test cases, the simulat ion was run for 70 seconds and the wind was increased gradually to its full speed in t he first second (t = 0.1 to 1.1 seconds). The fo llowing figures show the variat ion of the performance metrics wit h wind speed:

78


30

25

Error in the Focal Plane (mm)

20

15

Av g RM S Peak

10

5

0 0 5 10 15 20 25 30 35 Wind Speed (m/s)

Figure 6.6a - Error in the Focal Plane vs Wind Speed

6

5

4 Error out of the Focal Plane (mm)

3 Avg RMS Peak 2

1

0 0 5 10 15 20 25 30 35

-1 Wind Speed (m/s)

Figure 6.6b - Error Out of the Focal Plane vs Wind Speed

79


0.14

0.12

0.10

Tilt Angle (deg)

0.08 Avg RMS Peak 0.06

0.04

0.02

0.00 0 5 10 15 20 25 30 35 Wind Speed (m/s)

Figure 6.6c - Platform Tilt Angle vs Wind Speed

We notice that the error in the focal plane as well as the t ilt angle seem to fit a quadratic shaped curve wit h increasing wind speed, and that the error out the focal plane is rather insignificant. We can also conclude that the original Arecibo model, even at high wind speeds, is subject to motion on the order of centimeters or tens o f millimeters. The tilt angle is also very small at an average o f approximately 0.02° (or 3.5 â 10 -4 rad ) even at 20 m/s wind speeds. At this wind speed the tilt angle is in fact greater than the desired and "ambit ious goal" (but not required) of 2.4 â 10 -5 rad [5]. The peak tower deflect ion, even at the hurricane wind speed of 30 m/s, was found to be 6.2 mm which is well within the permissible limit of 2 inches, or 50.8 mm [13].

80


6.7.2 Effect of Wind Direction

The effect of wind direct ion was tested at a constant wind speed of 10 m/s and with no turbulent conditions. The wind direct ion was increased fro m 0° to 120° in increments of 15°. The fo llowing figures present the performance metrics versus wind direct ion for the original Arecibo model.

3. 5

5

4

3. 0
3

E rror out of the Focal Pl ane (mm)

2. 5 E rror i n t he Focal Pl ane (mm )

2

2. 0

1

0

1. 5

-1

1. 0

-2

-3

0. 5
-4

0. 0 0 20 40 60 80 100 120 140 Wind Di recti on (deg)

-5 0 20 40 60 80 100 120 140 Wind Di recti on ( deg)

Figure 6.7a - Error In the Focal Plane vs Wind Direction

Figure 6.7b - Error Out of the Focal Plane vs Wind Direction

0.014 Av g 0.012 RMS 0.010 Peak Ti lt Angle (deg) 0.008

0.006

0.004

0.002

0.000 0 20 40 60 80 100 120 140 Wind Directi on (deg)

Figure 6.7c - Platform Tilt Angle vs Wind Direction

81


The above figures show that the wind direction has little effect on the performance of the system. It is interest ing however to observe the cable tensio ns and tower-top deflect ions versus wind direction. The fo llowing figures show the symmetry of the system as the wind direct ion changes from 0° alo ng the T1 radial line and 120° along the T2 radial line. Notice that at exact ly 60° the tensio n in the first cable elements of T1 and T2 are equal while the tensio n in T3 is at its maximum value (demo nstrating a form o f symmetry). tension.
12460000

Also note that the tower-top deflect ions (denoted B1, B2, B3

corresponding to towers T1, T2, T3, respectively) fo llow an ident ical symmetry as the

12455000

12450000 Tensi on (N) T1 T2 T3 12445000

12440000

12435000 0 20 40 60 80 100 120 140 Win d Di rection (deg)

Figure 6.7d - Average Cable Tension vs Wind Direction

82


0.8

0.6

0.4

Tower d ef lecti on ( mm)

0.2 B1 B2 B3

0.0

-0.2

-0.4

-0.6

-0.8 0 20 40 60 80 100 120 140 Wind Di rect ion (deg)

