Design of Large Space Structures Derived from Line Geometry Principles

DESIGN OF LARGE SPACE STRUCTURES DERIVED FROM LINE GEOMETRY PRINCIPLES

PATRICK J. MCGINLEY

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2002

Patrick J. McGinley

I would like to dedicate this work to Dr. Joseph Duffy - mentor, friend, and inspiration. My thanks also go out to my parents, Thomas and Lorraine, for everything they have taught me, and my friends, especially Byron, Connie, and Richard, for all their support.

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Carl D. Crane, III, the members of my advisory committee, Dr. John Schueller and Dr. John Ziegert, as well as Dr. Joseph Rooney, for their help and guidance during my time at UF, and especially their understanding during a difficult last semester.

TABLE OF CONTENTS page

ACKNOWLEDGMENTS ...... iv

LIST OF TABLES...... vii

LIST OF FIGURES ...... viii

ABSTRACT...... xii

CHAPTER

1 INTRODUCTION ...... 1

2 BACKGROUND ...... 5

2.1 In-Parallel Robotic Manipulators...... 5 2.2 Definitions...... 9

3 THEORETICAL DEVELOPMENT ...... 14

3.1 Line Geometry Methods ...... 14 3.2 Axial Loading Approximations ...... 23 3.3 Lateral Loading Approximations...... 24 3.4 Torsional Loading Approximation ...... 24

4 DUALITY OF POLYHEDRA ...... 26

5 FINITE ELEMENT MODELING...... 34

6 DATA ANALYSIS AND RESULTS...... 46

6.1 Axial Specific Stiffness Performance ...... 48 6.2 Lateral Specific Stiffness Performance...... 51 6.3 Torsional Specific Stiffness Performance...... 53 6.4 Overall Performance Trends...... 55

7 NUMERICAL EXAMPLES...... 63

7.1 Dual Polyhedron ...... 63 7.2 Example Boom Design Problem...... 66

8 CONCLUSIONS...... 67

8.1 Line Geometry ...... 67 8.2 Duality of Polyhedra...... 68 8.3 Structural Simulation ...... 68

9 FUTURE WORK...... 72

APPENDIX

A MATLAB CODE FOR PSEUDO-INVERSE PROGRAM...... 74

B ALTERNATE LEAST SQUARES MODELS ...... 82

C DESIGN OPTIMIZATION CHARTS...... 87

D ADDITIONAL FIGURES ...... 95

LIST OF REFERENCES...... 97

BIOGRAPHICAL SKETCH ...... 99

LIST OF TABLES

Table page

2.1 Optimal 3-3 manipulator ratio conversions ...... 13

5.1 Comparison of AEC-Able specifications and SRTM model results [1]...... 37

6.1 Maximum specific stiffness performance for 300m x 5m CFRP booms...... 56

7.1 List of vertices for mutual moment calculations...... 65

B.1 Ranges of key design parameters used to fit regression models...... 83

vii

LIST OF FIGURES

Figure page

1.1 Highway traffic sign support truss...... 4

2.1 Top view of 3-3 in-parallel device platforms ...... 6

2.2 Top view of 3-3 in-parallel device with actuated legs added ...... 7

2.3 Construction of an octahedral 3-3 manipulator with Rtop:Rbottom = ½...... 8

2.4 Construction of a triangular truss unit cell...... 10

2.5 Triangular truss unit cell ...... 11

2.6 Octahedral truss unit cell ...... 11

2.7 Symmetric reinforced octahedral truss unit cell ...... 12

3.1 Unit cell geometries and two-cell boom structures – triangular truss (a,d), symmetric reinforced octahedral (b,e), and octahedral (c,f)...... 15

3.2 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=1 ...... 18

3.3 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=2 ...... 19

3.4 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=4 ...... 20

3.5 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=8 ...... 20

3.6 Corresponding length efficiency index plot to Figure 3.2 ...... 21

3.7 Corresponding length efficiency index plot to Figure 3.3 ...... 21

3.8 Corresponding length efficiency index plot to Figure 3.4 ...... 22

3.9 Corresponding length efficiency index plot to Figure 3.5 ...... 22

4.1 Regular octahedron to regular cube mapping ...... 27

4.2 Elongated octahedron dual to elongated cuboid ...... 30

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4.3 Optimal quality index octahedron configuration (w/perspective views) and dual (also with lines extended to accentuate intersections)...... 31

4.4 Limit of scaling one set of vertices to ward the limit at which the octahedron approaches a tetrahedron...... 32

5.1 Axial, lateral, torsional loads applied to center-plane constrained two-cell octahedral boom ...... 34

5.2 AEC-Able boom for NASA STS-99 Shuttle Radar Topography Mission [1]...... 36

5.3 Conversion of SRTM cell to stacked 4-4 platform cell ...... 37

5.4 AEC-Able SRTM boom finite element model, deformed under 10 lbf lateral end loading...... 38

5.5 FEM model of 16 cell, 300m x 5m octahedral boom ...... 39

5.6 FEM model of octahedron subjected to axial load ...... 40

5.7 Deformed FEM model of octahedron subjected to axial load ...... 41

T 5.8 Induced leg stresses in a single octahedron, wapp= [0 0 –3 0 0 0] ...... 42

5.9 Deformed FEM model of octahedron subjected to lateral load...... 43

T 5.10 Induced leg stresses in a single octahedron, wapp= [-3 0 0 3h 0 0] ...... 44

5.11 Deformed FEM model of octahedron subjected to torsional load...... 44

T 5.12 Induced leg stresses in a single octahedron, wapp= [0 0 0 0 0 3] ...... 45

6.1 Specific axial stiffness vs. Rtop:Rbottom for various numbers of cells per SYM boom..49

6.2 Specific axial stiffness vs. Rtop:Rbottom for various numbers of cells per OCT boom...50

6.3 Specific lateral stiffness vs. Rtop:Rbottom for various numbers of cells per SYM boom51

6.4 Specific lateral stiffness vs. Rtop:Rbottom for various numbers of cells per OCT boom 52

6.5 Specific torsional stiffness vs. Rtop:Rbottom for various numbers of cells per SYM boom ...... 55

6.6 Specific torsional stiffness vs. Rtop:Rbottom for various numbers of cells per OCT boom ...... 57

6.7 Mass vs. Rtop:Rbottom of 300m x 5m symmetric reinforced booms...... 58

6.8 Mass vs. Rtop:Rbottom of 300m x 5m octahedral booms...... 59

6.9 Design optimization chart, 300m x 5m, 16 cell symmetric reinforced octahedral boom vs. Rtop:Rbottom ...... 59

6.10 Comparison of kA, 300m x 5m octahedral boom vs. 24-cell triangular truss (CFRP) ...... 60

6.11 Comparison of kL, 300m x 5m octahedral boom vs. 24-cell triangular truss (CFRP) ...... 60

6.12 Comparison of kT, 300m x 5m octahedral boom vs. 24-cell triangular truss (CFRP) ...... 61

6.13 Comparison of kA, 300m x 5m symmetric reinforced octahedral boom vs. 24-cell triangular truss (CFRP)...... 61

6.14 Comparison of kL, 300m x 5m symmetric reinforced octahedral boom vs. 24-cell triangular truss (CFRP)...... 62

6.15 Comparison of kT, 300m x 5m, symmetric reinforced octahedral boom vs. 24-cell triangular truss (CFRP)...... 62

7.1 Octahedron elongated f = 2.5 in x-direction, translated to (2,0,1), and rotated 20° about z-axis ...... 64

7.2 Pseudo-Inverse dual of example octahedron, MATLAB™ program output (note: lines trimmed to enhance visibility) ...... 65

B.1 Comparison of FEM simulation results and regression model prediction for axial specific stiffness of OCT booms...... 83

B.2 Comparison of FEM simulation Results and regression model prediction for lateral specific stiffness of OCT booms...... 84

B.3 Comparison of FEM simulation results and regression model prediction for torsional specific stiffness of OCT booms...... 84

B.4 Comparison of FEM simulation results and regression model prediction for axial specific stiffness of SYM booms ...... 85

B.5 Comparison of FEM simulation results and regression model prediction for lateral specific stiffness of SYM booms ...... 85

B.6 Comparison of FEM simulation results and regression model prediction for torsional specific stiffness of SYM boom...... 86

C.1 Optimization chart for OCT 8, 300m x 5m, CFRP ...... 87

C.2 Optimization chart for OCT 16, 300m x 5m, CFRP ...... 88

C.3 Optimization chart for OCT 24, 300m x 5m, CFRP ...... 88

C.4 Optimization chart for OCT 32, 300m x 5m, CFRP ...... 89

C.5 Optimization chart for SYM 8, 300m x 5m, CFRP ...... 89

C.6 Optimization chart for SYM 16, 300m x 5m, CFRP ...... 90

C.7 Optimization chart for SYM 24, 300m x 5m, CFRP ...... 90

C.8 Optimization chart for SYM 32, 300m x 5m, CFRP ...... 91

C.9 Optimization chart for SYM 40, 300m x 5m, CFRP ...... 91

C.10 Comparison of kA for 300m x 5m, OCT vs BAS-32 (both CFRP)...... 92

C.11 Comparison of kL for 300m x 5m, OCT vs BAS-32 (both CFRP)...... 92

C.12 Comparison of kT for 300m x 5m, OCT vs BAS-32 (both CFRP)...... 93

C.13 Comparison of kA for 300m x 5m, SYM vs BAS-32 (both CFRP)...... 93

C.14 Comparison of kL for 300m x 5m, SYM vs BAS-32 (both CFRP)...... 94

C.15 Comparison of kT for 300m x 5m, SYM vs BAS-32 (both CFRP)...... 94

D.1 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=2 ...... 95

D.2 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=4 ...... 96

D.3 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=8 ...... 96

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

DESIGN OF LARGE SPACE STRUCTURES DERIVED FROM LINE GEOMETRY PRINCIPLES

Patrick J. McGinley

August 2002

Chair: Dr. Carl D. Crane III Department: Mechanical Engineering

This thesis develops the design procedures for a class of large truss structures derived from previous research on the line geometry, kinematics of tensegrity prisms, and in- parallel robot manipulators. The design relations are the results of a combined analysis approach, with the initial design based on line geometry and screw theory, and further optimized through finite element modeling.

The geometric properties of both the new boom designs and their more conventional counterparts were assessed. The properties of unit cells based on an octahedron, which has been shown to be an efficient design geometry for in-parallel robotic systems, are investigated. Based on a comparison between the simulation and actual performance of existing designs, the models are reasonable approximations of overall relative behavior, and not intended to be high-fidelity physical simulations.

xii

The optimal design geometry was shown to vary depending on the particular constraints imposed by system requirements, but booms constructed from a reinforced variant of the unit cell based on the 3-3 octahedral parallel robotic manipulator performed well in almost all cases compared to a conventional diagonal truss. Finite element modeling showed that improvements of a factor of two in axial and lateral specific stiffness and half an order of magnitude in torsional specific stiffness are potentially attainable. The octahedral unit cell showed significant improvement in torsional stiffness over diagonal truss designs, and exceeded the performance of the reinforced octahedron in that regard. From a performance standpoint, the reinforced octahedral cell is most

often the best choice.

xiii

CHAPTER 1 INTRODUCTION

This thesis builds upon the foundation of previous work on tensegrity structures, in-

parallel robotic manipulators, and basic kinematic principles to develop design guidelines to characterize a new class of lightweight structures with a wide range of applications and potentially superior performance to contemporary designs.

Tensegrity theory was originally studied by Fuller [7] to develop an efficient structure, one with a minimum number of its members in compression. The reason to favor tension over compression is that thin members can carry greater loads in tension before failure, since in compression buckling occurs before yield. Perhaps Stern [15], posing the concept as “a stable system made of compressive members connected only by tensile members,” gave the most concise definition of tensegrity. For several years, proponents of tensegrity structures have advocated their use in aerospace applications, due to their relatively small mass and innate deployability [9]. However, because of complexity and robustness concerns, the idea of tensegrity in space has also drawn criticism. Reliability aside, there is still something to be gleaned from the study of stable tensegrity geometries that can be extended to rigid structures as well.

Another important class of mechanism with many similarities to truss structures is the in-parallel platform manipulator. The Stewart-Gough device is generally considered the first true in-parallel robotic manipulator [16]. In 1947, Gough developed a parallel platform device to test the wear of vehicle tires in various orientations. Stewart adapted

1 2

the device, and envisioned it as a way to produce a more realistic flight simulator,

enabling a fixed base location to manipulate the position and orientation of a platform in

all six degrees of freedom, through the use of six actuated legs. The fixed base and

parallel actuation provides more load-bearing capacity and spatial rigidity than a

comparable six degree of freedom serial robot [18]. The lessons learned from the study

of parallel robots are quite useful in the design of structures, and directly applicable to

actuated, actively controlled “smart” structures.

The objective of this work is to bridge the gap between design methods involving only

geometric principles and those focusing on stress-strain based analyses. Many tensegrity

systems and robotic platforms are analyzed through geometric stability methods, with less emphasis placed on static or dynamic force analysis. A joint between structural members is often modeled as a ball-and-socket, which permits rotation, and does not allow for the transmission of moments.

Three methods were used to design the unit cells for the boom structures. The first was an initial investigation into cell geometry using the quality index and kinematic analysis. This provided some insight into a basic subset of possible geometries that should be studied further, and eliminated a large group of impractical choices (see

Chapter 3). Next, the geometric duals of these cells were determined, in hopes of producing an additional group of viable candidates. While the duality approach yielded no appropriate cell structures, some interesting theoretical results regarding duality of certain irregular polyhedra were obtained (see Chapter 4). Finite element analysis was used to optimize the designs previously identified. The results of the FEM simulations are presented graphically (see Chapter 5), as well as in a set of least-squares regression

3 models that are useful as approximate predictors of FEM results created to reduce time spent on detailed modeling during the early design stage (see Chapter 6). Further, a numerical example of the procedure to producing a dual is presented, and an optimization problem format for determining a candidate design from the least-squares relations and functional requirements is offered (see Chapter 7).

In contrast, contemporary space structure designs are optimized primarily through finite element modeling and experimental testing. The diagonal truss is a flight-proven, time-tested design that can provide reasonably high stiffness and specific stiffness in various configurations. It is also used in many terrestrial applications, from street sign mounts to cranes and bridges, and because of its simplicity, is a design that lends itself well to both manufacturing and deployment.

An applicable example of a truss structure subjected to somewhat similar loading is that of an overhead traffic sign mount used on interstate highways (Figure 1.1). The functional requirements are not drastically different from those imposed on a space structure. Many signs are supported by triangular and prismatic trusses, span a relatively long distance, and because of their horizontal orientation, are subjected to far more lateral and torsional than axial loading. The weight of the structure, sign, and lighting apparatus provide the primary lateral bending loads in the vertical direction, while aerodynamic forces on the face of the sign induce both horizontal lateral forces and torsional loads.

Instead, inertial forces that result from rotational and translational accelerations of a spacecraft replace the forces from weight and drag that are dominant at ground level.

Some of the conclusions drawn from the modeling and simulation are borne out by these real-world examples. The long, straight axial members have greater diameters than

their cross-member counterparts, often by ratios of 3:1 or more. Another relevant feature

found on some sign mounts is a cross section that decreases monotonically from the root

to the end of the boom. The primary motivation for this changing cross-section is to reduce the effects of the gravity load on the cantilever, but it proves effective in improving stiffness per unit mass for an end load as well.

Figure 1.1 Highway traffic sign support truss

As a result of this effort, a unique design approach is presented, which employs line

geometry to arrive at an initial design, and optimizes it based on finite element

simulations. The performance of the new design as a function of its major parameters is

captured in a set of simple numerical equations that allow for initial performance

prediction accurate within a few percent of that calculated by a detailed FEM simulation.

These design equations should enable initial design selection without the need for time

consuming detailed designs for a comprehensive set of candidates.

CHAPTER 2 BACKGROUND

The Center for Intelligent Machines and Robotics (CIMAR) in the University of

Florida Mechanical Engineering Department has been a leader in research involving line geometry and robotic mechanisms. Relevant work has focused on tensegrity antiprisms and in-parallel (Stewart platform) manipulators, which consist of two planar polygon platforms connected by a set of actuated legs, is presented.

2.1 In-Parallel Robotic Manipulators

Parallel manipulators are often designated P-Q, referring to the number of connection points on the top and bottom platforms, respectively. Plan views of a 3-3 in-parallel device are shown in Figures 2.1 and 2.2. A graphical depiction of the conceptual steps of its construction is presented in Figure 2.3.

A solution to the forward and reverse kinematic analyses for a 3-3 octahedral parallel manipulator was presented by Griffis and Duffy [8]. Subsequently, the mathematical formulations of screw theory were applied to the forward and reverse force analyses.

Lee, Duffy, and Hunt [11] proposed a dimensionless quality index to characterize the geometric stability of a 3-3 device. This quality index is based upon the principle that singularities should be avoided, and thus the optimal device is the one that is furthest from singular. The quality index, λ, is defined as

det(JJ T ) λ = T det()JJ max

5 6

where J is the matrix whose columns are the normalized Plücker line coordinates of the

lines bound to the legs of the robot. Mathematically, the argument can be made that the

configuration with the highest determinant of the matrix describing the lines in Plücker

line coordinates is furthest from a singularity condition. This work was later extended to

other platform manipulators, including the 3-6, 6-6, and redundant 4-4 and 4-8 devices.

Figure 2.1 Top view of 3-3 in-parallel device platforms

A singularity is defined as a position and orientation where the line coordinates of the

legs become linearly dependent, since with five or fewer linearly independent legs, there

are some force-moment combinations that cannot be resisted by any combination of

forces applied by the leg actuators. The forces in some actuators may spike, and there is

a risk of catastrophic collapse of the device [5]. The failure can be explained through the

concept of instant mobility. The mobility of a structure is the sum of the degrees of freedom it possesses minus the sum of constraints imposed on it. For instance, a single beam in free space has six degrees of freedom and zero constraints, and thus mobility equal to six. Attach it to a wall on a hinge, and translation (3-DOF) and two rotational

freedoms are removed, allowing only rotation about the hinge axis. The hinged beam has

a mobility of one. A structure, when fixed to ground, should have mobility zero. A

device with mobility greater than zero is a mechanism [4]. When a parallel robot enters a

position where the lines bound to its legs become linearly dependent, the legs cannot enforce six independent constraints, and the robot is free to move in one particular way

(perhaps a rotation, a translation, or a combination of the two) without constraint. The phenomenon of instant mobility occurs when a device reaches a singularity configuration

– named so because linear dependence of the leg lines will cause its Jacobian matrix

(formed from its Plücker line coordinates) to become singular.

Figure 2.2 Top view of 3-3 in-parallel device with actuated legs added

The 3-3 (octahedral) platform was chosen as the starting point for a geometric design

because it is the shape that allows a cell structure using only ball joints to be formed with

the minimum number of members. The 4-4 platform has two redundant legs, as only six are needed to constrain the six degrees of freedom for position and orientation of the

moving platform. However, in a boom cell, the platforms are not idealized rigid bodies,

8 but are created from additional members. Consequently, there is increased complexity in building a platform with four beams and four joints as opposed to one with three of each.

Figure 2.3 Construction of an octahedral 3-3 manipulator with Rtop:Rbottom = ½

There is a caveat to a designer when using the methods of line geometry and kinematics – results that apply to ball joints do not necessarily apply directly to rigid connections. In short, controlling the position and orientation of the moving platform requires control over all six degrees of freedom. This is most often accomplished through the use of six linearly actuated legs operating in parallel, controlling one degree of freedom each.

2.2 Definitions

Three main types of cell geometry are considered. The first is the conventional

diagonal truss cell. It consists of two congruent regular n-sided polygon ‘platforms’

connected by 2n additional members. The connecting members are divided equally into

two categories – n members are parallel, and connect similar vertices between the top and

bottom platforms, and the remaining n are diagonal members that connect the kth vertex

of one platform to the (k+1)th vertex of the other. The most common diagonal truss

types are the triangular and rectangular, where the platforms have three and four sides,

respectively. For a triangular truss with points T1, T2, T3, composing the upper platform

and B1, B2, B3 composing the similar lower platform, the axial members connect T1 to B1,

T2 to B2, and T3 to B3, while the diagonal braces connect T1 to B2, T2 to B3, and T3 to B1

(Figure 2.4).

A boom composed of diagonal truss cells can also be thought of as n long axial members (longerons) cross-braced by the platforms and diagonal struts. The longerons are typically the primary load-bearing members, with the platforms and diagonals providing rigidity against twisting motions. For comparison to the octahedral designs, a set of defining dimensions must be devised (Figure 2.5). Consider that a regular polygon can be inscribed within a circle by connecting points on the circle having angular spacing of 2π/n. This radius will be used to define the size of the platforms. For the triangular truss cells modeled, the radius of the top platform (Rtop) is equal to that of the bottom platform radius (Rbottom). The length of a side of the platform polygon will be denoted a, and the cell height, defined as the perpendicular distance between the platforms, will be denoted Hcell.

Figure 2.4 Construction of a triangular truss unit cell

The octahedral cell is a derivative of the symmetric 3-3 (octahedral) in-parallel device.

When octahedral cells are stacked upon each other to create a boom, every other octahedron must be inverted to maintain continuity for any design utilizing a

Rtop:Rbottom ratio other than unity. In order to facilitate accurate comparisons of boom structures with different cell geometries, the “unit cell” in an octahedral boom must be chosen in a way that ensures the end platforms of the booms are always at the same radius, as is the case for the diagonal truss cell. Therefore, the unit cell structure is defined as that of two 3-3 devices joined at the top platform (Figure 2.6). Congruent

“bottom” platforms reside at each end of the cell, and a third platform rotated relative to the end platforms exists halfway between them. Six legs connect each end platform to the middle platform, such that none cross the center area of the middle platform (Figure

2.7).

Figure 2.5 Triangular truss unit cell

The symmetric reinforced cell is a further extension of the octahedral cell. The inverted octahedral cell structure remains, but n parallel axial reinforcements are added between the two end platforms, bypassing the middle platform (Figure 2.7). While this design does not appear radically different from the other two unit cells, its redundant

Figure 2.6 Octahedral truss unit cell

nature does require different treatment from the point of view of certain analysis techniques. In fact, the octahedral cell can be seen as a subset of the symmetric reinforced cell having zero-thickness reinforcing members.

Figure 2.7 Symmetric reinforced octahedral truss unit cell

As a matter of convenience, Table 2.1 is provided as a reference to simplify comparison of different ratios used in the literature to characterize octahedra (and

platform manipulators in general). Each of the rows represents the same octahedron,

with the ratios normalized to a different parameter. The first four rows contain the exact

ratios; the last four are numerical approximations. When making a comparison between

different types of devices, such as a 4-4 to a 6-6, it is sometimes more useful to compare

Rtop and Rbottom instead of a and b. The same Rtop and Rbottom map out roughly equivalent

areas and approach the area of the same circle for an N, whereas figures with N sides of

length a and b do not.

Henceforth, the notation <> or will be used to refer to

octahedra defined by these respective characteristic dimensions. The abbreviations BAS,

OCT, and SYM will be used to represent the baseline diagonal cell truss, octahedral cell

truss, and symmetric reinforced truss, respectively, as a shorthand notation.

Table 2.1 Optimal 3-3 manipulator ratio conversions

a b h Rtop=Ra Rbottom=Rb Hcell ½ 1 ½ 1 ½ √3 /√3 1 1 2 1 2 1 /√3 /√3 2 √3 √3 /2 √3 /2 ½ 1 √3 √3 2√3 √3 1 2 2√3 .5000 1 .5000 .2887 .5773 1 1 2 1 .5773 1.1547 2 .8660 1.7321 .8660 .5000 1 1.7321 1.7321 3.4641 1.7321 1 2 3.4641

CHAPTER 3 THEORETICAL DEVELOPMENT

During the evaluation of the octahedral cell boom, it became necessary to define the octahedral and reinforced octahedral unit cells in a manner consistent with the diagonal truss unit cell. While a standard triangular truss cell could be stacked continuously end-

to-end, this was not the case with an octahedral cell. An inversion was required for

anything other than a 1:1 ratio of Rtop:Rbottom, so that top polygons connected with top

polygons, and bottom with bottom. From a system design perspective, it is preferable to

require as little information about other subsystems as possible, so that a minimum

number of artificial design constraints are imposed. Thus, a unit cell was defined as two

octahedra mated at the top platform (Figure 3.1), so that the geometry of the end of the

boom could be known with the interior represented as a "black box." Thus, only general

knowledge of boom performance would be required – maximum radius, mass, stiffness, etc. – by system integrators, as each geometry type would be otherwise interchangeable.

However, these structures differ from robot manipulators, and some of the previous

kinematic analyses are no longer applicable.

3.1 Line Geometry Methods

When the reinforced octahedral boom was created, reinforcements were added

between the two "bottom" platforms in an octahedral cell. As a result, a structure was created with two distinct types of vertices – those that connect only to the next platform layer, and those that join struts from multiple layers.

14 15

The effect such a change would have on the quality index needed to be studied – the quality index had been developed for robot manipulators that consisted of only one layer of platforms connected by a single set of legs [12], and its applicability to stacked configurations was not known or considered. And with the addition of reinforcing members between the extreme platforms, a fundamentally different problem arises. No longer is the structure an in-parallel manipulator, it is a multiply-parallel device. In addition, the optimal unit cell may not be the optimal stacked cell (either from geometric or static/dynamic structural point of view, and also depending on the performance requirements of the particular application).

Figure 3.1 Unit cell geometries and two-cell boom structures – triangular truss (a,d), symmetric reinforced octahedral (b,e), and octahedral (c,f)

Two separate methods of examining the geometric stability were entertained. The first was to determine the applied wrench-leg force transfer function for a multiply parallel cell. However, no closed-form solution was found to this problem.

The second was to apply a variant of the Quality Index. In the ideal case, this would involve the use of an analytic solution to the forward and reverse kinematic analysis for an in-parallel manipulator with all members being treated as load bearing, as opposed to the platforms being considered rigid bodies attached to the legs. With such a general result, a meaningful performance parameter could be found that would describe many different mechanism configurations and allow better comparisons.

Simply merging the twelve lines of two stacked octahedra into a single J matrix is insufficient to determine the geometric stability of the overall structure. It is possible using this concatenation method to arrive at a nonzero determinant for the stacked or multiply parallel combination even when one or both submatrices are singular. With two singular component 3-3 platforms, there must be at least one applied wrench that the device cannot resist, and will therefore cause motion. A quality index intended to keep designs from reaching singular configurations is meaningless rubbish if it cannot identify them. A numerical counter-example can easily be found to disprove its validity:

1 1 1 1 3 2 1 1 1 1 3 − 4     0 0 0 0 2 3 0 0 0 0 2 0  0 0 0 0 5 4 0 0 0 0 5 0  J 1 =   J 2 =   0 1 2 0 6 6 0 1 2 0 6 3  0 4 3 2 1 6 0 4 3 2 3 3      0 0 0 0 0 2 0 0 1 0 0 1 

det(J1 ) = 0 det(J 2 ) = 0

T det([J 1 J 2 ] [J 1 J 2 ] ) = 332220 ≠ 0

Among potential alternative solutions are a sum or a product of quality indices, with

subsequent segments calculated from a reference frame attached to the middle platform,

(which could be positioned or oriented in any way). Since one part of the sum could well

be zero without the total being zero, it suffers from some of the same problems as the

concatenation method. The product scheme is not without its faults either. For a boom

constructed from k simply stacked symmetric cells, the overall quality index would be

k (λcell )

This approach may or may not be the most appropriate, as long booms composed of many cells that differ only slightly from the maximum are heavily penalized by the power relationship.

It was shown by Zhang and Duffy [18] that the maximum value of |det J| (or √det(JJT) in the case of a redundant parallel manipulator) can be reduced to a function of the dimensions of the upper (moving) platform only. The dependencies on the height and dimension of the base platform can be removed by substitution, as those parameters can be expressed as multiples of a. In the 3-3 and 6-3 cases, the maximum value is ¾(√3)a3,

and for the redundant 4-4 it is 4(√2)a3. For a 6-6 platform the maximum increases to

54a3. Figure 3.2 shows the variation of √det(JJT) with respect to the number of polygon connection points N and rotation angle φ for given values of a, b, and h. As N increases,

more legs are added, and in the limit as N becomes infinitely large, the area of the top

platform becomes the area of the circle it is inscribed within, and the legs become so

tightly spaced that they approach a solid form.

The maximum values of √det(JJT) for each value of N correspond to a rotation of

φ=π/N. The two curves of singularities are found along the curves defined by

π π π π φ = + and φ = − N 2 N 2

The singularities identified are identical to those found for a stable N-sided tensegrity antiprism. As shown by Tran, [17] any tensegrity antiprism which is stable without an applied wrench must be in a singularity position. The value of √det(JJT) decreases quickly as the cell aspect ratio grows. As a comparison to the 3-3 optimal case,

<<1,2,1>>, a cell of dimensions <<1,2,2>> was analyzed, and the result is shown in

Figure 3.3. The cases <<1,2,4>> and <<1,2,8>> are shown in Figures 3.4 and 3.5, respectively.

Figure 3.2 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=1

Figure 3.3 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=2

It is also possible to introduce the total length of the legs as a normalizing factor for an

N-N platform device quality index. While a more complete investigation into this topic is necessary, it is interesting to note that a well-defined maximum has been observed in all cases tested, and a smaller secondary peak separated by a rotation of approximately π radians (Figures 3.6 through 3.9). Nevertheless, it helps to quantify the cost-benefit relationship between additional connecting points and quality index for a platform of given height and platform dimension. In and of itself, the length efficiency parameter does not prove sufficient in characterizing a system, but provides some semi-quantitative information as to the efficient number of connecting points for a device of a defined volume.

Figure 3.4 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=4

Figure 3.5 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=8

Figure 3.6 Corresponding length efficiency index plot to Figure 3.2

Figure 3.7 Corresponding length efficiency index plot to Figure 3.3

Figure 3.8 Corresponding length efficiency index plot to Figure 3.4

Figure 3.9 Corresponding length efficiency index plot to Figure 3.5

While there are a great number of geometric and material properties involved in structural design, a few were considered of greater importance. Different parameters were emphasized in each category of axial, lateral, and torsional stiffness. While none of these simplifications can be taken as a complete picture of boom performance, some physical insight is gained nonetheless.