Figure 6.7e - Average Tower Deflection vs Wind Direction

6.7.3 Effect of Turbulence

Both the wind and the turbulence model emplo yed herein are taken from the wind and turbulence model used in the LAR system. For details regarding the turbulence model, the reader is referred to [6] in which it is stated that the turbulent gusts, imposed on the mean wind, "were generated with the desired gust statist ical properties, including turbulence intensit y, scale length, and spectra". The turbulence model acts to vary the wind speed above and below its mean value in an intent ionally rando m manner. The effects of the turbulent wind condit io ns are evaluated by considering the wind direct ion constant at 0° and varying the mean wind speed from 0 to 15 m/s. The turbulent results are superimposed onto the results obtained for the same constant wind speed for a basis of comparison. condit ions. The following figures show the performance metrics in turbulent

83


14

12

10 Error in the Focal Plane (mm)

8

Steady Avg Turb Avg Steady RMS Turb RMS Steady Peak Turb Peak

6

4

2

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Mean Wind Speed (m/s)

Figure 6.8a - Error in the Focal Plane vs Turbulent Mean Wind Speed

30

20

10 Error ou t of the Fo cal Plane (mm)

0

-10

Steady Av g Turb Av g Steady RMS Turb RMS Steady Peak Turb Peak

-20

-30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-40 Mean Wind Sp eed (m/s)

Figure 6.8b - Error out of the Focal Plane vs Turbulent Mean Wind Speed

84


0.045

0.040

0.035

0.030 Tilt Angle (deg) Steady Avg 0.025 Turb Avg Turb RMS Steady RMS Steady Peak Turb Peak

0.020

0.015

0.010

0.005

0.000 0 2 4 6 8 10 12 14 16 Mean Wind Speed (m/s)

Figure 6.8c - Platform Tilt Angle vs Turbulent Mean Wind Speed Under turbulent conditions particularly at velocit ies greater than 10 m/s, the most obvious effect is an increase in the average and peak performance metrics. This can be reasoned by considering the non-linear relat ionship between mean wind speed and the performance metrics (as shown in Figures 6.8 a, b, c). Let us assume that over a long enough t ime, the turbulent model gusts the wind, in a rando m manner, above and below the mean wind speed by an equal amount. When the turbulence model gusts the wind above its mean value, the error is affected (increased) more than it is affected when the turbulence gusts the wind below its mean value. Hence, we see that when turbulence is applied the errors increase and the performance of the system is very slight ly degraded. However, even under these turbulent condit ions the original Arecibo model proves to be a well performing and stable system wit h very small posit ional and rotational errors.

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6.8

Upgraded Arecibo Configuration

One of the advantages o f having developed a model o f the original Arecibo configuration is that we may evaluate the effect of the so me o f the recent design changes [2]. After two big upgrades, the "upgraded" Arecibo configurat ion is very different fro m the "original" in terms o f the cable system layo ut, the mass o f the platform, and the receiver system. In this sect ion we would like to compare the "original" Arecibo mode l as per Table 6.1 with the "upgraded" Arecibo model. Again, we must be clear on exact ly what features of the upgraded system are being modeled. Table 6.3 gives the details o f the "upgraded" Arecibo configurat ion and how it differs fro m the actual Arecibo construction after the two major upgrades.
Model of the Upgraded Arecibo
815 000 kg Gregorian treated as a sphere 4 cables per tower d = 3" 5 cables per tower d = 3.25" 2 cables per tower d = 3.25" Added to the effective area of the mainstay cables. Attachment points not considered 2 cables per tower d = 3.625"

Actual Construction of the Upgraded Arecibo
platf orm mass receiver drag Mainstay Cables Backstay Cables 815 000 kg Gregorian System 4 cables per tower d = 3" 5 cables per tower d = 3.25" 2 cables per tower d = 3.25" Attached 2/3 of the way along the sides of the triangular truss 2 cables per tower d = 3.625" 6 cables total d = 1.5" Two cables run v ertically from each corner of the platform. Active Control: Jacks which can exert up to 60 tons of vertical force. [2]

Model of the Original Arecibo
550 000 kg Line f eed neglected 4 cables per tower d = 3" 5 cables per tower d = 3.25"