3.2 Axial Loading Approximations

In one sense, the boom subjected to a compressive axial load is akin to a slender column, and for a tensile load, it should exhibit some characteristics of an elastic tie. The elastic deformation due to axial tension or compression force F is described by

F = σ = E ε A where A is the cross-sectional area of the beam, σ is the stress induced by F, E is the elastic modulus, and ε is strain. Since strain ε can be expressed as the change in length ∆l normalized to the original length l0

(l − l ) ∆l ε = 0 = l0 l0

A simple substitution yields a relation for axial stiffness, KA,

F E A K A = = ∆l l0

While the Euler buckling stability criterion will yield a lower critical stress than the tensile yield strength, in the elastic region the structure will deform linearly about l0.

Stiffness will be governed by the behavior in the linear elastic region, and strength by the buckling regime. However, for a truss structure, local buckling of a single member is the primary mode of failure in compression. Thus, there is little need to calculate the

24 buckling strength for the global structure, but important to determine the critical limit for each type of member bearing a compressive load.

3.3 Lateral Loading Approximations

In another context, the structure was representative of an end-loaded cantilever beam, as it was rigidly supported at one end and required to support a force/moment combination at a distance that was long compared to the beam's width [2]. Bending stiffness of a cantilever beam is given by

F K L = δlateral

The end deflection can be found explicitly from

F L3 δ = lateral E I

From this expression, a linear dependence on the elastic modulus of the boom material can be reasonably assumed. Since a unit cell is not a continuous shape, there is no simple formula for its second moment of area, but since from first principles

I = ∫ y 2 da

It is obvious that I increases as radius, either top or bottom, increases, as the mass concentrations move further from the neutral axis. This increase occurs not only because of the wider mass distribution of the platform beams, but also the legs connecting them.

3.4 Torsional Loading Approximation

In its other capacity, the boom performs the duty of a shaft in torsion about its major axis. For an isotropic idealized shaft, elastic deflection is determined from,

l T θ = KG

25 where l is the shaft length, T is the applied torque, G is the shear modulus, and K is the polar moment of area. The shear modulus G of the material is related to its Young's modulus E and Poisson’s ratio υ through

E G = 2 ()1+υ

The polar moment Kpolar is defined as,

K = r 2 da polar ∫ and is essentially a function of r4 for thin-walled cross-sections. Deflection angle is directly proportional to length and inversely proportional to elastic modulus. Thus, a crude first approximation for the stiffness, KT, of a long boom under torsion would be

T K G r 4 E K = = polar ∝ T θ l l

This implies that in all three loading cases, the stiffness behavior should be linear with respect to E. Stiffness should increase rapidly as r increases, and decrease with increasing length.

CHAPTER 4 DUALITY OF POLYHEDRA

An alternative approach to developing an optimal cell geometry was to use relations involving duality. The duality of Platonic polyhedra is well known, and the a pseudo- inverse procedure that allows the determination of the dual figure mathematically was proposed by Lee, Duffy, and Rooney [13]. The premise is that a structure and its dual will exhibit some common characteristic that can be exploited to enhance the design's performance. In an instance where a tensegrity antiprism may not be appropriate, something may be learned from constructing the dual structure as a potential substitute.

Rooney has proposed the design of “compegrity” structures – a set of tensile members connected by a continuous network of compressive members – based upon duality to tensegrity structures [14].

Regular polyhedra may not always be the optimal building blocks for a particular structure design. Since the boom structures studied have a high aspect ratio, they seemed a natural fit for a slender column as a unit cell. Since a rectangular prism shape (4-4 platform) would serve well in this capacity, but have redundancy, its dual octahedron was investigated. The results obtained disagree with speculation from previous studies of duals of distorted polyhedra.

Based on the duality of points and planes, translating one point would cause a translation of its dual plane in a direction parallel and opposite. However, while it is important to note that while points and planes are dual in nature, the lines are being used

26 27 to define the polyhedron, and by moving one point, the incidence relations between several lines change.

By moving one vertex of the octahedron directly away from the origin, one does not obtain a closed polyhedron, but rather an open figure that approaches a trapezoidal prism

(Figure 4.1). Two opposing vertices must be moved an equal distance from the origin in opposite directions along the line through them to produce a closed polyhedron of the intended shape. In effect, this is a linear scaling of one of the coordinate axes, and this transformation appears to be valid regardless of the fixed coordinate system orientation

(for convenience, placing the six vertices of the octahedron at X= ±1, Y= ±1, Z= ±1 is common).

Figure 4.1 Regular octahedron to regular cube mapping

The method for realizing the duals of closed polyhedra through the use of a pseudo- inverse relation is relatively straightforward. The first step is to obtain the Plücker line coordinates of the lines, (in ray coordinates), which represent the edges of the polyhedron. The line coordinates are assembled from two vectors, S and S0. The S

28 vector is the vector in the direction parallel to the line, and can be found by subtracting the coordinates of any two points p1 and p2 that lie on the line,

x1 − x2  p − p = S =  y − y  1 2  1 2   z1 − z2 

The S0 vector represents the moment of a unit force along that line about the origin, and can be found with the S vector and any point on the line, r, using the following relationship,

S0 = r × S

The line defined by the vector pair S, S0 is denoted $. In Plücker ray coordinates, a

T line $(ray) is ordered [ S S0 ] , and Plücker axis coordinates are easily obtained by

T reversing the two component vectors of $(ray), giving [ S0 S ] . A simple transform between axis and ray coordinates can also be used.

0 0 0 1 0 0   0 0 0 0 1 0 0 0 0 0 0 1 S0   S      =   1 0 0 0 0 0  S  S0  0 1 0 0 0 0   0 0 1 0 0 0

i Throughout this paper, the symbol $i will denote line i in ray coordinates, and $ will be used to denote the same line in axis coordinates. Any scalar multiple of $ represents the same line, so when necessary the lines can be divided by the magnitude of S to create an $ with a unit vector S. (These are sometimes referred to as normalized or unitized line coordinates).

The edges of the polyhedron are represented as lines in normalized Plücker ray coordinates. Next, the Jacobian matrix J is constructed from the lines,

J = [ $1 $2 $3 ... $n-1 $n ]

where n is the number of edges (e.g. six for a tetrahedron, twelve for an octahedron or cuboid, thirty for an icosahedron or dodecahedron). Then, the pseudo-inverse relation is applied,

−1 M = J T (JJ T )

The rows of the M matrix that results express the lines along the edges of the dual polyhedron in Plücker axis coordinates [13].

Normally, the pseudo-inverse of a tetrahedron will result in a closed dual tetrahedron, since any four general planes in space will form a tetrahedron. There exists a maximum of 4 vertices that can be formed by these planes – since 3 are required per vertex, and

 4  = 4 . Special degenerate cases exist when planes are parallel, but in general a    3  tetrahedron will map to a dual. However, when the pseudo-inverse mapping of the octahedron is performed, there are 6 planes (from the 6 vertices) and in the general case a maximum of  6  = 20 vertices can exist. A cuboid is a special case where 6 vertices that    3  will form a solid closed figure can be extracted from the lines in the pseudo-inverse matrix. Most cases will yield open figures. In part, this is due to a maximum of 20 vertices being formed by 6 general planes in space. Some special cases have been found that will map to closed cuboids. A regular octahedron will map to a regular cube (Figure

4.1). If two opposite vertices are moved equal distances from the center of the octahedron (origin) along the line (axis) between them, an elongated cuboid is formed

(Figure 4.2). As long as opposite vertices are moved equal distances this behavior will hold, so that 2, 4, or 6 can be moved simultaneously. This result is somewhat intuitive, as moving one pair outward is the same as moving the other two pairs inward and rescaling the length of the sides.

Figure 4.2 Elongated octahedron dual to elongated cuboid

The third and most general case that maps to a cuboid is when three vertices, one from each axis, that single face of the octahedron are translated along their respective axes by the same amount. This is essentially a rescaling one of the triangular faces of the original octahedron (Figure 4.3a,b), while the opposite face remains at its original size. The result is a more general closed cuboid (Figure 4.3c). What is also interesting about this case is that four groups of four lines that make up this cuboid intersect at four points outside the dual figure, which can be seen by extending the line segments used to draw the dual

(Figure 4.3d). In retrospect, the dual of the regular octahedron exhibits the same characteristic, but the dual is composed groups of parallel lines, points of intersection are all at infinity.

Figure 4.3 Optimal quality index octahedron configuration (w/perspective views) and dual (also with lines extended to accentuate intersections)

Also worth noting is the behavior as one face is scaled to an even greater degree. The regular octahedron approaches a tetrahedron; one face is so much larger than its opposite face that in the limit the smaller one resembles a single point (Figure 4.4). The lines forming the cuboid also begin to form a dual tetrahedron. In the dual, the area of one face begins to converge to a point within a larger, coplanar face (noted by an arrow in

Figure 4.4).

A software program has been developed to perform these tasks and check for closure of the figure (Appendix A). It utilizes the concept of the mutual moment of two lines.

This is best described as the moment that a unit force acting along one line could exert about the point of closest approach on a second line. Consequently, the mutual moment

32 calculation allows one to find this closest approach distance. A concise formula for determining the mutual moment q of two lines $j and $k is

q = S j ⋅ S0k + S0 j ⋅ Sk

It can be proven that the mutual moment of any two intersecting lines is identically zero

[5]. If the figure is closed, the mutual moment of the lines forming each corner will be zero. This condition is checked both numerically and symbolically in the software (to avoid problems caused by machine roundoff error). It was also verified through the symbolic computations that the pseudo-inverse method will produce a closed polyhedron irrespective of the choice of position and orientation of the initial coordinate system.

Figure 4.4 Limit of scaling one set of vertices to ward the limit at which the octahedron approaches a tetrahedron

Further work in this area will include finding a more thorough explanation of precisely which particular changes can be made and why the mapping works under those conditions. To this point, the majority of the work has been focused on the octahedron- cuboid pair, but it would not be unreasonable to expect the nature of the results to carry

33 over to the icosahedron-dodecahedron pair, or to more complex polyhedra. The cube/octahedron pair seems to be a general enough case to draw conclusions from

(though the tetrahedron was not), but another extension is to verify that the same duality holds for the icosahedron-dodecahedron combination, and to see if a mapping can be found for semi-regular polyhedra, such as a truncated tetrahedron, etc. Further work also needs to be done in terms of finding applications of the results, either to platform mechanisms or tensegrity and compegrity structures [14]. As previously noted, the dual octahedron to a long, slender column did not exhibit any particularly beneficial structural characteristics, nor did the dual of a 3-3 in-parallel manipulator appear superior to other cuboids.

CHAPTER 5 FINITE ELEMENT MODELING

Finite element models were developed to test the predictions of the hypotheses based on line geometry. Each boom structure was primarily modeled as a rigid truss, cantilevered from a support holding each vertex of the mounted face. For the lower aspect ratio booms, they were modeled as a single cantilever mounted at one end. The very high aspect ratio booms were treated as a distributed system that would not be constrained at either end. A model comprised of two cantilevers attached to the rigidly supported (fixed-fixed-fixed) center plane was used to determine spring constant/stiffness. In this case, two modal analyses were performed, the first with the center plane constraint, and the other with a free body assumption. Simplified two-cell models of an octahedral boom are shown in Figure 5.1.

Figure 5.1 Axial, lateral, torsional loads applied to center-plane constrained two-cell octahedral boom

34 35

The design variables that appeared to be of greatest interest in characterizing the boom were: boom radius at the mount (Rbottom), total length (L), cell height (Hcell), ‘top’ platform radius (Rtop in octahedral and symmetric reinforced octahedral cells), elastic modulus (E), and material density (ρ), octahedral and axial reinforcing member cross section (Xoct,

Xrnf), and total material volume (V). Other derived parameters, such as number of cells in the boom (C#), can be formed easily from members of the previous list (e.g., C# = L /

Hcell). These can be reduced into more general, nondimensional variables, such as the overall boom aspect ratio A, (L:Rbottom), cell aspect ratio H (Hcell:Rbottom), and polygon platform radius ratio R (Rtop:Rbottom).

Sensitivity studies were performed on these parameters with the intent to produce a data set capable of describing a tradespace that would span the feasible ranges of each, with the goal of deriving comprehensive design rules for the various structure geometries, given a set of performance requirements and goals. Eight major performance parameters were identified: mass, axial/lateral/torsional stiffness (KA, KL, KT), axial/lateral/torsional stiffness per unit mass (kA, kL, kT), and first natural frequency (fn1), of which the stiffnesses per unit mass are most critical. In a microgravity environment, applied loads are generally small compared to those in terrestrial settings, so strength becomes a secondary performance measure, as long as an appropriate safety factor is maintained. In terms of loading magnitudes, launch is the most structurally stressing segment of the mission, and the assumption is made that if a structure is designed to survive launch, structural failures from on-orbit forces should be of little concern.

The deployable boom from the NASA STS-99 Shuttle Radar Topography Mission

(SRTM), developed by AEC-Able Engineering Company, Inc., was used as a flight-

36 proven example to assess the accuracy of the modeling (Figure 5.2). It was designed as part of the interferometric synthetic aperture radar (IfSAR) instrument used on the SRTM in February 2000, and was required to meet strict mechanical tolerances with regard to both bending and twisting motions.

Figure 5.2 AEC-Able boom for NASA STS-99 Shuttle Radar Topography Mission [1]

It is interesting to note that the SRTM boom is similar to a version of the symmetric reinforced square truss with the top platforms removed. Reinserting them would yield

Rbottom = √2·Rtop, the precise ratio that leads to optimal quality index for a 4-4 platform

[18], given Hcell = Rtop (Figure 5.3). In the SRTM case, however, Hcell = 0.878·Rtop, which equates to a quality index of approximately 0.9751, a value extremely close to unity, and thus very far from singularity. Such a design would get excellent marks from a geometric stability standpoint.

Figure 5.3 Conversion of SRTM cell to stacked 4-4 platform cell

The models were compared to a similarly constructed AEC-Able SRTM mast model to check their accuracy (Figure 5.4). The discrepancies observed between the quoted

SRTM performance and the model prediction can be attributed mainly to joint effects.

The simulations assume perfect, weightless, rigid joints, while the real latches have mass and allow some small motions, and the ball joints in the SRTM allow for rotation of one member relative to another, and limit the transmission of moments through the joint. The model was subjected to axial and lateral loads of 10 lbf each and a torque of 10 in-lb, all applied to the end plane. The performance comparison between the model and actual performance values provided by AEC-Able is summarized in Table 5.1.

Table 5.1 Comparison of AEC-Able specifications and SRTM model results [1] Parameter Specification Model Predicted Difference Factor Derived Value Value (Spec/Model) Axial Deflection, in .0012 2.5e-5 47.06 Lateral Deflection, in 28.07 2.94 9.548 Rotational Deflection, rad 4.46e-4 2.4e-7 1818 First Bending Mode, Hz .10 .092 1.087 First Twisting Mode, Hz .17 >15 Hz <.01

The first natural frequency calculated from the FEM model of the SRTM was 0.092

Hz, very close to the quoted 0.10 Hz. However, AEC-Able lists the first torsional mode

38 at 0.17 Hz, and this was not captured by the simulation. The first torsional frequency from the FEM is above 15 Hz. This is in line with the other results, as deflection was captured more accurately than twisting. In all cases, the models overpredict stiffness, but lateral and axial results are within 1-2 orders of magnitude, while torsional performance deviates by approximately 3 orders of magnitude.