Main Auxiliary Cables

N/A

Backstay Auxiliary Cables

N/A

Tie Down Cables

N/A

N/A

Table 6.3 - The Upgraded Arecibo Configuration

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In order to model the recent design changes, the mass o f the platform was increased fro m 550 tons to 815 tons. The new mass mo ment of inert ia o f the platform was recalculated using the CAD so ftware:
Mass kg 550000 815000 Ix kgm^2 1.0316E+08 1.5277E+08 Iy kgm^2 1.0316E+08 1.5277E+08 Iz kgm^2 1.9866E+08 2.9420E+07

Table 6.4 -Upgraded Platform Mass Moment of Inertias The new effect ive tower stiffness (with the new auxiliary backstay cables) was found using the method outlined in Sect ion 4.3 to be:
k
eff

= 1.977 â 107 N / m

(6.11)

The new auxiliary mainstay cables were added to the overall effect ive area of the exist ing mainstay cables. Important to note here are that the attachment points and any added rotational stabilit y achieved by t he auxiliary mainstay cables were not taken into account.
Aeff = 0.0289 m
2

(6.12)

Finally the drag of the Gregorian receiver (as outlined in Sect ion 5.5.5) was added to the overall drag of the plat form and treated as acting through the plat form's centre of mass. With the changes in place, we let the new upgraded configurat ion co me to a new equilibrium and we co mpare the average performance metrics of the two configurations over a range of wind speeds. Note that the equilibrium heights o f the original and upgraded configurations (even wit h the large mass and cable property changed) are within 0.6 m of each other. This was reassuring, since the number and sizes o f the auxiliary cables added during the upgrade had the purpose of keeping the platform exactly where it always had been, while using the three exist ing towers.

87


5.0

4.5

4.0

3.5 Error in the Focal Plane (mm)

3.0 Original Upgraded

2.5

2.0

1.5

1.0

0.5

0.0 0 2 4 6 8 10 12 14 Wind Speed (m/s)

Figure 6.9 - Average Error in the Focal Plane vs Wind Speed

0.012

0.010

0.008 Tilt Angle (deg)

0.006

Original Upgraded

0.004

0.002

0.000 0 2 4 6 8 10 12 14 Wi nd Speed (m/s)

Figure 6.10 - Average Platform Tilt Angle vs Wind Speed 88


The errors out of the focal plane in both cases proved to be insignificant (of the order of 0.01 mm) and are not shown here. The results for both the error in the foca l plane and the plat form tilt angle are shown to be significant ly less for the upgraded model than for the original model. performance. Thus we can conclude that the recent design changes (studied in the context of these models) were in fact beneficial to the system's

6.9

Selected Cases

Before moving on to the sensit ivit y analysis in the next chapter, it is of interest to study selected cases in more detail. In fact, two cases based on the original Arecibo model will be displayed here to better our understanding o f the system's behavior, if not for the purposes of demo nstrating the output capabilit ies o f the Arecibo model. The first case (case A) is for a wind speed of 10 m/s, wind direct ion of 60°, and wit h no turbulence effects. The second case (case B) is for a wind speed of 10 m/s, wind direct ion o f 0°, and with the turbulence active. The fo llowing output demonstrates the system's behavior over a 70 second period. Depending on the specific informat ion required; the Arecibo model is capable o f present ing a variet y o f data. constant wind speed. For Case A, we see how the performance metrics and plat form posit ion and rotation each vary wit h t ime under the

89


Figure 6.11 - Case A Performance Metrics

90


Figure 6.12 - Case A Platform Rotation

Figure 6.13 - Case A Platform Translation 91


In Figure 6.12, the y-axis angles alpha, beta, and gamma, refer to the Z-Y-X Euler angles o f , , respect ively. Notice that the wind direction o f 60° has caused the plat form to be displaced by a posit ive x and positive y co mponent. The init ial transient of the wind increasing to full speed in the first second may be seen (especially for the error in the focal plane). The fo llowing output is for Case B: Wind speed 10 m/s, wind direct ion 0°, with turbulence.

Figure 6.14 - Case B Performance Metrics

92


Figure 6.15 - Case B Platform Rotation

Figure 6.16 - Case B Platform Translation

93


Figure 6.17 - Case B: Tower-Top Motion For Case B, we have also included the tower-top motion, represented by the radius, fro m the inert ial Zi-axis, o f the tower-tops in their horizontal plane of mot ion. Note that in Case B, the random effects of turbulence are present and observed. However the system's mot ion, even under turbulent condit ions, is st ill on the order of millimeters. In the sensit ivit y analysis of the next chapter we will see figures that give the average, rootmean-square, and peak motion for a variet y of different system configurat ions.

94


Chapter 7

Sensitivity Anal ysis

Often one is interested in knowing which physical parameters of a given system, if changed or redesigned, could improve the system's performance. In our sensit ivit y analysis, the fo llowing six parameters are varied individually while keeping all else constant: (i) (ii) (iii) (iv) (v) (vi) Number of Mainstay Cables Tower Radius (Cable Length) Effect ive Tower Stiffness (Number of Backstay Cables) Mass of the Platform Cable-Plat form Attachment Points Mainstay Cable Properties (Plasma Rope) It should be clearly noted that the fo llowing sensit ivit y analysis has been performed on the original Arecibo model as described in Table 6.1. Also, each test case of this sensit ivit y analysis presents a new physic al configurat ion. Therefore, for each case, the system must be allowed to come to a new equilibrium condit ion before applying the given winds (see Section 6.6).

95


7.1

Test Matrix
The fo llowing Table 7.1 presents the test matrix emplo yed in our sensit ivit y

analysis, with the highlighted values referring to the no minal or original Arecibo model values.
T est # 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 Configuration Configuration No. of Cables = 3 No. of Cables = 4 No. of Cables = 5 No. of Cables = 6 No. of Cables = 7 rad = 150 m rad = 180 m rad = 213 m rad = 240 m rad = 270 m No. of Cables = 0 No. of Cables = 1 No. of Cables = 3 No. of Cables = 5 No. of Cables = 7 No. of Cables = 9 No. of Cables = 11 keff = infinity (No Tower Motion) mp = 400 tons mp = 550 tons mp = 700 tons mp = 850 tons mp = 1000 tons radp = 20 m radp = 30 m radp = 38 m radp = 50 m Speed m/s 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 2 4 6 8 10 12 14 Direction deg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Turbulence No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No No

Eff ective Mainstay Cable Area - incremented by the number of mainstay cables per tower

Tower Radius - increases cable length

Eff ective Tower Stiff ness - Incremented by the number of backstay cables per tower

Platf orm Mass

Cable-Platform Attachment Points

Cable Properties: Plasma Rope

dc = (six 2" cables)/tower = 840 kg/m^3 E = 37.4e9 N/m^2

Table 7.1 - Test Matrix for the Sensitivity Analysis 96


7.2

Number of Mainstay Cables
As per the test matrix, the number of mainstay cables has been incremented fro m

3 to 7 cables per tower in increments of 1. This was done by increasing the effect ive area of the mainstay cables using Equation 3.1. The fo llowing Table 7.2 presents the effect ive mainstay cable areas used for each test case.
No. of Mainstay Cables 3 4 5 6 7 Effective Area 2 [m ] 0.014 0.018 0.023 0.027 0.032

Table 7.2 - Mainstay Cable Effective Areas Figure 7.1 shows that as we increase the number of mainstay cables, the effect ive stiffness of the cables increases such that the equilibrium height of the plat form increases.

151.5

151.0

150.5

150.0 Height (m)

149.5

149.0

148.5

148.0

147.5 2 3 4 5 Number of Mainstay Cables per T ow er 6 7 8

Figure 7.1 - Equilibrium Platform Height vs No. of Mainstay Cables

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Figure 7.2 shows that the as the number o f mainstay cables are increased, the equilibriu m tensions per cable is decreased.

4500

4000

3500

3000

Tension (kN)

2500

2000

1500

1000

500

0 2 3 4 5 Number of Mai nst ay Cables per Tow er 6 7 8

Figure 7.2 ­ Equilibrium Tensions per Cable vs Number of Cables

In terms of the performance metrics when the system was subjected to a wind o f 10 m/s it was found that increasing the number of mainstay cables improves the system's performance by decreasing the error in the focal plane. By increasing the number o f cables fro m 3 to 7 the Errorin = -0.8 mm . There was no significant change to the error out of the focal plane or to the tilt angle.