Figure 5.4 AEC-Able SRTM boom finite element model, deformed under 10 lbf lateral end loading

The models were subjected to several static loading conditions. Pure axial, lateral, and torsional loads were applied. The lateral loads were applied both along the x- axis and at 60° from the x-axis toward the y-axis (but remaining in the xy-plane). This was to account for the triangular geometry of the platforms, so that loads both perpendicular and parallel to an edge of the end face were considered. While the stresses induced varied to some extent, the displacement was equivalent in both cases.

The assumption that the end platforms, where constraints and loads were applied, were rigid bodies was made in all cases. When this assumption was not made, the point

39 loading on the vertices produced behaviors that would be extremely unrealistic in a microgravity environment. There was extreme flexure in the top plane, while much of the remaining structure of the boom went essentially undeformed. Since it is likely that the end planes will be either rigid bodies or free ends, having large point loads limited to the vertices is nearly inconceivable.

Figure 5.5 FEM model of 16 cell, 300m x 5m octahedral boom

There was one major application difference that affected the construction of the models. The first use would be as a cantilever; the purpose would be to hold an object at a distance, and this was modeled as an applied load on one end plane and a fixed-fixed- fixed constraint on the other end. The second application, generally considered for longer booms only, is that of a distributed spacecraft bus. In this case, the center platform plane was constrained, with the assumption being that it would represent the center of mass, and the boom would act as two cantilevers on either side of the center plane (Figure 5.5). Thus, the effective length and mass of the structure was halved, and there were two instances of it connected together. This effect must be taken into account when assessing overall stiffness performance.

To quantify the differences between the rigid joint and ball joint modeling approaches, a finite element model was created to simulate a single octahedron under an applied load.

Two symmetric octahedra were tested, <½,1,1.73> and <½,1,3.46>. Then, the same octahedra were analyzed mathematically using screw theory methods. For purposes of consistency, all component beams were taken to have a cross-section of 0.0028 in2, based on a thin tube of outer radius 0.15 in. and a thickness of 0.003 in. In the kinematic models, the loads were applied as wrenches. A wrench is a resultant force-moment couple from all loads acting on a body. It is expressed in Plücker coordinates as

 f  wˆ =   m0  where f can be reduced to a scalar force f multiplying the line through the origin and a moment vector m0 about the origin of an infinitesimal force acting with infinite moment arm.

Figure 5.6 FEM model of octahedron subjected to axial load

An axial force of 3 lbf was applied, as a point load of 1 lbf on each of the top platform vertices in the FEM model (Figure 5.6), and as a wrench of [0 0 -3 0 0 0]T applied to the upper platform for the screw theory model. The results for tensile stress in each leg were in very close agreement; however, the FEM model showed significant bending stresses

41 for the <½,1,1.73> octahedron, up to 25% of the total stress at the ends of the beam. The results for the <½,1,3.46> octahedron showed only 10% of the total stress in bending, so the discrepancy is reduced as h increases and the legs become more axially aligned, as they support mostly tensile/compressive forces. The screw theory model simply did not account for the bending stress, but otherwise agreed with the tensile stress to within 5%.

The screw theory stress predictions are 252 psi and 196 psi for the <½,1,1.73> and

<½,1,3.46> models, and the FEM showed compressive stresses of 248 psi and 200 psi, but with additional bending stresses of 40 and 10 psi (average). The deformed FEM model is shown in Figure 5.7, and stress results from the model are shown in Figure 5.8.

Figure 5.7 Deformed FEM model of octahedron subjected to axial load

Next, a lateral force of 3 units [lbf] was applied, again as a point load of 1 unit on each of the top platform vertices in the FEM, and as a wrench of [-3 0 0 3h 0 0]T applied to the upper platform in the kinematic model. The FEM model showed significant bending stresses for both the <½,1,1.73> and <½,1,3.46> cases, up to 25% of the total stress in the

42 former and 30% in the latter. The screw theory model simply did not account for these bending stresses, but once again was within 5% agreement for the tensile component.

T Figure 5.8 Induced leg stresses in a single octahedron, wapp= [0 0 –3 0 0 0]

The kinematic stress predictions are 437 psi and 691 psi for the <½,1,1.73> and

<½,1,3.46> models, and the FEM showed compressive stresses of 428 and 672 psi, but with additional bending stresses of 80 and 150 psi (average). The deformed model is presented in Figure 5.9, and stress component results of the FEM are shown in Figure

5.10.

A moment of 3 units [lbf·in] was applied to the top platform vertices in the FEM and a wrench of [0 0 0 0 0 3]T was applied to the upper platform body in the kinematic model.

The torsional loading showed the greatest bending stresses, accounting for 30-35% of the total beam stress in each leg for either value of h. The discrepancy between the models in

43 terms of tensile stress components even increased under torsional loading, with agreement of only about 10% observed. The screw theory model stress predictions are

252 psi and 399 psi for the <½,1,1.73> and <½,1,3.46> models, and the FEM showed compressive stresses of 239 and 360 psi, but with additional bending stresses of 100 and

140 psi (average). The deformed model and FEM stress results are shown in Figures

5.11 and 5.12.

Figure 5.9 Deformed FEM model of octahedron subjected to lateral load

The fidelity of the kinematic model suffered from its lack of ability to transmit moments from one member to another and account for bending stresses in its component beams. It is nonetheless a fairly accurate predictor of the tensile and compressive stress components in the legs. In all cases, torsional shear stress on the structural members was noted to be two or more orders of magnitude less than bending or tensile/compressive stress components, and was assessed to have minimal impact on the relative convergence of the two models.

T Figure 5.10 Induced leg stresses in a single octahedron, wapp= [-3 0 0 3h 0 0]

Figure 5.11 Deformed FEM model of octahedron subjected to torsional load

T Figure 5.12 Induced leg stresses in a single octahedron, wapp= [0 0 0 0 0 3]

CHAPTER 6 DATA ANALYSIS AND RESULTS

The finite element simulations yielded a wealth of data. The challenge was extracting meaningful information to verify the hypotheses of which geometrical parameters were of greatest influence on the various performance metrics to develop design and optimization criteria.

A nonlinear least squares model was used to fit the data. A separate model was employed for each cell geometry and for each major output parameter, kA, kL, and kT. In each case, a logarithmic transform was used, as the influence of the parameters was better captured as the product of terms rather than their summation. The standard least squares form is,

Y = X β or β = (XTX)-1XTY where Y is a column vector of output values, X is a matrix whose columns are the values of each input parameter, with rows corresponding to each yi, and β is the least squares coefficient vector. Thus, rather than applying the least squares formulation directly to the output data, the Y vector was built from the values of log2(kA), log2(kL), or log2(kT). The

X matrix is created from columns of log2(xi). Therefore, by taking an antilog of the least squares model,

log2 yi = β1 log xi1 + β2 log xi2 + ... + βk log xik the result can be expressed more compactly as

β1 β2 βk yi = xi1 xi2 (...) xik

46 47

The multiplicative regression model approximates the observed behavior more accurately than the additive regression. It must be noted that the regression-based performance prediction models are data fits, not physical laws or even experimentally supported approximate engineering relationships. They should be used as a guideline only, preferably in the initial design stages. The functions predict the detailed simulation results accurately within a range of reasonable parameter changes, but may or may not lend themselves to extrapolation. It would be unrealistic to expect the functions to predict vastly different configurations than those studied.

For instance, theory would predict stiffness to increase linearly with elastic modulus.

Since this particular dependency was verified to hold in multiple sets of simulations as well as predicted by theory, the best-fit statistical models were constrained to match the conditions observed in the models. Such a constraint was deemed necessary because the true best-fit least squares regression models showed an inverse dependency on modulus.

In the symmetric reinforced case, kA and kL were observed to vary linearly with E, and kT varied linearly with E1.285. The octahedral boom models were observed to vary linearly with E in all cases, so this constraint was enforced on each of the functions.

Mathematically, this was accomplished by dividing yi by the xi corresponding to modulus, essentially solving the least squares for the kA/E ratio, then multiplying both sides by E. A similar procedure could be used to constrain the fit to any parameter with a known coefficient. It should be noted that this procedure did decrease the accuracy of the fit. The true best-fit functions to the complete least squares problem are listed in

Appendix B.

The multiple regression analysis of the axially and laterally loaded cases has shown that booms comprised of symmetric reinforced cells are fairly predictable in nature, with the least squares model prediction within 10% accuracy of over 90% of the data points.

The octahedral least squares functions accurately predict over 80% of the data points within 10% discrepancy from the detailed FEM model. In terms of torsional performance, the symmetric reinforced cell booms are most predictable, at 92%, and the octahedral booms follow at 81%. The curve fits were performed on a select data set in order to capture the asymptotic behavior of the boom stiffness performance, and extreme statistical outliers were removed. A range of cases was considered, varying Rtop, Rbottom,

Hcell, L, E, and the cross sections of the various octahedral and reinforcing members

(Xoct,, Xrnf). It is important to note that the function models were derived using units of in-lbm-s.

Figures 6.1-6.6 show the variation in axial, lateral, and torsional stiffness per unit mass for a 300m long boom with nominal 5m radius as a function of Rtop:Rbottom, composed of

6 3 Carbon Fiber Reinforced Polymer (CFRP) with E=3.62·10 psi, ρ =.061 lb./in. [3]. Each individual curve represents a different number of cells in the boom (and therefore a different Hcell:Rbottom ratio, as L is fixed). Figures 6.7 and 6.8 present the mass of the various designs as a reference to calculate absolute stiffness from the specific stiffness, defined as stiffness divided by mass.

6.1 Axial Specific Stiffness Performance

The booms constructed from high aspect ratio reinforced octahedral cells with low

Rtop:Rbottom ratios exhibited the highest absolute axial stiffness (Figure 6.1). Those booms were also the most desirable when stiffness per unit mass was used as the performance

49 metric. The octahedral booms exhibited similar trends, with high cell aspect ratio and low radius ratio providing maximum stiffness (Figure 6.2). The baseline triangular truss showed admirable axial stiffness performance, but the inherent twisting under load due to its asymmetric diagonals prevented it from eclipsing optimized reinforced octahedral cells.

Figure 6.1 Specific axial stiffness vs. Rtop:Rbottom for various numbers of cells per SYM boom

Axial stiffness is quite sensitive to both the Hcell:Rbottom and Rtop:Rbottom ratios, as they determine the axial alignment of the intermediate members. It is clear that axial stiffness would be maximized if all members were parallel to the major axis, as all force would be borne in tension or compression, as opposed to shear or bending. In axial loading for non-parallel members oriented at an angle θ from the boom axis, the axial force F becomes a component of the tensile force F/cos(θ) that results in the beam. Since cos(θ)

≤ 1, the resultant force F/cos(θ) is never less than F, and at best equal to F in the axially

50 aligned case. One of the major reasons the symmetric reinforced booms have flatter response curves with respect to Rtop:Rbottom than their octahedral counterparts is that the axial reinforcements bear a great deal of the load, and are always in line with the axis regardless of the platform dimensions.

Figure 6.2 Specific axial stiffness vs. Rtop:Rbottom for various numbers of cells per OCT boom

The least squares estimation functions for specific axial stiffness performance of an octahedral boom design spanning approximately 30-300 meters, with a nominal radius between 2 and 5 meters, as calculated from a subset of simulations is

251.8 1.28 49 3.68 2 R E ρ C # k A,oct = 2.65R 2.54 2.97 .007 .053 2 V L A X oct

The corresponding model performance prediction for a similarly sized symmetric reinforced boom is

563 .41 8.84 142.6−20.39 log2 ( A) 2 R E X oct A k A,sym = .22R .87 3.20 6.53 150.71−7.31log2 ( L) .67 2 ρ V X rnf L C #

6.2 Lateral Specific Stiffness Performance

As in the axial cases, the symmetric reinforced boom was the most predictable, and the top performer (Figure 6.3). The baseline triangular truss showed similar performance, due to its reliance on the longerons to resist bending moments, and the octahedral boom showed the greatest variance in performance, particularly for small

Rtop:Rbottom ratios (Figure 6.4).

Figure 6.3 Specific lateral stiffness vs. Rtop:Rbottom for various numbers of cells per SYM boom

The lateral bending performance was influenced greatly by two factors, the mass distribution from the central axis, and the axial alignment of the individual elements. The further the mass is distributed from the center of the structure, the greater the second

52 moment of area I, and therefore the greater the bending stiffness. Nevertheless, there is a practical limit to increasing the radius (either Rbottom or Rtop), as the forces are transmitted more efficiently at the joints when the members are at a small angle relative to each other. Thus, a high cell aspect ratio Hcell:Rbottom is desirable.

Figure 6.4 Specific lateral stiffness vs. Rtop:Rbottom for various numbers of cells per OCT boom

The octahedral boom performance is highly dependent on the cell aspect ratio. It is interesting to note that the maximum lateral stiffness as a function of Rtop:Rbottom shifts consistently to higher values as Hcell:Rbottom increases, which would serve to increase the quality index of the individual cells.

The numerical performance estimation functions for an octahedral boom design spanning approximately 30-300 meters, with a nominal radius between 2 and 5 meters, as calculated from a subset of simulations is

2282.3 R 2.18 E ρ 50 A.013 X .0025C 3.98 k = oct # L,oct 22.2R V 2.93 L 5.44

The corresponding model performance prediction for a similarly sized symmetric reinforced boom is

801 .26 11.306 205.5−29.46 log2 ( A) 2 R E X oct A k L,sym = .59R 1.27 4.33 3.808 215.74−10.46 log2 ( L) 1.35 2 ρ V X rnf L C #

6.3 Torsional Specific Stiffness Performance The torsional case proved the most difficult to predict numerically, though trends were apparent. The reinforced octahedral cell boom was the most predictable (Figure 6.5), but the best performance was delivered by the octahedral cell boom, due to its symmetry and lack of redundant members (Figure 6.6). While the longerons in the other two booms are the main load-bearers for axial and lateral loads, the legs of the octahedral structure and the diagonal braces are the workhorses in torsion. The octahedral cell shows better stiffness per unit mass than the symmetric reinforced octahedron, not only because the added axial members add material but carry little of the torsional load, they actually act almost as a separate shaft, and allow the octahedra inside to remain relatively undeformed, resulting in a decrease in overall torsional stiffness. Since nature prefers the minimum energy configuration even under stress, the forces will take the path of least resistance, and the deformation of the outside reinforcing tubes is certainly easier than transmitting force through the octahedral legs.

However, the symmetric reinforced boom does show substantial improvement over the baseline triangular truss in torsional loading. The three diagonal braces, which carry most of the load, since the longerons are not in a position to resist twisting motions, are stressed more heavily than and outnumbered by their octahedral leg counterparts. This

54 leads to a greater stress (and thus strain) in these diagonal members, and with a comparatively longer free length, results in even greater elongation. The intermediate members also are more effective in resisting torsion when they are oriented near 45° from the platforms. The more axially aligned they are, the smaller the contribution to torsional stiffness. The implication of this result is that maximizing axial and lateral stiffness performance will almost certainly degrade torsional stiffness. The torsional specific stiffness model prediction functions for octahedral and symmetric reinforced cells are

287.1 2.69 49 5.31 6.14 2 R E ρ L C # kT,oct = 1.59R 2.61 .009 .062 2 V A X oct

462 2.57 1.285 18.89 136.31−19.36 log2 ( A) 2 R E X rnf A kT,sym = 1.68R 1.54 1.1 9.19 133.46−6.41log2 (L) .57 2 ρ V X oct L C #

The octahedral cell booms again showed great variance in performance, both with

Rtop:Rbottom and with cell aspect ratio. However, unlike with the lateral and axial loads, the preferred geometry is one where each octahedron has a slightly wider base and less axial alignment of the legs. Increasing the cell aspect ratio magnifies this effect. Low aspect ratio cells work most efficiently near an Rtop:Rbottom ratio of unity, while higher aspect ratio cells tend to reach maximum performance at increased radius ratios. It is interesting to note that one of the better geometric ratios is <2, 1, 2√3>, which is the precisely the optimal platform radius ratio (½:1) and height to smaller triangle radius

(√3:1) predicted by line geometry, with the top and bottom platforms exchanging functions and dimensions.