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7.3

Tower Radius
Changing the radius of the towers (taken fro m the centre of the collector dish) is

essent ially one way o f also changing the total lengths of the cables. It was found that when increasing the tower radius, the only metric affected was the error in the focal plane. Figure 7.3 shows that for a tower radius increase fro m 150 m (right at the edge of the collector dish) to 270 m, the error in the focal plane is increased by approximately 0.5 mm. Each test case is for a constant mean wind of 10 m/s observed over 70 seconds. Of course, this could be looked at in the reverse manner, such that decreasing the tower radius would serve to better the performance of the system.
4.5

4.0

3.5

Error in the Focal Plane (mm)

3.0

2.5

2.0

Avg RMS Peak

1.5

1.0

0.5

0.0 120

150

180

210 Tower Radius (m)

240

270

300

Figure 7.3 - Error in the Focal Plane vs Tower Radius

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7.4

Effective Tower Stiffness

In order to evaluate the effect of the tower-top deflect ion on the performance o f the system, we vary the effective tower stiffness by increment ing the number of backstay cables and recalculat ing the effect ive tower stiffness for each case, as per Sect ion 4.3. Each test case is for a constant mean wind o f 10 m/s at 0° observed over 70 seconds. The fo llo wing Table 7.3 presents the average effective tower stiffnesses calculated for each increment of backstay cable. For the extreme case of no tower-top motion, we let the effect ive st iffness go to infinit y. Specifically, we use the value k
eff

= 1.0 â 10 20 N / m as

the infinite condit ion (any further increase has no effect to the system dynamics).
No. of Backstay Cables 0 1 3 5 7 9 Inf inite Effective 2 Area [m ] 0.0000 0.0054 0.0161 0.0268 0.0375 0.0482 Inf inite Average Effective Stiffness [N/m] 3.458E+05 2.946E+06 8.146E+06 1.335E+03 1.855E+07 2.375E+07 1.000E+20

Table 7.3 Average Tower Effective Stiffness It is observed that the effect ive tower stiffness plays a very important role in the performance of the system. Especially concerning the error in the focal plane and the t ilt angle metrics. The fo llowing figures present the results while indicat ing the ideal case o f
k
eff

= and zero tower-top deflection. As the effect ive tower stiffness is increased the

error and tilt angle drop rapidly fro m 0 to 3 backstay cables. The error then begins to asymptote toward the k
eff

= limit.

100


60

3.0
50

Error in the Focal Pl ane (mm)

2.5 2.0 1.5 1.0 0.5 0.0 3 5 7 9 11 13
Avg

Error in the Focal Plane (mm)

40

Inf inite Stiffness

30

Peak Avg RMS

20

10

0 0 1 2 3 4 5 6 7 8 9 10 Number of Backstay Cables

Figure 7.4a - Error in the Focal Plane vs Number of Backstay Cables
0.08

0.0 08
0.07

0.0 07 0.0 06
Tilt Angle (deg)

0.06

0.0 05 0.0 04 0.0 03 0.0 02
Av g Infinite Stif fness Avg RMS Peak

0.05 Tilt Angle (deg)

0.04

0.0 01 0.0 00 3 4 5 6 7 8 9 10 11 12 13 14 15

0.03

0.02

0.01

0.00 0 1 2 3 4 5 6 7 8 9 10 Number of Backstay Cables

Figure 7.4b -Platform Tilt Angle vs Number of Backstay Cables

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Also of interest is Figure 7.4c which shows the tower-top deflect ion as a funct ion of the number of backstay cables for a wind speed of 10 m/s at 0°. Notice that the towertop deflect ion approaches zero as keff approaches infinit y.

15.0

10.0

5.0

0.0 Tower deflection (mm)

-5.0
0. 0. 0. 0. 0. -0. -0. -0. -0. -1. -1. -1. 8 6 4 2 0 2 4 6 8 0 2 4 3 5 7 9 11 13 15 Tower deflection (mm)

-10.0

B1 B2 B3
B1 B2 B3 Inifinite Stiffness

-15.0

-20.0

-25.0

-30.0 0 1 2 3 4 5 6 7 8 9 10 Number of Backstay Cables

Figure 7.4c - Average Tower Deflection vs Number of Backstay Cables

7.5

Platform Mass

One o f the mo st astounding features of the Arecibo Radio Telescope is the mass of the suspended plat form structure. At 550 tons in the original Arecibo configurat ion it was later increased to 815 tons in 1997 [2]. Wit h the use of our original Arecibo mode l we will vary the platform mass in order to evaluate what effects it may have on the system's performance. Recall that in Section 5.3 the platform's mass mo ment of inertia (used in the rotational equat ions of mot ion) was found using CAD so ftware and a uniform densit y triangular shaped sect ion. In order to perform this sensit ivit y analysis,

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not only must we change the plat form mass, but we must also recalculate the mass mo ments of inertia for each case. Table 7.4 gives the mass mo ment of inertia for the various platform masses. Recall that the cross terms are in fact zero due to the symmetry of our modeled plat form.
Mass kg 550000 815000 400000 700000 850000 1000000 Ix kgm^2 0316E+08 5277E+08 5028E+07 3130E+08 5925E+08 8757E+08 Iy kgm^2 0316E+08 5277E+08 5028E+07 3130E+08 5925E+08 8757E+08 Iz kgm^2 9866E+08 9420E+07 4448E+08 5284E+08 0703E+07 6121E+07

1. 1. 7. 1. 1. 1.

1. 1. 7. 1. 1. 1.

1. 2. 1. 2. 3. 3.

Table 7.4 ­ Platform Mass-Moment of Inertia Figure 7.5 shows that as the mass o f the platform is increased, for the same number o f cables, the equilibrium height of the plat form decreases.
153 152 151 150 149 Height (m) 148 147 146 145 144 143 142 250

400

550

700 Platform Mass (kg)

850

1000

1150

Figure 7.5 - Equilibrium Platform Height vs Platform Mass

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Running the test cases for a wind speed o f 10 m/s, a wind direct ion o f 0° and no turbulence it was interest ing to find that increasing the platform mass had essent ially no effect on the platform's posit ional error. It did however reduce the plat form's t ilt angle error as shown in Figure 7.6, which makes sense in terms of an increasing mass-mo ment of inert ia.

0.016

0.014

0.012

0.010 Tilt Angle (deg) Avg RMS Peak

0.008

0.006

0.004

0.002

0.000 200

300

400

500

600

700

800

900

1000

1100

Platform Mass (tons)

Figure 7.6 ­ Platform Tilt Angle vs Platform Mass The fo llowing Table 7.5 gives the minimum cable breaking strength of the cables as found in the available AutoCAD drawings [14].
Mainstay Cables [in & kips] [m & N] 12 12 3 0.0762 1044 4.6439E+06 Backstay Cables [in & kips] [m & N] 5 5 3.25 0.08255 1212 5.3912E+06

Total Number of Cables Diameter of each cable Minimum breaking strength

Table 7.5 ­ Minimum Breaking Strength of Cables

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According to the original Arecibo model, the minimum breaking strength is reached at a plat form mass o f 1000 tons, giving a tensio n of approximately 4.5 â 106 N per cable.

7.6

Cable-Platform Attachment Points

Next we look to evaluate the effect of moving the cable-plat form attachment point closer in and further out from the plat form's centre of mass. The variable radp will be used to denote the radial distance at which the mainstay cables are attached to the triangular truss plat form (see Figure 7.7). Note that here we are only changing the attachment point and not the dimension o f the platform itself (as used in the drag model and mass mo ment of inertia calculat ions). Changing radp will have a direct effect on the mo ments act ing on the plat form, and our goal here is to find out how important this parameter is to the performance of the Arecibo Radio Telescope.

Figure 7.7 ­ Cable-Platform Attachment Points

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Setting up the test runs as per the test matrix in Table 7.1, we obtained so me interest ing results. Figure 7.8 shows there to be a relat ively significant increase to the error in the focal plane of the plat form wit h increasing attachment point radius (radp), which is obviously not good. However, Figure 7.9 shows there to be a decrease in the average tilt angle wit h increasing attachment point radius. Thus, the optimal attachment point distance would require so me sort of co mpromise between the performance metrics. Also, there was no significant change to the error out of the focal plane wit h increasing attachment point radius.