Figure 6.5 Specific torsional stiffness vs. Rtop:Rbottom for various numbers of cells per SYM boom

6.4 Overall Performance Trends

The maximum performance points of 300m x 5m class booms are characterized in

Table 6.1. Within the range of designs considered, the maximum specific axial stiffness of the octahedral cells decreases roughly by half for each change of aspect ratio caused by the addition of eight more cells. A similar trend is apparent in the specific lateral stiffness as well. The maximum specific torsional stiffness is less sensitive to changes in aspect ratio, with the best case being only a factor of three improvement over the worst performer. There also appears to be a trend in the optimal platform radius ratio with respect to performance. As Rtop:Rbottom for a given Rbottom:Hcell is increased from ½ to 1 to

1.5 and beyond, the best configuration is achieved for axial, lateral, and torsional applications, respectively. The implication is that the octahedral cell booms can be

56 optimized for only one function at a time, and a compromise must be made to ensure acceptable performance in two or three categories simultaneously.

Table 6.1 Maximum specific stiffness performance for 300m x 5m CFRP booms TYPE,# AXIAL LATERAL TORSIONAL Rtop:Rbottom:Hcell kA Rtop:Rbottom:Hcell kL Rtop:Rbottom:Hcell kT

OCT 8 .25:1:6.93 280 1.33:1:6.93 0.372 OCT 16 .50:1:3.47 157 1.00:1:3.47 0.155 2.00:1:3.47 2.90e6 OCT 24 .50:1:2.31 77 1.00:1:2.31 0.062 1.50:1:2.31 2.61e6 OCT 32 .60:1:1.73 38 .75:1:1.73 0.027 1.20:1:1.73 2.20e6

SYM 8 .25:1:6.93 321 1.25:1:6.93 0.475 SYM 16 .50:1:3.47 274 .88:1:3.47 0.405 2.00:1:3.47 2.28e6 SYM 24 .50:1:2.31 203 .75:1:2.31 0.314 1.63:1:2.31 2.13e6 SYM 32 .36:1:1.73 160 .50:1:1.73 0.265 1.38:1:1.73 1.87e6 SYM 40 .25:1:1.38 133 .25:1:1.38 0.233 1.25:1:1.38 1.58e6

BAS 16 1.00:1:3.47 155 1.00:1:3.47 0.298 1.00:1:3.47 1.90e5 BAS 20 1.00:1:2.78 149 1.00:1:2.78 0.284 1.00:1:2.78 2.56e5 BAS 24 1.00:1:2.78 140 1.00:1:2.78 0.270 1.00:1:2.78 3.11e5 BAS 32 1.00:1:1.73 126 1.00:1:1.73 0.243 1.00:1:1.73 3.83e5 BAS 40 1.00:1:1.38 98 1.00:1:1.38 0.188 1.00:1:1.38 3.50e5

However, upon closer inspection of the performance curves, the specific lateral stiffness performance is often far less sensitive than specific torsional stiffness performance to variations in platform radius ratio. An example design optimization chart has been created for the 16 cell symmetric octahedral boom to illustrate this point (Figure

6.9). It shows plots of the actual values kA, kL, and kT against Rtop:Rbottom, and then a combined plot of relative variations of performance as a fraction of its own maximum kA, kL, and kT. Clearly, one can see that axial and lateral stiffnesses vary little in absolute terms, as in the range of <.25,1,Hcell> to <2.5,1,Hcell> they drop to only 60% of their respective peaks. Torsional stiffness, on the other hand, drops as low as 5% of its maximum. Therefore, a designer could choose to sacrifice a small amount of performance in one parameter for a large gain in another. And if the performance curve of the parameter decreased is relatively flat, the damage done in an absolute sense could

57 be very minimal. A set of optimization charts is included in Appendix C that covering a wide range of potential designs.

Figure 6.6 Specific torsional stiffness vs. Rtop:Rbottom for various numbers of cells per OCT boom

Another key finding is that the specific torsional stiffness of the octahedral and reinforced octahedral booms is nearly an order of magnitude greater in some cases. The implication of this result is that a tradeoff can be made, sacrificing torsional stiffness for increased lateral and axial stiffness. The 24-cell diagonal truss boom was assessed to be an efficient design point due to its balanced performance with respect to lateral and torsional loading. Comparisons of performance among the various truss configurations

(Figures 6.10, 6.11, 6.12 for octahedral cells, Figures 6.13, 6.14, 6.15 for symmetric reinforced octahedral cells) show that there are designs that are capable of eclipsing the diagonal truss in all three specific stiffness categories, but they require high aspect ratios.

For instance, the 8-cell octahedral boom exceeds the baseline axial, lateral, and torsional stiffnesses by 81%, 27%, and 143% (factors of 1.81, 1.27, and 2.43). The 8-cell reinforced octahedral boom at Rtop:Rbottom=1.5 yields increases of 216%, 224%, and 93%, respectively for axial, lateral, and torsional performance. The 16-cell symmetric reinforced boom shows relative improvements of 61%, 40%, and 583% over the 24-cell baseline triangular truss boom. A set of comparison charts relating the octahedral and reinforced octahedral booms to the 32-cell triangular truss, a common design due to the

45° angle of its diagonals, is included in Appendix C.

Figure 6.7 Mass vs. Rtop:Rbottom of 300m x 5m symmetric reinforced booms

Figure 6.8 Mass vs. Rtop:Rbottom of 300m x 5m octahedral booms

Figure 6.9 Design optimization chart, 300m x 5m, 16 cell symmetric reinforced octahedral boom vs. Rtop:Rbottom

# cells

Figure 6.10 Comparison of kA, 300m x 5m octahedral boom vs. 24-cell triangular truss (CFRP)

Figure 6.11 Comparison of kL, 300m x 5m octahedral boom vs. 24-cell triangular truss (CFRP)

Figure 6.12 Comparison of kT, 300m x 5m octahedral boom vs. 24-cell triangular truss (CFRP)

Figure 6.13 Comparison of kA, 300m x 5m symmetric reinforced octahedral boom vs. 24- cell triangular truss (CFRP)

Figure 6.14 Comparison of kL, 300m x 5m symmetric reinforced octahedral boom vs. 24- cell triangular truss (CFRP)

Figure 6.15 Comparison of kT, 300m x 5m, symmetric reinforced octahedral boom vs. 24-cell triangular truss (CFRP)

CHAPTER 7 NUMERICAL EXAMPLES

Two numerical example problems are presented. The dual figure of a distorted octahedron is obtained, and an optimization problem for boom design is discussed.

7.1 Dual Polyhedron

As previously discussed, the pseudo-inverse method for producing dual polyhedra is valid irrespective of initial coordinate system position, orientation, and scale. This example will illustrate the procedure to find the dual cuboid of an octahedron elongated by a factor of 2.5 along its x-axis, translated from (0,0,0) to (2,0,1), and rotated 20° about its original z-axis. The six coordinates for the vertices are initially placed at

1 0 0 − 1  0   0  0 1 0  0  − 1  0  p =  , p =  , p =  , p =  , p =  , p =   1 0 2 0 3 1 4  0  5  0  6 − 1             1 1 1  1   1   1 

Then, p1 and p4 are multiplied by f=2.5,

2.5 − 2.5  0   0  p =  , p =   1  0  4  0       1   1 

Next, the coordinate transform matrix is created,

cos( 20°) − sin( 20°) 0 2 sin( 20°) cos( 20°) 0 0 T =    0 0 1 1    0 0 0 1

63 64

Each point is now multiplied by the coordinate transform T to yield the six new vertices vi.

 4.2286  − .4698 2.214  1.5374  1.8794  1.8794  −1.5391  .1710  .2557 −1.6237 − 6.840 − .6840 v =   v =   v =   v =   v =   v =   1  1  2  1  3  1  4  1  5  2  6  0               1   1   1   1   1   1 

The twelve lines along the edges of the octahedron are formed in Plücker ray coordinates from the six vertices vi. Next, the Jacobian matrix J is assembled from the twelve lines,

J = Columns 1 through 7 2.3492 2.3492 -2.3492 -2.3492 -2.0072 -2.6913 2.0072 -0.8551 -0.8551 0.8551 0.8551 1.7947 -0.0846 -1.7947 -1.0000 1.0000 1.0000 -1.0000 0 0 0 2.3941 1.0261 -0.6840 0.6840 -1.7947 0.0846 1.7947 6.5778 2.8191 -1.8794 1.8794 -2.0072 -2.6913 2.0072 0 0 0 0 4.5000 0.5000 0.5000

Columns 8 through 12 2.6913 -0.3420 -0.3420 0.3420 0.3420 0.0846 -0.9397 -0.9397 0.9397 0.9397 0 1.0000 -1.0000 -1.0000 1.0000 -0.0846 1.1953 2.5634 0.6840 -0.6840 2.6913 -2.5634 1.1953 1.8794 -1.8794 4.5000 -2.0000 -2.0000 2.0000 2.0000

The example octahedron drawn by the MATLAB program is shown in Figure 7.1.

Figure 7.1 Octahedron elongated f = 2.5 in x-direction, translated to (2,0,1), and rotated 20° about z-axis

The pseudo-inverse of J is now calculated, then rearranged into a Jcube matrix that represents the lines forming the cuboid as columns,

JCUBE = Columns 1 through 7 0.3420 0.3420 0.3420 0.3420 0.0000 0.0000 0.0000 0.9397 0.9397 0.9397 0.9397 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 1.0000 1.0000 1.0000 -0.4698 -0.4698 -1.4095 -1.4095 -0.7264 -1.5814 -0.6417 0.1710 0.1710 0.5130 0.5130 -0.5338 -2.8830 -3.2250 0.7500 3.2500 3.2500 0.7500 0.0000 0.0000 -0.0000

Columns 8 through 12 0.0000 0.9397 0.9397 0.9397 0.9397 -0.0000 -0.3420 -0.3420 -0.3420 -0.3420 1.0000 0.0000 0.0000 -0.0000 -0.0000 0.2133 0.1710 0.1710 0.5130 0.5130 -0.8758 0.4698 0.4698 1.4095 1.4095 -0.0000 0.5000 -0.5000 -0.5000 0.5000

It is left as an exercise for the reader to verify the mutual moment calculations for the eight vertices of the cuboid. The line combinations forming each of the eight new vertices are:

Table 7.1 List of vertices for mutual moment calculations Vertex 1 Vertex 2 Vertex 3 Vertex 4 Vertex 5 Vertex 6 Vertex 7 Vertex 8 $1, $5 $1, $9 $2, $6 $2, $9 $4, $5 $3, $6 $4, $8 $3, $7 $1, $8 $1, $10 $2, $7 $2, $10 $4, $12 $3, $12 $4, $11 $3, $11 $5, $8 $9, $10 $6, $7 $9, $10 $5, $12 $6, $12 $8, $11 $7, $11

A sketch of the resultant cuboid as generated by the software is shown in Figure 7.2.

Figure 7.2 Pseudo-Inverse dual of example octahedron, MATLAB™ program output (note: lines trimmed to enhance visibility)

7.2 Example Boom Design Problem

GIVEN: Performance Requirements – Mass, maximum (m) 700 lb -2 Specific Lateral stiffness, minimum (kL) .17 s -2 Specific Axial stiffness, minimum (kT) 200 s Geometric Requirements – Boom Length, (L) 2500 in Mount radius, maximum (Rbottom) 60 in Nominal radius, maximum (max of Rtop, Rbottom) 150 in Component beam cross-section .75 in2 Material Parameters (Al 2024) – Young’s Modulus (E) 10.5·106 psi Density (ρ) .100 lb/in3

FIND: Candidate design with optimal specific Torsional Stiffness (kT)

EXAMPLE PROCEDURE: First, take the log2 of both sides of each of the equations for kA, kL, kT, and m, reducing them to a linear combination of functions of the variables Rtop:Rbottom,, C#.

The design goal is maximum specific torsional stiffness kT, so use the performance equation for kT as the objective function. The other requirements can be written as simple inequality constraints. This problem can now be solved by standard linear programming techniques, such as the simplex method. The standard form of the problem in linear programming terms is:

Maximize log2 KT = β1 x1 + β2 x2 +… Subject to (c11 x1 + c12 x2 + …) >= log2 kA (c21 x1 + c22 x2 + …) >= log2 kL (c31 x1 + c32 x2 + …) <= log2 m all xi >= 0

The coefficients βk and cij are the powers of the original equations, and are known quantities. The variables xi are equal to log2(Xi), where Xi is the original variable, such as E, L, A, C#, etc. The problem remains fundamentally the same.

CHAPTER 8 CONCLUSIONS

The conclusions based upon each major portion of the effort are discussed. Results from the areas of line geometry as well as structural design are presented.

8.1 Line Geometry

Evidence was presented to support the claim that the results of Lee show the maximum value of √det(JJT), used in calculating the geometrical quality index for parallel platform manipulators, is limited by the size of the upper (moving) platform. As the area enclosed by the polygon created by connecting the vertices of the upper platform approaches that of the circle circumscribing them, a maximum value of √det(JJT) is observed. This upper bound implies that there is a diminishing return on the addition of connecting points between the platforms, and that a global maximum value for √det(JJT) exists for a given upper platform area.

An inherent limitation involved with using purely kinematic and geometric methods to model structures is the assumption of freedoms that must be made at the joints. These assumptions result in some discrepancies when calculating the forces and stresses in a structural component. The kinematic model has the ability to predict tensile and compressive components of stress in a rigid boom structure with 5-10% accuracy, but when bending stresses become significant the fidelity of the model wanes considerably.

The geometric analyses all produced results that heavily favored symmetry. The quality index of a rigidized version of an AEC-Able SRTM mast cell was assessed to by

0.9751. This result lends credence to the line geometry portion of the design

67 68 methodology. The SRTM mast is a flight-proven example that was successful in meeting strict design requirements. The fact that standard design methods arrived a near-optimal quality index is likely no accident.

8.2 Duality of Polyhedra

Meaningful results were obtained from the pseudo-inverse transform of the line coordinates of Platonic polyhedra. It was shown that the transform is valid for any choice of coordinate system. A discussion of the various distortions of the initial polyhedron that allow for a closed dual to result was presented. Scaling along any of the interior axes of an octahedron results in a rectangular prism. A simultaneous enlargement of one point on each interior axis (i.e. one face of the octahedron) by the same factor produces an even more general closed figure with six nonparallel faces. The extreme case of enlarging one face of an octahedron until the figure approaches a tetrahedron was examined, and it is interesting to note that its dual cuboid also approaches a tetrahedron.

8.3 Structural Simulation

While analytic closed form solutions were not obtained, the patterns that emerged from the combination of geometric stability analyses and finite element simulations are readily apparent.