4.0

3.5

3.0 Error in the Focal Plane (mm)

2.5 Avg RMS Peak

2.0

1.5

1.0

0.5

0.0 0 10 20 30 40 50 60 70 Attachment Point Radius (m)

Figure 7.8 - Error in the Focal Plane vs Platform-Cable Attachment Radius

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0.016

0.014

0.012

0.010 Tilt Angle (deg) Avg RMS Peak

0.008

0.006

0.004

0.002

0.000 0 10 20 30 40 50 60 70 Attachment Point Radius (m)

Figure 7.9 - Platform Tilt Angle vs Platform-Cable Attachment Radius

7.7

Mainstay Cable Properties: Plasma Rope

The fina l parameter of our sensit ivit y analys is sat isfied our curiosit y regarding the use of a co mpletely different type o f mainstay cable. We chose to replace the braided steel cables wit h the so-called "plasma rope" that is being used for the LAR system. The main advantage of plasma rope is that it is the "world's strongest rope for its weight" [26]. Presumably, a light weight material has no particular advantage in the Arecibo system in improving its performance, so it was interesting to investigate the feasibilit y o f using Plasma rope for the Arecibo mainstay cables. Table 7.6 gives the cable properties used for plasma rope in this analysis [26].

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Property

Plasma Rope Symbol

Value 840 37.4

Units kg/m3 GPa

Density Elastic Modulus Damping Ratio


E

0.015 Table 7.6 ­ Plasma Rope Properties

The cho ice of the effect ive area to be used for the plasma rope was next considered. It was decided to use an effect ive area of the plasma rope that would give the plasma cables the equivalent stiffness as the braided steel mainstay cables:
( EAeff )
steel

= ( EAeff )
seeel

plasma

(7.1)

Aeff

plasma

=

( EAeff ) E

plasma

(1.0 â 1011 )(4.56 â 10 -3 ) = 37.4 â 109 = 0.01219 m 2

This

effective

area

may

be

realized

by

emplo ying

six,

2

inch

tethers

( Aeff = 0.01216m 2 ) per tower or even three, 3 inch tethers ( Aeff = 0.01368m 2 ) per tower.

Finally to ensure that the plasma cables have enough strength to carry the load we perform a rough strength calculat ion: Ttotal = mp g sin
T

=

550000(9.81) = 25.950kN sin(12 o )
= Ttotal = 1.441 kN 3â 6

(7.2)

per cable

Where mp is the mass of the platform, g is the gravitat ional constant, and is the approximate angle of the cables at the cable-platform attachment. The tensio n per plasma cable is found to be approximately 1.441 kN which is under (but close to) the minimum tensile strength of 1.579 kN for a 2 inch stand [26]. Figure 7.10a shows that average error in the focal plane over a range o f wind speeds is significant ly higher when using plasma rope.

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12

10

Error in the Focal Plane (mm)

8

6

Steel Av g Plasm a Avg

4

2

0 0 2 4 6 8 10 12 14 Wind Speed (m/s)

Figure 7.10 a ­ Average Error in the Focal Plane vs. Wind Speed for Plasma Rope The same is true for the remaining performance metrics as is shown in the fo llowing figures. Therefore, we can conclude that using plasma rope for the Arecibo Radio Telescope degrades the system's performance.

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0.30

0.25

0.20 Error ou t of the Fo cal Plane (mm)

0.15 Steel Av g Plasma Av g 0.10

0.05

0.00 0 2 4 6 8 10 12 14

-0.05 Wind Sp eed (m/s)

Figure 7.10b ­ Average Error out the Focal Plane vs. Wind Speed for Plasma Rope
0. 025

0. 020

Tilt Angle (deg)

0. 015 Steel Av g Plasm a Av g 0. 010

0. 005

0. 000 0 2 4 6 8 10 12 14 Wind Speed (m/s)

Figure 7.10c ­ Tilt Angle vs. Wind Speed for Plasma Rope

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7.7

Summary
Having successfully co mpleted a sensit ivit y analysis on the original Arecibo

model, we present the Table 7.7 that qualitatively summarizes the results of the investigat ion. It should be kept in mind that these results are for a wind speed o f 10 m/s and a wind direction o f 0°, wit h no turbulence (except for the case o f the Plasma Rope which was tested over range of wind speeds). Also note that the general trends presented in this section are for the average values o f the performance metrics observed over a t ime of 70 seconds.

Variable Increase No. of Main Cables Increase Tower Radius (cable length) Increase Platf orm Mass

Error In FP

Error Out FP

Tilt Angle

T ension Tension per cable decreases Increases

TowerTop Deflection Constant

Decreases

Constant

Decreases

Increases

Constant

Constant

Constant

Constant

Constant

Decreases

Increases

Constant

Increase Tower Stiffness (No.of bacstay cables)

Decreases signif icantly

Constant

Decreases signif icantly

Mainstay cable tension increases

Decreases

Increase CablePlatf orm Attachment Radius Using Plasma Rope

Increases

Constant

Decreases

Decreases

Constant

Increases

Increases

Increases

Decreases

Constant

Table 7.7 ­ Qualitative Summary of Sensitivity Analysis

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Chapter 8 ­ Conclusions

8.1

Final Remarks
In summary, a basic model for the dynamics of the Arecibo Radio Telescope has

been developed. The development of the model successfully stemmed fro m an earlier versio n of a dynamics model o f the LAR system [6]. Upon developing the model, a successful performance evaluat ion and sensit ivit y analysis was carried out. evaluat ion: · · · The original Arecibo model, even at very high wind speeds, is subject to mot ion on the order of tens of millimeters. The t ilt angle is also very small, even at high wind speeds: approximately 0.02° (or 3.5 â 10 -4 rad ) at a wind speed of 20 m/s. The peak tower deflection, even at the hurricane wind speed o f 30 m/s, was found to be very small (6.2 mm), which is well within the permissible limit of 2 inches, or 50.8 mm [13] for the tower-top deflectio n. · · Wind direct ion has negligible effect on the performance o f the system Turbulent wind, particularly at mean wind speeds in excess of 10 m/s, has the effect of increasing the average and peak performance metrics, thus degrading the system's performance. · However the system's motion, even under turbulent condit ions, is still on the order of cent imeters. The recent design changes (studied in the context of these models) were in fact beneficial to the system's performance. The fo llo wing is a list o f the key conclusio ns that were drawn from the performance

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From the informat ion gathered in the sensit ivit y analysis, the fo llowing final remarks address the quest ion: What general trends or changes to the physica l construction of the Arecibo Radio Telescope would improve the system's performance? · · · · Increasing the number of mainstay cables Decreasing the tower radius Increasing the effect ive tower stiffness (i.e. the number of backstay cables) Increasing the plat form's mass

8.2

Recommendations for Future Work

The advantage o f co mputer modeling is once again demo nstrated in this sect ion. Upon development of a basic model, such as the Arecibo model, there is always room for improvement and added features. The key target areas of the model which should be considered in future development are: · Tiedown Cables: A large effort was in fact put forward in trying to implement the tie down cables (specifically the vertical t iedowns for the upgraded Arecibo configuration). Unfortunately, t ime was a limit ing factor and it was decided that the tie downs were beyo nd the scope of this thesis. For this reason, the next feature that should be added to the basic Arecibo model is the t ie down cables. There are two possible approaches:
o Develop a code that introduces 3 new cables that are self-contained. o Set the number of cables in the exist ing code to 6, with different base

point specificat ions for the mainstay and tiedown cables. In either case a new method for determining the unstretched length o f the tiedowns must be devised. Finally, it should be noted that the vert ical t iedowns o f the upgraded Arecibo and the off vert ical catenaries o f the original Arecibo present the same challenges or problems in terms of model development. Whe n one problem is so lved, the other follows. 113


·

Gregorian Posit ioning: Devise a feature, either in the dynamics model it self or offline, to take into account the different zenith and azimuth angles that the Gregorian system may take. To do this, the mass-mo ment of inertia o f the plat form may no longer be assumed as a uniform densit y triangular sect ion that includes the mass o f the Gregorian. Also, the drag of the Gregorian should no lo nger be taken at the centre of mass o f the plat form (this will introduce a moment force acting on the platform due to its own drag).

·

Platform Drag Model: Improvements to the triangular truss platform drag mode l should be a goal of future development. The drag through mult iple truss frame sect ions (i.e. more than a pair) should be researched and the platform drag coefficients should be improved.

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1. 2. 3. 4.

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116