A basic inefficiency exists in the conventional diagonal truss based on the asymmetry of its design. Only a part of the whole structure is in position to resist any particular loading. There is an inherent twisting motion induced by this asymmetry that reduces both axial and torsional stiffness. There is another important effect that becomes apparent upon close inspection of the stress-strain behavior. Under torsional loading, the triangular truss has only three primary load-bearers, which are the diagonal braces. The octahedral and reinforced octahedral cells distribute the load among six legs instead of

69 three, giving a reduction in force per member. However, the greater force leads to a greater stress, and thus, a greater strain. Strain, ∆l / l0, is a phenomenon that is proportional to initial length. The diagonal members are longer than the legs of the internal octahedra in the symmetric cases. These two factors coupled allow for much greater deformation of the diagonals in a conventional truss than the legs in the octahedrally braced cells. Given equal cell dimensions, there is also a lower critical stress for Euler buckling in torsion for the triangular truss, again due to the increased length of the diagonals.

The octahedral truss showed the greatest variability in terms of performance. At high cell aspect ratios, specifically Rbottom:Hcell from 2√3> to 4√3, and low platform radius ratios, especially between ¼ and ½, the axial performance was maximized. The peak torsional performance, generally a design with Rtop:Rbottom between 1 and 2, outdistanced the other two trusses by wide margins. However, the lateral stiffness was poor by comparison, particularly when torsional stiffness was high.

However, the geometric parameter ranges over which this performance can be achieved are relatively small, which could lead to problems for a deployable system, as the sizes of the constituent members must be kept within this tight tradespace of efficient values.

The symmetric reinforced octahedral truss was shown to be a viable alternative to the standard triangular diagonal truss in terms of stiffness and specific stiffness for a given mass constraint. Its axial and lateral stiffness performance is governed primarily by the stiffness of the axial reinforcements, as well as the axial alignment of the internal octahedra. The torsional stiffness is highly dependent on the ratio of top and bottom

platform radii (Rtop:Rbottom), and to a lesser degree on the cell aspect ratio (Hcell:Rtop). In terms of complexity and deployability, it is more tolerant of geometric changes with regard to performance degradation, so changes dictated by deployment mechanisms or manufacturing considerations can be more easily absorbed. While it does require a greater number of joints per cell than either of the other designs, this complexity may be justified by the additional performance it provides. The structural redundancy it offers may also be beneficial.

One of the keys to the design of a successful structure is the ability to optimize stress distribution amongst the various members. In a diagonal truss, one has five notable free choices, since the radius, aspect ratio, and the three component cross-sections can be chosen. However, the diagonals and longerons are inherently different, and it is difficult to equalize stresses under axial, lateral, and torsional loading in a single design. A better distribution can be created by variation of cross-section, but the optimum varies for each loading type. The octahedral cells, however, have an innate symmetry, which distributes the forces into the six legs evenly in the axial and torsional cases (in torsion, three are in tension and three are in compression of equal magnitude). Thus, one of the constraints on the optimization process is basically removed. Free choices still remain for leg, top, and bottom platform cross sections, as well as aspect ratio and platform radius ratio.

Since the same number of freedoms exist in both cases, but one limitation is removed in the latter, it becomes a more tractable problem to optimize for more than one performance metric.

The optimal cell geometry for pure axial stiffness is the reinforced octahedron with extremely high cell aspect ratio and Rtop:Rbottom ratio of ¼:1, so that it is composed of

71 nearly axial members. This result should come as no surprise, as almost all the mass in the structure is dedicated to resisting axial tension or compression, with the platforms added only to shorten the effective Euler buckling length of the legs and axial members and prevent failure.

High lateral stiffness is also observed in reinforced octahedral cells with high cell aspect ratio and platform radius ratio close to unity. In fact, there appears to be a relationship between the two parameters that optimizes the angle of the legs to distribute stress more evenly among the legs and reinforcing members.

Overall, by using higher aspect ratio cells than found to be optimal in a single octahedron, large performance gains can be realized. These gains are predicted from the modeling to be as high as 200% for the axial and lateral specific stiffness metrics, and up to an order of magnitude improvement in resistance to torsional loading when compared to a conventional diagonal truss.

CHAPTER 9 FUTURE WORK

While this effort is a first step toward a unified design methodology, more work needs to be done to continue along this useful path. There were several phenomena that were observed during this effort that could not be adequately explained at this time, and would likely have some contribution to design if they could be harnessed. High on that list is the determination the coefficient that drives the maximum quality index of a general N- sided polygon platform manipulator. Also of great importance is the solution to the forward and reverse analyses of multiply-parallel and devices.

The investigation into the duality of polyhedra yielded new theoretical results. While the duality of the icosahedron-dodecahedron pair was studied, a more extensive study of the distortions that allow a closed dual to result should be carried out. Another extension of this effort along the same lines would be to investigate semi-regular polyhedra.

While some trends about the optimal geometries for long booms, which is a one- dimensional stack of cells, have been observed, little is known about the performance of octahedral unit cells when aggregated in a two- or three-dimensional lattice structure.

This is an area that ought to be studied, as there are many applications for both planar and conical shapes, as well as volume filling support structures. Some insight into the optimal cells in these cases may be found in the study of strong materials, as the lattice structures formed by molecules may show some parallels to the placement of nodes in a structure. The strengths of the atomic bonds may prove useful in deciding the strengths of the component structural members.

72 73

The fact remains that the SRTM boom is the only flight-proven piece of hardware in this class of major dimension. Until larger structures can be physically tested, there still remains some doubt as to the accuracy of the model at points in the tradespace significantly distinct from the SRTM mast. Additional modeling and better development of the performance prediction functions would also benefit designers, as it could reduce time spent on initial design stages, providing an approach based on simple calculations rather than elaborate simulations.

An investigation into the stowage and on-orbit deployment is also critical before contemplating using such a structure in space. This would involve more significant development of the strengths of the structures, as from a mechanical perspective launch is strength-driven, and on-orbit performance tends to be stiffness-driven. The high aspect ratio cells suggested by this study may prove more difficult to package and deploy, and longer members typically will buckle more easily. Further study must involve system- level design philosophy, and compare the performance gains against the cost of additional complexity associated with it.

Another logical extension of this work is to construct prototype octahedral and reinforced octahedral cells and test their properties experimentally, as few revolutionary concepts become widely accepted without more practical evidence than theory and models.

APPENDIX A MATLAB CODE FOR PSEUDO-INVERSE PROGRAM

The files below should be stored as .m files in the MATLAB path. The command to begin the program is “octgui”

octgui.m %draw GUI interface for program. type "octgui" to run octgui1

%get handles from tags editXpos_ =findobj(gcf, 'Tag', 'editXpos'); sliderXpos_ =findobj(gcf, 'Tag', 'sliderXpos'); editYpos_ =findobj(gcf, 'Tag', 'editYpos'); sliderYpos_ =findobj(gcf, 'Tag', 'sliderYpos'); editZpos_ =findobj(gcf, 'Tag', 'editZpos'); sliderZpos_ =findobj(gcf, 'Tag', 'sliderZpos'); editXneg_ =findobj(gcf, 'Tag', 'editXneg'); sliderXneg_ =findobj(gcf, 'Tag', 'sliderXneg'); editYneg_ =findobj(gcf, 'Tag', 'editYneg'); sliderYneg_ =findobj(gcf, 'Tag', 'sliderYneg'); editZneg_ =findobj(gcf, 'Tag', 'editZneg'); sliderZneg_ =findobj(gcf, 'Tag', 'sliderZneg'); editL_ =findobj(gcf, 'Tag', 'editL'); sliderL_ =findobj(gcf, 'Tag', 'sliderL');

%set callback functions set(editXpos_, 'Callback', 'oct_edit'); set(sliderXpos_, 'Callback', 'oct_slider'); set(editYpos_, 'Callback', 'oct_edit'); set(sliderYpos_, 'Callback', 'oct_slider'); set(editZpos_, 'Callback', 'oct_edit'); set(sliderZpos_, 'Callback', 'oct_slider');

set(editXneg_, 'Callback', 'oct_edit'); set(sliderXneg_, 'Callback', 'oct_slider'); set(editYneg_, 'Callback', 'oct_edit'); set(sliderYneg_, 'Callback', 'oct_slider'); set(editZneg_, 'Callback', 'oct_edit'); set(sliderZneg_, 'Callback', 'oct_slider');

set(editL_, 'Callback', 'oct_edit'); set(sliderL_, 'Callback', 'oct_slider'); %end

74 75 octgui1.m function fig = octgui1() % This is the machine-generated representation of a Handle Graphics object % and its children. Note that handle values may change when these objects % are re-created. This may cause problems with any callbacks written to % depend on the value of the handle at the time the object was saved. % This problem is solved by saving the output as a FIG-file. % % To reopen this object, just type the name of the M-file at the MATLAB % prompt. The M-file and its associated MAT-file must be on your path. % % NOTE: certain newer features in MATLAB may not have been saved in this % M-file due to limitations of this format, which has been superseded by % FIG-files. Figures which have been annotated using the plot editor tools % are incompatible with the M-file/MAT-file format, and should be saved as % FIG-files. mat0 =[ 0 0 0.5625; 0 0 0.6250; 0 0 0.6875; 0 0 0.7500; 0 0 0.8125; 0 0 0.8750; 0 0 0.9375; 0 0 1.0000; 0 0.0625 1.0000; 0 0.1250 1.0000; 0 0.1875 1.0000; 0 0.2500 1.0000; 0 0.3125 1.0000; 0 0.3750 1.0000; 0 0.4375 1.0000; 0 0.5000 1.0000; 0 0.5625 1.0000; 0 0.6250 1.0000; 0 0.6875 1.0000; 0 0.7500 1.0000; 0 0.8125 1.0000; 0 0.8750 1.0000; 0 0.9375 1.0000; 0 1.0000 1.0000; 0.0625 1.0000 1.0000; 0.1250 1.0000 0.9375; 0.1875 1.0000 0.8750; 0.2500 1.0000 0.8125; 0.3125 1.0000 0.7500; 0.3750 1.0000 0.6875; 0.4375 1.0000 0.6250; 0.5000 1.0000 0.5625; 0.5625 1.0000 0.5000; 0.6250 1.0000 0.4375; 0.6875 1.0000 0.3750; 0.7500 1.0000 0.3125; 0.8125 1.0000 0.2500; 0.8750 1.0000 0.1875; 0.9375 1.0000 0.1250; 1.0000 1.0000 0.0625; 1.0000 1.0000 0; 1.0000 0.9375 0; 1.0000 0.8750 0; 1.0000 0.8125 0; 1.0000 0.7500 0; 1.0000 0.6875 0; 1.0000 0.6250 0; 1.0000 0.5625 0; 1.0000 0.5000 0; 1.0000 0.4375 0; 1.0000 0.3750 0; 1.0000 0.3125 0; 1.0000 0.2500 0; 1.0000 0.1875 0; 1.0000 0.1250 0; 1.0000 0.0625 0; 1.0000 0 0; 0.9375 0 0; 0.8750 0 0; 0.8125 0 0; 0.7500 0 0; 0.6875 0 0; 0.6250 0 0; 0.5625 0 0;]; h0 = figure('Color',[0.8 0.8 0.8], ... 'Colormap',mat0, ... 'FileName','C:\WINDOWS\Desktop\thesis\octgui1.m', ... 'PaperPosition',[18 180 576 432], ‘PaperUnits','points', ... 'Position',[417 285 560 420], 'Tag','Fig1', 'ToolBar','none'); h1 = uicontrol('Parent',h0, ... 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'FontName','Airal', 'FontSize',12, 'FontWeight','bold', ... 'ListboxTop',0, 'Position',[135.75 265.5 135 44.25], ... 'String','Octahedron Pseudo-Inverse Visualizer', 'Style','text', ... 'Tag','StaticText1'); h1 = uicontrol('Parent',h0, ... 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'Callback','oct_slider', 'FontWeight','bold', ... 'ListboxTop',0, 'Max',100, 'Position',[184.5 228.75 45 15], ... 'SliderStep',[1e-005 0.001],'String','1', 'Style','slider', ... 'Tag','sliderZpos', 'Value',1); h1 = uicontrol('Parent',h0, 'Units','points', 'BackgroundColor',[1 1 1], ... 'Callback','oct_edit', 'FontWeight','bold', 'ListboxTop',0, ... 'Max',100, 'Position',[184.5 243.75 45 15], 'SliderStep',[1e-005 0.001], ... 'String','1', 'Style','edit', 'Tag','editZpos', 'Value',1); h1 = uicontrol('Parent',h0, 'Units','points', 'BackgroundColor',[1 1 1], ... 'Callback','oct_edit', 'FontWeight','bold', 'ListboxTop',0, 'Max',100, ... 'Position',[246 173.25 45 15], 'SliderStep',[1e-005 0.001], ... 'String','1','Style','edit', 'Tag','editYpos', 'Value',1); h1 = uicontrol('Parent',h0, ... 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ...

'Callback','oct_slider', 'FontWeight','bold', 'ListboxTop',0, 'Max',100, ... 'Position',[246 158.25 45 15], 'SliderStep',[1e-005 0.001], 'String','1', ... 'Style','slider', 'Tag','sliderYpos', 'Value',1); h1 = uicontrol('Parent',h0, ... 'Units','points', 'BackgroundColor',[1 1 1], 'Callback','oct_edit', ... 'FontWeight','bold', 'ListboxTop',0, 'Max',100, 'Position',[229.5 120 45 15], ... 'SliderStep',[1e-005 0.001], 'String','1', 'Style','edit', 'Tag','editXpos', ... 'Value',1); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'Callback','oct_slider', 'FontWeight','bold', 'ListboxTop',0, 'Max',100, ... 'Position',[229.5 105 45 15], 'SliderStep',[1e-005 0.001], 'String','1', ... 'Style','slider', 'Tag','sliderXpos', 'Value',1); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'Callback','oct_slider', 'FontWeight','bold', 'ListboxTop',0, 'Max',0, ... 'Min',-100, 'Position',[120.75 160.5 45 15], 'SliderStep',[1e-005 0.001], ... 'String','-1', 'Style','slider', 'Tag','sliderXneg', 'Value',-1); h1 = uicontrol('Parent',h0, 'Units','points', 'BackgroundColor',[1 1 1], ... 'Callback','oct_edit', 'FontWeight','bold', 'ListboxTop',0, 'Max',0, ... 'Min',-100, 'Position',[120.75 175.5 45 15], 'SliderStep',[1e-005 0.001], ... 'String','-1', 'Style','edit', 'Tag','editXneg', 'Value',-1); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'Callback','oct_slider', 'FontWeight','bold', 'ListboxTop',0, 'Max',0, ... 'Min',-100, 'Position',[104.25 106.5 45 15], 'SliderStep',[1e-005 0.001], ... 'String','-1', 'Style','slider', 'Tag','sliderYneg', 'Value',-1); h1 = uicontrol('Parent',h0, 'Units','points', 'BackgroundColor',[1 1 1], ... 'Callback','oct_edit', 'FontWeight','bold', 'ListboxTop',0, 'Max',0, ... 'Min',-100, 'Position',[104.25 121.5 45 15], 'SliderStep',[1e-005 0.001], ... 'String','-1', 'Style','edit', 'Tag','editYneg', 'Value',-1); h1 = uicontrol('Parent',h0, 'Units','points', 'BackgroundColor',[1 1 1], ... 'Callback','oct_edit', 'FontWeight','bold', 'ListboxTop',0, ... 'Max',0, 'Min',-100, 'Position',[162 54.75 45 15], 'SliderStep',[1e-005 0.001], ... 'String','-1', 'Style','edit', 'Tag','editZneg', 'Value',-1); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'Callback','oct_slider', 'FontWeight','bold', 'ListboxTop',0, ... 'Max',0, 'Min',-100, 'Position',[162 39.75 45 15], 'SliderStep',[1e-005 0.001], ... 'String','-1', 'Style','slider', 'Tag','sliderZneg', 'Value',-1); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'ListboxTop',0, 'Position',[153.75 111 72 18], 'String','______', ... 'Style','text', 'Tag','StaticText2'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'ListboxTop',0, 'Position',[167.25 167.25 72 18], 'String','______', ... 'Style','text', 'Tag','StaticText2'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'ListboxTop',0, 'Position',[252 137.25 15 9], 'String','/', 'Style','text', ... 'Tag','StaticText3'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'ListboxTop',0, 'Position',[254.25 146.25 20.25 9.75], 'String','/', ... 'Style','text', 'Tag','StaticText3'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'ListboxTop',0, 'Position',[128.25 138.75 15 9], 'String','/','Style','text', ... 'Tag','StaticText3'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'ListboxTop',0, 'Position',[130.5 147.75 20.25 9.75], 'String','/', ... 'Style','text', 'Tag','StaticText3'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'Callback','octv3', 'FontAngle','italic', 'FontSize',10, 'FontWeight','bold', ... 'ListboxTop',0, 'Position',[14.25 9.75 134.25 19.5], 'String','FIND DUAL', ... 'Tag','run'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ...

'ListboxTop',0, 'Max',100000, 'Position',[333.75 43.5 45 15], ... 'SliderStep',[1e-005 0.0001], 'Style','slider', 'Tag','sliderL', 'Value',5); h1 = uicontrol('Parent',h0, 'Units','points', 'BackgroundColor',[1 1 1], ... 'FontWeight','bold', 'ListboxTop',0, 'Position',[333 58.5 45 15], ... 'String','5', 'Style','edit', 'Tag','editL'); h1 = uicontrol('Parent',h0, 'Units','points', ... 'BackgroundColor',[0.556862745098039 0.819607843137255 0.949019607843137], ... 'ListboxTop',0, 'Position',[330.75 74.25 53.25 12], 'String','Line Length', ... 'Style','text', 'Tag','StaticText4'); if nargout > 0, fig = h0; end %end

oct_edit.m %oct_edit.m %update all edit boxes in GUI xplus=get(editXpos_,'String');set(editXpos_,'String', xplus); xplus=str2num(xplus);set(sliderXpos_, 'Value',xplus); yplus=get(editYpos_,'String');set(editYpos_,'String', yplus); yplus=str2num(yplus);set(sliderYpos_, 'Value',yplus); zplus=get(editZpos_,'String');set(editZpos_,'String', zplus); zplus=str2num(zplus);set(sliderZpos_, 'Value',zplus); xminus=get(editXneg_,'String');set(editXneg_,'String', xminus); xminus=str2num(xminus);set(sliderXneg_, 'Value',xminus); yminus=get(editYneg_,'String');set(editYneg_,'String', yminus); yminus=str2num(yminus);set(sliderYneg_, 'Value',yminus); zminus=get(editZneg_,'String');set(editZneg_,'String', zminus); zminus=str2num(zminus);set(sliderZneg_, 'Value',zminus);

Ltemp=get(editL_,'String');set(editL_,'String', Ltemp); Ltemp=str2num(Ltemp);set(sliderL_, 'Value',Ltemp);

oct_slider.m %updates all coordinate sliders in GUI x1=get(sliderXpos_, 'Value');x1=num2str(x1);set(editXpos_,'String', x1); y1=get(sliderYpos_, 'Value');y1=num2str(y1);set(editYpos_,'String', y1); z1=get(sliderZpos_, 'Value');z1=num2str(z1);set(editZpos_,'String', z1); x2=get(sliderXneg_, 'Value');x2=num2str(x2);set(editXneg_,'String', x2); y2=get(sliderYneg_, 'Value');y2=num2str(y2);set(editYneg_,'String', y2); z2=get(sliderZneg_, 'Value');z2=num2str(z2);set(editZneg_,'String', z2);

Ltemp=get(sliderL_, 'Value');Ltemp=num2str(Ltemp);set(editL_,'String', Ltemp);

octv3.m %Pat McGinley %7/8/02 %ver 0.94GUI syms s r t x y z NML real; %LEN=Line LENGTH normalization constant for drawing %if the lines don't connect, increase LEN and rerun it, if they're too long reduce it LEN=get(sliderL_, 'Value'); %NML -> normalization factor for symbolic matrix %NOTE: the final symbolic output is in RAY coordinates, not AXIS coords like the paper % (it was just easier to visualize and to draw that way. the matrix after "J psuedo inverse" % in the output is in AXIS) NML=1; % -----THE POINTS ARE DEFINED BOTH NUMERICALLY AND SYMBOLICALLY------% YOU SHOULD CHANGE BOTH THE SAME WAY- THE SYMBOLIC AND NUMERIC OUTPUTS ARE % INDEPENDENT OF EACH OTHER, AND THE NUMERICAL VALUES ARE USED TO DRAW THE % FIGURES, SO IF THEY'RE DIFFERENT THE PSUEDO-INVERSE WON'T MATCH UP WITH THE % POLYGONS. JUST MAKE SURE THAT THE POINTS ARE IDENTICAL IN TERMS OF 'r' AND 'k'

%define points symbolically in terms of symbolic edge length %to change just edit the points [x y z];

%______% % NOTE- VALUES FOR -X, -Y, -Z VERTICES ARE % DEFINED AS NEGATIVE #'s IN GUI, SO % D,E,F MUST HAVE POSITIVE VALUES HERE %______r=s;%/sqrt(2);

A=[ r 0 01]; A(1)=A(1)*(get(sliderXpos_, 'Value'))+2; B=[02 r 01]; B(2)=B(2)*(get(sliderYpos_, 'Value')); C=[02 0 r]; C(3)=C(3)*(get(sliderZpos_, 'Value'))+1; D=[ r 0 01]; D(1)=D(1)*(get(sliderXneg_, 'Value'))+2; E=[02 r 01]; E(2)=E(2)*(get(sliderYneg_, 'Value')); F=[02 0 r]; F(3)=F(3)*(get(sliderZneg_, 'Value'))+1; % to graph it, using numerical points Rn k=1;%/sqrt(2); An=[ k 0 01]; An(1)=An(1)*(get(sliderXpos_, 'Value'))+2; Bn=[02 k 01]; Bn(2)=Bn(2)*(get(sliderYpos_, 'Value')); Cn=[02 0 k]; Cn(3)=Cn(3)*(get(sliderZpos_, 'Value'))+1; Dn=[ k 0 01]; Dn(1)=Dn(1)*(get(sliderXneg_, 'Value'))+2; En=[02 k 01]; En(2)=En(2)*(get(sliderYneg_, 'Value')); Fn=[02 0 k]; Fn(3)=Fn(3)*(get(sliderZneg_, 'Value'))+1; a=1*pi/9; %rotation angle about z-axis, degrees Rz=[cos(a) -sin(a) 0; sin(a) cos(a) 0; 0 0 1;];

An=An*Rz;Bn=Bn*Rz;Cn=Cn*Rz;Dn=Dn*Rz;En=En*Rz;Fn=Fn*Rz; A=A*Rz;B=B*Rz;C=C*Rz;D=D*Rz;E=E*Rz;F=F*Rz;

%determine direction vectors for line coordinates

S1= A-C; S2= C-D; S3= D-F; S4= F-A; S5= B-A; S6= D-B; S7= E-D; S8= A-E; S9= C-B; S10=E-C; S11=F-E; S12=B-F; %determine moments for line coordinates SO1= cross(A,S1); SO2= cross(C,S2); SO3= cross(D,S3); SO4= cross(F,S4); SO5= cross(B,S5); SO6= cross(D,S6); SO7= cross(E,S7); SO8= cross(A,S8); SO9= cross(C,S9); SO10=cross(E,S10); SO11=cross(F,S11); SO12=cross(B,S12);

J=[S1 SO1; S2 SO2; S3 SO3; S4 SO4; S5 SO5; S6 SO6; S7 SO7; S8 SO8; S9 SO9; S10 SO10; S11 SO11; S12 SO12]';

Jinv=J'*inv(J*J');

' J pseudo-inverse, symbolic' simplify(NML*Jinv) Jcube_ray=((2*s^2)*Jinv)'; ' Cuboid in symbolic Ray coordinates' Jcube_ray=[Jcube_ray(4,:); Jcube_ray(5,:); Jcube_ray(6,:);Jcube_ray(1,:); Jcube_ray(2,:); Jcube_ray(3,:);]; simplify(Jcube_ray)

S1n= An-Cn; S2n= Cn-Dn; S3n= Dn-Fn; S4n= Fn-An; S5n= Bn-An; S6n= Dn-Bn; S7n= En-Dn; S8n= An-En; S9n= Cn-Bn; S10n=En-Cn; S11n=Fn-En; S12n=Bn-Fn;

SO1n= cross(An,S1n); SO2n= cross(Cn,S2n); SO3n= cross(Dn,S3n); SO4n= cross(Fn,S4n); SO5n= cross(Bn,S5n); SO6n= cross(Dn,S6n); SO7n= cross(En,S7n); SO8n= cross(An,S8n); SO9n= cross(Cn,S9n); SO10n=cross(En,S10n); SO11n=cross(Fn,S11n); SO12n=cross(Bn,S12n);

Jn=[S1n SO1n; S2n SO2n; S3n SO3n; S4n SO4n; S5n SO5n; S6n SO6n; S7n SO7n; S8n SO8n; S9n SO9n; S10n SO10n; S11n SO11n; S12n SO12n]';

%NUMERICAL PSEUDO-INVERSE Jninv=Jn'*inv(Jn*Jn'); ' Jn pseudo-inverse' Jncube_ray=(Jninv)'; ' Cuboid in Ray coordinates, Numerical' Jncube_ray=[Jncube_ray(4,:); Jncube_ray(5,:); Jncube_ray(6,:);Jncube_ray(1,:); Jncube_ray(2,:); Jncube_ray(3,:);] %to avoid the numerical output, put a ; after the ]

%line vectors for cube C1=[Jncube_ray(1,1) Jncube_ray(2,1) Jncube_ray(3,1)]; C2=[Jncube_ray(1,2) Jncube_ray(2,2) Jncube_ray(3,2)]; C3=[Jncube_ray(1,3) Jncube_ray(2,3) Jncube_ray(3,3)]; C4=[Jncube_ray(1,4) Jncube_ray(2,4) Jncube_ray(3,4)]; C5=[Jncube_ray(1,5) Jncube_ray(2,5) Jncube_ray(3,5)]; C6=[Jncube_ray(1,6) Jncube_ray(2,6) Jncube_ray(3,6)]; C7=[Jncube_ray(1,7) Jncube_ray(2,7) Jncube_ray(3,7)]; C8=[Jncube_ray(1,8) Jncube_ray(2,8) Jncube_ray(3,8)]; C9=[Jncube_ray(1,9) Jncube_ray(2,9) Jncube_ray(3,9)]; C10=[Jncube_ray(1,10) Jncube_ray(2,10) Jncube_ray(3,10)]; C11=[Jncube_ray(1,11) Jncube_ray(2,11) Jncube_ray(3,11)]; C12=[Jncube_ray(1,12) Jncube_ray(2,12) Jncube_ray(3,12)];

CO1=[Jncube_ray(4,1) Jncube_ray(5,1) Jncube_ray(6,1)]; CO2=[Jncube_ray(4,2) Jncube_ray(5,2) Jncube_ray(6,2)]; CO3=[Jncube_ray(4,3) Jncube_ray(5,3) Jncube_ray(6,3)]; CO4=[Jncube_ray(4,4) Jncube_ray(5,4) Jncube_ray(6,4)]; CO5=[Jncube_ray(4,5) Jncube_ray(5,5) Jncube_ray(6,5)]; CO6=[Jncube_ray(4,6) Jncube_ray(5,6) Jncube_ray(6,6)]; CO7=[Jncube_ray(4,7) Jncube_ray(5,7) Jncube_ray(6,7)]; CO8=[Jncube_ray(4,8) Jncube_ray(5,8) Jncube_ray(6,8)]; CO9=[Jncube_ray(4,9) Jncube_ray(5,9) Jncube_ray(6,9)]; CO10=[Jncube_ray(4,10) Jncube_ray(5,10) Jncube_ray(6,10)]; CO11=[Jncube_ray(4,11) Jncube_ray(5,11) Jncube_ray(6,11)]; CO12=[Jncube_ray(4,12) Jncube_ray(5,12) Jncube_ray(6,12)];

CUBEJ=[C1 CO1; C2 CO2; C3 CO3; C4 CO4; C5 CO5; C6 CO6;... C7 CO7; C8 CO8; C9 CO9; C10 CO10; C11 CO11; C12 CO12;].';

%draw octahedron figure(2) clf hold on; Title('Original Octahedron') %face ABC x1=[An(1) Bn(1) Cn(1) An(1)];y1=[An(2) Bn(2) Cn(2) An(2)];z1=[An(3) Bn(3) Cn(3) An(3)]; patch(x1,y1,z1,[0 0 .33]) %face ABF x2=[An(1) Bn(1) Fn(1) An(1)];y2=[An(2) Bn(2) Fn(2) An(2)];z2=[An(3) Bn(3) Fn(3) An(3)]; patch(x2,y2,z2,[0 0 1]) %face AEF x3=[An(1) En(1) Fn(1) An(1)];y3=[An(2) En(2) Fn(2) An(2)];z3=[An(3) En(3) Fn(3) An(3)]; patch(x3,y3,z3,[0 1 0])

%face ACE x4=[An(1) Cn(1) En(1) An(1)];y4=[An(2) Cn(2) En(2) An(2)];z4=[An(3) Cn(3) En(3) An(3)]; patch(x4,y4,z4,[0 .4 0]) %face CDE x5=[Dn(1) Cn(1) En(1) Dn(1)];y5=[Dn(2) Cn(2) En(2) Dn(2)];z5=[Dn(3) Cn(3) En(3) Dn(3)]; patch(x5,y5,z5,[0 .5 .5]) %face BCD x6=[Bn(1) Cn(1) Dn(1) Bn(1)];y6=[Bn(2) Cn(2) Dn(2) Bn(2)];z6=[Bn(3) Cn(3) Dn(3) Bn(3)]; patch(x6,y6,z6,[.5 .5 .5]) %face BDF x7=[Bn(1) Dn(1) Fn(1) Bn(1)];y7=[Bn(2) Dn(2) Fn(2) Bn(2)];z7=[Bn(3) Dn(3) Fn(3) Bn(3)]; patch(x7,y7,z7,[.83 .83 .83]) %face DEF x8=[Dn(1) En(1) Fn(1) Dn(1)];y8=[Dn(2) En(2) Fn(2) Dn(2)];z8=[Dn(3) En(3) Fn(3) Dn(3)]; patch(x8,y8,z8,'c') axis equal grid view(3) hold off;

%find intersections of new lines

%DRAW CUBE AS LINES %find points on lines and draw them p1=(cross(C1,CO1)/dot(C1,C1));p2=(cross(C2,CO2)/dot(C2,C2)); p3=(cross(C3,CO3)/dot(C3,C3));p4=(cross(C4,CO4)/dot(C4,C4)); p5=(cross(C5,CO5)/dot(C5,C5));p6=(cross(C6,CO6)/dot(C6,C6)); p7=(cross(C7,CO7)/dot(C7,C7));p8=(cross(C8,CO8)/dot(C8,C8)); p9=(cross(C9,CO9)/dot(C9,C9));p10=(cross(C10,CO10)/dot(C10,C10)); p11=(cross(C11,CO11)/dot(C11,C11));p12=(cross(C12,CO12)/dot(C12,C12)); x1=[p1(1)-LEN*C1(1) (p1(1)+LEN*C1(1))];y1=[p1(2)-LEN*C1(2) (p1(2)+LEN*C1(2))]; z1=[p1(3)-LEN*C1(3) (p1(3)+LEN*C1(3))];x2=[p2(1)-LEN*C2(1) (p2(1)+LEN*C2(1))]; y2=[p2(2)-LEN*C2(2) (p2(2)+LEN*C2(2))];z2=[p2(3)-LEN*C2(3) (p2(3)+LEN*C2(3))]; x3=[p3(1)-LEN*C3(1) (p3(1)+LEN*C3(1))];y3=[p3(2)-LEN*C3(2) (p3(2)+LEN*C3(2))]; z3=[p3(3)-LEN*C3(3) (p3(3)+LEN*C3(3))];x4=[p4(1)-LEN*C4(1) (p4(1)+LEN*C4(1))]; y4=[p4(2)-LEN*C4(2) (p4(2)+LEN*C4(2))];z4=[p4(3)-LEN*C4(3) (p4(3)+LEN*C4(3))]; x5=[p5(1)-LEN*C5(1) (p5(1)+LEN*C5(1))];y5=[p5(2)-LEN*C5(2) (p5(2)+LEN*C5(2))]; z5=[p5(3)-LEN*C5(3) (p5(3)+LEN*C5(3))];x6=[p6(1)-LEN*C6(1) (p6(1)+LEN*C6(1))]; y6=[p6(2)-LEN*C6(2) (p6(2)+LEN*C6(2))];z6=[p6(3)-LEN*C6(3) (p6(3)+LEN*C6(3))]; x7=[p7(1)-LEN*C7(1) (p7(1)+LEN*C7(1))];y7=[p7(2)-LEN*C7(2) (p7(2)+LEN*C7(2))]; z7=[p7(3)-LEN*C7(3) (p7(3)+LEN*C7(3))];x8=[p8(1)-LEN*C8(1) (p8(1)+LEN*C8(1))]; y8=[p8(2)-LEN*C8(2) (p8(2)+LEN*C8(2))];z8=[p8(3)-LEN*C8(3) (p8(3)+LEN*C8(3))]; x9=[p9(1)-LEN*C9(1) (p9(1)+LEN*C9(1))];y9=[p9(2)-LEN*C9(2) (p9(2)+LEN*C9(2))]; z9=[p9(3)-LEN*C9(3) (p9(3)+LEN*C9(3))];x10=[p10(1)-LEN*C10(1) (p10(1)+LEN*C10(1))]; y10=[p10(2)-LEN*C10(2) (p10(2)+LEN*C10(2))];z10=[p10(3)-LEN*C10(3) (p10(3)+LEN*C10(3))]; x11=[p11(1)-LEN*C11(1) (p11(1)+LEN*C11(1))];y11=[p11(2)-LEN*C11(2) (p11(2)+LEN*C11(2))]; z11=[p11(3)-LEN*C11(3) (p11(3)+LEN*C11(3))];x12=[p12(1)-LEN*C12(1) (p12(1)+LEN*C12(1))]; y12=[p12(2)-LEN*C12(2) (p12(2)+LEN*C12(2))];z12=[p12(3)-LEN*C12(3) (p12(3)+LEN*C12(3))]; xlabel('x') ylabel('y') zlabel('z') figure(3) clf hold on title('Cube drawn as lines') plot3(x1,y1,z1,'b') plot3(x2,y2,z2,'y:',x2,y2,z2,'b-') plot3(x3,y3,z3,'c') plot3(x4,y4,z4,'c:') plot3(x5,y5,z5,'r:') plot3(x6,y6,z6,'r') plot3(x7,y7,z7,'y:',x7,y7,z7,'m-') plot3(x8,y8,z8,'m') plot3(x9,y9,z9,'g')

81 plot3(x10,y10,z10,'y:',x10,y10,z10,'g-') plot3(x11,y11,z11,'k:') plot3(x12,y12,z12,'k') view (-29,18) axis equal xlabel('x') ylabel('y') zlabel('z')

%mutual moments CO1=CO1/norm(C1);C1=C1/norm(C1);CO2=CO2/norm(C2);C2=C2/norm(C2); CO3=CO3/norm(C3);C3=C3/norm(C3);CO4=CO4/norm(C4);C4=C4/norm(C4); CO5=CO5/norm(C5);C5=C5/norm(C5);CO6=CO6/norm(C6);C6=C6/norm(C6); CO7=CO7/norm(C7);C7=C7/norm(C7);CO8=CO8/norm(C8);C8=C8/norm(C8); CO9=CO9/norm(C9);C9=C9/norm(C9);CO10=CO10/norm(C10);C10=C10/norm(C10); CO11=CO11/norm(C11);C11=C11/norm(C11);CO12=CO12/norm(C12);C12=C12/norm(C12); mm0105=dot(C1,CO5)+dot(C5,CO1);mm0108=dot(C1,CO8)+dot(C8,CO1); mm0109=dot(C1,CO9)+dot(C9,CO1);mm0110=dot(C1,CO10)+dot(C10,CO1); mm0206=dot(C2,CO6)+dot(C6,CO2);mm0207=dot(C2,CO7)+dot(C7,CO2); mm0209=dot(C2,CO9)+dot(C9,CO2);mm0210=dot(C2,CO10)+dot(C10,CO2); mm1205=dot(C12,CO5)+dot(C5,CO12);mm1204=dot(C12,CO4)+dot(C4,CO12); mm1206=dot(C12,CO6)+dot(C6,CO12);mm1203=dot(C12,CO3)+dot(C3,CO12); mm1104=dot(C11,CO4)+dot(C4,CO11);mm1108=dot(C11,CO8)+dot(C8,CO11); mm1107=dot(C11,CO7)+dot(C7,CO11);mm1103=dot(C11,CO3)+dot(C3,CO11);

' mutual moments - each row represents one vertex (NUMERICAL)' MM=[mm0105 mm0109; mm0108 mm0110; mm0206 mm0209; mm0207 mm0210; mm1204 mm1205; mm1203 mm1206; mm1104 mm1108; mm1103 mm1107;]

% SYMBOLIC line vectors for cube C1s=[Jcube_ray(1,1) Jcube_ray(2,1) Jcube_ray(3,1)]; C2s=[Jcube_ray(1,2) Jcube_ray(2,2) Jcube_ray(3,2)]; C3s=[Jcube_ray(1,3) Jcube_ray(2,3) Jcube_ray(3,3)]; C4s=[Jcube_ray(1,4) Jcube_ray(2,4) Jcube_ray(3,4)]; C5s=[Jcube_ray(1,5) Jcube_ray(2,5) Jcube_ray(3,5)]; C6s=[Jcube_ray(1,6) Jcube_ray(2,6) Jcube_ray(3,6)]; C7s=[Jcube_ray(1,7) Jcube_ray(2,7) Jcube_ray(3,7)]; C8s=[Jcube_ray(1,8) Jcube_ray(2,8) Jcube_ray(3,8)]; C9s=[Jcube_ray(1,9) Jcube_ray(2,9) Jcube_ray(3,9)]; C10s=[Jcube_ray(1,10) Jcube_ray(2,10) Jcube_ray(3,10)]; C11s=[Jcube_ray(1,11) Jcube_ray(2,11) Jcube_ray(3,11)]; C12s=[Jcube_ray(1,12) Jcube_ray(2,12) Jcube_ray(3,12)];

CO1s=[Jcube_ray(4,1) Jcube_ray(5,1) Jcube_ray(6,1)]; CO2s=[Jcube_ray(4,2) Jcube_ray(5,2) Jcube_ray(6,2)]; CO3s=[Jcube_ray(4,3) Jcube_ray(5,3) Jcube_ray(6,3)]; CO4s=[Jcube_ray(4,4) Jcube_ray(5,4) Jcube_ray(6,4)]; CO5s=[Jcube_ray(4,5) Jcube_ray(5,5) Jcube_ray(6,5)]; CO6s=[Jcube_ray(4,6) Jcube_ray(5,6) Jcube_ray(6,6)]; CO7s=[Jcube_ray(4,7) Jcube_ray(5,7) Jcube_ray(6,7)]; CO8s=[Jcube_ray(4,8) Jcube_ray(5,8) Jcube_ray(6,8)]; CO9s=[Jcube_ray(4,9) Jcube_ray(5,9) Jcube_ray(6,9)]; CO10s=[Jcube_ray(4,10) Jcube_ray(5,10) Jcube_ray(6,10)]; CO11s=[Jcube_ray(4,11) Jcube_ray(5,11) Jcube_ray(6,11)]; CO12s=[Jcube_ray(4,12) Jcube_ray(5,12) Jcube_ray(6,12)]; mm0105s=dot(C1s,CO5s)+dot(C5s,CO1s);mm0108s=dot(C1s,CO8s)+dot(C8s,CO1s); mm0109s=dot(C1s,CO9s)+dot(C9s,CO1s);mm0110s=dot(C1s,CO10s)+dot(C10s,CO1s); mm0206s=dot(C2s,CO6s)+dot(C6s,CO2s);mm0207s=dot(C2s,CO7s)+dot(C7s,CO2s); mm0209s=dot(C2s,CO9s)+dot(C9s,CO2s);mm0210s=dot(C2s,CO10s)+dot(C10s,CO2s); mm1205s=dot(C12s,CO5s)+dot(C5s,CO12s);mm1204s=dot(C12s,CO4s)+dot(C4s,CO12s); mm1206s=dot(C12s,CO6s)+dot(C6s,CO12s);mm1203s=dot(C12s,CO3s)+dot(C3s,CO12s); mm1104s=dot(C11s,CO4s)+dot(C4s,CO11s);mm1108s=dot(C11s,CO8s)+dot(C8s,CO11s); mm1107s=dot(C11s,CO7s)+dot(C7s,CO11s);mm1103s=dot(C11s,CO3s)+dot(C3s,CO11s);

' mutual moments - each row represents one vertex (SYMBOLIC)' MMs=[mm0105s mm0109s; mm0108s mm0110s; mm0206s mm0209s; mm0207s mm0210s; mm1204s mm1205s; mm1203s mm1206s; mm1104s mm1108s; mm1103s mm1107s;]; simplify(MMs)

APPENDIX B ALTERNATE LEAST SQUARES MODELS

True best-fit from least squares, freeing coefficient of E, octahedral cell boom

25 .9 8.71 6.25 2.33 1.03 2 R ρ L Rbottom C # k A,oct = R 4.54 1.63 .13 2 V X oct E

31.2 2.17 6.04 3.86 2.73 1.03 2 R ρ L Rbottom C # k L,oct = R 4.54 1.63 .13 2 V X oct E

33.9 2.45 2.78 3.10 3.48 3.24 2 R ρ L Rbottom C # kT,oct = .34R 4.17 1.77 .29 2 V X oct E

True best-fit from least squares, freeing coefficient of E, symmetric reinforced cell boom

17.7  X  2332 R.54 ρ 1.2  oct  E.52 A43−.65log2 ( A) C .057   #  X rnf  k A,sym = 2−.78R V1.92 L 69.7−3.38 log2 (L)

22.7  X  2415 R.48 ρ 2.2  oct  E .21 A39.6−6.27 log2 ( A) C .34   #  X rnf  k L,sym = .35R 1.92 3.37 80.7-3.9 log2 (L) 2 V X oct L

1108 2.2 16.6 2.61 414−58.2 log2 ( A) 2.27 2 R X rnf E A C # kT,sym = 34  X  2.108R ρ 7.35V 4.68  oct  L 359.7-17.4 log2 (L)    X rnf 

The regression model predictions and original data points are plotted in Figures B.1 through B.6. The ranges of parameters used are listed in Table B.1.

Table B.1 Ranges of key design parameters used to fit regression models Parameter Min. Value Max. Value Units L 1000 12000 in.

Rtop 19.5 540 in.

Rbottom 39 432 in. E 58.0·103 5.80·106 Psi ρ 0.0500 0.0979 lbm/in.3

C# 2 64 (none) 2 Xoct 0.1872 2.1598 in. 2 Xrnf 0.1872 2.1598 in.

Figure B.1 Comparison of FEM simulation results and regression model prediction for axial specific stiffness of OCT booms

83 84

Figure B.2 Comparison of FEM simulation Results and regression model prediction for lateral specific stiffness of OCT booms

Figure B.3 Comparison of FEM simulation results and regression model prediction for torsional specific stiffness of OCT booms

Figure B.4 Comparison of FEM simulation results and regression model prediction for axial specific stiffness of SYM booms

Figure B.5 Comparison of FEM simulation results and regression model prediction for lateral specific stiffness of SYM booms

Figure B.6 Comparison of FEM simulation results and regression model prediction for torsional specific stiffness of SYM boom

APPENDIX C DESIGN OPTIMIZATION CHARTS

Figure C.1 Optimization chart for OCT 8, 300m x 5m, CFRP

87 88

Figure C.2 Optimization chart for OCT 16, 300m x 5m, CFRP

Figure C.3 Optimization chart for OCT 24, 300m x 5m, CFRP

Figure C.4 Optimization chart for OCT 32, 300m x 5m, CFRP

Figure C.5 Optimization chart for SYM 8, 300m x 5m, CFRP

Figure C.6 Optimization chart for SYM 16, 300m x 5m, CFRP

Figure C.7 Optimization chart for SYM 24, 300m x 5m, CFRP

Figure C.8 Optimization chart for SYM 32, 300m x 5m, CFRP

Figure C.9 Optimization chart for SYM 40, 300m x 5m, CFRP

Figure C.10 Comparison of kA for 300m x 5m, OCT vs BAS-32 (both CFRP).

Figure C.11 Comparison of kL for 300m x 5m, OCT vs BAS-32 (both CFRP)

Figure C.12 Comparison of kT for 300m x 5m, OCT vs BAS-32 (both CFRP)

Figure C.13 Comparison of kA for 300m x 5m, SYM vs BAS-32 (both CFRP)

Figure C.14 Comparison of kL for 300m x 5m, SYM vs BAS-32 (both CFRP)

Figure C.15 Comparison of kT for 300m x 5m, SYM vs BAS-32 (both CFRP)

APPENDIX D ADDITIONAL FIGURES

Figures D.1 through D.3 are duplicates of the data presented in Figures 3.3 through

3.5, with the axes rescaled to show the localized variations more clearly.

Figure D.1 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=2

95 96

Figure D.2 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=4

Figure D.3 √det(JJT) vs. N {3,60} and φ {-180,180}, a=1 b=2 h=8

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BIOGRAPHICAL SKETCH

The author was born in 1978 in Norristown, Pennsylvania. He graduated with a

Bachelor of Science in Engineering with honors in 2000 from the Mechanical and

Aerospace Engineering Department at Princeton University, Princeton, NJ. He then received a Master of Science in mechanical engineering from the University of Florida in

August 2002.